# 3-D Integral Formulation for Thin Electromagnetic Shells Coupled with an External Circuit
## 4 Coupling Thin Shell with Circuit Equation
### Final System of Equations
These mutual inductances are expressed as follows:  
\[m_{ik}=\frac{\mu_{0}}{4\pi}\frac{1}{S_{i}S_{k}}\int_{\Omega\Sigma_{k}}\int_{ \Omega\Sigma_{i}}\frac{\mathbf{l}_{i}\mathbf{l}_{k}}{\mathrm{r}}\mathrm{d} \Omega_{ci}\mathrm{d}\Omega_{ck}, \tag{34}\]  
\[m^{f}_{ik}=\frac{\mu_{0}\overline{\mathrm{G}}}{4\pi}\frac{1}{S_{k}}\int_{ \Omega\Sigma_{k}}\int_{\Gamma_{i}}\frac{\mathbf{u}_{i}\times\mathbf{r}}{\mathrm {r}^{3}}\mathrm{d}\Gamma_{k}\mathrm{d}\Omega_{ck}, \tag{35}\]  
\[\mathrm{A}(i,k)=2j\omega\frac{\mu_{0}\mu_{r}}{\mu_{r}-1}\int_{\Gamma}w_{i} \cdot\mathbf{n}_{k}\mathrm{d}\Gamma=0, \tag{27}\]0  
where \(\mathbf{u}_{i}\) is a basis vector of magnetizations; \(\mathbf{n}_{i}\) is the normal vector of the element \(\Gamma_{i}\) and \(\mathbf{l}_{k}\) is the vector unit of the current direction in the conductor \(k\). Let us note that the mutual inductance terms \(\mathbf{m}_{ik}\) can be calculated by the well-known semi-analytic formulas in [7; 8; 9; 22; 23]. Whereas, the terms \(m^{f}_{ik^{\prime}}\), \(m^{q}_{ik}\) are computed with a standard numerical integration. Writing Equation (33) for all conductors, we get a matrix system known as the impedance matrix system:  
\[\mathrm{A}(i,k)=2j\omega\frac{\mu_{0}\mu_{r}}{\mu_{r}-1}\int_{\Gamma}w_{i} \cdot\mathbf{n}_{k}\mathrm{d}\Gamma=0, \tag{27}\]1  
where \([\mathbf{Z}]\) is a (\(m\times m\)) matrix; \([\mathbf{U}]\) is a (\(m\times 1\)) vector; \([\mathbf{L}^{f}]\) is a (\(m\times 3n\)) matrix and \([\mathbf{L}^{q}]\) is a (\(m\times p\)) matrix. Finally, the algebraic linear systems (26), (29) and (37) is considered as \(p+3n+m\) complex unknowns:  
\[\left[ \begin{array}{ccc}\mathbf{A}&\mathbf{B}&\mathbf{0}\\ \mathbf{MoM}&-\mathbf{C}&-\mathbf{D}\\ \mathbf{L}^{f}&\mathbf{L}^{q}&\mathbf{Z}\end{array} \right]\times\left[ \begin{array}{c}\mathbf{M}\\ \Delta\mathbf{\Phi}\\ \mathbf{I}\end{array} \right]=\left[ \begin{array}{c}0\\ 0\\ \mathbf{U}\end{array} \right] \tag{38}\]  
where \(\mathbf{MoM}=\frac{[\mathbf{L}]}{\mu-1}+[\mathbf{F}]\) is usually called the magnetic moment method matrix. Let us note that Equation (29) has a disadvantage when the permeability of the material is high. Indeed, in such cases, the matrix term \([\mathbf{I}_{d}]/(\mu_{r}-1)\) becomes very small in comparison with the [\(\mathbf{F}\)] matrix. This can lead to the singularity of matrix [**MoM**]. It should be noted that for the linear case, the simpler formulations can be developed. In such cases, Laplace equation is applied in the whole electromagnetic thin shell regions. Formulation (30) can be determined by a surface integral equation generated by charge distributions (Coulombian approach) or current distributions (Amperian approach) located on the surface area bounding the volume \(\Omega\) of the thin shell region. In such configuration, only this surface area must be discretized. Consequently, the degrees of freedom of the obtained matrix system is lower. Moreover, Newton-Raphson algorithm can be easily used for the non-linear case [25]. Besides, in order to solve the equation system (38) with a reduced number of degrees of freedom, a Kirchhoff's mesh rule is introduced. Consequently, the current and the voltage values become associated to each of the independent circuit mesh. The coupling formulation has been implemented for 3D geometry.