Q.The open loop gain and closed loop gain of an amplifier are 200 and 100 respectively. Calculate the feedback fraction.

When negative feedback is applied, a fraction $\beta V_o$ of the output is returned $180^\circ$ out of phase, so the net input to the basic amplifier is reduced to $V_i = V_s - V_f$. Working through the definitions of open-loop gain $A = V_o/V_i$, closed-loop gain $A_f = V_o/V_s$ and feedback factor $\beta = V_f/V_o$ gives the central result of the chapter:

The derivation follows from $V_i = V_s - \beta V_o = V_s - \beta A V_i$, which rearranges to $V_s = V_i(1+A\beta)$; dividing $V_o = A V_i$ by this gives $A_f$.

Since $(1+A\beta) > 1$, the closed-loop gain is always smaller than the open-loop gain — the gain decreases by the factor $(1+A\beta)$. This loss of raw gain is the price paid for the benefits of negative feedback. …