The area a curve cannot have

The inductance the current decides gave this collection’s core a curve instead of a wall. The material’s B(H) bends over the whole way, the flux per amp falls continuously, and there is no current at which the part stops being an inductor. It is the best magnetic model here, and it has a property nobody looked at.

It cannot get warm.

A single-valued B(H) is a function, and the energy a material takes from its winding over a closed cycle is ∮H dB around whatever path it traced. On a function that path is the same set of points going up and coming back, so the integral is zero exactly — not small, not approximately zero. Every core in this collection has been lossless since the day the field was written, and every core-loss figure it has drawn got its watts from somewhere else: a winding resistance in the resistance that grows with frequency, a skin depth in the copper that makes it worse, a switching event in the half that never arrives. The material has contributed nothing.

That is not a missing coefficient. A loss per cycle is the area of a loop, and a curve has no area, so nothing short of a second branch produces one.

It is worth being precise about how large an omission that is. The curve above is a good model. It predicts the collapse of inductance under bias, the three different saturation currents a catalogue might quote for one part, and a ripple current twice what the design expression gives. All of that is right. What it cannot do is answer how hot does it get, and for a magnetic component that is usually the question that sets its size.

The smallest object with two branches #

A branch is a memory. To have two of them a model needs a variable that depends on where the field has been and not only on where it is, and the smallest such thing is the play operator: a scalar that holds still until the field moves more than a threshold away from it, and then follows at that distance.

$$p \leftarrow \max(H - r,\ \min(H + r,\ p))$$

That is the whole of it — one line, one parameter, and no time in it anywhere. A play operator is rate-independent: it depends on the sequence of values the field takes and not at all on how fast it takes them, which is the defining property of hysteresis as distinct from a lag. A first-order lag also produces two branches on a cycle, and they close up when the drive slows down; a hysteresis loop does not.

One operator on its own is a parallelogram, which is not a hysteresis loop and looks nothing like one. A superposition of them, with thresholds spread over a density, is: the tips close smoothly rather than in a corner, because the operators stop moving at different fields, and a drive that reverses part-way traces a minor loop that returns exactly to the branch it left.

Pinning it to the core that was already here #

The superposition is put inside this site’s own material curve rather than beside it:

$$B(H) = \mu_0 H + f\,g(H) + (1 - f)\sum_k w_k\, g\big(p_k(H)\big), \qquad g(x) = B_\text{sat}\tanh(x/H_k)$$

with the weights summing to one. The vacuum term carries no hysteresis, because free space has none. The magnetisation term is the same tanh that the inductance the current decides uses, with the same saturation flux and the same knee field, evaluated at each operator’s lagged field instead of at the field itself.

Set every threshold to zero and every play operator becomes the identity. Both terms collapse to g(H), the sum collapses to $\mu_0 H + B_\text{sat}\tanh(H/H_k)$, and the object is exactly the core the two rungs below measured. The figure’s zero-coercivity slider position is not an approximation to that statement: the loop area there is 9.8 × 10⁻¹⁵ joules per cubic metre against 12.481 at the default, which is the floating-point noise of a trapezoid on a closed path.

That is what makes this the same part rather than a second one. No number in a boundary in volt-seconds or the flux that walks is invalidated by the second branch arriving; those rungs are the zero end of a slider that now runs the other way.

The fraction that follows, and what its absence looked like #

The split between a reversible fraction f and an irreversible remainder is the one part of the model that is not forced by the pinning, and the first version did without it. Every operator had a threshold, so nothing at all moved for a field smaller than the smallest one, and the initial permeability of the whole material was μ₀. It was a vacuum below about an ampere per metre.

The symptom was not a permeability anybody looked at. It was that the loss came out proportional to $B^{1.5}$ at small amplitude where the closed form says $B^3$, because down there B was rising as the square of the field rather than linearly. A wrong exponent two decades below anything a figure drew was the only visible trace of a material with the permeability of air — which is the shape of defect this collection keeps meeting, and the third time the symptom has been an absence rather than a wrong value.

With f = 0.55 the initial slope is $\mu_0 + f\,B_\text{sat}/H_k$, or 1,100 μ₀ against the anhysteretic curve’s 2,000. That gap is itself worth having: a hysteretic material’s initial permeability is below the slope of the curve through the middle of its loop, and by nearly a factor of two here, because half its magnetisation is waiting for a threshold.

What comes out that was not put in #

Nothing below takes a coercivity, a remanence or a loss coefficient as an input. The model takes a threshold density; the loop is marched; and the numbers are read off it.

The coercivity is where the descending branch crosses zero flux, located by interpolating between the two samples that straddle it. At a threshold spread of 20 amperes per metre it is 8.958 — not 20, and not 10 either. The reversible half of the magnetisation follows the field down and drags the crossing inwards, so the measured coercivity is a little under half the spread and the ratio itself depends on how hard the loop was driven.

The remanence is what the same branch holds at zero field: 22.30 millitesla, against a tip of 347.9. That is a squareness of 0.064, which is low even for a soft material, and it is the reversible fraction again — more than half the magnetisation goes home when the field does.

The loss is ∮H dB by trapezoid around the marched cycle: 12.481 joules per cubic metre per cycle. On this six-cubic-centimetre core at a tenth of a tesla and a hundred kilohertz that is 1.26 watts, which is the right order for a part of this size and is the first watt of core loss this collection has computed from a material.

The area is signed and the sign is kept rather than removed with an absolute value. A loop traced the other way round would be a core generating energy, and returning its magnitude would hide it.

Two integrals, one energy #

The area is a statement about the material’s own variables, and it knows nothing about the winding or the gap. There is an entirely different integral over the same cycle: the electrical energy ∮i dλ that the terminals of a sixty-turn winding deliver, with the current found from Ampère’s law around the whole path — core plus gap — and the flux linkage from the turns and the area.

The two share no arithmetic. One is in joules per cubic metre and integrates a field against a flux density; the other is in joules and integrates a current against a flux linkage; the gap and the turns appear in the second and in neither the first nor anything it is computed from.

They agree to ten parts in a thousand million million.

They agree because the gap is single-valued. Air stores energy on the way up and returns all of it on the way down, so it contributes exactly zero to a closed loop, whatever its length. That is worth having as a check rather than as a remark, because it settles a thing that is widely believed: a gap does not reduce core loss.

What the gap does is change the current. Half a millimetre in this path takes the inductance from 15.08 millihenries to 0.854 and the peak current for the same flux from 400 milliamperes to 2,707 — a factor of 6.8 — while the loss per cycle stays at 12.481 to fourteen figures. The energy is in the gap computed where a gapped core’s stored energy sits; this is the companion statement about the dissipated part, and the two go opposite ways. The gap takes almost all of the storage and none of the loss.

The state that is not a point on either branch #

A core is not described by its field. It is described by the state of every operator in it, and there are configurations of those that no single (H, B) pair identifies.

The virgin curve is the clearest case. A core that has never been anywhere starts at the origin, which is on neither branch, and climbs a curve that lies strictly between the two: at 200 amperes per metre it is at 0.3073 tesla where the descending branch is higher and the ascending one lower. It reaches the tip of the major loop at 400 A/m and after that it is gone. There is no operation that returns a core to it, and the nearest thing — a slowly decaying alternating field — leaves 0.170 millitesla behind rather than zero.

Every textbook picture of a B–H curve is this one, and no working core has been on it since the first time it was energised. That is not a pedantic point. It is why an inductance measured on a part straight out of its bag differs from the same measurement after the part has been used, and why the measurement after is the one that matters.

A minor loop that returns exactly #

The property that separates this model from a filter is what happens when the drive reverses part-way.

Five minor loops of 60 A/m at a bias of 100 leave the core at exactly the flux it was at when the first one began. The drift is 0 — the literal floating-point zero, not a residue below a tolerance — because each operator’s state is a maximum and a minimum of things it has already seen, and re-tracing a path it has already traced cannot move it.

That is the wiping-out property. It falls out of the play operators rather than being imposed on them, and it is the reason this model can be used for a converter’s ripple at all: a design that swings a small excursion on a large bias, ten thousand times a second, for years, does not creep.

A first-order lag put in the same place would creep, and the accumulated error would look exactly like a slow drift in the operating point — which is a real failure mode of magnetic models written without this property and is the reason it is asserted rather than assumed.

The minor loop’s area is 0.2371 joules per cubic metre, against 12.481 for the major loop it sits inside. A ripple costs about a fiftieth of a full swing, and what decides that ratio is the subject of the fourth rung of this ladder.

A rectangle that is nearly right, and where it stops being #

There is an old rule of thumb for a loop’s area: the rectangle through its coercivity and its tip, $4 H_c B_\text{max}$, times a “loop factor” near one. It is worth measuring against, because it is cheap and because it is the kind of rule that is quoted without its range.

At the default drive the factor is 1.0011 — right to a part in a thousand. It is also 0.846 at a quarter of the drive, and 1.060 at a spread of eighty amperes per metre. So the factor is a statement about how fully driven the loop is at least as much as about the material: a loop taken well past its knee is a rectangle to within a per cent, a lightly driven one is a lens and the rectangle overstates it by a sixth.

The panel on the first figure carries the ratio at every slider position, which is the form this collection prefers for a rule of thumb — not “it is about one” but the number, beside the drive it was measured at.

There is a second reason to keep it. The rectangle is the loop a square material would have — one that jumps from $-B_\text{max}$ to $+B_\text{max}$ at $\pm H_c$ and does nothing in between — so a loop factor of one says this material’s departures from that shape cancel, not that it has none. The squareness measured above is 0.064, which is about as far from square as a material gets, and the factor is still 1.0011. Two quite different loops can have the same area, and an area is all the loss is.

What the operator count is worth #

The threshold density is a continuum and the sum is finite, so the operator count is a numerical parameter and not a property of any material. Twenty-four of them put the loop area within 0.036 per cent of what forty-eight give; six give 0.57 per cent, and six is what a first version used.

There is a second consequence of the finite sum and it is not a small one. The smallest threshold in an n-cell midpoint rule is $r_\text{max}/2n$, and a drive that never exceeds it moves no operator at all. Below 1.15 millitesla, at twenty-four operators, this model is exactly lossless — not approximately, exactly, because nothing has moved. A continuum density has thresholds arbitrarily close to zero and no such floor, so that is the sum showing through.

It is stated rather than cropped away, and every loss measurement in the next rung starts well above it. The floor is drawn on that rung’s own figure as a rule with the words no operator moves below here beside it, and each doubling of the count halves it, which is what identifies it as arithmetic rather than physics. A discretisation that silently produces zero where the answer is small is the same class of defect as the tolerance that can only take away’s one-sided sampling: an artefact whose signature is an absence, which is what no assertion about a drawn number can catch.

Where this leaves the field #

The core now has the property every real one has and none of the models here did. Three things follow and each is a rung of its own.

The loss is an area, and an area has an amplitude law. What that law is — and whether the exponent in it is a constant, which is what every catalogue coefficient assumes — is the second rung.

The loop is rate-independent by construction, so its area is the same however fast it is traced and the power is exactly linear in frequency. Real cores are not, and the size of the disagreement is a measurement of what this model leaves out. That is the third.

The state is now an array rather than a point, so two cores at the same field and the same flux can behave differently on the next excursion. Nothing in this collection’s netlist machinery has carried a state like that before: an inductor’s current and a capacitor’s voltage are one number each, and a marched solve advances them. A core carries twenty-four, and the wiping-out property is what makes that manageable — the state is compressible, because operators that have been overwritten by a larger excursion no longer matter.

And the incremental inductance is now two numbers rather than one, because an operator held inside its own backlash contributes nothing to dB/dH, and which operators are moving depends on which way the excitation goes. That is the fourth, and it changes what a bridge reading on a biased inductor means.