## 7 Results and Discussion
### Testing the Efficiency of the ATATt Method  
In order to test the effectiveness of the ATATt method, it is necessary to analyze the behavior of the ATATt method on a model that is comparable in size to the actual open-type cores. Here, the ATATt method is compared with the homogenization method (9)-(14) that is based on the standard \(\vec{A},V-\vec{A}\) formulation (the AVA method below). The simulation model consists of a laminated open-type (I-shaped) core within the cylindrical coil, the frontal section of which is shown in Figure 11a. The lamination consists of 100 lamination sheets and has a fill factor \(k_{f}=0.95\). The magnetic reluctivity of the lamination is \(v_{\rm lam}=795\ {\rm m/H}\) and the electric resistivity is \(\rho_{\rm lam}=4\cdot 10^{-7}\ \Omega{\rm m}\). Figure 11b shows a homogenized cross-section of the core in the case of a standard, one-part core, and Figure 11c shows a homogenized cross-section of the core in the case of a multi-part core (a two-part core, in this case). The simulation was performed on both the one-part core model (hereinafter marked by 1C-model) and the two-part core model (hereinafter marked by 2C-model). Figure 10: The ratio of the eddy current losses obtained by the ATATt method to the eddy current losses obtained by the BF method. Figure 9: Joule field (density of the eddy current losses) in the homogenized core of the model that is shown in Figure 7. The simulation results were obtained using the ATATt method at 50 Hz. The source current is sinusoidal, with an amplitude of the uniform current density \(J_{s}\) given by the current vector potential \(\vec{T}_{s}\). Because the coil is cylindrical, the current vector potential \(\vec{T}_{s}\) is defined analytically as  
\[\vec{T}_{s}=\left\{ \begin{array}{ll}J_{s}w\cdot\vec{a}_{z},&\text{in the coil hole}\\ J_{s}\Big{(}\sqrt{x^{2}+y^{2}}-R_{2}\Big{)}\cdot\vec{a}_{z},&\text{inside the coil}\\ 0,&\text{elsewhere}\end{array} \right.\]  
where \(J_{s}=1\) A/mm\({}^{2}\), \(w=3\) cm and \(R_{2}=8.5\) cm. In the case of the 1C-model, the FE mesh is the same for both the ATATt and AVA simulation, whereas in the case of the 2C-model it is necessary to create the insulation layer \(\Omega_{\text{ins},1-2}\) between the two parts of the core, but the number of finite elements is similar for both types of FE mesh. The simulations were performed at four different levels of FE mesh density. There is a good agreement between the losses obtained by the ATATt and AVA methods in both the 1C-model case and the 2C-model case, as shown in Figure 12. As expected, the finer the FE mesh, the better the agreement. Figure 11: Homogenized real-size simulation model: (**a**) longitudinal section, (**b**) cross-section of the 1C model, (**c**) cross-section of the 2C model. All of the dimensions are in centimeters. Figure 12: Eddy current losses for the ATATt and AVA simulations at 50 Hz. The losses for 1C-ATATt and 1C-AVA are on the left axis. The losses for 2C-ATATt and 2C-AVA are on the right axis. of such a core is often not possible. Despite a significantly higher number of degrees of freedom, the \(\bar{A},\bar{T}-\bar{A}\) formulation offers the greatest flexibility in the optimization of the open-type transformer core with respect to eddy current losses. By creating an appropriate geometric model and FE mesh, it is possible to eliminate the redundant degrees of freedom of the current vector potential \(\bar{T}\), thus significantly improving the convergence rate of the eddy current simulations. Additionally, by eliminating the degrees of freedom in the \(\bar{A},\bar{T}-\bar{A}\) formulation, it is also possible to prescribe the anisotropic electrical conductivity of an open-type core with heterogeneous lamination stacking directions and/or curved lamination sheets, assuming that they are simply connected. In future work, the ATAIt method will be extended to the nonlinear case, and a comparison of the simulation results with the measurements will be made. Conceptualization, S.F. and B.T. ; Data curation, S.F., B.T. and I.Z.