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GATE 2016 EE – Question 42

Control Systems · Stability analysis using Routh-Hurwitz and Nyquist criteria, Bode plots, Root loci · 2 marks · Multiple choice

Loop transfer function of a feedback system is $G(s)H(s) = \frac{s + 3}{s^2(s - 3)}$. Take the Nyquist contour in the clockwise direction. Then, the Nyquist plot of $G(s)H(s)$ encircles $-1 + j0$

  1. once in clockwise direction
  2. twice in clockwise direction
  3. once in anticlockwise direction
  4. twice in anticlockwise direction

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Correct answer: (A) once in clockwise direction

Explanation

The open-loop function has one pole in the right half plane, at $s = 3$, so $P = 1$. The closed-loop characteristic equation is $s^2(s - 3) + (s + 3) = s^3 - 3s^2 + s + 3 = 0$. Its Routh array has first column 1, $-3$, 2, 3, which has two sign changes, so two closed-loop poles are in the right half plane, giving $Z = 2$. The number of clockwise encirclements is $N = Z - P = 2 - 1 = 1$.