Which picture sets the upper edge

The last two essays measured a transformer’s upper band edge and attributed it to the leakage inductance. That is right and it is not specific enough, because there are two ways the leakage can set an edge and they give answers a factor of fourteen apart.

The one every account names is a resonance: the leakage inductance and the capacitance between the windings form a series-resonant circuit, and above its frequency the response falls at forty decibels a decade. The one that actually binds at ordinary loads is duller and arrives much earlier: the leakage inductance is simply in series with the load, which is a first-order corner at R/2πL and has no resonance in it.

How the error was found #

This essay exists because of a gate failure, and the failure is the useful part.

edgePredictions originally returned one upper-edge candidate — the resonance, because that is the one the literature names — and the figure asserted that it predicted the measurement. It gave 1.13 MHz against a measured 81.3 kHz. A factor of fourteen, from a formula, in the confident direction.

What made it hard to notice by reading is that the formula is correct. The leakage and the winding capacitance really do resonate at 1.13 MHz, and above that frequency the response really does fall at forty decibels a decade. Nothing about the expression is wrong. It is answering a question about a mechanism that has already been overtaken by a different one, half a decade earlier, and a figure drawn from it would have marked an edge on a curve that had been falling since well before.

The repair is to compute both and take the lower, and to say which — edgePredictions returns a binding field naming the mechanism, so that a figure can state which picture it is drawing rather than implying there is only one.

The two mechanisms, written out #

Above the magnetising corner the primary’s inductance is effectively out of the circuit, and the signal path is a loop: source resistance, primary winding resistance, primary leakage, secondary leakage reflected through the turns ratio, secondary winding resistance, load.

The resistive corner is that loop’s total series inductance against its total series resistance:

$$f_{\mathrm{hi}} = \frac{R_{\mathrm{loop}}}{2\pi L_{\mathrm{leak}}}$$

with L_leak = L₁(1 − k) + L₂(1 − k)/n² and R_loop the whole loop’s, source included.

The resonance is the same leakage against the capacitance across the secondary:

$$f_{\mathrm{res}} = \frac{1}{2\pi\sqrt{L_{\mathrm{leak}}\,C_w}}$$

Both contain the leakage. Neither contains the other’s second element. And which of them arrives first depends entirely on the load, because the load is in one and not in the other.

The measurement #

Nine loads spanning four decades, at k = 0.99:

Two features carry the essay.

Below the crossing, the resistive corner predicts the measurement closely — within 1.1% wherever the resonance is at least eight times away, and within 0.6% at the heaviest load. The residual is the T-model’s own, and it tightens with the coupling: 6.2% at k = 0.9, 3.0% at 0.95, 0.6% at 0.99, 0.06% at 0.999. That is not scatter; the split of the leakage between the two windings is exact only as k → 1.

Above the crossing, the measured edge stops moving. At 5 kΩ and 20 kΩ the resistive corner is at 4 MHz and 16 MHz, and the measurement sits at 1.27 MHz and 1.19 MHz — pinned near the resonance, which does not contain the load. That flattening is the signature of the swap and it is visible in the figure as the measured curve leaving the resistive line and running alongside the resonant one.

Where the crossing is, and what moves it #

Setting the two expressions equal gives a load, and it is worth reading as a formula rather than a number because the number is not general.

$$R_{\mathrm{swap}} \;\approx\; \sqrt{\frac{L_{\mathrm{leak}}}{C_w}}$$

which is the characteristic impedance of the leakage and the winding capacitance taken together — the same quantity a transmission line has, arriving here as a property of a component rather than of a length of cable.

For this transformer that is about 1500 Ω. It moves with the square root of the leakage, so better coupling lowers it: at k = 0.999 the leakage is a tenth of its value at 0.99, so the crossing falls by about three, and the resonance becomes the binding mechanism at a lower load. That is the same reading as the previous essay’s closing observation, arriving from a different direction: a more tightly wound transformer behaves like the textbook picture over a wider range of loads, and behaves like a resonant circuit rather than a transformer over part of it.

The identity is checkable and the machinery checks it: for this transformer the leakage is 200 µH and the winding capacitance 100 pF, so √(L/C) is 1414 Ω, and the load at which the two candidate frequencies are equal comes out at 1367 Ω. The eight-decimal agreement a reader might expect is not there and should not be — the two expressions differ by the loop’s other resistances, which the crossing contains and the characteristic impedance does not — but three per cent is close enough to say that the crossing is that impedance, and the difference is the source resistance.

That is a satisfying place for the number to come from. A transformer’s own leakage and winding capacitance define an impedance, in exactly the way a length of cable’s inductance and capacitance per metre do, and the transformer behaves like a first-order low-pass into loads below it and like a resonator into loads above. The transmission-line field spent a whole essay on what that impedance means when it belongs to a length; here it belongs to a component, and it decides which of two pictures of that component is the right one.

What difference it makes #

It would be reasonable to ask whether this matters, given that both mechanisms are the leakage’s and that a designer wanting more bandwidth reduces the leakage either way. Three things follow that would not follow from the resonance alone.

The edge depends on the load, and steeply. From 2 Ω to 500 Ω the measured edge moves by a factor of twelve. A transformer characterised into one impedance and used into another has a bandwidth nobody stated, and the resonant picture — which contains no load at all — would predict that the bandwidth is a property of the part. It is not, and this is the fourth time this site has had to say so about a component.

The roll-off is first order, not second. Past a resistive corner the response falls at twenty decibels a decade; past a resonance it falls at forty. A system budgeting attenuation above the band — a rejection specification, a stopband, an interference requirement — gets half the slope it expected if it assumed the resonance.

There is no peaking below the crossing, and there is above it. A first-order corner cannot overshoot. A resonance can, and whether it does depends on how well the load damps it. That is the subject of the next essay, and it is the point at which the distinction stops being about a number and starts being about what kind of object the transformer is.

The same crossing, seen in time #

A first-order corner and a resonance differ in the frequency domain by a slope. They differ in the time domain by something a reader can see without a logarithmic axis: one of them rings and the other cannot.

That distinction is the whole subject of the filters field’s opening argument — a Butterworth, a Chebyshev and a Bessel at one corner differ in their step responses in ways their magnitudes barely show — and it arrives here as a property of the load rather than of a design choice.

Below the crossing, the transformer’s upper edge is a single pole and its step response has no overshoot at all, whatever the coupling. Above it, the leakage and the winding capacitance are a second-order pair whose damping is supplied by the load, and as the load lightens the damping falls and the overshoot grows. The 8.80× peak that the field’s fourth essay measures at 20 kΩ is that second-order pair with essentially nothing damping it.

So the two mechanisms are not merely two numbers for one edge. They are two different orders of response, with two different behaviours in time, and which one a circuit has depends on a resistor somewhere else. A pulse transformer specified by its rise time and used into a different load does not merely get a different rise time; it gets a different shape.

The shape of the mistake #

The site has now made this class of error four times and caught it four different ways, and the pattern is worth naming because it is not carelessness.

In each case a correct formula described a mechanism that was not the binding one:

• transferFunction drops leading coefficients that are small relative to the largest, which is right, and returns a network whose poles are 3.5 decades apart as first order.
• The quadrature rule for rise times is exact for Gaussian responses and gives 2.18× for every instrument, because it contains none — while a real chain of poles needs 2.79× to 5.62×.
• The conventional tenth-of-a-wavelength criterion marks a frequency at which the lumped model is already 30% wrong.
• And here, a resonance that is real, computable and half a decade past the edge that binds.

What the four have in common is that the formula’s own domain of validity is the thing nobody wrote down. The repair is the same each time and it is the reason this collection measures rather than derives: compute the mechanism, compute its competitors, and let the measurement say which one the object is actually doing.

What is not in the model #

Two honest limits on the numbers above.

The winding capacitance is one capacitor. A real winding’s capacitance is distributed along it, and a single element across the secondary is a coarse stand-in that puts the resonance in about the right place and says nothing reliable about what happens two octaves above it. The essay’s claims are about which mechanism arrives first, which the coarse model settles, rather than about the shape of the response past the second one, which it does not.

Nothing here is lossy. The core is treated as lossless and the winding resistance as constant with frequency. Both are wrong at the top of the range in the direction that would damp the resonance further and lower the measured edge slightly — so the resistive picture, which is the one that binds at ordinary loads, would if anything bind over a wider range than measured here. That is the comfortable direction for the essay’s conclusion and it is stated rather than relied on.

The instruction #

The transferable form of this is one sentence and it is not about transformers.

When a boundary has more than one candidate mechanism, compute all of them and take the lower — and record which one it was. A single formula, however correct, describes a single mechanism, and a model with two mechanisms has a boundary set by whichever arrives first. Which one that is will usually depend on something outside the component: a load, a source impedance, a temperature, a frequency range.

The machinery in lib/magnetics.js now returns both candidates, the one that binds, and the load at which they swap. That is three numbers where the literature offers one, and the reason for all three is that the one it offers is the wrong one over most of the range a transformer is used in.

What a data sheet would have to say instead #

The practical version of all this is a short list of what a transformer’s high-frequency specification would need to contain in order to be usable, and the gap between that list and what is actually printed is the essay’s last point.

The leakage inductance, which is one number and is measurable directly — short the secondary and measure the primary’s inductance, and what remains is the leakage. It is the quantity both mechanisms contain and neither can be computed without.

The winding capacitance, which is the other. Together the two give the crossing, and therefore tell a user which of the two pictures applies to their circuit before they build it.

The impedances the quoted bandwidth was measured at, without which the bandwidth is a number about somebody else’s circuit. A transformer whose data sheet says “3 dB bandwidth 100 kHz” has said almost nothing: into 2 Ω that same part is 42 kHz and into 500 Ω it is 500 kHz, a factor of twelve either side of the quoted figure, and the part has not changed.

What is usually printed instead is the bandwidth alone, occasionally the leakage, and almost never the capacitance. The consequence is that the most common way to find out which regime a transformer is in is to build the circuit and look — which is a perfectly good method and is the one this essay performs, at four hundred frequencies and nine loads, on a netlist.

That is the argument for the whole collection compressed into one component. The parameters that make a model predictive are usually available, usually cheap to measure, and usually not the ones quoted; and the difference between having them and not is the difference between knowing a number and knowing which of two mechanisms produced it.

The three rungs the load resistance decides between #

Both accounts of a transformer’s upper edge are current because both are sometimes right, and this ladder’s other essays are the same load resistance deciding other things.

The band a turns ratio holds over establishes what is being bounded: the flat part is not the turns ratio but the turns ratio times the coupling, times what the two winding resistances leave of the whole loop — and the first version of that last factor was a coincidence that held for every coupling and broke at a different load.

What coupling buys, and where it does not measures the effort side: across six designs from $k = 0.8$ to $k = 0.999$ the coupling moves the upper band edge by 168 times and the lower one by 1.083, so every hour spent on the winding buys bandwidth at one end and nothing at the other.

And where the band goes entirely is the case this essay’s swap point walks into. Past a few hundred ohms of load there is no band at all: 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. Asking for the band of a passive component with voltage gain returns the skirts of a resonance.

Which is the whole ladder in one sentence. The load resistance decides which mechanism sets the edge, how much the winding effort is worth, and whether there is an edge to find — and it is the one parameter a transformer’s data sheet cannot carry, because it belongs to the circuit.

Which is a general difficulty with two-port components rather than a complaint about transformers. The two resistors a ladder was designed between is the same problem in the filters field — a passive network that is not a transfer function with some resistors attached but a two-port designed between two stated resistances, whose passband moves by half a decibel outside a window of 0.886 to 1.137 times its design value. In both cases the component can only be specified against a stated termination, and in both cases the specification is usually printed without one. The difference is that a filter designer knows the termination is part of the design and a transformer’s user usually does not, which is why the factor of fourteen measured here reads as a surprise rather than as a condition.