The root a ring of equal stages takes

The gap a derivative needs drove two poles together and watched the derivative of a pole — the exact, step-free sensitivity a tolerance table is built on — climb as the reciprocal of the gap between them. At the meeting point there was no derivative to correct, and something else took its place: two coincident poles of a critically damped network, when the capacitor was moved by a fraction $\delta$, flew apart as $2\omega_0\sqrt\delta$, fitted to eight figures over six decades. A part a hundredth of a per cent out moved a simple pole by 1.15 radians a second and split a double one by 200.

That essay ended by noticing that every number in it was about one separation: two poles meeting. A third pole is a third separation, and a fourth a fourth, and the question it left is what the law becomes when more than two poles coincide. The answer is general and short — the m-th root — but the setting it arrives in is worth more than the answer, because the place in electronics where many equal poles actually sit together is a loop of equal stages, and there the law decides how much gain the loop can hold.

A loop of equal lags #

Take m buffered RC stages of equal time constant τ, a microsecond here, and close them into a loop with an inverting gain k: the output of the last stage, multiplied by k, is subtracted from the input of the first. That is a ring amplifier, a multi-stage feedback amplifier whose stages happen to be alike, a phase-shift oscillator with its stages buffered. With the loop open, $k = 0$, the network has an m-fold pole at $-1/\tau$. Close it a little and the pole splits.

The closed-loop poles are the roots of $(1 + s\tau)^m + k = 0$. Written as a distance from the old pole, $(1 + s\tau)^m = -k$, every root sits at the same distance $k^{1/m}/\tau$ from $-1/\tau$, at angles $\pi(2j + 1)/m$: an m-pointed star, its points sliding outward along straight lines as the gain rises. For three lags they leave at 60, 180 and 300 degrees; for two, straight up and straight down; for five, at 36 degrees either side of the positive real axis and three more behind. The solve recovers every root at every gain to two parts in ten thousand of that distance, from a netlist of controlled sources, resistors and capacitors in which no polynomial was ever written down.

Read as a sensitivity, the star is the essay’s point. The distance the poles move is $k^{1/m}$ times their own frequency. A loop gain of a hundredth moves a triple pole 21.5 per cent of its frequency; a single lag moved by the same hundredth would move one per cent. For five lags the same hundredth moves the poles 39.8 per cent. The more poles coincide, the more violently a small coupling between them moves them, and the less a first-order table of sensitivities describes it.

The m-th root, fitted #

The law should be measured rather than read off a factorisation, both because the factorisation assumes the netlist builds what it was meant to and because the earlier essay found that a law can be exact or merely asymptotic, and only a fit over decades says which.

Over six decades of loop gain the fitted exponents are 1.0000, 0.5000, 0.3333 and 0.2500 for one, two, three and four coincident poles. The first is the derivative — a simple pole moves in proportion to what moves it — and the second is the earlier essay’s square root, arriving here from a loop gain rather than from a capacitor. At a loop gain of a millionth, which is an extraordinarily small coupling, a single lag moves a millionth of its frequency, a double pole a thousandth, a triple pole a hundredth and a quadruple pole three hundredths.

That last column is the practical warning. A coupling of a part per million between stages is far below anything a schematic would show — a picofarad of stray capacitance between two stages’ wiring, a ground impedance shared by two buffers — and it moves four coincident poles by three per cent. Equal stages are not an innocent design choice from the point of view of tolerances: they place the poles at exactly the configuration where the poles are most sensitive to anything that couples the stages.

Where the star reaches the axis #

The star’s points slide outward along straight lines, and for three or more lags the two nearest the imaginary axis are pointed towards it. The gain at which they arrive is the gain at which the loop oscillates.

The branch at angle $\pi/m$ reaches the axis when $k^{1/m}\cos(\pi/m) = 1$, so the critical gain is $\sec(\pi/m)^m$. Bisected on the rightmost pole, the crossings are 8.000 for three lags, 4.000 for four, 2.885 for five, 2.370 for six, 2.075 for seven and 1.884 for eight, each matching the expression to a part in ten thousand. Two lags never cross, because their branches point straight up and down and stay at real part $-1/\tau$ for ever.

The sequence falls towards one, and the reason is the phase. At the frequency where the loop’s phase reaches 180 degrees each of the m lags contributes $180/m$ degrees, and that frequency is lower for more lags, where each lag has attenuated less; the loop gain needed to reach one there falls accordingly. The more identical stages a loop holds, the less gain it tolerates. A three-stage feedback amplifier with equal stages oscillates at a loop gain of eight — which is why nobody builds one — and the pole that crosses with the resistance met exactly that number from the other side: a bootstrap with an inverting source lagging through three equal stages oscillated at a gain of sixteen, because a divider halved its loop gain to eight.

Staggering the stages #

The repair for equal stages is to make them unequal. Give the three lags time constants of $\tau$, $r\tau$ and $r^2\tau$ — a geometric stagger — and the gain the loop tolerates rises.

Equal, the lags oscillate at 8. Staggered by 2 they tolerate 11.2; by 4, 26.6; by 10, 122.2, and the curve keeps rising. That is the mechanism of dominant-pole compensation, stated as a pole pattern: one lag made much slower than the others contributes its ninety degrees of phase at low frequency and very little more, so the loop’s phase at the frequencies where the fast lags act is set mostly by them, and they can tolerate a large gain before reaching 180 degrees. Every operational amplifier is compensated this way, with the stagger chosen so large that the slowest pole sits a factor of a million below the others.

The stagger also undoes the m-th root. With the poles distinct, a small loop gain moves each of them in proportion, the way a derivative says, and the violent sensitivity of the star belongs only to the configuration in which they coincide.

A gap restores the derivative #

Between exact coincidence and a generous stagger there is a range in which both laws hold, each over its own span of the gain, and the boundary between them is worth measuring.

Exactly equal, the poles move as the cube root at every gain. Staggered by one per cent, they move in proportion to the gain while it is small — a derivative exists, and it is very large — and join the cube-root law once the gain is large enough that the poles have moved further than the gap between them; by a stagger of ten per cent and of two the linear stretch extends to larger gains. The crossover sits where the distance the cube-root law would move the poles equals the gap, so at a gain of about the cube of the gap.

That is the earlier essay’s finding in its general form. There the derivative of a double pole grew as one over the gap; for a triple pole it grows as one over the gap squared, and for m poles as one over the gap to the power $m - 1$, because the derivative of a root of a polynomial divides by the product of its distances to the other roots. The range over which a derivative describes the pole’s motion shrinks as the gap to the power m. A tolerance table built from first-order sensitivities is valid for a pole only while the tolerance is smaller than the gap to the power m, and for nearly coincident poles that is a very small tolerance.

A gain margin is not a damping #

The stagger has bought something, and the step response is where a reader would see it. What it has bought depends on how it is spent.

At a loop gain of 4, half their critical 8, equal lags overshoot a step by 53.9 per cent. The same gain around lags staggered by four overshoots 20.6 per cent: the stagger has been spent on damping. Push the staggered loop to half its critical gain, 13.3, and it overshoots 60.7 per cent — the same 6 dB of gain margin as the equal loop and no better damped; the stagger has been spent on gain.

So a gain margin alone does not say which was taken. Six decibels of gain margin is a statement about the loop at the one frequency where its phase is 180 degrees; overshoot is decided by the phase margin at the frequency where its gain is one, and the two frequencies are close together when the poles are bunched and far apart when they are spread. What is left at crossover measured the phase margin as the number that decides the step, and where the behaviour is written down is the reason: the step is written in the closed-loop poles, and two loops with the same gain margin can put them in very different places.

Where equal stages are chosen on purpose #

The star is a hazard in an amplifier and a design tool in an oscillator, and the same arithmetic serves both.

A ring oscillator — an odd number of identical inverting stages in a loop, the commonest clock source on a chip — is this essay’s loop deliberately built above its critical gain. Each inverter is a gain and a lag; the loop oscillates because its gain exceeds $\sec(\pi/m)^m$, and since that number falls towards one as stages are added, a long ring of weak stages oscillates as readily as a short ring of strong ones. The frequency it starts at is the star’s crossing frequency, $\tan(\pi/m)/2\pi\tau$, which falls as stages are added, and that is how such a ring’s frequency is set: by counting stages. The m-th-root sensitivity is then a virtue, because it is what makes the oscillation start promptly from any disturbance.

A phase-shift oscillator of three buffered RC sections is the three-lag loop at a gain of eight, and its designer chooses the gain from the star: eight to start, slightly more to start reliably, and an amplitude control to keep it there, as the gain that is exactly one found for the Wien bridge’s equality on a set of measure zero. Unbuffered sections, which load each other, need a gain of twenty-nine instead, because loading staggers their poles — the stagger figure above, arrived at by accident rather than by design.

A multi-stage amplifier is the case where the star is the enemy. Three gain stages each with its own bandwidth, wrapped in overall feedback, are this loop, and if the stages were designed alike the feedback could use a loop gain of at most eight — eighteen decibels — before oscillating. Real three-stage amplifiers are compensated by making one stage dominant, which is the stagger, and nested-loop schemes that keep two stages fast are the art of placing poles neither coincident nor so spread that bandwidth is wasted.

What the fractional power does to a tolerance budget #

The practical consequence for anybody budgeting tolerances is that a first-order sensitivity analysis — the derivative of each pole with respect to each part, multiplied by each part’s tolerance, root-summed — is wrong by a factor that grows without bound as poles approach each other, and wrong in the dangerous direction.

Near a triple pole a tolerance of $\delta$ moves the poles by about $\delta^{1/3}$, and a derivative-based budget would report something proportional to $\delta$ times a large number. For one-per-cent parts the true movement is about a fifth of the frequency; a derivative computed at a configuration with a small gap between the poles reports a movement scaled by the reciprocal of the gap squared, which can be either larger or smaller depending on how small the gap is and has no reason to be right. The direction a response is most sensitive to replaced the first-order budget by the curvature of the response, which is the next term; near a coincidence even the curvature is not enough, because the motion is not analytic in the tolerance at all.

Three cliffs, and where they are found discontinuities in a settling time as the damping crossed particular values, and noted that all of them move when a third pole does. The star says how fast: a third pole near the other two moves them all by the cube root of whatever couples it, so the cliffs of a nearly critically damped third-order network move much further, for a given tolerance, than those of a second-order one.

How the poles were found, and the instrument’s own edge #

Every pole is a root of the loop’s determinant: the netlist of controlled sources — one summing, one per buffer, one for the gain — and resistors and capacitors for the lags, assembled into its matrices, the determinant sampled on a circle of radius $1/\tau$ and fitted as a polynomial, and the polynomial rooted. The star’s distance is checked at every gain against $k^{1/m}$, the critical gains are bisected on the rightmost root and checked against $\sec(\pi/m)^m$, and the exponents are least-squares slopes over six decades. Steps are the residues of the closed loop’s poles summed at a thousand instants and checked against the final value $1/(1 + k)$.

The rooting has its own edge, and it is the subject of this essay arriving inside the instrument. An m-fold root of a polynomial whose coefficients carry a double’s rounding is located only to the m-th root of that rounding: for three coincident poles, to about one part in a hundred thousand; for nine, to about one part in fifty. Past nine stages the determinant’s polynomial has more coefficients than a double resolves around an m-fold root and the rooting returns nothing at all, which is why the critical-gain figure stops at eight. Every derivative, and the one that is zero insisted on a second route for any quantity with no obvious way of looking wrong; here the second route is the closed form, and the closed form is what a fitted polynomial cannot reach.

Still open: a chain that is not buffered, the star in a filter, and tolerance near a crossing #

Unbuffered stages. Three RC sections connected directly load each other, and their poles are distinct however equal the parts: the loading staggers them by itself. The ratio it produces — about 2.6 between the outer poles of three equal sections — is a stagger nobody chose, and how much of the buffered loop’s sensitivity it removes, and at what cost in the gain the phase-shift oscillator needs, is the classic phase-shift network measured as a pole pattern.

The star inside a filter. A filter designed with repeated poles — a critically damped cascade, sometimes chosen because it cannot overshoot — sits at exactly the configuration of this essay, and its tolerance behaviour is the m-th root. Whether a fourth-order critically damped filter built from one-per-cent parts keeps its no-overshoot property, or whether the star’s fractional power splits its poles into a pair that rings, is the question the tolerance that is not on any part would answer with a distribution.

Tolerance near the crossing. At its critical gain a ring of equal stages has a complex pair on the axis, and a tolerance on any stage moves that pair to one side or the other. The probability that a ring built to sit a stated distance below its critical gain oscillates, given the stages’ tolerances, is a distribution of the rightmost pole’s real part — and near the crossing the poles are simple, so for once the derivative is the right tool.