## 4 Homogenization of the Laminated Open-Type Core  
An extremely dense finite element mesh, especially in the \(\gamma\)-direction, can be avoided by homogenizing the material characteristics of the \(\Omega_{\rm c}\) region, thus obtaining the \(\overline{\Omega}_{\rm c}\) region. The homogenization of the electrical and magnetic characteristics of a laminated medium with a fill factor \(k_{f}\) yields the anisotropic characteristics of the material, with the diagonal tensors of electrical resistivity \(\mathbf{\rho}\) and magnetic reluctivity \(\mathbf{v}\) being defined as  
\[\rho=\big{[}\ \rho_{\alpha}\ \ \ \rho_{\beta}\ \ \ \rho_{\gamma}\ \ \big{]}, \tag{9}\]  
\[\
u=\big{[}\ \
u_{\alpha}\ \ \ \
u_{\beta}\ \ \ \
u_{\gamma}\ \ \big{]}. \tag{10}\]  
As was already stated, only (2) and (7) are modeled in the first simulation step, whereas the fields in (8) are neglected. The values of all of the fields in (2) and (7) change monotonically in the \(\gamma\)-direction, and therefore the components of \(\mathbf{\rho}\) and \(\mathbf{v}\) defined in (9) and (10) are calculated using the following well-known Formula [11]:  
\[\rho_{\alpha}=\rho_{\beta}=\rho_{\rm lam}k_{f}^{-1}, \tag{11}\]  
\[\rho_{\gamma}^{-1}\approx\rho_{\rm ins}^{-1}\approx 0, \tag{12}\]  
\[\
u_{\alpha}^{-1}=\
u_{\beta}^{-1}=k_{f}\
u_{\rm lam}^{-1}+(1-k_{f})v_{\rm ins }^{-1}, \tag{13}\]  
\[\
u_{\gamma}=k_{f}\
u_{\rm lam}+(1-k_{f})v_{\rm ins}. \tag{14}\]  
Note that, due to (12), eddy current density \(\overline{J}_{\alpha\beta\gamma}\) goes approximately to zero, and consequently, the fields in (8) will approximately be equal to zero. In the case of a multipart open-type core, \(\Omega_{\rm c}\) region homogenization is carried out by applying (9)-(14) to each \(\Omega_{\rm c,i}\) part of the multipart core separately, with interface surfaces \(\Gamma_{\rm ins,i-j}\) instead of \(\Omega_{\rm ins,i-j}\) layers between them. Therefore, each homogenized part \(\overline{\Omega}_{\rm c,i}\) will be assigned its \(\rho_{i}\) and \(\
u_{i}\) tensors. However, for the thin insulating region \(\Omega_{\rm ins,i-j}\) to be homogenized and modeled as the interface surface \(\Gamma_{\rm ins,i-j}\) between the two adjacent core parts (\(\overline{\Omega}_{\rm c,i}\) and \(\overline{\Omega}_{\rm c,j}\) region), at least one of the two core parts must have a component of the tensor \(\rho_{\rm i}\) equal to zero in the direction perpendicular to the interface surface \(\Gamma_{\rm ins,i-j}\), in order to prevent the flow of eddy currents from \(\overline{\Omega}_{\rm c,i}\) to the \(\overline{\Omega}_{\rm c,j}\) region. If this is not the case, the direct modeling of \(\Omega_{\rm ins,i-j}\) is one of the options. However, a much denser mesh within and around the \(\Omega_{\rm ins,i-j}\) region is then needed, which significantly slows down the simulation convergence. For the multipart core shown in Figure 2b, i.e., in Figure 3b, regions \(\Omega_{\rm ins,1-4}\) and \(\Omega_{\rm ins,2-3}\) cannot be modeled as \(\Gamma_{\rm ins,1-4}\) and \(\Gamma_{\rm ins,2-3}\). A much more economical and simpler approach is to use a formulation based on the current vector potential \(\vec{T}\), approximated by the edge elements. In this case, instead of the direct modeling of the insulation regions \(\Omega_{\rm ins,i-j}\), it is possible to only prescribe the interface condition \(\vec{T}\times\vec{n}=0\) on the surface \(\Gamma_{\rm ins,i-j}\) between two parts of the multipart core, where \(\vec{n}\) is perpendicular to the surface \(\Gamma_{\rm ins,i-j}\). This ultimately prevents the penetration of eddy currents from one part of the core into another, because \(\vec{J}\cdot\vec{n}=\
abla\times\vec{T}\cdot\vec{n}=\
abla\cdot(\vec{T} \times\vec{n})=0\) holds. Hence, a \(\vec{T}\)-based formulation, such as a \(\vec{T},\varphi-\varphi\) formulation, is a better choice for the modeling of eddy currents in an open-type laminated core, where \(\varphi\) represents the magnetic scalar potential.