**Theory**

**Operational Amplifiers****(OAs)**

*Operational Amplifiers***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**

**on their input signal(s).**

*mathematical operations*
Figure 1 shows the symbol for an OA. There are two inputs, the

**(-) and the***inverting input***(+). These symbols have nothing to do with the polarity of the applied input signals.***non-inverting input***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**of the OA.***open loop gain**A*is infinite for the ideal amplifier, whereas for the various types of real OAs, it is usually within the range of 10^{4}to 10^{6}.
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**. Any result expected to be outside this range is clipped to the respective limit, and OA is in a***operational range***stage.***saturation*
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

**. 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***negative feedback loop***.***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

**, i.e. the mathematical equation describing the output signal (v***transfer function*_{o}) as a function of the input signal (v_{i}) or signals (v_{1}, v_{2}, …, 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 10

^{6}-10^{12}MΩ, for the ideal OA this is infinite), thus no current flows into them.

**Inverting Amplifier**
The basic circuit of the

**is shown in Figure 2.***inverting amplifier***Figure 2.**Inverting amplifier.

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

i

_{1}= (v_{i}- v_{s})/R_{i}and i_{2}= (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:
i

_{1}= v_{i}/R_{i}and i_{2}= - v_{o}/R_{f}
Since there is no current flow to any input (simplifying assumption #2), it is

i

_{1}= i_{2}
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_{o}are within the operational range of the OA.#
**Summing Amplifier**

The

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

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

v

_{o}= -(v_{1}/R_{1}+ v_{2}/R_{2}+ … + 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**

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} = (v_{2} - v_{1}) R_{f}/R_{i}

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

**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).***floating 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)]v_{2}- pv_{1}#
**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

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