One magnitude curve, two phase curves, and the one of them the magnitude decides
Drawn above at its default parameters, which is almost never how an essay calls it. A placement states the numbers that essay is arguing about, so the figure a reader meets is about that argument rather than about the generator — 98% of the placements on this site pass one, and the phase that raised that number from 12% found eight captions describing a figure the page was not showing.
At those defaults the edge it states is ±3 decades holds 99.9% of the weight — the right-hand slot
of the caption strip, which on this site is never used for anything else, and which is read
back out of the drawing above rather than out of the code that wrote it.
It belongs to Frequency, which is the same solve, which is to say a change to it is a
change to lib/figures/frequency.js.
It takes a slider on decades of magnitude used, either side with 7 settings,
and every one of them has passed the same assertions as the frame above — a figure whose
circuit stops doing what its caption says at any setting stops the build.
Called by 7 essays
which is the blast radius of changing it
The phase the magnitude already knows
For one class of network the phase is not an independent measurement: it is fixed everywhere by the magnitude, through an integral Bode wrote down in 1945. Fed nothing but the magnitudes of a solved lead network, that integral returns 39.289 degrees at the corner against a solved 39.289, and tracks the whole curve to two hundredths of a degree. Cascade an all-pass and the magnitudes agree to the last bits of a double while the phases part by 180 degrees — so the recovery is exact for one of the two and cannot see the other at all.
The straight lines, and where they are not the curve
Two straight lines through a corner is the most-used approximation in this subject and almost the only one with no number attached. It has one, and it is exact: at a single real pole the sketch is 3.0103 decibels high at the corner and nowhere worse, and its error is the same a factor above the corner as the same factor below — a symmetry the construction does not suggest. A pole pair has no such bound at all, and the best any two-slope sketch can do on one is 0.770 decibels, at a quality factor of 0.9152.
The phase a decibel buys
The area under a minimum-phase network's phase, counted in degrees across decades of frequency, is fixed by how far its gain moves from one end of the spectrum to the other: ninety degree-decades for every twenty decibels, however the network is arranged. One 100:1 lead peaks at 78.579 degrees and two 10:1 leads stacked together at 109.806, and both enclose 180.000. The peak is a design decision and the area is a bill — and a network that gives its gain back gives its phase back with it.
The energy that arrives first
Two networks with the same magnitude at every frequency have the same impulse-response energy, and Parseval's theorem says so before either is solved. They do not deliver it on the same schedule. A low-pass with poles at one and ten kilohertz and a zero at three delivers half its energy by 11.27 microseconds; the same network with its zero mirrored into the right half-plane, which changes no magnitude anywhere, takes 63.23, and at no instant has it delivered more. Its step response starts the wrong way, to −0.170 of the final value, before it turns round. Minimum phase is minimum delay, and the delay is in the energy rather than in any one number a frequency plot shows.
The level the phase cannot know
Bode's gain–phase relation runs both ways. Given only the phase of a lead network, its companion integral returns the gain's shape across six decades to 0.055 dB. It cannot return the gain's level, because multiplying a network by a constant changes no phase anywhere. The level is one number, −20.000 dB here, and it has to come from outside the phase: one gain reading at any frequency supplies it. The same integral fed an all-pass reads a flat gain as a double pole, 6.021 dB down at the all-pass's frequency and 40.086 dB down a decade above it.
The two mirrors that change places
A low-pass with two zeros has four networks that share its magnitude, one for each choice of side for each zero. The one with both zeros on the left delivers its energy first at every instant, and the one with both on the right delivers last. The two in between are not ranked at all. With zeros at 3 and 30 kHz, mirroring the higher zero delays half the energy more and mirroring the lower zero delays nine tenths more. They swap at 12.64 µs, and that instant depends only on the product of the two zeros: it is where the all-pole response is rising at a rate of √(ω₁ω₂). And the network that delivers last does not start the wrong way at all.
The zeros that have no side
A complex pair of zeros can be mirrored into the right half-plane without changing a magnitude anywhere, and what the mirror adds is a second-order all-pass whose delay is 4ζ/ω₀ at low frequency. Above the band that is the whole story: with the pair at 30 kHz on a 1 kHz low-pass, half the energy arrives 0.21 µs later at ζ = 0.01, 2.12 µs at 0.1 and 21.2 µs at 1, linear in the damping — so the zeros of a notch, which sit on the axis, have no side worth choosing. Inside the band the all-pass's delay peaks at a millisecond exactly where the notch has removed the energy, and the energy's centroid moves by the delay averaged over |H|², which for fifteen pairs from 2 to 100 kHz matches a quadrature over frequency to five parts in a hundred thousand. And unlike one mirrored zero, a mirrored pair never starts a step the wrong way: it is only late.