Two requirements pulling one capacitor

Every regulator datasheet has a paragraph about the output capacitor, and it usually says that the capacitor’s equivalent series resistance must lie inside a window. That is an unusual requirement — a parasitic that has to be large enough — and this essay measures both edges, finds them to be requirements of quite different kinds, and finds that the best of one of them lies inside the forbidden region of the other.

Why a parasitic is load-bearing #

The loop from the previous essay has two poles at similar frequencies — the error amplifier’s, at 15.9 Hz, and the output’s, set by the load resistance against the output capacitance. Two poles with a loop gain of ten thousand between them and the crossover means the phase reaches −180° before the gain reaches one, and the circuit oscillates.

What saves it is a zero. The output capacitor is not a capacitance; it is a capacitance with a resistance in series, so its impedance stops falling at $1/2\pi R_{esr}C$ and flattens out. That flattening is a zero in the loop gain, it contributes phase lead, and if it lands below the crossover frequency the phase is pulled back from −180° before the gain gets to one.

So the loop is stabilised by a component the manufacturer of the capacitor regards as a defect, and whose value is neither specified tightly nor stable with temperature. That is a strange arrangement, it is entirely standard, and the numbers below are why designs of this shape were replaced.

The lower edge, bisected #

Sweeping the series resistance and computing the phase margin of the return ratio at each value gives a curve that rises monotonically:

Bisecting for 45° puts the edge at 939 mΩ. Below it the design has less margin than anybody would accept; well below it — a few tens of milliohms, which is what a ceramic capacitor of this value actually has — the margin is heading for zero and the regulator rings on every load change and eventually sustains an oscillation of its own.

That is the first and best-known half of the story: replacing an electrolytic with a ceramic can make a working regulator oscillate, and the reason is not that the ceramic is worse but that it is better, in a parameter the loop was relying on being bad.

The upper edge is not a stability edge #

The datasheets say the window has a top as well. In this model it does not.

The margin column above keeps rising: 75.8° at 3 Ω, 87.9° at 10 Ω, and 89.6° at a hundred. The scan finds no upper edge anywhere in three decades, and the mechanism is visible in the loop gain: as the series resistance rises the zero moves down in frequency, the output stage’s pole and zero very nearly cancel each other, and what is left is a single-pole loop rolling off from the amplifier’s own corner. A single-pole loop is unconditionally stable.

So the honest statement is that this topology has a stability floor and no stability ceiling, and where a datasheet’s ceiling comes from is a different mechanism. Two candidates are worth naming. In parts whose pass device is fast and whose compensation relies on the output pole staying put, pushing the crossover up with a large zero eventually encloses a parasitic pole and the margin does fall again — that mechanism needs a faster driver pole than the 300 kHz modelled here. And in many parts the ceiling is not about oscillation at all; it is about what the resistance does to the output itself.

Which is the other half of this essay.

The upper edge that does exist #

Mark the moment a load step arrives. The capacitor cannot change its voltage in zero time and the loop has not moved yet, so the entire step current flows through the only thing that can carry it instantaneously: the capacitor’s series resistance.

The output therefore falls by $\Delta I \times R_{esr}$ at the first instant, before anything in the circuit has responded to anything. Marched, with a 100 mA step:

The second and third columns agree to better than nine per cent everywhere, and the gate holds that. The shortfall is not an error: the first sample after the step is one time step later, by which point the loop has begun to act and the capacitor has begun to discharge, and it shrinks as the step is shortened.

Read the fourth column and the two requirements are visible in one place. Above about a volt of resistance the worst droop is the first instant — the loop never gets a chance to make anything worse, and the transient is a step down and a fast recovery. Below it the first instant is small and the loop’s own ringing takes the output further down than the step did: at 100 mΩ the step itself is 12.5 mV and the excursion is 152 mV, twelve times larger, arriving a couple of hundred microseconds later and taking 1.2 milliseconds to recover.

The minimum is in the forbidden region #

Locating the minimum of that fourth column by golden-section search on the continuous measurement — not by reading the smallest of the twenty-five points drawn, which is a statement about the grid — puts it at 850 mΩ, with a droop of 129 mV.

The stability edge is at 939 mΩ.

The series resistance that gives the smallest droop is about ten per cent inside the region where the loop has less than 45° of margin. The best transient response this design can have is one it must not be built with, and the useful optimum is therefore not a minimum at all — it is the stability edge itself, which happens to sit almost exactly where the droop curve is flat.

That is a satisfying result and it is also a fragile one. Ten per cent is inside the tolerance of an electrolytic capacitor’s series resistance, well inside its variation with temperature — the resistance of an aluminium electrolytic roughly doubles between room temperature and −20 °C and falls as it warms — and comfortably inside the change over its life. A design sitting at the stability edge at 25 °C is sitting on the wrong side of it at 60 °C.

The capacitor is not the capacitance either #

Every sweep above moves one parameter and holds the other, which is the right way to measure two requirements against one component and is not what happens when somebody changes the part. An electrolytic replaced by a ceramic changes the series resistance by two decades and the capacitance by an amount nobody quotes.

The capacitance that is not one number measures that second change on a class II ceramic and it is not small: a ten-microfarad part at its rated five volts is 2.000 µF read as a slope, 5.814 µF read as a charge average and 2.105 µF as a bridge reads it — three answers to three questions, all correct, all printed as one number. So the row of the table above that a ceramic replacement lands on is not the 10 µF row. It is somewhere near the 2 µF row, where the stability floor is 657 mΩ rather than 939 and a part with tens of milliohms is further below it than the naive substitution suggests, and where the best droop available is 289 mV rather than 129. Two of this essay’s three quantities move at once and they move in the same direction.

The third number a data sheet gives does not settle it either. The coefficient that is about one reading shows that a printed temperature coefficient is a coefficient of the one capacitance a bridge reports at zero bias, that one number cannot determine the two parameters the part actually has, and that two parts a bridge cannot tell apart differ by a factor of 1.80 at the voltage they are used at. The tolerance argument at the end of the previous section — a ten per cent window against a resistance that doubles by −20 °C — is therefore the optimistic half of the problem, because it assumed the capacitance was fixed while the resistance drifted.

And the way the loss itself is written down has a range too. The same part written two ways finds the conversion between a series resistance and a parallel one exact at one frequency and nowhere else, with the width of the band around it set by the quality factor alone. The zero this whole essay depends on is at $1/2\pi R_{esr}C$, so it is placed by a series representation, and a part characterised in the other convention has to be converted at the frequency the zero lands on rather than at the frequency it was measured at.

A resistance bought for margin, elsewhere in the collection #

The arrangement here — a resistance that supplies phase lead and is paid for in something other than stability — is one the feedback field measures on a different circuit with a different answer, and the pair is worth reading together.

The resistor that buys the margin back puts ten ohms between an amplifier and a capacitive load, restores forty-five degrees of margin, and finds that it works for a reason that reads as a cheat: the feedback is taken from the wrong side of the resistor, so the resistor’s pole is outside the loop. What it costs is that the loop no longer regulates the load’s node at all — one per cent of error into a kilohm, at direct current, uncorrected. That is the same trade this essay makes and the price is charged in a different currency: here the resistance is inside the loop, so the regulation survives and the step response pays instead.

The load that gets inside the loop is where that ladder starts, and it names the quantity both essays turn on — fifty ohms of output resistance that no data sheet page puts next to the stability page, with forty-five degrees arriving at 905 picofarads, which is a metre of coaxial cable. And the path that buys the error back is the repair that keeps both: a second feedback path that restores the regulation the isolation resistor gave up, charging for it in a settling time that runs from 0.745 microseconds to 9.18 across the load range.

The other place a minimised series resistance turns out to be the wrong thing to minimise is the supply itself. The pair that is worse than either measures a bulk capacitor beside a ceramic and finds a frequency at which the pair presents six times the impedance either does alone — a parallel resonance whose height is one over the series resistance every data sheet asks to be minimised, and at which the two capacitors exchange 5.87 amps for every amp the load draws. Two essays in this collection now have a result of that shape, and both of them are about the same parasitic being treated as a defect by whoever manufactures the part.

Whether they conflict at all depends on the capacitance #

The crossing above is stated at 10 µF, and the obvious question is whether it is a general fact or a fact about that value. It is a fact about that value, and the sweep is worth having.

The two requirements cross at about five microfarads. Below that the resistance that minimises the droop is comfortably inside the stable region and a designer can simply have it; above it the optimum is forbidden and the stability edge is the best available.

Two other things in that table are worth reading. The droop falls steadily with capacitance, which is expected — a larger capacitor holds the output up for longer while the loop responds — so the reason to use a large one is real and the conflict is the price of it.

And the stability floor is not monotonic: it rises from 657 mΩ at 2 µF to 1054 at 4.7 and then falls. That is not noise. At small capacitance the output pole is at a high frequency — 15.9 kHz at 2 µF, above the loop’s own crossover — so the loop is nearly single-pole already and needs little help from the zero. As the capacitance grows the output pole comes down through the crossover, the loop needs the zero badly, and the required resistance peaks; past that the pole is far below crossover and what matters is only that the zero arrives before it, which a smaller resistance achieves as the capacitance grows.

So the shape of the floor is the output pole crossing the loop’s crossover frequency, seen through a requirement. It is the kind of non-monotonic curve that is easy to mistake for a numerical artefact and worth pushing on, which is the same lesson the power field recorded when a conduction-angle sweep turned out to have its optimum at neither extreme.

Why the whole arrangement was abandoned #

Modern low-dropout regulators do not work this way, and the measurements above are the reason. The standard answer is to put the dominant pole somewhere the designer controls — inside the part, with a compensation capacitor on the error amplifier’s output — and to make the output pole so high that it is not in the loop at all. Then the stability does not depend on the output capacitor’s parasitic and the part is specified as “stable with any capacitor above 1 µF”, which is a specification a user can meet.

What that costs is bandwidth, and therefore everything in the previous essay: a loop compensated to be stable with a ceramic capacitor has a lower crossover frequency and a higher output impedance across the audio band than one relying on the zero. The trade did not go away; it moved.

How much it moved is measurable, and it is measured. A source below a frequency drives this same output node and finds 0.43 milliohms at direct current, doubled by 4.8 hertz, ten times worse by 27 and 1.95 ohms at ten kilohertz — four and a half thousand times the number a data sheet quotes, and higher there than the same circuit with its loop cut. Every hertz taken off the crossover frequency moves the point at which that curve leaves the specification down with it. And what gets through from the rail shows that the loop is not even the whole account: the same netlist with one node moved, the same loop gain and the same 46.5 degrees of margin, rejects sixty decibels better at every frequency in six decades, because sixty of those decibels are the pass device’s own intrinsic gain rather than a design choice. So the compensation decision made for the capacitor’s sake is one of at least three things deciding what the regulator does, and it is the only one of the three the user of the part can see.

What the two measurements share #

It is worth stating plainly, because it is the reason to trust the crossing point.

The upper panel of the figure is a return ratio: the loop broken at the amplifier’s input, driven with a test voltage, and the returned signal read at the divider, solved at each of sixty frequencies in the complex plane. The lower panel is a march: the whole netlist stepped forward in time from its own operating point with a current source that changes value at one instant, four thousand steps of it, with the current law rebuilt from the element laws at every step.

They share the netlist. They share no arithmetic at all — one never leaves the frequency domain and the other never enters it — and the quantity they are both a function of is a single resistor. Where the first says the margin has fallen to 45° the second says the ringing has started to dominate the droop, and those two statements agreeing about the same value of one component is the check that neither is an artefact.

The gate #

Phase margin rises with series resistance at every step of the sweep — so there is a floor and the sweep is monotone, which is what makes bisecting for it meaningful.

The floor is found rather than quoted: 939 mΩ for 45°, bisected on the measured return ratio.

The first instant of the droop is the step current through the series resistance, at every value tested, to better than nine per cent.

The droop has a minimum inside the range, located by golden-section search rather than on the drawing grid.

And the minimum sits below the stability floor — 850 mΩ against 939 — which is the essay’s result and the reason its title has two requirements in it rather than one specification.