## 6 Simulation Results and Discussion
Figure 8: Magnetic flux density distribution in the transformer tank wall for \(\rho=\mathbf{0.25\times 10^{-6}}\)\(\Omega\cdot\)m: (a) \(\mathbf{n=1}\) (b) \(\mathbf{n=3}\). Figure 7: EF distribution in the transformer tank for \(\rho=\mathbf{0.75\times 10^{-6}}\)\(\Omega\cdot\)m: (a) \(\mathbf{n=1}\), (b) \(\mathbf{n=3}\). Figure 9: MF intensity in the mid-plane of the transformer tank for \(\rho=\mathbf{0.25\times 10^{-6}}\)\(\Omega\cdot\)m: (a) \(\mathbf{n=1}\), (b) \(\mathbf{n=3}\). To validate formula (29), total EC losses were analytically and numerically computed for eight different cases, where the radii, resistivity, and frequency were varied (see Table 1). The data are for a 17 kVA transformer with primary and secondary voltages of 120 and 240 volts, respectively that can be manufactured in the industry. Results of simulations are presented in Table 2. From Table 2, it can be observed that both the analytical and FEM-calculated results are very close. The relative error between analytical and FEM solutions is in the range 0.13% to 1.610%, which proves the high accuracy of Formulas (26), (27), and (29). Moreover, although the skin-effect depth decreases for higher frequencies, the MF density in the surface layer increases, which leads to higher losses. The geometry considered accurately represents the arrangement of the conductor carrying load current with harmonics passing through the transformer tank wall. The new formulas were successfully verified by comparing their performance with computationally expensive FEM simulations, proving their efficiency and efficacy. Hence, our results are useful for industrial applications, where transformer analysis and design demand accurate results and low-cost computational resources. It has been additionally shown that the presence of harmonics in the load current lowers the skin effect depth, they considerably increase stray losses and, therefore, must be taken into account for proper analysis and design of transformers. These formulas can be employed in the design algorithms for evaluating the impacts of harmonics on the transformer tank wall. The results can be used to improve the design of the transformer and thereby considerably reduce the possibility of hot spots in the bushing regions. Therefore, our formulas are an important advancement to the existing methods and formulas available in the literature. Our approach provides a basis for further developments such as analysis of stray losses in the case of three-phase currents (three conductors passing through the tank wall). Conceptualization, S.K., S.M., and J.C.O.-G.; methodology, S.K., S.M., and J.C.O.-G.; software, R.E.-P. and I.L.-G.; validation, S.K. ; formal analysis, S.K., S.M., and R.E.-P.; investigation, S.K. and S.M. ; resources, R.E.-P., I.L.-G., and J.C.O.-G.; data curation, S.K., S.M., and E.M.-V.; writing--original draft preparation, S.K. ; writing--review and editing, S.M., E.M.-V., and I.L.-G.; visualization, S.M., J.C.O.-G., R.E.-P., and E.M.-V.; supervision, S.M., J.C. O.-G., R.E.-P., and E.M.-V.; project administration, I.L.-G.; funding acquisition, I.L.-G. All authors have read and agreed to the published version of the manuscript. We want to acknowledge the support of Conacyt Mexico, under Project CB-2015/256519. The authors declare no conflict of interest.