2D FEM Modeling Comparison of Multi-Winding Magnetic Couplers for Power Electronics Applications

Floran Cassin, Adrien Mercier, Eric Laboure, Jeanne-Marie Dalbavie, Patrice Gomez

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Floran Cassin, Adrien Mercier, Eric Laboure, Jeanne-Marie Dalbavie, Patrice Gomez. 2D FEM Modeling Comparison of Multi-Winding Magnetic Couplers for Power Electronics Applications. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0173. hal-05107589

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# 2D FEM Modeling Comparison of Multi-Winding Magnetic Couplers for Power Electronics Applications

Floran Cassin*†, Adrien Mercier*, Eric Laboure*, Jeanne-Marie Dalbavie†, Patrice Gomez†

* Université Paris-Saclay, CentraleSupélec, Sorbonne Université, CNRS, Laboratoire de Génie Electrique et Electronique de Paris, 91192, Gif-sur-Yvette

† Ikos Consulting, 92300, Levallois-Perret

E-Mail: fcassin@ikosconsulting.com

**Index Terms**—Finite-element method, Solid-State Transformer, Magnetic device, Multi-active bridge, Isolated bidirectional DC-DC converters.

**Abstract**—Multiple-Active Bridge (MAB) converters, an extension of Dual-Active Bridge (DAB) converters, have emerged as a promising approach for multi-source, multi-load energy exchanges. A critical factor in their performance is the design of the multi-windings magnetic coupler, which determines the equivalent inductances and consequently the power flow between ports. This paper presents 2D FEM modeling techniques for multi-windings magnetic couplers, comparing various geometries models to balance precision and speed. The air gap-free FEM geometry reduces simulation time while maintaining accuracy, whereas the two stretched models involve a trade-off between precision on electrical model parameters.

## I. INTRODUCTION

Electrical converters play a key role in many sectors, such as transport, power grids and home automation, and are experiencing significant growth due to the energy transition. This has led to a multiplication of power chains within electrical energy conversion systems. Aggregating multiple sources and diverse energy consumers is a solution to reduce the number of converters needed in systems that require multiple energy transfers between different sources and loads.

Multiple-Active Bridge (MAB) converters (Fig. 1) - a generalization to multiple ports of the Dual-Active Bridge (DAB) - are a potential solution within the AC-coupled family [1], increasingly considered for this type of problem. These converters feature a multi-winding magnetic coupler, H-bridges for each port, and DC sources and loads. The technological bottleneck for this type of converter is, the control of the active bridges. There is an extensive body of literature on this topic,

<figure><img src="image_2.png" /><figcaption>Fig. 1: MAB converter with 4 port</figcaption></figure>

proposing control strategies aimed at maximizing efficiency by reducing the RMS current in the windings while maintaining zero-voltage switching for improved conversion efficiency [2] [3] [4]. In this family of converters, the power flow equations between port j and port k can be synthesized for the different control strategies in the following form :

$$P_{jk} = \frac{V_j V_k}{8 f n_j n_k l_{jk}} \times \Phi \quad (1)$$

where :

* $n_j$ and $n_k$ are the number of turns at port j and k;

* $V_j$ and $V_k$ the voltage across their respective windings;

* f is the switching frequency of the converter;

* $l_{jk}$ is the equivalent inductance between these two ports;

* $\Phi$ is a regulation function dependent on the control strategy used.

The common point across all these strategies is the key role played by the magnetic coupler, with the power exchange between two ports being directly dependent on the equivalent inductance connecting them (ie. $l_{jk}$ Eq 1).
The sizing of the magnetic coupler is therefore of primary importance, as it directly influences the values of the different coupling inductances between ports, which in turn define the energy transfers within the system. In the field of magnetic component design, the use of finite element method (FEM) is currently the preferred solution due to the complexity of the phenomena involved in these components. However, the disadvantage of this method is inherent in the computation time. Indeed, depending on the geometry and mesh being studied, 3D FEM simulations (in particular) can be particularly time-consuming. Modeling techniques that accelerate simulations by simplifying the geometry while maintaining a certain level of accuracy in the results are an important aspect of the design approach proposed in this article.

## II. PROBLEM POSITION

The device studied in this work is a current-reversible multi-input multi-output (MIMO) electrical converter, where the common energy node facilitating energy exchanges between the system's ports is implemented using a magnetic coupler with vertically distributed air-gap. The coupler is an EI-type with vertically distributed air gap on inner and outer legs [5] [6].

The features is that we are stacking magnetic bricks obtained by cutting an I-type ferrite. These magnetic bricks are not without flaws, the main one being their surface state at the interfaces. These surface states result in air gap at the interface between two bricks. To accurately represent the behavior of the final coupler, these air gap need to be included in the geometry of the 2D FEM simulation, which significantly increases the number of nodes required for meshing.

The aim of this article is to determine a method for accounting for these air gap without having to explicitly include them in the geometry, by adjusting the magnetic circuit and material characteristics. Initially, the study will focus on the MIMO converter, presenting the electrical model of the multi-winding coupler used. The second part will address the various geometries considered, based on simplifications that allow for accelerated simulations. Finally, the results obtained from different 2D FEM simulations of a 4-winding magnetic coupler will be compared.

## III. DISTRIBUTED MODEL

A section of the literature on magnetic couplers is dedicated to various multi-winding models. A relatively common model for applications involving more than three ports is the Cantilever model [7]. However, future work will consist on the production of a prototype and

the determination of the model thanks to measurement. Or, obtaining accurately the parameters of this model is challenging due to its sensitivity to measurements, particularly for strongly coupled windings.

<figure><img src="image_3.png" /><figcaption>Fig. 2: Distributed model</figcaption></figure>

To achieve this, we will use a distributed model (Fig. 2) constructed such that each k port contains an ideal coupler with a turn ratio of $n_k$, a series resistor $r_{k,DC}$ representing the static resistance of the k winding, a parallel impedance $Z_k$, which we will refer to as the magnetization impedance the k port, and link impedances $z_{jk}$ connecting each of the ports in pairs. Specifically, by considering this impedance $z_{jk}$ as a series combination of an inductance $l_{jk}$ and a resistor $r_{jk}$, we account for the impact of frequency on the model parameters. In our application, we aim to determine the equivalent inductance $l_{jk}$ simply to study the power flows between ports.

The distributed model has several advantages for our application:

* it allows for a straightforward analysis of energy transfer between two ports through the link impedance $z_{jk}$;

* it does not favor any particular port, as each port has the same parameters as the others;

* it includes the series static resistance as well as impedances instead of just inductances, allowing for the consideration of the dynamic effects of the coupler.

Moreover, the main advantage of this model in our case is the determination of its parameters. Indeed, the method used for obtaining the link impedances $z_{jk}$ involve injecting a current $n_j I$ into j port and a current $-n_k I$ at the k port in order to cancel the ampere-turns, which enables us to obtain:

$$z_{jk} = \frac{V_j}{n_j I} - r_{j,DC} - \frac{V_k}{n_k I} - r_{k,DC} \quad (2)$$

This method may present certain challenges in some cases, but future work will propose strategies to address them.
<figure><img src="image_4.png" /><figcaption>Fig. 3: Principle of the double 2D method : dividing a 3D geometry a) into two 2D parts : front view b) and side view c)</figcaption></figure>

Under the condition of a coupler with non-interleaved windings, we hypothesized that the link impedances undirectly connected between ports can be neglected (for instance $z_{13}$, $z_{14}$, $z_{24}$, ...). Indeed, the main factor of the coupling between winding non-interleaved is the distance between them. It means that the coupling between winding adjacent to each other will be stronger.

## IV. SIMULATION GEOMETRY FEM 2D

### A. Geometry with airgap

The coupler under study is similar to an EI-type coupler, with windings that can consist of either a single or double layer of conductors. Each winding contains a number $n_k$ of turns. The winding are stacked on top of each other, meaning that it is not possible to interleave the layers between the different windings.

A significant portion of the windings is located outside the winding window of the magnetic coupler. These coil overhang exhibit specific magnetic behavior, leading to a notable increase in magnetic leakage. In the case of a 2D FEM simulation, to account for the differing behavior between the parts of the winding inside the magnetic circuit's winding window and those outside, the "double 2D" study method will be used as shown on Fig. 3 and studied in [8] [9] [10].

This method allows for the avoidance of 3D FEM simulations, which are often much more time-consuming than 2D FEM simulations, while still accounting for the entire geometry of the magnetic coupler. This method involves dividing the magnetic circuit into two parts: a front view representing the winding section within the winding window, and a side view for the coil heads. This division is done by imposing the same current in both

<figure><img src="image_5.png" /><figcaption>Fig. 4: Parameters geometry scheme</figcaption></figure>

the front and side views of the winding. By analyzing the magnetic fields in both parts and summing their contributions, the electrical parameters of the complete coupler model can be obtained.

As previously mentioned, the feature of the coupler considered in our study arises from the presence of air gap at the interface between two magnetic blocks because of the surface state of each blocks after cutting. These air gap are represented by introducing a planar space between the different magnetic blocks, with a thickness $e_{airgap}$ estimated from measurements and modeled using a reluctance network. Since this thickness is much smaller compared to the other dimensions of the coupler, it is necessary to finely mesh this area in the FEM simulation to obtain accurate results. However, this significantly increases the simulation time.

### B. Geometry without airgap

To speed up FEM simulations, a straightforward solution would be to remove air gap from the simulation. However, to avoid significantly degrading the accuracy of the results, it is necessary to find a modeling method that accounts for their impact without including them directly in the geometry.

To achieve this, we will use a homogenized model that integrates both the magnetic material of the block and the air gap. An equivalent reluctance model for this combined system is provided below:

$$ \mathcal{R}_{brick+airgap} = \frac{h_{brick}}{\mu_r \mu_0 S_{interface}} + \frac{e_{airgap}}{\mu_0 S_{interface}} \quad (3) $$

where in Fig. 4:

* $h_{brique}$ is the length of the magnetic block in the direction normal to the magnetic material section (ferrite, in our case);
* $S_{interface}$ is the cross-sectional area of the magnetic material at the interface;

* $\mu_0$ is the permeability of free space (vacuum);

* $\mu_r$ is the relative permeability of the magnetic material;

* $e_{airgap}$ is the thickness of the air gap.

The equivalent homogeneous material, which has the same reluctance, must therefore have an equivalent relative permeability given by:

$$ \mu_{eq} = \mu_r \left( \frac{h_{brick} + e_{airgap}}{h_{brick} + \mu_r e_{airgap}} \right) \quad (4) $$

The removal of the airgap may have an effect on the distribution of electrical currents in the conductors, erroneously causing the fringe effect to disappear. However, as the winding is sufficiently far from the airgap, this effect will have no consequence.

By enabling a significant increase in the mesh refinement of the geometry, this method allows for a faster simulation time. However, to account for the contributions of the windings outside the winding window, it remains necessary to use the double 2D method, which requires twice the number of simulations needed to obtain the model.

### C. Stretched geometry without airgap

The double 2D method, by eliminating the need for 3D simulations, provides more representative results for a coupler with large coil heads. However, it requires doubling the number of simulations to obtain the model parameters. In keeping with the goal of reducing the necessary computation time, we propose a modeling

method that eliminates the need for the double 2D approach while still accounting for the behavior of the windings outside the winding window.

Here again, we will use a homogenization principle. To achieve this, we will define an equivalent geometry corresponding to an elongated version of the magnetic circuit, in which the windings are included along their entire length within an equivalent winding window. This equivalent length is defined as follows:

$$ l_{stretch} = b + 2e_j + a \quad (5) $$

where:

* b is the depth of the winding window of the actual component;

* a is the width of the central leg of the magnetic block;

* $e_j$ is the distance between the central leg and the outer leg of the magnetic circuit.

However, by elongating the magnetic circuit, we must preserve the original overall reluctance of the coupler. To achieve this, there are two options: reducing the permeability of the material $\mu_{r_{eq,stretch}}$ or reducing the width of the magnetic blocks $a_{stretch}$ to maintain the same ferrite cross-sectional area as the actual component. This results in the following two expressions depending on the chosen option:

$$ \mu_{r_{eq,stretch}} = \mu_{r_{eq}} \left( \frac{b}{l_{stretch}} \right) \quad (6) $$

$$ a_{stretch} = a \left( \frac{b}{l_{stretch}} \right) \quad (7) $$

<figure><img src="image_6.png" /><figcaption>Fig. 5: 2D FEM Geometries of the studied magnetic coupler: a) real geometry with airgap, b) equivalent geometry without airgap, c) equivalent stretched geometry without airgap with $\mu_{r_{eq,stretch}}$, d) equivalent stretched geometry without airgap with $a_{stretch}$</figcaption></figure>
Thus, we obtain two new geometries that do not require doubling the number of simulations required to obtain the electrical model of our multi-winding coupler.

## V. RESULTS

The parameters of the distributed model will be identified on the basis of the four proposed 2D FEM geometries presented above (Fig. 5) using FEMM 4.2 [11]. The results will be analyzed in comparison with the reference model of the geometry with airgap. In order to study the principles of multi-winding couplers and the identification of model parameters, all ports have the same number of turns.

### A. Comparison of FEM 2D simulations

<figure><img src="image_7.png" /><figcaption>Fig. 6: Link inductance $l_{12}$ simulated with the different FEM geometries, a) and b) quite identical</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7: Link inductance $l_{23}$ simulated with the different FEM geometries, a) and b) quite identical</figcaption></figure>

Fig. 6 and 7 shows the link inductances $l_{12}$ and $l_{23}$ over a frequency range $f\epsilon[40 \text{ kHz}, 280 \text{ kHz}]$. First of all, we notice that the simulated inductances are almost constant with frequency. This is due to the use of a constant permeability for the magnetic materials in the simulation. Secondly, the results obtained by the geometry with airgap and that without airgap are significantly equal for both inductances and the stretched simulation with $\mu_{r_{eq,stretch}}$ are very close to them. For the stretched simulation with $a_{stretch}$, it tends to underestimate the inductances compared to the reference geometry.

<figure><img src="image_9.png" /><figcaption>Fig. 8: Link resistance $r_{12}$ simulated with the different FEM geometries, a) and b) quite identical</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 9: Link resistance $r_{23}$ simulated with the different FEM geometries, a) and b) quite identical</figcaption></figure>

Fig. 8 and 9 shows the link resistances $r_{12}$ and $r_{23}$ over a frequency range $f\epsilon[40 \text{ kHz}, 280 \text{ kHz}]$. As with the link inductances, the link resistances with airgap and
without airgap are roughly equal. For all four simulations, we can see that the resistances curves has the same frequency-dependent trend, which is moreover an expected trend when dynamic effects in a resistor are taken into account. However, here it is the stretched simulation with $a_{stretch}$ that is closer to the reference when the stretched simulation with $\mu_{r_{eq,stretch}}$ is underestimating the resistance.

<figure><img src="image_11.png" /><figcaption>Fig. 10: Relative differences of the voltages with the different FEM geometries compared to the reference geometry with airgap 2\*2D</figcaption></figure>

Finally, Fig. 10 shows the relative differences of the absolute voltages with the multiple cases of current injection used to identify the distributed model. We can see that the geometry without airgap has a minor difference, less than 0.7% when the stretched simulation with $\mu_{r_{eq,stretch}}$ and the stretched simulation with $a_{stretch}$ has larger difference with the geometry with airgap, more than 7% and 6% respectively.

## B. Discussion

Based on comparisons of the results obtained with the various 2D FEM geometries, we can highlight several important points.

Firstly, it is known that the permeability for the magnetic materials is dependant of a several parameters including the frequency. A constant permeability is commonly used in FEM simulation based on the hypothesis that the frequencies studied is in the quasi-constant permeability frequency range.

Concerning link inductances, we note that they are mainly dependent on the position of the windings in the winding window. In fact, of the four simulations, the geometry with the greatest difference is the stretched width without airgap 1\*2D geometry, which is the only

one where the winding window was modified. It is showing an expected effect.

As shown on the different figures, the link parameters obtained with the geometry without airgap are nearly equal as the reference, further validating the hypothesis that the fringe effect can be neglected in our case. As the simulation time is more than four times lower, the method proposed to reduce is mesh complexity by making the airgap disappear is convincing.

However, the two stretched simulations are less convincing as they show contradictory results precision. Indeed, the third geometry with the permeability reduced by the stretching shows an higher precision on the link inductances and a lower precision on the link resistances while the fourth geometry with the width reduced by the stretching shows opposite results being more precise on link resistances. Knowing that the two stretched geometries are two times faster than the geometry without airgap, if we want to fully know the multi-winding electrical model with the same precision on the parameters, we dont save any time while losing precision. Nevertheless, if we only want to focus the link inductances between ports, the third geometry might be a good compromise.

<table><caption>TABLE I: Average simulation time of the different FEM geometries</caption><thead><tr><td>Geometry</td><td>Average simulation time</td></tr></thead><tbody><tr><td>With airgap 2\*2D (Reference)</td><td>2062 sec</td></tr><tr><td>Without airgap 2\*2D</td><td>483 sec</td></tr><tr><td>Stretched with μ<sub>r<sub>eq,stretch</sub></sub> 1\*2D</td><td>249 sec</td></tr><tr><td>Stretched with a<sub>stretch</sub> 1\*2D</td><td>241 sec</td></tr></tbody></table>

# VI. CONCLUSION

This study has explored various approaches for simulating and identifying the parameters of a multi-winding magnetic coupler with vertically distributed air gap used in a multi-input multi-output (MIMO) electrical converter. By proposing simplified geometries—both without air gap and using stretched models—the study demonstrated significant reductions in simulation time compared to the double 2D FEM method with explicit air gap.

The findings show that eliminating air gap from the geometry leads to comparable results in terms of link inductances and resistances, while reducing computation time by a factor of four. Although the stretched geometries further reduced simulation times, they introduced
trade-offs in precision. The third geometry with the permeability reduced by the stretching geometry provided accurate inductance estimates but underestimated resistances, whereas the alternative stretched geometry yielded the opposite behavior.

In summary, the simplified 2D FEM geometry without air gap offers a balanced solution for accurate simulations in reduced time, while the stretched models, particularly the third geometry with the permeability reduced by the stretching, present a valuable compromise when the focus is solely on link inductance calculations. These results open pathways for further optimization, especially in applications where precise inductance prediction is critical.

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