Two ports from two one-ports

Almost every component this collection has drawn is a one-port: two terminals, one impedance, one complex number at each frequency. A resistor, a capacitor, an inductor, a real capacitor with its lead inductance, a source with its internal resistance — all of them are completely described by what they do between two points.

Two coupled inductors are not. They have four terminals and two independent currents, and no single impedance describes them. The smallest complete description is a two-by-two matrix, and this essay measures its four entries by the definition rather than by an identity.

The definition, performed #

The impedance parameters of a two-port relate its two port voltages to its two port currents:

$$v_1 = z_{11}i_1 + z_{12}i_2, \qquad v_2 = z_{21}i_1 + z_{22}i_2$$

Each parameter has an operational definition and every one of them names an open circuit. z₁₁ is v₁/i₁ with port two open; z₂₁ is v₂/i₁ with port two open; z₂₂ and z₁₂ are the same two with the ports exchanged.

So there are four solves and each one is a netlist: drive one port through a large resistance so the current is set by it, leave the other port open through a very large resistance, and read two node voltages and one branch current. Nothing is derived; the ratios are taken directly.

For the pair drawn here — 10 mH and 40 mH at k = 0.8 — the four come out as

with the worst departure across four couplings at 2.2 × 10⁻¹⁶.

Reciprocity, which is the part worth measuring #

Three of those four rows are checks that the netlist contains what it was given. The fourth is not: it is a claim about the device.

z₁₂ = z₂₁ says that the voltage appearing at port two per amp into port one is exactly the voltage appearing at port one per amp into port two. That is reciprocity, and it is a genuine physical property — not every two-port has it. A circulator does not; a two-port containing a controlled source generally does not; and this site’s own E and G elements build non-reciprocal two-ports as a matter of course.

A coupled pair does have it, and the measurement here confirms it to 2.8 × 10⁻¹⁶ across four couplings from 0.2 to 0.95.

The reason that is worth doing rather than asserting is the reason the previous essay gives. A coupling stamped into one row of the matrix instead of two would be exactly a non-reciprocal two-port: energy would appear to flow more easily in one direction than the other, current law would be unmoved, the energy balance would be unmoved, and the response would look entirely plausible. The symmetry of the pair is the first property that error destroys, and this is the only measurement on the site that looks at it.

Why the open circuit is a large resistor #

A construction note that is not fussiness. There is no such thing as an open circuit in a nodal analysis: a node with no path to ground has no equation, and solveAt refuses such a network by name rather than returning a large plausible number.

So “open” is a 10¹² Ω resistor, which is a modelling decision with a range like everything else here. At the frequencies these measurements are taken at, the winding’s own reactance is a few hundred ohms against a terawatt-scale resistance, so the current diverted through it is a part in 10⁹ and the parameters come out to fifteen digits. At a frequency ten decades higher it would not, and the measurement would be of a divider rather than of a two-port.

The same reasoning applies to the drive. A 10⁷ Ω source resistance makes the port current independent of what the port does, which is the current source the netlist does not have. Both choices are stated because both are the kind of thing that quietly decides an answer, and because the site has been caught by exactly this before: a reactance of zero is the absence of an element, not an element of value zero, and stamping a 1 nΩ “wire” puts a 10⁹ beside a 10⁻² in one matrix and fails the site’s own current-law check by a part in 10⁴.

What the matrix says that the netlist does not #

Having four numbers rather than a netlist is worth something, and it is worth being precise about what.

The two-port is a complete description at one frequency. Anything connected to the pair — a source, a load, a feedback path, another two-port — can be analysed from the matrix alone, without knowing that the object inside is two windings rather than a bridge of resistors and controlled sources. That is the point of the representation and it is what makes cascaded systems tractable.

It makes the ideal transformer visible as a limit. As k → 1 with the inductances growing without bound, the four parameters all become infinite while their ratios stay finite — z₁₁/z₂₁ tends to √(L₁/L₂), which is the turns ratio. The ideal transformer is therefore not a member of this family of matrices at all; it is what the family approaches, and it is the reason K refuses a coupling of exactly one rather than clamping to it.

And it connects this field to one the site already has. The instruments field’s four-terminal essay is about the same distinction from the other side: a two-terminal measurement of a small resistance measures the leads as well, and the answer is to separate the current path from the voltage path — which is to treat the resistor as a four-terminal object rather than a two-terminal one. A coupled pair is the same lesson with the object genuinely having four terminals rather than being given them.

The four solves, and what each one costs #

It is worth counting what the definition actually requires, because the count is the reason two-port parameters are usually taken from a model rather than measured.

Four solves, each of a different netlist — the two drives differ in where the source is and where the open circuit is, so no two of the four share an assembly. On a bench that is four connections, four instrument settings and four readings, with the open circuit genuinely open and therefore genuinely sensitive to whatever stray capacitance is nearby.

That is the practical reason a data sheet gives an inductance and a coupling instead: two numbers, two readings, and the four parameters follow. This essay does it the long way round for the reason the site does everything the long way round — the short way is a rearrangement of the inputs, and cannot be wrong about the object because it never touches it.

There is a specific failure the short way would not catch and it is the one the field cares about. If the coupling had been stamped asymmetrically, an inductance-and-coupling description would still produce a symmetric matrix, because it constructs one; the four solves produce whatever the matrix actually is. That is why reciprocity is worth 2.8 × 10⁻¹⁶ of measurement rather than a sentence.

What the diagonal entries are not #

One correction worth making explicitly, because the notation invites it.

z₁₁ is measured with the secondary open, so it is the primary’s full self-inductance — 10 mH here, not the magnetising inductance and not the leakage. Neither of the two quantities the band essays are built on appears anywhere in the impedance matrix directly.

They are recoverable from it, and the recovery is the T-model: the magnetising inductance is M scaled by the turns ratio, and each leakage is the difference between a self-inductance and that. But they are a decomposition of these four numbers rather than more measurements of the pair, which is the same point the first essay in this field makes about deriving the band edges from them.

The practical version: an inductance meter reading a transformer’s primary with the secondary disconnected reads L₁, and a designer who takes that number as the magnetising inductance is out by 1/k², which is 56% at k = 0.8 and 2% at 0.99. The correct bench procedure is the one the previous essay describes — measure with the secondary open and again with it shorted, and the two together give the magnetising and the leakage separately.

What the matrix does when the coupling changes #

The slider on the figure moves the coupling and only two of the four entries move with it, which is the tidiest way to see what the coupling coefficient is.

The diagonal is fixed. Each winding’s own inductance is a property of that winding and its core and has nothing to say about the other one, which is obvious once written down and is worth writing down because the T-model’s “magnetising inductance” does move with the coupling and is easy to confuse with it.

The off-diagonal is linear in k, by construction — it is k√(L₁L₂) and √(L₁L₂) here is 20 mH, so the column reads 4, 8, 12, 16, 19. The perfectly coupled limit would be 20 mH, at which point the matrix becomes singular: its determinant L₁L₂ − M² is L₁L₂(1 − k²), which is zero at k = 1.

That determinant is the same quantity the energy argument uses, arriving from the two-port side. A singular impedance matrix means the two port currents are no longer independent — a current in one port forces one in the other — which is precisely what a perfect transformer does and precisely why it cannot be written as two coupled inductances with finite values.

The one thing four numbers cannot carry #

A two-port matrix is a complete description at one frequency, and every limitation of this essay follows from that clause.

It says nothing about saturation, which is an amplitude boundary and cannot appear in a linear description at all — the same reason a small-signal model has no distortion by construction. The core could be at 0.9 of its saturation flux or at a millionth of it and these four numbers would be identical.

It says nothing about the winding resistance’s frequency dependence, which would appear as the real parts of the diagonal terms growing with frequency in a way no two numbers evaluated at one frequency can express.

And it says nothing about the interwinding capacitance, which is not in this model at all: the four parameters here are purely inductive, and a real pair’s matrix has capacitive terms that dominate above the resonance the field’s third essay measures.

So the matrix is exact for what it describes and describes less than the netlist does. That is the ordinary relationship between a representation and an object on this site, and the useful discipline is to be able to say which one a given number came from — which for the four in this essay’s figure is four solves of a netlist, each performed by driving one port and opening the other.

Where two ports become three, and why the field stops at two #

A last note on scope, since the machinery would extend and deliberately does not.

K couples two inductors. Three coupled windings — a transformer with two secondaries, a three-phase transformer, an autotransformer with a tap — need three coupling coefficients, and the element as written handles that: three Ks, one per pair, each stamping its own two off-diagonal entries. The matrix stays affine in s, the verification’s rebuilt relation already sums over every partner a winding has, and nothing in lib/network.js would need changing.

What is missing is the constraint the three coefficients must satisfy. For two windings the condition is |k| < 1, which is the two-by-two inductance matrix being positive definite. For three it is that the three-by-three matrix is positive definite, which is not simply each pair being individually below one — three windings each coupled at 0.9 to the other two describe a pair of currents that would store negative energy, and the element as written would accept them.

So the refusal that makes the two-winding case safe does not generalise, and a three-winding field would have to check the determinant rather than the coefficients. That is a real piece of work, it is the natural next rung of this anchor, and it is stated here rather than discovered by whoever writes the essay that needs it.

The one-line version, for that person: the check is that the inductance matrix has all its leading principal minors positive, which for the two-by-two case reduces to exactly the |k| < 1 the solver already enforces — so the general test contains the specific one and would replace it rather than sit beside it.

Why the site did not need a two-port until now #

A closing observation about the collection rather than about the component.

Fifty-eight essays and eleven fields have been written without a two-port representation appearing anywhere, and that is not an omission. Modified nodal analysis works on netlists, and a netlist of any size is solved by the same code — there has never been a reason to reduce a subnetwork to four numbers and then reason about the four, because the whole thing can simply be solved.

Two-port parameters earn their place when the object inside is not available: a measured device, a component whose internals are a secret, a stage designed by somebody else. They are a way of carrying what a thing does without carrying what it is, which is the opposite of this site’s usual arrangement, where everything is built from primitives and nothing is quoted.

So this essay is unusual for the collection in taking a representation seriously rather than a circuit. What justifies it is that one of the four numbers carries a claim the netlist cannot state: reciprocity is a symmetry of the matrix, and a matrix is where a symmetry lives. The netlist has the property; only the two-port displays it.

That is a small argument for representations in general and it is bounded by the same clause as everything else here. The matrix is exact at one frequency, silent about amplitude, and blind to every mechanism the model it was measured on does not contain — which for this pair is saturation, the winding’s frequency-dependent resistance, and the capacitance between the windings, whose consequence which picture sets the upper edge measures.

Reciprocity, and the check it makes possible #

Two of the four parameters coming out equal to $2.8\times10^{-16}$ is the result on this page that does the most work elsewhere, because it is the first property a coupling stamped into one row instead of two would break — and nothing else in the machinery would notice.

One number from two measurements is the essay that explains why nothing else would. A coupling adds no current anywhere, so the current law is unmoved by it; a coupled pair dissipates nothing, so the energy balance is unmoved too. Both of this site’s standing verifications pass on a well-formed solution to a different circuit, which is exactly the failure mode what a network answers sets those two checks up to catch and cannot.

So this field has two independent guards on the same element, arrived at separately: the symmetry of the impedance matrix, measured here, and the series-aiding-against-series-opposing difference measured there. Neither is a refinement of the other — one is a property of the model’s algebra and one is a procedure a bench performs — and the coupling is the only element in this collection that needed both.

The reading that does not care which way round it is is where the property those guards rest on is measured in general, at five parts in $10^{14}$ over four decades, with the boundary being a transconductance of 0.78 femtosiemens.

Which is worth one closing note about what a two-port description is for. Four numbers at one frequency is more than a coupling coefficient and less than a netlist: it is exactly what a component can be characterised by without knowing what it will be connected to, and it is why an impedance matrix rather than a schematic is what a magnetic component’s measurement produces.