Where the band goes entirely

Three essays in this field have measured a transformer’s band, and every one of them rests on an assumption that has not been stated: that the response has a flat middle to measure.

It does not always. The leakage inductance that sets the upper edge is in series with the load, and the load is therefore what damps the resonance between that leakage and the winding capacitance. Lighten the load and the damping goes. Past a few hundred ohms the response stops being flat and acquires a peak; past a few thousand the peak is large.

The measurement #

The same 1:1 transformer as the rest of the field, at k = 0.99, into nine loads:

At the tightest coupling the effect is worse and arrives sooner. At k = 0.999 into 2 kΩ the peak is 3.03 times the turns ratio, at 3.47 MHz.

A 1:1 transformer with a voltage gain of three is not a defective transformer. It is a series resonant circuit — leakage inductance, winding capacitance, and whatever resistance is left to damp it — and the voltage across the capacitor at resonance is Q times the voltage driving the loop, which is what a resonant circuit does and what the resonance field measured directly.

What the machinery did before it was taught #

bandOf finds the midband by scanning for the maximum of the response, then bisects outward for the two frequencies at which it has fallen by a stated ratio. That works when the maximum is a plateau.

When the maximum is a peak, it returns two numbers and they are the resonance’s skirts. Worse, when the peak is narrow and near the top of the range, the “lower edge” comes out above the “upper edge” — at k = 0.999 into 2 kΩ it returned 2.83 MHz for the low one and 4.07 MHz for the high one, having found them on either side of a peak rather than on either side of a band.

Two numbers in the wrong order is a conspicuous failure and it would have been caught. What would not have been caught is the 1500 Ω case, where the peak is 1.18× and the “band” it produces is perfectly well ordered, plausible, and a description of a resonance.

So the function now decides which object it is looking at. If the maximum exceeds what the turns ratio allows by more than two per cent, the response is peaked and no band is reported at all — decades and ratio come back null rather than as numbers about the wrong thing.

That is the site’s standing habit applied to a measurement rather than to a solver. solveAt refuses a singular network by name instead of returning a large plausible number; refuses in the small-signal field declines a drive outside the model’s range; oversamplingFor in the digital field refuses a filter whose stopband floor is above the requirement. A measurement whose premise has failed should say so, and the difficulty is always the same — the failure produces an answer rather than an error.

Why better coupling makes it worse #

This is the part that inverts the previous essay’s advice, and it comes straight out of the arithmetic.

The quality factor of the resonance is

$$Q = \frac{1}{R_{\mathrm{loop}}}\sqrt{\frac{L_{\mathrm{leak}}}{C_w}}$$

Reducing the leakage reduces Q, which would suggest tighter coupling is safer. But reducing the leakage also raises the resonant frequency, and at a higher frequency the same load is a lighter damping relative to the reactances. The net effect measured on the solve is that the boundary moves down: at k = 0.99 the response is still flat into 500 Ω, and at k = 0.999 it is already peaking at 1.18×.

The gate holds exactly that: k = 0.99 and k = 0.995 are flat into 500 Ω, and k = 0.999 is not.

So the previous essay’s conclusion — that tighter coupling buys bandwidth at the top and costs nothing — has a boundary of its own, and it is this one. Tighter coupling buys bandwidth and narrows the range of loads over which the component is a transformer at all. Both statements come from the same measurement and neither is available from a figure that draws one response.

What it is good for #

A passive circuit with voltage gain sounds like a fault and is sometimes exactly what is wanted, so it is worth saying where.

A resonant transformer is how a small voltage becomes a large one without a large turns ratio. An ignition coil, a flyback, a Tesla coil and the output stage of a switched-mode supply are all operating in this regime deliberately: the peak is the design, the load is chosen light enough to let it happen, and the frequency is chosen at the resonance rather than inside a band.

It is also how a transformer becomes an unintentional oscillator’s resonant element. A winding whose secondary is open, or nearly so, presents a lightly damped resonance to whatever drives it, and a switching circuit that excites it once per cycle rings. The 8.80× measured here at 20 kΩ is a statement about a real hazard: the voltage on an unloaded secondary can be many times what its turns ratio says, and the insulation has to survive it.

And it is a reason a specification must state its load. A transformer measured into its intended load and used into an open circuit is not the same object. This site has said that about a divider, where two dividers of identical ratio give 5.970 V and 1.000 V into the same 100 kΩ load; a transformer’s version of the sentence is stronger, because the output can exceed the input.

What is damping it, exactly #

One clarification, because the previous section’s language is loose in a way the measurement is not.

The load damps the resonance through the leakage, not directly. The resonant loop is the leakage inductance and the winding capacitance; the resistance in that loop is the source resistance, the two winding resistances, and the load — all in series once the magnetising branch is out of the way.

That has a consequence worth stating: the source resistance damps it too. A transformer driven from a low impedance into a high one has only the winding resistances damping the loop, and is the worst case; driven from a high impedance into a high one it is better damped, at the cost of the midband gain. The measurements here use a 50 Ω source, which is a moderate case, and a voltage-source drive would peak higher.

That is also the answer to the obvious remedy. A resistor across the secondary damps it — which is what a load is — and a resistor in series with the primary damps it too, at a cost in the band. Both are the ordinary way of taming a lightly loaded transformer, and both are visible on this figure as moving left along the axis.

The other two edges, while this is happening #

A reader following the field will want to know what becomes of the lower edge once the upper one has stopped being an edge, and the answer is that nothing happens to it at all.

The magnetising inductance is still across the source, the reflected load is still what it works against, and the lower corner is still a resistance over an inductance. Lightening the load raises the reflected resistance and therefore raises the corner a little — at 5 Ω it is 371 Hz and at 500 Ω it is 729 Hz, which is a factor of two across two decades of load — and it goes on being a perfectly ordinary first-order corner while the top of the response has become a resonance.

So the object in the peaked regime is not a resonant circuit in place of a transformer. It is a transformer at the bottom and a resonator at the top, with nothing flat in between once the two have met. That is why the useful description stops being “a band” rather than becoming “a different band”: the two ends are still there and are still doing what they did; what has gone is the plateau that made one number out of them.

There is a tidy way to say when that happens, and it is the same crossing the previous essay computed. The plateau exists while the resonance’s frequency is above the resistive corner — which is the same condition as the load being below $\sqrt{L_\mathrm{leak}/C_w}$. When the two coincide there is no plateau left to have edges, and above it the resonance is the only feature.

The boundary this puts on the field #

Three of the eleven essays in this field measure a band, and this one says where they apply. That is worth stating as a limit rather than leaving implicit, because the collection’s whole discipline is that a model comes with the range it is true over — and “a transformer has a band” is itself a model.

The range is: loads below about $\sqrt{L_\mathrm{leak}/C_w}$, which for this component is 1.4 kΩ. Below it there is a passband, its edges are the two the field measures, and the midband is the turns ratio times the coupling. Above it there is a resonance, its frequency is the leakage against the winding capacitance, its height is a Q, and none of the three band statements applies.

That boundary is itself a resistance, which makes it the fourth kind of quantity this field has produced — after a frequency pair, a volt-second product and a count of cycles — and the second on this whole site whose independent variable is an impedance rather than a signal.

The instruction it generalises to is one the site has now met from several directions:

A measurement should be able to report that its own premise has failed. Not a wider tolerance, not a flag in a comment, and not a number that happens to look wrong to somebody who already knows what to expect. bandOf returns peaked and refuses to name a band, and that refusal is checked by feeding it a design that has one and a design that does not.

What a designer does about it #

The remedies are all resistors and all of them cost something measurable, which makes this a design trade rather than a hazard to be avoided.

Load it. The simplest and the one the figure draws: moving left along the axis. A resistor across the secondary that is small compared with √(L/C) puts the component back in its transformer regime, at the cost of the power it dissipates and of loading whatever the transformer was driving.

Damp it deliberately, with a series resistor and a capacitor across the secondary. A network chosen to present a low impedance only near the resonance, and a high one in the band, so that the damping costs nothing where the signal is. This is the standard answer and it is the same construction the filters field uses when a response needs one pole’s worth of correction and not a family’s.

Move the resonance out of the way. Reducing the winding capacitance raises the resonant frequency without changing the leakage, so the peak moves somewhere it does not matter — which is what sectioned and spaced windings buy, at the cost of the coupling, and therefore of the band’s upper edge. Every lever in this field pulls two things.

Or use it. If the resonance is where the signal is, the peak is gain and the component is a resonant transformer rather than a broadband one, with a bandwidth set by its Q and measured the way the resonance field measures one — where the half-power band is exactly f₀/Q at every Q tested, and is not centred on the resonance.

That last cross-reference is the reason this essay sits in the magnetics field rather than the resonance one. The object is a resonator and the site already knows what to do with resonators; what is new here is that it became one because of a resistor somewhere else, and that the same component is a transformer on the other side of that resistor.

One more thing the peak is not #

There is a reading of this figure that would be wrong and is easy to reach: that the transformer is somehow supplying energy, or that the peak is evidence of the coupling exceeding one.

Neither. The peak is a voltage magnification and it is exactly the same phenomenon as the voltage across a capacitor at series resonance being Q times the driving voltage — which the resonance field measured on a series RLC and which is a property of reactive energy circulating rather than of energy being created. The power delivered to the load is small, because the load is large; the current is small; and the whole thing is passive, lossless in its reactances and dissipative in its resistances, as everything on this site is required to be.

The site’s own machinery insists on it and would not have let the figure be drawn otherwise. Every solve behind it passed verifySolution, which requires the resistors’ |i|²R to equal the sources’ Re(v i*) — so at the 8.80× point the source is delivering exactly what the resistances consume, to within 10⁻⁷ of the larger. A figure showing a passive circuit with voltage gain is therefore an unusually well-checked one, and the check is the reason it can be believed.

That is worth ending on because it is the collection’s whole method in one place. A result that looks like it should not be true is either a fault or a finding, and the way to tell is that a fault will fail something the machinery already insists on. This one does not fail anything: the coupling is below one, the energy balances, the current law holds, the coupled relation is satisfied, and the output is nine times the input. All five statements are true at once, and the last of them is what a resonance does.

A passive component with voltage gain, elsewhere #

A transformer peaking at three times its turns ratio is startling and it is not unique, and the two other instances in this collection say what the three have in common.

The inductance that limits, and lifts finds a rectifier whose output sits 29 per cent above the peak of its own supply, permanently, once leakage inductance is added — which every steady-state expression in that field says cannot happen, and which is the same series resonance seen through a rectifier instead of through a load.

The far end that rises finds a transmission feeder whose far-end voltage goes above its source’s past a computable angle of leading load — 28.35 degrees for a line of fifty ohms and a hundred of reactance — and states the condition that makes it possible: with no reactance in the line there is no such angle at all, because the rise is a partial resonance and needs both halves.

Which is the common structure. In all three a network everybody thinks of as lossy and passive contains two reactances of opposite sign, and a lightly loaded resonance has a Q that nobody chose. The tell is the same in all three too: the gain appears at light load and disappears under load, because the load is the only damping in the circuit.

That is why this essay’s finding is a statement about a question rather than about a component. Asking for the band of a lightly loaded transformer returns the skirts of a resonance, and the honest answer is that the object has no band at that load rather than that its band is wide. Which is a refusal rather than a measurement, and this collection’s habit is that a refusal with a reason is worth more than a number with none.