Questions: Nyquist Plot and Encirclement Criterion

Question 1

A closed-loop system has no open-loop RHP poles (P = 0). Its Nyquist plot makes exactly one clockwise encirclement of the point −1. What can you conclude about closed-loop stability?

• A. The closed-loop system is stable — the plant is open-loop stable, so the closed-loop must also be stable
• B. The closed-loop system is unstable: Z = N + P = 1 + 0 = 1, meaning one RHP closed-loop pole exists
• C. The closed-loop system is marginally stable — one clockwise encirclement indicates a pole on the imaginary axis
• D. No conclusion can be drawn without also knowing the phase margin at the crossover frequency

Answer: (B)

The Nyquist criterion counts RHP closed-loop poles as Z = N + P, where N is the number of clockwise encirclements of −1 and P is the number of open-loop RHP poles. Here N = 1 (clockwise) and P = 0, so Z = 1 — one closed-loop pole in the right half-plane, meaning instability. The common misconception is that an open-loop stable plant (P = 0) guarantees a stable closed loop — it does not. Adding feedback can destabilize a stable plant, and the Nyquist criterion detects this via encirclement count.

Question 2

Why does the Nyquist stability criterion examine encirclements of the specific point −1 + 0j rather than the origin?

• A. The point −1 is where the open-loop transfer function G(jω) always reaches its maximum magnitude
• B. Closed-loop instability occurs when 1 + G(s) = 0, i.e., when G(s) = −1; the critical point −1 is exactly where the closed-loop characteristic equation has a root
• C. The origin is excluded because G(jω) always passes through the origin at ω = 0
• D. The argument principle can only be applied to contours that avoid the imaginary axis

Answer: (B)

The closed-loop transfer function has poles where 1 + G(s) = 0, i.e., where G(s) = −1. In the Nyquist analysis, the argument principle is applied to F(s) = 1 + G(s) — counting its zeros in the RHP gives the number of unstable closed-loop poles. Zeros of F(s) = 1 + G(s) correspond to points where G(s) = −1, which is the point (−1, 0) in the complex plane. So encircling −1 in the Nyquist plot of G(jω) is equivalent to encircling the origin in the Nyquist plot of F(jω) = 1 + G(jω). The −1 point is the direct read-off of when the closed-loop characteristic equation is satisfied.

Question 3

The Nyquist criterion can correctly determine the stability of a closed-loop system even when the open-loop plant has poles in the right half-plane (unstable open-loop poles), whereas Bode plot analysis cannot reliably handle this case.

• T. True
• F. False

Answer: True

For open-loop stable systems (P = 0), Bode-based gain and phase margin analysis is sufficient and intuitive. But when P > 0, stability requires the Nyquist plot to encircle −1 exactly P times counterclockwise — a requirement that looks like instability to the Bode-only intuition. The argument principle underlying the Nyquist criterion handles any number of open-loop RHP poles through the Z = N + P formula. Bode analysis silently fails for non-minimum-phase systems and unstable plants because it cannot track the encirclement count correctly.

Question 4

For a system with no open-loop RHP poles, the closed-loop system is stable as long as the Nyquist plot does not encircle or pass through the origin of the complex plane.

• T. True
• F. False

Answer: False

The critical point is −1, not the origin. Stability requires the Nyquist plot of G(jω) to not encircle the point (−1, 0j). The origin has no special significance in the Nyquist stability criterion — G(jω) can pass through or encircle the origin without affecting stability. This is a common confusion: the argument principle in its raw form counts encirclements of the origin, but the Nyquist criterion shifts this to −1 because we analyze F(s) = 1 + G(s) and translate back to G.

Question 5

Explain why the Nyquist criterion evaluates encirclements of the point −1 rather than the origin, and what physical condition the −1 point represents.

Model answer: The Nyquist criterion applies the argument principle to F(s) = 1 + G(s), whose zeros are the closed-loop poles. Zeros of F(s) in the RHP mean unstable closed-loop poles. The argument principle counts zeros minus poles of F inside the RHP contour by counting clockwise encirclements of the origin by F(jω). Since F(jω) = 1 + G(jω), an encirclement of the origin by F corresponds to an encirclement of −1 by G. Physically, the point −1 is where G(s) = −1, which satisfies 1 + G(s) = 0 — exactly the closed-loop characteristic equation. It represents the condition for closed-loop resonance or instability: the loop gain equals −1 (magnitude 1, phase −180°), meaning feedback reinforces rather than corrects disturbances.

The phase condition −180° is also where the Bode phase margin is defined: how many additional degrees of phase lag would bring the system to G(jω) = −1. Gain margin answers how much gain increase would bring the magnitude to 1 when the phase is already −180°. Both margins measure distance from the critical −1 point in different directions — confirming that −1 is the center of stability analysis in both the Nyquist and Bode frameworks.