Three cliffs, and where they are

The cliff before the fastest settling sweeps a second-order step’s damping ratio and finds that settling time is not a smooth curve with a minimum. It falls by a third in one step of a sweep of five thousandths, and the fastest damping sits on the edge of that step, so a design a hundredth of a damping ratio to the left of the optimum settles forty-eight per cent slower with a waveform that looks no different.

The work that produced it wrote down what it had not done and named the question this essay answers:

No second rung anywhere. Eleven ladders were started and none was climbed, deliberately… several are obvious already — what the settling cliff does to a third-order response…

Two things come out of asking. One is about the third order and one is about the second, and the second is the more embarrassing.

There were always three #

A settling time is the last time the response is outside the band. Not the envelope, not a time constant: the last crossing. So the quantity is decided by a single excursion — whichever peak or trough is the last one to poke out — and it changes discontinuously when that excursion stops poking out and the one before it takes over.

That happens once per excursion. A second-order step at a damping of 0.3 has several overshoots outside a two per cent band, and as the damping rises they leave the band one at a time.

Between damping ratios of 0.3 and 0.98 — which is the range anything is designed in, since a damping of 0.3 is forty per cent of overshoot — there are three cliffs, at ζ = 0.378, 0.522 and 0.773, with jumps of ×1.24, ×1.30 and ×1.42. Below 0.3 the staircase continues: from 0.15 upward there are eight. The rung below found the last and largest of them, which is the one next to the optimum and therefore the one that matters most, and reported it as the cliff. It is not; it is the top step of a staircase, and the two below it are the same phenomenon at two thirds and half the size.

That is not a correction to any number in the rung below. Every figure in it is right and the discontinuity it draws is the one at 0.773. What is new is that the shape is general: there is one cliff for every excursion, so a tighter band has more of them and a looser band has fewer, and the count is a property of the specification rather than of the circuit — which is the rung below’s own central claim, arriving as a count instead of as a position.

Finding them at all #

The cliffs are a few thousandths wide in damping and the jumps are tens of per cent, so the whole measurement is about not sampling.

Two things are bisected here rather than read off a grid. The settling time itself: the last crossing is located inside the grid interval that contains it, because a grid returns the grid’s own spacing and a discontinuity of a few per cent would be manufactured out of that. And the cliff position: a step between two grid points is refined by bisecting on which side of the midpoint the settling time falls, to thirty-four halvings, so the positions above are the response’s and not the sweep’s.

The step response itself is exact — a residue expansion of the transfer function, evaluated at whatever time is asked for, with no marching at all — which is what makes the second bisection affordable. One step, computed twice is where the two routes to a step response are compared and where the residue route’s exactness is established; this essay is one of the places that exactness is worth having, because a marched answer has a step size and the quantity being measured is smaller than one.

The routine also refuses. A residue expansion has no answer for a repeated pole, and the sweep runs through configurations where the third pole coincides with the pair — so the function that would return a plausible number there raises instead, and the sweep is kept away from those points deliberately rather than by luck.

What the third pole does #

A third-order response has two shape parameters rather than one: the pair’s damping, and where the real pole sits. Every cliff is therefore a curve rather than a point.

Bringing the real pole in from forty times the natural frequency to one and a half moves the last cliff from 0.7797 to 0.7437 and the other two with it, and makes all three shallower — the largest jump falls from ×1.48 to ×1.39. The best achievable settling rises from 3.630 to 4.372 in units of one over the natural frequency, which is a fifth slower, and that penalty is paid at every damping rather than only at the optimum.

So the classic number — ζ = 0.78 for the fastest two per cent settling — is a second-order number. At a third pole of 1.5 the fastest damping is 0.745, and 0.78 is 0.035 to the right of it, which sounds harmless and is not: the response there is on the flat top of the last step and settles at the same 4.37 the optimum does. The dangerous side is the other one, and a design that aims at 0.78 and lands at 0.74 through component tolerance has fallen off the cliff.

That asymmetry is worth stating on its own. The optimum sits at the left edge of a step, so the error that costs nothing is upward and the error that costs forty per cent is downward, and a tolerance analysis that reports a symmetric spread in damping is reporting a wildly asymmetric spread in settling time.

A boundary that was a number and is a curve #

A boundary that was a number and turns out to be a curve is the edge that is a region’s subject, and this is the third place in this collection it has come up.

There the ideal-amplifier model’s one per cent boundary is a contour in frequency and amplitude, and the corner where two mechanisms meet is a fifth inside where the two one-variable specifications cross. Here the cliff is a contour in damping and third-pole position, and the classic figure is one point on it — the point at infinity, which is the one case that cannot be built.

The general form is the same in both: a design rule stated as a number is a section through a surface, taken at a value of the other variable that was convenient rather than typical. A real amplifier has a third pole; a design rule computed with two is a rule about a system nobody has.

Which excursion is the last one #

The mechanism is worth drawing rather than described, because it makes the count obvious and it says what changes when the band changes.

At a damping of 0.35 the third overshoot is outside a two per cent band and sets the settling time. At 0.38 it is not, and the second one takes over — that is the first cliff, and the step is the interval between the two excursions, which is roughly half a period of the damped oscillation. At 0.52 the second overshoot leaves and the first takes over. At 0.77 the first overshoot leaves and what is left is the exponential approach, which is the fastest anything gets.

So the height of each step is about half a damped period and the position of each is where an excursion’s amplitude equals the band. That is why the jumps are all of a similar size — they are all the same half-period — and why the last is largest: past it there is no oscillation left at all, so the step is not to the previous excursion but to the envelope.

It also says exactly what a tighter band does. Counted from ζ = 0.15, a ±10 per cent band has four cliffs, ±5 has six, ±2 has eight and ±0.5 has eleven — the count is the count of excursions that exceed the band, which is roughly the logarithm of one over the band divided by the logarithmic decrement, a number with the specification in it and the circuit in it and neither on its own. And the cliffs crowd together at the bottom: the eight at two per cent are at 0.154, 0.175, 0.203, 0.242, 0.297, 0.383, 0.529 and 0.780, spaced by ratios that widen as the damping rises.

What this means for a design rule #

Three statements, each with a number.

Do not aim at a cliff. The optimum damping is the left edge of a step, so aim above it. At a third pole of three, aiming at 0.80 rather than 0.775 costs nothing measurable in settling time and buys 0.025 of damping ratio against the step.

Recompute the optimum for the poles the circuit has. The second-order figure is 0.780 and a third pole at one and a half times the natural frequency moves it to 0.745. That is 4.5 per cent, which is inside most component tolerances, which is exactly why it matters — a design that trims to the second-order optimum is trimming to a point on the wrong side of its own step.

And a third pole is not free at any damping. A fifth of settling time, from a pole one and a half times out. Where the behaviour is written down is where the poles are read off a network rather than assumed, and it is the measurement that says whether a real circuit’s third pole is at 1.5 or at 15.

Where a designer meets a third pole #

Nothing above is about a filter. It is about any step response with three poles, and the ordinary way to acquire a third one is to compensate a feedback loop.

What is left at crossover is where a loop’s phase margin is measured, and a two-pole loop is the textbook case that produces the second-order closed-loop response this essay’s back curve is. A real amplifier’s second pole is not at infinity, and the resistor that buys the margin back adds a zero and a pole on purpose. So a compensated stage’s closed-loop response is third order at best, and its third pole sits wherever the compensation put it, which is usually two to five times the crossover.

At three times, the measurement above says the optimum damping is 0.775 and the best settling is 3.99 rather than 3.61 — so the compensation that bought the margin cost ten per cent of settling time before any damping was chosen, and moved the damping to aim at.

This is also why the quantity is worth measuring rather than looked up. A phase margin translates to a damping ratio only for a second-order loop, and two measurements of one margin is the essay about how far apart the frequency-domain and time-domain readings of the same margin already are. Adding a third pole widens that gap and adds a discontinuity to the far side of it.

What the second-order sweep looked like from here #

Going back to the rung below’s own sweep with the finer instrument is worth one paragraph, because it says something about how the first two cliffs were missed.

They were in the data. The rung below sweeps damping in steps of five thousandths from 0.3 upward and its figure plots the whole range, so the steps at 0.383 and 0.529 are two of the points it drew. What it did not do was count them: the argument was about the discontinuity next to the optimum, the figure’s assertion was about that discontinuity’s size, and the two smaller steps a third of the way along the axis read as texture.

That is a recognisable failure and it is the same one the number that was wrong documents in a different field: an assertion that names one feature of a curve does not notice the others, and a figure drawn correctly is not the same as a figure read completely. What caught it here was asking a question about a different system — the third-order case — which required counting the cliffs to compare them, and counting is what found three.

What is not in this model #

No zeros. Every response here is all-pole with a unity direct-current gain. A zero in the left half plane speeds the approach and one in the right half plane produces an initial excursion the wrong way — which is a first excursion outside the band, at a time nothing else in the response controls, and would add a step at the bottom of the staircase rather than the top. The cancellation that leaves a tail is what a nearly-cancelled pole and zero do to the same quantity, and that is a fourth parameter.

No fourth pole, and the shape of the argument says what one would do: another parameter, another axis, and the cliffs become surfaces in a three-dimensional space rather than curves in a plane. The count of cliffs would not change, because it is set by the band and the decrement rather than by the order.

And one band. Everything is at ±2 per cent because that is the rung below’s band and the comparison has to be at the same one. The count of cliffs is a function of the band and the whole staircase moves with it, which is measurable with the same machinery and is left as the obvious next question rather than done.

The two ways to make the staircase go away #

Neither is free and both are worth naming, because a designer who has read this far will want one.

Loosen the band. Fewer excursions exceed a wider band, so there are fewer cliffs and they sit at lower damping: counted from ζ = 0.15 upward there are eleven at ±0.5 per cent, eight at ±2, six at ±5 and four at ±10, and the last of them moves from 0.865 down to 0.591. A settling specification that a circuit meets well inside its own margin has no cliff near it at all, and the count above is a statement about a two per cent specification rather than about a second-order system.

Or leave the last cliff behind entirely. Above the last step there is no excursion outside the band at any damping, and the settling time is the envelope’s — smooth, monotone and slowly worsening as the damping rises. A design at ζ = 0.9 settles nine per cent slower than the optimum and has nothing discontinuous anywhere near it, which for a circuit whose damping is set by component tolerances is a much better place to be than the optimum is.

That second option is the one the rung below’s finding argues for and does not quite say. The fastest damping is not the best damping: it is the most fragile one, sitting on an edge with forty per cent on the other side of it, and nine per cent is a cheap insurance premium.

The habit this belongs to #

A quantity defined as the last time something happens is discontinuous in every parameter it has, and the discontinuities are where the identity of “the last thing” changes hands. That is a general statement and it is the reason to expect a staircase rather than a step.

What the measurement adds is the count, the sizes and the movement — and the observation that the rung below, having found one cliff and drawn it carefully, had two more inside the range it was already sweeping.

Which leaves the classical 0.78 in an awkward position that is worth stating plainly. It is not wrong; it is the exact answer to a second-order question, and it is quoted for circuits that are not second-order. At a third pole one and a half times the natural frequency the answer is 0.745, and the distance between the two is not small — it is the width of a step, so a design at 0.78 is on the slow side of a cliff that a design at 0.745 is on the fast side of. Where the behaviour is written down is where “two numbers contain everything” is established, and this is its condition: everything, for a circuit that has only two.