The Q that peaks before resonance

The inductor that is a capacitor measured where a coil stops being an inductor because the capacitance between its turns appears across it, and found the ten-per-cent departure from $\omega L$ at a fixed fraction of the self-resonance, $f_0/3.317$, whatever the inductance and capacitance. It measured the impedance’s magnitude. The quantity that a coil is usually chosen by is not its magnitude but its quality factor — the ratio of its reactance to its resistance — and that ratio is affected by the winding capacitance far more strongly, and in a way that is easy to misread.

A coil’s Q is quoted as $\omega L/R$ and drawn as a line rising with frequency, which it is for a pure inductance in series with a fixed resistance. With the winding capacitance across the coil, the reactance seen at the terminals grows faster than $\omega L$ toward resonance — that was the departure the earlier measurement located — but the resistance seen at the terminals grows faster still. The Q therefore rises, peaks, and falls to zero at the self-resonance. Where it peaks turns out to be as fixed a fraction of the resonance as the departure was.

The coil, solved at each frequency #

The coil is the one the earlier measurement used: ten millihenries, 1.2 ohms of winding resistance and eight picofarads of capacitance between its turns, which puts its self-resonance at 563 kilohertz. It is built as a netlist — the resistance and the inductance in series, the capacitance across both — driven by a one-ampere source and solved at each frequency for the voltage, which is the impedance. Its Q at each frequency is the impedance’s imaginary part over its real part. Where a resistance that changes with frequency is needed, it is a different netlist at each frequency, not a correction applied afterwards.

The DC resistance alone gives this coil an implausibly high Q — tens of thousands — because a real coil’s resistance at hundreds of kilohertz is many times its DC value. That is the subject of the skin-effect figure below, and for the shape of the curve it does not matter: the position of the peak depends on the capacitance and the inductance, not on how large the resistance is.

Where the peak is, and why #

The impedance of a resistance and inductance in series, with a capacitance across them, can be split into a real and an imaginary part. Written in terms of $\omega_0 = 1/\sqrt{LC}$ and neglecting a term in $R^2C$ that is negligible whenever the coil’s Q is more than a few, the Q at the terminals is

$$Q(\omega) = \frac{\omega L}{R}\left(1 - \frac{\omega^2}{\omega_0^2}\right).$$

The first factor is the straight line; the second is the capacitance pulling it down, reaching zero at the self-resonance. Their product has a maximum where its derivative is zero, at $\omega = \omega_0/\sqrt3$, and the value there is $\frac{2}{3\sqrt3} \cdot \omega_0L/R$, or 0.3849 of what the straight line would reach at resonance. The solved coil peaks at 0.5773 of its resonance, at 0.3849 of the line’s value there.

The fraction is the same for every coil, because neither the inductance nor the capacitance appears in it except through the resonance they set.

Eighteen coils spanning two decades of inductance and more than two of capacitance, with straight-line Qs at resonance from fifteen hundred to a quarter of a million, all peak at 0.5774 of their resonance. So the rule is universal in the same way the earlier departure rule was: a coil is at its best at 58 per cent of its self-resonance, and its Q there is 38 per cent of what its DC resistance and inductance would promise at the resonance.

That has an immediate design consequence. A coil intended for use at a given frequency should have a self-resonance at least $\sqrt3$ times higher, not merely higher; above $f_0/\sqrt3$ every increase in frequency lowers the Q, and it has fallen to 85 per cent of its peak at three-quarters of the resonance and to 44 per cent at nine-tenths. A data sheet that quotes a Q “at” some frequency and a self-resonance somewhere above it is quoting two points of this curve, and the ratio of the two frequencies says which side of the peak the quoted Q is on.

Skin effect moves the peak down #

A real winding’s resistance rises with frequency as the current crowds toward the surface of the wire. Above the frequency at which the skin depth equals the wire’s radius, the resistance rises roughly as the square root of frequency, and that changes where the peak is.

With the resistance rising as $\sqrt\omega$, the Q is $\propto \sqrt\omega\,(1 - \omega^2/\omega_0^2)$, and its maximum moves to $\omega_0/\sqrt5$, 0.4472 of the resonance. The solved coil reaches 0.4472 when the skin-effect corner is far below the frequencies of interest, and sits between the two limits when the corner falls among them: 0.4763 with the corner at a tenth of the resonance, 0.4511 at a hundredth.

The height falls much more than the position moves. With a resistance that has risen by a factor of seventy at the peak frequency, the Q there is 158 rather than 11,340 — a realistic figure for a ten-millihenry coil at a couple of hundred kilohertz, and the reason the constant-resistance curve should be read for its shape and not its height. So a real coil peaks somewhere between 45 and 58 per cent of its self-resonance, nearer the lower figure the more its resistance is dominated by skin and proximity effects, and a designer allowing a margin of $\sqrt3$ in frequency is allowing slightly too little for a coil wound with thick wire. The copper that makes it worse measures how the proximity of neighbouring turns makes the resistance rise faster still, which pushes the peak lower again.

What a bridge reads #

The Q falls toward resonance because the resistance seen at the terminals rises faster than the reactance. That is worth isolating, because it is what an impedance bridge or an LCR meter reports, and it is easy to mistake for a physical loss.

The coil’s winding resistance is 1.2 ohms at every frequency in this figure, and nothing in the netlist dissipates more than that. Yet the series resistance a bridge reads rises as $R/(1-\omega^2/\omega_0^2)^2$ — 27 per cent higher at a third of the resonance, twenty times higher near it — while the inductance it reads rises only as $1/(1-\omega^2/\omega_0^2)$. Their ratio is the Q, and it falls as $1 - \omega^2/\omega_0^2$.

Nothing is lost that was not lost before; the capacitance has transformed the same resistance into a larger apparent one. Current circulates between the inductance and the winding capacitance near resonance, and the winding resistance carries that circulating current as well as the terminal current, so a terminal measurement attributes the heat of a larger current to a smaller one. Two parasitics, and the resonance neither of them has met the same transformation in a capacitor with an inductance in series, where it raised the apparent quality factor rather than lowering it.

The practical reading is a warning about measurement conditions: a coil’s “AC resistance” measured on a bridge near its self-resonance includes this transformation, and a model that adds that measured resistance in series with the inductance and also includes the winding capacitance has counted the effect twice.

In a tank the capacitance is tuning #

The figures so far describe the coil read alone, at its two terminals. A coil is rarely used alone: in a tuned circuit it is resonated by an external capacitor, and its own winding capacitance is then simply part of the tuning capacitance.

Tuned to 0.8 of the coil’s self-resonance with an external capacitor, the tank’s Q measured from its bandwidth is 23,570 — exactly $\omega L/R$ at that frequency — while the same coil read alone at the same frequency shows 8,485. The tank gets the full straight-line Q, and the falling curve of the bare coil does not apply to it.

The reason is that the winding capacitance and the external capacitor are in parallel, and in a tank both are part of one resonance. The circulating current that inflated the coil’s apparent resistance when it was read alone is, in the tank, simply the tank’s own resonant current, and the tank’s Q is set by the one resistance that current flows through. So the falling Q is a property of the coil as a two-terminal impedance — the right model when the coil is a choke, a filter element away from resonance or a load — and not of the coil as a resonator.

The two uses need different numbers from a data sheet. A choke’s designer needs the falling curve, since the choke must present a high impedance at frequencies where it is not resonated by anything. A tuned-circuit designer needs $\omega L/R$ at the tuning frequency and the winding capacitance as part of the tuning, and should not derate the coil’s Q for being near its self-resonance. The Q the components allow found that a resonator’s Q is limited by its worst component’s Q; the coil’s Q that enters that sum is the tank’s, not the bare part’s.

The capacitor has no such peak #

The dual comparison that the inductor that is a capacitor drew for the impedance’s magnitude breaks down for the quality factor, and the way it breaks is instructive. A capacitor’s parasitic inductance is in series with it, not across it, so its reactance at the terminals is $\frac{1}{\omega C}\left(1 - \omega^2/\omega_0^2\right)$ and its Q, over a constant series resistance, is $\frac{1}{\omega CR}\left(1 - \omega^2/\omega_0^2\right)$. Both factors fall with frequency. There is no maximum below resonance: a capacitor’s Q is highest at the lowest frequency and falls monotonically to zero at its self-resonance.

So the dual of “a coil is at its best at 58 per cent of its self-resonance” is not “a capacitor is at its best at some fraction of its own”; it is “a capacitor gets steadily worse all the way up”. The asymmetry comes from where each part’s useful reactance sits on the frequency axis. An inductor’s reactance rises with frequency and its parasitic capacitance eventually overtakes it, so there is a frequency at which rising reactance and rising parasitic effect balance. A capacitor’s reactance falls with frequency, the same direction its parasitic pushes it, and nothing balances. The capacitor that is an inductor found where the capacitor stops being one; the Q says it was getting worse long before that.

Reading the peak off a data sheet #

Most inductor data sheets give two numbers that this curve connects: a self-resonant frequency, and a Q at a stated test frequency. The ratio of the two frequencies says which side of the peak the quoted Q sits on. A test frequency below $f_0/\sqrt5$ is on the rising side and the quoted Q will improve if the coil is used a little higher; one above $f_0/\sqrt3$ is on the falling side and the coil is already being used past its best. A coil whose data sheet quotes its Q at a frequency close to its self-resonance is, by this curve, quoting a number that falls steeply with every per cent of frequency, and a small difference between the test fixture’s capacitance and the circuit’s will move it a long way.

The capacitance in that sentence matters because it adds to the winding’s own. A coil mounted on a board picks up pad and trace capacitance across its terminals, typically a picofarad or two, and that lowers its self-resonance and with it the frequency of the Q peak. For the eight-picofarad coil here, two picofarads of mounting capacitance lower the resonance by ten per cent and move the peak down by the same proportion; for a small surface-mount coil whose own winding capacitance is a fraction of a picofarad, the board can halve its self-resonance. The inductance the current decides adds the other condition a data sheet’s numbers are measured under — the current — and a coil chosen by its Q at a test frequency has to be checked at its operating current as well.

What it means for choosing a coil #

The rules that follow are short. A coil used as an impedance — a choke, an isolating element — is at its best at about half its self-resonance, between $f_0/\sqrt5$ and $f_0/\sqrt3$ depending on how much skin and proximity effect it has, and its Q falls to zero at the resonance. A coil used as a resonator gives its full $\omega L/R$ at any frequency below its self-resonance, provided the tuning capacitor is reduced by the winding capacitance. And a Q measured on a bridge near the self-resonance is the first kind of number, not the second, and must not be used as the second.

The interwinding capacitance that sets all of this is itself a design variable. The other half of the same window and interleaving is a choice, not an improvement measured how winding arrangement trades that capacitance against leakage inductance. For a choke, raising the self-resonance by winding in sections moves the Q peak up in proportion, which is often worth more than a better wire.

What the model leaves out #

The winding capacitance is one lumped capacitor across the whole coil. A real winding’s capacitance is distributed along it, which gives the coil a series of resonances above the first rather than one, and near the first the lumped model is accurate while above it it is not; the Q peak lies well below the first resonance, where the lumped model holds.

The core is absent. A coil on a magnetic core has core loss that rises with frequency, which acts as a resistance in parallel with the inductance and lowers the Q at high frequency in addition to the winding’s own rise. That moves the peak down further, by an amount that depends on the core material, and for a ferrite-cored coil at hundreds of kilohertz it may dominate the winding’s contribution.

And the skin effect is the simplest law, a resistance growing as $\sqrt{1 + f/f_s}$. A real winding has a more complicated law, set by the wire’s diameter, its neighbours and its strand structure; the qualitative result — the peak moving from $f_0/\sqrt3$ toward $f_0/\sqrt5$ as the resistance’s rise steepens — holds for any law between constant and square-root.

The number to keep #

A coil’s Q read at its terminals peaks at its self-resonance over $\sqrt3$, at $2/(3\sqrt3)$ of what $\omega L/R$ would reach at the resonance — 0.5774 of the resonance on eighteen coils spanning two decades of inductance — and falls to zero at the resonance. Skin effect moves the peak toward the resonance over $\sqrt5$. The fall is the winding capacitance transforming the winding resistance into a larger apparent one, $R/(1-\omega^2/\omega_0^2)^2$, while the apparent inductance rises only as $1/(1-\omega^2/\omega_0^2)$. In a tank the same capacitance is tuning, and the coil gives the full $\omega L/R$: 23,570 against 8,485 read alone at 0.8 of the resonance.

Still open: the core’s loss, the second resonance, and the coil that is also a transformer #

Core loss as a parallel resistance. A cored coil’s Q is limited above some frequency by its core rather than its winding, and the two limits combine. Where the peak sits when both a rising winding resistance and a falling parallel core resistance are present is the number a ferrite inductor’s designer actually needs.

Above the first resonance. A distributed winding has further resonances, alternately parallel and series, and the coil’s Q between them is not described by any lumped model. Solving a ladder network of turn-to-turn capacitances would show how far above the first resonance the coil is still a recognisable inductor.

Two windings on one core. A transformer’s winding capacitances couple its windings as well as shunting each, and the Q of each winding with the other loaded is a question about how the interwinding capacitance divides the circulating current between them.