The zero that lifts the lines

The margin the straight lines report found a rule for reading a phase margin off a sketch, and a proof of it in three steps. The loop’s gain never rises above its straight lines. So the loop crosses unity no higher than the lines do. And the phase falls with frequency, so a lower crossing has at least as much phase left. The margin the lines report can then only be short of the margin the loop has.

The rule held for integrators, for real poles in any number and for pole pairs up to Butterworth damping. It failed, without bound, for a pair that peaks, because a peaking pair’s response rises above its flat line and the first step goes. And it named one case it could not reach: a loop with a real zero in it, whose response sits above its asymptotes everywhere.

That case is not exotic. A lead section — a zero followed a decade or so later by a pole — is the standard way of buying phase at crossover, and the phase a decibel buys prices that phase at ninety degree-decades for every twenty decibels of gain the section moves. A compensated loop is a loop with a zero near crossover, placed there on purpose. So the question worth measuring is not whether the reading’s guarantee survives a zero — it cannot — but what replaces it.

The loop, and two broken steps #

The plant is the one the earlier measurement called marginal: an ideal integrator and two buffered real poles at 1.00 kHz, which with the straight lines crossing unity at the corner has 21.39° of phase margin and is reported at 0.00°. In front of it sits a lead section built as an inverting amplifier around an ideal operational amplifier: a resistor in parallel with a capacitor at its input, which sets the zero, and the same resistor with a smaller capacitor in its feedback, which sets the pole ten times higher, followed by an inversion. The integrator’s gain is set so that the straight lines cross unity at a stated frequency, and every figure checks the solved netlist’s gain and phase against the closed form at three frequencies before it reads anything off either.

The lead section breaks the first step directly. $|1 + jf/f_z|$ is at least one and at least $f/f_z$, so the zero’s response sits above both of its straight lines, by 3.0103 dB at its corner. Near the zero the loop gain can be above its lines, and then the loop crosses unity higher than the lines.

It breaks the third step too. Between the zero and its pole the section’s phase rises, peaking at their geometric mean, and in that stretch the loop’s total phase can rise with frequency rather than fall. A crossing moved higher is then not necessarily a crossing with less phase.

What is not obvious is which way the two breaks push, because they push in opposite directions. For the lines to report more margin than the loop has, the loop must cross higher than the lines — which needs the zero’s excess gain to outweigh the poles’ deficit at that frequency — and the phase must fall between the two crossings — which needs the poles’ lag to outweigh the zero’s lead over the same stretch. Both conditions ask something to dominate the same octave, and they ask it of opposite parts of the loop.

The compensated design, read off its lines #

Start with the placement a designer would use. The lead zero goes at 500 Hz and its pole at 5.00 kHz, and the lines cross unity at 1.58 kHz, near the geometric mean of the two where the lead section supplies most of its phase.

The lines report 29.56°. The loop crosses at 1.27 kHz, below the lines, and has 40.86°. The reading is short by 11.30°, in the safe direction, and by more than any all-pole loop managed: the worst single real pole gave 6.83°.

The size comes from where the crossing is. At 1.58 kHz the loop is past both plant poles, each of whose response is below its lines, and far from the lead zero, whose excess over its own line has mostly gone. The poles win the magnitude and the loop crosses lower. The lead section’s own phase is still rising at 1.27 kHz, but the two plant poles’ lag is rising faster, so the loop’s phase is falling and the lower crossing has more of it. Neither break has happened at this placement: it is the all-pole case, made larger because two poles past their corners pull the gain further under the lines than one does. With the phase read off the phase sketch as well, the reading is 27.12°, short by 13.74°.

Where the lines are over #

The figure that answers the question in general sweeps the crossing across four decades, two either side of the lead zero, and plots how far each reading is from the loop’s own margin.

The heavy line is the reading taken with the loop’s own phase, and it lies below zero almost everywhere. It rises above zero in one place: with the lines crossing just below the zero, within a hundredth of a decade of it. There the reading is over by at most 3.12°.

The gap in the curve is the stretch between the lead zero and the first plant pole, where the integrator’s fall is cancelled by the zero’s rise and the lines are flat. Placed there, the lines do not cross unity at a point — they sit on it — and no single reading exists; the figure leaves the gap rather than choosing a frequency for the lines.

With the lines crossing at 489 Hz the reading is 76.68° and the loop has 73.61°, over by 3.07°. The mechanism is exactly the pair of conditions. At 489 Hz the zero is 3.0103 dB above its line, and the two plant poles, half a decade away, are only a little below theirs; the zero wins and the loop crosses at 555 Hz. Over that stretch the zero’s phase lead is still rising, but the two plant poles at 1.00 kHz and the lead pole are adding lag faster than the zero adds lead, and the loop’s phase falls. Both conditions are met, and the over-report appears.

It is small because both conditions are only just met. The zero can only lift the gain by three decibels at most, and only near its own corner; the poles can only take phase away quickly when they are near. Anywhere the poles are near enough to make the phase fall steeply, they are also near enough to pull the gain down and cancel the zero’s lift. And a loop that crosses near a lead zero with its lead pole a decade away is a loop crossing in the lead section’s lower half, before the phase it supplies has built up: a loop that has plenty of margin to begin with. The over-report on this loop is a few degrees on seventy-odd.

The same sweep with the zero at 200 Hz reaches +4.71°, in the same place, on a loop with 101.9°; on loops with 60° or less it is never over, and it is short by as much as 49.12°. With the zero at 100 Hz or at 1.00 kHz the reading taken with the loop’s phase is never over at any placement of the crossing. With a single plant pole instead of two and the zero at 500 Hz, the most it is over is +1.02°, on a loop with 101.7°.

A narrower lead section raises every one of those numbers, because its pole is closer and takes phase away sooner. With the pole three times the zero rather than ten, the zero at 500 Hz gives +2.47° on a loop of 61.5°, and at 200 Hz +6.58° on a loop of 87.5° — the largest over-report in any sweep measured here, and still on a loop with nearly ninety degrees in hand.

The placement where the loop is thin #

There is one arrangement in which the small over-report arrives on a loop that cannot afford it: a lead zero above the plant’s corner, with the lines crossing at the zero.

With the zero at 2.00 kHz, its pole at 20.0 kHz and the lines crossing at 2.00 kHz, the lines report 2.42° and the loop has 1.17°. The error is 1.25°, which is small in degrees and large as a fraction: the reading says the loop has twice the margin it has. Neither number is one a designer would build, and the reading does say the loop is on the edge. But it is on the unsafe side of an edge, and it is the one placement in the whole sweep where the over-report and a thin loop coincide.

The reason is again the two conditions. Above the plant’s corner both plant poles are well past their own corners, so their lag is falling off slowly with frequency — each is short of ninety degrees by an amount that shrinks as the reciprocal of frequency — while the lead pole a decade above is only starting. The zero, at its own corner, supplies its 3.0103 dB of lift, and the poles, far past theirs, take comparatively little gain away over the same stretch. So the zero dominates the magnitude, the loop crosses higher, and the phase falls between the two crossings because the lead pole’s lag is growing faster there than the zero’s lead.

The phase sketch, which is where the error is large #

Everything so far takes the phase from the loop. On paper the phase is drawn too, and read off its own straight lines at the lines’ crossing. For an all-pole loop that already broke the guarantee: a single real pole, with the lines crossing above its corner, was reported 3.43° over. With a lead section it is worse, and the placement that was thin is where it is worst.

The phase sketch says 17.91°. The loop has 1.17°. The reading is over by 16.74°, and a loop that would ring for hundreds of cycles after any disturbance is reported with a margin that looks merely low.

This error is not the zero’s. At 2.00 kHz the zero’s sketch is exact: forty-five degrees, at its own corner, where the straight-line phase is always right. The error is the poles’. Each plant pole, at twice its corner, is sketched with 58.55° of lag and has 63.43°; together they are 9.78° under. The lead pole, a decade below its own corner, is sketched with none and has 5.71°. That is 15.49° of lag the sketch does not draw, and the 1.25° by which the magnitude lines misplace the crossing brings the total to 16.74°.

So the lead section’s role in this error is indirect, and it is worth stating exactly. Without the lead section, a loop made of an integrator and two poles crossing at twice their corner is not stable at all, and nobody reads a margin off it. The lead section makes that loop possible. It moves a viable crossover into the octave where two poles’ phase sketches are most wrong, and puts its own pole exactly a decade above it, where a pole’s sketch is at its worst.

The sweep with the zero at 2.00 kHz shows both readings together. The heavy line barely clears zero, at one point; the dashed line, the doubly sketched reading, rises well above it over the half decade either side of the zero. A lead pole three times the zero rather than ten times it moves the worst of the dashed line to +3.12° on a loop of 0.3°; thirty times puts it at +12.76° on a loop of 5.1°.

With one plant pole instead of two the pattern holds and the numbers shrink. With the loop’s phase, the most it is over anywhere is 0.12°, on a loop of 65.8°. With the phase sketch as well, it is over by as much as 5.72°, on a loop of 20.1°: one pole’s worth of the sketch’s lag deficit, where the loop with two poles had two.

The narrow lead section is the one case that puts a loop-phase over-report on a loop a designer might build. With one plant pole, a zero at 1.90 kHz and its pole three times higher, the reading is over by 1.52° on a loop of 52.8°: the magnitude lift of the zero at its own corner, with the lead pole close enough above it to make the phase fall across the few per cent between the two crossings. It is a degree and a half on fifty — within what any margin quoted to the nearest five degrees already hides — and the doubly sketched reading on the same loops is over by 7.20°, on a loop of 14.3°.

The pair of errors has a family resemblance worth naming. The straight lines, and where they are not the curve found a single real pole’s sketch error symmetric about its corner and bounded by 3.0103 decibels; a lead section is a zero and a pole whose sketch errors have opposite signs and each die away within a decade of their own corners, which is why its over-reports are confined to the octave around the zero. And a loop that crosses unity more than once, which a narrow lead section on a peaking plant can produce, is the case stable and unstable with less gain takes up — the sweeps here leave those placements as gaps rather than report a margin for them.

What replaces the guarantee #

For all-pole loops the reading had a proof. For lead-compensated loops it has a measurement, over the loops swept here — an integrator, one or two real poles, and a lead section whose pole sits three, ten or thirty times above its zero, with the zero at placements from a tenth of the plant’s corner to twice it.

Taken with the loop’s own phase, the reading can be over, and not by much. The largest over-report found is 6.58°, on a loop with 87.5° of margin. On loops with sixty degrees or less the largest is 1.52°, on a loop of 52.8°, and the thin case is 1.25° on 1.17°, which misstates a loop that the reading already calls marginal. The proof’s first step fails; the third step fails with it; and because they fail at opposite ends of the lead section, the failure mostly cancels.

Taken with the phase read off its straight lines, the reading is over by as much as 16.74°, on a loop with 1.17°. That error belongs to the poles’ phase sketches rather than to the zero, and the lead section’s contribution is to make a crossover viable exactly where those sketches are most wrong.

And the short errors are larger than any all-pole loop’s: 11.30° on the textbook placement, and up to 77.84° with the zero at a tenth of the corner, where the lines cross in a flat stretch of the sketch and mean very little.

The measurement’s reach is its limit. These are real zeros in the left half-plane, one lead section, real plant poles. A loop with a pair that peaks has already been shown to have no bound at all, and a lead section does not restore one. A zero in the right half-plane has a response above its lines and a phase that falls rather than rises, so its two breaks push the same way; the energy that arrives first measures what such a zero does in time, and what it does to a margin read off a sketch is not measured here. The safe practice is unchanged from what is left at crossover: read the crossing off the solved or measured loop and the phase there, and keep the straight lines for placing the zero.

The companion readings behave differently. The gain margin the straight lines get exactly wrong found an all-pole loop’s gain margin exact up to a known correction, and the corner error a filter hides in its sections found the same corner errors adding up inside a filter. A lead section’s zero has a corner error of its own, +3.0103 dB, and it adds to the poles’ in exactly the same way; it is the phase, not the gain, that the zero makes hard to read.

Still open: the gain margin with a lead, a zero on the other side, and the phase the magnitude recovers #

The gain margin of a lead-compensated loop. An all-pole loop’s phase crossed −180° at a corner, and its gain margin read off the lines was exact up to a correction. A lead section moves the −180° crossing off every corner, because it adds rising phase between its zero and its pole, and the frequency conversion that made the phase margin hard comes back into the gain margin. Whether the over-report there is also confined to comfortable loops, and whether the zero’s +3.0103 dB appears in it as a correction, is the next measurement on this loop.

A zero in the right half-plane. Its magnitude lines are the same as a left-half-plane zero’s, and its phase falls instead of rising. The first step of the proof fails as before, and now the third step holds, so the two no longer cancel: the loop crosses higher and has less phase there. That should make every reading of such a loop’s margin over rather than short, by an amount this sweep would bound. It has not been run.

The phase recovered from the magnitude, rather than sketched. The phase the magnitude already knows recovers a minimum-phase loop’s phase from its magnitude by an integral. The phase sketch here is a crude version of that integral, and most of its 16.74° is two poles’ lag missing a decade from their corners. Whether a sketch of the magnitude, fed through the integral, does better than a sketch of the phase would say which of the two drawings a designer should trust at crossover.