The damping a zero asks for

Every number about the fastest-settling damping so far has been worked out on a response with poles and nothing else. The cliff before the fastest settling found that settling time against damping is a staircase with the optimum on the edge of its last step; three cliffs, and where they are added a third pole and watched every step move; and the best damping is not the one to build measured how lopsided the penalty is on either side of that edge. In all three the numerator of the transfer function is a constant, chosen so the step ends at one.

Real second-order circuits rarely have a constant numerator. A feedback amplifier with a lead network in its loop has a zero in its closed-loop response; so does a filter whose output is taken across a resistor and a capacitor in series, a regulator whose output capacitor has series resistance, and a servo with derivative action. The zero does not change the poles at all, and the damping ratio is a property of the poles, so the habit is to quote the damping and read off the settling time as though the zero were not there. This essay asks whether that is safe, and the answer is that it is safe exactly as far as the zero is distant, and then it is not safe at all.

What a zero does to a step #

Take the response the earlier essays used, normalised to unit natural frequency, and give it a zero:

$$H(s) = \frac{1 + s/b}{s^2 + 2\zeta s + 1}.$$

The direct-current gain is still one, so the step still ends at one and a ±2 per cent band still means what it says. The zero sits at $s = -b$: in the left half-plane when $b$ is positive, in the right when it is negative, and further from the origin the larger $|b|$ is.

The useful thing about this particular numerator is that it splits. The term $1$ gives the ordinary second-order step, $y_0(t)$; the term $s/b$ gives $1/b$ times the step’s own derivative, because multiplying by $s$ differentiates. So

$$y(t) = y_0(t) + \frac{1}{b}\,\frac{dy_0}{dt}, \qquad \frac{dy_0}{dt} = \frac{e^{-\zeta t}\sin(\omega_d t)}{\omega_d},$$

with $\omega_d = \sqrt{1-\zeta^2}$. That gives two independent routes to the waveform. One is the residue expansion every settling calculation in these essays has used, run on the transfer function with its zero; the other is the closed form above, which needs no partial fractions. Every figure here computes both, and they agree to two parts in $10^{16}$, which is the last bit of a double.

The closed form also explains, before anything is computed, what the zero must do. The derivative of a rising step is positive while it rises and negative once it has passed its peak, so adding it makes the rise faster and the overshoot larger. A faster rise should shorten the settling time; a larger overshoot should push the first excursion out of the band and lengthen it. Which wins, and by how much damping the optimum has to move to stop the overshoot winning, is the question.

That last observation is the mechanism stated once and then measured. With no zero, the classic optimum is the damping at which the first overshoot is exactly two per cent: any less damping and the first peak leaves the band, so settling has to wait for the next crossing a half-period later; any more and the response creeps in more slowly. Add a zero and the peak is higher at every damping, so the damping must rise until the peak has come back down to two per cent. The optimum moves because the grazing condition moves, and the grazing condition is about the peak.

The rise is a separate effect and goes the other way. The derivative term pulls the whole step earlier, so at its own optimum the step with a zero at the natural frequency reaches the band at 2.38/ωₙ against 3.62 with no zero. More damping and faster settling at once looks like a contradiction only if damping is read as the thing that sets the speed. Here it is not; the zero sets the speed, and damping is spent holding the peak at the band.

The optimum, measured against the zero #

Each settling time comes from the residue expansion, with the crossing located by bisection inside the grid interval that contains it, and the damping is swept from 0.3025 in steps of 0.005. The offset keeps the sweep off ζ = 1 exactly, where the two poles coincide and the residue expansion stops applying. The optimum is therefore known to the grid’s resolution, five thousandths of damping, and not to the four figures it is printed with. The step the optimum sits on is sharp enough that this is the honest statement of what a sweep can locate.

Read from right to left, the curve has three regions. Beyond about five times the natural frequency the zero is invisible to the optimum: its derivative term is weighted by a fifth or less, the peak rises by less than the grid can resolve, and 0.78 remains the answer. Between five and about one and a half the optimum creeps, by a few hundredths. Below that it climbs steeply, and at a zero of 0.7 the fastest response is the critically damped one, then an overdamped one.

The last region deserves a sentence of its own, because “overdamped settles fastest” contradicts most of what is taught about settling. It happens because a zero below the natural frequency puts so much of the derivative into the response that even the critically damped step overshoots. The overshoot of a real-pole response with a slow zero is not a ringing peak at all — it is one hump with no oscillation after it — and the damping that holds that hump at two per cent is above one. The poles are real; the step still overshoots; the overshoot is the zero’s.

The settling times along the curve fall the whole way: 3.57/ωₙ at twenty, 3.39 at five, 3.03 at two, 2.38 at one, 1.73 at 0.6. A zero is a speed-up, paid for in damping. A design that is told to place its zero closer will always get a faster step at its optimum, and will always need more damping to get it.

A cliff on the side of a slow drift #

The optimum hardly moves for a distant zero, and that invites a conclusion that turns out to be the expensive mistake in this subject: since 0.78 is still nearly right with a zero at three times the natural frequency, keeping 0.78 must be nearly free. It is not, and the reason is what the earlier essays established about the shape of the optimum.

The fastest settling damping is the right-hand limit at a discontinuity. Just to its left, the first peak is outside the band, and settling has to wait for the response to come back down through the band’s edge rather than counting from when it first went in. On these normalised units that wait is about one and a half units of 1/ωₙ — 3.39/ωₙ against 4.84/ωₙ with the zero at four times the natural frequency. When the zero moves the edge to the right, even by a hundredth, a design held at the old optimum finds itself on the wrong side of a step it cannot see.

The numbers make the asymmetry concrete. With the zero at three times the natural frequency, the optimum has moved from 0.7825 to 0.7875, the smallest change the sweep can register. Keeping 0.7825 settles in 4.92/ωₙ against the best possible 3.24 — fifty-two per cent slower, for an error in the damping ratio that would vanish inside any capacitor’s tolerance. At four times, the optimum has moved by less than one grid step and the cost of not following it is forty-four per cent. These are not small sensitivities to a small cause. They are the whole of one excursion, gained or lost on one side of an edge.

The dashed curve is the useful one for design, because it says what a margin buys. A damping of 0.85 is never optimal for a distant zero — it is a steady fifteen or sixteen per cent slow — but it is bounded: it stays inside seventeen per cent of the best everywhere from twenty times the natural frequency down to 1.2. That is the same trade the best damping is not the one to build found for a component tolerance, now applied to an uncertain zero. The optimum is the wrong centre for a quantity that is not known exactly, and a modest step to the right of it, onto the shallow side of the edge, buys a guarantee in place of a gamble.

What the margin cannot do is cover the region below about 1.2, where the optimum itself is moving faster than any fixed margin. There the zero has to be known, and the damping chosen for it.

The waveform when the zero is close #

A slider makes the motion of the optimum visible as the zero approaches. At a zero of 0.7 times the natural frequency it has crossed into overdamping.

Two features of that picture carry over to any second-order system with a numerator. The staircase survives: every curve still falls in steps as each excursion stops leaving the band, and the optimum is still the right-hand limit of the last of them. And the steps move as a group, all to the right, because the zero raises every peak, not only the first. The structure the earlier essays found is a property of “settling time counts the last excursion”, and a zero changes where the excursions are without changing that.

What changes is the far right of the curve. With no zero, settling time climbs steadily past critical damping because an overdamped step creeps in. With a near zero the climb is postponed: the derivative term keeps the overdamped step fast for longer, and the best damping lands where the hump just grazes the band.

The zero on the other side #

A zero in the right half-plane is the case usually called non-minimum-phase, and the closed form predicts how it differs: with $b$ negative the derivative is subtracted, so the step starts the wrong way, dipping below zero before it rises. The zeros that have no side measured that difference in the frequency domain as an all-pass delay; here it is measured in settling.

The hypothesis going in was that a right-half-plane zero would remove overshoot, since it subtracts the derivative where the left-half-plane zero adds it, and so let the damping fall below 0.78. The measurement says otherwise. The derivative of a damped sinusoid is itself a damped sinusoid at the same frequency, so adding it with either sign changes the phase of the ringing and enlarges its amplitude. At the no-zero optimum a right-half-plane zero at the natural frequency gives an overshoot of 2.39 per cent against 1.93 with no zero, and the left-half-plane zero at the same distance gives 6.00. Both push the peak up, and the left-half-plane zero’s excess over 1.93 is nine times the right’s: added with a plus sign, the slope’s ringing lines up with the step’s own, while subtracted it spends most of itself at the start, in the initial dip — a 24 per cent undershoot at that distance.

This is the time-domain face of a result the energy that arrives first found by a different route: mirroring a zero into the right half-plane changes no magnitude anywhere and delivers the same total energy, but on a later schedule, and at no instant has the mirrored network delivered more. Settling time is one more reading of that schedule. The two responses ring with the same poles and nearly the same overshoot; the right-half-plane one simply starts later, because it spends its first moments going the wrong way, and no choice of damping recovers a delay that is written into the numerator. The phase the magnitude already knows is the frequency-domain statement of the same fact: for the minimum-phase network the phase, and so the timing, is fixed by the magnitude, and the mirrored zero adds lag the magnitude never reports.

So the asymmetry between the two sides is not about damping at all. A left-half-plane zero buys speed and costs damping; a right-half-plane zero costs speed and almost no damping. The 0.78 rule is nearly right for the second and badly wrong for the first, which is the opposite of what the frequency-domain reputation of the two would suggest, and it means that the zero to worry about for settling is the helpful one.

There is a reason it is the helpful one that matters most in practice. A designer adds a left-half-plane zero deliberately, as lead compensation, to make a loop faster — a capacitor across the upper resistor of a regulator’s feedback divider is the commonest example, and the lead a divider can give measures what that capacitor buys in phase — and then sets the damping from the poles. The lead that raises the phase margin also puts a zero in the closed-loop response, often within a factor of two of the natural frequency, which is the region where the optimum has already begun to climb. That is the precise combination in which keeping the textbook number costs half as much settling time again, or more.

What the closed form is worth #

It would be possible to treat all of this as the residue expansion’s output and stop. The closed form is worth having for a different reason than checking: it says what the zero is for. A numerator of $1 + s/b$ is a recipe — the step plus $1/b$ of its slope — and every effect here is a consequence of the slope’s sign at each moment of the step. The rise is faster because the slope is positive on the rise. The peak is higher because the slope is negative just after it. The right-half-plane dip is the slope subtracted at the start, when the slope is largest.

That reading generalises past this essay’s numerator. A zero further away contributes less slope; two zeros contribute the slope and the curvature; a zero that cancels a pole removes a mode altogether. Each is a statement about the derivatives of the plain step, and each can be predicted by asking where those derivatives are large. The step computed twice is the habit this continues: a waveform worth trusting comes by two routes that share nothing, and the closed form here is the second route as well as the explanation.

Where the classic number came from, and why it was right #

None of this undermines the textbook figure. The figure 0.78 for fastest two per cent settling is correct for the system it was derived on — a pure second-order response, poles and no zeros — and where the behaviour is written down found the same number from the pole plane. The difficulty is only that the number is quoted as a property of damping, when it is a property of damping and a numerator. A compensated amplifier almost always has a zero in its closed-loop response, and so the number has to be either re-derived for that numerator or protected by a margin.

The re-derivation is cheap. Find the damping at which the first peak is exactly at the band’s edge, including the zero’s contribution to that peak, and that is the optimum to within one grid step. The margin is cheaper and is what most designs actually do: choose a damping in the mid-0.8s and accept a sixth of the best settling time as the price of not knowing the zero to better than a factor of two.

The number worth carrying #

With a zero at the natural frequency, the fastest-settling damping is 0.873 and not 0.78, and keeping 0.78 costs 111 per cent. With the zero at three times the natural frequency the optimum moves by only 0.005 and keeping 0.78 still costs 52 per cent, because the optimum is the edge of a step.

The habit is to ask what the numerator does to the peak before trusting any rule stated for the poles. The damping ratio describes the poles exactly, and the settling time depends on the peak, and the peak belongs to the whole transfer function.

Still open: two zeros, and the band that is not symmetric #

Two questions follow and neither is answered here. The first is a pair of zeros, which is what a second-order filter with a notch or a bridged-tee network gives: the step plus its slope plus its curvature, with three coefficients rather than one, and the grazing condition depending on all three. The curvature term is largest exactly at the peak, which suggests the optimum moves faster with a pair than with a single zero, but that is a guess and the sweep is where it would be checked.

The second is a band that is not symmetric, which is the honest specification for a great many regulators — an overshoot limit tighter than the undershoot limit, because the load can tolerate a droop but not a spike. A zero raises the peak and a right-half-plane zero digs a dip, so an asymmetric band treats the two sides of the plane differently in a way a symmetric one hides, and the side of the plane that settles faster could change.