The band a turns ratio holds over

Fifty-eight essays into this collection, every boundary drawn has had one side. An ideal amplifier is right below 1.42 kHz. A capacitor is a capacitor below 4.69 MHz. A small-signal model holds below 7.30 mV; a noise floor holds above one. In each case a model is true on one side of a number and the figure’s job is to find the number.

A transformer is not like that. The turns ratio is wrong at low frequency, wrong at high frequency, and right in between — so the quantity the figure has to find is a pair, and what a designer buys is the distance between them.

One new element, and why only one #

The field needed exactly one addition to the solver, and the reason it needed only one is a decision made on this site’s first day.

lib/network.js gives every inductor a current unknown rather than an admittance of 1/(sL). Each inductor already owns a row of the matrix reading v − sL·i = 0, so coupling two of them is two off-diagonal entries in the same matrix:

$$v_1 - s(L_1 i_1 + M i_2) = 0, \qquad v_2 - s(M i_1 + L_2 i_2) = 0$$

The matrix stays affine in s, which is the property everything downstream depends on. The determinant is still a polynomial, so a transformer’s poles are recovered by the same sampling and rooting the rest of the site uses, and Bode, group delay, the step response by two routes and the Nyquist locus all work on it with no change at all. Written the other way — with admittances — a mutual inductance is a 1/s inside an inverse, and the entire pole machinery would have needed a second implementation.

The coupling refers to the two inductor objects rather than to node names, because there is no such thing as coupling between two places. That has a consequence the site has met before: sol.i(el) finds a branch current by object identity, so building the netlist twice gives a coupling that refers to inductors which are not in the network being solved. It is refused by name rather than ignored.

Three refusals, and one of them is an energy argument #

K declines three things, and the middle one is the interesting one.

A coupling naming an inductor the netlist does not contain is the identity trap above.

A coupling of an inductor to itself is not a coupling but a change of inductance, and asking for it means the caller meant something else.

A coupling coefficient of one or more is refused because of what a coupled pair stores:

$$W = \tfrac12\left(L_1 i_1^2 + 2M i_1 i_2 + L_2 i_2^2\right)$$

That quadratic form is positive for every pair of currents only while M² < L₁L₂ — which is |k| < 1. At exactly one the inductance matrix is singular, the two currents stop being independent unknowns, and the perfectly coupled transformer is not a netlist but a limit. Above one the pair stores negative energy at some pair of currents, which no component does.

It is refused rather than clamped, and the distinction matters here more than usual: k = 1 is the idealisation this whole field is about, and a solver that quietly returned 0.9999999 would have answered a different question very convincingly.

The check the new element needed #

The site’s standing promise is that every solve is verified twice before it returns — Kirchhoff’s current law rebuilt from the element relations, and the resistors’ |i|²R against the sources’ Re(v i*). Neither of them can see a mutual inductance.

A coupling adds no current anywhere, so current law is unmoved by it. A coupled pair dissipates nothing, so the energy balance is unmoved by it too. A coupling stamped into the wrong row, with the wrong sign, or not at all, would produce a well-formed solution to a different circuit and both standing checks would pass.

So verifySolution gained a third route: each winding’s own relation, v = s(L i + M i_other), rebuilt from the solved node voltages and branch currents with the stamps not consulted. The gate proves it has teeth by changing a coupling’s coefficient after the matrix was assembled — the stamps unchanged, both standing checks still passing — and requiring the rebuild to catch it. It does.

That is the same discipline currentThrough applies to a resistor, arriving four phases later for the element that needed it most.

The two edges, and what sets each #

The lower edge is the magnetising inductance. Below it the primary winding is simply an inductor across the source, its impedance falls with frequency, and it shorts out the drive; the load sees less and less. The corner is a resistance over an inductance, and neither of those has anything to do with how well the two windings are coupled.

The upper edge is the leakage — the part of each winding’s flux that does not reach the other — which appears as an inductance in series with the signal path. Above the corner it dominates the loop and the output falls. That corner is entirely the coupling’s, and it is the subject of the next essay.

What the flat part actually is #

Here is the result the field turns on, and it is not what “an ideal transformer of this ratio” would lead a reader to expect.

The midband gain is not the turns ratio. Measured across the whole slider:

Divide each of those percentages by its own coupling and the result is 0.99013, 0.99011, 0.99011, 0.99010, 0.99018, 0.99011 — constant to four parts in a hundred thousand. So there is a third factor, it is 0.990, and it is what the two half-ohm winding resistances take out.

Out of what, though, is the part this section exists for, because the numbers on this page cannot settle it. The obvious reading is the secondary’s own divider, $R_L/(R_L + R_2)$, which is $50/50.5 = 0.990099$. The right reading is the whole loop’s, $(R_L + R_s)/(R_s + R_1 + R_2 + R_L)$, which is $100/101 = 0.990099$. These are the same number here, to every digit the table shows, and they are the same number for the reason that they are both wrong to look at in isolation: the source is 50 Ω, the load is 50 Ω, and the two winding resistances are equal, so the ratio of the two expressions is exactly one.

The first version of the gate asserted the secondary’s form. It held at 4.6 × 10⁻⁵ across every coupling on the slider, which is precisely as convincing as it sounds, and it was still wrong — because the slider moves the coupling and the degeneracy is in the load. Rendering the same figure into 5 Ω is what broke it: the secondary’s divider says $5/5.5 = 0.909$ and the solve says 0.982. At a light load the source’s own winding resistance is a larger share of what is left, and only the loop form knows that.

So the flat part of a transformer’s response is

$$\text{turns ratio} \times k \times \frac{R_L + R_s}{R_s + R_1 + R_2 + R_L}$$

with nothing left over. The gate now asserts that product is one to 4.6 × 10⁻⁵ across the range of couplings and across three decades of load, 5 Ω to 500 Ω, which is the axis the first version never moved along. It is a stronger statement than any tolerance on the gain itself: it says the model has no residue.

The general moral is not about transformers. A claim checked only along the axis a slider moves is checked in one direction, and a coincidence in the other direction will hold still and look like agreement for as long as nobody varies it.

That the coupling appears linearly in the midband gain is worth pausing on. A designer told that a transformer is “95% coupled” might reasonably expect a few per cent of something to be wrong somewhere. What is wrong is the gain, by five per cent, everywhere in the band — which for a 1:1 transformer is a decibel of loss that no amount of bandwidth buys back.

Why the edges are bisected and not derived #

The T-model is the standard decomposition of a coupled pair: a magnetising inductance across the input, a leakage inductance in series with each winding, and an ideal transformer between them. It is exact — it is a change of variables, not an approximation — and deriving the two corners from it would have been quicker than solving the network at four hundred frequencies.

It is not what this site does, and the reason is worth one section because the same reason has come up in four fields now.

A decomposition’s parts are not measurements. The magnetising inductance is k²L₁ and the leakage is L(1 − k²) by definition, so a corner computed from them is a rearrangement of the inputs rather than a reading of the object. If the coupling had been stamped into one row instead of two, the T-model would go on producing the same two corners with complete confidence, because it never touches the matrix.

The decomposition’s own accuracy is a result. The gate measures how far the T-model’s upper-edge prediction is from the bisection, and it is 6.16% at k = 0.9, falling monotonically to 0.06% at 0.999. That is not noise: the split of the leakage between the two windings is exact only in the limit k → 1, and the residual is the decomposition’s error, visible only because the measurement was made independently of it.

The pattern is the collection’s oldest habit. The filters field computes each family’s poles from its definition and then measures the ripple on the built network, rather than quoting either. The transients field computes a step response from residues and again by walking the network forward in time. The power field computes a neutral current from the solve and again from the zero-sequence component. In every case the second route exists so that the first one can be wrong, and in every case it has been.

What is not in the flat part #

Two quantities that a reader might expect to matter here do not, and both are worth naming because each has an essay of its own further along.

The core is not in it. Nothing in this figure knows what the windings are wound on. The inductances and the coupling are the whole model, and a core enters only by deciding what those two numbers are — and by saturating, which is an amplitude boundary that this small-signal response cannot express at all.

The winding’s own frequency dependence is not in it. The 0.5 Ω resistances are constants here, and a real winding’s resistance rises with frequency once the skin depth is smaller than the conductor. That is a second-order effect on the band and a first-order one on the losses, and it belongs in its own figure rather than smuggled into this one.

Where this sits among the site’s other boundaries #

The collection now has boundaries in a frequency, an amplitude, a size, a duration, a level and a count. The transformer adds the first that is a pair, and the pair behaves differently from either of its parts.

Widening the band is not one problem. The bottom is fixed by inductance against resistance and is improved by winding more turns, using a higher-permeability core, or driving from a lower impedance into a lower load — all of which cost size, saturation headroom or power. The top is fixed by leakage and is improved by interleaving the windings, which costs capacitance between them and eventually moves the other upper-edge mechanism into the way.

That last point is the one the field’s third essay measures: there are two candidate mechanisms for the upper edge, the textbook names the one that usually does not bind, and there is a load at which they swap.

And where the band stops existing #

One more caution, because it is a boundary on this essay rather than in it.

Everything above assumes the response has a flat middle. That is true only while the load damps the leakage resonance. Lightly loaded, a well-coupled transformer is not a wide-band device with a small ripple — it is a resonant circuit, and it has voltage gain: at 2 kΩ and k = 0.999 the peak measured here is 3.03 times the turns ratio, at 3.47 MHz, from an input of one.

Asking for “the band” of that object returns the skirts of a resonance dressed as a passband, which is what the machinery did before it was taught otherwise. bandOf now reports that the response is peaked instead of returning two numbers in the wrong order, and the fourth essay in this field is about the boundary.

What the field is for #

The reason this belongs on a site about circuits rather than about electromagnetism is the same reason the instrument field does. A transformer here is not a flux linkage and a Faraday’s-law argument; it is two inductances and a number between them, in a netlist, solved by the same machinery as everything else, and every result above is a property of the pair it is connected to rather than of the component.

The turns ratio is a fact about a winding. The band is a fact about a circuit. And the flat part of the response is neither of those alone — it is the turns ratio times the coupling times a divider, which is three facts multiplied and no fourth one hiding.

The one thing an ideal transformer does that no band expresses #

There is a property of the ideal object that this essay has not measured and that the rest of the field depends on: a transformer transforms impedance, by the square of the turns ratio, and it does so in both directions.

That is what makes it the only passive element on this site that can match a source to a load without losing power — the maximum power transfer essay’s result is that the best load equals the source resistance and that the efficiency there is exactly one half, and a transformer is how a load which is not that resistance is made to look like it. The quarter-wave section does the same job at one frequency and over 17.1% of it; a transformer does it over the band this essay measures, which at k = 0.99 is two and a third decades.

Two qualifications, and both are measurable rather than rhetorical.

The impedance ratio inherits the coupling’s loss. If the midband voltage gain is k times the turns ratio, the impedance seen through the transformer is not exactly n² times the load either, and the departure is the same order. A matching network designed from the turns ratio alone is mismatched by a few per cent at k = 0.99 and by twenty at k = 0.8.

The band is a band in both directions. Outside it the impedance transformation fails the same way the voltage transformation does — below, because the magnetising inductance is in parallel with whatever is reflected; above, because the leakage is in series with it. A transformer used for matching therefore carries the same pair of numbers, and the useful statement about one is not its ratio but the two frequencies between which its ratio is what it says.

That is the shape of every result in this field, and it is why the caption strip’s right-hand slot — which on this site nearly always carries a single edge — holds a pair here, written as a number of decades.

What the three rungs above do with the pair #

A band with two edges invites three questions, and each of them is a rung.

What coupling buys, and where it does not asks which edge the winding effort moves, and the answer is lopsided: across six designs from $k = 0.8$ to $k = 0.999$ the coupling moves the upper edge by 168 times and the lower by 1.083. So the quantity a maker works on buys bandwidth at one end and, to within eight per cent, nothing at the other.

Which picture sets the upper edge asks what the upper edge is made of, and finds the mechanism every account names to be a factor of fourteen away from binding at fifty ohms: the resonance is at 1.13 MHz and the measured edge at 81.3 kHz, because what binds is the leakage in series with the load, a first-order corner with no resonance in it. The two accounts swap at about 1500 Ω.

And where the band goes entirely asks whether there is always a band, and finds there is not: past a few hundred ohms of load the leakage that sets the upper edge is also what damps the resonance behind it, so a lightly loaded, well-coupled transformer peaks at 3.03 times its own turns ratio and the question returns the skirts of a resonance instead of an edge.

Which makes the load resistance the parameter this whole ladder turns on — and it is the one quantity in the pair that belongs to the circuit rather than to the component.