Electrical equivalent models are necessary to take into account EMC constraints during the design process. According to the studied component, the modeling method that will be used to evaluate EMC performances will be different, depending on its nature, the required accuracy and the level of the model. Models have to consider heterogeneity of media (Magnetic components, copper tracks, substrates such as FR4 for example and air) and multilayer topologies.

On the one hand, Method of Moments (MoM) and method of Partial Element Equivalent Circuits (PEEC) are useful because they are not expensive in computational time. Indeed, they do not require meshing the whole considered space [16,17]. On the other hand, the Finite Element Method (FEM) takes into account inhomogeneous environments and magnetic materials [18]. Unfortunately, a fine meshing is needed to obtain accurate results. Moreover, all space is considered and computational time can become too costly for complex geometries. An alternative could be the development of coupling methods. As a consequence, works are carried out on the development of modeling methods making it possible to take into account these elements [5,6,7]. Moreover, due to the considered geometries, dedicated tools with adapted interfaces are really needed.

A complex system such as motor drivers is made of many thin conductors and ferromagnetic materials. Modeling them using FEM requires a lot of meshes, which becomes too problematic to find a solution. For power electronics applications, this step has to answer two very strict conditions:

Close to the conductors, variations of magnetic field is very important [19], so to take this aspect into account, the meshing must be dense;

Indeed, a good quality meshing requires two meshes into the skin depth for the skin effect to take into account.

On the other hand, because of three dimensional ferromagnetic materials, it is very complicated to solve this problem by PEEC method. Hence a hybrid method dealing with this type of problem was developed [7]. The idea of the hybrid method is to benefit from the main advantage both of FEM and PEEC method:

PEEC method easily takes into account interactions between complex 3D massive conductors;

One of the advantages of PEEC method is a simple meshing of conductors since assumptions on their current density can be applied. Skin and induced effects can be easily modeled. Each element of FEM meshing is considered as an inductor with a uniform current density. In other words, a conductor is represented by a set of inductors using this hybrid method. Air around conductors and magnetic regions are still meshed using finite elements.

A general problem with inductors and ferromagnetic material is shown in Figure 3.

To adapt FEM to the coupling, the magnetic scalar potential formulation is used. With this formulation, the magnetic field is calculated by a total scalar potential ϕ [20] and a reduced field T₀ due to all inductors as Equation (1).

(1)

Ω₀ is the region with air including the conductors. Ω₁ is the ferromagnetic region. T₀ can be expressed with t_0k representing the source of 1 A in the inductor k and its current I_k (2).

(2)

Because curl H = j in Ω₀ where j is local current density, so curl T₀ = j.

If j_0k is the current density of inductor k fed by a current of 1 A, t_0k has to satisfy (3) and (4).

(3)

(4)

Equation (4) is imposed because of H_t continuity on the interface Γ₀₁. t_0k is expressed by (5).

(5)

In Equation (5), h_0k is the Biot and Savart magnetic field and δϕ_k is the jump between scalar and reduced potentials [12] satisfying (4).

Note that this formulation does not require any meshing of the inductors but the source field t_0k has to be estimated before the finite element resolution. The weak form of T₀-ϕ formulation leads to the finite element matrix system (6).

(6)

In Equation (6), terms of matrices A and C are defined by (7).

(7)

Where α_i and α_j are the nodal finite element interpolation functions approximating ϕ. If the inductor k is fed by a voltage source U_k, the current I_k is unknown and a circuit relation must be added [21] (8).

(8)

The magnetic induction in Ω₀ (air) can be written using (9).

(9)

Combining (6) and (8) gives (10).

(10)

In equation (10), terms of matrices D and R are defined by (11).

(11)

Computations of D_kl require a fine meshing around inductors because the variation of t_0k and t_0l is very strong around them. To avoid this problem, a coupling of this formulation with PEEC method was proposed. D_kl is calculated by the mutual inductance M_kl that can be exactly determined by PEEC method. Substituting (5) into (11) gives (12).

(12)

The first term of right part of (12) represents the mutual inductance in vacuum between inductor k and l [20]. The second term can be transformed into surface integral by applying divergence theorem (13).

(13)

Note that the terms C_ik in (7) can be transformed as follow (14).

(14)

This avoids estimating t₀ in entire domain. t₀ is computed only on the interface Γ₀₁ and on a layer of elements (Ω₀₁, Figure 3). In conclusion, the proposed hybrid method avoids pre-calculating of t₀ in entire domain and relaxes the meshing around conductors. This was validated and gives stable results [7,19]. A case of study is defined in order to analyze this phenomenon using this original modeling method.

The chosen case of study is a simplified structure extracted from the geometry of an industrial variable speed drive. It is composed of a three-phase common mode filter with the floating potential tracks (U, V, W) of an inverter (Figure 4, Figure 5).

The toroidal core inductance material is the FT-1KM nanocrystalline, manufactured by Hitachi.

This material has a higher permeability than ferrite; it is first-rated for common mode filtering applications. The initial permeability µr is equal to 16,000 at 20 °C and for a frequency equal to 100 kHz. Each winding is composed of five turns. The theoretical value of the inductance is 1.2 mH. Due to the winding turns, the inductance is the main source in near field. Moreover, because of mutual couplings with the other parts of the structure, additional perturbations can occur.

Regarding the PCB, it is composed of two layers. Phase 1 and 2 are crossed. Phase 3 and the floating potentials U, V and W are routed on the top layer. The copper width is equal to 70 µm. The resolution frequency is chosen equal to the switching frequency (10.6 kHz).

Using the hybrid method that is described in the previous section, the cabling is modeled using PEEC approach and the magnetic material of the inductance using the FE method.

The aim of the study is to evaluate the interactions between the filter and the floating potentials tracks which are the most influent part regarding common mode current generation (ic = C.dV/dt) and by this way, far field emissions (Figure 5).

The currents of floating potentials tracks are computed with and without the common mode filter. Hence the influence of the couplings between the floating potential tracks U, V, W, and the common mode filter can be analyzed. In addition, Table 1 shows that the couplings are not negligible between the common mode filter and the floating potentials tracks with unbalanced currents of higher values.

Usually, for industrial devices, a well adapted solution is the shielding of wound components well known as the major source of near magnetic. Using the modeling process, a shielding has been easily added on the common mode inductance to evaluate its influence on the currents (Figure 6). It has been described in the FE part of the modeling method.

Since the shielding is made of thin plates of conducting material, the classical FE description is not the best way since the low thickness can lead to numerical errors during the solving stage. So an adapted meshing strategy with specific type of mesh has been applied. Its use and the associated formulations are detailed in [22,23].

As shown in Table 1, the shielding does not reduce the eddy currents. Its influence is localized around the inductance (Figure 7) and its efficiency decreasing is too fast to reach U, V, and W (Figure 8). In this case, the common mode inductance shielding is not an efficient solution to reduce the common mode generation.

Regarding the performances of such a modeling process using this hybrid method, its solving time as well as the necessary memory space have been compared with the case of the use of only FE method. With the proposed method, they have been divided by three. Therefore, it is faster for a same problem and is able to solve more complicated structures with higher number of unknowns.

The influence of couplings has been verified by measuring the far field of variable speed drives according to the studied configuration. The emitted field is compared with a topology where the inductance is placed closer to the floating potential tracks (Figure 9). As expected, the emitted field is more important in this case (Figure 10). The interactions are favored by the proximity between the inductance and the floating potential tracks. The common mode currents are more influent due to the adding of induced currents in U, V and W.

The geometry is influent and routing process with component placement need to take into account these elements in order to reduce the possible increase of the common mode current.

The couplings cannot be neglected in a near field computation. However, is it also the case in a far field context? As a consequence, it is now necessary to compare modeling methods in order to transpose the study in the far field domain. Works are in progress to compare FEM, PEEC and multipole expansion models in far field. For example, a static study of the inductor using FEM and multipole approach is presented on Figure 11 [15]. The magnetic field evaluation using a multipole approach instead of FEM is computed. This result shows the pertinence to use FEM for near field and multipole approach requiring less memory and time solving for far field studies.

In this part, the use of the proposed hybrid method on a test case has been detailed in order to underline its advantages and the real help it can bring to designers.

Nevertheless, it can be useful for very various types of applications where both magnetic and non-magnetic components and conductors are in the same structure such as power converters.

The possible parametric description, which is not detailed here, is a real advantage when designers want to evaluate the impact of geometrical but also physical parameters on the EMC performances of the studied structure.