The gap that is three gaps

The turns nearest the gap measured what a gap does to the copper standing beside it: a factor of six across three millimetres of clearance, with an exponential reach of 0.60 millimetres, and a worst turn dissipating twenty-seven times what the best turn does. The essay’s list of remedies put “split the gap” second, on an argument that costs one line — reluctances in series add, so the total gap decides the inductance and the arrangement is free.

That is a magnetic-circuit argument, and the assumption that is a geometry and the rungs above it have spent this whole ladder on the difference between a magnetic circuit and a field. So it is worth measuring rather than asserting.

The same circuit, five times #

One gap of 1.20 millimetres, then two of 0.60, three of 0.40, four of 0.30 and six of 0.20, spread over four millimetres of the centre leg. Same turns, same core, same total gap to the last decimal — which is a thing that had to be arranged rather than assumed, and the arranging is worth a paragraph.

The solver decides what is air by testing cell centres, so a slot’s height is a whole number of cells. A grid that divides 1.20 millimetres exactly and 0.40 only approximately makes the “constant” total vary by a few per cent between arrangements — and the inductance varies with it, so a comparison meant to hold the circuit fixed would have been comparing two different circuits. The cell here is stated rather than derived, at fifty microns, which divides every gap in the sweep and the conductor as well.

What it buys #

The winding’s average loss falls by a factor of two, and its worst turn by a factor of three and a half. The second number is the larger and it is the one that matters more: a hot spot is what a winding fails at — the degrees a thermocouple cannot see is the same distinction inside the core — and the mechanism that concentrates the loss is exactly the one splitting the gap attacks.

Both effects are stronger the closer the winding is. At six tenths of a millimetre of clearance the gain is 2.90; at 1.8 millimetres it is 1.41. That has to be so — a winding far enough out that the fringing field has already decayed has nothing to gain from a weaker one — and it means the remedy is worth most exactly where it is needed — the reverse of what the wire that is not a foil found about the porosity correction, whose error grows where the model is least trusted.

Why the square is the whole of it #

The mechanism is one sentence and it explains the size of the effect.

The total magnetomotive force across the gaps is NI whatever their number, because Ampère’s law does not care how the path is cut. Splitting into n gaps gives each of them NI/n, so each throws a field into the window that is 1/n of the single gap’s. Eddy loss goes as the square of the field driving it, so each gap’s contribution is 1/n² of the original — and there are n of them, so the total should be 1/n.

Measured, one to two gaps is 6.52 → 3.13, which is 2.08 rather than 2. Two to three is 3.13 → 3.00, which is 1.04 rather than 1.5. So the 1/n law is right for the first split and stops being right after it, and the reason is the second half of the geometry.

The arithmetic of the split, written out #

It is worth doing the algebra, because the algebra says what to expect and the measurement then says where the algebra stops.

The field a gap throws into the window at a point is proportional to the magnetomotive force it drops and falls off with distance over a reach the core sets — 0.60 millimetres for this geometry, measured in the turns nearest the gap. Write that as F·φ(y), with F the gap’s own share of the ampere-turns and φ a shape normalised to one at the gap.

The eddy loss in a conductor sitting in an external field goes as the square of the field, and as the fourth power of the conductor’s dimension across it. So the excess loss of the whole winding is proportional to F²·∫φ(y − y₀)² dy over the copper, with y₀ the gap’s own position up the leg.

With n gaps, F becomes NI/n and the integral becomes a sum of n shifted copies. If the copies overlap — if the gaps are close compared with the reach — the sum is n times one integral and the total goes as n·(NI/n)² = (NI)²/n. That is the 1/n law, and it is what the first split delivers: 6.52 to 3.13, a factor of 2.08 against a predicted 2.

If the copies do not overlap, the sum is still n integrals but each of them now covers copper the others did not, and the winding’s average stops falling — while its worst turn keeps falling, because no single turn sees more than one gap’s field. That is exactly the split the measurement shows past three gaps: the average turns back up and the worst turn keeps improving until it too saturates.

So the two numbers a designer might quote diverge, and which one to use depends on whether the part is limited by its total dissipation or by its hottest turn. Below about three gaps they agree; above it they do not.

Where the argument fails, and why there is a best number #

The 1/n law assumes the n gaps act on the same copper. They do not, because splitting a gap means spreading it along the leg, and the further apart the gaps are, the more of the winding is within reach of one of them.

A single gap concentrates a strong field on a few turns. Six gaps spread over four millimetres put a weak field on most of them. The loss is a sum over turns of the square of a local field, and moving from a tall narrow distribution to a low broad one at constant total field does not always lower that sum — past some point the broadening wins.

So there is an optimum, and here it is at three gaps, worth 2.18 times against one. That number is a property of this geometry rather than a law: it depends on how far the gaps are spread, on how long the winding is, and on the clearance. What is general is the shape — a gain that saturates, and then reverses.

Which makes the practical rule two gaps rather than many. Two is worth 2.08 of the 2.18 that three buys, needs one spacer instead of two, and is far less sensitive to how far apart they end up. A design that splits a gap in half has taken almost all of the available gain for the least manufacturing complexity, and that conclusion does not depend on where this particular optimum sits.

And it costs some inductance #

Not all of the circuit argument survives. The inductance falls the whole way across the sweep — 239.6 microhenries a metre with one gap, 224.9 with three, 216.9 with six — about ten per cent.

The reason is the previous rung’s subject. The gap that is bigger than it is measured the fringing correction: a single gap is 10 to 33 per cent easier to cross than g/µ₀A says, because the flux bulges out of its sides. Splitting the gap removes some of that. Three short gaps have less fringing between them than one long gap does, so the total reluctance rises and the inductance falls.

So the honest statement is that splitting the gap is nearly free rather than free: it costs about ten per cent of the inductance, which is recovered by making the total gap ten per cent shorter — and that in turn is a slightly stronger field per gap, which the sweep above already accounts for.

What the arrangement does not change #

The saturation current, to first order. The flux at which the core saturates is a property of the material and the area; the current that produces it is that flux times the total reluctance over the turns. Ten per cent on the reluctance is ten per cent on the saturation current, in the safe direction.

The energy split. Eighty-seven per cent of the stored energy is in the air whether the air is in one piece or three, because the share is a ratio of reluctances and both are the same — which is the energy is in the gap’s result, and it survives intact.

And the core loss. The flux density in the iron is the same, so the hysteresis loop is the same, so the loss per cycle is the same. This is a copper-side remedy only.

Why this is not the same as a distributed gap #

A powdered-iron or a gapped-ferrite composite takes the same idea to its limit: the gap is spread through the material, in the spaces between grains, so there is no discrete gap anywhere and no fringing field at all. By the argument above that ought to be the best possible arrangement, and in one respect it is — a distributed-gap inductor has essentially no gap-fringing loss in its winding.

It is not free, and the price is in the iron. A distributed gap means a low effective permeability throughout the core, which means more turns for the same inductance, which means more copper loss of the ordinary kind; and powdered materials have a core loss per unit volume several times a ferrite’s at the same flux and frequency, which the exponent nobody put in is the machinery for pricing.

So the three arrangements form a sequence rather than a ranking: one gap is the least core loss and the worst copper hot spot, a distributed gap is the reverse, and a split gap of two or three is the compromise that keeps a ferrite’s loss and takes most of the fringing improvement. Nothing in this essay says which of the three a design should use — it says what the middle one is worth, which was the number that was missing.

What it costs to build #

Three practical costs, and the third is the one that decides it.

Spacers. A split gap is made by stacking core pieces with non-magnetic spacers between them, or by grinding two centre-leg faces instead of one. Both are ordinary and both add parts.

Tolerance. Each gap has its own tolerance and they add, so a split gap’s inductance is a little less repeatable than a single gap’s — though rather more repeatable than the ten per cent this essay just found the fringing worth.

And the winding no longer has an obvious place to be. A single gap has a clear answer to “where should the copper not be”; several gaps spread along the leg do not, which removes the option of buying the same improvement by clearance alone. That is the real design tension: splitting the gap and standing away from it are the same remedy applied twice, and past a point they compete for the same window.

What the model does not say #

Two dimensions. A real gap fringes out of all four sides of the leg, so the field in the window is somewhat weaker than a two-dimensional cross-section gives and the ratios above are upper bounds. The comparison between arrangements is much more robust than either number, because both are computed the same way.

One layer of turns. The winding here is a single column of eight square conductors. A multi-layer winding has layers at different distances from the gaps, so the profile the splitting flattens is two-dimensional rather than one-dimensional, and the optimum number of gaps would shift.

And the spread is a free parameter. Four millimetres over a twelve-millimetre window is one choice among many, and it is the choice the optimum at three gaps is a property of. A design that can keep its gaps close together — a stack of thin spacers rather than a distributed grind — gets closer to the 1/n law and its optimum moves to a larger number of gaps.

The number worth carrying #

Two point one eight at three gaps, two point zero eight at two, and about ten per cent of the inductance for either.

The habit is the one this ladder keeps arriving at from new directions. A magnetic circuit is a correct statement about a scalar — reluctance, flux, inductance — and it is silent about everything that has a position. Two arrangements that a magnetic circuit cannot tell apart can differ by a factor of two in copper loss and a factor of three and a half in the hottest turn, and the only way to know which is which is to solve the field. What the circuit is good for is knowing what the change did not cost, which here is nine tenths of nothing.

Why there is a best number of gaps rather than a monotone gain #

Past three, spreading the gaps along the leg brings each one close to a different part of the winding, and that is the sentence worth unpacking because it explains why the obvious extrapolation fails.

The mechanism is the one the turns nearest the gap measures: the loss a gap’s fringing field drives falls off over a distance of 0.60 millimetres, which is not the gap length and does not scale with it. Dividing one gap into three divides each bulge’s strength and leaves three regions of copper affected instead of one — which is a gain while the three regions are separate and a loss once they overlap. The optimum is where the spacing between gaps is comparable with that 0.60 millimetres, and it is therefore a property of the material and the geometry rather than of the total gap length.

Which is why the answer is a count rather than a fraction. Two gaps and four gaps are different circuits in a way that a magnetic circuit says they are not, and the number that decides between them appears in no reluctance, no inductance and no energy split — the three quantities the design was made with. The energy is in the gap is where those three are computed, and its result — 87.0 per cent of the stored energy in a two-hundred-micron gap, from a ratio of two lengths containing neither the turns nor the area — is identical for every arrangement compared here.

What the manufacturer sells and what the designer needs #

Distributed gaps are sold, and they are sold as a material — a powdered core whose permeability is the effect of gaps distributed through its volume — rather than as an arrangement. That is this essay’s argument taken to its limit, and it has the consequence the argument predicts: the fringing loss disappears because no single gap is large enough to bulge anywhere, and what replaces it is a much larger core loss, because the material is now a composite rather than a ferrite.

Which is why the count measured here has an optimum rather than running to infinity. Three gaps are better than one because the bulges are separated; a thousand are worse than three not because the fringing returns but because the price of distributing them is paid in the material.

That is the trade the ladder ends on, and it is the same one the energy is in the gap opens with. A gap is put into a high-permeability core to store energy in air, and every step away from “one gap in a good material” towards “many gaps in a mediocre one” moves loss from the copper to the core. Neither the turns nearest the gap’s copper measurement nor a core-loss curve sees both halves, which is why the decision is usually made by trying two parts.