5. Extension using Hybridization
5.2. Inclusion of Radiation Loss
5.2.4. Validation and Examples
A rectangular power plane pair with a port of 10 mil radius at the center, as shown in Fig. 5.16, is simulated using the hybrid method. The power planes are configured as perfect electric conductors (PEC) and the dielectric is lossless, thus the real part of the input power is only dissipated by the radiation field. This is not captured by methods assuming the PMC boundary condition that prevents radiation from the structure. As a consequence, the real part of the input impedance is zero using CIM alone. However, the hybrid method predicts a nonzero real part of the input impedance, as shown in Fig. 5.17, owing to the radiation loss.
The same configuration is simulated using a MoM full-wave solver [43]. The proposed method shows a good agreement with the MoM solution except for the very low frequencies, possibly due to the instability of the CIM system matrix near DC in lossless cases. The efficiency of the proposed method is, however, much better due to the fact that only the boundary needs to be discretized, as shown in Fig. 5.18(b). In contrast, the whole plane has to be discretized in MoM, as shown in Fig. 5.18(a). The
Port q Port r
Port p
Z
pqZ
rrqq pq
Z − Z
pp pq
Z − Z
Figure 5.15 An equivalent circuit illustration of the parallel connection between radiation impedance and the power plane impedance. Port q can be assumed open (PMC boundary) only if Zrr ≫Zqq, which may not be true for low loss cavities at resonances.
Extension using Hybridization 85
0 1 2 3 4 5
10-4 10-3 10-2 10-1 100 101 102 103
Frequency [GHz]
Real Part of Input Impedance [Ω]
Hybrid method MoM full-wave solver
Figure 5.17 Real part of the input impedance computed for the case in Fig. 5.16, obtained with the proposed hybrid method and a full-wave MoM simulation. Real part of the input impedance indicates the radiated power for this case.
Port 1 (2, 1)
(0, 0)
(4, 3)
Port radius: 10 mil
εr= 1, tanδ= 0, d= 10 mil
∞
c → σ
d
y
x
x z
z
Figure 5.16 Rectangular plane pair example with one port at the center. Dimensions are given in inches (1 inch ≈ 2.54·10-2 m).
86 Extension using Hybridization
solving time of the MoM solver is about 22 second per frequency point (s/freq), whereas the hybrid approach takes less than 0.1 s/freq.
The second example is an irregular shaped copper plane pair with a 60 mil thick lossless dielectric layer, shown in Fig. 5.19. An FEM based full-wave solver [117] is used to validate the simulation results. In the FEM simulation, a coaxial lumped port is defined as excitation, leading to a capacitive parasitic via near field that is not taken into account by the proposed hybrid method. This extra capacitance is handled with the physics-based via model in the next section. Its value is analytically calculated to be about 85 fF [133]. An airbox is defined to account for the radiation loss in the FEM simulation.
The input impedances calculated by both the hybrid method and the FEM solver are plotted in Fig. 5.20, which shows a good agreement. The simulation time for the FEM solver is over 4 minutes per frequency, whereas about 0.6 s/freq is needed for the hybrid method. The magnitude of the correction impedance, defined in (5.15), is shown in Fig. 5.21. The FEM result is obtained by subtracting the input impedance simulated with a PMC boundary condition from the result with an airbox, which corresponds to the definition in (5.15). This correction impedance manifests the difference between the two boundary conditions seen by the input port. As shown in Fig. 5.21, it increases with the frequency and can be particularly high near and at the resonance frequencies.
Fig. 5.22 shows the radiated power of the structure in Fig. 5.19, obtained by an integral of the Poynting vector as in (5.4). The excitation is 1-Watt power at the input port.
y z x
y
x z
(a) MoM discretization (b) Hybrid method discretization
Figure 5.18 Discretization at 5 GHz of the power plane pair case in Fig. 5.16 by (a) MoM full-wave solver and (b) the hybrid method. The hybrid method needs much fewer unknowns since only the boundary is discretized.
Extension using Hybridization 87
Magnitude of Input Impedance [dB Ω]
Frequency [GHz]
Hybrid method FEM solver (airbox)
Figure 5.20 Magnitude of the input impedance computed for the case in Fig. 5.19, obtained with the proposed hybrid method and a full-wave FEM simulation.
Port 1 (3, 2)
88 Extension using Hybridization
0 1 2 3 4 5
10-4 10-3 10-2 10-1 100 101
Frequency [GHz]
Radiated power [Watt]
CIM (PMC) Hybrid method FEM solver (airbox)
Figure 5.22 Radiated power computed for the case in Fig. 5.19 excited by a 1-Watt power source, obtained with the proposed hybrid method, a full-wave FEM simulation, and a two-step approach using CIM.
0 1 2 3 4 5
10-3 10-2 10-1 100 101 102 103 104
Frequency [GHz]
Magnitude of Correction Impedance Zcorr [Ω]
Hybrid method FEM solver
Figure 5.21 Magnitude of the correction impedance, as defined in (5.15), computed for the case in Fig. 5.19, obtained with the proposed hybrid method and a full-wave FEM simulation.
Extension using Hybridization 89
The results are obtained with the hybrid method, a FEM solver, as well as a two-step approach where the boundary voltage distribution is first computed assuming a PMC boundary condition. The hybrid method shows a good correlation with the FEM simulation. The discrepancy at the low frequencies might be explained by an insufficient size of the airbox in the FEM simulation. The results from the two-step approach deviates from the other two at frequencies over 2 GHz. In particular, a higher radiated power is predicted at resonance frequencies and some of the peaks even exceed the 1 Watt input power. This may be explained by an overestimated boundary under the PMC condition. It is then necessary to consider the radiation loss in these cases.