The geometry is shown in Figure 1, and its SC conformal map is shown in Figure 2, left hand. In air, the aspect ratio base/height of the rectangle provides by inspection the capacitance of the line, C_{0sc}= 1.92249066·10^{–11} F/m. In Figures 1 and 2, solid lines represent conductor sides, the strip at the upper side and the metal case at the bottom, thick dashed lines Neumann sides, and thin dashed lines the two interfaces between different dielectrics (air, *ϵ*_{r} = 1, and alumina, *ϵ*_{r} = 9.5): plot1 maps the interface at the bottom plane of the dielectric, and plot2 maps that at the upper plane of it. To solve the Laplace equation in inhomogeneous dielectric, FD successive over-relaxation (SOR) procedures can be easily written using five-point equations.

Figure 1 A shielded dielectric-supported stripline, thin conductor: left: geometry with potential lines (from FEMs); right: the analyzed geometry

Figure 2 Left, the SC map of the half cross section of the thin strip geometry in Figure 1; right, the SC map for the thick strip geometry

Square meshes are used here for the sake of simplicity and speed, but the final results are corrected according to the aspect ratios of SC map and of the FD mesh outer rectangle.

The accuracy of direct FEM calculations on the original strip geometry and in vacuum can be preliminarily appreciated making comparison of the results in with the SCNI capacitance C_{0sc} = 1.92249066·10^{–11} F/m, to be considered virtually exact. Direct FEM capacitances (from the energy) agree with C_{0sc} within a 0.5% error with 6,308 nodes, and a 0.1% error with a locally dense mesh of only 1,856 nodes but geometrical progression, with rate 6 and 20 elements along the half stripline. The saving of degrees of freedom between case A and case B is impressive but it can be justified by the severe singularity at the end of the stripline. The capacitance value for case A computed in vacuum from the dielectric flux on the outer conductor was C_{0FEM} = 1.9227·10^{–11} F/m, and this seems to be the best result, with a percent error of about 0.01%.

Table 1 Accuracies derived in vacuum from FEMs

Table suggests that, proceeding with care, the effect of the singularity can be modest in inhomogeneous dielectric too, and is obtained with the same meshes as above.

Table 2 Thin conductor dielectric-supported stripline, *εr* = 9.5

Therefore, we assume the value C_{FEMA} = 3.29654·10^{–11} F/m obtained from a locally dense mesh as a reliable FEM reference in inhomogeneous dielectric. A very dense mesh of 23,800 nodes leads to C_{FEMB} = 3.301·10^{–11} F/m, different from C_{FEMA} of about 0.13%, and a mesh of 6,308 nodes leads to C_{FEMC} = 3.30852·10^{–11} F/m. We note that following one of the first proposed variational calculations [6], the capacitances in air, 1.9177·10^{–11} F/m, and in inhomogeneous dielectric, 3.2828610^{–11} F/m, agree within some 0.25% with C_{0sc} and with C_{FEMA}.

Considering and , we see that probably an uncertainty of about 0.1% can be considered the accuracy limit for the FEM results, both in the homogeneous and in the inhomogeneous dielectric cases. As for the dielectric interfaces in SC+FD calculations, experience shows that, using predictor-corrector (P-C) routines with suitable numbers of steps, the related errors can be made almost negligible. In the present case, they differ for less than some 0.02% using 4,000 instead of the 8,000 steps, the last leading to Figure 3.

Figure 3 Differences between SC+FD capacitance values and the reference FEM value for various dimensions of the FD mesh. Solid line: thin strip, 8000 P-C steps; dashed line: thick strip, 1000 P-C steps and, a little more oscillating, 8000 P-C steps

To apply the suitable five-point equation to any mesh node neighboring a dielectric interface, they are placed on suitable staircases obtained from the P-C traces, with minimum steps of one mesh cell both in the horizontal and the vertical directions. and show SC+FD results for various numbers of mesh cells and for 8,000 and 4,000 P-C steps, with various percent errors and differences.

Table 3 SC+FD Results compared to the FEM cases A and B

Table 4 Differences in SC+FD capacitances due to a different step number in the predictor-corrector interface mapping routine

Considering Figure 2, and shows that 4,000 or 8,000 P-C steps are well enough. Moreover, the differences between the corresponding values with large numbers of cells exhibit magnitudes analogous to C_{FEMA}-C_{FEMB}. However, and this is perhaps the most important evidence, a small but persistent difference from C_{FEMA} is found in the SC+FD values for large numbers of mesh cells, with a somewhat oscillating behavior.

In particular, with a large FD mesh, exhibiting 2,200·1,013 = 2286 cells, and with 8,000 P-C steps dielectric interfaces, the SC+FD capacitance was CSCFD = 3.29106·10^{–11} F/m, with differences from C_{FEMA} and C_{FEMB} of about –0.17% and –0.31%. This suggests that, despite of refined mesh discretization, the major cause of these errors remain discretization effects related, more than to the absolute dimensions of the meshes, to ratios of discretized lengths derived from the original geometries. In the current case, these appear unsuitable in particular to deal with the severe concentration of the dielectric flux lines in the short region of the bottom conductor facing the high permittivity dielectric. It was precisely this to suggest considering the SC+FEM procedures proposed in the following.

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.