The inductance that is a shape

What coupling buys and where treated the coupling coefficient as an input: a number between nought and one, handed to the two-port, moving the upper edge of the band and leaving the lower one alone. It is a good way to see what coupling does and it says nothing about where the number comes from.

Leakage inductance is where it comes from, and it is unlike every other parameter in this field because it is not a property of the core at all. Take the ferrite out and replace it with a piece of wood: the magnetising inductance collapses by three orders of magnitude, taking the lower edge the band a turns ratio holds over measures with it, and the leakage inductance barely moves. It is a property of the window — of how much space is between the two windings and how the copper is arranged in it — which makes it the one number in a magnetic design that a field solve is the natural way to get.

What it is, stated as an integral #

Short the secondary. The primary’s ampere-turns and the secondary’s are then equal and opposite, so the net magnetomotive force driving the core is nearly zero and there is nearly no flux in the iron. What flux there is runs through the window: it links one winding and not the other, which is what “leakage” names.

The energy stored in that field, per metre of mean turn, is ½∫|B|²/µ₀ over the window. Twice it, divided by the square of the primary current, is the leakage inductance referred to the primary.

That is a complete definition with no turns ratio in it, no permeability, and no core. It is also exactly what the window solver from the assumption that is a geometry computes, run at a frequency low enough that the current is uniform.

The closed form is the same integral in one dimension #

The expression every transformer designer uses is

L = µ₀ N² · MLT · (dᵢₙₛ + (bₚ + bₛ)/3) / h

— with dᵢₙₛ the insulation between the windings, bₚ and bₛ the two portions’ thicknesses, h the winding’s breadth up the bobbin and MLT the mean length of a turn — and it is not an empirical fit. It is that same energy integral, evaluated under the assumption that the magnetomotive force rises linearly across the primary, sits flat across the insulation, and falls linearly across the secondary — with the thirds coming from ∫F²dx over a linear ramp.

Which is to say: it is the same one-dimensional assumption this whole ladder is about, applied to a different quantity. Every objection raised against it for the alternating-current resistance applies here unchanged, and the size of the error is a separate question with a separate answer.

At full fill it is right to 1.3 per cent, which is the calibration and which says the field solve and the formula are computing the same object.

At half fill it is 36 per cent high, and the direction is the interesting part: unlike the alternating-current resistance, where the closed form ran high in the ratio and low in the loss, here there is only one quantity and the formula simply overstates it. Flux escaping round the ends of the windings finds a shorter path back and stores less energy than a field confined to the window’s full height would.

Two routes to the same microhenry #

The energy integral is one way to the number and it is not the way anybody measures a real part. The bench method is a short-circuit inductance: short the secondary, measure the primary’s inductance at a frequency well above the magnetising impedance, and read the answer off a bridge.

The two are the same quantity and they arrive from opposite ends. The bench measurement is a terminal one — a current, a voltage and a phase, with no geometry anywhere in it — and the field integral is a geometric one, with no terminals in it. That the two agree is what lets a designer transfer a measurement made on one part to a prediction about another, and it is why the closed form is written in millimetres rather than in henries.

There is a third route in the two-port itself. k√(L₁L₂) is the mutual inductance, and the leakage is what is left of the primary’s own inductance after the mutual is taken out; a coupling coefficient of 0.98 on a 10 mH primary is 200 µH of leakage — which is the same number this essay computes, written in the vocabulary what coupling buys and where uses. The reason the field is worth solving is that the two-port route starts from k, and k is exactly what a design cannot know in advance.

Interleaving, and the factor that is not four #

Splitting the primary in half and putting the secondary between the halves is the one change in transformer practice that costs nothing and is always quoted as a factor rather than a formula: it divides the leakage inductance by four.

The arithmetic behind the four is straightforward. Each portion now builds half the magnetomotive force it did, the energy goes as the square of that, so the energy quarters.

Measured, it is 3.11 times, not four. And the reason is in the formula the factor was derived from rather than in any subtlety of the field: the interleaved arrangement has two interwinding insulation gaps where the plain one has one, and the insulation term does not halve with the magnetomotive force. Writing the closed form out for both arrangements gives

plain: dᵢₙₛ + (bₚ + bₛ)/3 against interleaved: (2dᵢₙₛ + (bₚ + bₛ)/3)/4

and the ratio is four only in the limit of no insulation at all. With 0.6 mm between the windings — which is a single layer of tape, not a safety barrier — it is 3.03 by the formula and 3.11 measured.

That matters because the insulation between primary and secondary is the one dimension in the window a designer is least free to shrink. A reinforced-insulation barrier is three layers of tape and is specified by somebody else; the interleaving that would have bought four buys under three.

Why the window fill matters less here than it did for the loss #

Below full fill the closed form is high by 9 per cent at 90 per cent fill, 14 at 80, and 26 at 50 — sizeable, and about a third of what the same fill did to the alternating-current resistance ratio, which was 82 per cent high at a quarter.

The difference in sensitivity has a cause. The resistance ratio is a proximity effect and depends on the field inside the copper, which is where the escaping flux does most of its rearranging. The leakage inductance is an energy integrated over the whole window, most of which is not copper at all, so a change in the field near the winding’s ends is diluted by everything else in the integral.

The one place the window fill goes the other way #

Every measurement above says the closed form runs high below full fill, which sounds like good news and is not, for a reason that has nothing to do with the field.

A winding that fills less of its window has a larger leakage inductance, not a smaller one. Look at the absolute numbers rather than the ratio: 3.086 microhenries a metre at full fill, 3.372 at 80 per cent, 4.608 at half. The formula overstates how much it grows, but it is right that it grows — the same current, the same turns, and less window height means a higher magnetomotive force per unit breadth and more energy stored.

So the margin tape a safety standard requires costs leakage inductance twice: once by taking window height, and again by taking the interwinding distance in the other direction. A part that has to survive a reinforced-insulation specification has both, and no amount of interleaving recovers either.

What leakage inductance is for, and what it costs #

It is the parameter with the most contradictory requirements in the field, which is why measuring it properly is worth something.

It is a loss. Energy stored in the window at the end of each switching interval has to go somewhere, and in a flyback or a forward converter it goes into a snubber and becomes heat, at ½Lₗₑₐₖ I² per cycle — the same accounting the half that never arrives does for a capacitor charged through a resistor. That number is directly proportional to what this essay measures.

It is a resonance. With the winding capacitance it rings, which is what puts a spike on the switch and sets the voltage rating of the part that has to survive it — and which picture sets the upper edge shows that this is the mechanism people name and usually not the one that binds.

And it is sometimes wanted. A resonant converter uses the leakage inductance as the resonant element, which turns the least controllable parameter in the magnetic design into one the whole topology depends on. Designs that do this specify a leakage inductance with a tolerance, and the tolerance is where the 26 per cent above stops being an academic point.

Why it is the parameter that is always wrong first #

There is a reason leakage inductance is the number a magnetic design gets wrong more often than any other, and it is not that the formula is bad.

It is decided by the winder and not by the designer. Turns, wire gauge, core and gap all appear on a drawing. The distance between the last primary layer and the first secondary layer is whatever the tape and the tension produced, and it is the dominant term in the formula — 0.6 mm of insulation against 1.9 mm of winding thickness contributes more than half the answer here, because the insulation term enters at full weight and the winding terms enter divided by three.

It changes when nothing electrical changes. A safety agency asking for one more layer of tape moves the leakage by the thickness of that layer over the window height, and nothing in the schematic records it.

And it is the parameter every other one hides. Magnetising inductance is set by the gap and can be measured open-circuit; winding resistance is set by the wire and can be measured with a meter; the turns ratio is a count. Leakage is what is left, so every error in the others accumulates in the measurement of it.

That is the case for a geometric route to it. A formula written in millimetres can be evaluated on a drawing before anything is wound, and the field can say how much that formula is worth for the shape actually drawn — which is 1.3 per cent at full fill and 36 at half.

The measurement’s own checks #

The energy comes from the field and the comparison comes from a formula, and they share no arithmetic: the field integral never mentions a turns count and the formula never mentions a cell. At full fill they agree to 1.3 per cent, which is the whole basis for believing anything either says at other fills.

The magnetostatic limit is checked by construction. The solve is run at a frequency low enough that the current density is uniform to a part in ten thousand — verified, not assumed, by comparing the loss against the direct-current loss of the same copper — so the energy is a magnetostatic energy and no part of the answer is an eddy-current effect. That distinction matters because at a hundred kilohertz the leakage inductance genuinely does fall, as the current crowds towards the winding faces and the field can no longer penetrate the copper — the same crowding the copper that makes it worse prices as a resistance.

And the two arrangements are the same window. The interleaved and plain cases have identical window dimensions, identical copper, identical insulation and identical turns; the only difference is the order the foils are laid in. That is what makes 3.11 a measurement of interleaving rather than a comparison of two transformers.

Where the model stops #

Two dimensions, so no mean turn length. Everything above is per metre of turn. A real transformer’s leakage inductance is that times the mean length of a turn, which is a three-dimensional quantity and differs between the inner and outer windings by a factor that a two-dimensional cross-section cannot see.

No end effects along the leg. The winding wraps round the centre post and the field at the top and bottom of the window is genuinely three-dimensional. The cross-section taken here is through the middle, where the flux is most nearly in the plane.

And no capacitance. The window’s other parameter — the capacitance between the windings — is the same geometry read with a different equation, and it trades against leakage in the opposite direction: interleaving lowers the leakage inductance and raises the interwinding capacitance, because it puts more primary surface next to more secondary surface. Any real choice about interleaving is a choice between the two, and this essay prices only one of them.

The number worth carrying #

Three point one one, not four — and 1.3 per cent at full fill against 36 at half.

The habit is one this ladder has now produced three times in three different quantities. A closed form derived by integrating a one-dimensional field is exact when the field is one-dimensional and wrong by a geometric factor when it is not, and the size of the factor depends on the quantity rather than only on the geometry: 82 per cent for a resistance ratio, 36 for an energy, 7 for a porosity correction, all on windings of the same shape. There is no single “two-dimensional correction” for a window. There is one per question asked of it.

Where the interleaving factor is spent #

Interleaving being worth 3.11 times rather than four, with the missing 0.89 being insulation nobody puts in the formula, is the practical result on this page, and what it buys is measured one field over.

What coupling buys, and where it does not is the essay that prices it: across six designs from $k = 0.8$ to $k = 0.999$ the coupling moves a transformer’s upper band edge by 168 times and its lower one by 1.083. Leakage inductance and coupling coefficient are the same quantity written two ways, so a factor of three in leakage is most of the way across that range — and it is bought by rearranging the winding rather than by adding material.

Which picture sets the upper edge is where the same leakage decides which of two mechanisms binds. At fifty ohms the resonance everybody names is at 1.13 MHz and the measured edge is at 81.3 kHz — a factor of fourteen away — because what actually binds is the leakage in series with the load, a first-order corner with no resonance in it. The two accounts swap at about 1500 Ω, so reducing the leakage by three moves the edge in one regime and changes which regime the design is in.

And the inductor one mode cannot see is the component whose entire specification is this quantity: a common-mode choke presents a millihenry to one mode and a microhenry to the other, a ratio of $2/(1-k)$ containing no inductance at all — so a leakage computed as a geometry here is the whole of what that part is sold on.

Which is the unusual thing about this parameter and the reason it gets its own essay. Everything else a transformer is specified by is a property of its material or its turns; leakage is a property of the space between the windings, so it is decided by an insulation thickness and a winding order — two things chosen for reasons of safety and manufacturability, by people who are not computing an inductance.