3.2 Modeling for Microwave Applications
3.2.2 Novel Physics-based Model for SIWs
The method presented in the previous subsection allows to consider arbitrarily shaped SIWs. However, in practice SIWs are predominantly realized as straight segments and, hence, modeled as transmission lines. In this subsection, we propose the usage of a different near-field model to account for via transitions and use transmission line models for the SIW. We still consider the geometry illustrated in Figure 3.5 and assume the ground vias to form straight lines. All ground vias have the same radius ag and are placed with a pitch p on parallel lines of the distance t.
This novel PBV is proposed to significantly enhance computation efficiency for structures containing straight SIWs. The expected speedup compared to the conventional PBV is several orders of magnitude, but comes at the cost of flexibility. The conventional PBV can model arbitrary arrangements of ground vias, whereas the novel PBV is limited to straight SIWs.
Propagating Field Model
In the considered geometry, the ground via fences and the top and bottom metalization form a rectangular waveguide like structure. The fundamental mode of the SIW can be modeled by the mode of a rectangular waveguide with the same height and dielectric and an equivalent width. There are different approaches to obtain an equivalent width [147,154–157].
The empirical model proposed in [157] shows very good agreement and is widely used by other researchers. The equivalent width of the waveguide is given by [157]
teq =t
χ1+ χ2
p
2ag +χ1χ+χ3−χ2−χ1 3
(3.7)
3.2 Modeling for Microwave Applications
with the factors
χ1 = 1.0198 + 0.3465
t
p −1.0684, (3.8)
χ2 =−0.1183− 1.2729
t
p −1.2010, (3.9)
χ3 = 1.0082− 0.9163
t
p + 0.2152. (3.10)
Now, the SIW is modeled as a rectangular waveguide with the width teq. To represent the SIW using ABCD-matrices, the characteristic impedance ZSIW and the propagation constant βSIW are required. Adapting the equations for the TE01 mode of a rectangular waveguide [136] and using the equivalent width, they are given as
ZSIW = µ
2d teq
r
1−ωc,1ω 2
, (3.11)
βSIW=ω√ µ
s
1−
ωc,1 ω
2
. (3.12)
Like in the conventional PBV model, losses can be included by introducing a complex dielectric constant. Here, ωc,1 is the angular frequency of the firstcutoff frequency which is given by
ωc,1 = π teq√
µ. (3.13)
Near-field Model
Using the representation of a rectangular waveguide with equivalent width for the propa-gating field, the near-field model used in the conventional PBV model cannot be reused as the new propagating field model has rectangular ports instead of radial. A near-field model which provides an equivalent circuit representation for the transition between the coaxial via which excites the SIW and a rectangular port has been proposed by Williamson in [136]. The near-field model provides a three-port network where one port connect to the coaxial access and the other two refer to rectangular ports in the waveguide with different propagation directions. The model is used here as proposed in [136] with the difference that the width of the rectangular waveguide is substituted by the equivalent width of the SIW.
The proposed equivalent circuit is illustrated in Figure 3.8. Like in the near-field model used in the conventional PBV model, the element values consist of Bessel and modified
jXA jB
R:1
coaxial port
rectangular rectangular port
port
−jXb near-field
model
−jXb
Figure 3.8: Equivalent circuit model of a via SIW transition with port definitions like illustrated in Figure 3.5 according to [136]. Here,j is the imaginary unit. Figure and caption taken from [6].
Bessel functions as well as infinite sums of these. The computation of these values can be time consuming, as it has to be repeated for every frequency point. Therefore, a closer look is taken at the element values as a function of frequency for a typical geometry. Figure 3.9 shows the values of the elements as a function of frequency for typical dimensions for applications in X band. In the following, the behavior of the individual element values is discussed and some approximative forms are proposed to accelerate the computation.
These observations and approximations are valid for the observed frequency range of up to 20 GHz which is approximately three times the cutoff frequency of the SIW.
First, take a look at the value of the transformer turns ratio R in Figure 3.9. It does not vary much and monotonically decreases over the observed frequency range. A Taylor expansion forR (given by formula (30) in [136]) to the cubic term reads
R≈
1 + a2−b21 + lnab 4 lnba k2
sin πe teq
!
, (3.14)
where k = ω√
µ is the complex wavenumber. The expression mainly depends on the position of the access via relative to the SIW. The sinusoidal becomes one for a centered access via and decreases when the via is moved towards the via fences. Except for the position, the value does only depend on the radius of the access via and its antipad. The quadratic term is rather small and, therefore, the transformer turns ratio is approximately one in the observed frequency range in the case of a centered access via.
3.2 Modeling for Microwave Applications
0.0 -0.6 -1.2 -1.8
B(mS)
0 5 10 15 20
Frequency (GHz) 0
6 12 18
XA(Ω)
0.988 0.992 0.996 1.000
R
0.00 -0.12 -0.24 -0.36
XB(Ω)
Figure 3.9: Values of the elements of the equivalent circuit of Figure 3.8 in straight lines and the proposed approximations in dotted lines for a transition with a centered via ofa = 10 mil, b = 15 mil,t = 400 mil, d = 10 mil, ande = 200 mil. Adapted from [6].
The value of the element XB is given by formula (32) in [136]. It can be rewritten in a shorter form as
XB = 4πµd a teq
!
ω. (3.15)
The element value is frequency-dependent and behaves capacitive. The value depends on the radius of the access via, the equivalent width of the SIW, and the substrate thickness.
It is independent of the position of the access via and the radius of the antipad.
Approximation of the element value of XA fails using common techniques. As illustrated in Figure 3.9, it behaves inductive and depends on all geometric and material parameters.
Here, its value is determined by evaluating the formulas (29) and (31) in [136].
The element B arises from a parallel connection of two admittances. One representing the reflections at the coaxial port which are caused by the short circuit at the bottom metalization. The other represents the input impedances of the evanescent modes of the rectangular waveguide and the junction admittance. Similar to XB the element behaves capacitively and its value is frequency-dependent. As it can be seen in Figure 3.9, the value of B seems to depend linearly on the frequency. Hence, a linear approximation is done by sampling at the cutoff frequency
B ≈ωB(ω =ωc,1). (3.16)
The value of B depends on all geometric features. The proposed approximations are in excellent agreement with the fully evaluated formulas, see Figure 3.9.
This near-field model represents the junction illustrated in Figure 3.5 very accurately. To model more general kinds on junctions, near-field models could also be obtained using full-wave simulations. The application example in the next subsection serves as a validation example for the novel PBV model.