## 6 Simulation Results and Discussion  
Several simulations were carried out in order to compare the analytical results with FEM solutions. FEM computations were carried out using the Altair Flux finite element (FE) software. A disk with the characteristics: \(h=0.010\) m, \(a=0.0125\) m, \(b=0.150\) m, and \(\varepsilon_{r}=1\) is considered. A copper conductor with a radius \(r_{0}=2\) mm and length 1400 mm crosses the disk at the center of the hole. One quarter of the entire disc and a half of its height were considered in the simulations, taking advantage of the symmetry of the problem. The total system was enclosed in a cylinder with an inner radius of 255 mm, outer radius of 305 mm, inner half height of 700 mm, and outer half height of 800 mm. To mesh the geometry, 66,212 volume finite elements, 14,992 surface elements, and 1292 line elements were used, which gave as a result a total number of 82,496 elements. Of good quality were 96.36% of the elements. Moreover, to properly simulate the skin-effect in a thin boundary layer of the disk and to ensure a better accuracy of computations, nine layers of FE mesh of 0.255 mm of height in each one, were generated near the upper surface of the disk as shown in Figure 2(a). Additionally, a high-density mesh was created near the conductor (see Figure 2(b)). Figures 4 and 5 show the MF penetration in the transformer tank, computed for the first (60 Hz) and third harmonics (180 Hz) using formula (20) (Figures 4a and 5a for the first and Figures 4b and 5b for the third harmonics, respectively), for two different resistivities: \(\rho=0.25\times 10^{-6}\)\(\Omega\cdot\)m in Figure 4 and \(\rho=0.75\times 10^{-6}\)\(\Omega\cdot\)m in Figure 5. In order to compare the MF distribution in the magnetic disk for different harmonic numbers, both harmonics were studied with the same value of the RMS current \(I_{\text{rms,1}}=I_{\text{rms,3}}=141.42\) A. In both cases, the relative permeability was considered as \(\mu_{r}=200\). It can be observed in Figures 4 and 5 that the higher the harmonic number, the smaller is the penetration depth of the MF in the transformer tank. The EF distribution in the tank wall, computed using our analytical method (formulas (26) and (27)), is shown in Figures 6 and 7 for the first and third harmonics, under the same conditions (Figure 6 for \(\rho=0.25\times 10^{-6}\)\(\Omega\cdot\)m and Figure 7 for \(\rho=0.75\times 10^{-6}\)\(\Omega\cdot\)m, also \(I_{\text{rms,1}}=I_{\text{rms,3}}=141.42\) A and \(\mu_{r}=200\) in both cases). Since the EC density is proportional to the EF (\(\mathbf{j}_{n}=\sigma\mathbf{E}_{n}\)), Figures 6 and 7 also qualitatively depict EC density lines in the tank wall. It can be observed that the ECs are mainly concentrated near the wall surface (skin-effect), and the concentration of the current density is higher for the third harmonic than for the first. Figure 8a,b show the FEM-calculated magnetic flux density distribution in the transformer tank for \(\rho=0.25\times 10^{-6}\)\(\Omega\cdot\)m and RMS current \(I_{\text{rms}}=141.42\) A for the first and third harmonics, respectively. The maximum value of the magnetic flux density \(\text{B}=637.94\times 10^{-3}\) T was obtained on the hole surface for the first harmonic (see Figure 8a), whereas for the third harmonic it was \(\text{B}=638.19\times 10^{-3}\) T (Figure 8b). The absolute value wave-form for the MF on the central plane of the tank wall (at z = 0) is shown in Figures 9 and 10, where the blue line represents the analytical solution and the red points correspond to FEM simulations. Figure 9 is plotted for a resistivity of \(\rho=0.25\times 10^{-6}\)\(\Omega\cdot\)m, whereas Figure 10 shows the MF profile for \(\rho=0.75\times 10^{-6}\)\(\Omega\cdot\)m. Figures 9 and 10 show an excellent match between the analytical and FEM computations, which validates Equations (22)-(26). On the other hand, the time taken while executing the analytical model was 0.66 s, which is notably less than the several minutes the FE software takes in execution.