GATE 2015 EE – Question 64

Control Systems · Mathematical modelling and representation of systems, Feedback principle, Transfer function, Block diagrams and Signal flow graphs · 2 marks · Multiple choice

Find the transfer function $\frac{Y(s)}{X(s)}$ of the system given below.

The input $X(s)$ feeds two summing junctions. The upper one passes $X - HY$ through $G_1$, and the lower one passes $X - HY$ through $G_2$, where the output of the whole system $Y(s)$ is the sum of the two paths and is fed back through $H$.

$\frac{G_1}{1 - HG_1} + \frac{G_2}{1 - HG_2}$
$\frac{G_1}{1 + HG_1} + \frac{G_2}{1 + HG_2}$
$\frac{G_1 + G_2}{1 + H(G_1 + G_2)}$
$\frac{G_1 + G_2}{1 - H(G_1 + G_2)}$

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

Explanation

Both paths take the same signal $X - HY$, so $Y = (G_1 + G_2)(X - HY)$. Then $Y\left[1 + H(G_1 + G_2)\right] = (G_1 + G_2)X$, which gives $\frac{Y}{X} = \frac{G_1 + G_2}{1 + H(G_1 + G_2)}$.