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GATE 2015 EE – Question 35

Control Systems · Mathematical modelling and representation of systems, Feedback principle, Transfer function, Block diagrams and Signal flow graphs · 1 mark · Multiple choice

For the signal-flow graph shown in the figure, which one of the following expressions is equal to the transfer function $\frac{Y(s)}{X_2(s)}\Big|_{X_1(s) = 0}$?

A signal-flow graph with inputs $X_1(s)$ and $X_2(s)$ and output $Y(s)$ along a forward line with gains 1, $G_1$ and $G_2$, and two feedback arcs with gain $-1$.
  1. $\frac{G_1}{1 + G_2(1 + G_1)}$
  2. $\frac{G_2}{1 + G_2(1 + G_1)}$
  3. $\frac{G_1}{1 + G_1 G_2}$
  4. $\frac{G_2}{1 + G_1 G_2}$

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Correct answer: (B) $\frac{G_2}{1 + G_2(1 + G_1)}$

Explanation

With $X_1 = 0$, the only forward path from $X_2$ to $Y$ has gain $G_2$. The two feedback arcs give loops of gain $-G_2$ and $-G_1 G_2$, and both touch the forward path. By Mason's gain formula, $\frac{Y}{X_2} = \frac{G_2}{1 + G_2 + G_1 G_2} = \frac{G_2}{1 + G_2(1 + G_1)}$.