The open loop transfer function of a unity gain negative feedback system is given as
\[
G(s) = \frac{1}{s(s+1)}.
\]
The Nyquist contour in the s-plane encloses the entire right half plane and a small neighbourhood around the origin in the left half plane, as shown in the figure below. The number of encirclements of the point \( (-1 + j0) \) by the Nyquist plot of \( G(s) \), corresponding to the Nyquist contour, is denoted as \( N \). Then \( N \) equals to
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For stability analysis using the Nyquist criterion, the number of encirclements of the point \( (-1 + j0) \) is determined by the poles of the open-loop transfer function in the right half of the s-plane.
Step 1: Understanding the Nyquist Criterion.
The Nyquist criterion is used to determine the stability of a closed-loop control system. The number of encirclements \( N \) of the point \( (-1 + j0) \) on the Nyquist plot is related to the number of poles of the open-loop transfer function \( G(s) \) in the right half of the s-plane.
Step 2: Analysis of the Open-Loop Transfer Function.
The transfer function given is \( G(s) = \frac{1}{s(s+1)} \), which has poles at \( s = 0 \) and \( s = -1 \). Both of these poles are in the left half of the s-plane. Since there are no poles in the right half of the s-plane, the Nyquist plot will not encircle the point \( (-1 + j0) \).
Step 3: Conclusion.
Since there are no poles in the right half-plane, the Nyquist plot will not encircle the point \( (-1 + j0) \), and hence \( N = 0 \).
