Bode, Nyquist, Margins and Compensation

Syllabus: Frequency response; Nyquist stability criterion; Bode plots; Lag, lead and lag-lead compensation.

8 hand-picked visual resources on this topic, 3 of them little-known gems — under each card, look for See it. Every link is checked against the live site before it ships.

▶Interactive labThe Nyquist criterion, animated

Bode plot by inspection

Write $G(j\omega)$ in time-constant form. The low-frequency asymptote of a type- $N$ system has slope $-20N$ dB/dec and passes through $|K|$ dB at $\omega=1$ (for $N=0$ ) or through 0 dB at $\omega=K^{1/N}$ . Each pole then bends the slope by $-20$ dB/dec at its corner, each zero by $+20$ .

$\text{corner }\omega=1/\tau;\quad \text{pole}:-20\ \text{dB/dec},\ \text{zero}:+20$
$\text{at the corner of a simple pole: }-3\ \text{dB},\ -45^\circ$

Where marks are lost

Type-1 systems: the initial $-20$ dB/dec line crosses 0 dB at $\omega=K_v$ . Reading $K_v$

See it

Gain and phase margin

Gain margin is measured where the phase is $-180^\circ$ ; phase margin where the gain is 0 dB. Getting the two crossover frequencies the wrong way round is the single most common error in this topic.

$\text{PM}=180^\circ+\angle G(j\omega_{gc}),\qquad |G(j\omega_{gc})|=1$

Nyquist criterion

Count encirclements of the $-1$ point by the plot of $G(j\omega)H(j\omega)$ as $\omega$ runs from $-\infty$ to $+\infty$

Lead, lag and lag-lead

Lead adds phase near the crossover: faster response, more bandwidth, better transient — at the cost of amplifying noise. Lag adds gain at low frequency: better steady-state error, smaller bandwidth, slower. Lag-lead does both.

$G_c(s)=\dfrac{1+\alpha\tau s}{1+\tau s},\ \alpha>1\ \text{(lead)};\qquad \alpha<1\ \text{(lag)}$

← Routh–Hurwitz and Root Locus State Variable Models →

Loading revision…