The number of direction of encirclements around the point -1 + j0 in the complex plane by the Nyquist plot of \(\rm G(s)=\frac{1-s}{4+2 s}\) is:

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This question was previously asked in

SJVNL ET Electrical 2019 Official Paper

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Answer 1

Concept:

The Nyquist plot is used to assess the stability of a closed-loop control system using the open-loop transfer function \( G(s) \).

Given transfer function: \( G(s) = \frac{1}{s(s+2)} \)

Poles of G(s): \( s = 0 \) and \( s = -2 \) ⇒ Both in Left Half Plane (LHP), so open-loop system is stable.

According to the Nyquist criterion:

\( N = Z - P \), where:

N = number of encirclements of point \( -1 + j0 \) (clockwise = negative, anticlockwise = positive)
Z = number of right-half plane poles of closed-loop transfer function
P = number of right-half plane poles of open-loop transfer function (here, P = 0)

Since the open-loop poles are in LHP, the Nyquist plot of \( G(s) \) does not encircle \( -1 + j0 \).

\( Z = 0 \Rightarrow N = 0 \)

Hence, the number of encirclements is: zero

Final Answer: zero