# Multiple resonance frequencies of common-mode choke

In the datasheet of the CMC (common-mode choke) below (Würth Elektronik 744830017132), the impendence vs frequency graph has multiple peaks (resonances) which I marked.

The first one is the Self Resonance Frequency of the CMC. What about other the other peaks? What are they due to?

Here is the equivalent model of the common-mode choke that I found:

Image: Circuit model of a common mode choke (CMC) from ResearchGate -
C. Domínguez-Palacios, J. Mendez and M. M. Prats, "Characterization of Common Mode Chokes at High Frequencies With Simple Measurements," in IEEE Transactions on Power Electronics, vol. 33, no. 5, pp. 3975-3987, May 2018, doi: 10.1109/TPEL.2017.2724639

The model is just a model, and a simple one, maybe good up to the first resonating frequency, enough to maybe characterize the performance up to the rated frequency it is still useful.

In real world, the model will be inaccurate, as the CM choke does not consist of ideal capacitors, resistors, and inductances as shown, and electrical signals in conductors don't travel at infinite speed, but only slower than light, so even a piece of wire inside the choke will affect how the impedances affect each other.

The impedances are also not lumped or discrete like in the model, each millimeter of wire has a small inductance, each turn of a coiled wire has small capacitance between turns, etc.

So just like you cannot simulate a real word capacitor with a single capacitance, or a real world inductor with a single inductance, or a real world resistor with resistance, same applies to more complex devices like common-mode chokes.

From the datasheet:

they show a pair of two-layer windings. It's not clear how exactly they wired it for the test -- exact pinning, orientation, and even nearby materials, are critical at such frequencies -- also, it's surprising they went above 30MHz, most power CMCs are not swept this high -- but this does give us an uncommon insight into what lies beyond.

That is, the windings could be wired in series, or one can be shorted out and the other measured, for DM; and CM can be one open and the other measured, or both wired in parallel. These have no distinction at low frequencies, but at high frequencies (particularly over 10MHz), the capacitance of the core, the exact fields around and coupling between the windings, and coupling to other surrounding materials, all matter -- for example, if the test is done over a flat plane ground, or inside a shield box with the walls near the component, or an absorbent (anechoic) chamber, etc.

In any case, the resonances are due to the winding itself. I suspect the diagram shown is more of a worst-case, for high-inductance parts in the series; 1.7mH in this size probably uses a single layer winding. A single layer winding can be approximated as a helical waveguide, in this case with some manner of loading in the center (the core material doesn't look so much like core material at high frequencies, but a lossy, inductive ground plane of sorts).

I won't go into detail (also, I don't know the precise theory myself), but I will let ON4AA do so:
RF Inductance Calculator for Single‑Layer Helical Round‑Wire Coils, Serge Y. Stroobandt, ON4AA | hamwaves.com

For a more hand-waved explanation, consider the wire in a cylindrical solenoid (helix). We can consider one turn, against the adjacent turn, as a parallel-wire transmission line. This is in series with the next pair of turns, and so on down the length, so there is an immediate wave function, where at high frequencies, the signal propagates rapidly along the length of the coil. At lower frequencies, a signal propagates around the turns, but each turn is influenced by the next and so on; they form transmission lines of length one circumference, but they're stacked in series so that they add together, and couple and multiply (transformer action). The combined effect is that:

1. We have a transmission line structure of sorts, and the lowest resonant frequency is approximately the 1/4 wave length of the wire used;
2. The structure adds to itself, so that the impedance is not simply the characteristic impedance of a pair of turns, but many times higher;
3. The structure influences itself, so that the impedance and velocity factor are not constant with frequency, but vary (dispersion is strong).

If you play around with the calculator, you'll find the impedance of a solenoid winding has a peak (parallel resonance) at the expected SRF, and the frequency is close to the expected (wire length) value; as you go up, you'll find an antiresonance (series resonance, minimum impedance), then another peak, and so on and so forth, and the impedance and frequency of these peaks varies all over the place. In particular, the peaks are not spaced evenly: they are not harmonic. This illustrates the dispersion: each peak corresponds to a (2N+1)λ/4 resonant mode, but because velocity depends on wavelength, the frequencies are not simply proportional to N. The impedances similarly vary.

The key takeaway is that high-frequency behavior depends critically on the arrangement of wires. This is relevant for SMPS transformers, where square pulses excite stray resonances in the windings, and (often) where leakage inductance can be modeled as the distance between transmission lines; for RF amp design where well-behaved inductors are required for tuning, and inductive or high-impedance RFCs for supply; for power CMCs where the resonance may be desireable (the CM peak, roughly due to winding capacitance plus magnetizing inductance, can be designed to fall on harmonics of the main switching frequency); or for data CMCs where the DM resonances should be regularly spaced and symmetrical (i.e., exhibiting well-behaved transmission line behavior); etc.

• Thanks @Tim Williams for the explanation! When I am seeing the simulations in LTspice using the library file of this CMC, I am not seeing the other peaks only the 1st peak is visible. Commented Feb 6 at 10:41

Transistor models becoming less resembling subcircuits and more like fitted approximations, models of magnetics follow suit. Wurth's model of 744830017132 does not show high-frequency peaks of the datasheet, so I build a simplified CMC model to examine a transmission-line-like behavior of CMC windings. The transmission line has a number of cells, each cell represents one winding turn:

LBtm inductors represent winding turns adjacent to the core. For the turn-to-core capacitance of turns in this layer (Cts, Ctt), I use Takahashi-Maekava model, see Wideband Small-Signal Model of Common-Mode Inductors Based on Stray Capacitance Estimation Method. Turns in the layer above this layer are represented by LTop inductors. The respective capacitances are Cw (with the bottom layer) and Cwc (with the core). The purpose of adding L_air inductors will be explained later. I ran the first simulation without the unductor coupling, because I feared that mutual inductance directly connects the transmission line ends:

The graph with this circuit has little in common with the datasheet graph. It is not only because peaks are sharp; tinkering with losses, we can smoothen these; but the high-frequency peaks are too close to the principal one. I was surprised to discover that we need the coupling to place these peaks to where they belong:

Now, only the sharp self-resonance peak spoils the picture. No tinkering with parasitics helps. The model helped me to see that we need a permeability dispersion to effectively make this peak resemble the one from the datasheet. The simplest model of the core permeability dispersion is parasitic capacitance and resistance in parallel with the inductor:

And I added this simplest dispersion model to my simulation:

Now, the plot qualitatively resembles the datasheet one. For a more sophisticated model of core dispersion, see Straightforward Modeling of Complex Permeability for Common Mode Choke by Nomura, Kojima and Hattori.

Still, a trivial improvement is begged to be done. The simple dispersion model with parallel capacitance and resistance sets a high frequency permeability limit to zero, and, consequently, high-frequency inductance also tends to zero. This simplification results in a curious behaviour: the transmission-line-like peak positions can be translated to arbitrary great values as enabled by coupling coefficient. Theoretically, with a coupling coefficient of 1, these frequencies are infinite. We can limit the high-frequency permeability value with air permeability, adding a series inductor (L_air) with inductance of DC value divided by DC permeability of the core, like in this simulation:

Now, the coupling coefficient is 1, and the peak positions are defined by the circuit parameters.

Thanks to @Andr7 and @TimWilliams for inspiration.