The optimum a spectrum moves

The copper that makes it worse found the best thickness for a foil winding and the number that comes with it: past a certain thickness more copper is more resistance, the optimum sits at 0.663 skin depths for four layers, and the alternating-current resistance there is four thirds of what the same copper would have at direct current — the same four thirds at every layer count above one.

The optimum that does not move then pushed on it from another direction, solving the window as a two-dimensional field rather than as a stack of slabs, and found the constant almost immovable: 1.340, 1.351, 1.406 as the window emptied, while the resistance ratio those numbers are made of moved by eighty per cent. A magnitude was badly wrong and a balance was nearly right, which is the strongest thing that collection of essays has to say about closed forms.

Both of those results are for a winding carrying one sinusoid. A transformer in a converter does not carry one.

What the weighting is, and why it needed no new machinery #

The arithmetic is not the difficult part and it is worth writing down because its simplicity is the point. A current with harmonics Iₙ at frequencies nf dissipates Σ|Iₙ|²·Rac(nf) in a resistance that depends on frequency, and Rac at each harmonic is the same closed form the rung below uses, evaluated at nf instead of at f. The direct-current resistance is the same for all of them. So the whole optimisation is one sum inside the same minimisation that was already there.

This site’s own record of what it had not done said exactly that — that the machinery would take it without change and that the sweep would be a few times more expensive. Both were true. What the record could not say is what would come out, and three things did.

Where a square current’s loss actually is #

The reason the optimum moves so far is not that the harmonics add a little to the loss. It is that they are the loss.

A square wave’s harmonic amplitudes fall as 1/n, so their powers fall as 1/n² — which sounds like a rapidly convergent situation and is not, because the resistance those powers are multiplied by rises. Every harmonic sees the same copper as more skin depths than the last: at the ninth harmonic the foil is three times thicker in the units that matter, and the proximity term grows with it. The product falls as roughly $n^{-3/2}$, which converges, slowly, and leaves a long tail carrying real loss.

That is also why the optimum moves in the direction it does. Thicker copper is punished hardest at the high harmonics, so a current that keeps a large fraction of its loss up there wants thinner copper than one that does not — and the effect compounds with layer count, because the proximity term that punishes thickness is the term that grows as the square of the layer number. Measured across the range: a single layer’s optimum barely moves at all, two layers move by 1.25, four by 1.73, and eight by 2.45.

Why a single layer feels none of it #

The layer-count figures above are worth one more sentence, because the case where the effect vanishes explains the case where it does not.

At one layer the two optima are 1.5708 and 1.5568 skin depths — a difference of nine parts in a thousand, which is nothing. At two layers the gap is 1.253, at three 1.506, at four 1.728, at six 2.109 and at eight 2.451, monotone the whole way.

The reason is that a single layer has no proximity term at all. Its inner face sees no field, its outer face sees only its own, and its resistance ratio comes entirely from the skin term — which grows as ξ for thick copper and as 1 + ξ⁴/45 for thin. That is a weak enough dependence that pushing loss into the harmonics barely changes where the minimum sits. Add layers and the proximity term arrives carrying (m²−1), and it is the term that punishes thickness; a harmonic that sees the foil as three times more skin depths is punished by it three times harder than the fundamental is.

So the spectrum’s effect on the optimum is not an independent correction to be applied afterwards. It is a multiplier on the same mechanism the layer count controls, and the two compound. A deep winding carrying a rectangular current is the case where both are largest, and it is also the commonest construction in a converter transformer — the resistance that grows with frequency is the rung where that resistance was first measured on this site, at one frequency, on a winding of exactly that kind.

The result that is not a result, and it decides who needs this #

The strongest thing in the family is a case where nothing happens, and it is what tells a designer whether any of the above applies to the part in front of them.

A triangular ripple — the current in a buck inductor, a flyback’s magnetising current, anything whose waveform is the integral of a square voltage — has harmonic amplitudes falling as 1/n² rather than 1/n. Its powers fall as 1/n⁴, the rising resistance cannot catch that, and the sum is over almost as quickly as a single sinusoid’s. The optimum lands at 0.6432 skin depths against the sinusoid’s 0.6631: three per cent, which is nothing, and the ratio at it is 1.3461 against 1.3368, which is less.

So the answer is not “converters need a harmonic-weighted optimum”. It is that the waveform’s harmonic roll-off decides it, and there are two families of winding in an ordinary converter that fall on opposite sides of the line. An inductor carrying a triangular ripple can be designed with the single-frequency curve and lose nothing by it. A transformer carrying a rectangular current cannot, and the penalty for trying is 16.8 per cent more loss at a duty of one half — the duty cycle that costs nothing found the same split in the core’s loss, where a waveform’s shape enters through an exponent and a symmetric one costs nothing extra.

Four thirds becomes exactly two, and the axis between them is the edge #

If four thirds is a property of a sinusoid rather than of a winding, the obvious question is what replaces it. The answer is unusually clean.

Two is not a fitted number. Extrapolated from the harmonic sum at three different layer counts it comes out 1.9999, 1.9999 and 1.9999 — for two layers, four and eight — which is the same kind of statement four thirds is: a constant that does not know the geometry.

And that makes the pair of them one law rather than two. Four thirds is the limit of an infinitely slow edge and two is the limit of an infinitely fast one, and every real current is between them at a position its rise time fixes. A winding optimised without an edge rate has not been optimised badly; it has been optimised for a waveform nobody can generate, which is precisely the class of thing every model has an edge collects — except that here the boundary is not a frequency or an amplitude but the sharpness of a corner.

One sum, two answers, and only one of them can be quoted #

The last figure is about the instrument rather than the winding, and it is the one that changes how every number above should be reported.

Two numbers come out of one sum and they are not equally conditioned. The optimum thickness is settled to five figures by four hundred harmonics and never moves again. The loss ratio at that thickness is still moving in the third figure at twenty-five thousand, because the terms fall as $n^{-3/2}$ and the tail therefore goes as the inverse square root of the count — halving the residual costs four times the harmonics.

Nothing about the calculation announces which of its outputs is which. Both are computed from the same sum by the same code in the same call, both look equally converged at any single truncation, and a study that ran to four hundred harmonics and reported both would be right about one and wrong about the other. The only thing that separates them is running the truncation as a variable and looking — the same habit the digits the arithmetic did not have applies to a solve whose answer looks fine, arriving here in a sum rather than in a matrix.

And the divergence is physical rather than numerical. An ideal square edge has no rise time, so it has energy at every harmonic, so the loss integral has no natural end. Giving the current the rise time a real switch has puts the end back and the sum converges at once. The instrument is telling the truth: the quantity being asked for does not exist for the waveform it was asked about.

What a designer does with it #

The result is only useful if it collapses to something that can be done at a bench, and it does, because the three findings sort a design rather than merely correcting it.

Find the winding’s current, not the converter’s frequency. The single number that decides whether any of this applies is how fast the current’s harmonic amplitudes fall. Falling as 1/n — a rectangular current, which is what a transformer winding carries — means the correction is large. Falling as 1/n² — a triangular ripple, which is what a filter inductor carries — means it is three per cent and can be ignored. Nothing else about the circuit enters that decision.

Then use the duty, not the shape. For a rectangular current the optimum is a smooth function of duty and of nothing else, from 0.1800 skin depths at a twentieth to 0.3838 at a half for four layers. A primary conducting for a fifth of the period and a secondary conducting for four fifths are different windings on one core and want different foil, which is not a subtlety a single-frequency curve can express at all, since both see the same fundamental.

And do not quote the resistance ratio. The thickness is a design answer, converged and stable. The ratio at it is a number whose value depends on the rise time, which is usually not known to better than a factor of two — so the honest form of the result is a thickness with a tolerance on it and a loss computed afterwards from the edge rate the switch actually has. That is the same distinction the edge that is a region draws between a boundary that has a value and one that has a shape: here the optimum has a value and the loss at it has a range.

None of that requires the field solve or the harmonic sum at the bench. It requires knowing which of two waveform families the winding carries, which is available from the schematic.

What was found on the way #

Two things had to be repaired before any of the above could be measured, and both are the kind that would have shipped.

The closed form returned nothing at all for a thick foil at a high harmonic. Both of its terms are ratios of hyperbolic functions that grow like $e^{2\xi}$, and a double stops being able to hold the denominator at about 355 skin depths — past which the expression evaluates to infinity over infinity and returns a value that is not a number, silently. Asking for the 999th harmonic of a rectangular current at a thick foil reaches that easily, and it is an ordinary thing to want. The limit is written out now: the bounded trigonometric terms wash out and the ratio becomes ξ(1 + two thirds of (m²−1)), which agrees with the exact form to a part in 5×10¹⁰ at the point it takes over.

And the search has to scan before it refines. This loss curve is not single-humped: past its minimum it rises, turns over, and settles onto a plateau, because at great thickness every harmonic’s ratio has become proportional to ξ and the 1/ξ in front of the sum cancels it exactly. A golden section handed the whole range walks onto the plateau and reports it. Measured at a duty of 0.05 it returned an optimum of 5 skin depths and a loss ratio of 152 — both of them properties of the bracket rather than of the winding, and both perfectly plausible-looking numbers.

That second one is worth more than the repair. An optimiser that returns the edge of its own search is the same failure as an assertion that has stopped rejecting: it produces output, the output has the right units and the right order of magnitude, and nothing anywhere says the answer came from the apparatus rather than from the problem.

What this does not say #

It does not say the closed form is wrong. It is exact for the current it was derived for, it reproduces here to the last digit, and the harmonic-weighted answer is built entirely out of it. What has changed is that the current is now an input rather than an assumption.

It does not model the winding as a field, so everything the assumption that is a geometry says about a window that is not full applies to every number here as well. The two corrections are independent and they compose: the geometry moves the magnitude and the spectrum moves the optimum.

It does not treat the winding’s own capacitance, which at the harmonic orders that carry this loss is no longer negligible: the eleventh harmonic of a hundred-kilohertz square wave is above a megahertz, and the other half of the same window puts this window’s own resonance at 1.804. A harmonic above that frequency is not flowing where the low-frequency picture says it is, so the tail of the sum is optimistic in a way this model cannot see — which bounds how far into the harmonics the result should be trusted, and is a better reason than truncation for stopping.

And it says nothing about what the harmonics do to the core, which is a different loss with a different waveform dependence and its own exponent — the exponent nobody put in is about how badly that one is served by a fitted constant.

The number worth carrying #

Foil 1.728 times thinner for a square current than for a sinusoid at four layers, 2.45 at eight, and 3.68 for a narrow pulse; four thirds for one sinusoid and exactly two for an ideal square edge, with every real winding in between.

The habit that goes with it is about constants. Four thirds arrived in this collection as a measured number, was checked against a field solve, survived a geometry that moved everything around it by eighty per cent, and was reasonably treated as a property of windings. It is a property of sinusoids, and the way to find that out was not a better winding model but asking the same model a question with a different current in it. A constant that has only ever been evaluated under one condition has been measured once, however many times it has been confirmed.

The same sentence would have applied to the four-thirds law at any point in the last two years of this collection’s life, and the reason it went unexamined is not carelessness. It is that the constant had been confirmed by two genuinely independent routes — a closed form and a ladder solved on a netlist — and then by a third, a two-dimensional field. Three routes agreeing is normally the end of the argument. All three were asked about a sinusoid, so all three were one route, and the thing they shared was not an approximation anybody had made but a question nobody had thought to vary. That is the failure mode the assumption that is a geometry named for a geometry, arriving for an excitation, and it is worth stating in the general form: agreement between routes tests the arithmetic between them and nothing about the conditions they were all handed.