## 2 Model  
The conductor passing through the bushings and tank wall of a transformer is shown in Figure 1. The tank wall is considered as a disk of radius \(b\) with a hole of radius \(a\) in the center. An infinitely long conductor passes at a right angle through the hole. The axial geometry of the system requires the use of the cylindrical coordinates for solving [PERSON]'s equations. Figure 2 shows the schematic geometry of the system, which is divided into two regions. The hole, where the conductor is mounted, the medium above the tank wall, and the region below the wall are filled with air or any other dielectric. Region \(R_{I}=\{(r,z):r_{0}\leq r\}\backslash\{(r,z):a\leq r\leq b,|z|\leq h/2\}\), where \(r_{0}\) and \(h\) are the conductor half diameter and the transformer tank thickness, respectively. Region \(R_{II}=\{(r,z):a\leq r\leq b,|z|\leq h/2\}\) represents the tank wall, which is made up of a ferromagnetic material. In this paper, the magnitude of the magnetic flux density is considered not too high so that the permeability of the tank wall can be considered constant. The values of magnetic flux density found in [21] and [22] for the tank wall studies were 0.001 T and 22 mT, respectively, which are much lower than the maximum value (1.7 T maximum for steel material [23]). The conductor, passing through the hole, can be modeled as infinitely thin and long. With the advent of nonlinear loads in the distribution system, the current carries an infinite number of harmonics:  
\[I(t)=\sum_{n=-\infty}^{\infty}I_{n}e^{jn\omega t}, \tag{1}\]  
where \(n\) enumerates harmonics, \(I_{n}\) is the respective complex amplitude, and \(\omega\) is the fundamental angular frequency. The EF and MF produced by the current-carrying conductor, in both regions can be expanded to a Fourier series of the same form:  
\[\mathbf{E}=\sum_{n=-\infty}^{\infty}\mathbf{E}_{n}e^{jn\omega t}, \tag{2}\]  
\[\mathbf{H}=\sum_{n=-\infty}^{\infty}\mathbf{H}_{n}e^{jn\omega t}, \tag{3}\]  
Figure 1: Conductor passing through the bushing regions of a transformer. Figure 2: Tank wall geometry for calculating the electromagnetic field (EMF). where \(\mathbf{E}_{n}\) and \(\mathbf{H}_{n}\) are the EF and MF in the frequency domain (complex amplitudes of harmonics), respectively. Due to the linear permeability of the tank wall, the complete system of [PERSON]'s equations in the quasi-static approximation can be written for each harmonic as follows:  
\[ \begin{array}{ll}\
abla\times\mathbf{E}_{n}=-jn\omega\mu\mathbf{H}_{n},&\
abla \cdot\mathbf{H}_{n}=0,\\ \
abla\times\mathbf{H}_{n}=\mathbf{J}_{n}+\sigma\mathbf{E}_{n},&\
abla\cdot \mathbf{E}_{n}=0,\end{array}  \tag{4}\]  
where \(\mu=\mu_{n}\mu_{0}\). Due to the axial symmetry of the system, the solution to Equation (4) can be sought as follows:  
\[\mathbf{H}_{n}=H_{n,\varphi}(r,z)\mathbf{e}_{\varphi},\hskip 28.452756 pt\mathbf{E }_{n}=E_{n,r}(r,z)\mathbf{e}_{r}+E_{n,z}(r,z)\mathbf{e}_{z}, \tag{5}\]  
where the MF is an even function of the axial coordinate \(z\): \(H_{n,\varphi}(r,-z)=H_{n,\varphi}(r,z)\). The system of Equation (4) must be taken into account along with the boundary conditions given below in the interface of regions \(R_{I}\) and \(R_{II}\). \[ \begin{array}{c}H_{n,\varphi}^{(I)}\Big{|}_{r=a}=H_{n,\varphi}^{(II)}\Big{|} _{r=a},\\ \frac{\partial H_{n,\varphi}^{(I)}}{\partial z}\Big{|}_{r=a}=\left.\frac{ \partial H_{n,\varphi}^{(II)}}{\partial z}\right|_{r=a}=0,\end{array}  \tag{6}\]  
\[ \begin{array}{c}\left.\frac{\partial H_{n,\varphi}^{(I)}}{\partial r}\right| _{r=a}=-\left.\frac{H_{n,\varphi}^{(II)}}{a}\right|_{r=a},\\ H_{n,\varphi}^{(I)}\Big{|}_{r=b}=H_{n,\varphi}^{(II)}\Big{|}_{r=b},\\ \left.\frac{\partial H_{n,\varphi}^{(I)}}{\partial z}\right|_{r=b}=\left.\frac {\partial H_{n,\varphi}^{(II)}}{\partial z}\right|_{r=b}=0,\end{array}  \tag{7}\]  
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