Analog Electronics

Theory

Operational Amplifiers

Operational Amplifiers (OAs) 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 mathematical operations on their input signal(s).

 Figure 1 shows the symbol for an OA. There are two inputs, the inverting input (-) and the non-inverting input (+). These symbols have nothing to do with the polarity of the applied input signals.

 

 Figure 1. Symbol of the operational amplifier. Connections to power supplies are also shown.

The output signal (voltage), v_o, is given by: 

v_o = A(v₊ - v_-)

v₊ and v_- are the signals applied to the non-inverting and to the inverting input, respectively. Α represents the open loop gain of the OA. A is infinite for the ideal amplifier, whereas for the various types of real OAs, it is usually within the range of 10⁴ to 10⁶.

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_o) 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_o within the (approximately) -13 to +13 V range, called operational range. Any result expected to be outside this range is clipped to the respective limit, and OA is in a saturation stage.

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.

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 negative feedback loop. 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 closed loop gain.

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.

There is a plethora of circuits with OAs performing various mathematical operations. Each circuit is characterized by its own transfer function, i.e. the mathematical equation describing the output signal (v_o) as a function of the input signal (v_i) or signals (v₁, v₂, …, v_n). Generally, transfer functions can be derived by applying Kirchhoff’s rules and the following two simplifying assumptions:

#1. The output signal (v_o) acquires a value that (through the feedback circuits) practically equates the voltages applied to both inputs, i.e. v₊ ≈ v_-.

#2. The input resistance of both OA inputs is extremely high(usually within the range 10⁶-10¹² MΩ, for the ideal OA this is infinite), thus no current flows into them.

Inverting Amplifier

The basic circuit of the inverting amplifier is shown in Figure 2.

 

Figure 2. Inverting amplifier.

The transfer function is derived as follows: Considering the arbitrary current directions we have:

 i₁ = (v_i - v_s)/R_i   and   i₂ = (v_s - v_o)/R_f

 The non-inverting input is connected directly to the circuit common (i.e. v₊ = 0 V), therefore (considering simplifying assumption #1) v_s = v_- = 0 V, therefore:

i₁ = v_i/R_i   and   i₂ = - v_o/R_f

 Since there is no current flow to any input (simplifying assumption #2), it is

 i₁ = i₂

Therefore, the transfer function of the inverting amplifier is

v_o = -(R_f/R_i)v_i

 Thus, the closed loop gain of the inverting amplifier is equal to the ratio of R_f (feedback resistor) over R_i (input resistor). This transfer function describes accurately the output signal as long as the closed loop gain is much smaller than the open loop gain A of the OA used (e.g. it must not exceed 1000), and the expected values of v_oare within the operational range of the OA.

 

Summing Amplifier

The summing amplifier  is a logical extension of the previously described circuit, with two or more inputs. Its circuit is shown in Figure 3.

 

Figure 3.  Summing amplifier.

The transfer function of the summing amplifier (similarly derived) is:

 v_o = -(v₁/R₁  +  v₂/R₂  +  …  +  v_n/R_n)R_f

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.

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).  

 

Difference amplifier

Difference amplifier precisely amplifies the difference of two input signals. Its typical circuit is shown in Figure 4.

 

 

Figure 4. Difference amplifier.

 

If R_i = R_i΄ and R_f = R_f΄, then the transfer function of the difference amplifier is:

v_o = (v₂ - v₁) R_f/R_i

The difference amplifier is useful for handling signals referring not to the circuit common, but to other signals, known as floating signal 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).

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 p = R_f/R_i and q = R_f΄/R_i΄ must be precisely equal, otherwise the signal output will be:

v_o = [q(p+1)/(q+1)]v₂ - pv₁

Differentiator

The differentiator generates an output signal proportional to the first derivative of the input with respect to time. Its typical circuit is shown in Figure 5.

 

Figure 5. Differentiator.

 

The transfer function of this circuit is

v_o = -RC(dv_i/dt)

 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).

Integrator

The integrator generates an output signal proportional to the time integral of the input signal. Its typical circuit is shown in Figure 6.

Figure 6. Integrator

 v_o = -(1/RC)∫v_i(t)dt

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.  

If the input signal is stable then the output from the integrator will be given by the equation

v_o = -(v_i/RC) t

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.