Circuit Response
every model has an edge, and the figure carries it
Frequency, which is the same solve

The capacitor that is an inductor

A hundred-nanofarad capacitor follows 1/(2πfC) for four decades and then turns round and climbs. Above 14.5 MHz it is an inductor, and a decade past that its impedance is ninety-nine times what its capacitance predicts — all of it caused by about a nanohenry of lead and via that nobody chose and nobody drew.
14 min read 4 figures The parasitic is the componentEvery model has an edgeThe schematic is a label

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.

A 100 nF capacitor, and what it is above 14.5 MHzThe dashed line is 1/(ωC), which is what the symbol means. The solid line is the same part with 30 mΩ of series resistance and 1.2 nH of series inductance, solved. They part company at 4.69 MHz and by a decade above resonance the part's impedance is 101× what its capacitance predicts.10m1.0e+2m1101001k10k10k100k1M10M100M1Gfrequency (hertz)impedance magnitude (ohms)1/(ωC), the symbol's promise10% off above 4.69 MHzinductive above 14.5 MHz30 mΩ — the floor the resistance setssolved, then checked — the part as three elementsa capacitor below 14.5 MHz, an inductor above
Fig. 1 A 100 nF capacitor with 30 mΩ of series resistance and 1.2 nH of series inductance, solved as a three-element network at 221 frequencies. The dashed line is what the symbol means. The two part company at 4.69 MHz, reach a minimum of 30 mΩ at 14.5 MHz, and by 145 MHz the part’s impedance is 99 times what its capacitance predicts. The slider is the lead inductance.

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.

Impedance of a series RLC of Q = 4, measured by driving itOne ampere is forced into the terminals at each frequency and the resulting voltage is the impedance. The minimum is 7.91 Ω at 5.03 kHz.1101001k10k1001k10k100k1Mfrequency (hertz)impedance magnitude (ohms)reactances cancel at 5.03 kHz7.91 Ωsolved, then checked — one ampere in, 201 frequenciesnot a component value: what the pair does
Fig. 2 The general case behind the specific one: a resonant pair’s impedance falls to a minimum where its two reactances cancel and rises steeply either side. A real capacitor is exactly this figure with the resonance placed by parasitics that nobody chose.

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.

Where four of this site's models stop being trueIn order: the ideal operational amplifier at 1.42 kHz, a 10 V output at full amplitude at 7.96 kHz, Kirchhoff's laws on 10.0 cm at 3.97 MHz, the ideal 100 nF capacitor at 4.69 MHz. The fifth boundary is an amplitude rather than a frequency and cannot share this axis: a small-signal model is 1% wrong above 7.3 mV, at every frequency there is.101001k10k100k1M10M100M1G10Gfrequency (hertz)the ideal operational amplifier1.42 kHz — a gain of 100 from a 1 MHz part is 1% low herea 10 V output at full amplitude7.96 kHz — above this the output cannot move fast enoughthe ideal 100 nF capacitor4.69 MHz — 1.2 nH of lead makes it 10% wrong hereKirchhoff's laws on 10.0 cm3.97 MHz — the board is one degree long hereeach bar is where the model may be used; the rule at its end is the numbersolved, then checked — each boundary from its own modeland one that is not a frequency: 7.3 mV
Fig. 3 The capacitor’s boundary in company. Four models, four frequencies, one axis: the ideal amplifier first, then the output’s ability to move, then the capacitor at 4.69 MHz, then the assumption that the circuit board has no size. The capacitor is the third of four, not the first — which is not where most readers would put it.
10.0 cm of track, solved as a lumped circuit and as a lineThe two agree to 0.030% at 3.97 MHz, where the track is one degree long, and to 30.1% at 143 MHz, where it is a tenth of a wavelength. Above that the lumped model is not approximately right; it is describing a different object.1101001k10k100k1M10M100M1Gfrequency (hertz)impedance looking into 10.0 cm of track (ohms)the lumped model: one L, one C1° long at 3.97 MHza tenth of a wavelength at 143 MHzthe 200 Ω at the far endsolved, then checked — the line against a two-element modelKirchhoff's laws run out at 143 MHz
Fig. 4 The boundary just past this one. The lead inductance treated here as a number is itself an approximation, valid while the current path is short compared with a wavelength — and that assumption has its own frequency, set by nothing but physical size.

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.

Where this goes next

101001k10k100k1M10M100M1G10Gfrequency (hertz)the ideal operational amplifier1.42 kHz — a gain of 100 from a 1 MHz part is 1% low herea 10 V output at full amplitude7.96 kHz — above this the output cannot move fast enoughthe ideal 100 nF capacitor4.69 MHz — 1.2 nH of lead makes it 10% wrong hereKirchhoff's laws on 10.0 cm3.97 MHz — the board is one degree long hereeach bar is where the model may be used; the rule at its end is the numbersolved, then checked — each boundary from its own modeland one that is not a frequency: 7.3 mV Every model has an edge Four assumptions this collection runs on, with the frequency at which each stops being true, on one axis. The ordering is not the one most readers would guess — an ordinary amplifier circuit runs out of model at 1.42 kHz, three thousand times sooner than a ten-centimetre circuit board does. 00.200.400.600.8011001k10kfrequency (hertz)fraction of the source across the resistorhalf the power198.9 Hz measuredresonance 1.59 kHzsolved, then checked — half-power points by bisectionf₀/Q predicts 198.9 Hz — exactly Resonance, and the bandwidth it sets exactly The half-power bandwidth of a resonant circuit is f₀/Q — not approximately, but to every digit the arithmetic has, which is rare enough to be worth checking. What is not exact, and is drawn as though it were, is the idea that the band sits centred on the resonance. At a quality factor of one its middle is twelve per cent above. 1101001k10k100k1M10M100M1Gfrequency (hertz)impedance looking into 10.0 cm of track (ohms)the lumped model: one L, one C1° long at 3.97 MHza tenth of a wavelength at 143 MHzthe 200 Ω at the far endsolved, then checked — the line against a two-element modelKirchhoff's laws run out at 143 MHz Kirchhoff's own frequency The current law says the current entering a node equals the current leaving it at the same instant, which assumes the signal crosses the circuit in no time. It crosses at about two-thirds the speed of light, so the law has a frequency of its own — set by nothing but the physical size of the board.