Question 12.1: Objective: Calculate the feedback transfer function β, given...
Objective: Calculate the feedback transfer function β, given A and A_{f}
Case A. Assume that the open-loop gain of a system is A = 10^{5} and the closed-loop gain is A_{f}= 50
Case B. Now assume that the open-loop gain is A = – 10^{5} and the close-loop
gain is A_{f}= −50.
A. From Equation (12.5), the closed-loop gain is
A_{f} = \frac{S_{o}}{S_{i}} = \frac{A}{(1 + β A)} (12.5)
A_{f} = \frac{A}{(1 + β A)} or 50 = \frac{10^{5}}{1 + β(10^{5})}
which yields β = 0.01999 or 1/β = 50.025.
B. Again, from Equation (12.5), the closed-loop gain is
A_{f} = \frac{A}{(1 + β A)} or – 50 = \frac{- 10^{5}}{1 + β(- 10^{5})}
which yields β = −0.01999 or 1/β = −50.025.
Comment: From these typical parameter values, we see that A_{f} \cong 1/β, as Equation (12.8) predicts. We also see that if the open-loop gain A is negative, then the closed-loop gain A_{f} and feedback transfer function β will also be negative for a negative feedback network
A_{f} \cong \frac{A}{β A} = \frac{1}{β} (12.8)