The floor a second capacitor removes

Two requirements pulling one capacitor put a regulator’s output capacitor in an awkward position. Its series resistance is a stability requirement — below 939 milliohms the loop has less than forty-five degrees of margin — and a transient requirement, because the first instant of a load step falls entirely across that resistance. The droop is smallest at 885 milliohms, which is inside the region the margin forbids. That essay’s conclusion was that the best transient this design can have is one it must not be built with, and it left the conflict standing.

The capacitor across the upper resistor then added an element for an unrelated reason. The reference’s noise reaches the output multiplied by the divider’s four, and a capacitor across the upper divider resistor holds that gain down at high frequency; measured, the same capacitor turned out to be a lead pair in the loop, and 836.5 picofarads took the phase margin from 46.5 degrees to 90.9 — the element what gets through from the rail had recommended for the wrong resistor.

That element was never re-examined against the window. This essay does it, and the answer is that the conflict does not survive: at 836.5 picofarads the forty-five-degree floor falls off the bottom of a sweep that reaches a milliohm, and the droop improves by forty per cent at the same time.

What the capacitor is, in the loop #

The element is one capacitor from the output to the feedback node, across the upper of the two divider resistors. What it does to the loop is a zero and a pole.

At direct current it is not there and the divider is 30 kΩ over 10 kΩ, so the feedback fraction is a quarter. At high frequency it shorts the upper resistor, so the feedback fraction rises towards one, and the loop gain rises with it. Between the two there is a zero at $1/(2\pi R_1 C)$ and a pole at $1/(2\pi (R_1 \parallel R_2) C)$, separated by the ratio $R_1/(R_1 \parallel R_2)$, which for this divider is four. A zero and a pole four apart give a maximum phase lead of about thirty-seven degrees, placed where the geometric mean of the two falls.

The measurement says the lift is larger than that, and the reason is worth being explicit about. The lead pair raises the loop gain as well as its phase, which moves the crossover up — and up is where the output capacitor’s own zero and the pass device’s pole leave more phase. So the capacitor buys phase twice: directly, from the lead pair, and indirectly, by moving the crossover somewhere the loop was already healthier. Across the series-resistance sweep it is worth between 29.9 and 46.0 degrees, and it is worth most where the margin was worst.

Nothing about the output capacitor has changed while that happens. The same ten microfarads, the same load, the same pass device; one element added between two nodes that were already connected through a resistor.

The floor, against the capacitance #

Sweeping the divider capacitor and re-finding both numbers at every point is the measurement this essay exists to make, and it has a shape neither of the two earlier essays would have predicted.

The floor falls quickly at first: 939 milliohms with no capacitor, 640 at a hundred picofarads, 385 at two hundred, 119 at 336. Between five hundred picofarads and about a nanofarad it leaves the sweep altogether — there is no series resistance down to a milliohm at which the loop has less than forty-five degrees, so the output capacitor’s series resistance has stopped being a stability specification at all.

Then it comes back. At 1.5 nanofarads the floor is 128 milliohms again, at three nanofarads 331, at eight 484, at twenty 543. Too much divider capacitance is a lag as well as a lead — the pole of the pair moves down towards the crossover — and the margin falls away again from the other side.

The optimum series resistance for the transient moves too, and downwards: 885 milliohms with no capacitor, 638 at 836.5 picofarads, about 450 at eight nanofarads. That is the second half of why the conflict dissolves. The floor is coming down and the optimum is coming down more slowly, so for a range of capacitances the two cross over and the design a transient wants is one the loop permits.

That range runs from about a hundred picofarads to five nanofarads, at forty-five degrees. Two decades of tolerance on a part whose value is not critical is a comfortable place for a design to sit, and 836.5 picofarads — chosen for the phase margin and nothing else — lands in the middle of it.

The conflict is a property of the margin demanded #

Sliding the margin requirement turns the whole argument into something more useful than a fix for one design.

At forty degrees the floor with no capacitor is 742 milliohms, which is below the droop optimum of 885 — so there is no conflict to resolve and every divider capacitance drawn is legal. At forty-five it is 939 and the conflict is on. Bisecting between them — and a bisection on a phase margin is only as good as the margin, which the margin the straight lines report is about — the floor passes the optimum at 43.67 degrees: below that the two requirements happen to agree, and above it they pull apart.

That is a more honest statement of what two requirements pulling one capacitor found than the one it made. The conflict is not a property of this regulator; it is a property of this regulator and a forty-five-degree requirement, and a designer who would have accepted forty-three would never have met it. Raising the requirement makes it worse quickly — at sixty degrees the floor is 1.66 ohms, nearly twice the optimum, and at seventy it is 2.36 ohms, which is 2.7 times.

And the band of divider capacitances that resolves it narrows as the requirement rises: 100 pF to 5 nF at forty-five degrees, 336 pF to 1.5 nF at sixty, and a range too narrow to call a range at seventy. So the capacitor is not a way of getting arbitrary margin for nothing. It is a way of buying back about fifteen degrees of requirement, which happens to be exactly the fifteen degrees between where this design sat and where a cautious designer would want it.

The droop, marched #

The stability half of the argument is a frequency-domain measurement and the transient half is a march, and they have to be put side by side before the result means anything.

Marched from the operating point — so that what is drawn is the step and nothing else — a hundred-milliamp load step on the design with no divider capacitor and its own best series resistance of 885 milliohms droops 129 millivolts. The same step on the design with 836.5 picofarads and its own best resistance of 638 milliohms droops 76.8 millivolts, forty per cent less.

The first instant of each is the step current through that design’s own series resistance, and there the two are 90.8 and 65.3 millivolts — the difference is simply the smaller resistor. What the loop does after that instant is the rest, and it is where the second design gains again: the higher crossover recovers faster, so the output turns around before it has fallen as far below the initial step as the first design’s does.

Before the step the two outputs sit at the same voltage to within a millivolt, which is the check that the element has been added and nothing else has moved. At direct current a capacitor across a resistor is not there, so the divider is still 30 kΩ over 10 kΩ and the regulation is untouched, and the comb the ripple that arrives as a comb synthesises is unaffected at direct current for the same reason.

And at a resistance somebody actually has #

Comparing two designs each at its own optimum is the right comparison for a design question and the wrong one for a part that is already on the board, because the series resistance of a capacitor is not a knob.

At a fixed one ohm — a perfectly ordinary tantalum — the droop falls from 130 millivolts to 99.1, a quarter rather than forty per cent. The first instant is 102 and 99.1 millivolts, nearly the same because the resistance is the same, so the whole of the improvement is in what the loop does afterwards. That is the honest figure for retrofitting the capacitor to an existing design: a quarter off the droop and forty-five degrees of margin, for one part that costs nothing and changes no direct-current behaviour.

It also isolates which half of the earlier result was about the resistor and which about the loop. Of the forty per cent available when both are optimised, roughly twenty-four points come from the loop being faster and the rest from being allowed to use a smaller resistor — and the second is available only because the floor has moved.

What the transient argument would do on its own #

One figure in this essay exists to say what happens if the stability half is ignored, because it is the mistake this whole argument is about.

The smallest droop available falls monotonically with the divider capacitance — 129 millivolts with none, 94.4 at 336 picofarads, 76.6 at 836.5, 71.0 at 1.5 nanofarads, 65.3 at eight and 64.5 at twenty. A designer optimising the transient alone would keep going, and every step would look like an improvement.

Above five nanofarads the series resistance that produces those droops is back below the forty-five-degree floor. The last two and a half per cent of droop is therefore bought back exactly as it was before the capacitor was added: by building the design the loop refuses. Nothing in the transient measurement says so, which is the point — a load-step waveform of a marginally stable loop is a perfectly good-looking waveform until it is not.

So the useful summary of the whole sweep is a pair of edges rather than a value. Below about a hundred picofarads the capacitor has not done enough; above about five nanofarads it has done too much; between them the two requirements agree and the droop is within a few per cent of the best this design can reach. The value chosen for the reference’s noise sits inside that, and nothing about it was chosen for this.

Where else the phase could have come from #

A designer meeting the conflict for the first time reaches for the output capacitor, and it is worth saying what that does, because it looks like it works and does not.

Ten times the output capacitance improves the droop a great deal — 128.5 millivolts at ten microfarads, 86.7 at twenty-two, 59.4 at forty-seven, 40.7 at a hundred — and lowers the forty-five-degree floor as well, from 939 milliohms to 386. Both numbers move the right way, so the first two columns of a table built this way say the problem is solved.

The third column says it is not. At every one of those capacitances the best series resistance is still below the floor: 554 against 727 at twenty-two microfarads, 402 against 537 at forty-seven, 260 against 386 at a hundred. The ratio barely moves — 0.95, 0.76, 0.75, 0.68 — so the conflict is not something a larger capacitor grows out of. It cannot be, because both numbers are set by the same capacitor: the floor falls because the capacitor’s zero moves down, and the optimum resistance falls because the capacitor holds the output up for longer. They move together and stay in the same order.

That is the reason the fix has to come from somewhere other than the output node. Anything that changes the output capacitor changes both sides of the comparison; the divider capacitor changes the loop’s phase without touching the capacitor at all, so it moves one side and not the other.

Two other places would also work in principle and are worse in practice. A resistor deliberately put in series with the output capacitor meets the floor by force and pays for it in droop at the first instant, which is the trade two requirements pulling one capacitor already priced. Compensating the error amplifier further down — moving its dominant pole — lowers the crossover, which raises the margin and makes the transient recovery slower, so it trades the first instant for the recovery instead of improving both. The divider capacitor is the only one of the four that improves the droop and the margin at once, and that is because it is the only one that raises the crossover.

What this settles, and what it does not #

Three things are now decided and one of them is a correction.

The conflict was real and is removable. “The best transient this design can have is one it must not be built with” was true of the netlist as that essay had it, and one capacitor makes it false. The general statement that survives is the mechanism rather than the conclusion: a capacitor’s series resistance is a zero in the loop and a resistor in the transient, and those two roles are not automatically compatible — but the loop has other places to get phase from, and taking it from one of them relieves the output capacitor of having to supply it.

The divider capacitor is now justified three ways. It was added to hold the reference’s gain down, it turned out to give 90.9 degrees of margin, and it removes the output capacitor’s resistance floor. The three are not independent — all of them are the same lead pair — but they are three different design requirements satisfied by one part, which is the reason to prefer it to any of the other places the phase could have come from.

And the output impedance is not what this essay measured. A source below a frequency gives that regulator 0.43 milliohms at direct current rising to 1.95 ohms at ten kilohertz, and a droop after a step is a time-domain reading of the same quantity with the step’s whole spectrum in it. The two are not interchangeable: a design can have a lower peak output impedance and a worse droop, because the droop is dominated by the first instant and the peak is not. Which of them a load cares about is a question about the load, and neither this essay nor that one asks it.

Still open: the reference’s noise re-measured, a ceramic output capacitor, and the load that decides #

The reference’s noise, now that the series resistance has moved. The capacitor across the upper resistor measured the gain from the reference to the output through three arrangements and left it as a gain rather than a voltage. The floor moving means the output capacitor can now be a smaller-resistance part, which moves the loop’s crossover again, which moves that gain. Integrating a reference’s flat density and flicker corner through the loop as it now stands would give an output noise voltage for each arrangement — and would say whether the capacitor that bought the margin costs anything in noise.

A ceramic output capacitor, which is the case that made this question urgent. A few milliohms of series resistance is what a modern ceramic gives, and it is exactly the region the original floor forbade. With the divider capacitor the loop tolerates it, so the design becomes buildable — but a ceramic’s capacitance falls with bias and with temperature, which the capacitance that is not one number and the coefficient that is about one reading measure between them, and the window has not been re-solved against a capacitance that moves. The question is whether the band of divider capacitances stays open across a capacitor that is half its nominal value at the working voltage.

The load step that is not a step. Everything here is a hundred-milliamp step with an infinitely fast edge, so its spectrum reaches everywhere and the first instant is the whole of the series resistance. A real load changes over a rise time of its own, and once that rise time is long against the loop’s own response the first instant stops dominating and the droop becomes a reading of the output impedance at the load’s own frequencies instead. Where the crossover between those two regimes sits — in nanoseconds, for this loop — would say which of the two numbers above a designer should be budgeting against.