增益裕度

Gain Margin (GM)

定义为：

$$
\mbox{GM}=k=\frac{1}{|L(j\omega)|}
$$

which occurs at the phase crossover frequency, \(\omega_{cp}\) where

$$
\angle L(j\omega_{cp}) = -180^{\circ}
$$

相位裕度

Phase Margin (PM)

定义为：

$$
\mbox{PM} =180^{\circ} + \angle L(j\omega)
$$

which occurs at the unity gain crossover frequency, \(\omega_{cg}\) where

$$
|L(j\omega_{cg})| = 1
$$

裕度的计算

图像

GM 和 PM 可从 bode plot 和 Nyquist plot 中得到。

Bode plot:

Nyquist plot:

计算

For Gain Margin (GM)

  1. Find the phase crossover frequency, \(\omega_{cp}\), using \(\angle G(j\omega_{cp}) = −180^{\circ}\).

  2. Next find the magnitude \(|G(j\omega_{cp})|\).

  3. If \(|G(j\omega_{cp})| = −a\), then \(\mbox{GM} = \frac{1}{a}\ \mbox{or}\ −20 \log_{10}a\).

For Phase Margin (PM)

  1. Find the unity gain crossover frequency, \(\omega_{cg}\), using \(|G(j\omega_{cg})| = 1\).

  2. Next find the phase of \(G(j\omega_{cg})\).

  3. If \(G(j\omega_{cg}) = 1\angle \psi\), then \(\mbox{PM} = 180^{\circ} + \psi\).