Circuit Response
every model has an edge, and the figure carries it
Field

Feedback, and the margin

The site's strongest construction, and the subject supplies it free: a phase margin computed from the loop gain in the frequency domain, an overshoot measured from the step response in the time domain, one circuit, and a required agreement between two numbers that share no arithmetic.
-40-200204060loop gain (decibels)unity loop gaincrossover 5.73 kHz46.1 dB of gain margin-180-135-901101001k10k100k1M10M100Mfrequency (hertz)loop phase (degrees)−180°34.9° of marginsolved, then checked — the loop cut and injected34.9° of phase margin at 5.73 kHz

What is left at crossover

A feedback loop is stable or not according to one number read at one frequency — how much phase remains before −180° at the point where the loop gain passes unity. The loop gain here is obtained the way it is obtained on a bench: cut the loop, drive one side of the cut, and measure what comes back to the other.

 0501001500200400600closed-loop output (volts) for a 1 V stepthe 100× the divider asks for35.1% overphase margin, measured two waysfrom the loop gain34.9°from the overshoot35.0°apart by 0.1° — the relation assumes two poles and this loop has threesolved, then checked — margin against overshootthe second-order relation is 0.1° out here

Two measurements of one margin

A phase margin is computed from the loop gain in the frequency domain, without ever looking at a step. An overshoot is measured from the closed-loop step response in the time domain, without ever looking at a Bode plot. Inverting the standard relation on the second returns 34.9° against the first's 34.9°, and the residue is the third pole.

 010203040501101001k10k100k1Mfrequency (hertz)closed-loop gain (decibels)the ideal amplifier: two resistors, no frequencythe circuit+−1% low at 1.35 kHz3 dB down at 10.0 kHzsolved, then checked — a nullor against a real devicethe ideal answer is 1% wrong above 1.35 kHz

The ideal amplifier, and where it stops being one

An ideal operational amplifier's closed-loop gain is set by two resistors and nothing else — a horizontal line at every frequency. The real one is already a tenth of a per cent low at direct current, one per cent low by 1.35 kHz, and above 10 kHz has no loop gain left, at which point the ideal answer is not an approximation to anything.

All essays