The capacitor that is an inductor

The symbol for a capacitor is two parallel lines, and what it promises is an impedance of 1/(2πfC): a straight line of slope −1 on logarithmic axes, falling for ever. Real capacitors follow that line for three or four decades and then leave it, turn round, and climb with the same slope in the other direction. Above the turning point the component is an inductor.

Where the inductance comes from #

Nothing was added to the capacitor to produce this. A capacitor is a pair of conducting surfaces with an insulator between them, and a current has to reach those surfaces along something: a lead, a strip of metallisation, a via, a length of track on the board. Every one of those is a conductor carrying current, and every conductor carrying current has inductance — not as a defect, but because inductance is what a current-carrying conductor’s magnetic field is called.

The quantity involved is small and it is remarkably insensitive to the size of the capacitor. A surface-mount capacitor’s own structure contributes something under a nanohenry; the two vias connecting it to a ground plane contribute a few hundred picohenries each; a centimetre of track contributes about ten nanohenries. That is why the number in the figure is around a nanohenry and why moving the slider matters more than changing the part: the inductance is mostly the mounting, not the component.

The resistance has a similar story. A capacitor’s plates, its metallisation and its dielectric all dissipate a little, and the total shows up as a resistance in series. It is the reason the minimum in the figure has a floor of 30 mΩ rather than going to zero.

So the real part is three elements in series — a capacitance, a resistance and an inductance — and its impedance is the sum of theirs as complex numbers. That is the whole model, it has no free parameters, and every number in this essay comes out of solving it.

The turning point, and the number before it #

Two frequencies matter and they are not the same.

The self-resonant frequency is where the capacitive and inductive reactances are equal and cancel, leaving only the series resistance. It is 1/(2π√(LC)), which for 100 nF and 1.2 nH is 14.5 MHz. Below it the part is capacitive; above it, inductive. It is the number a datasheet quotes.

The frequency at which the part has already stopped behaving is much lower. The impedance departs from 1/(2πfC) by ten per cent at 4.69 MHz, about a third of the way to resonance in frequency — and by one per cent considerably lower still. A part chosen for its value alone is therefore already misbehaving well before the number on its datasheet, which is the same pattern as the ideal amplifier elsewhere in this collection: the quoted boundary is where the model has completely failed, not where it started to.

Past resonance the departure is spectacular rather than gradual. A decade above, the impedance is 99 times the capacitive prediction — two decades of error accumulated in one decade of frequency, because the true impedance is rising at the same rate the predicted one is falling.

Why the advice is per decade #

Anyone who has read about decoupling has met the instruction to place several capacitors of different values in parallel: a large one, a smaller one, a smaller one again. It is often presented as folklore, and it follows immediately from the figure.

A capacitor is only useful — only a low impedance — over the band between where it has enough capacitance to matter and where its own inductance has taken over. That band is a few decades wide, and it sits at a frequency determined by the capacitance. Choosing a larger capacitance moves the band down; choosing a smaller one moves it up. No single part covers the whole range, so several are used, each covering its own stretch.

There is a wrinkle that the naive version of this advice misses, and it is worth stating because it is a genuine failure mode rather than a subtlety. Two capacitors in parallel with different self-resonant frequencies produce, between those frequencies, a parallel resonance: one part is already inductive while the other is still capacitive, and together they form exactly the parallel LC combination that has an impedance peak. Placed carelessly, the pair is worse in a band than either would be alone.

The mitigation is the same in every treatment that gets it right: keep the values close enough that the peak is small, and — far more effective — reduce the inductance, since it is the inductance that creates the resonance in the first place. Which returns to the point above: the inductance is mostly the mounting.

The other reading of the same curve #

There is a second way to look at this figure that makes it more useful still.

A decoupling capacitor’s job is to be a low impedance between a supply rail and ground at whatever frequencies the circuit demands current. Below resonance, the impedance is set by the capacitance and can be improved by using more of it. Above resonance, the impedance is set by the inductance and using more capacitance does nothing at all — the two curves above resonance for a 100 nF part and a 10 µF part with the same mounting lie exactly on top of each other, because neither has any capacitive reactance left to speak of.

That is a rule with real teeth: above self-resonance, capacitance is not a variable. The only things that change the impedance there are the inductance and the number of parts in parallel, and the second works only because n inductances in parallel are n times smaller.

It is also the clearest example in this collection of a model whose failure inverts the design advice rather than merely degrading it. Below the boundary, more capacitance is better. Above it, more capacitance is irrelevant, and someone who has not located the boundary will keep turning the wrong knob and see no improvement whatsoever.

Choosing a capacitor by its impedance #

The consequence for choosing parts is worth setting out, because it inverts the usual procedure.

The usual procedure picks a capacitance: enough charge storage for the job, with some margin. That is the right criterion below the part’s resonance and the wrong one above it, and the frequency the circuit actually cares about is usually the one where a current step has to be supplied — which is set by an edge rate and is often far higher than any nominal clock.

The criterion that works across the whole range is an impedance: what is the largest impedance acceptable between the rail and ground, at the frequencies where current is demanded? Below resonance that requirement converts to a capacitance and the usual procedure follows. Above it, it converts to an inductance, and the answer is a mounting geometry and a number of parts in parallel rather than a value.

Two consequences fall out of that immediately. A larger capacitance in the same package does not help above resonance at all, because the impedance there is entirely inductive. And n parts in parallel help by a factor of n, because n inductances in parallel are n times smaller — which is why a board with twenty small decoupling capacitors outperforms one with a single large one at high frequency by a factor of twenty and at low frequency not at all.

The resistance, which is not a defect either #

The series resistance has been treated as the thing that sets the floor of the impedance curve, and it has a second role worth naming.

A capacitor with no series resistance and some series inductance is a perfect series resonant circuit with an infinite quality factor. Put two of them in parallel with different resonant frequencies and the parallel resonance between them is a genuine singularity — an impedance that goes to infinity at one frequency, in a network intended to be a low impedance everywhere.

The series resistance is what limits that peak, and it limits it in proportion to the quality factor of the resonance. Which means that a better capacitor — lower resistance, everything else equal — produces a worse peak when combined with another, and the standard advice to use several values in parallel is more dangerous with the best parts than with ordinary ones.

That is a genuine inversion and it is the kind of thing the impedance plot makes obvious and a component table cannot. The remedy in practice is either to keep the values close enough that the peak is small and lands where nothing demands current, or to include a part with deliberately higher series resistance to damp it. Both are choices that look wrong from a datasheet and right from the curve.

What the two curves have in common #

A closing observation, because it is the general form of the essay’s argument.

The two lines in the figure at the top of this page agree over four decades and then diverge without limit, and there is no frequency at which the ideal one becomes “approximately” right or wrong. That is the shape of every boundary in this collection: an idealisation and a better model, indistinguishable across a range, and describing different objects outside it.

What makes this instance the clearest is that the divergence is not a degradation but a reversal. The ideal curve falls with frequency and the real one rises; above resonance they are not merely different in size but opposite in sign of slope. A model that is wrong by a factor is a model with an error; a model whose derivative has the wrong sign is a model of something else.

The inductor has the same disease #

Symmetry is worth noting here because it makes the phenomenon easier to remember. An inductor also has a self-resonant frequency, produced by the capacitance between its turns, and above that frequency an inductor is a capacitor. The same figure, reflected.

The consequence for a filter is that an LC network built from real parts has, at high enough frequency, both of its elements behaving as the other one, at which point the filter’s stopband stops descending and can even come back up. That is the mechanism behind the “leakage” that appears at the top of a real filter’s measured stopband and is invisible in its design — and it is the reason that measuring a filter over a much wider band than it was designed for is standard practice.

Where the schematic stops helping #

This essay is the strongest case on the site for the drawing convention it follows.

A schematic of a decoupling capacitor is two parallel lines between a rail and ground, and it is the same two parallel lines whether the part is soldered directly across the pins of the device it serves or connected by two centimetres of track. Those two circuits differ by a factor of twenty in inductance, which is a factor of twenty in impedance at every frequency above resonance and a factor of four and a half in where the resonance sits. The schematic is identical and the behaviour is not.

So the schematic is not merely uninformative here; it is actively misleading, in the specific sense that it presents two very different objects as the same one. What the response plot shows and the schematic cannot is the whole of the difference.

This is also the moment to note what the model in this essay still leaves out. The inductance was treated as a number, which requires the current path to be small compared with a wavelength. At high enough frequency that stops being true as well, and the component and its mounting become a structure with modes of its own. There is an essay on that boundary too, and it is set by nothing but physical size.

That inductance is a mounting rather than a component, and it is worth separating from the other parasitic before the section ends, because the two do quite different things to the same picture.

Why the package decides more than the value #

The lead inductance in this page’s model is 1.2 nH, and where that number comes from is worth a sentence because it is the parameter the whole result turns on.

It is geometry. A conductor’s inductance depends on the area of the loop its current encloses, so it is set by the length of the leads and their separation rather than by anything about the dielectric between the plates. A component in a small surface-mount package has a fraction of the inductance of the same capacitance in a leaded one, and two capacitors of the same value in different packages have resonances a decade apart.

The practical consequence is the reverse of the intuition most people start with. Choosing a capacitor for high-frequency work is mostly choosing a package, and a smaller capacitance in a smaller package is frequently the better decoupling component — which is why a board carries a hundred nanofarads and a hundred picofarads side by side rather than one large capacitor.

What the figure asserts #

Since the argument rests entirely on the three-element model being solved correctly, the generator that draws it checks itself in two ways before returning.

The impedance is computed by forcing one ampere into the terminals of the assembled network and reading the resulting voltage — the definition, put to the nodal solver — and separately from the closed-form series expression for the same three elements. The two agree to about a part in 10¹⁵ at every one of the 221 frequencies drawn. That is not a check on the physics; it is a check that the network being solved is the network being described, which is the failure a picture cannot show.

The second assertion is the essay’s headline: that a decade above resonance the part’s impedance is at least fifty times what its capacitance predicts. It comes out at ninety-nine, it is required to hold at every position of the slider, and at the low end of the slider — where the lead inductance is only 0.3 nH — it holds because the resonance has simply moved up, not because the effect has gone away.

That is the site’s usual arrangement, and it is worth naming once more here. A slider is not a demonstration; it is a range over which the claim has already been tested. Every frame a reader can reach was produced by the same generator, with the same assertions, at build time. A capacitor that behaved itself could not appear on it.

What the climbing half is for #

The self-resonance measured here is usually presented as the end of a capacitor’s usefulness, and two essays in this collection are about what happens after it — in one case because the rising branch is the whole problem and in the other because it is exploited.

The pair that is worse than either is the problem. A bulk capacitor and a ceramic are fitted together because each is good where the other is not, and between them is a frequency at which the pair presents six times the impedance either does alone — a parallel resonance between one part’s inductance and the other’s capacitance, 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. The rising branch measured here is one half of that resonance; the other capacitor’s falling branch is the other half. The capacitor that is not where the load is then adds the wiring — three nanohenries of ordinary copper, one of mounting loop and two of plane — and finds the anti-resonance moving down to 5.63 megahertz and up to 1.29 ohms, with the twentieth capacitor worse than the second.

Which reframes the number on this page. A hundred nanofarads turning inductive at 14.5 MHz is not a component failing; it is a component becoming the inductive half of whatever resonance the next component supplies, and the frequency at which it does so is what decides where that resonance lands. A designer who reads the self-resonance as an upper limit fits a smaller capacitor beside it and moves the problem; a designer who reads it as the crossover between two branches asks what the pair does there, which is the question the power field answers.

The other place the rising branch is the point is the resistor that is only a resistor, where the two kinds of parasitic are in one part and cancel: at 91.02 ohms a package’s series inductance and shunt capacitance produce an impedance flat to fourth order, which is the same two branches this essay separates, arranged so that neither wins.

Where the component’s own boundary is met #

This is one of four boundaries Every model has an edge draws on one frequency axis, and the only one of them that belongs to a component rather than to an amplifier or to a size. Its two neighbours are Kirchhoff’s own frequency, which is a length, and The resistor that is only a resistor, which is the same argument run on the component that is supposed to have no reactance at all. The same part written two ways is where the series resistance drawn here becomes a quality factor, and The capacitor that is not where the load is is where the inductance drawn here decides a rail impedance rather than a component’s own. Resonance, and the bandwidth it sets exactly is the deliberate version of the same resonance, wanted rather than inherited.