## 3 Results
### Case Study 2: Scattering from a Cable Bundle in Air  
As pointed out in the introduction, the study of the near-field scattering from unshielded bundled cables is of great interest for the EMC analysis of a wide variety of electronic applications, from automotive on-board systems to buried power lines. Here, we refer to the geometry shown in Figure 4a, where three conducting wires are bundled in an unshielded cable, surrounded by air (\(\epsilon_{r}=~{}1\)). The frequency is again assumed to be equal to \(3\) GHz, with the wire radius \(a=\lambda/6\), and the cable radius \(b=\lambda\). The three wires are reciprocally rotated by \(120^{\circ}\) and their centers are placed at a distance equal to \(c=\lambda/2\) from the cable center. The DtN solution is calculated by choosing again a circular boundary of radius \(R=1.2\lambda\). Here, the reference solution is given by the numerical solution provided by commercial code COMSOL Multiphysics(r)with PML approach, stabilized to an accuracy below \(10^{-4}\). For this example, this requires about \(1.1\times 10^{6}\) mesh elements. The obtained distribution of the amplitude of the scattered field is plotted in Figure 4b. The evaluation of the maximum scattered field gives an important information in EMC analysis, since it could provide the worst-case estimation of unwanted radiated emission and/or crosstalk noise between the conductors in the cable. Therefore, in the following we study the maximum value of the scattered field in the near-field region, namely inside the circle of radius \(R=1.2\lambda\). According to the solution in Figure 5b, the maximum value of the scattered field, normalized to the amplitude of the incident wave, is equal to \(\left|E^{S}\right|/E_{0}=1.604\). In Figure 5a we compare the DtN and PML numerical solution for increasing value number, \(N\), of the mesh nodes: The DtN solution converges to the reference value much faster than the PML one; the required values of \(N\) for reaching a difference of \(1\%\) between the numerical solutions and the reference one are reported in Table 1. The use of the DtN provides a reduction in \(N\) of a factor of about \(5\). Figure 4: Case study 2 (cable bundle in air). (**a**) Geometry and references for evaluating the scattering from the cable, illuminated by a TM\({}_{x}\) plane wave; (**b**) Map of the amplitude of the scattered field, normalized to that of the incident field. The axes \(x\) and \(y\) are normalized to the wavelength value. Finally, in order to better estimate the gain in terms of computational cost, beside the number of the mesh nodes, \(N\), it is of interest to compare the condition number, \(k\), of the resulting stiffness matrix \(\mathbf{M}\) associated with the two numerical models, defined in our case as:  
\[k(\mathbf{M})\equiv\
ormalsize\big{\|}\mathbf{M}\big{\|}\big{\|}\mathbf{M}^{-1}\big{\|}, \tag{24}\]  
where \(\mathbf{M}\) is the 2-norm of \(\mathbf{M}\). Indeed, the computational cost of the numerical solution (the number of iterations required for convergence in an iterative solver), is inversely related to the condition number. In Figure 6b, the condition number, \(k\), versus the number of the mesh nodes, \(N\), is reported. The DtN scheme provides a lower condition number, hence, outperforming the PML one also in this sense. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Case** & **DtN** & **PML** \\ \hline w/o dielectric & 511 & 2638 \\ \hline w dielectric & 11,573 & 59,534 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Required number of mesh nodes for 1% accuracy, Case study 3. Figure 5: Case study 2 (cable bundle in air). (**a**) Maximum value of the scattered field, normalized to amplitude of the incident field, versus the number of mesh nodes: The DtN and PML solutions are compared to the reference one. (**b**) Condition number of the stiffness matrices associated with the two numerical solutions, versus the number of mesh nodes. Figure 6: Case study 3 (cable bundle with dielectric).