What coupling buys, and where it does not

The previous essay established that a transformer is a two-sided model and measured both of its edges. This one asks the question a person winding one actually has: what does the effort buy?

The parameter a winder controls most directly is the coupling — how much of each winding’s flux reaches the other. Interleaving the windings, using a bobbin that keeps them close, choosing a core that closes the magnetic path: all of it is work on k. The answer to what it buys is unusually lopsided.

The measurement #

Six designs, identical except for the coupling, at k = 0.8, 0.9, 0.95, 0.99, 0.995 and 0.999. Each is a pair of 10 mH windings driven from 50 Ω into 50 Ω with half an ohm of winding resistance either side, and each has both band edges bisected on its own solved response.

The lower edge spans 1.083× across the whole range. The upper spans 168×. That is the essay.

The gate holds the ratio of the two spreads rather than either alone, at every load on the slider, because the ratio is the claim: fifty times more movement at the top than at the bottom. It also holds that the upper edge rises monotonically with the coupling, so that the statement is “better coupling is better” rather than “better coupling is different”.

Why the bottom does not move #

The lower corner is a resistance divided by an inductance, and the inductance is the magnetising inductance — the part of the primary’s flux that links the primary. In the T-model that is k²L₁, so it does depend on the coupling, and at k = 0.8 it is 6.4 mH against 10 mH at unity.

That is a 36% change in the inductance and it produces an 8% change in the corner, because the resistance in the numerator moves the other way at the same time: with poorer coupling, less of the load is reflected into the primary, so the resistance across the magnetising inductance falls too. The two effects very nearly cancel, which is why the lower edge is flat to within a tenth over a range where the coupling’s complement — one minus k — changes by a factor of two hundred.

Nothing about that cancellation is exact and this essay does not claim it is. What is measured is that the corner moves by 8% while the other one moves by 16,800%, and the useful reading of that is about where to spend effort rather than about a coincidence.

What does move the bottom #

Three things do, and none of them is the winding technique.

More inductance. The corner is inversely proportional to it, so doubling the turns quarters the corner — inductance goes as the square of the turns. That is the cheapest lever and it is bounded by two other essays in this field: more turns is more winding resistance, which rises with frequency and eats the midband gain; and more turns is more winding capacitance, which moves the upper edge down.

A higher-permeability core, or a smaller gap. Same effect on the inductance, without the extra copper. It is bounded by saturation, which is a volt-second limit that a higher inductance makes worse for a given core, because the flux for a given applied voltage is unchanged while the headroom is not.

A lower impedance level. The corner is R/2πL, so halving both the source and load resistances halves it. This is the lever a systems designer has and a winder does not, and it is the reason a transformer specified into 600 Ω and the same transformer into 50 Ω are different components as far as their low-frequency response is concerned.

Why the top moves so much #

The upper corner is the leakage inductance against the loop resistance, and the leakage is L(1 − k) — the complement of the coupling.

That is the whole explanation and it is worth stating as arithmetic. Going from k = 0.9 to k = 0.99 changes the coupling by ten per cent and the complement by a factor of ten, so the leakage falls by ten and the corner rises by ten. Going from 0.99 to 0.999 does it again. The coupling coefficient is a number that looks like it is approaching a limit and behaves like a logarithm: the useful variable is 1 − k, and it is the one that spans four orders of magnitude in a table whose first column spans twenty per cent.

The measured spread of 168× against a coupling complement spanning 200× is the check on that reading, and the small shortfall is the loop resistance’s — at the tightest coupling the leakage has fallen far enough that other terms begin to matter.

The consequence for how a transformer is specified #

Two practical statements follow, and both are of the kind a data sheet does not make.

A transformer’s bandwidth is not a property of the transformer. Both of its corners contain the impedance it is working between: the lower one directly, the upper one through the loop resistance. The same component into 5 Ω and into 500 Ω has upper edges a decade apart, and neither number belongs to the part. The site has made this observation about a probe, about a converter and about an instrument’s rise time; a transformer is the fourth.

Coupling is one of two numbers and the field usually quotes the other. A specification saying “95% coupled” sounds like a five-per-cent imperfection. In the band it is a five per cent imperfection — the midband gain is k times the turns ratio, which the previous essay measured — but in the bandwidth it is a factor of twenty, because what enters there is 1 − k. The two facts come from the same number and read completely differently.

What happens if the coupling is pushed too far #

There is a limit to the strategy, and it is not the one an extrapolation of the table suggests.

Interleaving the windings raises the coupling by putting them physically closer, which also raises the capacitance between them. That capacitance is the other mechanism competing to set the upper edge, and the next essay measures which of the two binds at a given load. Past a point, tightening the coupling stops moving the edge because the edge has stopped being the leakage’s.

There is a sharper version of the same limit, and the field’s fourth essay is about it. The leakage inductance is also what damps the resonance between itself and the winding capacitance. Reduce it far enough at a light enough load and the response stops having a flat middle and acquires a peak: at k = 0.999 into 500 Ω the peak is already 1.18 times the turns ratio, and into 2 kΩ it is 3.03.

So better coupling narrows the range of loads over which the component behaves as a transformer at all. That is a genuinely counter-intuitive consequence and it comes straight out of the measurement: the gate records that k = 0.99 and 0.995 are flat into 500 Ω and k = 0.999 is not.

The one number a winder should be given #

Putting the three essays’ measurements together suggests a specification that is not usually offered and that the machinery here produces for free.

A transformer’s useful description is two frequencies and one gain, all three at a stated source and load: the lower corner, the upper corner, and the midband as a fraction of the turns ratio. Every one of them is measurable on a bench with a signal generator and a voltmeter, and every one of them is a property of the pair rather than of the part — which is why they must be quoted with the impedances and almost never are.

From those three, both of the underlying quantities can be recovered. The magnetising inductance follows from the lower corner and the loop resistance; the leakage follows from the upper one; and the coupling follows from the midband gain, since it is the gain divided by the turns ratio and by what the winding resistances leave of the loop — the loop’s divider and not the secondary’s, which the band essay shows are the same number at a matched load and eight per cent apart at a light one. So a measurement a technician can make in five minutes determines the model completely, and the model then predicts the behaviour into any other impedance.

That is the argument for measuring a transformer rather than modelling one, and it is the same argument the series-connection essay makes in a different currency: the coupling comes out of two inductance readings that never look at the geometry, to 2.4 × 10⁻¹⁵.

The same measurement at four impedance levels #

The slider on the figure is the load, and it is there because the asymmetry could in principle be an artefact of one operating point. It is not.

Across a factor of thirty in load impedance the ratio between the two spreads stays between 155 and 178 — which is what the gate holds, rather than either column, because a claim that survives a change of operating point is worth more than one measured at a convenient one.

The axis stops at 150 Ω and the reason is not tidiness. Past a few hundred ohms the highest coupling on the axis stops having a band at all: at k = 0.999 into 500 Ω the response peaks at 1.18 times the turns ratio, and a “band” measured on a peaked response is the skirts of a resonance. Truncating the slider silently would have made a figure whose claim quietly stopped applying at one end; the boundary has a figure of its own instead.

What this essay does not settle #

Two things are deliberately outside it.

Nothing here is about how to achieve a coupling. Interleaving, bifilar winding, sectioned bobbins and gapped versus ungapped cores are the craft of the subject and they are questions about geometry. This site owns the circuit; the coupling coefficient is where the geometry enters and it enters as a number.

Nothing here is about loss in the core. Every figure treats the magnetic path as lossless, which is a model with a range like everything else here: eddy currents and hysteresis in the material dissipate power that rises steeply with frequency and with flux, and above some frequency they, and not the leakage, decide what a transformer is good for. That is a real boundary, it is measurable, and it is not measured here — which is stated rather than left for a reader to discover from a curve that is too optimistic at its right-hand end.

The band as a single number, and why it is not one #

It is tempting to reduce the table to its last column and call the band a figure of merit: 1.12 decades at k = 0.8, 3.31 at 0.999, and a transformer is three times better when it is well wound.

The reduction loses the thing the essay is about. Two designs with the same number of decades are not interchangeable if one has them from 400 Hz to 5 kHz and the other from 4 kHz to 50 kHz, and a transformer’s job is almost always stated as a range rather than as a ratio — an audio transformer must cover 20 Hz to 20 kHz, a switching one must work at 100 kHz and nowhere else, a pulse one must pass an edge of a stated width. What matters is whether the required range fits inside the measured band, and a ratio cannot answer that.

There is also a real asymmetry in what the two edges cost to move, which a single figure hides:

Two of those four are the same lever pulling in opposite directions — more turns lowers the bottom and lowers the top — so a transformer that must be wide has a design problem rather than a manufacturing one, and the site’s habit of drawing both ends is what makes that visible.

That is the last of the field’s three statements about the band. The first was that there are two edges; the second is that one number moves one of them; and the third, which the next essay measures, is that the edge everyone names is often not the one that binds.

A note on where the coupling coefficient comes from #

Everything above treats k as an input, which on this site means it is a number somebody has to supply and therefore a number somebody could supply wrongly. It is worth saying how it is obtained, because the answer is unusually satisfying and it is the subject of the essay after next.

It is not measured with a flux meter and it is not computed from the geometry. It comes from two inductance readings: connect the two windings in series one way and the total is L₁ + L₂ + 2M; reverse one and it is L₁ + L₂ − 2M. The difference is four times the mutual inductance, so M — and therefore k — falls out of two measurements a bench instrument makes directly, with no knowledge of the core, the turns, the interleaving or the geometry entering anywhere.

That is worth having in this field for the same reason the site keeps two routes to everything else: the coupling is the one number in the model that could not otherwise be checked. The gate performs the two solves and recovers the coupling that was stamped into the netlist to 2.4 × 10⁻¹⁵ across eighteen combinations of inductance and coupling, by a path that never reads the coupling’s own value.

So the number this essay has been spending is measurable, and one number from two measurements is the field’s answer to a question the rung below raised: what checks a mutual inductance, when neither the current law nor the energy balance can see one. A coupling adds no current anywhere and a coupled pair dissipates nothing, so a coupling stamped into the wrong row would produce a well-formed solution to a different circuit with both checks passing — and the third route is the bench method, series-aiding against series-opposing, with the difference four times the mutual inductance.

The asymmetry, and what it says about where to spend #

A coupling coefficient moving the upper band edge by 168 times and the lower by 1.083 is a strong result and it is worth saying which quantity each edge belongs to, because the two are not the same kind of thing at all.

The lower edge is the magnetising inductance against the source and load resistances — a first-order corner, decided by the number of turns and the core’s permeability, and improved by winding more turns or choosing a better material. Coupling barely enters it, which is why an hour spent on the winding pattern buys nothing there.

The upper edge is the leakage inductance, and the inductance that is a shape is where its provenance is established: twice the magnetic energy in the window under equal and opposite ampere-turns divided by the square of the current — a geometry rather than a material, coming out at 3.086 microhenries a metre against a closed form’s 3.128 when the copper fills the window and 4.608 against 6.255 when it fills half of it, with interleaving worth 3.11 times and not the four it is quoted as.

So the 168 is the winding pattern and the 1.083 is the winding count, and they are bought with different work. Which is the practical form of this essay’s asymmetry: coupling is what interleaving buys, interleaving is worth three times rather than four, and none of it touches the low end. Which is a useful thing to be able to tell somebody winding a transformer: the effort has a direction, and if the design is short of bandwidth at the bottom the answer is more turns or a better core rather than a better winding pattern.