Short note: Gain margin and phase margin

Gain margin and phase margin are measures of relative stability: they show how close the closed-loop system is to instability, using the open-loop frequency response $G(j\omega)H(j\omega)$.

Gain margin (GM)

The factor by which the open-loop gain can be increased before the closed-loop system becomes marginally stable.

$$GM = \frac{1}{|G(j\omega_{pc})H(j\omega_{pc})|},\qquad GM_{dB} = -20\log|G(j\omega_{pc})H(j\omega_{pc})|$$

• $\omega_{pc}$ = phase crossover frequency, where $\angle GH = -180^\circ$.
• On the polar plot: if the curve cuts the negative real axis at $-a$, then $GM = 1/a$.

Phase margin (PM)

The additional phase lag that can be added at the gain crossover frequency before the system becomes marginally stable.

$$PM = 180^\circ + \angle G(j\omega_{gc})H(j\omega_{gc})$$

• $\omega_{gc}$ = gain crossover frequency, where $|GH|=1$ (0 dB).

        Im
         |   unit circle
     ----+----
   /     |     \
 -1 o----+------+--- Re
   \ PM /|
     \_/ |   GM = 1/a

Stability from margins (minimum-phase systems)

• Typical good values: GM ≥ 6 dB, PM 30°–60°.
• A larger PM means less overshoot (approximately PM ≈ 100ζ).

Example: if $|GH|=0.25$ at $\omega_{pc}$ then GM $=4=12$ dB; if $\angle GH=-140^\circ$ at $\omega_{gc}$ then PM $=40^\circ$.