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GATE 2017 EC – Question 59

Control Systems · Routh-Hurwitz and Nyquist Stability Criteria · 2 marks · Multiple choice

The Nyquist plot of the transfer function

$$G(s) = \frac{K}{(s^2 + 2s + 2)(s + 2)}$$

does not encircle the point $(-1 + j0)$ for $K = 10$ but does encircle the point $(-1 + j0)$ for $K = 100$. Then the closed loop system (having unity gain feedback) is

  1. stable for $K = 10$ and stable for $K = 100$
  2. stable for $K = 10$ and unstable for $K = 100$
  3. unstable for $K = 10$ and stable for $K = 100$
  4. unstable for $K = 10$ and unstable for $K = 100$

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Show answer and explanation

Correct answer: (B) stable for $K = 10$ and unstable for $K = 100$

Explanation

All the open-loop poles ($-1 \pm j$ and $-2$) are in the left half plane, so there are no open-loop poles in the right half plane. By the Nyquist criterion the closed loop is stable when the plot does not encircle $-1 + j0$, and unstable when it does. So it is stable for $K = 10$ and unstable for $K = 100$.