Question

Loop transfer function of a feedback system is: \[ G(s)H(s) = \frac{10}{s - 2} \] Assume the Nyquist contour in the clockwise direction. Then the Nyquist plot of \( G(s) \) encircles \(-1 + j0\):

• A. Once in clockwise direction
• B. Twice in clockwise direction
• C. Once in anti-clockwise direction
• D. Twice in anti-clockwise direction

Hint:

The Nyquist plot helps determine system stability by analyzing encirclements of \(-1\). If the plot encircles \(-1\) in clockwise direction \( P \) times, the system is unstable.

Answer 1

The Correct Option is A

Step 1: The Nyquist criterion determines the stability of a closed-loop system based on the open-loop transfer function \( G(s)H(s) \). 
Step 2: The given transfer function: \[ G(s)H(s) = \frac{10}{s - 2} \] has a single pole at \( s = 2 \), which is in the right-half plane (unstable region). 
Step 3: The Nyquist contour encloses the entire right half-plane, and the Nyquist plot is obtained by substituting: \[ s = j\omega \] 
Step 4: Evaluating at \( s = j\omega \): \[ G(j\omega)H(j\omega) = \frac{10}{j\omega - 2} \] - At high frequencies (\( \omega \to \infty \)), \( G(j\omega) \to 0 \).
- At low frequencies (\( \omega \to 0 \)), \( G(j\omega) \to -5 \).
- As \( s \) moves along the Nyquist contour, the plot encircles -1 in a clockwise direction exactly once. 
Step 5: According to Nyquist stability criterion: \[ N = P - Z \] where:
- \( N \) = Number of encirclements of \(-1\) (to be determined)
- \( P = 1 \) (one pole in the right-half plane)
- \( Z = 0 \) (no closed-loop unstable poles for stability) 
Step 6: Since \( N = 1 \), the Nyquist plot encircles \(-1\) once in a clockwise direction.