4. Extension using Analytical Solutions
4.5. Modeling of Anisotropic Modes
4.5.4. Port Array Analysis
Based on the extended circular port definitions for taking into account the anisotropic propagating modes, analytical modal expressions have been derived in the last section to improve the efficiency of CIM. They have been validated using the conventional CIM for simple cases and the mode conversion effect and its impact on the Zpp has been analyzed. In this section, the validation and comparison are extended to an 8 by 8 array 64-port case, as shown in Fig. 4.19. Simulation results with regard to ports 1, 2, and 3, as depicted in Fig. 4.19, will be shown for both the infinite and finite plane cases.
z
(0, 0)
(2, 1)
Port radius: 10 mil Pitch size: 40 mil
σc= 5.8.107S/m, εr= 4.2, tan δ= 0.02, d= 10 mil z
x y
x Port 1
(0.5, 0.3)
(0.78, 0.58)
Port 2
(0.62, 0.42)
Port 3
(0.62, 0.46)
Figure 4.19 An 8 by 8 port array example. Both infinite and finite plane cases are studied.
Dimensions are given in inches (1 inch ≈ 2.54·10-2 m).
60 Extension using Analytical Solutions
0 20 40 60 80 100
0 5 10 15 20 25
Magnitude of Transfer Impedance (Z 23) 00 [Ω]
Frequency [GHz]
Modal K = 0 Modal K = 1 Modal K = 2
Numerical with 32 segments
{
On top of each other
Figure 4.21 Magnitude of transfer impedance of the 0th mode between ports 2 and 3 for the infinite plane case in Fig. 4.19, obtained with CIM modal solutions and the conventional numerical CIM considering discretized ports.
0 20 40 60 80 100
0 5 10 15 20 25 30 35
Magnitude of Input Impedance (Z 11) 00 [Ω]
Frequency [GHz]
Modal K = 0 Modal K = 1 Modal K = 2
Numerical with 32 segments
On top of
{
each other
Figure 4.20 Magnitude of input impedance of the 0th mode of port 1 in the infinite plane case, shown in Fig. 4.19, obtained with CIM modal solutions and the conventional CIM considering discretized ports.
Extension using Analytical Solutions 61
The infinite plane case is first studied. The input impedance of the 0th mode for port 1 and the transfer impedance of the 0th mode between port 2 and 3, obtained by the modal solutions and the conventional CIM, are plotted in Fig. 4.20 and Fig. 4.21, respectively. The results of the modal solution converge to the one calculated by the conventional CIM as more anisotropic modes are included in the computation, and they become visually indistinguishable when the mode truncation number K is increased to 2. With an isotropic port assumption (K = 0), the impedance results are inaccurate even at frequencies below 10 GHz, whereas adding one higher order mode significantly improves the correlation with the numerical CIM result by discretizing ports into 32 segments up to 60 GHz, as shown in Fig. 4.20 and 4.21.
The mode conversion transfer impedance between port 2 and 3 is shown in Fig. 4.22, where the conversion from the 2nd order mode to the fundamental mode is much less that from the 1st order mode for frequencies below 60 GHz. However, it shows a resonance peak at about 90 GHz, which causes the deviation of the K = 1 solution from the numerical CIM result near that frequency.
0 20 40 60 80 100
Magnitude of Mode Conversion Transfer Impedance [Ω]
Frequency [GHz] considering discretized ports. The modal solution (solid line) overlaps with the numerical solution (dashed line).
62 Extension using Analytical Solutions
Emax
Emin Emin Emax
(a) K = 0 (b) K = 1
Emax
Emin Emin Emax
(c) K = 2 (d) with 32 segments
Figure 4.23 Electric field distribution (linear scale) at 50 GHz for the infinite plane case in Fig. 4.19, obtained with modal solution K = 0 (a), K = 1 (b), K = 2 (c), and numerical solution for ports with 32 segments (d). Port 2 is excited with a 1-mA isotropic current source and all other ports are left open.
Extension using Analytical Solutions 63
0 5 10 15 20 25 30
0 5 10 15 20 25 30
Frequency [GHz]
0 20 40 60 80 100
0 5 10 15 20 25 30 35
Magnitude of Input Impedance (Z 11) 00 [Ω]
Frequency [GHz]
Modal K = 0 Modal K = 1 Modal K = 2
Numerical with 32 segments
{
On top of each other
Figure 4.24 Magnitude of input impedance of the 0th mode of port 1 for the finite plane case, as shown in Fig. 4.19, obtained with CIM modal solutions and the pure numerical approach with circular ports discretized into 32 segments.
64 Extension using Analytical Solutions
0 5 10 15 20 25 30
0 5 10 15 20 25
Frequency [GHz]
0 20 40 60 80 100
0 5 10 15 20 25
Magnitude of Transfer Impedance (Z 23) 00 [Ω]
Frequency [GHz]
modal K = 0 modal K = 1 modal K = 2
Numerical with 32 segments
on top of
{
each other
Figure 4.25 Magnitude of transfer impedance of the 0th mode between ports 2 and 3 for the finite plane case in Fig. 4.19, obtained with CIM modal solutions and the pure numerical approach with circular ports discretized into 32 segments.
Extension using Analytical Solutions 65
In comparison to the 2-port results in Fig. 4.12, where the curves are relatively smooth, the impedance results for the array case in Fig. 4.20 and 4.21 oscillate around a baseline due to multiple reflections from other open ports. The intensity and period of the fluctuation depend on the number of ports and pitch of the array. Disregarding the interference among anisotropic modes leads to an underestimation of the magnitude of this oscillation, as can be observed in Fig. 4.21. Therefore, including the anisotropic modes becomes even more important for analysis of large and dense arrays that can contain more than thousands of ports, e.g., in modern BGA via pin fields. The electric field distributions at 50 GHz for the 64-port array of infinite planes are demonstrated in Fig. 4.23 with an excitation of 1 mA isotropic current source at port 2 and all other ports open. Good agreement is achieved between the numerical CIM (d) and modal solution with K > 0 (b, c). A strong anisotropic field distribution is observed around the input port 2 and overall inside the array, which is absent in the K = 0 solution (a).
The port array analysis is next extended to a finite plane pair with its boundary shown in Fig. 4.19. Simulation results of the input impedance of the 0th mode for port 1 and the transfer impedance of the 0th mode between ports 2 and 3 are plotted in Fig. 4.24 and Fig. 4.25, respectively. In contrast to the one-port case in Fig. 4.17, where the difference between whether or not to include anisotropic modes is small, the results for the port array case show a significant difference between the solutions of K = 0 and K
> 0, especially for frequencies over 20 GHz. This is because reflections from the port array itself become considerably large in comparison to reflections from the plane boundary, which again manifests the importance of considering anisotropic modes in modeling of large and dense arrays.
To evaluate the efficiency gain by extending CIM with analytical modal solutions, the simulation time used for both the infinite and finite plane cases is compared, as shown in Table 4.1. It can be concluded that the efficiency is improved by at least 300 times for the infinite plane case using the analytical solutions in contrast to the numerical one
TABLE 4.1. CPU Time for Computation of the 64-Port Array Case
Cases MODAL
K=0
MODAL
K=1
MODAL
K=2
NUMERICAL
16 SEGMENTS
NUMERICAL
32 SEGMENTS
Infinite Plane 0.005 0.11 0.44 19.7 159
Finite Plane 5.2 7.8 11.2 56.0 210
Quantities are specified in second per frequency point.
Simulation was run on a 64-bit PC with a single 2.8-GHz CPU.
66 Extension using Analytical Solutions
with 32 segments. For the finite planes, the improvement decreases to about 20 times, since the number of unknowns needed for modeling of the plane boundary is not reduced by the extension. In general, the efficiency gain can be expected to be more than 100 when modeling only the circular ports.