Two windings, and the band between them

The optimum that does not move

A foil winding has a best thickness — past it, more copper is more resistance — and at that thickness the alternating-current resistance is four thirds of the direct-current resistance, whatever the layer count. Both of those are one-dimensional results, and this ladder has spent three rungs finding that the one-dimensional picture is 82 per cent wrong about the resistance ratio. Solved as a field, the optimum drifts by 8.3 per cent between a full window and a quarter-full one, and four thirds becomes 1.34, 1.35, 1.41. The trade barely moves while everything it is made of moves a great deal.

15 min read 8 figures Measured, not tabulated Two routes to a number Every model has an edge

Assumes: The resistance that grows with frequency · The assumption that is a geometry

The copper that makes it worse found the strangest result in this field: a foil winding has a thickness past which more copper is more resistance. Below it, adding metal lowers the direct-current resistance faster than the proximity effect raises the alternating-current one; above it, the trade reverses. And at the optimum the ratio between the two is 4/3 — the same number for every layer count above one, which is the kind of coincidence that usually means something.

Three rungs of the assumption that is a geometry have since found that the one-dimensional model those results came from is wrong about this winding by 82 per cent in the resistance ratio, wrong in both directions per layer, and wrong by a factor of six once a gap is nearby.

So the obvious question is what is left of the optimum — and it is the same question the wire that is not a foil asked of the porosity factor, put to the one result of the family that a designer actually acts on.

Alternating-current resistance against foil thickness, 4 layers. computed by solving, not by drawing. The falling dashed curve is the direct-current resistance, which is what more copper buys. The solid curve is the alternating-current resistance at 100 kHz for a portion of 4 layers, and it turns over: past ξ = 0.663 skin depths, thicker foil has MORE resistance, not less. The minimum sits at 1.3368 times the direct-current resistance of the same foil, which is four thirds and is the same number for every layer count above one. The resistance per turn there is 2.016 against √m = 2.000, which is the law the layer count obeys. — Fig. 1 The result being tested, from the rung below. A four-layer portion at 100 kHz has a minimum alternating-current resistance at 0.663 skin depths of foil, and there the ratio is 1.3368 — four thirds. Both numbers come from Dowell’s closed form, whose derivation assumes the field runs along the layers everywhere.

Solving for it rather than differentiating it

The one-dimensional route to the optimum is a differentiation: write F(ξ, m)/ξ, set the derivative to zero, and use the small-ξ expansions to get ξ⁴ = 1/(m² − m). It is exact in the limit of many layers and 0.4 per cent out by the fourth.

The field has no closed form to differentiate, so the optimum is swept and bracketed: eight thicknesses from a third of a skin depth to 1.7, the loss computed for each by a two-dimensional solve, and the minimum located by a parabola through the three lowest points in log ξ. That last step matters more than it looks — a grid of eight can only say which of eight is least, and the whole question is where the minimum is rather than which sample is nearest it.

The one number the window does not move. computed by solving, not by drawing. Loss against foil thickness, at three window fills, with each curve's minimum located by a parabola through its three lowest points rather than read off the grid. The optimum sits at 0.654 skin depths at full fill, 0.708 at forty per cent, against the closed form's 0.663 — a drift of 8.3 per cent while the ratio the same winding carries moves by eighty. The alternating-current resistance at the optimum is 1.340, 1.351, 1.406, against four thirds. What did move is the loss it costs: 0.0555 watts a metre at full fill and 0.1383 at forty per cent, for the same current in the same number of layers. — Fig. 2 Three curves, three window fills, each minimum bracketed. At full fill the optimum is at 0.654 skin depths against the closed form’s 0.663 — 1.4 per cent, which is the grid’s own error. At forty per cent fill it is 0.708. The whole drift is 8.3 per cent.

The optimum drifts thicker as the window empties, which is the direction the mechanism predicts: a winding that fills less of its window has flux escaping round its ends, so the proximity term — the one that penalises thickness — is weaker, and thicker foil is affordable.

And the ratio at the optimum drifts from 1.340 to 1.406. Four thirds is 1.333, so the closed form’s value survives at full fill to a quarter of a per cent and is 5 per cent low at forty per cent fill.

Where the four thirds comes from

It is worth deriving, because the derivation is what says whether the number should have survived.

Write the portion’s alternating-current resistance as Rac = Rdc·(S + P), where S is the skin term and P the proximity term, both functions of ξ = h/δ. For small ξ the two behave very differently: S → 1 + O(ξ⁴), so a thin foil has no skin effect at all, while P → (2/3)(m² − 1)·ξ⁴/6 in the same limit. And Rdc itself goes as 1/ξ, because a thicker foil is more copper.

So the loss at fixed current goes as (1 + cξ⁴)/ξ, and setting its derivative to zero gives 3cξ⁴ = 1 — which is to say the proximity term is one third of the direct-current term at the optimum, so the total is four thirds of it. The layer count and the frequency are both inside c and both cancel out of the ratio, which is why the number is the same for every count above one.

That derivation makes the survival of the result almost inevitable. The window’s correction is a factor on the proximity term, so it is a factor on c — and c does not appear in the four thirds at all. It appears only in where ξ⁴ = 1/3c puts the optimum, and a factor on c is a fourth root of that factor on ξ. A 45 per cent change in the proximity term is an 8 per cent change in the thickness, which is exactly the 8.3 measured, arrived at from the other end.

Why a ratio survives what its parts do not

Both of those numbers are small, and beside the rest of this ladder they are startlingly small. The same three fills move the resistance ratio from 16.20 to 9.00 — 45 per cent — and the loss from 0.2322 to 0.5278 watts a metre.

Fig. 3 What the same geometry does to the quantity everybody quotes. The closed form is flat because it has no window in it; the field falls by 45 per cent while the loss more than doubles. That is the comparison that makes 8.3 per cent worth noticing.

The reason is that an optimum is a balance between two terms, and the window’s correction acts on both of them in nearly the same proportion.

Write the loss as a sum: a skin term that is a slab in its own field, and a proximity term proportional to the external magnetomotive force squared. Shortening the winding lowers the field that reaches the layers, so it scales the proximity term down. The skin term is untouched — a conductor’s own field does not care what the window looks like. So the balance point moves, but only by however much the ratio of the two terms moved, and that is a much smaller number than either term’s own change.

The four-thirds law has the same structure. It says that at the optimum the proximity term is exactly one third of the total, which is a statement about the shape of the trade rather than about its magnitude — and a shape survives a scaling.

What each layer really costs at 70% window fill. computed by solving, not by drawing. The pale bars are the per-layer form of Dowell's expression, in which layer p carries a proximity term proportional to 3p² − 3p; the solid bars are what the field gives for the same copper in a window it does not fill. The closed form does not merely run high: it redistributes. The first layer comes out 10.4 per cent WORSE than it predicts, because in one dimension that layer's inner face sees no field at all and in two it sees the flux that curled round the end of the winding; every layer behind it comes out 24 to 20 per cent better. At full fill the same comparison agrees to 0.62 per cent. — Fig. 4 The redistribution the window does inside the portion, at seventy per cent fill. The first layer is above the closed form and the rest are below it. The optimum is computed from the portion’s total, so these partly cancel — which is a second reason the balance point is more robust than any of the numbers it is computed from.

What is worth taking from a robust optimum

The design rule survives. A foil winding should be about two thirds of a skin depth thick per layer for a four-layer portion — the skin depth being the resistance that grows with frequency’s own length — and ∜3/√m thick in general, and nothing in this ladder changes that. A designer using the closed form to choose a foil gauge is using it for the one thing it is reliable about.

The number it promises does not. “Rac/Rdc = 4/3 at the optimum” is a promise about a ratio, and a ratio has a denominator. The direct-current resistance of a winding that fills 40 per cent of its window is two and a half times that of one that fills it, so the loss at the optimum goes from 0.0555 watts a metre to 0.1383 — and the ratio stayed near four thirds the whole way.

That is the same trap the rung below found in a different quantity, and it is worth stating as a habit: a dimensionless figure of merit can be excellent while the thing it is a ratio of is terrible, and every design that specifies Rac/Rdc without specifying Rdc has left the door open for it. It is the same reading error every model has an edge is written against, applied to a figure of merit rather than to a model.

The best foil thickness and what it costs, against the layer count. computed by solving, not by drawing. Every layer count is given the thickness that suits it, so what is left is the cost of stacking and nothing else. The best thickness falls as 3^(1/4)/√m — the dashed line — and the resistance per turn rises as the square root of the layer count, fitted at 0.518. The ratio at the optimum is 1.3368 and it is the same for every count above one: the layer count moves the thickness, not the penalty. One layer is the exception, at 1.4407 and exactly π/2 skin depths, because it has no proximity term to trade against. — Fig. 5 The scaling law the optimum obeys, from the rung below: give every layer count its own best thickness and the resistance per turn still rises as the square root of the count. That comparison is between portions rather than against an absolute, which is why it is another quantity this ladder leaves intact.

Two things that do move, and one that does not

Set the three geometric corrections of this ladder against the optimum, because the contrast is the result.

The porosity substitution is worth up to 7.2 per cent, in the safe direction, and it acts on the proximity term — so it moves the optimum slightly and the four-thirds law slightly, in the same way and for the same reason as the window fill.

What the porosity factor is worth, measured against a field. computed by solving, not by drawing. The dashed curve is Dowell's expression reached through the porosity substitution; the solid one is the same copper solved as a two-dimensional field. They agree to 1.10 per cent at η = 1, where the layer is a foil and the substitution is an identity — which is what says the rest of the gap is the substitution and not the grid. From there the substitution runs high, monotonically, reaching 7.2 per cent at η = 0.40. It errs on the safe side and by an amount worth knowing rather than an amount worth ignoring. — Fig. 6 The second correction, priced. Exact at η = 1, where the layer is a foil, and 7.2 per cent high at η = 0.40. It errs the other way from the window fill, so a real winding gets some of one and some of the other.

A gap in the core is worth a factor of six — the turns nearest the gap measures it — and it does not act on the proximity term at all — it adds a field perpendicular to the layers, which is a third mechanism the trade above has no term for. An optimum computed without it is an optimum for a different winding.

And the optimum itself moves by 8.3 per cent, which for a foil gauge chosen from a stock list is no move at all.

What was checked, and what the numbers are worth

The full-fill end is the calibration. The optimum there comes out at 0.654 skin depths against a closed form’s 0.663, and the closed form is exact for that geometry — so the 1.4 per cent is the solver’s own error and everything else is quoted against it.

The field solver reproducing the closed form it is about to break. computed by solving, not by drawing. Foils spanning the window from yoke to yoke make the field parallel to the layers everywhere, which is the assumption Dowell's expression is derived under — so the two-dimensional solve must return Dowell's number, and how nearly it does is a statement about the grid rather than about the physics. It converges at order 2.00 in the cell size and reaches 0.155 per cent at 32 cells across a foil. Every departure measured in the other modes is larger than this by two orders of magnitude. — Fig. 7 Where that error comes from, measured. Second order in the cell size, fitted at 2.00, reaching 0.155 per cent at thirty-two cells across a foil. The optimum sweep runs at sixteen, which is where the error is about a third of a per cent — small against 8.3 and not against 1.4.

And the sweep is over the loss rather than the ratio. Those have different minima: the ratio Rac/Rdc rises monotonically with thickness and has no interior minimum at all, so a sweep over it would find nothing. The quantity with an optimum is the loss at fixed current, which is Rac — and getting that distinction wrong is the commonest way to mis-state this result. It is the same distinction between a ratio and the thing it is a ratio of that exact outside and wrong within draws about an equivalent circuit.

The fourth root, which is the practical form of all of this

The relationship the derivation gives is the one worth carrying out of this essay, because it prices every uncertainty in the problem at once.

The optimum thickness goes as the fourth root of the reciprocal of the proximity coefficient. So:

A 45 per cent error in the proximity term — the window fill, at a quarter — is 8 per cent on the thickness. A 7 per cent error — the porosity substitution — is 1.8 per cent. A factor of two in the layer count is 16 per cent. And the frequency, which enters through δ, is the only variable the optimum is a strong function of, because it scales the thickness directly rather than through a fourth root.

That is a comfortable place for a design to be. It means the foil gauge can be chosen from a stock list with the closed form and a rough layer count, and that no amount of geometric correction will move it to the next size up — while the loss that gauge produces has to be computed properly, because it moves by a factor.

What it does not settle

The optimum is for a portion at one frequency. A converter’s winding carries a fundamental and a great many harmonics — the exponent nobody put in prices the core’s side of the same problem — and the thickness that is best for one is not best for another; the honest optimisation is over the whole spectrum weighted by the current in it, and this essay’s is over a single sinusoid.

Grading is untouched. The per-layer form of the closed form prescribes different thicknesses for different layers — π/2 skin depths for the first and 0.37 for the eighth — and this ladder has already shown the per-layer form is wrong in both directions once the window is not full. The optimum grade is therefore not as robust as the optimum uniform thickness, and measuring it needs an eight-dimensional sweep rather than a one-dimensional one.

And the loss is copper only. A thicker foil is more copper in the same window, so it displaces insulation, changes the window fill and raises the interwinding capacitance. Those are constraints on the same variable and none of them is in the curve above — and the last of them is what decides the inductance that is a shape, which trades against this one through the same window height.

Fig. 8 The object underneath all of it: iron on four sides, copper in the middle, and the flux lines that decide every term in the trade. Drag the fill and watch the flux stop being parallel to the layers — which changes almost everything about this winding, and not where the best thickness is.

The question the sweep could not answer, and why it was not asked

There is an obvious next move and it is deliberately not made here: sweep the thickness at every fill, at every porosity, with and without a gap, and produce a corrected optimum for the general case.

It is not made because the arithmetic says it would not be worth reading. Each point on each of the three curves above is a two-dimensional field solve of ten to twenty thousand unknowns, and a bracketed minimum costs a curve; three fills is twenty-seven solves and about sixteen seconds. A four-dimensional sweep at the same resolution is tens of thousands of them, and the answer at the end would be a table whose largest entry differs from the closed form’s single number by under ten per cent — a great deal of computation to confirm that the thing being computed does not vary.

That is a judgement rather than a rule, and the thing it turns on is the fourth root above. A quantity that responds to its inputs as a fourth root is a quantity to compute once and tabulate; a quantity that responds linearly is one to compute per design. The loss is the second kind and the optimum is the first, and the whole value of measuring the optimum was finding out which it is.

The number worth carrying

8.3 per cent on the optimum, and 1.340 to 1.406 against four thirds.

The habit is about which results to trust when a model’s assumption fails. A closed form derived under a geometry produces two kinds of number: magnitudes, which are wrong by whatever the geometry is wrong by, and balances — optima, ratios at optima, scaling exponents — which are wrong by very much less, because the error acts on both sides of the balance. This ladder has now measured the same model’s magnitudes as 45 to 82 per cent out and its balance as 8 per cent out, on identical copper. Knowing which kind of number is in hand is worth more than knowing either error.

The same distinction, in four other fields

That division is worth carrying because it decides which published numbers are usable outside their own conditions, and this collection has instances of both kinds everywhere.

The exponent nobody put in is a magnitude failing and a balance holding, in the same measurement: a Steinmetz flux exponent is a local slope running from 2.94 to 1.46 so five windows on one curve give predictions three times apart, while the frequency exponent is exactly one — a theorem rather than a fit, because a rate-independent locus has the same area however fast it is traced.

The load that takes the most is a pure balance: the optimum load is the source resistance and the efficiency there is exactly a half, both independent of every absolute value in the circuit. The floor a circuit has is another — an optimum source resistance that is the ratio of two noise generators, so it survives a change of device and a change of temperature that either generator alone does not.

And the same filter a thousand times larger is the limiting case, where the balance is exact to a part in $10^{15}$ and every magnitude in the design is free. Which is the generalisation this essay’s finding belongs to: a ratio survives a systematic error that both of its terms share, and most of what a designer wants from a model is a ratio.

The condition on that is worth stating, because it is what makes the finding a measurement rather than a principle. Both terms have to share the error, and whether they do is not decidable from the algebra. Here they do: the optimum balances a direct-current loss against an eddy loss, and the two-dimensional geometry misrepresents both by comparable factors, so the balance point barely moves. Where they would not is a balance between a loss and something that is not a loss — a temperature rise, a saturation limit, a cost — and there a systematic error in the loss moves the optimum by its full size.

The gap that is three gaps is the case in this field where the condition fails. Its optimum is a count of gaps balancing a fringing loss against a spacing, and the two sides are not two versions of one quantity — which is why that optimum is at three rather than at a number the one-dimensional picture could have estimated, and why the one-dimensional picture has no term for it at all.

So the rule to carry is a question rather than a licence: are both sides of this balance the same kind of quantity, computed by the same model? If they are, the optimum survives the model’s errors and the magnitude does not. If they are not, neither does. The question costs nothing to ask and it is the only thing standing between a ratio that transfers and one that does not.