# Op amp equations and formula

Operational amplifiers (op-amps) are fundamental building blocks in the design of analog electronic circuits. They have wide range of applications due to their high gain, higher slew rate, high CMRR and other desirable characteristics. Op-amp formulas and equations are crucial to design, analyze and troubleshoot op-amp circuits.

## Op amp gain

Figure-1 mentions op amp gain for inverting type and non inverting type operational amplifier(op amp). As mentioned, Op amp gain for inverting and non inverting type in terms of voltage is ratio of output by input.

It is expressed as follows.
op amp gain = Vout/Vin
op amp gain (inverting type) = - Rf/Rin
op amp gain (non inverting type) = 1 + Rf/Rin

Let us understand difference between various op amp gain types such as closed-loop gain, open-loop gain, signal gain, loop gain and noise gain. Signal gain is same as mentioned in above op amp equations. Noise gain for both inverting and non inverting op amp is expressed as follows.
Noise gain = 1 + Rf/Rin
Figure-2 mentions comparison between various op amp gain types.

In practice, op-amp gains can be very high, often in the range of tens of thousands to hundreds of thousands. Higher op amp gain can lead to greater sensitivity to noise and stability issues. Moreover lower gain might not provide required signal amplification. Hence higher gain is derirable.

## op amp CMRR

The full form of CMRR is common mode rejection ratio. If the signal is applied with equal amount to both the inputs of op amp such that differential input voltage is un-affected then output should not be affected as well. In general changes in common mode voltage will produce change in the output. The op amp CMRR is the ratio of the common mode gain to differential mode gain.
CMRR = common mode gain/differential mode gain
It is expressed in dB.

A higher CMRR is desirable, as it indicates better rejection of unwanted common-mode signals.

## op amp Full Power Bandwidth

It is the max. output frequency at which slew limiting happens. It is expressed as follows.
FPBW( Full Power Bandwidth ) = Slew rate/(2*pi*Vp)

The op amp FPBW should be approx. 5 to 10 times than max. output frequency This is needed to obtain acceptable distortion performance using op amp circuit.

Ideally, this bandwidth would be infinite, which indicates that the op-amp can handle signals of any frequency without distortion. However, in practice, op-amps have limited bandwidth due to internal limitations. For high-frequency applications, a higher bandwidth is preferable.

## op amp Slew rate formula

The rate of change in the output voltage with respect to time for a step change at the input of op amp is referred as slew rate. It is expressed as V/s or V/µs
Typically op amps will have different slew rates during the negative and positive transitions.

Let us assume sine wave with amplitude of 2Vp and frequency f
Output voltage V(t) = Vp*sin(2*pi*f*t)

Maximum slew rate for this is expressed as
dv/dt(max.) = 2* pi *f*Vp

The slew rate is generally measured between 10% and 90% of the final value. In practice, the slew rate of op amp should be high so that op-amp can respond to rapid changes of input signal such as high frequency applications or pulse amplification.