# 3-D Integral Formulation for Thin Electromagnetic Shells Coupled with an External Circuit
## 2 Thin Shell Equation  
A thin electromagnetic shell (\(\Omega\)) with thickness \(e\), conductivity \(\sigma\) and linear permeability \(\mu_{\rm r}\) in this study is illustrated in Figure 1, where \(\Gamma_{1}\) and \(\Gamma_{2}\) are the surface boundaries of the shell with the air region. The electromagnetic behavior for the side \"1\" of the shell regions is represented by [15; 16]:  
\[\int_{\Gamma_{1}}\mathbf{grad}_{\rm s}w\cdot(\alpha\mathbf{H}_{1 \rm s}-\beta\mathbf{H}_{2\rm s})\mathrm{d}\Gamma+j\omega\int_{\Gamma_{1}}w \cdot\mathbf{B}_{1}\cdot\mathbf{n}_{1}\mathrm{d}\Gamma=0, \tag{1}\]  
where \(\delta\) is the skin depth associated to the shell; \(a=(1+j)/\delta\); \(\alpha=a/\sigma\mathrm{th}(ae)\), \(\beta=a/\sigma\mathrm{sh}(ae)\); \(w\) is a set of nodal surface weighting functions; \(\mathbf{H}_{1\rm s}\) and \(\mathbf{H}_{2\rm s}\) are the tangential magnetic fields on both sides \(\Gamma_{1}\), \(\Gamma_{2}\) and \(\mathbf{n}_{1}\) is the normal vector of the side \(\Gamma_{1}\). Figure 1: Thin region and associated notations [16]. The symmetrical equation corresponding to the other side of the shell is obtained by simply exchanging the indices \"1\" and \"2\":  
\[\int_{\Gamma_{2}}\mathbf{grad}_{\mathrm{s}}w\cdot(\alpha\mathbf{H}_{2\mathrm{s}} -\beta\mathbf{H}_{1\mathrm{s}})\mathrm{d}\Gamma+j\omega\int_{\Gamma_{2}}w\cdot \mathbf{B}_{2}\cdot\mathbf{n}_{2}\mathrm{d}\Gamma=0, \tag{2}\]  
where \(\mathbf{n}_{2}\) is the normal vector corresponding the side \"2\" of the shell. Equations (1) and (2) are written on the averaged surface \(\Gamma\) of the thin shell region. Subtracting Equations (2) and (1) leads to:  
\[\left(\alpha+\beta\right)\int_{\Gamma}\mathbf{grad}_{\mathrm{s}}w(\mathbf{H}_ {2\mathrm{s}}-\mathbf{H}_{1\mathrm{s}})\mathrm{d}\Gamma+j\omega\int_{\Gamma}w (2\cdot\mathbf{B}_{\mathrm{a}}\cdot\mathbf{n})\mathrm{d}\Gamma=0, \tag{3}\]  
where \(\mathbf{B}_{\mathrm{a}}=\left(\mathbf{B}_{1}+\mathbf{B}_{2}\right)/2\) is defined as the averaged induction and \(\mathbf{n}\) denotes the normal vector of the surface \(\Gamma\) (Figure 1). The expressions related to the tangential magnetic fields on both sides and the outside of the shell are:  
\[\mathbf{H}_{1\mathrm{s}}=\mathbf{H}_{01\mathrm{s}}-\mathbf{grad}_{\mathrm{s}} \phi_{1};\;\mathbf{H}_{2\mathrm{s}}=\mathbf{H}_{02\mathrm{s}}-\mathbf{grad}_ {\mathrm{s}}\phi_{2} \tag{4}\]  
where \(\mathbf{H}_{01\mathrm{s}}\) and \(\mathbf{H}_{02\mathrm{s}}\) are the tangential fields generated by the inductors at side \(\Gamma_{1}\), \(\Gamma_{2}\) and \(\phi_{1}\), \(\phi_{2}\) denote the corresponding reduced scalar magnetic potentials. By using (4) and assuming the small variations of \(\mathbf{H}_{0\mathrm{s}}\) through the thickness of the thin shell, Equation (3) becomes:  
\[\left(\alpha+\beta\right)\int_{\Gamma}\mathbf{grad}_{\mathrm{s}}w\cdot\mathbf{ grad}_{\mathrm{s}}\Delta\phi\mathrm{d}\Gamma+2j\omega\int_{\Gamma}w\cdot \mathbf{B}_{\mathrm{a}}\cdot\mathbf{n}\mathrm{d}\Gamma=0, \tag{5}\]  
where \(\Delta\phi=\phi_{1}-\phi_{2}\) is the scalar magnetic potential discontinuity. Using magnetization law of the linear material, equation (5) can be rewritten as:  
\[\left(\alpha+\beta\right)\int_{\Gamma}\mathbf{grad}_{\mathrm{s}}w\cdot \mathbf{grad}_{\mathrm{s}}\Delta\phi\mathrm{d}\Gamma+2j\omega\frac{\mu_{0}\mu_ {r}}{\mu_{r}-1}\int_{\Gamma}w\cdot\mathbf{M}_{\mathrm{a}}\cdot\mathbf{n} \mathrm{d}\Gamma=0, \tag{6}\]  
where \(\mathbf{M}_{\mathrm{a}}=\left(\mathbf{M}_{1}+\mathbf{M}_{2}\right)/2\) is defined as the averaged magnetization. On the averaged surface \(\Gamma\), the total magnetic field \(\mathbf{H}_{\mathrm{a}}\) is the sum of the inductor field \(\mathbf{H}_{0}\), the field created by magnetization \(\mathbf{H}_{\mathrm{M}}\) and the field created by the eddy currents flowing in the shell \(\mathbf{H}_{\mathrm{EC}}\):  
\[\mathbf{H}_{a}(P)=\mathbf{H}_{0}(P)+\mathbf{H}_{\mathrm{M}}(P)+\mathbf{H}_{ \mathrm{EC}}(P) \tag{7}\]  
where \(P\) is a point located in the surface region \(\Gamma\).