The best damping is not the one to build

Three cliffs, and where they are sweeps a third-order step’s damping ratio, finds three discontinuities in the settling time rather than the one the rung below it reported, and shows all three moving when the third pole moves. Both essays are about the discontinuities. Neither is about the quantity the machinery returns beside them on every call — the smallest settling time in the sweep, and the damping at which it happens.

That is not an oversight in the reporting so much as an assumption about what kind of object an optimum is. A minimum is normally a stationary point: flat at the top, forgiving in proportion to the square of the error, and the sort of thing a design can sit on with a tolerance around it. This one is not a stationary point at all. It is the right-hand limit at a discontinuity, and everything a designer would do with an optimum has to be done differently because of it.

What is being solved #

A normalised third-order response: a complex pair at unit natural frequency with damping ratio ζ, and a real pole at −a in the same units, with the numerator set to the product of the poles’ negatives so the direct-current gain is one and a settling band means what it says. The step response comes from the residue expansion, which is exact, and the settling time is the last time the response is outside the band — a property of the last excursion rather than of an envelope, so it is located by bisecting inside the grid interval that contains the crossing rather than read off the grid.

The reason the smallest value sits there is worth stating as mechanism rather than as coincidence. The fastest damping is the one whose first overshoot peaks exactly on the band’s edge: the response reaches the band as early as the damping allows and then never leaves it again. One step of damping lower and the peak pokes through, the settling time is set by the decay of that excursion instead, and the number jumps. So the optimum and the last cliff are the same point approached from two sides, and there is no interval between them in which the curve is flat.

Why the last cliff, and not one of the others #

There is more than one discontinuity in the ordinary range, and the optimum is on the last of them every time. That is a fact about how the staircase is built rather than a coincidence, and the ladder of jumps says why.

The steps grow as the damping rises. At a ±2 per cent band with a third pole at three times the natural frequency the three cliffs are worth 1.237, 1.297 and 1.423 in settling time; at ±0.2 per cent the six are worth 1.112, 1.145, 1.167, 1.178, 1.234 and 1.308. In both ladders the last is the largest, and between any two cliffs the curve rises again — so a damping ratio sitting just above an earlier cliff has given up all the later steps and is climbing towards the next one. Only the right-hand limit at the final cliff has taken every step and has no rise ahead of it before the curve turns upward for good.

How many steps there are is decided almost entirely by the band. Counting the cliffs between ζ = 0.3 and 0.98 at three third-pole positions gives two, two and three at ±5 per cent, then three, four, five and six at ±2, ±1, ±0.5 and ±0.2 per cent with no variation across the poles at all. Each tightening of the band by a factor of two to two and a half admits one more excursion into the count, because one more of the response’s ringing peaks now pokes outside a narrower band. The third pole moves where the steps are and does not change how many there are — which is the same division of labour the surface above shows, arriving from the discrete side.

The optimum is a surface, and its two axes are not worth the same #

The classic figure — ζ = 0.78 for the fastest two per cent settling — is quoted as a property of second-order systems. It is a property of second-order systems and of the two per cent, and the second half is missing from every statement of it.

The asymmetry between the two axes is the practical reading, and it goes the useful way round. The band is a specification: a designer knows it exactly, because it is what the circuit was asked for. The third pole is a parasitic — an extra stage’s rolloff, a load capacitance, an amplifier’s second corner — and it is known to perhaps a factor of two. The axis that is uncertain is the one worth 0.047 of damping ratio across its whole range, and the axis worth 0.231 is the one written on the requirement.

That is the opposite of the usual situation with a design constant and it makes the constant more useful rather than less: a settling optimum computed for the wrong third pole is nearly right, and a settling optimum computed for the wrong band is not. Which is exactly how the textbook number is normally misused, because the band is the half that gets dropped when the number is repeated.

The cost of dropping it is not small. For a second-order response ζ = 0.78 settles in 1.0004 times the fastest available at ±2 per cent — right to four figures, which is why the number survived — and in 1.5512 times the fastest available at ±1 per cent. One factor of two in the accuracy asked for, no change to the system, and a value that was exact is now 55 per cent slow. The Butterworth damping ζ = 0.707, which is what most second-order sections are actually built at, takes 1.6547 times the optimum at the very band the 0.78 figure is quoted for.

What the third pole costs, since it is not the decision #

Reading the surface as the third pole hardly matters would be reading half of it. It hardly matters to the choice; it matters a great deal to the result.

The fastest settling a third-order response can reach, at each band, with the pole at 1.5 times the natural frequency against a pure second-order response: 3.5631 against 2.8605 at ±5 per cent, 4.3638 against 3.6042 at ±2, 4.9910 against 4.1937 at ±1, 5.6319 against 4.8013 at ±0.5, and 6.4947 against 5.6262 at ±0.2. In units of 1/ωₙ, and the ratios are 1.2456, 1.2108, 1.1901, 1.1730 and 1.1544.

So a third pole only half again the pair’s own natural frequency costs between 15 and 25 per cent of the settling time, at every band, whatever damping is chosen — and it costs it in the quantity a specification is written in, while moving the design decision by two to five hundredths of a damping ratio, which is the same order as the tolerance window the design rule below is built around. The two halves of that sentence are usually run together into “the third pole degrades settling”, and they behave completely differently: one of them is a term in the budget and the other is not a term in anything.

The ratios also fall as the band tightens, which is the third pole becoming relatively cheaper the more accuracy is asked for. At a loose band the response arrives quickly and the extra pole’s own decay is a large fraction of the total; at a tight band the pair’s own ringing dominates the last excursion and the third pole is a smaller part of it. That direction is worth knowing because it is the opposite of the intuition that extra poles hurt most when the requirement is hardest.

The shape of the penalty #

Being wrong about the optimum by a given amount is not one number. It is two, and they differ by two orders of magnitude.

A stationary point has a vanishing first derivative, so an error of ε costs order ε² in either direction and costs the same either way. This has a first derivative on one side, a jump on the other, and nothing that behaves like a curvature at all. The consequence for a design is immediate: the usual instinct to centre a nominal value on an optimum is exactly wrong here, because half the tolerance band lands on the wrong side of a step.

The direction of that trend is worth having because it says which designs the effect matters most to. A loose band has the sharpest cliffs and the widest forgiving interval above them; a tight band has shallower cliffs and less room. So the circuit most exposed to sitting a hundredth below the optimum is the one asked for a loose settling specification, which is not where anybody looks for a precision problem.

What to build, given that ζ has a tolerance #

No design has a damping ratio. It has components with tolerances, and a damping ratio that lives somewhere inside a range — which turns the question from where is the optimum into where should the range be put.

The searched answer has a closed form behind it and the search was not told about it. The window’s lower limit is ζ(1 − x); the disaster is having that limit below the cliff; so the smallest nominal that avoids the disaster is ζ_c/(1 − x), and above that the worst case is the window’s upper edge, which rises. The minimum is therefore the boundary approached from above — 0.7734/0.98 = 0.7892, against a search that returns 0.7927, the first value it tried that clears it. At ±10 per cent the same construction gives 0.7734/0.90 = 0.8593 against a searched 0.8585.

Two things about that agreement are worth separating. The closed form and the search share the settling-time function and nothing else: one evaluates it at eighty-five nominal values and takes a minimum of maxima, and the other never evaluates it at all except to locate the cliff. And the rule it produces is a one-sided design rule, which is unusual enough to state plainly — the nominal is not centred on anything, it is pushed up against a boundary by exactly the tolerance, and the whole tolerance band lives above the optimum rather than around it. The optimum being a right-hand limit propagates: so is the design centre.

The rule also has an edge, and it is reached from the same direction as everything else here.

That is not a failure of the closed form; it is the closed form’s own range, and it has a recognisable shape. Clearing the cliff buys a fixed saving — the height of the step, which at this band is a factor of 1.299 — and costs a rising amount as the window is pushed up. The rule is the answer while the saving exceeds the cost, and the crossing arrives sooner the tighter the band, because the steps shrink while the rising branch does not. That is a boundary in a design rule rather than in a model, of the kind the edge that is a region collects, and on a ±2 per cent band it has not arrived by ±10 per cent on ζ while on a ±0.5 per cent band it has arrived by then.

What else moves a damping ratio #

A tolerance on ζ is not a manufacturing tolerance on one part. The damping ratio of a second-order section is a combination of several components, and the arithmetic that turns their tolerances into its own is the subject of the tolerance that is not on any part — where the spread of a response’s own quantity is smaller than the parts’ when the sensitivities partly cancel and larger when they add. Two per cent on ζ is therefore a statement about a realisation rather than about a bill of materials, and it is a computed number rather than a stated one, of exactly the kind the derivative of a root produces.

Two other things move it and neither is symmetric either. A temperature coefficient slides ζ in one direction over the operating range, so the effective window is not centred on the nominal at all; building at the closed form’s value with a one-sided drift below it puts the drift straight over the cliff. And a load, an amplifier’s own gain-bandwidth or a stray capacitance moves the third pole, which the surface above says moves the cliff’s position by up to 0.047 — three times the half-width of a ±2 per cent window, and in a direction that depends on the sign of the parasitic.

The design rule survives all of that, because it is a rule about which side of a boundary the whole range sits on rather than about where its centre is. The window to clear the cliff with is the total excursion of ζ from every cause, and the nominal is that window’s lower edge divided by one minus its half-width. What does not survive is the habit of quoting a tolerance as ± something and centring on the middle of it.

The waveform at the edge #

Nothing in any of these pictures is visible in a step response, which is the reason the effect survived being quoted for decades.

The peak sitting exactly on the band is the whole mechanism, drawn as a waveform rather than as a statistic, and the picture is completely unremarkable. A response at ζ = 0.85 looks the same to any eye and to any measurement of overshoot, rise time or peak time; what has changed is which excursion is the last one outside the band, and that is a fact about a threshold crossing rather than about a shape. It is the same reason where the behaviour is written down puts the poles rather than the waveform at the centre of the argument: the quantity a specification names is often not a feature of the curve.

What the pictures refuse #

Every figure here is built to fail on a claim it might have got wrong, and three of those refusals are worth naming because each rules out a different way of being fooled.

The surface refuses a sweep in which the optimum does not rise monotonically with the tightening band at every third pole and with the receding pole at every band. Either curve turning back would mean the last cliff had been mis-identified — that the bisection had locked onto a different step at one point of the grid — and it is the kind of error that produces a plausible surface with one point wrong in it.

The asymmetry figure refuses a run in which the damping ratios above the optimum are not monotone. A second discontinuity inside the two tenths of a damping ratio drawn above the edge would mean the optimum was not the last cliff after all, and the picture would be about something else entirely; requiring monotonicity there is what makes “the last one” a measured statement rather than a label.

And the tolerance figure refuses a set of curves in which a wider tolerance guarantees a faster worst case. That cannot happen — the wider window contains the narrower one, so its maximum cannot be smaller — and it is the cheapest check in the essay. It also found both of the defects the construction shipped with, and neither was visible to anything else.

The first was a candidate silently dropped. Damping ratios within a few thousandths of ζ = 1 are stepped over throughout this collection, because the residue expansion has no answer at a repeated pole; the worst-case routine was discarding them instead of stepping over them, so any window whose upper edge landed there had its largest candidate removed. A maximum with its largest candidate removed is not a maximum, and a ±10 per cent window reported 4.296 where the answer was near six.

The second was an argument that was simply wrong. The worst case was originally evaluated at the window’s two edges and just under any cliff inside it, on the reasoning that a staircase of rising treads can only peak at one of those places. The treads do not all rise: between two cliffs the settling time falls as the damping increases, so a window’s interior carries maxima that are neither an edge nor a cliff. That shortcut reported 7.93 where the answer was 8.47 — and a scan of the window would not have caught it either, because the scan had the same ζ = 1 hole in it. Both routes were wrong together, which is exactly the case one step, computed twice warns that two routes cannot detect. The window is sampled at twenty points now, plus the cliffs, which agrees with a hundred and sixty to 0.014 per cent.

What this does not say #

It does not say the optimum is not worth computing. It is the reference every number above is quoted against, and without it there is no way to know that ζ = 0.707 costs 65 per cent at ±2 per cent or that the design centre should sit 2.5 per cent above the optimum.

It does not say the settling time is discontinuous in any physical quantity, in the sense every model has an edge means by one. The response is perfectly continuous in ζ; what jumps is a functional of it — the last time outside a band — because the last excursion changes identity. Any quantity defined as the last crossing of a threshold has this structure, which is why the same shape turns up in the cliff before the fastest settling and is absent from every smooth measure of the same waveform.

And it does not extend past the model that produced it. Everything here is a three-pole linear response with an exact step: no slewing, no finite output impedance, no thermal tail. A real amplifier asked for a large step does not follow this curve at all until it is back inside its linear range, which is what the step that is too big measures, and a settling specification below the hundred parts per million that a millisecond of hold costs on a 0.2 per cent dielectric meets a different mechanism again — the hold capacitor itself, whose own figure is a property of the test that produced it and is not quoted in a form a settling budget can use.

The band figure below is the one to read against a specification, because it is the axis worth the most and the one most often left out.

Two things in that picture are worth putting beside the surface. The first is that its optimum, found on a 260-point grid over a netlist’s own components, is ζ = 0.830 against the normalised third-order machinery’s bisected 0.8261 for a distant third pole — agreement to 0.9 per cent between a solved network and a pole-placement, which is what entitles the rest of these numbers to be quoted to four figures. The second is that the overshoot at that damping is 0.93 per cent against a band of 1: the mechanism stated at the top of this essay, measured on a circuit rather than on a normalisation.

The number worth carrying #

0.7927 rather than 0.7734, for a ±2 per cent tolerance on a ±2 per cent settling band — a nominal 2.5 per cent above the optimum, which turns a guaranteed 5.943/ωₙ into a guaranteed 4.243/ωₙ.

The habit that goes with it is about what kind of object an optimum is before anything is done with it. A minimum located by a search comes back as a coordinate and a value, and both of those are the same whether the minimum is a smooth basin or the edge of a step. The question that separates them is cheap — evaluate the objective a little either side and compare the two penalties — and until it is asked, centring a tolerance on the answer is a guess about a curvature nobody measured. Here the two penalties differ by a factor of 120, and every consequence in this essay follows from that one comparison rather than from the optimum itself.