The window that moves

The leakage a series reading cannot see found that shorting the other winding reads a transformer’s leakage directly, to a meter’s own 0.1 per cent, where the series-aiding and series-opposing readings leave it 133 per cent uncertain at $k = 0.999$. It also found the first limit on the reading: a shorted winding is never a perfect short, so the reading is the leakage only above a frequency set by that winding’s resistance, and that frequency rises as $1/\sqrt{1-k^2}$. It closed by anticipating a second limit. The reading must be wrong again near the resonance of the leakage with the winding capacitance, which it took to fall as the leakage falls. So a tighter coupling would narrow the band in which the leakage can be measured at all, from both ends, and there would be a coupling at which the window closes.

That was a reasonable picture to arrive at, and it has the direction of the second edge the wrong way round. A smaller inductance resonates with the same capacitance at a higher frequency. The first few lines of the calculation show it, and what they show is more interesting than a closing window. The window does not close. It moves, and it keeps its width while it moves. That width turns out to have no leakage in it at all, and it can be written down in one line. It is a property of the winding’s construction and hardly depends on how well the windings are coupled, which is about as far as it could be from what was expected.

A reading with both limits in it #

The pair is the earlier essay’s: 10 mH driven and 40 mH shorted, with 1 Ω in the shorted winding and 0.5 Ω in the driven one. To that is added 100 pF across the driven terminals, as the lumped sum of what a winding has across itself and to its neighbour. A real pair has its capacitance distributed through its layers, as the inductor that is a capacitor measured for a single coil, but a shorted secondary removes one winding’s capacitance from the circuit, and what stands across the driven winding is a reasonable lumped value for the first resonance.

The impedance at the terminals has an exact form. The pair, without the capacitor, is

$$Z_p = R_1 + j\omega L_1 + \frac{(\omega M)^2}{R_2 + j\omega L_2},$$

and the capacitor then sits across that. The reading is the reactance divided by the frequency, which is what an LCR meter reports as an inductance. A netlist of two inductors and a coupling, solved at three frequencies either side of the band, returns the same reactance as the closed form to a part in a thousand, so nothing is being lost in the algebra.

Below the band, the reading is too large by the term the earlier essay found: $L_1k^2x^2/(1+x^2)$ with $x = R_2/\omega L_2$, the shared flux that the short has failed to cancel. Above it, the reading is too large for a different reason. The leakage resonates with the capacitor, and below that resonance the terminal reactance is raised by $1/(1 - (f/f_r)^2)$. The reading climbs without limit towards the resonance, and past it it goes negative, which an LCR meter displays as a capacitance.

Both edges go the same way #

With a 1 per cent tolerance the two edges have simple forms. The lower is ten times the shorted winding’s corner, $R_2/2\pi L_2$, multiplied by $k/\sqrt{1-k^2}$. The upper is a tenth of the leakage’s resonance, $0.1/(2\pi\sqrt{L_1(1-k^2)C})$.

Both contain $1/\sqrt{1 - k^2}$. As the coupling tightens, the leakage shrinks, and that does two things that happen to have the same size. It makes one per cent of the leakage a smaller target, so the short has to be better, and this raises the lower edge. It also makes the inductance that resonates with $C$ smaller, and this raises the upper edge. Bisected on the reading, the band runs 82.1 Hz to 36.3 kHz at $k = 0.9$, 279 Hz to 112 kHz at 0.99, 889 Hz to 354 kHz at 0.999 and 2.81 kHz to 1.12 MHz at 0.9999. The ratio is 443 at 0.9 and 402, 399 and 399 at the others.

The practical reading is the reverse of the anticipated difficulty. The measurement on a tighter transformer is not harder because a band has closed. It is a measurement at a higher frequency, and a meter that tops out at 10 kHz can do it at $k = 0.99$ and $k = 0.999$, and it is near its limit at 0.9999. That is a real constraint, but it is a constraint on the meter. A designer choosing the frequency for this reading should choose it from the coupling expected, near the geometric middle of the band: about 5.6 kHz at 0.99 and 56 kHz at 0.9999. A fixed 1 kHz, a common default on a bench meter, is inside the band for one of those and outside it for the other, and at 0.9999 the reading it gives is too high, by the amount the earlier essay drew.

A width with no leakage in it #

Dividing the two laws makes the coupling almost disappear:

$$\frac{f_{hi}}{f_{lo}} = \frac{0.01\,L_2}{R_2\,k\,\sqrt{L_1C}}.$$

The factor $\sqrt{1-k^2}$ has cancelled. Only the $k$ remains in the denominator, which is close to one on anything anyone would call a transformer. The band’s width is a property of the windings alone: their inductances, the resistance of the one being shorted and the capacitance across the other.

The lines fall as one over the winding’s resistance and each capacitance a factor of ten down gives a width $\sqrt{10}$ times larger. The law agrees within 12 per cent everywhere the width is above about three. Then the window closes, and it does so at a definite resistance: 404 Ω for this pair with 100 pF, or 128 Ω with 1 nF. At that point there is no frequency at which the shorted reading is within 1 per cent of the leakage, for any coupling.

So a window does close, as expected, but on a different axis from the one expected. A pair with no band at $k = 0.9$ has no band at 0.9999 either, and a pair that has one keeps it. For a real 40 mH winding a closing resistance of hundreds of ohms is not a practical issue, and at audio-transformer sizes the width is in the hundreds. But the formula can be turned round to give something useful. It is a kind of Q: $L_2/(R_2\sqrt{L_1C})$ is the shorted winding’s reactance over its resistance at the frequency where $L_1$ would resonate with $C$. Winding that has this number high has room to read its leakage. Winding that has it low does not, and no amount of coupling helps.

What turns do #

An engineer’s next question is how this scales from the small winding on a bench to a real one. Holding the winding window fixed, the number of turns sets the inductance as its square, and the resistance as its square too, because more turns in the same space use a thinner wire.

The lower edge is the ratio $R_2/L_2$, which contains no turns, and it stays at 889 Hz over a range of a hundred in turns, until the upper edge starts to approach it. The upper edge falls in proportion to the turns, 354 kHz at the reference and 71.1 kHz at five times, because the leakage rises as their square and the capacitance is held. So the band is 398 times wide at the reference, 80 at five times and 22 at twenty.

This is why large power transformers and high-turns transformers are harder to read than the small ones. A mains transformer with many turns has the band’s lower edge unchanged and its upper edge brought down near it, because its capacitance has not fallen in the way its inductance has risen. Holding the capacitance fixed is a simplification, because adding layers changes it, and the other half of the same window discusses that in more detail. But if a winding’s capacitance rises at all as its turns rise, it takes the band down faster than this line.

Which picture sets the upper edge discussed a transformer’s band edges in transmission, set by its own leakage and capacitance, and that is the same resonance. A reading’s upper edge is a tenth of it. So a transformer’s useful band and its measurement band share a physical limit, and a transformer that carries a wide band is one that can be measured.

The real part, which knows what the short missed #

The earlier essay also asked a different question about the lower edge: a meter reporting real and imaginary parts could separate what the short has not cancelled. It can do this exactly. From the impedance formula, the real part above $R_1$ is $(\omega M)^2R_2/(R_2^2+\omega^2L_2^2)$. The error in the reactance is the same thing multiplied by $R_2/(\omega L_2)$. So

$$L_{leak} = \frac{\mathrm{Im}\,Z}{\omega} - (\mathrm{Re}\,Z - R_1)\frac{R_2}{\omega^2L_2}$$

is exact at every frequency, and it needs only the two winding resistances and $L_2$, all of which are measurable beforehand.

On a perfect meter the correction works to the arithmetic’s limit. On a real one the correction’s usefulness is decided by a small number: the meter’s phase error. Close to the lower edge, the terminal impedance’s reactance is small next to its real part, since the reading of a 20 µH leakage at a kilohertz is 0.13 Ω with half an ohm of winding in series. So a phase error of 0.1° on an impedance of about half an ohm, which is a small and very ordinary specification, rotates about a millohm of real part into the reactance: nearly one per cent of the reading. With the correction applied, the reading on such a meter reaches 1 per cent at 993 Hz, which is no better than the plain 906 Hz on a perfect one. The plain reading on the same meter reaches 1 per cent earlier, at 572 Hz, but that is an accident of the sign of the meter’s error here, and the plain reading’s own loss term has been cancelled by it. It would be worse at the other sign.

A meter ten times better, 0.01 per cent and 0.01°, extends the corrected reading down to 109 Hz, a factor of eight below the plain edge. So the correction is a good idea on a good instrument, and on an ordinary one it does not move the edge. The way to do this measurement on an ordinary one is simply to raise the frequency, since the band has plenty of room on its upper side. The meter that read a difference found that a measurement’s arithmetic can look far more capable than its reading allows, and a phase accuracy is the specification that decides it here.

The same two numbers in a pulse #

The two edges are not specific to a measurement. The band counted in rise times described a transformer by its pulse’s rise and droop, with a rise time of $0.35/f_H$ set by the same leakage and the same capacitance, and that leakage is what a shorted reading reads. A pair with $k = 0.99$ and these parts resonates at 1.13 MHz, and a pulse through it has a rise near 300 ns, and an engineer can read off the rise a design will have by measuring its leakage once. The reverse is also useful. A pulse’s rise, seen on a bench, is an independent estimate of the leakage and capacitance together, and it needs no short at all. The rise’s reading and the shorted reading should agree to the extent that the lumped resonance is the true first one, and a disagreement between them says how far the winding’s capacitance has failed to be a single lump. That is a cheap cross-check, and it uses only the two instruments any bench already has.

Sizing a measurement #

Together, the laws give a simple procedure. Estimate the coupling from the winding type, and from that the lower edge. Measure $R_2$ and $L_2$ directly, for the corner. Estimate $C$ from the winding’s first self-resonance, for the upper edge. Pick a frequency near the geometric mean. Then a second reading at a tenth or ten times that frequency checks the result, since a reading in the middle of a four-hundred-fold band moves by very little over a decade. If a second reading moves by more than a per cent the first was out of the band.

There is also a useful number in a failure. A reading that falls steadily with frequency, then flattens, then rises, has given its own band without any formula, and an instrument that can sweep will read the leakage off the flat part. The flat part’s length then measures the width formula’s combination of parameters, and its upper end gives the capacitance. A sweep is a richer measurement than one spot reading, and on a meter that offers both the sweep is what should be used.

Where the band goes entirely showed a pair with the transmission band gone, by the leakage and capacitance reaching the magnetising limit. The measurement band here has an analogue of the same event, in which the width reaches one. Both happen at a stated combination of the winding’s parts, and neither is a function of the coupling in the way a first look suggests.

What this leaves out #

The capacitance is one lumped element, and a real winding has a distribution with several resonances, of which the first is what is meant here. The shorted winding’s resistance is taken as constant, with skin effect absent; at the upper end of a wide band, $R_2$ rises, the width formula’s numerator stays fixed and the band narrows a little. The core is linear and lossless, and core loss appears in the real part of the reading in a way this correction would attribute to the short: on a real transformer that would enter the corrected reading directly. A gapped core is also absent, which is the third of the earlier essay’s questions. Its result needs a field solve to say how the leakage moves, and a measurement through this window can answer it from outside.

The numbers to keep #

A shorted reading is the leakage within 1 per cent from $10(R_2/2\pi L_2)k/\sqrt{1-k^2}$ up to a tenth of the leakage’s resonance with the capacitance across the driven winding. Both rise as $1/\sqrt{1-k^2}$, so the band is 279 Hz to 112 kHz at $k = 0.99$ and 2.81 kHz to 1.12 MHz at 0.9999, with the width unmoved. It is $0.01L_2/(R_2k\sqrt{L_1C})$: about four hundred here, closing only if $R_2$ reaches 404 Ω. Turns in a fixed window hold the lower edge and bring down the upper, in proportion. Correcting from the real part is exact on paper, and with a 0.1° meter it does not move the lower edge at all. What a tighter coupling costs is a higher measuring frequency, and the meter has to reach it.

Still open: core loss in the real part, the gapped core, and the second resonance #

Core loss in the correction. A core’s loss appears as a real part of the driven winding’s reading, and at a shorted winding’s small flux it should be small. Whether it is small enough for the correction to survive on a real core, or whether it adds its own edge, is a measurement that would decide if the correction is usable at all outside a netlist.

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. The window’s lower edge rises with $k/\sqrt{1-k^2}$ and falls with a smaller $L_2$ in its corner, and those move together in a way the shorted reading can read from outside.

The second resonance. A winding with distributed capacitance has further resonances, and below the first of them the reactance is already affected by them. A reading a tenth below the first resonance is far from the second, but a winding with a low-frequency interwinding resonance could put one in the band. How far a real winding’s lumped first resonance can be trusted for the upper edge is a question for the winding-capacitance measurements.