Why are the shapes of the impedance curves different for cores made of the same ferrite material?

The impedance (Z) of an inductor wound on a ferrimagnetic core is: $$Z=j\omega L_s + R_s = j\omega L_0 (\mu_{s}^{'} -j\mu_{s}^{"})$$ where $L_0$ is the inductance of the winding in vaccuum and $\mu_{s}^{'}$,$\mu_{s}^{"}$ comprise the material's complex permeability, representing a core's lossless reactance and resistive dissipation, respectively. The variation of $\mu_{s}^{'}$,$\mu_{s}^{"}$ for a material vs. frequency is provided by the core's manufacturer:

43 material complex permeability

$L_0$ is calculated as: $$L_0 = \frac{4 \pi N^2 10^{-9}}{C_1}$$ where N is the number of winding turns and $C_1$ is the core's "structure constant" or "form factor". $C_1$ is $\frac{l_e}{A_e}$, the ratio of the effective magnetic path length to the effective magnetic cross-sectional area of the core, physical characteristics that are constants for a given core's geometry. Since all of the parameters are constant, $L_0$ should also be a constant.

Since $L_0$ is a constant for a given core geometry and $\mu_{s}^{'}$,$\mu_{s}^{"}$ are inherent properties of the core material, the shapes of the reactance, resistance and impedance curves vs. frequency should be the same for all cores of the same material, but this is not the case as shown below for two 43-material cores from Fair-Rite:

enter image description here

What is the cause of the differences in the shapes of the two sets of curves?

  • 1
    $\begingroup$ But but they're different size cores - one is 19 mm long, the other is 13 mm. Why would you expect $L_0$ to be the same? The big difference happens near 1 GHz where 20 mm is starting to matter too. $\endgroup$
    – tomnexus
    Dec 11, 2019 at 21:06
  • $\begingroup$ @tomnexus he's not saying that L0 should be the same, but rather that it should only cause a proportional difference between the two cores, and the two curves should be the same other than their heights, because u' and u'' are bulk properties. $\endgroup$ Dec 11, 2019 at 22:10
  • $\begingroup$ (The answer, I believe, is "dimensional resonance" but that's not something I can talk about in enough detail for an answer). $\endgroup$ Dec 11, 2019 at 22:11
  • $\begingroup$ interwinding capacitance and winding resistance may also not change proportionally as the size changes $\endgroup$ Dec 13, 2019 at 19:32
  • $\begingroup$ @PhilFrost-W8II But, these curves are for a single winding on a binocular core. $\endgroup$
    – Brian K1LI
    Dec 15, 2019 at 15:12


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