The exponent nobody put in

The area a curve cannot have gave this collection’s core a second branch and therefore a loss: 12.481 joules per cubic metre per cycle, measured as the area of a marched loop and confirmed by an independent integral at the winding’s terminals.

One number is not a design tool. What a catalogue prints is a law,

$$P = k\,f^{\alpha} B^{\beta}$$

with α near 1.3 and β near 2.5, and the two exponents sitting beside the material’s name as though they belonged to it in the way the saturation flux does. Neither of them does, and the model built in the rung below is in an unusual position to say so, because it can be asked what its own exponents are without anybody fitting anything.

The frequency exponent is one, and that is a theorem #

Start with the easy half, because it is the half that is exact.

A play operator has no time in it. Its output depends on the sequence of fields it has been shown and not at all on how fast it was shown them, so a superposition of them traces the same locus in the (H, B) plane whether the cycle takes a microsecond or an hour. The area of that locus is the loss per cycle, the power is the area times the frequency, and therefore

$$\alpha = 1 \quad\text{exactly.}$$

Not approximately, not over some range: it is a property of the model’s construction and no measurement of it can return anything else. A figure that swept frequency and drew a line of slope one would be drawing an identity.

So every measured α above one is a second mechanism. The obvious one is the classical eddy current: the changing flux drives a current in the conducting material itself, that current is proportional to dB/dt, and the dissipation is proportional to its square. For a lamination of thickness d and resistivity ρ that is π²d²f²B²/6ρ per cubic metre per second — going as f², not as $f^{1.3}$.

Their sum has a local slope that is 1 where the first dominates, 2 where the second does, and every value in between where they are comparable. α is a mixture ratio being read as an exponent.

The slider makes the point better than any argument. At a ferrite’s five ohm-metres the two terms cross at ten megahertz and α is 1.0000 at a hundred hertz; at silicon steel’s 4.5 × 10⁻⁷ they cross at 631 hertz and α is already 1.256 at a hundred. Same law, same fitting procedure, same range of frequencies, and an exponent that differs by a quarter of a whole number — because the material’s resistivity decided where the crossover was relative to the decade somebody measured.

A number that moves when the measurement range moves is not a constant of the material. It is a description of the range.

There is a third term in common use — an excess or anomalous loss going as $(fB)^{1.5}$, which comes from domain walls moving in bursts rather than smoothly — and it is included in the model here for the same reason as the eddy term: to be a separate mechanism rather than a correction. A fitted α between one and two is then a mixture of three things, and reading it as a property of the material is reading a weighted average of three mechanisms’ worth of physics as one number.

The same trap catches a reader of this collection in a familiar place. The mismatch that the cable hides is a loss that looks like a match; here an exponent looks like a material. In both cases what has happened is that two independent quantities have been summarised by one, and the summary is stable only while their ratio is.

The flux exponent is a derivative #

The other half is worse, because there is no second mechanism to blame. β moves within one mechanism.

The local exponent is d ln W / d ln B, computed by central difference along the measured curve, and it is drawn on the same frame as the curve itself. It runs from 2.80 at 4.6 millitesla up to 2.88 at eight and then down to 1.46 at a hundred and fifty-four. There is no plateau anywhere on it — no amplitude at which the curve is locally a power law and stays one — and the fall is monotonic once past the first few points.

And how far it moves is itself a property of the material rather than of the law. Over the same amplitudes a soft core’s exponent moves by 1.73 and a hard one’s by 0.21: the second sits above 2.8 across the whole drawable range and rises before it falls, because with a wide threshold spread the loop’s area is dominated by the spread rather than by the envelope’s curvature, and the cubic small-signal law persists much further up.

So two materials can disagree about whether a single exponent describes them at all. For the hard one a constant β near 2.9 is a fair description over a decade and a half; for the soft one no constant is, anywhere. Nothing on either data sheet says which case a reader is in.

Two ends are worth taking separately, because they have different explanations.

The small-amplitude end is Rayleigh’s law, and it comes out in closed form #

At amplitudes below the largest threshold, only the operators with thresholds under the drive amplitude move at all. A single play hysteron of threshold r driven to ±H encloses a parallelogram of area 4r(H − r) while the envelope is still linear, so a density ρ® gives

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

and a uniform density makes that $2H^3/(3r_\text{max})$ — cubic in the amplitude, which is Rayleigh’s law, which is what soft ferrites actually do at small excitation.

That is a derivation and not a fit, and it is the reason the threshold density in the rung below is uniform. A density concentrated at one threshold gives a quadratic and a loop with corners in it.

The measured exponent approaches three from below and does not reach it, and the figure exists to keep two different limits apart. In the amplitude, the exponent rises towards three as the drive falls, because the tanh envelope’s curvature and the operators already near saturation both pull it down at any finite amplitude. In the operator count, each discretisation gives out at its own floor — the smallest threshold, below which nothing moves and the model is exactly lossless — and doubling the count halves that floor without moving the limit.

At three hundred and eighty-four operators the exponent reaches 2.943 at half a millitesla, which is Rayleigh’s three to within two per cent. At a fixed eight millitesla the counts converge instead on 2.865, because eight millitesla is not infinitesimal. A figure showing only the second convergence would suggest the material’s exponent is 2.865; a figure showing only the first would suggest the discretisation does not matter. Both are drawn.

The large-amplitude end is the material running out #

Past about seventy millitesla the exponent falls quickly: 2.32, 2.05, 1.76, 1.58, 1.46. The loop is approaching the tip of the tanh and there is less magnetisation left to reverse, so the area grows more slowly than the flux does. In the limit it would stop growing altogether and the exponent would go to zero.

That is not a small correction at the amplitudes anyone uses. A power ferrite is run at a tenth to a fifth of a tesla, and the exponent moves by a whole number across exactly that band.

It is also the amplitude range where the first rung’s rule of thumb starts to give way. The area of a loop is the rectangle through its coercivity and its tip to within a part in a thousand at full drive; a loop driven to a fifth of that is a lens and the rectangle overstates it by a sixth. Both statements are about the same thing — the loop’s shape changes with how hard it is driven — and the exponent is the derivative of that change.

What five windows on one curve give #

The practical consequence is best shown by doing what a catalogue does: fit the law over a range and print the answer.

Five windows, five exponents, from 1.578 to 2.843, and five coefficients spanning a factor of thirty-five. Each is a good fit inside its own window: the worst residual over three to ten millitesla is 0.85 per cent and over a hundred to a hundred and eighty it is 0.86. These are not bad measurements.

Extrapolated to a hundred and fifty millitesla they give 12.92, 9.26, 6.19 and 4.05 joules per cubic metre against the curve’s own 4.04. The three-to-ten-millitesla fit is out by a factor of 3.20 — and it is the fit with the lowest residual of the four.

That is the whole trap, and it has the shape this collection keeps finding. A residual measures how well a law describes the data it was fitted to. It says nothing whatever about the law’s behaviour outside them, and a power law extrapolated past its window is exactly the object whose error grows as a power.

The window that contains a hundred and fifty millitesla is right to 0.3 per cent. So the practice is defensible and the extrapolation is not, and the difference between the two is the one thing the printed coefficients do not carry — because k, α and β are printed and the range they were measured over usually is not.

The fit over everything, which is what looks most reasonable #

There is a fifth row on the figure and it is the one worth ending on: the same law fitted over the whole measured range at once. It gives β = 2.52, which is a sensible-looking number in the middle of what the four windows found, and a coefficient of 656.

Its worst residual is 72.2 per cent.

Fitting across a range the exponent moves through does not average the exponent; it produces a line through a curve, and the residual reports what that costs. Nothing about the returned β = 2.52 tells anybody that. It is the same shape as the coefficient that is about one reading’s class II ceramic — a specification honoured to six digits about a quantity nobody uses — and as the capacitance that is not one number’s measurement-dependent capacitance: a number that is exactly right about a question that was not asked.

A single fitted exponent quoted without its window is one of those, and the resemblance to every model has an edge’s central claim is exact: a model with no stated range is not a simpler model, it is one whose range the reader has to guess. The residual is the only thing that separates a description from a slogan, and it is not on the data sheet.

What the two exponents are worth to a design #

The two failures compound, and they compound in the direction that matters.

A converter’s core is chosen by working backwards: pick a flux swing, look up the loss density, multiply by the volume, check the temperature rise, and if it is too high pick a larger core and go round again. The loop is short and every pass through it uses the printed coefficients twice — once in β, at a flux the design will actually run at, and once in α, at a frequency that may not be the one the coefficients were measured at.

Take the four fits above at face value and design at a hundred and fifty millitesla. The three-to-ten-millitesla row predicts 12.92 joules per cubic metre where the material gives 4.04. On the six-cubic-centimetre core of the rung below, at a hundred kilohertz, that is 7.75 watts predicted against 2.42 delivered — and a designer who believed it would reach for a core three times too large, which is a real cost and the wrong direction.

Going the other way is worse rather than better. The hundred-to-one-hundred-and-eighty-millitesla fit extrapolated down to ten millitesla predicts 0.0564 joules per cubic metre where the curve gives 0.00588 — over by a factor of 9.6, at the amplitude a signal transformer runs at.

Both errors have the same sign, and that is worth being clear about, because the instinct is that extrapolating up and extrapolating down should err in opposite directions. They do not. The exponent falls with amplitude, so the measured curve is concave in the log–log plane, and every straight line drawn through part of a concave curve lies above it everywhere else. Whichever window a coefficient came from, extrapolating outside it overestimates the loss — by 3.2 times going up here and 9.6 times going down.

Overestimating is the safe direction for a temperature rise and the expensive one for a bill of materials, so the failure is quiet: it produces designs that work and cores that are too big, and nothing in the finished product says so.

This is the same structure as the tolerance that can only take away, where a one-sided error turned out to be worse than a symmetric one of the same size because a design margin sized for scatter does not cover a bias. Here the bias is in the extrapolation, its sign is set by the curvature rather than by which side of the window the design sits on, and it is the same sign every time.

Two routes to the same watt #

Everything above rests on the loop area being right, and the rung below gave that two independent routes: ∮H dB over the material’s own variables, and ∮i dλ at the terminals of a winding, agreeing to ten parts in a thousand million million.

That agreement is what lets the amplitude sweep be trusted at its ends. At the small-amplitude end the loop is a sliver — 5.7 × 10⁻⁴ joules per cubic metre, four orders below the major loop — and a trapezoid on a nearly-closed path is exactly the arithmetic that returns noise there. The terminal integral is a different arithmetic on a different path, and it returns the same sliver.

At the large-amplitude end the risk is the other way: the loop is fat, the drive is bisected to hit a stated peak flux, and an error in the bisection would put every point of the curve at the wrong amplitude by a per cent or two. That is not visible in the loss values and it is visible in the exponent, which is a ratio of logarithmic differences and would develop a wobble. The measured exponent is smooth to three decimal places along the whole sweep, which is the check that the amplitudes are where they are said to be.

Both routes are what the area a curve cannot have put in place, and this rung is the first thing that needed them.

What this rung does not claim #

Two limits are worth stating plainly, because both bear on the next rung.

The loss numbers are this model’s, not a material’s. The threshold density is uniform because that is what makes the small-signal law come out cubic; a real distribution is not uniform, and a real material’s exponent curve would have a different shape. What transfers is not the value of β at any amplitude but the fact that it is a derivative, which follows from the loop having a shape rather than from which shape it has.

The eddy term is a closed form and not a solve. The classical expression assumes the flux is uniform across the lamination, which is the statement that the induced current does not push the field out of the middle of the sheet — and that is only true below the frequency at which the sheet is a skin depth thick. Above it the real loss falls short of f², and this collection has the machinery to say where: it is the same skin depth the resistance that grows with frequency computes for a round conductor, applied to a slab. Nothing here does that solve, and the α curve’s approach to 2 is therefore an upper bound rather than a measurement.

And α = 1 is a property of a rate-independent model, which no core is. The eddy current above is put in by hand, as a separate closed-form term, precisely because the play superposition cannot produce one. Everything that makes a real α exceed one lives outside this model, and the size of what lives outside it is measurable — by asking the model to price a waveform that is not a sinusoid, and watching it charge nothing. That is the third rung.