Phase Margin⚠ unverified
Physics / Controls · Compute the phase margin from the phase at the gain crossover frequency
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| phase_at_gain_crossover | phaseatgaincrossover | — | 1.0 | Open-loop phase at the gain crossover frequency, in degrees |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | PM | — | Phase margin, in degrees |
The science & history
Understanding the Parameters
-
phase_at_gain_crossover — e.g. $-120$ for a common stable design; must match the degree/radian convention of the formula’s $180$.
-
PM — positive margin indicates the additional negative phase that would bring the loop to $-180^{\circ}$ at $\omega_{gc}$.
Derivation (Approaching a Proof)
At gain crossover $|G|=1$. Distance in phase from the critical $-180^{\circ}$ is $\mathrm{PM} = \angle G - (-180^{\circ}) = 180^{\circ} + \angle G$ when $\angle G$ is in degrees.
History
Phase margin is the most used single robustness number in classical control (rule of thumb: 30–60 deg).
Related Concepts: Gain Margin, Phase Margin From Damping, Gain Crossover Frequency, Bode Magnitude
Notes: Registry calculator phase-margin (unverified). Arithmetic form only — phase must be
precomputed at $\omega_{gc}$.