tag:blogger.com,1999:blog-30058623924043771112024-03-12T17:31:29.215-07:00ElectronicsUnknownnoreply@blogger.comBlogger35125tag:blogger.com,1999:blog-3005862392404377111.post-76091229315975507272017-12-02T04:27:00.010-08:002023-06-25T02:13:49.271-07:00Kirchhoffs Circuit Law<div dir="ltr" style="text-align: left;" trbidi="on">
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<b><i><span style="color: blue;"><u>Kirchhoffs</u> <u>Law</u></span></i></b></div>
<span style="background-color: white; color: #414042; font-family: Martel, serif; font-size: 16px;"> In 1845, a German physicist, </span><span style="background-color: white; box-sizing: border-box; color: #414042; font-family: Martel, serif; font-size: 16px; font-weight: 700;">Gustav Kirchhoff</span><span style="background-color: white; color: #414042; font-family: Martel, serif; font-size: 16px;"> developed a pair or set of rules or laws which deal with the conservation of current and energy within electrical circuits. These two rules are commonly known as:</span><div>
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<span style="background-color: white; color: #414042; font-family: Martel, serif; font-size: 16px; font-weight: 700;"> Kirchhoffs Current Law, (KCL)</span></div>
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<span style="background-color: white; color: #414042; font-family: Martel, serif; font-size: 16px;"> </span><span style="background-color: white; box-sizing: border-box; color: #414042; font-family: Martel, serif; font-size: 16px; font-weight: 700;">Kirchhoffs Voltage Law, (KVL)</span></div>
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<span style="background-color: white; box-sizing: border-box; color: #414042; font-family: Martel, serif; font-size: 16px; font-weight: 700;"> </span><span style="background-color: white; box-sizing: border-box; font-family: Martel, serif; font-size: 16px; font-weight: 700;"><span style="color: blue;"><u>Kirchhoffs</u> <u>Current</u> <u>Law</u>, (KCL)</span></span></div>
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<span style="color: blue; font-family: Martel, serif;"><span style="background-color: white;"><b> </b></span></span><span style="background-color: white; box-sizing: border-box; color: #414042; font-family: Martel, serif; font-size: 16px; font-weight: 700;">Kirchhoffs Current Law</span><span style="background-color: white; color: #414042; font-family: Martel, serif; font-size: 16px;"> or KCL, states that the “</span><em style="background-color: white; box-sizing: border-box; color: #414042; font-family: Martel, serif; font-size: 16px;">total current or charge entering a junction or node is exactly equal to the charge leaving the node as it has no other place to go except to leave, as no charge is lost within the node</em><span style="background-color: white; color: #414042; font-family: Martel, serif; font-size: 16px;">“.</span></div>
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<span style="background-color: white; color: #414042; font-family: Martel, serif; font-size: 16px;"> </span><span style="background-color: white; color: #414042; font-family: Martel, serif; font-size: 16px;"> </span><span style="background-color: white; color: #414042; font-family: Martel, serif; font-size: 16px;">In other words the algebraic sum of ALL the currents entering and leaving a node must be equal to zero,</span><span style="background-color: white; color: #414042; font-family: Martel, serif; font-size: 16px;"> </span></div>
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<span style="background-color: white; color: #414042; font-family: Martel, serif; font-size: 16px;"> </span></div>
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<span style="background-color: white; color: #414042; font-family: Martel, serif; font-size: 16px;"> </span><span style="font-size: large;"><span style="background-color: white; color: #414042; font-family: Martel, serif;">I</span><span style="background-color: white; bottom: -0.3em; box-sizing: border-box; color: #414042; font-family: Martel, serif; line-height: 0; position: relative; vertical-align: baseline;">(exiting)</span><span style="background-color: white; color: #414042; font-family: Martel, serif;"> + I</span><span style="background-color: white; bottom: -0.3em; box-sizing: border-box; color: #414042; font-family: Martel, serif; line-height: 0; position: relative; vertical-align: baseline;">(entering)</span><span style="background-color: white; color: #414042; font-family: Martel, serif;"> = 0</span></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgK5CqUPoopMS1m_1wtc5q99DFhWw2a7vKL_Pa8RUjH_hjxVUf8YiJyM-EEnlKrxfW4NWzS0Jb01GBmnMND_2TddfkJYdDQKliOWLWbEJENxAgyAy57U4N4eTN7afnUUHWfLrfIjqeoUf8/s1600/kcl-300x268.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="268" data-original-width="300" height="178" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgK5CqUPoopMS1m_1wtc5q99DFhWw2a7vKL_Pa8RUjH_hjxVUf8YiJyM-EEnlKrxfW4NWzS0Jb01GBmnMND_2TddfkJYdDQKliOWLWbEJENxAgyAy57U4N4eTN7afnUUHWfLrfIjqeoUf8/s200/kcl-300x268.png" width="200" /></a></div>
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<span style="background-color: white; color: #414143; font-family: Martel, serif; font-size: 22px; text-align: center;"> I</span><span style="background-color: white; color: #414143; font-family: Martel, serif; font-size: 16.5px; text-align: center;">2</span><span style="background-color: white; color: #414143; font-family: Martel, serif; font-size: 22px; text-align: center;"> + I</span><span style="background-color: white; color: #414143; font-family: Martel, serif; font-size: 16.5px; text-align: center;">4</span><span style="background-color: white; color: #414143; font-family: Martel, serif; font-size: 22px; text-align: center;"> + I</span><span style="background-color: white; color: #414143; font-family: Martel, serif; font-size: 16.5px; text-align: center;">5</span><span style="background-color: white; color: #414143; font-family: Martel, serif; font-size: 22px; text-align: center;"> + I</span><span style="background-color: white; color: #414143; font-family: Martel, serif; font-size: 16.5px; text-align: center;">6</span><span style="background-color: white; font-family: Martel, serif; font-size: 22px; text-align: center;"><span style="color: #414143;"> </span><span style="color: red;">+ I</span></span><span style="color: red;"><span style="background-color: white; font-family: Martel, serif; font-size: 16.5px; text-align: center;">3 + </span><span style="background-color: white; font-family: Martel, serif; font-size: 22px; text-align: center;">I</span><span style="background-color: white; font-family: Martel, serif; font-size: 16.5px; text-align: center;">1</span></span><span style="background-color: white; color: #414143; font-family: Martel, serif; font-size: 22px; text-align: center;"> = 0</span></div>
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<span style="background-color: white; color: #414143; font-family: Martel, serif; font-size: 22px; text-align: center;"><u><br /></u></span></div>
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<span style="background-color: white; font-family: Martel, serif; font-size: 16px; font-weight: 700;"><span style="color: blue;"><u>Kirchhoffs</u> Voltage <u>Law</u>, (KVL)</span></span></div>
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<span style="background-color: white; font-family: Martel, serif; font-size: 16px; font-weight: 700;"><span style="color: blue;"> </span></span><span style="background-color: white; box-sizing: border-box; color: #414042; font-family: Martel, serif; font-size: 16px; font-weight: 700;">Kirchhoffs Voltage Law</span><span style="background-color: white; color: #414042; font-family: Martel, serif; font-size: 16px;"> or KVL, states that “</span><em style="background-color: white; box-sizing: border-box; color: #414042; font-family: Martel, serif; font-size: 16px;">in any closed loop network, the total voltage around the loop is equal to the sum of all the voltage drops within the same loop</em><span style="background-color: white; color: #414042; font-family: Martel, serif; font-size: 16px;">” which is also equal to zero.</span></div>
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<span style="background-color: white; color: #414042; font-family: Martel, serif; font-size: 16px;"> </span><span style="background-color: white; color: #414042; font-family: Martel, serif; font-size: 16px;"> </span><span style="background-color: white; color: #414042; font-family: Martel, serif; font-size: 16px;">In other words the algebraic sum of all voltages within the loop must be equal to zero.</span></div>
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<span style="background-color: white; box-sizing: border-box; color: #414042; font-family: Martel, serif; font-size: 16px; font-weight: 700;"> </span></div>
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<span style="background-color: white; box-sizing: border-box; color: #414042; font-family: Martel, serif; font-size: 16px; font-weight: 700;"> </span><img alt="Image result for kvl" src="https://upload.wikimedia.org/wikipedia/commons/thumb/4/40/Kirchhoff_voltage_law.svg/200px-Kirchhoff_voltage_law.svg.png" /></div>
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<span style="background-color: white; box-sizing: border-box; color: #414042; font-family: Martel, serif; font-size: 16px; font-weight: 700;"> </span></div>
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<span style="background-color: white; box-sizing: border-box; color: #414042; font-family: Martel, serif; font-size: 16px; font-weight: 700;"> </span><span style="font-size: 20pt;">V</span><sub>AB</sub><span style="font-size: 20pt;">+V</span><sub>BC</sub><span style="font-size: 20pt;">+V</span><sub>CD</sub><span style="font-size: 20pt;">+V</span><sub>DA</sub><span style="font-size: 20pt;">=0</span></div>
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<b style="font-family: "Times New Roman", serif; font-size: 16px;"><span style="font-size: 16pt;"><br /></span></b></div>
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<b style="font-family: "Times New Roman", serif; font-size: 16px;"><span style="font-size: 16pt;"> <span style="color: blue;"><u>Sign</u> <u>Convention</u> <u>in</u> <span class="SpellE"><u>Kirchhoff’s</u></span> <u>Rules</u>:</span></span></b></div>
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<span style="font-family: "Times New Roman", serif; font-size: 14pt;">1.</span><span style="font-family: "Times New Roman", serif; font-size: 6pt;"> </span><span style="font-family: "Times New Roman", serif; font-size: 14pt;">Move through EMF device in assumed direction of polarity (– to +):</span><span style="font-family: "Times New Roman", serif; font-size: 12pt;"><o:p></o:p></span></div>
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<span style="font-family: "Times New Roman", serif; font-size: 14pt;"> </span><span style="font-family: "Times New Roman", serif; font-size: 12pt;">
</span><span style="font-family: Symbol; font-size: 14pt;">e</span><span style="font-family: "Times New Roman", serif; font-size: 14pt;"> > 0<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Times New Roman", serif; font-size: 14pt;"> </span><span style="font-family: "Times New Roman", serif; font-size: 12pt;"> </span><span style="font-family: "Times New Roman", serif; font-size: 14pt;">2.</span><span style="font-family: "Times New Roman", serif; font-size: 6pt;"> </span><span style="font-family: "Times New Roman", serif; font-size: 14pt;">Move through EMF device in opposite
direction of polarity (+ to –):</span><span style="font-family: "Times New Roman", serif; font-size: 12pt;"><o:p></o:p></span></div>
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<span style="font-family: "Times New Roman", serif; font-size: 14pt;"> </span><span style="font-family: Symbol; font-size: 14pt;">e</span><span style="font-family: "Times New Roman", serif; font-size: 14pt;"> < 0</span><span style="font-family: "Times New Roman", serif; font-size: 12pt;"><o:p></o:p></span></div>
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<span style="font-family: "Times New Roman", serif; font-size: 14pt;"> </span><span style="font-family: "Times New Roman", serif; font-size: 12pt;"> </span><span style="font-family: "Times New Roman", serif; font-size: 14pt;">3.</span><span style="font-family: "Times New Roman", serif; font-size: 6pt;"> </span><span style="font-family: "Times New Roman", serif; font-size: 14pt;">Move through resistor in assumed direction of current (with current </span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;">
<span style="font-family: "Times New Roman", serif; font-size: 14pt;"> arrow):</span><span style="font-family: "Times New Roman", serif; font-size: 12pt;"><o:p></o:p></span></div>
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<span style="font-family: "Times New Roman", serif; font-size: 14pt;"> </span><span style="font-family: "Times New Roman", serif; font-size: 12pt;"> </span><i><span style="font-family: Symbol; font-size: 14pt;">D</span></i><i><span style="font-family: "Times New Roman", serif; font-size: 14pt;">V</span></i><span style="font-family: "Times New Roman", serif; font-size: 14pt;"> = –<i>IR</i></span><span style="font-family: "Times New Roman", serif; font-size: 12pt;"><o:p></o:p></span></div>
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<span style="font-family: "Times New Roman", serif; font-size: 14pt;"> </span><span style="font-family: "Times New Roman", serif; font-size: 12pt;"> </span><span style="font-family: "Times New Roman", serif; font-size: 14pt;">4.</span><span style="font-family: "Times New Roman", serif; font-size: 6pt;"> </span><span style="font-family: "Times New Roman", serif; font-size: 14pt;">Move through resistor in opposite
direction of current (against current <o:p></o:p></span></div>
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<span style="font-family: "Times New Roman", serif; font-size: 14pt;"> arrow):</span><span style="font-family: "Times New Roman", serif; font-size: 12pt;"><o:p></o:p></span></div>
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<span style="font-family: "Times New Roman", serif; font-size: 14pt;">
</span><i><span style="font-family: Symbol; font-size: 14pt;">D</span></i><i><span style="font-family: "Times New Roman", serif; font-size: 14pt;">V</span></i><span style="font-family: "Times New Roman", serif; font-size: 14pt;"> = +<i>IR<o:p></o:p></i></span></div>
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<span style="font-family: "Times New Roman", serif; font-size: 14pt;"> 5.</span><span style="font-family: "Times New Roman", serif; font-size: 6pt;"> </span><span style="font-family: "Times New Roman", serif; font-size: 14pt;">Move through capacitor from + plate to
– plate (with <b>E</b>-field):</span><span style="font-family: "Times New Roman", serif; font-size: 12pt;"><o:p></o:p></span></div>
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<span style="font-family: "Times New Roman", serif; font-size: 14pt;"> </span><span style="font-family: "Times New Roman", serif; font-size: 12pt;"> </span><i><span style="font-family: Symbol; font-size: 14pt;">D</span></i><i><span style="font-family: "Times New Roman", serif; font-size: 14pt;">V</span></i><span style="font-family: "Times New Roman", serif; font-size: 14pt;"> = –<i>q/C</i></span><span style="font-family: "Times New Roman", serif; font-size: 12pt;"><o:p></o:p></span></div>
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<span style="font-family: "Times New Roman", serif; font-size: 14pt;"> </span><span style="font-family: "Times New Roman", serif; font-size: 12pt;"> <o:p></o:p></span></div>
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<span style="font-family: "Times New Roman", serif; font-size: 12pt;"> </span><span style="font-family: "Times New Roman", serif; font-size: 14pt;">6.</span><span style="font-family: "Times New Roman", serif; font-size: 6pt;"> </span><span style="font-family: "Times New Roman", serif; font-size: 14pt;">Move through capacitor from – plate to + plate (against <b>E</b>-field):</span><span style="font-family: "Times New Roman", serif; font-size: 12pt;"><o:p></o:p></span></div>
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<span style="font-family: "Times New Roman", serif; font-size: 14pt;"> </span><i><span style="font-family: Symbol; font-size: 14pt;">D</span></i><i><span style="font-family: "Times New Roman", serif; font-size: 14pt;">V</span></i><span style="font-family: "Times New Roman", serif; font-size: 14pt;"> = +<i>q/C</i></span></div>
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<span style="font-size: 20pt; line-height: 107%; mso-bidi-font-size: 11.0pt;"><o:p></o:p></span></div>
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</head>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3005862392404377111.post-36909723405413952362017-12-02T01:38:00.005-08:002023-06-24T02:43:57.913-07:00Important Question 2<div dir="ltr" style="text-align: left;" trbidi="on">
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<b><span face=""helvetica" , "sans-serif"" style="color: #1d2129; font-size: 10.5pt;">1. What is engineering?</span></b><span face=""helvetica" , "sans-serif"" style="color: #1d2129; font-size: 10.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; line-height: 21.4pt; margin-bottom: 4.5pt; vertical-align: top;">
<span face=""helvetica" , "sans-serif"" style="color: #1d2129; font-size: 10.5pt;">The <span style="border: 1pt none windowtext; mso-border-alt: none windowtext 0cm; padding: 0cm;">application</span> of science to the needs of humanity and a
profession in which a knowledge of the mathematical and natural sciences gained
by study, experience, and practice is applied with judgment to develop ways to
use economically the materials and forces of nature for the benefit of mankind.<o:p></o:p></span></div>
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<b><span face=""helvetica" , "sans-serif"" style="color: #1d2129; font-size: 10.5pt;">2. What is the difference between Electronics and Electrical?</span></b><span face=""helvetica" , "sans-serif"" style="color: #1d2129; font-size: 10.5pt;"><br />
Electronics work on DC and with a voltage range of -48vDC to +48vDC. If the
electronic device is plugged into a standard wall outlet, there will be a
transformer inside which will convert the AC voltage you are supplying to the
required DC voltage needed by the device. Examples: Computer, radio, T.V,
etc...<br />
Electric devices use line voltage (120vAC, 240vAC, etc...). Electric devices
can also be designed to operate on DC sources, but will be at DC voltages above
48v. Examples: are incandescent lights, heaters, fridge, stove, etc...<o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; line-height: 21.4pt; margin-bottom: 4.5pt; vertical-align: top;">
<b><span face=""helvetica" , "sans-serif"" style="color: #1d2129; font-size: 10.5pt;">3. What is Electronics?</span></b><span face=""helvetica" , "sans-serif"" style="color: #1d2129; font-size: 10.5pt;"><br />
The study and use of electrical devices that operate by controlling the flow of
electrons or other electrically charged particles.<o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; line-height: 21.4pt; margin-bottom: 4.5pt; vertical-align: top;">
<b><span face=""helvetica" , "sans-serif"" style="color: #1d2129; font-size: 10.5pt;">4. What is Communication?</span></b><span face=""helvetica" , "sans-serif"" style="color: #1d2129; font-size: 10.5pt;"><br />
Communication means transferring a signal from the transmitter which passes
through a <span style="border: 1pt none windowtext; mso-border-alt: none windowtext 0cm; padding: 0cm;">medium</span> then the output is obtained at the receiver.
(or)communication says as transferring of message from one place to another
place called communication.<o:p></o:p></span></div>
<span face=""helvetica" , "sans-serif"" style="color: #1d2129; font-size: 10.5pt; line-height: 107%;"><br />
<b>5. Define the terms?</b></span><br />
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<span face=""helvetica" , "sans-serif"" style="color: #1d2129; font-size: 10.5pt; line-height: 107%;"><b><br /></b>
1.Resistor: A resistor is an electrical component that limits or regulates the
flow of electrical current in an electronic circuit.<br />
2.Capacitor : A capacitor is a passive electronic component that stores energy
in the form of an electrostatic field.<br />
3.Diode: A diode is a semiconductor device with two terminals, typically
allowing the flow of current in one direction only.<br />
4.Transistor: A transistor is a semiconductor device that regulates current or
voltage flow and acts as a switch or gate for electronic signals.</span><script async="" src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>
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<div><b style="background-color: white; color: #1d2129; font-size: 10.5pt;"><span style="font-size: 10.5pt;"><br /></span></b></div><div><b style="background-color: white; color: #1d2129; font-size: 10.5pt;"><span style="font-size: 10.5pt;">6. What is microprocessor?</span></b></div><div><div class="MsoNormal" style="background: white; line-height: 21.4pt; margin-bottom: 4.5pt; vertical-align: top;"><span face=""helvetica" , "sans-serif"" style="color: #1d2129; font-size: 10.5pt; line-height: 107%;"><span style="font-size: 10.5pt;">
Microprocessor is a program controlled semi conductor device (IC), which
fetches, decodes and execute instructions.<o:p></o:p></span></span></div>
<span face=""helvetica" , "sans-serif"" style="color: #1d2129; font-size: 10.5pt; line-height: 107%;">
<div class="MsoNormal" style="background: white; line-height: 21.4pt; margin-bottom: 4.5pt; vertical-align: top;">
<b><span style="font-size: 10.5pt;">7. Why is NAND gate preferred over NOR gate for fabrication?</span></b></div><div class="MsoNormal" style="background: white; line-height: 21.4pt; margin-bottom: 4.5pt; vertical-align: top;"><span style="font-size: 10.5pt;">NAND is a better gate for design than NOR because at the transistor
level the mobility of electrons is normally three times that of holes compared
to NOR and thus the NAND is a faster gate.</span></div><div class="MsoNormal" style="background: white; line-height: 21.4pt; margin-bottom: 4.5pt; vertical-align: top;"><b style="font-size: 10.5pt;"><span style="font-size: 10.5pt;">8. What are the basic units of microprocessor?</span></b></div><div class="MsoNormal" style="background: white; line-height: 21.4pt; margin-bottom: 4.5pt; vertical-align: top;"><span style="font-size: 10.5pt;">
The basic units or blocks of microprocessor are ALU, an array of registers and
control unit.<o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; line-height: 21.4pt; margin-bottom: 4.5pt; vertical-align: top;">
<b><span style="font-size: 10.5pt;">9. What is a bus?<o:p></o:p></span></b></div>
<div class="MsoNormal" style="background: white; line-height: 21.4pt; margin-bottom: 4.5pt; vertical-align: top;">
<span style="font-size: 10.5pt;">Bus is a group of conducting lines that carries data, address and
control signals.<o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; line-height: 21.4pt; margin-bottom: 4.5pt; vertical-align: top;">
<b><span style="font-size: 10.5pt;">10. What is a signal?</span></b><span style="font-size: 10.5pt;"><br />
Signal is an electric current or <span style="border: 1pt none windowtext; mso-border-alt: none windowtext 0cm; padding: 0cm;">electromagnetic</span> field
used to convey data from one place to another.<o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; line-height: 21.4pt; margin-bottom: 4.5pt; vertical-align: top;">
<b><span style="font-size: 10.5pt;">11. What is sampling?</span></b><span style="font-size: 10.5pt;"><br />
The process of obtaining a set of samples from a continuous function of time
x(t) is referred to as sampling.<o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; line-height: 21.4pt; margin-bottom: 4.5pt; vertical-align: top;">
<b><span style="font-size: 10.5pt;">12. State sampling theorem.<o:p></o:p></span></b></div>
<div class="MsoNormal" style="background: white; line-height: 21.4pt; margin-bottom: 4.5pt; vertical-align: top;">
<span style="font-size: 10.5pt;">It states that, while taking the samples of a continuous signal, it has
to be taken care that the sampling rate is equal to or greater than twice the
cut off <span style="border: 1pt none windowtext; mso-border-alt: none windowtext 0cm; padding: 0cm;">frequency</span> and the minimum sampling rate is known as
the Nyquist rate.<o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; line-height: 21.4pt; margin-bottom: 4.5pt; vertical-align: top;">
<b><span style="font-size: 10.5pt;">13. What is cut-off frequency?</span></b><span style="font-size: 10.5pt;"><br />
The frequency at which the response is -3dB with respect to the
maximum response.<o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; line-height: 21.4pt; margin-bottom: 4.5pt; vertical-align: top;">
<b><span style="font-size: 10.5pt;">14. What is pass band?</span></b><span style="font-size: 10.5pt;"><br />
Passband is the range of <span style="border: 1pt none windowtext; mso-border-alt: none windowtext 0cm; padding: 0cm;">frequencies</span> or
wavelengths that can pass through a filter without being attenuated.<o:p></o:p></span></div>
<b><span style="font-size: 10.5pt; line-height: 107%;">15. What is the principle
of microwave?</span></b><span style="font-size: 10.5pt; line-height: 107%;"><br />
Microwave essentially means very short wave. The
microwave frequency spectrum is usually</span></span></div>
<div>
<span face=""helvetica" , "sans-serif"" style="color: #1d2129; font-size: 10.5pt; line-height: 107%;"><span style="font-size: 10.5pt; line-height: 107%;"><span face=""helvetica" , sans-serif" style="font-size: 10.5pt;">taken to extend from </span><span face=""helvetica" , sans-serif" style="border: 1pt none; font-size: 10.5pt; padding: 0cm;">1GHZ</span><span face=""helvetica" , sans-serif" style="font-size: 10.5pt;"> to 30GHZ. The
main reason why we have to go in for microwave frequency for
communication is that lower frequency band are congested and demand
for point to point communication continue to increase. The propagation of the
microwave takes place in spacewave in v</span></span></span></div>
<div>
<span face=""helvetica" , "sans-serif"" style="color: #1d2129; font-size: 10.5pt; line-height: 107%;"><span style="font-size: 10.5pt; line-height: 107%;"><br /></span></span></div>
<div class="MsoNormal" style="background: white; line-height: 21.4pt; margin-bottom: 4.5pt; vertical-align: top;">
<b><span face=""helvetica" , "sans-serif"" style="color: #1d2129; font-size: 10.5pt;">Q. What is latch up?</span></b><span face=""helvetica" , "sans-serif"" style="color: #1d2129; font-size: 10.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; line-height: 21.4pt; margin-bottom: 4.5pt; vertical-align: top;">
<span face=""helvetica" , "sans-serif"" style="color: #1d2129; font-size: 10.5pt;">Latch-up pertains to a failure mechanism wherein a parasitic thyristor
(such as a parasitic silicon controlled rectifier, or SCR) is inadvertently
created within a circuit, causing a high amount of current to continuously flow
through it once it is accidentally triggered or turned on. Depending on the
circuits involved, the amount of current flow produced by this mechanism
can <span style="border: 1pt none windowtext; mso-border-alt: none windowtext 0cm; padding: 0cm;">be large</span> enough to result in permanent destruction of
the device due to electrical overstress (EOS) .<o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; line-height: 21.4pt; margin-bottom: 4.5pt; vertical-align: top;">
<b><span face=""helvetica" , "sans-serif"" style="color: #1d2129; font-size: 10.5pt;">Q. What is microcontroller?</span></b><span face=""helvetica" , "sans-serif"" style="color: #1d2129; font-size: 10.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; line-height: 21.4pt; margin-bottom: 4.5pt; vertical-align: top;">
<span face=""helvetica" , "sans-serif"" style="color: #1d2129; font-size: 10.5pt;">A microcontroller is a single chip that contains the processor (the
CPU), non-volatile memory for the program (ROM or flash), volatile memory for
input and output (RAM), a clock and an I/O control unit.<o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; line-height: 21.4pt; margin-bottom: 4.5pt; vertical-align: top;">
<b><span face=""helvetica" , "sans-serif"" style="color: #1d2129; font-size: 10.5pt;">. What is the principle of microwave?</span></b><span face=""helvetica" , "sans-serif"" style="color: #1d2129; font-size: 10.5pt;"><br /><o:p></o:p></span></div>
<div>
<span face=""helvetica" , "sans-serif"" style="color: #1d2129; font-size: 10.5pt; line-height: 107%;"><span style="font-size: 10.5pt; line-height: 107%;"><span face=""helvetica" , sans-serif" style="background-color: white; font-size: 10.5pt;">Microwave essentially means very short wave. The
microwave frequency spectrum is usually taken to extend
from 1GHZ to 30GHZ. The main reason why we have to go in for
microwave frequency for communication is that
lower frequency band are congested and demand for point to point
communication continue to increase. The propagation of the microwave takes
place in spacewave in view of high gain and directivity in the form of a </span><span face=""helvetica" , sans-serif" style="background-color: white; border: 1pt none; font-size: 10.5pt; padding: 0cm;">bean</span><span face=""helvetica" , sans-serif" style="background-color: white; font-size: 10.5pt;"> and is similar to that of light.</span> </span></span></div>
</div>
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</head>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3005862392404377111.post-5524348268403339732017-11-20T17:28:00.004-08:002023-06-23T23:34:11.326-07:00Important Question 1<div dir="ltr" style="text-align: left;" trbidi="on">
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
<span style="box-sizing: border-box; font-weight: 700;">1. What is Electronic?</span></div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
The study and use of electrical devices that operate by controlling the flow of electrons or other electrically charged particles.</div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
<br /></div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
<span style="box-sizing: border-box; font-weight: 700;">2. What is communication?</span></div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
Communication means transferring a signal from the transmitter which passes through a medium then the output is obtained at the receiver. (or)communication says as transferring of message from one place to another place called communication.</div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
<br style="box-sizing: border-box;" />
<span style="box-sizing: border-box; font-weight: 700;">3. Different types of communications? Explain.</span></div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
Analog and digital communication.<br />
As a technology, analog is the process of taking an audio or video signal (the human voice) and translating it into electronic pulses. Digital on the other hand is breaking the signal into a binary format where the audio or video data is represented by a series of "1"s and "0"s.<br />
Digital signals are immune to noise, quality of transmission and reception is good, components used in digital communication can be produced with high precision and power consumption is also very less when compared with analog signals.</div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
<br style="box-sizing: border-box;" />
<span style="box-sizing: border-box; font-weight: 700;">4. What is sampling?</span></div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
The process of obtaining a set of samples from a continuous function of time x(t) is referred to as sampling.</div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
<br style="box-sizing: border-box;" />
<span style="box-sizing: border-box; font-weight: 700;">5. State sampling theorem.</span></div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
It states that, while taking the samples of a continuous signal, it has to be taken care that the sampling rate is equal to or greater than twice the cut off frequency and the minimum sampling rate is known as the Nyquist rate.</div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
<br style="box-sizing: border-box;" />
<span style="box-sizing: border-box; font-weight: 700;">6. What is cut-off frequency?</span></div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
The frequency at which the response is -3dB with respect to the maximum response.</div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
<br style="box-sizing: border-box;" />
<span style="box-sizing: border-box; font-weight: 700;">7. What is pass band?</span></div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
Passband is the range of frequencies or wavelengths that can pass through a filter without being attenuated.</div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
<br style="box-sizing: border-box;" />
<span style="box-sizing: border-box; font-weight: 700;">8. What is stop band?</span></div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
A stopband is a band of frequencies, between specified limits, in which a circuit, such as a filter or telephone circuit, does not let signals through, or the attenuation is above the required stopband attenuation level.</div>
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<br style="box-sizing: border-box;" />
<span style="box-sizing: border-box; font-weight: 700;">9. Explain RF?</span></div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
Radio frequency (RF) is a frequency or rate of oscillation within the range of about 3 Hz to 300 GHz. This range corresponds to frequency of alternating current electrical signals used to produce and detect radio waves. Since most of this range is beyond the vibration rate that most mechanical systems can respond to, RF usually refers to oscillations in electrical circuits or electromagnetic radiation.</div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
<br style="box-sizing: border-box;" />
<span style="box-sizing: border-box; font-weight: 700;">10. What is modulation? And where it is utilized?</span></div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
Modulation is the process of varying some characteristic of a periodic wave with an external signals.<br />
Radio communication superimposes this information bearing signal onto a carrier signal.<br />
These high frequency carrier signals can be transmitted over the air easily and are capable of travelling long distances.<br />
The characteristics (amplitude, frequency, or phase) of the carrier signal are varied in accordance with the information bearing signal.<br />
Modulation is utilized to send an information bearing signal over long distances.</div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
<br style="box-sizing: border-box;" />
<span style="box-sizing: border-box; font-weight: 700;">11. What is demodulation?</span></div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
Demodulation is the act of removing the modulation from an analog signal to get the original baseband signal back. Demodulating is necessary because the receiver system receives a modulated signal with specific characteristics and it needs to turn it to base-band.</div>
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<br style="box-sizing: border-box;" />
<span style="box-sizing: border-box; font-weight: 700;">12. Name the modulation techniques.</span></div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
For Analog modulation--AM, SSB, FM, PM and SM<br />
Digital modulation--OOK, FSK, ASK, Psk, QAM, MSK, CPM, PPM, TCM, OFDM</div>
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<br style="box-sizing: border-box;" />
<span style="box-sizing: border-box; font-weight: 700;">13. Explain AM and FM.</span></div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
AM-Amplitude modulation is a type of modulation where the amplitude of the carrier signal is varied in accordance with the information bearing signal.<br />
FM-Frequency modulation is a type of modulation where the frequency of the carrier signal is varied in accordance with the information bearing signal.</div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
<br style="box-sizing: border-box;" />
<span style="box-sizing: border-box; font-weight: 700;">14. Where do we use AM and FM?</span></div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
AM is used for video signals for example TV. Ranges from 535 to 1705 kHz.<br />
FM is used for audio signals for example Radio. Ranges from 88 to 108 MHz.</div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
<br style="box-sizing: border-box;" />
<span style="box-sizing: border-box; font-weight: 700;">15. What is a base station?</span></div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
Base station is a radio receiver/transmitter that serves as the hub of the local wireless network, and may also be the gateway between a wired network and the wireless network.</div>
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<span style="box-sizing: border-box; font-weight: 700;">16. How many satellites are required to cover the earth?</span></div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
3 satellites are required to cover the entire earth, which is placed at 120 degree to each other. The life span of the satellite is about 15 years.</div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
<br style="box-sizing: border-box;" />
<span style="box-sizing: border-box; font-weight: 700;">17. What is a repeater?</span></div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
A repeater is an electronic device that receives a signal and retransmits it at a higher level and/or higher power, or onto the other side of an obstruction, so that the signal can cover longer distances without degradation.</div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
<span style="font-weight: 700;">18. What is an Amplifier?</span></div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
An electronic device or electrical circuit that is used to boost (amplify) the power, voltage or current of an applied signal.</div>
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<br style="box-sizing: border-box;" />
<span style="box-sizing: border-box; font-weight: 700;">19. Example for negative feedback and positive feedback?</span></div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
Example for ve feedback is ---Amplifiers And for +ve feedback is Oscillators</div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
<br style="box-sizing: border-box;" />
<span style="box-sizing: border-box; font-weight: 700;">20. What is Oscillator?</span></div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
An oscillator is a circuit that creates a waveform output from a direct current input. The two main types of oscillator are harmonic and relaxation. The harmonic oscillators have smooth curved waveforms, while relaxation oscillators have waveforms with sharp changes.</div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
<br style="box-sizing: border-box;" />
<span style="box-sizing: border-box; font-weight: 700;">21. What is an Integrated Circuit?</span></div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
An integrated circuit (IC), also called a microchip, is an electronic circuit etched onto a silicon chip. Their main advantages are low cost, low power, high performance, and very small size.</div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
<br style="box-sizing: border-box;" />
<span style="box-sizing: border-box; font-weight: 700;">22. What is crosstalk?</span></div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
Crosstalk is a form of interference caused by signals in nearby conductors. The most common example is hearing an unwanted conversation on the telephone. Crosstalk can also occur in radios, televisions, networking equipment, and even electric guitars.</div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
<br style="box-sizing: border-box;" />
<span style="box-sizing: border-box; font-weight: 700;">23. What is resistor?</span></div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
A resistor is a two-terminal electronic component that opposes an electric current by producing a voltage drop between its terminals in proportion to the current, that is, in accordance with Ohm's law:<br />
V = IR.</div>
<div style="background-color: white; box-sizing: border-box; color: #333333; font-family: verdana, arial, sans-serif; font-size: 13px; margin-bottom: 10px;">
<br style="box-sizing: border-box;" />
<span style="box-sizing: border-box; font-weight: 700;">25. What is inductor?</span><br />
An inductor is a passive electrical device employed in electrical circuits for its property of inductance. An inductor can take many forms.</div>
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<span style="box-sizing: border-box; font-weight: 700;">26. What is conductor?</span></div>
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A substance, body, or device that readily conducts heat, electricity, sound, etc. Copper is a good conductor of electricity.</div>
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<span style="box-sizing: border-box; font-weight: 700;">27. What is a semi conductor?</span></div>
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A semiconductor is a solid material that has electrical conductivity in between that of a conductor and that of an insulator(An Insulator is a material that resists the flow of electric current. It is an object intended to support or separate electrical conductors without passing current through itself); it can vary over that wide range either permanently or dynamically.</div>
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<span style="box-sizing: border-box; font-weight: 700;">28. What is diode?</span></div>
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In electronics, a diode is a two-terminal device. Diodes have two active electrodes between which the signal of interest may flow, and most are used for their unidirectional current property.</div>
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<span style="box-sizing: border-box; font-weight: 700;">29. What is transistor?</span></div>
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In electronics, a transistor is a semiconductor device commonly used to amplify or switch electronic signals. The transistor is the fundamental building block of computers, and all other modern electronic devices. Some transistors are packaged individually but most are found in integrated circuits</div>
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<span style="box-sizing: border-box; font-weight: 700;">30. What is op-amp?</span></div>
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An operational amplifier, often called an op-amp , is a DC-coupled high-gain electronic voltage amplifier with differential inputs[1] and, usually, a single output. Typically the output of the op-amp is controlled either by negative feedback, which largely determines the magnitude of its output voltage gain, or by positive feedback, which facilitates regenerative gain and oscillation.</div>
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<span style="box-sizing: border-box; font-weight: 700;">31. What is a feedback?</span></div>
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Feedback is a process whereby some proportion of the output signal of a system is passed (fed back) to the input. This is often used to control the dynamic behaviour of the system.</div>
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<span style="box-sizing: border-box; font-weight: 700;">32. Advantages of negative feedback over positive feedback.</span></div>
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Much attention has been given by researchers to negative feedback processes, because negative feedback processes lead systems towards equilibrium states. Positive feedback reinforces a given tendency of a system and can lead a system away from equilibrium states, possibly causing quite unexpected results.</div>
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<span style="box-sizing: border-box; font-weight: 700;">33. What is Barkhausen criteria?</span></div>
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Barkhausen criteria, without which you will not know which conditions, are to be satisfied for oscillations.<br />
Oscillations will not be sustained if, at the oscillator frequency, the magnitude of the product of the<br />
transfer gain of the amplifier and the magnitude of the feedback factor of the feedback network ( the magnitude of the loop gain ) are less than unity.<br />
The condition of unity loop gain -A? = 1 is called the Barkhausen criterion. This condition implies that<br />
A?= 1and that the phase of - A? is zero.</div>
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<span style="box-sizing: border-box; font-weight: 700;">34. What is CDMA, TDMA, FDMA?</span></div>
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Code division multiple access (CDMA) is a channel access method utilized by various radio communication technologies. CDMA employs spread-spectrum technology and a special coding scheme (where each transmitter is assigned a code) to allow multiple users to be multiplexed over the same physical channel. By contrast, time division multiple access (TDMA) divides access by time, while frequency-division multiple access (FDMA) divides it by frequency.<br />
An analogy to the problem of multiple access is a room (channel) in which people wish to communicate with each other. To avoid confusion, people could take turns speaking (time division), speak at different pitches (frequency division), or speak in different directions (spatial division). In CDMA, they would speak different languages. People speaking the same language can understand each other, but not other people. Similarly, in radio CDMA, each group of users is given a shared code. Many codes occupy the same channel, but only users associated with a particular code can understand each other.</div>
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<span style="box-sizing: border-box; font-weight: 700;">35. explain different types of feedback</span></div>
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Types of feedback:<br />
Negative feedback: This tends to reduce output (but in amplifiers, stabilizes and linearizes operation). Negative feedback feeds part of a system's output, inverted, into the system's input; generally with the result that fluctuations are attenuated.<br />
Positive feedback: This tends to increase output. Positive feedback, sometimes referred to as "cumulative causation", is a feedback loop system in which the system responds to perturbation (A perturbation means a system, is an alteration of function, induced by external or internal mechanisms) in the same direction as the perturbation. In contrast, a system that responds to the perturbation in the opposite direction is called a negative feedback system.<br />
Bipolar feedback: which can either increase or decrease output.</div>
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<span style="box-sizing: border-box; font-weight: 700;">36. What are the main divisions of power system?</span></div>
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The generating system,transmission system,and distribution system</div>
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<span style="box-sizing: border-box; font-weight: 700;">37. </span><b><span style="font-family: "verdana" , sans-serif; font-size: 10pt;">What is the need for base values? </span></b></div>
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<span style="color: #333333; font-family: "verdana" , "sans-serif"; font-size: 10pt;">The components of power system may operate at different voltage and power levels. It will be convenient for analysis of power system if the voltage, power, current ratings of the components of the power system is expressed with referance to a common value called base value.<o:p></o:p></span></div>
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<span style="box-sizing: border-box; font-weight: 700;">38. What is meant by impedance diagram. </span></div>
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The equivalent circuit of all the components of the power system are drawn and they are interconnected is called impedance diagram.</div>
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<span style="box-sizing: border-box; font-weight: 700;">39. What is the need for load flow study.</span><br />
The load flow study of a power system is essential to decide the best operation existing system and for planning the future expansion of the system. It is also essential for designing the power system.</div>
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<span style="box-sizing: border-box; font-weight: 700;">40. </span><b><span style="font-family: "verdana" , sans-serif; font-size: 10pt;">What is Instrumentation Amplifier (IA) and what are all the advantages?</span></b></div>
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<span style="color: #333333; font-family: "verdana" , "sans-serif"; font-size: 10pt;">An instrumentation amplifier is a differential op-amp circuit providing high input impedances with ease of gain adjustment by varying a single resistor.</span><br />
<span style="color: #333333; font-family: "verdana" , "sans-serif"; font-size: 10pt;"><br /></span>
<span style="color: #333333; font-family: "verdana" , sans-serif;"><span style="font-size: 13.3333px;"><b>41. </b></span></span><span style="color: #484848; font-family: "arial" , sans-serif;"><b>Explain the concept
of frequency re-use.</b></span><br />
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<span style="font-family: "arial" , sans-serif; font-size: 10pt;">The whole of the geographical area is divided into
hexagonal shape geometrical area called cell and each cell having its own
transceiver. Each BTS (cell site) allocated different band of frequency or
different channel. Each BTS antenna is designed in such a way that i cover cell
area in which it is placed with frequency allotted without interfering other
cell signals. The design process of selecting and allocating channel groups for
all of the cellular base station within system is called frequency reuse.<o:p></o:p></span></div>
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<b>42. </b><span style="color: #484848; font-family: "arial" , sans-serif;"><b>What is CDMA?</b></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 10pt;">CDMA stands for Code Division Multiple Access which
uses digital format. In CDMA systems several transmissions via the radio
interface take place simultaneously on the same frequency bandwidth. User data
is combined at the transmitter’s side with a code, then transmitted. On air,
all transmission get mixed. At the receiver's side the same code is used as in
the transmitter’s side. The code helps the receiver to filter the user
information of the transmitter from incoming mixture of all transmissions on
the same frequency band and same time.<o:p></o:p></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 10pt;"><b>43. </b></span><span style="color: #484848; font-family: "arial" , sans-serif;"><b>Explain Bluetooth.</b></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 10pt;">Bluetooth is designed to be a personal area
network, where participating entities are mobile and require sporadic
communication with others. It is omni directional i.e. it does not have line of
sight limitation like infra-red does. Ericsson started the work on Bluetooth
and named it after the Danish king Harold Bluetooth. Bluetooth operates in the
2.4 GHz area of spectrum and provides a range of 10 metres. It offers transfer
speeds of around 720 Kbps.<o:p></o:p></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 10pt;"><b>44. </b></span><span style="color: #333333; font-family: "verdana" , sans-serif; font-size: 10pt;"> </span><b style="color: #484848; font-family: Arial, sans-serif;">What are different categories of antenna and give an example of each?</b></div>
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<br /></div>
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<span style="font-family: "arial" , sans-serif; font-size: 10pt;">Different categories of antenna are as follows: <br />
<br />
1. Wire Antennas - Short Dipole Antenna<br />
2. Microstrip Antennas - Rectangular Microstrip (Patch) Antennas<br />
3. Reflector Antennas - Corner Reflector<br />
4. Travelling Wave Antennas - Helical Antennas<br />
5. Aperture Antennas - Slot Antenna<br />
6. Other Antennas - NFC Antennas<o:p></o:p></span></div>
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<span style="color: #484848; font-family: "arial" , sans-serif;"><b style="font-size: 13.5pt;">45. </b><b>What is handover
and what are its types?</b><span style="font-size: 13.5pt;"><o:p></o:p></span></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 10pt;">Handover in mobile communication refers to the
process of transferring a call from one network cell to another without
breaking the call.<br />
There are two types of handover which are as follows: <br />
<br />
Hard Handoff: hard handoff is the process in which the cell connection is
disconnected from the previous cell before it is made with the new one.<br />
<br />
Soft Handoff: It is the process in which a new connection is established first
before disconnecting the old one. It is thus more efficient and smart.<o:p></o:p></span></div>
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<span style="color: #484848; font-family: "arial" , sans-serif;"><b>46.What is ionospheric
bending?</b><span style="font-size: 13.5pt;"><o:p></o:p></span></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 10pt;">When a radio wave travels into the ionospheric
layer it experiences refraction due to difference in density. The density of
ionospheric layer is rarer than the layer below which causes the radio wave to
be bent away from the normal. Also the radio wave experiences a force from the
ions in the ionospheric layer. If incident at the correct angle the radio wave
is completely reflected back to the inner atmosphere due to total internal
reflection. This phenomenon is called ionospheric reflection and is used in
mobile communication for radio wave propagation also known as ionospheric
bending of radio waves.<o:p></o:p></span></div>
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<span style="font-family: "arial" , sans-serif;"><b>47. </b></span><span style="color: #484848; font-family: "arial" , sans-serif; font-size: 13.5pt; line-height: 107%;"><b>What are GPRS services?</b></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 10pt;">GPRS services are defined to fall in one of the two categories :</span><span style="color: #484848; font-family: "arial" , "sans-serif"; font-size: 13.5pt; font-weight: normal;"><o:p></o:p></span></h2>
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<span style="color: #484848; font-family: "arial" , sans-serif; font-size: 13.5pt; line-height: 107%;">
</span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 10pt;">- PTP ( Point to point)<br />
- PTM ( Point to Multipoint)<br />
Some of the GPRS services are not likely to be provided by network operators
during early deployment of GPRS due in part to the phased development of
standard. Market demand is another factor affecting the decision of operators
regarding which services to offer first.<o:p></o:p></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 10pt;"><br /></span></div>
<h2>
<span style="color: #484848; font-family: "arial" , "sans-serif"; font-size: 13.5pt;">48. </span><span style="color: #484848; font-family: "arial" , "sans-serif"; font-size: 12.0pt;">What are the advantages of
CDMA?</span></h2>
<h2>
<span style="font-weight: normal;"><span style="font-size: small;">Advantages
of CDMa are as follows :</span></span></h2>
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<span style="font-family: "arial" , sans-serif; font-size: 10pt;">1. Frequency diversity : Transmission is spread out over a large bandwidth due
to that less affected by noise. If bandwidth is increased S/N ratio increases,
which means noise will be reduced.<br />
2. Multiplication Resistance : Chipping codes used for CDMA not only exhibit
low correlation but also low autocorrelation. Hence a version of the signal
that is delayed by more than one chip interval does not interfere with dominant
signal as in other multipath environments.<br />
3. Privacy : Due to spread spectrum is obtained by the use of noise like
signals, where each user has a unique code, so privacy is inherent.<br />
4. Graceful Degradation. In CDMA, more users access the system simultaneously
as compared to FDMa, TDMA.<o:p></o:p></span></div>
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<br /></div>
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<o:p></o:p></div>
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<span style="color: #484848; font-family: "arial" , "sans-serif"; font-size: 13.5pt;">49. What are the advantages of
spread spectrum?<o:p></o:p></span></h2>
<div>
<span style="font-family: "arial" , sans-serif; font-size: 10pt; line-height: 107%;">SPread spectrum has the following advantages :<br />
<br />
1. No crosstalk interference.<br />
2. Better voice quality/data integrity and less static noise.<br />
3. Lowered susceptibility to multipath fading.<br />
4. Inherent security.<br />
5. Co-existence.<br />
6. Longer operating distances.<br />
7. Hard to detect.<br />
8. Hard to intercept or demodulate.<br />
9. Harder to jam than narrow bands.<br />
10. Use of ranging and radar</span></div>
<div>
<span style="font-family: "arial" , sans-serif; font-size: 10pt; line-height: 107%;"><br /></span></div>
<div>
<span style="font-family: "arial" , sans-serif; font-size: 10pt; line-height: 107%;"></span><br />
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<span style="font-family: "arial" , sans-serif; font-size: 10pt; line-height: 107%;">
<span style="color: #484848; font-family: "arial" , "sans-serif"; font-size: 13.5pt;">50. Explain the steps involved in
demodulating a signal.<o:p></o:p></span></span></h2>
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</span>
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<span style="font-family: "arial" , sans-serif; font-size: 10pt; line-height: 107%;"><span style="font-family: "arial" , sans-serif; font-size: 10pt;">Once
the signal is coded, modulated and then sent, the receiver must demodulate the
signal. This is usually done in two steps :<br />
1. Spectrum spreading (e.g., direct sequence or frequency hopping) modulation
is removed.<br />
2. The remaining information bearing signal is demodulated by multiplying with
a local reference identical in structure and synchronised with received signal.<o:p></o:p></span></span></div>
<span style="font-family: "arial" , sans-serif; font-size: 10pt; line-height: 107%;">
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<span style="font-family: "arial" , sans-serif; font-size: 10pt;"><br /></span></div>
<h2 style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial;">
<span style="color: #484848; font-family: "arial" , "sans-serif"; font-size: 13.5pt;">51. How can a Pseudo Random Noise
COde be usable?<o:p></o:p></span></h2>
<div style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial;">
<span style="font-family: "arial" , sans-serif; font-size: 10pt; line-height: 107%;">To
be usable for direct sequence spreading, a PN code must meet the following
conditions :<br />
<br />
1. Sequence must be built from 2 levelled numbers.<br />
2. The codes must have sharp autocorrelation peak to enable code
synchronization.<br />
3. Codes must have a low cross-correlation value, the lower it is, more are the
number of users which can be allowed in the system.<br />
4. The codes should be “balanced” i.e. the difference between ones and zeros in
code may only be 1.</span></div>
</span></div>
</div>
</div>
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</head>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3005862392404377111.post-11494859070090488692017-10-05T07:16:00.003-07:002023-06-23T23:34:11.452-07:00Temperature Controlled DC Fan using Microcontroller<div dir="ltr" style="text-align: left;" trbidi="on">
<div class="toc_title" style="box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 15.2px; font-weight: 700; padding: 0px; text-align: center;">
<br /></div>
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<span style="color: #666666; font-family: "raleway" , sans-serif;"><span style="font-size: 15.2px;"><br /></span></span></div>
<div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
<span style="box-sizing: border-box; color: black;">A Temperature Controlled DC Fan is a system which automatically turns on a DC Fan when the ambient temperature increases above a certain limit.</span></div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
<span style="box-sizing: border-box; color: black;">Generally, electronic devices produce more heat. So this heat should be reduced in order to protect the device. There are many ways to reduce this heat. One way is to switch on the fan spontaneously.</span></div>
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<span style="box-sizing: border-box; color: black;">This article describes two such circuits that automatically, switches the fan when it detects the temperature inside the device greater than its threshold value. </span></div>
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<span style="box-sizing: border-box; color: black;"><br /></span></div>
<h3 style="background-color: white; box-sizing: border-box; font-family: Raleway, sans-serif; font-size: 24px; line-height: 1.2; margin: 0px 0px 16px;">
<span id="Circuit_1_Temperature_Controlled_DC_Fan_using_8051" style="box-sizing: border-box;">Circuit 1 Temperature Controlled DC Fan using 8051</span></h3>
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<span style="box-sizing: border-box; color: black;"></span></div>
<h4 style="background-color: white; box-sizing: border-box; font-family: Raleway, sans-serif; font-size: 20px; line-height: 1.2; margin: 0px 0px 16px;">
<span id="Circuit_Diagram" style="box-sizing: border-box;">Circuit Diagram</span></h4>
</div>
<div>
<span style="box-sizing: border-box;"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhbltGga6ST_xaE6lsDyUJtxuZYQfH8jojnEtZ9TGJW9EtgMpCiWwjgjiFoYaS7SMimALoi73Jribi74_6snspjDaR0XjY8hCWE9k9Quu2ilitTLeQYdhX3Yljwa-k4S_AUKbddQNj_BuM/s1600/Temperature-Controlled-DC-Fan-using-8051-Circuit-Diagram.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="418" data-original-width="750" height="355" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhbltGga6ST_xaE6lsDyUJtxuZYQfH8jojnEtZ9TGJW9EtgMpCiWwjgjiFoYaS7SMimALoi73Jribi74_6snspjDaR0XjY8hCWE9k9Quu2ilitTLeQYdhX3Yljwa-k4S_AUKbddQNj_BuM/s640/Temperature-Controlled-DC-Fan-using-8051-Circuit-Diagram.jpg" width="640" /></a></div>
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<span style="box-sizing: border-box;"><br /></span></div>
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<span style="box-sizing: border-box;"></span><br />
<h4 style="background-color: white; box-sizing: border-box; font-family: Raleway, sans-serif; font-size: 20px; line-height: 1.2; margin: 0px 0px 16px;">
<span style="box-sizing: border-box;">
<span id="Principle" style="box-sizing: border-box;">Principle</span></span></h4>
<span style="box-sizing: border-box;">
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
The project works on the principle of Analog to Digital Conversion. The Analog data from the LM35 temperature sensor is given to the analog to digital converter ADC0804.</div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
The analog output of the temperature sensor will vary at 10mV per degree Celsius.</div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
ADC0804 is an 8-bit ADC. For a reference voltage of 5V, we’ll get a resolution of 5V/28 = 20mV. Which means, this is the minimum change in the analog value from the sensor which is recognisable by the ADC IC.</div>
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As per the changes in the temperature, the output of the ADC is generated. The digital output of the ADC is given to Microcontroller to calculate the temperature and control the fan accordingly.</div>
<h4 style="background-color: white; box-sizing: border-box; font-family: Raleway, sans-serif; font-size: 20px; line-height: 1.2; margin: 0px 0px 16px;">
<span id="Components" style="box-sizing: border-box;">Components</span></h4>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
<span style="box-sizing: border-box; font-weight: 700;">Microcontroller Section</span></div>
<ul style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin: 0px 0px 26px 40px; padding: 0px;">
<li style="box-sizing: border-box; list-style-type: disc;">AT89C51 Microcontroller</li>
<li style="box-sizing: border-box; list-style-type: disc;">AT89C51 Programmer Board</li>
<li style="box-sizing: border-box; list-style-type: disc;">11.0592 MHz Quartz Crystal</li>
<li style="box-sizing: border-box; list-style-type: disc;">33pF Ceramic Capacitor</li>
<li style="box-sizing: border-box; list-style-type: disc;">2 x 10KΩ Resistor</li>
<li style="box-sizing: border-box; list-style-type: disc;">10µF Electrolytic Capacitor</li>
<li style="box-sizing: border-box; list-style-type: disc;">Push Button</li>
<li style="box-sizing: border-box; list-style-type: disc;">16 X 2 LCD Display</li>
<li style="box-sizing: border-box; list-style-type: disc;">10KΩ POT</li>
</ul>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
<span style="box-sizing: border-box; font-weight: 700;">Temperature Sensor Section</span></div>
<ul style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin: 0px 0px 26px 40px; padding: 0px;">
<li style="box-sizing: border-box; list-style-type: disc;">LM35</li>
<li style="box-sizing: border-box; list-style-type: disc;">ADC0804</li>
<li style="box-sizing: border-box; list-style-type: disc;">10KΩ Resistor</li>
<li style="box-sizing: border-box; list-style-type: disc;">150pF Ceramic Capacitor</li>
<li style="box-sizing: border-box; list-style-type: disc;">1KΩ x 8 Resistor Pack</li>
</ul>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
<span style="box-sizing: border-box; font-weight: 700;">Load Section</span></div>
<ul style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin: 0px 0px 26px 40px; padding: 0px;">
<li style="box-sizing: border-box; list-style-type: disc;">2N2222 NPN Transistor</li>
<li style="box-sizing: border-box; list-style-type: disc;">1N4007 Diode</li>
<li style="box-sizing: border-box; list-style-type: disc;">12V Relay</li>
<li style="box-sizing: border-box; list-style-type: disc;">1KΩ Resistor</li>
<li style="box-sizing: border-box; list-style-type: disc;">Fan</li>
</ul>
<div>
<h4 style="background-color: white; box-sizing: border-box; font-family: Raleway, sans-serif; font-size: 20px; line-height: 1.2; margin: 0px 0px 16px;">
<span id="Configuring_ADC0804_for_this_Project" style="box-sizing: border-box;">Configuring ADC0804 for this Project</span></h4>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
The configuration of the ADC0804 is explained here. First, we need to connect the 5V regulated power supply to the Vcc pin (Pin 20). Then, connect the analog and digital ground pins (Pins 8 and 10) to the GND.</div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
In order to use the internal clock, we need to connect a 10KΩ resistor between CLK IN (Pin 4 and CLK R (Pin 19) and then, connect a 150pF cap between pins 4 and GND to complete the oscillator circuit.</div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
The CS pin (Pin 1) is connected to GND to enable the ADC.</div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
In order to read the data from the ADC continuously by the microcontroller, we need to connect the RD pin (Pin 2) to the GND.</div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
For the ADC to continuously read the analog data from the sensor, we need to short the Write pin (Pin 3) with the Interrupt pin (Pin 5).</div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
The analog output of the sensor (LM35) is connected to the Vin+ (Pin 6) of the ADC. The negative analog input pin i.e. Vin- of the ADC is connected to the GND.</div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
The converted digital data which is an 8-bit data will be available at DB0 to DB7 (Pins 18 to 11).</div>
<h4 style="background-color: white; box-sizing: border-box; font-family: Raleway, sans-serif; font-size: 20px; line-height: 1.2; margin: 0px 0px 16px;">
<span id="Circuit_Design" style="box-sizing: border-box;">Circuit Design</span></h4>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
The main components of the project are 8051 Microcontroller, 16×2 LCD Display, LM35 Temperature Sensor, ADC0804, Relay and Fan.</div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
The basic connections with respect to microcontroller include clock, reset and EA. Clock consists of an 11.0592 MHz crystal and two 33pF capacitors. The reset circuit consists of a 10µF capacitor, 10KΩ resistor and a push button. The EA pin is pulled high with a 10KΩ resistor.</div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
Now we’ll see the connections with respect to other components.</div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
For the LCD display, a 10KΩ pot is connected to contrast adjust pin. The three control pins of the LCD are connected to the pins P3.6, GND and P3.7.</div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
The 8 data pins of the LCD are connected to PORT1 of the microcontroller.</div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
The basic connections with respect to ADC are explained in its configuration. The 8 digital outputs of the ADC must be connected to PORT 2 of the microcontroller.</div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
The next component we are going to connect is LM35. Connect the data pin of the LM35 to the analog input pin i.e. Pin 6 of ADC.</div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
Finally, we need to connect the Relay circuit consisting of resistor, transistor and relay to the PORT0 of the microcontroller with PORT 0 pulled-up externally.</div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
Connect the input of relay i.e. base of the transistor to P0.0 pin of the microcontroller.</div>
<h4 style="background-color: white; box-sizing: border-box; font-family: Raleway, sans-serif; font-size: 20px; line-height: 1.2; margin: 0px 0px 16px;">
<span id="Working" style="box-sizing: border-box;">Working</span></h4>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
The aim of this project is to design a temperature controlled fan using 8051 microcontroller, in which the fan is automatically turned ON or OFF according to the temperature. The working of the project is explained here.</div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
In this circuit, the LM35 temperature sensor will give continuous analog output corresponding to the temperature sensed by it. This analog signal is given to the ADC, which converts the analog values to digital values.</div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
The digital output of the ADC is equivalent to sensed analog voltage.</div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
In order to get the temperature from the sensed analog voltage, we need to perform some calculations in the programming for the microcontroller.</div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
Once the calculations are done by the microcontroller according to the logic, the temperature is displayed on the LCD. Like this, the microcontroller will continuously monitor the temperature.</div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
If the temperature exceeds more than 50 deg Celsius (as per the code), the microcontroller will turn on the relay to start the fan. If the temperature drops below 40 deg Celsius (as per the </div>
</div>
</span></div>
</div>
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</head>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3005862392404377111.post-87032397795825821922017-10-05T07:07:00.002-07:002023-06-23T23:34:12.510-07:00Water Level Indicator Mini Project<div dir="ltr" style="text-align: left;" trbidi="on">
<div class="toc_title" style="box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-weight: 700; padding: 0px; text-align: center;">
<br /></div>
<div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px; text-align: justify;">
<span style="box-sizing: border-box; color: black;"> The Water Level Indicator employs a simple mechanism to detect and indicate the water level in an overhead tank or any other water container.</span></div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px; text-align: justify;">
<span style="box-sizing: border-box; color: black;">The sensing is done by using a set of nine probes which are placed at nine different levels on the tank walls (with probe 9 to probe 1 placed in increasing order of height, common probe (i.e. a supply carrying probe) is placed at the base of the tank). The level 8 represents the “tank full” condition while level 0 represents the “tank empty” condition.</span></div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
<span style="box-sizing: border-box; color: black;">When the water-level is below the minimum detectable level (MDL), the seven segment display is arranged to show the digit 0, indicating that the tank is empty, when the water reaches level1 (but is below level2) the connection between the probes gets completed (through the conducting medium – water) and the base voltage of transistor increases.</span></div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
<span style="box-sizing: border-box; color: black;">This causes the base-emitter junction of transistor to get forward biased, this switches transistor from cut-off to conduction mode thus PIN (B7) of microcontroller is pulled to ground hence, the corresponding digit displayed by the seven segment display is 1.</span></div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
<span style="box-sizing: border-box; color: black;">The similar mechanism applies to the detection of all the other levels. When the tank is full, all input pins of microcontroller become low. This causes the display to show 8 and also in this case a buzzer sound is given, thereby indicating a “tank full” condition.</span></div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
<span style="box-sizing: border-box; color: black;">Most water level indicators are equipped to indicate and detect only a single level. The Water Level Indicator implemented here can indicate up to nine such levels and the microcontroller displays the level number on a seven segment display.</span></div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px; text-align: justify;">
<span style="box-sizing: border-box; color: black;"></span></div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
<span style="box-sizing: border-box; color: black;">So, the circuit not only capable of cautioning a person that the water tank has been filled up to certain level, but also indicates that the water level has fallen below the minimum detectable level. This circuit is important in appliances such as the water cooler where there is a danger of motor-burnout when there is no water in the radiator used up also it can be used in fuel level indication.</span></div>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
<span style="box-sizing: border-box; color: black;">In this project we show the water level indicator using eight transistors which conducts as level rises, a buzzer is also added which will automatically start as the water level becomes full, auto buzzer start with the help of microcontroller. With the help of this project we not only show the level of water on seven segment display but also indicate the water full condition using a buzzer.</span></div>
<h3 style="background-color: white; box-sizing: border-box; font-family: Raleway, sans-serif; font-size: 24px; line-height: 1.2; margin: 0px 0px 16px;">
<span id="Water_Level_Indicator_Project_Circuit_Features" style="box-sizing: border-box;">Water Level Indicator Project Circuit Features:</span></h3>
<div style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin-bottom: 26px; padding: 0px;">
<span style="box-sizing: border-box; color: black;"></span></div>
<ol style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin: 0px 0px 26px 40px; padding: 0px;">
<li style="box-sizing: border-box; list-style-type: decimal;"><span style="box-sizing: border-box; color: black;">Easy installation.</span></li>
<li style="box-sizing: border-box; list-style-type: decimal;"><span style="box-sizing: border-box; color: black;">Low maintenance.</span></li>
<li style="box-sizing: border-box; list-style-type: decimal;"><span style="box-sizing: border-box; color: black;">Compact elegant design.</span></li>
<li style="box-sizing: border-box; list-style-type: decimal;"><span style="box-sizing: border-box; color: black;">The Automatic water level controller ensures no overflows or dry running of pump there by saves electricity and water.</span></li>
<li style="box-sizing: border-box; list-style-type: decimal;"><span style="box-sizing: border-box; color: black;">Avoid seepage of roofs and walls due to overflowing tanks.</span></li>
<li style="box-sizing: border-box; list-style-type: decimal;"><span style="box-sizing: border-box; color: black;">Fully automatic, saves man power.</span></li>
<li style="box-sizing: border-box; list-style-type: decimal;"><span style="box-sizing: border-box; color: black;">Consume very little energy, ideal for continuous operation.</span></li>
<li style="box-sizing: border-box; list-style-type: decimal;"><span style="box-sizing: border-box; color: black;">Automatic water level controller provides you the flexibility to decide for yourself the water levels for operations of pump set.</span></li>
<li style="box-sizing: border-box; list-style-type: decimal;"><span style="box-sizing: border-box; color: black;">Shows clear indication of water levels in the overhead tank.</span></li>
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Water Level Indicator Project Block Diagram</h3>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj8BCnAwAc9J_anYuXUv3NFzgr0-CMCoxgK2I54iDr6nOU_5aQCPekCFby3XRTanMahqkGdVJaRISkM7C9kWP6RCmadCp7FBVP5s0gA9BuF3EjFI5Lctz3QaA6x1ZdeB0VvZtm2PRIkMOI/s1600/Water-Level-Indicator-Block-Diagram.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="482" data-original-width="663" height="232" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj8BCnAwAc9J_anYuXUv3NFzgr0-CMCoxgK2I54iDr6nOU_5aQCPekCFby3XRTanMahqkGdVJaRISkM7C9kWP6RCmadCp7FBVP5s0gA9BuF3EjFI5Lctz3QaA6x1ZdeB0VvZtm2PRIkMOI/s320/Water-Level-Indicator-Block-Diagram.jpg" width="320" /></a></div>
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<span id="Water_Level_Indicator_Circuit_Diagram" style="box-sizing: border-box;">Water Level Indicator Circuit Diagram:</span></h3>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhb1lZUcyuNO-MaH0kRaOT7mPRIO_P7B2qsU83ytivSOGEnlYrVtgaVbfT-XUy1xDjsuQosI4xdWUMEmX9-h4eXpr91-dKJh4MDIXv_LJTaOjt4xcETlx0isJG3EmyvJ1pfh5TWyLIJElA/s1600/Water-Level-Indicator-Schematic.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1049" data-original-width="1190" height="282" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhb1lZUcyuNO-MaH0kRaOT7mPRIO_P7B2qsU83ytivSOGEnlYrVtgaVbfT-XUy1xDjsuQosI4xdWUMEmX9-h4eXpr91-dKJh4MDIXv_LJTaOjt4xcETlx0isJG3EmyvJ1pfh5TWyLIJElA/s320/Water-Level-Indicator-Schematic.jpg" width="320" /></a></div>
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<span id="How_to_Design_Water_Level_Indicator_Project_using_AVR_Microcontroller" style="box-sizing: border-box;">How to Design Water Level Indicator Project using AVR Microcontroller?</span></span></h3>
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<ul style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin: 0px 0px 26px 40px; padding: 0px;"><span style="box-sizing: border-box;">
<li style="box-sizing: border-box; list-style-type: disc;"><span style="box-sizing: border-box; color: black;">A constant 5v power supply is given to the microcontroller and rest of the circuit from a battery.</span></li>
<li style="box-sizing: border-box; list-style-type: disc;"><span style="box-sizing: border-box; color: black;">The tank has 9 conductive type sensors (other types of sensors have been mentioned earlier but in our project only conductive type are used) embedded into it and 8 wires of sensors out of 9 are connected to transistors and the 9th is connected to 5v+ supply.</span></li>
<li style="box-sizing: border-box; list-style-type: disc;"><span style="box-sizing: border-box; color: black;">The use of transistor is it acts as inverter (i.e. in on state gives low voltage at output and in non conducting state gives high voltage at its output), all transistors outputs are connected to PORTB of microcontroller.</span></li>
<li style="box-sizing: border-box; list-style-type: disc;"><span style="box-sizing: border-box; color: black;">Seven segment display is connected to PORTD. It is connected in common cathode fashion.</span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; color: black;">The Output for the 7th level is not only shown on seven segment display but also indicated with a discontinuous buzzer sound.</span></li>
<li style="box-sizing: border-box; list-style-type: disc;"><span style="box-sizing: border-box; color: black;">Output for the 8th level (i.e. tank full condition) is not only shown in seven segment display but also indicated with a continuous buzzer sound.</span></li>
</span></ul>
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<h3 style="background-color: white; box-sizing: border-box; font-family: Raleway, sans-serif; font-size: 24px; line-height: 1.2; margin: 0px 0px 16px;">
<span id="HowWater_Level_Indicator_Project_Circuit_Works" style="box-sizing: border-box;"><span style="box-sizing: border-box;"><span style="box-sizing: border-box;">How Water Level Indicator Project Circuit Works?</span></span></span></h3>
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<span style="box-sizing: border-box; color: black;">The operation of this project is very simple and can be understood easily. In our project “water level indicator” there are 3 main conditions:</span></div>
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<li style="box-sizing: border-box; list-style-type: decimal;"><span style="box-sizing: border-box; color: black;">There is no water available in the source tank.</span></li>
<li style="box-sizing: border-box; list-style-type: decimal;"><span style="box-sizing: border-box; color: black;">Intermediate level i.e. either of 3rd to 7th level.</span></li>
<li style="box-sizing: border-box; list-style-type: decimal;"><span style="box-sizing: border-box; color: black;">There is ample amount of water available in the source tank.</span></li>
</ol>
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<span style="box-sizing: border-box; color: black;">So let us discuss more about these 3 conditions</span></div>
<h4 style="background-color: white; box-sizing: border-box; font-family: Raleway, sans-serif; font-size: 20px; line-height: 1.2; margin: 0px 0px 16px;">
<span id="CONDITION_1_Water_not_available" style="box-sizing: border-box;"><span style="box-sizing: border-box;">CONDITION 1: Water not available</span></span></h4>
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<span style="box-sizing: border-box; color: black;">When the tank is empty there is no conductive path between any of the 8 indicating probes and the common probe (which is connected to 5v+ supply) so the transistor base emitter region will not have sufficient biasing voltage hence it remains in cut off region and the output across its collector will be Vc approximately 4.2v.</span></div>
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<span style="box-sizing: border-box; color: black;">As in this case the microcontroller is used in the active low region (which means it considers 0-2 volts for HIGH and 3-5 volts for LOW) now the output of transistor which is 4.2v approximately will be considered as LOW by the microcontroller and hence the default value given by microcontroller to the seven segment display is 0 which indicates as the tank is empty.</span></div>
<h4 style="background-color: white; box-sizing: border-box; font-family: Raleway, sans-serif; font-size: 20px; line-height: 1.2; margin: 0px 0px 16px;">
<span id="CONDITION_2_Intermediate_levels" style="box-sizing: border-box;"><span style="box-sizing: border-box;">CONDITION 2: Intermediate levels</span></span></h4>
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<span style="box-sizing: border-box; color: black;">Now as the water starts filling in the tank a conductive path is established between the sensing probes and the common probe and the corresponding transistors get sufficient biasing at their base, they starts conducting and now the outputs will be Vce (i.e. 1.2v-1.8v) approximately which is given to microcontroller.</span></div>
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<span style="box-sizing: border-box; color: black;">Here the microcontroller is programmed as a priority encoder which detects the highest priority input and displays corresponding water level in the seven segment display.</span></div>
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<span style="box-sizing: border-box; color: black;">In this project while the water level reaches the 7th level i.e. last but one level along with display in seven segment a discontinuous buzzer is activated which warns user that tank is going to be full soon.</span></div>
<h4 style="background-color: white; box-sizing: border-box; font-family: Raleway, sans-serif; font-size: 20px; line-height: 1.2; margin: 0px 0px 16px;">
<span id="CONDITION_3_Water_full" style="box-sizing: border-box;"><span style="box-sizing: border-box;">CONDITION 3: Water full</span></span></h4>
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<span style="box-sizing: border-box; color: black;">When the tank becomes full, the top level probe gets the conductive path through water and the corresponding transistor gets into conduction whose output given to microcontroller with this input microcontroller not only displays the level in seven segment display but also activates the continuous buzzer by which user can understand that tank is full and can switch off the motor and save water.</span></div>
<h4 style="background-color: white; box-sizing: border-box; font-family: Raleway, sans-serif; font-size: 20px; line-height: 1.2; margin: 0px 0px 16px;">
<span id="Water_Level_Indicator_Project_Working_Flow_Chart" style="box-sizing: border-box;"><span style="box-sizing: border-box;">Water Level Indicator Project Working Flow Chart:</span></span></h4>
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<a href="http://www.electronicshub.org/wp-content/uploads/2013/05/Water-Level-Indicator-Flow-Chart.jpg" style="box-sizing: border-box; color: #27a3d1; text-decoration-line: none; transition: all 0.1s ease-in-out;"><img alt="Water Level Indicator Flow Chart" class="aligncenter wp-image-19 size-full" src="http://www.electronicshub.org/wp-content/uploads/2013/05/Water-Level-Indicator-Flow-Chart.jpg" height="797" style="border: 0px; box-sizing: border-box; display: block; height: auto; margin: 0px auto 20px; max-width: 100%;" width="628" /></a></div>
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<span style="box-sizing: border-box; color: black;">Flow chart gives the clear and easy understanding of the project. The process goes on as follows:</span><br />
<span style="box-sizing: border-box; color: black;">The microcontroller checks for tank full condition, if the condition is satisfied it indicates the same on display unit and also sounds a buzzer if the condition fails it checks again and this process continues and the corresponding level is indicated in the display unit.</span></div>
<blockquote style="background-color: white; box-sizing: border-box; color: #999999; font-family: Raleway, sans-serif; font-size: 16px; margin: 40px 40px 24px;">
<div style="box-sizing: border-box; margin-bottom: 26px; padding: 0px;">
<span style="box-sizing: border-box; color: black;">Recommend</span><span style="box-sizing: border-box; color: black;"> Reading –</span> <span style="box-sizing: border-box; color: blue;"><a href="http://www.electronicshub.org/interfacing-7-segment-display-8051/" style="box-sizing: border-box; color: #27a3d1; text-decoration-line: none; transition: all 0.1s ease-in-out;" target="_blank" title="Interfacing Segment Display to 8051 Microcontroller"><span style="box-sizing: border-box; color: blue;">How to Interface Seven Segment Display to 8051 Microcontroller?</span></a></span></div>
</blockquote>
<h4 style="background-color: white; box-sizing: border-box; font-family: Raleway, sans-serif; font-size: 20px; line-height: 1.2; margin: 0px 0px 16px;">
<span id="Water_Level_Indicator_ProjectApplications" style="box-sizing: border-box;"><span style="box-sizing: border-box;"><span style="box-sizing: border-box;">Water Level Indicator Project Applications:</span></span></span></h4>
<ul style="background-color: white; box-sizing: border-box; color: #666666; font-family: Raleway, sans-serif; font-size: 16px; margin: 0px 0px 26px 40px; padding: 0px;">
<li style="box-sizing: border-box; list-style-type: disc;"><span style="box-sizing: border-box; color: black;">Automatic Water level Controller can be used in Hotels, Factories, Homes Apartments, Commercial Complexes, Drainage, etc., It can be fixed for single phase motor, Single Phase Submersibles, Three Phase motors. (For 3Æ and Single Phase Submersible Starter is necessary) and open well, Bore well and Sump. We can control two motor and two sumps and two overhead tanks by single unit.</span></li>
<li style="box-sizing: border-box; list-style-type: disc;"><span style="box-sizing: border-box; color: black;">Automatic water level controller will automatically START the pump set as soon as the water level falls below the predetermined level (usually 1/2 tank) and shall SWITCH OFF the pump set as soon as tank is full.</span></li>
<li style="box-sizing: border-box; list-style-type: disc;"><span style="box-sizing: border-box; color: black;">Fuel level indicator in vehicles.</span></li>
<li style="box-sizing: border-box; list-style-type: disc;"><span style="box-sizing: border-box; color: black;">Liquid level indicator in the huge containers in the companies.</span></li>
</ul>
</span></div>
</span></div>
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<img src="https://www.electronicshub.org/wp-content/uploads/2013/05/Water-Level-Indicator-Block-Diagram.jpg" /></div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3005862392404377111.post-89847585043592098582017-06-13T00:43:00.004-07:002023-06-23T23:34:12.912-07:00741 operational amplifier<div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">An </span><b style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">operational amplifier</b><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> (often </span><b style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">op-amp</b><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> or </span><b style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">opamp</b><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">) is a </span><a href="https://en.wikipedia.org/wiki/Direct_coupling" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Direct coupling">DC-coupled</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> high-</span><a href="https://en.wikipedia.org/wiki/Gain_(electronics)" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Gain (electronics)">gain</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> electronic voltage </span><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Electronic_amplifier" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Electronic amplifier">amplifier</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> with a </span><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Differential_input" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Differential input">differential input</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> and, usually, a </span><a href="https://en.wikipedia.org/wiki/Single-ended_signaling" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Single-ended signaling">single-ended</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> output.</span><sup class="reference" id="cite_ref-1" style="background-color: white; color: #222222; font-family: sans-serif; font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Operational_amplifier#cite_note-1" style="background: none; color: #0b0080; text-decoration-line: none;">[1]</a></sup><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> In this configuration, an op-amp produces an output potential (relative to circuit ground) that is typically hundreds of thousands of times larger than the potential difference between its input terminals. Operational amplifiers had their origins in </span><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Analog_computers" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Analog computers">analog computers</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">, where they were used to perform mathematical operations in many linear, non-linear and frequency-dependent circuits. The popularity of the op-amp as a building block in </span><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Analog_circuit" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Analog circuit">analog circuits</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> is due to its versatility. Due to </span><a href="https://en.wikipedia.org/wiki/Negative_feedback" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Negative feedback">negative feedback</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">, the characteristics of an op-amp circuit, its gain, input and </span><a href="https://en.wikipedia.org/wiki/Output_impedance" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Output impedance">output impedance</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">, </span><a href="https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Bandwidth (signal processing)">bandwidth</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> etc. are determined by external components and have little dependence on temperature coefficients or manufacturing variations in the op-amp itself.</span></div>
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<span class="mw-headline" id="Operation">Operation</span><span class="mw-editsection" style="display: inline-block; font-family: sans-serif; font-size: x-small; line-height: 1em; margin-left: 1em; unicode-bidi: isolate; user-select: none; vertical-align: baseline; white-space: nowrap;"><span class="mw-editsection-bracket" style="color: #555555; margin-right: 0.25em;">[</span><a href="https://en.wikipedia.org/w/index.php?title=Operational_amplifier&action=edit&section=1" style="background: none; color: #0b0080; text-decoration-line: none;" title="Edit section: Operation">edit</a><span class="mw-editsection-bracket" style="color: #555555; margin-left: 0.25em;">]</span></span></h2>
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<span style="font-size: 12.3704px;">An op-amp without negative feedback (a comparator)</span></div>
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The amplifier's differential inputs consist of a non-inverting input (+) with voltage <i>V</i><sub style="font-size: 11.2px; line-height: 1;">+</sub> and an inverting input (–) with voltage <i>V</i><sub style="font-size: 11.2px; line-height: 1;">−</sub>; ideally the op-amp amplifies only the difference in voltage between the two, which is called the <i>differential input voltage</i>. The output voltage of the op-amp <i>V</i><sub style="font-size: 11.2px; line-height: 1;">out</sub> is given by the equation</div>
<dl style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px; text-align: justify;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle V_{\text{out}}=A_{\text{OL}}(V_{+}-V_{-}),}</annotation></semantics></math></span><img alt="{\displaystyle V_{\text{out}}=A_{\text{OL}}(V_{+}-V_{-}),}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/283725e5e699d1e4e188a28fac17e3a31610cb0e" style="border: 0px; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 22.517ex;" /></span></dd></dl>
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where <i>A</i><sub style="font-size: 11.2px; line-height: 1;">OL</sub> is the <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Electronic_feedback_loops" style="background: none; color: #0b0080; text-decoration-line: none;" title="Electronic feedback loops">open-loop</a> gain of the amplifier (the term "open-loop" refers to the absence of a feedback loop from the output to the input).</div>
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<span class="mw-headline" id="Open-loop_amplifier">Open-loop amplifier</span><span class="mw-editsection" style="display: inline-block; font-size: x-small; font-weight: normal; line-height: 1em; margin-left: 1em; unicode-bidi: isolate; user-select: none; vertical-align: baseline; white-space: nowrap;"><span class="mw-editsection-bracket" style="color: #555555; margin-right: 0.25em;">[</span><a href="https://en.wikipedia.org/w/index.php?title=Operational_amplifier&action=edit&section=2" style="background: none; color: #0b0080; text-decoration-line: none;" title="Edit section: Open-loop amplifier">edit</a><span class="mw-editsection-bracket" style="color: #555555; margin-left: 0.25em;">]</span></span></h3>
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The magnitude of <i>A</i><sub style="font-size: 11.2px; line-height: 1;">OL</sub> is typically very large (100,000 or more for integrated circuit op-amps), and therefore even a quite small difference between <i>V</i><sub style="font-size: 11.2px; line-height: 1;">+</sub> and <i>V</i><sub style="font-size: 11.2px; line-height: 1;">−</sub> drives the amplifier output nearly to the supply voltage. Situations in which the output voltage is equal to or greater than the supply voltage are referred to as <i>saturation</i> of the amplifier. The magnitude of <i>A</i><sub style="font-size: 11.2px; line-height: 1;">OL</sub> is not well controlled by the manufacturing process, and so it is impractical to use an open-loop amplifier as a stand-alone <a href="https://en.wikipedia.org/wiki/Differential_amplifier" style="background: none; color: #0b0080; text-decoration-line: none;" title="Differential amplifier">differential amplifier</a>.</div>
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Without <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Negative_feedback_amplifier" style="background: none; color: #0b0080; text-decoration-line: none;" title="Negative feedback amplifier">negative feedback</a>, and perhaps with <a href="https://en.wikipedia.org/wiki/Positive_feedback" style="background: none; color: #0b0080; text-decoration-line: none;" title="Positive feedback">positive feedback</a> for <a href="https://en.wikipedia.org/wiki/Regenerative_circuit" style="background: none; color: #0b0080; text-decoration-line: none;" title="Regenerative circuit">regeneration</a>, an op-amp acts as a <a href="https://en.wikipedia.org/wiki/Comparator" style="background: none; color: #0b0080; text-decoration-line: none;" title="Comparator">comparator</a>. If the inverting input is held at ground (0 V) directly or by a resistor <i>R</i><sub style="font-size: 11.2px; line-height: 1;">g</sub>, and the input voltage <i>V</i><sub style="font-size: 11.2px; line-height: 1;">in</sub> applied to the non-inverting input is positive, the output will be maximum positive; if <i>V</i><sub style="font-size: 11.2px; line-height: 1;">in</sub> is negative, the output will be maximum negative. Since there is no feedback from the output to either input, this is an <i><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Electronic_feedback_loops" style="background: none; color: #0b0080; text-decoration-line: none;" title="Electronic feedback loops">open-loop</a></i> circuit acting as a <a href="https://en.wikipedia.org/wiki/Comparator" style="background: none; color: #0b0080; text-decoration-line: none;" title="Comparator">comparator</a>.</div>
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<span class="mw-headline" id="Closed-loop_amplifier">Closed-loop amplifier</span><span class="mw-editsection" style="display: inline-block; font-size: x-small; font-weight: normal; line-height: 1em; margin-left: 1em; unicode-bidi: isolate; user-select: none; vertical-align: baseline; white-space: nowrap;"><span class="mw-editsection-bracket" style="color: #555555; margin-right: 0.25em;">[</span><a href="https://en.wikipedia.org/w/index.php?title=Operational_amplifier&action=edit&section=3" style="background: none; color: #0b0080; text-decoration-line: none;" title="Edit section: Closed-loop amplifier">edit</a><span class="mw-editsection-bracket" style="color: #555555; margin-left: 0.25em;">]</span></span></h3>
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<span style="font-size: 12.3704px;">An op-amp with negative feedback (a non-inverting amplifier)</span></div>
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If predictable operation is desired, negative feedback is used, by applying a portion of the output voltage to the inverting input. The <i>closed-loop</i> feedback greatly reduces the gain of the circuit. When negative feedback is used, the circuit's overall gain and response becomes determined mostly by the feedback network, rather than by the op-amp characteristics. If the feedback network is made of components with values small relative to the op amp's input impedance, the value of the op-amp's open-loop response <i>A</i><sub style="font-size: 11.2px; line-height: 1;">OL</sub> does not seriously affect the circuit's performance. The response of the op-amp circuit with its input, output, and feedback circuits to an input is characterized mathematically by a <a href="https://en.wikipedia.org/wiki/Transfer_function" style="background: none; color: #0b0080; text-decoration-line: none;" title="Transfer function">transfer function</a>; designing an op-amp circuit to have a desired transfer function is in the realm of <a href="https://en.wikipedia.org/wiki/Electrical_engineering" style="background: none; color: #0b0080; text-decoration-line: none;" title="Electrical engineering">electrical engineering</a>. The transfer functions are important in most applications of op-amps, such as in <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Analog_computers" style="background: none; color: #0b0080; text-decoration-line: none;" title="Analog computers">analog computers</a>. High input <a href="https://en.wikipedia.org/wiki/Electrical_impedance" style="background: none; color: #0b0080; text-decoration-line: none;" title="Electrical impedance">impedance</a> at the input terminals and low output impedance at the output terminal(s) are particularly useful features of an op-amp.</div>
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In the non-inverting amplifier on the right, the presence of negative feedback via the <a href="https://en.wikipedia.org/wiki/Voltage_divider" style="background: none; color: #0b0080; text-decoration-line: none;" title="Voltage divider">voltage divider</a> <i>R</i><sub style="font-size: 11.2px; line-height: 1;">f</sub>, <i>R</i><sub style="font-size: 11.2px; line-height: 1;">g</sub> determines the <i>closed-loop gain</i><i>A</i><sub style="font-size: 11.2px; line-height: 1;">CL</sub> = <i>V</i><sub style="font-size: 11.2px; line-height: 1;">out</sub> / <i>V</i><sub style="font-size: 11.2px; line-height: 1;">in</sub>. Equilibrium will be established when <i>V</i><sub style="font-size: 11.2px; line-height: 1;">out</sub> is just sufficient to "reach around and pull" the inverting input to the same voltage as <i>V</i><sub style="font-size: 11.2px; line-height: 1;">in</sub>. The voltage gain of the entire circuit is thus 1 + <i>R</i><sub style="font-size: 11.2px; line-height: 1;">f</sub>/<i>R</i><sub style="font-size: 11.2px; line-height: 1;">g</sub>. As a simple example, if <i>V</i><sub style="font-size: 11.2px; line-height: 1;">in</sub> = 1 V and R<sub style="font-size: 11.2px; line-height: 1;">f</sub> = <i>R</i><sub style="font-size: 11.2px; line-height: 1;">g</sub>, <i>V</i><sub style="font-size: 11.2px; line-height: 1;">out</sub> will be 2 V, exactly the amount required to keep <i>V</i><sub style="font-size: 11.2px; line-height: 1;">−</sub> at 1 V. Because of the feedback provided by the <i>R</i><sub style="font-size: 11.2px; line-height: 1;">f</sub>, <i>R</i><sub style="font-size: 11.2px; line-height: 1;">g</sub> network, this is a <i>closed-loop</i> circuit.</div>
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Another way to analyze this circuit proceeds by making the following (usually valid) assumptions:<sup class="reference" id="cite_ref-3" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Operational_amplifier#cite_note-3" style="background: none; color: #0b0080; text-decoration-line: none;">[3]</a></sup></div>
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<li style="margin-bottom: 0.1em; text-align: justify;">When an op-amp operates in linear (i.e., not saturated) mode, the difference in voltage between the non-inverting (+) pin and the inverting (−) pin is negligibly small.</li>
<li style="margin-bottom: 0.1em; text-align: justify;">The input impedance between (+) and (−) pins is much larger than other resistances in the circuit.</li>
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The input signal <i>V</i><sub style="font-size: 11.2px; line-height: 1;">in</sub> appears at both (+) and (−) pins, resulting in a current <i>i</i> through <i>R</i><sub style="font-size: 11.2px; line-height: 1;">g</sub> equal to <i>V</i><sub style="font-size: 11.2px; line-height: 1;">in</sub>/<i>R</i><sub style="font-size: 11.2px; line-height: 1;">g</sub>:</div>
<dl style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px; text-align: justify;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle i={\frac {V_{\text{in}}}{R_{g}}}.}</annotation></semantics></math></span><img alt="{\displaystyle i={\frac {V_{\text{in}}}{R_{g}}}.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67efb6626c058bc429a14f604a76fcd7444b59e8" style="border: 0px; display: inline-block; height: 6.176ex; vertical-align: -2.671ex; width: 8.399ex;" /></span></dd></dl>
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Since Kirchhoff's current law states that the same current must leave a node as enter it, and since the impedance into the (−) pin is near infinity, we can assume practically all of the same current <i>i</i> flows through <i>R</i><sub style="font-size: 11.2px; line-height: 1;">f</sub>, creating an output voltage</div>
<dl style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px; text-align: justify;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle V_{\text{out}}=V_{\text{in}}+i\times R_{f}=V_{\text{in}}+\left({\frac {V_{\text{in}}}{R_{g}}}\times R_{f}\right)=V_{\text{in}}+{\frac {V_{\text{in}}\times R_{f}}{R_{g}}}=V_{\text{in}}\left(1+{\frac {R_{f}}{R_{g}}}\right).}</annotation></semantics></math></span><img alt="{\displaystyle V_{\text{out}}=V_{\text{in}}+i\times R_{f}=V_{\text{in}}+\left({\frac {V_{\text{in}}}{R_{g}}}\times R_{f}\right)=V_{\text{in}}+{\frac {V_{\text{in}}\times R_{f}}{R_{g}}}=V_{\text{in}}\left(1+{\frac {R_{f}}{R_{g}}}\right).}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2672ff6aad4000b2568939dbf5ff6d671939fb87" style="border: 0px; display: inline-block; height: 6.509ex; vertical-align: -2.671ex; width: 78.774ex;" /></span></dd></dl>
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By combining terms, we determine the closed-loop gain <i>A</i><sub style="font-size: 11.2px; line-height: 1;">CL</sub>:</div>
<dl style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px; text-align: justify;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle A_{\text{CL}}={\frac {V_{\text{out}}}{V_{\text{in}}}}=1+{\frac {R_{f}}{R_{g}}}.}</annotation></semantics></math></span><img alt="{\displaystyle A_{\text{CL}}={\frac {V_{\text{out}}}{V_{\text{in}}}}=1+{\frac {R_{f}}{R_{g}}}.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34b617fb189f39719daae9d019380d3d41f18592" style="border: 0px; display: inline-block; height: 6.509ex; vertical-align: -2.671ex; width: 23.7ex;" /></span></dd></dl>
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<span class="mw-headline" id="Op-amp_characteristics">Op-amp characteristics</span><span class="mw-editsection" style="display: inline-block; font-family: sans-serif; font-size: x-small; line-height: 1em; margin-left: 1em; unicode-bidi: isolate; user-select: none; vertical-align: baseline; white-space: nowrap;"><span class="mw-editsection-bracket" style="color: #555555; margin-right: 0.25em;">[</span><a href="https://en.wikipedia.org/w/index.php?title=Operational_amplifier&action=edit&section=4" style="background: none; color: #0b0080; text-decoration-line: none;" title="Edit section: Op-amp characteristics">edit</a><span class="mw-editsection-bracket" style="color: #555555; margin-left: 0.25em;">]</span></span></h2>
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<span class="mw-headline" id="Ideal_op-amps">Ideal op-amps</span><span class="mw-editsection" style="display: inline-block; font-size: x-small; font-weight: normal; line-height: 1em; margin-left: 1em; unicode-bidi: isolate; user-select: none; vertical-align: baseline; white-space: nowrap;"><span class="mw-editsection-bracket" style="color: #555555; margin-right: 0.25em;">[</span><a href="https://en.wikipedia.org/w/index.php?title=Operational_amplifier&action=edit&section=5" style="background: none; color: #0b0080; text-decoration-line: none;" title="Edit section: Ideal op-amps">edit</a><span class="mw-editsection-bracket" style="color: #555555; margin-left: 0.25em;">]</span></span></h3>
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<a class="image" href="https://en.wikipedia.org/wiki/File:Op-Amp_Internal.svg" style="background: none; color: #0b0080; font-size: 13.16px; text-decoration-line: none;"><img alt="" class="thumbimage" data-file-height="300" data-file-width="400" height="188" src="https://upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Op-Amp_Internal.svg/250px-Op-Amp_Internal.svg.png" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Op-Amp_Internal.svg/375px-Op-Amp_Internal.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Op-Amp_Internal.svg/500px-Op-Amp_Internal.svg.png 2x" style="background-color: white; border: 1px solid rgb(200, 204, 209); vertical-align: middle;" width="250" /></a></div>
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<a class="internal" href="https://en.wikipedia.org/wiki/File:Op-Amp_Internal.svg" style="background: linear-gradient(transparent, transparent), url("data:image/svg+xml,%3C%3Fxml%20version%3D%221.0%22%20encoding%3D%22UTF-8%22%20standalone%3D%22no%22%3F%3E%0A%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20viewBox%3D%220%200%2011%2015%22%20width%3D%2215%22%20height%3D%2211%22%3E%0A%20%20%20%20%3Cg%20id%3D%22magnify-clip%22%20fill%3D%22%23fff%22%20stroke%3D%22%23000%22%3E%0A%20%20%20%20%20%20%20%20%3Cpath%20id%3D%22bigbox%22%20d%3D%22M1.509%201.865h10.99v7.919h-10.99z%22%2F%3E%0A%20%20%20%20%20%20%20%20%3Cpath%20id%3D%22smallbox%22%20d%3D%22M-1.499%206.868h5.943v4.904h-5.943z%22%2F%3E%0A%20%20%20%20%3C%2Fg%3E%0A%3C%2Fsvg%3E%0A"); color: #0b0080; display: block; height: 11px; overflow: hidden; text-align: justify; text-decoration-line: none; text-indent: 15px; user-select: none; white-space: nowrap; width: 15px;" title="Enlarge"></a></div>
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<span style="font-size: 12.3704px;">An equivalent circuit of an operational amplifier that models some resistive non-ideal parameters.</span></div>
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An ideal op-amp is usually considered to have the following characteristics:<sup class="reference" id="cite_ref-4" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Operational_amplifier#cite_note-4" style="background: none; color: #0b0080; text-decoration-line: none;">[4]</a></sup><sup class="reference" id="cite_ref-5" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Operational_amplifier#cite_note-5" style="background: none; color: #0b0080; text-decoration-line: none;">[5]</a></sup></div>
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<li style="margin-bottom: 0.1em; text-align: justify;">Infinite <a href="https://en.wikipedia.org/wiki/Open-loop_gain" style="background: none; color: #0b0080; text-decoration-line: none;" title="Open-loop gain">open-loop gain</a> G = <i>v</i><sub style="font-size: 11.2px; line-height: 1;">out</sub> / <i>v</i><sub style="font-size: 11.2px; line-height: 1;">in</sub></li>
<li style="margin-bottom: 0.1em; text-align: justify;">Infinite <a href="https://en.wikipedia.org/wiki/Input_impedance" style="background: none; color: #0b0080; text-decoration-line: none;" title="Input impedance">input impedance</a> R<sub style="font-size: 11.2px; line-height: 1;">in</sub>, and so zero input current</li>
<li style="margin-bottom: 0.1em; text-align: justify;">Zero <a href="https://en.wikipedia.org/wiki/Input_offset_voltage" style="background: none; color: #0b0080; text-decoration-line: none;" title="Input offset voltage">input offset voltage</a></li>
<li style="margin-bottom: 0.1em; text-align: justify;">Infinite output voltage range</li>
<li style="margin-bottom: 0.1em; text-align: justify;">Infinite <a href="https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Bandwidth (signal processing)">bandwidth</a> with zero <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Phase_shift" style="background: none; color: #0b0080; text-decoration-line: none;" title="Phase shift">phase shift</a> and infinite <a href="https://en.wikipedia.org/wiki/Slew_rate" style="background: none; color: #0b0080; text-decoration-line: none;" title="Slew rate">slew rate</a></li>
<li style="margin-bottom: 0.1em; text-align: justify;">Zero <a href="https://en.wikipedia.org/wiki/Output_impedance" style="background: none; color: #0b0080; text-decoration-line: none;" title="Output impedance">output impedance</a> R<sub style="font-size: 11.2px; line-height: 1;">out</sub></li>
<li style="margin-bottom: 0.1em; text-align: justify;">Zero <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Electronic_noise" style="background: none; color: #0b0080; text-decoration-line: none;" title="Electronic noise">noise</a></li>
<li style="margin-bottom: 0.1em; text-align: justify;">Infinite <a href="https://en.wikipedia.org/wiki/Common-mode_rejection_ratio" style="background: none; color: #0b0080; text-decoration-line: none;" title="Common-mode rejection ratio">common-mode rejection ratio</a> (CMRR)</li>
<li style="margin-bottom: 0.1em; text-align: justify;">Infinite <a href="https://en.wikipedia.org/wiki/Power_supply_rejection_ratio" style="background: none; color: #0b0080; text-decoration-line: none;" title="Power supply rejection ratio">power supply rejection ratio</a>.</li>
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These ideals can be summarized by the two "golden rules":</div>
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<li style="margin-bottom: 0.1em; text-align: justify;"><b>In a closed loop</b> the output attempts to do whatever is necessary to make the voltage difference between the inputs zero.</li>
<li style="margin-bottom: 0.1em; text-align: justify;">The inputs draw no current.<sup class="reference" id="cite_ref-AoE_6-0" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Operational_amplifier#cite_note-AoE-6" style="background: none; color: #0b0080; text-decoration-line: none;">[6]</a></sup><sup class="reference" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;">:177</sup></li>
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The first rule only applies in the usual case where the op-amp is used in a closed-loop design (negative feedback, where there is a signal path of some sort feeding back from the output to the inverting input). These rules are commonly used as a good first approximation for analyzing or designing op-amp circuits.<sup class="reference" id="cite_ref-AoE_6-1" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Operational_amplifier#cite_note-AoE-6" style="background: none; color: #0b0080; text-decoration-line: none;">[6]</a></sup><sup class="reference" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;">:177</sup></div>
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None of these ideals can be perfectly realized. A real op-amp may be modeled with non-infinite or non-zero parameters using equivalent resistors and capacitors in the op-amp model. The designer can then include these effects into the overall performance of the final circuit. Some parameters may turn out to have negligible effect on the final design while others represent actual limitations of the final performance that must be evaluated.</div>
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</head>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3005862392404377111.post-46487652958780531622017-06-13T00:35:00.003-07:002023-06-23T23:34:11.881-07:00TIMER IC 555<div dir="ltr" style="text-align: left;" trbidi="on">
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TIMER IC 555</h2>
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The <b>555 timer IC</b> is an <a href="https://en.wikipedia.org/wiki/Integrated_circuit" style="background: none; color: #0b0080; text-decoration-line: none;" title="Integrated circuit">integrated circuit</a> (chip) used in a variety of <a href="https://en.wikipedia.org/wiki/Timer" style="background: none; color: #0b0080; text-decoration-line: none;" title="Timer">timer</a>, pulse generation, and <a href="https://en.wikipedia.org/wiki/Electronic_oscillator" style="background: none; color: #0b0080; text-decoration-line: none;" title="Electronic oscillator">oscillator</a> applications. The 555 can be used to provide time delays, as an <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Oscillator" style="background: none; color: #0b0080; text-decoration-line: none;" title="Oscillator">oscillator</a>, and as a <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Flip-flop_element" style="background: none; color: #0b0080; text-decoration-line: none;" title="Flip-flop element">flip-flop element</a>. Derivatives provide two or four timing circuits in one package.</div>
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Introduced in 1972<sup class="reference" id="cite_ref-fuller_1-0" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/555_timer_IC#cite_note-fuller-1" style="background: none; color: #0b0080; text-decoration-line: none;">[1]</a></sup> by <a href="https://en.wikipedia.org/wiki/Signetics" style="background: none; color: #0b0080; text-decoration-line: none;" title="Signetics">Signetics</a>,<sup class="reference" id="cite_ref-Sinetics1973Databook_2-0" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/555_timer_IC#cite_note-Sinetics1973Databook-2" style="background: none; color: #0b0080; text-decoration-line: none;">[2]</a></sup> the 555 is still in widespread use due to its low price, ease of use, and stability. It is now made by many companies in the original <a href="https://en.wikipedia.org/wiki/Bipolar_junction_transistor" style="background: none; color: #0b0080; text-decoration-line: none;" title="Bipolar junction transistor">bipolar</a> and in low-power <a href="https://en.wikipedia.org/wiki/CMOS" style="background: none; color: #0b0080; text-decoration-line: none;" title="CMOS">CMOS</a>. As of 2003, it was estimated that 1 <a href="https://en.wikipedia.org/wiki/Billion" style="background: none; color: #0b0080; text-decoration-line: none;" title="Billion">billion</a> units were manufactured every year.<sup class="reference" id="cite_ref-semiconductormuseum.com_3-0" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/555_timer_IC#cite_note-semiconductormuseum.com-3" style="background: none; color: #0b0080; text-decoration-line: none;">[3]</a></sup> The 555 is the most popular integrated circuit ever manufactured</div>
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<span style="font-size: large;"><b>HISTORY:</b></span></div>
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The IC was designed in 1971 by <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Hans_R._Camenzind" style="background: none; color: #0b0080; text-decoration-line: none;" title="Hans R. Camenzind">Hans R. Camenzind</a> under contract to <a href="https://en.wikipedia.org/wiki/Signetics" style="background: none; color: #0b0080; text-decoration-line: none;" title="Signetics">Signetics</a> (later acquired by <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Philips_Semiconductors" style="background: none; color: #0b0080; text-decoration-line: none;" title="Philips Semiconductors">Philips Semiconductors</a>, and now <a class="mw-redirect" href="https://en.wikipedia.org/wiki/NXP" style="background: none; color: #0b0080; text-decoration-line: none;" title="NXP">NXP</a>).</div>
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In 1962, Camenzind joined PR Mallory's Laboratory for Physical Science in <a href="https://en.wikipedia.org/wiki/Burlington,_Massachusetts" style="background: none; color: #0b0080; text-decoration-line: none;" title="Burlington, Massachusetts">Burlington, Massachusetts</a>.<sup class="reference" id="cite_ref-semiconductormuseum.com_3-1" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/555_timer_IC#cite_note-semiconductormuseum.com-3" style="background: none; color: #0b0080; text-decoration-line: none;">[3]</a></sup> He designed a <a href="https://en.wikipedia.org/wiki/Pulse-width_modulation" style="background: none; color: #0b0080; text-decoration-line: none;" title="Pulse-width modulation">pulse-width modulation</a> (PWM) amplifier for audio applications,<sup class="reference" id="cite_ref-6" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/555_timer_IC#cite_note-6" style="background: none; color: #0b0080; text-decoration-line: none;">[6]</a></sup> but it was not successful in the market because there was no power transistor included. He became interested in tuners such as a <a href="https://en.wikipedia.org/wiki/Gyrator" style="background: none; color: #0b0080; text-decoration-line: none;" title="Gyrator">gyrator</a> and a <a href="https://en.wikipedia.org/wiki/Phase-locked_loop" style="background: none; color: #0b0080; text-decoration-line: none;" title="Phase-locked loop">phase-locked loop</a> (PLL). He was hired by Signetics to develop a PLL IC in 1968. He designed an oscillator for PLLs such that the frequency did not depend on the power supply voltage or temperature. However, Signetics laid off half of its employees, and the development was frozen due to a recession.<sup class="reference" id="cite_ref-:0_7-0" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/555_timer_IC#cite_note-:0-7" style="background: none; color: #0b0080; text-decoration-line: none;">[7]</a></sup></div>
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Camenzind proposed the development of a universal circuit based on the oscillator for PLLs, and asked that he would develop it alone, borrowing their equipment instead of having his pay cut in half. Other engineers argued the product could be built from existing parts, but the marketing manager bought the idea. Among 5xx numbers that were assigned for analogue ICs, the special number "555" was chosen.<sup class="reference" id="cite_ref-semiconductormuseum.com_3-2" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/555_timer_IC#cite_note-semiconductormuseum.com-3" style="background: none; color: #0b0080; text-decoration-line: none;">[3]</a></sup><sup class="reference" id="cite_ref-:0_7-1" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/555_timer_IC#cite_note-:0-7" style="background: none; color: #0b0080; text-decoration-line: none;">[7]</a></sup></div>
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Camenzind also taught circuit design at his nearby university<sup class="noprint Inline-Template" style="font-size: 11.2px; line-height: 1; white-space: nowrap;">[<i><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Wikipedia:Avoid_weasel_words" style="background: none; color: #0b0080; text-decoration-line: none;" title="Wikipedia:Avoid weasel words"><span title="The material near this tag possibly uses too vague attribution or weasel words. (December 2016)">which?</span></a></i>]</sup> in the morning, and went to the <a href="https://en.wikipedia.org/wiki/Northeastern_University" style="background: none; color: #0b0080; text-decoration-line: none;" title="Northeastern University">Northeastern University</a> to get the master's degree at night. The first design was reviewed in the summer of 1971. There was no problem, so it had gone to the layout design. A few days later, he got the idea of using a direct resistance instead of a constant current source, and found that it worked. The change decreased the required 9 pins to 8, so the IC could be fit in an 8-pin package instead of a 14-pin package. This design passed the second design review, and the prototype was completed in October 1971. Its 9-pin copy had been already released by another company founded by an engineer who attended the first review and retired from Signetics, but they withdrew it soon after the 555 was released. The 555 timer was manufactured by 12 companies in 1972 and it became the best selling product.</div>
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<span class="mw-headline" id="Design">Design</span></h2>
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<span style="font-size: 12.3704px;">Internal schematic (bipolar version)</span></div>
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<span style="font-size: 12.3704px;">Internal schematic (CMOS version)</span></div>
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Depending on the manufacturer, the standard 555 package includes 25 <a href="https://en.wikipedia.org/wiki/Transistor" style="background: none; color: #0b0080; text-decoration-line: none;" title="Transistor">transistors</a>, 2 <a href="https://en.wikipedia.org/wiki/Diode" style="background: none; color: #0b0080; text-decoration-line: none;" title="Diode">diodes</a> and 15 <a href="https://en.wikipedia.org/wiki/Resistor" style="background: none; color: #0b0080; text-decoration-line: none;" title="Resistor">resistors</a> on a <a href="https://en.wikipedia.org/wiki/Silicon" style="background: none; color: #0b0080; text-decoration-line: none;" title="Silicon">silicon</a> chip installed in an 8-pin mini dual-in-line package (<a class="mw-redirect" href="https://en.wikipedia.org/wiki/DIP-8" style="background: none; color: #0b0080; text-decoration-line: none;" title="DIP-8">DIP-8</a>).<sup class="reference" id="cite_ref-8" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/555_timer_IC#cite_note-8" style="background: none; color: #0b0080; text-decoration-line: none;">[8]</a></sup> Variants available include the 556 (a 14-pin DIP combining two 555s on one chip), and the two 558 & 559s (both a 16-pin DIP combining four slightly modified 555s with DIS & THR connected internally, and TR is falling edge sensitive instead of level sensitive).</div>
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The <b>NE555</b> parts were commercial temperature range, 0 °C to +70 °C, and the <b>SE555</b> part number designated the military temperature range, −55 °C to +125 °C. These were available in both high-reliability metal can (T package) and inexpensive epoxy plastic (V package) packages. Thus the full part numbers were NE555V, NE555T, SE555V, and SE555T. It has been hypothesized that the 555 got its name from the three 5 <a href="https://en.wikipedia.org/wiki/Ohm" style="background: none; color: #0b0080; text-decoration-line: none;" title="Ohm">kΩ</a> resistors used within,<sup class="reference" id="cite_ref-9" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/555_timer_IC#cite_note-9" style="background: none; color: #0b0080; text-decoration-line: none;">[9]</a></sup> but Hans Camenzind has stated that the number was arbitrary.<sup class="reference" id="cite_ref-semiconductormuseum.com_3-3" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/555_timer_IC#cite_note-semiconductormuseum.com-3" style="background: none; color: #0b0080; text-decoration-line: none;">[3]</a></sup></div>
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Low-power versions of the 555 are also available, such as the 7555 and CMOS TLC555.<sup class="reference" id="cite_ref-10" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/555_timer_IC#cite_note-10" style="background: none; color: #0b0080; text-decoration-line: none;">[10]</a></sup> The 7555 is designed to cause less supply noise than the classic 555 and the manufacturer claims that it usually does not require a "control" capacitor and in many cases does not require a <a href="https://en.wikipedia.org/wiki/Decoupling_capacitor" style="background: none; color: #0b0080; text-decoration-line: none;" title="Decoupling capacitor">decoupling capacitor</a> on the power supply. Those parts should generally be included, however, because noise produced by the timer or variation in power supply voltage might interfere with other parts of a circuit or influence its threshold voltages.</div>
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<span class="mw-headline" id="Pins">Pins</span><span class="mw-editsection" style="display: inline-block; font-size: x-small; font-weight: normal; line-height: 1em; margin-left: 1em; unicode-bidi: isolate; user-select: none; vertical-align: baseline; white-space: nowrap;"><span class="mw-editsection-bracket" style="color: #555555; margin-right: 0.25em;">[</span><a href="https://en.wikipedia.org/w/index.php?title=555_timer_IC&action=edit&section=3" style="background: none; color: #0b0080; text-decoration-line: none;" title="Edit section: Pins">edit</a><span class="mw-editsection-bracket" style="color: #555555; margin-left: 0.25em;">]</span></span></h3>
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<a class="image" href="https://en.wikipedia.org/wiki/File:555_Pinout.svg" style="background: none; color: #0b0080; font-size: 13.16px; text-decoration-line: none;"><img alt="" class="thumbimage" data-file-height="145" data-file-width="220" height="171" src="https://upload.wikimedia.org/wikipedia/commons/thumb/c/c7/555_Pinout.svg/260px-555_Pinout.svg.png" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/555_Pinout.svg/390px-555_Pinout.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c7/555_Pinout.svg/520px-555_Pinout.svg.png 2x" style="background-color: white; border: 1px solid rgb(200, 204, 209); vertical-align: middle;" width="260" /></a></div>
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<span style="font-size: 12.3704px;">Pinout diagram</span></div>
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The connection of the pins for a DIP package is as follows:</div>
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<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">Pin</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">Name</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">Purpose</th></tr>
<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">GND</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">Ground reference voltage, low level (0 V)</td></tr>
<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">TRIG</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">The OUT pin goes high and a timing interval starts when this input falls below 1/2 of CTRL voltage (which is typically 1/3 <i>V</i><sub style="font-size: 11.2px; line-height: 1;">CC</sub>, CTRL being 2/3 <i>V</i><sub style="font-size: 11.2px; line-height: 1;">CC</sub> by default if CTRL is left open). More simply we can say that OUT will be high as long as the trigger is kept at low voltage. Output of the timer totally depends upon the amplitude of the external trigger voltage applied to this pin.</td></tr>
<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">OUT</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">This output is driven to approximately 1.7 V below <a href="https://en.wikipedia.org/wiki/IC_power-supply_pin" style="background: none; color: #0b0080; text-decoration-line: none;" title="IC power-supply pin">+<i>V</i><sub style="font-size: 11.2px; line-height: 1;">CC</sub></a>, or to GND.</td></tr>
<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">4</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"><span style="text-decoration-line: overline;">RESET</span></td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">A timing interval may be reset by driving this input to GND, but the timing does not begin again until RESET rises above approximately 0.7 volts. Overrides TRIG which overrides THR. (THR instead overrides TRIG on the LM555)</td></tr>
<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">5</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">CTRL</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">Provides "control" access to the internal voltage divider (by default, 2/3 <i>V</i><sub style="font-size: 11.2px; line-height: 1;">CC</sub>).</td></tr>
<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">6</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">THR</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">The timing (OUT high) interval ends when the voltage at THR ("threshold") is greater than that at CTRL (2/3 <i>V</i><sub style="font-size: 11.2px; line-height: 1;">CC</sub> if CTRL is open). Overrides TRIG on the LM555</td></tr>
<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">7</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">DIS</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"><a href="https://en.wikipedia.org/wiki/Open_collector" style="background: none; color: #0b0080; text-decoration-line: none;" title="Open collector">Open collector</a> output which may discharge a capacitor between intervals. In phase with output.</td></tr>
<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">8</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"><i>V</i><sub style="font-size: 11.2px; line-height: 1;">CC</sub></td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">Positive supply voltage, which is usually between 3 and 15 V depending on the variation.</td></tr>
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Pin 5 is also sometimes called the CONTROL VOLTAGE pin. By applying a voltage to the CONTROL VOLTAGE input one can alter the timing characteristics of the device. In most applications, the CONTROL VOLTAGE input is not used. It is usual to connect a 10 nF capacitor between pin 5 and 0 V to prevent interference. The CONTROL VOLTAGE input can be used to build an astable multivibrator with a frequency-modulated output.</div>
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<span class="mw-headline" id="Modes">Modes</span></h3>
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The IC 555 has three operating modes:</div>
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<li style="margin-bottom: 0.1em; text-align: justify;"><b><a href="https://en.wikipedia.org/wiki/Flip-flop_(electronics)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Flip-flop (electronics)">Bistable</a></b> mode or <a href="https://en.wikipedia.org/wiki/Schmitt_trigger" style="background: none; color: #0b0080; text-decoration-line: none;" title="Schmitt trigger">Schmitt trigger</a> – the 555 can operate as a <a href="https://en.wikipedia.org/wiki/Flip-flop_(electronics)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Flip-flop (electronics)">flip-flop</a>, if the DIS pin is not connected and no capacitor is used. Uses include bounce-free latched switches.</li>
<li style="margin-bottom: 0.1em; text-align: justify;"><b><a href="https://en.wikipedia.org/wiki/Monostable" style="background: none; color: #0b0080; text-decoration-line: none;" title="Monostable">Monostable</a></b> mode – in this mode, the 555 functions as a "one-shot" pulse generator. Applications include timers, missing pulse detection, bounce-free switches, touch switches, frequency divider, capacitance measurement, <a href="https://en.wikipedia.org/wiki/Pulse-width_modulation" style="background: none; color: #0b0080; text-decoration-line: none;" title="Pulse-width modulation">pulse-width modulation</a> (PWM) and so on.</li>
<li style="margin-bottom: 0.1em; text-align: justify;"><b><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Astable" style="background: none; color: #0b0080; text-decoration-line: none;" title="Astable">Astable</a></b> (free-running) mode – the 555 can operate as an <a href="https://en.wikipedia.org/wiki/Electronic_oscillator" style="background: none; color: #0b0080; text-decoration-line: none;" title="Electronic oscillator">electronic oscillator</a>. Uses include <a href="https://en.wikipedia.org/wiki/Light-emitting_diode" style="background: none; color: #0b0080; text-decoration-line: none;" title="Light-emitting diode">LED</a> and lamp flashers, pulse generation, logic clocks, tone generation, security alarms, <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Pulse_position_modulation" style="background: none; color: #0b0080; text-decoration-line: none;" title="Pulse position modulation">pulse position modulation</a> and so on. The 555 can be used as a simple <a href="https://en.wikipedia.org/wiki/Analog-to-digital_converter" style="background: none; color: #0b0080; text-decoration-line: none;" title="Analog-to-digital converter">ADC</a>, converting an analog value to a pulse length (e.g., selecting a <a href="https://en.wikipedia.org/wiki/Thermistor" style="background: none; color: #0b0080; text-decoration-line: none;" title="Thermistor">thermistor</a> as timing resistor allows the use of the 555 in a temperature sensor and the period of the output pulse is determined by the temperature). The use of a microprocessor-based circuit can then convert the pulse period to temperature, linearize it and even provide calibration means.</li>
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<span class="mw-headline" id="Bistable">Bistable</span></h4>
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<span style="font-size: 12.3704px;">Schematic of a 555 in bistable mode</span></div>
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In bistable (also called <a href="https://en.wikipedia.org/wiki/Schmitt_trigger" style="background: none; color: #0b0080; text-decoration-line: none;" title="Schmitt trigger">Schmitt trigger</a>) mode, the 555 timer acts as a basic flip-flop. The trigger and reset inputs (pins 2 and 4 respectively on a 555) are held high via <a href="https://en.wikipedia.org/wiki/Pull-up_resistor" style="background: none; color: #0b0080; text-decoration-line: none;" title="Pull-up resistor">pull-up resistors</a> while the threshold input (pin 6) is simply floating. Thus configured, pulling the trigger momentarily to ground acts as a 'set' and transitions the output pin (pin 3) to Vcc (high state). Pulling the reset input to ground acts as a 'reset' and transitions the output pin to ground (low state). No timing capacitors are required in a bistable configuration. Pin 5 (control voltage) is connected to ground via a small-value capacitor (usually 0.01 to 0.1 μF). Pin 7 (discharge) is left unconnected, or may be used as an <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Open-collector" style="background: none; color: #0b0080; text-decoration-line: none;" title="Open-collector">open-collector</a> output.<sup class="reference" id="cite_ref-11" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/555_timer_IC#cite_note-11" style="background: none; color: #0b0080; text-decoration-line: none;">[11]</a></sup></div>
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<span class="mw-headline" id="Monostable">Monostable</span></h4>
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<span style="font-size: 12.3704px;">Schematic of a 555 in monostable mode</span></div>
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The output pulse ends when the voltage on the capacitor equals 2/3 of the supply voltage. The output pulse width can be lengthened or shortened to the need of the specific application by adjusting the values of R and C.<sup class="reference" id="cite_ref-12" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/555_timer_IC#cite_note-12" style="background: none; color: #0b0080; text-decoration-line: none;">[12]</a></sup></div>
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The output pulse width of time <i>t</i>, which is the time it takes to charge C to 2/3 of the supply voltage, is given by</div>
<dl style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px; text-align: justify;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle t=\ln(3)\cdot RC\approx 1.1RC}</annotation></semantics></math></span><img alt="t=\ln(3)\cdot RC\approx 1.1RC" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a288d5c3dc3e272f5faccc979168acd907d2e02" style="border: 0px; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 23.826ex;" /></span></dd></dl>
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where t is in seconds, R is in <a href="https://en.wikipedia.org/wiki/Ohm" style="background: none; color: #0b0080; text-decoration-line: none;" title="Ohm">ohms</a> (resistance) and C is in <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Farads" style="background: none; color: #0b0080; text-decoration-line: none;" title="Farads">farads</a> (capacitance).</div>
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While using the timer IC in monostable mode, the main disadvantage is that the time span between any two triggering pulses must be greater than the RC time constant.<sup class="reference" id="cite_ref-13" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/555_timer_IC#cite_note-13" style="background: none; color: #0b0080; text-decoration-line: none;">[13]</a></sup> Conversely, ignoring closely spaced pulses is done by setting the RC time constant to be larger than the span between spurious triggers. (Example: ignoring switch contact bouncing.)</div>
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<span class="mw-headline" id="Astable">Astable</span></h4>
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<span style="font-size: 12.3704px;">Schematic of a 555 in astable mode</span></div>
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In astable mode, the 555 timer puts out a continuous stream of rectangular pulses having a specified frequency. Resistor R<sub style="font-size: 11.2px; line-height: 1;">1</sub> is connected between V<sub style="font-size: 11.2px; line-height: 1;">CC</sub> and the discharge pin (pin 7) and another resistor (R<sub style="font-size: 11.2px; line-height: 1;">2</sub>) is connected between the discharge pin (pin 7), and the trigger (pin 2) and threshold (pin 6) pins that share a common node. Hence the capacitor is charged through R<sub style="font-size: 11.2px; line-height: 1;">1</sub> and R<sub style="font-size: 11.2px; line-height: 1;">2</sub>, and discharged only through R<sub style="font-size: 11.2px; line-height: 1;">2</sub>, since pin 7 has low impedance to ground during output low intervals of the cycle, therefore discharging the capacitor.</div>
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In the astable mode, the frequency of the pulse stream depends on the values of R<sub style="font-size: 11.2px; line-height: 1;">1</sub>, R<sub style="font-size: 11.2px; line-height: 1;">2</sub> and C:</div>
<dl style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px; text-align: justify;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle f={\frac {1}{\ln(2)\cdot C\cdot (R_{1}+2R_{2})}}}</annotation></semantics></math></span><img alt="f={\frac {1}{\ln(2)\cdot C\cdot (R_{1}+2R_{2})}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d78d6ad9efbc78e3b0533284a4836bc166bc255" style="border: 0px; display: inline-block; height: 6.009ex; vertical-align: -2.671ex; width: 26.879ex;" /></span><sup class="reference" id="cite_ref-14" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/555_timer_IC#cite_note-14" style="background: none; color: #0b0080; text-decoration-line: none;">[14]</a></sup></dd></dl>
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The high time from each pulse is given by:</div>
<dl style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px; text-align: justify;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \mathrm {high} =\ln(2)\cdot C\cdot (R_{1}+R_{2})}</annotation></semantics></math></span><img alt="\mathrm {high} =\ln(2)\cdot C\cdot (R_{1}+R_{2})" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3eb7249717598553fdd927f117dbef57d26e516b" style="border: 0px; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 28.018ex;" /></span></dd></dl>
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and the low time from each pulse is given by:</div>
<dl style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px; text-align: justify;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \mathrm {low} =\ln(2)\cdot C\cdot R_{2}}</annotation></semantics></math></span><img alt="\mathrm {low} =\ln(2)\cdot C\cdot R_{2}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7983402458bd6fbb7e837ac1b3ed6c97afdad83a" style="border: 0px; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 19.583ex;" /></span></dd></dl>
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where R<sub style="font-size: 11.2px; line-height: 1;">1</sub> and R<sub style="font-size: 11.2px; line-height: 1;">2</sub> are the values of the resistors in ohms and C is the value of the capacitor in farads.</div>
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The power capability of R<sub style="font-size: 11.2px; line-height: 1;">1</sub> must be greater than <span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\frac {V_{cc}^{2}}{R_{1}}}}</annotation></semantics></math></span><img alt="{\frac {V_{cc}^{2}}{R_{1}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd7b6a81483eae33615d4199ace2ba7bc9771bb6" style="border: 0px; display: inline-block; height: 6.176ex; margin: 0px; vertical-align: -2.338ex; width: 3.873ex;" /></span>.</div>
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Particularly with bipolar 555s, low values of <span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle R_{1}}</annotation></semantics></math></span><img alt="R_{1}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1d63c96f59d98589d923c4f0b04222feaa7283e" style="border: 0px; display: inline-block; height: 2.509ex; margin: 0px; vertical-align: -0.671ex; width: 2.836ex;" /></span> must be avoided so that the output stays saturated near zero volts during discharge, as assumed by the above equation. Otherwise the output low time will be greater than calculated above. The first cycle will take appreciably longer than the calculated time, as the capacitor must charge from 0V to 2/3 of V<sub style="font-size: 11.2px; line-height: 1;">CC</sub> from power-up, but only from 1/3 of V<sub style="font-size: 11.2px; line-height: 1;">CC</sub> to 2/3 of V<sub style="font-size: 11.2px; line-height: 1;">CC</sub> on subsequent cycles.</div>
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To have an output high time shorter than the low time (i.e., a <a href="https://en.wikipedia.org/wiki/Duty_cycle" style="background: none; color: #0b0080; text-decoration-line: none;" title="Duty cycle">duty cycle</a> less than 50%) a small diode (that is fast enough for the application) can be placed in parallel with R<sub style="font-size: 11.2px; line-height: 1;">2</sub>, with the cathode on the capacitor side. This bypasses R<sub style="font-size: 11.2px; line-height: 1;">2</sub> during the high part of the cycle so that the high interval depends only on R<sub style="font-size: 11.2px; line-height: 1;">1</sub> and C, with an adjustment based the voltage drop across the diode. The voltage drop across the diode slows charging on the capacitor so that the high time is a longer than the expected and often-cited ln(2)*R<sub style="font-size: 11.2px; line-height: 1;">1</sub>C = 0.693 R<sub style="font-size: 11.2px; line-height: 1;">1</sub>C. The low time will be the same as above, 0.693 R<sub style="font-size: 11.2px; line-height: 1;">2</sub>C. With the bypass diode, the high time is</div>
<dl style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px; text-align: justify;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \mathrm {high} =R_{1}\cdot C\cdot \ln \left({\frac {2V_{\textrm {cc}}-3V_{\textrm {diode}}}{V_{\textrm {cc}}-3V_{\textrm {diode}}}}\right)}</annotation></semantics></math></span><img alt="\mathrm {high} =R_{1}\cdot C\cdot \ln \left({\frac {2V_{\textrm {cc}}-3V_{\textrm {diode}}}{V_{\textrm {cc}}-3V_{\textrm {diode}}}}\right)" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56f965b28e5e94badfa18a09133d6a1d36c960a7" style="border: 0px; display: inline-block; height: 6.176ex; vertical-align: -2.505ex; width: 35.517ex;" /></span></dd></dl>
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where V<sub style="font-size: 11.2px; line-height: 1;">diode</sub> is when the diode's "on" current is 1/2 of V<sub style="font-size: 11.2px; line-height: 1;">cc</sub>/R<sub style="font-size: 11.2px; line-height: 1;">1</sub> which can be determined from its datasheet or by testing. As an extreme example, when V<sub style="font-size: 11.2px; line-height: 1;">cc</sub>= 5 and V<sub style="font-size: 11.2px; line-height: 1;">diode</sub>= 0.7, high time = 1.00 R<sub style="font-size: 11.2px; line-height: 1;">1</sub>C which is 45% longer than the "expected" 0.693 R<sub style="font-size: 11.2px; line-height: 1;">1</sub>C. At the other extreme, when V<sub style="font-size: 11.2px; line-height: 1;">cc</sub>= 15 and V<sub style="font-size: 11.2px; line-height: 1;">diode</sub>= 0.3, the high time = 0.725 R<sub style="font-size: 11.2px; line-height: 1;">1</sub>C which is closer to the expected 0.693 R<sub style="font-size: 11.2px; line-height: 1;">1</sub>C. The equation reduces to the expected 0.693 R<sub style="font-size: 11.2px; line-height: 1;">1</sub>C if V<sub style="font-size: 11.2px; line-height: 1;">diode</sub>= 0.</div>
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The operation of RESET in this mode is not well-defined. Some manufacturers' parts will hold the output state to what it was when RESET is taken low, others will send the output either high or low.</div>
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The astable configuration, with two resistors, cannot produce a 50% duty cycle. To produce a 50% duty cycle, eliminate R1, disconnect pin 7 and connect the supply end of R2 to pin 3, the output pin. This circuit is similar to using an inverter gate as an oscillator, but with fewer components than the astable configuration, and a much higher power output than a TTL or CMOS gate. The duty cycle for either the 555 or inverter-gate timer will not be precisely 50% and will change based off any load that the output is also driving while high (longer duty cycles for greater loads) due to the fact the timing network is supplied from the devices output pin, which has different internal resistances depending on whether it is in the high or low state (high side drivers tend to be more resistive).</div>
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<span class="mw-headline" id="Specifications">Specifications</span></h2>
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These specifications apply to the NE555. Other 555 timers can have different specifications depending on the grade (military, medical, etc.).</div>
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<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">Supply voltage (<i>V</i><sub style="font-size: 11.2px; line-height: 1;">CC</sub>)</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">4.5 to 15 V</td></tr>
<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">Supply current (<i>V</i><sub style="font-size: 11.2px; line-height: 1;">CC</sub> = +5 V)</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3 to 6 mA</td></tr>
<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">Supply current (<i>V</i><sub style="font-size: 11.2px; line-height: 1;">CC</sub> = +15 V)</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">10 to 15 mA</td></tr>
<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">Output current (maximum)</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">200 mA</td></tr>
<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">Maximum Power dissipation</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">600 mW</td></tr>
<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">Power consumption (minimum operating)</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">30 mW@5V, 225 mW@15V</td></tr>
<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"><a href="https://en.wikipedia.org/wiki/Operating_temperature" style="background: none; color: #0b0080; text-decoration-line: none;" title="Operating temperature">Operating temperature</a></td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0 to 75 °C<br /></td></tr>
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</head>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3005862392404377111.post-38369878989102307192017-06-13T00:23:00.004-07:002023-06-23T23:34:11.649-07:00Integrated circuit<div dir="ltr" style="text-align: left;" trbidi="on">
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An <b>integrated circuit</b> or <b>monolithic integrated circuit</b> (also referred to as an <b>IC</b>, a <b>chip</b>, or a <b>microchip</b>) is a set of <a href="https://en.wikipedia.org/wiki/Electronic_circuit" style="background: none; color: #0b0080; text-decoration-line: none;" title="Electronic circuit">electronic circuits</a> on one small flat piece (or "chip") of <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Semiconductor_material" style="background: none; color: #0b0080; text-decoration-line: none;" title="Semiconductor material">semiconductor material</a>, normally <a href="https://en.wikipedia.org/wiki/Silicon" style="background: none; color: #0b0080; text-decoration-line: none;" title="Silicon">silicon</a>. The integration of large numbers of tiny <a href="https://en.wikipedia.org/wiki/Transistor" style="background: none; color: #0b0080; text-decoration-line: none;" title="Transistor">transistors</a> into a small chip resulted in circuits that are orders of magnitude smaller, cheaper, and faster than those constructed of discrete <a href="https://en.wikipedia.org/wiki/Electronic_component" style="background: none; color: #0b0080; text-decoration-line: none;" title="Electronic component">electronic components</a>. The IC's <a href="https://en.wikipedia.org/wiki/Mass_production" style="background: none; color: #0b0080; text-decoration-line: none;" title="Mass production">mass production</a> capability, reliability and building-block approach to <a href="https://en.wikipedia.org/wiki/Integrated_circuit_design" style="background: none; color: #0b0080; text-decoration-line: none;" title="Integrated circuit design">circuit design</a> ensured the rapid adoption of standardized ICs in place of designs using discrete transistors. ICs are now used in virtually all electronic equipment and have revolutionized the world of <a href="https://en.wikipedia.org/wiki/Electronics" style="background: none; color: #0b0080; text-decoration-line: none;" title="Electronics">electronics</a>. <a href="https://en.wikipedia.org/wiki/Computer" style="background: none; color: #0b0080; text-decoration-line: none;" title="Computer">Computers</a>, <a href="https://en.wikipedia.org/wiki/Mobile_phone" style="background: none; color: #0b0080; text-decoration-line: none;" title="Mobile phone">mobile phones</a>, and other digital <a href="https://en.wikipedia.org/wiki/Home_appliance" style="background: none; color: #0b0080; text-decoration-line: none;" title="Home appliance">home appliances</a> are now inextricable parts of the structure of modern societies, made possible by the small size and low cost of ICs.</div>
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ICs were made possible by experimental discoveries showing that <a href="https://en.wikipedia.org/wiki/Semiconductor_device" style="background: none; color: #0b0080; text-decoration-line: none;" title="Semiconductor device">semiconductor devices</a> could perform the functions of <a href="https://en.wikipedia.org/wiki/Vacuum_tube" style="background: none; color: #0b0080; text-decoration-line: none;" title="Vacuum tube">vacuum tubes</a>, and by mid-20th-century technology advancements in <a href="https://en.wikipedia.org/wiki/Semiconductor_device_fabrication" style="background: none; color: #0b0080; text-decoration-line: none;" title="Semiconductor device fabrication">semiconductor device fabrication</a>. Since their origins in the 1960s, the size, speed, and capacity of chips have progressed enormously, driven by technical advances that allow more and more transistors on chips of the same size - a modern chip may have several billion <a href="https://en.wikipedia.org/wiki/Transistor" style="background: none; color: #0b0080; text-decoration-line: none;" title="Transistor">transistors</a> in an area the size of a human fingernail. These advances, roughly following <a href="https://en.wikipedia.org/wiki/Moore%27s_law" style="background: none; color: #0b0080; text-decoration-line: none;" title="Moore's law">Moore's law</a>, allow a computer chip of today to have millions of times the capacity and thousands of times the speed of the computer chips of the early 1970s.</div>
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<span class="mw-headline" id="Terminology">Terminology</span></h2>
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<span class="mw-headline"><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">A circuit in which all or some of the circuit elements are inseparably associated and electrically interconnected so that it is considered to be indivisible for the purposes of construction and commerce.</span></span></div>
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<span class="mw-headline" id="Types">Types</span></h2>
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Digital integrated circuits can contain anywhere from one<sup class="reference" id="cite_ref-33" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-33" style="background: none; color: #0b0080; text-decoration-line: none;">[33]</a></sup> to billions<sup class="reference" id="cite_ref-Pascal_18-1" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-Pascal-18" style="background: none; color: #0b0080; text-decoration-line: none;">[18]</a></sup> of <a href="https://en.wikipedia.org/wiki/Logic_gate" style="background: none; color: #0b0080; text-decoration-line: none;" title="Logic gate">logic gates</a>, <a href="https://en.wikipedia.org/wiki/Flip-flop_(electronics)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Flip-flop (electronics)">flip-flops</a>, <a href="https://en.wikipedia.org/wiki/Multiplexer" style="background: none; color: #0b0080; text-decoration-line: none;" title="Multiplexer">multiplexers</a>, and other circuits in a few square millimeters. The small size of these circuits allows high speed, low power dissipation, and reduced <a href="https://en.wikipedia.org/wiki/Manufacturing_cost" style="background: none; color: #0b0080; text-decoration-line: none;" title="Manufacturing cost">manufacturing cost</a> compared with board-level integration. These digital ICs, typically <a href="https://en.wikipedia.org/wiki/Microprocessor" style="background: none; color: #0b0080; text-decoration-line: none;" title="Microprocessor">microprocessors</a>, <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Digital_signal_processors" style="background: none; color: #0b0080; text-decoration-line: none;" title="Digital signal processors">DSPs</a>, and <a href="https://en.wikipedia.org/wiki/Microcontroller" style="background: none; color: #0b0080; text-decoration-line: none;" title="Microcontroller">microcontrollers</a>, work using <a href="https://en.wikipedia.org/wiki/Boolean_algebra" style="background: none; color: #0b0080; text-decoration-line: none;" title="Boolean algebra">boolean algebra</a> to process "one" and "zero" signals.</div>
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<span style="font-size: 12.3704px;">The</span><span style="font-size: 12.3704px;"> </span><a href="https://en.wikipedia.org/wiki/Die_(integrated_circuit)" style="background: none; color: #0b0080; font-size: 12.3704px; text-decoration-line: none;" title="Die (integrated circuit)">die</a><span style="font-size: 12.3704px;"> </span><span style="font-size: 12.3704px;">from an Intel</span><span style="font-size: 12.3704px;"> </span><a href="https://en.wikipedia.org/wiki/Intel_MCS-48" style="background: none; color: #0b0080; font-size: 12.3704px; text-decoration-line: none;" title="Intel MCS-48">8742</a><span style="font-size: 12.3704px;">, an 8-bit</span><span style="font-size: 12.3704px;"> </span><a href="https://en.wikipedia.org/wiki/Microcontroller" style="background: none; color: #0b0080; font-size: 12.3704px; text-decoration-line: none;" title="Microcontroller">microcontroller</a><span style="font-size: 12.3704px;"> </span><span style="font-size: 12.3704px;">that includes a</span><span style="font-size: 12.3704px;"> </span><a class="mw-redirect" href="https://en.wikipedia.org/wiki/CPU" style="background: none; color: #0b0080; font-size: 12.3704px; text-decoration-line: none;" title="CPU">CPU</a><span style="font-size: 12.3704px;"> </span><span style="font-size: 12.3704px;">running at 12 MHz, 128 bytes of</span><span style="font-size: 12.3704px;"> </span><a class="mw-redirect" href="https://en.wikipedia.org/wiki/RAM" style="background: none; color: #0b0080; font-size: 12.3704px; text-decoration-line: none;" title="RAM">RAM</a><span style="font-size: 12.3704px;">, 2048 bytes of</span><span style="font-size: 12.3704px;"> </span><a href="https://en.wikipedia.org/wiki/EPROM" style="background: none; color: #0b0080; font-size: 12.3704px; text-decoration-line: none;" title="EPROM">EPROM</a><span style="font-size: 12.3704px;">, and</span><span style="font-size: 12.3704px;"> </span><a href="https://en.wikipedia.org/wiki/Input/output" style="background: none; color: #0b0080; font-size: 12.3704px; text-decoration-line: none;" title="Input/output">I/O</a><span style="font-size: 12.3704px;"> </span><span style="font-size: 12.3704px;">in the same chip</span></div>
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Among the most advanced integrated circuits are the <a href="https://en.wikipedia.org/wiki/Microprocessor" style="background: none; color: #0b0080; text-decoration-line: none;" title="Microprocessor">microprocessors</a> or "<b>cores</b>", which control everything from computers and cellular phones to digital <a href="https://en.wikipedia.org/wiki/Microwave_oven" style="background: none; color: #0b0080; text-decoration-line: none;" title="Microwave oven">microwave ovens</a>. Digital <a href="https://en.wikipedia.org/wiki/Computer_memory" style="background: none; color: #0b0080; text-decoration-line: none;" title="Computer memory">memory chips</a> and <a href="https://en.wikipedia.org/wiki/Application-specific_integrated_circuit" style="background: none; color: #0b0080; text-decoration-line: none;" title="Application-specific integrated circuit">application-specific integrated circuits</a> (ASICs) are examples of other families of integrated circuits that are important to the modern <a href="https://en.wikipedia.org/wiki/Information_society" style="background: none; color: #0b0080; text-decoration-line: none;" title="Information society">information society</a>.</div>
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In the 1980s, <a href="https://en.wikipedia.org/wiki/Programmable_logic_device" style="background: none; color: #0b0080; text-decoration-line: none;" title="Programmable logic device">programmable logic devices</a> were developed. These devices contain circuits whose logical function and connectivity can be programmed by the user, rather than being fixed by the integrated circuit manufacturer. This allows a single chip to be programmed to implement different LSI-type functions such as <a href="https://en.wikipedia.org/wiki/Logic_gate" style="background: none; color: #0b0080; text-decoration-line: none;" title="Logic gate">logic gates</a>, <a href="https://en.wikipedia.org/wiki/Adder_(electronics)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Adder (electronics)">adders</a> and <a href="https://en.wikipedia.org/wiki/Processor_register" style="background: none; color: #0b0080; text-decoration-line: none;" title="Processor register">registers</a>. Current devices called <a href="https://en.wikipedia.org/wiki/Field-programmable_gate_array" style="background: none; color: #0b0080; text-decoration-line: none;" title="Field-programmable gate array">field-programmable gate arrays</a> (FPGAs) can (as of 2016) implement the equivalent of millions of gates in parallel and operate up to 1 GHz.<sup class="reference" id="cite_ref-Altera_34-0" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-Altera-34" style="background: none; color: #0b0080; text-decoration-line: none;">[34]</a></sup></div>
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Analog ICs, such as sensors, <a href="https://en.wikipedia.org/wiki/Power_network_design_(IC)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Power network design (IC)">power management circuits</a>, and <a href="https://en.wikipedia.org/wiki/Operational_amplifier" style="background: none; color: #0b0080; text-decoration-line: none;" title="Operational amplifier">operational amplifiers</a>, work by processing continuous signals. They perform functions like <a href="https://en.wikipedia.org/wiki/Amplifier" style="background: none; color: #0b0080; text-decoration-line: none;" title="Amplifier">amplification</a>, <a href="https://en.wikipedia.org/wiki/Active_filter" style="background: none; color: #0b0080; text-decoration-line: none;" title="Active filter">active filtering</a>, <a href="https://en.wikipedia.org/wiki/Demodulation" style="background: none; color: #0b0080; text-decoration-line: none;" title="Demodulation">demodulation</a>, and <a href="https://en.wikipedia.org/wiki/Frequency_mixer" style="background: none; color: #0b0080; text-decoration-line: none;" title="Frequency mixer">mixing</a>. Analog ICs ease the burden on circuit designers by having expertly designed analog circuits available instead of designing a difficult analog circuit from scratch.</div>
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ICs can also combine analog and digital circuits on a single chip to create functions such as <a href="https://en.wikipedia.org/wiki/Analog-to-digital_converter" style="background: none; color: #0b0080; text-decoration-line: none;" title="Analog-to-digital converter">A/D converters</a> and <a href="https://en.wikipedia.org/wiki/Digital-to-analog_converter" style="background: none; color: #0b0080; text-decoration-line: none;" title="Digital-to-analog converter">D/A converters</a>. Such mixed-signal circuits offer smaller size and lower cost, but must carefully account for signal interference. Prior to the late 1990s, radios could not be fabricated in the same low-cost CMOS processes as microprocessors. But since 1998, a large number of radio chips have been developed using CMOS processes. Examples include Intel's DECT cordless phone, or <a class="mw-redirect" href="https://en.wikipedia.org/wiki/802.11" style="background: none; color: #0b0080; text-decoration-line: none;" title="802.11">802.11</a> (<a href="https://en.wikipedia.org/wiki/Wi-Fi" style="background: none; color: #0b0080; text-decoration-line: none;" title="Wi-Fi">Wi-Fi</a>) chips created by <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Atheros" style="background: none; color: #0b0080; text-decoration-line: none;" title="Atheros">Atheros</a> and other companies.<sup class="reference" id="cite_ref-IEEE-CMOS-dualband-n_35-0" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-IEEE-CMOS-dualband-n-35" style="background: none; color: #0b0080; text-decoration-line: none;">[35]</a></sup></div>
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Modern <a href="https://en.wikipedia.org/wiki/Category:Electronic_component_distributors" style="background: none; color: #0b0080; text-decoration-line: none;" title="Category:Electronic component distributors">electronic component distributors</a> often further sub-categorize the huge variety of integrated circuits now available:</div>
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<li style="margin-bottom: 0.1em; text-align: justify;">Digital ICs are further sub-categorized as logic ICs, <a href="https://en.wikipedia.org/wiki/Semiconductor_memory" style="background: none; color: #0b0080; text-decoration-line: none;" title="Semiconductor memory">memory</a> chips, interface ICs (<a href="https://en.wikipedia.org/wiki/Logic_level" style="background: none; color: #0b0080; text-decoration-line: none;" title="Logic level">level shifters</a>, <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Serializer/deserializer" style="background: none; color: #0b0080; text-decoration-line: none;" title="Serializer/deserializer">serializer/deserializer</a>, etc.), <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Power_Management_IC" style="background: none; color: #0b0080; text-decoration-line: none;" title="Power Management IC">Power Management ICs</a>, and programmable devices.</li>
<li style="margin-bottom: 0.1em; text-align: justify;">Analog ICs are further sub-categorized as linear ICs and RF ICs.</li>
<li style="margin-bottom: 0.1em; text-align: justify;"><a href="https://en.wikipedia.org/wiki/Mixed-signal_integrated_circuit" style="background: none; color: #0b0080; text-decoration-line: none;" title="Mixed-signal integrated circuit">mixed-signal integrated circuits</a> are further sub-categorized as data acquisition ICs (including A/D converters, D/A converter, <a href="https://en.wikipedia.org/wiki/Digital_potentiometer" style="background: none; color: #0b0080; text-decoration-line: none;" title="Digital potentiometer">digital potentiometers</a>) and clock/timing ICs.</li>
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<span class="mw-headline" id="Manufacturing">Manufacturing</span></h2>
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The <a href="https://en.wikipedia.org/wiki/Semiconductor" style="background: none; color: #0b0080; text-decoration-line: none;" title="Semiconductor">semiconductors</a> of the <a href="https://en.wikipedia.org/wiki/Periodic_table" style="background: none; color: #0b0080; text-decoration-line: none;" title="Periodic table">periodic table</a> of the <a href="https://en.wikipedia.org/wiki/Chemical_element" style="background: none; color: #0b0080; text-decoration-line: none;" title="Chemical element">chemical elements</a> were identified as the most likely materials for a <i><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Solid-state_(electronics)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Solid-state (electronics)">solid-state</a> <a href="https://en.wikipedia.org/wiki/Vacuum_tube" style="background: none; color: #0b0080; text-decoration-line: none;" title="Vacuum tube">vacuum tube</a></i>. Starting with <a href="https://en.wikipedia.org/wiki/Copper(I)_oxide" style="background: none; color: #0b0080; text-decoration-line: none;" title="Copper(I) oxide">copper oxide</a>, proceeding to <a href="https://en.wikipedia.org/wiki/Germanium" style="background: none; color: #0b0080; text-decoration-line: none;" title="Germanium">germanium</a>, then <a href="https://en.wikipedia.org/wiki/Silicon" style="background: none; color: #0b0080; text-decoration-line: none;" title="Silicon">silicon</a>, the materials were systematically studied in the 1940s and 1950s. Today, <a href="https://en.wikipedia.org/wiki/Monocrystalline_silicon" style="background: none; color: #0b0080; text-decoration-line: none;" title="Monocrystalline silicon">monocrystalline silicon</a> is the main <a href="https://en.wikipedia.org/wiki/Substrate_(printing)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Substrate (printing)">substrate</a> used for ICs although some III-V compounds of the periodic table such as <a href="https://en.wikipedia.org/wiki/Gallium_arsenide" style="background: none; color: #0b0080; text-decoration-line: none;" title="Gallium arsenide">gallium arsenide</a> are used for specialized applications like <a href="https://en.wikipedia.org/wiki/Light-emitting_diode" style="background: none; color: #0b0080; text-decoration-line: none;" title="Light-emitting diode">LEDs</a>, <a href="https://en.wikipedia.org/wiki/Laser" style="background: none; color: #0b0080; text-decoration-line: none;" title="Laser">lasers</a>, <a href="https://en.wikipedia.org/wiki/Solar_cell" style="background: none; color: #0b0080; text-decoration-line: none;" title="Solar cell">solar cells</a> and the highest-speed integrated circuits. It took decades to perfect methods of creating <a href="https://en.wikipedia.org/wiki/Crystal" style="background: none; color: #0b0080; text-decoration-line: none;" title="Crystal">crystals</a> without defects in the <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Crystalline_structure" style="background: none; color: #0b0080; text-decoration-line: none;" title="Crystalline structure">crystalline structure</a> of the semiconducting material.</div>
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<a href="https://en.wikipedia.org/wiki/Semiconductor" style="background: none; color: #0b0080; text-decoration-line: none;" title="Semiconductor">Semiconductor</a> ICs are fabricated in a <a href="https://en.wikipedia.org/wiki/Planar_process" style="background: none; color: #0b0080; text-decoration-line: none;" title="Planar process">planar process</a> which includes three key process steps – imaging, deposition and etching. The main process steps are supplemented by doping and cleaning.</div>
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<a href="https://en.wikipedia.org/wiki/Monocrystalline_silicon" style="background: none; color: #0b0080; text-decoration-line: none;" title="Monocrystalline silicon">Mono-crystal silicon</a> <a href="https://en.wikipedia.org/wiki/Wafer_(electronics)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Wafer (electronics)">wafers</a> (or for special applications, <a href="https://en.wikipedia.org/wiki/Silicon_on_sapphire" style="background: none; color: #0b0080; text-decoration-line: none;" title="Silicon on sapphire">silicon on sapphire</a> or gallium arsenide wafers) are used as the <i>substrate</i>. <a href="https://en.wikipedia.org/wiki/Photolithography" style="background: none; color: #0b0080; text-decoration-line: none;" title="Photolithography">Photolithography</a> is used to mark different areas of the substrate to be <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Doping_(Semiconductors)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Doping (Semiconductors)">doped</a> or to have polysilicon, insulators or metal (typically aluminium or copper) tracks deposited on them.</div>
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<li style="margin-bottom: 0.1em; text-align: justify;">Integrated circuits are composed of many overlapping layers, each defined by photolithography, and normally shown in different colors. Some layers mark where various dopants are diffused into the substrate (called diffusion layers), some define where additional ions are implanted (implant layers), some define the conductors (polysilicon or metal layers), and some define the connections between the conducting layers (via or contact layers). All components are constructed from a specific combination of these layers.</li>
<li style="margin-bottom: 0.1em; text-align: justify;">In a self-aligned <a href="https://en.wikipedia.org/wiki/CMOS" style="background: none; color: #0b0080; text-decoration-line: none;" title="CMOS">CMOS</a> process, a <a href="https://en.wikipedia.org/wiki/Transistor" style="background: none; color: #0b0080; text-decoration-line: none;" title="Transistor">transistor</a> is formed wherever the gate layer (polysilicon or metal) crosses a diffusion layer.</li>
<li style="margin-bottom: 0.1em; text-align: justify;"><a href="https://en.wikipedia.org/wiki/Capacitor" style="background: none; color: #0b0080; text-decoration-line: none;" title="Capacitor">Capacitive structures</a>, in form very much like the parallel conducting plates of a traditional electrical capacitor, are formed according to the area of the "plates", with insulating material between the plates. Capacitors of a wide range of sizes are common on ICs.</li>
<li style="margin-bottom: 0.1em; text-align: justify;">Meandering stripes of varying lengths are sometimes used to form on-chip <a href="https://en.wikipedia.org/wiki/Resistor" style="background: none; color: #0b0080; text-decoration-line: none;" title="Resistor">resistors</a>, though most logic circuits do not need any resistors. The ratio of the length of the resistive structure to its width, combined with its sheet resistivity, determines the resistance.</li>
<li style="margin-bottom: 0.1em; text-align: justify;">More rarely, <a href="https://en.wikipedia.org/wiki/Inductor" style="background: none; color: #0b0080; text-decoration-line: none;" title="Inductor">inductive structures</a> can be built as tiny on-chip coils, or simulated by <a href="https://en.wikipedia.org/wiki/Gyrator" style="background: none; color: #0b0080; text-decoration-line: none;" title="Gyrator">gyrators</a>.</li>
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Since a CMOS device only draws current on the <i>transition</i> between <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Boolean_algebra_(logic)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Boolean algebra (logic)">logic</a> <a href="https://en.wikipedia.org/wiki/State_(computer_science)" style="background: none; color: #0b0080; text-decoration-line: none;" title="State (computer science)">states</a>, CMOS devices consume much less current than <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Bipolar_transistor" style="background: none; color: #0b0080; text-decoration-line: none;" title="Bipolar transistor">bipolar</a> devices.</div>
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A <a href="https://en.wikipedia.org/wiki/Random-access_memory" style="background: none; color: #0b0080; text-decoration-line: none;" title="Random-access memory">random-access memory</a> is the most regular type of integrated circuit; the highest density devices are thus memories; but even a <a href="https://en.wikipedia.org/wiki/Microprocessor" style="background: none; color: #0b0080; text-decoration-line: none;" title="Microprocessor">microprocessor</a> will have memory on the chip. (See the regular array structure at the bottom of the first image.) Although the structures are intricate – with widths which have been shrinking for decades – the layers remain much thinner than the device widths. The layers of material are fabricated much like a photographic process, although light <a href="https://en.wikipedia.org/wiki/Wave" style="background: none; color: #0b0080; text-decoration-line: none;" title="Wave">waves</a> in the <a href="https://en.wikipedia.org/wiki/Visible_spectrum" style="background: none; color: #0b0080; text-decoration-line: none;" title="Visible spectrum">visible spectrum</a> cannot be used to "expose" a layer of material, as they would be too large for the features. Thus <a href="https://en.wikipedia.org/wiki/Photon" style="background: none; color: #0b0080; text-decoration-line: none;" title="Photon">photons</a> of higher frequencies (typically <a href="https://en.wikipedia.org/wiki/Ultraviolet" style="background: none; color: #0b0080; text-decoration-line: none;" title="Ultraviolet">ultraviolet</a>) are used to create the patterns for each layer. Because each feature is so small, <a href="https://en.wikipedia.org/wiki/Electron_microscope" style="background: none; color: #0b0080; text-decoration-line: none;" title="Electron microscope">electron microscopes</a> are essential tools for a <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Industrial_process" style="background: none; color: #0b0080; text-decoration-line: none;" title="Industrial process">process</a> engineer who might be <a href="https://en.wikipedia.org/wiki/Debugging" style="background: none; color: #0b0080; text-decoration-line: none;" title="Debugging">debugging</a> a fabrication process.</div>
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Each device is tested before packaging using automated test equipment (ATE), in a process known as <a href="https://en.wikipedia.org/wiki/Wafer_testing" style="background: none; color: #0b0080; text-decoration-line: none;" title="Wafer testing">wafer testing</a>, or wafer probing. The wafer is then cut into rectangular blocks, each of which is called a <i>die</i>. Each good <a href="https://en.wikipedia.org/wiki/Die_(integrated_circuit)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Die (integrated circuit)">die</a> (plural <i>dice</i>, <i>dies</i>, or <i>die</i>) is then connected into a package using aluminium (or gold) <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Bond_wire" style="background: none; color: #0b0080; text-decoration-line: none;" title="Bond wire">bond wires</a> which are <a href="https://en.wikipedia.org/wiki/Thermosonic_bonding" style="background: none; color: #0b0080; text-decoration-line: none;" title="Thermosonic bonding">thermosonically bonded</a><sup class="reference" id="cite_ref-36" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-36" style="background: none; color: #0b0080; text-decoration-line: none;">[36]</a></sup> to <i>pads</i>, usually found around the edge of the die. . <a href="https://en.wikipedia.org/wiki/Thermosonic_bonding" style="background: none; color: #0b0080; text-decoration-line: none;" title="Thermosonic bonding">Thermosonic bonding</a> was first introduced by A. Coucoulas which provided a reliable means of forming these vital electrical connections to the outside world. After packaging, the devices go through final testing on the same or similar ATE used during wafer probing. <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Industrial_CT_scanning" style="background: none; color: #0b0080; text-decoration-line: none;" title="Industrial CT scanning">Industrial CT scanning</a> can also be used. Test cost can account for over 25% of the cost of fabrication on lower-cost products, but can be negligible on low-yielding, larger, or higher-cost devices.</div>
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As of 2016, a <a href="https://en.wikipedia.org/wiki/Semiconductor_fabrication_plant" style="background: none; color: #0b0080; text-decoration-line: none;" title="Semiconductor fabrication plant">fabrication facility</a> (commonly known as a <i>semiconductor fab</i>) can cost over US$8 billion to construct.<sup class="reference" id="cite_ref-37" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-37" style="background: none; color: #0b0080; text-decoration-line: none;">[37]</a></sup> The cost of a fabrication facility rises over time (<a href="https://en.wikipedia.org/wiki/Rock%27s_law" style="background: none; color: #0b0080; text-decoration-line: none;" title="Rock's law">Rock's law</a>) because much of the operation is automated. Today, the most advanced processes employ the following techniques:</div>
<ul style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; list-style-image: url("data:image/svg+xml,%3C%3Fxml%20version%3D%221.0%22%20encoding%3D%22UTF-8%22%3F%3E%0A%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20version%3D%221.1%22%20width%3D%225%22%20height%3D%2213%22%3E%0A%3Ccircle%20cx%3D%222.5%22%20cy%3D%229.5%22%20r%3D%222.5%22%20fill%3D%22%2300528c%22%2F%3E%0A%3C%2Fsvg%3E%0A"); margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em; text-align: justify;">The wafers are up to 300 mm in diameter (wider than a common dinner plate).</li>
<li style="margin-bottom: 0.1em; text-align: justify;">As of 2016, a state of the art foundry can produce 14 nm transistors, as implemented by <a href="https://en.wikipedia.org/wiki/Intel" style="background: none; color: #0b0080; text-decoration-line: none;" title="Intel">Intel</a>, <a href="https://en.wikipedia.org/wiki/TSMC" style="background: none; color: #0b0080; text-decoration-line: none;" title="TSMC">TSMC</a>, <a href="https://en.wikipedia.org/wiki/Samsung" style="background: none; color: #0b0080; text-decoration-line: none;" title="Samsung">Samsung</a>, and <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Global_Foundries" style="background: none; color: #0b0080; text-decoration-line: none;" title="Global Foundries">Global Foundries</a>. The next step, to 10 nm devices, is expected in 2017.<sup class="reference" id="cite_ref-38" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-38" style="background: none; color: #0b0080; text-decoration-line: none;">[38]</a></sup></li>
<li style="margin-bottom: 0.1em; text-align: justify;"><a href="https://en.wikipedia.org/wiki/Copper_interconnect" style="background: none; color: #0b0080; text-decoration-line: none;" title="Copper interconnect">Copper interconnects</a> where copper wiring replaces aluminium for interconnects.</li>
<li style="margin-bottom: 0.1em; text-align: justify;"><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Low-K" style="background: none; color: #0b0080; text-decoration-line: none;" title="Low-K">Low-K</a> dielectric insulators.</li>
<li style="margin-bottom: 0.1em; text-align: justify;"><a href="https://en.wikipedia.org/wiki/Silicon_on_insulator" style="background: none; color: #0b0080; text-decoration-line: none;" title="Silicon on insulator">Silicon on insulator</a> (SOI).</li>
<li style="margin-bottom: 0.1em; text-align: justify;"><a href="https://en.wikipedia.org/wiki/Strained_silicon" style="background: none; color: #0b0080; text-decoration-line: none;" title="Strained silicon">Strained silicon</a> in a process used by <a href="https://en.wikipedia.org/wiki/IBM" style="background: none; color: #0b0080; text-decoration-line: none;" title="IBM">IBM</a> known as <a href="https://en.wikipedia.org/wiki/Strained_silicon_directly_on_insulator" style="background: none; color: #0b0080; text-decoration-line: none;" title="Strained silicon directly on insulator">strained silicon directly on insulator</a> (SSDOI).</li>
<li style="margin-bottom: 0.1em; text-align: justify;"><a href="https://en.wikipedia.org/wiki/Multigate_device" style="background: none; color: #0b0080; text-decoration-line: none;" title="Multigate device">Multigate devices</a> such as tri-gate transistors being manufactured by <a href="https://en.wikipedia.org/wiki/Intel" style="background: none; color: #0b0080; text-decoration-line: none;" title="Intel">Intel</a> from 2011 in their 22 nm process.</li>
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<span class="mw-headline" id="Packaging">Packaging</span></h3>
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<span style="font-size: 12.3704px;">A Soviet MSI nMOS chip made in 1977, part of a four-chip calculator set designed in 1970</span><sup class="reference" id="cite_ref-39" style="font-size: 9.89632px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-39" style="background: none; color: #0b0080; text-decoration-line: none;">[39]</a></sup></div>
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The earliest integrated circuits were packaged in ceramic <a href="https://en.wikipedia.org/wiki/Flatpack_(electronics)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Flatpack (electronics)">flat packs</a>, which continued to be used by the military for their reliability and small size for many years. Commercial circuit packaging quickly moved to the <a href="https://en.wikipedia.org/wiki/Dual_in-line_package" style="background: none; color: #0b0080; text-decoration-line: none;" title="Dual in-line package">dual in-line package</a> (DIP), first in ceramic and later in plastic. In the 1980s pin counts of VLSI circuits exceeded the practical limit for DIP packaging, leading to <a href="https://en.wikipedia.org/wiki/Pin_grid_array" style="background: none; color: #0b0080; text-decoration-line: none;" title="Pin grid array">pin grid array</a> (PGA) and <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Leadless_chip_carrier" style="background: none; color: #0b0080; text-decoration-line: none;" title="Leadless chip carrier">leadless chip carrier</a> (LCC) packages. <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Surface_mount" style="background: none; color: #0b0080; text-decoration-line: none;" title="Surface mount">Surface mount</a> packaging appeared in the early 1980s and became popular in the late 1980s, using finer lead pitch with leads formed as either gull-wing or J-lead, as exemplified by the <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Small-outline_integrated_circuit" style="background: none; color: #0b0080; text-decoration-line: none;" title="Small-outline integrated circuit">small-outline integrated circuit</a> (SOIC) package – a carrier which occupies an area about 30–50% less than an equivalent DIP and is typically 70% thinner. This package has "gull wing" leads protruding from the two long sides and a lead spacing of 0.050 inches.</div>
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In the late 1990s, <a class="mw-redirect" href="https://en.wikipedia.org/wiki/PQFP" style="background: none; color: #0b0080; text-decoration-line: none;" title="PQFP">plastic quad flat pack</a> (PQFP) and <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Thin_small-outline_package" style="background: none; color: #0b0080; text-decoration-line: none;" title="Thin small-outline package">thin small-outline package</a> (TSOP) packages became the most common for high pin count devices, though PGA packages are still often used for high-end <a href="https://en.wikipedia.org/wiki/Microprocessor" style="background: none; color: #0b0080; text-decoration-line: none;" title="Microprocessor">microprocessors</a>. Intel and AMD are currently<sup class="noprint Inline-Template" style="font-size: 11.2px; line-height: 1; white-space: nowrap;">[<i><a href="https://en.wikipedia.org/wiki/Wikipedia:Manual_of_Style/Dates_and_numbers#Chronological_items" style="background: none; color: #0b0080; text-decoration-line: none;" title="Wikipedia:Manual of Style/Dates and numbers"><span title="The time period mentioned near this tag is ambiguous. (May 2017)">when?</span></a></i>]</sup> transitioning from PGA packages on high-end microprocessors to <a href="https://en.wikipedia.org/wiki/Land_grid_array" style="background: none; color: #0b0080; text-decoration-line: none;" title="Land grid array">land grid array</a> (LGA) packages.</div>
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<a href="https://en.wikipedia.org/wiki/Ball_grid_array" style="background: none; color: #0b0080; text-decoration-line: none;" title="Ball grid array">Ball grid array</a> (BGA) packages have existed since the 1970s. <a href="https://en.wikipedia.org/wiki/Flip_chip" style="background: none; color: #0b0080; text-decoration-line: none;" title="Flip chip">Flip-chip Ball Grid Array</a> packages, which allow for much higher pin count than other package types, were developed in the 1990s. In an FCBGA package the die is mounted upside-down (flipped) and connects to the package balls via a package substrate that is similar to a printed-circuit board rather than by wires. FCBGA packages allow an array of input-output signals (called Area-I/O) to be distributed over the entire die rather than being confined to the die periphery.</div>
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Traces going out of the die, through the package, and into the <a href="https://en.wikipedia.org/wiki/Printed_circuit_board" style="background: none; color: #0b0080; text-decoration-line: none;" title="Printed circuit board">printed circuit board</a> have very different electrical properties, compared to on-chip signals. They require special design techniques and need much more electric power than signals confined to the chip itself.</div>
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When multiple dies are put in one package, the result is a <a class="mw-redirect" href="https://en.wikipedia.org/wiki/System_in_Package" style="background: none; color: #0b0080; text-decoration-line: none;" title="System in Package">System in Package</a>, or SiP. A <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Multi-Chip_Module" style="background: none; color: #0b0080; text-decoration-line: none;" title="Multi-Chip Module">Multi-Chip Module</a>, or MCM, is created by combining multiple dies on a small substrate often made of ceramic. The distinction between a big MCM and a small printed circuit board is sometimes fuzzy.</div>
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<span class="mw-headline" id="Chip_labeling_and_manufacture_date">Chip labeling and manufacture date</span></h3>
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Most integrated circuits are large enough to include identifying information. Four common sections are the manufacturer's name or logo, the part number, a part production batch number and serial number, and a four-digit date-code to identify when the chip was manufactured. Extremely small <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Surface_mount_technology" style="background: none; color: #0b0080; text-decoration-line: none;" title="Surface mount technology">surface mount technology</a> parts often bear only a number used in a manufacturer's lookup table to find the chip characteristics.</div>
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The manufacturing date is commonly represented as a two-digit year followed by a two-digit week code, such that a part bearing the code 8341 was manufactured in week 41 of 1983, or approximately in October 1983.</div>
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<span class="mw-headline" id="Intellectual_property">Intellectual property</span></h2>
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The possibility of copying by photographing each layer of an integrated circuit and preparing <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Photomasks" style="background: none; color: #0b0080; text-decoration-line: none;" title="Photomasks">photomasks</a> for its production on the basis of the photographs obtained is a reason for the introduction of legislation for the protection of layout-designs. The <a href="https://en.wikipedia.org/wiki/Semiconductor_Chip_Protection_Act_of_1984" style="background: none; color: #0b0080; text-decoration-line: none;" title="Semiconductor Chip Protection Act of 1984">Semiconductor Chip Protection Act of 1984</a> established intellectual property protection for photomasks used to produce integrated circuits.<sup class="reference" id="cite_ref-USC-circ100_40-0" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-USC-circ100-40" style="background: none; color: #0b0080; text-decoration-line: none;">[40]</a></sup></div>
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A diplomatic conference was held at Washington, D.C., in 1989, which adopted a <a class="new" href="https://en.wikipedia.org/w/index.php?title=Treaty_on_Intellectual_Property_in_Respect_of_Integrated_Circuits&action=edit&redlink=1" style="background: none; color: #a55858; text-decoration-line: none;" title="Treaty on Intellectual Property in Respect of Integrated Circuits (page does not exist)">Treaty on Intellectual Property in Respect of Integrated Circuits</a> (IPIC Treaty).</div>
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The Treaty on Intellectual Property in respect of Integrated Circuits, also called Washington Treaty or IPIC Treaty (signed at Washington on 26 May 1989) is currently not in force, but was partially integrated into the <a class="mw-redirect" href="https://en.wikipedia.org/wiki/TRIPS" style="background: none; color: #0b0080; text-decoration-line: none;" title="TRIPS">TRIPS</a> agreement.<sup class="reference" id="cite_ref-41" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-41" style="background: none; color: #0b0080; text-decoration-line: none;">[41]</a></sup></div>
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National laws protecting IC layout designs have been adopted in a number of countries, including Japan,<sup class="reference" id="cite_ref-42" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-42" style="background: none; color: #0b0080; text-decoration-line: none;">[42]</a></sup> the EC,<sup class="reference" id="cite_ref-43" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-43" style="background: none; color: #0b0080; text-decoration-line: none;">[43]</a></sup> the UK, Australia, and Korea.<sup class="reference" id="cite_ref-44" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-44" style="background: none; color: #0b0080; text-decoration-line: none;">[44]</a></sup></div>
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<span class="mw-headline" id="Other_developments">Other developments</span></h2>
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Future developments seem to follow the <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Multi-core" style="background: none; color: #0b0080; text-decoration-line: none;" title="Multi-core">multi-core</a> multi-microprocessor paradigm, already used by Intel and AMD multi-core processors. Rapport Inc. and IBM started shipping the <a href="https://en.wikipedia.org/wiki/Kilocore" style="background: none; color: #0b0080; text-decoration-line: none;" title="Kilocore">KC256</a> in 2006, a 256-core microprocessor. Intel, as recently as February–August 2011, unveiled a prototype, "not for commercial sale" chip that bears 80 cores. Each core is capable of handling its own task independently of the others. This is in response to the <a class="new" href="https://en.wikipedia.org/w/index.php?title=Heat-versus-speed_limit&action=edit&redlink=1" style="background: none; color: #a55858; text-decoration-line: none;" title="Heat-versus-speed limit (page does not exist)">heat-versus-speed limit</a> that is about to be reached using <a href="https://en.wikipedia.org/wiki/CMOS" style="background: none; color: #0b0080; text-decoration-line: none;" title="CMOS">existing transistor technology</a> (see: <a href="https://en.wikipedia.org/wiki/Thermal_design_power" style="background: none; color: #0b0080; text-decoration-line: none;" title="Thermal design power">thermal design power</a>). This design provides a new challenge to chip programming. <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Parallel_programming_language" style="background: none; color: #0b0080; text-decoration-line: none;" title="Parallel programming language">Parallel programming languages</a> such as the open-source <a href="https://en.wikipedia.org/wiki/X10_(programming_language)" style="background: none; color: #0b0080; text-decoration-line: none;" title="X10 (programming language)">X10</a> programming language are designed to assist with this task.<sup class="reference" id="cite_ref-45" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-45" style="background: none; color: #0b0080; text-decoration-line: none;">[45]</a></sup></div>
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<span class="mw-headline" id="Generations">Generations</span></h2>
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In the early days of simple integrated circuits, the technology's large scale limited each chip to only a few transistors, and the low degree of integration meant the design process was relatively simple. Manufacturing yields were also quite low by today's standards. As the technology progressed, millions, then billions<sup class="reference" id="cite_ref-46" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-46" style="background: none; color: #0b0080; text-decoration-line: none;">[46]</a></sup> of transistors could be placed on one chip, and good designs required thorough planning, giving rise to the field of <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Electronic_Design_Automation" style="background: none; color: #0b0080; text-decoration-line: none;" title="Electronic Design Automation">Electronic Design Automation</a>, or EDA.</div>
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<tr><th class="headerSort" role="columnheader button" style="background-color: #eaecf0; background-image: linear-gradient(transparent, transparent), url("data:image/svg+xml,%3C%3Fxml%20version%3D%221.0%22%20encoding%3D%22UTF-8%22%3F%3E%0A%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20viewBox%3D%220%200%2021%209%22%20height%3D%229%22%20width%3D%2221%22%3E%0A%20%20%20%20%3Cpath%20d%3D%22M14.5%205l-4%204-4-4zM14.5%204l-4-4-4%204z%22%2F%3E%0A%3C%2Fsvg%3E%0A"); background-position: right center; background-repeat: no-repeat; border: 1px solid rgb(162, 169, 177); cursor: pointer; padding: 0.2em 21px 0.2em 0.4em; text-align: center;" tabindex="0" title="Sort ascending">Name</th><th class="headerSort" role="columnheader button" style="background-color: #eaecf0; background-image: linear-gradient(transparent, transparent), url("data:image/svg+xml,%3C%3Fxml%20version%3D%221.0%22%20encoding%3D%22UTF-8%22%3F%3E%0A%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20viewBox%3D%220%200%2021%209%22%20height%3D%229%22%20width%3D%2221%22%3E%0A%20%20%20%20%3Cpath%20d%3D%22M14.5%205l-4%204-4-4zM14.5%204l-4-4-4%204z%22%2F%3E%0A%3C%2Fsvg%3E%0A"); background-position: right center; background-repeat: no-repeat; border: 1px solid rgb(162, 169, 177); cursor: pointer; padding: 0.2em 21px 0.2em 0.4em; text-align: center;" tabindex="0" title="Sort ascending">Signification</th><th class="headerSort" role="columnheader button" style="background-color: #eaecf0; background-image: linear-gradient(transparent, transparent), url("data:image/svg+xml,%3C%3Fxml%20version%3D%221.0%22%20encoding%3D%22UTF-8%22%3F%3E%0A%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20viewBox%3D%220%200%2021%209%22%20height%3D%229%22%20width%3D%2221%22%3E%0A%20%20%20%20%3Cpath%20d%3D%22M14.5%205l-4%204-4-4zM14.5%204l-4-4-4%204z%22%2F%3E%0A%3C%2Fsvg%3E%0A"); background-position: right center; background-repeat: no-repeat; border: 1px solid rgb(162, 169, 177); cursor: pointer; padding: 0.2em 21px 0.2em 0.4em; text-align: center;" tabindex="0" title="Sort ascending">Year</th><th class="headerSort" role="columnheader button" style="background-color: #eaecf0; background-image: linear-gradient(transparent, transparent), url("data:image/svg+xml,%3C%3Fxml%20version%3D%221.0%22%20encoding%3D%22UTF-8%22%3F%3E%0A%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20viewBox%3D%220%200%2021%209%22%20height%3D%229%22%20width%3D%2221%22%3E%0A%20%20%20%20%3Cpath%20d%3D%22M14.5%205l-4%204-4-4zM14.5%204l-4-4-4%204z%22%2F%3E%0A%3C%2Fsvg%3E%0A"); background-position: right center; background-repeat: no-repeat; border: 1px solid rgb(162, 169, 177); cursor: pointer; padding: 0.2em 21px 0.2em 0.4em; text-align: center;" tabindex="0" title="Sort ascending"><a href="https://en.wikipedia.org/wiki/Transistor" style="background: none; color: #0b0080; text-decoration-line: none;" title="Transistor">Transistors</a> number<sup class="reference" id="cite_ref-47" style="font-size: 11.2px; font-weight: normal; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-47" style="background: none; color: #0b0080; text-decoration-line: none;">[47]</a></sup></th><th class="headerSort" role="columnheader button" style="background-color: #eaecf0; background-image: linear-gradient(transparent, transparent), url("data:image/svg+xml,%3C%3Fxml%20version%3D%221.0%22%20encoding%3D%22UTF-8%22%3F%3E%0A%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20viewBox%3D%220%200%2021%209%22%20height%3D%229%22%20width%3D%2221%22%3E%0A%20%20%20%20%3Cpath%20d%3D%22M14.5%205l-4%204-4-4zM14.5%204l-4-4-4%204z%22%2F%3E%0A%3C%2Fsvg%3E%0A"); background-position: right center; background-repeat: no-repeat; border: 1px solid rgb(162, 169, 177); cursor: pointer; padding: 0.2em 21px 0.2em 0.4em; text-align: center;" tabindex="0" title="Sort ascending"><a href="https://en.wikipedia.org/wiki/Logic_gate" style="background: none; color: #0b0080; text-decoration-line: none;" title="Logic gate">Logic gates</a> number<sup class="reference" id="cite_ref-48" style="font-size: 11.2px; font-weight: normal; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-48" style="background: none; color: #0b0080; text-decoration-line: none;">[48]</a></sup></th></tr>
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<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">SSI</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"><i>small-scale integration</i></td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1964</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1 to 10</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1 to 12</td></tr>
<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">MSI</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"><i>medium-scale integration</i></td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1968</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">10 to 500</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">13 to 99</td></tr>
<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">LSI</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"><i>large-scale integration</i></td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1971</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">500 to 20,000</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">100 to 9,999</td></tr>
<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">VLSI</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"><i>very large-scale integration</i></td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1980</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">20,000 to 1,000,000</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">10,000 to 99,999</td></tr>
<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">ULSI</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"><i>ultra-large-scale integration</i></td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1984</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1,000,000 and more</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">100,000 and more</td></tr>
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<span class="mw-headline" id="SSI.2C_MSI_and_LSI">SSI, MSI and LSI </span></h3>
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The first integrated circuits contained only a few transistors. Early digital circuits containing tens of transistors provided a few logic gates, and early linear ICs such as the <a href="https://en.wikipedia.org/wiki/Plessey" style="background: none; color: #0b0080; text-decoration-line: none;" title="Plessey">Plessey</a> SL201 or the <a href="https://en.wikipedia.org/wiki/Philips" style="background: none; color: #0b0080; text-decoration-line: none;" title="Philips">Philips</a> TAA320 had as few as two transistors. The number of transistors in an integrated circuit has increased dramatically since then. The term "large scale integration" (LSI) was first used by <a href="https://en.wikipedia.org/wiki/IBM" style="background: none; color: #0b0080; text-decoration-line: none;" title="IBM">IBM</a> scientist <a href="https://en.wikipedia.org/wiki/Rolf_Landauer" style="background: none; color: #0b0080; text-decoration-line: none;" title="Rolf Landauer">Rolf Landauer</a> when describing the theoretical concept;<sup class="noprint Inline-Template Template-Fact" style="font-size: 11.2px; line-height: 1; white-space: nowrap;">[<i><a href="https://en.wikipedia.org/wiki/Wikipedia:Citation_needed" style="background: none; color: #0b0080; text-decoration-line: none;" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (March 2011)">citation needed</span></a></i>]</sup> that term gave rise to the terms "small-scale integration" (SSI), "medium-scale integration" (MSI), "very-large-scale integration" (VLSI), and "ultra-large-scale integration" (ULSI). The early integrated circuits were SSI.</div>
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SSI circuits were crucial to early <a href="https://en.wikipedia.org/wiki/Aerospace" style="background: none; color: #0b0080; text-decoration-line: none;" title="Aerospace">aerospace</a> projects, and aerospace projects helped inspire development of the technology. Both the <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Minuteman_missile" style="background: none; color: #0b0080; text-decoration-line: none;" title="Minuteman missile">Minuteman missile</a> and <a href="https://en.wikipedia.org/wiki/Apollo_program" style="background: none; color: #0b0080; text-decoration-line: none;" title="Apollo program">Apollo program</a> needed lightweight digital computers for their inertial guidance systems. Although the <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Apollo_guidance_computer" style="background: none; color: #0b0080; text-decoration-line: none;" title="Apollo guidance computer">Apollo guidance computer</a> led and motivated integrated-circuit technology,<sup class="reference" id="cite_ref-49" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-49" style="background: none; color: #0b0080; text-decoration-line: none;">[49]</a></sup> it was the Minuteman missile that forced it into mass-production. The Minuteman missile program and various other Navy programs accounted for the total $4 million integrated circuit market in 1962, and by 1968, U.S. Government space and defense spending still accounted for 37% of the $312 million total production.</div>
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The demand by the U.S. Government supported the nascent integrated circuit market until costs fell enough to allow IC firms to penetrate first the industrial and eventually the consumer markets. The average price per integrated circuit dropped from $50.00 in 1962 to $2.33 in 1968.<sup class="reference" id="cite_ref-50" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-50" style="background: none; color: #0b0080; text-decoration-line: none;">[50]</a></sup> Integrated circuits began to appear in consumer products by the turn of the decade, a typical application being <a href="https://en.wikipedia.org/wiki/Frequency_modulation" style="background: none; color: #0b0080; text-decoration-line: none;" title="Frequency modulation">FM</a> inter-carrier sound processing in television receivers.</div>
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The first <a href="https://en.wikipedia.org/wiki/MOSFET" style="background: none; color: #0b0080; text-decoration-line: none;" title="MOSFET">MOS</a> chips were small-scale integration chips for <a href="https://en.wikipedia.org/wiki/NASA" style="background: none; color: #0b0080; text-decoration-line: none;" title="NASA">NASA</a> satellites.<sup class="reference" id="cite_ref-forging_51-0" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-forging-51" style="background: none; color: #0b0080; text-decoration-line: none;">[51]</a></sup></div>
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The next step in the development of integrated circuits, taken in the late 1960s, introduced devices which contained hundreds of transistors on each chip, called "medium-scale integration" (MSI).</div>
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In 1964, <a href="https://en.wikipedia.org/wiki/Frank_Wanlass" style="background: none; color: #0b0080; text-decoration-line: none;" title="Frank Wanlass">Frank Wanlass</a> demonstrated a single-chip 16-bit shift register he designed, with an incredible (at the time) 120 transistors on a single chip.<sup class="reference" id="cite_ref-forging_51-1" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-forging-51" style="background: none; color: #0b0080; text-decoration-line: none;">[51]</a></sup><sup class="reference" id="cite_ref-52" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-52" style="background: none; color: #0b0080; text-decoration-line: none;">[52]</a></sup></div>
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MSI devices were attractive economically because while they cost a little more to produce than SSI devices, they allowed more complex systems to be produced using smaller circuit boards, less assembly work (because of fewer separate components), and a number of other advantages.</div>
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Further development, driven by the same economic factors, led to "large-scale integration" (LSI) in the mid-1970s, with tens of thousands of transistors per chip.</div>
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The masks used to process and manufacture SSI, MSI and early LSI and VLSI devices (such as the microprocessors of the early 1970s) were mostly created by hand, often using <a href="https://en.wikipedia.org/wiki/Rubylith" style="background: none; color: #0b0080; text-decoration-line: none;" title="Rubylith">Rubylith</a>-tape or similar.<sup class="reference" id="cite_ref-53" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-53" style="background: none; color: #0b0080; text-decoration-line: none;">[53]</a></sup> For large or complex ICs (such as memories or processors), this was often done by specially hired layout people under supervision of a team of engineers, who would also, along with the circuit designers, inspect and verify the correctness and completeness of each mask. However, modern VLSI devices contain so many transistors, layers, interconnections, and other features that it is no longer feasible to check the masks or do the original design by hand. The engineer depends on computer programs and other hardware aids to do most of this work.<sup class="reference" id="cite_ref-54" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-54" style="background: none; color: #0b0080; text-decoration-line: none;">[54]</a></sup></div>
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Integrated circuits such as 1K-bit RAMs, calculator chips, and the first microprocessors, that began to be manufactured in moderate quantities in the early 1970s, had under 4,000 transistors. True LSI circuits, approaching 10,000 transistors, began to be produced around 1974, for computer main memories and second-generation microprocessors.</div>
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Some SSI and MSI chips, like <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Discrete_transistor" style="background: none; color: #0b0080; text-decoration-line: none;" title="Discrete transistor">discrete transistors</a>, are still mass-produced, both to maintain old equipment and build new devices that require only a few gates. The <a href="https://en.wikipedia.org/wiki/7400_series" style="background: none; color: #0b0080; text-decoration-line: none;" title="7400 series">7400 series</a> of <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Transistor-transistor_logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Transistor-transistor logic">TTL</a> chips, for example, has become a <a href="https://en.wikipedia.org/wiki/De_facto_standard" style="background: none; color: #0b0080; text-decoration-line: none;" title="De facto standard">de facto standard</a> and remains in production.</div>
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<span class="mw-headline" id="VLSI">VLSI</span></h3>
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<a class="image" href="https://en.wikipedia.org/wiki/File:80486DX2_200x.png" style="background: none; color: #0b0080; font-size: 13.16px; text-decoration-line: none;"><img alt="" class="thumbimage" data-file-height="1024" data-file-width="1024" height="220" src="https://upload.wikimedia.org/wikipedia/commons/thumb/2/2b/80486DX2_200x.png/220px-80486DX2_200x.png" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/80486DX2_200x.png/330px-80486DX2_200x.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2b/80486DX2_200x.png/440px-80486DX2_200x.png 2x" style="background-color: white; border: 1px solid rgb(200, 204, 209); vertical-align: middle;" width="220" /></a></div>
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<a class="internal" href="https://en.wikipedia.org/wiki/File:80486DX2_200x.png" style="background: linear-gradient(transparent, transparent), url("data:image/svg+xml,%3C%3Fxml%20version%3D%221.0%22%20encoding%3D%22UTF-8%22%20standalone%3D%22no%22%3F%3E%0A%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20viewBox%3D%220%200%2011%2015%22%20width%3D%2215%22%20height%3D%2211%22%3E%0A%20%20%20%20%3Cg%20id%3D%22magnify-clip%22%20fill%3D%22%23fff%22%20stroke%3D%22%23000%22%3E%0A%20%20%20%20%20%20%20%20%3Cpath%20id%3D%22bigbox%22%20d%3D%22M1.509%201.865h10.99v7.919h-10.99z%22%2F%3E%0A%20%20%20%20%20%20%20%20%3Cpath%20id%3D%22smallbox%22%20d%3D%22M-1.499%206.868h5.943v4.904h-5.943z%22%2F%3E%0A%20%20%20%20%3C%2Fg%3E%0A%3C%2Fsvg%3E%0A"); color: #0b0080; display: block; height: 11px; overflow: hidden; text-align: justify; text-decoration-line: none; text-indent: 15px; user-select: none; white-space: nowrap; width: 15px;" title="Enlarge"></a></div>
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<span style="font-size: 12.3704px;">Upper interconnect layers on an</span><span style="font-size: 12.3704px;"> </span><a href="https://en.wikipedia.org/wiki/Intel_80486" style="background: none; color: #0b0080; font-size: 12.3704px; text-decoration-line: none;" title="Intel 80486">Intel 80486</a><span style="font-size: 12.3704px;">DX2 microprocessor die</span></div>
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The final step in the development process, starting in the 1980s and continuing through the present, was "very-large-scale integration" (<a class="mw-redirect" href="https://en.wikipedia.org/wiki/VLSI" style="background: none; color: #0b0080; text-decoration-line: none;" title="VLSI">VLSI</a>). The development started with hundreds of thousands of transistors in the early 1980s, and continues beyond ten billion transistors as of 2016.</div>
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Multiple developments were required to achieve this increased density. Manufacturers moved to smaller design rules and cleaner fabrication facilities, so that they could make chips with more transistors and maintain adequate yield. The path of process improvements was summarized by the <a href="https://en.wikipedia.org/wiki/International_Technology_Roadmap_for_Semiconductors" style="background: none; color: #0b0080; text-decoration-line: none;" title="International Technology Roadmap for Semiconductors">International Technology Roadmap for Semiconductors</a> (ITRS). <a href="https://en.wikipedia.org/wiki/Electronic_design_automation" style="background: none; color: #0b0080; text-decoration-line: none;" title="Electronic design automation">Design tools</a> improved enough to make it practical to finish these designs in a reasonable time. The more energy-efficient <a href="https://en.wikipedia.org/wiki/CMOS" style="background: none; color: #0b0080; text-decoration-line: none;" title="CMOS">CMOS</a> replaced <a href="https://en.wikipedia.org/wiki/NMOS_logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="NMOS logic">NMOS</a> and <a href="https://en.wikipedia.org/wiki/PMOS_logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="PMOS logic">PMOS</a>, avoiding a prohibitive increase in power consumption.</div>
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In 1986 the first one-megabit <a href="https://en.wikipedia.org/wiki/Random-access_memory" style="background: none; color: #0b0080; text-decoration-line: none;" title="Random-access memory">RAM</a> chips were introduced, containing more than one million transistors. Microprocessor chips passed the million-transistor mark in 1989 and the billion-transistor mark in 2005.<sup class="reference" id="cite_ref-55" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-55" style="background: none; color: #0b0080; text-decoration-line: none;">[55]</a></sup> The trend continues largely unabated, with chips introduced in 2007 containing tens of billions of memory transistors.<sup class="reference" id="cite_ref-56" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-56" style="background: none; color: #0b0080; text-decoration-line: none;">[56]</a></sup></div>
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<span class="mw-headline" id="ULSI.2C_WSI.2C_SOC_and_3D-IC">ULSI, WSI, SOC and 3D-IC</span></h3>
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To reflect further growth of the complexity, the term <i>ULSI</i> that stands for "ultra-large-scale integration" was proposed for chips of more than 1 million transistors.<sup class="reference" id="cite_ref-57" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-57" style="background: none; color: #0b0080; text-decoration-line: none;">[57]</a></sup></div>
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<a href="https://en.wikipedia.org/wiki/Wafer-scale_integration" style="background: none; color: #0b0080; text-decoration-line: none;" title="Wafer-scale integration">Wafer-scale integration</a> (WSI) is a means of building very large integrated circuits that uses an entire silicon wafer to produce a single "super-chip". Through a combination of large size and reduced packaging, WSI could lead to dramatically reduced costs for some systems, notably massively parallel supercomputers. The name is taken from the term Very-Large-Scale Integration, the current state of the art when WSI was being developed.<sup class="reference" id="cite_ref-58" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-58" style="background: none; color: #0b0080; text-decoration-line: none;">[58]</a></sup></div>
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A <a class="mw-redirect" href="https://en.wikipedia.org/wiki/System-on-a-chip" style="background: none; color: #0b0080; text-decoration-line: none;" title="System-on-a-chip">system-on-a-chip</a> (SoC or SOC) is an integrated circuit in which all the components needed for a computer or other system are included on a single chip. The design of such a device can be complex and costly, and building disparate components on a single piece of silicon may compromise the efficiency of some elements. However, these drawbacks are offset by lower manufacturing and assembly costs and by a greatly reduced power budget: because signals among the components are kept on-die, much less power is required (see <a href="https://en.wikipedia.org/wiki/Integrated_circuit#Packaging" style="background: none; color: #0b0080; text-decoration-line: none;">Packaging</a>).<sup class="reference" id="cite_ref-59" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-59" style="background: none; color: #0b0080; text-decoration-line: none;">[59]</a></sup></div>
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A <a href="https://en.wikipedia.org/wiki/Three-dimensional_integrated_circuit" style="background: none; color: #0b0080; text-decoration-line: none;" title="Three-dimensional integrated circuit">three-dimensional integrated circuit</a> (3D-IC) has two or more layers of active electronic components that are integrated both vertically and horizontally into a single circuit. Communication between layers uses on-die signaling, so power consumption is much lower than in equivalent separate circuits. Judicious use of short vertical wires can substantially reduce overall wire length for faster operation.<sup class="reference" id="cite_ref-60" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Integrated_circuit#cite_note-60" style="background: none; color: #0b0080; text-decoration-line: none;">[60]</a></sup></div>
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<span class="mw-headline" id="Silicon_labelling_and_graffiti">Silicon labelling and graffiti</span></h2>
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To allow identification during production most silicon chips will have a serial number in one corner. It is also common to add the manufacturer's logo. Ever since ICs were created, some chip designers have used the silicon surface area for surreptitious, non-functional images or words. These are sometimes referred to as <a href="https://en.wikipedia.org/wiki/Chip_art" style="background: none; color: #0b0080; text-decoration-line: none;" title="Chip art">chip art</a>, silicon art, silicon graffiti or silicon doodling.</div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3005862392404377111.post-60039273295979275812017-06-13T00:16:00.004-07:002023-06-23T23:34:12.104-07:00De Morgan's laws<div dir="ltr" style="text-align: left;" trbidi="on">
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In <a href="https://en.wikipedia.org/wiki/Propositional_calculus" style="background: none; color: #0b0080; text-decoration-line: none;" title="Propositional calculus">propositional logic</a> and <a href="https://en.wikipedia.org/wiki/Boolean_algebra" style="background: none; color: #0b0080; text-decoration-line: none;" title="Boolean algebra">boolean algebra</a>, <b>De Morgan's laws</b><sup class="reference" id="cite_ref-1" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/De_Morgan%27s_laws#cite_note-1" style="background: none; color: #0b0080; text-decoration-line: none;">[1]</a></sup><sup class="reference" id="cite_ref-2" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/De_Morgan%27s_laws#cite_note-2" style="background: none; color: #0b0080; text-decoration-line: none;">[2]</a></sup><sup class="reference" id="cite_ref-3" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/De_Morgan%27s_laws#cite_note-3" style="background: none; color: #0b0080; text-decoration-line: none;">[3]</a></sup> are a pair of transformation rules that are both <a href="https://en.wikipedia.org/wiki/Validity" style="background: none; color: #0b0080; text-decoration-line: none;" title="Validity">valid</a> <a href="https://en.wikipedia.org/wiki/Rule_of_inference" style="background: none; color: #0b0080; text-decoration-line: none;" title="Rule of inference">rules of inference</a>. They are named after <a href="https://en.wikipedia.org/wiki/Augustus_De_Morgan" style="background: none; color: #0b0080; text-decoration-line: none;" title="Augustus De Morgan">Augustus De Morgan</a>, a 19th-century British mathematician. The rules allow the expression of <a href="https://en.wikipedia.org/wiki/Logical_conjunction" style="background: none; color: #0b0080; text-decoration-line: none;" title="Logical conjunction">conjunctions</a> and <a href="https://en.wikipedia.org/wiki/Logical_disjunction" style="background: none; color: #0b0080; text-decoration-line: none;" title="Logical disjunction">disjunctions</a> purely in terms of each other via <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Logical_negation" style="background: none; color: #0b0080; text-decoration-line: none;" title="Logical negation">negation</a>.</div>
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The rules can be expressed in English as:</div>
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the negation of a conjunction is the disjunction of the negations; and</div>
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the negation of a disjunction is the conjunction of the negations;</div>
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or</div>
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the <a href="https://en.wikipedia.org/wiki/Complement_(set_theory)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Complement (set theory)">complement</a> of the union of two sets is the same as the intersection of their complements; and</div>
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the complement of the intersection of two sets is the same as the union of their complements.</div>
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In set theory and <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Boolean_algebra_(logic)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Boolean algebra (logic)">Boolean algebra</a>, these are written formally as</div>
<dl style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px; text-align: justify;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\overline {A\cup B}}&={\overline {A}}\cap {\overline {B}},\\{\overline {A\cap B}}&={\overline {A}}\cup {\overline {B}},\end{aligned}}}</annotation></semantics></math></span><img alt="{\displaystyle {\begin{aligned}{\overline {A\cup B}}&={\overline {A}}\cap {\overline {B}},\\{\overline {A\cap B}}&={\overline {A}}\cup {\overline {B}},\end{aligned}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b55ab78fcd4c3b617df4e2195d487dda13c09e7d" style="border: 0px; display: inline-block; height: 7.176ex; vertical-align: -3.005ex; width: 17.105ex;" /></span></dd></dl>
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where</div>
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<li style="margin-bottom: 0.1em; text-align: justify;"><i>A</i> and <i>B</i> are sets,</li>
<li style="margin-bottom: 0.1em; text-align: justify;"><span style="text-decoration-line: overline;"><i>A</i></span> is the complement of A,</li>
<li style="margin-bottom: 0.1em; text-align: justify;">∩ is the <a href="https://en.wikipedia.org/wiki/Intersection_(set_theory)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Intersection (set theory)">intersection</a>, and</li>
<li style="margin-bottom: 0.1em; text-align: justify;">∪ is the <a href="https://en.wikipedia.org/wiki/Union_(set_theory)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Union (set theory)">union</a>.</li>
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In <a href="https://en.wikipedia.org/wiki/Formal_language" style="background: none; color: #0b0080; text-decoration-line: none;" title="Formal language">formal language</a>, the rules are written as</div>
<dl style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px; text-align: justify;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \neg (P\land Q)\iff (\neg P)\lor (\neg Q)}</annotation></semantics></math></span><img alt="\neg (P\land Q)\iff (\neg P)\lor (\neg Q)" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a70d27523ab4226f661e0b4e98ce668817b9e756" style="border: 0px; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 29.476ex;" /></span>,</dd></dl>
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and</div>
<dl style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px; text-align: justify;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \neg (P\lor Q)\iff (\neg P)\land (\neg Q),}</annotation></semantics></math></span><img alt="\neg (P\lor Q)\iff (\neg P)\land (\neg Q)," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5b5f9f8696e764f9cba78b0d5182f320ec0b876" style="border: 0px; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 30.133ex;" /></span></dd></dl>
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where</div>
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<li style="margin-bottom: 0.1em; text-align: justify;"><i>P</i> and <i>Q are propositions,</i></li>
<li style="margin-bottom: 0.1em; text-align: justify;"><a href="https://en.wikipedia.org/wiki/Negation" style="background: none; color: #0b0080; text-decoration-line: none;" title="Negation"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \neg }</annotation></semantics></math></span><img alt="\neg " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa78fd02085d39aa58c9e47a6d4033ce41e02fad" style="border: 0px; display: inline-block; height: 1.343ex; margin-bottom: -0.355ex; vertical-align: 0.183ex; width: 1.561ex;" /></span></a> is the negation logic operator (NOT),</li>
<li style="margin-bottom: 0.1em; text-align: justify;"><a href="https://en.wikipedia.org/wiki/Logical_conjunction" style="background: none; color: #0b0080; text-decoration-line: none;" title="Logical conjunction"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \land }</annotation></semantics></math></span><img alt="\land " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" style="border: 0px; display: inline-block; height: 2.009ex; vertical-align: -0.338ex; width: 1.561ex;" /></span></a> is the conjunction logic operator (AND),</li>
<li style="margin-bottom: 0.1em; text-align: justify;"><a href="https://en.wikipedia.org/wiki/Logical_disjunction" style="background: none; color: #0b0080; text-decoration-line: none;" title="Logical disjunction"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \lor }</annotation></semantics></math></span><img alt="\lor " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47f6b1f589aedcf14638df1d63049d233d851a" style="border: 0px; display: inline-block; height: 2.009ex; vertical-align: -0.338ex; width: 1.561ex;" /></span></a> is the disjunction logic operator (OR),</li>
<li style="margin-bottom: 0.1em; text-align: justify;"><a href="https://en.wikipedia.org/wiki/If_and_only_if" style="background: none; color: #0b0080; text-decoration-line: none;" title="If and only if"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \iff }</annotation></semantics></math></span><img alt="\iff " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff942842a50b24e7585cc42c5b50c34650e3aa99" style="border: 0px; display: inline-block; height: 1.843ex; vertical-align: -0.338ex; width: 5.617ex;" /></span></a> is a <a href="https://en.wikipedia.org/wiki/Metalogic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Metalogic">metalogical</a> symbol meaning "can be replaced in a <a href="https://en.wikipedia.org/wiki/Formal_proof" style="background: none; color: #0b0080; text-decoration-line: none;" title="Formal proof">logical proof</a> with".</li>
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Applications of the rules include simplification of logical <a href="https://en.wikipedia.org/wiki/Expression_(computer_science)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Expression (computer science)">expressions</a> in <a href="https://en.wikipedia.org/wiki/Computer_program" style="background: none; color: #0b0080; text-decoration-line: none;" title="Computer program">computer programs</a> and digital circuit designs. De Morgan's laws are an example of a more general concept of <a href="https://en.wikipedia.org/wiki/Duality_(mathematics)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Duality (mathematics)">mathematical duality</a>.</div>
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</head>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3005862392404377111.post-65500831457351608582017-06-13T00:15:00.003-07:002023-06-23T23:34:12.175-07:00COMBINATIONAL CIRCUIT<div dir="ltr" style="text-align: left;" trbidi="on">
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In <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Digital_circuit" style="background: none; color: #0b0080; text-decoration-line: none;" title="Digital circuit">digital circuit</a> theory, <b>combinational logic</b> (sometimes also referred to as <b>time-independent logic</b><sup class="reference" id="cite_ref-1" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Combinational_logic#cite_note-1" style="background: none; color: #0b0080; text-decoration-line: none;">[1]</a></sup> ) is a type of <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Digital_logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Digital logic">digital logic</a> which is implemented by <a href="https://en.wikipedia.org/wiki/Boolean_circuit" style="background: none; color: #0b0080; text-decoration-line: none;" title="Boolean circuit">Boolean circuits</a>, where the output is a <a href="https://en.wikipedia.org/wiki/Pure_function" style="background: none; color: #0b0080; text-decoration-line: none;" title="Pure function">pure function</a> of the present input only. This is in contrast to <a href="https://en.wikipedia.org/wiki/Sequential_logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Sequential logic">sequential logic</a>, in which the output depends not only on the present input but also on the history of the input. In other words, sequential logic has <i><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Computer_storage" style="background: none; color: #0b0080; text-decoration-line: none;" title="Computer storage">memory</a></i> while combinational logic does not.</div>
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Combinational logic is used in <a href="https://en.wikipedia.org/wiki/Computer" style="background: none; color: #0b0080; text-decoration-line: none;" title="Computer">computer</a> circuits to perform <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Boolean_algebra_(logic)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Boolean algebra (logic)">Boolean algebra</a> on input signals and on stored data. Practical computer circuits normally contain a mixture of combinational and sequential logic. For example, the part of an <a href="https://en.wikipedia.org/wiki/Arithmetic_logic_unit" style="background: none; color: #0b0080; text-decoration-line: none;" title="Arithmetic logic unit">arithmetic logic unit</a>, or ALU, that does mathematical calculations is constructed using combinational logic. Other circuits used in computers, such as <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Half_adder" style="background: none; color: #0b0080; text-decoration-line: none;" title="Half adder">half adders</a>, <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Full_adder" style="background: none; color: #0b0080; text-decoration-line: none;" title="Full adder">full adders</a>, <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Half_subtractor" style="background: none; color: #0b0080; text-decoration-line: none;" title="Half subtractor">half subtractors</a>, <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Half_subtractor" style="background: none; color: #0b0080; text-decoration-line: none;" title="Half subtractor">full subtractors</a>, <a href="https://en.wikipedia.org/wiki/Multiplexer" style="background: none; color: #0b0080; text-decoration-line: none;" title="Multiplexer">multiplexers</a>, <a href="https://en.wikipedia.org/wiki/Multiplexer" style="background: none; color: #0b0080; text-decoration-line: none;" title="Multiplexer">demultiplexers</a>, <a href="https://en.wikipedia.org/wiki/Encoder" style="background: none; color: #0b0080; text-decoration-line: none;" title="Encoder">encoders</a> and <a href="https://en.wikipedia.org/wiki/Binary_decoder" style="background: none; color: #0b0080; text-decoration-line: none;" title="Binary decoder">decoders</a> are also made by using combinational logic.</div>
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<span class="mw-headline" id="Representation">Representation</span></h2>
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Combinational logic is used to build circuits that produce specified outputs from certain inputs. The construction of combinational logic is generally done using one of two methods: a sum of products, or a product of sums. Consider the following truth table:</div>
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<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle A}</annotation></semantics></math></span><img alt="A" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" style="border: 0px; display: inline-block; height: 2.343ex; vertical-align: -0.338ex; width: 1.754ex;" /></span></th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle B}</annotation></semantics></math></span><img alt="B" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" style="border: 0px; display: inline-block; height: 2.176ex; vertical-align: -0.338ex; width: 1.774ex;" /></span></th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle C}</annotation></semantics></math></span><img alt="C" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" style="border: 0px; display: inline-block; height: 2.176ex; vertical-align: -0.338ex; width: 1.777ex;" /></span></th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">Result</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;"><a href="https://en.wikipedia.org/wiki/Logical_equivalence" style="background: none; color: #0b0080; text-decoration-line: none;" title="Logical equivalence">Logical equivalent</a></th></tr>
<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">F</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">F</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">F</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">F</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \neg A\cdot \neg B\cdot \neg C}</annotation></semantics></math></span><img alt="\neg A\cdot \neg B\cdot \neg C" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14e91a1da66417ac47e8bfc978068637d01cc4f2" style="border: 0px; display: inline-block; height: 2.343ex; vertical-align: -0.338ex; width: 13.366ex;" /></span></td></tr>
<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">F</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">F</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">T</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">F</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \neg A\cdot \neg B\cdot C}</annotation></semantics></math></span><img alt="\neg A\cdot \neg B\cdot C" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f04ff771e35de4249b8b0089dcb389ebbf6b745" style="border: 0px; display: inline-block; height: 2.343ex; vertical-align: -0.338ex; width: 11.805ex;" /></span></td></tr>
<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">F</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">T</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">F</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">F</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \neg A\cdot B\cdot \neg C}</annotation></semantics></math></span><img alt="\neg A\cdot B\cdot \neg C" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c7daf706b9970a023ffd5086be305efd5e2b5e3" style="border: 0px; display: inline-block; height: 2.343ex; vertical-align: -0.338ex; width: 11.805ex;" /></span></td></tr>
<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">F</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">T</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">T</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">F</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \neg A\cdot B\cdot C}</annotation></semantics></math></span><img alt="\neg A\cdot B\cdot C" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f78bc398fe597e15902316913a1bc3a00fdc6735" style="border: 0px; display: inline-block; height: 2.343ex; vertical-align: -0.338ex; width: 10.245ex;" /></span></td></tr>
<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">T</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">F</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">F</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">T</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle A\cdot \neg B\cdot \neg C}</annotation></semantics></math></span><img alt="A\cdot \neg B\cdot \neg C" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/566b3926a08cd991d630b5258d8629181e748e76" style="border: 0px; display: inline-block; height: 2.343ex; vertical-align: -0.338ex; width: 11.805ex;" /></span></td></tr>
<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">T</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">F</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">T</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">F</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle A\cdot \neg B\cdot C}</annotation></semantics></math></span><img alt="A\cdot \neg B\cdot C" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cde8c725807782eb02430c749dcf2bd1eee09e06" style="border: 0px; display: inline-block; height: 2.343ex; vertical-align: -0.338ex; width: 10.245ex;" /></span></td></tr>
<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">T</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">T</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">F</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">F</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle A\cdot B\cdot \neg C}</annotation></semantics></math></span><img alt="A\cdot B\cdot \neg C" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ff01292fa353f55439c177a05668cf68b519fd4" style="border: 0px; display: inline-block; height: 2.343ex; vertical-align: -0.338ex; width: 10.245ex;" /></span></td></tr>
<tr><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">T</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">T</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">T</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">T</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle A\cdot B\cdot C}</annotation></semantics></math></span><img alt="A\cdot B\cdot C" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f45e37d635311e73acebdc2337bd6f3c5c56c426" style="border: 0px; display: inline-block; height: 2.343ex; vertical-align: -0.338ex; width: 8.684ex;" /></span></td></tr>
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Using sum of products, all logical statements which yield true results are summed, giving the result:</div>
<dl style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px; text-align: justify;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle A\cdot \neg B\cdot \neg C+A\cdot B\cdot C\,}</annotation></semantics></math></span><img alt="A\cdot \neg B\cdot \neg C+A\cdot B\cdot C\," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f955bd1c381aeda7375d6f5afc6131915717740" style="border: 0px; display: inline-block; height: 2.509ex; vertical-align: -0.505ex; width: 23.727ex;" /></span></dd></dl>
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Using <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Boolean_algebra_(logic)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Boolean algebra (logic)">Boolean algebra</a>, the result simplifies to the following equivalent of the truth table:</div>
<dl style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px; text-align: justify;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle A\cdot (\neg B\cdot \neg C+B\cdot C)\,}</annotation></semantics></math></span><img alt="A\cdot (\neg B\cdot \neg C+B\cdot C)\," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc23609c133d08c0fc4d08118de0495f78bfb868" style="border: 0px; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 22.114ex;" /></span></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><h2 style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border-bottom: 1px solid rgb(162, 169, 177); color: black; font-family: "Linux Libertine", Georgia, Times, serif; font-weight: normal; line-height: 1.3; margin: 1em 0px 0.25em; overflow: hidden; padding: 0px; text-align: justify;">
<span class="mw-headline" id="Logic_formula_minimization">Logic formula minimization</span></h2>
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Minimization (simplification) of combinational logic formulas is done using the following rules based on the <a href="https://en.wikipedia.org/wiki/Boolean_algebra#Laws" style="background: none; color: #0b0080; text-decoration-line: none;" title="Boolean algebra">laws of Boolean algebra</a>:</div>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px; text-align: justify;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(A+B)\cdot (A+C)&=A+(B\cdot C)\\(A\cdot B)+(A\cdot C)&=A\cdot (B+C)\end{aligned}}}</annotation></semantics></math></span><img alt="{\begin{aligned}(A+B)\cdot (A+C)&=A+(B\cdot C)\\(A\cdot B)+(A\cdot C)&=A\cdot (B+C)\end{aligned}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cced43e3b747917b886b432cea98f79348bf2159" style="border: 0px; display: inline-block; height: 6.176ex; vertical-align: -2.505ex; width: 33.646ex;" /></span></dd></dl>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px; text-align: justify;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}A+(A\cdot B)&=A\\A\cdot (A+B)&=A\end{aligned}}}</annotation></semantics></math></span><img alt="{\begin{aligned}A+(A\cdot B)&=A\\A\cdot (A+B)&=A\end{aligned}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28e56e18d578752c060e978f52275439531ec7dd" style="border: 0px; display: inline-block; height: 6.176ex; vertical-align: -2.505ex; width: 17.266ex;" /></span></dd></dl>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px; text-align: justify;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}A+(\lnot A\cdot B)&=A+B\\A\cdot (\lnot A+B)&=A\cdot B\end{aligned}}}</annotation></semantics></math></span><img alt="{\begin{aligned}A+(\lnot A\cdot B)&=A+B\\A\cdot (\lnot A+B)&=A\cdot B\end{aligned}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d11351a7058043459e3be71aad421c04cc64a557" style="border: 0px; display: inline-block; height: 6.176ex; vertical-align: -2.505ex; width: 23.452ex;" /></span></dd></dl>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px; text-align: justify;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(A+B)\cdot (\lnot A+B)&=B\\(A\cdot B)+(\lnot A\cdot B)&=B\end{aligned}}}</annotation></semantics></math></span><img alt="{\begin{aligned}(A+B)\cdot (\lnot A+B)&=B\\(A\cdot B)+(\lnot A\cdot B)&=B\end{aligned}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b0643bda0a64579ace860e87860d9189791b4e2" style="border: 0px; display: inline-block; height: 6.176ex; vertical-align: -2.505ex; width: 25.303ex;" /></span></dd></dl>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px; text-align: justify;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(A\cdot B)+(\lnot A\cdot C)+(B\cdot C)&=(A\cdot B)+(\lnot A\cdot C)\\(A+B)\cdot (\lnot A+C)\cdot (B+C)&=(A+B)\cdot (\lnot A+C)\end{aligned}}}</annotation></semantics></math></span><img alt="{\begin{aligned}(A\cdot B)+(\lnot A\cdot C)+(B\cdot C)&=(A\cdot B)+(\lnot A\cdot C)\\(A+B)\cdot (\lnot A+C)\cdot (B+C)&=(A+B)\cdot (\lnot A+C)\end{aligned}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/301c4673eecca2a6fa9cec8d02834097dec813aa" style="border: 0px; display: inline-block; height: 6.176ex; vertical-align: -2.505ex; width: 53.124ex;" /></span></dd></dl>
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With the use of minimization (sometimes called <a href="https://en.wikipedia.org/wiki/Logic_optimization" style="background: none; color: #0b0080; text-decoration-line: none;" title="Logic optimization">logic optimization</a>), a simplified logical function or circuit may be arrived upon, and the logic <a class="new" href="https://en.wikipedia.org/w/index.php?title=Combinational_circuit&action=edit&redlink=1" style="background: none; color: #a55858; text-decoration-line: none;" title="Combinational circuit (page does not exist)">combinational circuit</a> becomes smaller, and easier to analyse, use, or build.</div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3005862392404377111.post-43991546953963383542017-06-13T00:08:00.001-07:002023-06-23T23:34:11.767-07:00THREE VALUED LOGIC<div dir="ltr" style="text-align: left;" trbidi="on">
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In <a href="https://en.wikipedia.org/wiki/Logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Logic">logic</a>, a <b>three-valued logic</b> (also <b>trinary logic</b>, <b>trivalent</b>, <b>ternary</b>, or <b>trilean</b>,sometimes abbreviated <b>3VL</b>) is any of several <a href="https://en.wikipedia.org/wiki/Many-valued_logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Many-valued logic">many-valued logic</a> systems in which there are three <a href="https://en.wikipedia.org/wiki/Truth_value" style="background: none; color: #0b0080; text-decoration-line: none;" title="Truth value">truth values</a> indicating <i>true</i>, <i>false</i> and some indeterminate third value. This is contrasted with the more commonly known <a href="https://en.wikipedia.org/wiki/Principle_of_bivalence" style="background: none; color: #0b0080; text-decoration-line: none;" title="Principle of bivalence">bivalent</a> logics (such as classical sentential or <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Boolean_logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Boolean logic">Boolean logic</a>) which provide only for <i>true</i> and <i>false</i>. Conceptual form and basic ideas were initially created by <a href="https://en.wikipedia.org/wiki/Jan_%C5%81ukasiewicz" style="background: none; color: #0b0080; text-decoration-line: none;" title="Jan Łukasiewicz">Jan Łukasiewicz</a> and <a class="mw-redirect" href="https://en.wikipedia.org/wiki/C._I._Lewis" style="background: none; color: #0b0080; text-decoration-line: none;" title="C. I. Lewis">C. I. Lewis</a>. These were then re-formulated by <a href="https://en.wikipedia.org/wiki/Grigore_Moisil" style="background: none; color: #0b0080; text-decoration-line: none;" title="Grigore Moisil">Grigore Moisil</a> in an axiomatic algebraic form, and also extended to <i>n</i>-valued logics in 1945.</div>
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<span class="mw-headline" id="Representation_of_values">Representation of values</span></h2>
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As with bivalent logic, truth values in ternary logic may be represented numerically using various representations of the <a href="https://en.wikipedia.org/wiki/Ternary_numeral_system" style="background: none; color: #0b0080; text-decoration-line: none;" title="Ternary numeral system">ternary numeral system</a>. A few of the more common examples are:</div>
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<li style="margin-bottom: 0.1em; text-align: justify;">in <a href="https://en.wikipedia.org/wiki/Balanced_ternary" style="background: none; color: #0b0080; text-decoration-line: none;" title="Balanced ternary">balanced ternary</a>, each digit has one of 3 values: −1, 0, or +1; these values may also be simplified to −, 0, +, respectively;<sup class="reference" id="cite_ref-1" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Three-valued_logic#cite_note-1" style="background: none; color: #0b0080; text-decoration-line: none;">[1]</a></sup></li>
<li style="margin-bottom: 0.1em; text-align: justify;">in the <a href="https://en.wikipedia.org/wiki/Redundant_binary_representation" style="background: none; color: #0b0080; text-decoration-line: none;" title="Redundant binary representation">redundant binary representation</a>, each digit can have a value of −1, 0, 0/1 (the value 0/1 has two different representations);</li>
<li style="margin-bottom: 0.1em; text-align: justify;">in the <a href="https://en.wikipedia.org/wiki/Ternary_numeral_system" style="background: none; color: #0b0080; text-decoration-line: none;" title="Ternary numeral system">ternary numeral system</a>, each <a href="https://en.wikipedia.org/wiki/Numerical_digit" style="background: none; color: #0b0080; text-decoration-line: none;" title="Numerical digit">digit</a> is a <i><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Trit_(computing)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Trit (computing)">trit</a></i> (trinary digit) having a value of: 0, 1, or 2;</li>
<li style="margin-bottom: 0.1em; text-align: justify;">in the <a href="https://en.wikipedia.org/wiki/Skew_binary_number_system" style="background: none; color: #0b0080; text-decoration-line: none;" title="Skew binary number system">skew binary number system</a>, only most-significant non-zero digit has a value 2, and the remaining digits have a value of 0 or 1;</li>
<li style="margin-bottom: 0.1em; text-align: justify;">1 for <i>true</i>, 2 for <i>false</i>, and 0 for <i>unknown</i>, <i>unknowable</i>/<i><a href="https://en.wikipedia.org/wiki/Undecidable_problem" style="background: none; color: #0b0080; text-decoration-line: none;" title="Undecidable problem">undecidable</a></i>, <i>irrelevant</i>, or <i>both</i>;<sup class="reference" id="cite_ref-2" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Three-valued_logic#cite_note-2" style="background: none; color: #0b0080; text-decoration-line: none;">[2]</a></sup><sup class="noprint Inline-Template" style="font-size: 11.2px; line-height: 1; white-space: nowrap;">[<i><a href="https://en.wikipedia.org/wiki/Wikipedia:Verifiability" style="background: none; color: #0b0080; text-decoration-line: none;" title="Wikipedia:Verifiability"><span title="The material near this tag failed verification of its source citation(s). (October 2012)">not in citation given</span></a></i>]</sup></li>
<li style="margin-bottom: 0.1em; text-align: justify;">0 for <i>false</i>, 1 for <i>true</i>, and a third non-integer "maybe" symbol such as ?, #, ½,<sup class="reference" id="cite_ref-3" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Three-valued_logic#cite_note-3" style="background: none; color: #0b0080; text-decoration-line: none;">[3]</a></sup> or xy.</li>
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Inside a <a href="https://en.wikipedia.org/wiki/Ternary_computer" style="background: none; color: #0b0080; text-decoration-line: none;" title="Ternary computer">ternary computer</a>, ternary values are represented by <a href="https://en.wikipedia.org/wiki/Ternary_signal" style="background: none; color: #0b0080; text-decoration-line: none;" title="Ternary signal">ternary signals</a>.</div>
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This article mainly illustrates a system of ternary <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Propositional_logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Propositional logic">propositional logic</a> using the truth values {false, unknown, true}, and extends conventional Boolean <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Connectives" style="background: none; color: #0b0080; text-decoration-line: none;" title="Connectives">connectives</a> to a trivalent context. Ternary <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Predicate_logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Predicate logic">predicate logics</a> exist as well;<sup class="noprint Inline-Template Template-Fact" style="font-size: 11.2px; line-height: 1; white-space: nowrap;">[<i><a href="https://en.wikipedia.org/wiki/Wikipedia:Citation_needed" style="background: none; color: #0b0080; text-decoration-line: none;" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2007)">citation needed</span></a></i>]</sup> these may have readings of the <a href="https://en.wikipedia.org/wiki/Quantifier_(logic)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Quantifier (logic)">quantifier</a> different from classical (binary) predicate logic and may include alternative quantifiers as well.</div>
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<span class="mw-headline" id="Logics">Logics</span></h2>
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Where <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Boolean_logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Boolean logic">Boolean logic</a> has 2<sup style="font-size: 11.2px; line-height: 1;">2</sup> = 4 <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Unary_operator" style="background: none; color: #0b0080; text-decoration-line: none;" title="Unary operator">unary operators</a>, the addition of a third value in ternary logic leads to a total of 3<sup style="font-size: 11.2px; line-height: 1;">3</sup> = 27 distinct operators on a single input value. Similarly, where Boolean logic has 2<sup style="font-size: 11.2px; line-height: 1;">2<sup style="font-size: 8.96px; line-height: 1;">2</sup></sup> = 16 distinct binary operators (operators with 2 inputs), ternary logic has 3<sup style="font-size: 11.2px; line-height: 1;">3<sup style="font-size: 8.96px; line-height: 1;">2</sup></sup> = 19,683 such operators. Where we can easily name a significant fraction of the Boolean operators (<a class="mw-redirect" href="https://en.wikipedia.org/wiki/Not_(logic)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Not (logic)">not</a>, <a class="mw-redirect" href="https://en.wikipedia.org/wiki/And_(logic)" style="background: none; color: #0b0080; text-decoration-line: none;" title="And (logic)">and</a>, <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Or_(logic)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Or (logic)">or</a>, <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Logical_NAND" style="background: none; color: #0b0080; text-decoration-line: none;" title="Logical NAND">nand</a>, <a href="https://en.wikipedia.org/wiki/Logical_NOR" style="background: none; color: #0b0080; text-decoration-line: none;" title="Logical NOR">nor</a>, <a href="https://en.wikipedia.org/wiki/Exclusive_or" style="background: none; color: #0b0080; text-decoration-line: none;" title="Exclusive or">exclusive or</a>, <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Material_equivalence" style="background: none; color: #0b0080; text-decoration-line: none;" title="Material equivalence">equivalence</a>, <a href="https://en.wikipedia.org/wiki/Material_conditional" style="background: none; color: #0b0080; text-decoration-line: none;" title="Material conditional">implication</a>), it is unreasonable to attempt to name all but a small fraction of the possible ternary operators.<sup class="reference" id="cite_ref-4" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Three-valued_logic#cite_note-4" style="background: none; color: #0b0080; text-decoration-line: none;">[4]</a></sup></div>
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<span class="mw-headline" id="Kleene_and_Priest_logics">Kleene and Priest logics</span></h3>
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Below is a set of <a href="https://en.wikipedia.org/wiki/Truth_table" style="background: none; color: #0b0080; text-decoration-line: none;" title="Truth table">truth tables</a> showing the logic operations for <a href="https://en.wikipedia.org/wiki/Stephen_Cole_Kleene" style="background: none; color: #0b0080; text-decoration-line: none;" title="Stephen Cole Kleene">Kleene</a>'s "strong logic of indeterminacy" and Priest's "logic of paradox".</div>
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<tr><td><table border="0" cellspacing="12px" style="font-size: 14px;"><caption>(F, false; U, unknown; T, true)</caption><tbody>
<tr valign="bottom"><td><table class="wikitable" style="background-color: #f8f9fa; border-collapse: collapse; border: 1px solid rgb(162, 169, 177); color: black; font-size: 14px; margin: 1em 0px; text-align: center;"><caption style="font-weight: bold;">NOT(A)</caption><tbody>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">A</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">¬A</th></tr>
<tr><th class="table-no" scope="row" style="background: rgb(255, 153, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center; vertical-align: middle;" width="25">F</th><td class="table-yes" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">T</td></tr>
<tr><th class="unknown table-unknown" scope="row" style="background: rgb(236, 236, 236); border: 1px solid rgb(162, 169, 177); color: #2c2c2c; font-size: smaller; padding: 0.2em 0.4em; text-align: center; vertical-align: middle;" width="25">U</th><td class="unknown table-unknown" style="background: rgb(236, 236, 236); border: 1px solid rgb(162, 169, 177); color: #2c2c2c; font-size: smaller; padding: 0.2em 0.4em; vertical-align: middle;">U</td></tr>
<tr><th class="table-yes" scope="row" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center; vertical-align: middle;" width="25">T</th><td class="table-no" style="background: rgb(255, 153, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">F</td></tr>
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</td><td><table class="wikitable" style="background-color: #f8f9fa; border-collapse: collapse; border: 1px solid rgb(162, 169, 177); color: black; font-size: 14px; margin: 1em 0px; text-align: center;"><caption style="font-weight: bold;">AND(A, B)</caption><tbody>
<tr><th colspan="2" rowspan="2" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">A ∧ B</th><th colspan="3" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">B</th></tr>
<tr><th class="table-no" style="background: rgb(255, 153, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center; vertical-align: middle;" width="25">F</th><th class="unknown table-unknown" style="background: rgb(236, 236, 236); border: 1px solid rgb(162, 169, 177); color: #2c2c2c; font-size: smaller; padding: 0.2em 0.4em; text-align: center; vertical-align: middle;" width="25">U</th><th class="table-yes" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center; vertical-align: middle;" width="25">T</th></tr>
<tr><th rowspan="3" scope="row" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0px 10px; text-align: center;">A</th><th class="table-no" scope="row" style="background: rgb(255, 153, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center; vertical-align: middle;" width="25">F</th><td class="table-no" style="background: rgb(255, 153, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">F</td><td class="table-no" style="background: rgb(255, 153, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">F</td><td class="table-no" style="background: rgb(255, 153, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">F</td></tr>
<tr><th class="unknown table-unknown" scope="row" style="background: rgb(236, 236, 236); border: 1px solid rgb(162, 169, 177); color: #2c2c2c; font-size: smaller; padding: 0.2em 0.4em; text-align: center; vertical-align: middle;" width="25">U</th><td class="table-no" style="background: rgb(255, 153, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">F</td><td class="unknown table-unknown" style="background: rgb(236, 236, 236); border: 1px solid rgb(162, 169, 177); color: #2c2c2c; font-size: smaller; padding: 0.2em 0.4em; vertical-align: middle;">U</td><td class="unknown table-unknown" style="background: rgb(236, 236, 236); border: 1px solid rgb(162, 169, 177); color: #2c2c2c; font-size: smaller; padding: 0.2em 0.4em; vertical-align: middle;">U</td></tr>
<tr><th class="table-yes" scope="row" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center; vertical-align: middle;" width="25">T</th><td class="table-no" style="background: rgb(255, 153, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">F</td><td class="unknown table-unknown" style="background: rgb(236, 236, 236); border: 1px solid rgb(162, 169, 177); color: #2c2c2c; font-size: smaller; padding: 0.2em 0.4em; vertical-align: middle;">U</td><td class="table-yes" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">T</td></tr>
</tbody></table>
</td><td><table class="wikitable" style="background-color: #f8f9fa; border-collapse: collapse; border: 1px solid rgb(162, 169, 177); color: black; font-size: 14px; margin: 1em 0px; text-align: center;"><caption style="font-weight: bold;">OR(A, B)</caption><tbody>
<tr><th colspan="2" rowspan="2" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">A ∨ B</th><th colspan="3" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">B</th></tr>
<tr><th class="table-no" style="background: rgb(255, 153, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center; vertical-align: middle;" width="25">F</th><th class="unknown table-unknown" style="background: rgb(236, 236, 236); border: 1px solid rgb(162, 169, 177); color: #2c2c2c; font-size: smaller; padding: 0.2em 0.4em; text-align: center; vertical-align: middle;" width="25">U</th><th class="table-yes" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center; vertical-align: middle;" width="25">T</th></tr>
<tr><th rowspan="3" scope="row" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0px 10px; text-align: center;">A</th><th class="table-no" scope="row" style="background: rgb(255, 153, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center; vertical-align: middle;" width="25">F</th><td class="table-no" style="background: rgb(255, 153, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">F</td><td class="unknown table-unknown" style="background: rgb(236, 236, 236); border: 1px solid rgb(162, 169, 177); color: #2c2c2c; font-size: smaller; padding: 0.2em 0.4em; vertical-align: middle;">U</td><td class="table-yes" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">T</td></tr>
<tr><th class="unknown table-unknown" scope="row" style="background: rgb(236, 236, 236); border: 1px solid rgb(162, 169, 177); color: #2c2c2c; font-size: smaller; padding: 0.2em 0.4em; text-align: center; vertical-align: middle;" width="25">U</th><td class="unknown table-unknown" style="background: rgb(236, 236, 236); border: 1px solid rgb(162, 169, 177); color: #2c2c2c; font-size: smaller; padding: 0.2em 0.4em; vertical-align: middle;">U</td><td class="unknown table-unknown" style="background: rgb(236, 236, 236); border: 1px solid rgb(162, 169, 177); color: #2c2c2c; font-size: smaller; padding: 0.2em 0.4em; vertical-align: middle;">U</td><td class="table-yes" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">T</td></tr>
<tr><th class="table-yes" scope="row" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center; vertical-align: middle;" width="25">T</th><td class="table-yes" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">T</td><td class="table-yes" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">T</td><td class="table-yes" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">T</td></tr>
</tbody></table>
</td></tr>
</tbody></table>
</td><td><table border="0" cellspacing="12px" style="font-size: 14px;"><caption>(−1, false; 0, unknown; +1, true)</caption><tbody>
<tr valign="bottom"><td><table class="wikitable" style="background-color: #f8f9fa; border-collapse: collapse; border: 1px solid rgb(162, 169, 177); color: black; font-size: 14px; margin: 1em 0px; text-align: center;"><caption style="font-weight: bold;">NEG(A)</caption><tbody>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">A</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">¬A</th></tr>
<tr><th scope="row" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">−1</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">+1</td></tr>
<tr><th scope="row" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">0</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td></tr>
<tr><th scope="row" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">+1</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">−1</td></tr>
</tbody></table>
</td><td><table class="wikitable" style="background-color: #f8f9fa; border-collapse: collapse; border: 1px solid rgb(162, 169, 177); color: black; font-size: 14px; margin: 1em 0px; text-align: center;"><caption style="font-weight: bold;">MIN(A, B)</caption><tbody>
<tr><th colspan="2" rowspan="2" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">A ∧ B</th><th colspan="3" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">B</th></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">−1</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">0</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">+1</th></tr>
<tr><th rowspan="3" scope="row" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0px 10px; text-align: center;">A</th><th scope="row" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">−1</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">−1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">−1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">−1</td></tr>
<tr><th scope="row" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">0</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">−1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td></tr>
<tr><th scope="row" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">+1</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">−1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">+1</td></tr>
</tbody></table>
</td><td><table class="wikitable" style="background-color: #f8f9fa; border-collapse: collapse; border: 1px solid rgb(162, 169, 177); color: black; font-size: 14px; margin: 1em 0px; text-align: center;"><caption style="font-weight: bold;">MAX(A, B)</caption><tbody>
<tr><th colspan="2" rowspan="2" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">A ∨ B</th><th colspan="3" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">B</th></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">−1</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">0</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">+1</th></tr>
<tr><th rowspan="3" scope="row" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0px 10px; text-align: center;">A</th><th scope="row" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">−1</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">−1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">+1</td></tr>
<tr><th scope="row" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">0</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">+1</td></tr>
<tr><th scope="row" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">+1</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">+1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">+1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">+1</td></tr>
</tbody></table>
</td></tr>
</tbody></table>
</td></tr>
</tbody></table>
<div style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: justify;">
In these truth tables, the <i>unknown</i> state can be thought of as neither true nor false in Kleene logic, or thought of as both true and false in Priest logic. The difference lies in the definition of tautologies. Where Kleene logic's only designated truth value is T, Priest logic's designated truth values are both T and U. In Kleene logic, the knowledge of whether any particular <i>unknown</i> state secretly represents <i>true</i> or <i>false</i> at any moment in time is not available. However, certain logical operations can yield an unambiguous result, even if they involve at least one <i>unknown</i> operand. For example, since <i>true</i> OR <i>true</i> equals <i>true</i>, and <i>true</i> OR <i>false</i> also equals <i>true</i>, one can infer that <i>true</i> OR <i>unknown</i> equals <i>true</i>, as well. In this example, since either bivalent state could be underlying the <i>unknown</i> state, but either state also yields the same result, a definitive <i>true</i> results in all three cases.</div>
<div style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: justify;">
If numeric values, e.g. <a href="https://en.wikipedia.org/wiki/Balanced_ternary" style="background: none; color: #0b0080; text-decoration-line: none;" title="Balanced ternary">balanced ternary</a> values, are assigned to <i>false</i>, <i>unknown</i> and <i>true</i> such that <i>false</i> is less than <i>unknown</i> and <i>unknown</i> is less than <i>true</i>, then A AND B AND C... = MIN(A, B, C …) and A OR B OR C … = MAX(A, B, C...).</div>
<div style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: justify;">
Material implication for Kleene logic can be defined as:</div>
<div style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: justify;">
<span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle A\rightarrow B\ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\mbox{NOT}}(A)\ {\mbox{OR}}\ B}</annotation></semantics></math></span><img alt="A\rightarrow B\ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\mbox{NOT}}(A)\ {\mbox{OR}}\ B" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5c897342031a1c69ac1223f35bd602706f39ccb" style="border: 0px; display: inline-block; height: 3.843ex; margin: 0px; vertical-align: -0.838ex; width: 25.483ex;" /></span>, and its truth table is</div>
<table style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; text-align: center; width: 60%px;"><tbody>
<tr><td><table class="wikitable" style="background-color: #f8f9fa; border-collapse: collapse; border: 1px solid rgb(162, 169, 177); color: black; font-size: 14px; margin: 1em 0px; text-align: center;"><caption style="font-weight: bold;">IMP<sub style="font-size: 11.2px; line-height: 1;">K</sub>(A, B), OR(¬A, B)</caption><tbody>
<tr><th colspan="2" rowspan="2" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">A → B</th><th colspan="3" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">B</th></tr>
<tr><th class="table-yes" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center; vertical-align: middle;" width="25">T</th><th class="unknown table-unknown" style="background: rgb(236, 236, 236); border: 1px solid rgb(162, 169, 177); color: #2c2c2c; font-size: smaller; padding: 0.2em 0.4em; text-align: center; vertical-align: middle;" width="25">U</th><th class="table-no" style="background: rgb(255, 153, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center; vertical-align: middle;" width="25">F</th></tr>
<tr><th rowspan="3" scope="row" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0px 10px; text-align: center;">A</th><th class="table-yes" scope="row" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center; vertical-align: middle;" width="25">T</th><td class="table-yes" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">T</td><td class="unknown table-unknown" style="background: rgb(236, 236, 236); border: 1px solid rgb(162, 169, 177); color: #2c2c2c; font-size: smaller; padding: 0.2em 0.4em; vertical-align: middle;">U</td><td class="table-no" style="background: rgb(255, 153, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">F</td></tr>
<tr><th class="unknown table-unknown" scope="row" style="background: rgb(236, 236, 236); border: 1px solid rgb(162, 169, 177); color: #2c2c2c; font-size: smaller; padding: 0.2em 0.4em; text-align: center; vertical-align: middle;" width="25">U</th><td class="table-yes" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">T</td><td class="unknown table-unknown" style="background: rgb(236, 236, 236); border: 1px solid rgb(162, 169, 177); color: #2c2c2c; font-size: smaller; padding: 0.2em 0.4em; vertical-align: middle;">U</td><td class="unknown table-unknown" style="background: rgb(236, 236, 236); border: 1px solid rgb(162, 169, 177); color: #2c2c2c; font-size: smaller; padding: 0.2em 0.4em; vertical-align: middle;">U</td></tr>
<tr><th class="table-no" scope="row" style="background: rgb(255, 153, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center; vertical-align: middle;" width="25">F</th><td class="table-yes" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">T</td><td class="table-yes" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">T</td><td class="table-yes" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">T</td></tr>
</tbody></table>
</td><td><table class="wikitable" style="background-color: #f8f9fa; border-collapse: collapse; border: 1px solid rgb(162, 169, 177); color: black; font-size: 14px; margin: 1em 0px; text-align: center;"><caption style="font-weight: bold;">IMP<sub style="font-size: 11.2px; line-height: 1;">K</sub>(A, B), MAX(−A, B)</caption><tbody>
<tr><th colspan="2" rowspan="2" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">A → B</th><th colspan="3" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">B</th></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">+1</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">0</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">−1</th></tr>
<tr><th rowspan="3" scope="row" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0px 10px; text-align: center;">A</th><th scope="row" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">+1</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">+1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">−1</td></tr>
<tr><th scope="row" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">0</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">+1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td></tr>
<tr><th scope="row" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">−1</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">+1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">+1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">+1</td></tr>
</tbody></table>
</td></tr>
</tbody></table>
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which differs from that for Łukasiewicz logic (described below).</div>
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Kleene logic has no tautologies (valid formulas) because whenever all of the atomic components of a well-formed formula are assigned the value Unknown, the formula itself must also have the value Unknown. (And the only <i>designated</i> truth value for Kleene logic is True.) However, the lack of valid formulas does not mean that it lacks valid arguments and/or inference rules. An argument is semantically valid in Kleene logic if, whenever (for any interpretation/model) all of its premises are True, the conclusion must also be True. (Note that the <a href="https://en.wikipedia.org/wiki/Paraconsistent_logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Paraconsistent logic">Logic of Paradox</a> (LP) has the same truth tables as Kleene logic, but it has two <i>designated</i> truth values instead of one; these are: True and Both (the analogue of Unknown), so that LP does have tautologies but it has fewer valid inference rules.)<sup class="reference" id="cite_ref-5" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Three-valued_logic#cite_note-5" style="background: none; color: #0b0080; text-decoration-line: none;">[5]</a></sup></div>
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<span class="mw-headline" id=".C5.81ukasiewicz_logic">Łukasiewicz logic</span><span class="mw-editsection" style="display: inline-block; font-size: x-small; font-weight: normal; line-height: 1em; margin-left: 1em; unicode-bidi: isolate; user-select: none; vertical-align: baseline; white-space: nowrap;"><span class="mw-editsection-bracket" style="color: #555555; margin-right: 0.25em;">[</span><a href="https://en.wikipedia.org/w/index.php?title=Three-valued_logic&action=edit&section=4" style="background: none; color: #0b0080; text-decoration-line: none;" title="Edit section: Łukasiewicz logic">edit</a><span class="mw-editsection-bracket" style="color: #555555; margin-left: 0.25em;">]</span></span></h3>
<div class="hatnote navigation-not-searchable" role="note" style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; font-style: italic; margin-bottom: 0.5em; padding-left: 1.6em; text-align: justify;">
Further information: <a href="https://en.wikipedia.org/wiki/%C5%81ukasiewicz_logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Łukasiewicz logic">Łukasiewicz logic</a></div>
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The Łukasiewicz Ł3 has the same tables for AND, OR, and NOT as the Kleene logic given above, but differs in its definition of implication in that "unknown implies unknown" is <b>true</b>. This section follows the presentation from Malinowski's chapter of the <i>Handbook of the History of Logic</i>, vol 8.<sup class="reference" id="cite_ref-6" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Three-valued_logic#cite_note-6" style="background: none; color: #0b0080; text-decoration-line: none;">[6]</a></sup></div>
<table style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; text-align: center; width: 60%px;"><tbody>
<tr><td><table class="wikitable" style="background-color: #f8f9fa; border-collapse: collapse; border: 1px solid rgb(162, 169, 177); color: black; font-size: 14px; margin: 1em 0px; text-align: center;"><caption style="font-weight: bold;">IMP<sub style="font-size: 11.2px; line-height: 1;">Ł</sub>(A, B)</caption><tbody>
<tr><th colspan="2" rowspan="2" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">A → B</th><th colspan="3" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">B</th></tr>
<tr><th class="table-yes" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center; vertical-align: middle;" width="25">T</th><th class="unknown table-unknown" style="background: rgb(236, 236, 236); border: 1px solid rgb(162, 169, 177); color: #2c2c2c; font-size: smaller; padding: 0.2em 0.4em; text-align: center; vertical-align: middle;" width="25">U</th><th class="table-no" style="background: rgb(255, 153, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center; vertical-align: middle;" width="25">F</th></tr>
<tr><th rowspan="3" scope="row" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0px 10px; text-align: center;">A</th><th class="table-yes" scope="row" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center; vertical-align: middle;" width="25">T</th><td class="table-yes" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">T</td><td class="unknown table-unknown" style="background: rgb(236, 236, 236); border: 1px solid rgb(162, 169, 177); color: #2c2c2c; font-size: smaller; padding: 0.2em 0.4em; vertical-align: middle;">U</td><td class="table-no" style="background: rgb(255, 153, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">F</td></tr>
<tr><th class="unknown table-unknown" scope="row" style="background: rgb(236, 236, 236); border: 1px solid rgb(162, 169, 177); color: #2c2c2c; font-size: smaller; padding: 0.2em 0.4em; text-align: center; vertical-align: middle;" width="25">U</th><td class="table-yes" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">T</td><td class="table-yes" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">T</td><td class="unknown table-unknown" style="background: rgb(236, 236, 236); border: 1px solid rgb(162, 169, 177); color: #2c2c2c; font-size: smaller; padding: 0.2em 0.4em; vertical-align: middle;">U</td></tr>
<tr><th class="table-no" scope="row" style="background: rgb(255, 153, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center; vertical-align: middle;" width="25">F</th><td class="table-yes" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">T</td><td class="table-yes" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">T</td><td class="table-yes" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">T</td></tr>
</tbody></table>
</td><td><table class="wikitable" style="background-color: #f8f9fa; border-collapse: collapse; border: 1px solid rgb(162, 169, 177); color: black; font-size: 14px; margin: 1em 0px; text-align: center;"><caption style="font-weight: bold;">IMP<sub style="font-size: 11.2px; line-height: 1;">Ł</sub>(A, B)</caption><tbody>
<tr><th colspan="2" rowspan="2" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">A → B</th><th colspan="3" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">B</th></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">+1</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">0</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">−1</th></tr>
<tr><th rowspan="3" scope="row" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0px 10px; text-align: center;">A</th><th scope="row" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">+1</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">+1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">−1</td></tr>
<tr><th scope="row" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">0</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">+1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">+1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td></tr>
<tr><th scope="row" style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25">−1</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">+1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">+1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">+1</td></tr>
</tbody></table>
</td></tr>
</tbody></table>
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In fact, using Łukasiewicz's implication and negation, the other usual connectives may be derived as:</div>
<ul style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; list-style-image: url("data:image/svg+xml,%3C%3Fxml%20version%3D%221.0%22%20encoding%3D%22UTF-8%22%3F%3E%0A%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20version%3D%221.1%22%20width%3D%225%22%20height%3D%2213%22%3E%0A%3Ccircle%20cx%3D%222.5%22%20cy%3D%229.5%22%20r%3D%222.5%22%20fill%3D%22%2300528c%22%2F%3E%0A%3C%2Fsvg%3E%0A"); margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em; text-align: justify;"><span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: 'lnum' 1, 'tnum' 1, 'kern' 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;"><i>A</i> ∨ <i>B</i> = (<i>A</i> → <i>B</i>) → <i>B</i></span></li>
<li style="margin-bottom: 0.1em; text-align: justify;"><span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: 'lnum' 1, 'tnum' 1, 'kern' 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;"><i>A</i> ∧ <i>B</i> = ¬(¬<i>A</i> ∨ ¬ <i>B</i>)</span></li>
<li style="margin-bottom: 0.1em; text-align: justify;"><span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: 'lnum' 1, 'tnum' 1, 'kern' 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;"><i>A</i> ↔ <i>B</i> = (<i>A</i> → <i>B</i>) ∧ (<i>B</i> → <i>A</i>)</span></li>
</ul>
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It's also possible to derive a few other useful unary operators (first derived by Tarski in 1921):</div>
<ul style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; list-style-image: url("data:image/svg+xml,%3C%3Fxml%20version%3D%221.0%22%20encoding%3D%22UTF-8%22%3F%3E%0A%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20version%3D%221.1%22%20width%3D%225%22%20height%3D%2213%22%3E%0A%3Ccircle%20cx%3D%222.5%22%20cy%3D%229.5%22%20r%3D%222.5%22%20fill%3D%22%2300528c%22%2F%3E%0A%3C%2Fsvg%3E%0A"); margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em; text-align: justify;"><span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: 'lnum' 1, 'tnum' 1, 'kern' 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;"><b>M</b><i>A</i> = ¬<i>A</i> → <i>A</i></span></li>
<li style="margin-bottom: 0.1em; text-align: justify;"><span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: 'lnum' 1, 'tnum' 1, 'kern' 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;"><b>L</b><i>A</i> = ¬<b>M</b>¬<i>A</i></span></li>
<li style="margin-bottom: 0.1em; text-align: justify;"><span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: 'lnum' 1, 'tnum' 1, 'kern' 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;"><b>I</b><i>A</i> = <b>M</b><i>A</i> ∧ ¬<b>L</b><i>A</i></span></li>
</ul>
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They have the following truth tables:</div>
<table border="0" cellspacing="12px" style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; text-align: justify;"><tbody>
<tr><td valign="top"><table class="wikitable" style="background-color: #f8f9fa; border-collapse: collapse; border: 1px solid rgb(162, 169, 177); color: black; font-size: 14px; margin: 1em 0px; text-align: center;"><tbody>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25"><span class="texhtml mvar" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: 'lnum' 1, 'tnum' 1, 'kern' 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">A</span></th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25"><span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: 'lnum' 1, 'tnum' 1, 'kern' 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">M<i>A</i></span></th></tr>
<tr><th class="table-no" scope="row" style="background: rgb(255, 153, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center; vertical-align: middle;">F</th><td class="table-no" style="background: rgb(255, 153, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">F</td></tr>
<tr><th class="unknown table-unknown" scope="row" style="background: rgb(236, 236, 236); border: 1px solid rgb(162, 169, 177); color: #2c2c2c; font-size: smaller; padding: 0.2em 0.4em; text-align: center; vertical-align: middle;">U</th><td class="table-yes" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">T</td></tr>
<tr><th class="table-yes" scope="row" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center; vertical-align: middle;">T</th><td class="table-yes" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">T</td></tr>
</tbody></table>
</td><td><table class="wikitable" style="background-color: #f8f9fa; border-collapse: collapse; border: 1px solid rgb(162, 169, 177); color: black; font-size: 14px; margin: 1em 0px; text-align: center;"><tbody>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25"><span class="texhtml mvar" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: 'lnum' 1, 'tnum' 1, 'kern' 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">A</span></th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25"><span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: 'lnum' 1, 'tnum' 1, 'kern' 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">L<i>A</i></span></th></tr>
<tr><th class="table-no" scope="row" style="background: rgb(255, 153, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center; vertical-align: middle;">F</th><td class="table-no" style="background: rgb(255, 153, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">F</td></tr>
<tr><th class="unknown table-unknown" scope="row" style="background: rgb(236, 236, 236); border: 1px solid rgb(162, 169, 177); color: #2c2c2c; font-size: smaller; padding: 0.2em 0.4em; text-align: center; vertical-align: middle;">U</th><td class="table-no" style="background: rgb(255, 153, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">F</td></tr>
<tr><th class="table-yes" scope="row" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center; vertical-align: middle;">T</th><td class="table-yes" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">T</td></tr>
</tbody></table>
</td><td><table class="wikitable" style="background-color: #f8f9fa; border-collapse: collapse; border: 1px solid rgb(162, 169, 177); color: black; font-size: 14px; margin: 1em 0px; text-align: center;"><tbody>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25"><span class="texhtml mvar" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: 'lnum' 1, 'tnum' 1, 'kern' 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">A</span></th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;" width="25"><span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: 'lnum' 1, 'tnum' 1, 'kern' 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">I<i>A</i></span></th></tr>
<tr><th class="table-no" scope="row" style="background: rgb(255, 153, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center; vertical-align: middle;">F</th><td class="table-no" style="background: rgb(255, 153, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">F</td></tr>
<tr><th class="unknown table-unknown" scope="row" style="background: rgb(236, 236, 236); border: 1px solid rgb(162, 169, 177); color: #2c2c2c; font-size: smaller; padding: 0.2em 0.4em; text-align: center; vertical-align: middle;">U</th><td class="table-yes" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">T</td></tr>
<tr><th class="table-yes" scope="row" style="background: rgb(153, 255, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center; vertical-align: middle;">T</th><td class="table-no" style="background: rgb(255, 153, 153); border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; vertical-align: middle;">F</td></tr>
</tbody></table>
</td></tr>
</tbody></table>
<div style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: justify;">
M is read as "it is not false that..." or in the (unsuccessful) Tarski–Łukasiewicz attempt to axiomatize <a href="https://en.wikipedia.org/wiki/Modal_logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Modal logic">modal logic</a> using a three-valued logic, "it is possible that..." L is read "it is true that..." or "it is necessary that..." Finally I is read "it is unknown that..." or "it is contingent that..."</div>
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In Łukasiewicz's Ł3 the <a class="new" href="https://en.wikipedia.org/w/index.php?title=Designated_value&action=edit&redlink=1" style="background: none; color: #a55858; text-decoration-line: none;" title="Designated value (page does not exist)">designated value</a> is True, meaning that only a proposition having this value everywhere is considered a <a href="https://en.wikipedia.org/wiki/Tautology_(logic)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Tautology (logic)">tautology</a>. For example, <span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: 'lnum' 1, 'tnum' 1, 'kern' 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;"><i>A</i> → <i>A</i></span> and <span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: 'lnum' 1, 'tnum' 1, 'kern' 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;"><i>A</i> ↔ <i>A</i></span> are tautologies in Ł3 and also in classical logic. Not all tautologies of classical logic lift to Ł3 "as is". For example, the <a href="https://en.wikipedia.org/wiki/Law_of_excluded_middle" style="background: none; color: #0b0080; text-decoration-line: none;" title="Law of excluded middle">law of excluded middle</a>, <span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: 'lnum' 1, 'tnum' 1, 'kern' 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;"><i>A</i> ∨ ¬<i>A</i></span>, and the <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Law_of_non-contradiction" style="background: none; color: #0b0080; text-decoration-line: none;" title="Law of non-contradiction">law of non-contradiction</a>, <span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: 'lnum' 1, 'tnum' 1, 'kern' 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">¬(<i>A</i> ∧ ¬<i>A</i>)</span> are not tautologies in Ł3. However, using the operator <span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: 'lnum' 1, 'tnum' 1, 'kern' 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;"><b>I</b></span> defined above, it is possible to state tautologies that are their analogues:</div>
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<li style="margin-bottom: 0.1em; text-align: justify;"><span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: 'lnum' 1, 'tnum' 1, 'kern' 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;"><i>A</i> ∨ <b>I</b><i>A</i> ∨ ¬<i>A</i></span> (<a class="mw-redirect" href="https://en.wikipedia.org/wiki/Law_of_excluded_fourth" style="background: none; color: #0b0080; text-decoration-line: none;" title="Law of excluded fourth">law of excluded fourth</a>)</li>
<li style="margin-bottom: 0.1em; text-align: justify;"><span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: 'lnum' 1, 'tnum' 1, 'kern' 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">¬(<i>A</i> ∧ ¬<b>I</b><i>A</i> ∧ ¬<i>A</i>)</span> (<a class="new" href="https://en.wikipedia.org/w/index.php?title=Extended_contradiction_principle&action=edit&redlink=1" style="background: none; color: #a55858; text-decoration-line: none;" title="Extended contradiction principle (page does not exist)">extended contradiction principle</a>).</li>
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<span class="mw-headline" id="Bochvar_logic">Bochvar logic</span></h3>
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Main article: <a href="https://en.wikipedia.org/wiki/Many-valued_logic#Bochvar.27s_internal_three-valued_logic_.28also_known_as_Kleene.27s_weak_three-valued_logic.29" style="background: none; color: #0b0080; text-decoration-line: none;" title="Many-valued logic">Many-valued_logic § Bochvar.27s_internal_three-valued_logic_.28also_known_as_Kleene.27s_weak_three-valued_logic.29</a></div>
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<span class="mw-headline" id="Modular_algebras"><br /><span style="font-size: 1.2em;">Application in SQL</span><span style="color: #555555; font-size: x-small;"><span style="font-weight: normal; white-space: nowrap;"> </span></span><span style="font-size: 1.2em;">Modular algebras</span></span></h3>
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Some 3VL <a class="new" href="https://en.wikipedia.org/w/index.php?title=Modular_algebra&action=edit&redlink=1" style="background: none; color: #a55858; text-decoration-line: none;" title="Modular algebra (page does not exist)">modular algebras</a> have been introduced more recently, motivated by circuit problems rather than philosophical issues:<sup class="reference" id="cite_ref-7" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Three-valued_logic#cite_note-7" style="background: none; color: #0b0080; text-decoration-line: none;">[7]</a></sup></div>
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<li style="margin-bottom: 0.1em; text-align: justify;">Cohn algebra</li>
<li style="margin-bottom: 0.1em; text-align: justify;">Pradhan algebra</li>
<li style="margin-bottom: 0.1em; text-align: justify;">Dubrova and Muzio algebra</li>
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Main article: <a href="https://en.wikipedia.org/wiki/Null_(SQL)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Null (SQL)">Null (SQL)</a></div>
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The database structural query language <a href="https://en.wikipedia.org/wiki/SQL" style="background: none; color: #0b0080; text-decoration-line: none;" title="SQL">SQL</a> implements ternary logic as a means of handling comparisons with <a href="https://en.wikipedia.org/wiki/Null_(SQL)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Null (SQL)">NULL</a> field content. The original intent of NULL in SQL was to represent missing data in a database, i.e. the assumption that an actual value exists, but that the value is not currently recorded in the database. SQL uses a common fragment of the Kleene K3 logic, restricted to AND, OR, and NOT tables. In SQL, the intermediate value is intended to be interpreted as UNKNOWN. Explicit comparisons with NULL, including that of another NULL yields UNKNOWN. However this choice of semantics is abandoned for some set operations, e.g. UNION or INTERSECT, where NULLs are treated as equal with each other. Critics assert that this inconsistency deprives SQL of intuitive semantics in its treatment of NULLs.<sup class="reference" id="cite_ref-Meyden_8-0" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Three-valued_logic#cite_note-Meyden-8" style="background: none; color: #0b0080; text-decoration-line: none;">[8]</a></sup> The SQL standard defines an optional feature called F571, which adds some unary operators, among which is <code style="background-color: #f8f9fa; border-radius: 2px; border: 1px solid rgb(234, 236, 240); color: black; font-family: monospace, Courier; padding: 1px 4px;">IS UNKNOWN</code> corresponding to the Łukasiewicz <b>I</b> in this article. The addition of <code style="background-color: #f8f9fa; border-radius: 2px; border: 1px solid rgb(234, 236, 240); color: black; font-family: monospace, Courier; padding: 1px 4px;">IS UNKNOWN</code> to the other operators of SQL's three-valued logic makes the SQL three-valued logic <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Functionally_complete" style="background: none; color: #0b0080; text-decoration-line: none;" title="Functionally complete">functionally complete</a>,<sup class="reference" id="cite_ref-9" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Three-valued_logic#cite_note-9" style="background: none; color: #0b0080; text-decoration-line: none;">[9]</a></sup> meaning its logical operators can express (in combination) any conceivable three-valued logical function.</div>
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</head>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3005862392404377111.post-52330816482506960402017-06-13T00:03:00.005-07:002023-06-23T23:34:11.958-07:00LOGIC DESIGN<div dir="ltr" style="text-align: left;" trbidi="on">
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In electronics, <b>logic synthesis</b> is a process by which an abstract form of desired circuit behavior, typically at <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Register_transfer_level" style="background: none; color: #0b0080; text-decoration-line: none;" title="Register transfer level">register transfer level</a> (RTL), is turned into a design implementation in terms of <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Logic_gates" style="background: none; color: #0b0080; text-decoration-line: none;" title="Logic gates">logic gates</a>, typically by a <a href="https://en.wikipedia.org/wiki/Computer_program" style="background: none; color: #0b0080; text-decoration-line: none;" title="Computer program">computer program</a> called a <i>synthesis tool</i>. Common examples of this process include synthesis of <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Hardware_Description_Language" style="background: none; color: #0b0080; text-decoration-line: none;" title="Hardware Description Language">HDLs</a>, including <a href="https://en.wikipedia.org/wiki/VHDL" style="background: none; color: #0b0080; text-decoration-line: none;" title="VHDL">VHDL</a> and <a href="https://en.wikipedia.org/wiki/Verilog" style="background: none; color: #0b0080; text-decoration-line: none;" title="Verilog">Verilog</a>.<sup class="reference" id="cite_ref-1" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Logic_synthesis#cite_note-1" style="background: none; color: #0b0080; text-decoration-line: none;">[1]</a></sup>Some synthesis tools generate <a href="https://en.wikipedia.org/wiki/Bitstream" style="background: none; color: #0b0080; text-decoration-line: none;" title="Bitstream">bitstreams</a> for <a href="https://en.wikipedia.org/wiki/Programmable_logic_device" style="background: none; color: #0b0080; text-decoration-line: none;" title="Programmable logic device">programmable logic devices</a> such as <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Programmable_array_logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Programmable array logic">PALs</a> or <a href="https://en.wikipedia.org/wiki/Field-programmable_gate_array" style="background: none; color: #0b0080; text-decoration-line: none;" title="Field-programmable gate array">FPGAs</a>, while others target the creation of <a class="mw-redirect" href="https://en.wikipedia.org/wiki/ASIC" style="background: none; color: #0b0080; text-decoration-line: none;" title="ASIC">ASICs</a>. Logic synthesis is one aspect of <a href="https://en.wikipedia.org/wiki/Electronic_design_automation" style="background: none; color: #0b0080; text-decoration-line: none;" title="Electronic design automation">electronic design automation</a>.</div>
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<span class="mw-headline" id="History_of_logic_synthesis">History of logic synthesis</span></h2>
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The roots of logic synthesis can be traced to the treatment of logic by <a href="https://en.wikipedia.org/wiki/George_Boole" style="background: none; color: #0b0080; text-decoration-line: none;" title="George Boole">George Boole</a> (1815 to 1864), in what is now termed <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Boolean_algebra_(logic)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Boolean algebra (logic)">Boolean algebra</a>. In 1938, <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Claude_Elwood_Shannon" style="background: none; color: #0b0080; text-decoration-line: none;" title="Claude Elwood Shannon">Claude Shannon</a> showed that the two-valued <a href="https://en.wikipedia.org/wiki/Boolean_algebra" style="background: none; color: #0b0080; text-decoration-line: none;" title="Boolean algebra">Boolean algebra</a> can describe the operation of switching circuits. In the early days, <b>logic design</b> involved manipulating the <a href="https://en.wikipedia.org/wiki/Truth_table" style="background: none; color: #0b0080; text-decoration-line: none;" title="Truth table">truth table</a> representations as <a href="https://en.wikipedia.org/wiki/Karnaugh_map" style="background: none; color: #0b0080; text-decoration-line: none;" title="Karnaugh map">Karnaugh maps</a>. The Karnaugh map-based minimization of logic is guided by a set of rules on how entries in the maps can be combined. A human designer can typically only work with Karnaugh maps containing up to four to six variables.</div>
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The first step toward automation of <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Logic_minimization" style="background: none; color: #0b0080; text-decoration-line: none;" title="Logic minimization">logic minimization</a> was the introduction of the <a href="https://en.wikipedia.org/wiki/Quine%E2%80%93McCluskey_algorithm" style="background: none; color: #0b0080; text-decoration-line: none;" title="Quine–McCluskey algorithm">Quine–McCluskey algorithm</a> that could be implemented on a computer. This exact minimization technique presented the notion of prime implicants and minimum cost covers that would become the cornerstone of <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Two-level_minimization" style="background: none; color: #0b0080; text-decoration-line: none;" title="Two-level minimization">two-level minimization</a>. Nowadays, the much more efficient <a href="https://en.wikipedia.org/wiki/Espresso_heuristic_logic_minimizer" style="background: none; color: #0b0080; text-decoration-line: none;" title="Espresso heuristic logic minimizer">Espresso heuristic logic minimizer</a> has become the standard tool for this operation.<sup class="noprint Inline-Template" style="font-size: 11.2px; line-height: 1; white-space: nowrap;">[<i><a href="https://en.wikipedia.org/wiki/Wikipedia:Manual_of_Style/Dates_and_numbers#Chronological_items" style="background: none; color: #0b0080; text-decoration-line: none;" title="Wikipedia:Manual of Style/Dates and numbers"><span title="This correctly describes the situation around 1995. We need to expand this to include the changes of the past twenty years, however. (May 2017)">needs update</span></a></i>]</sup> Another area of early research was in state minimization and encoding of <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Finite_state_machine" style="background: none; color: #0b0080; text-decoration-line: none;" title="Finite state machine">finite state machines</a> (FSMs), a task that was the bane of designers. The applications for logic synthesis lay primarily in digital computer design. Hence, <a href="https://en.wikipedia.org/wiki/IBM" style="background: none; color: #0b0080; text-decoration-line: none;" title="IBM">IBM</a> and <a href="https://en.wikipedia.org/wiki/Bell_Labs" style="background: none; color: #0b0080; text-decoration-line: none;" title="Bell Labs">Bell Labs</a> played a pivotal role in the early automation of logic synthesis. The evolution from <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Discrete_logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Discrete logic">discrete logic</a> components to <a href="https://en.wikipedia.org/wiki/Programmable_logic_array" style="background: none; color: #0b0080; text-decoration-line: none;" title="Programmable logic array">programmable logic arrays</a> (PLAs) hastened the need for efficient two-level minimization, since minimizing terms in a two-level representation reduces the area in a PLA.</div>
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However, two-level logic circuits are of limited importance in a <a href="https://en.wikipedia.org/wiki/Very-large-scale_integration" style="background: none; color: #0b0080; text-decoration-line: none;" title="Very-large-scale integration">very-large-scale integration</a> (VLSI) design; most designs use multiple levels of logic. As a matter of fact, almost any circuit representation in RTL or Behavioural Description is a multi-level representation. An early system that was used to design multilevel circuits was LSS from IBM. It used local transformations to simplify logic. Work on LSS and the Yorktown Silicon Compiler spurred rapid research progress in logic synthesis in the 1980s. Several universities contributed by making their research available to the public, most notably SIS from <a href="https://en.wikipedia.org/wiki/University_of_California,_Berkeley" style="background: none; color: #0b0080; text-decoration-line: none;" title="University of California, Berkeley">University of California, Berkeley</a>, RASP from <a href="https://en.wikipedia.org/wiki/University_of_California,_Los_Angeles" style="background: none; color: #0b0080; text-decoration-line: none;" title="University of California, Los Angeles">University of California, Los Angeles</a> and BOLD from <a class="mw-redirect" href="https://en.wikipedia.org/wiki/University_of_Colorado,_Boulder" style="background: none; color: #0b0080; text-decoration-line: none;" title="University of Colorado, Boulder">University of Colorado, Boulder</a>. Within a decade, the technology migrated to commercial logic synthesis products offered by electronic design automation companies.</div>
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<span class="mw-headline" id="Logic_elements">Logic elements</span></h2>
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<i>Logic design</i> is a step in the standard design cycle in which the <a href="https://en.wikipedia.org/wiki/Functional_design" style="background: none; color: #0b0080; text-decoration-line: none;" title="Functional design">functional design</a> of an <a href="https://en.wikipedia.org/wiki/Electronic_circuit" style="background: none; color: #0b0080; text-decoration-line: none;" title="Electronic circuit">electronic circuit</a> is converted into the representation which captures <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Boolean_algebra_(logic)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Boolean algebra (logic)">logic operations</a>, <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Arithmetic_operations" style="background: none; color: #0b0080; text-decoration-line: none;" title="Arithmetic operations">arithmetic operations</a>, <a href="https://en.wikipedia.org/wiki/Control_flow" style="background: none; color: #0b0080; text-decoration-line: none;" title="Control flow">control flow</a>, etc. A common output of this step is <a class="mw-redirect" href="https://en.wikipedia.org/wiki/RTL_description" style="background: none; color: #0b0080; text-decoration-line: none;" title="RTL description">RTL description</a>. Logic design is commonly followed by the <a href="https://en.wikipedia.org/wiki/Circuit_design" style="background: none; color: #0b0080; text-decoration-line: none;" title="Circuit design">circuit design</a> step. In modern <a href="https://en.wikipedia.org/wiki/Electronic_design_automation" style="background: none; color: #0b0080; text-decoration-line: none;" title="Electronic design automation">electronic design automation</a> parts of the logical design may be automated using <a href="https://en.wikipedia.org/wiki/High-level_synthesis" style="background: none; color: #0b0080; text-decoration-line: none;" title="High-level synthesis">high-level synthesis</a> tools based on the behavioral description of the circuit.<sup class="reference" id="cite_ref-Sherwani1999_2-0" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Logic_synthesis#cite_note-Sherwani1999-2" style="background: none; color: #0b0080; text-decoration-line: none;">[2]</a></sup></div>
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<span style="font-size: 12.3704px;">Various representations of Boolean operations</span></div>
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Logic operations usually consist of boolean AND, OR, XOR and NAND operations, and are the most basic forms of operations in an electronic circuit. Arithmetic operations are usually implemented with the use of logic operators. Circuits such as a <a href="https://en.wikipedia.org/wiki/Binary_multiplier" style="background: none; color: #0b0080; text-decoration-line: none;" title="Binary multiplier">binary multiplier</a> or a <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Binary_adder" style="background: none; color: #0b0080; text-decoration-line: none;" title="Binary adder">binary adder</a> are examples of more complex binary operations that can be implemented using basic logic operators.</div>
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<span class="mw-headline" id="High-level_synthesis_or_behavioral_synthesis">High-level synthesis or behavioral synthesis</span></h2>
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With a goal of increasing designer productivity, research efforts on the synthesis of circuits specified at the behavioral level have led to the emergence of commercial solutions in 2004,<sup class="reference" id="cite_ref-EETimes_3-0" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Logic_synthesis#cite_note-EETimes-3" style="background: none; color: #0b0080; text-decoration-line: none;">[3]</a></sup> which are used for complex ASIC and FPGA design. These tools automatically synthesize circuits specified using high-level languages, like ANSI C/C++ or SystemC, to a register transfer level (RTL) specification, which can be used as input to a gate-level logic synthesis flow.<sup class="reference" id="cite_ref-EETimes_3-1" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Logic_synthesis#cite_note-EETimes-3" style="background: none; color: #0b0080; text-decoration-line: none;">[3]</a></sup> Using high-level synthesis, also known as ESL synthesis, the allocation of work to clock cycles and across structural components, such as floating-point ALUs, is done by the compiler using an optimisation procedure, whereas with RTL logic synthesis (even from behavioural Verilog or VHDL, where a thread of execution can make multiple reads and writes to a variable within a clock cycle) those allocation decisions have already been made.</div>
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<span class="mw-headline" id="Multi-level_logic_minimization">Multi-level logic minimization</span></h2>
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Typical practical implementations of a logic function utilize a multi-level network of logic elements. Starting from an RTL description of a design, the synthesis tool constructs a corresponding multilevel <a href="https://en.wikipedia.org/wiki/Boolean_network" style="background: none; color: #0b0080; text-decoration-line: none;" title="Boolean network">Boolean network</a>.</div>
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Next, this network is optimized using several technology-independent techniques before technology-dependent optimizations are performed. The typical cost function during technology-independent optimizations is total <a href="https://en.wikipedia.org/wiki/Propositional_formula#Literal.2C_term_and_alterm" style="background: none; color: #0b0080; text-decoration-line: none;" title="Propositional formula">literal</a> count of the factored representation of the logic function (which correlates quite well with circuit area).</div>
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Finally, technology-dependent optimization transforms the technology-independent circuit into a network of gates in a given technology. The simple cost estimates are replaced by more concrete, implementation-driven estimates during and after technology mapping. Mapping is constrained by factors such as the available gates (logic functions) in the technology library, the drive sizes for each gate, and the delay, <a href="https://en.wikipedia.org/wiki/Electric_power" style="background: none; color: #0b0080; text-decoration-line: none;" title="Electric power">power</a>, and area characteristics of each gate.</div>
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</head>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3005862392404377111.post-44607250018815497582017-06-13T00:01:00.001-07:002023-06-23T23:34:11.807-07:00Boolean algebra (structure)<div dir="ltr" style="text-align: left;" trbidi="on">
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In <a href="https://en.wikipedia.org/wiki/Abstract_algebra" style="background: none; color: #0b0080; text-decoration-line: none;" title="Abstract algebra">abstract algebra</a>, a <b>Boolean algebra</b> or <b>Boolean lattice</b> is a <a href="https://en.wikipedia.org/wiki/Complemented_lattice" style="background: none; color: #0b0080; text-decoration-line: none;" title="Complemented lattice">complemented</a> <a href="https://en.wikipedia.org/wiki/Distributive_lattice" style="background: none; color: #0b0080; text-decoration-line: none;" title="Distributive lattice">distributive lattice</a>. This type of <a href="https://en.wikipedia.org/wiki/Algebraic_structure" style="background: none; color: #0b0080; text-decoration-line: none;" title="Algebraic structure">algebraic structure</a> captures essential properties of both <a href="https://en.wikipedia.org/wiki/Set_(mathematics)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Set (mathematics)">set</a> operations and <a href="https://en.wikipedia.org/wiki/Logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Logic">logic</a> operations. A Boolean algebra can be seen as a generalization of a <a href="https://en.wikipedia.org/wiki/Power_set" style="background: none; color: #0b0080; text-decoration-line: none;" title="Power set">power set</a> algebra or a <a href="https://en.wikipedia.org/wiki/Field_of_sets" style="background: none; color: #0b0080; text-decoration-line: none;" title="Field of sets">field of sets</a>, or its elements can be viewed as generalized <a href="https://en.wikipedia.org/wiki/Truth_value" style="background: none; color: #0b0080; text-decoration-line: none;" title="Truth value">truth values</a>. It is also a special case of a <a href="https://en.wikipedia.org/wiki/De_Morgan_algebra" style="background: none; color: #0b0080; text-decoration-line: none;" title="De Morgan algebra">De Morgan algebra</a> and a <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Kleene_algebra_(with_involution)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Kleene algebra (with involution)">Kleene algebra (with involution)</a>.</div>
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Every Boolean algebra <a href="https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Boolean_rings" style="background: none; color: #0b0080; text-decoration-line: none;" title="Boolean algebra (structure)">gives rise</a> to a <a href="https://en.wikipedia.org/wiki/Boolean_ring" style="background: none; color: #0b0080; text-decoration-line: none;" title="Boolean ring">Boolean ring</a>, and vice versa, with ring multiplication corresponding to <a href="https://en.wikipedia.org/wiki/Logical_conjunction" style="background: none; color: #0b0080; text-decoration-line: none;" title="Logical conjunction">conjunction</a> or <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Meet_(mathematics)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Meet (mathematics)">meet</a> ∧, and ring addition to <a href="https://en.wikipedia.org/wiki/Exclusive_or" style="background: none; color: #0b0080; text-decoration-line: none;" title="Exclusive or">exclusive disjunction</a> or <a href="https://en.wikipedia.org/wiki/Symmetric_difference" style="background: none; color: #0b0080; text-decoration-line: none;" title="Symmetric difference">symmetric difference</a> (not <a href="https://en.wikipedia.org/wiki/Logical_disjunction" style="background: none; color: #0b0080; text-decoration-line: none;" title="Logical disjunction">disjunction</a> ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the axioms and theorems of Boolean algebra express the symmetry of the theory described by the <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Duality_principle_(Boolean_algebra)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Duality principle (Boolean algebra)">duality principle</a>.</div>
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<span style="font-size: large;"><b>HISTORY</b></span></div>
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<span style="font-size: large;"><b> </b></span><span style="font-size: 14px; text-align: left;">The term "Boolean algebra" honors </span><a href="https://en.wikipedia.org/wiki/George_Boole" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-size: 14px; text-align: left; text-decoration-line: none;" title="George Boole">George Boole</a><span style="font-size: 14px; text-align: left;"> (1815–1864), a self-educated English mathematician. He introduced the </span><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Algebraic_system" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-size: 14px; text-align: left; text-decoration-line: none;" title="Algebraic system">algebraic system</a><span style="font-size: 14px; text-align: left;"> initially in a small pamphlet, </span><i style="font-size: 14px; text-align: left;">The Mathematical Analysis of Logic</i><span style="font-size: 14px; text-align: left;">, published in 1847 in response to an ongoing public controversy between </span><a href="https://en.wikipedia.org/wiki/Augustus_De_Morgan" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-size: 14px; text-align: left; text-decoration-line: none;" title="Augustus De Morgan">Augustus De Morgan</a><span style="font-size: 14px; text-align: left;"> and </span><a href="https://en.wikipedia.org/wiki/Sir_William_Hamilton,_9th_Baronet" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-size: 14px; text-align: left; text-decoration-line: none;" title="Sir William Hamilton, 9th Baronet">William Hamilton</a><span style="font-size: 14px; text-align: left;">, and later as a more substantial book, </span><i style="font-size: 14px; text-align: left;"><a href="https://en.wikipedia.org/wiki/The_Laws_of_Thought" style="background: none; color: #0b0080; text-decoration-line: none;" title="The Laws of Thought">The Laws of Thought</a></i><span style="font-size: 14px; text-align: left;">, published in 1854. Boole's formulation differs from that described above in some important respects. For example, conjunction and disjunction in Boole were not a dual pair of operations. Boolean algebra emerged in the 1860s, in papers written by </span><a class="mw-redirect" href="https://en.wikipedia.org/wiki/William_Jevons" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-size: 14px; text-align: left; text-decoration-line: none;" title="William Jevons">William Jevons</a><span style="font-size: 14px; text-align: left;"> and </span><a href="https://en.wikipedia.org/wiki/Charles_Sanders_Peirce" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-size: 14px; text-align: left; text-decoration-line: none;" title="Charles Sanders Peirce">Charles Sanders Peirce</a><span style="font-size: 14px; text-align: left;">. The first systematic presentation of Boolean algebra and </span><a href="https://en.wikipedia.org/wiki/Distributive_lattice" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-size: 14px; text-align: left; text-decoration-line: none;" title="Distributive lattice">distributive lattices</a><span style="font-size: 14px; text-align: left;"> is owed to the 1890 </span><i style="font-size: 14px; text-align: left;">Vorlesungen</i><span style="font-size: 14px; text-align: left;"> of </span><a href="https://en.wikipedia.org/wiki/Ernst_Schr%C3%B6der" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-size: 14px; text-align: left; text-decoration-line: none;" title="Ernst Schröder">Ernst Schröder</a><span style="font-size: 14px; text-align: left;">. The first extensive treatment of Boolean algebra in English is </span><a class="mw-redirect" href="https://en.wikipedia.org/wiki/A._N._Whitehead" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-size: 14px; text-align: left; text-decoration-line: none;" title="A. N. Whitehead">A. N. Whitehead</a><span style="font-size: 14px; text-align: left;">'s 1898 </span><i style="font-size: 14px; text-align: left;">Universal Algebra</i><span style="font-size: 14px; text-align: left;">. Boolean algebra as an axiomatic algebraic structure in the modern axiomatic sense begins with a 1904 paper by </span><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Edward_V._Huntington" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-size: 14px; text-align: left; text-decoration-line: none;" title="Edward V. Huntington">Edward V. Huntington</a><span style="font-size: 14px; text-align: left;">. Boolean algebra came of age as serious mathematics with the work of </span><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Marshall_Stone" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-size: 14px; text-align: left; text-decoration-line: none;" title="Marshall Stone">Marshall Stone</a><span style="font-size: 14px; text-align: left;"> in the 1930s, and with </span><a href="https://en.wikipedia.org/wiki/Garrett_Birkhoff" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-size: 14px; text-align: left; text-decoration-line: none;" title="Garrett Birkhoff">Garrett Birkhoff</a><span style="font-size: 14px; text-align: left;">'s 1940 </span><i style="font-size: 14px; text-align: left;">Lattice Theory</i><span style="font-size: 14px; text-align: left;">. In the 1960s, </span><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Paul_Cohen_(mathematician)" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-size: 14px; text-align: left; text-decoration-line: none;" title="Paul Cohen (mathematician)">Paul Cohen</a><span style="font-size: 14px; text-align: left;">, </span><a href="https://en.wikipedia.org/wiki/Dana_Scott" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-size: 14px; text-align: left; text-decoration-line: none;" title="Dana Scott">Dana Scott</a><span style="font-size: 14px; text-align: left;">, and others found deep new results in </span><a href="https://en.wikipedia.org/wiki/Mathematical_logic" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-size: 14px; text-align: left; text-decoration-line: none;" title="Mathematical logic">mathematical logic</a><span style="font-size: 14px; text-align: left;"> and </span><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Axiomatic_set_theory" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-size: 14px; text-align: left; text-decoration-line: none;" title="Axiomatic set theory">axiomatic set theory</a><span style="font-size: 14px; text-align: left;"> using offshoots of Boolean algebra, namely </span><a href="https://en.wikipedia.org/wiki/Forcing_(mathematics)" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-size: 14px; text-align: left; text-decoration-line: none;" title="Forcing (mathematics)">forcing</a><span style="font-size: 14px; text-align: left;"> and </span><a href="https://en.wikipedia.org/wiki/Boolean-valued_model" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-size: 14px; text-align: left; text-decoration-line: none;" title="Boolean-valued model">Boolean-valued models</a><span style="font-size: 14px; text-align: left;">.</span></div>
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<span style="font-size: large;"><b>DEFINATION:</b></span></div>
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<span style="font-size: large;"><b> </b></span><span style="font-size: 14px;">A</span><span style="font-size: 14px;"> </span><b style="font-size: 14px;">Boolean algebra</b><span style="font-size: 14px;"> </span><span style="font-size: 14px;">is a six-</span><a href="https://en.wikipedia.org/wiki/Tuple" style="background: none; color: #0b0080; font-size: 14px; text-decoration-line: none;" title="Tuple">tuple</a><span style="font-size: 14px;"> </span><span style="font-size: 14px;">consisting of a</span><span style="font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/Set_(mathematics)" style="background: none; color: #0b0080; font-size: 14px; text-decoration-line: none;" title="Set (mathematics)">set</a><span style="font-size: 14px;"> </span><i style="font-size: 14px;">A</i><span style="font-size: 14px;">, equipped with two</span><span style="font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/Binary_operation" style="background: none; color: #0b0080; font-size: 14px; text-decoration-line: none;" title="Binary operation">binary operations</a><span style="font-size: 14px;"> </span><span style="font-size: 14px;">∧ (called "meet" or "and"), ∨ (called "join" or "or"), a</span><span style="font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/Unary_operation" style="background: none; color: #0b0080; font-size: 14px; text-decoration-line: none;" title="Unary operation">unary operation</a><span style="font-size: 14px;"> </span><span style="font-size: 14px;">¬ (called "complement" or "not") and two elements 0 and 1 (called "bottom" and "top", or "least" and "greatest" element, also denoted by the symbols ⊥ and ⊤, respectively), such that for all elements</span><span style="font-size: 14px;"> </span><i style="font-size: 14px;">a</i><span style="font-size: 14px;">,</span><span style="font-size: 14px;"> </span><i style="font-size: 14px;">b</i><span style="font-size: 14px;"> </span><span style="font-size: 14px;">and</span><span style="font-size: 14px;"> </span><i style="font-size: 14px;">c</i><span style="font-size: 14px;"> </span><span style="font-size: 14px;">of</span><span style="font-size: 14px;"> </span><i style="font-size: 14px;">A</i><span style="font-size: 14px;">, the following</span><span style="font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/Axiom" style="background: none; color: #0b0080; font-size: 14px; text-decoration-line: none;" title="Axiom">axioms</a><span style="font-size: 14px;"> </span><span style="font-size: 14px;">hold:</span><sup class="reference" id="cite_ref-2" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#cite_note-2" style="background: none; color: #0b0080; text-decoration-line: none;">[2]</a></sup></div>
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<tr><td><i>a</i> ∨ (<i>b</i> ∨ <i>c</i>) = (<i>a</i> ∨ <i>b</i>) ∨ <i>c</i></td><td><i>a</i> ∧ (<i>b</i> ∧ <i>c</i>) = (<i>a</i> ∧ <i>b</i>) ∧ <i>c</i></td><td><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Associativity" style="background: none; color: #0b0080; text-decoration-line: none;" title="Associativity">associativity</a></td></tr>
<tr><td><i>a</i> ∨ <i>b</i> = <i>b</i> ∨ <i>a</i></td><td><i>a</i> ∧ <i>b</i> = <i>b</i> ∧ <i>a</i></td><td><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Commutativity" style="background: none; color: #0b0080; text-decoration-line: none;" title="Commutativity">commutativity</a></td></tr>
<tr><td><i>a</i> ∨ (<i>a</i> ∧ <i>b</i>) = <i>a</i></td><td><i>a</i> ∧ (<i>a</i> ∨ <i>b</i>) = <i>a</i></td><td><a href="https://en.wikipedia.org/wiki/Absorption_law" style="background: none; color: #0b0080; text-decoration-line: none;" title="Absorption law">absorption</a></td></tr>
<tr><td><i>a</i> ∨ 0 = <i>a</i></td><td><i>a</i> ∧ 1 = <i>a</i></td><td><a href="https://en.wikipedia.org/wiki/Identity_element" style="background: none; color: #0b0080; text-decoration-line: none;" title="Identity element">identity</a></td></tr>
<tr><td><i>a</i> ∨ (<i>b</i> ∧ <i>c</i>) = (<i>a</i> ∨ <i>b</i>) ∧ (<i>a</i> ∨ <i>c</i>) </td><td><i>a</i> ∧ (<i>b</i> ∨ <i>c</i>) = (<i>a</i> ∧ <i>b</i>) ∨ (<i>a</i> ∧ <i>c</i>) </td><td><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Distributivity" style="background: none; color: #0b0080; text-decoration-line: none;" title="Distributivity">distributivity</a></td></tr>
<tr><td><i>a</i> ∨ ¬<i>a</i> = 1</td><td><i>a</i> ∧ ¬<i>a</i> = 0</td><td><a href="https://en.wikipedia.org/wiki/Complemented_lattice" style="background: none; color: #0b0080; text-decoration-line: none;" title="Complemented lattice">complements</a></td></tr>
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Note, however, that the absorption law can be excluded from the set of axioms as it can be derived by the other axioms.</div>
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A Boolean algebra with only one element is called a <b>trivial Boolean algebra</b> or a <b>degenerate Boolean algebra</b>. (Some authors require 0 and 1 to be <i>distinct</i> elements in order to exclude this case.)</div>
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It follows from the last three pairs of axioms above (identity, distributivity and complements), or from the absorption axiom, that</div>
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<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px; text-align: justify;"><i>a</i> = <i>b</i> ∧ <i>a</i> if and only if <i>a</i> ∨ <i>b</i> = <i>b</i>.</dd></dl>
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The relation ≤ defined by <i>a</i> ≤ <i>b</i> if these equivalent conditions hold, is a <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Partial_order" style="background: none; color: #0b0080; text-decoration-line: none;" title="Partial order">partial order</a> with least element 0 and greatest element 1. The meet <i>a</i> ∧ <i>b</i> and the join <i>a</i> ∨ <i>b</i> of two elements coincide with their <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Infimum" style="background: none; color: #0b0080; text-decoration-line: none;" title="Infimum">infimum</a> and <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Supremum" style="background: none; color: #0b0080; text-decoration-line: none;" title="Supremum">supremum</a>, respectively, with respect to ≤.</div>
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The first four pairs of axioms constitute a definition of a <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Bounded_lattice" style="background: none; color: #0b0080; text-decoration-line: none;" title="Bounded lattice">bounded lattice</a>.</div>
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It follows from the first five pairs of axioms that any complement is unique.</div>
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The set of axioms is <a href="https://en.wikipedia.org/wiki/Duality_(order_theory)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Duality (order theory)">self-dual</a> in the sense that if one exchanges ∨ with ∧ and 0 with 1 in an axiom, the result is again an axiom. Therefore, by applying this operation to a Boolean algebra (or Boolean lattice), one obtains another Boolean algebra with the same elements; it is called its <b>dual</b>.</div>
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<span style="font-size: large;"><b>EXAMPLE:</b></span></div>
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<li style="margin-bottom: 0.1em; text-align: justify;">The simplest non-trivial Boolean algebra, the <a href="https://en.wikipedia.org/wiki/Two-element_Boolean_algebra" style="background: none; color: #0b0080; text-decoration-line: none;" title="Two-element Boolean algebra">two-element Boolean algebra</a>, has only two elements, 0 and 1, and is defined by the rules:</li>
</ul>
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<tr><td width="70"></td><td><table border="1" cellpadding="4" cellspacing="0" class="wikitable" style="background-color: #f8f9fa; border-collapse: collapse; border: 1px solid rgb(162, 169, 177); color: black; font-size: 14px; margin: 1em 0px;"><tbody>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">∧</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">0</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">1</th></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">0</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">1</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td></tr>
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</td><td></td><td width="30"></td><td><table border="1" cellpadding="4" cellspacing="0" class="wikitable" style="background-color: #f8f9fa; border-collapse: collapse; border: 1px solid rgb(162, 169, 177); color: black; font-size: 14px; margin: 1em 0px;"><tbody>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">∨</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">0</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">1</th></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">0</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">1</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td></tr>
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</td><td></td><td width="40"></td><td><table border="1" cellpadding="4" cellspacing="0" class="wikitable" style="background-color: #f8f9fa; border-collapse: collapse; border: 1px solid rgb(162, 169, 177); color: black; font-size: 14px; margin: 1em 0px;"><tbody>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;"><i>a</i></th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">0</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">1</th></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">¬<i>a</i></th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td></tr>
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<li style="margin-bottom: 0.1em; text-align: justify;">It has applications in <a href="https://en.wikipedia.org/wiki/Logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Logic">logic</a>, interpreting 0 as <i>false</i>, 1 as <i>true</i>, ∧ as <i>and</i>, ∨ as <i>or</i>, and ¬ as <i>not</i>. Expressions involving variables and the Boolean operations represent statement forms, and two such expressions can be shown to be equal using the above axioms if and only if the corresponding statement forms are <a href="https://en.wikipedia.org/wiki/Logical_equivalence" style="background: none; color: #0b0080; text-decoration-line: none;" title="Logical equivalence">logically equivalent</a>.</li>
</ul>
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<dl style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><ul style="list-style-image: url("data:image/svg+xml,%3C%3Fxml%20version%3D%221.0%22%20encoding%3D%22UTF-8%22%3F%3E%0A%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20version%3D%221.1%22%20width%3D%225%22%20height%3D%2213%22%3E%0A%3Ccircle%20cx%3D%222.5%22%20cy%3D%229.5%22%20r%3D%222.5%22%20fill%3D%22%2300528c%22%2F%3E%0A%3C%2Fsvg%3E%0A"); margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em; text-align: justify;">The two-element Boolean algebra is also used for circuit design in <a href="https://en.wikipedia.org/wiki/Electrical_engineering" style="background: none; color: #0b0080; text-decoration-line: none;" title="Electrical engineering">electrical engineering</a>; here 0 and 1 represent the two different states of one <a href="https://en.wikipedia.org/wiki/Bit" style="background: none; color: #0b0080; text-decoration-line: none;" title="Bit">bit</a> in a <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Digital_circuit" style="background: none; color: #0b0080; text-decoration-line: none;" title="Digital circuit">digital circuit</a>, typically high and low <a href="https://en.wikipedia.org/wiki/Voltage" style="background: none; color: #0b0080; text-decoration-line: none;" title="Voltage">voltage</a>. Circuits are described by expressions containing variables, and two such expressions are equal for all values of the variables if and only if the corresponding circuits have the same input-output behavior. Furthermore, every possible input-output behavior can be modeled by a suitable Boolean expression.</li>
</ul>
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<li style="margin-bottom: 0.1em;"><div style="text-align: justify;">
The two-element Boolean algebra is also important in the general theory of Boolean algebras, because an equation involving several variables is generally true in all Boolean algebras if and only if it is true in the two-element Boolean algebra (which can be checked by a trivial <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Brute_force_search" style="background: none; color: #0b0080; text-decoration-line: none;" title="Brute force search">brute force</a> algorithm for small numbers of variables). This can for example be used to show that the following laws (<i>Consensus theorems</i>) are generally valid in all Boolean algebras:</div>
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<li style="margin-bottom: 0.1em; text-align: justify;">(<i>a</i> ∨ <i>b</i>) ∧ (¬<i>a</i> ∨ <i>c</i>) ∧ (<i>b</i> ∨ <i>c</i>) ≡ (<i>a</i> ∨ <i>b</i>) ∧ (¬<i>a</i> ∨ <i>c</i>)</li>
<li style="margin-bottom: 0.1em; text-align: justify;">(<i>a</i> ∧ <i>b</i>) ∨ (¬<i>a</i> ∧ <i>c</i>) ∨ (<i>b</i> ∧ <i>c</i>) ≡ (<i>a</i> ∧ <i>b</i>) ∨ (¬<i>a</i> ∧ <i>c</i>)</li>
</ul>
</li>
</ul>
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<ul style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; list-style-image: url("data:image/svg+xml,%3C%3Fxml%20version%3D%221.0%22%20encoding%3D%22UTF-8%22%3F%3E%0A%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20version%3D%221.1%22%20width%3D%225%22%20height%3D%2213%22%3E%0A%3Ccircle%20cx%3D%222.5%22%20cy%3D%229.5%22%20r%3D%222.5%22%20fill%3D%22%2300528c%22%2F%3E%0A%3C%2Fsvg%3E%0A"); margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em; text-align: justify;">The <a href="https://en.wikipedia.org/wiki/Power_set" style="background: none; color: #0b0080; text-decoration-line: none;" title="Power set">power set</a> (set of all subsets) of any given nonempty set <i>S</i> forms a Boolean algebra, an <a href="https://en.wikipedia.org/wiki/Algebra_of_sets" style="background: none; color: #0b0080; text-decoration-line: none;" title="Algebra of sets">algebra of sets</a>, with the two operations ∨ := ∪ (union) and ∧ := ∩ (intersection). The smallest element 0 is the <a href="https://en.wikipedia.org/wiki/Empty_set" style="background: none; color: #0b0080; text-decoration-line: none;" title="Empty set">empty set</a> and the largest element 1 is the set <i>S</i> itself.</li>
</ul>
<dl style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><ul style="list-style-image: url("data:image/svg+xml,%3C%3Fxml%20version%3D%221.0%22%20encoding%3D%22UTF-8%22%3F%3E%0A%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20version%3D%221.1%22%20width%3D%225%22%20height%3D%2213%22%3E%0A%3Ccircle%20cx%3D%222.5%22%20cy%3D%229.5%22%20r%3D%222.5%22%20fill%3D%22%2300528c%22%2F%3E%0A%3C%2Fsvg%3E%0A"); margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em; text-align: justify;">After the two-element Boolean algebra, the simplest Boolean algebra is that defined by the <a href="https://en.wikipedia.org/wiki/Power_set" style="background: none; color: #0b0080; text-decoration-line: none;" title="Power set">power set</a> of two atoms:</li>
</ul>
</dd></dl>
<table style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; text-align: justify;"><tbody>
<tr><td width="70"></td><td><table border="1" cellpadding="4" cellspacing="0" class="wikitable" style="background-color: #f8f9fa; border-collapse: collapse; border: 1px solid rgb(162, 169, 177); color: black; font-size: 14px; margin: 1em 0px;"><tbody>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">∧</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">0</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">a</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">b</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">1</th></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">0</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">a</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">a</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">a</td></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">b</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">b</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">b</td></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">1</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">a</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">b</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td></tr>
</tbody></table>
</td><td></td><td width="30"></td><td><table border="1" cellpadding="4" cellspacing="0" class="wikitable" style="background-color: #f8f9fa; border-collapse: collapse; border: 1px solid rgb(162, 169, 177); color: black; font-size: 14px; margin: 1em 0px;"><tbody>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">∨</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">0</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">a</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">b</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">1</th></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">0</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">a</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">b</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">a</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">a</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">a</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">b</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">b</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">b</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">1</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td></tr>
</tbody></table>
</td><td></td><td width="40"></td><td><table border="1" cellpadding="4" cellspacing="0" class="wikitable" style="background-color: #f8f9fa; border-collapse: collapse; border: 1px solid rgb(162, 169, 177); color: black; font-size: 14px; margin: 1em 0px;"><tbody>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;"><i>x</i></th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">0</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">a</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">b</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">1</th></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">¬<i>x</i></th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">b</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">a</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td></tr>
</tbody></table>
</td></tr>
</tbody></table>
<ul style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; list-style-image: url("data:image/svg+xml,%3C%3Fxml%20version%3D%221.0%22%20encoding%3D%22UTF-8%22%3F%3E%0A%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20version%3D%221.1%22%20width%3D%225%22%20height%3D%2213%22%3E%0A%3Ccircle%20cx%3D%222.5%22%20cy%3D%229.5%22%20r%3D%222.5%22%20fill%3D%22%2300528c%22%2F%3E%0A%3C%2Fsvg%3E%0A"); margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em; text-align: justify;">The set of all subsets of <i>S</i> that are either finite or <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Cofinite" style="background: none; color: #0b0080; text-decoration-line: none;" title="Cofinite">cofinite</a> is a Boolean algebra, an <a href="https://en.wikipedia.org/wiki/Algebra_of_sets" style="background: none; color: #0b0080; text-decoration-line: none;" title="Algebra of sets">algebra of sets</a>.</li>
<li style="margin-bottom: 0.1em; text-align: justify;">Starting with the <a href="https://en.wikipedia.org/wiki/Propositional_calculus" style="background: none; color: #0b0080; text-decoration-line: none;" title="Propositional calculus">propositional calculus</a> with κ sentence symbols, form the <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Lindenbaum-Tarski_algebra" style="background: none; color: #0b0080; text-decoration-line: none;" title="Lindenbaum-Tarski algebra">Lindenbaum algebra</a> (that is, the set of sentences in the propositional calculus modulo <a href="https://en.wikipedia.org/wiki/Tautology_(logic)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Tautology (logic)">tautology</a>). This construction yields a Boolean algebra. It is in fact the <a href="https://en.wikipedia.org/wiki/Free_Boolean_algebra" style="background: none; color: #0b0080; text-decoration-line: none;" title="Free Boolean algebra">free Boolean algebra</a> on κ generators. A truth assignment in propositional calculus is then a Boolean algebra homomorphism from this algebra to the two-element Boolean algebra.</li>
<li style="margin-bottom: 0.1em; text-align: justify;">Given any <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Linearly_ordered" style="background: none; color: #0b0080; text-decoration-line: none;" title="Linearly ordered">linearly ordered</a> set <i>L</i> with a least element, the interval algebra is the smallest algebra of subsets of <i>L</i> containing all of the half-open intervals [<i>a</i>, <i>b</i>) such that <i>a</i> is in <i>L</i> and <i>b</i> is either in <i>L</i> or equal to ∞. Interval algebras are useful in the study of <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Lindenbaum-Tarski_algebra" style="background: none; color: #0b0080; text-decoration-line: none;" title="Lindenbaum-Tarski algebra">Lindenbaum-Tarski algebras</a>; every countable Boolean algebra is isomorphic to an interval algebra.</li>
</ul>
<div class="thumb tright" style="background-color: white; clear: right; color: #222222; float: right; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px 1.3em 1.4em; width: auto;">
<div class="thumbinner" style="background-color: #f8f9fa; border: 1px solid rgb(200, 204, 209); font-size: 13.16px; min-width: 100px; overflow: hidden; padding: 3px; text-align: center; width: 152px;">
<div style="text-align: justify;">
<a class="image" href="https://en.wikipedia.org/wiki/File:T_30.svg" style="background: none; color: #0b0080; font-size: 13.16px; text-decoration-line: none;"><img alt="" class="thumbimage" data-file-height="100" data-file-width="100" height="150" src="https://upload.wikimedia.org/wikipedia/commons/thumb/1/10/T_30.svg/150px-T_30.svg.png" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/T_30.svg/225px-T_30.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/T_30.svg/300px-T_30.svg.png 2x" style="background-color: white; border: 1px solid rgb(200, 204, 209); vertical-align: middle;" width="150" /></a></div>
<div class="thumbcaption" style="border: 0px; font-size: 12.3704px; line-height: 1.4em; padding: 3px; text-align: left;">
<div class="magnify" style="float: right; margin-left: 3px; margin-right: 0px;">
<a class="internal" href="https://en.wikipedia.org/wiki/File:T_30.svg" style="background: linear-gradient(transparent, transparent), url("data:image/svg+xml,%3C%3Fxml%20version%3D%221.0%22%20encoding%3D%22UTF-8%22%20standalone%3D%22no%22%3F%3E%0A%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20viewBox%3D%220%200%2011%2015%22%20width%3D%2215%22%20height%3D%2211%22%3E%0A%20%20%20%20%3Cg%20id%3D%22magnify-clip%22%20fill%3D%22%23fff%22%20stroke%3D%22%23000%22%3E%0A%20%20%20%20%20%20%20%20%3Cpath%20id%3D%22bigbox%22%20d%3D%22M1.509%201.865h10.99v7.919h-10.99z%22%2F%3E%0A%20%20%20%20%20%20%20%20%3Cpath%20id%3D%22smallbox%22%20d%3D%22M-1.499%206.868h5.943v4.904h-5.943z%22%2F%3E%0A%20%20%20%20%3C%2Fg%3E%0A%3C%2Fsvg%3E%0A"); color: #0b0080; display: block; height: 11px; overflow: hidden; text-align: justify; text-decoration-line: none; text-indent: 15px; user-select: none; white-space: nowrap; width: 15px;" title="Enlarge"></a></div>
<div style="background: none; color: #0b0080; text-align: justify; text-decoration-line: none;">
<a href="https://en.wikipedia.org/wiki/Hasse_diagram" style="background: none; color: #0b0080; text-decoration-line: none;" title="Hasse diagram"></a><a href="https://en.wikipedia.org/wiki/Hasse_diagram" style="background: none; color: #0b0080; font-size: 12.3704px; text-decoration-line: none;" title="Hasse diagram">Hasse diagram</a><span style="color: #222222; font-size: 12.3704px;"> </span><span style="color: #222222; font-size: 12.3704px;">of the Boolean algebra of divisors of 30.</span></div>
</div>
</div>
</div>
<ul style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; list-style-image: url("data:image/svg+xml,%3C%3Fxml%20version%3D%221.0%22%20encoding%3D%22UTF-8%22%3F%3E%0A%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20version%3D%221.1%22%20width%3D%225%22%20height%3D%2213%22%3E%0A%3Ccircle%20cx%3D%222.5%22%20cy%3D%229.5%22%20r%3D%222.5%22%20fill%3D%22%2300528c%22%2F%3E%0A%3C%2Fsvg%3E%0A"); margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em; text-align: justify;">For any <a href="https://en.wikipedia.org/wiki/Natural_number" style="background: none; color: #0b0080; text-decoration-line: none;" title="Natural number">natural number</a> <i>n</i>, the set of all positive <a href="https://en.wikipedia.org/wiki/Divisor" style="background: none; color: #0b0080; text-decoration-line: none;" title="Divisor">divisors</a> of <i>n</i>, defining <i>a</i>≤<i>b</i> if <i>a</i> <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Divides" style="background: none; color: #0b0080; text-decoration-line: none;" title="Divides">divides</a> <i>b</i>, forms a <a href="https://en.wikipedia.org/wiki/Distributive_lattice" style="background: none; color: #0b0080; text-decoration-line: none;" title="Distributive lattice">distributive lattice</a>. This lattice is a Boolean algebra if and only if <i>n</i> is <a href="https://en.wikipedia.org/wiki/Square-free_integer" style="background: none; color: #0b0080; text-decoration-line: none;" title="Square-free integer">square-free</a>. The bottom and the top element of this Boolean algebra is the natural number 1 and <i>n</i>, respectively. The complement of <i>a</i> is given by <i>n</i>/<i>a</i>. The meet and the join of <i>a</i> and <i>b</i> is given by the <a href="https://en.wikipedia.org/wiki/Greatest_common_divisor" style="background: none; color: #0b0080; text-decoration-line: none;" title="Greatest common divisor">greatest common divisor</a> (gcd) and the <a href="https://en.wikipedia.org/wiki/Least_common_multiple" style="background: none; color: #0b0080; text-decoration-line: none;" title="Least common multiple">least common multiple</a> (lcm) of <i>a</i> and <i>b</i>, respectively. The ring addition <i>a</i>+<i>b</i> is given by lcm(<i>a</i>,<i>b</i>)/gcd(<i>a</i>,<i>b</i>). The picture shows an example for <i>n</i> = 30. As a counter-example, considering the non-square-free <i>n</i>=60, the greatest common divisor of 30 and its complement 2 would be 2, while it should be the bottom element 1.</li>
<li style="margin-bottom: 0.1em; text-align: justify;">Other examples of Boolean algebras arise from <a href="https://en.wikipedia.org/wiki/Topology" style="background: none; color: #0b0080; text-decoration-line: none;" title="Topology">topological spaces</a>: if <i>X</i> is a topological space, then the collection of all subsets of <i>X</i> which are <a href="https://en.wikipedia.org/wiki/Clopen_set" style="background: none; color: #0b0080; text-decoration-line: none;" title="Clopen set">both open and closed</a> forms a Boolean algebra with the operations ∨ := ∪ (union) and ∧ := ∩ (intersection).</li>
<li style="margin-bottom: 0.1em;"><div style="text-align: justify;">
If <i>R</i> is an arbitrary <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Mathematical_ring" style="background: none; color: #0b0080; text-decoration-line: none;" title="Mathematical ring">ring</a> and we define the set of <i>central idempotents</i> by</div>
<i><div style="text-align: justify;">
<i>A</i> = { <i>e</i> ∈ <i>R</i> : <i>e</i><sup style="font-size: 11.2px; line-height: 1;">2</sup> = <i>e</i>, <i>ex</i> = <i>xe</i>, ∀<i>x</i> ∈ <i>R</i> }</div>
</i><div style="text-align: justify;">
then the set <i>A</i> becomes a Boolean algebra with the operations <i>e</i> ∨ <i>f</i> := <i>e</i> + <i>f</i> - <i>ef</i> and <i>e</i> ∧ <i>f</i> := <i>ef</i>.</div>
</li>
</ul>
</div>
<head>
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</head>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3005862392404377111.post-35708544703397415172017-06-12T23:48:00.001-07:002023-06-23T23:34:11.570-07:00Boolean algebra<div dir="ltr" style="text-align: left;" trbidi="on">
<div style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: justify;">
In <a href="https://en.wikipedia.org/wiki/Mathematics" style="background: none; color: #0b0080; text-decoration-line: none;" title="Mathematics">mathematics</a> and <a href="https://en.wikipedia.org/wiki/Mathematical_logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Mathematical logic">mathematical logic</a>, <b>Boolean algebra</b> is the branch of <a href="https://en.wikipedia.org/wiki/Abstract_algebra" style="background: none; color: #0b0080; text-decoration-line: none;" title="Abstract algebra">algebra</a> in which the values of the <a href="https://en.wikipedia.org/wiki/Variable_(mathematics)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Variable (mathematics)">variables</a> are the <a href="https://en.wikipedia.org/wiki/Truth_value" style="background: none; color: #0b0080; text-decoration-line: none;" title="Truth value">truth values</a> <i>true</i> and <i>false</i>, usually denoted 1 and 0 respectively. Instead of <a href="https://en.wikipedia.org/wiki/Elementary_algebra" style="background: none; color: #0b0080; text-decoration-line: none;" title="Elementary algebra">elementary algebra</a> where the values of the variables are numbers, and the prime operations are addition and multiplication, the main operations of Boolean algebra are the <a href="https://en.wikipedia.org/wiki/Logical_conjunction" style="background: none; color: #0b0080; text-decoration-line: none;" title="Logical conjunction">conjunction</a> <i>and</i> denoted as the <a href="https://en.wikipedia.org/wiki/Logical_disjunction" style="background: none; color: #0b0080; text-decoration-line: none;" title="Logical disjunction">disjunction</a> <i>or</i> denoted as ∨, and the <a href="https://en.wikipedia.org/wiki/Negation" style="background: none; color: #0b0080; text-decoration-line: none;" title="Negation">negation</a> <i>not</i> denoted as. It is thus a formalism for describing logical relations in the same way that ordinary algebra describes numeric relations.</div>
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Boolean algebra was introduced by <a href="https://en.wikipedia.org/wiki/George_Boole" style="background: none; color: #0b0080; text-decoration-line: none;" title="George Boole">George Boole</a> in his first book <i>The Mathematical Analysis of Logic</i> (1847), and set forth more fully in his <i><a href="https://en.wikipedia.org/wiki/The_Laws_of_Thought" style="background: none; color: #0b0080; text-decoration-line: none;" title="The Laws of Thought">An Investigation of the Laws of Thought</a></i> (1854).According to <a href="https://en.wikipedia.org/wiki/Edward_Vermilye_Huntington" style="background: none; color: #0b0080; text-decoration-line: none;" title="Edward Vermilye Huntington">Huntington</a>, the term "Boolean algebra" was first suggested by <a href="https://en.wikipedia.org/wiki/Henry_M._Sheffer" style="background: none; color: #0b0080; text-decoration-line: none;" title="Henry M. Sheffer">Sheffer</a> in 1913.</div>
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Boolean algebra has been fundamental in the development of <a href="https://en.wikipedia.org/wiki/Digital_electronics" style="background: none; color: #0b0080; text-decoration-line: none;" title="Digital electronics">digital electronics</a>, and is provided for in all modern programming languages. It is also used in <a href="https://en.wikipedia.org/wiki/Set_theory" style="background: none; color: #0b0080; text-decoration-line: none;" title="Set theory">set theory</a> and <a href="https://en.wikipedia.org/wiki/Statistics" style="background: none; color: #0b0080; text-decoration-line: none;" title="Statistics">statistics</a></div>
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<span style="font-size: large;"><b>HISTORY:</b></span></div>
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Boole's algebra predated the modern developments in <a href="https://en.wikipedia.org/wiki/Abstract_algebra" style="background: none; color: #0b0080; text-decoration-line: none;" title="Abstract algebra">abstract algebra</a> and mathematical logic; it is however seen as connected to the origins of both fields.<sup class="reference" id="cite_ref-DunnHardegree2001_4-0" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Boolean_algebra#cite_note-DunnHardegree2001-4" style="background: none; color: #0b0080; text-decoration-line: none;">[4]</a></sup> In an abstract setting, Boolean algebra was perfected in the late 19th century by <a href="https://en.wikipedia.org/wiki/William_Stanley_Jevons" style="background: none; color: #0b0080; text-decoration-line: none;" title="William Stanley Jevons">Jevons</a>, <a href="https://en.wikipedia.org/wiki/Ernst_Schr%C3%B6der" style="background: none; color: #0b0080; text-decoration-line: none;" title="Ernst Schröder">Schröder</a>, <a href="https://en.wikipedia.org/wiki/Edward_Vermilye_Huntington" style="background: none; color: #0b0080; text-decoration-line: none;" title="Edward Vermilye Huntington">Huntington</a>, and others until it reached the modern conception of an (abstract) <a href="https://en.wikipedia.org/wiki/Mathematical_structure" style="background: none; color: #0b0080; text-decoration-line: none;" title="Mathematical structure">mathematical structure</a>.<sup class="reference" id="cite_ref-DunnHardegree2001_4-1" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Boolean_algebra#cite_note-DunnHardegree2001-4" style="background: none; color: #0b0080; text-decoration-line: none;">[4]</a></sup> For example, the empirical observation that one can manipulate expressions in the <a href="https://en.wikipedia.org/wiki/Algebra_of_sets" style="background: none; color: #0b0080; text-decoration-line: none;" title="Algebra of sets">algebra of sets</a> by translating them into expressions in Boole's algebra is explained in modern terms by saying that the algebra of sets is <i>a</i> <a href="https://en.wikipedia.org/wiki/Boolean_algebra_(structure)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Boolean algebra (structure)">Boolean algebra</a> (note the <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Indefinite_article" style="background: none; color: #0b0080; text-decoration-line: none;" title="Indefinite article">indefinite article</a>). In fact, <a class="mw-redirect" href="https://en.wikipedia.org/wiki/M._H._Stone" style="background: none; color: #0b0080; text-decoration-line: none;" title="M. H. Stone">M. H. Stone</a> <a href="https://en.wikipedia.org/wiki/Stone%27s_representation_theorem_for_Boolean_algebras" style="background: none; color: #0b0080; text-decoration-line: none;" title="Stone's representation theorem for Boolean algebras">proved in 1936</a> that every Boolean algebra is <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Isomorphic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Isomorphic">isomorphic</a> to a <a href="https://en.wikipedia.org/wiki/Field_of_sets" style="background: none; color: #0b0080; text-decoration-line: none;" title="Field of sets">field of sets</a>.</div>
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<span style="font-size: 14px;">In the 1930s, while studying</span><span style="font-size: 14px;"> </span><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Switching_circuit" style="background: none; color: #0b0080; font-size: 14px; text-decoration-line: none;" title="Switching circuit">switching circuits</a><span style="font-size: 14px;">,</span><span style="font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/Claude_Shannon" style="background: none; color: #0b0080; font-size: 14px; text-decoration-line: none;" title="Claude Shannon">Claude Shannon</a><span style="font-size: 14px;"> </span><span style="font-size: 14px;">observed that one could also apply the rules of Boole's algebra in this setting, and he introduced</span><span style="font-size: 14px;"> </span><b style="font-size: 14px;">switching algebra</b><span style="font-size: 14px;"> </span><span style="font-size: 14px;">as a way to analyze and design circuits by algebraic means in terms of</span><span style="font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/Logic_gate" style="background: none; color: #0b0080; font-size: 14px; text-decoration-line: none;" title="Logic gate">logic gates</a><span style="font-size: 14px;">. Shannon already had at his disposal the abstract mathematical apparatus, thus he cast his switching algebra as the</span><span style="font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/Two-element_Boolean_algebra" style="background: none; color: #0b0080; font-size: 14px; text-decoration-line: none;" title="Two-element Boolean algebra">two-element Boolean algebra</a><span style="font-size: 14px;">. In circuit engineering settings today, there is little need to consider other Boolean algebras, thus "switching algebra" and "Boolean algebra" are often used interchangeably.</span><sup class="reference" id="cite_ref-BalabanianCarlson2001_5-0" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Boolean_algebra#cite_note-BalabanianCarlson2001-5" style="background: none; color: #0b0080; text-decoration-line: none;">[5]</a></sup><sup class="reference" id="cite_ref-Radhakrishnan_6-0" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Boolean_algebra#cite_note-Radhakrishnan-6" style="background: none; color: #0b0080; text-decoration-line: none;">[6]</a></sup><sup class="reference" id="cite_ref-Camara2010_7-0" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Boolean_algebra#cite_note-Camara2010-7" style="background: none; color: #0b0080; text-decoration-line: none;">[7]</a></sup><span style="font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/Logic_optimization" style="background: none; color: #0b0080; font-size: 14px; text-decoration-line: none;" title="Logic optimization">Efficient implementation</a><span style="font-size: 14px;"> </span><span style="font-size: 14px;">of</span><span style="font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/Boolean_function" style="background: none; color: #0b0080; font-size: 14px; text-decoration-line: none;" title="Boolean function">Boolean functions</a><span style="font-size: 14px;"> </span><span style="font-size: 14px;">is a fundamental problem in the</span><span style="font-size: 14px;"> </span><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Logic_design" style="background: none; color: #0b0080; font-size: 14px; text-decoration-line: none;" title="Logic design">design</a><span style="font-size: 14px;"> </span><span style="font-size: 14px;">of</span><span style="font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/Combinational_logic" style="background: none; color: #0b0080; font-size: 14px; text-decoration-line: none;" title="Combinational logic">combinational logic</a><span style="font-size: 14px;"> </span><span style="font-size: 14px;">circuits. Modern</span><span style="font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/Electronic_design_automation" style="background: none; color: #0b0080; font-size: 14px; text-decoration-line: none;" title="Electronic design automation">electronic design automation</a><span style="font-size: 14px;"> </span><span style="font-size: 14px;">tools for</span><span style="font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/Very-large-scale_integration" style="background: none; color: #0b0080; font-size: 14px; text-decoration-line: none;" title="Very-large-scale integration">VLSI circuits</a><span style="font-size: 14px;"> </span><span style="font-size: 14px;">often rely on an efficient representation of Boolean functions known as (reduced ordered)</span><span style="font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/Binary_decision_diagram" style="background: none; color: #0b0080; font-size: 14px; text-decoration-line: none;" title="Binary decision diagram">binary decision diagrams</a><span style="font-size: 14px;"> </span><span style="font-size: 14px;">(BDD) for</span><span style="font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/Logic_synthesis" style="background: none; color: #0b0080; font-size: 14px; text-decoration-line: none;" title="Logic synthesis">logic synthesis</a><span style="font-size: 14px;"> </span><span style="font-size: 14px;">and</span><span style="font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/Formal_verification" style="background: none; color: #0b0080; font-size: 14px;" title="Formal verification">formal verification</a><span style="font-size: 14px;">.</span></div>
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<span style="font-size: large;"><b>VALUES:</b></span></div>
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Whereas in elementary algebra expressions denote mainly <a href="https://en.wikipedia.org/wiki/Number" style="background: none; color: #0b0080; text-decoration-line: none;" title="Number">numbers</a>, in Boolean algebra they denote the <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Truth_values" style="background: none; color: #0b0080; text-decoration-line: none;" title="Truth values">truth values</a> <i>false</i> and <i>true</i>. These values are represented with the <a href="https://en.wikipedia.org/wiki/Bit" style="background: none; color: #0b0080; text-decoration-line: none;" title="Bit">bits</a> (or binary digits), namely 0 and 1. They do not behave like the <a href="https://en.wikipedia.org/wiki/Integer" style="background: none; color: #0b0080; text-decoration-line: none;" title="Integer">integers</a> 0 and 1, for which 1 + 1 = 2, but may be identified with the elements of the <a href="https://en.wikipedia.org/wiki/GF(2)" style="background: none; color: #0b0080; text-decoration-line: none;" title="GF(2)">two-element field GF(2)</a>, that is, <a href="https://en.wikipedia.org/wiki/Modular_arithmetic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Modular arithmetic">integer arithmetic modulo 2</a>, for which 1 + 1 = 0. Addition and multiplication then play the Boolean roles of XOR (exclusive-or) and AND (conjunction) respectively, with disjunction <i>x</i>∨<i>y</i> (inclusive-or) definable as <i>x</i> + <i>y</i> + <i>xy</i>.</div>
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Boolean algebra also deals with <a href="https://en.wikipedia.org/wiki/Function_(mathematics)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Function (mathematics)">functions</a> which have their values in the set {0, 1}. A <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Bit_vector" style="background: none; color: #0b0080; text-decoration-line: none;" title="Bit vector">sequence of bits</a> is a commonly used such function. Another common example is the subsets of a set <i>E</i>: to a subset <i>F</i> of <i>E</i> is associated the <a href="https://en.wikipedia.org/wiki/Indicator_function" style="background: none; color: #0b0080; text-decoration-line: none;" title="Indicator function">indicator function</a> that takes the value 1 on <i>F</i> and 0 outside <i>F</i>. The most general example is the elements of a <a href="https://en.wikipedia.org/wiki/Boolean_algebra_(structure)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Boolean algebra (structure)">Boolean algebra</a>, with all of the foregoing being instances thereof.</div>
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As with elementary algebra, the purely equational part of the theory may be developed without considering explicit values for the variables.</div>
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<span class="mw-headline" id="Basic_operations">Basic operations</span></h3>
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The basic operations of Boolean calculus are as follows.</div>
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<li style="margin-bottom: 0.1em; text-align: justify;">AND (<a href="https://en.wikipedia.org/wiki/Logical_conjunction" style="background: none; color: #0b0080; text-decoration-line: none;" title="Logical conjunction">conjunction</a>), denoted <i>x</i>∧<i>y</i> (sometimes <i>x</i> AND <i>y</i> or K<i>xy</i>), satisfies <i>x</i>∧<i>y</i> = 1 if <i>x</i> = <i>y</i> = 1 and <i>x</i>∧<i>y</i> = 0 otherwise.</li>
<li style="margin-bottom: 0.1em; text-align: justify;">OR (<a href="https://en.wikipedia.org/wiki/Logical_disjunction" style="background: none; color: #0b0080; text-decoration-line: none;" title="Logical disjunction">disjunction</a>), denoted <i>x</i>∨<i>y</i> (sometimes <i>x</i> OR <i>y</i> or A<i>xy</i>), satisfies <i>x</i>∨<i>y</i> = 0 if <i>x</i> = <i>y</i> = 0 and <i>x</i>∨<i>y</i> = 1 otherwise.</li>
<li style="margin-bottom: 0.1em; text-align: justify;">NOT (<a href="https://en.wikipedia.org/wiki/Negation" style="background: none; color: #0b0080; text-decoration-line: none;" title="Negation">negation</a>), denoted ¬<i>x</i> (sometimes NOT <i>x</i>, N<i>x</i> or !<i>x</i>), satisfies ¬<i>x</i> = 0 if <i>x</i> = 1 and ¬<i>x</i> = 1 if <i>x</i> = 0.</li>
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Alternatively the values of <i>x</i>∧<i>y</i>, <i>x</i>∨<i>y</i>, and ¬<i>x</i> can be expressed by tabulating their values with <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Truth_tables" style="background: none; color: #0b0080; text-decoration-line: none;" title="Truth tables">truth tables</a> as follows.</div>
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<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle x}</annotation></semantics></math></span><img alt="x" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" style="border: 0px; display: inline-block; height: 1.676ex; vertical-align: -0.338ex; width: 1.34ex;" /></span></th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle y}</annotation></semantics></math></span><img alt="y" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" style="border: 0px; display: inline-block; height: 2.009ex; vertical-align: -0.671ex; width: 1.166ex;" /></span></th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle x\wedge y}</annotation></semantics></math></span><img alt="x\wedge y" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e39c22fab294b953b40e439378be357dea68150" style="border: 0px; display: inline-block; height: 2.343ex; vertical-align: -0.671ex; width: 5.099ex;" /></span></th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle x\vee y}</annotation></semantics></math></span><img alt="x\vee y" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/304b00d1f1cf4a707c7863e8fae02a2dff7d5a8a" style="border: 0px; display: inline-block; height: 2.343ex; vertical-align: -0.671ex; width: 5.099ex;" /></span></th></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">0</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">0</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">1</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">0</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">0</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">1</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">1</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">1</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td></tr>
</tbody></table>
</td><td style="vertical-align: top;"><table class="wikitable" style="background-color: #f8f9fa; border-collapse: collapse; border: 1px solid rgb(162, 169, 177); color: black; font-size: 14px; margin: 1em 0px; text-align: center;"><tbody>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle x}</annotation></semantics></math></span><img alt="x" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" style="border: 0px; display: inline-block; height: 1.676ex; vertical-align: -0.338ex; width: 1.34ex;" /></span></th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \neg x}</annotation></semantics></math></span><img alt="\neg x" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4737ada4f1bfb57e805dec52f6e30a82873304e" style="border: 0px; display: inline-block; height: 1.676ex; vertical-align: -0.338ex; width: 2.901ex;" /></span></th></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">0</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">1</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td></tr>
</tbody></table>
</td></tr>
</tbody></table>
</dd></dl>
<div style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: justify;">
If the truth values 0 and 1 are interpreted as integers, these operations may be expressed with the ordinary operations of arithmetic, or by the minimum/maximum functions:</div>
<dl style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px; text-align: justify;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x\wedge y&=x\times y=\min(x,y)\\x\vee y&=x+y-(x\times y)=\max(x,y)\\\neg x&=1-x\end{aligned}}}</annotation></semantics></math></span><img alt="{\displaystyle {\begin{aligned}x\wedge y&=x\times y=\min(x,y)\\x\vee y&=x+y-(x\times y)=\max(x,y)\\\neg x&=1-x\end{aligned}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c50ef62354662f621569d80601b1fd797583a2d1" style="border: 0px; display: inline-block; height: 9.176ex; vertical-align: -4.005ex; width: 37.201ex;" /></span></dd></dl>
<div style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: justify;">
One may consider that only the negation and one of the two other operations are basic, because of the following identities that allow to define the conjunction in terms of the negation and the disjunction, and vice versa:</div>
<dl style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px; text-align: justify;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x\wedge y&=\neg (\neg x\vee \neg y)\\x\vee y&=\neg (\neg x\wedge \neg y)\end{aligned}}}</annotation></semantics></math></span><img alt="{\begin{aligned}x\wedge y&=\neg (\neg x\vee \neg y)\\x\vee y&=\neg (\neg x\wedge \neg y)\end{aligned}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a1628f610033d1ba2dff543ddf5aef5acc8fea3" style="border: 0px; display: inline-block; height: 6.176ex; vertical-align: -2.505ex; width: 20.571ex;" /></span></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px; text-align: justify;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><br /></span></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><h3 style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border-bottom: 0px; color: black; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em; text-align: justify;">
<span class="mw-headline" id="Secondary_operations">Secondary operations</span><span class="mw-editsection" style="display: inline-block; font-size: x-small; font-weight: normal; line-height: 1em; margin-left: 1em; unicode-bidi: isolate; user-select: none; vertical-align: baseline; white-space: nowrap;"><span class="mw-editsection-bracket" style="color: #555555; margin-right: 0.25em;">[</span><a href="https://en.wikipedia.org/w/index.php?title=Boolean_algebra&action=edit&section=5" style="background: none; color: #0b0080; text-decoration-line: none;" title="Edit section: Secondary operations">edit</a><span class="mw-editsection-bracket" style="color: #555555; margin-left: 0.25em;">]</span></span></h3>
<div style="line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em; text-align: justify;">
The three Boolean operations described above are referred to as basic, meaning that they can be taken as a basis for other Boolean operations that can be built up from them by <b>composition,</b> the manner in which operations are combined or compounded. Operations composed from the basic operations include the following examples:</div>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px; text-align: justify;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle x\rightarrow y=\neg {x}\vee y}</annotation></semantics></math></span><img alt="x\rightarrow y=\neg {x}\vee y" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02ca5f896cdc45c89977e3da32d23468fb085537" style="border: 0px; display: inline-block; height: 2.343ex; vertical-align: -0.671ex; width: 15.899ex;" /></span></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px; text-align: justify;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle x\oplus y=(x\vee y)\wedge \neg {(x\wedge y)}}</annotation></semantics></math></span><img alt="x\oplus y=(x\vee y)\wedge \neg {(x\wedge y)}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d11bc0cb9a7df01617b4b1630babbb32accef1c" style="border: 0px; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 26.478ex;" /></span></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px; text-align: justify;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle x\equiv y=\neg {(x\oplus y)}}</annotation></semantics></math></span><img alt="x\equiv y=\neg {(x\oplus y)}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c218dfeebb2ffac776ea836686b4b3bf144cfb72" style="border: 0px; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 17.472ex;" /></span></dd></dl>
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These definitions give rise to the following truth tables giving the values of these operations for all four possible inputs.</div>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><table class="wikitable" style="background-color: #f8f9fa; border-collapse: collapse; border: 1px solid rgb(162, 169, 177); color: black; font-size: 14px; margin: 1em 0px; text-align: justify;"><tbody>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle x}</annotation></semantics></math></span><img alt="x" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" style="border: 0px; display: inline-block; height: 1.676ex; vertical-align: -0.338ex; width: 1.34ex;" /></span></th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle y}</annotation></semantics></math></span><img alt="y" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" style="border: 0px; display: inline-block; height: 2.009ex; vertical-align: -0.671ex; width: 1.166ex;" /></span></th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle x\rightarrow y}</annotation></semantics></math></span><img alt="x\rightarrow y" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fce077cec2b56644f63a641afc4266677f1238e7" style="border: 0px; display: inline-block; height: 2.176ex; vertical-align: -0.671ex; width: 6.131ex;" /></span></th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle x\oplus y}</annotation></semantics></math></span><img alt="x\oplus y" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10fc94462e7622639c0c464161a1f0c8fc057999" style="border: 0px; display: inline-block; height: 2.343ex; vertical-align: -0.671ex; width: 5.357ex;" /></span></th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle x\equiv y}</annotation></semantics></math></span><img alt="x\equiv y" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/344d00ea5455ff5454015cc1f00cf534be450a03" style="border: 0px; display: inline-block; height: 2.009ex; vertical-align: -0.671ex; width: 5.615ex;" /></span></th></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">0</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">0</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">1</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">0</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">0</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">1</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td></tr>
<tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">1</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">1</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td></tr>
</tbody></table>
</dd></dl>
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The first operation, <i>x</i> → <i>y</i>, or C<i>xy</i>, is called <b>material implication</b>. If <i>x</i> is true then the value of <i>x</i> → <i>y</i> is taken to be that of <i>y</i>. But if <i>x</i> is false then the value of <i>y</i> can be ignored; however the operation must return <i>some</i> truth value and there are only two choices, so the return value is the one that entails less, namely <i>true</i>. (<a href="https://en.wikipedia.org/wiki/Relevance_logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Relevance logic">Relevance logic</a> addresses this by viewing an implication with a <a href="https://en.wikipedia.org/wiki/False_premise" style="background: none; color: #0b0080; text-decoration-line: none;" title="False premise">false premise</a> as something other than either true or false.)</div>
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The second operation, <i>x</i> ⊕ <i>y</i>, or J<i>xy</i>, is called <b><a href="https://en.wikipedia.org/wiki/Exclusive_or" style="background: none; color: #0b0080; text-decoration-line: none;" title="Exclusive or">exclusive or</a></b> (often abbreviated as XOR) to distinguish it from disjunction as the inclusive kind. It excludes the possibility of both <i>x</i> and <i>y</i>. Defined in terms of arithmetic it is addition mod 2 where 1 + 1 = 0.</div>
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The third operation, the complement of exclusive or, is <b>equivalence</b> or Boolean equality: <i>x</i> ≡ <i>y</i>, or E<i>xy</i>, is true just when <i>x</i> and <i>y</i> have the same value. Hence <i>x</i> ⊕ <i>y</i> as its complement can be understood as <i>x</i> ≠ <i>y</i>, being true just when <i>x</i> and <i>y</i> are different. Equivalence counterpart in arithmetic mod 2 is <i>x</i> + <i>y</i> + 1.</div>
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Given two operands, each with two possible values, there are 2<sup style="font-size: 11.2px; line-height: 1;">2</sup> = 4 possible combinations of inputs. Because each output can have two possible values, there are a total of 2<sup style="font-size: 11.2px; line-height: 1;">4</sup> = <a href="https://en.wikipedia.org/wiki/Boolean_algebras_canonically_defined#Truth_tables" style="background: none; color: #0b0080; text-decoration-line: none;" title="Boolean algebras canonically defined">16 possible binary Boolean operations</a>.</div>
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<span class="mw-headline" id="Laws">Laws</span></h2>
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A <b>law</b> of Boolean algebra is an <a href="https://en.wikipedia.org/wiki/Identity_(mathematics)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Identity (mathematics)">identity</a> such as <i>x</i>∨(<i>y</i>∨<i>z</i>) = (<i>x</i>∨<i>y</i>)∨<i>z</i> between two Boolean terms, where a <b>Boolean term</b> is defined as an expression built up from variables and the constants 0 and 1 using the operations ∧, ∨, and ¬. The concept can be extended to terms involving other Boolean operations such as ⊕, →, and ≡, but such extensions are unnecessary for the purposes to which the laws are put. Such purposes include the definition of a <a href="https://en.wikipedia.org/wiki/Boolean_algebra#Boolean_algebras" style="background: none; color: #0b0080; text-decoration-line: none;" title="Boolean algebra">Boolean algebra</a> as any <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Model_(logic)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Model (logic)">model</a> of the Boolean laws, and as a means for deriving new laws from old as in the derivation of <i>x</i>∨(<i>y</i>∧<i>z</i>) = <i>x</i>∨(<i>z</i>∧<i>y</i>) from <i>y</i>∧<i>z</i> = <i>z</i>∧<i>y</i> as treated in the section on <a href="https://en.wikipedia.org/wiki/Boolean_algebra#Axiomatizing_Boolean_algebra" style="background: none; color: #0b0080; text-decoration-line: none;" title="Boolean algebra">axiomatization</a>.</div>
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<span class="mw-headline" id="Monotone_laws">Monotone laws</span></h3>
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Boolean algebra satisfies many of the same laws as ordinary algebra when one matches up ∨ with addition and ∧ with multiplication. In particular the following laws are common to both kinds of algebra:<sup class="reference" id="cite_ref-O.27Regan_p33_13-0" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Boolean_algebra#cite_note-O.27Regan_p33-13" style="background: none; color: #0b0080; text-decoration-line: none;">[13]</a></sup></div>
<dl style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><table style="font-size: 14px;"><tbody>
<tr><td>Associativity of <span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \vee }</annotation></semantics></math></span><img alt="\vee " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" style="border: 0px; display: inline-block; height: 2.009ex; vertical-align: -0.338ex; width: 1.561ex;" /></span>:</td><td style="width: 2em;"></td><td style="text-align: right;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle x\vee (y\vee z)}</annotation></semantics></math></span><img alt="{\displaystyle x\vee (y\vee z)}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e43f6b6bb6feceba01569e15016d6a8af4ada801" style="border: 0px; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 10.621ex;" /></span></td><td><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle =(x\vee y)\vee z}</annotation></semantics></math></span><img alt="{\displaystyle =(x\vee y)\vee z}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d25a96fc01de59a76cfb5302b067b7f02da8f2ca" style="border: 0px; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 13.085ex;" /></span></td></tr>
<tr><td>Associativity of <span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation></semantics></math></span><img alt="\wedge " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" style="border: 0px; display: inline-block; height: 2.009ex; vertical-align: -0.338ex; width: 1.561ex;" /></span>:</td><td></td><td style="text-align: right;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle x\wedge (y\wedge z)}</annotation></semantics></math></span><img alt="{\displaystyle x\wedge (y\wedge z)}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f985d5acedc3560c9dfbe76622058a49bbb4cc2a" style="border: 0px; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 10.621ex;" /></span></td><td><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle =(x\wedge y)\wedge z}</annotation></semantics></math></span><img alt="{\displaystyle =(x\wedge y)\wedge z}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ddbf9998bf875ecf6948b9b871cdd9a02fe015" style="border: 0px; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 13.085ex;" /></span></td></tr>
<tr><td>Commutativity of <span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \vee }</annotation></semantics></math></span><img alt="\vee " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" style="border: 0px; display: inline-block; height: 2.009ex; vertical-align: -0.338ex; width: 1.561ex;" /></span>:</td><td></td><td style="text-align: right;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle x\vee y}</annotation></semantics></math></span><img alt="x\vee y" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/304b00d1f1cf4a707c7863e8fae02a2dff7d5a8a" style="border: 0px; display: inline-block; height: 2.343ex; vertical-align: -0.671ex; width: 5.099ex;" /></span></td><td><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle =y\vee x}</annotation></semantics></math></span><img alt="{\displaystyle =y\vee x}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84a116ff6ae01fd596a4a2952803a8e5bb3b8571" style="border: 0px; display: inline-block; height: 2.343ex; vertical-align: -0.671ex; width: 7.563ex;" /></span></td></tr>
<tr><td>Commutativity of <span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation></semantics></math></span><img alt="\wedge " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" style="border: 0px; display: inline-block; height: 2.009ex; vertical-align: -0.338ex; width: 1.561ex;" /></span>:</td><td></td><td style="text-align: right;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle x\wedge y}</annotation></semantics></math></span><img alt="x\wedge y" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e39c22fab294b953b40e439378be357dea68150" style="border: 0px; display: inline-block; height: 2.343ex; vertical-align: -0.671ex; width: 5.099ex;" /></span></td><td><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle =y\wedge x}</annotation></semantics></math></span><img alt="{\displaystyle =y\wedge x}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/612e28d9c887bf17b78deeb8b6690827b6bf7dab" style="border: 0px; display: inline-block; height: 2.343ex; vertical-align: -0.671ex; width: 7.563ex;" /></span></td></tr>
<tr><td>Distributivity of <span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation></semantics></math></span><img alt="\wedge " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" style="border: 0px; display: inline-block; height: 2.009ex; vertical-align: -0.338ex; width: 1.561ex;" /></span> over <span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \vee }</annotation></semantics></math></span><img alt="\vee " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" style="border: 0px; display: inline-block; height: 2.009ex; vertical-align: -0.338ex; width: 1.561ex;" /></span>:</td><td></td><td style="text-align: right;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle x\wedge (y\vee z)}</annotation></semantics></math></span><img alt="{\displaystyle x\wedge (y\vee z)}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3acd78dcc7481cd258b13f2f22a13bf5894bfe19" style="border: 0px; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 10.621ex;" /></span></td><td><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle =(x\wedge y)\vee (x\wedge z)}</annotation></semantics></math></span><img alt="{\displaystyle =(x\wedge y)\vee (x\wedge z)}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49e4bd21a491b4a9653a0636a36b1c4276e9a347" style="border: 0px; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 18.848ex;" /></span></td></tr>
<tr><td>Identity for <span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \vee }</annotation></semantics></math></span><img alt="\vee " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" style="border: 0px; display: inline-block; height: 2.009ex; vertical-align: -0.338ex; width: 1.561ex;" /></span>:</td><td></td><td style="text-align: right;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle x\vee 0}</annotation></semantics></math></span><img alt="{\displaystyle x\vee 0}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cbf63019e961cfde4abddd64640ca3f7b712fe0" style="border: 0px; display: inline-block; height: 2.176ex; vertical-align: -0.338ex; width: 5.106ex;" /></span></td><td><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle =x}</annotation></semantics></math></span><img alt="=x" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c66e79d9cccef418849534769b5e3c6ffeb9d52" style="border: 0px; display: inline-block; height: 1.676ex; vertical-align: -0.338ex; width: 3.804ex;" /></span></td></tr>
<tr><td>Identity for <span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation></semantics></math></span><img alt="\wedge " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" style="border: 0px; display: inline-block; height: 2.009ex; vertical-align: -0.338ex; width: 1.561ex;" /></span>:</td><td></td><td style="text-align: right;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle x\wedge 1}</annotation></semantics></math></span><img alt="{\displaystyle x\wedge 1}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e13cfe3c5eea983f9f6f14fa71ee27beee2f1d5" style="border: 0px; display: inline-block; height: 2.176ex; vertical-align: -0.338ex; width: 5.106ex;" /></span></td><td><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle =x}</annotation></semantics></math></span><img alt="=x" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c66e79d9cccef418849534769b5e3c6ffeb9d52" style="border: 0px; display: inline-block; height: 1.676ex; vertical-align: -0.338ex; width: 3.804ex;" /></span></td></tr>
<tr><td>Annihilator for <span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation></semantics></math></span><img alt="\wedge " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" style="border: 0px; display: inline-block; height: 2.009ex; vertical-align: -0.338ex; width: 1.561ex;" /></span>:</td><td></td><td style="text-align: right;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle x\wedge 0}</annotation></semantics></math></span><img alt="{\displaystyle x\wedge 0}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f53bbfb4f7678ad702506b628799febe354f8a83" style="border: 0px; display: inline-block; height: 2.176ex; vertical-align: -0.338ex; width: 5.106ex;" /></span></td><td><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle =0}</annotation></semantics></math></span><img alt="=0" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dc9e66de468806365c20e32e83456cc526ce29e" style="border: 0px; display: inline-block; height: 2.176ex; vertical-align: -0.338ex; width: 3.637ex;" /></span></td></tr>
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</dd></dl>
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The following laws hold in Boolean Algebra, but not in ordinary algebra:</div>
<dl style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><table style="font-size: 14px;"><tbody>
<tr><td>Annihilator for <span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \vee }</annotation></semantics></math></span><img alt="\vee " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" style="border: 0px; display: inline-block; height: 2.009ex; vertical-align: -0.338ex; width: 1.561ex;" /></span>:</td><td></td><td style="text-align: right;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle x\vee 1}</annotation></semantics></math></span><img alt="{\displaystyle x\vee 1}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d179f5e174104545342b7ae195cabaebd31f86d8" style="border: 0px; display: inline-block; height: 2.176ex; vertical-align: -0.338ex; width: 5.106ex;" /></span></td><td><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle =1}</annotation></semantics></math></span><img alt="=1" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/282a76fe69ce05e31352dfd19b7700eb784fb3f8" style="border: 0px; display: inline-block; height: 2.176ex; vertical-align: -0.338ex; width: 3.637ex;" /></span></td></tr>
<tr><td>Idempotence of <span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \vee }</annotation></semantics></math></span><img alt="\vee " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" style="border: 0px; display: inline-block; height: 2.009ex; vertical-align: -0.338ex; width: 1.561ex;" /></span>:</td><td></td><td style="text-align: right;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle x\vee x}</annotation></semantics></math></span><img alt="{\displaystyle x\vee x}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18299516620d66d42505a46089684d22e18cae58" style="border: 0px; display: inline-block; height: 2.009ex; vertical-align: -0.338ex; width: 5.273ex;" /></span></td><td><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle =x}</annotation></semantics></math></span><img alt="=x" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c66e79d9cccef418849534769b5e3c6ffeb9d52" style="border: 0px; display: inline-block; height: 1.676ex; vertical-align: -0.338ex; width: 3.804ex;" /></span></td></tr>
<tr><td>Idempotence of <span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation></semantics></math></span><img alt="\wedge " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" style="border: 0px; display: inline-block; height: 2.009ex; vertical-align: -0.338ex; width: 1.561ex;" /></span>:</td><td></td><td style="text-align: right;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle x\wedge x}</annotation></semantics></math></span><img alt="{\displaystyle x\wedge x}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61c3e1f66a763d5a708485438e3f6d952e0bc4f4" style="border: 0px; display: inline-block; height: 2.009ex; vertical-align: -0.338ex; width: 5.273ex;" /></span></td><td><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle =x}</annotation></semantics></math></span><img alt="=x" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c66e79d9cccef418849534769b5e3c6ffeb9d52" style="border: 0px; display: inline-block; height: 1.676ex; vertical-align: -0.338ex; width: 3.804ex;" /></span></td></tr>
<tr><td>Absorption 1:</td><td></td><td style="text-align: right;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle x\wedge (x\vee y)}</annotation></semantics></math></span><img alt="{\displaystyle x\wedge (x\vee y)}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7317a18981f4f50ddd5281621860434d32a2055c" style="border: 0px; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 10.862ex;" /></span></td><td><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle =x}</annotation></semantics></math></span><img alt="=x" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c66e79d9cccef418849534769b5e3c6ffeb9d52" style="border: 0px; display: inline-block; height: 1.676ex; vertical-align: -0.338ex; width: 3.804ex;" /></span></td></tr>
<tr><td>Absorption 2:</td><td></td><td style="text-align: right;"><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle x\vee (x\wedge y)}</annotation></semantics></math></span><img alt="{\displaystyle x\vee (x\wedge y)}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da75106eaa322a67bccf2daf277d8e74eabbab36" style="border: 0px; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 10.862ex;" /></span></td><td><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle =x}</annotation></semantics></math></span><img alt="=x" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c66e79d9cccef418849534769b5e3c6ffeb9d52" style="border: 0px; display: inline-block; height: 1.676ex; vertical-align: -0.338ex; width: 3.804ex;" /></span></td></tr>
<tr><td>Distributivity of <span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \vee }</annotation></semantics></math></span><img alt="\vee " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" style="border: 0px; display: inline-block; height: 2.009ex; vertical-align: -0.338ex; width: 1.561ex;" /></span> over <span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation></semantics></math></span><img alt="\wedge " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" style="border: 0px; display: inline-block; height: 2.009ex; vertical-align: -0.338ex; width: 1.561ex;" /></span>:</td><td></td><td><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle x\vee (y\wedge z)}</annotation></semantics></math></span><img alt="{\displaystyle x\vee (y\wedge z)}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6e9658b0b6387c317cfb8ad6da05af1095f6351" style="border: 0px; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 10.621ex;" /></span></td><td><span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle =(x\vee y)\wedge (x\vee z)}</annotation></semantics></math></span><img alt="{\displaystyle =(x\vee y)\wedge (x\vee z)}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c90090bce4eaa5a183fdb45215c63b98f2ed08e1" style="border: 0px; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 18.848ex;" /></span></td></tr>
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</dd></dl>
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Taking x = 2 in the third law above shows that it is not an ordinary algebra law, since 2×2 = 4. The remaining five laws can be falsified in ordinary algebra by taking all variables to be 1, for example in Absorption Law 1 the left hand side would be 1(1+1) = 2 while the right hand side would be 1, and so on.</div>
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All of the laws treated so far have been for conjunction and disjunction. These operations have the property that changing either argument either leaves the output unchanged or the output changes in the same way as the input. Equivalently, changing any variable from 0 to 1 never results in the output changing from 1 to 0. Operations with this property are said to be <b>monotone</b>. Thus the axioms so far have all been for monotonic Boolean logic. Nonmonotonicity enters via complement as follows.</div>
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<span class="mw-headline" id="Applications_2">Applications</span><span class="mw-editsection" style="display: inline-block; font-family: sans-serif; font-size: x-small; line-height: 1em; margin-left: 1em; unicode-bidi: isolate; user-select: none; vertical-align: baseline; white-space: nowrap;"><span class="mw-editsection-bracket" style="color: #555555; margin-right: 0.25em;">[</span><a href="https://en.wikipedia.org/w/index.php?title=Boolean_algebra&action=edit&section=25" style="background: none; color: #0b0080; text-decoration-line: none;" title="Edit section: Applications">edit</a><span class="mw-editsection-bracket" style="color: #555555; margin-left: 0.25em;">]</span></span></h2>
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Boolean algebra as the calculus of two values is fundamental to computer circuits, computer programming, and mathematical logic, and is also used in other areas of mathematics such as set theory and statistics.<sup class="reference" id="cite_ref-givhal_3-5" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Boolean_algebra#cite_note-givhal-3" style="background: none; color: #0b0080; text-decoration-line: none;">[3]</a></sup></div>
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<span class="mw-headline" id="Computers">Computers</span><span class="mw-editsection" style="display: inline-block; font-size: x-small; font-weight: normal; line-height: 1em; margin-left: 1em; unicode-bidi: isolate; user-select: none; vertical-align: baseline; white-space: nowrap;"><span class="mw-editsection-bracket" style="color: #555555; margin-right: 0.25em;">[</span><a href="https://en.wikipedia.org/w/index.php?title=Boolean_algebra&action=edit&section=26" style="background: none; color: #0b0080; text-decoration-line: none;" title="Edit section: Computers">edit</a><span class="mw-editsection-bracket" style="color: #555555; margin-left: 0.25em;">]</span></span></h3>
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In the early 20th century, several electrical engineers intuitively recognized that Boolean algebra was analogous to the behavior of certain types of electrical circuits. <a href="https://en.wikipedia.org/wiki/Claude_Shannon" style="background: none; color: #0b0080; text-decoration-line: none;" title="Claude Shannon">Claude Shannon</a> formally proved such behavior was logically equivalent to Boolean algebra in his 1937 master's thesis, <i><a href="https://en.wikipedia.org/wiki/A_Symbolic_Analysis_of_Relay_and_Switching_Circuits" style="background: none; color: #0b0080; text-decoration-line: none;" title="A Symbolic Analysis of Relay and Switching Circuits">A Symbolic Analysis of Relay and Switching Circuits</a></i>.</div>
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Today, all modern general purpose <a href="https://en.wikipedia.org/wiki/Computer" style="background: none; color: #0b0080; text-decoration-line: none;" title="Computer">computers</a> perform their functions using two-value Boolean logic; that is, their electrical circuits are a physical manifestation of two-value Boolean logic. They achieve this in various ways: as voltages on wires in high-speed circuits and capacitive storage devices, as orientations of a magnetic domain in ferromagnetic storage devices, as holes in punched cards or paper tape, and so on. (Some early computers used decimal circuits or mechanisms instead of two-valued logic circuits.)</div>
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Of course, it is possible to code more than two symbols in any given medium. For example, one might use respectively 0, 1, 2, and 3 volts to code a four-symbol alphabet on a wire, or holes of different sizes in a punched card. In practice, the tight constraints of high speed, small size, and low power combine to make noise a major factor. This makes it hard to distinguish between symbols when there are several possible symbols that could occur at a single site. Rather than attempting to distinguish between four voltages on one wire, digital designers have settled on two voltages per wire, high and low.</div>
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Computers use two-value Boolean circuits for the above reasons. The most common computer architectures use ordered sequences of Boolean values, called bits, of 32 or 64 values, e.g. 01101000110101100101010101001011. When programming in <a href="https://en.wikipedia.org/wiki/Machine_code" style="background: none; color: #0b0080; text-decoration-line: none;" title="Machine code">machine code</a>, <a href="https://en.wikipedia.org/wiki/Assembly_language" style="background: none; color: #0b0080; text-decoration-line: none;" title="Assembly language">assembly language</a>, and certain other <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Programming_languages" style="background: none; color: #0b0080; text-decoration-line: none;" title="Programming languages">programming languages</a>, programmers work with the low-level digital structure of the <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Word_(data_type)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Word (data type)">data registers</a>. These registers operate on voltages, where zero volts represents Boolean 0, and a reference voltage (often +5V, +3.3V, +1.8V) represents Boolean 1. Such languages support both numeric operations and logical operations. In this context, "numeric" means that the computer treats sequences of bits as <a href="https://en.wikipedia.org/wiki/Binary_number" style="background: none; color: #0b0080; text-decoration-line: none;" title="Binary number">binary numbers</a> (base two numbers) and executes arithmetic operations like add, subtract, multiply, or divide. "Logical" refers to the Boolean logical operations of disjunction, conjunction, and negation between two sequences of bits, in which each bit in one sequence is simply compared to its counterpart in the other sequence. Programmers therefore have the option of working in and applying the rules of either numeric algebra or Boolean algebra as needed. A core differentiating feature between these families of operations is the existence of the <a href="https://en.wikipedia.org/wiki/Carry_(arithmetic)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Carry (arithmetic)">carry</a> operation in the first but not the second.</div>
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<span class="mw-headline" id="Two-valued_logic">Two-valued logic</span></h3>
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Other areas where two values is a good choice are the law and mathematics. In everyday relaxed conversation, nuanced or complex answers such as "maybe" or "only on the weekend" are acceptable. In more focused situations such as a court of law or theorem-based mathematics however it is deemed advantageous to frame questions so as to admit a simple yes-or-no answer—is the defendant guilty or not guilty, is the proposition true or false—and to disallow any other answer. However much of a straitjacket this might prove in practice for the respondent, the principle of the simple yes-no question has become a central feature of both judicial and mathematical logic, making two-valued logic deserving of organization and study in its own right.</div>
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A central concept of set theory is membership. Now an organization may permit multiple degrees of membership, such as novice, associate, and full. With sets however an element is either in or out. The candidates for membership in a set work just like the wires in a digital computer: each candidate is either a member or a nonmember, just as each wire is either high or low.</div>
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Algebra being a fundamental tool in any area amenable to mathematical treatment, these considerations combine to make the algebra of two values of fundamental importance to computer hardware, mathematical logic, and set theory.</div>
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<a class="mw-redirect" href="https://en.wikipedia.org/wiki/Two-valued_logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Two-valued logic">Two-valued logic</a> can be extended to <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Multi-valued_logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Multi-valued logic">multi-valued logic</a>, notably by replacing the Boolean domain {0, 1} with the unit interval [0,1], in which case rather than only taking values 0 or 1, any value between and including 0 and 1 can be assumed. Algebraically, negation (NOT) is replaced with 1 − <i>x</i>, conjunction (AND) is replaced with multiplication (<span class="mwe-math-element" style="max-width: 100%; overflow-x: auto;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle xy}</annotation></semantics></math></span><img alt="xy" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72eb345e496513fb8b2fa4aa8c4d89b855f9a01" style="border: 0px; display: inline-block; height: 2.009ex; margin: 0px; vertical-align: -0.671ex; width: 2.506ex;" /></span>), and disjunction (OR) is defined via <a class="mw-redirect" href="https://en.wikipedia.org/wiki/De_Morgan%27s_law" style="background: none; color: #0b0080; text-decoration-line: none;" title="De Morgan's law">De Morgan's law</a>. Interpreting these values as logical <a href="https://en.wikipedia.org/wiki/Truth_value" style="background: none; color: #0b0080; text-decoration-line: none;" title="Truth value">truth values</a> yields a multi-valued logic, which forms the basis for <a href="https://en.wikipedia.org/wiki/Fuzzy_logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Fuzzy logic">fuzzy logic</a> and <a href="https://en.wikipedia.org/wiki/Probabilistic_logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Probabilistic logic">probabilistic logic</a>. In these interpretations, a value is interpreted as the "degree" of truth – to what extent a proposition is true, or the probability that the proposition is true.</div>
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<span class="mw-headline" id="Boolean_operations">Boolean operations</span></h3>
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The original application for Boolean operations was <a href="https://en.wikipedia.org/wiki/Mathematical_logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Mathematical logic">mathematical logic</a>, where it combines the truth values, true or false, of individual formulas.</div>
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Natural languages such as English have words for several Boolean operations, in particular conjunction (<i>and</i>), disjunction (<i>or</i>), negation (<i>not</i>), and implication (<i>implies</i>). <i>But not</i> is synonymous with <i>and not</i>. When used to combine situational assertions such as "the block is on the table" and "cats drink milk," which naively are either true or false, the meanings of these <a href="https://en.wikipedia.org/wiki/Logical_connective" style="background: none; color: #0b0080; text-decoration-line: none;" title="Logical connective">logical connectives</a> often have the meaning of their logical counterparts. However, with descriptions of behavior such as "Jim walked through the door", one starts to notice differences such as failure of commutativity, for example the conjunction of "Jim opened the door" with "Jim walked through the door" in that order is not equivalent to their conjunction in the other order, since <i>and</i> usually means <i>and then</i> in such cases. Questions can be similar: the order "Is the sky blue, and why is the sky blue?" makes more sense than the reverse order. Conjunctive commands about behavior are like behavioral assertions, as in <i>get dressed and go to school</i>. Disjunctive commands such <i>love me or leave me</i> or <i>fish or cut bait</i> tend to be asymmetric via the implication that one alternative is less preferable. Conjoined nouns such as <i>tea and milk</i> generally describe aggregation as with set union while <i>tea or milk</i> is a choice. However context can reverse these senses, as in <i>your choices are coffee and tea</i> which usually means the same as <i>your choices are coffee or tea</i> (alternatives). Double negation as in "I don't not like milk" rarely means literally "I do like milk" but rather conveys some sort of hedging, as though to imply that there is a third possibility. "Not not P" can be loosely interpreted as "surely P", and although <i>P</i> necessarily implies "not not <i>P</i>" the converse is suspect in English, much as with <a href="https://en.wikipedia.org/wiki/Intuitionistic_logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Intuitionistic logic">intuitionistic logic</a>. In view of the highly idiosyncratic usage of conjunctions in natural languages, Boolean algebra cannot be considered a reliable framework for interpreting them.</div>
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Boolean operations are used in <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Digital_logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Digital logic">digital logic</a> to combine the bits carried on individual wires, thereby interpreting them over {0,1}. When a vector of <i>n</i> identical binary gates are used to combine two bit vectors each of <i>n</i> bits, the individual bit operations can be understood collectively as a single operation on values from a <a href="https://en.wikipedia.org/wiki/Boolean_algebra_(structure)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Boolean algebra (structure)">Boolean algebra</a> with 2<sup style="font-size: 11.2px; line-height: 1;"><i>n</i></sup> elements.</div>
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<a href="https://en.wikipedia.org/wiki/Naive_set_theory" style="background: none; color: #0b0080; text-decoration-line: none;" title="Naive set theory">Naive set theory</a> interprets Boolean operations as acting on subsets of a given set <i>X</i>. As we saw earlier this behavior exactly parallels the coordinate-wise combinations of bit vectors, with the union of two sets corresponding to the disjunction of two bit vectors and so on.</div>
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The 256-element free Boolean algebra on three generators is deployed in <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Computer_displays" style="background: none; color: #0b0080; text-decoration-line: none;" title="Computer displays">computer displays</a> based on <a href="https://en.wikipedia.org/wiki/Raster_graphics" style="background: none; color: #0b0080; text-decoration-line: none;" title="Raster graphics">raster graphics</a>, which use <a href="https://en.wikipedia.org/wiki/Bit_blit" style="background: none; color: #0b0080; text-decoration-line: none;" title="Bit blit">bit blit</a> to manipulate whole regions consisting of <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Pixels" style="background: none; color: #0b0080; text-decoration-line: none;" title="Pixels">pixels</a>, relying on Boolean operations to specify how the source region should be combined with the destination, typically with the help of a third region called the <a href="https://en.wikipedia.org/wiki/Mask_(computing)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Mask (computing)">mask</a>. Modern <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Video_cards" style="background: none; color: #0b0080; text-decoration-line: none;" title="Video cards">video cards</a> offer all 2<sup style="font-size: 11.2px; line-height: 1;">2<sup style="font-size: 8.96px; line-height: 1;">3</sup></sup> = 256 ternary operations for this purpose, with the choice of operation being a one-byte (8-bit) parameter. The constants SRC = 0xaa or 10101010, DST = 0xcc or 11001100, and MSK = 0xf0 or 11110000 allow Boolean operations such as (SRC^DST)&MSK (meaning XOR the source and destination and then AND the result with the mask) to be written directly as a constant denoting a byte calculated at compile time, 0x60 in the (SRC^DST)&MSK example, 0x66 if just SRC^DST, etc. At run time the video card interprets the byte as the raster operation indicated by the original expression in a uniform way that requires remarkably little hardware and which takes time completely independent of the complexity of the expression.</div>
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<a href="https://en.wikipedia.org/wiki/Solid_modeling" style="background: none; color: #0b0080; text-decoration-line: none;" title="Solid modeling">Solid modeling</a> systems for <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Computer_aided_design" style="background: none; color: #0b0080; text-decoration-line: none;" title="Computer aided design">computer aided design</a> offer a variety of methods for building objects from other objects, combination by Boolean operations being one of them. In this method the space in which objects exist is understood as a set <i>S</i> of <a href="https://en.wikipedia.org/wiki/Voxel" style="background: none; color: #0b0080; text-decoration-line: none;" title="Voxel">voxels</a> (the three-dimensional analogue of pixels in two-dimensional graphics) and shapes are defined as subsets of <i>S</i>, allowing objects to be combined as sets via union, intersection, etc. One obvious use is in building a complex shape from simple shapes simply as the union of the latter. Another use is in sculpting understood as removal of material: any grinding, milling, routing, or drilling operation that can be performed with physical machinery on physical materials can be simulated on the computer with the Boolean operation <i>x</i> ∧ ¬<i>y</i> or <i>x</i> − <i>y</i>, which in set theory is set difference, remove the elements of <i>y</i> from those of <i>x</i>. Thus given two shapes one to be machined and the other the material to be removed, the result of machining the former to remove the latter is described simply as their set difference.</div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3005862392404377111.post-75018562293449055152017-06-12T23:39:00.001-07:002023-06-23T23:34:12.285-07:00Digital Electronics<div dir="ltr" style="text-align: left;" trbidi="on">
<b style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">Digital electronics</b><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> or </span><b style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">digital (electronic) circuits</b><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> are </span><a href="https://en.wikipedia.org/wiki/Electronics" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Electronics">electronics</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> that handle </span><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Digital_signal_(electronics)" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Digital signal (electronics)">digital signals</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> (discrete bands of </span><a href="https://en.wikipedia.org/wiki/Analog_signal" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Analog signal">analog</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> levels) rather than by </span><a href="https://en.wikipedia.org/wiki/Continuous_function" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Continuous function">continuous</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> ranges as used in </span><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Analog_electronics" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Analog electronics">analog electronics</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">. All levels within a band of values represent the same </span><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Information_state" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Information state">information state</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">. Because of this </span><a href="https://en.wikipedia.org/wiki/Discretization" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Discretization">discretization</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">, relatively small changes to the analog signal levels due to </span><a href="https://en.wikipedia.org/wiki/Engineering_tolerance" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Engineering tolerance">manufacturing tolerance</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">, </span><a href="https://en.wikipedia.org/wiki/Path_loss" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Path loss">signal attenuation</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> or </span><a href="https://en.wikipedia.org/wiki/Noise_(electronics)" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Noise (electronics)">noise</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> do not leave the discrete envelope, and as a result are ignored by signal state sensing circuitry.</span><br />
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<span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> </span><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">Digital </span><a href="https://en.wikipedia.org/wiki/Electronic_circuit" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Electronic circuit">electronic circuits</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> are usually made from large assemblies of </span><a href="https://en.wikipedia.org/wiki/Logic_gate" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Logic gate">logic gates</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">, simple electronic representations of </span><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Boolean_logic#Digital_electronic_circuit_design" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Boolean logic">Boolean logic functions</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">.</span><br />
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<span style="background-color: white; color: #222222; font-family: sans-serif;"><b><span style="font-size: large;">HISTORY:</span></b></span><br />
<span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">The</span><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> </span><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Binary_number_system" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Binary number system">binary number system</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> </span><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">was refined by</span><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> </span><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">(published in 1705) and he also established that by using the binary system, the principles of arithmetic and logic could be combined. Digital logic as we know it was the brain-child of</span><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/George_Boole" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="George Boole">George Boole</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">, in the mid 19th century. Boole died young, but his ideas lived on. In an 1886 letter,</span><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/Charles_Sanders_Peirce" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Charles Sanders Peirce">Charles Sanders Peirce</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> </span><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">described how logical operations could be carried out by electrical switching circuits.</span><sup class="reference" id="cite_ref-P2M_2-0" style="color: #222222; font-family: sans-serif; font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Digital_electronics#cite_note-P2M-2" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; text-decoration-line: none;">[2]</a></sup><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> </span><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">Eventually,</span><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/Vacuum_tube" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Vacuum tube">vacuum tubes</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> </span><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">replaced relays for logic operations.</span><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> </span><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Lee_De_Forest" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Lee De Forest">Lee De Forest</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">'s modification, in 1907, of the</span><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/Vacuum_tube" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Vacuum tube">Fleming valve</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> </span><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">can be used as an AND logic gate.</span><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/Ludwig_Wittgenstein" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Ludwig Wittgenstein">Ludwig Wittgenstein</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> </span><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">introduced a version of the 16-row</span><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/Truth_table" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Truth table">truth table</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> </span><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">as proposition 5.101 of</span><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> </span><i style="color: #222222; font-family: sans-serif; font-size: 14px;"><a href="https://en.wikipedia.org/wiki/Tractatus_Logico-Philosophicus" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; text-decoration-line: none;" title="Tractatus Logico-Philosophicus">Tractatus Logico-Philosophicus</a></i><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> </span><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">(1921).</span><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/Walther_Bothe" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Walther Bothe">Walther Bothe</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">, inventor of the</span><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/Coincidence_circuit" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Coincidence circuit">coincidence circuit</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">, got part of the 1954</span><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/Nobel_Prize" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Nobel Prize">Nobel Prize</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> </span><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">in physics, for the first modern electronic AND gate in 1924.</span><br />
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<a href="https://en.wikipedia.org/wiki/Mechanical_computer" style="background: none; color: #0b0080; text-decoration-line: none;" title="Mechanical computer">Mechanical</a> <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Analog_computers" style="background: none; color: #0b0080; text-decoration-line: none;" title="Analog computers">analog computers</a> started appearing in the first century and were later used in the medieval era for astronomical calculations. In <a href="https://en.wikipedia.org/wiki/World_War_II" style="background: none; color: #0b0080; text-decoration-line: none;" title="World War II">World War II</a>, mechanical analog computers were used for specialized military applications such as calculating torpedo aiming. During this time the first electronic <a href="https://en.wikipedia.org/wiki/Digital_data" style="background: none; color: #0b0080; text-decoration-line: none;" title="Digital data">digital</a> computers were developed. Originally they were the size of a large room, consuming as much power as several hundred modern <a href="https://en.wikipedia.org/wiki/Personal_computer" style="background: none; color: #0b0080; text-decoration-line: none;" title="Personal computer">personal computers</a> (PCs).<sup class="reference" id="cite_ref-3" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Digital_electronics#cite_note-3" style="background: none; color: #0b0080; text-decoration-line: none;">[3]</a></sup></div>
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The <a href="https://en.wikipedia.org/wiki/Z3_(computer)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Z3 (computer)">Z3</a> was an <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Electromechanical" style="background: none; color: #0b0080; text-decoration-line: none;" title="Electromechanical">electromechanical</a> <a href="https://en.wikipedia.org/wiki/Computer" style="background: none; color: #0b0080; text-decoration-line: none;" title="Computer">computer</a> designed by <a href="https://en.wikipedia.org/wiki/Konrad_Zuse" style="background: none; color: #0b0080; text-decoration-line: none;" title="Konrad Zuse">Konrad Zuse</a>, finished in 1941. It was the world's first working <a href="https://en.wikipedia.org/wiki/Computer_programming" style="background: none; color: #0b0080; text-decoration-line: none;" title="Computer programming">programmable</a>, fully automatic digital computer.<sup class="reference" id="cite_ref-4" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Digital_electronics#cite_note-4" style="background: none; color: #0b0080; text-decoration-line: none;">[4]</a></sup> Its operation was facilitated by the invention of the vacuum tube in 1904 by <a href="https://en.wikipedia.org/wiki/John_Ambrose_Fleming" style="background: none; color: #0b0080; text-decoration-line: none;" title="John Ambrose Fleming">John Ambrose Fleming</a>.</div>
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Purely <a href="https://en.wikipedia.org/wiki/Electronic_circuit" style="background: none; color: #0b0080; text-decoration-line: none;" title="Electronic circuit">electronic circuit</a> elements soon replaced their mechanical and electromechanical equivalents, at the same time that digital calculation replaced analog. The <a href="https://en.wikipedia.org/wiki/Bipolar_junction_transistor" style="background: none; color: #0b0080; text-decoration-line: none;" title="Bipolar junction transistor">bipolar junction transistor</a> was invented in 1947. From 1955 onwards transistors replaced <a href="https://en.wikipedia.org/wiki/Vacuum_tube" style="background: none; color: #0b0080; text-decoration-line: none;" title="Vacuum tube">vacuum tubes</a> in computer designs, giving rise to the "second generation" of computers.</div>
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Compared to vacuum tubes, transistors have many advantages: they are smaller, and require less power than vacuum tubes, so give off less heat. Silicon junction transistors were much more reliable than vacuum tubes and had longer, indefinite, service life. Transistorized computers could contain tens of thousands of binary logic circuits in a relatively compact space.</div>
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At the <a href="https://en.wikipedia.org/wiki/University_of_Manchester" style="background: none; color: #0b0080; text-decoration-line: none;" title="University of Manchester">University of Manchester</a>, a team under the leadership of <a href="https://en.wikipedia.org/wiki/Tom_Kilburn" style="background: none; color: #0b0080; text-decoration-line: none;" title="Tom Kilburn">Tom Kilburn</a> designed and built a machine using the newly developed <a href="https://en.wikipedia.org/wiki/Transistor" style="background: none; color: #0b0080; text-decoration-line: none;" title="Transistor">transistors</a> instead of valves.<sup class="reference" id="cite_ref-5" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Digital_electronics#cite_note-5" style="background: none; color: #0b0080; text-decoration-line: none;">[5]</a></sup> Their first <a href="https://en.wikipedia.org/wiki/Transistor_computer" style="background: none; color: #0b0080; text-decoration-line: none;" title="Transistor computer">transistorised computer</a> and the first in the world, was <a href="https://en.wikipedia.org/wiki/Manchester_computers#Transistor_Computer" style="background: none; color: #0b0080; text-decoration-line: none;" title="Manchester computers">operational by 1953</a>, and a second version was completed there in April 1955.</div>
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While working at <a href="https://en.wikipedia.org/wiki/Texas_Instruments" style="background: none; color: #0b0080; text-decoration-line: none;" title="Texas Instruments">Texas Instruments</a>, <a href="https://en.wikipedia.org/wiki/Jack_Kilby" style="background: none; color: #0b0080; text-decoration-line: none;" title="Jack Kilby">Jack Kilby</a> recorded his initial ideas concerning the <a href="https://en.wikipedia.org/wiki/Integrated_circuit" style="background: none; color: #0b0080; text-decoration-line: none;" title="Integrated circuit">integrated circuit</a> in July 1958, successfully demonstrating the first working integrated example on 12 September 1958.<sup class="reference" id="cite_ref-TIJackBuilt_6-0" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Digital_electronics#cite_note-TIJackBuilt-6" style="background: none; color: #0b0080; text-decoration-line: none;">[6]</a></sup> This new technique allowed for quick, low-cost fabrication of complex circuits by having a set of <a href="https://en.wikipedia.org/wiki/Electronic_circuit" style="background: none; color: #0b0080; text-decoration-line: none;" title="Electronic circuit">electronic circuits</a> on one small plate ("chip") of <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Semiconductor_material" style="background: none; color: #0b0080; text-decoration-line: none;" title="Semiconductor material">semiconductor material</a>, normally <a href="https://en.wikipedia.org/wiki/Silicon" style="background: none; color: #0b0080; text-decoration-line: none;" title="Silicon">silicon</a>.</div>
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<span style="font-size: large;"><b>PROPERTIES:</b></span></div>
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<b><span style="font-size: x-small;"> </span></b><span style="font-size: 14px;">An advantage of digital circuits when compared to analog circuits is that signals represented digitally can be transmitted without degradation due to</span><span style="font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/Noise" style="background: none; color: #0b0080; font-size: 14px; text-decoration-line: none;" title="Noise">noise</a><span style="font-size: 14px;">.</span><sup class="reference" id="cite_ref-8" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Digital_electronics#cite_note-8" style="background: none; color: #0b0080; text-decoration-line: none;">[8]</a></sup><span style="font-size: 14px;"> </span><span style="font-size: 14px;">For example, a continuous audio signal transmitted as a sequence of 1s and 0s, can be reconstructed without error, provided the noise picked up in transmission is not enough to prevent identification of the 1s and 0s. An hour of music can be stored on a</span><span style="font-size: 14px;"> </span><a href="https://en.wikipedia.org/wiki/Compact_disc" style="background: none; color: #0b0080; font-size: 14px; text-decoration-line: none;" title="Compact disc">compact disc</a><span style="font-size: 14px;"> </span><span style="font-size: 14px;">using about 6 billion binary digits.</span></div>
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In a digital system, a more precise representation of a signal can be obtained by using more binary digits to represent it. While this requires more digital circuits to process the signals, each digit is handled by the same kind of hardware, resulting in an easily <a href="https://en.wikipedia.org/wiki/Scalability" style="background: none; color: #0b0080; text-decoration-line: none;" title="Scalability">scalable</a> system. In an analog system, additional resolution requires fundamental improvements in the linearity and noise characteristics of each step of the <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Signal_chain_(signal_processing_chain)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Signal chain (signal processing chain)">signal chain</a>.</div>
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<span style="font-size: large;"><b>CONSTRUCTION:</b></span></div>
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A digital circuit is typically constructed from small electronic circuits called <a href="https://en.wikipedia.org/wiki/Logic_gate" style="background: none; color: #0b0080; text-decoration-line: none;" title="Logic gate">logic gates</a> that can be used to create <a href="https://en.wikipedia.org/wiki/Combinational_logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Combinational logic">combinational logic</a>. Each logic gate is designed to perform a function of <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Boolean_logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Boolean logic">boolean logic</a> when acting on logic signals. A logic gate is generally created from one or more electrically controlled switches, usually <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Transistors" style="background: none; color: #0b0080; text-decoration-line: none;" title="Transistors">transistors</a> but <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Thermionic_valves" style="background: none; color: #0b0080; text-decoration-line: none;" title="Thermionic valves">thermionic valves</a> have seen historic use. The output of a logic gate can, in turn, control or feed into more logic gates.</div>
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<a href="https://en.wikipedia.org/wiki/Integrated_circuit" style="background: none; color: #0b0080; text-decoration-line: none;" title="Integrated circuit">Integrated circuits</a> consist of multiple transistors on one silicon chip, and are the least expensive way to make large number of interconnected logic gates. Integrated circuits are usually designed by engineers using <a href="https://en.wikipedia.org/wiki/Electronic_design_automation" style="background: none; color: #0b0080; text-decoration-line: none;" title="Electronic design automation">electronic design automation</a> software (see below for more information) to perform some type of function.</div>
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<span style="font-size: large;"><b>DESIGN:</b></span></div>
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<span style="font-size: large;"><b> </b></span><span style="font-size: 14px;">Each logic symbol is represented by a different shape. The actual set of shapes was introduced in 1984 under IEEE/ANSI standard 91-1984. "The logic symbol given under this standard are being increasingly used now and have even started appearing in the literature published by manufacturers of digital integrated circuits."</span><sup class="reference" id="cite_ref-9" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Digital_electronics#cite_note-9" style="background: none; color: #0b0080; text-decoration-line: none;">[9]</a></sup></div>
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Another form of digital circuit is constructed from lookup tables, (many sold as "<a href="https://en.wikipedia.org/wiki/Programmable_logic_device" style="background: none; color: #0b0080; text-decoration-line: none;" title="Programmable logic device">programmable logic devices</a>", though other kinds of PLDs exist). Lookup tables can perform the same functions as machines based on logic gates, but can be easily reprogrammed without changing the wiring. This means that a designer can often repair design errors without changing the arrangement of wires. Therefore, in small volume products, programmable logic devices are often the preferred solution. They are usually designed by engineers using electronic design automation software.</div>
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When the volumes are medium to large, and the logic can be slow, or involves complex algorithms or sequences, often a small <a href="https://en.wikipedia.org/wiki/Microcontroller" style="background: none; color: #0b0080; text-decoration-line: none;" title="Microcontroller">microcontroller</a> is <a href="https://en.wikipedia.org/wiki/Computer_program" style="background: none; color: #0b0080; text-decoration-line: none;" title="Computer program">programmed</a> to make an <a href="https://en.wikipedia.org/wiki/Embedded_system" style="background: none; color: #0b0080; text-decoration-line: none;" title="Embedded system">embedded system</a>. These are usually programmed by <a href="https://en.wikipedia.org/wiki/Software_engineering" style="background: none; color: #0b0080; text-decoration-line: none;" title="Software engineering">software engineers</a>.</div>
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When only one digital circuit is needed, and its design is totally customized, as for a factory production line controller, the conventional solution is a <a href="https://en.wikipedia.org/wiki/Programmable_logic_controller" style="background: none; color: #0b0080; text-decoration-line: none;" title="Programmable logic controller">programmable logic controller</a>, or PLC. These are usually programmed by electricians, using <a href="https://en.wikipedia.org/wiki/Ladder_logic" style="background: none; color: #0b0080; text-decoration-line: none;" title="Ladder logic">ladder logic</a>.</div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3005862392404377111.post-10346313522767977132017-04-20T08:30:00.002-07:002023-06-23T23:34:11.411-07:00Analog Electronics<div dir="ltr" style="text-align: left;" trbidi="on">
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<b><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">Theory</span></b></div>
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<span style="font-family: "tahoma"; font-size: x-small;"><span lang="EN-US"><i><b>Operational Amplifiers</b></i></span></span></div>
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<b><i><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">Operational Amplifiers</span></i></b><span lang="EN-GB" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"> (</span><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">OAs</span><span lang="EN-GB" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">)<b><i> </i></b></span><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">are highly stable, high gain dc difference amplifiers. Since there is no capacitive coupling between their various amplifying stages, they can handle signals from zero frequency (dc signals) up to a few hundred kHz. Their name is derived by the fact that they are used for performing <b><i>mathematical operations</i></b> on their input signal(s).</span></div>
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<span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt; line-height: 5.33333px;"> </span><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">Figure 1 shows the symbol for an OA. There are two inputs, the <b><i>inverting input</i></b> (-) and the <b><i>non-inverting input</i></b> (+). These symbols have nothing to do with the polarity of the applied </span><span lang="EN-GB" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">input </span><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">signals.</span><span lang="EN-GB" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"></span></div>
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<span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"> </span><span style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"><img alt="Image result for operational amplifier" class="rg_ic rg_i" data-sz="f" jsaction="load:str.tbn" name="FTGqq-FKBGRbtM:" src="data:image/png;base64,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" style="font-family: arial; height: 150px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: left; width: 200px;" /></span></div>
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<span style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt; line-height: 6.66667px;"> </span><b><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">Figure</span><span lang="EN-GB" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"> 1</span><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">.</span></b><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"> Symbol of the operational amplifier</span><span lang="EN-GB" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">.</span><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"> Connections to power supplies are also shown.</span></div>
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<span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">The output signal (voltage), v<sub>o</sub>, is given by</span><span lang="EN-GB" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">: </span></div>
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<span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">v<sub>o</sub> = <i>A</i>(v<sub>+</sub> - v</span><sub><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">-</span></sub><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">)</span></div>
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<span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">v<sub>+</sub> and v</span><sub><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">-</span></sub><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"> are the signals applied to the non-inverting and to the inverting input, respectively. </span><i><span style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">Α</span></i><span style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"> </span><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">represents the <b><i>open loop gain</i></b> of the OA</span><span lang="EN-GB" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">. </span><i><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">A</span></i><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"> is infinite for the ideal amplifier, whereas for the various types of real OAs, it is usually within the range of </span><span lang="EN-GB" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">10</span><sup><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">4</span></sup><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"> to</span><span lang="EN-GB" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"> 10<sup>6</sup></span><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">.</span></div>
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<span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">OAs require two power supplies to operate, supplying a positive voltage (+V) and a negative voltage (-V) with respect to circuit common. This bipolar power supply allows OAs to generate output signals (results) of either polarity. The output signal (v<sub>o</sub>) range is not unlimited. The voltages of the power supplies determine its actual range. Thus, a typical OA fed with -15 and +15 V, may yield a v<sub>o</sub> within the (approximately) -13 to +13 V range, called <b><i>operational range</i></b>. Any result expected to be outside this range is clipped to the respective limit, and OA is in a <b><i>saturation</i></b> stage.</span></div>
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<span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">The connections to the power supplies and to the circuit common symbols, shown in Figure 1, hereafter will be implied, and they will be not shown in the rest of the circuits for simplicity.</span></div>
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<span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">Because of their very high open loop gain, OAs are almost exclusively used with some additional circuitry (mostly with resistors and capacitors), required to ensure a <b><i>negative feedback loop</i></b>. Through this loop a tiny fraction of the output signal is fed back to the inverting input. The negative feedback stabilizes the output within the operational range and provides a much smaller but precisely controlled gain, the so-called <b><i>closed loop gain</i></b>.</span></div>
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<span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">Circuits of OAs have been used in the past as analog computers, and they are still in use for mathematical operations and modification of the input signals in real time. A large variety of OAs is commercially available in the form of low cost integrated circuits.</span></div>
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<span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">There is a plethora of circuits with OAs performing various mathematical operations. Each circuit is characterized by its own <b><i>transfer function</i></b>, i.e. the mathematical equation describing the output signal (v<sub>o</sub>) as a function of the input signal (v<sub>i</sub>) or signals (v<sub>1</sub>, v<sub>2</sub>, …, v<sub>n</sub>). Generally, transfer functions can be derived by applying Kirchhoff’s rules and the following two simplifying assumptions:</span></div>
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<span style="color: #cc3300;"><span style="font-family: "tahoma"; letter-spacing: 0.1pt;"><span lang="EN-GB">#1. </span><span lang="EN-US" style="font-size: 10pt; letter-spacing: 0.1pt;">The output signal (v<sub>o</sub>) acquires a value that (through the feedback circuits) practically equates the voltages applied to both inputs, i.e. v<sub>+</sub> </span><span lang="EN-US" style="letter-spacing: 0.1pt;"><span style="font-size: x-small;">≈</span></span><span lang="EN-US" style="font-size: 10pt; letter-spacing: 0.1pt;"> v</span><sub><span lang="EN-US" style="font-size: 10pt; letter-spacing: 0.1pt;">-</span></sub><span lang="EN-US" style="font-size: 10pt; letter-spacing: 0.1pt;">.</span></span></span></div>
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<span style="color: #cc3300;"><span style="font-family: "tahoma"; letter-spacing: 0.1pt;"><span lang="EN-GB">#</span><span lang="EN-US" style="font-size: 10pt; letter-spacing: 0.1pt;">2. The input resistance of both OA inputs is extremely high</span><span lang="EN-GB" style="font-size: 10pt; letter-spacing: 0.1pt;">(</span><span lang="EN-US" style="font-size: 10pt; letter-spacing: 0.1pt;">usually within the range 10<sup>6</sup>-10<sup>12</sup> M</span><span style="font-size: 10pt; letter-spacing: 0.1pt;">Ω</span><span lang="EN-GB" style="font-size: 10pt; letter-spacing: 0.1pt;">, </span><span lang="EN-US" style="font-size: 10pt; letter-spacing: 0.1pt;">for the ideal OA this is infinite), thus no current flows into them.</span></span></span></div>
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<span style="font-family: "tahoma"; font-size: x-small;"><span lang="EN-US" style="letter-spacing: 0.3pt;"><i><b>Inverting Amplifier</b></i></span></span></div>
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<span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">The basic circuit of the <b><i>inverting amplifier</i></b> is shown in Figure</span><span lang="EN-GB" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"> 2.</span></div>
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<span lang="EN-US" style="color: red; font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"> </span><img alt="Image result for inverting amplifier" class="rg_ic rg_i" data-sz="f" jsaction="load:str.tbn" name="BeaWlqdlIzUWIM:" src="data:image/png;base64,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" style="height: 152px; margin-left: -24px; margin-right: -23px; margin-top: 0px; text-align: left; width: 259px;" /></div>
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<b><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">Figure</span><span lang="EN-GB" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"> 2</span><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">.</span></b><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"> Inverting amplifier.</span></div>
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<span style="font-family: "tahoma"; font-size: x-small;"><span lang="EN-US">The transfer function is derived as follows: Considering the arbitrary current directions we have:</span></span></div>
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<span style="font-family: "tahoma"; font-size: x-small;"><span lang="EN-US"> </span></span><span style="font-family: "tahoma";"><span lang="EN-US" style="letter-spacing: 0.3pt;"><span style="font-size: x-small;">i<sub>1</sub> = (v<sub>i</sub> </span></span></span><span style="font-size: x-small;"><span lang="EN-US" style="font-family: "tahoma"; letter-spacing: 0.3pt;">-</span><span style="font-family: "tahoma";"><span lang="EN-US" style="letter-spacing: 0.3pt;"> v<sub>s</sub>)/R<sub>i</sub> and i<sub>2</sub> = (v<sub>s</sub> </span></span><span lang="EN-US" style="font-family: "tahoma"; letter-spacing: 0.3pt;">-</span></span><span style="font-family: "tahoma";"><span lang="EN-US" style="letter-spacing: 0.3pt;"><span style="font-size: x-small;"> v<sub>o</sub>)/R</span><sub><span style="font-size: x-small;">f</span></sub></span></span></div>
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<span style="font-family: "tahoma"; font-size: x-small;"><span lang="EN-US"> </span></span><span style="font-family: "tahoma";"><span lang="EN-US"><span style="font-size: x-small;">The non-inverting input is connected directly to the circuit common (i.e. v<sub>+</sub> = 0 V), therefore (considering simplifying assumption #1) v<sub>s</sub> = v</span></span></span><span style="font-size: x-small;"><sub><span lang="EN-US" style="font-family: "tahoma";">-</span></sub></span><span style="font-family: "tahoma";"><span lang="EN-US"><span style="font-size: x-small;"> = 0 V, therefore:</span></span></span></div>
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<span style="font-family: "tahoma";"><span lang="EN-US" style="letter-spacing: 0.3pt;"><span style="font-size: x-small;">i<sub>1</sub> = v<sub>i</sub>/R<sub>i</sub> and i<sub>2</sub> = </span></span></span><span style="font-size: x-small;"><span lang="EN-US" style="font-family: "tahoma"; letter-spacing: 0.3pt;">-</span></span><span style="font-family: "tahoma";"><span lang="EN-US" style="letter-spacing: 0.3pt;"><span style="font-size: x-small;"> v<sub>o</sub>/R</span><sub><span style="font-size: x-small;">f</span></sub></span></span></div>
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<span style="font-family: "tahoma"; font-size: x-small;"><span lang="EN-US"> Since there is no current flow to any input (simplifying assumption #2), it is</span></span></div>
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<span style="font-family: "tahoma"; font-size: x-small;"><span lang="EN-US"> i<sub>1</sub> = i<sub>2</sub></span></span></div>
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<span style="font-family: "tahoma"; font-size: x-small;"><span lang="EN-US">Therefore, the transfer function of the inverting amplifier is</span></span></div>
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<span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.3pt;">v<sub>o</sub> = -(R<sub>f</sub>/R<sub>i</sub>)v<sub>i</sub></span></div>
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<span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"> Thus, the closed loop gain of the inverting amplifier is equal to the ratio of R<sub>f</sub> (feedback resistor</span><span lang="EN-GB" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">)</span><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"> over R<sub>i</sub> (input resistor</span><span lang="EN-GB" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">). </span><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">This transfer function describes accurately the output signal as long as the closed loop gain is much smaller than the open loop gain <i>A</i> of the OA used </span><span lang="EN-GB" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">(</span><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">e.g. it must not exceed </span><span lang="EN-GB" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">1000), and the expected values of v<sub>o</sub>are within the operation</span><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">al</span><span lang="EN-GB" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"> range of the OA.</span></div>
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<span style="font-family: "tahoma"; font-size: x-small;"><span lang="EN-GB"> </span></span></h1>
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<span style="font-family: "tahoma"; font-size: x-small;"><span lang="EN-US"><b>Summing Amplifier</b></span></span></h1>
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<span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">The </span><b><i><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">summing amplifier </span></i></b><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"> is a logical extension of the previously described circuit, with two or more inputs. Its circuit is shown in Figure 3.</span></div>
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<span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"> </span><span style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"><img alt="Image result for summing amplifier" class="rg_ic rg_i" data-sz="f" jsaction="load:str.tbn" name="G67q4QemO5LMeM:" 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<b><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">Figure</span><span lang="EN-GB" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"> 3</span><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">.</span></b><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"> Summing amplifier</span><span lang="EN-GB" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">.</span></div>
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<span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">The transfer function of the summing amplifier (similarly derived) is:</span></div>
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<span lang="EN-GB" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"> </span><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.3pt;">v<sub>o</sub> = -(v<sub>1</sub>/R<sub>1</sub> + v<sub>2</sub>/R<sub>2</sub> + … + v<sub>n</sub>/R<sub>n</sub>)R<sub>f</sub></span></div>
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<span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">Thus if all input resistors are equal, the output is a scaled sum of all inputs, whereas, if they are different, the output is a weighted linear sum of all inputs.</span></div>
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<span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">The summing amplifier is used for combining several signals. The most common use of a summing amplifier with two inputs is the amplification of a signal combined with a subtraction of a constant amount from it (dc offset). </span></div>
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<span style="font-family: "tahoma"; font-size: x-small;"><span lang="EN-US"><i><b>Difference amplifier</b></i></span></span></h1>
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<span style="font-family: "tahoma"; font-size: x-small;"><span lang="EN-US">Difference amplifier precisely amplifies the difference of two input signals. Its typical circuit is shown in Figure</span> 4.</span></div>
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<span style="font-family: "tahoma"; font-size: x-small;"><span style="font-style: normal;"> <img alt="Image result for difference amplifier" class="rg_ic rg_i" data-sz="f" jsaction="load:str.tbn" name="gpQJaG9vW5K-AM:" 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<span style="font-family: "tahoma"; font-size: x-small;"><span style="font-style: normal;"> </span></span></h1>
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<b><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt;">Figure</span><span lang="EN-GB" style="font-family: "tahoma"; font-size: 10pt;"> 4</span><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt;">.</span></b><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt;"> Difference amplifier.</span></div>
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<span style="font-family: "tahoma"; font-size: x-small;"><span lang="EN-GB" style="font-style: normal;"> </span></span></h1>
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<span style="font-family: "tahoma"; font-size: x-small;"><span lang="EN-US" style="font-style: normal;">If R<sub>i</sub> = R<sub>i</sub></span><span style="font-style: normal;">΄ </span><span lang="EN-US" style="font-style: normal;">and R<sub>f</sub> = R<sub>f</sub></span><span style="font-style: normal;">΄</span><span lang="EN-US" style="font-style: normal;">, then the transfer function of the difference amplifier is</span><span lang="EN-GB" style="font-style: normal;">:</span></span></h1>
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<span style="font-family: "tahoma";"><span lang="EN-US" style="font-style: normal; letter-spacing: 0.3pt;"><span style="font-size: x-small;">v<sub>o</sub> = (v<sub>2</sub> </span></span></span><span style="font-size: x-small;"><span lang="EN-US" style="font-family: "tahoma"; font-style: normal; letter-spacing: 0.3pt;">-</span></span><span style="font-family: "tahoma";"><span lang="EN-US" style="font-style: normal; letter-spacing: 0.3pt;"><span style="font-size: x-small;"> v<sub>1</sub>) R<sub>f</sub>/R</span><sub><span style="font-size: x-small;">i</span></sub></span></span></h1>
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<span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt;">The difference amplifier is useful for handling signals referring not to the circuit common, but to other signals, known as <b><i><span style="letter-spacing: 0.3pt;">floating signal</span></i></b> sources. Its capability to reject a common signal makes it particularly valuable for amplifying small voltage differences contaminated with the same amount of noise (common signal).</span></div>
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<span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">In order for the difference amplifier to be able to reject a large common signal and to generate at the same time an output precisely proportional to the two signals difference, the two ratios </span><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.3pt;">p = R<sub>f</sub>/R<sub>i</sub> and q = R<sub>f</sub></span><span style="font-family: "tahoma"; font-size: x-small;"><i><span style="letter-spacing: 0.3pt;">΄</span></i></span><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.3pt;">/R<sub>i</sub></span><span style="font-family: "tahoma"; font-size: x-small;"><i><span style="letter-spacing: 0.3pt;">΄</span></i><span style="letter-spacing: 0.1pt;"> </span></span><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">must be precisely equal</span><span lang="EN-GB" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">, otherwise the signal output will be:</span></div>
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<span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.3pt;">v<sub>o</sub> = [q(p+1)/(q+1)]v<sub>2</sub> - pv<sub>1</sub></span></div>
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<span style="font-family: "tahoma"; font-size: x-small;"><span lang="EN-US"><b>Differentiator</b></span></span></h1>
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<span style="font-family: "tahoma"; font-size: x-small;"><span lang="EN-US">The </span><b><span lang="EN-US">differentiator</span></b><span lang="EN-US"> generates an output signal proportional to the first derivative of the input with respect to time. Its typical circuit is shown in Figure</span><span lang="EN-US"> </span><span lang="EN-US">5</span>.</span></div>
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<span style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"> <img alt="Image result for Differentiator circuit" class="rg_ic rg_i" data-sz="f" jsaction="load:str.tbn" name="rGjJMeZHv5w9gM:" src="data:image/png;base64,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" style="font-family: arial; height: 138px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: left; width: 240px;" /></span></div>
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<b><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">Figure</span><span lang="EN-GB" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"> 5</span><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">.</span></b><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"> Differentiator</span><span lang="EN-GB" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">.</span></div>
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<span style="font-family: "tahoma"; font-size: x-small;"><span lang="EN-GB"> </span></span></h1>
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<span style="font-family: "tahoma"; font-size: x-small;"><span lang="EN-GB">The transfer function of this circuit is</span></span></div>
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<span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.3pt;">v<sub>o</sub> = -RC(dv<sub>i</sub>/dt)</span></div>
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<span style="font-family: "tahoma"; font-size: x-small;"><span lang="EN-GB"> </span><span lang="EN-US" style="letter-spacing: 0pt;">Obviously, a constant input (regardless of its magnitude) generates a zero output signal. A typical usage of the differentiator in the field of chemical instrumentation is obtaining the first derivative of a potentiometric titration curve for the easier location of the titration final points (points of maximum slope).</span></span></div>
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<span style="font-family: "tahoma"; font-size: x-small;"><span lang="EN-US"><b>Integrator</b></span></span></h1>
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<span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">The <b><i>integrator</i></b> generates an output signal proportional to the time integral of the input signal.</span><span lang="EN-GB" style="font-family: "tahoma"; font-size: 10pt;"> Its typical circuit is shown in Figure 6</span><span style="font-family: "tahoma"; font-size: 10pt;">.</span></div>
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<img alt="Image result for integrator circuit" class="rg_ic rg_i" data-sz="f" jsaction="load:str.tbn" name="V59p9_Q-Rs2NmM:" src="data:image/png;base64,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" style="height: 160px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: left; width: 314px;" /><b><span style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"></span></b></div>
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<b><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">Figure</span><span lang="EN-GB" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"> 6</span><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">.</span></b><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;"> Integrator</span></div>
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<span style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.3pt;"> </span><span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.3pt;">v<sub>o</sub> = -(1/RC)∫v<sub>i</sub>(t)dt</span></div>
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<span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">The output remains zero as far as switch S remains closed. The integration starts (t = 0) when S opens. The output is proportional to the charge accumulated in capacitor C, which serves as the integrating device. A typical application of the (analog) integrator in chemical instrumentation is the integration of chromatographic peaks, since its output will be proportional to the peak area. </span></div>
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<span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">If the input signal is stable then the output from the integrator will be given by the equation</span></div>
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<span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.3pt;">v<sub>o</sub> = -(v<sub>i</sub>/RC) t</span></div>
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<span lang="EN-US" style="font-family: "tahoma"; font-size: 10pt; letter-spacing: 0.1pt;">i.e. the output signal will be a voltage ramp. Voltage ramps are commonly used for generating the linear potential sweep required in polarography and many other voltammetric techniques.</span></div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3005862392404377111.post-90701753256587250322017-04-20T08:19:00.005-07:002023-06-23T23:34:12.962-07:00Memory Devices<div dir="ltr" style="text-align: left;" trbidi="on">
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A memory is just like a human brain. It is used to store data and instruction. Computer memory is the storage space in computer where data is to be processed and instructions required for processing are stored.</div>
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The memory is divided into large number of small parts. Each part is called a cell. Each location or cell has a unique address which varies from zero to memory size minus one.</div>
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For example if computer has 64k words, then this memory unit has 64 * 1024 = 65536 memory location. The address of these locations varies from 0 to 65535.</div>
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Memory is primarily of two types</div>
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<b style="box-sizing: border-box;">Internal Memory</b> − cache memory and primary/main memory</div>
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<b style="box-sizing: border-box;">External Memory</b> − magnetic disk / optical disk etc.</div>
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<img alt="Memory Hiearchy" src="https://www.tutorialspoint.com/computer_logical_organization/images/memory_hiearchy.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Characteristics of Memory Hierarchy are following when we go from top to bottom.</div>
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<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Capacity in terms of storage increases.</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Cost per bit of storage decreases.</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Frequency of access of the memory by the CPU decreases.</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Access time by the CPU increases.</li>
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RAM</h2>
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A RAM constitutes the internal memory of the CPU for storing data, program and program result. It is read/write memory. It is called random access memory (RAM).</div>
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Since access time in RAM is independent of the address to the word that is, each storage location inside the memory is as easy to reach as other location & takes the same amount of time. We can reach into the memory at random & extremely fast but can also be quite expensive.</div>
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RAM is volatile, i.e. data stored in it is lost when we switch off the computer or if there is a power failure. Hence, a backup uninterruptible power system (UPS) is often used with computers. RAM is small, both in terms of its physical size and in the amount of data it can hold.</div>
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RAM is of two types</div>
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<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Dynamic RAM (DRAM)</li>
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Static RAM (SRAM)</h3>
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The word <b style="box-sizing: border-box;">static</b> indicates that the memory retains its contents as long as power remains applied. However, data is lost when the power gets down due to volatile nature. SRAM chips use a matrix of 6-transistors and no capacitors. Transistors do not require power to prevent leakage, so SRAM need not have to be refreshed on a regular basis.</div>
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Because of the extra space in the matrix, SRAM uses more chips than DRAM for the same amount of storage space, thus making the manufacturing costs higher.</div>
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Static RAM is used as cache memory needs to be very fast and small.</div>
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Dynamic RAM (DRAM)</h3>
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DRAM, unlike SRAM, must be continually <b style="box-sizing: border-box;">refreshed</b> in order for it to maintain the data. This is done by placing the memory on a refresh circuit that rewrites the data several hundred times per second. DRAM is used for most system memory because it is cheap and small. All DRAMs are made up of memory cells. These cells are composed of one capacitor and one transistor.</div>
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ROM</h2>
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ROM stands for Read Only Memory. The memory from which we can only read but cannot write on it. This type of memory is non-volatile. The information is stored permanently in such memories during manufacture.</div>
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A ROM, stores such instruction as are required to start computer when electricity is first turned on, this operation is referred to as bootstrap. ROM chip are not only used in the computer but also in other electronic items like washing machine and microwave oven.</div>
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Following are the various types of ROM −</div>
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MROM (Masked ROM)</h3>
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The very first ROMs were hard-wired devices that contained a pre-programmed set of data or instructions. These kind of ROMs are known as masked ROMs. It is inexpensive ROM.</div>
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PROM (Programmable Read Only Memory)</h3>
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PROM is read-only memory that can be modified only once by a user. The user buys a blank PROM and enters the desired contents using a PROM programmer. Inside the PROM chip there are small fuses which are burnt open during programming. It can be programmed only once and is not erasable.</div>
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EPROM (Erasable and Programmable Read Only Memory)</h3>
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The EPROM can be erased by exposing it to ultra-violet light for a duration of upto 40 minutes. Usually, an EPROM eraser achieves this function. During programming an electrical charge is trapped in an insulated gate region. The charge is retained for more than ten years because the charge has no leakage path. For erasing this charge, ultra-violet light is passed through a quartz crystal window (lid). This exposure to ultra-violet light dissipates the charge. During normal use the quartz lid is sealed with a sticker.</div>
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EEPROM (Electrically Erasable and Programmable Read Only Memory)</h3>
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The EEPROM is programmed and erased electrically. It can be erased and reprogrammed about ten thousand times. Both erasing and programming take about 4 to 10 ms (millisecond). In EEPROM, any location can be selectively erased and programmed. EEPROMs can be erased one byte at a time, rather than erasing the entire chip. Hence, the process of re-programming is flexible but slow.</div>
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Serial Access Memory</h2>
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Sequential access means the system must search the storage device from the beginning of the memory address until it finds the required piece of data. Memory device which supports such access is called a Sequential Access Memory or Serial Access Memory. Magnetic tape is an example of serial access memory.</div>
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Direct Access Memory</h2>
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Direct access memory or Random Access Memory, refers to conditions in which a system can go directly to the information that the user wants. Memory device which supports such access is called a Direct Access Memory. Magnetic disks, optical disks are examples of direct access memory.</div>
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Cache Memory</h2>
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Cache memory is a very high speed semiconductor memory which can speed up CPU. It acts as a buffer between the CPU and main memory. It is used to hold those parts of data and program which are most frequently used by CPU. The parts of data and programs, are transferred from disk to cache memory by operating system, from where CPU can access them.</div>
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Advantages</h3>
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<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Cache memory is faster than main memory.</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">It consumes less access time as compared to main memory.</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">It stores the program that can be executed within a short period of time.</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">It stores data for temporary use.</li>
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Disadvantages</h3>
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<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Cache memory has limited capacity.</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">It is very expensive.</li>
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Virtual memory is a technique that allows the execution of processes which are not completely available in memory. The main visible advantage of this scheme is that programs can be larger than physical memory. Virtual memory is the separation of user logical memory from physical memory.</div>
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This separation allows an extremely large virtual memory to be provided for programmers when only a smaller physical memory is available. Following are the situations, when entire program is not required to be loaded fully in main memory.</div>
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User written error handling routines are used only when an error occurred in the data or computation.</div>
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Certain options and features of a program may be used rarely.</div>
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Many tables are assigned a fixed amount of address space even though only a small amount of the table is actually used.</div>
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The ability to execute a program that is only partially in memory would counter many benefits.</div>
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Less number of I/O would be needed to load or swap each user program into memory.</div>
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A program would no longer be constrained by the amount of physical memory that is available.</div>
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Each user program could take less physical memory, more programs could be run the same time, with a corresponding increase in CPU utilization and throughput.</div>
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Auxiliary Memory</h2>
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Auxiliary memory is much larger in size than main memory but is slower. It normally stores system programs, instruction and data files. It is also known as secondary memory. It can also be used as an overflow/virtual memory in case the main memory capacity has been exceeded. Secondary memories cannot be accessed directly by a processor. First the data/information of auxiliary memory is transferred to the main memory and then that information can be accessed by the CPU. Characteristics of Auxiliary Memory are following −</div>
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<b style="box-sizing: border-box;">Non-volatile memory</b> − Data is not lost when power is cut off.</div>
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<b style="box-sizing: border-box;">Reusable</b> − The data stays in the secondary storage on permanent basis until it is not overwritten or deleted by the user.</div>
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<b style="box-sizing: border-box;">Reliable</b> − Data in secondary storage is safe because of high physical stability of secondary storage device.</div>
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<b style="box-sizing: border-box;">Convenience</b> − With the help of a computer software, authorised people can locate and access the data quickly.</div>
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<b style="box-sizing: border-box;">Capacity</b> − Secondary storage can store large volumes of data in sets of multiple disks.</div>
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<b style="box-sizing: border-box;">Cost</b> − It is much lesser expensive to store data on a tape or disk than primary memory.</div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3005862392404377111.post-38926035713270144092017-04-20T08:19:00.004-07:002023-06-23T23:34:11.729-07:00CPU Architecture<div dir="ltr" style="text-align: left;" trbidi="on">
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Microprocessing unit is synonymous to central processing unit, CPU used in traditional computer. Microprocessor (MPU) acts as a device or a group of devices which do the following tasks.</div>
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<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">communicate with peripherals devices</li>
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<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">direct data flow</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">perform computer tasks as specified by the instructions in memory</li>
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8085 Microprocessor</h2>
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The 8085 microprocessor is an 8-bit general purpose microprocessor which is capable to address 64k of memory. This processor has forty pins, requires +5 V single power supply and a 3-MHz single-phase clock.</div>
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Block Diagram</h3>
<img alt="8080 Mircroprocessor block diagram" src="https://www.tutorialspoint.com/computer_logical_organization/images/microprocessor_blockdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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ALU</h2>
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The ALU perform the computing function of microprocessor. It includes the accumulator, temporary register, arithmetic & logic circuit & and five flags. Result is stored in accumulator & flags.</div>
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Block Diagram</h3>
<img alt="ALU" src="https://www.tutorialspoint.com/computer_logical_organization/images/alu.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Accumulator</h2>
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It is an 8-bit register that is part of ALU. This register is used to store 8-bit data & in performing arithmetic & logic operation. The result of operation is stored in accumulator.</div>
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Diagram</h3>
<img alt="Accumulator" src="https://www.tutorialspoint.com/computer_logical_organization/images/accumulator.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Flags</h2>
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Flags are programmable. They can be used to store and transfer the data from the registers by using instruction. The ALU includes five flip-flops that are set and reset according to data condition in accumulator and other registers.</div>
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<b style="box-sizing: border-box;">S (Sign) flag</b> − After the execution of an arithmetic operation, if bit D<sub style="box-sizing: border-box;">7</sub>of the result is 1, the sign flag is set. It is used to signed number. In a given byte, if D<sub style="box-sizing: border-box;">7</sub> is 1 means negative number. If it is zero means it is a positive number.</div>
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<b style="box-sizing: border-box;">Z (Zero) flag</b> − The zero flag is set if ALU operation result is 0.</div>
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<b style="box-sizing: border-box;">AC (Auxiliary Carry) flag</b> − In arithmetic operation, when carry is generated by digit D3 and passed on to digit D<sub style="box-sizing: border-box;">4</sub>, the AC flag is set. This flag is used only internally BCD operation.</div>
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<b style="box-sizing: border-box;">P (Parity) flag</b> − After arithmetic or logic operation, if result has even number of 1s, the flag is set. If it has odd number of 1s, flag is reset.</div>
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<b style="box-sizing: border-box;">C (Carry) flag</b> − If arithmetic operation result is in a carry, the carry flag is set, otherwise it is reset.</div>
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Register section</h2>
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It is basically a storage device and transfers data from registers by using instructions.</div>
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<b style="box-sizing: border-box;">Stack Pointer (SP)</b> − The stack pointer is also a 16-bit register which is used as a memory pointer. It points to a memory location in Read/Write memory known as stack. In between execution of program, sometime data to be stored in stack. The beginning of the stack is defined by loading a 16-bit address in the stack pointer.</div>
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<b style="box-sizing: border-box;">Program Counter (PC)</b> − This 16-bit register deals with fourth operation to sequence the execution of instruction. This register is also a memory pointer. Memory location have 16-bit address. It is used to store the execution address. The function of the program counter is to point to memory address from which next byte is to be fetched.</div>
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<b style="box-sizing: border-box;">Storage registers</b> − These registers store 8-bit data during a program execution. These registers are identified as B, C, D, E, H, L. They can be combined as register pair BC, DE and HL to perform some 16 bit operations.</div>
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Time and Control Section</h2>
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This unit is responsible to synchronize Microprocessor operation as per the clock pulse and to generate the control signals which are necessary for smooth communication between Microprocessor and peripherals devices. The RD bar and WR bar signals are synchronous pulses which indicates whether data is available on the data bus or not. The control unit is responsible to control the flow of data between microprocessor, memory and peripheral devices.</div>
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PIN diagram</h3>
<img alt="PIN diagram" src="https://www.tutorialspoint.com/computer_logical_organization/images/pin_diagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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All the signal can be classified into six groups</div>
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The 8085 microprocessor has 8 signal line, A<sub style="box-sizing: border-box;">15</sub> - A<sub style="box-sizing: border-box;">8</sub> which are uni directional and used as a high order address bus.</div>
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The signal line AD7 - AD0 are bi-directional for dual purpose. They are used as low order address bus as well as data bus.</div>
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Control Signal</div>
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<b style="box-sizing: border-box;">RD bar</b> − It is a read control signal (active low). If it is active then memory read the data.</div>
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<b style="box-sizing: border-box;">WR bar</b> − It is write control signal (active low). It is active when written into selected memory.</div>
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Status signal</div>
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<b style="box-sizing: border-box;">ALU (Address Latch Enable)</b> − When ALU is high. 8085 microprocessor use address bus. When ALU is low. 8085 microprocessor is use data bus.</div>
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<b style="box-sizing: border-box;">IO/M bar</b> − This is a status signal used to differentiate between i/o and memory operations. When it is high, it indicate an i/o operation and when it is low, it indicate memory operation.</div>
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<b style="box-sizing: border-box;">S<sub style="box-sizing: border-box;">1</sub> and S<sub style="box-sizing: border-box;">0</sub></b> − These status signals, similar to i/o and memory bar, can identify various operations, but they are rarely used in small system.</div>
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<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: middle;">4</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: middle;"><b style="box-sizing: border-box;">Power supply and frequency signal</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><div style="box-sizing: border-box; color: black; font-size: 15px !important; line-height: 24px; margin: 0em 0.2em 1em; padding: 0px; text-align: justify; word-wrap: break-word;">
<b style="box-sizing: border-box;">V<sub style="box-sizing: border-box;">cc</sub></b> − +5v power supply.</div>
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<b style="box-sizing: border-box;">V<sub style="box-sizing: border-box;">ss</sub></b> − ground reference.</div>
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<b style="box-sizing: border-box;">X, X</b> − A crystal is connected at these two pins. The frequency is internally divided by two operate system at 3-MHz, the crystal should have a frequency of 6-MHz.</div>
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<b style="box-sizing: border-box;">CLK out</b> − This signal can be used as the system clock for other devices.</div>
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<b style="box-sizing: border-box;">INTR (i/p)</b> − Interrupt request.</div>
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<b style="box-sizing: border-box;">INTA bar (o/p)</b> − It is used as acknowledge interrupt.</div>
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<b style="box-sizing: border-box;">TRAP (i/p)</b> − This is non maskable interrupt and has highest priority.</div>
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<b style="box-sizing: border-box;">HOLD (i/p)</b> − It is used to hold the executing program.</div>
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<b style="box-sizing: border-box;">HLDA (o/p)</b> − Hold acknowledge.</div>
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<b style="box-sizing: border-box;">READY (i/p)</b> − This signal is used to delay the microprocessor read or write cycle until a slow responding peripheral is ready to accept or send data.</div>
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<b style="box-sizing: border-box;">RESET IN bar</b> − When the signal on this pin goes low, the program counter is set to zero, the bus are tri-stated, & MPU is reset.</div>
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<b style="box-sizing: border-box;">RESET OUT</b> − This signal indicate that MPU is being reset. The signal can be used to reset other devices.</div>
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<b style="box-sizing: border-box;">RST 7.5, RST 6.5, RST 5.5 (Request interrupt)</b> − It is used to transfer the program control to specific memory location. They have higher priority than INTR interrupt.</div>
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The 8085 microprocessor has two signals to implement the serial transmission serial input data and serial output data.</div>
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Instruction Format</h2>
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Each instruction is represented by a sequence of bits within the computer. The instruction is divided into group of bits called field. The way instruction is expressed is known as instruction format. It is usually represented in the form of rectangular box. The instruction format may be of the following types.</div>
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Variable Instruction Formats</h2>
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These are the instruction formats in which the instruction length varies on the basis of opcode & address specifiers. For Example, VAX instruction vary between 1 and 53 bytes while X86 instruction vary between 1 and 17 bytes.</div>
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Format</h3>
<img alt="Variable Instruction Format" src="https://www.tutorialspoint.com/computer_logical_organization/images/variable_instruction_format.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Advantage</h3>
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These formats have good code density.</div>
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Drawback</h3>
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These instruction formats are very difficult to decode and pipeline.</div>
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Fixed Instruction Formats</h2>
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In this type of instruction format, all instructions are of same size. For Example, MIPS, Power PC, Alpha, ARM.</div>
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Format</h3>
<img alt="Fixed Instruction Format" src="https://www.tutorialspoint.com/computer_logical_organization/images/fixed_instruction_format.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Advantage</h3>
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They are easy to decode & pipeline.</div>
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Drawback</h3>
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They don't have good code density.</div>
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Hybrid Instruction Formats</h2>
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In this type of instruction formats, we have multiple format length specified by opcode. For example, IBM 360/70, MIPS 16, Thumb.</div>
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Format</h3>
<img alt="Hybrid Instruction Format" src="https://www.tutorialspoint.com/computer_logical_organization/images/hybrid_instruction_format.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Advantage</h3>
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These compromise between code density & instruction of these type are very easy to decode.</div>
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Addressing Modes</h2>
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Addressing mode provides different ways for accessing an address to given data to a processor. Operated data is stored in the memory location, each instruction required certain data on which it has to operate. There are various techniques to specify address of data. These techniques are called Addressing Modes.</div>
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<b style="box-sizing: border-box;">Direct addressing mode</b> − In the direct addressing mode, address of the operand is given in the instruction and data is available in the memory location which is provided in instruction. We will move this data in desired location.</div>
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<b style="box-sizing: border-box;">Indirect addressing mode</b> − In the indirect addressing mode, the instruction specifies a register which contain the address of the operand. Both internal RAM and external RAM can be accessed via indirect addressing mode.</div>
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<b style="box-sizing: border-box;">Immediate addressing mode</b> − In the immediate addressing mode, direct data is given in the operand which move the data in accumulator. It is very fast.</div>
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<b style="box-sizing: border-box;">Relative addressing mode</b> − In the relative address mode, the effective address is determined by the index mode by using the program counter in stead of general purpose processor register. This mode is called relative address mode.</div>
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<b style="box-sizing: border-box;">Index addressing mode</b> − In the index address mode, the effective address of the operand is generated by adding a content value to the contents of the register. This mode is called index address mode.</div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3005862392404377111.post-51580701399000872682017-04-20T08:18:00.002-07:002023-06-23T23:34:11.920-07:00Digital Counters<div dir="ltr" style="text-align: left;" trbidi="on">
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Counter is a sequential circuit. A digital circuit which is used for a counting pulses is known counter. Counter is the widest application of flip-flops. It is a group of flip-flops with a clock signal applied. Counters are of two types.</div>
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<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Asynchronous or ripple counters.</li>
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Asynchronous or ripple counters</h2>
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The logic diagram of a 2-bit ripple up counter is shown in figure. The toggle (T) flip-flop are being used. But we can use the JK flip-flop also with J and K connected permanently to logic 1. External clock is applied to the clock input of flip-flop A and Q<sub style="box-sizing: border-box;">A</sub> output is applied to the clock input of the next flip-flop i.e. FF-B.</div>
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Logical Diagram</h3>
<img alt="Logic Diagram of Asynchronous or ripple counters" src="https://www.tutorialspoint.com/computer_logical_organization/images/ripple_counter_diagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Operation</h3>
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<tr style="box-sizing: border-box;"><th style="background: rgb(238, 238, 238); border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; text-align: left; vertical-align: top;">S.N.</th><th style="background: rgb(238, 238, 238); border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; text-align: left; vertical-align: top; width: 301px;">Condition</th><th style="background: rgb(238, 238, 238); border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; text-align: left; vertical-align: top;">Operation</th></tr>
<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">1</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><b style="box-sizing: border-box;">Initially let both the FFs be in the reset state</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">Q<sub style="box-sizing: border-box;">B</sub>Q<sub style="box-sizing: border-box;">A</sub> = 00 initially</td></tr>
<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">2</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><b style="box-sizing: border-box;">After 1st negative clock edge</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><div style="box-sizing: border-box; color: black; font-size: 15px !important; line-height: 24px; margin: 0em 0.2em 1em; padding: 0px; text-align: justify; word-wrap: break-word;">
As soon as the first negative clock edge is applied, FF-A will toggle and Q<sub style="box-sizing: border-box;">A</sub> will be equal to 1.</div>
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Q<sub style="box-sizing: border-box;">A</sub> is connected to clock input of FF-B. Since Q<sub style="box-sizing: border-box;">A</sub> has changed from 0 to 1, it is treated as the positive clock edge by FF-B. There is no change in Q<sub style="box-sizing: border-box;">B</sub>because FF-B is a negative edge triggered FF.</div>
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Q<sub style="box-sizing: border-box;">B</sub>Q<sub style="box-sizing: border-box;">A</sub> = 01 after the first clock pulse.</div>
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<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">3</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><b style="box-sizing: border-box;">After 2nd negative clock edge</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><div style="box-sizing: border-box; color: black; font-size: 15px !important; line-height: 24px; margin: 0em 0.2em 1em; padding: 0px; text-align: justify; word-wrap: break-word;">
On the arrival of second negative clock edge, FF-A toggles again and Q<sub style="box-sizing: border-box;">A</sub> = 0.</div>
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The change in Q<sub style="box-sizing: border-box;">A</sub> acts as a negative clock edge for FF-B. So it will also toggle, and Q<sub style="box-sizing: border-box;">B</sub>will be 1.</div>
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Q<sub style="box-sizing: border-box;">B</sub>Q<sub style="box-sizing: border-box;">A</sub> = 10 after the second clock pulse.</div>
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<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">4</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><b style="box-sizing: border-box;">After 3rd negative clock edge</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><div style="box-sizing: border-box; color: black; font-size: 15px !important; line-height: 24px; margin: 0em 0.2em 1em; padding: 0px; text-align: justify; word-wrap: break-word;">
On the arrival of 3rd negative clock edge, FF-A toggles again and Q<sub style="box-sizing: border-box;">A</sub> become 1 from 0.</div>
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Since this is a positive going change, FF-B does not respond to it and remains inactive. So Q<sub style="box-sizing: border-box;">B</sub> does not change and continues to be equal to 1.</div>
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Q<sub style="box-sizing: border-box;">B</sub>Q<sub style="box-sizing: border-box;">A</sub> = 11 after the third clock pulse.</div>
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<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">5</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><b style="box-sizing: border-box;">After 4th negative clock edge</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><div style="box-sizing: border-box; color: black; font-size: 15px !important; line-height: 24px; margin: 0em 0.2em 1em; padding: 0px; text-align: justify; word-wrap: break-word;">
On the arrival of 4th negative clock edge, FF-A toggles again and Q<sub style="box-sizing: border-box;">A</sub> becomes 1 from 0.</div>
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This negative change in Q<sub style="box-sizing: border-box;">A</sub>acts as clock pulse for FF-B. Hence it toggles to change Q<sub style="box-sizing: border-box;">B</sub>from 1 to 0.</div>
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Q<sub style="box-sizing: border-box;">B</sub>Q<sub style="box-sizing: border-box;">A</sub> = 00 after the fourth clock pulse.</div>
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Truth Table</h3>
<img alt="Truth Table of Asynchronous or ripple counters" src="https://www.tutorialspoint.com/computer_logical_organization/images/ripple_counter_truthtable.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Synchronous counters</h2>
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If the "clock" pulses are applied to all the flip-flops in a counter simultaneously, then such a counter is called as synchronous counter.</div>
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2-bit Synchronous up counter</h3>
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The J<sub style="box-sizing: border-box;">A</sub> and K<sub style="box-sizing: border-box;">A</sub> inputs of FF-A are tied to logic 1. So FF-A will work as a toggle flip-flop. The J<sub style="box-sizing: border-box;">B</sub> and K<sub style="box-sizing: border-box;">B</sub> inputs are connected to Q<sub style="box-sizing: border-box;">A</sub>.</div>
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Logical Diagram</h3>
<img alt="Logic Diagram of Synchronous counter" src="https://www.tutorialspoint.com/computer_logical_organization/images/synchronous_counter_diagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Operation</h3>
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<tr style="box-sizing: border-box;"><th style="background: rgb(238, 238, 238); border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; text-align: left; vertical-align: top;">S.N.</th><th style="background: rgb(238, 238, 238); border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; text-align: left; vertical-align: top; width: 301px;">Condition</th><th style="background: rgb(238, 238, 238); border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; text-align: left; vertical-align: top;">Operation</th></tr>
<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">1</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><b style="box-sizing: border-box;">Initially let both the FFs be in the reset state</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">Q<sub style="box-sizing: border-box;">B</sub>Q<sub style="box-sizing: border-box;">A</sub> = 00 initially.</td></tr>
<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">2</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><b style="box-sizing: border-box;">After 1st negative clock edge</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><div style="box-sizing: border-box; color: black; font-size: 15px !important; line-height: 24px; margin: 0em 0.2em 1em; padding: 0px; text-align: justify; word-wrap: break-word;">
As soon as the first negative clock edge is applied, FF-A will toggle and Q<sub style="box-sizing: border-box;">A</sub> will change from 0 to 1.</div>
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But at the instant of application of negative clock edge, Q<sub style="box-sizing: border-box;">A</sub> , J<sub style="box-sizing: border-box;">B</sub> = K<sub style="box-sizing: border-box;">B</sub> = 0. Hence FF-B will not change its state. So Q<sub style="box-sizing: border-box;">B</sub> will remain 0.</div>
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Q<sub style="box-sizing: border-box;">B</sub>Q<sub style="box-sizing: border-box;">A</sub> = 01 after the first clock pulse.</div>
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<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">3</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><b style="box-sizing: border-box;">After 2nd negative clock edge</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><div style="box-sizing: border-box; color: black; font-size: 15px !important; line-height: 24px; margin: 0em 0.2em 1em; padding: 0px; text-align: justify; word-wrap: break-word;">
On the arrival of second negative clock edge, FF-A toggles again and Q<sub style="box-sizing: border-box;">A</sub> changes from 1 to 0.</div>
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But at this instant Q<sub style="box-sizing: border-box;">A</sub> was 1. So J<sub style="box-sizing: border-box;">B</sub> = K<sub style="box-sizing: border-box;">B</sub>= 1 and FF-B will toggle. Hence Q<sub style="box-sizing: border-box;">B</sub> changes from 0 to 1.</div>
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Q<sub style="box-sizing: border-box;">B</sub>Q<sub style="box-sizing: border-box;">A</sub> = 10 after the second clock pulse.</div>
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<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">4</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><b style="box-sizing: border-box;">After 3rd negative clock edge</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><div style="box-sizing: border-box; color: black; font-size: 15px !important; line-height: 24px; margin: 0em 0.2em 1em; padding: 0px; text-align: justify; word-wrap: break-word;">
On application of the third falling clock edge, FF-A will toggle from 0 to 1 but there is no change of state for FF-B.</div>
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Q<sub style="box-sizing: border-box;">B</sub>Q<sub style="box-sizing: border-box;">A</sub> = 11 after the third clock pulse.</div>
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<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">5</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><b style="box-sizing: border-box;">After 4th negative clock edge</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><div style="box-sizing: border-box; color: black; font-size: 15px !important; line-height: 24px; margin: 0em 0.2em 1em; padding: 0px; text-align: justify; word-wrap: break-word;">
On application of the next clock pulse, Q<sub style="box-sizing: border-box;">A</sub> will change from 1 to 0 as Q<sub style="box-sizing: border-box;">B</sub> will also change from 1 to 0.</div>
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Q<sub style="box-sizing: border-box;">B</sub>Q<sub style="box-sizing: border-box;">A</sub> = 00 after the fourth clock pulse.</div>
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Classification of counters</h2>
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Depending on the way in which the counting progresses, the synchronous or asynchronous counters are classified as follows −</div>
<ul class="list" style="box-sizing: border-box; color: #313131; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px;">
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Up counters</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Down counters</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Up/Down counters</li>
</ul>
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UP/DOWN Counter</h2>
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Up counter and down counter is combined together to obtain an UP/DOWN counter. A mode control (M) input is also provided to select either up or down mode. A combinational circuit is required to be designed and used between each pair of flip-flop in order to achieve the up/down operation.</div>
<ul class="list" style="box-sizing: border-box; color: #313131; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px;">
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Type of up/down counters</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">UP/DOWN ripple counters</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">UP/DOWN synchronous counter</li>
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UP/DOWN Ripple Counters</h2>
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In the UP/DOWN ripple counter all the FFs operate in the toggle mode. So either T flip-flops or JK flip-flops are to be used. The LSB flip-flop receives clock directly. But the clock to every other FF is obtained from (Q = Q bar) output of the previous FF.</div>
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<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;"><div style="box-sizing: border-box; font-size: 15px !important; line-height: 24px; margin: 0em 0.2em 1em; padding: 0px; text-align: justify; word-wrap: break-word;">
<b style="box-sizing: border-box;">UP counting mode (M=0)</b> − The Q output of the preceding FF is connected to the clock of the next stage if up counting is to be achieved. For this mode, the mode select input M is at logic 0 (M=0).</div>
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<b style="box-sizing: border-box;">DOWN counting mode (M=1)</b> − If M = 1, then the Q bar output of the preceding FF is connected to the next FF. This will operate the counter in the counting mode.</div>
</li>
</ul>
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Example</h3>
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3-bit binary up/down ripple counter.</div>
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3-bit − hence three FFs are required.</div>
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UP/DOWN − So a mode control input is essential.</div>
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For a ripple up counter, the Q output of preceding FF is connected to the clock input of the next one.</div>
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For a ripple up counter, the Q output of preceding FF is connected to the clock input of the next one.</div>
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For a ripple down counter, the Q bar output of preceding FF is connected to the clock input of the next one.</div>
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Let the selection of Q and Q bar output of the preceding FF be controlled by the mode control input M such that, If M = 0, UP counting. So connect Q to CLK. If M = 1, DOWN counting. So connect Q bar to CLK.</div>
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Block Diagram</h3>
<img alt="Block Diagram of Up/Down counters" src="https://www.tutorialspoint.com/computer_logical_organization/images/updown_counter_diagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Truth Table</h3>
<img alt="Truth Table of Up/Down counters" src="https://www.tutorialspoint.com/computer_logical_organization/images/updown_counter_truthtable.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Operation</h3>
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<tr style="box-sizing: border-box;"><th style="background: rgb(238, 238, 238); border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; text-align: left; vertical-align: top;">S.N.</th><th style="background: rgb(238, 238, 238); border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; text-align: left; vertical-align: top; width: 301px;">Condition</th><th style="background: rgb(238, 238, 238); border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; text-align: left; vertical-align: top;">Operation</th></tr>
<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">1</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><b style="box-sizing: border-box;">Case 1 − With M = 0 (Up counting mode)</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><div style="box-sizing: border-box; color: black; font-size: 15px !important; line-height: 24px; margin: 0em 0.2em 1em; padding: 0px; text-align: justify; word-wrap: break-word;">
If M = 0 and M bar = 1, then the AND gates 1 and 3 in fig. will be enabled whereas the AND gates 2 and 4 will be disabled.</div>
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Hence Q<sub style="box-sizing: border-box;">A</sub> gets connected to the clock input of FF-B and Q<sub style="box-sizing: border-box;">B</sub>gets connected to the clock input of FF-C.</div>
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These connections are same as those for the normal up counter. Thus with M = 0 the circuit work as an up counter.</div>
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<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">2</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><b style="box-sizing: border-box;">Case 2: With M = 1 (Down counting mode)</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><div style="box-sizing: border-box; color: black; font-size: 15px !important; line-height: 24px; margin: 0em 0.2em 1em; padding: 0px; text-align: justify; word-wrap: break-word;">
If M = 1, then AND gates 2 and 4 in fig. are enabled whereas the AND gates 1 and 3 are disabled.</div>
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Hence Q<sub style="box-sizing: border-box;">A</sub> bar gets connected to the clock input of FF-B and Q<sub style="box-sizing: border-box;">B</sub> bar gets connected to the clock input of FF-C.</div>
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These connections will produce a down counter. Thus with M = 1 the circuit works as a down counter.</div>
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Modulus Counter (MOD-N Counter)</h2>
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The 2-bit ripple counter is called as MOD-4 counter and 3-bit ripple counter is called as MOD-8 counter. So in general, an n-bit ripple counter is called as modulo-N counter. Where, MOD number = 2<sup style="box-sizing: border-box;">n</sup>.</div>
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Type of modulus</h3>
<ul class="list" style="box-sizing: border-box; color: #313131; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px;">
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">2-bit up or down (MOD-4)</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">3-bit up or down (MOD-8)</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">4-bit up or down (MOD-16)</li>
</ul>
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Application of counters</h2>
<ul class="list" style="box-sizing: border-box; color: #313131; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px;">
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Frequency counters</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Digital clock</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Time measurement</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">A to D converter</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Frequency divider circuits</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Digital triangular wave generator.</li>
</ul>
</div>
<head>
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</head>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3005862392404377111.post-70792367982070767982017-04-20T08:17:00.004-07:002023-06-23T23:34:12.212-07:00Sequential Circuits<div dir="ltr" style="text-align: left;" trbidi="on">
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The combinational circuit does not use any memory. Hence the previous state of input does not have any effect on the present state of the circuit. But sequential circuit has memory so output can vary based on input. This type of circuits uses previous input, output, clock and a memory element.</div>
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Block diagram</h2>
<img alt="Block Diagram of sequential circuit" src="https://www.tutorialspoint.com/computer_logical_organization/images/sequential_circuit_blockdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Flip Flop</h2>
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Flip flop is a sequential circuit which generally samples its inputs and changes its outputs only at particular instants of time and not continuously. Flip flop is said to be edge sensitive or edge triggered rather than being level triggered like latches.</div>
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S-R Flip Flop</h2>
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It is basically S-R latch using NAND gates with an additional <b style="box-sizing: border-box;">enable</b> input. It is also called as level triggered SR-FF. For this, circuit in output will take place if and only if the enable input (E) is made active. In short this circuit will operate as an S-R latch if E = 1 but there is no change in the output if E = 0.</div>
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Block Diagram</h3>
<img alt="Block Diagram of SR Flip Flop" src="https://www.tutorialspoint.com/computer_logical_organization/images/srflipflop_blockdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Circuit Diagram</h3>
<img alt="Circuit Diagram of SR Flip Flop" src="https://www.tutorialspoint.com/computer_logical_organization/images/srflipflop_circuitdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Truth Table</h3>
<img alt="Truth Table of SR Flip Flop" src="https://www.tutorialspoint.com/computer_logical_organization/images/srflipflop_truthtable.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Operation</h3>
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<tr style="box-sizing: border-box;"><th style="background: rgb(238, 238, 238); border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; text-align: left; vertical-align: top; width: 49px;">S.N.</th><th style="background: rgb(238, 238, 238); border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; text-align: left; vertical-align: top; width: 168px;">Condition</th><th style="background: rgb(238, 238, 238); border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; text-align: left; vertical-align: top;">Operation</th></tr>
<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">1</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><b style="box-sizing: border-box;">S = R = 0 : No change</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><div style="box-sizing: border-box; color: black; font-size: 15px !important; line-height: 24px; margin: 0em 0.2em 1em; padding: 0px; text-align: justify; word-wrap: break-word;">
If S = R = 0 then output of NAND gates 3 and 4 are forced to become 1.</div>
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Hence R' and S' both will be equal to 1. Since S' and R' are the input of the basic S-R latch using NAND gates, there will be no change in the state of outputs.</div>
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<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">2</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><b style="box-sizing: border-box;">S = 0, R = 1, E = 1</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><div style="box-sizing: border-box; color: black; font-size: 15px !important; line-height: 24px; margin: 0em 0.2em 1em; padding: 0px; text-align: justify; word-wrap: break-word;">
Since S = 0, output of NAND-3 i.e. R' = 1 and E = 1 the output of NAND-4 i.e. S' = 0.</div>
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Hence Q<sub style="box-sizing: border-box;">n+1</sub> = 0 and Q<sub style="box-sizing: border-box;">n+1</sub> bar = 1. This is reset condition.</div>
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<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">3</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><b style="box-sizing: border-box;">S = 1, R = 0, E = 1</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><div style="box-sizing: border-box; color: black; font-size: 15px !important; line-height: 24px; margin: 0em 0.2em 1em; padding: 0px; text-align: justify; word-wrap: break-word;">
Output of NAND-3 i.e. R' = 0 and output of NAND-4 i.e. S' = 1.</div>
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Hence output of S-R NAND latch is Q<sub style="box-sizing: border-box;">n+1</sub> = 1 and Q<sub style="box-sizing: border-box;">n+1</sub> bar = 0. This is the reset condition.</div>
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<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">4</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><b style="box-sizing: border-box;">S = 1, R = 1, E = 1</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><div style="box-sizing: border-box; color: black; font-size: 15px !important; line-height: 24px; margin: 0em 0.2em 1em; padding: 0px; text-align: justify; word-wrap: break-word;">
As S = 1, R = 1 and E = 1, the output of NAND gates 3 and 4 both are 0 i.e. S' = R' = 0.</div>
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Hence the <b style="box-sizing: border-box;">Race</b> condition will occur in the basic NAND latch.</div>
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Master Slave JK Flip Flop</h2>
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Master slave JK FF is a cascade of two S-R FF with feedback from the output of second to input of first. Master is a positive level triggered. But due to the presence of the inverter in the clock line, the slave will respond to the negative level. Hence when the clock = 1 (positive level) the master is active and the slave is inactive. Whereas when clock = 0 (low level) the slave is active and master is inactive.</div>
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Circuit Diagram</h3>
<img alt="Circuit Diagram of J-K Flip Flop" src="https://www.tutorialspoint.com/computer_logical_organization/images/jkflipflop_circuitdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Truth Table</h3>
<img alt="Truth Table of J-K Flip Flop" src="https://www.tutorialspoint.com/computer_logical_organization/images/jkflipflop_truthtable.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Operation</h3>
<table class="table table-bordered" style="border-collapse: collapse; border-spacing: 0px; border: 1px solid rgb(221, 221, 221); box-sizing: border-box; color: #313131; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-bottom: 20px; max-width: 100%; width: 604px;"><tbody style="box-sizing: border-box;">
<tr style="box-sizing: border-box;"><th style="background: rgb(238, 238, 238); border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; text-align: left; vertical-align: top; width: 49px;">S.N.</th><th style="background: rgb(238, 238, 238); border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; text-align: left; vertical-align: top; width: 168px;">Condition</th><th style="background: rgb(238, 238, 238); border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; text-align: left; vertical-align: top;">Operation</th></tr>
<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">1</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><b style="box-sizing: border-box;">J = K = 0 (No change)</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><div style="box-sizing: border-box; color: black; font-size: 15px !important; line-height: 24px; margin: 0em 0.2em 1em; padding: 0px; text-align: justify; word-wrap: break-word;">
When clock = 0, the slave becomes active and master is inactive. But since the S and R inputs have not changed, the slave outputs will also remain unchanged. Therefore outputs will not change if J = K =0.</div>
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<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">2</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><b style="box-sizing: border-box;">J = 0 and K = 1 (Reset)</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><div style="box-sizing: border-box; color: black; font-size: 15px !important; line-height: 24px; margin: 0em 0.2em 1em; padding: 0px; text-align: justify; word-wrap: break-word;">
Clock = 1 − Master active, slave inactive. Therefore outputs of the master become Q<sub style="box-sizing: border-box;">1</sub> = 0 and Q<sub style="box-sizing: border-box;">1</sub> bar = 1. That means S = 0 and R =1.</div>
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Clock = 0 − Slave active, master inactive. Therefore outputs of the slave become Q = 0 and Q bar = 1.</div>
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Again clock = 1 − Master active, slave inactive. Therefore even with the changed outputs Q = 0 and Q bar = 1 fed back to master, its output will be Q1 = 0 and Q1 bar = 1. That means S = 0 and R = 1.</div>
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Hence with clock = 0 and slave becoming active the outputs of slave will remain Q = 0 and Q bar = 1. Thus we get a stable output from the Master slave.</div>
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<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">3</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><b style="box-sizing: border-box;">J = 1 and K = 0 (Set)</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><div style="box-sizing: border-box; color: black; font-size: 15px !important; line-height: 24px; margin: 0em 0.2em 1em; padding: 0px; text-align: justify; word-wrap: break-word;">
Clock = 1 − Master active, slave inactive. Therefore outputs of the master become Q<sub style="box-sizing: border-box;">1</sub> = 1 and Q<sub style="box-sizing: border-box;">1</sub> bar = 0. That means S = 1 and R =0.</div>
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Clock = 0 − Slave active, master inactive. Therefore outputs of the slave become Q = 1 and Q bar = 0.</div>
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Again clock = 1 − then it can be shown that the outputs of the slave are stabilized to Q = 1 and Q bar = 0.</div>
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<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">4</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><b style="box-sizing: border-box;">J = K = 1 (Toggle)</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><div style="box-sizing: border-box; color: black; font-size: 15px !important; line-height: 24px; margin: 0em 0.2em 1em; padding: 0px; text-align: justify; word-wrap: break-word;">
Clock = 1 − Master active, slave inactive. Outputs of master will toggle. So S and R also will be inverted.</div>
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Clock = 0 − Slave active, master inactive. Outputs of slave will toggle.</div>
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These changed output are returned back to the master inputs. But since clock = 0, the master is still inactive. So it does not respond to these changed outputs. This avoids the multiple toggling which leads to the race around condition. The master slave flip flop will avoid the race around condition.</div>
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Delay Flip Flop / D Flip Flop</h2>
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Delay Flip Flop or D Flip Flop is the simple gated S-R latch with a NAND inverter connected between S and R inputs. It has only one input. The input data is appearing at the output after some time. Due to this data delay between i/p and o/p, it is called delay flip flop. S and R will be the complements of each other due to NAND inverter. Hence S = R = 0 or S = R = 1, these input condition will never appear. This problem is avoid by SR = 00 and SR = 1 conditions.</div>
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Block Diagram</h3>
<img alt="Block Diagram of D Flip Flop" src="https://www.tutorialspoint.com/computer_logical_organization/images/dflipflop_blockdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Circuit Diagram</h3>
<img alt="Circuit Diagram of D Flip Flop" src="https://www.tutorialspoint.com/computer_logical_organization/images/dflipflop_circuitdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Truth Table</h3>
<img alt="Truth Table of D Flip Flop" src="https://www.tutorialspoint.com/computer_logical_organization/images/dflipflop_truthtable.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Operation</h3>
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<tr style="box-sizing: border-box;"><th style="background: rgb(238, 238, 238); border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; text-align: left; vertical-align: top; width: 49px;">S.N.</th><th style="background: rgb(238, 238, 238); border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; text-align: left; vertical-align: top; width: 120px;">Condition</th><th style="background: rgb(238, 238, 238); border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; text-align: left; vertical-align: top;">Operation</th></tr>
<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">1</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><b style="box-sizing: border-box;">E = 0</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><div style="box-sizing: border-box; color: black; font-size: 15px !important; line-height: 24px; margin: 0em 0.2em 1em; padding: 0px; text-align: justify; word-wrap: break-word;">
Latch is disabled. Hence no change in output.</div>
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<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">2</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><b style="box-sizing: border-box;">E = 1 and D = 0</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><div style="box-sizing: border-box; color: black; font-size: 15px !important; line-height: 24px; margin: 0em 0.2em 1em; padding: 0px; text-align: justify; word-wrap: break-word;">
If E = 1 and D = 0 then S = 0 and R = 1. Hence irrespective of the present state, the next state is Q<sub style="box-sizing: border-box;">n+1</sub> = 0 and Q<sub style="box-sizing: border-box;">n+1</sub> bar = 1. This is the reset condition.</div>
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<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">3</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><b style="box-sizing: border-box;">E = 1 and D = 1</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><div style="box-sizing: border-box; color: black; font-size: 15px !important; line-height: 24px; margin: 0em 0.2em 1em; padding: 0px; text-align: justify; word-wrap: break-word;">
If E = 1 and D = 1, then S = 1 and R = 0. This will set the latch and Q<sub style="box-sizing: border-box;">n+1</sub> = 1 and Q<sub style="box-sizing: border-box;">n+1</sub> bar = 0 irrespective of the present state.</div>
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Toggle Flip Flop / T Flip Flop</h2>
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Toggle flip flop is basically a JK flip flop with J and K terminals permanently connected together. It has only input denoted by <b style="box-sizing: border-box;">T</b> as shown in the Symbol Diagram. The symbol for positive edge triggered T flip flop is shown in the Block Diagram.</div>
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Symbol Diagram</h3>
<img alt="Symbol Diagram of T Flip Flop" src="https://www.tutorialspoint.com/computer_logical_organization/images/tflipflop_symboldiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Block Diagram</h3>
<img alt="Block Diagram of T Flip Flop" src="https://www.tutorialspoint.com/computer_logical_organization/images/tflipflop_blockdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Truth Table</h3>
<img alt="Truth Table of T Flip Flop" src="https://www.tutorialspoint.com/computer_logical_organization/images/tflipflop_truthtable.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Operation</h3>
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<tr style="box-sizing: border-box;"><th style="background: rgb(238, 238, 238); border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; text-align: left; vertical-align: top; width: 49px;">S.N.</th><th style="background: rgb(238, 238, 238); border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; text-align: left; vertical-align: top; width: 120px;">Condition</th><th style="background: rgb(238, 238, 238); border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; text-align: left; vertical-align: top;">Operation</th></tr>
<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">1</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><b style="box-sizing: border-box;">T = 0, J = K = 0</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">The output Q and Q bar won't change</td></tr>
<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">2</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><b style="box-sizing: border-box;">T = 1, J = K = 1</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">Output will toggle corresponding to every leading edge of clock signal.</td></tr>
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</head>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3005862392404377111.post-86450539517891037282017-04-20T08:17:00.003-07:002023-06-23T23:34:12.473-07:00Digital Registers<div dir="ltr" style="text-align: left;" trbidi="on">
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Flip-flop is a 1 bit memory cell which can be used for storing the digital data. To increase the storage capacity in terms of number of bits, we have to use a group of flip-flop. Such a group of flip-flop is known as a <b style="box-sizing: border-box;">Register</b>. The <b style="box-sizing: border-box;">n-bit register</b> will consist of <b style="box-sizing: border-box;">n</b> number of flip-flop and it is capable of storing an <b style="box-sizing: border-box;">n-bit</b> word.</div>
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The binary data in a register can be moved within the register from one flip-flop to another. The registers that allow such data transfers are called as <b style="box-sizing: border-box;">shift registers</b>. There are four mode of operations of a shift register.</div>
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<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Serial Input Serial Output</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Serial Input Parallel Output</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Parallel Input Serial Output</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Parallel Input Parallel Output</li>
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Serial Input Serial Output</h2>
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Let all the flip-flop be initially in the reset condition i.e. Q<sub style="box-sizing: border-box;">3</sub> = Q<sub style="box-sizing: border-box;">2</sub> = Q<sub style="box-sizing: border-box;">1</sub> = Q<sub style="box-sizing: border-box;">0</sub> = 0. If an entry of a four bit binary number 1 1 1 1 is made into the register, this number should be applied to <b style="box-sizing: border-box;">D<sub style="box-sizing: border-box;">in</sub></b> bit with the LSB bit applied first. The D input of FF-3 i.e. D<sub style="box-sizing: border-box;">3</sub> is connected to serial data input <b style="box-sizing: border-box;">D<sub style="box-sizing: border-box;">in</sub></b>. Output of FF-3 i.e. Q<sub style="box-sizing: border-box;">3</sub> is connected to the input of the next flip-flop i.e. D<sub style="box-sizing: border-box;">2</sub> and so on.</div>
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Block Diagram</h3>
<img alt="Block Diagram of SISO Register" src="https://www.tutorialspoint.com/computer_logical_organization/images/siso_blockdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Operation</h3>
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Before application of clock signal, let Q<sub style="box-sizing: border-box;">3</sub> Q<sub style="box-sizing: border-box;">2</sub> Q<sub style="box-sizing: border-box;">1</sub> Q<sub style="box-sizing: border-box;">0</sub> = 0000 and apply LSB bit of the number to be entered to D<sub style="box-sizing: border-box;">in</sub>. So D<sub style="box-sizing: border-box;">in</sub> = D<sub style="box-sizing: border-box;">3</sub> = 1. Apply the clock. On the first falling edge of clock, the FF-3 is set, and stored word in the register is Q<sub style="box-sizing: border-box;">3</sub> Q<sub style="box-sizing: border-box;">2</sub> Q<sub style="box-sizing: border-box;">1</sub>Q<sub style="box-sizing: border-box;">0</sub> = 1000.</div>
<img alt="SISO Operation step 1" src="https://www.tutorialspoint.com/computer_logical_organization/images/siso_operation1.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Apply the next bit to D<sub style="box-sizing: border-box;">in</sub>. So D<sub style="box-sizing: border-box;">in</sub> = 1. As soon as the next negative edge of the clock hits, FF-2 will set and the stored word change to Q<sub style="box-sizing: border-box;">3</sub> Q<sub style="box-sizing: border-box;">2</sub> Q<sub style="box-sizing: border-box;">1</sub> Q<sub style="box-sizing: border-box;">0</sub> = 1100.</div>
<img alt="SISO Operation step 2" src="https://www.tutorialspoint.com/computer_logical_organization/images/siso_operation2.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Apply the next bit to be stored i.e. 1 to D<sub style="box-sizing: border-box;">in</sub>. Apply the clock pulse. As soon as the third negative clock edge hits, FF-1 will be set and output will be modified to Q<sub style="box-sizing: border-box;">3</sub> Q<sub style="box-sizing: border-box;">2</sub> Q<sub style="box-sizing: border-box;">1</sub> Q<sub style="box-sizing: border-box;">0</sub> = 1110.</div>
<img alt="SISO Operation step 3" src="https://www.tutorialspoint.com/computer_logical_organization/images/siso_operation3.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Similarly with D<sub style="box-sizing: border-box;">in</sub> = 1 and with the fourth negative clock edge arriving, the stored word in the register is Q<sub style="box-sizing: border-box;">3</sub> Q<sub style="box-sizing: border-box;">2</sub> Q<sub style="box-sizing: border-box;">1</sub> Q<sub style="box-sizing: border-box;">0</sub> = 1111.</div>
<img alt="SISO Operation step 4" src="https://www.tutorialspoint.com/computer_logical_organization/images/siso_operation4.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Truth Table</h3>
<img alt="Truth Table of SISO Register" src="https://www.tutorialspoint.com/computer_logical_organization/images/siso_truthtable.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Waveforms</h3>
<img alt="Waveform of SISO Register" src="https://www.tutorialspoint.com/computer_logical_organization/images/siso_waveform.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Serial Input Parallel Output</h2>
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In such types of operations, the data is entered serially and taken out in parallel fashion.</div>
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Data is loaded bit by bit. The outputs are disabled as long as the data is loading.</div>
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As soon as the data loading gets completed, all the flip-flops contain their required data, the outputs are enabled so that all the loaded data is made available over all the output lines at the same time.</div>
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4 clock cycles are required to load a four bit word. Hence the speed of operation of SIPO mode is same as that of SISO mode.</div>
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Block Diagram</h3>
<img alt="Block Diagram of SIPO Register" src="https://www.tutorialspoint.com/computer_logical_organization/images/sipo_blockdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Parallel Input Serial Output (PISO)</h2>
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Data bits are entered in parallel fashion.</div>
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The circuit shown below is a four bit parallel input serial output register.</div>
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Output of previous Flip Flop is connected to the input of the next one via a combinational circuit.</div>
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The binary input word B<sub style="box-sizing: border-box;">0</sub>, B<sub style="box-sizing: border-box;">1</sub>, B<sub style="box-sizing: border-box;">2</sub>, B<sub style="box-sizing: border-box;">3</sub> is applied though the same combinational circuit.</div>
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There are two modes in which this circuit can work namely - shift mode or load mode.</div>
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Load mode</h3>
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When the shift/load bar line is low (0), the AND gate 2, 4 and 6 become active they will pass B<sub style="box-sizing: border-box;">1</sub>, B<sub style="box-sizing: border-box;">2</sub>, B<sub style="box-sizing: border-box;">3</sub> bits to the corresponding flip-flops. On the low going edge of clock, the binary input B<sub style="box-sizing: border-box;">0</sub>, B<sub style="box-sizing: border-box;">1</sub>, B<sub style="box-sizing: border-box;">2</sub>, B<sub style="box-sizing: border-box;">3</sub> will get loaded into the corresponding flip-flops. Thus parallel loading takes place.</div>
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Shift mode</h3>
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When the shift/load bar line is low (1), the AND gate 2, 4 and 6 become inactive. Hence the parallel loading of the data becomes impossible. But the AND gate 1,3 and 5 become active. Therefore the shifting of data from left to right bit by bit on application of clock pulses. Thus the parallel in serial out operation takes place.</div>
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Block Diagram</h3>
<img alt="Block Diagram of PISO Register" src="https://www.tutorialspoint.com/computer_logical_organization/images/piso_blockdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Parallel Input Parallel Output (PIPO)</h2>
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In this mode, the 4 bit binary input B<sub style="box-sizing: border-box;">0</sub>, B<sub style="box-sizing: border-box;">1</sub>, B<sub style="box-sizing: border-box;">2</sub>, B<sub style="box-sizing: border-box;">3</sub> is applied to the data inputs D<sub style="box-sizing: border-box;">0</sub>, D<sub style="box-sizing: border-box;">1</sub>, D<sub style="box-sizing: border-box;">2</sub>, D<sub style="box-sizing: border-box;">3</sub> respectively of the four flip-flops. As soon as a negative clock edge is applied, the input binary bits will be loaded into the flip-flops simultaneously. The loaded bits will appear simultaneously to the output side. Only clock pulse is essential to load all the bits.</div>
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Block Diagram</h3>
<img alt="Block Diagram of PIPO Register" src="https://www.tutorialspoint.com/computer_logical_organization/images/pipo_blockdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Bidirectional Shift Register</h2>
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If a binary number is shifted left by one position then it is equivalent to multiplying the original number by 2. Similarly if a binary number is shifted right by one position then it is equivalent to dividing the original number by 2.</div>
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Hence if we want to use the shift register to multiply and divide the given binary number, then we should be able to move the data in either left or right direction.</div>
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Such a register is called bi-directional register. A four bit bi-directional shift register is shown in fig.</div>
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There are two serial inputs namely the serial right shift data input DR, and the serial left shift data input DL along with a mode select input (M).</div>
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Block Diagram</h3>
<img alt="Block Diagram of Bidirectional Shift Register" src="https://www.tutorialspoint.com/computer_logical_organization/images/bidirectional_shiftregister_blockdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Operation</h3>
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<tr style="box-sizing: border-box;"><th style="background: rgb(238, 238, 238); border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; text-align: left; vertical-align: top; width: 49px;">S.N.</th><th style="background: rgb(238, 238, 238); border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; text-align: left; vertical-align: top; width: 253px;">Condition</th><th style="background: rgb(238, 238, 238); border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; text-align: left; vertical-align: top;">Operation</th></tr>
<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">1</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><b style="box-sizing: border-box;">With M = 1 − Shift right operation</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><div style="box-sizing: border-box; color: black; font-size: 15px !important; line-height: 24px; margin: 0em 0.2em 1em; padding: 0px; text-align: justify; word-wrap: break-word;">
If M = 1, then the AND gates 1, 3, 5 and 7 are enabled whereas the remaining AND gates 2, 4, 6 and 8 will be disabled.</div>
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The data at D<sub style="box-sizing: border-box;">R</sub> is shifted to right bit by bit from FF-3 to FF-0 on the application of clock pulses. Thus with M = 1 we get the serial right shift operation.</div>
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<tr style="box-sizing: border-box;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;">2</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><b style="box-sizing: border-box;">With M = 0 − Shift left operation</b></td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: border-box; line-height: 1.42857; padding: 8px; vertical-align: top;"><div style="box-sizing: border-box; color: black; font-size: 15px !important; line-height: 24px; margin: 0em 0.2em 1em; padding: 0px; text-align: justify; word-wrap: break-word;">
When the mode control M is connected to 0 then the AND gates 2, 4, 6 and 8 are enabled while 1, 3, 5 and 7 are disabled.</div>
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The data at D<sub style="box-sizing: border-box;">L</sub> is shifted left bit by bit from FF-0 to FF-3 on the application of clock pulses. Thus with M = 0 we get the serial right shift operation.</div>
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Universal Shift Register</h2>
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A shift register which can shift the data in only one direction is called a uni-directional shift register. A shift register which can shift the data in both directions is called a bi-directional shift register. Applying the same logic, a shift register which can shift the data in both directions as well as load it parallely, is known as a universal shift register. The shift register is capable of performing the following operation −</div>
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<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Parallel loading</li>
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The mode control input is connected to logic 1 for parallel loading operation whereas it is connected to 0 for serial shifting. With mode control pin connected to ground, the universal shift register acts as a bi-directional register. For serial left operation, the input is applied to the serial input which goes to AND gate-1 shown in figure. Whereas for the shift right operation, the serial input is applied to D input.</div>
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Block Diagram</h3>
<img alt="Block Diagram of Universal Shift Register" src="https://www.tutorialspoint.com/computer_logical_organization/images/universal_shiftregister_blockdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3005862392404377111.post-85211196120866931622017-04-20T08:16:00.001-07:002023-06-23T23:34:12.435-07:00Combinational Circuits<div dir="ltr" style="text-align: left;" trbidi="on">
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Combinational circuit is a circuit in which we combine the different gates in the circuit, for example encoder, decoder, multiplexer and demultiplexer. Some of the characteristics of combinational circuits are following −</div>
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The output of combinational circuit at any instant of time, depends only on the levels present at input terminals.</div>
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The combinational circuit do not use any memory. The previous state of input does not have any effect on the present state of the circuit.</div>
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A combinational circuit can have an n number of inputs and m number of outputs.</div>
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Block diagram</h3>
<img alt="Block Diagram of combinational circuit" src="https://www.tutorialspoint.com/computer_logical_organization/images/combinational_circuit_blockdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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We're going to elaborate few important combinational circuits as follows.</div>
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Half Adder</h2>
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Half adder is a combinational logic circuit with two inputs and two outputs. The half adder circuit is designed to add two single bit binary number A and B. It is the basic building block for addition of two <b style="box-sizing: border-box;">single</b> bit numbers. This circuit has two outputs <b style="box-sizing: border-box;">carry</b> and <b style="box-sizing: border-box;">sum</b>.</div>
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Block diagram</h3>
<img alt="Block Diagram of Half Adder" src="https://www.tutorialspoint.com/computer_logical_organization/images/halfadder_blockdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Truth Table</h3>
<img alt="Half Adder Truth Table" src="https://www.tutorialspoint.com/computer_logical_organization/images/halfadder_truthtable.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Circuit Diagram</h3>
<img alt="Half Adder Circuit Diagram" src="https://www.tutorialspoint.com/computer_logical_organization/images/halfadder_circuitdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Full Adder</h2>
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Full adder is developed to overcome the drawback of Half Adder circuit. It can add two one-bit numbers A and B, and carry c. The full adder is a three input and two output combinational circuit.</div>
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Block diagram</h3>
<img alt="Block Diagram of Full Adder" src="https://www.tutorialspoint.com/computer_logical_organization/images/fulladder_blockdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Truth Table</h3>
<img alt="Full Adder Truth Table" src="https://www.tutorialspoint.com/computer_logical_organization/images/fulladder_truthtable.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Circuit Diagram</h3>
<img alt="Full Adder Circuit Diagram" src="https://www.tutorialspoint.com/computer_logical_organization/images/fulladder_circuitdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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N-Bit Parallel Adder</h2>
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The Full Adder is capable of adding only two single digit binary number along with a carry input. But in practical we need to add binary numbers which are much longer than just one bit. To add two n-bit binary numbers we need to use the n-bit parallel adder. It uses a number of full adders in cascade. The carry output of the previous full adder is connected to carry input of the next full adder.</div>
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4 Bit Parallel Adder</h3>
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In the block diagram, A<sub style="box-sizing: border-box;">0</sub> and B<sub style="box-sizing: border-box;">0</sub> represent the LSB of the four bit words A and B. Hence Full Adder-0 is the lowest stage. Hence its C<sub style="box-sizing: border-box;">in</sub> has been permanently made 0. The rest of the connections are exactly same as those of n-bit parallel adder is shown in fig. The four bit parallel adder is a very common logic circuit.</div>
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Block diagram</h3>
<img alt="Block Diagram of Four bit Adder" src="https://www.tutorialspoint.com/computer_logical_organization/images/fourbitadder_blockdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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N-Bit Parallel Subtractor</h2>
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The subtraction can be carried out by taking the 1's or 2's complement of the number to be subtracted. For example we can perform the subtraction (A-B) by adding either 1's or 2's complement of B to A. That means we can use a binary adder to perform the binary subtraction.</div>
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4 Bit Parallel Subtractor</h3>
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The number to be subtracted (B) is first passed through inverters to obtain its 1's complement. The 4-bit adder then adds A and 2's complement of B to produce the subtraction. S<sub style="box-sizing: border-box;">3</sub> S<sub style="box-sizing: border-box;">2</sub> S<sub style="box-sizing: border-box;">1</sub> S<sub style="box-sizing: border-box;">0</sub> represents the result of binary subtraction (A-B) and carry output C<sub style="box-sizing: border-box;">out</sub> represents the polarity of the result. If A > B then Cout = 0 and the result of binary form (A-B) then C<sub style="box-sizing: border-box;">out</sub> = 1 and the result is in the 2's complement form.</div>
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Block diagram</h3>
<img alt="Block Diagram of Four bit Substrator" src="https://www.tutorialspoint.com/computer_logical_organization/images/fourbitsubstractor_blockdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Half Subtractors</h2>
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Half subtractor is a combination circuit with two inputs and two outputs (difference and borrow). It produces the difference between the two binary bits at the input and also produces an output (Borrow) to indicate if a 1 has been borrowed. In the subtraction (A-B), A is called as Minuend bit and B is called as Subtrahend bit.</div>
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Truth Table</h3>
<img alt="Half Substractor Truth Table" src="https://www.tutorialspoint.com/computer_logical_organization/images/halfsubstrator_truthtable.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Circuit Diagram</h3>
<img alt="Half Substractor Circuit Diagram" src="https://www.tutorialspoint.com/computer_logical_organization/images/halfsubstrator_circuitdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Full Subtractors</h2>
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The disadvantage of a half subtractor is overcome by full subtractor. The full subtractor is a combinational circuit with three inputs A,B,C and two output D and C'. A is the 'minuend', B is 'subtrahend', C is the 'borrow' produced by the previous stage, D is the difference output and C' is the borrow output.</div>
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Truth Table</h3>
<img alt="Full Substractor Truth Table" src="https://www.tutorialspoint.com/computer_logical_organization/images/fullsubstrator_truthtable.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Circuit Diagram</h3>
<img alt="Full Substractor Circuit Diagram" src="https://www.tutorialspoint.com/computer_logical_organization/images/fullsubstrator_circuitdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Multiplexers</h2>
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Multiplexer is a special type of combinational circuit. There are n-data inputs, one output and m select inputs with 2m = n. It is a digital circuit which selects one of the n data inputs and routes it to the output. The selection of one of the n inputs is done by the selected inputs. Depending on the digital code applied at the selected inputs, one out of n data sources is selected and transmitted to the single output Y. E is called the strobe or enable input which is useful for the cascading. It is generally an active low terminal that means it will perform the required operation when it is low.</div>
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Block diagram</h3>
<img alt="Block Diagram of n:1 Multiplexer" src="https://www.tutorialspoint.com/computer_logical_organization/images/n_onemultiplexer_blockdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Multiplexers come in multiple variations</div>
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<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">2 : 1 multiplexer</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">4 : 1 multiplexer</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">16 : 1 multiplexer</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">32 : 1 multiplexer</li>
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Block Diagram</h3>
<img alt="2:1 Multiplexer Block Diagram" src="https://www.tutorialspoint.com/computer_logical_organization/images/two_onemultiplexer_blockdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Truth Table</h3>
<img alt="2:1 Multiplexer Truth Table" src="https://www.tutorialspoint.com/computer_logical_organization/images/two_onemultiplexer_truthtable.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Demultiplexers</h2>
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A demultiplexer performs the reverse operation of a multiplexer i.e. it receives one input and distributes it over several outputs. It has only one input, n outputs, m select input. At a time only one output line is selected by the select lines and the input is transmitted to the selected output line. A de-multiplexer is equivalent to a single pole multiple way switch as shown in fig.</div>
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Demultiplexers comes in multiple variations.</div>
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<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">1 : 2 demultiplexer</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">1 : 4 demultiplexer</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">1 : 16 demultiplexer</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">1 : 32 demultiplexer</li>
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Block diagram</h3>
<img alt="Block Diagram of 1:2 Demultiplexer" src="https://www.tutorialspoint.com/computer_logical_organization/images/one_twodemultiplexer_blockdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Truth Table</h3>
<img alt="1:2 Demultiplexer Truth Table" src="https://www.tutorialspoint.com/computer_logical_organization/images/one_twodemultiplexer_truthtable.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Decoder</h2>
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A decoder is a combinational circuit. It has n input and to a maximum m = 2n outputs. Decoder is identical to a demultiplexer without any data input. It performs operations which are exactly opposite to those of an encoder.</div>
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Block diagram</h3>
<img alt="Block Diagram of Decoder" src="https://www.tutorialspoint.com/computer_logical_organization/images/decoder_blockdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Examples of Decoders are following.</div>
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<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Code converters</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">BCD to seven segment decoders</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Nixie tube decoders</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Relay actuator</li>
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2 to 4 Line Decoder</h2>
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The block diagram of 2 to 4 line decoder is shown in the fig. A and B are the two inputs where D through D are the four outputs. Truth table explains the operations of a decoder. It shows that each output is 1 for only a specific combination of inputs.</div>
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Block diagram</h3>
<img alt="Block Diagram of 2 to 4 Decoder" src="https://www.tutorialspoint.com/computer_logical_organization/images/two_fourdecoder_blockdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Truth Table</h3>
<img alt="Truth Table of 2 to 4 Decoder" src="https://www.tutorialspoint.com/computer_logical_organization/images/two_fourdecoder_truthtable.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Logic Circuit</h3>
<img alt="Logic Circuit of 2 to 4 Decoder" src="https://www.tutorialspoint.com/computer_logical_organization/images/two_fourdecoder_logiccircuit.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Encoder</h2>
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Encoder is a combinational circuit which is designed to perform the inverse operation of the decoder. An encoder has n number of input lines and m number of output lines. An encoder produces an m bit binary code corresponding to the digital input number. The encoder accepts an n input digital word and converts it into an m bit another digital word.</div>
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Block diagram</h3>
<img alt="Block Diagram of encoder" src="https://www.tutorialspoint.com/computer_logical_organization/images/encoder_blockdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Examples of Encoders are following.</div>
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<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Priority encoders</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Decimal to BCD encoder</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Octal to binary encoder</li>
<li style="background-attachment: initial; background-clip: initial; background-image: url("/images/icon-bullet.png"); background-origin: initial; background-position: 0px 4px; background-repeat: no-repeat; background-size: initial; box-sizing: border-box; color: black; line-height: 24px; list-style: none; margin-bottom: 5px; padding: 0px 0px 0px 19px;">Hexadecimal to binary encoder</li>
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Priority Encoder</h2>
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This is a special type of encoder. Priority is given to the input lines. If two or more input line are 1 at the same time, then the input line with highest priority will be considered. There are four input D<sub style="box-sizing: border-box;">0</sub>, D<sub style="box-sizing: border-box;">1</sub>, D<sub style="box-sizing: border-box;">2</sub>, D<sub style="box-sizing: border-box;">3</sub> and two output Y<sub style="box-sizing: border-box;">0</sub>, Y<sub style="box-sizing: border-box;">1</sub>. Out of the four input D<sub style="box-sizing: border-box;">3</sub> has the highest priority and D<sub style="box-sizing: border-box;">0</sub> has the lowest priority. That means if D<sub style="box-sizing: border-box;">3</sub> = 1 then Y<sub style="box-sizing: border-box;">1</sub> Y<sub style="box-sizing: border-box;">1</sub> = 11 irrespective of the other inputs. Similarly if D<sub style="box-sizing: border-box;">3</sub> = 0 and D<sub style="box-sizing: border-box;">2</sub> = 1 then Y<sub style="box-sizing: border-box;">1</sub> Y<sub style="box-sizing: border-box;">0</sub> = 10 irrespective of the other inputs.</div>
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Block diagram</h3>
<img alt="Block Diagram of Priority Encoder" src="https://www.tutorialspoint.com/computer_logical_organization/images/priorityencoder_blockdiagram.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Truth Table</h3>
<img alt="Truth Table of Priority Encoder" src="https://www.tutorialspoint.com/computer_logical_organization/images/priorityencoder_truthtable.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Logic Circuit</h3>
<img alt="Logic Circuit of Priority Encoder" src="https://www.tutorialspoint.com/computer_logical_organization/images/priorityencoder_logiccircuit.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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</head>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3005862392404377111.post-89503220511592887052017-04-20T08:14:00.004-07:002023-06-23T23:34:12.545-07:00Logic Gates<div dir="ltr" style="text-align: left;" trbidi="on">
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Logic gates are the basic building blocks of any digital system. It is an electronic circuit having one or more than one input and only one output. The relationship between the input and the output is based on a <b style="box-sizing: border-box;">certain logic</b>. Based on this, logic gates are named as AND gate, OR gate, NOT gate etc.</div>
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AND Gate</h2>
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A circuit which performs an AND operation is shown in figure. It has n input (n >= 2) and one output.</div>
<img alt="AND gate" src="https://www.tutorialspoint.com/computer_logical_organization/images/and1.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Logic diagram</h3>
<img alt="AND Logical Diagram" src="https://www.tutorialspoint.com/computer_logical_organization/images/and_logic.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Truth Table</h3>
<img alt="AND Truth Table" src="https://www.tutorialspoint.com/computer_logical_organization/images/and_truthtable.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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OR Gate</h2>
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A circuit which performs an OR operation is shown in figure. It has n input (n >= 2) and one output.</div>
<img alt="OR gate" src="https://www.tutorialspoint.com/computer_logical_organization/images/or1.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Logic diagram</h3>
<img alt="OR Logical Diagram" src="https://www.tutorialspoint.com/computer_logical_organization/images/or_logic.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Truth Table</h3>
<img alt="OR Truth Table" src="https://www.tutorialspoint.com/computer_logical_organization/images/or_truthtable.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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NOT Gate</h2>
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NOT gate is also known as <b style="box-sizing: border-box;">Inverter</b>. It has one input A and one output Y.</div>
<img alt="NOT gate" src="https://www.tutorialspoint.com/computer_logical_organization/images/not1.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Logic diagram</h3>
<img alt="NOT Logical Diagram" src="https://www.tutorialspoint.com/computer_logical_organization/images/not_logic.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Truth Table</h3>
<img alt="NOT Truth Table" src="https://www.tutorialspoint.com/computer_logical_organization/images/not_truthtable.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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NAND Gate</h2>
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A NOT-AND operation is known as NAND operation. It has n input (n >= 2) and one output.</div>
<img alt="NAND gate" src="https://www.tutorialspoint.com/computer_logical_organization/images/nand.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Logic diagram</h3>
<img alt="NAND Logical Diagram" src="https://www.tutorialspoint.com/computer_logical_organization/images/nand_logic.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Truth Table</h3>
<img alt="NAND Truth Table" src="https://www.tutorialspoint.com/computer_logical_organization/images/nand_truthtable.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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NOR Gate</h2>
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A NOT-OR operation is known as NOR operation. It has n input (n >= 2) and one output.</div>
<img alt="NOR gate" src="https://www.tutorialspoint.com/computer_logical_organization/images/nor.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Logic diagram</h3>
<img alt="NOR Logical Diagram" src="https://www.tutorialspoint.com/computer_logical_organization/images/nor_logic.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Truth Table</h3>
<img alt="NOR Truth Table" src="https://www.tutorialspoint.com/computer_logical_organization/images/nor_truthtable.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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XOR Gate</h2>
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XOR or Ex-OR gate is a special type of gate. It can be used in the half adder, full adder and subtractor. The exclusive-OR gate is abbreviated as EX-OR gate or sometime as X-OR gate. It has n input (n >= 2) and one output.</div>
<img alt="XOR gate" src="https://www.tutorialspoint.com/computer_logical_organization/images/xor.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Logic diagram</h3>
<img alt="XOR Logical Diagram" src="https://www.tutorialspoint.com/computer_logical_organization/images/xor_logic.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Truth Table</h3>
<img alt="XOR Truth Table" src="https://www.tutorialspoint.com/computer_logical_organization/images/xor_truthtable.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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XNOR Gate</h2>
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XNOR gate is a special type of gate. It can be used in the half adder, full adder and subtractor. The exclusive-NOR gate is abbreviated as EX-NOR gate or sometime as X-NOR gate. It has n input (n >= 2) and one output.</div>
<img alt="XNOR gate" src="https://www.tutorialspoint.com/computer_logical_organization/images/xnor.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Logic diagram</h3>
<img alt="XNOR Logical Diagram" src="https://www.tutorialspoint.com/computer_logical_organization/images/xnor_logic.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Truth Table</h3>
<img alt="XNOR Truth Table" src="https://www.tutorialspoint.com/computer_logical_organization/images/xnor_truthtable.jpg" style="border: 0px; box-sizing: border-box; color: #313131; display: block; font-family: Verdana, Geneva, Tahoma, Arial, Helvetica, sans-serif; font-size: 14px; margin-left: auto; margin-right: auto; max-width: 100%; padding-bottom: 4px; vertical-align: middle;" /><br />
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3005862392404377111.post-71914997718768681642017-04-20T08:10:00.002-07:002023-06-23T23:34:12.249-07:00Boolean Expression<div dir="ltr" style="text-align: left;" trbidi="on">
<h3 style="color: #4286f4; font-family: Verdana, Geneva, sans-serif; font-size: 18px; margin: 10px 0px 0px; padding: 0px;">
Minterm Solution of K Map</h3>
<span style="color: #505050; font-family: "verdana" , "geneva" , sans-serif; font-size: 18px; text-align: justify;">The following are the steps to obtain simplified minterm solution using </span><a href="https://www.electrical4u.com/k-map/" style="color: #cc0000; font-family: Verdana, Geneva, sans-serif; font-size: 18px; margin: 0px; padding: 0px; text-align: justify; text-decoration-line: none;">K-map</a><span style="color: #505050; font-family: "verdana" , "geneva" , sans-serif; font-size: 18px; text-align: justify;">. </span><span class="Green" style="color: #005256; font-family: "verdana" , "geneva" , sans-serif; font-size: 18px; font-weight: bold; margin: 0px; padding: 0px; text-align: justify;">Step 1: Initiate</span><span style="color: #505050; font-family: "verdana" , "geneva" , sans-serif; font-size: 18px; text-align: justify;"> Express the given expression in its canonical form </span><span class="Green" style="color: #005256; font-family: "verdana" , "geneva" , sans-serif; font-size: 18px; font-weight: bold; margin: 0px; padding: 0px; text-align: justify;">Step 2: Populate the K-map</span><span style="color: #505050; font-family: "verdana" , "geneva" , sans-serif; font-size: 18px; text-align: justify;">Enter the value of 'one' for each product-term into the K-map cell, while filling others with zeros. </span><span class="Green" style="color: #005256; font-family: "verdana" , "geneva" , sans-serif; font-size: 18px; font-weight: bold; margin: 0px; padding: 0px; text-align: justify;">Step 3: Form Groups</span><span style="color: #505050; font-family: "verdana" , "geneva" , sans-serif; font-size: 18px; text-align: justify;"></span><br />
<ol style="color: #505050; font-family: Verdana, Geneva, sans-serif; font-size: 18px; margin: 0px; padding: 0px; text-align: justify;">
<li style="margin: 0px 0px 0px 20px; padding: 0px;">Consider the consecutive 'ones' in the K-map cells and group them (green boxes).<img alt="" src="https://www.electrical4u.com/images/march16/1460802136.PNG" style="display: block; margin: 15px auto; max-width: 100%; padding: 0px;" /></li>
<li style="margin: 0px 0px 0px 20px; padding: 0px;">Each group should contain the largest number of 'ones' and no blank cell.<img alt="" src="https://www.electrical4u.com/images/march16/1460802187.PNG" style="display: block; margin: 15px auto; max-width: 100%; padding: 0px;" /></li>
<li style="margin: 0px 0px 0px 20px; padding: 0px;">The number of 'ones' in a group must be a power of 2 i.e. a group can contain<img alt="" src="https://www.electrical4u.com/images/march16/1460725662.GIF" style="display: block; margin: 15px auto; max-width: 100%; padding: 0px;" /><img alt="" src="https://www.electrical4u.com/images/march16/1460802230.PNG" style="display: block; margin: 15px auto; max-width: 100%; padding: 0px;" /></li>
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<li style="margin: 0px 0px 0px 20px; padding: 0px;">Grouping has to be carried-on in decreasing order meaning, one has to try to group for 8 (octet) first, then for 4 (quad), followed by 2 and lastly for 1 (isolated 'ones').<img alt="" src="https://www.electrical4u.com/images/march16/1460802289.PNG" style="display: block; margin: 15px auto; max-width: 100%; padding: 0px;" /></li>
<li style="margin: 0px 0px 0px 20px; padding: 0px;">Grouping is to done either horizontally or vertically or interms of squares or rectangles. Diagonal grouping of 'ones' is not permitted.<img alt="" src="https://www.electrical4u.com/images/march16/1460802468.PNG" style="display: block; margin: 15px auto; max-width: 100%; padding: 0px;" /></li>
<li style="margin: 0px 0px 0px 20px; padding: 0px;">The same element(s) may repeat in multiple groups only if this increases the size of the group.<img alt="" src="https://www.electrical4u.com/images/march16/1460802520.PNG" style="display: block; margin: 15px auto; max-width: 100%; padding: 0px;" /></li>
<li style="margin: 0px 0px 0px 20px; padding: 0px;">The elements around the edges of the table are considered to be adjacent and can be grouped together.<img alt="" src="https://www.electrical4u.com/images/march16/1460802606.PNG" style="display: block; margin: 15px auto; max-width: 100%; padding: 0px;" /></li>
<li style="margin: 0px 0px 0px 20px; padding: 0px;">Don’t care conditions are to be considered only if they aid in increasing the group-size (else neglected).<img alt="" src="https://www.electrical4u.com/images/march16/1460802645.PNG" style="display: block; margin: 15px auto; max-width: 100%; padding: 0px;" /></li>
</ol>
<span class="Green" style="color: #005256; font-family: "verdana" , "geneva" , sans-serif; font-size: 18px; font-weight: bold; margin: 0px; padding: 0px; text-align: justify;">Step 4: Obtain Boolean Expression for Each Group</span><br />
<span style="color: #505050; font-family: "verdana" , "geneva" , sans-serif; font-size: 18px; text-align: justify;">Express each group interms of input variables by looking at the common variables seen in cell-labelling. For example in the figure shown below there are two groups with two and one number of 'ones' in them (Group 1 and Group 2, respectively). All the 'ones' in the Group 1 of the K-map are present in the row for which A = 0. Thus they contain the variable A̅. Further these two 'ones' are present in adjacent columns which have only B term in common as indicated by the pink arrow in the figure.</span><br />
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Hence the next term is B. This yields the product term corresponding to this group as A̅B. Similarly the 'one' in the Group 2 of the K-map is present in the row for which A = 1. Further the variables corresponding to its column are B̅C̅. Thus one gets the overall product-term for this group as AB̅C̅.<img alt="" src="https://www.electrical4u.com/images/march16/1460802807.PNG" style="display: block; margin: 15px auto; max-width: 100%; padding: 0px;" /><span class="Green" style="color: #005256; font-weight: bold; margin: 0px; padding: 0px;">Step 5: Obtain Boolean Expression for the Output</span> The product-terms obtained for individual groups are to be combined to form sum-of-product (SOP) form which yields the overall simplified Boolean expression. This means that for the K-map shown in Step 4, the overall simplified output expression is<img alt="" src="https://www.electrical4u.com/images/march16/1460727678.GIF" style="display: block; margin: 15px auto; max-width: 100%; padding: 0px;" />A few more examples elaborating K-map simplification process are shown below.<img alt="" src="https://www.electrical4u.com/images/march16/1460727522.GIF" style="display: block; margin: 15px auto; max-width: 100%; padding: 0px;" /><img alt="" src="https://www.electrical4u.com/images/march16/1460802954.PNG" style="display: block; margin: 15px auto; max-width: 100%; padding: 0px;" /><img alt="" src="https://www.electrical4u.com/images/march16/1460727774.GIF" style="display: block; margin: 15px auto; max-width: 100%; padding: 0px;" /><img alt="" src="https://www.electrical4u.com/images/march16/1460728318.GIF" style="display: block; margin: 15px auto; max-width: 100%; padding: 0px;" /><img alt="" src="https://www.electrical4u.com/images/march16/1460728026.PNG" style="display: block; margin: 15px auto; max-width: 100%; padding: 0px;" /><img alt="" src="https://www.electrical4u.com/images/march16/1460728146.GIF" style="display: block; margin: 15px auto; max-width: 100%; padding: 0px;" /><img alt="" src="https://www.electrical4u.com/images/march16/1460728623.GIF" style="display: block; margin: 15px auto; max-width: 100%; padding: 0px;" /><img alt="" src="https://www.electrical4u.com/images/march16/1460804154.PNG" style="display: block; margin: 15px auto; max-width: 100%; padding: 0px;" /><img alt="" src="https://www.electrical4u.com/images/march16/1460728789.GIF" style="display: block; margin: 15px auto; max-width: 100%; padding: 0px;" /></div>
<h3 style="color: #4286f4; font-family: Verdana, Geneva, sans-serif; font-size: 18px; margin: 10px 0px 0px; padding: 0px;">
Maxterm Solution of K Map</h3>
<span style="color: #505050; font-family: "verdana" , "geneva" , sans-serif; font-size: 18px; text-align: justify;">The method to be followed in order to obtain simplified </span><strong style="color: #505050; font-family: Verdana, Geneva, sans-serif; font-size: 18px; margin: 0px; padding: 0px; text-align: justify;">maxterm solution</strong><span style="color: #505050; font-family: "verdana" , "geneva" , sans-serif; font-size: 18px; text-align: justify;"> using K-map is similar to that for </span><strong style="color: #505050; font-family: Verdana, Geneva, sans-serif; font-size: 18px; margin: 0px; padding: 0px; text-align: justify;">minterm solution</strong><span style="color: #505050; font-family: "verdana" , "geneva" , sans-serif; font-size: 18px; text-align: justify;"> except minor changes listed below.</span><br />
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<li style="margin: 0px 0px 0px 20px; padding: 0px;">K-map cells are to be populated by 'zeros' for each sum-term of the expression instead of 'ones'.</li>
<li style="margin: 0px 0px 0px 20px; padding: 0px;">Grouping is to be carried-on for 'zeros' and not for 'ones'.</li>
<li style="margin: 0px 0px 0px 20px; padding: 0px;">Boolean expressions for each group are to be expressed as sum-terms and not as product-terms.</li>
<li style="margin: 0px 0px 0px 20px; padding: 0px;">Sum-terms of all individual groups are to be combined to obtain the overall simplified Boolean expression in product-of-sums (POS) form.</li>
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<img alt="" src="https://www.electrical4u.com/images/march16/1460729091.GIF" style="color: #505050; display: block; font-family: Verdana, Geneva, sans-serif; font-size: 18px; margin: 15px auto; max-width: 100%; padding: 0px; text-align: justify;" /><img alt="" src="https://www.electrical4u.com/images/march16/1460804261.PNG" style="color: #505050; display: block; font-family: Verdana, Geneva, sans-serif; font-size: 18px; margin: 15px auto; max-width: 100%; padding: 0px; text-align: justify;" /><img alt="" src="https://www.electrical4u.com/images/march16/1460729239.GIF" style="color: #505050; display: block; font-family: Verdana, Geneva, sans-serif; font-size: 18px; margin: 15px auto; max-width: 100%; padding: 0px; text-align: justify;" /></div>
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</head>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3005862392404377111.post-38519135177957437852017-04-20T07:33:00.001-07:002023-06-23T23:34:11.690-07:00Binary Arithmetic<div dir="ltr" style="text-align: left;" trbidi="on">
<span style="color: #505050; font-family: "verdana" , "geneva" , sans-serif; font-size: 18px; text-align: justify;">In </span><a href="https://www.electrical4u.com/binary-number-system-binary-to-decimal-and-decimal-to-binary-conversion/" style="color: #cc0000; font-family: Verdana, Geneva, sans-serif; font-size: 18px; margin: 0px; padding: 0px; text-align: justify; text-decoration-line: none;">binary number system</a><span style="color: #505050; font-family: "verdana" , "geneva" , sans-serif; font-size: 18px; text-align: justify;"> there are only 2 digits 0 and 1, and any number can be represented by these two digits. The </span><strong style="color: #505050; font-family: Verdana, Geneva, sans-serif; font-size: 18px; margin: 0px; padding: 0px; text-align: justify;">arithmetic of binary numbers</strong><span style="color: #505050; font-family: "verdana" , "geneva" , sans-serif; font-size: 18px; text-align: justify;"> means the operation of addition, subtraction, multiplication and division. </span><strong style="color: #505050; font-family: Verdana, Geneva, sans-serif; font-size: 18px; margin: 0px; padding: 0px; text-align: justify;">Binary arithmetic</strong><span style="color: #505050; font-family: "verdana" , "geneva" , sans-serif; font-size: 18px; text-align: justify;"> operation starts from the least significant bit i.e. from the right most side. We will discuss the different operations one by one in the following article.</span><br />
<h2 style="color: #4286f4; font-family: Verdana, Geneva, sans-serif; font-size: 20px; margin: 10px 0px 0px; padding: 0px;">
Binary Addition:-</h2>
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<span style="color: #505050; font-family: "verdana" , "geneva" , sans-serif; font-size: 18px; text-align: justify;">There are four steps in </span><a href="https://www.electrical4u.com/binary-addition/" style="color: #cc0000; font-family: Verdana, Geneva, sans-serif; font-size: 18px; margin: 0px; padding: 0px; text-align: justify; text-decoration-line: none;">binary addition</a><span style="color: #505050; font-family: "verdana" , "geneva" , sans-serif; font-size: 18px; text-align: justify;">, they are written below</span><br />
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<li style="margin: 0px 0px 0px 20px; padding: 0px;">0 + 0 = 0</li>
<li style="margin: 0px 0px 0px 20px; padding: 0px;">0 + 1 = 1</li>
<li style="margin: 0px 0px 0px 20px; padding: 0px;">1 + 0 = 1</li>
<li style="margin: 0px 0px 0px 20px; padding: 0px;">1 + 1 = 0 (carry 1 to the next significant bit)</li>
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<span style="color: #505050; font-family: "verdana" , "geneva" , sans-serif;"><span style="font-size: 18px;"><br /></span></span>
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<span style="color: blue; font-family: "verdana" , "geneva" , sans-serif;"><span style="font-size: 18px;"><i>Example:-</i></span></span></div>
<div>
<span style="color: blue; font-family: "verdana" , "geneva" , sans-serif;"><span style="font-size: 18px;"><i> </i></span></span><span class="bigbul" style="font-size: 27.2px; white-space: nowrap;">•</span><span style="white-space: nowrap;"> </span><span style="white-space: nowrap;">10001 + 11101 = 101110:</span></div>
<table class="sum" style="border-collapse: collapse; margin-left: 20pt;"><tbody>
<tr class="carries"><td style="font-size: 12.8px; width: 18pt;"></td><td class="cry" style="font-size: 12.8px; width: 18pt;">1</td><td class="cry" style="font-size: 12.8px; width: 18pt;"></td><td class="cry" style="font-size: 12.8px; width: 18pt;"></td><td class="cry" style="font-size: 12.8px; width: 18pt;"></td><td class="cry" style="font-size: 12.8px; width: 18pt;">1</td><td class="cry" style="font-size: 12.8px; width: 18pt;"></td></tr>
<tr class="addin1"><td style="width: 18pt;"></td><td class="tb" style="width: 18pt;"></td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">1</td></tr>
<tr class="addin2"><td style="width: 18pt;">+</td><td class="tb" style="width: 18pt;"></td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">1</td></tr>
<tr class="sum" style="border-top: 1px solid black;"><td style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;"></td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td></tr>
</tbody></table>
<br />
<span style="white-space: nowrap;"><span class="bigbul" style="font-size: 27.2px;">•</span> 101101 + 11001 = 1000110:</span><br />
<table class="sum" style="border-collapse: collapse; margin-left: 20pt;"><tbody>
<tr class="carries"><td style="font-size: 12.8px; width: 18pt;"></td><td class="cry" style="font-size: 12.8px; width: 18pt;">1</td><td class="cry" style="font-size: 12.8px; width: 18pt;">1</td><td class="cry" style="font-size: 12.8px; width: 18pt;">1</td><td class="cry" style="font-size: 12.8px; width: 18pt;"></td><td class="cry" style="font-size: 12.8px; width: 18pt;"></td><td class="cry" style="font-size: 12.8px; width: 18pt;">1</td><td class="cry" style="font-size: 12.8px; width: 18pt;"></td></tr>
<tr class="addin1"><td style="width: 18pt;"></td><td class="tb" style="width: 18pt;"></td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">1</td></tr>
<tr class="addin2"><td style="width: 18pt;">+</td><td class="tb" style="width: 18pt;"></td><td class="tb" style="width: 18pt;"></td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">1</td></tr>
<tr class="sum" style="border-top: 1px solid black;"><td style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;"></td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td></tr>
</tbody></table>
<br />
<span style="white-space: nowrap;"><span class="bigbul" style="font-size: 27.2px;">•</span> 1011001 + 111010 = 10010011:</span><br />
<table class="sum" style="border-collapse: collapse; margin-left: 20pt;"><tbody>
<tr class="carries"><td style="font-size: 12.8px; width: 18pt;"></td><td class="cry" style="font-size: 12.8px; width: 18pt;">1</td><td class="cry" style="font-size: 12.8px; width: 18pt;">1</td><td class="cry" style="font-size: 12.8px; width: 18pt;">1</td><td class="cry" style="font-size: 12.8px; width: 18pt;">1</td><td class="cry" style="font-size: 12.8px; width: 18pt;"></td><td class="cry" style="font-size: 12.8px; width: 18pt;"></td><td class="cry" style="font-size: 12.8px; width: 18pt;"></td><td class="cry" style="font-size: 12.8px; width: 18pt;"></td></tr>
<tr class="addin1"><td style="width: 18pt;"></td><td class="tb" style="width: 18pt;"></td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">1</td></tr>
<tr class="addin2"><td style="width: 18pt;">+</td><td class="tb" style="width: 18pt;"></td><td class="tb" style="width: 18pt;"></td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td></tr>
<tr class="sum" style="border-top: 1px solid black;"><td style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;"></td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td></tr>
</tbody></table>
<br />
<span style="white-space: nowrap;"><span class="bigbul" style="font-size: 27.2px;">•</span> 1110 + 1111 = 11101:</span><br />
<table class="sum" style="border-collapse: collapse; margin-left: 20pt;"><tbody>
<tr class="carries"><td style="font-size: 12.8px; width: 18pt;"></td><td class="cry" style="font-size: 12.8px; width: 18pt;">1</td><td class="cry" style="font-size: 12.8px; width: 18pt;">1</td><td class="cry" style="font-size: 12.8px; width: 18pt;">1</td><td class="cry" style="font-size: 12.8px; width: 18pt;"></td><td class="cry" style="font-size: 12.8px; width: 18pt;"></td></tr>
<tr class="addin1"><td style="width: 18pt;"></td><td class="tb" style="width: 18pt;"></td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td></tr>
<tr class="addin2"><td style="width: 18pt;">+</td><td class="tb" style="width: 18pt;"></td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">1</td></tr>
<tr class="sum" style="border-top: 1px solid black;"><td style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;"></td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td></tr>
</tbody></table>
<br />
<span style="white-space: nowrap;"><span class="bigbul" style="font-size: 27.2px;">•</span> 10111 + 110101 = 1001100:</span><br />
<table class="sum" style="border-collapse: collapse; margin-left: 20pt;"><tbody>
<tr class="carries"><td style="font-size: 12.8px; width: 18pt;"></td><td class="cry" style="font-size: 12.8px; width: 18pt;">1</td><td class="cry" style="font-size: 12.8px; width: 18pt;">1</td><td class="cry" style="font-size: 12.8px; width: 18pt;"></td><td class="cry" style="font-size: 12.8px; width: 18pt;">1</td><td class="cry" style="font-size: 12.8px; width: 18pt;">1</td><td class="cry" style="font-size: 12.8px; width: 18pt;">1</td><td class="cry" style="font-size: 12.8px; width: 18pt;"></td></tr>
<tr class="addin1"><td style="width: 18pt;"></td><td class="tb" style="width: 18pt;"></td><td class="tb" style="width: 18pt;"></td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">1</td></tr>
<tr class="addin2"><td style="width: 18pt;">+</td><td class="tb" style="width: 18pt;"></td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">1</td></tr>
<tr class="sum" style="border-top: 1px solid black;"><td style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;"></td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td></tr>
</tbody></table>
<br />
<span style="white-space: nowrap;"><span class="bigbul" style="font-size: 27.2px;">•</span> 11011 + 1001010 = 1100101:</span><br />
<table class="sum" style="border-collapse: collapse; margin-left: 20pt;"><tbody>
<tr class="carries"><td style="font-size: 12.8px; width: 18pt;"></td><td class="cry" style="font-size: 12.8px; width: 18pt;"></td><td class="cry" style="font-size: 12.8px; width: 18pt;">1</td><td class="cry" style="font-size: 12.8px; width: 18pt;">1</td><td class="cry" style="font-size: 12.8px; width: 18pt;"></td><td class="cry" style="font-size: 12.8px; width: 18pt;">1</td><td class="cry" style="font-size: 12.8px; width: 18pt;"></td><td class="cry" style="font-size: 12.8px; width: 18pt;"></td></tr>
<tr class="addin1"><td style="width: 18pt;"></td><td class="tb" style="width: 18pt;"></td><td class="tb" style="width: 18pt;"></td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">1</td></tr>
<tr class="addin2"><td style="width: 18pt;">+</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td></tr>
<tr class="sum" style="border-top: 1px solid black;"><td style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;"></td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td></tr>
</tbody></table>
<br /><h2 style="color: #4286f4; font-family: verdana, geneva, sans-serif; font-size: 20px; margin: 10px 0px 0px; padding: 0px;">
Binary Subtraction:-</h2>
<div>
<br /></div>
<div>
<span style="color: #505050; font-family: Verdana, Geneva, sans-serif; font-size: 18px; text-align: justify;">Here are too four simple steps to keep in memory</span><br />
<ul style="color: #505050; font-family: Verdana, Geneva, sans-serif; font-size: 18px; margin: 0px; padding: 0px; text-align: justify;">
<li style="margin: 0px 0px 0px 20px; padding: 0px;">0 - 0 = 0</li>
<li style="margin: 0px 0px 0px 20px; padding: 0px;">0 - 1 = 1, borrow 1 from the next more significant bit</li>
<li style="margin: 0px 0px 0px 20px; padding: 0px;">1 - 0 = 1</li>
<li style="margin: 0px 0px 0px 20px; padding: 0px;">1 - 1 = 0</li>
</ul>
<span style="color: #505050; font-family: Verdana, Geneva, sans-serif; font-size: 18px; text-align: justify;">A binary arithmetic example is given to understand the operation more clearly.</span><br />
<br />
<div style="text-align: justify;">
<span style="color: blue; font-family: Verdana, Geneva, sans-serif;"><span style="font-size: 18px;"><i>Example:-</i></span></span></div>
<span style="color: #505050; font-family: Verdana, Geneva, sans-serif; font-size: 18px; text-align: justify;"><br /></span>
<span style="color: #505050; font-family: Verdana, Geneva, sans-serif; font-size: 18px; text-align: justify;"><br /></span>
<span style="white-space: nowrap;"><span class="bigbul" style="font-size: 27.2px;">•</span> 1011011 − 10010 = 1001001:</span><br />
<table class="diff" style="border-collapse: collapse; margin-left: 20pt;"><tbody>
<tr class="borrows"><td style="font-size: 12.8px; width: 18pt;"></td><td class="btop" style="font-size: 12.8px; width: 18pt;"></td><td class="btop" style="font-size: 12.8px; width: 18pt;"></td><td class="btop" style="font-size: 12.8px; width: 18pt;"></td><td class="btop" style="font-size: 12.8px; width: 18pt;"></td><td class="btop" style="font-size: 12.8px; width: 18pt;"></td><td class="btop" style="font-size: 12.8px; width: 18pt;"></td><td class="btop" style="font-size: 12.8px; width: 18pt;"></td></tr>
<tr class="mhend"><td style="width: 18pt;"></td><td class="menorm" style="width: 18pt;">1</td><td class="menorm" style="width: 18pt;">0</td><td class="menorm" style="width: 18pt;">1</td><td class="menorm" style="width: 18pt;">1</td><td class="menorm" style="width: 18pt;">0</td><td class="menorm" style="width: 18pt;">1</td><td class="menorm" style="width: 18pt;">1</td></tr>
<tr class="sbhend"><td style="width: 18pt;">−</td><td class="tb" style="width: 18pt;"></td><td class="tb" style="width: 18pt;"></td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td></tr>
<tr class="diff"><td style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;"></td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td></tr>
</tbody></table>
<br /><span style="white-space: nowrap;"><span class="bigbul" style="font-size: 27.2px;">•</span> 1010110 − 101010 = 101100:</span><br />
<table class="diff" style="border-collapse: collapse; margin-left: 20pt;"><tbody>
<tr class="borrows"><td style="font-size: 12.8px; width: 18pt;"></td><td class="btop" style="font-size: 12.8px; width: 18pt;">0</td><td class="btop" style="font-size: 12.8px; width: 18pt;"></td><td class="btop" style="font-size: 12.8px; width: 18pt;">0</td><td class="btop" style="font-size: 12.8px; width: 18pt;"></td><td class="btop" style="font-size: 12.8px; width: 18pt;"></td><td class="btop" style="font-size: 12.8px; width: 18pt;"></td><td class="btop" style="font-size: 12.8px; width: 18pt;"></td></tr>
<tr class="mhend"><td style="width: 18pt;"></td><td class="mecross" style="width: 18pt;"><span class="exout" style="font-size: 17.6px; position: absolute;">×</span>1</td><td class="menorm" style="width: 18pt;"><span class="raise" style="bottom: 5px; font-size: 12.8px; position: relative;">1</span>0</td><td class="mecross" style="width: 18pt;"><span class="exout" style="font-size: 17.6px; position: absolute;">×</span>1</td><td class="menorm" style="width: 18pt;"><span class="raise" style="bottom: 5px; font-size: 12.8px; position: relative;">1</span>0</td><td class="menorm" style="width: 18pt;">1</td><td class="menorm" style="width: 18pt;">1</td><td class="menorm" style="width: 18pt;">0</td></tr>
<tr class="sbhend"><td style="width: 18pt;">−</td><td class="tb" style="width: 18pt;"></td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td></tr>
<tr class="diff"><td style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;"></td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;"></td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td></tr>
</tbody></table>
<br /><span style="white-space: nowrap;"><span class="bigbul" style="font-size: 27.2px;">•</span> 1000101 − 101100 = 11001:</span><br />
<table class="diff" style="border-collapse: collapse; margin-left: 20pt;"><tbody>
<tr class="borrows"><td style="font-size: 12.8px; width: 18pt;"></td><td class="btop" style="font-size: 12.8px; width: 18pt;">0</td><td class="btop" style="font-size: 12.8px; width: 18pt;">1</td><td class="btop" style="font-size: 12.8px; width: 18pt;">1</td><td class="btop" style="font-size: 12.8px; width: 18pt;"></td><td class="btop" style="font-size: 12.8px; width: 18pt;"></td><td class="btop" style="font-size: 12.8px; width: 18pt;"></td><td class="btop" style="font-size: 12.8px; width: 18pt;"></td></tr>
<tr class="mhend"><td style="width: 18pt;"></td><td class="mecross" style="width: 18pt;"><span class="exout" style="font-size: 17.6px; position: absolute;">×</span>1</td><td class="mecross" style="width: 18pt;"><span class="exout" style="font-size: 17.6px; position: absolute;">×</span><span class="raise" style="bottom: 5px; font-size: 12.8px; position: relative;">1</span>0</td><td class="mecross" style="width: 18pt;"><span class="exout" style="font-size: 17.6px; position: absolute;">×</span><span class="raise" style="bottom: 5px; font-size: 12.8px; position: relative;">1</span>0</td><td class="menorm" style="width: 18pt;"><span class="raise" style="bottom: 5px; font-size: 12.8px; position: relative;">1</span>0</td><td class="menorm" style="width: 18pt;">1</td><td class="menorm" style="width: 18pt;">0</td><td class="menorm" style="width: 18pt;">1</td></tr>
<tr class="sbhend"><td style="width: 18pt;">−</td><td class="tb" style="width: 18pt;"></td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">0</td></tr>
<tr class="diff"><td style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;"></td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;"></td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;"></td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td></tr>
</tbody></table>
<br /><span style="white-space: nowrap;"><span class="bigbul" style="font-size: 27.2px;">•</span> 100010110 − 1111010 = 10011100:</span><br />
<table class="diff" style="border-collapse: collapse; margin-left: 20pt;"><tbody>
<tr class="borrows"><td style="font-size: 12.8px; width: 18pt;"></td><td class="btop" style="font-size: 12.8px; width: 18pt;">0</td><td class="btop" style="font-size: 12.8px; width: 18pt;">1</td><td class="btop" style="font-size: 12.8px; width: 18pt;">1</td><td class="btop" style="font-size: 12.8px; width: 18pt;">1</td><td class="btop" style="font-size: 12.8px; width: 18pt;"><span class="raise" style="bottom: 5px; font-size: 10.24px; position: relative;">1</span>0</td><td class="btop" style="font-size: 12.8px; width: 18pt;"></td><td class="btop" style="font-size: 12.8px; width: 18pt;"></td><td class="btop" style="font-size: 12.8px; width: 18pt;"></td><td class="btop" style="font-size: 12.8px; width: 18pt;"></td></tr>
<tr class="mhend"><td style="width: 18pt;"></td><td class="mecross" style="width: 18pt;"><span class="exout" style="font-size: 17.6px; position: absolute;">×</span>1</td><td class="mecross" style="width: 18pt;"><span class="exout" style="font-size: 17.6px; position: absolute;">×</span><span class="raise" style="bottom: 5px; font-size: 12.8px; position: relative;">1</span>0</td><td class="mecross" style="width: 18pt;"><span class="exout" style="font-size: 17.6px; position: absolute;">×</span><span class="raise" style="bottom: 5px; font-size: 12.8px; position: relative;">1</span>0</td><td class="mecross" style="width: 18pt;"><span class="exout" style="font-size: 17.6px; position: absolute;">×</span><span class="raise" style="bottom: 5px; font-size: 12.8px; position: relative;">1</span>0</td><td class="mecross" style="width: 18pt;"><span class="exout" style="font-size: 17.6px; position: absolute;">×</span>1</td><td class="menorm" style="width: 18pt;"><span class="raise" style="bottom: 5px; font-size: 12.8px; position: relative;">1</span>0</td><td class="menorm" style="width: 18pt;">1</td><td class="menorm" style="width: 18pt;">1</td><td class="menorm" style="width: 18pt;">0</td></tr>
<tr class="sbhend"><td style="width: 18pt;">−</td><td class="tb" style="width: 18pt;"></td><td class="tb" style="width: 18pt;"></td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td></tr>
<tr class="diff"><td style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;"></td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;"></td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td></tr>
</tbody></table>
<br /><span style="white-space: nowrap;"><span class="bigbul" style="font-size: 27.2px;">•</span> 101101 − 100111 = 110:</span><br />
<table class="diff" style="border-collapse: collapse; margin-left: 20pt;"><tbody>
<tr class="borrows"><td style="font-size: 12.8px; width: 18pt;"></td><td class="btop" style="font-size: 12.8px; width: 18pt;"></td><td class="btop" style="font-size: 12.8px; width: 18pt;"></td><td class="btop" style="font-size: 12.8px; width: 18pt;">0</td><td class="btop" style="font-size: 12.8px; width: 18pt;"><span class="raise" style="bottom: 5px; font-size: 10.24px; position: relative;">1</span>0</td><td class="btop" style="font-size: 12.8px; width: 18pt;"></td><td class="btop" style="font-size: 12.8px; width: 18pt;"></td></tr>
<tr class="mhend"><td style="width: 18pt;"></td><td class="menorm" style="width: 18pt;">1</td><td class="menorm" style="width: 18pt;">0</td><td class="mecross" style="width: 18pt;"><span class="exout" style="font-size: 17.6px; position: absolute;">×</span>1</td><td class="mecross" style="width: 18pt;"><span class="exout" style="font-size: 17.6px; position: absolute;">×</span>1</td><td class="menorm" style="width: 18pt;"><span class="raise" style="bottom: 5px; font-size: 12.8px; position: relative;">1</span>0</td><td class="menorm" style="width: 18pt;">1</td></tr>
<tr class="sbhend"><td style="width: 18pt;">−</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">1</td></tr>
<tr class="diff"><td style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;"></td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;"></td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;"></td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;"></td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">0</td></tr>
</tbody></table>
<br /><span style="white-space: nowrap;"><span class="bigbul" style="font-size: 27.2px;">•</span> 1110110 − 1010111 = 11111:</span><br />
<table class="diff" style="border-collapse: collapse; margin-left: 20pt;"><tbody>
<tr class="borrows"><td style="font-size: 12.8px; width: 18pt;"></td><td class="btop" style="font-size: 12.8px; width: 18pt;"></td><td class="btop" style="font-size: 12.8px; width: 18pt;">0</td><td class="btop" style="font-size: 12.8px; width: 18pt;"><span class="raise" style="bottom: 5px; font-size: 10.24px; position: relative;">1</span>0</td><td class="btop" style="font-size: 12.8px; width: 18pt;">1</td><td class="btop" style="font-size: 12.8px; width: 18pt;"><span class="raise" style="bottom: 5px; font-size: 10.24px; position: relative;">1</span>0</td><td class="btop" style="font-size: 12.8px; width: 18pt;"><span class="raise" style="bottom: 5px; font-size: 10.24px; position: relative;">1</span>0</td><td class="btop" style="font-size: 12.8px; width: 18pt;"></td></tr>
<tr class="mhend"><td style="width: 18pt;"></td><td class="menorm" style="width: 18pt;">1</td><td class="mecross" style="width: 18pt;"><span class="exout" style="font-size: 17.6px; position: absolute;">×</span>1</td><td class="mecross" style="width: 18pt;"><span class="exout" style="font-size: 17.6px; position: absolute;">×</span>1</td><td class="mecross" style="width: 18pt;"><span class="exout" style="font-size: 17.6px; position: absolute;">×</span><span class="raise" style="bottom: 5px; font-size: 12.8px; position: relative;">1</span>0</td><td class="mecross" style="width: 18pt;"><span class="exout" style="font-size: 17.6px; position: absolute;">×</span>1</td><td class="mecross" style="width: 18pt;"><span class="exout" style="font-size: 17.6px; position: absolute;">×</span>1</td><td class="menorm" style="width: 18pt;"><span class="raise" style="bottom: 5px; font-size: 12.8px; position: relative;">1</span>0</td></tr>
<tr class="sbhend"><td style="width: 18pt;">−</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">0</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">1</td><td class="tb" style="width: 18pt;">1</td></tr>
<tr class="diff"><td style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;"></td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;"></td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;"></td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td><td class="tb" style="border-top: 1px solid black; padding-top: 0pt; width: 18pt;">1</td></tr>
</tbody></table>
<h2 id="3" style="color: #4286f4; font-family: Verdana, Geneva, sans-serif; font-size: 20px; margin: 10px 0px 0px; padding: 0px;">
Binary Multiplication:-</h2>
<div>
<br /></div>
<span style="color: #505050; font-family: Verdana, Geneva, sans-serif; font-size: 18px; text-align: justify;">Here are also four steps to be followed, which are</span><ul style="color: #505050; font-family: Verdana, Geneva, sans-serif; font-size: 18px; margin: 0px; padding: 0px; text-align: justify;">
<li style="margin: 0px 0px 0px 20px; padding: 0px;">0×0=0</li>
<li style="margin: 0px 0px 0px 20px; padding: 0px;">1×0=0</li>
<li style="margin: 0px 0px 0px 20px; padding: 0px;">0×1=0</li>
<li style="margin: 0px 0px 0px 20px; padding: 0px;">1×1=1 (there is no carry or borrow for this)</li>
</ul>
<div style="text-align: justify;">
<span style="color: #505050; font-family: Verdana, Geneva, sans-serif;"><span style="font-size: 18px;"><br /></span></span></div>
</div>
<div style="text-align: justify;">
<i style="color: blue; font-family: Verdana, Geneva, sans-serif; font-size: 18px;">Example:-</i></div>
<div style="text-align: justify;">
<i style="color: blue; font-family: Verdana, Geneva, sans-serif; font-size: 18px;"> </i></div>
<div style="text-align: justify;">
<table cellpadding="0" cellspacing="0" style="background-color: white;"><tbody>
<tr align="right" valign="top"><td rowspan="6"><kbd>00101001 × 00000110 = 11110110</kbd></td><td rowspan="6"> </td></tr>
</tbody></table>
</div>
<div style="text-align: justify;">
<i style="color: blue; font-family: Verdana, Geneva, sans-serif; font-size: 18px;"><br /></i></div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<table cellpadding="0" cellspacing="0" style="background-color: white;"><tbody>
<tr align="right" valign="top"><td><kbd>0 0 1 0 1 0 0 1</kbd></td><td> = </td><td>41<sub>(base 10)</sub></td></tr>
<tr align="right" valign="top"><td><kbd>× 0 0 0 0 0 1 1 0</kbd><hr noshade="" />
</td><td> = </td><td>6<sub>(base 10)</sub></td></tr>
<tr align="right" valign="top"><td><kbd>0 0 0 0 0 0 0 0</kbd></td><td colspan="2"> </td></tr>
<tr align="right" valign="top"><td><kbd>0 0 1 0 1 0 0 1 </kbd></td><td colspan="2"> </td></tr>
<tr align="right" valign="top"><td><kbd>0 0 1 0 1 0 0 1 </kbd><hr noshade="" />
</td><td colspan="2"> </td></tr>
<tr align="right" valign="top"><td><kbd>0 0 1 1 1 1 0 1 1 0</kbd></td><td> = </td><td>246<sub>(base 10)</sub></td></tr>
<tr><td colspan="5"> </td></tr>
</tbody></table>
<br /><span style="background-color: white; font-family: monospace; text-align: -webkit-right;">00010111 × 00000011 = 01000101</span></div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<sub><br /></sub></div>
<div style="text-align: justify;">
<sub><br /></sub></div>
<div style="text-align: justify;">
<table cellpadding="0" cellspacing="0" style="background-color: white;"><tbody>
<tr align="right" valign="top"><td><kbd>0 0 0 1 0 1 1 1</kbd></td><td> = </td><td>23<sub>(base 10)</sub></td></tr>
<tr align="right" valign="top"><td><kbd>× 0 0 0 0 0 0 1 1</kbd><hr noshade="" />
</td><td> = </td><td>3<sub>(base 10)</sub></td></tr>
<tr align="right" valign="top"><td><kbd><em> 1 1 1 1 1 </em></kbd></td><td> </td><td align="left"><em>carries</em></td></tr>
<tr align="right" valign="top"><td><kbd>0 0 0 1 0 1 1 1</kbd></td><td colspan="2"> </td></tr>
<tr align="right" valign="top"><td><kbd>0 0 0 1 0 1 1 1 </kbd><hr noshade="" />
</td><td colspan="2"> </td></tr>
<tr align="right" valign="top"><td><kbd>0 0 1 0 0 0 1 0 1</kbd></td><td> = </td><td>69<sub>(base 10</sub></td></tr>
</tbody></table>
</div>
<div style="text-align: justify;">
<sub><br /></sub></div>
<div style="text-align: justify;">
<sub><br /></sub></div>
<div style="text-align: justify;">
<h2 style="color: #4286f4; font-family: Verdana, Geneva, sans-serif; font-size: 20px; margin: 10px 0px 0px; padding: 0px; text-align: left;">
Binary Division:-</h2>
<a href="https://www.electrical4u.com/binary-division/" style="color: #cc0000; font-family: Verdana, Geneva, sans-serif; font-size: 18px; margin: 0px; padding: 0px; text-decoration-line: none;">Binary division</a><span style="color: #505050; font-family: Verdana, Geneva, sans-serif; font-size: 18px;"> is comprised of other two binary arithmetic operations, multiplication and subtraction; an example will explain the operation more easily.</span></div>
<div style="text-align: justify;">
<sub><br /></sub></div>
<div style="text-align: justify;">
<span style="color: blue;"><i>Example:-</i></span></div>
<div style="text-align: justify;">
<span style="color: blue;"><i><br /></i></span></div>
<div style="text-align: justify;">
<table cellpadding="0" cellspacing="0" style="background-color: white;"><tbody>
<tr align="right" valign="top"><td rowspan="13"><kbd>00101010 ÷ 00000110 = 00000111</kbd></td><td rowspan="13"> </td><td rowspan="2"> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td><kbd> 1 </kbd></td><td><kbd> 1 </kbd></td><td><kbd> 1 </kbd></td><td rowspan="2"> = </td><td rowspan="2">7<sub>(base 10)</sub></td></tr>
<tr align="right" valign="top"><td colspan="9"><hr noshade="" />
</td></tr>
<tr align="right" valign="middle"><td><kbd>1 1 0 </kbd></td><td><span>)</span></td><td><kbd> 0 </kbd></td><td><kbd> 0 </kbd></td><td><kbd> <strike>1</strike> </kbd></td><td><kbd><sup><em>1</em></sup>0 </kbd></td><td><kbd> 1 </kbd></td><td><kbd> 0 </kbd></td><td><kbd> 1 </kbd></td><td><kbd> 0 </kbd></td><td> = </td><td>42<sub>(base 10)</sub></td></tr>
<tr align="right" valign="top"><td rowspan="10"> </td><td rowspan="10"> </td><td> </td><td> </td><td><kbd>- </kbd></td><td><kbd> 1 </kbd></td><td><kbd> 1 </kbd></td><td><kbd> 0 </kbd></td><td> </td><td> </td><td rowspan="2"> = </td><td rowspan="2">6<sub>(base 10)</sub></td></tr>
<tr align="right" valign="top"><td colspan="2"> </td><td colspan="4"><hr noshade="" />
</td><td colspan="2"> </td></tr>
<tr align="left" valign="top"><td> </td><td> </td><td> </td><td> </td><td><kbd> <em>1</em> </kbd></td><td> </td><td> </td><td> </td><td rowspan="8"> </td><td rowspan="8"><em>borrows</em></td></tr>
<tr align="right" valign="top"><td> </td><td> </td><td> </td><td><kbd><strike>1</strike> </kbd></td><td><kbd> <strike>0</strike> </kbd></td><td><kbd><sup><em>1</em></sup>0 </kbd></td><td><kbd> 1 </kbd></td><td> </td></tr>
<tr align="right" valign="top"><td> </td><td> </td><td> </td><td><kbd>- </kbd></td><td><kbd> 1 </kbd></td><td><kbd> 1 </kbd></td><td><kbd> 0 </kbd></td><td> </td></tr>
<tr align="right" valign="top"><td colspan="3"> </td><td colspan="4"><hr noshade="" />
</td><td> </td></tr>
<tr align="right" valign="top"><td> </td><td> </td><td> </td><td> </td><td><kbd> </kbd></td><td><kbd> 1 </kbd></td><td><kbd> 1 </kbd></td><td><kbd> 0 </kbd></td></tr>
<tr align="right" valign="top"><td> </td><td> </td><td> </td><td> </td><td><kbd>- </kbd></td><td><kbd> 1 </kbd></td><td><kbd> 1 </kbd></td><td><kbd> 0 </kbd></td></tr>
<tr align="right" valign="top"><td colspan="4 "> </td><td colspan="4"><hr noshade="" />
</td></tr>
<tr align="right" valign="top"><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td><kbd> 0 </kbd></td></tr>
<tr><td colspan="14"><br /><table cellpadding="0" cellspacing="0"><tbody>
<tr align="right" valign="top"><td rowspan="18"><kbd>10000111 ÷ 00000101 = 00011011</kbd></td><td rowspan="18"> </td><td rowspan="2"> </td><td> </td><td> </td><td> </td><td> </td><td><kbd> 1 </kbd></td><td><kbd> 1 </kbd></td><td><kbd> 0 </kbd></td><td><kbd> 1 </kbd></td><td><kbd> 1 </kbd></td><td rowspan="2"> = </td><td rowspan="2">27<sub>(base 10)</sub></td></tr>
<tr align="right" valign="top"><td colspan="9"><hr noshade="" />
</td></tr>
<tr align="right" valign="middle"><td><kbd>1 0 1 </kbd></td><td><span>)</span></td><td><kbd> <strike>1</strike> </kbd></td><td><kbd> <strike>0</strike> </kbd></td><td><kbd> <strike>0</strike> </kbd></td><td><kbd><sup><em>1</em></sup>0 </kbd></td><td><kbd> 0 </kbd></td><td><kbd> 1 </kbd></td><td><kbd> 1 </kbd></td><td><kbd> 1 </kbd></td><td> = </td><td>135<sub>(base 10)</sub></td></tr>
<tr align="right" valign="top"><td rowspan="15"> </td><td rowspan="15"> </td><td><kbd>- </kbd></td><td><kbd> 1 </kbd></td><td><kbd> 0 </kbd></td><td><kbd> 1 </kbd></td><td> </td><td> </td><td> </td><td> </td><td rowspan="15"> = </td><td rowspan="15">5<sub>(base 10)</sub></td></tr>
<tr align="right" valign="top"><td colspan="4"><hr noshade="" />
</td><td colspan="4"> </td></tr>
<tr align="right" valign="top"><td> </td><td> </td><td><kbd> 1 </kbd></td><td><kbd> <strike>1</strike> </kbd></td><td><kbd><sup><em>1</em></sup>0 </kbd></td><td> </td><td> </td><td> </td></tr>
<tr align="right" valign="top"><td> </td><td><kbd>- </kbd></td><td><kbd> 1 </kbd></td><td><kbd> 0 </kbd></td><td><kbd> 1 </kbd></td><td> </td><td> </td><td> </td></tr>
<tr align="right" valign="top"><td> </td><td colspan="4"><hr noshade="" />
</td><td colspan="3"> </td></tr>
<tr align="right" valign="top"><td> </td><td> </td><td> </td><td> </td><td><kbd> 1 </kbd></td><td><kbd> 1 </kbd></td><td> </td><td> </td></tr>
<tr align="right" valign="top"><td> </td><td> </td><td> </td><td><kbd>- </kbd></td><td><kbd> </kbd></td><td><kbd> 0 </kbd></td><td> </td><td> </td></tr>
<tr align="right" valign="top"><td colspan="3"> </td><td colspan="3"><hr noshade="" />
</td><td colspan="2"> </td></tr>
<tr align="right" valign="top"><td> </td><td> </td><td> </td><td> </td><td><kbd> 1 </kbd></td><td><kbd> 1 </kbd></td><td><kbd> 1 </kbd></td><td> </td></tr>
<tr align="right" valign="top"><td> </td><td> </td><td> </td><td><kbd>- </kbd></td><td><kbd> 1 </kbd></td><td><kbd> 0 </kbd></td><td><kbd> 1 </kbd></td><td> </td></tr>
<tr align="right" valign="top"><td colspan="3"> </td><td colspan="4"><hr noshade="" />
</td><td> </td></tr>
<tr align="right" valign="top"><td> </td><td> </td><td> </td><td> </td><td> </td><td><kbd> 1 </kbd></td><td><kbd> 0 </kbd></td><td><kbd> 1 </kbd></td></tr>
<tr align="right" valign="top"><td> </td><td> </td><td> </td><td> </td><td><kbd>- </kbd></td><td><kbd> 1 </kbd></td><td><kbd> 0 </kbd></td><td><kbd> 1 </kbd></td></tr>
<tr align="right" valign="top"><td colspan="4"> </td><td colspan="4"><hr noshade="" />
</td></tr>
<tr align="right" valign="top"><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td><kbd> 0<br /></kbd></td></tr>
</tbody></table>
</td></tr>
</tbody></table>
</div>
<div style="text-align: justify;">
<sub><br /></sub></div>
<div>
<sub><br /></sub></div>
</div>
<head>
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