The wire that is not a foil

The assumption that is a geometry put a field into the winding window and found the one-dimensional closed form 82 per cent high on a winding that fills a quarter of its bobbin. That measurement was made on foils, because foils are what Dowell’s expression is about and because comparing like with like is the only way to attribute a difference.

Almost no winding is made of foil.

A layer of a real transformer is a row of round or square wires with insulation and air between them, and the bridge from that to a closed form about foils is a substitution that everybody makes and nobody measures. It is worth stating in full, because every step in it is a decision:

One. Replace each round conductor by a square of the same cross-sectional area, side a = d√π/2.

Two. Push the layer’s N conductors together into a solid foil of thickness a and breadth N·a.

Three. Stretch that foil back across the full breadth b so that the geometry is a stack of full-width layers again, and divide its conductivity by the porosity η = N·a/b so that the direct-current resistance is unchanged.

The whole of the correction is then one factor: the skin depth in the spread foil is δ/√η, so ξ becomes a√η/δ, and Dowell’s expression is evaluated at that.

Step three is the interesting one. It replaces a geometry in which the field runs between the conductors with one in which it cannot, and it compensates for the missing copper with a fictitious resistivity. That is exactly the shape of statement a field can be pointed at.

The choice that carries the whole substitution #

Three steps, and only one of them is doing real work.

Step one, round to square of equal area, is a shape approximation and nothing more. Step two, pushing the conductors together, changes no geometry at all, since step three undoes it. It is step three — the stretch, with the conductivity divided by η — that decides everything, and the constraint it is built around is that the direct-current resistance must not change.

That constraint is the right one, and it is worth seeing why. The alternative would be to preserve the conductivity and let the resistance fall, which would make the substituted winding a different winding; or to preserve the thickness and let the copper area change, which would make it a different amount of metal. Preserving Rdc means the substituted object and the real one agree exactly at zero frequency and disagree only in the mechanism this collection is about — so the whole of the error is in Rac/Rdc and none of it is in the denominator. That is a well-posed substitution, which is more than can be said for most.

The price is the fictitious resistivity, ρ/η, and with it a fictitious skin depth δ/√η that is larger than any real skin depth in the problem. A winding of copper is being analysed as a winding of a metal that does not exist, and the justification is that the two share the one number the answer is normalised by. Stated that way it sounds worse than it turns out to be, and the measurement above is the reason it can be stated that way at all — the same relationship between a claim and its check that every model has an edge sets out for the whole collection.

The porosity a winding actually has #

η is copper divided by breadth in one layer, and it is a number nobody writes down, so it is worth being concrete about where it comes from.

A layer of N turns of round wire of bare diameter d and outside diameter dₒ, wound on a bobbin of breadth b, has η = N·(d√π/2)/b — the equal-area square, not the wire. With the turns touching, N = b/dₒ, so η = (√π/2)·(d/dₒ), which for a grade-2 enamel on a 0.5 mm wire is 0.886 × 0.926 = 0.82. That is the best case: turns touching, no margin tape, no layer insulation, perfect winding.

A margin for creepage takes breadth away from N without changing b, and three millimetres each side of a twenty-millimetre bobbin is a factor of 0.7 on η immediately. A deliberate space wind, used to lower interwinding capacitance, takes another factor. A litz bundle has a packing factor of its own inside the bundle, which multiplies again.

So 0.4 to 0.8 is the range this essay swept for a reason, and the low end of it is an ordinary part rather than a pathological one. What is not ordinary is a data sheet that states η, which is why the correction is usually applied with a number somebody guessed.

The calibration is free, which is what makes the measurement worth anything #

At η = 1 the conductors touch and the layer is a foil. Step two does nothing, step three divides by one, and the substitution is an identity rather than an approximation. So the two-dimensional solve must return the closed form’s number there, and how nearly it does is a statement about the grid.

That 1.1 per cent is the instrument’s own error, measured on the one geometry where the answer is known, and every departure below is quoted against it rather than against zero. Without such a point a five per cent gap between two routes is uninterpretable: it could be the substitution, the discretisation, or the staircase the grid makes of a curved boundary. That is the difference between two routes and two independent routes, which one step computed twice draws for a transient and which applies without change here.

Which is why the conductors here are square rather than round. A staircase circle on a finite grid has an area that jitters by a per cent or two as the resolution changes, so refining the grid changes the copper as well as the field, and the convergence stops being readable — measured, four grids gave 10.30, 10.65, 10.26 and a fourth value with no trend at all. A square conductor is exact on the grid at every resolution. Step one of the substitution, round to square, is a separate approximation with its own error, and nothing here is a claim about it.

What it is worth #

A porosity of 0.40 is not an unusual winding, in the sense the band a turns ratio holds over uses the word about a transformer’s useful range: it is inside what ordinary parts do rather than at an extreme. Round wire on a bobbin with a build tolerance, with insulation and the space a winder leaves, lands between 0.5 and 0.7; a wire wound with deliberate spacing for interwinding capacitance, or a layer of litz whose bundles do not pack, goes lower.

So the honest summary is: the substitution is worth a few per cent over the range real windings occupy, and it is a few per cent in the safe direction. That is a range attached to a model, which is what this collection asks of one, and it is a range in a variable — a packing fraction — that no other boundary here is written in.

Both halves of that matter and neither could have been assumed. The size is not obvious — a substitution that deletes the field between the conductors could plausibly have been out by a factor, and the loosely-wound case is where any one-dimensional argument is weakest. And the direction is not obvious at all, because the mechanism that goes missing is field crowding, which sounds like something that would make the loss larger rather than smaller.

Why it errs high #

The spread foil and the real layer have the same direct-current resistance by construction, so the difference is entirely in how the alternating current arranges itself.

In the spread foil the external field is uniform across the breadth and drives a circulating current through the full width of the layer, which is the picture the copper that makes it worse solves as a ladder. In the real layer the same total flux passes between the conductors as well as through them: the field is stronger in the gaps and weaker inside the copper than the smeared version says, and it is the field inside the conductor that drives the eddy current. Spreading the copper out flatters the external field’s access to it.

That last observation is the sharpest thing here. Dowell’s expression returns its two terms separately — a skin term that is a slab in its own field, and a proximity term proportional to m² − 1 — and the error is almost entirely in the second. The first layer, whose proximity term is identically zero, is right to a few parts in a thousand at every porosity tested.

What “conservative” is and is not worth #

A model that runs high is better than one that runs low, and it is a long way from being free.

A conservative loss is an unconservative design. Seven per cent of extra predicted copper loss is seven per cent of extra predicted temperature rise, which on a part designed to a temperature limit buys larger wire, a larger core or a lower current rating — and pays for it in cost, in window area and in direct-current resistance. The margin is real; so is what it costs — and a margin whose size is unknown is spent twice, once by the model and once by the designer who does not trust it. This is the case exact outside and wrong within makes for knowing which side of a model a design is sitting on.

And the direction is not guaranteed outside the range measured. Everything above holds for square conductors at four layers and a hundred kilohertz. The step this essay does not test — round to square — has been reported both ways in the literature, and its error grows in the same regime as this one. The safe statement is the narrow one: for the geometry solved, over 0.4 ≤ η ≤ 1, the substitution is high by nought to seven per cent.

The one place the substitution’s direction changes the design #

Almost everything above is a margin story, and margins are a matter of degree. There is one number where the direction matters qualitatively, and it is the optimum foil thickness.

The turnover in the rung below exists because two terms fight: the direct-current resistance falls as a conductor gets thicker and the proximity term rises. The optimum is where they balance, and because the substitution overstates the proximity term specifically — not the skin term, and not the resistance — it puts the optimum at a conductor slightly thinner than the true one.

That is a small error in a shallow minimum, which is the saving grace: the curve is at four thirds of the direct-current resistance at its bottom, and it is within a per cent of four thirds over a sizeable range of thickness either side. A design landing on the substitution’s optimum is not noticeably worse than one landing on the true optimum. But the direction is now known rather than guessed, and it says that erring towards thicker wire is erring the right way, which is convenient because thicker wire is also what the direct-current resistance and the winder both want.

The three geometric corrections, ranked #

This ladder has now measured every departure from “the field is parallel to the layers” that a transformer window contains, and they are not the same size.

A gap in the core is worth a factor. The turns nearest the gap measures 14.07 against a far-away 2.40 — the field is perpendicular to the layers rather than approximately parallel, and no correction to a one-dimensional model can express it.

A winding that does not fill its window is worth tens of per cent. The rung below measures the closed form 82 per cent high at a quarter fill, with the loss moving the other way.

And the porosity substitution is worth a few per cent, in the safe direction.

That ranking is the practical result of the whole ladder. A designer with one afternoon should spend it on the gap clearance, then on the window fill, and should leave the porosity factor alone — it is the one of the three that is doing its job.

What is still assumed #

The layers are alike and equally spaced. The substitution and the field solve here both use a regular lattice of identical conductors. A real winding’s second layer sits in the valleys of the first, and the turns of the last layer are wherever the wire ended up.

The frequency is one hundred kilohertz. The error grows with ξ because it lives in the proximity term, so the numbers above are a statement at one ratio of conductor to skin depth rather than a correction curve — the same caution kirchhoff’s own frequency attaches to every number that has a size in it.

And the conductors are solid. Litz is the standard answer to everything in this essay and it has no place in this model at all: a bundle of insulated strands transposed along the winding is a three-dimensional object whose whole mechanism is that no strand stays in one place. Two-dimensional cross-sections cannot see a transposition, and a model that cannot see the mechanism cannot price it — the same refusal the answer that is perfect and absurd makes about a network that does not determine its own answer.

The number worth carrying #

Nought to seven per cent, high, over the porosities a winding actually has.

The habit that goes with it is about how the number was obtained rather than about the number. A substitution has one geometry at which it is an identity rather than an approximation, and that geometry is the calibration: it separates the model’s error from the instrument’s, and without it a comparison between two routes is a difference with two possible causes and no way to tell them apart. Every measurement above is quoted against 1.1 per cent at η = 1, and the reason the conclusion can be stated as “a few per cent” rather than “under ten” is that the 1.1 was measured rather than assumed.

Erring on the safe side, which was not the likely answer #

A substitution 7.2 per cent high at a porosity of 0.40 is the best possible outcome for a rule that is going to be used without checking, and it is worth saying that the sign was not predictable in advance.

The two adjacent measurements in this ladder both err the other way. The assumption that is a geometry finds the same closed form returning 9.00 against an expression’s 16.38 when the copper fills a quarter of its window, and dissipating 0.528 watts a metre against 0.232 — a resistance ratio 45 per cent low and a loss more than doubled. The turns nearest the gap finds a gapped inductor’s worst turn at 37.5 times its direct-current loss where the one-dimensional picture has no mechanism for it at all.

So a designer applying the standard corrections in sequence gets one that overstates by seven per cent and two that understate by tens of per cent, and the net is an underestimate — which is the useful thing to know, because it says that the porosity correction is not where the error is and that checking it more carefully would buy nothing.

That ordering is what the optimum that does not move then makes usable: the magnitudes are wrong by 45 to 82 per cent and the balance — a best foil thickness, a ratio of four thirds at it — is wrong by 8. A design sized on the optimum rather than on the loss inherits the small error, and a thermal budget written from the loss inherits the large one.