The depth a resonance does not have

The inductor one mode cannot see measured a common-mode choke as a netlist and found three things no catalogue prints: two corners a factor of two thousand apart, set by the coupling alone; a conversion from the signal into common mode, set by the balance of the windings; and a ceiling, set by the capacitance across each winding, above which the part stops attenuating.

Its figure for the ceiling printed a number: the best the choke ever does is 102 decibels, at 1.59 megahertz, and by ten gigahertz its hundred decibels have come back down to six. Both halves of that sentence were wrong, in different ways, and neither could be seen from the figure.

The six was a reference. The decibels were measured against the one-volt source, and with no choke fitted at all the load already reads half of it — minus 6.02 decibels — so a curve arriving at minus six is a choke doing nothing. Against the circuit with no choke in it, the part attenuates 0.02 decibels at ten gigahertz. The hundred and two was not a property of the part at all, and the rest of this essay is about what it was, what the depth is when there is one, and what the depth depends on.

A number that moved when the sweep did #

The earlier figure found its best attenuation the way a sweep does: it computed the common mode at 2,000 frequencies spaced logarithmically from a kilohertz to ten gigahertz, and kept the smallest.

Repeating that with other numbers of samples gives other answers. Measured as insertion loss — the attenuation against no choke — 500 samples report 87.81 decibels, 1,000 report 110.04, 1,999 report 110.04, 2,000 report 96.31, 2,001 report 103.64 and 4,000 report 107.22. The figure’s 102.33 was the 2,000-sample reading plus the 6.02 decibels of reference. The six readings span 22.2 decibels, the widest part of that spread lies between three sweeps a single sample apart, and more samples do not converge on anything: 2,000 is shallower than 1,000.

Two of the readings agree, and the reason is not reassuring. A sweep of 1,999 logarithmic points contains every point of a sweep of 1,000 — every second one of them is the same frequency — so the two found the same nearest sample. The agreement is between two grids, not between two measurements.

What is being measured is the distance from the nearest sample to the resonance. Two windings in parallel for the common mode make an inductance of about a millihenry; with five picofarads across each winding they make a parallel resonance at 1.5919 megahertz. With nothing in the netlist that dissipates, the impedance of that resonance at its own frequency is infinite, and near it the impedance grows as one over the distance to it in frequency. Every tenfold approach adds twenty decibels. A sweep’s smallest sample reports how close its grid happened to land.

The repair is to find the minimum rather than sample it: search between the two samples either side of the smallest one until the bracket closes. Done that way, the 1,999-, 2,000- and 2,001-sample sweeps land on the same frequency to the last digit printed, and the depth runs to 350 decibels, which is the arithmetic’s floor. There was never a depth to find.

The failure has been met before in another form, in a chain of inductors and capacitors. A ladder is not a line warns that a measurement taken at a finite value of whatever is going to infinity returns the approach and looks exactly like an arrival. This is the version without even an approach: the quantity being sampled has no limit, so each sample count returns a different finite number and each of them looks like a result.

What is left at the bottom of a resonance #

At a parallel resonance the inductance and the capacitance cancel exactly, and what remains is whatever dissipates. In the netlist the earlier essay used, nothing does, which is the whole of the defect. A real choke has a core, and the core is lossy — deliberately so in a part meant to absorb interference — so the physical question is what that loss does to the depth.

The loss is represented here as a resistance, Rp, across each winding. For the common mode the two windings are in parallel, so the resonance is left with Rp/2. The circuit either side of the choke has twenty-five ohms in each conductor, which the common mode sees in parallel, twelve and a half ohms at each end. At the resonance the insertion loss is then

$$20\log_{10}\left(1 + \frac{R_p/2}{Z_s + Z_l}\right)$$

and at a hundred kilohms of core loss that is twenty times the logarithm of 2,001, 66.02 decibels. The solve finds the same minimum without being told the expression, to 4×10⁻¹⁵ decibels, and it finds it at the same frequency as the lossless resonance.

The expression is short, and what is missing from it is the finding. There is no inductance in it, no coupling and no winding capacitance. Those three decide where the resonance is. The core loss and the circuit decide how deep it is, and nothing else does.

The same structure turns up in what actually fills a null, with the sign reversed. There a notch built from ideal parts has a null three hundred decibels deep, a component error moves it without filling it, and what fills it is loss, at twenty decibels per decade of arm resistance. Here a resonance built from ideal parts has a peak of infinite impedance, a change of capacitance moves it without bounding it, and what bounds it is loss, at twenty decibels per decade of core resistance. In both cases an ideal model’s extreme is a feature of the model, and the number worth quoting is the one the dissipation sets.

The loss has a second reading, as the resonance’s quality factor. Rp/2 against the reactance of the common-mode inductance at the resonance is a Q of 5.0 at a hundred kilohms, 50 at a megohm and 0.5 at ten kilohms. Resonance and its bandwidth makes the band of a resonance f₀/Q exactly, so a lossier core buys a shallower floor and a wider notch at once — and a choke’s usefulness is the width over which it is deep, not the depth at one frequency.

Twenty decibels for every decade of loss #

Once Rp/2 is large against the twenty-five ohms of circuit, the one inside the logarithm stops mattering and the depth is twenty decibels per decade of loss: 66.02 at a hundred kilohms, 86.02 at a megohm, 106.02 at ten.

The one inside the logarithm is where that rule stops, and it stops at losses a real part can have. Read as twenty decibels a decade from the large-loss end, three hundred ohms across each winding would give 15.56 decibels; the solved depth is 16.90, and the difference is that one. At a kilohm it is 26.44 against 26.02. And when Rp/2 equals the twenty-five ohms of circuit the choke is worth exactly 6.02 decibels at the one frequency it is best at. A core whose loss at its resonance is a few hundred ohms sits at that end of the curve, and for it the whole expression is the design equation while the decade rule is only its asymptote.

That gives the lossless readings a meaning they did not have. Each of them is a depth this expression produces for some core loss, and the loss can be solved for. The 500-sample sweep describes a choke with 1.23 megohms across each winding; the 2,000-sample sweep — the one that was printed — describes 3.27 megohms, a resonance with a Q of about 160; the 1,000- and 1,999-sample sweeps describe 15.9 megohms. A sweep of a lossless netlist reports a lossy component, and the component it reports changes with the number of points.

The practical content is the other way round. A choke is deep at its resonance because its core is lossy there, and a part whose common-mode impedance is drawn against frequency with a peak in it has, in this model, Rp/2 as the height of that peak. The depth is that peak divided by the circuit, and the inductance printed beside it is the quantity that decides the frequency.

The capacitance decides where, and not how deep #

The earlier essay drew the same choke with twenty picofarads across each winding and wrote that four times the winding capacitance halves the frequency of the best attenuation and leaves its depth alone. Its figures printed 102 decibels for five picofarads and 96 for twenty, both sampled from resonances without a floor, so the sentence was a statement about two numbers that were not depths.

With a loss in the netlist it is true, and exactly. Twenty picofarads puts the resonance at 796 kilohertz, half of 1,592, and the depth at 66.0249 decibels, the same as five picofarads to a thousandth of a decibel. A coupling of 0.99 instead of 0.999 leaves the depth alone to the same precision.

That separates two design moves that are usually made together. More turns on the same core is more inductance and more capacitance, which moves the resonance down, and — as the inductor that is a capacitor found for an ordinary inductor — it moves the part’s useful range down with it. What it does not do is make the floor any deeper. The depth belongs to the core material’s loss at whatever frequency the resonance lands on, and to nothing about the winding.

It also bounds the hazard the earlier essay named for two chokes of different sizes in series, a large one for low frequencies and a small one for high. Between them sits a series resonance — one part’s capacitance against the other’s inductance — which the pair that is worse than either measures in a decoupling bank and finds limited by one over the series resistance. The same netlist with core loss in it gives that resonance a finite height too; without the loss it would print whatever the sweep landed on, exactly as the single choke did.

Half of the depth is the circuit #

An insertion loss is a ratio of two circuits, the one with the part in and the one without, and the circuit without the part is in that ratio as much as the part is. The expression says so directly: the choke contributes Rp/2 at its resonance, and how much Rp/2 matters depends on what it is added to.

With ten kilohms of core loss the same part is 67.96 decibels deep between ends of one ohm, 46.06 between the ends this essay has used so far, and 15.56 between ends of five hundred. That is fifty-two decibels from one component, with nothing about the component changed. A depth measured on one fixture and quoted as a property of the part carries the fixture in it.

The limit of the expression makes the point sharply. As the load’s common-mode impedance goes to infinity the insertion loss goes to nothing, because a part in series with an open circuit carries no current and so drops no voltage. And an open circuit to the common mode is not exotic. The millimetre that becomes common mode found that the most common differential termination, a single resistor across the pair, is an exact open circuit to the even mode whatever its value. A choke fitted in front of that termination is working into the one load against which it can do least.

What a real cable presents to the common mode is a harder question than this essay answers. A cable above a ground is itself a line to that ground, so its common-mode impedance swings with frequency between low and high, and the depth measured here swings with it. The five fixed values above bracket the range; the cable is not modelled.

Where the choke is worth six decibels #

The earlier figure’s caption strip made a second claim about the top of the range: that the choke is useless above 47.6 megahertz. That was thirty times the sampled resonance, and the thirty was never measured. At 47.6 megahertz the lossless part still attenuates the common mode by 22.56 decibels.

Bisected on the solve instead, the frequency above which the insertion loss falls below six decibels is 369 megahertz, for a coupling of 0.999 and for 0.99 alike, since above its resonance the part is a capacitance across the path and the coupling no longer enters. And at ten gigahertz it is 0.02 decibels, not the six that was printed. Both corrections are made in the essay that printed them, and both figures now draw against the no-choke circuit rather than against the source.

The combined picture is a part that is deep at one frequency to the extent its core dissipates, worth less than six decibels two and a half decades above that frequency, and worth nothing measurable by ten gigahertz. What coupling buys, and where found the coupling coefficient buying bandwidth at one end of a transformer’s band and almost nothing at the other; for a choke the coupling buys the ratio of the two corners, the capacitance places the resonance, and the loss sets how deep that resonance is. Three quantities, three consequences, and a data sheet that prints an inductance names none of them.

What a resistor across a winding does not say #

The core loss here is one resistance across each winding, the same at every frequency. That is a fair description of a lossy core over the narrow band a resonance occupies, and a poor one across the seven decades the figures draw. A ferrite’s loss is a function of frequency — it is usually given as the imaginary part of its permeability — so the Rp that sets a real part’s depth is the loss at the frequency its resonance happens to land on, and moving the resonance moves the loss that sets it.

Nothing else is added. There is no series resistance in the windings, which would matter at direct current and not at the resonance; no saturation of the core, which the earlier essay records as out of scope; and the two windings are equal, so the conversion it measured is absent here by construction.

And a lossless netlist is now refused rather than answered. Asked for its depth, it declines, with the reason: every number it could return would be a grid’s. That refusal is the claim this essay is built on, turned into something that has to keep rejecting. The pairing of a two-drive netlist with the mutual inductance built as two entries in one matrix, from one number from two measurements, is unchanged; what has changed is that the netlist can now say it has no answer.

Still open: the loss against frequency, a load that is a line, and two chokes in series #

The loss as a function of frequency. With the core’s loss given as a permeability rather than a resistance, more turns does not merely move the resonance down; it moves it into a part of the material’s curve where the loss is different, and the depth follows. That is a distinct claim — a choke can get deeper or shallower by adding turns, depending on the material — and it needs only the same netlist with Rp replaced by a frequency-dependent element.

A common-mode load that is a line. The terminations above are fixed resistances. A cable over a ground is a transmission line to that ground, and the lines field already computes the impedance at the input of one; putting that impedance at the choke’s far end makes the depth a function of the cable’s length as well as of the part, with frequencies at which a quarter wavelength of cable turns a good choke into none.

Two chokes in series. The earlier essay named the series resonance between a large part and a small one. With core loss in both netlists that resonance has a height that can be measured rather than sampled, and whether the pair is worse than either alone becomes a question with a number in it.