# 3-D Integral Formulation for Thin Electromagnetic Shells Coupled with an External Circuit
## 4 Coupling Thin Shell with Circuit Equation
### Final System of Equations  
Equations (6), (20) and (25) are solved by using a numerical method. The most suitable way is to mesh the averaged surface \(\Gamma\) into \(n\) triangular elements and \(p\) nodes. Let us assume that the tangential component of the eddy current and the magnetization in each element are uniform. The [PERSON] projection method is then applied to Equation (6), and we get the following matrix system [11; 15; 16]:  
\[[\mathbf{A}]\cdot[\mathbf{M}]+[\mathbf{B}]\cdot[\Delta\mathbf{\Phi}]=0, \tag{26}\]  
where:  
**[M]** is a complex vector of dimension \(3n\); \([\Delta\mathbf{\Phi}]\) is a complex vector of dimension \(p\);  
**[A]** is a \((p\times 3n)\) matrix expressed as follows:  
\[\mathrm{A}(i,k)=2j\omega\frac{\mu_{0}\mu_{r}}{\mu_{r}-1}\int_{\Gamma}w_{i} \cdot\mathbf{n}_{k}\mathrm{d}\Gamma=0, \tag{27}\]  
**[B]** is a \((p\times p)\) sparse matrix and can be written as:  
\[\mathrm{B}(i,k)=(\alpha+\beta)\int_{\Gamma}\mathbf{grad}_{s}w_{i}\cdot\mathbf{ grad}_{s}w_{k}\mathrm{d}\Gamma. \tag{28}\]  
A matrix system representing Equation (20) can also be determined thanks to a point matching approach at the element centroids. This linear matrix system is expressed as [10; 11; 16]:  
\[\left(\frac{[\mathbf{I}_{d}]}{\mu_{r}-1}+[\mathbf{F}]\right)[\mathbf{M}]-[ \mathbf{C}]\cdot[\Delta\mathbf{\Phi}]-[\mathbf{D}]\cdot[\mathbf{I}]=\mathbf{0}, \tag{29}\]  
where \([\mathbf{I}_{d}]\) represents the identity matrix; \([\mathbf{F}]\) is a \((3n\times 3n)\) dense matrix; \([\mathbf{C}]\) is a \((3n\times p)\) also dense matrix; \([\mathbf{D}]\) is a \((3n\times m)\) matrix and \([\mathbf{I}]\) is a \((m\times 1)\) vector. The coefficients of these matrices are expressed as follows:  
\[\mathrm{F}(i,\ j)=\left[\left(\frac{\overline{\mathrm{G}}}{4\pi}\mathbf{grad} \int_{\Gamma_{j}}\frac{\mathbf{u}_{j}\cdot\mathbf{r}_{i}}{\mathbf{r}_{i}^{3}} dT\right)\mathbf{u}_{i}\right] \tag{30}\]  
where \([,]\) is defined as the scalar product operator and \(\mathbf{u}_{i}\) is the vector basis of the element \(i\) and \(\mathbf{r}_{i}\) is the vector between the integration point on \(\Gamma_{j}\) to the centroid of the element \(i\). \[\mathrm{C}(i,j)=\frac{1}{4\pi}\int_{\Gamma}\frac{\mathbf{n}_{j}\times\mathbf{ grad}w_{i}\times\mathbf{r}_{i}}{\mathbf{r}_{i}^{3}}\mathrm{d}\Gamma. \tag{31}\]  
\[\mathrm{D}(i,j)=\frac{1}{4\pi}\frac{1}{\overline{S}_{j}}\int_{\Gamma\mathbf{c }_{j}}\frac{\mathbf{l}_{j}\times\mathbf{r}_{i}}{\mathbf{r}_{i}^{3}}\mathrm{d}\Omega. \tag{32}\]  
By integrating Equation (25) on each conductor, we have the link between the partial voltages of the conductors to the currents flowing in them. Thus, the voltage appearing on the conductor \(k\) is given by:  
\[\mathrm{U}_{k}=\mathrm{R}_{k}\mathrm{I}_{k}+j\omega\sum\limits_{i=1}^{m}m_{ik} \mathrm{I}_{i}+j\omega\sum\limits_{i=1}^{n}m_{ik}^{\prime}\mathbf{M}_{ai}+j \omega\sum\limits_{i=1}^{p}m_{ik}^{q}\phi_{i}, \tag{33}\]where \(\mathbf{R}_{k}\) is the resistance of the \(k^{th}\) conductor; \(\mathbf{m}_{ik}\) is the mutual inductance between the two conductors \(k\), \(i\); \(m^{f}_{ik}\) is defined as the mutual inductance between the magnetization \(\mathbf{M}_{ai}\) of the shell element and the conductor \(k\); \(m^{q}_{ik}\) is defined as the mutual inductance between the scalar magnetic potential discontinuity \(\Delta\phi\) of the shell node and the conductor \(k\).