Two windings, and the band between them

Two inductances at one current

A data sheet prints one L(i) curve and there are two. An operator sitting inside its own backlash contributes nothing to dB/dH, so a small excitation sees only the operators still moving — every one at the tip of a loop, none just after a reversal — and the same core at zero bias measures 14.90 millihenries pushed downward and 8.30 pushed upward, a factor of 1.80. The same split decides what a converter's ripple costs: held at the flux swing volt-seconds actually fix, a twenty-millitesla ripple costs twelve times more at a hundred and seventy-five amperes per metre of bias than at none, and above a hundred and eighty-three the question has no answer at all.

Assumes: The area a curve cannot have · A boundary in volt-seconds

The inductance the current decides drew the curve a catalogue prints: the small-signal inductance of a gapped core against the direct current already flowing in it, falling continuously, with three different saturation currents marked because ten, twenty and thirty per cent drops are all in use.

Every point on that curve is a function of the current alone. That was true of the model it came from and it is not true of a core, and the difference is not a correction — it is a second curve.

Fig. 1 The same measurement made twice at every bias: once with the excitation pushing the current up, once pushing it down. The single-valued curve is drawn above both. The slider is how far the core was driven before it was walked back.

The virgin curve is inside the loop everywhere but at its tip, and nothing returns a core to it. computed by solving, not by drawing. A demagnetised core taken up to ±400 A/m for the first time, drawn inside the major loop it then traces for ever. The two meet only at the tip: at half the drive the virgin curve is at 0.0098 tesla, strictly between the ascending branch's -0.0057 and the descending branch's 0.0388, because on the virgin curve no domain has yet been moved in either direction. Once a core has been anywhere there is no operation that puts it back, except a slowly decaying alternating field — which leaves -0.170 millitesla behind rather than zero. — Fig. 2 Why a bridge reading depends on history at all. A core is described by the state of every operator in it, not by a point, and the virgin curve is the one state no working core is ever in twice.

Why there are two

A play operator held inside its own backlash contributes nothing whatever to dB/dH. Its output does not move when the field does, so its slope is exactly zero, and it stays exactly zero until the field has travelled the whole width of the backlash.

So a small excitation applied to a core sees only the operators that are still moving, and which those are depends on the direction. A core walked down to a bias and then pushed further down finds every operator already travelling that way: they all contribute. Pushed back up instead, every one of them has to cross its own backlash first, and near the reversal almost none of them contributes at all.

That is the whole mechanism, and it produces numbers that are not small. At zero bias, on a core walked back from four hundred amperes per metre, the inductance is

14.899 millihenries measured with the excitation going down,
8.297 millihenries measured with it going up,

a factor of 1.796 on one part at one current. The ratio narrows with bias but not much: 1.726 at fifty amperes per metre, 1.679 at a hundred, 1.615 at three hundred. It does not converge to one anywhere on the axis.

A major hysteresis loop at 9.0 A/m of coercivity, and the anhysteretic curve it closes onto. computed by solving, not by drawing. The B–H loop of a core driven sinusoidally to ±400 A/m, marched through a superposition of twenty-four play operators and drawn over the single-valued curve the two rungs below this one measured. The loop encloses 12.481 joules per cubic metre per cycle, which is the core loss and which no single-valued model can produce, because a curve has no area. The coercivity is 8.96 amperes per metre and the remanence 22.3 millitesla; both are read off the marched descending branch rather than handed in. The slider takes the threshold spread down to zero, where the two branches become one, the area falls to 9.8e-15 J/m³, and the object is exactly the core the field already had. — Fig. 3 Where the two slopes are. On the descending branch at any field, the tangent is steep; immediately after a reversal the core has to cross the width of the loop before anything irreversible moves, and the tangent there is the reversible part alone.

There is a limiting case worth naming, because it makes the mechanism concrete. At the very tip of a major loop every operator is moving, so both directions agree and the two branches meet. Immediately after any reversal none of the irreversible operators is moving, so the slope is the reversible fraction alone — $\mu_0 + f\,B_\text{sat}/H_k$ , which for this material is 1,100 μ₀ against the anhysteretic curve’s 2,000. The two branches are therefore not two parallel curves; they are one curve and a family of re-entry slopes that depend on how far back the reversal was.

The same core at the same current has two inductances, 1.80 times apart. computed by solving, not by drawing. The small-signal inductance of a sixty-turn winding on a core walked down from 800 amperes per metre, measured by pushing the excitation up and by pushing it down at each bias. They are never the same: 14.899 millihenries against 8.297 at zero bias, a factor of 1.796, and up to 1.796 across the sweep. The mechanism is that an operator held inside its own backlash contributes nothing to dB/dH, so a reversal is measured by whichever operators are still moving, and that is a different set in each direction. The single-valued curve the field already had is drawn above both, which is what it is: an optimistic reading of an object with two answers. — Fig. 4 A deeper excursion: walked back from eight hundred amperes per metre the branches are further apart again, because more of the distribution was driven and therefore more of it is pinned on the way back.

What a bridge is reporting

An inductance bridge applies a small alternating excitation and reports one number. The excitation goes both ways, so what comes back is a blend of the two branches — weighted by the waveform, by the amplitude, and by how far the core had been driven before the measurement started.

That last one is worth separating out, because it is the parameter nobody records. Walk the core back from four hundred amperes per metre and the zero-bias pair is 14.899 and 8.297. Walk it back from a hundred instead and at a bias of fifty the pair is 10.799 and 7.315 — a ratio of 1.476 against 1.726 for the same bias reached from further out. The measurement depends on where the core has been, and a bench does not log that.

This is the same shape as the capacitance that is not one number’s class II ceramic, where the capacitance a bridge reports is a function of the test amplitude and the bias as much as of the part, and the coefficient that is about one reading, where a specification turned out to be exactly true about a quantity nobody uses. Here the uncontrolled variable is the history, which is a worse one to have loose because it is not on any instrument’s front panel.

The single-valued curve sits above both branches everywhere — 15.080 millihenries at zero bias against 14.899 and 8.297. So a design taking its inductance from the curve the rung below drew is optimistic in both directions, by 1.2 per cent against the favourable branch and by 82 per cent against the unfavourable one.

Held at the flux swing volt-seconds actually set, a ripple costs 4.6× more on a bias. computed by solving, not by drawing. The energy a minor loop dissipates against the direct bias it sits on, measured twice. Held at a constant field excursion the loss falls, from 0.1479 to 0.0045 joules per cubic metre — a true measurement of the wrong thing, since what is collapsing is the flux swing as the material stiffens. Held at the ±35 millitesla a converter's volt-seconds actually fix, it rises instead, by 4.59 times across the axis, because the field needed to move that much flux has grown from 42.8 to 129.7 amperes per metre. Above 142.5 A/m the measurement refuses: the excursion the ripple needs would carry the core past the loop it was characterised on, so there is no minor loop to have an area. That is a direct-current ceiling with no volt-seconds in it at all. — Fig. 5 Thirty-five millitesla of ripple. The refusal ceiling has come in from a hundred and eighty-three amperes per metre to a hundred and forty-two, which is the same statement about room inside the major loop.

The ripple, and which quantity to hold

The other half of what the branches decide is what a ripple costs, and getting the question right turns out to matter more than the answer.

Held at the flux swing volt-seconds actually set, a ripple costs 12.3× more on a bias. computed by solving, not by drawing. The energy a minor loop dissipates against the direct bias it sits on, measured twice. Held at a constant field excursion the loss falls, from 0.1479 to 0.0045 joules per cubic metre — a true measurement of the wrong thing, since what is collapsing is the flux swing as the material stiffens. Held at the ±20 millitesla a converter's volt-seconds actually fix, it rises instead, by 12.28 times across the axis, because the field needed to move that much flux has grown from 25.8 to 151.8 amperes per metre. Above 183.1 A/m the measurement refuses: the excursion the ripple needs would carry the core past the loop it was characterised on, so there is no minor loop to have an area. That is a direct-current ceiling with no volt-seconds in it at all. — Fig. 6 The energy a minor loop dissipates against the bias it sits on, measured two ways. The two go opposite directions. The slider is the size of the ripple.

Hold the field excursion constant — forty amperes per metre, at every bias — and the loss falls monotonically, from 0.1479 joules per cubic metre at zero to 0.00454 at three hundred and twenty. A factor of thirty-three. It is a true measurement, and it is worthless, because what is collapsing is the flux swing: 32.4 millitesla at zero bias and 1.03 at the top. The material has stiffened and the same field moves almost no flux.

Hold the flux excursion constant instead and the answer reverses. Twenty millitesla of ripple costs 0.0405 joules per cubic metre at zero bias and 0.4971 at a hundred and seventy-five — twelve times more. The field needed to move that flux has grown from 25.8 amperes per metre to 151.8, and the loop that field traces is correspondingly larger.

The second is the design question, because a boundary in volt-seconds settled that the volt-seconds fix ΔB and not ΔH. A converter’s ripple is set by the input voltage, the duty cycle and the period; the material does not get a vote. So the flux swing is held and the field does whatever it must.

A ripple gets more expensive as the direct current rises, and it is not a mild effect. That is the opposite conclusion from the constant-field measurement, and the constant-field measurement is the one that is easier to make.

Held at the flux swing volt-seconds actually set, a ripple costs 65.3× more on a bias. computed by solving, not by drawing. The energy a minor loop dissipates against the direct bias it sits on, measured twice. Held at a constant field excursion the loss falls, from 0.1479 to 0.0045 joules per cubic metre — a true measurement of the wrong thing, since what is collapsing is the flux swing as the material stiffens. Held at the ±5 millitesla a converter's volt-seconds actually fix, it rises instead, by 65.25 times across the axis, because the field needed to move that much flux has grown from 7.0 to 68.6 amperes per metre. Above 271.6 A/m the measurement refuses: the excursion the ripple needs would carry the core past the loop it was characterised on, so there is no minor loop to have an area. That is a direct-current ceiling with no volt-seconds in it at all. — Fig. 7 The smallest ripple the slider carries. At five millitesla the loss rises sixty-five times across the axis rather than twelve, because a smaller excursion is further into the region where the loop area goes as the cube.

Why the two readings disagree in sign

It is worth being explicit about why one measurement rises and the other falls, because both are correct and the pair is a small lesson in what “hold everything else constant” means.

The loss of a minor loop is roughly cubic in its own excursion at small amplitude — Rayleigh’s law, which the exponent nobody put in derived from the threshold density in closed form. Holding the field excursion fixed holds the argument of that cube fixed in the wrong variable: the flux excursion falls as the material stiffens, and the loop area falls with it. Holding the flux excursion fixed forces the field excursion up as the material stiffens, and the cube then works the other way.

So the two curves are the same law read along two different sections through the same surface, and which section a measurement takes is decided by what the apparatus holds constant. A bench that drives a fixed current ripple into a biased inductor is taking the first section. A converter takes the second.

That is the same trap as the three tolerances that do nothing’s: a derivative taken with the wrong thing held fixed is a true derivative of the wrong function.

The bias above which there is no answer

Push the bias further and the measurement stops working, and the way it stops is the interesting part.

A minor loop lives inside a major one. The excursion may reach the tip of the loop the core was characterised on and no further; past that it is not a minor loop and the whole vocabulary — the reversal it returns to, the branch it left — has nothing to refer to. So there is a bias at which a stated flux ripple simply cannot be had, because the material will not give that much flux for any field that stays inside the loop.

For twenty millitesla on this core it is 183.08 amperes per metre, bisected. Above it the routine raises rather than returning a number.

That ceiling moves with the ripple, and it moves the way it should: 271.6 amperes per metre for a five-millitesla ripple, 229.6 for ten, 183.1 for twenty, 142.5 for thirty-five, 114.8 for fifty. A bigger ripple runs out of room sooner.

It is a direct-current limit with no volt-seconds in it. That is what makes it worth a section. A boundary in volt-seconds established the field’s first ceiling and it is an integral of voltage; the flux that walks found a second, which is an asymmetry accumulating into that integral. This is a third and it is neither. The core here is nowhere near its flux ceiling — the mean flux at the last measurable point is 0.323 tesla against a saturation flux of 0.35, and the ripple is what cannot be accommodated.

Compare it against the numbers the rung below quoted for the same part. The ten, twenty and thirty per cent inductance drops sit at 45.6, 67.1 and 85.7 amperes per metre. The ripple ceiling is 183 — more than twice the most generous of them — so a part driven to its own catalogue rating has plenty of room for the ripple, and a part driven to twice it does not. The two limits do not coincide and neither is derivable from the other.

The refusal, and what it was protecting

The routine refuses rather than answering, and it does so because of what the first version returned.

Bracketing the search for the field depth at ten thousand amperes per metre gave an answer at every bias. Above two hundred the bisection ran to its own ceiling, drove the core five thousand amperes per metre past a tip characterised at four hundred, and came back with a negative loop area — a core generating energy.

Nothing would have caught that. Every assertion on the figure asks whether a drawn number is right; none asks whether it has the right sign, because a loop area’s sign is not a thing anybody thinks to check. It would have been a row on a table, in a plausible font, in a figure whose other seven rows were correct.

The repair is not a clamp. It is the bracket saying what it means: the depth may run to the tip of the major loop and no further, and if the requested flux is not reachable inside that bracket then what is being asked for is not a minor loop. The message names the bias and the flux the material actually gives there.

That is the same discipline as the refusal, and what it was protecting’s current mirror declining below a fifth of a volt, and an exact answer to a different question’s refusal to interpolate across an alias. A model that returns a number outside its range is worse than one that returns nothing, because the number is indistinguishable from a measurement.

Held at the flux swing volt-seconds actually set, a ripple costs 2.8× more on a bias. computed by solving, not by drawing. The energy a minor loop dissipates against the direct bias it sits on, measured twice. Held at a constant field excursion the loss falls, from 0.1479 to 0.0045 joules per cubic metre — a true measurement of the wrong thing, since what is collapsing is the flux swing as the material stiffens. Held at the ±50 millitesla a converter's volt-seconds actually fix, it rises instead, by 2.76 times across the axis, because the field needed to move that much flux has grown from 58.8 to 147.9 amperes per metre. Above 114.8 A/m the measurement refuses: the excursion the ripple needs would carry the core past the loop it was characterised on, so there is no minor loop to have an area. That is a direct-current ceiling with no volt-seconds in it at all. — Fig. 8 A fifty-millitesla ripple instead of twenty. The loss is larger everywhere and the ceiling has come in from a hundred and eighty-three amperes per metre to a hundred and fifteen, so a bigger ripple runs out of room at a lower bias.

The two effects compound, and in the same direction

A converter design meets both of these at once, and they push the same way.

Raising the direct current raises the bias, which does two things. It lowers the inductance — that is the curve the rung below drew, and on a hysteretic core it is whichever of the two branches the ripple’s own direction selects. And a lower inductance means more ripple, because the volt-seconds are fixed by the supply and the duty cycle: ΔB is unchanged but Δi is not, since the same flux excursion now costs more current.

Then the larger ripple costs more per cycle, because the ripple sits on a higher bias and the loss at constant flux swing rises with bias — by twelve times across the axis above. And a bigger ripple brings the refusal ceiling down: fifty millitesla runs out at 114.8 amperes per metre where twenty runs out at 183.1.

None of the three steps is large on its own and all three go the same way, which is the shape a design margin is worst at covering. The margin is usually taken on the inductance — pick a part with thirty per cent more than the calculation asks for — and none of the other two is in the calculation at all.

What would put them in is not more margin. It is holding the right quantity: state the flux swing the volt-seconds produce, ask the material what field that needs at the bias it will actually sit at, and read the loss off the loop that field traces. All three steps are then one measurement rather than three corrections, which is what the routine behind the figure above does.

What the four rungs add up to

The core this field started with was a reluctance and a number. What it is now:

A loop rather than a curve, so it has an area, so it dissipates — which the area a curve cannot have established and confirmed by two integrals that share no arithmetic.

An amplitude law that is a derivative rather than a constant, so a catalogue’s β is a description of the decade it was measured over, and extrapolating outside that decade overestimates in both directions because the curve is concave.

A locus with no clock in it, so the waveform correction every converter design applies is carried entirely by a frequency exponent that is itself a mixture ratio between two mechanisms.

And a state rather than a point, which is this rung: two inductances at one current, a ripple cost that depends on the bias in the opposite direction to the obvious measurement, and a direct-current ceiling that has nothing to do with saturation.

What is still missing is the thing the third rung named. The model has no rate dependence at all, so the eddy current and the excess loss are both bolted on as closed forms rather than solved. Solving the eddy current properly means a field inside the lamination rather than a coefficient — which is the same two-dimensional problem the winding field of the copper that makes it worse left open, met from the other side.

And there is a second gap, smaller and more specific. The threshold density here is uniform, which is what makes the small-signal law cubic and which is the only shape justified by anything. A real material’s density is not uniform, and the quantity most sensitive to its shape is the one this rung measures: the ratio between the two branch slopes is set by how much of the distribution is inside the backlash at any moment, which is a statement about the density’s shape rather than its width. The factor of 1.80 above is therefore this density’s number and not a material’s. What transfers is that there are two, that neither is the printed curve, and that the printed curve is above both.

Where a design meets the wrong one of the two

The consequence that matters is the second half of this essay — a twenty-millitesla ripple costing twelve times more at a hundred and seventy-five amperes per metre of bias than at none, and the question having no answer at all above a hundred and eighty-three — and it is worth saying which design decisions walk into it.

The inductance the current decides is the one that puts a converter at a bias in the first place: the volt-seconds fix the flux swing whatever the material does, so the ripple current is what moves — 514 mA by the design expression against 1 022 mA at 2.56 A of load, with the peak reaching 3.57 A where the part is at a fifth of its nameplate inductance. The bias is not a design choice; it is the load current, and the loss measured here is a function of it.

The duty cycle that costs nothing then shows what does not move the answer: 2.1006 joules per cubic metre at every duty and for a sinusoid of the same peak, to three parts in ten thousand, because a rate-independent locus depends on where the flux went and not on how fast. So a designer has a loss that is indifferent to waveform and strongly dependent on bias, which is the opposite of the way core loss is usually presented — as a function of frequency and peak flux, with the direct component nowhere in the expression.

Part 4 on magnetic loss

One argument about Magnetic loss, and one of 8 essays on it so far, each part numbered by how much of the idea it assumes. What sits either side of it: