# 3-D Integral Formulation for Thin Electromagnetic Shells Coupled with an External Circuit
## 5 Numerical Examples
### A Pratical Device Example  
The second test problem is the modelling of a practical device proposed by the EDF (Electricite de France) [26]. The power station \"Folies\" is equipped with a three-phase reactance to limit short-circuit currents. In this case, the current in each reactance is 1000 A phase-shifted with 120 degrees, the frequency is 50 Hz (Figure 5). \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**FEM Axisymmetric**} \\ \hline Number of elements & 15,000 & 30,000 & 55,000 \\ \hline Current values (A) & 1905.95 \(-\) j493.12 & 1905.78 \(-\) j492.87 & **1905.75 \(-\) j492.86** \\ \hline \multicolumn{3}{c}{**FEM 3D with shell elements**} \\ \hline Number of elements & 130,000 & 500,000 & 950,000 \\ \hline Current values (A) & 1866.75 \(-\) j460.52 & 1841.78 \(-\) j505.48 & 1839.23 \(-\) j509.75 \\ \hline \multicolumn{3}{c}{**Integral method**} \\ \hline Number of elements & 300 & 800 & 1000 \\ \hline Current values (A) & 1908.21 \(-\) j495.44 & 1907.85 \(-\) j495.37 & 1907.76 \(-\) j495.35 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Current values in the conductor, where j is an imaginary unit. Figure 4: Magnetic field on path AB. Figure 5: Three phase reactance in the power station “Folies”. The parameters are presented as below (Figure 6):  
* Outside diameter: \(\emptyset\)e = 1.6 m
* Inside diameter: \(\emptyset\)i = 0.74 m
* Winding thickness: ep = 0.5 m
* Center distance: Ent = 1.4 m  
In nominal conditions, a three-phase reactance generate a leakage induction in the neighborhoods. In order to minimize this electromagnetic disturbance, an electromagnetic shielding and a passive loop are added between the magnetic field source and the protected area (Figure 7). The device parameters are presented as below:  
* Passive loop: \(\bigcirc\) & Permeability \(\mu_{\text{r}}\) = 1, \(\bigcirc\) & Electrical conductivity \(\sigma\) = 3.03 \(\times\) 10\({}^{7}\) S/m \(\bigcirc\) & Section radius r\({}_{\text{s}}\) = 9.25 mm \(\bigcirc\) & Z\({}_{\text{loop}}\) = 3.85 m
* Shielding: \(\bigcirc\) & Permeability \(\mu\)r = 20,000, \(\bigcirc\) & Electrical conductivity \(\sigma\) = 2.2 \(\times\) 106 S/m \(\bigcirc\) & Thickness = 3.5 mm \(\bigcirc\) & Zshielding = 5.15 m \(\bigcirc\) & The skin depth \(\delta\) = 0.339 mm is thinner than the shielding's thickness. The last case is tested by our integral method and the FEM 3D with shell elements. Let us note that this example is in 3-dimensional space and cannot be modelled by the 2D FEM. The current values in the passive loop and the current distribution in the shells are also compared. The obtained results from the coupling method converge quite close to the current values presented in Table 2. Figure 8 also shows that the surface distribution of the current in the shell is quite similar to the two methods. The results achieved by the coupling integral method are very encouraging. The convergence is reached with few elements (about 1000 elements). Figure 6: Parameters of three reactances. However, some matrices of Equation (38) are fully populated and compression algorithms must be applied if there is a large number of elements. The coupling of the model order reduction techniques or the matrix compression algorithms with some integral methods clearly demonstrates its efficiency. For example, the coupling of a matrix compression algorithm like the FMM and the MoM or the PEEC method has reduced the computation time and the memory requirements down to more than 10 times and the compressed ratio is more than 80 percent [18; 27]. Moreover, the acquired model can be reused to build a real circuit, which is easy to employ in all conventional SPICEs-like circuit solvers [28; 29].