The distribution the bench cannot see

Every hysteresis figure in this collection is built from a superposition of play operators over a spread of thresholds, and the spread is uniform. That choice was made once, in the area a curve cannot have, and it was defended: a uniform density makes the small-signal loss law come out cubic, which is Rayleigh’s law, which is what soft ferrites actually do at small excitation. A density concentrated at one threshold gives a quadratic and a loop with corners in it, and both faults were visible in one figure.

That is a real constraint on the shape, and it is far weaker than it sounds. The loop area a single hysteron of threshold r encloses at amplitude H is 4r(H − r) while the envelope is still linear, so

$W(H) = \int_0^H 4r(H-r)\,\rho(r)\,dr$,

and when H is small that integral only ever sees ρ near the origin. Every density that is finite and non-zero at zero threshold gives a cubic, by Taylor’s theorem and nothing else. The measurement that pins Rayleigh’s coefficient pins ρ(0) and says nothing whatever about the rest of the distribution.

So five rungs of measurement have been taken on one shape out of a family, and the question this rung exists for is which of those results are about magnetism and which are about the shape.

Matching on what a bench reads, not on what the model was handed #

The coercivity handed to the core model is nominal, and the docstring beside it has always said so: the measured value comes out of the march and is not the number put in. For a uniform density the two differ by a factor near two point two, and that factor is a property of the shape — so two densities given the same nominal number produce two different measured coercivities, and any comparison between them is a comparison of two things at once.

Each shape here is therefore bisected until the coercivity read off its own descending branch lands on nine amperes per metre. It is the same discipline two inductances at one current applies to which quantity is held, moved from a measurement to a model: match on what the bench reads.

The spreads that result run from thirty point one to seventy-six point zero amperes per metre. A factor of two and a half in the width of the distribution, arriving at one number on the instrument.

Four materials one bench cannot separate #

Drawn over one another, the four major loops lie on top of each other.

The remanence runs from twenty-two point three two to twenty-two point four five millitesla — six parts in a thousand. The loop area runs from twelve point five one four to twelve point six four four joules per cubic metre — one per cent. Both spreads are smaller than the sample-to-sample variation of a single part number, against distributions whose widths differ by a factor of two and a half.

The agreement is not uniform across the family and it is worth saying where it weakens. At a coercivity of three amperes per metre the four remanences differ by a fifth of a per cent; at thirty they differ by four and a half, and the areas by ten. A harder loop spends more of its excursion in the part of the distribution the four shapes disagree about, so the discrimination improves with coercivity — and even at the hard end it is a tenth against a factor of two and a half.

A conjecture from the rung below, checked and refused #

Two inductances at one current measured the ratio between the two branch slopes at zero bias — fourteen point nine millihenries pushing one way and eight point three the other, a factor of one point eight — and it ended by attributing the number to the density: the factor of 1.80 is this density’s number and not a material’s.

That was a conjecture, it was stated as one, and it is wrong.

Matched on the coercivity, the four distributions give ratios of one point seven eight nine to one point seven nine nine. Half a per cent, across shapes whose spreads differ by two and a half times. The factor is a property of the model class — of a rate-independent superposition of play operators driven to the tip of a major loop — and not of how the thresholds are distributed within it.

The reason is the same reason the loops agree. Which operators are moving at the tip of a loop, and which are pinned inside their own backlash, depends on the excursion the core has just made; and the excursion has been fixed by matching the coercivity. What is left for the shape to decide is a second moment of a distribution whose first moment is already pinned, and the branch ratio is not sensitive to it.

Where they do differ, and it is a whole exponent #

At flux amplitudes a thousandth of the ones the loops above were drawn at, the four separate completely.

The uniform density gives a local exponent of two point eight one — Rayleigh’s cubic law, which is what the choice was made for and which the rungs below take as read. The density proportional to the threshold gives three point eight one: a quartic. It vanishes at the origin, so the Taylor argument that produced the cubic has no leading term to work with and the next one takes over. The two-population density gives no clean power law at all over the window, because its two humps enter the integral at different amplitudes.

So the two ends of one distribution are measured by two different experiments and neither experiment sees the other’s end. The major loop pins the bulk of ρ and says nothing about ρ(0). The small-signal law pins ρ(0) and says nothing about the bulk. A material characterised at one of them is characterised on half a distribution, and the half that is missing is exactly the half the other kind of design depends on.

The one shape that has no exponent at all #

Three of the four densities have a power law over the window. The two-population one does not, and it is the most instructive of the four because it is also the most physical.

Two separated humps of thresholds — a soft population and a hard one, which is what a bonded composite has, or a core that was annealed unevenly, or any material with two distinct kinds of pinning site — enter the loop-area integral at two different amplitudes. Below the soft hump nothing moves. Between the humps the soft population is fully engaged and the hard one is untouched, so the area grows the way a single population’s would, with its own exponent. Above the hard hump both are in play and the exponent changes again.

The local exponent therefore runs up and back down across the window, and no straight line through the logarithms is a description of it. A fitted Steinmetz exponent on such a material is a number about the window rather than about the material, which is the same complaint the exponent nobody put in made about a single-population curve — sharper here, because there the curve was smoothly concave and a fit at least tracked it locally, and here the curvature changes sign.

There is a visible consequence on the major loop as well, and it is the one thing on the loops figure that distinguishes the four by eye at a hard enough coercivity: a two-population loop has a waist, a shoulder where the soft population has finished switching and the hard one has not started. It is small at the coercivity these are matched at and it is not zero, and it is the only feature on the page that a curve-tracer could have caught.

What the model refuses to guess #

The obvious next question is which of the four a real ferrite is, and the model declines to answer it, correctly.

Nothing on this site measures a threshold distribution. What it measures is a loop, and the whole finding above is that a loop does not determine one. Choosing a shape and then reporting quantities computed from it as though the shape had been established is the failure this rung exists to prevent, and it is a failure that had already half happened: six rungs of results carried a uniform density that nobody had defended since the day it was chosen, and one of them had a warning attached to it that turned out to point the wrong way.

What can be said is which measurement would settle it. The small-signal exponent separates the four by a whole number, so a loss measurement at three decades below the usual characterisation amplitude discriminates between them cleanly — and that is exactly the measurement nobody takes, because it is slow, the signal is tiny, and the answer is not on any specification. The distribution is not unmeasurable. It is unmeasured.

What a floor does to a measurement of a floor #

There is a numerical trap in the paragraph above, and it is worth stating because it is the same trap the exponent nobody put in recorded in a different form.

A superposition of a finite number of operators has a smallest threshold, and an excursion below it moves nothing at all. The loop area is then not small — it is zero, plus floating-point noise — and a local exponent taken across two such points is a ratio of two noises. With twenty-four operators the first three amplitudes tried here returned exponents of three point one, two point seven and two point three that were identical for all four densities, because all four were reporting the same nothing.

The floor scales with the threshold spread, which the matching has just made different for every shape. So a window that is above the floor for the narrow density is below it for the wide one, and a comparison drawn on a fixed window compares one measurement with one artefact.

The window here starts at four times the largest smallest-threshold on the page, and the operator count is raised until the floor is out of the way. That is not a tolerance; it is a statement about where the model has resolution, and the same class of error at the other end of the same ladder is what a discretisation floor did to a loss curve five rungs ago.

What this does and does not license #

It licenses the six rungs below. Every quantity they measured at or near a major loop — the loss, the coercivity, the remanence, the squareness, the loop factor, the branch ratio, the Steinmetz exponents and their drift — is robust to the choice of density, at the one-per-cent level over a family that spans two and a half times in width. Those results are about the model class and about magnetism, not about a choice made in one line of one file six rungs ago.

It does not license the small-signal end. Rayleigh’s coefficient, the approach to a cubic, and anything computed from a permeability at very small excitation are statements about ρ(0), and ρ(0) is the one part of the distribution nothing else on this site has constrained.

And it says something about the habit rather than about the ferrite. A model parameter chosen for one reason acquires, over six rungs, the appearance of having been measured — not because anybody claimed it was, but because everything downstream of it stopped mentioning it. The cheapest way to find out which conclusions were resting on it is to change it to three other things and see which numbers move.

Two numbers moved. Everything else stayed inside a per cent, and one of the things that stayed was a factor the rung below had explicitly warned was probably an artefact of the choice. Warning about the right thing and being wrong about it is a better outcome than not warning — but only because somebody went back and checked.

The same shape of check is available almost everywhere in this collection and is almost never made. The tolerance that is not on any part makes it for component values, the three tolerances that do nothing makes it for which values matter, and every model has an edge makes it for the models themselves. What is different here is that the parameter under test is not a component and not an approximation: it is a modelling choice, made for a stated reason, which turned out to be load-bearing for one conclusion out of eight and decorative for the other seven.

Which of this ladder’s results the finding protects #

Saying that seven conclusions out of eight survive is only useful if the seven are named, and they are the ones the rest of this ladder is built on.

The exponent nobody put in is the largest and it is entirely safe: the frequency exponent being exactly one is a theorem about a rate-independent locus rather than a fit, and the flux exponent running from 2.94 at half a millitesla to 1.46 near saturation is a local slope of a measured loop, so neither depends on the density of thresholds behind it. The duty cycle that costs nothing is safe for the same reason — 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.

Two inductances at one current and the core the solver has to remember both depend on the existence of backlash rather than on its distribution — an operator sitting inside its own backlash contributes nothing to $dB/dH$ whatever the spread of thresholds is — so the factor of 1.80 and the twenty-four extra state variables stand.

What does not stand is the small-signal exponent, and it is worth being blunt about how badly: 2.81 to 3.81 across four distributions matched on the same coercivity. Any argument in this collection that quoted a cubic small-signal law as a property of the material was quoting a property of the uniform density, and this essay is where that is withdrawn.

What would distinguish the four #

Four distributions agreeing on everything a major loop can show and disagreeing by a whole exponent on the small signal is a strong statement about what a bench cannot see, and it invites the obvious question: what would see it?

The measurement that would is a small-signal one at several bias points, which is exactly what two inductances at one current makes: two branch slopes at one bias, 14.90 millihenries pushed downward and 8.30 pushed upward, a factor of 1.80 — and its own closing note says the ratio between them 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. So the pair of essays brackets the problem: the shape is invisible in a major loop, visible in a small-signal exponent, and visible again in a branch-slope ratio.

Which makes the practical instruction a short one. A core characterised by its major loop — coercivity, remanence, loop area, the three numbers a bench produces — is characterised well enough for the duty cycle that costs nothing’s result and for the exponent nobody put in’s, and not well enough for anything about small excitations on a bias. That second class is where a converter’s ripple loss lives, which is why the gap matters rather than being a modelling curiosity. A core chosen on a catalogue’s major loop and used at a bias with a small ripple on it is being used in the one regime the catalogue’s own measurement cannot constrain.