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GATE 2022 EC – Question 21

Control Systems · Routh-Hurwitz and Nyquist Stability Criteria · 1 mark · Multiple choice

Consider a closed-loop control system with unity negative feedback and $KG(s)$ in the forward path, where the gain $K=2$. The complete Nyquist plot of the transfer function $G(s)$ is shown in the figure. Note that the Nyquist contour has been chosen to have the clockwise sense. Assume $G(s)$ has no poles on the closed-right-half of the complex plane. The number of poles of the closed-loop transfer function in the closed-right-half of the complex plane is ________.

Diagram for GATE 2022 EC question 21
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Correct answer: (C) 2

Explanation

With $K=2$ the critical point in the $G(s)$ plane is $-1/K=-0.5$. The plot crosses the negative real axis at $-0.8$ and $-0.4$, so $-0.5$ lies inside both the large circle and the left small loop, giving $N=2$ clockwise encirclements. With $P=0$ open-loop poles in the right half plane, $Z=P+N=2$ closed-loop poles in the right half plane.