GATE 2015 EC – Question 56
The open-loop transfer function of a plant in a unity feedback configuration is given as $G(s) = \frac{K(s + 4)}{(s + 8)(s^2 - 9)}$. The value of the gain $K$ ($> 0$) for which $-1 + j2$ lies on the root locus is ________.
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Correct answer: 25 to 26
Explanation
A point is on the root locus when $|G(s)| = 1$ there. At $s = -1 + j2$: $|s + 4| = |3 + j2| = \sqrt{13}$, $|s + 8| = |7 + j2| = \sqrt{53}$ and $s^2 - 9 = (1 - 4j - 4) - 9 = -12 - 4j$, whose magnitude is $\sqrt{160}$. So $K = \frac{\sqrt{53}\sqrt{160}}{\sqrt{13}} = \sqrt{652} = 25.5$.