In many practical cases, the transverse propagation constant is small. In these cases, the approximation from Section 4.3.4 can be applied. As pointed out in [24], it is applicable over a large parameter range. This parameter range is given by the configurations and frequencies which lead to only quasi-TEM and slow-wave propagation.
Next, structure B which has two oxide layers and dimensions similar to a typical interposer is investigated. The results are shown in Fig. 4.7. They are very similar to the results for structure A when using the same silicon conductivity which is to be expected from the approximate formula. When using a very large resistivity of 500Ωcm (0.02 S/m), the transition region is shifted towards lower frequencies by about two decades. The effect of a oxide layer of only10 nmfor structure B is also shown. The main effect is an even stronger slow-wave effect and an increased width (frequency range) of the transition region.
As a further demonstration of the applicability of the TRM, a structure with a larger num-ber of layers is investigated. Structure C has a sinusoidal resistivity profile with 25 complete periods of variation of the resistivity between1Ωcmand100Ωcm. A discretization of the 100µm thick substrate into 500 layers, each with constant resistivity, is used. An average of about 143 ms per frequency point is needed for the computation of the wave number in this case. As can be observed in Fig. 4.7a, slow-wave and quasi-TEM mode can also be observed but the transition region is significantly larger than in the cases considered before. The results in Fig. 4.7b show that the attenuation at low frequencies is significantly higher. In all considered examples it can be observed that the proposed methods, i.e., the TRM and the simplified calculation of the wave number/propagation constant are in good agreement.
4.7 Application to Computation of Planar Microwave Circuits
4.7.1 Application with the CIM
The CIM is the adopted approach for the determination of network parameters of the paral-lel plate structures. As already mentioned, the wave number computed with the transverse resonance method can be used to adapt the CIM to compute the properties of planar cir-cuits with either infinite planes or finite planes and with arbitrarily shaped contours. Some details of the CIM are given in Appendix D. Instructions on where the adaptations need to be applied are discussed in Section 4.7.3
Frequency in GHz
PhaseVelocity/FreeSpacePhaseVelocity
Bulk silicon TRM solution
Simplified calcualtion of γ B with500Ωcm
B with10Ωcm and dSiO2 = 10 nm B with10Ωcm
C
C
(a)
Frequency in GHz
AttenuationindB/mm
Bulk silicon loss TRM solution
B with500Ωcm B with10Ωcm
C 10Ωcm C
10Ωcm
500Ωcm 500Ωcm
Simplified calcualtion ofγ B with10Ωcm and dSiO2 = 10 nm
(b)
Figure 4.7:(a) Results for phase velocity of structures B and C. (b) Results for attenuation of structures B and C. Figure and text taken from [8].
4.7 Application to Computation of Planar Microwave Circuits 4.7.2 Application with Radial Line Segment
Another propagating field model that is of interest in the following is the radial line. In the near field modeling technique presented in the following chapter that is based on a finite-difference frequency domain method, the superposition of localized and propagating field is computed. In order to separate the near and far field models, a de-embedding is required which uses a radial line segment and the network parameter description in terms of the fundamental isotropic mode.
Solution for Homogeneous Filling
The following two-port description of a homogeneously filled waveguide from [13] can be derived with the help of [96, Ch. 5]. For two radial ports with index 1 of the inner port at the smaller radius and with index 2 at the outer radial port with the larger radius, an admittance parameter matrix that relates the fundamental isotropic modes can be given as where ◦ denotes the Hadamard product, i.e. an element-wise multiplication of matrices, and
h is the thickness of the substrate that is equal to the plane separation, kTM is the wave number of the (transverse magnetic) mode, r1 and r2 are the inner and outer radius, respectively, of the waveguide.
4.7.3 Adaptation of Wave Impedances and Characteristic Impedances to Inho-mogeneous Fillings
As has is shown in Appendix B.3, the wave number of propagation in the radial direction is equal to that of the planar waveguide with plane wave propagation and the same stackup.
An effective wave impedance is defined in the following which allows the application of the CIM and (4.35e) with inhomogeneously filled waveguides. It is defined analogously to the approximation of the wave number in Section 4.3.4 as
ηeff = kx
ωεeff. (4.36)
where
εeff = P3
i=1ti P3
i=1(ti/εi), (4.37)
The derivation is presented in Appendix B.5.2.
It is known that the values of the wave impedances and characteristic impedances in the case of inhomogeneous media depend on the definition [106]. In the applications that follow, waveguides with the same stackup are concatenated in terms of their network parameters and completely equivalent definitions are applied in all cases. Therefore, the concatenation corresponds to enforcing the continuity of the tangential field components at the interfaces.
Any influence of the definition vanishes because the overall results of the characterizations are given at the coaxial via ports on homogeneously filled antipads for which the usualy definitions of characteristic impedance can be applied.