The leakage a series reading cannot see

One number from two measurements needed a way to check a mutual inductance that Kirchhoff’s law and the energy balance could not see, and found it on the bench: connect the two windings in series one way, then the other, and the difference between the two readings is four times the mutual inductance. On a netlist the method returned every coupling from 0.1 to 0.99 to fifteen figures. That essay also noticed, in passing, which end of the range the method is good at: it subtracts two numbers that are close together when the coupling is weak, so its precision degrades as the coupling falls, “and it is at its best exactly where a transformer lives”.

That last remark is right about the coupling and wrong about the transformer, and the difference is the subject of this essay. A transformer’s coupling is 0.99 or 0.999, and nobody designing one wants to know it to three figures; they want the leakage inductance, $L_1(1 - k^2)$, the part of each winding’s inductance that the other winding does not share. That is the quantity that sets a transformer’s high-frequency band, its regulation under load, the spike on a switching converter’s drain and the current a rectifier draws in its first cycle. At $k = 0.999$ it is two thousandths of the winding’s own inductance, and the series method measures the winding’s own inductance.

So this essay measures a coupled pair the other common way — read one winding with the other open, then with the other shorted — and compares the two routes not on a netlist, where both are exact, but on a bench, where every reading has a meter’s error in it.

Shorted, a winding reads its own leakage #

Take the same pair: ten millihenries and forty, so a turns ratio of two, read at a kilohertz by driving the first winding through ten megohms and dividing the voltage by the current. Load the second winding with a resistance and sweep it from a milliohm to thirty kilohms.

Open, the reading is 10.000 millihenries, the winding’s own inductance: the second winding carries no current and might as well not be there. Shorted, the reading is 199.00 microhenries, which is $L_1(1 - k^2)$ with $k = 0.99$. The shorted winding carries whatever current it takes to keep its own flux linkage at zero, and in doing so it cancels every line of flux the two windings share; what the first winding still sees is the flux that links it alone. The transition between the two readings is centred on a load equal to the second winding’s reactance, 251 ohms at a kilohertz.

Move the slider through the couplings and the open reading never moves while the shorted one does nearly all the moving: 7,500 µH at k = 0.5, 1,900 at 0.9, 199.00 at 0.99 and 19.99 at 0.999. The ratio of the two readings is $1 - k^2$ exactly, and so

$$k = \sqrt{1 - \frac{L_\text{short}}{L_\text{open}}}.$$

That is a second bench route to the same coefficient, with two readings rather than four and no reconnection of the windings at all — which matters on a transformer whose windings cannot be reconnected because they are potted, or whose ends are a hundred turns apart.

Two routes, judged by a real meter #

On a netlist both routes are exact, so the netlist cannot choose between them. A bench can, because every reading has an error, and the routes combine their readings differently.

Give every reading an independent error of one part in a thousand and propagate it. The series route computes $k = (L_\text{aid} - L_\text{opp})/4\sqrt{L_1 L_2}$ from four readings, and its uncertainty is about 0.001 at every coupling — slightly less at weak coupling, slightly more at strong, but essentially flat. The short-circuit route computes $k$ from a ratio that is itself $1 - k^2$, so its uncertainty is proportional to $1 - k^2$: 0.0011 at $k = 0.5$, where the two routes are equally good, and 1.4 × 10⁻⁶ at 0.999 — 943 times better than the series route at the coupling of an ordinary transformer.

The crossover, at $k = 0.513$, says the two routes divide the range between them. Below it the series route is the better instrument, which is the weak-coupling regime of two coils on a bench, an antenna’s matching loop, a crosstalk path. Above it the short-circuit route is better, and above 0.9 it is better by orders of magnitude. The two essays together therefore describe one measurement strategy rather than two alternatives: series for loose, shorted for tight.

What a designer wants is the leakage #

The comparison is sharper still when the target is the leakage inductance itself rather than $k$.

The shorted reading is the leakage seen from the first winding, so the short-circuit route knows it to the meter’s own 0.1 per cent at every coupling. The series route has to compute it as $L_1(1 - k^2)$ from its $k$, and an error $\delta k$ becomes an error $2k\,\delta k$ in $1 - k^2$ — which, divided by a $1 - k^2$ that is small, is enormous. The leakage from the series route is 1.2 per cent uncertain at $k = 0.9$, 13 per cent at 0.99 and 133 per cent at 0.999.

At a hundred and thirty-three per cent a measurement has stopped measuring. The number makes the point, but a distribution makes it vividly.

Four hundred benches, each with its own seeded 0.1 per cent errors on every reading, measure a pair whose leakage is 19.99 microhenries. Shorted, they report it with a spread of 0.10 per cent — a spike. Through the series route, they report a spread of 131 per cent, against the 133 per cent the propagation predicted, and 88 of the 400 report a negative leakage: a coupling coefficient greater than one, which is not a measurement of a transformer but of a meter. A designer reading the series difference on a good bench, with a meter better than most, would be about as likely to get a nonsensical answer as a right one.

That is why the leakage is specified the way it is on every transformer datasheet that quotes it: “measured at the primary with the secondary shorted”. The specification is the measurement, and this figure is the reason the measurement is that one.

A short is a short only above a frequency #

The shorted winding is never a perfect short. Its own resistance is in series with whatever shorts it, and it decides whether the winding’s current is large enough to cancel the shared flux.

With the winding’s resistance $R$ in the short, the shorted reading is

$$L_\text{short} = L_1\left[(1 - k^2) + k^2\frac{x^2}{1 + x^2}\right], \qquad x = \frac{R}{\omega L_2},$$

and the solve matches it at every frequency drawn. The second term is the shared flux the short failed to cancel. It falls as the square of frequency, but it is compared with $1 - k^2$, so the tighter the coupling the further it must fall. With an ohm in the forty-millihenry winding the reading is within one per cent of the leakage above 82.2 Hz at $k = 0.9$, 279 Hz at 0.99 and 889 Hz at 0.999 — a frequency rising as $k/\sqrt{1 - k^2}$, because one per cent of a small number needs $\omega L_2$ more than ten times $R\,k/\sqrt{1-k^2}$.

That is a practical constraint with no counterpart in the series route. A 50 Hz mains transformer measured shorted at its own line frequency, with the resistance real windings carry, reports a leakage that is partly the winding’s resistance seen through the coupling; the measurement belongs at a kilohertz or above. The thresholds scale with the resistance: a quarter of an ohm moves every threshold down by four, four ohms moves them up by four.

Which winding to short #

The leakage of a pair belongs to the pair, not to either winding, and the ratio of the shorted to the open reading should be the same from both sides. It is — and the sides still differ, in a way the previous section predicts.

Read from the first winding with the second shorted, and from the second with the first shorted, the ratio of shorted to open readings tends to the same $0.0199 = 1 - k^2$. The leakage seen from each side is that side’s own inductance times this ratio — 199 microhenries from the ten-millihenry side, 796 from the forty — and the two are the same leakage referred through the square of the turns ratio.

What differs is the frequency above which each side’s reading can be trusted, and it is set by the shorted winding’s time constant $L/R$. Shorting the forty-millihenry winding, whose time constant is 40 milliseconds, gives a reading within one per cent above 282 Hz; shorting the ten-millihenry winding, whose half-ohm gives it 20 milliseconds, needs 562 Hz. The winding to short is the one with the longer time constant, which on most transformers is the one wound with the heavier wire. When both windings fill equal shares of a window, their time constants come out equal, and it does not matter.

Where an error in the leakage goes #

The leakage is not a curiosity of the measurement. It is a component, and in most circuits that use a transformer it is the component that decides a specification, so an error in it propagates into a number somebody signs off.

What coupling buys, and where measured six transformers from $k = 0.8$ to $0.999$ and found the upper band edge moving by a factor of 168 across them while the lower edge moved by eight per cent. The upper edge is set by the leakage, so a leakage known to 133 per cent is an upper band edge known to within a factor of two either way — a bandwidth specification that a series measurement cannot support and a shorted one supports to a tenth of a per cent. The same is true in time rather than frequency. The inductance that limits, and lifts found leakage dividing a rectifier’s first current peak by seven and lifting its output above the supply’s own peak; a leakage the bench cannot pin down is an inrush current and a regulation figure it cannot pin down either.

In a switching converter the leakage is the energy that has to go somewhere every cycle. A primary current of a few amperes through a few microhenries of leakage stores tens of microjoules that the secondary never receives, and a snubber or clamp is sized to absorb it; sized from a series measurement with 88 chances in 400 of a negative leakage, it would be sized from noise. That is the practical reason the shorted reading is what transformer manufacturers quote and what a converter designer asks for.

Where the leakage is the component #

There is one class of coupled pair in which the leakage is not an error at all but the reason the part is bought. The inductor one mode cannot see measured a common-mode choke: two windings on one core, presenting a millihenry to current flowing the same way through both and a microhenry to current flowing opposite ways. The microhenry is the leakage. A current in opposite directions through two tightly coupled windings is exactly the shorted measurement, because each winding’s current cancels the other’s shared flux, and the ratio of the two inductances the choke presents is $2/(1-k)$ — two thousand at $k = 0.999$.

So the shorted reading of a common-mode choke is its differential-mode inductance, the number that decides how much the choke filters the wanted signal along with the unwanted one. The series-opposing reading of the same choke measures nearly the same quantity, because opposing series connection of a tightly coupled pair is also a leakage measurement — but it measures it as a small difference of two large terms, with exactly the precision problem drawn above. The bench practice of reading a choke’s differential inductance with one winding driven and the other shorted is this essay’s conclusion, arrived at by people who needed the number before anybody drew the figure.

Interleaving is a choice, not an improvement is the other side of the same quantity: sectioning a transformer’s windings divides its leakage by twenty-one and multiplies its winding capacitance by eleven. Verifying a factor of twenty-one in the leakage between two builds of one transformer is exactly the measurement the series route cannot make, since both builds sit above $k = 0.999$, and the shorted route makes to a tenth of a per cent.

Where the leakage lives, and why the split is not on the bench #

The shorted reading measures the leakage as a total, referred to whichever side is driven. Every transformer model splits it into a primary and a secondary part, and the inductance that is a shape computed where it physically lives — in the window between the windings, as the field that links one and not the other. Terminal measurements cannot make that split. The pair’s terminal behaviour is fixed by three numbers, $L_1$, $L_2$ and $M$, and a T-shaped equivalent with an ideal transformer of any ratio can be fitted to those three; each choice of ratio puts a different share of the leakage on each side and gives an identical impedance matrix. Two ports from two one-ports measured that matrix and found it reciprocal to $2.8 \times 10^{-16}$; reciprocity is exactly why the ratio of shorted to open reading is the same from both sides, and the same symmetry is why no bench measurement of the terminals can divide the leakage between them.

That is not a failing of the bench. For a circuit, only the total matters: which picture sets the upper edge and the band a turns ratio holds over each depend on the leakage referred to one side, and the shorted reading supplies it. The split matters to a winding designer moving the field around the window, and that is a field computation rather than a measurement.

How the readings were made, and what they leave out #

Every reading is a netlist: two inductors with a coupling element between them, each winding with its own resistance in series, the driven winding fed from a voltage source through ten megohms so that the drive is a current, and the other winding loaded by a resistor that is a nanohm for a short and a gigohm for an open. The inductance read is the imaginary part of voltage over current divided by the angular frequency, which is what an inductance meter reports. The shorted reading is held against $L_1[(1-k^2) + k^2x^2/(1+x^2)]$ to two parts in ten thousand at every frequency drawn, and the open reading against $L_1$ to a part in a million.

The bench uncertainty is propagated from independent 0.1 per cent errors on every reading, by perturbing one reading at a time; the four hundred benches perturb all of them at once from a seeded generator, and their spread is held against the propagation to twelve per cent, which is the sampling error four hundred benches allow.

The model has no core loss, no winding capacitance and no frequency dependence in the windings’ resistance. Each of those changes the shorted reading at the top of the frequency range: a winding capacitance resonates with the leakage, so the shorted reading rises towards a resonance the inductor that is a capacitor found in a single winding; skin and proximity effect raise the shorted winding’s resistance with frequency, which pushes the threshold above up again. There is a window between the frequency the resistance sets and the one the capacitance sets, and for a pair with tight coupling and much capacitance it can close.

Still open: the window that closes, a meter’s phase error, and the gapped core #

The window between two edges. The shorted reading is right above a frequency set by the shorted winding’s L/R, rising as $1/\sqrt{1-k^2}$, and wrong again near the resonance of the leakage with the winding capacitance, which falls as the leakage falls. So a tighter coupling narrows the band in which the leakage can be measured at all, from both ends. The coupling at which the window closes, for a stated winding resistance and capacitance, is a solve on this pair with a capacitor added.

A meter that reads phase as well as magnitude. The shorted winding’s resistance appears in the shorted reading’s real part as well as its imaginary part, and an LCR meter that reports both could separate them. Whether reading the real part lets the measurement be made below the frequency the resistance sets — fitting $x$ rather than waiting for it to vanish — is a question about how well the two parts are measured, which is a phase error specification rather than a magnitude one.

A core with a gap. Gapping a core lowers both windings’ inductance and the coupling with them, and the leakage that matters in a flyback converter is then a larger share of a smaller inductance. Whether the leakage itself moves with the gap, or only the magnetising inductance does, is a field question that the shorted reading can answer from outside: read it at several gaps and see which part of the pair moved.