The cliff before the fastest settling

“Critically damped is the fastest response without overshoot” is true and is almost always read as “critically damped is the fastest response”. Those are different claims, and the second is false — which is unsurprising, and would not be worth an essay. What is worth an essay is how it is false: the quantity it is false about does not vary smoothly, the optimum sits on the edge of a discontinuity, and where that discontinuity is depends on something that is not a property of the circuit at all.

The circuit and the quantity #

A series RLC driven by a step, with the output taken across the capacitor: ten millihenries, a microfarad, and a resistance chosen to give whatever damping ratio is asked for. The natural frequency is 1591.5 Hz and stays there; only the resistance moves.

The quantity is settling time — the last moment the response is outside a stated band around its final value. That is the thing a data converter’s specification means by settling, the thing a mechanical positioner means by it, and the thing a control loop’s step response is judged on. It is not the same as rise time and it is not the same as the time constant, and it is the only one of the three with a discontinuity in it.

The band matters and is usually left implicit. “Settling time” with no band is not a specification.

The one point the exact route cannot compute #

The responses come from the residue expansion — the transfer function’s poles are recovered from the network’s own matrix and the inverse transform is written out — which is exact.

It has no answer at ζ = 1. Two coincident poles have no partial-fraction expansion; the residue of a double pole is not a residue of two simple ones, and asking for it divides by the distance between poles that are not apart. The routine refuses by name: the network has a repeated pole; the residue expansion does not apply to it.

That is worth pausing on. The single damping ratio that the entire textbook claim is about is the single point this method has nothing to say at. So the sweep steps around it, and the point is filled by the other route — marching the netlist forward in time with the trapezoidal rule, which does not know or care whether the poles coincide.

The two routes are compared where both exist. At ζ = 0.6 the marched settling time is 5.9430/ω₀ and the expanded one is 5.9430/ω₀, agreeing to better than two parts in a thousand, which is the check that the marched value at ζ = 1 can be trusted.

The cliff #

At a ±2% band the curve falls from 5.346/ω₀ at ζ = 0.775 to 3.605/ω₀ at ζ = 0.780. That is a thirty-three per cent fall for a change in the damping ratio of five thousandths, and it is not a steep slope drawn at coarse resolution — the sweep resolves it to that step and the response either side of it is a perfectly ordinary second-order step.

Read the other way, the number a designer cares about: a circuit built at ζ = 0.775 settles forty-eight per cent slower than one built at ζ = 0.780. Component tolerances of a few per cent move a damping ratio by considerably more than five thousandths, so this is not a hypothetical distinction; it is a lottery that a design sitting near the optimum is entering.

Why it is there, and why the optimum moves #

The mechanism is one sentence, and it explains everything else in the essay.

At the fastest damping, the first overshoot’s peak sits exactly on the edge of the band.

The overshoot at the ±2% optimum is 2.0%. At ±5% it is 4.8%; at ±1%, 0.9%; at ±0.5%, 0.4%; at ±0.2%, 0.2%. Every one of them is the band. That is not a coincidence and it is not fitted — it is the condition that defines the optimum, arrived at from below.

A little overshoot is helpful, because the response reaches the band’s edge sooner going up than it would arriving from underneath. It stops being helpful the instant the peak pokes out through the top of the band, because then the response has left the band and settling is no longer decided by when it first arrived: it is decided by when that excursion decays back inside, which is roughly another half-cycle plus a decay. That step, from “the peak is inside” to “the peak is outside”, is the cliff.

And it explains the thing that makes this more than a curiosity. The best damping ratio is not a property of the circuit. It is a property of the circuit and the accuracy demanded of it together:

As the band tightens the optimum walks towards critical damping, which is what a reader would expect — at a band of zero, nothing that overshoots at all can help. What is less expected is how slowly it walks: at two parts in a thousand the optimum is still ζ = 0.895, and critical damping is still forty-nine per cent slower than it.

Reading the discontinuity off the response itself #

It is worth checking that the cliff is a property of the response and not of the way the settling time is extracted from it, because that is exactly the kind of artefact this collection keeps finding.

Two things were done about it. The crossing of the band’s edge is interpolated between samples rather than being taken as the sample at which the trace was last outside — a quantised crossing turns a smooth dependence into a staircase and puts steps of the sampling grid’s own size next to the real one, which is the artefact. And the sweep was run at five thousand time points per trace over thirty-four time constants, so a step of the grid is a hundredth of the smallest quantity being reported.

What is left after both is a curve that is smooth to the resolution of the damping grid everywhere except at one place, where it falls by a third between adjacent points. Refining the damping grid narrows the interval the fall happens in without removing it: bisected rather than sampled, the jump sits at ζ = 0.77970 and goes from 5.068/ω₀ to 3.603/ω₀, a fall of 28.9% across a damping interval that can be made as small as the arithmetic allows. That is what a genuine discontinuity does. A sampling artefact does the opposite — it shrinks with the grid and disappears.

The thirty-three per cent quoted from the sweep is therefore slightly larger than the true jump, because the sampled point to the left of the cliff is not right at it. The figure reports what the sweep measured and the bisection is what the number means.

The other check is the mechanism itself: the overshoot at the fastest damping equals the band, at all five bands, to within a grid step. An artefact of the extraction has no reason to satisfy that.

The value everybody reaches for #

ζ = $1/\sqrt2$ is the most quoted damping ratio in the subject, on the entirely good ground that it is the one with a maximally flat frequency response. Judged on settling it is a lottery ticket.

At a ±5% band, 0.707 gives 2.921/ω₀ against the best available 2.879 — within one and a half per cent of optimal, and about as good as it gets.

At a ±2% band, 0.707 gives 5.968/ω₀, which is worse than critical damping’s 5.834 and sixty-six per cent worse than the 3.605 available at 0.780. It sits just to the wrong side of the cliff.

At ±1% and below it is back to being reasonable — 6.583 against 6.638 for critical damping — because by then the cliff has moved to a higher ζ and 0.707 is on the ordinary part of the curve again.

So the same damping ratio is near-optimal, worse than critical, and near-critical, on the same circuit, depending on nothing but how tightly the answer is required to arrive. There is no way to read that off a frequency response.

The other side, where nothing interesting happens #

Everything above is about the region between 0.6 and 1. To the right of it the curve is dull and it is worth saying how dull, because the dullness is the argument for not going there.

Past critical damping there is no overshoot at all, so settling is simply the slower of the two real poles decaying through the band, and the time grows steadily as the damping ratio does. At ζ = 1.6 — a resistance sixty per cent above critical, which is a perfectly ordinary thing to find in a circuit somebody wanted to be safe — the ±2% settling is 11.520/ω₀, three and a fifth times the 3.605 available at the optimum and twice what critical damping costs.

That is the price of the usual instinct. Asked to make a response “well behaved”, the reflex is more damping, and more damping is monotonically worse from ζ = 1 onwards with nothing to show for it: the overshoot was already zero. The whole of the interesting behaviour is in the four tenths of a damping ratio below one, and the reflex walks away from it.

A worked number #

The circuit here has ω₀ = 10 000 rad/s, so the times above convert directly.

A sample-and-hold in front of a converter, an actuator moving between positions, a supply’s output recovering from a load step — all of them are this table, and the spread across it is a factor of 3.2 on a network whose components differ by a single resistor.

The row that costs the most is not the slowest. It is the second one: 534.6 µs from a design aimed at 360.5, missed by five thousandths of a damping ratio, with a step response that a scope shows as slightly less overshoot than intended. There is no measurement of that waveform, short of measuring the settling itself, that says anything is wrong.

What this does to a specification #

Three consequences, and they are practical rather than curious.

Do not specify a damping ratio; specify a settling band and let the damping fall out. A design handed “ζ = 0.7” and a 2% requirement is being handed the wrong number by two thirds of the time available.

Sit to the right of the cliff, not on it. The optimum is the cliff edge, so a design at the optimum has zero margin against component tolerance in one direction and very little cost in the other: at ±2% moving from 0.780 to 0.85 costs 4.189 against 3.605, sixteen per cent, and buys complete immunity to the discontinuity. Sixteen per cent of settling time is usually a much better trade than a coin flip on forty-eight.

And measure the settling rather than inferring it. Overshoot, rise time and bandwidth are all smooth in the damping ratio. Settling to a band is not, and no measurement of the other three locates the step.

What the sweep does not cover #

Only second order. Everything here has two poles. A real settling problem usually has three or more, and while the mechanism generalises — settling is set by whichever excursion last leaves the band — the arithmetic does not, and the number of cliffs grows.

No zeros. A zero in the response changes the overshoot without changing the poles, which moves the cliff without moving the damping ratio. That is the case where reading the damping off the poles and looking up a settling time is most wrong.

And nothing about the source of the step. The drive is an ideal step. A real one has a rise time, and a rise time comparable with the natural period changes the excursions substantially. What is measured here is a property of the network’s response to an idealisation, which is the honest description of every settling number in every datasheet.

Where the discontinuity is met again #

A settling time that jumps rather than curving is the same shape as three other results in this collection. The cancellation that leaves a tail is a settling time set by a residue rather than by a pole, which no damping ratio reaches. Two measurements of one margin is the frequency-domain reading of the quantity this page measures in time, and the two agree to a tenth of a degree only when the margin is comfortable. Where the behaviour is written down is the pair of poles behind both. All three depend on One step, computed twice, because a cliff a third of a per cent wide is not a thing to read off an approximation, and A sum that is exact, and the estimate that is not is the shortcut that would have missed it entirely.

The gate #

The discontinuity is asserted to exist, at more than ten per cent in one step of the sweep, and its size and position are reported rather than assumed. At ±2% it is thirty-three per cent between ζ = 0.775 and 0.780.

The optimum is asserted to be faster than critical damping, at every band on the slider, which is the claim the essay is named for.

The overshoot at the optimum is asserted to equal the band, within thirty per cent of the band — because that is the mechanism, and an assertion on the mechanism is worth more than one on the number it produces.

The residue expansion is required to refuse ζ = 1 by name. An assertion that has stopped rejecting has stopped testing, and this one guards the only place in the essay where a second method was needed.

And the two routes are checked where both exist, at ζ = 0.6, agreeing on a settling time to two parts in a thousand.

Why a discontinuity is the right shape for this quantity #

A settling time falling by a third in one step of a five-thousandth sweep looks like a numerical artefact and is not, and the reason is worth stating because it recurs.

Settling time is defined as the last instant at which the response leaves a band, so it is the maximum over a set of excursions rather than a smooth function of anything. As the damping changes, one excursion shrinks continuously until it stops leaving the band at all — and at that instant the maximum jumps to whatever the previous excursion was, which is an entirely different time. The underlying waveform moves smoothly; the statistic taken over it does not.

Three cliffs, and where they are is the essay that counts them and finds three between 0.3 and 0.98, one for each excursion that stops leaving the band, with a third pole moving all three left and making all three shallower. Which is why the classical 0.78 for fastest two per cent settling is a second-order number: at a third pole one and a half times the natural frequency the answer is 0.745 and 0.78 is on the wrong side of the step.

The general instruction is the one this collection applies to every optimum. The load that takes the most locates a smooth maximum by search rather than on a drawing grid; here the quantity is not smooth, so a search finds the right answer and a grid finds whichever cliff it happened to straddle — and the number that matters is not the optimum but the distance from it at which the design falls off.

Which makes this the one place in the collection where the usual advice is exactly backwards. Nearly every optimum here is worth sitting at — a burden voltage at a geometric mean, a source resistance at a ratio of two noise generators, an impedance level in the middle of a band — because the curves around them are flat and a tolerance costs little. Here the curve is not a curve, the optimum is on the edge of a step, and the right design is a deliberate distance to the safe side of it. Which side that is depends on the pole count, and three cliffs, and where they are is where the answer stops being obvious: a third pole moves every step left, so a design placed conservatively above the second-order optimum can be sitting on the wrong side of a cliff that has moved underneath it. So the safe distance has to be computed on the circuit that exists rather than on the second-order idealisation, which is one more sweep of a slider and is the difference between a design that settles and one that settles half as fast for no reason anybody can see.

That last clause is the part worth insisting on. A circuit forty-eight per cent slower to settle looks identical on an oscilloscope, has the same overshoot to within a per cent, and passes every check a frequency response could make — so the cost of being on the wrong side of the step is invisible to every measurement except the one this essay makes.