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GATE 2020 EC – Question 21

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

The pole-zero map of a rational function $G(s)$ is shown below. When the closed contour $\Gamma$ is mapped into the $G(s)$-plane, then the mapping encircles

Diagram for GATE 2020 EC question 21
  1. the origin of the $G(s)$-plane once in the counter-clockwise direction.
  2. the origin of the $G(s)$-plane once in the clockwise direction.
  3. the point $-1+j0$ of the $G(s)$-plane once in the counter-clockwise direction.
  4. the point $-1+j0$ of the $G(s)$-plane once in the clockwise direction.

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Correct answer: (B) the origin of the $G(s)$-plane once in the clockwise direction.

Explanation

A contour traversed clockwise in the $s$-plane encircles the origin of the $G(s)$-plane $N=Z-P$ times in the clockwise direction, where $Z$ and $P$ are the numbers of zeros and poles of $G(s)$ inside the contour. The contour encloses three zeros and two poles, so $N=3-2=1$ clockwise encirclement of the origin. The point $-1+j0$ is relevant only for $1+G(s)$, not for $G(s)$ itself.