Question 1

What is the Nyquist stability criterion?

Accepted Answer

The Nyquist stability criterion states that the number of unstable closed-loop poles Z equals N + P, where N is the number of counter-clockwise encirclements of the (-1, 0) point by the Nyquist plot of the open-loop transfer function G(jω)H(jω), and P is the number of open-loop right-half-plane poles. Z = 0 means the closed-loop system is stable. It generalizes the Bode-based gain/phase margin concept to systems with open-loop instabilities.

Question 2

What do gain margin and phase margin mean intuitively?

Accepted Answer

Gain margin (GM) is 'how much can I multiply the gain before the system goes unstable?' At the phase crossover frequency ωpc where G(jωpc) has -180° phase, GM = 1/|G(jωpc)| (or 20log10 of that in dB). Phase margin (PM) is 'how many extra degrees of phase lag until I reach -180°?' At the gain crossover frequency ωgc where |G(jωgc)| = 1, PM = 180° + ∠G(jωgc). Design targets: GM ≥ 6 dB and PM ≥ 30°.

Question 3

What happens to the Nyquist plot when I increase the gain K?

Accepted Answer

Increasing K scales the entire Nyquist locus outward. As K grows, the locus moves closer to the (-1, 0) critical point and eventually encircles it, causing closed-loop instability. You can observe this directly using the gain slider in this tool — notice how GM decreases and PM decreases as K increases, until the system becomes unstable.

Question 4

How does a plant with an integrator 1/s affect the Nyquist plot?

Accepted Answer

A plant containing an integrator K/(s(s+a)) has a pole at the origin, causing the gain to diverge to infinity as ω→0 and the phase to start at -90°. On the Nyquist plot, the locus approaches infinity along the negative imaginary axis at low frequencies, forming a half-infinite curve. This requires special treatment at ω = 0 by using a small-radius semicircular detour in the Nyquist contour. In this tool, the sweep starts at ω = 0.01 rad/s to handle this numerically.

Nyquist Diagram & Stability Margin Calculator

Nyquist Stability Criterion