# Analytically Regularized Evaluation of the Coupling of Planar Concentric Conducting Rings
## 3 Results  
As a first task, the convergence of the method has been checked. As outlined in the Introduction, a blind application of numerical methods can lead to slow convergence, or could even not converge at all. On the contrary MAR allows to achieve fast convergence and error control. In order to verify the effectiveness of the method illustrated in previous Section the following error is introduced  
\[e(N)=\frac{\left\|\mathbf{c}^{(\mathbf{N}+1)}-\mathbf{c}^{(\mathbf{N})}\right\| }{\left\|\mathbf{c}^{(\mathbf{N})}\right\|}+\frac{\left\|\mathbf{d}^{(\mathbf{ N}+1)}-\mathbf{d}^{(\mathbf{N})}\right\|}{\left\|\mathbf{d}^{(\mathbf{N})}\right\|} \tag{19}\]  
where \(N\) is the truncation order (actually, two different truncation orders should be introduced for the two components of the current. In the present work they have been taken equal for simplicity), namely the number of terms retained in expansion (13.), \(\mathbf{c}^{(\mathbf{N})}\) and \(\mathbf{d}^{(\mathbf{N})}\) are the coefficients vectors corresponding to that truncation error, and the standard euclidean norm in \(\ell^{2}\) is employed. The error is plotted in Figure 2 for all geometries listed in Table 1 in TE case for an incidence angle \(\theta_{inc}=45^{\circ}\). In Figures 3 and 4 the error is plotted for geometry #2, for different incidence angles, in TE and TM case, respectively. As can be seen, in all the cases shown, as well as in any other case considered and not displayed for the sake of brevity, the error is exponentially decaying, thus confirming the effectiveness of the method. In the following all the reported examples have been calculated taking into account seven expansion terms, thus ensuring an accuracy not larger than \(10^{-3}\).  
Figure 3: Plot of the error for the geometry #2 of Table 1 (sketched in the inset) for TE incidence and different incidence angles \(\theta_{inc}\).  
Figure 2: Plot of the error for the geometries of Table 1 for TE incidence and an incidence angle \(\theta_{inc}=45^{\circ}\).  
Once the unknown coefficients have been calculated, the currents and the fields can be evaluated. As an example, the reconstructed current density behavior is plotted in Figures 5 and 6 for the case of geometry #2 of Table 1 for an incidence angle \(\theta_{inc}=0\). As can be seen, the edge behavior of both current components is perfectly reconstructed. As expected, the radial component vanishes at the edges, whereas the azimuthal component diverges, as prescribed by [PERSON]'s conditions [48].  
Figure 4: Plot of the error for the geometry #2 of Table 1 (sketched in the inset) for TM incidence and different incidence angles \(\theta_{inc}\).  
Figure 5: Behavior of the radial component of the current density for the geometry #2 of Table 1 (sketched in the inset) for an incidence angle \(\theta_{inc}=0^{\circ}\).  
Finally, the bistatic radar cross section is plotted in Figures 7-9 for different geometries and different incidence angles. As expected, the more oblique the incidence, the less smooth, the radar cross section. In the same plots, a comparison with CST Microwave Studio is also shown, exhibiting a very good agreement in any case.  
Figure 6: Behavior of the azimuthal component of the current density for the geometry #2 of Table 1 (sketched in the inset) for an incidence angle \(\theta_{inc}=0^{\circ}\).  
Figure 7: Bistatic radar cross section for geometry #1 (sketched in the inset) for different incidence angles \(\theta_{inc}\). Solid line: this method; dashed line: CST Microwave Studio.