The cancellation that leaves a tail

There is a construction that appears in three different places in this collection under three different names, and the third name is the one that says what it is.

A compensated attenuator, where a capacitor across each resistor makes the division the same at every frequency. A lag network cancelled by a lead so the loop sees neither. A feedback path trimmed until its own corner disappears. All three are a pole and a zero placed on top of each other, and when they land together the network behaves as though neither existed.

They never land exactly together. What a small miss costs is the essay, and the answer is not where anybody looks for it.

What a doublet is, in one expression #

Take a transfer function with a pole at $p$ and a zero at $z$, both far below the circuit’s own bandwidth:

$$H(s) = H_0 \cdot \frac{1 + s/z}{1 + s/p} \cdot \frac{1}{1 + s\tau_f}$$

with $\tau_f$ the fast circuit. If $z = p$ the first factor is one at every frequency and the response is the fast circuit alone. If $z = p(1 + \epsilon)$ it is not, and the interesting question is how it is not.

In the frequency domain the answer is: barely. The first factor’s magnitude departs from unity by at most a fraction of order $\epsilon$, reached somewhere between the two frequencies and returning to unity on both sides. At $\epsilon = 1\%$ the figure measures the worst disagreement anywhere below the fast corner as 0.0777 dB, against the exactly cancelled network. That is well inside the accuracy of any real gain measurement, and it is also inside the tolerance the components themselves were bought to.

In the time domain the answer is: entirely. The step response acquires a term

$$A\,e^{-t/\tau_s}, \qquad A \approx \epsilon\cdot\frac{R_1}{R_1+R_2},\qquad \tau_s = 1/p$$

— an exponential of small amplitude and slow time constant. The figure measures $A = 0.909\%$ for $\epsilon = 1\%$ at its own resistor ratio, and $\tau_s$ is a hundred times the fast circuit’s own by construction.

That is the whole mechanism. A doublet does not distort a response; it adds a small, slow tail to everything the circuit does.

Why the settling time is where it shows #

The reason a small slow term matters more than a small fast one is that settling is a question about the last part of a response, not the largest.

Settling to a band $B$ means the error has fallen below $B$ and stays there. With a single fast pole that takes $\tau_f\ln(1/B)$: 4.6 time constants to one per cent, 6.9 to a thousandth, 9.2 to a ten-thousandth. Those are the numbers the figure measures with the cancellation exact, and they are what a designer has in mind.

With the tail present, the error is eventually the tail alone, and the tail decays with $\tau_s$. So

$$t_{\text{settle}} = \tau_s \ln\!\left(\frac{A}{B}\right)$$

and the figure asserts this against the marched response to four digits: 245.2 fast time constants measured against 245.2 predicted, at one per cent of miss and a thousandth of band.

Three things about that expression are worth reading off it.

The fast circuit does not appear. Making the amplifier ten times faster changes $\tau_f$ and leaves $\tau_s$ and $A$ alone, so the settling time is unchanged. This is the observation that makes doublets notorious: a stage that is not fast enough cannot be fixed by making it faster.

The dependence on the miss is logarithmic, so halving $\epsilon$ buys $\tau_s\ln 2$ of settling time and no more. Trimming a doublet by a factor of two is worth about seven tenths of a slow time constant, which on this circuit is seventy fast ones — real, but not the order-of-magnitude win the trim looks like it should be.

And the whole thing goes away when $A < B$. If the tail is already smaller than the band being settled to, the logarithm is negative and the doublet costs nothing at all. That is the boundary, and it is unusually blunt.

The boundary: the miss must be smaller than the band #

Written as a design rule, $A < B$ says: the cancellation must be as good as the accuracy being asked for. Settling to 0.1% needs the pole and zero matched to about 0.1%. Settling to 0.01% needs them matched to 0.01%.

That is a much harsher statement than it first sounds, because the two are usually bought from different budgets. The 0.01% is a specification on the converter the stage is driving, and it is met with a good amplifier and a fast settling time. The 0.01% on the doublet is a specification on two components in a compensation network matching each other, which nobody wrote down, and which for a resistor-and-capacitor pair means a capacitor accurate to a part in ten thousand against a resistor.

Nothing about the amplifier helps. The measurement in the figure holds the fast circuit fixed and sweeps only the miss, and the settling time moves by a factor of thirty-six while the fast circuit’s own settling is unchanged at 6.9 of its time constants.

The residue is what changed, not the poles #

That last sentence is the structural point and is worth making carefully, because it explains why the frequency response is so quiet.

A step response is a sum of exponentials, one per pole, and each carries a residue — how much of that pole appears in this particular response. Cancelling a pole with a zero does not remove the pole from the network. The pole is still there; the zero sets its residue to zero.

So a doublet with a miss is a network whose poles are exactly where they were and one of whose residues is small instead of zero. Nothing has moved on the pole plane. The settling time, which depends on the slowest pole with a non-negligible residue, has changed by a factor of thirty-six because a residue crossed a threshold set by the accuracy being demanded.

This is why every frequency-domain instinct fails here. Bandwidth, roll-off, phase margin, group delay — all of them are dominated by the poles, and the poles are unchanged. The only quantity that notices is one that asks about a small part of the response a long time after the event, and settling time is the only common measurement that does.

The instance this collection already had #

The compensated attenuator in the instruments field is a doublet, and reading it that way explains something that essay measured and did not name.

A ten-to-one probe is nine megohms over one, with a capacitor across each. Match the two time constants and the division is ten to one at every frequency. Mismatch them and the division is one ratio at low frequency and another at high — which is what that essay draws, and it is the frequency-domain face of the same object.

The time-domain face is the one every oscilloscope user has met: the flat top of a square wave that tilts, or overshoots and creeps back. That tilt is the tail. Its amplitude is the compensation error and its time constant is the probe’s own, about a hundred microseconds — so a “settled” reading taken a few microseconds after an edge can be a fraction of a per cent wrong for a millisecond.

Where doublets come from when nobody put one there #

The compensated divider is a doublet somebody built on purpose. Most of the ones that cause trouble are not built at all.

A feedback network with stray capacitance. A resistive divider around an amplifier acquires a few picofarads across its upper resistor from the board, and that is a zero; the amplifier’s own input capacitance across the lower one is a pole. Neither was designed, and their ratio is whatever the layout gave.

An amplifier’s internal compensation. A two-stage design frequently has a pole-zero pair from its compensation network that is nearly, not exactly, cancelled — which is why settling-time specifications on precision amplifiers are quoted separately from bandwidth and are so much worse than bandwidth suggests.

A coupling capacitor with a leaky dielectric. The dielectric absorption of a film or ceramic part is a set of slow time constants in parallel with the intended one — a doublet, or several, at milliseconds to seconds, with amplitudes of a few tenths of a per cent. Nothing in a data sheet’s capacitance value contains them.

In every case the signature is the same: a response that is right in the frequency domain and creeps in the time domain, with a time constant far longer than anything in the circuit’s own bandwidth.

How to find one #

The measurement that finds a doublet is not a frequency sweep and not a step response at its usual scale. It is a step response with the vertical axis expanded by a hundred and the horizontal axis expanded by a hundred in the other direction — which is what the lower panel of the figure is.

Plotting the error rather than the response, in units of the tail rather than of the step, turns an invisible feature into the whole picture. The usual view of a step response is dominated by the fast edge and shows a flat top; the error view shows the fast edge as a spike at the left and then the entire tail, and a doublet is unmistakable in it.

The second signature is the ratio test. Settling to $B$ and to $B/10$ should differ by $\tau\ln 10 = 2.30$ time constants of whatever is limiting. If the two settling times differ by 230 fast time constants rather than 2.3, the thing limiting is not the fast pole, and its time constant is the difference divided by $\ln 10$. That measurement identifies $\tau_s$ without knowing anything about the circuit.

The theorem that gives the answer and not the time #

There is a neat statement of what a doublet costs in terms of machinery this field already has.

The final-value theorem returns the settled value of a step response from the transfer function at $s = 0$, with no marching at all, and it is exact. It is also completely silent about when. A doublet does not move the final value by a hair — the cancellation is exact at direct current whatever $\epsilon$ is, because both the pole and the zero contribute unity there — so the theorem’s answer is unchanged by the fault this essay is about.

That is the same relationship the value-theorems essay found between a pole condition and an observation window, arriving from the other side. There the theorem was satisfied and sixty cycles was not long enough to see the answer; here the theorem is satisfied and a hundred fast time constants is not long enough. In both cases the theorem is right and the measurement is not finished, and what separates them is a number the theorem does not contain.

So the pair of questions a transient measurement has to keep apart is: what does it settle to, which is a statement about the network at one point on the complex plane, and when is it settled, which is a statement about every pole with a residue. A doublet is the cleanest case in which the first is perfect and the second is thirty-six times worse than it looks.

Where the tail is met, and what it is measured with #

A doublet is invisible in the frequency domain and expensive in the time domain, so everything on this page depends on being able to compute a step exactly. One step, computed twice is where that exactness is established — a residue expansion against a marched network, agreeing to the integration error and no further. The cliff before the fastest settling is the other place where settling time refuses to be a smooth function of a design parameter, and A sum that is exact, and the estimate that is not is the shortcut that this page’s residue is exactly the kind of thing to defeat. A divider with two ratios is the doublet a reader meets first, deliberately introduced and adjustable, and The capacitor that remembers is one nobody asked for at all.

What is checked #

Three assertions, and the middle one is the essay.

That the settling time is $\tau_s\ln(A/B)$ — the marched response against the closed form, at every setting of the slider. The tolerance is one per cent rather than a part in ten thousand, and the slack is the closed form’s own range rather than the march’s: $A$ is the amplitude of the first-order tail, and at a tenth of a miss the second-order term is worth half a per cent of the settling time. At one per cent the two agree to four digits, which is the setting quoted above.

That the worst departure in the magnitude is smaller than the miss that caused it, measured against the exactly cancelled network across the passband. That is the asymmetry stated as a comparison rather than as a decibel figure, so that it holds at every setting rather than at the one it was written against.

And that a miss that is invisible in the magnitude multiplies the settling time several times over — the ratio against the exactly cancelled circuit, asserted to be at least three at every non-zero setting. It is the refusal that keeps the first two assertions from being read as a curiosity about a small term.

A magnitude that says nothing about a time #

0.078 decibels of magnitude change against a settling time thirty-six times longer is the largest discrepancy in this collection between what a frequency response reports and what a step does, and it is worth saying why the two come apart so far here when they usually do not.

They usually do not because a settling time is dominated by the slowest pole, and a slow pole is a large feature of a magnitude response. A doublet is the one arrangement where a pole is nearly cancelled by a zero: the residue is tiny, which is why the magnitude barely moves, and the time constant is unchanged, which is why the tail lasts as long as it does. A response with a small residue at a long time constant is exactly the response a magnitude measurement cannot see.

Two measurements of one margin is the case where the two domains agree — a phase margin computed from a loop gain returning 34.9° against an overshoot inverted out of a step response returning 34.9° — and its agreement holds because the quantity being compared is dominated by the pole pair near crossover. This essay is the same pair of measurements applied to a circuit where the dominant feature of one domain is invisible in the other, and the conclusion is practical: settling to a part in ten thousand needs a cancellation good to a part in ten thousand, however fast the amplifier in front of it is, and no magnitude specification on that amplifier constrains it.

Where a doublet comes from #

Nobody designs one deliberately, so it is worth naming the arrangements that produce a nearly cancelled pole and zero without anybody choosing to.

A compensation network is the commonest: any lead or lag section placed to cancel an existing pole is a doublet if the cancellation is imperfect, and the cancellation is imperfect by whatever the two components’ tolerances allow. The tolerance that is not on any part gives the arithmetic — a worst case that does not improve with the number of parts, a measured spread of 0.29 per cent for one per cent components — and a per cent of mismatch is, on the numbers measured here, thirty-six times the settling time.

A second feedback path is the other, and this field has one measured. The path that buys the error back crosses over between two paths at a frequency where each is exact in its own half of the spectrum, and finds its cost to be a settling time running from 0.745 microseconds to 9.18 across the load range — a tail whose time constant is the feedback network’s own, fitted rather than rooted, which is a doublet by another name.

Which is the useful warning. A pole cancelled on purpose is a doublet with a tolerance on it, and the tolerance that matters is not the one on the response.

Which is why the closed form on this page is worth having in the shape it has. It contains neither the fast circuit nor the doublet’s separation as such, so it converts a component tolerance directly into a settling time — and a design that has to settle to a part in ten thousand can read off the matching it needs before choosing an amplifier, rather than discovering afterwards that the amplifier was never the limit.