Gain margin (GM) is the factor by which the open-loop gain can be increased before the closed-loop system becomes unstable. It is measured in decibels as GM = −20 log₁₀|G(jωpc)|, where ωpc is the phase crossover frequency. A positive GM means the system is stable; a negative GM means it is already unstable.

What is phase margin?

Phase margin (PM) is the additional phase lag (in degrees) that the open-loop system can tolerate before the closed-loop becomes unstable. PM = 180° + ∠G(jωgc), where ωgc is the gain crossover frequency. Typical well-designed systems target PM ≥ 30°–60°.

What are good stability margin targets?

Industry guidelines commonly recommend GM ≥ 6 dB and PM ≥ 30°. Higher margins mean a more robust, less oscillatory response but often slower speed of response. For most servo and process control applications, PM of 45°–60° provides a good balance.

Can this tool handle non-minimum-phase systems?

This tool applies the simplified gain/phase-margin approach which assumes a minimum-phase open-loop system with a single gain and phase crossover. Non-minimum-phase systems (those with right-half-plane zeros or time delays) can have multiple crossovers — in those cases, use a full Nyquist plot for reliable conclusions.

What does a gain margin of infinity mean?

Infinite gain margin occurs when the open-loop frequency response never reaches a phase of −180°. This is common in first- and second-order systems, which cannot become unstable by increasing gain alone.

Nyquist Stability Helper

Name: Nyquist Stability Helper
Availability: InStock
Author: Nham Vu

Enter open-loop gain, phase crossover frequency, and gain crossover frequency to compute gain margin, phase margin, and closed-loop stability verdict.

Open-Loop Parameters

|G(jω_pc)| — Magnitude at Phase Crossover

Open-loop gain magnitude where phase = −180°. Enter as a linear ratio (e.g. 0.25 means the gain is ¼ at that frequency).

∠G(jω_gc) — Phase at Gain Crossover (°)

Open-loop phase angle (degrees) where |G(jω)| = 1. Typically a negative value, e.g. −130°.

Optional: Crossover Frequencies

ω_gc (rad/s)

ω_pc (rad/s)

Stability Rule of Thumb

GM > 6 dB and PM > 30° — robust stable
GM 0–6 dB or PM 0–30° — stable but low margin
GM ≤ 0 dB or PM ≤ 0° — unstable

Gain Margin (GM)

—

GM = −20 log₁₀|G(jω_pc)|

Phase Margin (PM)

—

PM = 180° + ∠G(jω_gc)

Summary

Enter open-loop gain, phase crossover frequency, and gain crossover frequency to compute gain margin, phase margin, and closed-loop stability verdict.

How it works

Enter the open-loop gain at the phase crossover frequency (|G(jωpc)|).
Enter the open-loop phase at the gain crossover frequency (∠G(jωgc)) in degrees.
Optionally adjust the gain crossover and phase crossover frequencies.
The tool computes gain margin = −20 log₁₀|G(jωpc)| and phase margin = 180° + ∠G(jωgc).
A stability verdict is shown: stable (both margins positive), marginally stable (either margin ≈ 0), or unstable.
Adjust parameters to explore how gain changes affect stability margins.

Use cases

Verify a PID-tuned loop has adequate stability margins before deployment.
Understand why a controller oscillates and find the minimum gain reduction needed.
Teach Nyquist/Bode stability theory with an interactive numerical example.
Quick sanity-check during early design before running full frequency-response simulation.
Compare stability margins across different controller gains.
Determine how much additional gain a system can tolerate before going unstable.

Frequently Asked Questions

Last updated: 2026-07-01 · Reviewed by Nham Vu