# Modelling of Nanocrystalline Common-Mode Chokes with Custom Shape by Simulation-Based Extraction of Effective Permeability

Herbert Hackl, Werner Konrad, Martin Stoiber, Martin Heinisch, Christian Riener, Bernhard Auinger

<figure><img src="image_1.png" /></figure>

To cite this version:

Herbert Hackl, Werner Konrad, Martin Stoiber, Martin Heinisch, Christian Riener, et al. Modelling of Nanocrystalline Common-Mode Chokes with Custom Shape by Simulation-Based Extraction of Effective Permeability. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0201. hal-05101042

HAL Id: hal-05101042

https://utc.hal.science/hal-05101042v1

Submitted on 6 Jun 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Modelling of Nanocrystalline Common-Mode Chokes with Custom Shape by Simulation-Based Extraction of Effective Permeability

Herbert Hackl, Werner Konrad, Martin Stoiber,
Christian Riener, Bernhard Auinger
Silicon Austria Labs GmbH / CEMC
Graz, Austria
herbert.hackl@silicon-austria.com

Martin Heinisch
BMW GmbH
Steyr, Austria
Martin.Heinisch@bmw.com

## ACKNOWLEDGMENT

This work has been jointly supported by BMW GmbH, Steyr, Austria and by Silicon Austria Labs (SAL), owned by the Republic of Austria, the Styrian Business Promotion Agency (SFG), the federal state of Carinthia, the Upper Austrian Research (UAR), and the Austrian Association for the Electric and Electronics Industry (FEEI).

*Index Terms*—Nano-crystalline core, Filtering, Permeability, Modelling, Simulation

*Abstract*—Knowing the complex relative permeability of a ferrite or nanocrystalline core material is crucial to predict the insertion impedance of common-mode chokes. In practice, however, the material properties are often not known across the full desired frequency range. Additionally, factors like core shape, lamination and strip thickness influence the effective permeability and consequently the choke's impedance behaviour. This paper presents a workflow to create core material models that can be used in electromagnetic simulations to estimate the impedance behaviour of common-mode chokes. The core's material characteristics are determined by iteratively fitting model parameters to an impedance measurement using a generic laboratory setup. The resulting core model is then employed to predict the impedance of a common-mode choke in the intended application.

## I. INTRODUCTION

For nanocrystalline cores, the frequency dependence of the permeability is affected by the material's electrical resistivity, the thickness of the strip or ribbon and the intrinsic relative permeability [1]. Overall, it is not practical to model common-mode chokes (CMC) at the nano-crystalline microscale when the goal is to predict filter performance, i.e. insertion impedance, over

a specific frequency range. It is computationally more efficient to replace the core body with an equivalent magnetic material that accounts only for the macroscopic behaviour. Various analytical material models, i.e. Debye model [2], Cole-Cole model [3], Havriliak and Negami model [4], are commonly used to model the frequency dependence of these equivalent materials [5]. They ensure a smooth and causal behaviour, which may not be the case when material properties are extracted directly from measurements [6]. In practical applications, the magnetic material properties are needed for:

1. Developing a simulation model to assess a choke's performance for its intended application.

2. Predicting a choke's characteristic for alternative core geometries.

To assess a choke's performance in terms of simulation, one can use equivalent circuit models fitted to measurements [7] or 3D finite element (FE) models where material parameters are identified analytically from dedicated measurements [8], [9]. To perform these measurements, test fixtures are required, such as shown in [9]–[16], to connect measurement equipment, i.e. impedance analyzer and vector network analyzer, to the device-under-test. Unfortunately, predicting performance for a core geometry different from the measured device is generally not possible because the material parameters depend on its geometric features [10], [17]. This is especially true for nanocrystalline cores, because the manufacturing process and internal structure might differ with geometry. Another issue with nanocrystalline cores is that they require coating or housing for mechanical stability, such that the material can typically not be accessed for measurement-based characterization.

Consequently, this paper presents a workflow for extracting electromagnetic material parameters of nanocrystalline cores of custom shape which might not
fit into commercially available test fixtures. The approach is based on iterative fitting the material properties of a simulation model mimicking the measurement setup [18]. The novel contribution of this work is to amend the fitting process with a physical material model and to discuss the influence of permittivity alongside permeability. Section II explains how the complex relative permeability of the core is determined from impedance analyzer measurements using analytical equations. Section III demonstrates how the accuracy of the core's material properties can be improved through FE simulations and parameter fitting. Finally, the derived material parameters are applied in FE simulations to predict the common-mode impedance of a CMC intended for traction inverter applications, with results compared to experimental measurements.

## II. INITIAL SOLUTION: ANALYTIC DERIVATION OF COMPLEX PERMEABILITY FROM IMPEDANCE MEASUREMENT

The inductance induced by arbitrary core shapes can be derived from the self inductance of a generalized magnetic circuit [19] as

$$L = \frac{N\phi}{i} = \frac{N^2BA_c}{Ni} = \frac{N^2BA_c}{Hl_c} = \frac{N^2\mu_0\mu_rA_c}{l_c} \quad (1)$$

where $l_c$ is the average magnetic path length and $A_c$ is the effective core area.

The complex relative permeability $\mu_r$ of a core can be obtained by subtracting measurements of the loop impedance of a test fixture with and without a core, neglecting mutual inductance, as

$$\begin{align}
\underline{Z}_{\text{Fixture}} &= R_{\text{Cu}} + j\omega L_{\text{Cu}} \nonumber \\
\underline{Z}_{\text{Core}} &= R_{\text{Cu}} + j\omega L_{\text{Cu}} + j\omega L_{\text{Core}} \tag{2} \\
j\omega L_{\text{Core}} &= \underline{Z}_{\text{Core}} - \underline{Z}_{\text{Fixture}} \nonumber
\end{align}$$

Applying (1) with $N=1$ allows to express the core's relative permeability as

$$\underline{\mu}_r = \frac{(\underline{Z}_{\text{Core}} - \underline{Z}_{\text{Fixture}}) \cdot l_c}{j\omega \cdot \mu_0 \cdot A_c} \quad (3)$$

To assess the inductance induces by customized nanocrystalline cores, a generic PCB fixture has been used which allows cores of many shapes to be inserted, see Fig. 1. As an example, in this paper we extract $\underline{\mu}_r$ of the O-core of Fig. 1b which dimensions are given in Fig. 2, resulting to $A_c$ and $l_c$ according (4). A Keysight E4990A impedance analyzer was used for measurements between 20 Hz to 120 MHz.

$$\begin{align}
A_c &= (72 \text{ mm} - 59.1 \text{ mm})/2 \cdot 20.6 \text{ mm} \nonumber \\
l_c &= 2 \cdot 33.5 \text{ mm} + 2\pi \cdot 15.9 \text{ mm} \tag{4}
\end{align}$$

<figure><img src="image_3.png" /></figure>

(a)

<figure><img src="image_4.png" /></figure>

(b)

<caption>Figure 1: PCB fixture to measure the effect of cores with arbitrary shape on loop impedance. The impedance analyzer is connected at the left SMA socket. The other SMA is shorted, closing the loop to the common GND plane on the PCB bottom layer. Top: Measurement of a toroid. Bottom: Oval nanocrystalline core. Measurement of $\underline{Z}_{\text{Fixture}}$ without core is not shown.</caption>

## III. EXTRACTING COMPLEX PERMEABILITY FOR EM SIMULATION

A model of the impedance test fixture of Fig. 1 is constructed for 3D electromagnetic simulation using the full-wave FEM solver Ansys HFSS [20], see Fig. 3. Starting with the analytic solution from (3) as initial material definition, the core's permeability is iteratively adjusted until the simulated loop impedance equals the measurement result within the frequency range of interest. The workflow is visualized by Fig. 4.

Assuming that the system is mainly inductive, i.e. neglecting resistive and capacitive components, the impedance is expressed proportional to the complex permeability in consistence with (3) as

$$\underline{Z} = \frac{\mu_0 A_c}{l_c} \cdot j\omega(\mu' - j\mu'') = \frac{\omega\mu_0 A_c}{l_c}(\mu'' + j\mu') \quad (5)$$
<figure><img src="image_5.png" /><figcaption>Figure 2: Dimensions of the magnetic material inside the O-Core, excluding housing. The height is $h = 20.6$ mm.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 3: Electromagnetic simulation setup replicating the impedance measurement setup of Fig. 1b.</figcaption></figure>

Applying (5) we define the ratios $A_{Re,i}$, $A_{Im,i}$ between measured core impedance $Z_{meas}$ and simulation result $Z_{sim}$ of iteration $i$ which give the relation between actual permeability $\mu_{core}$ and the values used for simulation (6). For identical values the ratio is 1. Otherwise the permeability for the next simulation iteration $i+1$ is adjusted according to (7).

$$A_{Re,i} = \frac{Re(Z_{meas})}{Re(Z_{sim})} = \frac{\mu''_{core}}{\mu''_{sim,i}} \qquad (6)$$

$$A_{Im,i} = \frac{Im(Z_{meas})}{Im(Z_{sim})} = \frac{\mu'_{core}}{\mu'_{sim,i}}$$

$$\mu'_{sim,i+1} = \mu'_{sim,i} \cdot A_{Im,i} \qquad (7)$$

$$\mu''_{sim,i+1} = \mu''_{sim,i} \cdot A_{Re,i}$$

The correction factor is computed and applied at every frequency point individually. This can lead to discontinuous curves for $\mu(\omega)$, hence nonphysical material properties and bad convergence of the fitting process. To avoid such, for each iteration the data is smoothed and fitted to a physical model, namely (8) which is the so-called Havriliak-Negami model [4].

$$\hat{\mu}(\omega) = \hat{\mu}(\omega)' - j\hat{\mu}(\omega)'' = \mu_{\infty} + \frac{\mu_{s} - \mu_{\infty}}{(1 + (j\omega\tau)^{\alpha})^{\beta}} \quad (8)$$

<figure><img src="image_7.png" /><figcaption>Figure 4: Workflow to extract $\mu(\omega)$ for electromagnetic simulation. This work uses Ansys HFSS [20], controlled via Python using pyAEDT [21].</figcaption></figure>

Here $\hat{\mu}$ is an empiric estimate of the complex relative permeability, $\mu_s$ is the real static permeability, $\mu_{\infty}$ is the real permeability at infinite frequency, $\omega$ is the angular frequency and $\tau$ is the magnetic relaxation time. The exponents $\alpha$ and $\beta$ describe the asymmetry and broadness of the complex permeability. For $\alpha = 1$ and $\beta = 1$ this formula reduces to the Debye equation. For $\alpha = 1$ this formula reduces to the Cole-Davidson equation. For $\beta = 1$ this formula reduces to the Cole-Cole equation.

## A. Discussion of Material Extraction Results

Fig. 5 compares the complex permeability of the O-core from Fig. 3 computed with (3) or fitted with described workflow, while applying different material models. Additionally, Fig. 6 gives the corresponding simulated loop impedance compared to measurement. The analytic solution from (3) leads to good prediction of the loop impedance. However, the permeability data is nonphysical for low and high frequencies, because it reflects effects which are actually not originating from the core's magnetic properties. Below 1 kHz $\mu''$ becomes negative, overcompensating ohmic losses. Above 10 MHz capacitive effects lead to impedance phase shift. This is due to the core material's permittivity $\epsilon(\omega)$ which is not accounted for in (3). In contrast, the complex permeability found by applying the Havriliak-Negami model (8) makes physical sense over the complete frequency range, but does not directly lead to correct impedance simulation. It requires additional adjustment of the complex permittivity in the simulation model, as illustrated by Fig. 7. The zoomed view in Fig. 6 highlights the effect of complex permittivity assigned to the fitted magnetic material (green trace = original results
<figure><img src="image_8.png" /><figcaption>Figure 5: Extracted complex permeability for different material models.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 6: Simulated loop impedance for the setup of Fig. 3. Before the core's impact can be assessed, it is crucial to ensure that the simulation model of the fixture accurately reflects the measurement setup. For that matter, the lower traces show the measured and simulated fixture impedance without core.</figcaption></figure>

without permittivity; blue trace = with complex permittivity). In this work we used a frequency-independent value for the permittivity $\epsilon$ found by manual adjustment of the simulation model.

## IV. APPLICATION TO COMMON-MODE CHOKE

After having extracted a simulation model for the oval nanocrystalline core, it should be applied to predict the insertion loss of a common-mode choke (CMC) composed of suchlike cores and with application-typical current rails. The experiment of Fig. 8 enables to change

<figure><img src="image_10.png" /><figcaption>Figure 7: The effect of the core's complex permittivity $\epsilon$ on the first resonance around 100 MHz is best assessed by examining Re(Z) and Im(Z). Increasing $\epsilon'$ mainly shifts the resonance to lower frequencies, best visible in Im(Z). Increasing $\epsilon''$ damps the resonance peak, best visible in Re(Z). The blue traces give the original simulation result with $\epsilon_r = 1$. The green traces correspond to $\epsilon' = 200$ and a dielectric loss tangent of 0.9.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 8: 4-port impedance measurement setup for high-current CMC. The photo shows S-parameter measurement with a 2-port VNA.</figcaption></figure>

the number of cores used for the CMC. The common-mode impedance $Z_{CM}$ is obtained by post-processing 4-port S-parameter measurement made with a vector network analyzer (VNA). The investigated frequency range is 100 kHz-200 MHz. The material data used for the simulation model was extrapolated accordingly. The complex permittivity was fitted to the Havriliak-Negami model using a nonlinear least-squares method. Fig. 10 shows the measured and simulated magnitudes of $Z_{CM}$ for CMCs composed of either one or two cores. In the frequency range presented, the prediction by simulation offers reasonable accuracy. At higher frequencies capacitive effects become dominant, which are not well reflected by the simulation model.
<figure><img src="image_12.png" /><figcaption>Figure 9: Ansys HFSS simulation model reflecting the measurement setup of Fig. 8 with different CMC configurations.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Figure 10: Measured and simulated magnitude of common-mode impedance for the setups from Fig. 9. Again, the dotted traces show the fixture's impedance without core.</figcaption></figure>

## V. CONCLUSION

This paper showed a workflow to extract the frequency-dependent complex permeability of a nanocrystalline core from impedance measurement. The obtained material data was then used to derive a simulation model enabling the prediction of the core's performance in its final application. In the presented scenario, the core material could be sufficiently described for frequencies up to 10 MHz by specifying permeability only. By adding a constant complex permittivity the range could be extended up to 200 MHz. However, the applicable frequency range is product-specific and cannot be generalized. It is known from literature that

core properties are function of manufacturing process, geometry, temperature, DC-bias and other factors.

For future work the authors propose a nested fitting process: First, fitting of permeability as shown here, with a physical material model but no relative permittivity. In a second iteration, apply the found permeability and do the same fitting process for the complex relative permittivity, again with a physical material model. Repeat both steps till the model converges.

## REFERENCES

[1] R. Hilzinger and W. Rodewald, *Magnetic Materials: Fundamentals, Products, Properties, Applications*. Publicis Publishing, Erlangen, 2013.

[2] C. A. Balanis, *Advanced Engineering Electromagnetics*, 2nd ed. New York, United States of America: John Wiley & Sons, Inc, 2012.

[3] K. S. Cole and R. H. Cole, "Dispersion and absorption in dielectrics i. alternating current characteristics," *The Journal of chemical physics*, vol. 9, no. 4, pp. 341-351, 1941.

[4] S. Havriliak and S. Negami, "A complex plane representation of dielectric and mechanical relaxation processes in some polymers," *Polymer*, vol. 8, pp. 161-210, 1967. [Online]. Available: https://doi.org/10.1016/0032-3861(67)90021-3

[5] M. Y. Koledintseva, "Soft ferrites for emc applications," in *2018 IEEE Symposium on Electromagnetic Compatibility, Signal Integrity and Power Integrity (EMC, SI & PI)*, 2018.

[6] R. Suárez, M. Tijero, R. Moreno, A. Arriola, and J. M. González, "Influence of Complex Magnetic Permeability on 3-D Simulation of MnZn Common-Mode Chokes," in *2023 International Symposium on Electromagnetic Compatibility – EMC Europe*, 2023, pp. 1–6.

[7] K. Nomura, T. Kojima, and Y. Hattori, "Straightforward modeling of complex permeability for common mode chokes," *IEEE Journal of Industry Applications*, vol. 7, no. 6, pp. 462-472, 2018.

[8] N. Moonen, R. Vogt-Ardatjew, A. Roc'h, and F. Leferink, "3-d full-wave high frequency common mode choke modeling," *IEEE Transactions on Electromagnetic Compatibility*, vol. 62, no. 3, pp. 707-714, 2020.

[9] W. Konrad, H. Hackl, and M. Stoiber, "Extraction of ferrite material properties and application to simulation-based common-mode filter design," in *2024 IEEE International Conference on Electrical Systems for Aircraft, Railway, Ship Propulsion and Road Vehicles & International Transportation Electrification Conference (ESARS-ITEC)*, 2024, pp. 1-6.

[10] M. Kacki, M. S. Rylko, J. G. Hayes, and C. R. Sullivan, "Measurement methods for high-frequency characterizations of permeability, permittivity, and core loss of mn-zn ferrite cores," *IEEE Transactions on Power Electronics*, vol. 37, no. 12, pp. 15152-15162, 2022.

[11] J. Xu, M. Y. Koledintseva, Y. Zhang, Y. He, B. Matlin, R. E. DuBroff, J. L. Drewniak, and J. Zhang, "Complex permittivity and permeability measurements and finite-difference time-domain simulation of ferrite materials," *IEEE Transactions on Electromagnetic Compatibility*, vol. 52, no. 4, pp. 878-887, 2010.
[12] R. Huang and D. Zhang, "Using a single toroidal sample to determine the intrinsic complex permeability and permittivity of mnzn ferrites," *IEEE Transactions on Magnetics*, vol. 43, no. 10, pp. 3807-3815, 2007.

[13] B. Sun, D. Ma, G. Bai, M. Zeng, T. Cheng, Y. Ma, X. Xu, W. Quan, B. Han, and Y. Zhai, "A novel measurement method of complex magnetic permeability of ferrites materials and analysis of its influencing factors," *Measurement*, vol. 221, p. 113419, 2023. [Online]. Available: https://doi.org/10.1016/j.measurement.2023.113419

[14] M. Kovacic, Z. Hanic, S. Stipetic, S. Krishnamurthy, and D. Zarko, "Analytical wideband model of a common-mode choke," *IEEE Transactions on Power Electronics*, vol. 27, no. 7, pp. 3173-3185, 2012.

[15] C. Cuellar, W. Tan, X. Margueron, A. Benabou, and N. Idir, "Measurement method of the complex magnetic permeability of ferrites in high frequency," in *2012 IEEE International Instrumentation and Measurement Technology Conference Proceedings*, 2012, pp. 63-68.

[16] C. R. Sullivan and A. Muetze, "Simulation model of common-mode chokes for high-power applications," in *2007 IEEE Industry Applications Annual Meeting*, 2007, pp. 1810-1815.

[17] C. Cuellar and N. Idir, "Stray capacitances determination methods of emi filter inductors," in *IECON 2017 - 43rd Annual Conference of the IEEE Industrial Electronics Society*, 2017, pp. 7040-7045.

[18] M. Heinisch, "Messtechnische Bestimmung und simulative Vorhersage der Dämpfungseigenschaften von Magnetkernen in Traktionsinvertern," in *OVE Schriftenreihe Nr. 110: 20. EMV Fachtagung Österreichischer Verband für Elektrotechnik (OVE)*, 2023.

[19] W. G. Hurley and W. H. Wölfle, *Transformers and Inductors for Power Electronics*. John Wiley & Sons, Ltd, 2013.

[20] ANSYS Inc. ANSYS HFSS. [Online]. Available: www.ansys.com/products/electronics/ansys-hfss

[21] ——. PyAnsys. [Online]. Available: https://aedt/docs.pyansys.com/version/stable/