The assumption that is a geometry

The copper that makes it worse computed a winding’s alternating-current resistance three separate ways: Dowell’s closed form, the per-layer version of it that the average hides, and a slab of copper solved as a ladder on this site’s own netlist solver. The three agree, and the essay treated that agreement as a verification.

It is not one. All three are the same assumption written down three times, and the assumption is a single sentence that appears in the ladder’s own description and is then never mentioned again: the field is parallel to the layers everywhere.

That sentence is not an approximation with a small parameter attached. It is a statement about the shape of the window, and it is exactly true only when the copper spans the window from one yoke to the other, so that the flux between the layers has nowhere to go but along them. Real windings do not do that. A bobbin has cheeks, a layer stops short of the flange, a safety winding leaves six millimetres of creepage at each end, and a gapped centre leg puts a field across the copper rather than along it.

Nothing in this collection could say what that costs, because saying it needs a field.

What is being solved #

The magnetic vector potential, z-directed, complex, at one frequency, over the window’s cross section:

∇·(ν∇A) = −J, with J = σ(Eₖ − jωA) inside conductor k and zero elsewhere.

Field lines are contours of A and the flux between two points is the difference of A at them, so the picture and the arithmetic are the same object rather than an illustration of it. There is one unknown per cell of the grid and one more per conductor — the axial electric field its terminals impose along it — because a conductor is an equipotential in the plane and what a circuit prescribes is its total current, not its current density.

The iron is a boundary condition, not a material. Infinite permeability carries no tangential magnetic field, so Bₜ = 0 at the surface; in two dimensions B = (∂A/∂y, −∂A/∂x), which makes Bₜ = 0 identically ∂A/∂n = 0. Neumann is the iron. That is the opposite of the condition most people reach for, and getting it the wrong way round gives a window whose flux refuses to enter the core. An infinite permeability is of course an idealisation, and it is the kind exact outside and wrong within is about: it is exact for the question asked here, because a ferrite’s two thousand makes the iron’s own reluctance a half-thousandth of the window’s.

The reproduction, which comes before the departure #

A field solver is exactly the kind of machinery that returns a plausible number for a mis-stated problem, so the first thing asked of it is a question whose answer is already known.

Foils spanning the window from yoke to yoke make the field parallel to the layers everywhere by construction. That is Dowell’s assumption, 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.

Two more checks run on every solve and neither is decoration — the same pairing one step computed twice insists on for a transient. The loss integrated over the copper — a cell-by-cell sum of |J|²/σ — equals the loss the terminals deliver, Σ Re(Eₖ Iₖ*), to a part in 10¹⁴; the two share the solved potential and nothing else, one of them never mentioning a terminal and the other never mentioning a cell. And the residual of the banded factorisation against the original operator is at machine precision, which matters because the factorisation does no pivoting. Not pivoting is justified for the class of matrix a five-point stencil produces, and that justification is a statement about matrices in general rather than about this one, so it is checked rather than believed — the same habit the digits the arithmetic did not have applies to a solve whose answer looks fine.

The departure #

Now shorten the foils. Same copper thickness, same layer count, same frequency, same current; the winding simply stops short of the yokes, as every real winding on a bobbin does.

The one-dimensional answer has no window in it at all. Dowell’s expression takes a layer count and a thickness in skin depths and returns 16.382, and it returns 16.382 whether the copper covers the whole bobbin or a quarter of it, because there is no term in it for the thing that has changed.

That is the shape of the result and it is worth stating plainly, because the two halves of it point in opposite directions. The alternating-current resistance ratio falls, monotonically, from 16.20 to 9.00. The alternating-current loss rises, from 0.2322 watts a metre to 0.5278. A designer who computes Rac/Rdc, finds it lower than the handbook said, and concludes the winding is better than predicted has read the one number that improved and missed the one that matters — the same failure mode every model has an edge is about, arriving from an unusual direction, since here the model is conservative in a quantity nobody pays for and optimistic in the quantity they do.

A minimum that is not at the end #

There is one more feature in that curve and it is easy to miss because it is worth under two per cent. The loss is not least at full fill. It is least at about ninety-two per cent, by 1.8 per cent, and it climbs on both sides.

The mechanism is a race. Shortening the winding raises its direct-current resistance in exact proportion — the same current in less copper — and lowers the proximity field by letting flux escape round the ends. Over a narrow range the second wins. Past it, the first does, and by a quarter fill the loss has more than doubled.

The stationary point is only there for three layers and more. A two-layer portion has almost no proximity term to give away, so the copper’s cost dominates from the first millimetre and the loss rises from full fill onwards. That is not an exception to work around; it is the mechanism visible in its own absence, and it is why the assertion in the figure is written per layer count rather than as a law.

The closed form does not merely run high — it redistributes #

The portion’s average hides something sharper. Dowell’s per-layer form gives layer p a proximity term proportional to 3p² − 3p, which is zero for the first layer: in one dimension the innermost layer’s inner face sees no field at all, whatever is stacked on top of it, and that is the reason the first layer’s best thickness is π/2 skin depths where the eighth layer’s is 0.37.

This matters more than the portion average does, because a winding fails where it is hottest and not where it is on average. The per-layer form is what a designer uses to grade foil thickness through a stack — thick where the field is weak, thin where it is strong — and the grading it prescribes is derived from a distribution of loss that is wrong at both ends.

Where the assumption came from, and why it survived #

Dowell’s expression was written for a transformer wound on a bobbin whose windings fill the window, and inside that geometry it is not an approximation at all — it is the exact solution of the one-dimensional problem, which is why it reproduces to a part in a thousand above. Every subsequent use of it inherited the geometry without inheriting the sentence.

There is a second reason it survived, and it is the reason this collection could not have found the error either: every check available was a check against another one-dimensional model. The ladder in the copper that makes it worse is a genuinely independent computation — a netlist of forty-eight cells with a series inductance and a shunt resistance each, solved by the same machinery that answers what a network answers — and it agrees with the closed form to four figures. It agrees because it makes the same assumption. Two routes that share an assumption are one route, however different their arithmetic, and the site’s own habit of demanding two routes is not proof against it.

What a window fill really is #

“Fill” is a share of the window height and it stands in for several different things a real design does.

A bobbin’s cheeks take one to two millimetres from a twelve-millimetre window and are not optional. A creepage distance for reinforced insulation takes six to eight millimetres of combined clearance, which on a small core is most of the window; that is a safety requirement and it is not negotiable either. A partly wound layer — the last layer of a winding whose turn count is not a multiple of the turns per layer — is a fill less than one for that layer alone, and it is the commonest case of all.

So the range swept above is not an academic extreme. Sixty per cent is a bobbin with modest cheeks; a quarter is a mains transformer with proper creepage — a part whose useful band the band a turns ratio holds over measures at both ends, with the winding resistance setting one of them.

What it does not say #

It does not say the closed form should be abandoned. Inside its own geometry it is exact, it costs nothing to evaluate, and the field solve behind every number here is a banded complex factorisation of some ten thousand unknowns — three orders of magnitude more expensive, and not something to put inside an optimiser loop over turn counts.

It says the expression has a range, which is what this collection asks of every model, and the range is a geometric one rather than a frequency or an amplitude — a boundary of the kind the edge that is a region collects. Above ninety per cent fill the error is under two per cent and the closed form is simply right. Below about seventy it is worth more than ten, and it is worth it in both directions at once: the ratio too high, the per-layer distribution wrong at both ends, and the actual dissipation — the number a thermal design is built on — not addressed at all, because a ratio without its denominator is not a loss.

The two things it opens #

The window is now a place where a field can be asked a question, and there are two obvious ones.

The first is what happens when the flux crossing the copper does not come from the copper at all but from a gap in the core, which puts a field across the layers rather than along them and is the one case where “parallel to the layers” is not approximately true but exactly false. That is the turns nearest the gap, and the numbers there are larger than anything above.

The second is that almost no winding is made of foil. A layer of round or square wires is reached through a substitution — squeeze the conductors into a foil, spread it back over the breadth, and divide the conductivity by the porosity — and that substitution replaces a two-dimensional geometry with a one-dimensional one in exactly the way this essay is about. It is measured in the wire that is not a foil, and the answer is more flattering than the one here.

The number worth carrying #

Two per cent above ninety per cent fill, ten per cent below seventy, and a factor of 1.8 in the ratio at a quarter — with the loss moving the other way the whole time.

The habit that goes with it is shorter. A closed form derived under a geometry keeps working when the geometry changes; it returns the same number, with the same confidence, and nothing about the number says which winding it belongs to. The way to find out is to solve the geometry once and compare, and the comparison is only worth anything if the solver has first been made to reproduce the closed form where the closed form is right.

What the field solver then found, three times #

The ladder above this one is the field being pointed at the three assumptions the one-dimensional picture makes, and each returned a number the closed form could not have.

The turns nearest the gap is the largest: a gapped inductor’s flux does not turn a corner into the iron on its way out of the gap but bulges into the window and crosses the copper at right angles to the layers — the exact opposite of the assumption this essay tests. Four tenths of a millimetre from a one-millimetre gap the worst turn of an eight-turn winding dissipates 37.5 times its direct-current loss and the winding as a whole 14.1 times; three millimetres further out those become 2.9 and 2.4, and the distance governing it is 0.60 millimetres, which is not the gap length and does not scale with it.

The wire that is not a foil tests the substitution that lets a foil result be used on a round-wire winding, and finds it exact where it must be — at a porosity of one — and 7.2 per cent high at a porosity of 0.40. It errs on the safe side, which is the half of the answer nobody could have assumed.

And the optimum that does not move asks what happens to the conclusion rather than to the number, and finds the best foil thickness drifting by 8.3 per cent between a full window and a quarter-full one while four thirds becomes 1.34, 1.35, 1.41. The trade barely moves while everything it is made of moves a great deal.

Which is the ladder’s finding in one line: the one-dimensional picture is badly wrong about magnitudes, nearly right about balances, and wrong in kind about a gapped core.

What “returns Dowell’s expression to 0.155 per cent” is doing #

The full-window case agreeing to a sixth of a per cent is the least interesting number on this page and the one everything else depends on, and it is worth saying why.

A field solver and a closed form disagreeing by nine against sixteen at a quarter-full window is a difference with two possible causes: the expression is wrong about that geometry, or the solver is. A comparison between two routes cannot tell them apart. What separates them is a geometry where the expression is exactly right by construction — a full window, where the field genuinely is parallel to the layers — and agreement there is a calibration of the instrument rather than a result.

The wire that is not a foil makes the same move for the same reason, and says so: its substitution is exact at a porosity of one, that geometry is the calibration, and every measurement is quoted against 1.1 per cent there — which is why its conclusion can be stated as “a few per cent” rather than “under ten”.

The habit generalises past magnetics. The frequency a sample rate invents is an identity generated twice and differenced at the arithmetic’s floor, and three phases, and the wire that carries nothing is a cancellation at $5\times10^{-15}$ amperes. In every case the exact case is not the finding; it is what entitles the inexact one to be quoted.

The corollary is worth stating too, because it decides what is worth building. A field solver with no case where it must agree exactly with something else is an instrument with no calibration, and its disagreements with a closed form are unattributable. Which is to say that finding the geometry where the old model is exactly right is the first job rather than a formality — and on a winding it is the full window, which is also the geometry nobody builds.