The current inside the iron

The exponent nobody put in showed that the frequency exponent in a core-loss law is a mixture ratio: a hysteresis term rising as f, an eddy-current term rising as f², and a fitted α somewhere between them that moves when the fitting window moves.

It put the eddy term in as a closed form and said so: the α curve’s approach to 2 is an upper bound rather than a measurement. This rung makes it a measurement, and the upper bound turns out to be reached over a narrower range than the expression suggests.

It also closes the gap the duty cycle that costs nothing left. That rung found the play-operator model exactly indifferent to a waveform’s shape and named the eddy current as one of the two mechanisms that are not — so the size of a real core’s duty-cycle penalty depends on how large the eddy term is relative to the hysteresis term, which depends on the exponent this rung measures rather than on the one the closed form asserts.

What the expression assumes #

The eddy term everyone uses is

$$P = \frac{\pi^2 d^2 f^2 B^2}{6\rho}$$

for a lamination of thickness d and resistivity ρ. It is derived by supposing the flux density is uniform across the sheet — that the current the changing flux induces is too small to push the field out of the middle of the iron — computing the induced current on that basis, and integrating its dissipation.

The assumption is a frequency, and the expression does not carry one. That is the same shape of omission as every other one this collection collects, and it is worse than usual here because the term is quoted beside a hysteresis term that has its own quite separate range, in a law whose two exponents are already doing more work than they can support.

The equation it is actually a limit of #

Inside a conducting sheet the field diffuses:

$$\frac{d^2H}{dx^2} = j\omega\mu\sigma H, \qquad H\!\left(\pm\tfrac{d}{2}\right) = H_s$$

with the field at the two faces pinned by Ampère’s law and the winding. That has a solution, $H(x) = H_s\cosh(\gamma x)/\cosh(\gamma d/2)$, with $\gamma = (1+j)/\delta$ and δ the skin depth — the same skin depth the resistance that grows with frequency computes for a round conductor, and the same equation the copper that makes it worse solves for a foil in a winding window.

This is that equation solved a third time, for the core material rather than for the copper, and it is the one place in the field where the two subjects turn out to be the same mathematics. A lamination is a slab of conductor sitting in a field it is itself producing, which is exactly what a winding layer is.

There is one difference and it is a big one. In copper the quantity of interest is the current distribution and the material’s permeability is that of free space, so the skin depth at a hundred kilohertz is 0.209 millimetres. In iron the permeability is four thousand times larger and the resistivity only twenty-six times larger, and the depth goes as the square root of their ratio: 0.0169 millimetres, twelve times smaller at the same frequency. That is most of the reason a core has to be cut into sheets and a wire does not.

At fifty hertz the profile across a 0.35 millimetre sheet is flat to nine parts in ten thousand. At a kilohertz it is not: the sheet keeps 76.9 per cent of the flux the uniform assumption would put in it, and at ten kilohertz 21.5 per cent. The middle of the iron is carrying almost nothing and contributing nothing but its share of a current flowing near the faces.

Two routes, because one of them is a transcription #

The closed form above is four lines of hyperbolic trigonometry and every one of them is a place to put a factor of two in the wrong place. So it is computed twice.

The first route is the expression. The second is a finite-difference solve of the same equation: the field on a grid across the half-sheet, a Neumann condition at the centre where symmetry makes the derivative zero, a Dirichlet condition at the face, a tridiagonal complex system solved by elimination, and the loss integrated as ∫ρ|dH/dx|²/2 dx.

They share no arithmetic. One is a cosh and a tanh; the other is a linear solve and a quadrature. They agree to 7.5 × 10⁻³ per cent across six decades of frequency, and that agreement is what says the closed form was transcribed correctly.

It also caught the error it was there for. The first version of the closed form was out by exactly two — an integral of cosh(2x/δ) over the half-sheet done as δ rather than δ/2 — and the numerical route reported it as a factor of 5,714 against the classical low-frequency limit, which is a number nobody would mistake for a modelling subtlety.

And the grid caught a second thing, which is subtler #

A fixed grid of two hundred cells across the half-sheet is forty cells per skin depth at ξ = 1 and three at ξ = 32. The symptom of that was not a wrong loss — the loss stayed within a per cent — but a wrong exponent: the measured slope came out 1.460 where the equation’s is exactly 1.5, because a slope is a difference of two under-resolved numbers and the error does not cancel.

The loss is what a check would look at. The exponent is what the figure draws. The grid now follows the skin depth, at forty cells per depth everywhere, and a tridiagonal solve is linear in the count so the deepest case costs nothing.

The two exponents, both exact #

Below the frequency at which the sheet is two skin depths thick, the exponent is 2.0000. Above it, 1.5000. Both to four decimal places, and neither is fitted.

The second one has a short explanation. Once the flux is excluded from the middle, it is confined to a layer of thickness about δ at each face, and δ falls as one over the square root of the frequency. Holding the total flux constant while the layer carrying it thins means the flux density in that layer rises, and the arithmetic comes out as $f^{1.5}$.

For a 0.35 millimetre sheet at the material’s four thousand relative permeability, the crossing is at 465 hertz. So a mains transformer at fifty hertz is comfortably inside the uniform-flux range — the classical term is 0.998 of the truth — and a four-hundred-hertz aircraft transformer is not, at 0.893. At a kilohertz the classical term is 1.74 times the truth and at ten kilohertz it is forty-seven.

That last number is the one worth carrying. The classical term overstates the loss above its range, not understates it, because it assumes flux in iron that has no flux in it.

The undershoot, which the asymptote has no room for #

The exponent does not arrive at three halves monotonically. It falls from 2, undershoots to 1.4847 at ξ = 3.28, and comes back up to 1.5 from below.

That is small and it is real. The closed form’s denominator carries a cos(2ξ) beside its cosh(2ξ), and the asymptotic statement drops it — correctly, because it is bounded while the other term grows. A bounded term contributes nothing to a value at large argument and it can still contribute to a derivative, because a derivative is a difference and the growth cancels out of it.

That distinction turns up whenever a limit is taken and then differentiated in the wrong order, and this collection has met it before: the straight lines and where they are not the curve is the same complaint about asymptotes on a Bode plot, where the slopes are right in the limit and the curve between them is where every design actually sits.

Which quantity is held #

There are two ways to sweep frequency on this problem and they give different answers.

Hold the surface field constant and the loss appears to level off, because the flux the sheet carries is collapsing at the same time. That is a true measurement of a quantity nobody controls: the surface field is set by the winding current, and a winding driven at constant current is not what a transformer is.

Hold the mean flux constant — which is what the volt-seconds fix, exactly as a boundary in volt-seconds established for the material’s own limit — and the loss goes on rising, at $f^{1.5}$ rather than f². The figure holds the flux, and the surface field is whatever the material needs to produce it.

This is the third time in this ladder that the choice of held quantity has decided the answer. Two inductances at one current found a ripple’s loss falling with bias at constant field and rising at constant flux, an order of magnitude apart in the same direction as here, and for the same underlying reason: the material stiffens, and holding a field while it does is holding the input to a relationship that has changed.

What a lamination thickness buys #

The slider on the second figure is the thing a designer actually chooses, and the arithmetic is clean at low frequency and not at high.

Below the crossing, halving the thickness quarters the loss — the d² in the classical expression. Above it, the loss does not depend on the thickness at all: the flux is in a layer of thickness δ at each face and the rest of the sheet might as well not be there. So thinner laminations buy nothing above their own crossing frequency, and the crossing itself moves up as the reciprocal square of the thickness: 57 hertz for a millimetre, 465 for 0.35 millimetres, 5.7 kilohertz for a tenth.

That is why a mains transformer is laminated at a third of a millimetre and a switching converter is not laminated at all. A ferrite has a resistivity seven decades higher, so the same calculation on a ten-millimetre ferrite core puts its crossing at 12.7 megahertz — and a hundred-kilohertz converter is two decades below it with a solid core.

Where the boundary sits for the things people build #

The crossing frequency is δ = d/2 rearranged, and it is worth having as a table because the answer is different by five decades across the parts in ordinary use.

Two readings come out of it. A one-millimetre lamination at fifty hertz is already past its own crossing — ξ is 0.66 there and the classical term is 1.12 times the truth — which is most of the reason mains cores are cut thinner than a millimetre, and the reason is usually given as “eddy losses” without the observation that the expression predicting them has stopped applying.

And a ferrite needs no lamination at all below about a megahertz, which is why they are moulded solid. That is not because ferrites have no eddy currents; it is because seven decades of resistivity move the crossing seven half-decades up, and the geometry that would otherwise have to be cut into sheets is inside the range where the classical term still holds and is small.

There is a design consequence in that pair of readings which is easy to state backwards. Thinner laminations do not reduce loss by moving the crossing up; they reduce loss because below the crossing the loss goes as the square of the thickness. Moving the crossing up is what makes that reduction available. Cut a sheet in half above its own crossing and nothing happens at all, because the flux was never in the middle of it — so the same change that is worth a factor of four at fifty hertz is worth nothing at ten kilohertz, and a data sheet quoting a loss per kilogram at one frequency carries no information about the other.

What this leaves #

The eddy term now has a range and it is a computable one, so the exponent nobody put in’s frequency figure can be read more carefully than it was drawn: the α it plots rises towards 2 only while the sheet is thin, and above that it is heading for a mixture of 1 and 1.5 rather than 1 and 2. The ceiling on α is not 2 at all frequencies.

Two things are still asserted rather than solved. The excess or anomalous loss — domain walls moving in bursts rather than smoothly — is a $(fB)^{1.5}$ term with a coefficient and no mechanism here, and it is the same power as the skin-limited eddy term, which means a fit cannot separate them at all above the crossing. That is a real ambiguity in the standard three-term decomposition and it is worth knowing it exists.

There is a third, and it is the one that makes the standard decomposition uncomfortable. The three terms — hysteresis as f, eddy as f², excess as $f^{1.5}$ — are separated in practice by fitting P/f against √f and f, which is a two-parameter linear regression that assumes the three powers are distinct. Above the crossing frequency the eddy term is $f^{1.5}$, so two of the three basis functions coincide and the regression is singular in exactly the way the matrix that is ill and the answer that is not describes: the fit still returns numbers, the residual is still small, and the split between the two coefficients is decided by rounding.

That does not make the total wrong. It makes the attribution wrong, and an attribution is what a designer uses when deciding whether to buy thinner laminations or a different alloy.

And the permeability in the skin depth is the anhysteretic one. It should be the incremental permeability on the branch the material is standing on, which two inductances at one current measured to be two different numbers 1.80 apart. So δ is itself history-dependent, by up to a factor of 1.34 in the square root, and nothing here accounts for it.

Two exponents on one loss, and which one moves #

This ladder has now measured both exponents in the expression a catalogue prints, and it is worth setting them side by side because they fail in completely different ways.

The exponent nobody put in finds the frequency exponent exactly one — a theorem rather than a fit, because a rate-independent locus has the same area however fast it is traced — and the flux exponent a local slope running from 2.94 at half a millitesla to 1.46 near saturation, so five windows on one measured curve give β from 1.58 to 2.84 and predictions three times apart at a hundred and fifty millitesla.

This essay is where the frequency exponent stops being one, and the reason is that the eddy term is not rate-independent: exactly 2 below the frequency at which the sheet is two skin depths thick, exactly 1.5 above it, undershooting to 1.485 on the way. For a 0.35 mm sheet the crossing is 465 hertz, so at a kilohertz the classical expression is 1.74 times the truth and at ten kilohertz forty-seven times.

Which is the useful decomposition. A catalogue’s single power law is a fit to a sum of two mechanisms with different exponents in both variables, one of which is exactly one in frequency and locally variable in flux, the other of which is exactly two in flux and moves from two to one and a half in frequency. A fitted pair of exponents is therefore a weighted average of four, valid over the range the fit was made in and nowhere else.

Which is why the crossing frequency is the number worth printing beside a core-loss curve. It is computable from two quantities already on the data sheet — a resistivity and a lamination thickness — it separates a regime where the frequency exponent is one from one where it is two and then one and a half, and it is the single piece of information that would tell a reader which of the fitted exponents their own operating point is inside.