Inhomogeneous dielectrics: conformal mapping and finite-element models

2.1 SC+FD and FEM thin strip results

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 Table 1 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 suggests that, proceeding with care, the effect of the singularity can be modest in inhomogeneous dielectric too, and Table 2 is obtained with the same meshes as above.

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 Table 1 and Table 2, 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. Table 3 and 4 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.

Considering Figure 2, Table 2 and Table 3 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.

2.2 SC+FD and FEM thick strip results

When the stripline model exhibits a finite thickness, the geometry of the problem is slightly different, although hardly perceived at the scale of Figure 1, being the ratio between the total height and the thickness equal to 130. The field singularities at the left end of the conductor are now two, but the effect on the overall capacitance is milder because of the different exponent at the π/2 corners. The SC+FD procedure leads (Figure 2, right hand) to a rectangle with a slightly smaller aspect ratio than the thin strip, i.e., to a slightly large capacitance in vacuum, and the maps of the dielectric interfaces are something more complex.

The SC capacitance in vacuum is now C_0scT = 1.97509295·10^–11 F/m and two FEM capacitance values have been provided, both in vacuum and in inhomogeneous dielectric, with two meshes having both triangle densities matched to the needs in proximity of the end of the stripline. Accordingly, Table 5 and Table 6 report the same evaluations as Tables 1 and 2 did for the thin stripline case, and we note immediately that the FEM accuracy seems to be somewhat better than with the more severe singularity of Figure 1. Due to the matched mesh densities, we can immediately appreciate the good uniformity and accuracy of these results, despite of the modest number of triangle for the case TB.

Table 7 shows the comparison of the SC+FD calculations in inhomogeneous dielectric with the capacitance of the FEM case TA.

It is analogous to the previous Table 3, but the last column reports the differences in the SC+FD capacitances obtained with 1000 instead of 4000 steps in the P-C mappings of the dielectric interfaces, instead of the comparison of a second FEM values. As in vacuum, the accuracies are perhaps somewhat better than in Table 3, but show in Figure 3 (dashed line) a similar behavior, leading again to the proposal of SC+FEM calculations, although limited to smaller numbers of FD cells and perhaps less worrying.

2.3 The SC+FEM procedure and its accuracy

To make it easier to represent the details of Figure 1 at the input of FEM codes, we will continue the discussion using a second stage map analogous to Figure 2, but in which plot1 was replaced with a perfect circle, and plot2 by a vertical straight segment, being unchanged the boundary vertices and the end points of the interfaces. Of course, the new geometry is no longer the map of a strip, but this is unimportant when discussing the possible accuracy of second stage calculations. This trial geometry is shown in Figure 4.

Figure 4

A second stage trial geometry simulating that obtained from the thin stripline geometry but with simpler interface shapes

The FEM second stage results, with number of nodes ranging from 276 to 6351, ranged from 3.32221·10^–11 to 3.32187·10^–11 F/m, showing immediately differences of the order of only 0.01%, provide good reference values. In contrast, the behavior of the FD capacitances was again unsatisfactory. As for the thin strip, oscillatory differences are found for large numbers of cells ranging from zero to some –0.2%. Therefore, we find the same conditions we had when comparing direct FEM and SC + FD results.

This suggested a new step of analysis. Starting from the same geometry, the lengths of the various segments of the boundary have been replaced by numbers with integer ratios among them, to secure the possibility of positioning any vertex or interface ending point on the nodes of the FD mesh. This was possible with special care of the segments of the bottom conductor and with modest errors in regions of negligible influence.

This time, a reliable FEM reference was C_FEMstage2 = 3.3224·10^–11 F/m, from the energy, with 25,884 nodes, locally dense mesh with nonlinear rule of the second order (and, with a very similar value, C = 3.32270·10^–11 F/m with a linear rule). The equipotential lines and field magnitudes are shown in Figure 5. The FD results have been very satisfactory too, as the results were, for instance, C_FDstage2 = 3.32176·10^–11 F/m with the quarter circle modeled with 8,000 P-C segments, and 3.32200·10^–11 F/m with 4000. The maximum difference among these second stage FEM or FD values is of about 0.03%.

Figure 5

Equipotential lines and field magnitudes for the geometry in Figure 4

This allows us to draw important conclusions. Assuming FEM results on the original geometry as the reference, we can expect that the results obtained from a FEM second stage geometry, i.e., a virtually exact rectangular boundary and very accurate maps of the dielectric interfaces, will be very similar to the reference. At the same time, it was shown that the FD results for the same second stage geometry can be good in general and in addition that they are very good, and very similar to the FEM results, if the details of the maps of the interfaces are well represented by the FD mesh.