One dissipation, two exponents

The area a curve cannot have gave this collection’s magnetic material a second branch, because a single-valued B(H) gives back on the way down exactly what it took on the way up and a transformer built out of one runs cold. The object it built is a superposition of play operators — each holding its own output until the field moves more than a threshold away from it — added to the saturating curve the field already had.

That object takes two numbers. One is the threshold spread, quoted as a coercivity, which sets how far the operators lag. The other is the reversible fraction $f$: the share of the magnetisation that follows the field with no threshold at all, written into the model because without it the small-signal permeability is that of free space and the loss law comes out with the wrong exponent two decades below anything a figure drew.

Seven rungs have moved the first of those. Not one has moved the second. It has sat at 0.55 through every essay on this ladder, and the reason given for the value is the defect it was introduced to repair rather than any statement about what it does.

It turns out to do something quite different from what the spread does, and the difference is the whole of this essay: the spread decides how much loss there is, and the reversible fraction decides what exponent a measurement of that loss reports. The two are not two settings of one knob.

What is being solved #

The material is

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

with $p_k$ the play operator of threshold $r_k$ and the thresholds spread uniformly over $[0, r_{max}]$, $r_{max}$ being twice the nominal coercivity. The vacuum term carries no hysteresis because free space has none. The magnetisation term is split between a reversible share $f$ that follows the field instantly and an irreversible remainder that lags.

Two consequences follow from the form alone and both are worth stating before any number appears. The first is that at zero spread every play operator becomes the identity and the whole expression collapses to $\mu_0 H + B_{sat}\tanh(H/H_k)$ — the curve the field already had, exactly. The second is that the vacuum term and the reversible term are both single-valued, and a single-valued function of $H$ encloses no area under $\oint H\,dB$. So whatever loss the model has comes from the irreversible sum alone, and the reversible fraction can only scale it.

Both of those are checkable rather than merely true, and the first is the calibration everything else here is quoted against.

The lossless end, and the axis leading away from it #

At a spread of zero the marched loop encloses $9.8\times10^{-15}$ joules per cubic metre against 12.481 at a spread of 20 amperes per metre. That is not a small number; it is floating-point noise, a part in $1.3\times10^{15}$ of the loss at the default material. The core at that end of the axis is not an approximation to the single-valued one the earlier rungs used. It is that core, to the last bit, and the rung that built the model asserts as much.

What nobody has done is read the axis in between. The spread is a slider from a lossless material to a lossy one, and the three quantities a loop is quoted by have never been measured along it.

The coefficients are worth having on their own, and one of them is worth more than the others. The measured coercivity is 0.4499 of the spread the model was given, not equal to it and not half of it, and the number is a limit rather than a coincidence of one setting — it reads 0.449869 at spreads of a ten-thousandth, a thousandth and a hundredth of an ampere per metre and 0.449831 at a spread of one, which is four figures across four decades and five across three of them. A model that took a coercivity as an input and reported it back would have no such number in it.

Where the three lines end is a statement about the tanh rather than about the operators, and which one ends first depends on how hard the loop is driven. At 400 amperes per metre the remanence gives way well before the area; at 900 the area is still proportional to within 0.02 per cent while the remanence is 38.8 per cent low. That ordering is a property of the measurement, and it is measured and printed rather than asserted, because an assertion about it fails on one of the figure’s own drive settings.

Every loop above is checked twice, as the core the solver has to remember checks its own: $\oint H\,dB$ over the material’s variables against $\oint i\,d\lambda$ at the terminals of a winding on it — two integrals of different quantities, in different units, over different variables. At the default material they agree to two parts in $10^{15}$.

Two of the three coefficients are closed forms #

A slope measured over four decades is a good thing to have and a poor thing to stop at, because a straight line on a logarithmic pair of axes is what almost everything looks like over a short enough range. Two of these three have an arithmetic behind them, and finding it is what turns a fitted number into a checked one.

The remanence is the easier of the two and it is exact. On the way down from a drive large enough to have moved every operator, an operator of threshold $r$ arrives at $H = 0$ holding exactly $r$ — because a play operator descending from above settles at $H + r$ and $H$ is zero. So the remanence is $(1-f)\sum w_k g(r_k)$, and for thresholds small enough that $g$ is still linear that is $(1-f)(B_{sat}/H_k)$ times the mean threshold, which for a uniform density on $[0, 2H_c]$ is $H_c$ itself. The arithmetic gives 1.130408 millitesla per ampere-metre of spread and the marched loop gives 1.130408.

The interesting part of that derivation is what is missing from it: the drive. The operators arrive at their own thresholds however far above them the excursion went, so the remanence coefficient has no drive in it at all — and measured at 150, 400 and 900 amperes per metre it is 1.130408 every time, to seven figures. The area coefficient is not like that. It is 0.498906 at a drive of 150 and 0.629989 at 900, because the area is an integral over the whole excursion and the top of that excursion is 2.9 times $H_k$ up the tanh, where nothing is linear.

The coercivity yields to the same treatment with one more term. On the descending branch near zero field each operator holds $H + r$, so the total flux is $\mu_0 H + (B_{sat}/H_k)(H + (1-f)H_c)$ and it crosses zero at $H_c(1-f)(B_{sat}/H_k)/(\mu_0 + B_{sat}/H_k)$, which is 0.449775 of the spread. The marched figure reads 0.449869 at the 720 points it draws its loop with, 0.449781 at 2,880 and 0.449775 at 11,520 — so the discrepancy is the grid the crossing is interpolated on, and it converges onto the closed form rather than towards some other number. That is the difference between a coefficient and a coincidence, and it is why the figure’s own reading is quoted to four figures and not to six.

The third coefficient, the area, has no such form at this drive and is honestly a measurement.

The other parameter does nothing to the loss #

The reversible fraction is where the surprise is, and the first half of it is a negative result stated exactly.

The closed form in that caption is the threshold density integrated rather than fitted. A single play operator of threshold $r$ driven to $\pm H$ encloses $4r(H-r)$ while the envelope is still linear, and a uniform density on $[0, r_{max}]$ integrates that to $(2/3)H^3/r_{max}$. Cubic, with no free constant in it, and with the reversible fraction appearing only as the factor $(1-f)$ in front.

The factor is exact and not approximate. At a field amplitude of 5 amperes per metre the loop areas at $f = 0$, 0.55 and 0.95 stand in the ratios 20.000000000 and 2.222222222 to one another — which are $1/0.05$ and $1/0.45$ to every digit double precision has. The reversible half of the magnetisation contributes nothing whatever to the area, because it is a single-valued function of the field and a closed path over one encloses none.

So the loss mechanism is one mechanism, obeying one law, at every setting of the parameter.

There is a reading of that result which makes it sound trivial and it is worth refusing. The factor $(1-f)$ is visible in the model’s own definition, so of course the area carries it — but “of course” is doing work here that the arithmetic has not done. A superposition whose operators all sit inside the same saturating envelope could perfectly well have had the reversible term change where on that envelope the irreversible one operates, in which case the area would have scaled by something near $(1-f)$ and not by it. It does not, and the reason it does not is that the two terms are added rather than composed: the reversible share is a separate single-valued path, not a modification of the operators’ own. Twelve identical decimal places is what says the model has that structure and not the other one.

What the parameter does move is the magnetisation #

The mechanism is short. With $f = 0$ nothing moves at all until the field exceeds the smallest threshold; above it, each operator that has moved contributes $H - r$, and the number of operators that have moved is itself proportional to $H$. The sum of a linear thing over a linearly growing count is quadratic. With $f > 0$ the reversible term follows the field from zero, is linear in it, and dominates everything the operators do at small amplitude.

The initial permeability is the same statement read at the origin: $\mu_0 + f B_{sat}/H_k$, which is $\mu_0$ exactly at $f = 0$ — a material with the permeability of air — and 1100 times $\mu_0$ at the 0.55 the ladder has been using.

That a material could have no reversible magnetisation is an idealisation of the same kind as the infinitely permeable iron in the assumption that is a geometry, and it is being used the same way: as the end of an axis that makes the axis legible, rather than as a claim about a ferrite.

Composing the two #

A catalogue does not print loss against field. It prints loss against flux, because flux is what a winding’s volt-seconds fix and field is not — which is a boundary in volt-seconds seen from the material’s side. So the exponent a data sheet reports is the cubic above composed with the magnetisation law, and there are two magnetisation laws.

The composition is arithmetic that can be done on the back of the caption. If $W \propto H^3$ and $B \propto H^n$ then $W \propto B^{3/n}$, so a linear magnetisation gives three and a quadratic one gives three halves. Nothing else enters. The measured pair is 2.9860 and 1.5042 against the predicted 3 and 1.5, and the shortfall in each is the same shortfall the magnetisation figure already accounted for — a share of the flux at those amplitudes is vacuum rather than material, and vacuum is linear whatever the operators are doing.

That is the finding in one picture. A single dissipation mechanism, unchanged and provably unchanged, reports two Steinmetz exponents a whole unit and a half apart, and which one it reports is decided by the half of the magnetisation that never dissipates at all.

The corollary matters more than the number. β is not a property of the loss. It is a property of the loss divided by a property of the magnetisation, and a measurement of it cannot separate the two. The exponent nobody put in established that β is a local slope rather than a constant and that two data sheets for one material disagree because they fitted over different decades; this adds that two materials with identical dissipation and different magnetisation curves disagree at the same decade.

The axis, and why nothing noticed it #

The shape of that curve explains the silence. Above about a half the exponent is within a few per cent of three whatever the fraction is, so every rung of this ladder has been sitting on the flat part of a sensitivity curve and reporting numbers that would have been almost the same at 0.7 or at 0.9. A parameter with no visible effect is a parameter nobody has to justify, which is exactly how it survived seven essays without one.

It also means the inverse problem has no solution in that region. A bench measurement of β near three constrains the reversible fraction to somewhere above a half and no further, so the parameter cannot be recovered from the quantity it decides. That is the same shape of finding as the distribution the bench cannot see, which matched four threshold densities to one coercivity and found the major loop unable to tell them apart while the small signal told them apart completely — and it points the other way. There, the small signal was the discriminating measurement. Here, the small signal saturates against this parameter while the model’s own internals are still moving.

What it does not say #

It does not say that Rayleigh’s cube is an artefact. Soft ferrites do have a large reversible magnetisation, the cube is what they show, and the duty cycle that costs nothing and two inductances at one current are unaffected: neither depends on the exponent, and both were measured at a fraction on the flat.

It does not say the three-halves law belongs to any real material either. A material with literally no reversible magnetisation is the end of an axis, and what the axis is for is to show which parameter carries which behaviour — the habit every model has an edge collects, applied to a parameter rather than to a frequency.

Nor does it license reading a measured β backwards into a claim about how much of a material is pinned. The exponent is flat against the fraction over most of the range and the two amplitudes it is read at disagree by up to a tenth over the same range, so the ordinary amplitude dependence — the subject of the second rung — is comparable with the whole signal this parameter produces above $f = 0.3$. Anything inferred about the fraction from a single measured exponent is inferred from a difference smaller than the systematic effect sitting on top of it.

The one operational consequence is narrower and it is about model fitting rather than about materials. A hysteresis model fitted to a measured loop is normally matched on the quantities the loop shows: coercivity, remanence, area, and the shape between them. Three of those four are set by the spread and the fourth by the drive, and none of them constrains the reversible fraction at all — so a fit that matches a major loop perfectly can still be wrong by a whole unit and a half about the small-signal exponent, which is the regime a converter’s ripple actually sits in.

And it does not overturn the limit at the origin.

Two boundaries on the arithmetic are worth naming, since both were found rather than assumed. Below about one ampere per metre at 192 operators the discrete sum sits 1.7 per cent above the integral it approximates, and at 768 operators it sits 0.1 per cent above — which is what identifies that end as the sum. At 24 amperes per metre — a sixth of $H_k = 139.3$ — the departure is 0.3 per cent at both counts, which identifies the other end as the material rather than the arithmetic. Neither is visible in a curve that has not been drawn at two discretisations, and the habit of drawing it at two is what the digits the arithmetic did not have is about.

What it opens #

The eddy-current mechanism of the current inside the iron also produces a flux exponent, and it produces it by an entirely different composition — a diffusion whose frequency exponent runs from 2 to 1.5 across the thickness at which the sheet is two skin depths. Two mechanisms, two compositions, and a bench that measures their sum. Whether the pair can be separated by their amplitude dependence rather than by their frequency dependence is a question this ladder has the machinery for and has not asked.

The nearer question is the one the axis above makes obvious. The reversible fraction and the threshold density are both descriptions of the same physical thing — how much of the magnetisation is pinned and how strongly — and the model treats them as independent. A density with a large weight at very small thresholds is nearly reversible, so the two parameters overlap, and how far they overlap is measurable by matching a fraction against a shaped density on the small-signal exponent. That is a rung, not a remark.

The number worth carrying #

The loss is cubic in the field at every setting of both parameters, to twelve decimal places. The exponent a data sheet prints is 1.5042 or 2.9860 depending on a parameter that dissipates nothing.

The habit that goes with it is about what a fitted exponent is. An exponent measured on a composed quantity belongs to the composition and not to either part, and there is no amount of care in the measurement that recovers the parts — the same reason exact outside and wrong within insists on knowing which of two nested idealisations a discrepancy belongs to. A material constant that is really a ratio of two laws will sit still while both of them move, and will move when neither of them does.