A feedback loop can be stable at its nominal parameters yet be close to instability. Gain margin and phase margin measure how much additional loop change can be tolerated before the frequency-domain stability boundary is reached.

For loop transfer function $L(i\omega)$, the gain crossover frequency satisfies

$$|L(i\omega_{gc})|=1.$$

The phase margin is the additional phase lag required at that frequency to reach $-180^\circ$.

The phase crossover frequency satisfies

$$\arg L(i\omega_{pc})=-180^\circ.$$

The gain margin is the reciprocal of $|L(i\omega_{pc})|$, commonly expressed in decibels.

Positive margins indicate distance from the classical instability boundary for the usual negative-feedback setting. Larger margins often correspond to greater tolerance of modelling error and delay, although they do not summarize every form of robustness.

Bode plots make these margins readable directly from the loop frequency response and turn frequency-domain plots into quantitative design tools.