Validation

After derivation of the formulas required to treat a circular dielectric inclusion in an analytic fashion, the formulas are validated with examples and comparison to commercial implementations of full-wave simulators. The simulations shown in this section were all conducted on the same Intel Core i7 CPU 960 (8×3.2 GHz) with 24 GB RAM. The commercial FEM code ran as a multi-thread application and the CIM as a single-thread implementation.

First, we compare the analytic formulas for circular inclusions with the discretization of a circle with linear contour segments and the approximation with an equivalent circuit.

Consider the transfer impedance between two vias with a radius of 5 mil, a distance of 100 mil, in a substrate with ε_r = 4.4 and a loss tangent of tanδ = 0.02, and a cavity height of d = 10 mil. The top and bottom metalizations are assumed to have the finite conductivity of copper (κ = 5.8·10⁷ S/m, [113]). The finite conductivity of the metal planes and the dielectric losses are taken into account by using an effective wavenumber as proposed in [54]. In the center between both vias, we consider a circular dielectric inclusion of radius a_o = 20 mil and a lossless dielectric with ε_r= 100. Figure 2.10 depicts the geometry, as well as, the magnitude of the transfer impedance for the analytical solution using (2.49) and (2.48), discretization of the inclusion using 8 and 32 linear contour segments, respectively, and the approximation using the lumped capacitance (2.54). For the given geometry, the equivalent capacitance is C₀⁰ = 2.826 pF. Up to a frequency of 6 GHz, all approaches are in a good agreement, beyond that, the discretized inclusion with 8 linear segments and the result for the equivalent capacitance deviate. Nevertheless, the

2.4 Extension to Outer Circular Contours capacitance gives a good approximation for the low-frequency behavior. The discretized inclusion with 32 linear contour segments is in a very good agreement over the complete frequency range. This validates the analytic formulas and illustrates the applicability of the approximation by a capacitance. The computation times for the analytical formulas and the discretized boundaries are listed in Table 2.3. Clearly, the analytical solution is superior in terms of computational efficiency and accuracy.

In the next examples, the proposed extension of the CIM with an analytical description for circular inclusions is compared to full-wave simulations using a commercial solver. The geometries of interest are illustrated in Figure 2.11. Consider a substrate with ε_r = 4.4, tanδ = 0.02, a cavity height ofd= 10 mil, and a finite conductivity of the top and bottom metalization of κ = 5.8·10⁷ S/m. The substrate is square shaped with an edge length of 2000 mil and bounded by a PMC. Again, we look at the transfer impedance between two vias. On the substrate there are circular inclusions made of lossless dielectric arranged as a fence, see Figure 2.11a, and as lattice filling the complete substrate, see Figure 2.11b.

Figures 2.12 and 2.13 show the transfer impedance for inclusions withε_r= 10 and ε_r = 100 and no inclusions (reference) for the fence and lattice arrangement, respectively. The solid lines refer to results obtained with the CIM and the analytic formulation of dielectric inclusions, the dots where obtained with a 3-D FEM using [123]. In all cases, the CIM and the FEM are in excellent agreement. This validates the analytic treatment of circular inclusions and the methodology to incorporate inhomogeneous substrates as described in Section 2.3. The computation times are summarized in Table 2.4. The speedup of the CIM compared to the commercial solver is about one order of magnitude. Increasing the number of inclusions increases the number of unknowns and hence the computation time in the CIM. However, due to the contrast in the dielectric, the FEM is subject to the same phenomenon and the computation time increases likewise.

Table 2.3: Computation times for the CIM simulation (single frequency point) for the geometry depicted in Figure 2.10 with the different methods.

Method Computation Time (ms)

Analytical 2.59

8 linear contour segments 4.00 32 linear contour segments 63.5

0 2 4 6 8 10 12 14 Frequency (GHz)

0 1 2 3 4 5

|Z21|(Ω)

Analytical K = 1 8 Segments 32 Segments C₀⁰

ε_r = 4.4

ε_r = 100 100 mil

Port 1 Port 2

20 mil

50 mil 5 mil

d = 10 mil

Figure 2.10: Transfer impedance for a structure with two ports in an infinitely extended substrate with ε_r = 4.4 and a cylindrical inclusion with _r = 100 in between using the analytical solution with K = 1, an approximation with a lumped capacitance C₀⁰, and approximated circular shapes with 8 and 32 line segments, respectively. Figure and caption adapted from [5].

250 mil ε_r,c

ε_r = 4.4

55 mil

Port 1

PMC Port 2

tanδ = 0.02

tanδ= 0

2000 mil

(a) (b)

Figure 2.11: Top view of the substrate with (a) a fence of dielectric rods and (b) a lattice covering the whole board. Both boards are bounded by PMC and the substrate hasεr= 4.4 and tanδ = 0.02. Figure and caption taken from [5].

2.4 Extension to Outer Circular Contours

0 2 4 6 8 10

Frequency (GHz) 60

40 20 0 20

|Z21|dBΩ

CIM FEM

Reference εr = 11 ε_r = 100

Reference εr = 11 ε_r = 100

Figure 2.12: Transfer impedance computed with the CIM (solid lines) and the FEM (dots) of the structure shown in Figure 2.11a, for the reference case without dielectric rods and with dielectric rods for two different dielectric constants. The values ofεr refer to the dielectric rods. Figure and caption adapted from [5].

0 2 4 6 8 10

Frequency (GHz) 120

100 80 60 40 20 0 20

|Z21|dBΩ

CIM FEM

Reference ε_r= 11 ε_r= 100

Reference ε_r = 11 ε_r = 100

Figure 2.13: Transfer impedance computed with the CIM (solid lines) and the FEM (dots) of the structure shown in Figure 2.11b, for the reference case without dielectric rods and with dielectric rods for two different dielectric constants. The values ofε_r refer to the dielectric rods. The deviation between the results obtained by the CIM and the FEM in the band gap regime for theεr= 100 can be explained by numerical inaccuracies. Figure and caption adapted from [5].

Table 2.4: Computation time per frequency step and discretization for the three test cases. Rectangular reference board with homogeneous substrate, with fence, and with lattice.

Adapted from [5].

FEM CIM

Reference 6,328 Tetrahedrons 150 Basis Functions

1.76 s 0.36 s

Fence 21,777 Tetrahedrons 352 Basis Functions

5.76 s 0.53 s

Lattice 32,870 Tetrahedrons 652 Basis Function

8.13 s 0.91 s