The inductor that is a capacitor

The second essay this collection ever published measures where a capacitor stops being a capacitor. A 100 nF part with 1.2 nH of lead inductance is ten per cent wrong at 4.69 MHz and ninety-nine times its nominal impedance a decade above its 14.5 MHz self-resonance — and above that resonance it is an inductor, which is the essay’s title.

This is the dual, and the site’s habit is to build the dual rather than assert it.

Where the capacitance comes from #

A capacitor’s parasitic inductance is easy to point at: it is in the leads, and it is the reason a surface-mounted part is better than a leaded one.

An inductor’s parasitic capacitance is less localised and more inevitable. Every turn of the winding sits next to the turns either side of it, at a potential difference equal to the voltage across one turn; every layer sits next to the layer beneath it, at a potential difference equal to the voltage across a whole layer. Adding turns raises the inductance as N² and raises the capacitance too, so the self-resonance falls as the winding grows.

That is why the parasitic cannot be designed away by choosing better parts. It is a property of the geometry of a coil, and the levers are the same ones that move everything else in this field: fewer turns, more spacing, a sectioned bobbin — each of which costs inductance, size, or coupling.

What is measured, and against what #

The measurement here needs a decision that the capacitor essay also needed, and getting it wrong produces no answer at all.

The departure could be measured against ωL — the ideal inductor — or against |R + jωL|, the inductor with its series resistance and without its capacitance. The first is the more natural thing to write and it is the wrong reference, because at low frequency the series resistance dominates ωL completely: at a millihertz the resistance is thirty times the reactance, so the “departure from an ideal inductor” starts enormous, falls, and rises again, and a bisection assuming it is monotone returns nothing.

So the reference is the resistive one, and what the measurement isolates is the effect of the capacitance and nothing else. The series resistance is a real effect with an essay of its own — it is the equivalent series resistance’s dual, and it rises with frequency — and it belongs in the reference rather than in the answer.

That is the same treatment the capacitor essay gives its own series resistance, and the first version of this measurement did it the other way and returned null.

The number, and the thing about it #

The last column does not move. Nor does it move with the inductance: at 1 mH, 10 mH and 100 mH with the same capacitance it is 3.317 every time, and the gate holds both invariances to a part in 10⁶.

The reason is that the departure has no scale of its own. Below the resonance the impedance of the pair is ωL/(1 − (f/f₀)²), so the fractional departure is (f/f₀)²/(1 − (f/f₀)²) — a function of f/f₀ alone. Setting it to a tenth gives f/f₀ = √(1/11) = 0.3015, whose reciprocal is 3.317.

So the number is √11, and it is √11 for every inductor there is. That is unusual on this site, where almost every boundary carries the component values that produced it, and it is worth saying what it means practically: an inductor’s usable range is a fixed fraction of its self-resonance, and the self-resonance is the only number a data sheet needs to give.

The same expression at a different tolerance #

Since the fraction is a closed form, it can be read at any tolerance, and the family of answers is worth tabulating because it makes the shape of the departure legible.

Setting (f/f₀)²/(1 − (f/f₀)²) = ε gives f₀/f = √(1 + 1/ε), so:

The gate checks two of those against the measurement — 3.3166 against √11 and 10.0499 against √101 — because a constant that is secretly a root is exactly the sort of thing this collection is supposed to recognise rather than tabulate.

The practical reading of the table is the last row. An inductor is within a tenth of a per cent of ωL only below a thirtieth of its self-resonance, which for the 10 mH part drawn here is 17.8 kHz. Above that it is still an inductor for most purposes and it is not an inductor for a measurement, and the difference between those two statements is the tolerance somebody chose.

That is the same structure as the small-signal boundary, whose 1% and 0.1% amplitudes differ by √10 because a second-harmonic distortion is first order in the drive and a gain error is second. Here the two tolerances differ by √(1/ε), and for the same kind of reason: the quantity being bounded goes as a power of the variable, and inverting a power law is what turns a tolerance into a frequency.

Against the capacitor’s version #

The capacitor essay’s equivalent measurement gives 2.883, 3.096 and 3.146 across three lead inductances — not a constant, and the site’s own gate records why: the spread is the series resistance, which in that circuit is large enough relative to the reactance near the corner to move the departure.

Here the same quantity is exactly √11 at every setting, because the reference used isolates the capacitance completely. The two essays therefore measure the same shape and report differently, and the difference is a modelling choice rather than a fact about the components.

That is worth stating rather than presenting the tidier number as the better one. What the capacitor essay measures is the departure of a real part from its nominal value — which is the question a designer asks, and which contains the resistance. What this essay measures is the departure caused by one named parasitic, which is a cleaner question with a cleaner answer and slightly less use.

Both are on the site; neither supersedes the other; and the pair is a small demonstration that a “boundary” is defined by the reference it is measured against as much as by the circuit.

What sets the winding capacitance, in practice #

The essay has treated the capacitance as a given, which is what the netlist needs, so it is worth one section on where the number comes from — because unlike the coupling coefficient it is not recoverable from two easy measurements.

The self-resonance is measurable directly: sweep the impedance and find the peak. From it and a known inductance the capacitance follows, which is the ordinary bench procedure and is why data sheets quote a self-resonant frequency rather than a capacitance.

What decides it is geometry, and three levers move it in the directions this field has met before.

Fewer turns. Capacitance falls roughly with the turn count while inductance falls as its square, so fewer turns raises the resonance — at the cost of the inductance, and therefore of the lower band edge and of the volt-second headroom.

More spacing. Spaced turns have less capacitance between them, at the cost of a longer winding and therefore more resistance — which rises with frequency anyway.

Sectioning. Splitting the winding into sections in series puts the largest voltage differences between sections rather than between adjacent turns, which lowers the effective capacitance substantially. The cost is a bulkier bobbin and, in a transformer, worse coupling between primary and secondary — which is the upper band edge again.

Every one of those three trades a high-frequency property against a low-frequency one, which is the sentence this whole field has been making in eleven different ways.

Above the resonance #

Past f₀ the impedance falls, and the component is a capacitor of the value of its winding capacitance. That is the sentence the title is about and it has two practical consequences.

A choke stops choking. An inductor used to block high-frequency current — a supply-line choke, a common-mode choke, a filter element — has an impedance that rises to a peak and then falls, so above its self-resonance it is less effective than it was an octave below. A filter designed on ωL alone predicts a stopband that keeps improving; the real one has a best frequency.

A resonance is a high impedance, and that is sometimes what is wanted. At f₀ the impedance is not ωL but Q times it, which for a lightly damped winding is a very large number. A parallel resonant circuit is exactly this object used deliberately, and everything the resonance field measures about the quality factor and the half-power bandwidth applies to it.

The second point connects to this field’s own fourth essay in a way worth noticing. A transformer with a light load becomes a resonant circuit and produces voltage gain; an inductor with its own winding capacitance becomes a resonant circuit and produces impedance gain. Both are the same phenomenon — a reactance pair with insufficient damping — arriving in two of this field’s eleven essays from different directions.

What limits the measurement #

Two model boundaries, both of them the kind this site prefers to state.

The winding capacitance is one capacitor. It is distributed along the winding, so a real coil has a series of resonances rather than one, and the first is the one this model describes. Above it, the single-capacitor model is qualitatively right — the impedance falls — and quantitatively unreliable. That is the same limitation the transformer figures’ single interwinding capacitance carries and it is stated for the same reason.

The core is linear and lossless. A real inductor’s core has losses that rise with frequency, and those damp the resonance — so the measured peak here is higher and sharper than a real component’s. The 10% departure frequency, which is the essay’s number, is far below the resonance and is barely affected; the peak height is not a number this essay quotes for that reason.

Where the field ends #

This is the eleventh and last essay of the magnetics field, and it is worth one paragraph on what the eleven established, because the field turned out to be about a different thing from what it was scoped as.

It was planned as transformers, coupled inductors, mutual inductance and saturation — a list of objects. What it produced is a list of new kinds of boundary: a band rather than an edge, a volt-second product, a count of cycles with no safe side, a load resistance at which a component stops being the object its name says, and a rule of thumb that names a frequency the effect has already passed. Five of those had no precedent in the previous fifty-eight essays.

And it needed exactly one new element in the solver, because a decision made on the site’s first day — that an inductor carries a current unknown rather than an admittance — left the matrix affine in s and made a mutual inductance two off-diagonal entries. Every piece of machinery the site already had worked on a transformer without modification: the pole recovery, the group delay, the step response by two routes, the Nyquist locus.

What that element cost was a third verification route, because neither of the site’s two standing checks can see a coupling at all. That is the field’s most transferable finding and it is not about magnetics: an element that adds a constraint rather than a current is invisible to a check that sums currents, and needs a witness of its own before anything is built on it.

One more dual worth drawing #

The capacitor essay’s other headline is that a 100 nF part is ninety-nine times its nominal impedance a decade above resonance — which is the statement that the parasitic does not merely degrade the component but replaces it.

The inductor’s version is the same factor with the sign of the exponent flipped. A decade above 563 kHz, ωL would be 3.5 kΩ and the actual impedance is about 35 Ω: a hundredth rather than a hundred times, because above resonance the pair is a capacitor and a capacitor’s impedance falls where an inductor’s rises.

So both parts, a decade past their own resonance, are off by two orders of magnitude, in opposite directions, for the same reason. That symmetry is the whole content of the two essays’ titles and it is why this field’s last essay is a restatement of the frequency field’s second.

What the fifty-seven essays in between add is the vocabulary to say why it happens. In the frequency field it was a surprise about a component; by the eleventh field it is an instance of a pattern the collection has drawn a dozen times — a model with one element in it, a parasitic with another, and a frequency at which the second overtakes the first. The number changes; the shape does not.

The eleven, in one table #

Since this is the field’s last essay, the boundaries it measured are worth putting in one place. Each row is a figure and a number that came out of a solve.

Ten boundaries, five kinds of unit, and three of those kinds — a volt-second, a count, a load resistance — had never appeared in the previous fifty-eight essays.

That is more than a curiosity about vocabulary. The site’s premise is that no model is drawn without the frequency, amplitude or size at which it stops being true, and this field has made that phrasing inadequate twice over: once for a picosecond, read as bits’s duration, and again here for a product, a count and an impedance. What the rule is really saying is state the quantity and the value at which the model gives out — and the quantity is not always one of the three it happens to have started with.

Why the departure sits at the same fraction of resonance #

$f_0/3.317$ at every winding capacitance and every inductance tried is the result that makes this essay more than a companion to the capacitor’s, and it is worth saying what it means.

It means the departure is a property of the resonance rather than of either component. Above and below, the curve is $\omega L$ and $1/\omega C$; near it, the shape of the transition is the same shape whatever the two values are, because a lossless resonance has only one dimensionless parameter and the ten-per-cent point is a fixed point of it. So a designer who knows an inductor’s self-resonant frequency knows where it is ten per cent wrong without knowing either the inductance or the capacitance, which is exactly the useful form.

The capacitor that is an inductor is the same statement with the components exchanged, and the resistor that is only a resistor is the third member of the family — the one whose edge is not monotone in its own value, because it has both parasitics and they bind at opposite ends. Three components, one mechanism, and in each case a frequency a data sheet either prints or implies.

Where the three differ is in what the parasitic is. A capacitor’s is a lead inductance, which a layout can shorten; a resistor’s is a package geometry, which a designer chooses by choosing a part; and an inductor’s is the capacitance between its own turns, which is a consequence of the winding that produced the inductance in the first place. That is the one of the three that cannot be reduced without reducing the quantity being bought — which is why an inductor’s self-resonance is the least improvable of the three and the one most worth checking before a design depends on it. What can be done about it is a winding arrangement rather than a value — spacing the turns, splitting the winding into sections — which is the same kind of decision the inductance that is a shape finds deciding a transformer’s leakage, made for the same reason and by the same person.