The Q a core allows

The Q that peaks before resonance found the one number a coil’s own capacitance imposes on its quality factor. Quoted as $\omega L/R$ and drawn rising with frequency, a coil’s Q is turned over by its winding capacitance: it peaks at the self-resonance over $\sqrt3$, at $2/(3\sqrt3)$ of the straight line’s value there, for every coil from a millihenry with three hundred picofarads to a hundred millihenries with one. The coil was air-cored, so the winding was the only thing that dissipated.

It closed on the coil a designer is more likely to hold: one wound on a core, where the core dissipates as well. A core’s loss is usually drawn as a resistance across the inductance, and a ferrite inductor’s data sheet gives its Q peaking at a frequency that has nothing obvious to do with the self-resonance. This essay puts the core into the solved coil and finds where the peak goes, what sets it, and which of the coil’s two losses the designer should spend effort on.

The coil, with a core across it #

The coil is the earlier one — ten millihenries, 1.2 ohms of winding, eight picofarads of winding capacitance, self-resonant at 563 kilohertz — built as a netlist, driven by a current and read as an impedance at each frequency, with its Q the reactance over the resistance. The core is a resistance across the inductance. That is the standard lumped model of a magnetic core’s loss: the core dissipates in proportion to the square of the flux, the flux follows the voltage across the inductance, and a resistance across the inductance dissipates the square of that voltage.

What the resistance does with frequency is a property of the material, and two cases bound most real ones. A core of constant loss tangent dissipates the same fraction of its stored energy every cycle, so its parallel resistance is $\omega L Q_c$ and the core alone has the same Q, $Q_c$, at every frequency. A core whose loss is a fixed resistance — eddy currents in a lamination at low frequency come close — has a Q of $R_c/\omega L$ that falls with frequency. The hero figure uses the first.

The peak falls to the core and moves down #

Without any core loss the Q climbs along $\omega L/R$ and turns over at 325 kilohertz at 11,340. A core whose own Q is a thousand cuts that to 825.6 at 136 kilohertz; three hundred to 274.9 at 94; a hundred to 95.9 at 66; fifty to 48.7 at 53. The peak is always below the core’s own Q and close to it — 0.83 of it for the best core and 0.97 for the worst — and it moves down in frequency as the core gets worse.

The reason is the two ways the Q can be limited. Far below resonance the coil’s Q is the combination of two Qs in parallel, the winding’s $\omega L/R$ rising with frequency and the core’s constant $Q_c$; the smaller one wins. Near resonance the capacitance takes the reactance away, as it did in the air-cored coil. So the curve rises along the winding’s line, flattens against the core’s ceiling, and then falls to the capacitance; the peak is where the ceiling meets the capacitive fall, which for a core much worse than the winding is well below the self-resonance.

Where the peak sits, in closed form #

The combination has a closed form good enough to locate the peak. With $x = f/f_0$, $q_0 = \omega_0 L/R_w$ the winding’s Q at resonance, and $a = Q_c/q_0$ the core’s Q in units of that, the coil’s Q is close to $Q_c\,x(1 - x^2)/(x + a)$: the two Qs combined in parallel, multiplied by the $(1 - x^2)$ the capacitance takes away. For a core much worse than the winding, $a$ is small and the peak sits at $x^3 = a/2$, reaching $Q_c(1 - 3(a/2)^{2/3})$.

The solved coil agrees with that to under two per cent in position and one in height wherever $a$ is below a hundredth. At a core Q of ten the peak is at 0.0554 of the self-resonance, where the closed form says 0.0554; at a hundred, 0.1176 against 0.1193. The cube root is the useful part. It says the peak moves slowly with the core: a core ten times better moves the peak up by a factor of 2.15, and a capacitance that halves the self-resonance moves the peak down by less than the resonance itself, because $q_0$ falls with it.

The coil this describes is ordinary. A ferrite pot core with a core Q of a hundred or two and a winding that could reach tens of thousands on its own is in the small-$a$ regime by a factor of a hundred, and its peak frequency is set by the core and the capacitance together, at a cube root of their ratio, rather than by either.

A fixed resistance instead: the geometric mean #

A core whose loss is a fixed resistance behaves completely differently, and more simply. Its own Q, $R_c/\omega L$, falls with frequency while the winding’s rises, so the two cross, and the coil’s Q peaks there: at the geometric mean of the two corners, $\sqrt{R_wR_c}/2\pi L$, with a value of $\tfrac12\sqrt{R_c/R_w}$. For ten kilohms across this coil that is 45.64 at 1.74 kilohertz, and the solved coil agrees to half a per cent at every resistance. The capacitance plays no part; the peak is far below the self-resonance.

This is the textbook’s version of a coil’s Q peak, and the two models show how much the material decides. With a fixed resistance the peak is where the two losses are equal and its height is the square root of their ratio. With a constant loss tangent there is no crossing at all — the core’s Q never falls — and the peak is where the capacitance ends a plateau. A data sheet’s Q curve is somewhere between, and its shape near the peak says which model the core is closer to.

Which loss to spend effort on #

The split of the loss answers the designer’s question directly. Transformed into series, the core’s parallel resistance is $(\omega L)^2R_c/(R_c^2 + (\omega L)^2)$, which for a constant loss tangent rises in proportion to frequency, while the winding’s 1.2 ohms stays put. The two are equal at 1.91 kilohertz, where the winding’s $\omega L/R$ reaches the core’s hundred. At the Q peak, 66.2 kilohertz, the core carries 97.2 per cent of the loss.

So at the frequency the coil is best at, its winding hardly matters. Halving the winding resistance — thicker wire, fewer turns of a better material — moves the 97.2 per cent to 98.2 and the Q by under two per cent, through the $a$ in the cube root. The effort that moves the peak goes into the core: a better material, or a gap. The Q the components allow found a resonator’s Q capped by the worst of its components; inside one coil, the same rule picks the core.

A loss that rises with frequency #

A real ferrite’s loss tangent is flat at low frequency and rises above a corner set by the material, so its Q holds and then falls. Drawn as $Q_c/(1 + f/f_1)$, a core of Q 300 with its corner at a hundred kilohertz pulls the peak from 94 kilohertz down to 23.7 and from 274.9 to 202.5; with the corner at ten kilohertz, to 7.56 kilohertz and 119.3. Above its corner the core behaves like a fixed resistance — its Q falling as one over frequency — and the peak moves toward the fixed-resistance case’s geometric mean. The coil’s best frequency is then the material’s, and choosing a ferrite grade is choosing where the coil’s Q will peak.

Why a fixed resistance is the eddy current’s model #

The two core models are not arbitrary, and the fixed resistance has a physical owner. A lamination’s eddy-current loss at low frequency goes as the square of frequency times the square of the flux density, and the flux density at a fixed voltage across the winding goes as one over frequency, so the eddy loss at a fixed voltage is independent of frequency: a resistance across the inductance that does not change. The current inside the iron measured where that stops being true — when the lamination is thick enough against its own skin depth that the flux no longer fills it — and above that frequency the eddy resistance rises and the core’s Q stops falling.

A constant loss tangent is the hysteresis model’s shape: a fixed energy lost per cycle at a fixed flux, so a loss proportional to frequency and a Q that holds. Ferrites at the frequencies they are used at are mostly hysteresis and residual loss with little eddy current, which is why their data sheets quote a loss tangent rather than a resistance, and why the hero figure’s model is the one that fits them. A laminated iron core at mains frequencies is closer to the fixed resistance, and its coil’s Q peaks low, at the geometric mean of two corners a few hundred hertz apart.

A worked coil #

A designer wants a ten-millihenry coil for a tuned circuit at sixty kilohertz with a Q above eighty, and has two ferrite grades: one with a loss tangent giving a core Q of a hundred, the other of three hundred, at the cost of a lower permeability and so more turns. On the first grade, with this winding and capacitance, the coil peaks at 95.9 near 66 kilohertz — almost exactly where the tuned circuit needs it, and above eighty. On the second it peaks at 274.9 near 94 kilohertz and is still well above eighty at sixty, but the extra turns raise the winding’s resistance and its capacitance, which move the peak by the cube root of their ratio and the self-resonance with it.

The cube root is what makes that decision forgiving. A grade three times better triples the core’s Q and moves the peak up by only 1.44 in frequency; a winding with twice the resistance moves it up by 1.26 and lowers the peak by under three per cent. The coil’s best frequency is set mostly by the core and its capacitance, and a designer choosing a grade to place the peak near the working frequency is choosing almost entirely by the core.

The capacitance still decides the top #

Even when the core sets the height of the peak, the capacitance sets where the curve ends. The inductor that is a capacitor found the coil’s impedance turning capacitive above its self-resonance, and a cored coil has the same resonance: the core’s loss widens it and lowers its peak impedance, but the inductance and the winding capacitance still put it at 563 kilohertz here. A tuned circuit built on the coil must sit below the peak and well below the resonance, and the core decides how much of that span is usable. The tuning capacitor has the mirror-image problem: the capacitor that is an inductor found its own leads turning it inductive above a resonance of its own, and a tank built from the two parts is good only where both are still what they are called — the lower of their two edges, and for a cored coil the core’s peak is usually the lower.

Two routes, and the edge of the closed form #

The peaks have been found two ways. The solved coil — winding, inductance, capacitance and core as a netlist at each frequency — is searched on a fine logarithmic grid and the peak golden-sectioned on the impedance itself. The closed forms, $x^3 = a/2$ for a constant loss tangent and the geometric mean for a fixed resistance, come from the coil’s parallel-Q combination and the capacitance’s $(1 - x^2)$ without any netlist. They agree to half a per cent for the fixed resistance, which is exact, and to two per cent for the loss tangent wherever $a$ is below a hundredth.

Above that the closed form fails, and the figure shows where. As the core approaches the winding the approximation $x \gg a$ inside the cube root stops holding, and the peak moves toward the air-cored coil’s $1/\sqrt3$ by a route the small-$a$ formula cannot follow. The edge is where the core’s Q is a hundredth of the winding’s at resonance — for this coil a core Q of about three hundred — which is the boundary between the coils a ferrite makes and the coils an air core makes.

What a measurement reads #

A Q meter or an impedance analyser measuring a cored coil reads the total Q and a series resistance, and the series resistance is mostly core above a kilohertz or two. Five readings of one Q found the different ways of reading a resonator’s Q agreeing only above a Q of three and a half; a cored coil’s Q is high enough for that to hold, but what the reading measures is the core’s loss transformed through the coil, and a designer who reads a series resistance of a few ohms at sixty kilohertz and blames the winding is looking at the wrong part. The same part, written two ways found a component’s loss described exactly by a series and a parallel resistance at one frequency only; the core’s parallel resistance is the description that holds across a band, which is why the model puts it there.

What the model leaves out #

The core’s loss is linear: a resistance across the inductance, independent of the flux. A real core’s loss rises faster than the square of the flux, so its Q depends on the drive level as well as the frequency, and one dissipation, two exponents measured how the loss law’s exponent moves with the flux. The winding resistance is fixed; skin and proximity effects raise it with frequency, which for an air-cored coil moved the peak to $f_0/\sqrt5$ and for a cored coil matters only near the self-resonance, where the core has already taken the loss. And the inductance is the core’s at low level; a core’s permeability falls with frequency near its own resonance, which moves $L$ and so the whole curve.

The number to keep #

A core whose own Q is $Q_c$ caps the coil’s Q just under itself and moves its peak down from the self-resonance over $\sqrt3$ to $x^3 = a/2$, with $a$ the core’s Q over the winding’s at resonance — 95.9 at 66 kHz for a core of Q 100 on a coil that peaked at 11,340 near 325 kHz. At that peak the core carries 97 per cent of the loss. A fixed core resistance puts the peak at $\sqrt{R_wR_c}/2\pi L$, $\tfrac12\sqrt{R_c/R_w}$ high; a loss tangent that rises above a corner drags the peak toward the material’s corner.

Still open: the gap, the drive level, and the second resonance #

A gapped core. A gap lowers the effective permeability and with it the core’s share of the stored energy, so the core’s effective Q rises by the ratio of the material’s permeability to the gapped one — at the cost of more turns for the same inductance and so more winding resistance. Where the gap that maximises the coil’s peak Q sits, for a stated material, is the optimisation the cube root makes cheap.

The drive level. A core’s loss grows faster than the square of the flux, so the core’s Q falls as the drive rises. A coil driven harder peaks lower and at a different frequency, and how far is the question for a resonant converter’s tank inductor, which runs at large flux by design.

Above the first resonance. A distributed winding has further resonances, alternately parallel and series, and a lossy core damps them. Whether a cored coil is still a recognisable inductor between its first and second resonances — and how much the core’s loss widens that window — is the measurement a ladder network of turn-to-turn capacitances would make.