How does op-amp feedback work?

Fig. 1 shows an ideal linear time invariant feedback amplifier where the basic amplifier A (an amplifier with intrinsic or basic gain A) is in a circuit with an ideal negative feedback block B = beta. The circuit is ideal with no time delays and the feedback signal received at the summing node is exactly180 degrees out of phase with the input signal Vin. The feedback therefore subtracts from the input signal Vin --instead of in-phase with the input which would add to the signal strength and be an unstable condition (if A is unity or greater.) In-phase feedback is called positive feedback and is generally not found in op-amp or amplifier circuits but is commonly employed in comparator circuits to provide hysteresis. Vs is the voltage sum of the input signal and the signal A* B = A(beta).

We can now do some simple algebra to find the overall amplifier gain including the feedback.

Vout = Vs *A

Vs = Vin – B*Vout

Vin = Vs + B*Vout

The gain G can be written as

G = Vout / Vin = A*Vs/(Vs + B*Vs*A)

Canceling out Vs from the numerator and denominator (Vs not = 0) we have

G = A / (1 + B *A)

The last equation is the classical ideal feedback amplifier gain equation.

We can use the same equation for op-amps when we equate B with beta. Recall that beta for a simple op-amp circuit is

beta = Rin / (Rin + Rf)

So then we can write from the above that


G = A/ (1+A*(Rin / (Rin+Rf))

Dividing the numerator and denominator be A (A not = 0), we have


G =1/ ((1/A)+Rin/(Rin+Rf))

Now since A is usually a very large value we have an amplifier for which gain is not dependent on A but the external components Rin and Rf:

G = (Rf + Rin) / Rin = 1 / beta

where beta is the quantity for op-amps we discussed in the previous article.