The margin the straight lines report

The straight-line sketch has its error written down. The straight lines, and where they are not the curve measured it exactly: 3.0103 dB above the response at a real pole’s corner, the same a factor above the corner as the same factor below, and $20\log_{10}Q$ at the corner of a pair. Those are errors in decibels, at the frequencies where the sketch is drawn.

The use the sketch is most often put to is not a reading in decibels at all. A phase margin is a reading of the phase, taken at a frequency the magnitude chooses — the frequency where the loop gain passes unity — and on paper that frequency is found where the asymptotes cross the unity line. So the question worth asking is not how many decibels the lines are wrong by. It is what those decibels do to the frequency, and what the frequency then does to the degrees, at exactly the point where a stability judgement is made.

The answer turns out to depend on one thing, and it is not the size of the sketch’s error. It is whether the loop’s gain ever rises above its own straight lines.

The loop, and the two places it crosses unity #

The loop in every figure here is the plainest one that has a margin worth reading. An ideal integrator — a transconductance charging a capacitor, with a buffer after it — supplies −90° of phase at every frequency and a gain falling at twenty decibels a decade. A buffered resistor–capacitor pole at 1.00 kHz supplies the rest. Closed, this loop is a second-order system and stable at every gain; its margin says how well it is damped, not whether it survives.

The integrator’s gain is chosen so that the asymptotes cross unity at a stated multiple of the corner, which is how each figure names where the reading is taken. The solve then says where the loop itself crosses, the phase is taken from the solve at both frequencies, and the closed loop is built from the same netlist so that its poles can be asked directly whether it is stable.

With the lines crossing unity at the corner, the reading is 45.00°. The loop has 51.83°.

The mechanism is easy to follow once the two crossings are drawn. At 1.00 kHz the lines say the loop gain is one, and the true gain there is 3.0103 dB lower, because that is what a real pole’s corner does. So the true gain has already fallen through unity by the time the lines reach it, and the loop crosses at 786 Hz, a tenth of a decade lower. At 786 Hz the pole has contributed 38.17° of lag rather than the 45° it contributes at its corner, and the difference is the 6.83° by which the reading is short.

The slider shows how the error falls away on either side. With the lines crossing at a quarter of the corner the reading is 75.96° against 76.35°, short by 0.38°. At half the corner, 63.43° against 65.53°, short by 2.10°. At twice the corner, 26.57° against 28.02°, short by 1.46°. At four times, 14.04° against 14.25°, short by 0.21°. The worst reading sits with the crossing on the corner, which is where the sketch’s own error is worst.

The error is not symmetric about the corner, although the sketch’s error in decibels is. At half the corner the lines cross at 500 Hz and the loop at 455 Hz, 0.041 of a decade apart; at twice the corner it is 2.00 kHz against 1.88 kHz, 0.027 of a decade. The decibel error is the same at both, but above the corner the lines fall at forty decibels a decade rather than twenty, and the same error moves the crossing roughly half as far.

Why a real pole can only make the reading short #

Every one of those readings is short, and the reason is a three-line argument rather than a pattern in five numbers.

The true gain of a real pole never rises above either of its asymptotes. Its magnitude is $1/\sqrt{1+x^2}$, which is below one and below $1/x$ at every frequency. An integrator has no corner and no error at all. So the true loop gain lies at or below the straight lines everywhere.

So the loop crosses unity no higher than the lines do. Where the lines reach unity the true gain is at most one, and since the gain falls with frequency, the true crossing is at or below that frequency.

And the phase falls monotonically with frequency. A lower crossing has at least as much phase left before −180°. The margin the lines report cannot exceed the margin the loop has.

Nothing in that argument used a single pole. It holds for an integrator with any number of real poles, and for a pole pair whose damping keeps its response under its asymptotes, which is every pair with a quality factor at or below $1/\sqrt2$. There $|H|^2 = 1/\bigl((1-x^2)^2 + x^2/Q^2\bigr)$ is at most $1/(1+x^4)$, below both lines. The inequality holds, solved, at the setting of every figure in this essay whose loop meets that condition.

What the argument does not reach is a loop with a zero in it. A real zero’s response sits above its asymptotes, so the first step fails, and nothing measured here says which way such a loop’s reading errs, or by how much. That is a gap in what has been shown, not a claim that the reading is safe.

Reading the phase off its lines too #

The guarantee belongs to one particular reading: crossover from the straight lines, phase from the loop. On paper the phase is usually sketched as well — flat to a tenth of the corner, a straight line to ten times it, flat after — and read off that sketch at the lines’ crossover.

Below the corner the phase sketch has more lag than the pole, so its error falls on the same side as the crossing error and the two add. With the lines crossing at 617 Hz the reading taken from both sketches is 54.44° against a solved 61.53°, short by 7.09° — to a hundredth of a degree the worst this loop gives with both halves read off paper.

Above the corner the phase sketch has less lag than the pole — the sign of its error reverses about the corner, as the straight lines’ own error curve shows — and that is enough to beat the crossing error.

With the lines crossing at 2.00 kHz, the reading that takes its phase from the loop is 26.57°, short of the loop’s 28.02° as the argument above requires. The reading that takes its phase from the phase sketch is 31.45°. It is over by 3.43°, on a loop built of nothing but an integrator and one real pole.

So the safe direction is a property of one reading and not of the sketch as a whole. With both halves read off paper, even the plainest loop can be reported with more margin than it has. With the crossing read off the lines and the phase read off anything that is the loop’s own phase — a solve, a measurement, or the integral that recovers phase from magnitude — it cannot. At the corner itself the distinction vanishes, because the phase sketch is exact there: both readings of the first figure are 45.00°.

Two poles, and a margin of nothing that is twenty-one degrees #

A second real pole at the same corner keeps the direction of the error and multiplies its size.

Two coincident poles put the sketch 6.02 dB above the response at their corner, twice the single pole’s error, and the true crossing moves correspondingly further down: to 682 Hz, a sixth of a decade below the lines’. At 1.00 kHz the lines’ crossover sits exactly where the phase reaches −180°, so the reading is a margin of 0.00°. The loop crosses where two poles have spent 68.61° of lag between them, and has 21.39°.

The lines have called a stable loop marginal, and the closed loop agrees with the solve rather than with the lines: its poles sit in the left half-plane, the least damped of them a real part of −0.123 times the corner’s angular frequency. The error is the whole of the margin. It is on the safe side, and it is expensive — a designer who believes it adds compensation the loop never needed, and two measurements of one margin is a reminder of how directly a margin of twenty-one degrees shows up in a step, which is a long way from the undamped ringing a margin of zero would mean.

A pair that stays under its lines #

The bound on the error comes from the response never rising above its asymptotes, and a pole pair can honour that too, up to a damping.

At a quality factor of 0.7071 — the Butterworth pair — the response is $1/\sqrt{1+x^4}$, the flattest that never peaks, and it lies under both of its lines. The reading at the corner is again 0.00° against a loop that has 15.21°, crossing at 826 Hz. It is short, as the argument says it must be.

That is the last damping for which the argument holds. Above a quality factor of $1/\sqrt2$ the pair’s response rises above its flat asymptote somewhere below the corner, and the first step of the argument is gone.

The pair that rises above its lines #

A pair at a quality factor of one is the best-behaved pair at its own corner: the sketch is exactly right there, 0.0000 dB out. Its worst error is elsewhere, 1.249 dB at 0.708 times the corner, and the sign of that error is the one that matters here — the response is above the flat line.

Put the lines’ crossing at half the corner, inside the band where the response has risen above its asymptote, and every step of the argument runs backwards. The true gain at 500 Hz is above one, so the loop crosses higher, at 565 Hz. Higher means more lag. The lines report 56.31° and the loop has 50.29°.

The error is 6.02°, about the size of a single real pole’s worst, and it is in the direction that lets a loop through that should not be. It arrives at the damping where the sketch looks most trustworthy at the corner. The quantity that decides the sign of the reading is not the sketch’s error at the corner. It is which side of the lines the response lies on where the loop crosses.

On the edge, and past it #

Raise the quality factor and there is no longer any bound on the error at all.

At a quality factor of two, with the lines crossing at half the corner, the lines report 71.57°. The loop crosses unity at exactly 1.00 kHz with a margin of 0.00°.

The arithmetic is exact rather than approximate. At its own frequency the pair’s gain is its quality factor, 2, and the integrator’s gain there is 0.5, so the loop gain is exactly one at the one frequency where the integrator’s −90° and the pair’s −90° make −180°. The closed loop confirms it by a separate route: the largest real part among its poles is $-5.87\times10^{-17}$ of the corner’s angular frequency, which is zero to the arithmetic, and Routh’s condition on its characteristic cubic — stable only while the integrator’s unity frequency is below the pair’s frequency divided by its quality factor — is met with equality. The loop is on the edge of oscillation, and the sketch reports a margin most designers would call generous.

Neither asymptote contains the quality factor, and that is the whole of why. Where the behaviour is written down puts it in its general form: a pair’s distance from the origin is its frequency and its angle is its damping. The straight lines keep the distance, discard the angle, and the angle is exactly what decides whether this loop rings.

At a quality factor of five, the lines report 82.41° of margin. The loop crosses unity at 1.17 kHz, past the pair’s frequency, where its phase has gone beyond −180°, and its margin is −57.05°.

That loop is unstable. A negative margin is not a thin one. Closed, this loop has a pole in the right half-plane, with a real part of +0.125 times the corner’s angular frequency. A disturbance in it does not ring down; it grows, by a factor of e roughly every 1.3 milliseconds, until something in a real circuit — a supply rail, a limit on current — stops it. The straight lines report more margin for this loop than for any other in this essay.

Going further up the same line of settings stops producing a margin at all. A pair at a quality factor of ten with the lines crossing at a tenth of the corner is refused by the figure: its loop gain passes unity three times, at 101 Hz, 990 Hz and 1.00 kHz, and no single margin describes it. Stable and unstable with less gain is about loops like that, and about why reading one crossing of several is a true statement about a frequency and a false one about the loop.

For this family of loops there is a closed form for how far wrong the lines can be. With a quality factor above one, a loop on the edge has its integrator’s unity frequency at the pair’s frequency over its quality factor, below the corner, and there the lines report $90^\circ - \arctan\bigl(Q^{-2}/(1-Q^{-2})\bigr)$ — 71.57° at a quality factor of two, and rising towards ninety as the quality factor grows. So once a pair peaks near crossover, no margin read off the straight lines, short of ninety degrees, rules out a loop that is on the point of oscillating.

Where a margin off the lines can be trusted #

The measurements sort into a rule that is short enough to use.

Where the loop’s gain stays at or below its straight lines, the reading is safe. That covers integrators, real poles in any number, and pole pairs up to Butterworth damping, provided the crossing is read off the lines and the phase off the loop. The error is then an under-report — at most 6.83° for one pole, and 21.39° for two poles sharing a corner — and a loop that passes on the sketch passes in fact.

Where the phase is read off its own sketch as well, the direction is lost, even for one real pole: 3.43° over when the crossing falls above the corner.

Where a pair peaks near crossover, the reading has no bound and the wrong sign. The band in which the response rises above its lines is narrow when the quality factor is high — resonance, and the bandwidth it sets exactly measures the half-power width of a resonance as $f_0/Q$ — which is what makes it easy to miss on a plot and decisive when a crossover falls inside it. A pair’s quality factor is also rarely the number its components suggest: the Q the amplifier decides finds a section’s quality factor 1.97 per cent high because of the amplifier it was built with. A loop’s margin read from a sketch of such a pair is a statement about a curve that is not in the circuit.

And the remedy is not a better sketch. It is to read the margin where what is left at crossover reads it — at the crossing located on the solved or measured loop gain, by bisection rather than by eye — and to keep the straight lines for the job one solve read four ways gives them, which is finding the shape.

Still open: the gain margin, the sections, and the zeros #

Three continuations follow from what has been measured, and each carries its own argument.

The gain margin read off the lines. A gain margin is read where the phase passes −180°, and for an integrator with a pair that is the pair’s own frequency, where the sketch’s error is exactly $20\log_{10}Q$. So a gain margin read off the lines should be wrong by exactly that amount: short below a quality factor of one, over above it, and unbounded in both directions — the pair’s corner error turned directly into decibels of margin, with none of the frequency-to-phase conversion this essay had to measure. The same loops would check that identity.

A cascade whose sketch is right as a whole and wrong in every part. An eighth-order Butterworth filter built as four sections has pairs at quality factors from 0.5098 to 2.5629. The sketch of the first is 5.852 dB out at the corner and of the last 8.1746 dB. The sketch of the whole filter is 3.0103 dB out at the same corner, because the sections’ errors must sum to it: the product of the four quality factors is $1/\sqrt2$. The straight lines of a filter are then a good description of the filter and a poor description of every section it is built from, and the section that most needs a tight tolerance is the one the sketch misrepresents most. Measuring the sections against the whole would say by how much.

Loops with zeros in them. The argument that makes the reading safe fails at its first step for a real zero, whose response sits above its asymptotes, and a lead-compensated loop is exactly a loop with a zero near crossover — the zero whose phase the phase a decibel buys prices at ninety degree-decades per twenty decibels. Whether such a reading errs safe or unsafe, and by how much, is unmeasured here, and it is the case a compensated design actually meets.