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

Crossover frequency — where it appears

The frequency at which a loop's gain passes unity, where everything that loop was doing stops and where its phase margin is quoted. Everything a loop does for its circuit stops there, so the desensitivity a design relies on is bounded by it and not by the open-loop gain.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

Loop gain of a three-pole amplifier closed for a gain of 100. Unity loop gain at 5.73 kHz, where 34.9° of phase remains before −180°. The phase reaches −180° at 89.6 kHz, where the loop gain is 46.1 dB below unity.

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.

feedback · Loop gain How much of the amplifier reaches the answer, at 35° of phase margin. computed by solving, not by drawing by multiplying the forward path's gain by 1 + 10⁻⁵ and solving again. At low frequency 0.0999% of a fractional change in the device reaches the closed-loop gain, which is one over the loop gain of 995.04. It changes sign at 258 Hz — above there a better amplifier gives a smaller gain — and reaches 1.61 at 8.08 kHz, which is worse than having no feedback, and settles at exactly 1 above crossover (5.73 kHz), where there is no loop gain left to spend. The second route, the real part of 1/(1 + T) from the cut loop, agrees to 9.6e-6.

How much of the amplifier gets through

The reason to build an amplifier from a bad amplifier and two resistors is that a change in the device barely reaches the answer — a tenth of a per cent of it at low frequency, measured by making the device better and solving again. Near crossover the same fraction rises above one, so feedback makes the gain more device-dependent than no feedback at all, and below fifty degrees of margin it changes sign on the way.

feedback · Desensitivity What the summing junction of an inverting amplifier actually is, at 1.00 MHz of gain–bandwidth. computed by solving, not by drawing by driving a current into the node and reading the voltage. It is 100 mΩ at direct current, rises 1.000 decades per decade of frequency, and settles at 909.5 Ω — which is the 1 kΩ and 10 kΩ in parallel, with the amplifier contributing nothing. It passes one per cent of the input resistor at 995 Hz, a factor of 1,005 below the gain–bandwidth. The second route — the open-loop impedance over one plus the return ratio from the cut loop — agrees to 0.045%.

The node that is at ground for a while

An inverting amplifier's summing junction is held at ground by the loop, so it is at ground exactly as well as the loop is strong. Driven with a current and measured, it is a tenth of an ohm at direct current, ten ohms at a kilohertz, and 909 ohms above a megahertz — which is the two feedback resistors in parallel, with the amplifier contributing nothing.

feedback · Virtual earth The straight lines report 45.00° of margin, and the solved loop has 51.83°. computed by solving, not by drawing. The gain and phase of a loop made of an integrator and one pole, the corner at 1.00 kHz, with the integrator set so that the straight-line asymptotes cross unity at 1.00 kHz. The solved loop crosses at 786 Hz instead. Read at the lines' crossover, with the phase taken from the solve, the margin is 45.00°; at the loop's own crossover it is 51.83°. Closed, the loop is stable: the largest real part among its closed-loop poles is -5.00e-1 of the corner's angular frequency. The solved gain never rises above its asymptotes, so the lines cannot report more margin than the loop has.

The margin the straight lines report

A phase margin read off a sketch is read where the straight lines cross unity, and that is not where the loop does. For an integrator and one real pole the lines report 45.00 degrees on a loop that has 51.83 — short, and always short, because a real pole's response never rises above its asymptotes. A pair that peaks reverses the sign and removes the bound: at a quality factor of two the lines report 71.57 degrees on a loop that has no margin at all, and at five they report 82.41 on a loop that is unstable.

frequency · Asymptotic approximation Lead zero at 500 Hz: the lines over-report by at most 3.12° with the loop's phase, and short by up to 20.6°. computed by solving, not by drawing. The error in the phase margin read off the straight lines of an integrator, two poles at 1.00 kHz and a lead section with its zero at 500 Hz and its pole 10 times higher, against where the lines cross unity, measured in decades from the zero. Gaps are placements where the lines are flat at unity or the loop crosses more than once. Read with the loop's own phase the reading is over by at most +3.12°, with the lines crossing −0.01 decades from the zero on a loop with 73.4° of margin, and never over on any loop with 60° or less; it is short by as much as 20.65°. Read with the phase off its straight lines too, on loops with 60° or less, it is never over.

The zero that lifts the lines

A phase margin read off the straight lines of an all-pole loop can only be short, and the proof takes three steps. A lead section breaks two of them: a zero's response lies above its lines, and between a zero and its pole the phase rises. The two breaks pull opposite ways, and measured across every placement of the crossing, in every sweep drawn, they leave an over-report of at most 6.58°, on a loop with 87.5°; on loops with sixty degrees or less it never exceeds 1.52°. The phase sketch is another matter: with the lines crossing at a lead zero above the plant's corner, it reports 17.91° on a loop with 1.17°.

frequency · Asymptotic approximation

Named alongside it

The objects these essays reach for when they reach for this one.

Loop gainPhase marginModel rangeAsymptotic approximationBode plotClosed-loop gainClosed-loop responseComponent sensitivityDamping ratioGain marginLead compensationLoading

All concepts