The turns nearest the gap

The energy is in the gap established what a gap is for: at two hundred microns on a core of relative permeability two thousand, 87.2 per cent of the stored energy is in the air, and the share is lg/(lg + le/µᵣ) — a formula with no turns in it and no area, which is why a gap is the part of a magnetic circuit whose properties do not drift.

That essay is a magnetic circuit, and a magnetic circuit has no geometry. It says how much energy is in the gap. It cannot say where in the window the flux goes on its way there, and the answer is not “along the leg”. Flux leaving a gap bulges sideways, into the window, across whatever is standing there — and what is standing there is the copper.

The field the copper is standing in #

The assumption that is a geometry gave this collection a two-dimensional solve of the window: vector potential, complex, at one frequency, with the iron as a Neumann boundary and one unknown per conductor for the axial field its terminals impose. It was used there to measure how much a winding’s own field departs from being parallel to the layers.

A gap is the case where the departure is not a departure. The field a gap puts into the window is not approximately parallel to the layers; it is approximately perpendicular to them, which is the one direction Dowell’s expression, the per-layer form of it and the slab-as-a-ladder route all treat as having no field in it at all.

Modelling it needs one thing the previous window did not have. With iron all round, the line integral of H along the boundary is identically zero and so is the current it encloses — the field-theoretic statement of “an ungapped infinitely-permeable inductor has infinite inductance” — so a window with iron on four sides cannot carry a net current. The gap is where the iron stops: a slot cut into the centre leg, whose far end is the core’s own symmetry plane, where the flux runs along the leg axis and the potential is therefore constant. That single boundary segment is what lets an inductor exist.

Turn by turn #

The number a data sheet has is one resistance. A winding has one wire, one current and one temperature coefficient, so one resistance is what anybody would quote.

It is not one number here.

The distribution matters more than the total for two reasons that have nothing to do with each other.

The insulation fails at the hot spot. A wire’s enamel has a temperature index, and it is a statement about the hottest part of the wire rather than about the winding’s mean. A design that budgets a thirty-kelvin rise from an averaged loss, in a winding whose worst turn makes twenty-seven times what its best turn makes, has not budgeted for the part that will fail — the same distinction the pulse the heatsink does not feel draws in time rather than in space.

And the loss is not where the cooling is. The turns level with the gap are, in most geometries, the turns in the middle of the winding — furthest from the bobbin’s flange, furthest from the core’s outer surface, and with the longest thermal path to anywhere. The mechanism concentrates the heat exactly where it is hardest to remove.

The clearance, and what it is really a function of #

The trade rule is to keep the winding two or three gap lengths away from the gap. It is quoted that way consistently enough that the scaling in it looks like a result.

The exponential is the useful part. Fit the excess — the winding’s ratio minus the value it settles to far away — against clearance, and it is a straight line on a logarithm to better than a per cent of variance explained. That gives a reach: one length, in millimetres, over which the gap’s influence decays by a factor of e.

And the reach is not the gap. Sweep the gap from four tenths of a millimetre to two and the fitted reach moves from 0.575 millimetres to 0.628 — nine per cent, while the gap moves by a factor of five. Measured on three grids from eight to twelve cells across a conductor the three numbers move by under two per cent each, so it is not the discretisation talking.

The rule of thumb has the wrong variable in it. Two gap lengths is 0.8 mm on a small gap and 4 mm on a large one, and the field does not care: the distance the fringing field reaches into the window is set by the leg, which is what the flux has to spread over before it can leave, and not by the gap, which sets only how much magnetomotive force is dropped across it. A designer following the rule over-builds a large-gap part and under-builds a small-gap one, which is the worse of the two errors, because a small gap concentrates the field: at a fixed clearance the ratio is higher for a shorter gap, since the same ampere-turns are dropped across less distance.

What the far-away floor is not #

The curve settles at 2.40 and not at 1, and the reason is worth being explicit about because it is the whole of the previous rung.

Move the winding far enough and the gap’s field stops reaching it. What is left is the winding’s own field — the layer-to-layer proximity effect, plus each conductor’s own skin effect — and that is a number the closed forms do compute, because in that limit the field really is parallel to the layers. So the floor of this curve is the previous essay’s subject and the excess above it is this one’s, and the two mechanisms add rather than competing.

Why this is the mechanism that surprises people #

Every other loss in a magnetic component scales with something the designer is already watching. Core loss goes as a power of the flux density, which is on the drawing, and the area a curve cannot have computes it from a loop. The voltage a winding may carry is a volt-second product, which is in the specification. Copper loss goes as the square of the current, which is the load.

Gap fringing scales with a position, and a position is not a number that appears anywhere in the electrical design. Two parts with identical turns, identical wire, identical core and identical gap can differ by a factor of six in copper loss because one was wound with a millimetre of former under the first layer and the other was not — and nothing in the schematic, the bill of materials or the magnetic circuit distinguishes them.

That is also why it is a manufacturing problem rather than only a design one. The clearance is set by whatever the winder does, and a reach of six tenths of a millimetre means a half-millimetre of variation is a factor of two in the excess loss.

What the model does not include #

The turns are square in section. A staircase circle on a finite grid has an area that jitters with the resolution by a per cent or two, so a round wire and the porosity question would have been measured together with no way to separate them. A square conductor is exact on the grid at every resolution. The round-to-square step is its own approximation and it is the subject of the wire that is not a foil.

The winding is one layer. A real inductor has several, and the layers further from the wall are further from the gap, so the mechanism here interacts with the stacking mechanism of the previous rung rather than adding to it cleanly. What is measured above is the shape of the dependence on distance; the multi-layer arithmetic is a superposition of that shape with the one the copper that makes it worse computes.

There is no third dimension. A real gap fringes out of the sides of the leg as well as into the window, and a real winding wraps round the leg rather than lying flat beside it. The two-dimensional solve is the cross-section through the middle of the window, which is where the effect is largest and where the hot spot is.

And the core has infinite permeability. A ferrite’s two thousand makes the iron’s own reluctance a half-thousandth of the gap’s, so the idealisation costs a tenth of a per cent of the flux — an approximation of the kind exact outside and wrong within is about, exact for the question and not for every question; it would cost more in a powdered-iron part at µᵣ = 60, where the distributed gap is the whole mechanism and there is no discrete gap to stand away from. That is a different component with a different answer, and it is the reason distributed-gap materials exist.

A shorter gap is worse, which is the opposite of the rule #

Hold the clearance fixed and change the gap instead. At four tenths of a millimetre of clearance the winding’s ratio is 17.0 for a 0.38 mm gap, 14.1 for a 0.98 mm gap and 9.9 for a 2.03 mm gap — the shortest gap is the worst by a factor of 1.7.

That is straightforward once the mechanism is stated. The ampere-turns are the same in all three cases, because the same current flows through the same turns; the whole of the magnetomotive force is dropped across the gap; and a shorter gap therefore has a higher field in it. The fringing field that spills into the window is proportional to that field, so it is largest for the gap a designer chose because it was small.

The rule of thumb inverts this. “Two gap lengths” asks for 0.76 mm of clearance from the short gap and 4 mm from the long one, which is more clearance where the field is weaker and less where it is stronger — precisely backwards. A rule in millimetres, from a reach that belongs to the core, asks for the same clearance from both and is right about both.

This is also why splitting a gap helps twice over. Two gaps of half the length in series have the same total reluctance and therefore the same inductance, but each has half the magnetomotive force across it, so each fringes half as hard — and the two fringing fields are in different places, so neither of them finds the whole winding. The arithmetic of the split is the energy is in the gap’s: reluctances add, so the inductance is untouched.

What can be done about it, in the order the numbers rank them #

Move the winding. A reach of six tenths of a millimetre means three millimetres of clearance is five reaches, and five reaches is a factor of e⁵ on the excess — which is why the curve is flat past about two and a half. This is the cheapest fix and it costs window area, which is the same currency everything else in the window is paid in.

Split the gap. Two half-gaps in series, or a distributed gap, for the reasons above. A powdered-iron or gapped-ferrite part with a distributed gap has no discrete fringing field at all and pays for it in core loss, which is the trade the exponent nobody put in prices from the other side.

Change the conductor. The mechanism is an induced circulating current in a conductor sitting in an external field, and it goes as the fourth power of the conductor’s dimension across that field. Litz wire, or a foil turned edge-on, attacks the exponent rather than the coefficient. It also costs copper area, so the direct-current resistance rises — the same trade the assumption that is a geometry measures as a minimum in the fill.

And do not change the gap to fix it. Lengthening the gap lowers the fringing field but also lowers the inductance, so the turns go up to compensate and the ampere-turns with them. That is a loop, not a fix.

The two numbers worth carrying #

The reach is 0.60 millimetres and it is not a number of gap lengths. It moves by a seventh while the gap moves fivefold, so it belongs to the leg and the window rather than to the gap. The rule of thumb should be a clearance in millimetres for a given core, not a multiple of a variable it does not depend on.

And the spread inside the winding is larger than the winding’s own excess. At a millimetre the winding averages 4.93 times its direct-current loss and its worst turn is at 14.4; at four tenths those are 14.07 and 37.5. The average is what a loss budget uses and the worst turn is what fails, and between them is a factor of 2.7 that grows as the winding gets closer.

The habit that goes with both is one line. A magnetic circuit prices a gap; it does not locate one. Two designs with the same reluctance, the same energy split and the same inductance can differ by a factor of six in copper loss, and the difference is a distance that appears nowhere in the circuit.

A distance that is not the gap length #

The 0.60 millimetres is the number worth carrying and it is the one hardest to guess, because every instinct says a bulge from a one-millimetre gap should reach about a millimetre. It does not scale with the gap at all, which means a design that halves its gap to reduce fringing has not moved the distance over which the copper is affected — only the strength of what reaches it.

That puts this boundary in a class the collection keeps separately. The edges that are lengths is where it belongs: boundaries whose axis is a distance, set by whoever builds the thing, appearing in no netlist. What is unusual about this one is that it is a length the designer controls — where the winding starts, relative to the gap — rather than one the assembly decides, which makes it the rare member of that class with a schematic-side remedy. Three millimetres of clearance takes the worst turn from 37.5 times its direct-current loss to 2.9.

The clearance is not free, and the gap that is three gaps is where the alternative is priced: dividing one gap into several brings each bulge close to a different part of the winding, which is a gain to three gaps and a loss beyond — so a designer has two knobs, distance and count, and both act on the same 0.60 millimetres.

Why the worst turn is the number to design against #

The winding as a whole dissipating 14.1 times its direct-current loss and the worst turn 37.5 times is a spread of nearly three to one within one component, and the second number is the one a design lives or dies by — because copper fails locally.

That is a distinction this collection has made once before in a thermal context. The degrees a thermocouple cannot see finds a core 2.59 kelvin hotter in the middle than on the outside in still air and 3.37 on a cold plate, and the question it leaves open — whether an ignition threshold should be compared against the centre or the mean — is this essay’s question in the other material. A winding’s insulation is rated against the hottest turn, and a loss figure averaged over the winding is the wrong number to compare it with by a factor of two and a half.

Which is the strongest form of what a one-dimensional model costs. It gives a total, and a total is what a thermal budget wants and not what an insulation rating wants — and the field solve that produces the distribution is the only route to the second.