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.

One of the key aspects of the performance of operational amplifiers is their gain. Operational amplifiers on their own offer huge levels of gain when used in what is termed an open loop configuration.

Under open loop conditions, the op amp gain may be anything upwards of 10 000, with some operational amplifier gain levels extending to well over ten times this figure. Even with op amps of the same type there may be large gain variations as a result of the fabrication processes used.

Whilst op amps themselves offer huge levels of gain, this gain is seldom used in this form to provide signal amplification - it would be hugely difficult to utilise if an analogue signal needed to retain its integrity with minimal distortion.

By using a technique known as negative feedback, the huge levels of gain can be used to good effect providing flat frequency responses, low distortion, and very defined levels of gain for the overall circuit, not dependent upon the actual gain of the IC, but on that of the external components whose values can be accurately chosen.

In some circumstances positive feedback may be used, but this is normally undertaken in a particular way to achieve a particular effect.

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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 convenient 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

One aspect closely associated with operational amplifier gain is the bandwidth. The huge gain of operational amplifiers can lead to instability if steps are not taken to ensure that the op amp and its circuit remain stable, even with negative feedback applied.

A technique known as compensation is used. In early op amps, external components were used to add the compensation, but in later chips, it was added internally. In its basic terms a small capacitor is added to the internal elements of the op amp. This has the effect of reducing tendency to oscillate, but it also reduces the open loop bandwidth.

Although the open loop bandwidth of the op amp is reduced, once negative feedback has been applied, a sufficient level gain with a flat frequency response can be achieved for most purposes.

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.

The use of feedback and controlling the gain of operational amplifiers is key to their use. The open loop gain is not normally used because it is undefined and not controlled.

By applying negative feedback, the very high levels of gain can be exchanged for a lower level of gain, but with much higher levels of performance in terms of distortion, frequency response and the functions that can be attained, e.g. filters, etc..

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

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

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

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

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 amp 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° out of phase with the input and also provides a virtual earth input.

The circuit is quite straightforward using a single feedback resistor from the output to the inverting input, and a resistor from the inverting input to the input of the circuit. The non-inverting input is taken a ground point.

It is easy to derive the op-amp gain equation. The input to the op-amp itself draws no current as far as our calculations are concerned as the impedance of each input both e amplifier will be well above 100kΩ and possibly well over 1MΩ. This means that any current flowing into the chip can be ignored.

From this we can see that the current flowing in the resistors R1 and R2 is the same, because no current is flowing out of the junction between the two resistors.

Using ohms law Vout /R2 = -Vin/R1. Hence the voltage gain of the circuit Av can be taken as:

$Av=-\frac{\mathrm{R2}}{\mathrm{R1}}$

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.

The non-inverting amplifier also has the characteristic that the input and output are in the same phase as a result of the signal being applied to the non-inverting input of the op amp.

The gain of the non-inverting circuit for the operational amplifier is also 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.

We can assume that fort he purpose of our calculation, the input to the op-amp draws no current as the input impedance both e chip inputs will be well above the resistor values used.

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:

$Av=1+\frac{\mathrm{R2}}{\mathrm{R1}}$

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, although this is not always the case. This utilises the very high gain of the open loop amplifier to provide repeatable performance governed by the external components.

Examples of these circuits include amplifiers, filters, differentiators and integrators.

However it is also possible to use operational amplifiers with other forms of feedback to produce other effects.

One of the applications of using positive feedback within an op amp circuit 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. However the basic principles of feedback and gain still apply to this type of IC or circuit block.

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

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