The other half of the same window

The assumption that is a geometry solved a winding window in two dimensions and settled a sentence that every alternating-current resistance in this collection had rested on. It computed a loss, a resistance ratio, a per-layer distribution and a leakage inductance, and it did all of that from one field.

A winding window has two parameters, and that solve computed one of them.

The other is the capacitance between the windings, and it is not a different geometry or a different part. It is the same cross-section — the same copper, the same films, the same core — read with a different equation. Where the magnetic solve asks how the current arranges itself, the electrostatic one asks how the voltage does, and the two answers together are what a transformer actually is above the frequency at which either one alone stops describing it.

Two things are the opposite way round #

The magnetic problem and the electrostatic one are written on the same grid and differ in exactly two places, and both differences are the kind that give a plausible answer to the wrong question if they are taken the wrong way round.

The iron is a Dirichlet boundary here and a Neumann one there. An infinitely permeable core carries no tangential magnetic field, so the magnetic solve meets it with ∂A/∂n = 0 and the flux runs along it. An earthed core is an equipotential, so the electrostatic solve meets it with φ = 0 and the field arrives at right angles to it. The condition that is right in one problem is the one that is wrong in the other, and the notation gives no help at all: both are “the core”, both are drawn as the same grey rectangle, and getting either backwards produces a converged solution to a different transformer.

And a conductor carries a prescribed potential rather than a prescribed current. The magnetic solve gives each conductor an unknown axial electric field and constrains its total current, because a circuit prescribes current and not current density. The electrostatic solve does the reverse: each foil is an equipotential at a voltage the winding decides, and what is unknown is the charge that sits on it. That is why the copper in the picture is shaded rather than uniform. A foil eight turns along a winding is at eight turns’ worth of voltage, and the capacitance is a weighted sum over those, not a single plate pair.

What the two solves share, and the check that comes out of it #

Both problems are solved on the same cells, and that is a convenience rather than a result. What is a result is that each of them can be read twice.

A capacitance is an energy, and the energy can be had two ways from one solved potential: integrate ½ε|∇φ|² over every cell of the window, or collect the charge on each electrode from the flux leaving it and form ½ΣQV at the terminals. The first never mentions an electrode and the second never mentions a cell. They agree to the last digits the arithmetic has, and the agreement is worth something specific: the volume integral is sensitive to the interior stencil and the surface sum is sensitive to how the electrode boundary is walked, so a mistake in either one moves only one of them.

That is the same pairing the magnetic solve uses when it puts the loss integrated over the copper against the loss the terminals deliver, and the same one one step computed twice insists on for a transient. It is not a tolerance and it is not decoration. Two routes that share an assumption are one route, however different their arithmetic — which is the trap the assumption that is a geometry had to be written to escape, where three independent-looking computations of a winding’s resistance turned out to be one sentence written down three times.

The calibration, which comes first #

A field solver returns a plausible number for a mis-stated problem, so the first thing to ask of one is a question whose answer is already known. The magnetic solve was entitled to its disagreements with a closed form because it first reproduced that closed form to 0.155 per cent in the geometry where the closed form is exact. This one needs the same entitlement and it needs it three times, because there are three separate things that can be wrong: the interior stencil, the treatment of the dielectric interfaces, and the geometry.

The flat line is the interesting one and it is worth being clear about why it is flat. A finite-volume scheme that takes the harmonic mean of the permittivity at each face is exact for a one-dimensional stack of dielectrics, whatever the cell size, because the harmonic mean is precisely the rule that makes series capacitances add correctly. The obvious alternative — the arithmetic mean of the two cells’ permittivities — is not exact, and the difference is not a rounding matter: it reads 1.85, 0.92 and 0.46 per cent as the grid is refined, which is wrong, plausible, and first order. A solver built the second way converges, publishes a convergence study, and is a per cent out on every dielectric interface it ever crosses. That is the failure this calibration exists to exclude, and the only thing that excludes it is a case with an exact answer.

And the winding’s own order of 1.34 is not a defect in the scheme. The manufactured field gives 1.97, so the interior is second order as designed. What the winding adds is a corner: a conductor’s right angle leaves the surrounding dielectric a 270-degree wedge, and the potential near the tip of such a wedge goes as $r^{2/3}$ rather than analytically. An energy — which is what a capacitance is — converges at twice that exponent, and twice two thirds is four thirds. The measured 1.34 is that number, not an accident of the mesh.

That distinction decides what to do about it, which is nothing. Refining the grid buys convergence at order 4/3 and no faster, because the limit is the geometry rather than the stencil, and the working grid used everywhere below sits 0.21 per cent from the extrapolated answer. A per cent is not what this essay’s findings turn on; factors of forty are. The same reasoning appears in the digits the arithmetic did not have, where the question is likewise which of two candidate limits a number has run into, and the answer decides whether more effort would help.

Where a winding keeps its electric energy #

Every capacitance in this family is an energy — half ε|∇φ|² integrated over the window — so the question “what is the capacitance” has a companion question that is more useful: which material is holding it.

The mechanism is not subtle and it is worth stating in one line, because it inverts the intuition a schematic gives. A permittivity of three and a half pulls the field into the layer that has it, and a thin layer holding a given voltage carries the strongest field in the problem, since field is volts per metre and the metres are few. The two effects multiply. So the electric energy of a transformer is kept almost entirely in films that nobody specifies, chose, or measures.

That is the shape of finding this collection keeps arriving at from different directions. The parasitic is the component is one way to say it; the sharper way is that the quantity is decided by a part of the object that does not appear in its description at all. A foil is bought on its thickness and its conductivity. The film coated on it is whatever the supplier coats it with, and that film is the capacitor.

The one parameter that changes no dimension #

If the energy is in the films, then the films’ permittivity is a design variable — and it is the only one available that changes no dimension, costs no window and can be applied to a design already wound.

Two findings, and the small one is the useful one.

The scaling holds, which means a designer who has solved one window can price a change of tape by multiplication rather than by solving again. That is worth having precisely because the solve is expensive and the question is asked often.

And the shortfall is a measurement rather than an error. The missing 1.7 per cent is the share of the coupling that leaves the edge of one winding, goes round through the air above and below the copper, and comes back — a path that does not get cheaper when the tape does. So the shortfall grows as the permittivity rises rather than shrinking, which is the signature that says it is a second path rather than a numerical residue. A discrepancy that behaves like an error would fall.

A core is a third electrode, and somebody decides what it is with a wire #

The magnetic problem treats the core as a boundary and is finished with it. The electrostatic problem cannot be, because a conductor at an unspecified potential is not a boundary condition at all — it is a third terminal, and what it is connected to is a wire that a schematic does not usually draw.

Three of those numbers are close together and one is not, and the one that is not is the one a designer chooses without noticing. Bonding the core to the quiet end of a winding and bonding it to the noisy end give the same capacitance at the terminals and differ by 26 per cent in the primary’s own self-capacitance, because a core bonded to the moving end of a winding is a large electrode being driven, and everything it faces is now being driven through it.

A floating core is not neutral either. It settles at 0.500 volts — the average of what surrounds it, which for a symmetric window is exactly half — and is then a coupling path between the two windings that nothing in the circuit description mentions. The distinction between a model’s parameter and a model’s boundary is the same one every model has an edge is about, arriving here as a piece of hardware: the core stops being a boundary and becomes a node the moment the equation changes.

The capacitance is not one number, and a bridge cannot tell #

The last figure is the one that decides how every number above should be quoted, and it is the reason this rung exists before the trade the next one is about.

The bridge is not wrong. It measures a real quantity — the capacitance between two equipotential electrodes — and that quantity is the same for all six arrangements because strapping each winding’s ends together is exactly what makes the arrangement stop mattering. It is simply not the quantity the circuit has. In the circuit the voltage varies along each winding, the two variations either add or subtract, and the energy is a quadratic form over the whole distribution rather than a plate pair.

This is the same species of defect as the one in the capacitance that is not one number, where a ceramic’s small-signal, charge-average and bridge readings are three different numbers, all correct, all called the capacitance. There the difference is nonlinearity; here it is geometry. In both cases the measurement condition is part of the answer, and in both cases the specified condition is the one least like the circuit.

And there is no rule to carry away, which is itself the finding. Reversing the secondary is worth 1.69 times at one section and costs 8.1 times at six, so the ordering of the arrangements is not even stable. What replaces the rule is arithmetic: once the capacitance matrix of the cross-section exists, evaluating any particular arrangement is a product of three numbers, and the matrix is solved once.

Two numbers are a frequency and an impedance #

A capacitance beside an inductance is not two facts about a transformer. It is one fact written in two halves, because what a circuit does with them is form their product and their ratio, and those are the only two combinations that mean anything.

The product gives a frequency. This window’s 8.399 microhenries and 926.9 picofarads a metre put a resonance at 1.804 megahertz — and that is not an abstraction, it is the mechanism that closes the useful band at the top. The band a turns ratio holds over measures both edges of that band and finds the lower one nearly immovable while the upper one moves by three decades; this is what the upper one is made of, and it has now been computed from the geometry rather than fitted to a measurement.

The ratio gives an impedance. √(L/C) is 95.19 ohms for this window, and it is the impedance the winding presents to a disturbance that arrives faster than the transformer can respond as a transformer — a switching edge, an electrostatic discharge, the common-mode step that the millimetre that becomes common mode is about. Below that frequency the part is a turns ratio; above it, it is a piece of transmission line with this impedance, and the two descriptions have to be joined somewhere.

The reason this matters for the design decisions that follow is that the two combinations move differently. Thickening the interlayer from a third of a millimetre to a whole one takes the capacitance from 927 to 295 picofarads a metre and raises the leakage from 8.399 to 10.991 microhenries — because the same millimetre that separates the electrodes also separates the ampere-turns. The product falls to 0.42 of what it was, so the resonance rises to 2.795 megahertz; the ratio rises to 193 ohms, more than doubling. One action, both quantities, and they do not move together.

That is the ordinary case rather than the exception. Almost every action available to a winding designer moves leakage and capacitance in opposite directions, because both are decided by the same distance between the same two pieces of copper. The inductance that is a shape measured the first half of that and stopped there, and the reason it stopped is that it had no way to compute the second.

What this does not say #

It does not say a lumped interwinding capacitance is useless. Above the frequency where it matters at all, a transformer is a two-port with a leakage inductance in series and a capacitance across it, and that model predicts the upper band edge the band a turns ratio holds over measures. What the field adds is which capacitance to put in it, and the answer is that the choice depends on the wiring rather than on the part.

It does not extend to three dimensions. Every number here is per metre of conductor, as every number in the magnetic solve is, and a real winding has a mean turn length, an outer winding longer than an inner one, and end regions where the field is neither of the two things solved here.

And it says nothing yet about the trade. A capacitance of 927 picofarads a metre beside a leakage of 8.399 microhenries a metre is two numbers, not a decision. The decision is what happens to both when the windings are split, and it goes in opposite directions — which is the next rung.

The number worth carrying #

Eighty-nine per cent of the energy in 14.1 per cent of the area, and a factor of forty between the largest and smallest capacitance the same transformer can present depending on how it is wired.

The habit that goes with it is the one the calibration is about. Three exact cases were needed before any of those numbers could be quoted, and each excludes a different failure: the layered slab excludes a wrong dielectric-interface rule that would have been a plausible one per cent, the manufactured field excludes a wrong interior stencil, and the geometry’s own 4/3 says which of the remaining error is worth chasing. A solver with one calibration has an instrument whose disagreements are unattributable — the same argument one step computed twice makes for a transient and exact outside and wrong within makes for an equivalent circuit, arriving here in a problem where the closed form and the geometry fail in different places and a single check would have caught neither.