# 3-D Integral Formulation for Thin Electromagnetic Shells Coupled with an External Circuit
## 3 Conductor System Modelling  
We now consider \(m\) volume conductors fed with alternating sources placed in an air region. The external electric field incident at point \(P\) can be written [7, 8, 9]:  
\[\mathbf{E}_{\mathrm{ext}}(P)=\frac{\mathbf{J}_{c}(P)}{\sigma}+j\omega\mathbf{ A}(P)+\mathbf{grad}\mathrm{V}(P), \tag{16}\]  
where \(\mathbf{A}(P)\) is the vector potential and \(\mathrm{V}(P)\) is the scalar potential.  
The magnetic vector potential generated by the current density \(\mathbf{J}_{\mathrm{c}}\) is:  
\[\mathbf{A}(P)=\sum_{k=1}^{m}\frac{\mu_{0}}{4\pi}\int_{\Omega\chi_{\mathrm{c}}} \frac{\mathbf{J}_{\mathrm{c}}}{\tau}\mathrm{d}\Omega, \tag{17}\]where \(\Omega c_{k}\) is the volume of the conductor \(k\); r is the distance between the integration point on \(\Omega c_{k}\) and the point \(P\).  
In the context of the quasi-stationary regime operated at the frequency range up to ten MHz, it is possible to neglect the capacitive effect. Equation (16) is written in the form:  
\[\mathbf{E}_{\text{ext}}(P)=\frac{\mathbf{J}_{c}(P)}{\sigma}+j\omega\sum_{k=1} ^{m}\frac{\mu_{0}}{4\pi}\int_{\Omega c_{k}}\frac{\mathbf{J}_{c}}{\mathbf{r}} \mathrm{d}\Omega. \tag{18}\]  
The PEEC method is particularly pertinent and reliable to solve this kind of problem. Let us assume that the current density in each conductor is uniform. For each conductor, Equation (18) can be associated to the electrical equivalent circuit presented by the self and mutual inductances in series with the resistances [8; 9]. In most respects, what we know about this approach is difficult to the magnetic material or the magnetic conductive material. However, the mesh of the air region can be neglected and some conventional SPICES-like or Saber circuit solvers can be employed to analyze the equivalent circuit. As a result, the behavior of several electrical devices that have electrical interconnections can be studied in only one system. With all of the advantages described above, the PEEC is well-suited for modelling real industrial devices [17; 18; 19; 20; 21].  
Despite these clear advantages, this technique has a major drawback due to the necessity to store a fully dense matrix inherent to the use of the integral formulation in Maxwell's equation. Consequently, this approach is strongly limited in modelling large-scale electrical devices requiring substantial meshes. Let us denote N as the number of unknown then the needed memory for the fully dense matrix storage leads to O(N2) and the computational costs of a direct solver (LU-decomposition) increases in proportion to O(N3). For example, a large size modelling problem containing 50,000 unknowns requires at least 50,000 \(\times\) 50,000 \(\times\) 8 bytes or approximately 20 GB in the memory to store a fully dense matrix and the computational costs of finding one solution by direct solver are roughly several weeks. In order to study the characteristic of the devices in a frequency range or in a period of time by the PEEC method, the total computational costs must be multiplied with the number of frequency or the time steps. Moreover, none of the circuit solvers can handle the equivalent circuit composing of 50,000 \(\times\) 50,000 basic circuit elements RLMC (resistor, inductor, partial inductor, and capacitor). Overall, in both cases the computational time tends to infinity, or it is impossible to solve. To solve this problem, the algorithms coupling different matrix compression algorithms or the model order reduction technique with integral methods have been developed with the purpose of limiting matrix storage [19; 21].  
In addition, the expansion of the PEEC method for more general problems has been investigated [11; 18; 20]. However, this method is still difficult for considering special structures such as a circular coil, thick circular coil, and thin disk coil. This drawback can be easily improved by using the semi-analytic methods developed in [22; 23].