Glossary term
Nyquist Plot
A complex-plane frequency-response plot used to assess feedback-loop stability, encirclements, gain margin, and phase margin.
Definition
methodA Nyquist plot is a complex-plane plot of frequency response used with the Nyquist stability criterion to assess closed-loop stability.
A Nyquist plot traces the complex frequency response of an open-loop transfer function and uses its relationship to the critical point -1 to infer closed-loop stability. It is especially useful for systems with delay, non-minimum-phase behaviour, measured frequency-response data, or dynamics that are awkward to interpret from time response alone.
A Nyquist plot maps the open-loop frequency response L(j\omega) into the complex plane as frequency varies. The horizontal axis is real part, the vertical axis is imaginary part, and the curve shows both magnitude and phase at once. For a negative-feedback loop, the critical point is -1+0j because the closed-loop characteristic equation is:
The Nyquist stability criterion relates encirclements of the critical point to the number of unstable open-loop poles and unstable closed-loop poles. In practical terms, the plot tells whether closing the feedback loop will stabilize or destabilize the system and how much margin remains before instability.
Engineering use
Nyquist plots are used in servo control, power electronics, process loops, structural control, amplifier stability, and systems with transport delay. They complement Bode plots: Bode diagrams make gain and phase trends easy to read separately, while Nyquist plots show how the loop approaches the critical point in the complex plane.
Gain margin and phase margin can be interpreted geometrically from the distance and crossing behaviour near the critical point. More advanced robustness measures consider minimum distance to -1, sensitivity peaks, and uncertainty envelopes. If experimental frequency-response data is available, a Nyquist plot can be built without a full parametric transfer-function model, although data quality and frequency coverage then become critical.
Reading the plot
The direction of increasing frequency, the treatment of negative frequencies, and the handling of poles on the imaginary axis must be clear. For systems with time delay, the plot can spiral toward the origin and may pass near the critical point multiple times. For systems with right-half-plane open-loop poles, simple “do not encircle -1” rules are insufficient; the full Nyquist criterion must account for those poles.
Common mistakes
A common mistake is to plot only positive frequencies and then apply the criterion as if the complete contour had been considered. Another is to ignore open-loop unstable poles, delays, sign conventions, or units in measured frequency response. A good stability review states the loop transfer function, feedback sign, frequency range, open-loop pole count, contour convention, and how gain and phase margins were extracted.