Frequency Response Method

Bode plots show $|G(j\omega)|$ (dB) and $\angle G(j\omega)$ vs $\log\omega$. Stability margins: the gain margin is $-|G|$ (dB) at the phase-crossover ($\angle=-180^\circ$); the phase margin is $180^\circ+\angle G$ at the gain-crossover ($|G|=0\,dB$). Positive margins $\Rightarrow$ stable. The Nyquist criterion relates encirclements of $-1$ to closed-loop stability.

Note

Example — Reading a phase margin

If at the gain-crossover frequency the open-loop phase is $-135^\circ$, then $\text{PM}=180^\circ-135^\circ=45^\circ$, indicating a well-damped, comfortably stable closed loop.