The Characteristic Basis Function Method is employed in conjunction with the parallel-plate dyadic Green's function method to obtain the impedance characteristics of electrically large gap-waveguide structures. Numerical results are shown for the groove gap waveguide demonstrating reduced execution times relative to the HFSS software, while the solution accuracy is barely compromised

Kildal, Per-Simon

Maaskant, Rob

Takook, Pegah

Chalmers Publication Library

Chalmers Publication Library
Fast analysis of gap waveguides using the characteristic basis function method and
the parallel-plate Green’s function
This document has been downloaded from Chalmers Publication Library (CPL). It is the author´s
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2012 International Conference on Electromagnetics in Advanced Applications (ICEAA),
Cape Town, WP, South Africa, 2-8 Sept 2012
Citation for the published paper:
Maaskant, R. ; Takook, P. ; Kildal, P. (2012) "Fast analysis of gap waveguides using the
characteristic basis function method and the parallel-plate Green’s function". 2012
International Conference on Electromagnetics in Advanced Applications (ICEAA), Cape
Town, WP, South Africa, 2-8 Sept 2012 pp. 788-791.
http://dx.doi.org/10.1109/ICEAA.2012.6328737
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(article starts on next page)
Fast Analysis of Gap Waveguides Using the
Characteristic Basis Function Method and the
Parallel-Plate Green’s Function
R. Maaskant∗ P. Takook† P.-S. Kildal‡
Abstract — The Characteristic Basis Function
Method is employed in conjunction with the parallel-
plate dyadic Green’s function method to obtain the
impedance characteristics of electrically large gap-
waveguide structures. Numerical results are shown
for the groove gap waveguide demonstrating reduced
execution times relative to the HFSS software, while
the solution accuracy is barely compromised.
1 INTRODUCTION
To anticipate to future trends in high-frequency
electronics, a novel low-loss low-cost waveguide and
transmission-line technology has been proposed –
referred to as the gap waveguide technology [1].
The basic structure consists of a pair of perfect
electric conducting (PEC) top and bottom ground
planes in conjunction with a bed of nails for realiz-
ing a perfect magnetic conductor (PMC) boundary
condition. Fig. 1 illustrates an example of a groove
PEC
PEC
Figure 1: Coaxial-probe excited groove gap waveg-
uide.
gap waveguide, where two coaxial probes are used
to excite the waveguide ﬁelds. The groove is bor-
dered by only a few rows of pins to prevent the ﬁelds
from leaking out (sideways) when operating in the
stop band. The pins are electrically interconnected
∗Chalmers University of Technology, Dept. of Sig-
nals and Systems, 412 96, Gothenburg , Sweden, e-mail:
rob.maaskant@chalmers.se
†Chalmers University of Technology, Dept. of Sig-
nals and Systems, 412 96, Gothenburg , Sweden, e-mail:
takook@student.chalmers.se
‡Chalmers University of Technology, Dept. of Sig-
nals and Systems, 412 96, Gothenburg , Sweden, e-mail:
per-simon.kildal@chalmers.se
to the bottom PEC plate, while no electrical con-
tact between the pins and the upper PEC plate is
required. The so-formed capacitive gap in the “gap
waveguide” is not only needed to realize the PMC,
but represents a mechanical advantage as well.
To allow for a fast analysis and optimization
of gap waveguides, it is required to develop tai-
lored computational electromagnetic methods that
are numerically eﬃcient. In this paper, we pro-
pose to employ the Characteristic Basis Func-
tion Method (CBFM, [2]) in conjunction with the
parallel-plate Green’s function [3]. This methodol-
ogy enhances the computational eﬃciency of con-
ventional method of moment (MoM) codes signiﬁ-
cantly. Furthermore, the CBFM ismemory eﬃcient
and allows us to solve problems that are typically
1–3 orders larger in size.
2 PARALLEL PLATE DYADIC
GREEN’S FUNCTION
Consider Fig. 2, where the EM ﬁeld of a dipole
point source inside a parallel-plate region is com-
puted with the help of the image principle. De-
pending upon whether the dipole has an x-, y-,
or z-orientation, the ﬁeld for each polarization is
regarded as being radiated by two superimposed
1D-line arrays of dipole point sources with the
original and possibly mirrored polarization, respec-
tively. Each line array has an inter-dipole spacing of
twice the plate distance. In Dyadic form, the spa-
tial parallel-plate Green’s function G is expressed
as G(r, r′) = GN
xx
xˆxˆT +GN
yy
yˆyˆT +GN
zz
zˆzˆT , where
GN
xx
=
N∑
n=−N
e−jk0
√
ρ2+(z−z′+2dn)2
4π
√
ρ2 + (z − z′ + 2dn)2
−
N∑
n=−N
e−jk0
√
ρ2+(z+z′+2d(n−1))2
4π
√
ρ2 + (z + z′ + 2d(n− 1))2
GN
yy
= GN
xx
GN
zz
=
N∑
n=−N
e−jk0
√
ρ2+(z−z′+2dn)2
4π
√
ρ2 + (z − z′ + 2dn)2
+
N∑
n=−N
e−jk0
√
ρ2+(z+z′+2d(n−1))2
4π
√
ρ2 + (z + z′ + 2d(n− 1))2
(1)
xˆr
PEC
PEC
J = Iδ(r − r′)zˆ
r′
zˆ d
r
PEC
PEC
r′
J = Iδ(r − r′)xˆ
(a) (b)
r
2d
zˆ
xˆ
n = 0
n = 1
r
2d
n = 0
n = 1
(c) (d)
Figure 2: (a) z-polarized and (b) x-polarized elec-
tric Hertzian dipole currents in between two in-
ﬁnitely large parallel PEC plates separated by a
distance d; (c) and (d), the repeated application of
the image principle for both dipole polarizations,
respectively.
for N → ∞, and where we have deﬁned that
ρ2 = (x − x′)2 + (y − y′)2. It is well-known that
each summation in Eq. (1) is performed over a
slowly convergent series. However, the ﬁeld of a line
array can be computed rapidly using the Shanks-
accelerated spatial Green’s functionmethod. Other
methods, such as the Ewald summation method,
or the spectral-domain summation method, were
found to be less eﬃcient for our type of problems [3].
For instance, the Ewald summation method re-
quires very few terms in the near-ﬁeld region of
the source, but the computation time of a single
term is orders larger than that of a spatial Green’s
function term. Furthermore, if only a few digits ac-
curacy is required, which is often suﬃciently good,
only a few terms are needed. In fact, we found that
for our type of problems, the overall series eval-
uation time is shortest for the Shanks-accelerated
spatial Green’s function method. Finally, we point
out that the Shanks transformation is easy to im-
plement. For example, when considering the new
series GN
xx
, for N = 1, 2, . . ., the extrapolated value
for N → ∞ is
G∞xx ≈
GN+1
xx
GN−1
xx
−
(
GN
xx
)2
GN+1xx − 2GNxx +G
N−1
xx
(2)
where N is taken suﬃciently large. We concluded
that the double Shanks tranformation on the orig-
nal spatial Green’s function series represents the
best trade oﬀ between the solution accuracy and
the total series evaluation time [3].
3 CHARACTERISTIC BASIS FUNC-
TION METHOD
To minimize the total number of basis functions,
entire-domain basis functions are employed for the
electric surface currents on the conductors inside
the gap waveguide; in this speciﬁc case on the pins
and coaxial probes only. Each of these so-called
Characteristic Basis Functions (CBFs) is expanded
in terms of a ﬁxed combination of lower-level ba-
sis functions, which are herein chosen to be the
Rao-Wilton-Glisson (RWG) basis functions. The
CBFs are generated for a small part of the struc-
ture only, namely by solving for the current in the
corresponding subdomain. As an example, consider
Fig. 3, where the CBFs are generated for one of
the pins. Here, each pin consists of 382 RWGs. A
2 4 6 8 10 12 14 16 18 20
10−15
10−10
10−5
100
n    [−]
σ
n
/σ
m
ax
 
 
 
 
[−]
 
 
Singular Values
Threshold
PWS
Δθi = 900
Δφi = 900
Figure 3: Generation of CBFs on a pin using a
plane wave spectrum (PWS). The singular value
spectrum is shown where a threshold is used to limit
the maximum number of employed CBFs.
plane wave spectrum is incident on the pin with
Δθi = Δφi = 900 (and two polarizations). This
generates 16 currents on the pin. The currents are
stored as RWG expansion coeﬃcients vectors and
make up the columns of the matrix J = UDVH ,
where the right-hand side is the SVD of J. The
magnitude of the normalized singular values in D
are plotted in Fig. 3. With an appropriate thresh-
olding procedure on the singular values, only the
ﬁrst 12 column vectors of the left-singular matrix U
are retained and subsequently used as CBFs. The
reduction in the number of unknowns per pin is
therefore 382/12=31.8 (@ 12 GHz).
Since the set of CBFs is identical for each pin,
we use symmetry to generate a set for all the pins
through translation [4]. The same procedure is fol-
lowed for the coaxial probes. More details on the
generation of CBFs can be found in [2], where also
electrically interconnected subdomains are consid-
ered.
After generating the CBFs for allM subdomains,
i.e., Jm, for m = 1, 2, . . . ,M , one can construct the
reduced moment matrix equation ZredIred = Vred,
where
Z
red =
⎡⎢⎣ (J1)
T
Z11J1 · · · (J1)
T
Z1MJM
...
. . .
...
(JM )
T
ZM1J1 · · · (JM )
T
ZMMJM
⎤⎥⎦
(3)
and
V
red =
[
(J1)
T
V1, · · · , (JM )
T
VM
]T
(4)
and where the elements of the matrix block Zmn are
the reaction integrals between the RWG basis func-
tions on the nth source and mth observation sub-
domain, respectively. Likewise, Vm is the original
RWG voltage excitation vector for the mth subdo-
main, where the incident electric ﬁeld is generated
by the impressed magnetic frill currents that are
supported by the coaxial apertures. This CBFM
procedure leads to a reduced MoM matrix equation
which can be solved in-core using standard Gaus-
sian elimination techniques.
One observes that the CBFM can be regarded as
an enhancement technique for conventional MoM
approaches, since the original MoM-matrix blocks
and excitation vectors are reduced with the help of
the above-described CBFs. Although the method-
ology is very general and independent of the
Green’s function, in this work the MoM is used
to discretize an Electric Field Integral Equation
(EFIE) for the electric current inside a parallel-
plate waveguide region using Galerkinsmethod and
RWG basis and test functions.
Finally, the CBFM ismemory eﬃcient and allows
us to solve problems that are many wavelengths in
size, even though the structure exhibits very ﬁne ge-
ometrical features requiring detailed discretizations
(multi-scale features).
4 Numerical Results
In this section the numerical results on the groove
gap waveguide are presented, whose geometrical di-
mensions are as described in Table 1. Additionally,
Table 1: Geometrical dimensions of the Groove gap
waveguide in [mm].
Wpin Hpin L1 L2 L3 L4
1 6.25 15.8 43.7 5.4 2
Wpin
L3
L4
dprobe
L2
L1
the radius and length of the coaxial probes are 0.25
and 5 mm, respectively. The parallel-plate distance
d = 7.25 mm.
The CBFM computations have been carried out
on a 64 bit (x86-64) Linux – openSUSE (v.11.4)
server equipped with 2 Intel Xeon E5640 CPUs
operating at 2.67 GHz (each CPU has 4 cores/8
threads), with access to 144 GB RAM memory and
2 TB harddisk space. The HFSS simulations were
performed on a 64 bit Windows XP server equipped
with 2 Intel Xeon 5130 CPUs operating at 2 GHz
(each CPU has 2 cores/2 threads), 8 GB RAM, and
300 GB harddisk space.
Firstly, to verify the accuracy of CBFM, the 100-
Ohm S-parameters for the gap waveguide structure
(cf. Table 1) have been computed, both by our
CBFM-enhanced MoM procedure and the HFSS
software. In HFSS, the adaptive meshing termi-
nates if the relative ﬁeld error is less than 1%. The
computed S-parameters for CBFM and HFSS are
shown in Fig. 4 and are in good agreement. The
solve times for this relatively small problem is 1
min. 41 sec. and 1 min. 24 sec. for CBFM and
HFSS, respectively. The simulation times are com-
parable due to the relatively large overhead needed
by the CBFM to determine the CBFs. The CBFM
employed 37088 RWGs (1112 CBFs), while HFSS
employed 42158 tetrahedra. Owing to the array
structure, the meshing time is only 5 sec. for the
CBFM, while it takes 3 min. 57 sec. for HFSS.
By changing L2 one can examine the total solve
time as a function of the problem size. The results
are shown in Fig. 5, where the scaling of the solve
time for CBFM outforms that of HFSS by about a
factor of three. Also, for equal simulation times, the
problem size for CBFM can be about twice larger
than for the HFSS software. The HFSS software
−0.03
−0.02
−0.01
0
0.01
0.02
−0.01
0
0.01
0
0.005
0.01
 
x    [m]
Magnitude of Averaged Current Distribution
[dBA/m or dBV/m] at f=12GHz
y    [m]
 
z 
  
 [m
]
−27.0303
−7.0303
12.9697
32.9697
52.9697
72.9697
10 12 14 16 18 20
−60
−50
−40
−30
−20
−10
0
Zo=100 Ω
Frequency    [GHz]
|S|
    
[dB
]
 
 
|S11| CBFM
|S21| CBFM
|S11| HFSS
|S21| HFSS
(a) (b)
Figure 4: (left) the magnitude of the computed electric current and S-parameters of a groove gap-
waveguide when excited by a pair of coaxial probes; (right) the numerical results are computed by the
CBFM employing the parallel-plate Greens function. The numerically computed results are validated
through the HFSS software.
could not handle problems for dprobe > 1.3 m, due
to memory constraints for that server. The non-
gradual increase in simulation time is caused by
the adaptive meshing of HFSS.
0 0.5 1 1.5 2 2.5 3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x 104
dProbe    [m]
Si
m
ul
at
io
n 
tim
e 
   
[se
c]
 
 
CBFM
HFSS
Mesh for larger problems
could not be saved by HFSS
Figure 5: Scaling of the method: the total solve
time as a function of problem size (dprobe).
5 Conclusions
The CBFM has been supplemented with the
parallel-plate Green’s function to enable the fast
analysis of gap waveguide structures. By com-
paring the computed S-parameters obtained from
CBFM and the HFSS software, it has been con-
cluded that the solution accuracy for the CBFM
is hardly compromised, while it is at least a factor
three faster than the HFSS software.
Acknowledgments
This work is part of the research programme Ru-
bicon, which is partly ﬁnanced by the Netherlands
Organization for Scientiﬁc Research (NWO).
References
[1] P.-S. Kildal, E. Alfonso, A. Valero-Nogueira,
and E. Rajo-Iglesias, “Local metamaterial-
based waveguides in gaps between parallel
metal plates,” IEEE Antennas Wireless Propag.
Lett., vol. 8, no. 1, pp. 84–87, 2009.
[2] R. Maaskant, R. Mittra, and A. G. Tijhuis,
“Application of trapezoidal-shaped character-
istic basis functions to arrays of electrically
interconnected antenna elements,” in Proc.
Int. Conf. on Electromagn. in Adv. Applicat.
(ICEAA), Torino, Sep. 2007, pp. 567–571.
[3] P. Takook, R. Maaskant, and P.-S. Kildal,
“Comparison of parallel-plate greens function
acceleration techniques,” in Proc. European
Conference on Antennas and Propag. (Eu-
CAP), Prague, Czech Republic, Mar. 2012, pp.
1–5.
[4] R. Maaskant, R. Mittra, and A. G. Tijhuis,
“Fast solution of multi-scale antenna prob-
lems for the square kilomtre array (SKA) ra-
dio telescope using the characteristic basis
function method (CBFM),” Applied Computa-
tional Electromagnetics Society (ACES) Jour-
nal, vol. 24, no. 2, pp. 174–188, Apr. 2009.


Fast analysis of gap waveguides using the characteristic basis function method and the parallel-plate Green’s function

The Characteristic Basis Function Method is employed in conjunction with the parallel-plate dyadic Green\u27s function method to obtain the impedance characteristics of electrically large gap-waveguide structures. Numerical results are shown for the groove gap waveguide demonstrating reduced execution times relative to the HFSS software, while the solution accuracy is barely compromised

Fast analysis of gap waveguides using the characteristic basis function method and the parallel-plate Green’s function

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