GATE EC 2022 Set 1 — Question 21
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Control Systems → Stability Analysis → Nyquist Criterion
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Question
Consider a closed-loop control system with unity negative feedback and in the forward path, where the gain . The complete Nyquist plot of the transfer function is shown in the figure. Note that the Nyquist contour has been chosen to have the clockwise sense. Assume 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 __________.
Correct answer
(C) 2
Solution
The Nyquist stability criterion states that , where is the number of clockwise encirclements of the critical point by the Nyquist plot of , is the number of open-loop poles in the right-half plane (RHP), and is the number of closed-loop poles in the RHP.Given:
.Thus, there are 2 poles of the closed-loop transfer function in the closed right-half of the complex plane. Option (C) is correct.
1.Gain , so the critical point is .
2. (no open-loop poles in the RHP).
From the provided Nyquist plot of , we observe the encirclements of the point on the real axis. The plot consists of a large clockwise circle and two smaller clockwise loops on the left. The point lies inside both of these smaller loops. Tracing the path, the point is encircled twice in the clockwise direction ().Using the formula :.Thus, there are 2 poles of the closed-loop transfer function in the closed right-half of the complex plane. Option (C) is correct.
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