Chapter 12, Problem 12.1EP

(a) The open-loop gain of an amplifier is $A=5\times {10}^4$ and the closed-loop gain is $A_f=50$ . (i) What is the feedback transfer function? (ii) What is the ratio of $A_f$ to $1/\beta ?$ (b) Repeat part (a) for $A=100$ and $A_f=20.$ (Ans. (a) (i) 0.01998, (ii) 0.9990; (b) (i) 0.04, (ii) 0.80 )

To determine

1. The feedback transfer function for a given closed-loop gain and open-loop gain
2. The ratio of closed loop gain to $1/\beta$

Answer to Problem 12.1EP

(i) The feedback transfer function is $\beta =0.01998$

(ii) The ratio of closed loop gain to $1/\beta$ is $0.999$

Explanation of Solution

Given Information:

The open-loop gain of a feedback amplifier is $A=5\times {10}^4$ and its closed loop gain is $A_f=50$ .

Calculation:

(i) Consider the general feedback amplifier representation in Figure 1. Here, a feedback section with feedback transfer function $\beta$ is connected to an amplifier having open-loop gain $A$ is connected to form a closed-loop amplifier with gain $A_f$ , given by,

  $A_f=\frac{A}{1+A\beta }$

  

Substituting the open-loop gain and the closed-loop gain , the feedback transfer function can be obtained as,

  $\begin{array}{c}\beta =\frac{\frac{5\times {10}^4}{50}-1}{5\times {10}^4}\\ =0.01998\end{array}$

(ii) Substituting the value of $\beta$ obtained from the previous part, the ratio of closed loop gain to $1/\beta$ can be calculated as,

  $\begin{array}{c}\frac{A_f}{1/\beta }=A_f\beta \\ =50\times 0.01998\\ =0.999\end{array}$

To determine

(i) The feedback transfer function for a given closed-loop gain and open-loop gain

(ii) The ratio of closed loop gain to $1/\beta$

Answer to Problem 12.1EP

(i) The feedback transfer function is $\beta =0.04$

(ii) The ratio of closed loop gain to $1/\beta$ is $0.08$

Explanation of Solution

Given Information:

The open-loop gain of a feedback amplifier is $A=100$ and its closed loop gain is $A_f=20$ .

Calculation:

(i) Consider the general feedback amplifier representation in Figure 1. Here, a feedback section with feedback transfer function $\beta$ is connected to an amplifier having open-loop gain $A$ is connected to form a closed-loop amplifier with gain $A_f$ , given by,

  $A_f=\frac{A}{1+A\beta }$

Substituting the open-loop gain and the closed-loop gain , the feedback transfer function can be obtained as,

  $\begin{array}{c}\beta =\frac{\frac{100}{20}-1}{100}\\ =0.04\end{array}$

(ii) Substituting the value of $\beta$ obtained from the previous part, the ratio of closed loop gain to $1/\beta$ can be calculated as,

  $\begin{array}{c}\frac{A_f}{1/\beta }=A_f\beta \\ =20\times 0.04\\ =0.8\end{array}$