The path that buys the error back

The rung below this one repaired a capacitively loaded amplifier and reported the repair’s cost honestly, which is what made the next question obvious.

The repair is a resistor between the amplifier’s output and the load. It moves the load’s pole outside the loop, and on the solved and injected loop transmission it restores exactly forty-five degrees of phase margin at a value found by bisection rather than computed from the pole it makes. What it costs is that the loop no longer regulates the node the load is on: the feedback comes from the amplifier’s own output, so any current the load draws produces a drop across the isolation resistor that the feedback has no way of seeing. Into a kilohm that drop is 0.990 per cent of the output, it is invisible to every stability measurement, and it is the reason the repair is a trade rather than a fix.

There is a standard repair for that, and it is the subject of this essay. Take the feedback resistor to the load, so that direct current is regulated where it matters, and add a capacitor from the amplifier’s own output to the summing node so that the fast feedback never passes through the isolation resistor at all. Two paths, each exact in its own half of the spectrum.

It works, and the direct-current error falls by more than three orders. What it costs is not a number of the kind the previous two rungs produced. It is a range.

Three arrangements, and what each is wrong about #

Everything below is one netlist with one thing changed, so the comparisons are between arrangements rather than between designs.

Feedback from the amplifier. The isolation resistor’s pole is outside the loop and the margin is repairable: ten ohms takes it from 30.1 degrees with no resistor to 45.1. The direct-current error into a kilohm is 0.990 per cent, and it settles to a hundredth of a per cent in 1.419 microseconds.

Feedback from the load. The loop regulates the right node — the direct-current error falls to $1.2\times10^{-4}$ per cent, which is the amplifier’s own finite gain and nothing else — and the isolation resistor is now inside the loop, where its pole makes everything worse. The margin is 27.6 degrees, below the 30.1 it started with. Every value of the resistor makes it worse, which is the assertion the rung below this one used to separate “the resistor damps something” from “the resistor moves a pole out of the loop”.

Both. The resistor goes to the load and a capacitor goes from the amplifier’s output to the summing node. Below the crossover of the two paths the feedback comes from the load; above it, from the amplifier. The direct-current error is the second arrangement’s, $1.2\times10^{-4}$ per cent, and the margin is the first arrangement’s or better.

The capacitor has a range and both ends of it are real #

The second path’s capacitor is not a free parameter, and the two things bounding it point in opposite directions.

It must be large enough to carry the loop at crossover. Its job is to bypass the isolation resistor where the loop’s phase is at stake, so its impedance at the crossover frequency has to be small compared with the feedback resistor. With ten kilohms of feedback resistance and a crossover near five megahertz, that means a few picofarads at least.

It must be small enough not to become the dominant pole. It sits across the feedback resistor, and a capacitor across a feedback resistor is an integrator: the closed-loop response rolls off at $1/2\pi R_fC_f$ whatever else is happening. At a hundred picofarads that is 159 kilohertz, thirty times below the amplifier’s own crossover, and the circuit has been slowed down by a factor of thirty for nothing.

Measured, to a hundredth of a per cent of final value:

The best value settles in half the time the isolation resistor alone takes, with four orders less direct-current error and seventeen more degrees of margin. So the answer to the question the previous rung left — what does the second path cost in settling — is, at the right capacitor, nothing. It is better in every quantity measured.

What the margin does not report #

The row at a hundred picofarads is the one worth staring at, because it is where the two measurements disagree about which circuit is better.

Its phase margin is 47.9 degrees. That is a perfectly respectable number — better than the 45.1 the isolation resistor alone achieves, better than the 43.8 at 3.3 picofarads — and a designer working from margins alone would accept it without a second look. Its settling time is twelve times the best available and six times worse than the circuit it was supposed to be repairing.

The two numbers are not measuring the same thing and there is no contradiction. A phase margin is a statement about the loop transmission at one frequency: how far from instability the circuit is. A settling time is a statement about the closed-loop step response over four decades of amplitude, and what dominates it is whichever pole is slowest — which, at a hundred picofarads, is a pole the loop put there deliberately.

That is the general lesson and it is worth stating in its own sentence: a margin bounds the ringing and says nothing about the slow tail. Which is the same statement the transient field makes about a pole-zero doublet, arrived at from the other side.

The tail, fitted rather than rooted #

What the over-compensated circuit is settling by can be identified, and the identification is a better check than a settling number alone because it names the mechanism.

Fitted as a straight line on a logarithmic error axis over the window from three to twenty microseconds, the remaining error at a hundred picofarads is $A\,e^{-t/\tau}$ with $\tau = 0.9925$ microseconds and $A = 1.0401$, at $r^2 = 1.000000$. And $R_fC_f$ is 1.0000 microseconds.

So the tail’s time constant is the feedback network’s own $RC$ to within three parts in a thousand, and it contains no amplifier, no load capacitance and no isolation resistor. The settling time to a band $B$ is then $\tau\ln(A/B)$ — the transient field’s expression — which gives 9.181 microseconds against 9.180 measured.

The fit is done rather than the roots because the roots do not survive. The closed-loop polynomial has its roots five decades apart with a pair that nearly cancel, which is the case a companion-matrix rooting is worst at: the recovered pair came back with an imaginary part a sixth of its real part, on a network whose poles must be real or conjugate. A straight line on a logarithmic axis, over a window chosen after the fast response has gone, is a more robust instrument than a polynomial factorisation here — and it reports its own straightness, so it can be refused.

It refuses the good case, which is the point. At twelve picofarads the same fit returns $r^2 = 0.84$ and a time constant with no meaning, because there is no exponential tail: the circuit is settling by a damped ring, not by a doublet, and the fitter says so rather than quoting a number for it.

Where the crossover of the two paths sits #

The frequency at which the feedback hands over from one path to the other is $1/2\pi R_fC_f$, and putting a number on it makes the whole trade legible in one quantity rather than two.

At twelve picofarads and ten kilohms it is 1.33 megahertz. The loop’s own crossover is a little under five, so the handover happens comfortably below the frequency where the phase is at stake — which is what it has to do, since the capacitor’s job is to be carrying the feedback by the time the isolation resistor’s pole matters.

At three picofarads the handover is at 5.3 megahertz, above the loop crossover, so at crossover the feedback is still coming through the isolation resistor and the margin is barely improved: 43.8 degrees against the isolation resistor’s own 45.1.

At a hundred picofarads it is 159 kilohertz, which is thirty times below the crossover and, more to the point, below the closed-loop bandwidth the amplifier could otherwise deliver. Past that frequency the feedback network is an integrator and the circuit’s speed is set by two passive components.

So the range this essay is about can be stated in one sentence with no capacitors in it: the handover must be below the loop’s crossover and above the bandwidth the circuit needs. Those two frequencies are about a factor of thirty apart here, and the useful capacitor values are the factor of thirty between them — narrowed at both ends by wanting margin at one and speed at the other.

The measurement, and what would have hidden the result #

Two details of how this was measured decide whether the numbers mean anything, and both were got wrong first.

The settling is measured on the closed loop marched in time, not from a residue expansion of recovered poles — for the reason above: the polynomial’s roots do not survive this network. Marching costs thirty thousand steps and a few seconds; factorising costs nothing and returns a complex number where a real one has to be.

And the settling band is a hundredth of a per cent, not one per cent. At one per cent the three arrangements are 0.725, 1.122 and 0.332 microseconds, and the over-compensated case is 4.61 — differences of a factor of a few, where at a hundredth of a per cent they are 1.419, 2.372, 0.745 and 9.18. The tail this essay is about is an exponential, so what it costs grows linearly in the number of decades being settled to, and the gap between a good capacitor and a bad one grows with it. A measurement taken at one per cent would have reported a factor of fourteen where the honest figure is twenty-eight.

Why the arrangement is not simply better #

Three costs are real and none of them is in the table above.

A second path is a second thing to get wrong. The capacitor’s value depends on the load capacitance, the isolation resistor and the amplifier’s crossover, none of which a designer controls tightly. The band is about a factor of four wide at a tenth of a microsecond of settling, so a part substitution that changes the load by ten times leaves the design outside it.

It adds a zero to the noise gain. A capacitor from the output to the summing node raises the noise gain above the corner it forms with the feedback resistor, so the amplifier’s own voltage noise is amplified more at high frequencies than the signal is. The gain the loop closes against is where that quantity is separated from the signal gain and measured: an inverting amplifier with two equal resistors has a gain of one and a loop that closes against two, so it has half the bandwidth of a follower built from the same part, and a hundred picofarads at the summing junction puts twelve decibels of peaking on a response whose designed gain is nought decibels and whose measured gain at a kilohertz has not moved by three parts in a million. A capacitor placed as this essay places it is on the same node, doing the same thing to the same quantity, on purpose.

Whether the cost is real was worth measuring rather than assuming, and the rung above did. What the second path costs at the floor suspected this arrangement of paying for itself in noise, because that is how compensations usually pay, and found that it does not: there is no peak in its noise gain at all, and the total at the load falls from 54.9 microvolts to 28.9 as the capacitor is added. What it costs instead is settling, and the capacitor that is quietest is eighteen times the capacitor that settles fastest — a trade the phase margin says nothing about, because the margin is comfortable at both.

The loop still regulates a node the load is not on — at frequencies above the crossover of the two paths. So a load current step faster than 159 kilohertz sees the same error the first arrangement had, and only the slow part of it is corrected. That is not a defect: it is what “two paths, each exact in its own half of the spectrum” means, read carefully. It is also the reason the arrangement is a repair for a capacitive load and not for a load that draws current, since a capacitor takes nothing at the frequencies where the slow path is in charge and a resistor takes the same current at every frequency there is.

What the range means for the rest of the ladder #

The result this essay ends on is a range rather than a number, and that is the unusual part: the two rungs below it end on a margin and an error, both of which are single quantities a design either has or does not. A range is a different kind of specification and the ladder above this one is largely about what escapes it.

What the load sees looking back turns the arrangement round and measures the impedance a load-drawn current meets, finding the two-path circuit recovering to a thousandth of what the isolation resistor alone leaves — which is this essay’s direct-current repair confirmed from the other terminal, and charged for in a quantity none of the rungs below measured. The step too large to have an impedance then finds where that impedance stops being a number at all: give the amplifier the differential pair’s own tanh in place of a linear transconductor and the ratio of voltage to current is a constant up to 10.6 milliamps and not above, where the input error is 3.63 thermal voltages and the slew rate deciding it is twice the thermal voltage times the gain–bandwidth in radians, containing no design choice whatever. And the current above which there is no impedance adds the second limit, which is nothing but design choice, and finds the two binding at different loads — 19.4 per cent against 4.2 at 0.47 nF, 0.9 against 2.3 at 22 nF, and above the output stage’s rating an excursion that does not come back at all.

Read together with this page, those say that the settling range measured here is the good case: it is what the arrangement costs while everything in it is still linear. The rungs above measure what happens at the currents where it is not, and the capacitor value chosen from the table on this page is chosen without reference to any of them.

The three repairs, side by side #

A capacitive load inside a feedback loop has three standard repairs and this field measures all three. The resistor that buys the margin back is the isolation resistor, which costs a load error linear in its value. This page is the second path, which costs nothing at direct current and buys a settling time that is not monotone in the capacitor. The load that gets inside the loop is the problem all three address, and What the second path costs at the floor is the noise the second path charges instead. The cancellation that leaves a tail is the residue that makes the settling time refuse to follow the margin.

What is checked #

Four assertions, and the last is the one that identifies the mechanism rather than the number.

That the second path removes the direct-current error the isolation resistor left, by more than three orders, measured on the closed loop at a hundredth of a hertz with and without a kilohm of load.

That at its best capacitor it settles faster than the arrangement it repairs, which is the answer to the question the rung below this one asked and is not the answer that was expected.

That a capacitor an order too large costs an order of settling, with the phase margin there reported beside it, so that the two measurements are seen disagreeing rather than described as disagreeing.

And that what the slow case settles by is one exponential whose time constant is the feedback network’s own, to a per cent, with the straightness of the fit reported — because a fitted time constant on a ringing response is a number with no referent, and the same instrument has to be able to say so.