GENERALIZED SCATTERING MATRIX MODELING OF DISTRIBUTED MICROWAVE AND MILLIMETER-WAVE SYSTEMS by AHMED IBRAHIM KHALIL A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy ELECTRICAL ENGINEERING Raleigh 1999 APPROVED BY: Chair of Advisory Committee
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GENERALIZED SCATTERING MATRIX
MODELING OF DISTRIBUTED MICROWAVE
AND MILLIMETER-WAVE SYSTEMS
by
AHMED IBRAHIM KHALIL
A dissertation submitted to the Graduate Faculty ofNorth Carolina State University
in partial fulfillment of therequirements for the Degree of
Doctor of Philosophy
ELECTRICAL ENGINEERING
Raleigh
1999
APPROVED BY:
Chair of Advisory Committee
Abstract
KHALIL, AHMED IBRAHIM. Generalized Scattering Matrix Modeling of Dis-tributed Microwave and Millimeter-Wave Systems. (Under the direction of MichaelB. Steer.)
A full-wave electromagnetic simulator is developed for the analysis of
transverse multilayered shielded structures as well as waveguide-based spatial power-
combining systems. The electromagnetic simulator employs the method of moments
(MoM) in conjunction with the generalized scattering matrix (GSM) approach. The
Kummer transformation is applied to accelerate slowly converging double series ex-
pansions of Green’s functions that occur in evaluating the impedance (or admit-
tance) matrix elements. In this transformation the quasi-static part is extracted
and evaluated to speed up the solution process resulting in a dramatic reduction of
terms in a double series summation. The formulation incorporates electrical ports
as an integral part of the GSM formulation so that the resulting model can be
integrated with circuit analysis.
The GSM-MoM method produces a scattering matrix that represents
the relationship between waveguide modes and device ports. The scattering matrix
can then be converted to port-based admittance or impedance matrix. This allows
the modeling of a waveguide structure that can support multiple electromagnetic
modes by a circuit with defined coupling between the modes. Since port-based
representations are not suited for most circuit simulation tools, a circuit theory
based on the local reference node concept, is developed. The theory adapts modified
nodal analysis to accommodate spatially distributed circuits allowing conventional
harmonic balance and transient simulators to be used.
To show the flexibility of the modeling technique, results are obtained
for general shielded microwave and millimeter-wave structures as well as various
spatial power combining systems.
Biographical Summary
Ahmed Ibrahim Khalil was born in Cairo, Egypt, on November 15, 1969.
He received the B.S. (with honors) and M.S. degrees from Cairo University, Giza,
Egypt, both in electronics and communications engineering, in 1992 and 1996, re-
spectively. From 1992 to 1996 he worked at Cairo University as a Research and
Teaching Assistant. While working towards his Ph.D. degree in electrical engi-
neering at North Carolina State University, since 1996, he held a Research Assis-
tantship with the Electronics Research Laboratory in the Department of Electrical
and Computer Engineering. Interests include numerical modeling of microwave and
millimeter-wave passive and active circuits, MMIC design, quasi-optical power com-
bining, and waveguide discontinuities. He is a student member of the Institute of
Electrical and Electronic Engineers and the honor society Phi Kappa Phi.
ii
Acknowledgments
This dissertation would have never been finished without the will and
blessing of God, the most gracious, the most merciful. AL HAMDU LELLAH.
I would like to express my gratitude to my advisor Dr. Michael Steer for
his support and guidance during my graduate studies. I would also like to express
my sincere appreciation to Dr. James Mink, Dr. Frank Kauffman, and Dr. Pierre
Gremaud for showing an interest in my research and serving on my Ph.D. committee
and to Dr. Amir Mortazawi for helping me with the measurements and many useful
discussions.
A very big thanks go to my colleagues, Mr. Mostafa N. Abdulla for many
useful suggestions regarding my work, Mr. Mete Ozkar for working with me on the
excitation horn, Mr. Carlos E. Christoffersen for his computer skills which came
in handy many times, Dr. Todd W. Nuteson for his encouragement while starting
my PhD. degree, Mr. Satoshi Nakazawa for sharing the same cubical, Dr. Hector
Gutierrez for many useful advice, Mr. Usman Mughal, Mr. Rizwan Bashirullah,
Mr. Adam Martin, Mr. Chris W. Hicks, and Dr. Huan-sheng Hwang.
Also, I would like to thank my professors and colleagues at Cairo Uni-
versity, Egypt, for the part they played in my academic career. They are truly
outstanding.
And finally, I wish to thank my wife and two sons Omar and Ali for
their support, understanding and encouragement and my parents whom without
their total love, guidance, and dedication I would not have made it this far.
iii
Contents
List of Figures viii
1 Introduction 1
1.1 Motivation For and Objective of This Study . . . . . . . . . . . . . . 1
Figure 6.13: Calculated input impedance for centered and off-centered positions.
on its input impedance, the antenna is placed at X1 = 3 mm away from the vertical
waveguide wall. Considerable variation in the input impedance is observed in Fig.
6.13 due to the close proximity to the waveguide wall.
6.2.5 Shielded microstrip filter
The GSM-MoM method can also be applied to completely shielded microwave and
millimeter wave structures. Numerical results have been obtained for the specific
example of the shielded microstrip filter shown in Fig. 6.14. The filter is contained
in a box of dimensions 92× 92× 11.4 mm (a× b× c). The substrate height is 1.57
mm and it has a relative permitivity of 2.33.
In analysis, the structure is decomposed into three layers as shown in
CHAPTER 6. RESULTS 99
Fig. 6.15, with layers 1 and 3 being the top and bottom covers, respectively. The
covers are perfect conductors and hence their GSMs are diagonal matrices with −1
as diagonal elements. Layer 2 is a metal layer with ports.
Port 1
Port 2
23 mm
92 m
m
92 mm
18.4
mm
4.6 mm
4.6 mm
Figure 6.14: Geometry of a microstrip stub filter showing the triangular basis func-
tions used. Shaded basis indicate port locations.
The excitation ports are modeled by the delta-gap voltage model pro-
posed by Eleftheriades and Mosig [88] (the current basis functions for the excitation
ports are shaded in Fig. 6.14). This serves two purposes, to ensure the current
continuity at the edges and to allow the direct computation of network parameters
without the need to extend the line beyond its physical length. It should be noted
that these half-basis functions can only be used, for direct port computation as de-
scribed in [88], at the microstrip-wall intersection. The equivalent circuit model of
the port representation using half basis function is shown in Fig. 6.16. The voltage
source V is the delta gap voltage source accompanying the half basis function.
CHAPTER 6. RESULTS 100
TOP COVER
METAL LAYER
BOTTOM COVER
Figure 6.15: Three dimensional view illustrating the layers of the stub filter.
-V+
Figure 6.16: Port definition using half basis functions.
CHAPTER 6. RESULTS 101
The GSM of layer 2 is computed using the method described in Chapters
3 and 4. The number of modes considered in the GSM for layers 1 and 3 is 287. Layer
2 has 289 ports, 287 modes and 2 circuit ports. After cascading the three layers the
modes are augmented. The final scattering matrix has rank two representing the
circuit ports of the filter. This is illustrated in Fig. 6.17.
SHORTCIRCUIT
WAVEGUIDE SECTION
..
. ... WAVEGUIDE
SECTION... ..
. SHORTCIRCUIT
V1 V2+ - + -
MICROSTRIP
V1 V2
+
-
+
-FILTERSHIELDED
CASCADING
Figure 6.17: Block diagram for the GSM-MoM analysis of shielded stub filter.
The reflection and transmission coefficients S11 and S21 are calculated
in Figs. 6.18 and 6.19, respectively. The transmission from port 1 to port 2 is
approximately −37 dB at 2.7 GHz and compares favorably with previously reported
results [88].
To explain the box effect appearing in the reflection and transmission
coefficients, a plot of the propagation constant diagram is shown in Fig. 6.20. The
CHAPTER 6. RESULTS 102
solid and dashed curves represent the air filled and the dielectric substrate regions,
respectively. It is observed that the notches in the S11 and S21 curves (at 2.2 and
3.4 GHz) correspond to the cut off frequencies of certain modes in the dielectric
substrate.11
S
1.5 2 2.5 3 3.5 4
Frequency (GHz)
(dB
)
-10
-12
-8
-6
-4
-2
0
1
Figure 6.18: Scattering parameter S11: solid line GSM-MoM, dotted line from [88].
Convergence curves for the scattering parameters are shown for various
numbers of modes in Figs. 6.21 and 6.22. As desired, convergence to a result
is asymptotically approached as the number of modes considered increases. The
need for a large number of modes is in intuitive agreement since dimensions are
small compared to the guide wavelength and so evanescent mode coupling should
dominate. This example represents an extreme test of the method developed here
and it also verifies the calculation of the GSM with circuit ports technique.
CHAPTER 6. RESULTS 103
21(d
B)
S
1.5 2 2.5 3 3.5 4Frequency (GHz)
0
-5
-10
-15
-20
-25
-30
-35
-40
-451
Figure 6.19: Scattering parameter S21: solid line GSM-MoM; dotted line from [88].
1 1.5 2 2.5 3 3.5 40
0.5
1
1.5
FREQUENCY (GHz)
RO
PA
GA
TIO
N C
ON
ST
AN
T (
Bet
a)
Figure 6.20: Propagation constant: solid lines for air, dashed lines for dielectric.
CHAPTER 6. RESULTS 104
71 modes
127 modes
(dB
)11
199 modes
S
1.5 2 2.5 3 3.5 4-14
-12
-10
-8
-6
-4
-2
0
Frequency (GHz)1
Figure 6.21: Various cascading modes showing convergence of S11.
21(d
B)
S
1 1.5 2 2.5 3 3.5 4
-5
-10
-15
-20
-25
-30
-35
-40
-45
0
Frequency (GHz)
199 modes
127 modes
71 modes
Figure 6.22: Various cascading modes showing convergence of S21.
CHAPTER 6. RESULTS 105
6.3 Patch-Slot-Patch Array
In this section a spatial power combiner structure is simulated and measured. The
system shown in Fig. 6.23 is divided into three blocks, transmitting horn, receiving
horn and a double layer array. Each block is simulated by a separate EM routine.
To reduce the coupling between the receiving and the transmitting patch antennas,
a strip-slot-strip transition is designed to couple energy from one patch to the other.
The patch antenna used is shown in Fig. 6.24 along with the amplifying unit.
Figure 6.23: A patch-slot-patch waveguide-based spatial power combiner.
6.3.1 Array simulation
The double layer array consists of three interfaces (patch-slot-patch). Each interface
is modeled separately using the Generalized Scattering Matrix-Method of Moment
technique. The method first calculates the MoM impedance matrix for an interface
from which a GSM matrix is calculated directly without the intermediate step of
current calculation. This enables the modeling of arbitrary shaped structures and
CHAPTER 6. RESULTS 106
18
4632.5
75130
99
66
44
5
AMPLIFIER
Figure 6.24: Geometry of the patch-slot-patch unit cell, all dimensions are in mils.
the calculation of large number of modes needed in the cascade to obtain the required
accuracy. The nonuniform meshing scheme described in Chapter 4 is used here to
reduce the number of basis functions required in case of the uniform meshing scheme.
6.3.2 Horn simulation
The GSMs for the transmitting and receiving horns are calculated using the mode
matching technique [18]. The mode matching technique is known to be an efficient
method for calculating the GSM of horn antennas. For horns used in this study
the length of the Ka to X band waveguide transition is 16.51 cm (Fig. 6.25). This
long transition is to insure minimum higher order mode excitations. The GSM of
the waveguide transition is obtained using the mode matching technique program
described in [18]. The two important parameters in using the program are the num-
CHAPTER 6. RESULTS 107
ber of steps and the number of modes considered. The number of sections needed
depends on the flaring angle of the transition and on the frequency of operation [19].
In choosing the step size, the λ/32 criteria can be used.
Y
XY
X
2
1
1
2
16.51 cm
Ka-bandWaveguide
X-bandWaveguide
Figure 6.25: Ka band to X band transition.
CHAPTER 6. RESULTS 108
A typical double step plane junction section is shown in Fig. 6.25. The
smaller waveguide dimensions are X1 and Y1, and the larger waveguide dimensions
are X2 and Y2. At the double plane step discontinuity, incident and reflected waves
for all modes (evanescent and propagating) are excited, thus the total field can be
expressed as a superposition of an infinite number of modes. The total power in
all modes on both sides of the junction is matched according to the mode matching
technique. The GSM for the whole waveguide transition is obtained by cascading
the GSMs for all sections.
6.3.3 Numerical results
Numerical results are obtained for two cases a single cell and a 2×2 array. In the first
example a single unit cell (Fig. 6.24) is centered in an X-band waveguide. The circuit
was fabricated on a 0.381-mm-thick Duroid substrate with relative permitivity ε =
6.15. The GSM, for each layer, is calculated for 512 modes. The horns are simulated
using 80 modes. The calculated magnitude and phase of the transmission coefficient
S21 for the dominant TE10 mode is shown in Figs. 6.26 and 6.27, respectively. It is
shown that a transmission of approximately −13 dB is achieved at 32.25 GHz.
The second example is a two by two patch array. The same number
of modes is considered as in the first example. The results for the transmission
coefficient S21 is shown in Fig. 6.29. The maximum transmission obtained is −6
dB at 32.5 GHz and agrees well with our measured results. Again, this example is
for an overmoded waveguide, where many modes can be excited (approximately 18
modes in the air-filled sections of the X-band waveguide).
CHAPTER 6. RESULTS 109
31.5 32 32.5 33 33.5 34−23
−22
−21
−20
−19
−18
−17
−16
−15
−14
−13
S21
(dB
)
FREQUENCY (GHz)
Figure 6.26: Magnitude of transmission coefficient S21.
31.5 32 32.5 33 33.5 34−200
−150
−100
−50
0
50
100
150
200
FREQUENCY (GHz)
PH
AS
E S
21 (
degr
ees)
Figure 6.27: Angle of transmission coefficient S21.
CHAPTER 6. RESULTS 110
Figure 6.28: A two by two patch-slot-patch array in metal waveguide.
31.5 32 32.5 33 33.5 34−14
−13
−12
−11
−10
−9
−8
−7
−6
−5
FREQUENCY GHz
S21
dB
measured simulated
Figure 6.29: Magnitude of transmission coefficient S21.
CHAPTER 6. RESULTS 111
6.4 CPW Array
Three examples, based on the magnetic current interface, are numerically investi-
gated in this section. The first two examples concentrate on the effect of waveguide
walls on the antenna input impedance. Two antennas are proposed, a folded slot
and a five slot antenna. The third example is a 3× 3 slot antenna array for the use
in spatial waveguide power combiners. The self and mutual impedances of the array
elements are calculated.
6.4.1 Folded slot antenna
The input impedance of a folded slot antenna shielded in a waveguide, shown in Fig.
6.30, is calculated. The dimensions of the waveguide are a = b = 20 cm and the
antenna dimensions are c = 7.8 cm and d = 0.9 cm. The dielectric constant is 2.2
and of thickness 0.0813 cm. The structure is decomposed into two layers. These are
a magnetic layer with ports, and a dielectric interface. The GSM of the magnetic
layer with ports is calculated using the GSM-MoM method and cascaded with the
dielectric interface to give the composite GSM.
The results are compared with an algorithm utilizing only MoM calcu-
lation by using the Green’s function for the composite structure (slot backed by
dielectric slab). In implementation of the MoM scheme piecewise testing and basis
functions are used [86]. The port impedance in this case is claculated directly from
the MoM impedance matrix.
The real and imaginary parts of the input impedance at the center of the
CHAPTER 6. RESULTS 112
folded slot are shown in Figs. 6.31 and 6.32, respectively. The antenna resonates
at 1.5 GHz and has input resistance of 365 Ω at the resonant frequency. It can be
seen that the GSM-MoM solution agrees favorably with the MoM port calculation.
This verifies the cascading of the GSM for a magnetic layer with ports.
The high input resistance at resonance makes the design for a 50 Ω
match more challenging. For this reason York et al. suggested the use of multiple
slot antenna configurations for spatial power combining applications. The following
example is a demonstration of this idea.
2
2
X
Y
a
b
b
a
c
d
Figure 6.30: Geometry of the folded slot in a waveguide.
CHAPTER 6. RESULTS 113
1 1.5 2 2.50
50
100
150
200
250
300
350
400
Frequency (GHz)
Rea
l−pa
rt (
Ohm
s)
GSM−MoM MoM
Figure 6.31: Real part of the input impedance for folded slot.
1 1.5 2 2.5−200
−150
−100
−50
0
50
100
150
200
250
Frequency (GHz)
Imag
inar
y−pa
rt (
Ohm
s)
GSM−MoM MoM
Figure 6.32: Imaginary part of the input impedance for folded slot.
CHAPTER 6. RESULTS 114
6.4.2 Five slot antenna
In an effort to design a CPW antenna system matched to 50 Ω, York [4] suggested
a five slot configuration shown in Fig. 6.33. Since the input impedance is inversely
proportional to the square of the number of turns, increasing the number of slots
will automatically reduce the input impedance. Free space measurements [4] show
that an input return loss of −28 dB is observed at 10.5 GHz for the 5-slot antenna
shown in Fig. 6.33.
7.2 mm
2.7
mm 0.3mm
Figure 6.33: Five-slot antenna [4].
The GSM-MoM technique is used to calculate the input impedance of
the five-slot antenna inside a square waveguide. This gives an insight on the change
of the input impedance of the antenna when operating inside shielded environment.
In analysis, the CPW cell is composed of two layers. A magnetic layer
with ports and a dielectric interface. The scattering parameters for the magnetic
CHAPTER 6. RESULTS 115
8 8.5 9 9.5 10 10.5 11 11.5 12−9
−8
−7
−6
−5
−4
−3
−2
−1
0
FREQUENCY (GHz)
S11
(dB
)
Figure 6.34: Magnitude of input return loss for 5 folded slots.
8 8.5 9 9.5 10 10.5 11 11.5 12−200
−150
−100
−50
0
50
100
150
200
FREQUENCY (GHz)
S11
(de
gree
s)
Figure 6.35: Phase of input return loss for 5 folded slots.
CHAPTER 6. RESULTS 116
layer are computed as described in Chapter 3. The dielectric interface has a diagonal
scattering submatrices representing the transmission and reflection coefficients.
The antenna is placed in the center of a square waveguide of dimensions
22.86 × 22.86 mm. The geometry and dimensions of the antenna are shown in
Fig. 6.33. The dielectric thickness is 0.635 mm and its dielectric constant εr is 9.8.
The simulated input returned loss and its phase are shown in Figs. 6.34 and 6.35,
respectively. The input return loss has in fact increased from −28 in free space to
−8 dB when placed in the waveguide. This might result in less achievable gain when
using matched MMIC devices (to 50 Ω).
6.4.3 Slot antenna array
A 3 × 3 slot antenna array fed by CPW transmission lines is shown in Fig. 6.36.
The array consists of nine unit cells. Each unit cell is composed of two orthogonal
slot antennas, one for receiving and the other for transmitting. The amplifying
unit is a single ended amplifier. To properly design the amplifiers, it is essential to
calculate the driving point impedances of each antenna. This impedance depends
on self as well as mutual coupling between the antennas. The array is placed in a
square waveguide (a = b = 4 cm) and the antenna length is 0.72 cm.
The self impedance matrix (18 × 18) is calculated for the slot array for
the frequency range 8–12 GHz. The real and imaginary parts of the self impedances
of the antenna elements 1, 2, and 5 are shown in Figs. 6.37 and 6.38, respectively.
Resonance is achieved at 9.25 GHz for the self impedances. The impedance at
resonance is very high (1700 Ω) which make it difficult to match to 50 Ω. The value
CHAPTER 6. RESULTS 117
of the self impedances are much less away from resonance as shown in Figs 6.39 and
6.40. Operating at 10 GHz is more appealing than operating at resonance since it is
easier to compensate for the imaginary part while designing the amplifier matching
circuit.
When designing an amplifier, the feedback from the output to the input
is very critical. Positive feedback might result in amplifier oscillations. The mutual
coupling between the input and output antennas provides that feedback path and
so it is essential to account for that kind of coupling. The mutual impedance for
the center unit cell is shown in Fig. 6.41. Ideally the coupling should be zero. To
minimize the coupling, the antennas should be at right angles.
1 2 3
4 5 6
7 8 9
10
11
12
13
14
15
16
17
18
Figure 6.36: A 3 × 3 slot antenna array shielded by rectangular waveguide.
CHAPTER 6. RESULTS 118
8 8.5 9 9.5 10 10.5 11 11.5 120
200
400
600
800
1000
1200
1400
1600
1800
Frequency (GHz)
Rea
l−pa
rt (
Ohm
s)
Z11
Z22
Z55
Figure 6.37: Real part of self impedances.
8 8.5 9 9.5 10 10.5 11 11.5 12−1500
−1000
−500
0
500
1000
1500
Frequency (GHz)
Imag
inar
y−pa
rt (
Ohm
s)
Z11
Z22
Z55
Figure 6.38: Imaginary part of self impedances.
CHAPTER 6. RESULTS 119
9.5 10 10.5 11 11.5 120
50
100
150
200
250
300
350
400
Frequency (GHz)
Rea
l−pa
rt (
Ohm
s)
Z11
Z22
Z55
Figure 6.39: Real part of self impedances.
9.5 10 10.5 11 11.5 12−900
−800
−700
−600
−500
−400
−300
−200
−100
0
Frequency (GHz)
Imag
inar
y−pa
rt (
Ohm
s)
Z11
Z22
Z55
Figure 6.40: Imaginary part of self impedances.
CHAPTER 6. RESULTS 120
8 8.5 9 9.5 10 10.5 11 11.5 12−10
0
10
20
30
40
50
60
70
Frequency (GHz)
Mut
ual I
mpe
danc
e (O
hms)
Real Z5,14
Imaginary Z
5,14
Figure 6.41: Real and imaginary parts for the mutual impedance Z5,14.
6.5 Grid Array
Perhaps most of the early design efforts for spatial power combiners have been ori-
ented towards grid structures. The grid array systems are easy to build and fabricate.
Analysis and design techniques have emerged specifically for these structures, all for
free space case. In this section we will investigate grid arrays when constructed in
a shielded environment.
A 3 × 3 grid array is shown in Fig. 6.42. The array is composed of
nine unit cells. Each unit cell consists of two perpendicular dipole antennas, one
for receiving and the other for transmitting. The grid structure uses a differential
pair amplifying unit as that shown in Fig. 1.4. To accurately design the differential
pair, the driving point impedance of the antennas must be accurately calculated.
CHAPTER 6. RESULTS 121
In this example, the impedance of the center cell is calculated. The
magnitude and phase of the input return loss are plotted in Figs. 6.43 and 6.44,
respectively. Resonance is achieved at approximately 31 GHz with −15.7dB return
loss.
Port
0.7 cm
1 cm
1 cm
Figure 6.42: A grid array inside a metal waveguide.
CHAPTER 6. RESULTS 122
27 28 29 30 31 32 33 34−16
−14
−12
−10
−8
−6
−4
−2
0
FREQUENCY (GHz)
S11
(dB
)
Figure 6.43: Magnitude of input return loss.
27 28 29 30 31 32 33 34−200
−150
−100
−50
0
50
100
150
200
FREQUENCY (GHz)
Pha
se S
11
Figure 6.44: Angle of input return loss.
CHAPTER 6. RESULTS 123
6.6 Cavity Oscillator
A multiple device oscillator using dipole arrays was proposed in [89, 90]. In both
referenced papers, a dedicated Green’s function was developed to model the cavity
oscillator and predict the coupling effects. In this section we will analyze a cavity
oscillator of the type described in [90] and shown in Fig 6.45.
b
c
a
B
A
Dipole Array
Figure 6.45: Geometry of a dipole array cavity oscillator.
6.6.1 Single dipole
The first example is a single dipole antenna inside a cavity. The cavity dimensions
are 22.6×10.2×5.0 mm (a×b×c) and the patch is centered in the transverse plane.
The dipole length and width are 6 mm and 1 mm, respectively. The frequency of
operation is chosen to be from 30 to 33.5 GHz. This means that the X band
waveguide is overmoded. The calculated input impedance of the dipole is shown
in Fig. 6.46. The dipole goes through resonance at 32.25 GHz. Below resonance
it is capacitive and above resonance it becomes inductive. A negative resistance
diode can be placed at the center of the dipole antenna by properly choosing the
CHAPTER 6. RESULTS 124
30 30.5 31 31.5 32 32.5 33 33.5−800
−600
−400
−200
0
200
400
600
800
Frequency (GHz)
Inpu
t Im
peda
nce
(Ohm
s)
Real−Part Imaginary−Part
Figure 6.46: Input impedance of a dipole antenna inside a cavity.
resistive part. Also, since the diode is usually capacitive in nature, an inductive
input impedance might be chosen for the dipole antenna.
In the analysis, the structure is decompsed into three layers. These are
short-circuit, electric current interface with ports (dipole), and magnetic current
interface (patch). A block diagram illustrating the modeling process using the GSM-
MoM technique is shown in Fig. 6.47. After cascading all GSMs, the composit GSM
with ports will describe the relationship between the device ports and the output
modes. The composit GSM can be represented in terms of a scattering matrix,
impedance matrix, or an admittance matrix. Any of these forms may be employed
in nonlinear analysis using a nonlinear frequency-domain circuit simulator.
It is interesting to note that if a multilayer array (more than one trans-
verse active dipole array) of the same structure is used, the modeling scheme will
CHAPTER 6. RESULTS 125
only require the analysis of one of these arrays. The analysis would then proceed
with cascading all sections to obtain the composite GSM. This is in comparison with
the direct MoM technique, where the coupling between all arrays must be accounted
for numerically. Hence the number of elements and the size of the MoM matrix are
increased.
SHORT
CIRCUIT
WAVEGUIDE
SECTION
.
.
.
.
.
.
.
.
. DIPOLE
WAVEGUIDE
SECTION
.
.
.PATCH
.
.
.
+ -V
MODES
(CIRCUIT-PORT)
.
.
.
+ -V
CASCADING
MODES
(CIRCUIT-PORT)
COMPOSITEGSM WITH
PORTS
Figure 6.47: Block diagram for the GSM-MoM analysis of cavity oscillator.
CHAPTER 6. RESULTS 126
6.6.2 A 3× 1 dipole antenna array
As a second example a 3 × 1 dipole antenna array is placed inside a similar cavity
of the one described in the previous example. The antennas are shown in Fig.
6.48. The mutual and self scattering coefficients are calculated when all antennas
are of same lengths and width (6 × 1 mm) and the separations X1 = X2. Due to
symmetry there are only four distinct scattering coefficients (S11, S12, S13, and S22).
The magnitude and phase of the self and mutual scattering coefficients are shown
in Figs. 6.49 and 6.50, respectively. It is observed that the scattering coefficient
S22 has changed considerably, from the previouse example, due to coupling to the
other two antennas. This is illustrated by the nonresonant behaviour of S22 which is
now purely capacitive within the frequency range. The port scattering coefficients
calculated in this example is essential for designing an active array oscillator.
V1 V2 V3
X X1 2
X
Y
a
b
1 2 3
Figure 6.48: Dipole antenna array in a cavity.
CHAPTER 6. RESULTS 127
30 30.5 31 31.5 32 32.5 33 33.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
frequency (GHz)
Mag
nitu
de o
f Sca
tterin
g C
oeffi
cien
ts
S11
S12
S13
S22
Figure 6.49: Magnitude of scattering coefficients for a dipole antenna array inside a
cavity.
30 30.5 31 31.5 32 32.5 33 33.5
−150
−100
−50
0
50
100
150
frequency (GHz)
Pha
se o
f Sca
tterin
g C
oeffi
cien
ts (
degr
ees)
S11
S12
S13
S22
Figure 6.50: Phase of scattering coefficients for a dipole antenna array inside a
cavity.
Chapter 7
Conclusions and Future Research
7.1 Conclusions
A generalized scattering matrix technique is developed based on a method of mo-
ments formulation to model multilayer structures with circuit ports. Four general
building blocks are considered. These are electric interface with ports, magnetic in-
terface with ports, dielectric interface, and perfect conductor. With these blocks the
method is applicable to almost all shielded transverse active or passive structures.
The GSM for each block is derived separately. The method explicitly incorporates
device ports and circuit ports in the formulation of both the electric and magnetic
current interfaces. The scattering parameters are derived for all modes in a single
step without the need to calculate the current distribution as an intermediate step.
Two cascading formulas are presented to calculate the composite scat-
tering matrix of a multilayer structure. This matrix is a complete description of
128
CHAPTER 7. CONCLUSIONS AND FUTURE RESEARCH 129
the structure. The technique can be applied to general structures as well as to
waveguide-based spatial power combiners. Various general type structures are sim-
ulated. These are a wide strip in waveguide, patch array on a dielectric slab, a
strip-slot transition module, shielded dipole antenna, and a shielded microstrip stub
filter. Spatial power combiners such as patch-slot-patch array, CPW array, grid
array, and cavity oscillator array are also simulated. Results are verified by either
comparisons to measurements or to other numerical techniques.
The interaction of layers is handled using a GSM method where an evolv-
ing composite GSM matrix must be stored to which only the GSM of one layer at a
time is evaluated and then cascaded. Thus the computation increases approximately
linearly as the number of layers increases. Memory requirements are determined by
the number of modes and so is independent of the number of layers. The result-
ing composite matrix can be reduced in rank to the number of circuit ports to be
interfaced to a circuit simulator.
An acceleration procedure based on the Kummer transformation is im-
plemented to speed up the MoM matrix elements. The quasistatic terms are ex-
tracted and evaluated using fast decaying modified Bessel functions of the second
kind. The convergence as well as the accuracy of the acceleration scheme are demon-
strated. In implementation two discretization schemes are used, uniform and nonuni-
form. Although simple, uniform discretization can not accurately represent struc-
tures with high aspect ratios nor can it capture fine geometrical details without
a gross increase in the number of elements. With the nonuniform scheme, finer
resolutions can be obtained for selective areas as desired.
The port parameters obtained from the electromagnetic simulator are
CHAPTER 7. CONCLUSIONS AND FUTURE RESEARCH 130
converted to nodal parameters using the localized reference node concept described
in Chapter 5. A scheme for the augmentation of a nodal admittance matrix by
a port-based matrix with a number of local reference nodes permits field derived
models to be incorporated in a general purpose circuit simulator based on nodal
formulation. The method is immediately applicable to modified nodal admittance
(MNA) analysis as the additional rows and columns of the MNA matrix are unaf-
fected by the augmentation.
7.2 Future Research
There are still many new ideas to be explored in the modeling of waveguide based
spatial power combiners. One feature that can be added to the current program
is the implementation of nonuniform triangular basis functions. This will enable
the modeling of geometrical curves and bends with much better accuracy. Another
feature is to include the losses due to dielectrics and metal portions.
It is well known that as the separation between layers decreases, in terms
of guide wavelength, the number of modes required in the GSM representation
will increase to achieve the required accuracy. This might render the procedure
impractical for very small separations (less than 0.01 λg). In this case it is more
efficient to construct a separate analysis module based on the MoM that takes into
account both layers in the Green’s function. Then a GSM is constructed for the
closely spaced layer using the calculated MoM matrix. We adopted this methodology
and implemented it for strip and slot layers [86]. There are other combinations to
be considered such as strip and strip, slot and slot layers, and even three layer
CHAPTER 7. CONCLUSIONS AND FUTURE RESEARCH 131
combinations.
Different types of Green’s functions such as the potential Green’s func-
tions and the complex images can be used instead of the electric- and magnetic-
type Green’s functions used here. This may reduce the CPU time, eventhough an
acceleration procedure might still be necessary.
Diakoptics in conjunction with the GSM-MoM scheme is another area
to be investigated. If the structure can be decomposed in the transverse plane into
separate structures related by a matrix then a three dimensional segmentation is
achieved (GSM and Diakoptics).
Furthermore, when the waveguide dimensions are several wavelengths,
then the number of modes involved in the modal expansion of the electromagnetic
fields becomes very large and approaches the free space case. It would be interesting
to see when the free space solution approaches the waveguide solution and if a hybrid
analysis can be employed.
In terms of applications to spatial power combining design, the structures
to be modeled are endless. Many novel designs can be thought of and perhaps
achieve the desired power combining efficiencies.
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Appendix A
Usage of GSM-MoM Code
Two steps are involved to run GSM-MoM:
• Converting layout CIF into input geometry file
• Running GSM-MoM with input parameter file
To convert the CIF file into the geometry file standard format a program calledyomoma is used. The command is
yomoma ’file.cif’ a.d
This will cause yomoma to convert the ’file.cif’ into a geometry file and give it thename ’geometry’. The input parameter file contains the necessary information torun GSM-MoM. These information are:
• Frequency range (start-stop-number of points)
• Waveguide dimensions
• Number and type of layers
• Separation between Layers
• Dielectric constants
140
APPENDIX A. USAGE OF GSM-MOM CODE 141
• Input geometry files
• Number of cascading modes
• Number of modes involved in the MoM matrix element calculation
• Output file names
A.1 Example
In this section a sample run of GSM-MoM is illustrated. The input file, geometryfile, and output file are listed bellow.
A.1.1 Input file
The input file used in the simulation is described as follows:
”FREQUENCY:””———————””Start at Frequency:” 1.d9”Stop at Frequency:” 2.5d9”Number of Frequency Points:” 31”GREEN:””————””m max:” 450”n max:” 450” ””WAVEGUIDE””——————-””a:(x direction):” 20.d0”b:(y direction):” 20.d0”xmax:(maximum x-dimensions)” 2.5d-1”number of units in maximum x-dimension:” 1” ””LAYERS:””————-””Number of Layers:” 2”Type of Layer 1:” 2 0
Where X-center and Y-center are the center coordinates for the basisfunctions, ci and di are the x and y dimensions of the basis function described inChapter 4, direction is either 1 or 2 representing either x or y direction.
A.1.3 Output file
A sample of the output file ports-s.dat is shown bellow
The program consists of the following subroutines:
• scatter main.f: This subroutine is the main program. It reads in the geometryfiles and the input data file and calls all other programs.
• mom layer.f: This is the MoM calculation subroutine. It calls the approperiatefunctions to calculate the MoM impedance and admittance matrix elements.
• empty guide.f: Contains functions used for the MoM matrix element calcula-tions.
• matrix.f: Contains the math routine for matrix inversion.
• scatter layer.f: calculates scattering parameters for each layer.