The inductor one mode cannot see

The millimetre that becomes common mode measured what a pair of conductors of unequal length does to a signal: a tenth of it arrives as common mode by 3.04 GHz, from 1.5 mm of difference, and the differential signal has lost five thousand parts per million of itself getting there. The pair passes its own eye diagram and fails an emissions test.

The part fitted to fix that is a common-mode choke, and the reason it works is that it is not one component. Two windings on one core, wound so that the signal’s two conductors go round in opposite senses: a current that goes the same way round both meets the sum of the self-inductance and the mutual, and a current that goes opposite ways meets the difference. With a coupling of 0.999 those two numbers are three orders apart.

Nothing here computes (1 ± k)L #

The expressions are in every catalogue and this collection does not use them. Two inductors and one K element go into the netlist, the source pair is driven either differentially or in common, and the answer is read off the same solve. That matters for one specific reason: two windings that are not the same inductance have no mode inductances to write down at all, and the last section of this essay is about exactly that case.

The elements are the ones the magnetics field already has. One number from two measurements built the mutual inductance as two off-diagonal entries in the same matrix the self-inductances are on, which keeps the whole system affine in s — so a choke’s poles are recovered by the same sampling and rooting everything else on this site uses, and no symbolic transfer function appears anywhere.

The common-mode corner is 3.98 kHz and the differential corner is 7.86 MHz. Their ratio is 1,975; two over one minus the coupling is 2,000. The signal loses nothing until eight megahertz and the common mode is down three decibels by four kilohertz, from one part.

The useful band is opened from one end only #

Sweeping the coupling shows which of the two corners a designer actually controls. The common-mode corner barely moves — it is set by (1 + k)L, and k is nearly one whatever else is true, so the common-mode inductance is very nearly twice the self-inductance no matter how the part is wound.

The differential corner moves by three decades over the same range, because the leakage the signal meets is (1 − k)L and one minus a number near one is entirely at the mercy of how near. A choke wound at k = 0.99 costs the signal a corner at 786 kHz; the same core wound bifilar at k = 0.9999 costs it a corner at 78 MHz. The catalogue number that changed is not the inductance.

That is the same quantity what coupling buys, and where found deciding a transformer’s band, arriving here to decide something else: there, the leakage inductance sets the upper edge of a wanted response; here it sets the upper edge of a cost.

What the signal pays, in the units a signal is specified in #

The differential corner at 7.86 MHz is a statement about a sinusoid, and a data link is not one. What the leakage inductance actually does to a signal is lengthen its edges, and the quantity a receiver’s specification is written about is the rise time rather than the corner.

Two microhenries in the loop against a hundred ohms of source and load is a time constant of 20 ns, so a step arrives with a 10–90% rise of 44 ns added in quadrature to whatever it had. For a hundred-megabit link with a 3 ns edge that is not a cost, it is the whole of the edge. For a one-megabit link it is nothing.

That is the trade in one line: the coupling coefficient buys common-mode attenuation at low frequency and sells differential bandwidth at high frequency, and neither of the two numbers a catalogue prints — the inductance and the rated current — says where the exchange rate is. The measurement that says so is the pair of corners, and it takes two drives of one netlist.

Where it stops working, which is where it was needed #

An inductance alone gives an attenuation that grows without limit at twenty decibels a decade. That is the answer a data sheet’s impedance curve implies at its right-hand end, and it is wrong, because a winding has capacitance across it.

With five picofarads the choke’s attenuation turns over at 1.59 MHz, and by ten gigahertz it is worth 0.02 dB against the same circuit with no choke fitted. At the turn itself this netlist has no depth to quote. Nothing in it dissipates, so the resonance’s impedance is unbounded and a sweep reports only how close its nearest sample fell; the depth a resonance does not have measures that and finds the depth a real core sets instead — 66 decibels for a hundred kilohms of loss across each winding, and twenty more for every decade of it. Above the winding’s own resonance the part is a capacitor bridging exactly the path it was fitted to block. A component bought for radiated emissions is transparent at the frequencies that radiate.

This is the same object the inductor that is a capacitor measured on an ordinary inductor, and the reason it is worth measuring again is that the consequence is inverted. A signal inductor above its self-resonance stops doing something useful. A choke above its self-resonance stops doing the only thing it was fitted for, in the band where the requirement is hardest.

The repair is not a better choke, because a bigger inductance on the same core is more turns and more turns is more capacitance. It is two chokes of different sizes in series, which is the same argument the decoupling field makes about capacitors — and which brings the same hazard, since two parts each excellent in their own decade have a frequency between them where they are worse than either.

Two chokes, and the frequency between them #

The repair named above — a large choke for the low frequencies and a small one for the high — has a defect of exactly the shape the pair that is worse than either measures in a decoupling bank, and for the same reason. Above its own resonance the large choke is a capacitance; the small one is still an inductance; a capacitance and an inductance in series across the same path is a series resonance, which is a short circuit rather than an open one, and a short circuit in a common-mode path is the choke absent.

The arithmetic is different from the capacitor case — series rather than parallel, so the anomaly is a dip in impedance rather than a peak — and the structure is identical: two parts fitted because each is good where the other is not, and a frequency between them where the pair is worse than either. The number that damps it is again the one nobody wanted, which for a choke is the core’s own loss.

Four times the winding capacitance halves the frequency of the resonance. Its depth is not in this netlist at all; with a core loss added the depth is set by the loss and not by the capacitance, and it stays at 66.02 decibels to a thousandth. So a choke chosen by inductance alone — more turns, more millihenries, better catalogue number — moves its useful band downward while the emissions requirement stays where the regulator put it.

What an unequal pair converts #

The last measurement is the one that needs the netlist rather than the expressions. Give the two windings different inductances — different numbers of turns, or the same turns at different radii on a toroid — and drive the pair differentially. The common-mode component at the far end is read from the same solve.

One per cent of mismatch between the windings converts at −15.6 dB at ten megahertz. A tenth of a per cent converts at −35.6. The relationship is first order — twenty decibels for every factor of ten in the winding tolerance — which is the same order the skew in the millimetre that becomes common mode produced, and for the same structural reason: a difference between two things that were supposed to be identical appears in the sum at first order while its effect on the difference is second order.

So the component fitted downstream to remove common mode is upstream of the common mode it makes. A choke wound by hand, or wound on a core whose two halves see different permeability, converts the signal into exactly the quantity it was bought to attenuate — and then attenuates that quantity by the ratio measured above, which is why the arrangement works at all despite the defect.

The two currents are not two signals #

It is worth being exact about what a mode is, because the phrase common-mode current invites the reading that there are two currents in the pair. There are two currents — one in each conductor — and the modes are two linear combinations of them: half the sum and half the difference. Nothing in the circuit knows about modes; the decomposition is a change of basis chosen because the choke is diagonal in it.

That the choke is diagonal in that basis is the whole of the trick and it is exactly true only for identical windings. The conversion above is what off-diagonal terms look like when they are small, and the reason a real part is specified with a balance as well as an inductance.

What the netlist does not have in it #

No core. The inductance here is a number rather than a saturating flux, which is generous in one direction and pessimistic in another. A common-mode choke’s core sees the common-mode ampere-turns only — the differential ones cancel by construction — so it can be a small high-permeability ferrite carrying the full signal current without saturating. That is the design’s other reason for existing and it is not measured here. A boundary in volt-seconds is the machinery it would need.

No loss. A real common-mode choke’s ferrite is deliberately lossy at high frequency, so its common-mode impedance is resistive rather than reactive over most of its useful range — it dissipates the common mode rather than reflecting it, which matters because a reflected common mode is still in the room. The curves here are lossless and therefore describe a filter rather than an absorber.

And no source or load imbalance. Both modes are driven and terminated symmetrically here. A real cable is terminated by a receiver whose two inputs do not present the same impedance to ground, and that imbalance converts as surely as the winding mismatch does — the rejection the parts have measured the same mechanism in an instrumentation amplifier and found the resistors’ tolerance setting the answer.

Measuring one of these, and what the measurement is of #

A choke is specified by its common-mode impedance against frequency, measured with the two windings in parallel, and by its differential-mode inductance, measured with them in series opposition. Both are two-terminal measurements of a four-terminal part, and each throws away exactly the information the other one carries.

The consequence is that the balance — the quantity that decides the conversion above — is measured by neither. It is a difference between two windings, and both standard measurements are made on a combination of the windings in which that difference either cancels or is swamped. A part can meet both published numbers to a per cent and convert at −20 dB, and nothing on the certificate says so.

There is a four-terminal measurement that would catch it — drive differentially, read the common-mode output, which is a mixed-mode transmission parameter — and it is the same measurement the rejection four resistors decide makes on an instrumentation amplifier under a different name. In both cases the interesting number is a conversion between modes rather than a gain within one, and in both cases the part’s published specification is written about the gains.

The shape of the result #

The three numbers this essay measured are a corner ratio, a ceiling and a conversion, and none of them is on a data sheet. The ratio is set by the coupling coefficient, which is not printed. The ceiling is set by the interwinding capacitance, which is not printed. The conversion is set by the balance between the two windings, which is not printed and is not measured by either of the two measurements that are.

What is printed — an inductance and a rated current — describes the part’s behaviour in the one regime where it is uncontroversial: low frequency, matched windings, one mode at a time. Every question this essay asked is about what happens outside that, and every answer came from the same two-drive solve of a four-element netlist.

One component, and the inductance it has is a function of the question. Not of frequency, not of amplitude, not of temperature — of which linear combination of its two terminal currents is being asked about. A millihenry and a microhenry, from one part, at the same instant, with a ratio that depends on a coupling coefficient nobody prints.

The two things it has to be measured for are therefore in opposite directions: the inductance it presents, which wants to be large, and the leakage it presents, which wants to be small, and both are the same core wound in the same way. A number on a data sheet that names only the first is naming the half of the part that costs nothing.

Both halves are measured in the magnetics field #

The two quantities this component is sold on are exactly the two the magnetics field spends its time on, which makes this the one place in the lines field where the numbers come from somewhere else.

The coupling coefficient that decides the ratio of two thousand is measured in what coupling buys, and where it does not: across six designs from $k = 0.8$ to $k = 0.999$ it 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 of the band and, to within eight per cent, nothing at all at the other. For a choke that arithmetic is inverted — the quantity being bought is the ratio $2/(1-k)$, which is where all of the winding effort goes and which is why a common-mode choke is wound bifilar when a transformer need not be.

The leakage half is measured directly in the inductance that is a shape, and its result is the one that matters to the differential mode here: leakage is twice the magnetic energy in the window under equal and opposite ampere-turns divided by the square of the current — a geometry rather than a material — coming out at 3.086 microhenries a metre against a closed form’s 3.128 when the copper fills the window, and 4.608 against 6.255 when it fills half of it. Interleaving is worth 3.11 times and not the four it is quoted as.

And the five picofarads across each winding that leave the part worth 0.02 decibels by ten gigahertz are the same parasitic that which picture sets the upper edge finds resonating with the leakage — where the mechanism everybody names for a transformer’s upper edge turns out to be a factor of fourteen away from binding at fifty ohms, with the two accounts swapping at about 1500 Ω. A choke works into a line impedance rather than into a load, so which of the two accounts applies to it is a question its designer has to ask rather than inherit.

The core is also carrying two modes at once, which is a condition none of the magnetics essays assumes. A common-mode choke’s differential current produces almost no flux in the core by construction, and its common-mode current produces almost all of it — so the same part is operating near zero flux for the signal it is passing and near its volt-second limit for the interference it is blocking. A boundary in volt-seconds is the quantity that decides when the second of those saturates, and it has no frequency in it at all, which means a choke rated for a mains-frequency common-mode disturbance is rated for a very different one at a switching converter’s fundamental.