The pair that is worse than either

A supply rail is asked for a current that changes, and what decides how far the rail moves when it does is an impedance: volts per amp, at whatever frequency the change contains. Fitting capacitors across the rail is how that impedance is made small, and the instruction everyone is given is to fit two kinds — a large electrolytic or tantalum for the low frequencies, and a small ceramic for the high ones, because neither is good over the whole range.

The instruction is right about the parts and wrong about the arithmetic. Two impedances in parallel are smaller than either, which is true of two resistors and true of two capacitors and is not true of these, because neither of them is a capacitor over the range in question. Above its own self-resonance a capacitor is an inductor, and an inductor across a capacitor is a parallel resonance — a frequency at which the pair presents more impedance than either part alone.

Three elements, and only two of them are the capacitor #

Every capacitor in this collection since the frequency field has been three elements rather than one: its capacitance, a series resistance and a series inductance. That is not a refinement, it is what the part is, and the capacitor that is an inductor measured where the third element takes over. A 10 µF electrolytic with 5 nH of lead and terminal inductance is an inductor above 711 kHz. A 100 nF ceramic with 1 nH is an inductor above 15.9 MHz.

Between those two frequencies the bulk part is inductive and the ceramic is still capacitive, and they are in parallel. That is the definition of a parallel resonant circuit, and it is assembled here out of two parts fitted for entirely unrelated reasons, neither of which is drawn as an inductor on any schematic.

The netlist is the honest statement. Each branch is a series R–L–C from the rail to ground, a one-amp source drives the rail, and the potential the rail reaches is the impedance. Writing it as a parallel combination of three-term expressions would give the same magnitude and would throw away what the argument needs, which is the branch currents.

At 6.52 MHz the pair reaches 1.187 Ω. The bulk part alone at that frequency is 0.2023 Ω and the ceramic alone is 0.2055 Ω. The two together are 5.87 times worse than either of them, and the load sees the pair.

The current that is not going anywhere #

The second reading from the same solve says what is happening. At the peak the bulk capacitor carries 5.87 amps for every amp the load draws, and the ceramic carries 5.78 in the opposite direction. Almost none of it reaches the load: it is a current going round a loop made of one part’s inductance and the other’s capacitance, and the load’s own amp is a small perturbation on it.

That circulating current is the same number as the impedance ratio, to three decimals, and it is not a coincidence — both are the quality factor of the same resonance. A parallel resonant circuit multiplies the current in its reactive branches by Q relative to the current fed into it, and it multiplies its own impedance by Q relative to the resistance that damps it. Resonance and its bandwidth is where that quantity was measured on a circuit built to be one; here it appears in a network nobody designed as a resonator.

Two consequences follow immediately and neither is visible from the impedance curve alone. The circulating current heats both capacitors, at a frequency where their series resistance is the whole of their loss. And it flows in a loop of board copper whose area is whatever the layout happened to be, which is a magnetic dipole radiating at 6.5 MHz — a quantity where the current comes back is about, and one that no impedance target can express.

The peak is one over the resistance that was to be minimised #

A parallel resonance is damped by loss and by nothing else. The only loss in this network is the equivalent series resistance of the two capacitors, which is the number a data sheet quotes as a figure of demerit and the number a low-impedance polymer part is sold on.

Quartering the resistance takes the peak from 1.187 Ω to 4.699 Ω. The fitted exponent is −0.906 — one over, to within the shift in where the peak sits — so the better part makes the worse network. A polymer electrolytic with a fifth of an ordinary one’s resistance is a better capacitor at its own self-resonance, which is what it was bought for, and it is five times worse at the pair’s anti-resonance, which nobody measured.

This is the same shape of result as the capacitor that was right once, where a correction capacitor exact at one operating point is wrong everywhere else: a part improved against one specification is being asked to serve in a network that has a different one. The difference is that here the two specifications are the same quantity — impedance — measured at two frequencies.

Adding capacitors moves both ends of the curve #

The next instruction after fit two kinds is fit more of the small ones, and it works, and it is not free.

Twenty ceramics take the floor from 5.000 mΩ to 1.500 mΩ — a factor of 3.33, not the twenty the part count suggests, because for the first six of them the floor still belongs to the bulk capacitor’s own resistance and does not move at all. Which part owns the floor, and the count at which the ownership changes hands, is the floor and the ceiling move apart. The peak falls too, from 1.19 Ω to 0.32 Ω — and it moves down in frequency by a factor of 3.8, from 6.5 MHz to 1.7 MHz, into the decade the bulk capacitor was there to cover. Whether that is an improvement depends entirely on what the load’s current does at 1.7 MHz, which is a property of the digital circuit and not of the bank.

The impedance a load sees is not a number to be minimised part by part. It is a curve, and parts move it in both directions at once.

The cure is a capacitor bought for its resistance #

The standard repair is to add a third part whose series resistance is chosen rather than minimised, so that it damps the resonance without joining it.

There is an optimum and it is not zero. Too little resistance and the damping capacitor is simply a third branch in the resonance; too much and it is not connected at high frequency at all. Golden section on the peak itself — bisected on the solve rather than read off a sweep, for the reason the load that takes the most gives — puts it at 93.9 mΩ, which takes the peak down by a factor of 8.6.

A resistance added on purpose to make a network behave is not a new idea in this collection. The resistor that buys the margin back puts one in series with a capacitive load to recover phase margin, and the reactance cancelled and the resonance it buys finds the same shape in a power system: cancelling a reactance builds a resonator, and something has to damp it. What is unusual here is that the damping element is a capacitor, chosen against the specification its own data sheet is written to.

What the parts are, at the voltage they are used at #

One number above is generous, and it is worth saying which. The 10 µF is 10 µF at no bias and at a small test signal. A class II ceramic loses most of its capacitance under the direct voltage it is decoupling: the capacitance that is not one number measures a 10 µF part at 2.0 µF at its rated voltage, with the charge-average value over the same range at 5.8 µF and a bridge reading 2.1.

Which of those three numbers belongs in this network is decided by the question. The floor and the anti-resonance are small-signal properties, so the small-signal value at the working bias is the right one — and it moves the ceramic’s self-resonance up by the square root of the derating, which moves the anti-resonance with it. The result is that a bank’s impedance curve depends on the supply voltage, which is not a variable any of it is drawn with.

The quality factor, arriving where nobody built one #

The number 5.87 has appeared three times now — as the ratio of the pair’s impedance to either part’s, as the circulating current per amp of load current, and implicitly as the sharpness of the peak. It is the quality factor of a resonance that nobody designed.

The parts that set it are not the ones a resonator’s designer would name. Its frequency is the bulk part’s inductance against the ceramic part’s capacitance — one parasitic and one nominal value, from two different components — and its damping is the sum of two series resistances that exist because copper and electrolyte are not superconductors.

That is the whole of why the effect is under-reported. Everything about the arrangement is individually excellent. Each capacitor has a low impedance at its own resonance; each has a high quality factor; each is the right part for its own decade. The defect belongs to the pair, and no measurement of a part can find it.

Where the impedance has to be, and where it is not #

A load that draws a step of current while tolerating some stated movement on its rail is asking for an impedance below the ratio of the two, over the band its current step contains — a target impedance, flat, from direct current to some frequency set by the edge rate. The bank above meets a 50 mΩ target everywhere except a band around 6.5 MHz, where it misses by a factor of twenty-four.

Whether that matters is a question about the load’s spectrum and not about the bank. A processor clocked at 200 MHz draws current at 200 MHz and at every harmonic, and also at the frequency its power management switches between states, which may be tens of kilohertz. A gap at 6.5 MHz is harmless if nothing asks for current there and is the whole problem if a switching regulator’s control loop does.

That is the same reasoning a source below a frequency applies to a regulator: an output impedance is a curve, the load has a spectrum, and the specification is a statement about the two of them together rather than about either.

The two curves join, and the joint is where the argument is. A regulator’s loop gives out somewhere between ten kilohertz and a megahertz; the bulk capacitor takes over there and gives out at its own self-resonance; the ceramic takes over and gives out at its. Each handover is a place where two things are comparable, and every one of them is a candidate anti-resonance. The bank measured here has one because it has two parts. A rail with a regulator, a bulk part, a ceramic and a package has three.

What a measurement of this looks like #

The peak is easy to see and easy to miss, and which of those it is depends on how the measurement is made. A network analyser looking into the rail with two ports sees it directly, and sees it at whatever point the probes touch — which is not the same point the load is at, because the copper between them has inductance of its own.

A time-domain measurement sees it differently and more usefully: a load current step whose edge contains 6.5 MHz rings at 6.5 MHz, with an envelope decaying at the resonance’s own rate, and the ring is on the rail rather than on the signal. Six and a half megahertz is a slow enough ring to be mistaken for a control-loop instability in the regulator, which is a component that has a loop and can be blamed. The bank has no loop and cannot.

The overlap with that argument is worth stating precisely, because the two are easy to conflate. A regulator can be destabilised by a capacitor whose series resistance is below the window its loop needs, and the cure is the same physical part — a small resistance in series. But that is a statement about a feedback loop’s phase margin and it has a loop gain in it. The peak in this essay is in a passive network with no active device anywhere near it, and it would be there if the rail were fed by a battery.

What is not in this model #

No plane. Both capacitors here connect to the load through nothing. A real board has spreading inductance between the part and the load, of the same order as the part’s own, and it is in series with the branch — so the bank a load sees is not the bank a network analyser sees at the capacitor. That is a distributed problem above a few hundred megahertz and Kirchhoff’s own frequency is where the lumped description stops.

No load capacitance. The die’s own on-chip capacitance is across the rail too, and above the frequency where the package inductance isolates it, it is the only thing across the rail. The curves here rise without limit at the top of the range because nothing stops them, which is a statement about the model rather than about a supply.

And no temperature. The series resistance of an aluminium electrolytic rises by an order of magnitude between room temperature and −40 °C, and the peak here is one over that resistance. A bank whose anti-resonance is damped at 25 °C is a bank whose anti-resonance is eight times higher on a cold morning, and nothing in the impedance curve says so.

The habit this belongs to #

Two parts, each correct, each measured, and a network property that is not a property of either. The bulk capacitor is a good capacitor. The ceramic is a good capacitor. The pair is worse than either over a decade of frequency, and the quantity that makes it worse — the series resistance — is the one both data sheets are competing to reduce.

Nothing here needed a new element or a new method. It needed the parasitics to be in the netlist rather than in a footnote, and the branch currents to be read rather than discarded. The rest is the solver doing what it does on every page of this collection: one solve, and the answer read from it rather than assembled from expressions about the parts.

The parasitic that is asked to be minimised #

The height of the peak being one over the series resistance is the part of this result that changes a design decision, because that resistance is the number every capacitor data sheet leads with and every selection guide asks to be as small as possible.

It is the second time in this collection that the same parasitic turns out to be load-bearing. Two requirements pulling one capacitor finds a regulator’s loop stabilised by the zero that same resistance puts in the loop gain — with a stability floor bisected at 939 milliohms, and a design that replaces an electrolytic with a ceramic oscillating not because the ceramic is worse but because it is better, in a parameter the loop was relying on being bad.

And the component most likely to be fitted as the improvement is not the component the printed number describes. The capacitance that is not one number measures a class II ceramic at its rated voltage as 2.000 µF read as a slope on a part printed as ten, so replacing a bulk capacitor with a bank of ceramics changes the capacitance, the series resistance and the inductance at once — three of the four quantities that decide where this essay’s anti-resonance sits and how tall it is.

Which is why the useful reading of this page is not “fit fewer capacitors” but “compute the impedance of what is fitted”. The peak’s frequency is set by one part’s inductance against another’s capacitance, its height by the total series resistance, and both are computable from a netlist that has the parasitics in it — which is a smaller ask than the design rules that are usually offered instead.