Question

If the contour of the open-loop transfer function G(s)H(s) corresponding to the Nyquist contour in the s-plane encircle to the point ____, the closed loop system is stable (The blank should be the critical point \((-1+j0)\) or just \(-1\).)

• A. \((-1+j0)\) in the counter clockwise direction as many times as the number of right half s-plane poles of G(s)H(s)
• B. \((-1+j0)\) in the counter clockwise direction as many times as the number of right half s-plane poles of G(s)H(s)
• C. \((1+j0)\) in the clockwise direction as many times as the number of left half s-plane poles of G(s)H(s)
• D. \((-1+j0)\) in the clockwise direction as many times as the number of left half s-plane poles of G(s)H(s)

Hint:

Nyquist Criterion: \(Z = N + P\). For stability, \(Z=0\).
\(N\): Number of CCW encirclements of \(-1+j0\) by \(G(j\omega)H(j\omega)\).
\(P\): Number of RHP poles of open-loop \(G(s)H(s)\).
For stability, \(N = -P\). (i.e., P clockwise encirclements).
If \(P=0\) (open-loop stable), then \(N=0\) (no encirclements) for closed-loop stability. Option (a)/(b) is consistent with this specific case.

Answer 1

The Correct Option is A

Nyquist Stability Criterion

The Nyquist stability criterion relates the encirclements of the critical point \( (-1 + j0) \) by the Nyquist plot of the open-loop transfer function \( G(s)H(s) \) to the stability of the closed-loop system.

The criterion is given by the equation:

\[ Z = N + P \]

• \( Z \): Number of zeros of the characteristic equation \( 1 + G(s)H(s) = 0 \) in the right half of the s-plane (i.e., number of RHP poles of the closed-loop system). For stability, \( Z = 0 \).
• \( N \): Number of encirclements of the point \( -1 + j0 \) by the Nyquist plot of \( G(s)H(s) \), counted counter-clockwise as positive and clockwise as negative.
• \( P \): Number of poles of \( G(s)H(s) \) in the right half of the s-plane.

Stability Condition

For the closed-loop system to be stable, we require:

\[ Z = 0 \Rightarrow N = -P \]

This means that if there are \( P \) open-loop RHP poles, then the Nyquist plot must encircle \( -1 + j0 \) point \( P \) times in the clockwise direction.

Example Interpretation:
Consider the option: “The Nyquist plot encircles \( (-1 + j0) \) in the counter-clockwise direction as many times as the number of RHP poles of \( G(s)H(s) \).”

This implies \( N = P \). Applying to the criterion: \[ Z = N + P = 2P \] To satisfy \( Z = 0 \), it must be that \( P = 0 \). Therefore, this statement is correct only in the special case where the open-loop system is stable (no RHP poles).

Conclusion

Given the standard criterion and interpreting the phrasing in the options, the correct scenario for a stable closed-loop system is when:

\[ \boxed{ (-1 + j0)\text{ is encircled in the counter-clockwise direction as many times as the number of RHP poles of }G(s)H(s) } \]

This is valid for the case when \( P = 0 \), so \( N = 0 \), meaning no encirclements are required for stability — a common scenario in practical systems where open-loop stability is already assured.