The copper that makes it worse

The rung below this one measured a single round conductor’s resistance against frequency. Its own field pushes its own current towards its rim, the exact answer comes from the Kelvin functions, and the rule of thumb names a frequency the effect has already passed. It is a clean result and it contains nothing a designer can act on. The curve rises monotonically, there is no minimum anywhere on it, and the only instruction that falls out is the one everybody already follows: for a given current, use more copper.

A winding is not a conductor. It is a stack of them, and the difference is not a matter of degree.

The field a layer sits in is not its own #

Take a portion of foil layers between two surfaces at which the magnetomotive force is zero — the core on one side, the boundary with the other winding on the other. The current in each layer is the same, because the layers are turns of one winding in series. The field is not the same, because the field at any depth is the current enclosed: zero at the inner boundary, one layer-current after the first layer, two after the second, and so on.

So the third layer of a portion sits in two layers’ worth of external field, and that field is not its own. It drives a circulating current in the metal that adds to the useful current on one face and subtracts on the other. The layer still carries the same net current; it carries it in a smaller part of its own cross-section, and it dissipates more doing so.

That is the whole mechanism, and its consequence is the one the rung below has no way to reach: the loss the external field drives grows without bound in the thickness, while the useful conduction the thickness buys does not. Something has to turn over.

At 100 kHz the skin depth in copper is 208.7 µm, and the four-layer portion’s best foil is 0.663 of it — about 138 µm. Below that thickness the direct-current resistance dominates and adding copper helps. Above it the proximity loss dominates and adding copper hurts: at twice the optimum the alternating-current resistance is higher than at the optimum, not lower, with more metal in the window and more money spent on it.

This is the one place in this collection where the naive design move is wrong in sign rather than merely suboptimal. Everywhere else — a larger gap stores more energy, a tighter coupling widens the band — the obvious lever points the obvious way and the interest is in how far it goes. Here it reverses.

The optimum is always the same ratio #

Sweep the layer count and give each count the thickness that suits it, and two numbers come out that are worth more than the curve.

The first is that the optimum sits at 4/3 of the foil’s own direct-current resistance, whatever the layer count is. Two layers, four, eight, sixteen: 1.3487, 1.3368, 1.3342, 1.3335, converging on four thirds from above. The layer count moves the thickness — 0.9613, 0.6631, 0.4662, 0.3292 skin depths — and not the penalty.

The reason is visible in the small-thickness expansions. Writing ξ for the thickness in skin depths, the resistance per unit of copper behaves as 1/ξ + m²ξ³/9: the first term is the direct-current resistance falling with thickness, the second is the proximity loss rising. Setting the derivative to zero gives ξ⁴ = 3/m², so

$$\xi_{\text{best}} = \frac{3^{1/4}}{\sqrt{m}}$$

and substituting back gives a ratio of exactly 4/3, with the layer count cancelling out of it entirely. Both are asymptotes and both are measured here rather than assumed: the fitted best thickness follows $3^{1/4}/\sqrt{m}$ to under a per cent by the fourth layer, and the ratio is inside a fiftieth of four thirds by the third.

The second number is the cost. At its own best thickness, a portion’s alternating-current resistance per turn grows as the square root of the layer count — fitted at 0.518 over the range drawn. Eight layers cost 2.86 units per turn against one layer’s 0.917, a factor of 3.1, and the best foil for eight layers is a third the thickness of the best foil for one. So a deep winding uses less copper per turn than a shallow one and dissipates more in it.

The one-layer case is the exception and it is instructive. With no external field there is no proximity term to trade against, so the resistance falls monotonically with thickness and the “optimum” is only the point of diminishing return — which lands at exactly π/2 skin depths, with a ratio of 1.4407. Above four thirds, and for a different reason: it is the skin effect’s own asymptote rather than a balance between two mechanisms.

Two routes, and the second one is a transmission line #

Everything above comes from Dowell’s expression, which is a closed form and therefore an assumption stated as a result. It assumes the field is parallel to the layer everywhere, that the layer is a foil rather than a row of round wires, and — the assumption that matters — that the layers of a portion are identical and equally spaced.

The site’s habit is to compute a thing twice by routes that share no arithmetic, and the second route here is not an analogy. The one-dimensional eddy-current problem in a slab is a transmission line into the conductor’s own depth. Faraday’s law along the conductor says the electric field falls by jωµH per unit depth, which is a series inductance carrying H. Ampère’s law says H falls by J per unit depth, which is a shunt conductance drawing E. So a layer of thickness h is a ladder of n cells, each with a series inductance µ·dy·ℓ/b and a shunt resistance ρ·ℓ/(b·dy), and the whole of this site’s solver applies to it unchanged — including the two checks every solve here passes before it is returned.

The variables are the ones the analogy names. The current in the ladder is H·b, the magnetomotive force at that depth. The voltage is E·ℓ, the volts per turn at that depth. A layer’s own current is the difference in magnetomotive force between its two faces, so imposing that difference is imposing a current source at each end — which is what makes a layer with external field a different network from a layer without one, rather than the same network with a correction applied.

The two routes agree to 0.07 per cent at 64 cells, and the disagreement is the discretisation rather than a difference of opinion: at 12, 24, 48, 96 and 192 cells the gap is 0.210, 0.0526, 0.0132, 0.0033 and 0.00082 of the answer, falling by a factor of four each time the cell count doubles. That is second order, which is what a trapezoidal discretisation of a smooth field ought to give, and it is the same exponent the transients field measures for its own marching.

What the ladder buys is not accuracy. It is the questions Dowell cannot be asked.

Where the loss is, inside the metal #

The first of them is where the dissipation sits.

The first layer has zero field on its inner face, so its current leans towards the outer one and the profile is monotone: this is the rung below’s picture, a skin effect, with the current crowding towards a single surface.

The last layer of the portion is a different shape. Its loss has a minimum inside the metal — at 100 µm and 100 kHz the middle of the layer carries 58 per cent of what its faces carry — because the external field pushes current to both faces and the interior is left carrying the difference between two circulating currents that nearly cancel. Nothing in the closed form says this. Dowell’s expression returns a ratio for the layer and is silent on where the ratio came from.

It also explains the fourth-power law. The proximity loss is driven by a field that is proportional to the layer’s depth in the stack, and the loss goes as the square of that field and the cube of the thickness — which is where m²ξ³ comes from, and therefore where the fourth root in the optimum comes from.

Every layer wants a different thickness #

The second question the ladder can be asked is the one Dowell’s average forbids: what if the layers are not all the same?

They should not be. Layer p sits between magnetomotive forces of p−1 and p, so its own optimum depends on p alone — and the first layer’s proximity term is identically zero whatever is stacked on top of it, because its inner face sees no field. Its best thickness is a one-layer problem however deep the winding is.

Optimising each layer separately on the ladder gives, for the first eight layers, thicknesses of 1.571, 0.824, 0.634, 0.535, 0.472, 0.427, 0.392 and 0.365 skin depths. The first is π/2, exactly, for the reason just given. The rest follow

$$\xi_p = \frac{1}{\sqrt[4]{p(p-1)}}$$

to 0.4 per cent by the fourth layer and to 0.07 per cent by the eighth, and 2 per cent out at the second, where the small-thickness expansion the law comes from is not yet good. That is the shape of an asymptote measured rather than quoted, which is what this field’s first essays do to every expression they meet.

A graded winding built that way is worth 12.0 per cent of the alternating-current resistance at eight layers, against the best uniform foil. It uses 40 per cent more copper to get it. That is the inversion worth having: the trouble with a thick foil is not that it is copper, it is that it is copper in the wrong layer, and putting more of it where the field is small is worth paying for.

A real winding cannot be graded continuously — foil comes in gauges — but the direction is actionable, and it is not the direction anybody’s intuition supplies.

The worst mistake is the one that looks careful #

Put the two results together and a specific, expensive error falls out.

A designer who has read the rung below knows about the skin effect and sizes the foil accordingly: one skin depth, or the π/2 the single-conductor optimum gives. That is a careful choice. It is correct for one layer. Used for eight layers it gives an alternating-current resistance of 35.7 times the direct-current resistance, against 1.33 for the right foil — and because the wrong foil is also thicker, the resistance in ohms is 7.9 times what the right foil would have given.

Eight times the copper loss, from a choice that was made deliberately and with the right physics in mind, because the right physics was about a different object.

The cure that costs nothing #

There is one lever in this whole subject that is free, and it is the layer count itself.

Dowell’s m is the number of layers between two zero-field surfaces, not the number on the bobbin. Winding half the primary, then the secondary, then the other half of the primary puts a zero-field surface in the middle of the primary and halves m — at no cost in copper, turns, window area or part count. At 200 µm and 100 kHz, eight layers in one portion give 6.78 times the direct-current resistance; two portions of four give 2.43; four portions of two give 1.34. A factor of five for nothing but the order the wire was laid down in.

The ladder and the closed form agree to 0.01 per cent at every arrangement, which is the check that the argument is about the field and not about the expression.

This is also why the interleaving instruction is stated the way it is in practice, and why it is so often followed without being understood: the benefit is real, it is large, and the quantity it acts on — the number of layers between zero-field surfaces — is not the number anybody counts when they look at a bobbin.

What the model here stops being true at #

Three boundaries, and the third is the interesting one.

Foil, not wire. Every number above is for a rectangular conductor filling the window’s breadth. A layer of round wires has a different geometry and Dowell’s usual patch for it — an equivalent foil thickness with a porosity factor — is a fit rather than a solution. The ladder route can be given a non-uniform layer but not a two-dimensional one, so neither route here is right for a real wound inductor to better than a factor that this essay has not measured. The shape of the answer — an interior optimum, a four-thirds ratio, a square-root cost in layer count — survives, and the thickness does not.

One dimension. Both routes assume the field is parallel to the layers everywhere, which is false near the ends of the window and false everywhere in a gapped core, where the gap’s fringing field sprays across the winding at right angles. That field drives eddy currents in the plane of the foil and there is no version of this calculation that contains them.

And the frequency is one number. A converter’s winding does not carry a sinusoid. It carries a trapezoid whose harmonics run out to many times the switching frequency, and the alternating-current resistance rises with frequency, so the loss is a sum over harmonics with a rising weight on each. A winding optimised at the fundamental is not optimised for the waveform, and the correction goes the same way every correction in this essay goes: towards thinner copper. That calculation needs the spectrum the power field computes for its own currents, applied to a resistance that is a function of frequency, and it is not done here.

What the gate checks #

The site’s gate carries the claims a single figure cannot make.

The two routes are required to agree, at every layer count and every thickness on the sweep, to within the discretisation — and the discretisation is asserted to fall as the square of the cell count rather than merely to be small, because a fixed cell count agreeing to a fixed tolerance is consistent with two expressions that are both wrong.

The ratio at the optimum is asserted to be four thirds for every layer count above one, and the best thickness to follow $3^{1/4}/\sqrt{m}$, and the resistance per turn to follow $\sqrt{m}$ — three separate claims, because a realisation that got one of them right and the others wrong would pass a weaker assertion that only checked the curve had a minimum.

The per-layer law is asserted with its own departure: within a part in a thousand at the eighth layer, within a per cent at the fourth, and outside five per cent at the second, so a version whose asymptote had been quietly promoted to an identity would fail.

And the first layer’s optimum is asserted to be π/2 skin depths independently of how many layers are above it, which is the claim that the zero-field face is a boundary condition rather than an approximation.

The two rungs, side by side #

The rung below is about a conductor and this one is about a stack, and it is worth being exact about what changed.

Both are eddy currents and both raise resistance with frequency. In the rung below the field driving them is the conductor’s own, so the effect is bounded — the current can be pushed no further than onto the surface, and the resistance rises as √f for ever after. In this one the field is external, its size is set by how many layers are underneath, and there is no surface to be pushed onto: the loss rises as the square of the layer’s depth in the stack, without limit.

That difference is why one has no optimum and the other does, why one is a property of the wire and the other of the winding, and why the cure for one is thicker copper and the cure for the other is splitting the portion — or, failing that, less copper in the places where the field is largest.