Op Amp Gain: explanation & equations

Operational amplifiers have very different levels of gain when used open or closed circuit. Using negative feedback provides very controllable gain and characteristics.

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When used without feedback, i.e. open loop, operational amplifiers have exceedingly high levels of gain.

By using feedback negative feedback, it is possible to precisely control the gain and other characteristics of the amplifier.

By using negative feedback, the very high gain with high levels of variation of the open loop op-amp, can be used to provide a very repeatable level of gain, not dependent upon the basic open loop amp gain.

Op amp gain basics

Operational amplifiers are normally used with feedback around the amplifier element itself. This tailors the performance to what is needed. There are two scenarios for which the gain can be considered:

  • Open loop gain:   This form of gain is measured when no feedback is applied to the op amp. In other words it is running in an open loop format. Gain figures for the op amp in this configuration are normally very high, typically between 10 000 and 100 000. This is the gain of the operational amplifier on its own.

    Figures are often quoted in the op amp datasheets in terms of volts per millivolt, V/mV. Quoting the the gain in these terms enables the gain to be written in a more convenienet format. 10 V/mV corresponds to a voltage gain of 10 000. It saves writing many zeros.
  • Closed loop gain:   This form of gain is measured when the feedback loop is operation, i.e. a closed loop. By applying negative feedback, the overall gain of the circuit is much reduced, and can be accurately tailored to the required level or to produce the required output format as in the case of filters, integrators, etc.. The gain is measured with the loop closed and provided there is a sufficient difference between the open loop and closed loop gain, the circuit will operate according to the feedback placed around it. Although negative feedback is normally used for analogue circuits, there are instances where positive feedback is used. The most common application is for comparators where the output is required at one of two levels. The Schmitt trigger is one example where hysteresis is introduced into the system

Generalised op-amp gain

Negative feedback is used to control the gain of the overall op amp circuit. There are many ways in which the feedback can be applied - it may be independent of frequency, or it may be frequency dependent to produce filters for example.

However it is possible to produce a generalised concept for applying negative feedback. From this the more specific scenarios can be developed.

Generic op amp gain feedback configuration
Generic op amp negative feedback configuration

It is possible to calculate a general formula for the op amp gain in the circuit:

V sum = V in - B   V out

The output voltage can then be calculated from a knowledge of the input voltage, gain and feedback:

V out = A   V sum = A   V in - A   B   V out

This can now be used to generate the generic closed loop op amp gain equation.

V out V in = G = A 1   +   A B

Using this generic equation it is possible to develop equations for more specific scenarios. The feedback can be frequency dependent, or flat as required.

The two simplest examples of op am circuits using feedback are the formats for inverting and non-inverting amplifiers.

Inverting op-amp gain

The circuit for the inverting op-amp circuit is shown below. This circuit has the output 180 degrees out of phase with the input and also provides a virtual earth input.

Basic inverting operational amplifier circuit

Basic inverting operational amplifier circuit

It is easy to derive the op-amp gain equation. The input to the op-amp itself draws no current and this means that the current flowing in the resistors R1 and R2 is the same. Using ohms law Vout /R2 = -Vin/R1. Hence the voltage gain of the circuit Av can be taken as:

A v = - R2 R1

As an example, an amplifier requiring a gain of ten could be built by making R2 47 k ohms and R1 4.7 k ohms.

Read more about . . . . the inverting op amp circuit.

Non-Inverting op-amp gain

The circuit for the non-inverting op-amp is shown below. It offers a higher input impedance than the inverting op amp circuit.

Basic non-inverting operational amplifier circuit

Basic non-inverting operational amplifier circuit

The gain of the non-inverting circuit for the operational amplifier is easy to determine. The calculation hinges around the fact that the voltage at both inputs is the same. This arises from the fact that the gain of the amplifier is exceedingly high. If the output of the circuit remains within the supply rails of the amplifier, then the output voltage divided by the gain means that there is virtually no difference between the two inputs.

As the input to the op-amp draws no current this means that the current flowing in the resistors R1 and R2 is the same. The voltage at the inverting input is formed from a potential divider consisting of R1 and R2, and as the voltage at both inputs is the same, the voltage at the inverting input must be the same as that at the non-inverting input. This means that Vin = Vout x R1 / (R1 + R2). Hence the op amp gain equation for the voltage gain of the circuit Av can be taken as:

A v = 1 + R2 R1

As an example, an amplifier requiring a gain of eleven could be built by making R2 47 k ohms and R1 4.7 k ohms.

Op-amp gain is very easy to determine. The calculations for the different circuits is slightly different, but essentially both circuits are able to offer similar levels of gain, although the resistor values will not be the same for the same levels of op amp gain.

Read more about . . . . the non-inverting op amp circuit.

Op amp gain in other situations

It is normal to use operational amplifiers in linear applications with negative feedback. This utilises the very high gain of the open loop amplifier to provide repeatable performance governed by the external components.

However it is also possible to use operational amplifiers with other forms of feedback to produce other effects. Normally this type of feedback is used to provide switching, for which comparators provide much better performance as they operator much faster and do not suffer from latching issues, but that does not mean that the basic principles of positive feedback do not apply.

That said, negative feedback is by the most widely used form of feedback for analogue, linear applications.

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