Where the behaviour is written down

A second-order circuit — one capacitor, one inductor, one resistor, or any of the dozens of arrangements that behave the same way — is completely described by two complex numbers. Not approximately, not for a range of inputs, but completely: given the two poles, every response the circuit will ever produce to every input is determined.

That is a strong claim and it is worth making precisely, because it is the reason the s-plane is drawn at all.

Recovered, not placed #

Almost every drawing of a pole diagram is made by choosing a natural frequency and a damping ratio and marking the two points that follow. That is fine as an illustration and it cannot be wrong, which is the problem: a figure that cannot be wrong is not evidence of anything.

The poles here arrive the other way round. A netlist is assembled into a matrix that depends linearly on the complex frequency; its determinant is therefore a polynomial; that polynomial is recovered by sampling the determinant at a few dozen points around a circle in the complex plane and transforming back; and its roots are found by an iteration that refines every root against every other simultaneously. Nothing in that chain knows what a damping ratio is.

The circle and the ray are then drawn from the components — the natural frequency is 1/√(LC) and the damping ratio is (R/2)√(C/L) — and the assertion is that the recovered poles sit on them. They do, to six digits at every position of the slider. Two routes to the same two numbers: one through a matrix and a polynomial rooting, one through two component formulas, sharing nothing.

That check is not decoration. The sampling radius, the polynomial recovery and the root finder are each capable of quiet failure, and a wrong pole produces a step response that is smooth, plausible and about a different circuit.

What each coordinate means #

The correspondence between position and behaviour is exact and worth stating carefully, because it is the reason engineers think in this plane at all.

Distance from the origin is the natural frequency. A pole at radius ω₀ produces oscillation and decay whose combined scale is ω₀; moving both poles outward along the same rays speeds everything up proportionally without changing the shape of the response at all.

The real part is the decay rate. A pole at −σ ± jω contributes e^−σt, so the further left the poles are, the faster the transient dies. A pole on the imaginary axis never decays; a pole in the right half-plane grows without bound, which is instability.

The imaginary part is the ringing frequency. Not the natural frequency — the damped frequency, which is lower, and at a damping ratio of 0.3 is 95% of it. The two are often conflated and the difference is small until it is not: at a damping ratio of 0.707 the ringing frequency is 71% of the natural one.

The angle carries the damping ratio. The cosine of the angle from the negative real axis is ζ, which means every circuit with the same damping ratio has its poles on the same pair of rays regardless of speed, and therefore the same shape of response. Two circuits differing by a factor of a thousand in frequency and identical in damping produce identical curves with different axes.

That last one is why the shape and the speed of a response can be designed independently, and it is the single most useful thing the plane offers.

The relation to overshoot, checked rather than quoted #

A lightly damped step response overshoots, and the amount is a function of the damping ratio alone. The standard relation is

overshoot = exp( −πζ / √(1 − ζ²) )

and it appears in every text on the subject. Here it is used as a prediction to test rather than a result to quote: the overshoot is measured by finding the maximum of the computed response, that measurement is inverted to give a damping ratio, and the answer is required to match the ratio the components were built with. It does, to three per cent or better at every setting where there is an overshoot to measure.

The measured family, with the settling time to within two per cent:

The row that is not where it is supposed to be #

Read down the settling-time column and something happens that the usual account does not prepare a reader for.

Settling time falls as damping increases — 7.60 ms, 3.83, 1.95, 1.12, 807 µs, 596 µs — and then keeps falling past the “no overshoot” boundary to 524 µs at a damping ratio of 0.95, and then turns round and rises again to 787 µs at 1.6.

The minimum is not at 0.707, which is the value usually named as the good one, and it is not at 1.0, which is the value usually named as critical damping and described as the fastest response without overshoot. It is near 0.9, and the circuit there settles thirteen per cent faster than the one at 0.707 and thirty-three per cent faster than the one at 1.6.

None of that is mysterious once it is measured. Below the minimum the response is fast to arrive and slow to stop ringing; above it there is no ringing left to wait for, and the limit is a single slow exponential which gets slower the more damping is added. The optimum is where those two costs balance, and it is not at either of the two round numbers the literature names.

The reason it is worth drawing attention to is not that a textbook is wrong — 0.707 is the maximally flat frequency response and 1.0 is genuinely the boundary of oscillation, and both are correct answers to their own questions. It is that neither of them is the answer to “which settles fastest”, and the three questions get run together often enough that the distinction disappears. Measuring is what separates them.

The same two numbers, seen in frequency #

The poles do not belong to the time domain. They are the frequencies at which the network’s matrix loses rank, which is to say the frequencies at which it can support a non-zero response with no source driving it — and that is a statement about the circuit rather than about a domain.

Read in frequency, the same two numbers give the resonance. A pole pair close to the imaginary axis means a sharp peak in the frequency response, because the response is inversely proportional to the distance from the driving frequency to the pole, and along the imaginary axis that distance is smallest opposite the pole. The quality factor and the damping ratio are two names for the angle: Q = 1/(2ζ).

So every statement in the previous field about bandwidth has a partner here about settling time, and they are not two facts but one. A circuit’s half-power bandwidth is f₀/Q; the number of cycles its transient takes to fall below a hundredth is about 1.47 Q. Multiply them together and the Q cancels: the product of bandwidth and settling time is a constant, and no arrangement of components changes it.

Two poles from two elements #

A brief accounting, because the correspondence between component count and pole count is exact and useful.

Each independent energy-storing element contributes one pole. A capacitor stores energy in an electric field, an inductor in a magnetic one, and each adds one state to the differential equation that describes the circuit — so a network with one of each has two poles, a network with three capacitors has three, and so on.

“Independent” is doing work in that sentence. Two capacitors in parallel are one energy store and contribute one pole; a loop consisting only of capacitors and voltage sources has one fewer state than it has capacitors, because the loop’s voltage law fixes one of them. Those degenerate cases are exactly the ones where a naive count goes wrong, and they are why the poles here are recovered from the matrix — the degree of the determinant polynomial counts the independent states correctly by construction, with no case analysis.

The upper bound is what makes it useful in practice. A network cannot ring in more ways than it has places to store energy, so a circuit with one capacitor cannot overshoot at all, and a circuit that does overshoot has at least two energy stores whether or not both were drawn. That is often the quickest diagnosis available for an unexpected ringing: something is storing energy that is not on the schematic, and the usual candidates are the inductance of a wire and the capacitance of a load.

Reading a response off the plane #

Once the poles are placed, the frequency response can be read off the diagram directly, and the construction is worth knowing because it explains the shapes rather than merely producing them.

The magnitude of the response at a frequency ω is one over the product of the distances from the point jω on the imaginary axis to each pole. Slide a point up the imaginary axis and watch those distances: far below the poles both are large and roughly constant, so the response is flat. Level with a pole, the distance to it reaches its minimum, so the response peaks — and how sharp the peak is depends entirely on how close the pole is to the axis, which is the damping.

That single construction accounts for everything in this field and the previous one. A lightly damped pair gives a tall narrow peak because the minimum distance is small. A heavily damped pair gives no peak at all, because the distance is dominated by the real part and barely changes as the point slides. And far above every pole the distances grow in proportion to ω, so an n-pole response falls as ω⁻ⁿ, which is 20n decibels per decade — the slope from the filter field, derived here by moving a point up a line.

The same construction with zeros in the numerator explains what a zero does: the response is multiplied by the distances to the zeros, so a zero on the imaginary axis puts the response to exactly nothing at its own frequency. That is a notch, and it is the mechanism the elliptic filter uses.

What the diagram cannot say #

Two things a pole diagram does not contain are worth naming, because “the poles describe the system” is a claim that gets over-extended.

The zeros. A network with the same poles and different zeros behaves quite differently, and the most striking case is a zero in the right half-plane, which produces a step response that moves backwards before it recovers. Nothing about the pole positions predicts it, and a shorthand that describes a system by its poles alone has silently dropped half the information.

The gain. Two networks with identical poles and zeros can differ by an overall constant, which is not visible in the plane at all. That sounds trivial and is exactly the mistake behind a filter whose shape is right and whose level is wrong by a factor of two — a category of error the pole diagram cannot detect and a step response can.

Which is the general point this collection keeps arriving at from different directions. Every representation of a circuit discards something, and knowing what a particular picture cannot show is as useful as knowing what it does. A pole diagram cannot show a zero or a gain; a Bode plot cannot show an amplitude limit; a schematic cannot show a size.

Where two poles stop being enough #

The obvious question is what happens with more than two, and the answer is the reason the feedback field exists.

Nothing in the machinery changes: an nth-order network has n poles, the determinant is a polynomial of degree n, and the response is a sum of n exponentials. What changes is that the tidy correspondence between position and behaviour stops holding, because the response is now a sum in which terms can interfere.

The practical consequence is that the second-order relations above become approximations whose error has to be stated. A third-order loop with a phase margin of 35° does not overshoot by exactly what the second-order relation predicts for 35°; it overshoots by nearly that, and the discrepancy is the third pole’s contribution. Measuring that discrepancy is the subject of an essay in the feedback field, and it comes out at a fraction of a degree for a well-separated third pole and several degrees for a close one.

There is a second complication that a two-pole circuit cannot show at all. A network with a zero as well as poles — a numerator root, a frequency at which the response vanishes — behaves quite differently even with the same poles, and a zero in the right half-plane produces a step response that initially moves the wrong way before recovering. No pole diagram alone predicts that, and the common shorthand of describing a system by its poles has quietly dropped half the information.

The refusal worth naming #

One case in the family above is not drawn, and it is the one a reader would expect to find in the middle of the slider: exactly critical damping, ζ = 1, where the two poles coincide.

The residue expansion divides by the distance between poles, so at coincidence it divides by zero. The generator refuses rather than perturbing the damping slightly to make the arithmetic work — a response computed at ζ = 0.9999 would be correct-looking, correct to about four digits, and about a circuit nobody asked about. The slider therefore runs to 0.95 and picks up again at 1.6, and the gap in it is a fact about the method rather than about the circuit.

That is a small thing and it is the site’s habit in miniature. A model that cannot answer a question should say so, and the alternative — answering a nearby question quietly — is how a collection of figures stops being trustworthy one convenience at a time.

Two numbers, and everything that is read out of them #

The claim that two numbers in the complex plane contain everything a second-order circuit will ever do is the field’s opening statement, and four later essays are that statement being spent.

The cliff before the fastest settling reads a settling time out of the angle and finds it is not a smooth function of it: a fall of a third in one step of a five-thousandth sweep, with the fastest damping on the edge of the step, so a design a hundredth of a damping ratio to the left settles forty-eight per cent slower with a waveform that looks no different. Three cliffs, and where they are counts them — one for each excursion that stops leaving the band — and finds a third pole moving all three, which is the statement that “second-order” is a condition rather than a description.

Two measurements of one margin reads the same angle out of a loop instead of a step, inverting an overshoot to get 34.9° against a frequency-domain 34.9°. And the derivative of a root asks how far the two numbers move when a component does, and finds that a different question from how far the response moves — a ladder’s advantage being a factor of 2.17 in pole position against eight orders of magnitude in magnitude.

Which is the useful frame for this essay. Everything a second-order circuit does is in the pair; not everything is a smooth function of it, and not every question about the circuit is a question about the pair rather than about a response. And the condition on the whole claim is that the circuit has two poles and not three, which is a condition three cliffs, and where they are shows costing a real design decision: a third pole one and a half times the natural frequency moves the fastest-settling damping from 0.78 to 0.745, which is across a step rather than along a curve.

What the pair is read as #

Two numbers in the complex plane are read four ways across this collection. One step, computed twice reads them as a step response, exactly and then by marching. Resonance, and the bandwidth it sets exactly reads them as a centre and a width. The cliff before the fastest settling reads them as a settling time, which turns out not to be a smooth function of the angle. What is left at crossover reads them as a phase margin, and The straight lines, and where they are not the curve is where the asymptotic construction that stands in for all four is measured against the curve.