How it works
The Nyquist criterion states that N = Z − P, where N is the number of clockwise encirclements of the −1+j0 point by the complete Nyquist plot (ω from −∞ to +∞), Z is the number of closed-loop poles in the right half-plane, and P is the number of open-loop poles in the right half-plane. For stability, Z = 0, so N = −P. For an open-loop stable plant (P = 0), the closed-loop is stable if the Nyquist plot does not encircle −1. Gain margin is 1/|G(jω_pc)| and phase margin is the angle from −180° at the gain crossover.
Key points to remember
The polar plot of G(jω) for a Type 1 system starts at phase −90° (infinite magnitude) as ω→0 and ends at the origin as ω→∞. The real axis crossing gives the gain and phase margins directly: GM = distance from origin to the point where the plot crosses the negative real axis; PM = angle at the unit circle crossing. For G(s) = 1/[s(1+s)(1+2s)], the real-axis crossing of the polar plot occurs at ω = 1/√2 rad/s. The Nyquist plot for a minimum-phase system with positive gain margin and phase margin is always stable. Semi-infinite indentations around poles at the origin follow a clockwise arc of radius → 0 in the s-plane.
Exam tip
The examiner always asks you to sketch the Nyquist/polar plot for a Type 1 or Type 2 system, identify the −1+j0 encirclements, and determine GM and PM — mark the gain crossover and phase crossover frequencies clearly on the plot.