## 7 Results and Discussion
### Validation of the ATATt Method  
For this purpose, a small-scale model consisting of an open-type core inside the coil, as shown in Figure 7, was used. The core is a laminated cuboid with a fill factor \(k_{f}=0.96\). It consists of 10 lamination sheets of thickness \(d=0.48\) mm. The magnetic reluctivity of the lamination is \(v_{\mathrm{lam}}=795\) m/H and the electric resistivity is \(\rho_{\mathrm{lam}}=4\cdot 10^{-7}\)\(\Omega\)m. The source current density \(\overline{j}_{s}\) is sinusoidal, and is given by the source current vector potential \(\overline{T}_{s}\). Because the coil has a regular geometry, the source current vector potential \(\overline{T}_{s}\) can be expressed analytically as  
\[\overline{T}_{s}=\left\{\begin{array}{ll}J_{s}w\cdot\overline{a}_{z},& \text{in the coil hole}\\ T_{s,coil}\cdot\overline{a}_{z},&\text{inside the coil}\\ 0,&\text{elsewhere,}\end{array}\right.\]  
where \(J_{s}=1\) A/mm\({}^{2}\) represents the source current density amplitude, \(w\) represents the cross-sectional width of the coil, and \(T_{s,coil}\) represents the current vector potential function that depends on the position inside the coil; for this model, it is calculated as:  
\[T_{s,coil}=\left\{\begin{array}{ll}J_{s}(R_{2}-x),&|y|<l_{c}/2\\ J_{s}\bigg{(}R_{2}-\sqrt{x^{2}+\left(y+l_{c}/2\right)^{2}}\bigg{)},&y<-l_{c}/2 \\ J_{s}\bigg{(}R_{2}-\sqrt{x^{2}+\left(y-l_{c}/2\right)^{2}}\bigg{)},&y>l_{c}/2,\end{array}\right.\]  
where \(R_{2}=0.85\) cm represents the outer radius of the coil, as shown in Figure 7b.  
The BF simulation was performed on the laminated model shown in Figure 7. In order to achieve a sufficiently fine discretization of the lamination sheets and insulation layers, an extremely dense FE mesh was required, with a total of 5 million elements. In contrast, the ATATt method was performed on the homogenized core model (calculated with (9)-(14)), with a total of 100,000 elements. The visualization of the Joule field (the eddy current loss density field) at 50 Hz for the BF simulation is shown in Figure 8, and visualization of the Joule field at 50 Hz for the ATATt simulation is shown in Figure 9. There is a very good agreement of the Joule fields in Figures 8 and 9.  
The ratio of the eddy current losses obtained by the ATATt method and BF method, at different frequencies, is shown in Figure 10. Curve \(Pa\beta\) shows the ratio for \(P_{a\beta}\) losses, and curve \(Pa\beta\gamma\) shows the ratio for \(P_{a\beta\gamma}\) losses, whereas the ratio for the total eddy current losses \(P\) is represented by the \(P\) curve. According to the \(Pa\beta\) curve in Figure 8, there is a very good agreement in the values of \(P_{a\beta}\) between the BF and ATATt simulation results, thus validating the ATATt method. It is important to emphasize that the curve \(Pa\beta\) is the same even in the case when the elimination of the degrees of freedom on \(\Gamma_{\mathrm{t}}\) surfaces is not performed, i.e., the elimination of the degrees of freedom has no effect on the simulation  
In the post-processing phase of the ATATt simulation, the losses \(P_{\alpha\beta}\) are directly calculated from \(\vec{J}_{\alpha\beta}\), whereas the losses due to \(\vec{J}_{\alpha\beta\gamma}\) are calculated indirectly from the tangential component of induction \(\vec{B}_{\alpha\beta}\) using the well-known formula \(P_{\alpha\beta\gamma}=\sigma\pi^{2}f^{2}d^{2}B_{\alpha\beta}^{2}/6\)[21]. For a more accurate calculation of \(P_{\alpha\beta\gamma}\) losses, it is possible to use a different method that is compatible with the \(\vec{A},\vec{T}-\vec{A}\) formulation [14; 15]; in this paper, the emphasis is on eddy currents \(\vec{J}_{\alpha\beta}\) and the associated losses \(P_{\alpha\beta}\).