One margin, and the gain under it

Two measurements of one margin cut a three-pole loop, bisected for its phase margin, closed it, measured the overshoot of its step, and inverted the textbook second-order relation on the overshoot to get a margin back. The two agreed: 34.9° against 34.9°. It named the residue — the third pole — and found it small on that loop.

That loop had a great deal of gain: a hundred thousand open-loop, a closed-loop gain of a hundred, so a loop gain of a thousand below its first corner. The agreement it found is a statement about loops like that. A phase margin is one number read at one frequency, the crossover, and the second-order relation that turns it into an overshoot assumes the loop’s phase behaves near crossover the way a two-pole loop’s does. Whether it does depends on what the loop is doing on either side of crossover, and the clearest way to find out is to hold the margin fixed and change everything else.

This essay builds loops with exactly 45° of margin from three lags at different spacings, and finds the overshoot moving from 23.6 to 41.0 per cent while the margin does not move at all.

Loops built to one margin #

The loop is the one the earlier measurement used — an amplifier modelled as a gain and three RC lags, each lag buffered, closed through a resistive divider for a gain of a hundred — with two changes. The three lags are placed at one kilohertz, r kilohertz and $r^2$ kilohertz, so a single number sets how far apart they are. And the amplifier’s open-loop gain is no longer fixed: for each spacing it is bisected until the loop’s phase margin is 45.000° exactly, measured by cutting the loop at the amplifier’s input and bisecting the return ratio for unity.

From each closed loop three things are read. The step’s overshoot, from the closed loop’s poles and residues. The closed-loop peak — the largest magnitude of the closed-loop response over its low-frequency value — by golden-section search on the solved network. And the gain margin, from the same return ratio at the frequency where its phase reaches −180°.

With the lags a hundredfold apart the bisection finds an open-loop gain giving a loop gain of 141 below the first lag. With the lags coincident, three equal poles at one kilohertz, it finds a loop gain of 2.83 — the classic result that three equal lags reach a given phase margin with very little gain.

The overshoot moves; the margin does not #

Across nine spacings the overshoot falls smoothly from 41.0 per cent at coincident lags to 23.6 per cent at a spacing of a hundred. The second-order relation, which turns 45° into a damping ratio of 0.42 and that into an overshoot, predicts 23.3 per cent for every one of them. At wide spacing it is right to three-tenths of a point, which is the agreement the earlier measurement found. At close spacing it is wrong by nearly eighteen points.

The reason is visible in what the bisection had to do. With widely spaced lags the loop crosses unity well above the first lag and well below the third, where the loop behaves like an integrator and one pole: its phase sits near −90° plus whatever the second lag adds, and it changes slowly with frequency. That is a second-order loop near crossover, and the second-order relation applies. With coincident lags the loop gain is so low that crossover happens near the lags themselves, where all three are contributing phase at once and the phase falls steeply through the crossover. The margin, read at exactly the crossover frequency, is 45°; a few per cent above crossover the phase is already much worse. The step’s ringing is made across the whole band around crossover, not at a single frequency, and a loop whose phase falls steeply through that band rings more than its margin admits.

What is left at crossover measured how much of a loop’s gain and phase remain at the crossover frequency, and this is the other half of that reading: what the loop does just above crossover, which the margin does not see.

The closed-loop peak sees the band #

A single frequency-domain number that does read the band around crossover is the closed-loop peak: how far the closed-loop response rises above its low-frequency value before it falls.

At a fixed 45° margin the closed-loop peak moves from 1.985 — six decibels of peaking — at coincident lags to 1.319 at a spacing of a hundred. It moves with the overshoot, because the peak is the largest response anywhere in the band, and the closed-loop response near its peak is where the poles that ring live.

The same second-order model can be run backwards from the peak instead of from the margin: a peak of $M_p$ implies a damping ratio, and that implies an overshoot. The question is then which reading predicts the measured overshoot better.

Over twenty-seven loops — the nine spacings at margins of 35°, 45° and 55° — the margin’s prediction misses by up to 22.2 percentage points and the peak’s by at most 3.1. Both use the same second-order model; the peak simply applies it to the right part of the response. A designer who must predict a step response from a frequency-domain measurement should read the closed-loop peak, and should read the phase margin as a stability statement rather than a damping one.

The gain margin is where the difference shows #

The two margins together say what the phase margin alone does not.

At 45° of phase margin the gain margin runs from 37.2 decibels with widely spaced lags down to 9.0 decibels with coincident ones. Phase margin is read at the frequency where the loop gain is one; gain margin at the frequency where the phase reaches −180°. Close lags put those two frequencies close together, because the phase falls through 180° only a little above the crossover, so the loop is near instability in the gain direction even though it is comfortably stable in the phase direction. The overshoot follows the gain margin, and a loop with both margins healthy rings much as the second-order relation says.

A single gain margin is also cheaper to measure than either the peak or the step: it needs only the frequency at which the loop’s phase reaches −180° and the loop’s magnitude there, both of which a loop-gain measurement already provides. That is why practical stability specifications state both margins, and why a loop specified by phase margin alone can ring well beyond what its designer expected. Stable and unstable with less gain found a loop that becomes unstable when its gain is reduced, a case where the gain margin has two sides; here the single gain margin is enough to separate the loops the phase margin cannot.

Why three equal lags need so little gain #

The number 2.83 for coincident lags is not an accident of the bisection, and working it out shows exactly why such a loop rings. Three equal lags at one frequency together contribute three times the phase of one. For a margin of 45° the loop’s total phase at crossover must be −135°, so each lag contributes 45°, which happens at exactly the lags’ own corner frequency. There each lag’s magnitude is $1/\sqrt2$, so the three together attenuate by $2\sqrt2$, and for the loop gain to be one at that frequency it must be $2\sqrt2 = 2.83$ below the corner. The bisection found 2.83 to the third figure.

The ringing follows from where that puts the −180° frequency. Each lag contributes 60° at $\sqrt3$ times its corner, so the loop’s phase reaches −180° only 1.73 times above the crossover, and there the three lags attenuate by eight. The loop gain is 2.83 over eight, a gain margin of 9.0 decibels — again what the measurement found. Between crossover and −180° the phase falls 45° in a factor of 1.73 in frequency, which is the steep fall the second-order relation does not expect. A two-pole loop with a 45° margin never reaches −180° at all.

With lags a hundredfold apart, by contrast, the loop crosses unity a decade or more above the first lag, where that lag has already contributed its 90° and is no longer changing; the second lag supplies most of the remaining 45°, and the third is so far above that its phase is still small an octave above crossover. The loop is, for practical purposes, two poles, and the relation built for two poles describes it.

The same story in the gain a margin costs #

The two ends of the spacing also say something about the price of a margin. The gain added to buy a margin found a decompensated amplifier made stable by raising its noise gain — spending loop gain to move crossover to where the phase is better. On widely spaced lags that trade is efficient: a little less loop gain moves crossover a long way down the integrator-like slope and buys margin quickly. On closely spaced lags it is not: the phase falls so steeply near the lags that reducing the loop gain moves the crossover only slightly in phase, and the margin it buys comes without the damping the margin promises.

The resistor that buys the margin back isolated a capacitive load behind a resistor to separate a lag from the loop. In the language of this essay that spreads the lags apart, and the measurement here says that spreading them is worth more than the margin number alone records: the same 45° with a wider spacing rings less.

Reading a step back into the frequency domain #

The earlier measurement inverted the second-order relation on an overshoot to recover a margin, and found agreement. The results here say when that inversion is safe and when it is not. On a loop with widely spaced lags, the overshoot gives back the margin. On a loop with close lags it gives back a margin that is far too small — a 41 per cent overshoot inverts to about 30° on a loop that actually has 45° — and a designer who measured the overshoot and adjusted the compensation to raise that inferred margin would be over-compensating a loop that was already where it was meant to be.

The closed-loop peak inverts better because it is the frequency-domain quantity that is the overshoot’s counterpart: both are the response of the closed loop to everything near crossover. The settling time, which the cliff before the fastest settling found discontinuous in the damping ratio, inherits the same dependence on the band rather than on the single frequency, and a loop with close lags settles as its peak predicts, not as its margin does.

When a loop has little gain under its crossover #

The loops that ring beyond their margin are the ones that reach crossover with little loop gain to spare, and they are common. A loop closed at a high closed-loop gain from an amplifier of modest open-loop gain has a small loop gain at low frequency. A multi-stage amplifier whose stages have similar bandwidths has its lags close together. A regulator or a servo whose plant has several lags of similar size — a thermal mass, a motor and its driver, a filter — reaches its crossover in the region where all of them act. The phase a delay never stops adding is the extreme case: a pure delay’s phase falls without limit through crossover, and a delay-dominated loop rings well beyond any margin read at one frequency.

The opposite extreme is worth placing on the same scale. The ideal amplifier and its bandwidth treats an amplifier with one dominant pole and enormous gain, which is the widest spacing of all: a loop with a single pole and a distant second has a phase margin near 90° and a step with no overshoot, and the second-order relation describes it perfectly because it is second order. The loop that never crosses is the other end, a loop whose gain never reaches unity and has no crossover to read a margin at. Real loops sit between, and where on that line a loop sits is set by how much gain it has to spend under its crossover, which is the quantity this measurement varied and the margin does not report. A designer can read it directly: the loop gain at low frequency, divided by the factor by which the lags have attenuated it at crossover, says how many lags are acting there, and two or more acting together is the warning.

For loops like these the second-order relation should not be used to set the margin. The design target should be the closed-loop peak or the gain margin, and the phase margin should be read as the minimum necessary rather than as the specification of the damping. The earlier measurement’s agreement — 34.9° against 34.9° — was a statement about a loop with a thousand of loop gain; at a loop gain of three the same procedure would have reported a margin from the overshoot well below the true one.

What the measurement leaves out #

The loops are three lags and nothing else: no zeros, no delay. A loop with a zero near crossover — a lead network, or the zero of an output capacitor’s series resistance — has a closed-loop zero that adds overshoot independently of the poles’ damping, and for such loops neither margin nor peak predicts the overshoot without also knowing the zero. The peak still reads the band, so it remains the better of the two, but neither is enough alone.

The lags are spaced geometrically, so one number sets the spacing. Real loops have lags at arbitrary positions, and the measure that matters is how much phase the lags above crossover contribute within an octave of it — which a geometric spacing captures with one parameter and a real loop needs more than one to describe.

And the step is small. A real amplifier’s response to a large step is limited by its slew rate, which flattens the first overshoot and changes the apparent damping; two measurements of one margin measured on a small step, and so does this.

The number to keep #

A phase margin fixes the overshoot only when the loop has plenty of gain under its crossover. Three lags a hundredfold apart, closed at 45°, overshoot 23.6 per cent — the second-order relation’s 23.3 — at a loop gain of 141. The same three lags coincident reach 45° at a loop gain of 2.83 and overshoot 41.0 per cent, 1.73 times as much. The closed-loop peak follows the overshoot where the margin does not: from the peak the worst prediction error over twenty-seven loops is 3.1 points, from the margin 22.2. The gain margin moves with the ringing, from 37.2 to 9.0 decibels at the same phase margin.

Still open: the loop with a zero, the delay, and the margin that is a disc #

A loop with a zero near crossover. A lead network or an output capacitor’s series resistance puts a zero in the closed loop that adds overshoot of its own. How much of the overshoot then belongs to the poles and how much to the zero, at a fixed margin, separates two effects every compensation design mixes.

A loop with a delay. A delay’s phase falls linearly with frequency and never stops. Building loops of fixed margin from a delay and a lag in varying proportion would extend the spacing result to the case that limits sampled and digital loops.

The disc margin. The distance of the loop’s Nyquist locus from the critical point, minimised over all frequencies, combines gain and phase margin into one number that reads the whole band. Whether it predicts the overshoot as well as the closed-loop peak does — it is closely related — is a measurement on these same loops.