3.2 Modeling for Microwave Applications
3.2.1 Modeling of Substrate Integrated Waveguides (SIWs)
Substrate Integrated Waveguides (SIWs) are frequently used in radio Frequency (RF) and microwave applications [146]. In SIW technology, rectangular waveguide-like structures are implemented on a conventional PCB or Low-Temperature Co-fired Ceramic (LTCC) technology. Top and bottom of the rectangular waveguide are represented by the top and bottom metalization on the substrate. For sidewalls via fences are used. The radius and spacing of the vias have a strong impact on the electromagnetic behavior of the SIW. If the fence is sufficiently dense, an SIW can be considered as a rectangular waveguide of the cavity height and an equivalent width [147]. There are different ways to couple an electromagnetic wave into an SIW. To couple from a microstrip line to an SIW, usually tapered transitions [148,149] or perforations in the metalization [150] are used. In multilayer applications, it is possible to couple into an SIW by exciting a via [151–153]. Here we will focus on the excitation with vias, an example with microstrip feed will be shown in Section 3.2.4.
The geometry investigated is illustrated in Figure 3.5. The access vias that meant to excite the SIW have an antipad clearance in the top metalization. All remaining vias are ground vias forming the ground via fences. By means of physics-based modeling the problem can be dissected into the 2-D wave propagation inside the cavity and the near-fields in the
2Please note that the notation has been changed in order to fit in the notation used in this thesis.
Furthermore, the units have been changed to mil for the sake of readability.
d b a
p
t teq
ag coaxial port rectangular ports
e x
z y
gound vias access via
for novel PBV
metal plates
Figure 3.5: Illustration of a single cavity via fed SIW. The coaxial port is used by both PBV models and the rectangular ports are used in the novel PBV. Figure and caption taken from [6].
vicinity of the access vias. In this subsection, the problem shall be modeled with the CIM and an appropriate near-field model. To distinguish this approach from the one proposed in the next subsection, we will call it conventional PBV model .
The idea of physics-based modeling applied to SIWs is illustrated in Figure 3.6. The propagating fields inside the cavity, which represent the rectangular waveguide modes supported by the SIW are modeled by the CIM and represented by the parallel-plate impedance Zpp. The terminals of this network block refer to the circular ports (on the cylindrical surface) defined on all access (p) and ground (q) vias inside the cavity. The ports at the ground vias are short circuited representing a PEC boundary condition. This is implemented by employing the methods proposed in Section 2.3.3. To model the transition between the circular ports on the cylindrical surface of the access vias to the coaxial access at the top metalization a near-field model is required.
The near-field model required in this scenario is very similar to the one used in the previous section with the difference that there is only one coaxial port on the top metalization. The via is short circuited on the bottom. A near-field model for this scenario is also proposed by Williamson in [135] as an intermediate result for the near-field model used in the previous section. The equivalent circuit model for this near-field model is illustrated in Figure 3.7.
The formulas to determine the element values consist of Bessel, Hankel, and modified Bessel and Hankel functions, the explicit expressions are given in [135]. This near-field model
3.2 Modeling for Microwave Applications
coaxial
Zpp port
radial ports near-field
model
coaxial
port near-field model
access vias cavity ground vias
V1p
VNpp INpp I1p
INqq I1q V1q = 0
VNqq = 0 propagating
field model
Figure 3.6: Equivalent circuit for an SIW in a single cavity modeled by the conventional PBV. On the left side are the coaxial ports of the vias that excite the antipad, like in Figure 3.5.
On the right side are the ground vias which build the SIW. In the propagating field model all vias are represented as circular ports and connected with the parallel-plate impedanceZpp. Figure and caption taken from [6].
represents the transition between the coaxial and radial port under consideration of the higher order evanescent modes.
Having the propagating field model – computed with the CIM– and the near-field model as illustrated in Figure 3.7 available, the SIW can be modeled completely. To connect the different models, the following strategy is chosen: first, consider the PEC boundary condition at the radial ports referring to ground vias using the formulas presented in Section 2.3.3.
Second, cast all network parameter blocks in ABCD-matrix form and concatenate the blocks by multiplication. Finally, the result is cast into scattering parameters in order to provide comparable results.
With this modeling approach arbitrarily shaped SIWs can be modeled, in the sense that ground via fences are not required to form straight lines but can be placed arbitrarily in the cavity. Furthermore, the usage of the CIM allows for the consideration of inhomogeneous substrates which may be used to model dielectric inclusions. In Section 3.2.4 a brief example with a circular inclusion will be presented. Furthermore, it is possible to exchange the near-field model to take slightly different transitions into account – e.g. where the via is not short-circuited, but open at the bottom.
near-field model coaxial
port
radial port jB1
R: 1
jB2 jB3
Figure 3.7: Equivalent circuit representation of near-field model used in the conventional PBV, see Figure 3.5. According to [135]. The formulas to determineB1,B2,B3, andR are given by the formulas (21), (29), (26), and (28) in [135], respectively.