The gap that is bigger than it is

The energy is in the gap is a magnetic circuit: reluctances in series, the inductance N²/ℛ, and the share of the stored energy in each element proportional to its share of the reluctance. Every number in it is exact for the model it is in, and the model has one unstated assumption holding it up — that the flux crossing the gap stays inside the gap’s own cross-section.

It does not. Flux leaving one iron face and arriving at the other takes every path available, including the ones that bulge out into the surrounding air, and the reluctance of a bundle of paths in parallel is lower than the reluctance of any one of them. So the gap is easier to cross than g/µ₀A says, the flux is larger, and the inductance is higher than the formula.

Everybody knows this. What nobody has is a number for it, because getting one requires solving the field around the gap, and the substitute is a rule of thumb: add one gap length to each dimension of the gap’s area. A 6 mm × 6 mm leg with a 1 mm gap is treated as a 7 mm × 7 mm gap, which is a correction of 36 per cent in three dimensions and 17 in two.

Measuring it #

The window solver from the turns nearest the gap already has the geometry: a slot cut through the centre leg to the core’s own symmetry plane, iron everywhere else as a Neumann boundary, and the winding in the window. Run it at one hertz, where the current is uniform and nothing is an eddy current, and it is a magnetostatic solve.

The quantity read out is the energy, integrated over the whole domain from the solved field — ½∫|B|²/µ₀, with B taken as the centred difference of the potential across each face, so no interpolation enters it. Twice that energy over the square of the winding current is the inductance per metre, and comparing it with N²µ₀A/g is the fringing factor.

That is a comparison between a field integral and a circuit formula, which share nothing at all: one of them never mentions a reluctance and the other never mentions a cell.

Why the flux does not stay in the gap #

The argument is one line and it is worth having, because it also fixes the sign.

A reluctance is a resistance, and the flux takes every path in parallel. Add a path — any path, of any reluctance — and the total falls. The fringing paths are additional routes from one pole face to the other, so the gap’s reluctance is always lower than g/µ₀A and the inductance is always higher than N²µ₀A/g. There is no geometry in which the bare formula is high, which is more than can be said for most approximations in this collection and is the reason this correction is applied by people who have never measured it.

What the argument does not give is a magnitude, because the fringing paths are longer and narrower than the direct one and how much they help depends on how much room there is beside the gap. That is a shape question, and shape questions are what a field answers.

The solver’s own checks #

Three of them, and the first two are the ones that decide whether the number means anything.

The grid. Refining from seven to fifteen cells across a conductor moves the fringing factor at a 1 mm gap from 1.2133 to 1.2198 — half a per cent, monotonically, converging on about 1.222. The correction being measured is 22 per cent, so the instrument is forty times finer than the quantity.

The gap has to be an exact number of cells. The inductance goes as 1/g, so a gap that lands at 0.98 mm when it was meant to be 1.00 mm reports a two per cent error in the fringing factor — and because the rounding falls whichever way it falls, a sweep over the gap length came out 1.02, 1.27, 1.29, 1.16, 1.29: arithmetic in the grid reading as structure in the physics. The grid is built around the gap for this measurement and around the conductor for the loss measurement in the rung below, and they are different grids for that reason.

And the solve is checked, not trusted. The banded factorisation does no pivoting, which is justified for the class of matrix a five-point stencil makes and not for a particular one, so the residual against the original operator is computed on every solve and it sits at 10⁻¹³. The habit is the digits the arithmetic did not have’s: a plausible answer from an unstable factorisation looks exactly like a right one.

The rule’s accuracy is a function of what it is correcting #

At a short gap the true correction is 1.100 and the rule offers 1.050. At a long one the true correction is 1.333 and the rule offers 1.282. In absolute terms the rule is about five points of inductance short at both ends — but as a fraction of the correction it is supplying, it goes from half to 85 per cent.

Both readings are true and they suggest opposite things. The absolute error is roughly constant, so a designer who cares about inductance tolerance sees a five per cent bias whatever gap they use. A designer who cares about why the formula is wrong sees a rule that captures most of the physics for a large gap and half of it for a small one.

The mechanism behind the second reading is that the rule is dimensional and the physics is not. “Add one gap length” says the fringing path is a semicircle of radius g round each edge, which is the right shape when the gap is comparable with the pole face and wrong when it is much smaller — a short gap’s fringing field spreads out over a distance set by how far the flux has to travel along the iron before it can turn, and that distance is a property of the leg.

That is the same finding the previous rung reached from the loss side, where the fitted reach of the fringing field moved by nine per cent while the gap moved by a factor of five. Two independent measurements, one about stored energy at one hertz and one about dissipation at a hundred kilohertz, both say the gap length is not the length that governs the fringing field.

What the correction does to everything downstream #

The gap’s reluctance is not one number among several in a magnetic design. It is very nearly the whole of the reluctance — 87 per cent of the energy at 200 µm on a ferrite — so an error in it propagates undiluted.

The inductance is 10 to 33 per cent high relative to the uncorrected formula, which is a first-pass design coming out with too few turns or too long a gap.

The peak flux density is correspondingly wrong, because Φ = F/ℛ and ℛ has just changed. A design that computes Bₚₖ from the volt-second product and the core area is unaffected, since that route does not use the reluctance at all — it is a boundary in volt-seconds, and it has no gap in it; a design that computes it from the current and the reluctance is out by the same 10 to 33 per cent — and it is out in the direction of more flux than predicted, which is the direction that saturates.

And the current at which the core saturates moves, because that is the current whose ampere-turns push the flux to Bₛₐₜ, and the reluctance is what converts one to the other.

The three ways a gap is priced, and what each is worth #

The bare formula, g/µ₀A. Low by 10 to 33 per cent on the geometries measured here. Its virtue is that it costs one division, and its second virtue is that it is a bound — the true reluctance is always lower, never higher, because adding a parallel path cannot raise a reluctance.

The rule of thumb, (a + g)(b + g). Recovers half to 85 per cent of the correction, with the smaller share at the small gaps that most designs use. Costs two additions.

The field. A banded complex factorisation of some twenty thousand unknowns, giving the answer for the geometry actually built. It is not something to put inside a design loop — the same objection a ladder is not a line raises about solving a transmission line by brute force — and it is what says how much the first two are worth.

Which of the three to use is the ordinary question every model has an edge asks of every model, and it has an ordinary answer: the formula for a first pass, the rule for a design, and the field once — to find out how far off the other two are for the family of cores being used, after which the answer is a number written on a page rather than a computation.

What a designer does with it #

Nothing, if the inductance is trimmed. Many gapped parts are ground to inductance rather than to gap, so the correction is absorbed into the grinding and never appears. That is the commonest case and it is why the rule of thumb has survived: it is applied to a first-pass turn count that will be adjusted anyway.

Something, if the gap is set by a spacer. A shim of known thickness gives a known gap and an unknown inductance, and the 10 to 33 per cent is then the tolerance the design has to live with — which matters for a resonant converter whose operating frequency is set by that inductance.

And rather a lot, if the gap is on the outer legs. The correction here is measured for a centre-leg gap, where the fringing field spills into the window and is bounded by it. A gap on the outer legs fringes into open air on one side and its correction is larger — which is one of the two reasons the centre leg is where gaps go. The other is the turns nearest the gap: an outer-leg gap fringes into the window from both ends of the winding instead of the middle, and the loss goes with it.

Where the model stops #

Two dimensions. A real gap fringes out of all four sides of a rectangular leg, and out of the whole circumference of a round centre post. The two-dimensional solve captures one pair of sides, so its correction is the correction per pair rather than the whole of it — which is why the numbers above are compared against the two-dimensional form of the rule, (2d + g)/2d, rather than against the three-dimensional (a + g)(b + g)/ab. A round centre leg is a different problem again and its correction is larger, because a circle has more perimeter per unit area than a square of the same size.

The winding is in the window. The copper is not magnetic and does not change the field, but it does occupy space, and the fringing paths that would otherwise pass through the window at that distance from the gap are unaffected. What a real winding does change is the field a little further out, once the current in it is large enough for its own magnetomotive force to matter — which at one hertz and this geometry is a fraction of a per cent, and at a hundred kilohertz is the whole of the previous rung.

And the iron is infinite. A ferrite’s two thousand puts the core’s own reluctance at a half-thousandth of the gap’s, so the idealisation is worth about that much. A powdered-iron part at µᵣ = 60 is a different object entirely: its gap is distributed through the material, there is no discrete gap to fringe from, and the whole of this essay is about a mechanism it does not have.

The number worth carrying #

Ten per cent at a short gap, a third at a long one, and the standard rule supplies half of the first.

The habit is the older one this collection keeps returning to. A correction that everybody applies and nobody has measured is not a correction; it is a convention, and its error is unknown in both size and sign until somebody computes the thing it is standing in for. The pleasant surprise here is that the sign is known in advance — a parallel path cannot raise a reluctance, so the bare formula is always low and the rule always undershoots — and the unpleasant one is that the fraction of the correction the rule recovers is worst exactly where designs cluster.

The fringing field, seen from the copper #

The flux this essay counts as an inductance correction is the same flux that the turns nearest the gap measures as a loss, and reading the two together is the argument for solving the field once rather than correcting two formulas.

There the fringing is not a small addition to a reluctance; it is a field crossing the copper at right angles to the layers, and the worst turn of an eight-turn winding four tenths of a millimetre from a one-millimetre gap dissipates 37.5 times its direct-current loss. The governing distance is 0.60 millimetres, which is not the gap length and does not scale with it — so the correction on this page and the loss on that one are functions of the same bulge and of different features of it.

The gap that is three gaps then shows the two coming apart completely. One gap of 1.2 millimetres and three of 0.4 are the same magnetic circuit — the same inductance, the same saturation current, the same energy in the air — and they are not the same field: the winding beside the single gap dissipates 6.52 times its direct-current loss and its worst turn 13.3, while beside three gaps those are 3.00 and 3.7. Everything this essay computes is identical between the two, and the quantity a designer cares about is not.

Which is the practical reading. The correction here is worth 10 per cent of an inductance at a 0.3 mm gap and 33 per cent at 1.7 mm, and getting it exactly right by a better formula would still leave untouched the larger consequence of the same flux, which is where it goes and what it crosses.

A rule whose accuracy depends on what it is correcting #

That the rule supplies half the correction at 0.3 mm and 85 per cent at 1.7 mm is the finding, and its shape is unusual enough to be worth naming: an approximation that gets better as the quantity it approximates gets larger.

That inverts the usual reading of a correction term. Most of them are first terms of an expansion, so they are excellent where the correction is small and poor where it is large — what the fourth order says about the third is the worked example, with the leading expression 6.89 per cent optimistic and its error growing as the square of the drive. Here the opposite holds, because the rule is a geometric construction rather than a truncated series: adding one gap length to each dimension is close to the true bulge when the bulge is large compared with the gap’s own dimensions and a poor description of it when the gap is a narrow slot.

The consequence for a design is stated in the paragraph above and is worth the emphasis: the fraction recovered is worst exactly where designs cluster. A ferrite gapped to a few tenths of a millimetre — the ordinary case — is the case where the rule is least reliable, and the case where it is nearly right is a gap so large that the turns nearest the gap’s loss problem has become the binding one anyway.