Nyquist Stability Calculator

• •Use the Nyquist Stability Calculator to check your hand calculations on homework or exam problems — comparing its result with your own working helps identify arithmetic errors before they propagate.
• •Use it during the early design phase to rapidly iterate on parameters and narrow down feasible configurations before committing time to detailed finite element simulations or full design packages.
• •Use it when reviewing a colleague's calculation or checking a vendor's data sheet for plausibility — a quick sanity check can prevent costly downstream errors.
• •Use it to generate reference data for a technical report or presentation without manual computation, ensuring consistent, reproducible numbers throughout the document.
• •Use it in the field when a quick estimate is needed and a full engineering software package is not available.

The Nyquist Stability Calculator is a precision engineering calculation tool designed for students, engineers, and technical professionals. Compute gain margin (dB), phase margin (degrees), gain crossover frequency, and phase crossover frequency from open-loop transfer function All calculations are performed using established engineering formulas from the relevant scientific literature and standards. Follow the units printed beside each field. Where a unit selector is provided, the selected unit is converted internally. Results are computed in your browser without sending data to a server, which keeps your inputs private. This calculator is intended as a supplementary tool for learning and design exploration; always verify results against authoritative references for safety-critical applications.

The Nyquist stability criterion uses the frequency response plot G(jω)·H(jω) (mapping of the jω axis through the loop transfer function) to determine closed-loop stability. The key statement: Z = N + P, where Z is the number of closed-loop poles in the right half plane, N is the number of clockwise encirclements of the point −1 by the Nyquist plot (counter-clockwise encirclements count as negative) and P is the number of open-loop right-half-plane poles; the closed loop is stable only when Z = 0, i.e. N = −P. For typical open-loop stable systems (P = 0), closed-loop stability requires that the Nyquist plot does not encircle −1. The Nyquist plot also reveals the gain margin and the phase margin. Where the plot crosses the negative real axis (at ω_180, where ∠G = −180°), GM = 1/|G(jω_180)|, the reciprocal of that crossing's distance from the origin (GM_dB = −20·log₁₀|G(jω_180)|); where it crosses the unit circle (at ω_gc, where |G| = 1), PM = 180° + ∠G(jω_gc), the angle from the negative real axis to that point. These margins quantify stability robustness. Nyquist analysis is especially useful for systems with time delay (which cause phase lag and can destabilize feedback loops), non-minimum-phase systems (with right-half-plane zeros), and open-loop unstable systems that are stabilized by feedback. The criterion works for all linear time-invariant systems and provides a more general stability check than Bode plots alone, though it is less intuitive for beginners.

• •Feedback loop stability analysis including processes with time delay: traditional Bode analysis is tricky for delays; Nyquist handles them naturally.
• •Non-minimum-phase system design: systems with right-half-plane zeros have unusual frequency responses that are best analyzed using Nyquist.
• •Open-loop unstable system stabilization: some aerospace and chemical processes are open-loop unstable (requiring feedback to stabilize). Nyquist correctly handles the required encirclements.
• •Multivariable system analysis: generalized Nyquist for MIMO systems uses characteristic loci or diagonal elements of the return difference matrix.
• •Robustness analysis: distance from the Nyquist plot to the −1 point quantifies stability margin beyond the simpler gain/phase margin concepts.