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LlBRARY
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This is to certify that the
dissertation entitled

A TROUGH LEAKY-WAVE ANTENNA

presented by

CARLOS ALBERTO JARAMILLO

has been accepted towards fulfillment
of the requirements for the

PhD degree in Electrical Engineering

 

 

Major Professor’s Signature

W¢ 2.00?

Date

 

 

MSU is an Affirmative Action/Equal Opportunity Employer

 

 

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE

DATE DUE

DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5/08 KzlProleccaiPres/CIRCIDateDue.indd

7——___

A Trough Leaky-Wave Antenna
By

Carlos Alberto J aramillo

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Electrical Engineering

2009

ABSTRACT

A Trough Leaky—Wave Antenna
By

Carlos Alberto Jaramillo

Leaky-wave antennas have been an on—going research interest of US. government,
academic and industrial research groups. One significant feature of these antennas is
that the voltage standing wave ratio (VSWR) bandwidth can be greater than that of
resonant antennas. This is principally due to the fact that the radiation mechanism
is attributed to a traveling-wave as compared to the standing-wave that is responsible
for radiation in resonant antennas. Another important aspect of leaky-wave antennas
is that the radiation pattern bandwidth is regrettably somewhat narrow since the
main-lobe direction varies with operational frequency. This dissertation deals with
the effective design of trough leaky—wave (TLW) antennas. A TLW antenna is an
electrically narrow trough in a ground plane excited using a coaxial probe feed and
terminated in a lumped load. In this antenna, a traveling wave propagates along
the aperture with a complex propagation constant which can be computed using the
transverse resonance method (TRM) assuming an infinitely long antenna. An alter—
native approach, that considers the aperture length, is the finite element boundary
integral (FE—BI) method. In this work, both formulations are developed and im-
plemented numerically not only to compute the propagation constants but also the
antenna impedance and radiation pattern. The major contribution of this work lies

in the balun used to feed in concert with the load to terminate the antenna.

To My Parents and Wife

iii

ACKNOWLEDGMENTS

First of all, I would like to thank God for His favor and grace.

During my years at MSU I had the opportunity to meet excellent people that help
me to accomplish one of my dreams in my life, a Ph.D in Electrical Engineering.

I would like to express my gratitude to my advisor Dr. Leo C. Kempel and
the members of my Ph.D. committee, Dr. B. Shanker, Dr. Edward Rothwell and
Dr. Gouwei Wei. Thankyou for participating in this committee and providing their
guidance and support when it was needed. I would also like to acknowledge Dr.
Stephen W. Scheneider of the Air Force Research Laboratory for motivating this
research and collaboration with Dr. Kempel and myself.

Special thanks to the people which through their finnacial assitance and support
made it possible for me to complete this work. Thanks to Dr. Barbara O’Kelly and
Dr. Percy Pierre for their confidence during my years at MSU. I would like to thank
to Kathleen Kreh and the former graduate secretary Sheryl Hulet for their devotion
to their work

I also want to express my gratitude to my fellow graduate students that in a way
or another help me over the years at MSU.

A very special thanks to my friends in Puerto Rico Manuel and Carmen Garcia,
they were always present during this journey.

Finally, I would like to thank to my Parents Oliva and Pablo, my sisters Martha

and Patricia, my brother Pablo and my lovely wife Yanira for their love and sacrifice.

iv

TABLE OF CONTENTS

LIST OF TABLES .................................

LIST OF FIGURES ................................

KEY TO SYMBOLS AND ABBREVIATIONS .................

CHAPTER 1

Introduction .....................................
1.1 Leaky wave antennas background ....................
1.2 Surface wave and leaky waves ......................
1.3 Operation modes of leaky-wave antennas ................
1.4 Radiation characteristics of a TLW antenna ..............

CHAPTER 2

Analitical solution for the TLW antenna .....................
2.1 Transverse resonance method ......................
2.2 TLW antenna driving point impedance .................

CHAPTER 3

Numerical solution for the TLW antenna .....................
Finite element boundary integral method ................
Finite element formulation ........................
Boundary integral formulation ......................
Antenna feed and load ..........................
FE matrix entries using tetrahedrons ..................
BI matrix entries using triangles .....................

3.1
3.2
3.3
3.4
3.5
3.6

3.7

3.6.1
3.6.2

Numerical integration of BI equations ..............
BI self term evaluation ......................

FE—BI program structure .........................

3.7.1
3.7.2
3.7.3
3.7.4
3.7.5

3.7.6

Pre—processing ...........................
Meshing ..............................
Pre-assembly of F E-BI matrices for the TLW antenna .....
Matrix assembly .........................
Matrix solution of F E—BI linear system using biconjugate gra-
dient (BiCG) method with preconditioning ...........
Post-processing ..........................
3.7.6.1 Input Impedance, reflection coefficient, and VSWR .
3.7.6.2 Radiation pattern ...................
3.7.6.3 Propagation constants .................

vii

H
Awoowh-

H

22
22
33

42
42
43
48
64
65
74
82
84
88
88
89
91
92

92
93
94
94
98

CHAPTER 4

Antenna design and results ............................ 101
4.1 Design of the RT/Duroid 5880—filled TLW antenna .......... 101
4.2 Code validation .............................. 106
4.3 Air-filled TLW antenna .......................... 111

CHAPTER 5

Conclusions and future work ............................ 135

APPENDIX A

Review of some complex variable functions .................... 138
Al The exponential function 11) (z) = ez ................... 138
A2 The square root function w (z) = fl .................. 139

APPENDIX B

Thin-substrate approximation for microstrip patch antennas .......... 142

APPENDIX C

Proof of some properties of RWG basis function ................. 145
CI Proof of expression (3.114) ........................ 145
C2 RWG basis function are free of charge accumulation over its support . 148

APPENDIX D

Analytical computation of integrals with singularities .............. 151
DJ Integrals with uniform source distribution ............... 151
D2 Integrals with linearly varying source distribution ........... 153

BIBLIOGRAPHY ................................. 156

vi

Table 3.1

Table 3.2

Table 4.1

Table 4.2

Table 4.3

Table 4.4

Table 4.5

LIST OF TABLES

Tetrahedron edge—node local conection ................ 73
Triangular coordinates and integration weights [38] ......... 85

RT/Duroid 5880-filled T LVV antenna banwidth for three different

values of h. .............................. 101
RT/Duroid 5880-filled TLW antenna banwidth for three different

values of w. .............................. 104
RT/Duroid 5880-filled TLW antenna design parameters. ..... 104
Air-filled TLW antenna design parameters .............. 120
Air-filled TLW antenna circular termination design parameters. . 123

vii

Figure 1.1

Figure 1.2
Figure 1.3
Figure 1.4
Figure 1.5

Figure 1.6

Figure 1.7

Figure 1.8

Figure 2.1

Figure 2.2

Figure 2.3
Figure 2.4

Figure 2.5

Figure 2.6

Figure 2.7

Figure 2.8

Figure 2.9

LIST OF FIGURES

(a) Geometrical description of the TLW antenna (b) Feeding tech—
nique. .................................

Electric field for the E H 10 mode ...................
Illustration of HWLVV antenna. ...................
Field structure for a TLW antenna ..................
Leakage from a closed wave guide opened at the top (after [3]).

Typical dispersion characteristics of the hybrid leaky-wave EH 10
mode (after [24]) ............................

Equivalence models for the trough leaky wave antenna mounted on
an infinite flat electric ground plane. ................

H-plane amplitude pattern for a forward traveling wave distribu-
tion aperture mounted on an infinite ground plane. ........

Transmission line model of the transverse cross-section of the TLW
antenna .................................

Transmission line used to illustrate the TRM. ...........

Shifting reference planes in the transmission line. .........

Complex plane representation of kg = :lz‘ / kg — kg .........

Propagation constant for a TLW antenna filled with RT/Duroid
5880 ...................................

Propagation constants for a TLW antenna filled with RT/Duroid
5880 and a TLW antenna filled with air. ..............

Real and imaginary part of the driving point impedance for an
infinitely long TLW antenna (RT/Duroid 5880-filled) ........

11

13

19

25

26

28

31

32

34

38

VSWR for an infinitely long TLW antenna (RT/Duroid 5880-filled). 39

Real and imaginary part of the driving point impedance for an
infinitely long TLW antenna (air-filled). ..............

Figure 2.10 VSWR for an infinitely long TLW antenna (air-filled). ......

viii

41

Figure 3.1

Figure 3.2
Figure 3.3
Figure 3.4

Figure 3.5

Figure 3.6

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7
Figure 4.8

Figure 4.9

Figure 4.10
Figure 4.11
Figure 4.12

Figure 4.13

Geometry of a three-dimensional cavity-backed aperture in an in-

finite ground plane ........................... 43
Tetrahedron element and local edge-node numeration ........ 66
Two triangles sharing an edge and their geometrical parameters. . 80
Local coordinates for triangle Tq ................... 83
Projections of F, 7"", and 7"} onto the plain that contains Tq. . . . 87

Different configurations of surfacemesh: (a)Rectangular, (b)Bow—
tie, (c)Circular. ............................ 90

Effect of the variation of h in the propagation constants for the

RT / Duroid 5880-filled TLW antenna ................. 102
Effect of the variation of w in the propagation constants for the

RT/Duroid 5880—filled TLW antenna ................. 103
TLW antenna with terminations and design parameters. (a)Front

view. (b)Side view ........................... 105
Real part of the input impedance of the RT/Duroid 5880—filled

TLW antenna simulated using ctets and tet. ............ 107
Imaginary part of the input impedance of the RT/Duroid 5880-

filled TLW antenna simulated using ctets and tet .......... 108
Radiation pattern at F=8.0 GHZ of the RT/Duroid 5880—filled

TLW antenna simulated using ctets and tet. ............ 109
VSWR of the RT/Duroid 5880—filled TLW antenna ......... 110
Magnitude of Ey of the RT/Duroid 5880-filled TLW antenna. 112
Phase of Ey at 6 GHz of the RT/Duroid 5880—filled TLW antenna. 113
Phase of Ey at 7 GHz of the RT/Duroid 5880—filled TLW antenna. 114
Phase of Ey at 8 GHz of the RT/Duroid 5880-filled TLW antenna. 115
Slope term computed via FE—BI and the least square method. . . 116
Constant term computed via FE—BI and the least square method. 117

ix

Figure 4.14

Figure 4.15

Figure 4.16

Figure 4.17

Figure 4.18

Figure 4.19

Figure 4.20

Figure 4.21

Figure 4.22

Figure 4.23

Figure 4.24

Figure 4.25

Figure 4.26

Figure 4.27
Figure 4.28

Figure A.1

Ratio constant to slope term computed via FEBI and least square
method.

Propagation constant of the RT/Duroid 5880—filled TLW antenna
computed using FE—BI and T RM ...................

Effect of the variation of h in the propagation constants for the
air-filled TLW antenna.

Effect of the variation of w in the prOpagation constants for the
air—filled TLW antenna.

Different terminations: (a) Rectangular (b)Bow—tie (c)Circular
(d)Side View. .............................
VSWR for the air filled TLW antenna with different terminations.

Radiation pattern at 12 GHz for the air filled TLW antenna with
different terminations.

oooooooooooooooooooooooo

Input impedance of the air—filled TLW antenna with circular ter-
mination.

ooooooooooooooooooooooooooooooo

Magnitude of Ey of the air-filled TLW antenna with circular ter-
mination.

Phase of Ey at 6 GHz of the air-filled TLW antenna with circular
termination ...............................

Phase of Ey at 9 GHz of the air-filled TLW antenna with circular
termination ...............................

Phase of E3) at 12 GHz of the RT/Duroid 5880-filled TLW antenna
with circular termination.

oooooooooooooooooooooo

Propagation constant of the air filled TLW antenna with circular
termination ...............................

Efficiency of the TLW antenna with circular termination ......
Ratio PR L to Pin of the TLW antenna with circular termination.

(a) A circular contour in the 2 plane. (b) The mapping of the 2
plane into 211 plane by the function 62. ...............

118

119

121

122

124

125

126

127

128

129

130

131

132

133

134

140

Figure A.2

Figure B. 1
Figure C .1

Figure D.1

(a) A circular contour in the 2 plane. (b) The mapping of the 2
plane into it) plane by the function J? ................ 141
Convergence for equation (8.6), (Er = 2.33) ............. 144
Current interpolation between two triangles ............. 150
Geometrical parameters associated with the evaluation of potential

integrals over the triangle Tq (after [50]) ............... 154

xi

KEY TO SYMBOLS AND ABBREVIATIONS

TLW: Trough Leaky-VVave

TRM: Transverse Resonance Method

TE: Transverse Electric

TM: Transverse Magnetic

VSWR: Voltage Standing Wave Ratio

HWLW: Half—Width Leaky-Wave

FE-BI: Finite Element-Boundary Integral

CHAPTER 1

INTRODUCTION

1.1 Leaky wave antennas background

A geometrical description of the Trough Leaky-Wave (TLW) antenna considered in
this research is shown in Figure 1.1. The TLW antenna is excited by a coaxial probe
and terminated by a lumped load. The trough may be filled with a dielectric (er),
and the infinite ground plane is a perfect electric conductor. The aperture width is
less than one tenth the dielectric wavelength (i.e., w < Am/ 10), where Am 2 ADA/5,
A0 is the free space wavelength. The aperture length is on the order of seven times
the dielectric wavelength (i.e., L z 7/\m); and the thickness of the trough is on the
order of one fourth the dielectric wavelength (i.e., h % Am / 4).

In the 1950’s, structures similar to the TLW antenna were considered. These
structures were based on closed waveguides with a cut along the side of the
waveguide to radiate power. Rumsey [1] develOped a variational procedure to
derivate approximate formulas for the complex propagation constant and field
configuration of an infinite-length, traveling-wave-slot antenna. His configuration
was a rectangular wave-guide. At the same time, Harrington [2] formulated another
variational method to determine the attenuation and phase constants for the fields
along a slotted circular wave-guide. In both works, analytic results were validated
with experimental data. Goldstone et a1. [3] used the Transverse Resonance Method

(TRM) approach to compute the complex propagation constants in terms of the

Infinite ground plane

 

 

 

 

 

 

 

 

 

 

 

[‘v
.J
f‘v

 

 

probe h’ 8r

Coax ]
h

 

6—}
W

(b)

Figure 1.1. (a) Geometrical description of the TLW antenna (b) Feeding technique.

transverse dimensions of the leaky-wave antenna. Hines et a1. [4] identified three
categories of slotted rectangular guides depending on wether the excitation mode was
TE, TM or hybrid TE—TM. In addition, they presented analytic and experimental
data for these three cases. Oliner [5] has made significant contributions explaining
the leaky wave phenomena not only for rectangular slot waveguides but also for

dielectric waveguides (i.e., microstrip leaky—wave antennas).

Microstrip leaky—wave antennas are wide-microstrip lines that operate primarily
as traveling wave antennas. As a wave propagates along the guiding structure
energy leaks out in the form of radiation. Ideally, the structure is designated to
be electrically long, by the time the wave reaches the end termination most of the
energy has leaked out into free space and therefore, the reflected power does not
affect the input match. This leads to a wide input impedance bandwidth and a
large effective aperture illumination with a narrow radiated beamwidth. If the feed
end of the antenna is properly designed, and a suitable termination load is placed
at the opposite end of the antenna, a microstrip leaky—wave antenna is essentially a
printed version of the classic Beverage antenna commonly used by the amateur radio
community [6]. Such antennas typically can be designed for 25% or more VSWR
bandwidth (e.g., VSWR of two or less over a fractional bandwidth of 25% of the
center frequency) and have good gain. Unfortunately this antenna diverges from
the “perfect antenna” in that it must be relatively long to achieve high radiation
efficiency, and the radiation pattern main lobe steers from near broadside to near

end-fire across the operational bandwidth of the antenna. Nevertheless, this antenna

is well-suited for applications requiring wide bandwidth and a thin profile. Hence,
there is interest in characterizing its operational characteristics and develOping

effective design procedures.

Microstrip leaky-wave antennas have been the subject of significant research by a
number of investigators [7, 8, 9, 10, 11]. Their radiation mechanism is well known,
and different feeding and termination techniques have been proposed [10, 11]. Figure
1.2 illustrates the electric field for the EH 10 mode used by a microstrip leaky-wave

antenna.

PEC

 

 

 

EH10

Er

 

 

F!

 

 

 

 

Ground plane

Figure 1.2. Electric field for the EH 10 mode.

In Figure 1.2, the arrow length is an indication of the electric field strength. Hence,
the highest field strength is near the edge of the antenna while a null is formed along
the centerline. Since the potential between the ground plane and the microstrip is
proportional to the substrate thickness (assumed much less than a wavelength for
this work) and to the normal component electric field strength, feeding the antenna
near an edge will result in a high driving-point impedance (assuming a matched
load at the other end of the antenna). On the other hand, attempting to feed the
antenna along the centerline would lead to a low driving-point impedance. Hence, by
adjusting the transverse feed position, the driving-point impedance can be adjusted

as needed.

An alternative to the traditional microstrip leaky-wave antenna design is the
so—called half-width leaky-wave (HWLW) antenna that makes use of the observation
that a physical short may be placed along the centerline of the antenna (Figure 1.2)
without perturbing the modal fields. This antenna has undergone considerable recent
work [12], [14], [15]; one of the principal advantages of this design is the fact that
the feeding structure is considerably simplified. Traditional, full-width microstrip
leaky-wave antennas, usually require a rather complex feeding structure (for example
[16]) to preferentially excite the EH 10 mode over the fundamental EH00 mode
(i.e., the usual microstrip Transmission line mode). On the other hand, the HWLW
antenna will automatically suppress the EH00 mode since it is quasi-TEM, and
therefore cannot be supported by the structure shown in Figure 1.3. Hence, the

HWLW antenna can be fed with a relatively simple feed, such as a coaxial probe.

\
I

PEC

 

 

 

EH10

  
 

Sr

Shorting wall

  

 

 

 

 

 

Ground plane

Figure 1.3. Illustration of HWLW antenna.

If the HWLW antenna is rotated by 90°, it is possible to obtain the trough
leaky—wave (TLW) antenna as shown in Figure 1.4. One main advantage of this
configuration is that the antenna can be integrated with a vehicle and does not
require a stand—off dielectric (e. g., the trough can be air-filled to increase bandwidth),
thereby reducing drag. A rough sketch of the anticipated electric field inside the
cavity of the trough antenna is also shown on Figure 1.4. Due to the narrow trough
width, the polarization of the electric field is solely in the y direction, and due to

the metallic walls at the start and end of the trough, the tangential component

of the electric field is zero at both ends of the cavity. Therefore some type of end
termination design feature is necessary to maintain the bandwidth of the antenna.
Should termination features be omitted, the antenna’s VSWR bandwidth will be
severely limited due to the reflection from the nearby vertical wall of the cavity. In
addition, placing the feed near such a vertical wall will “short-out” the feed, leading
to an inefficient radiator. The major contribution of this dissertation is the design

and analysis of one of such termination scheme.

 

   

     
     
 

Coax
probe

—-—) E-field lines
H-field lines not shown

 

 

 

 

 

Figure 1.4. Field structure for a TLW antenna.

The objective of this research is to analyze the properties of the TLW antenna
using the transverse resonance method (TRM) and the finite element boundary

integral (FE-BI) method [17]. Using these two methods, the propagation constant,

the impedance and the radiation characteristics of the TLW antenna are computed.
In this research, it is vital to understand how the propagation constant is affected
by geometry and electric changes in the structure in order to improve the antenna’s
far field pattern and bandwidth. Based on FE—BI simulations the TLW antenna
terminations are designed. In order to validate the analytic and numerical studies a

prototype is fabricated and tested.

This dissertation is organized as follows: the rest of Chapter 1 presents a theory
review about leaky-wave antennas. Chapter 2 deals with analytic considerations of
an infinitely long TLW antenna in order to compute its propagation constants and
driving point impedance. Chapter 3 presents the formulation of the FE—BI method
used to perform numerical analysis of the TLW antenna. In this chapter, details of
code implementation are given. Chapter 4 validates the code implementation and
shows analytic and numerical results. In addition, it contains details about the TLW
antenna design, and presents simulation results of the TLW antenna. Conclusions

and future work are presented in Chapter 5.

1.2 Surface wave and leaky waves

Surface-wave antennas and leaky wave antennas are related because they may be
analyzed as traveling wave type antennas [25], [26]. Consider the TLW antenna
shown on Figure 1.4. The wave number in free space for an antenna is related to the

propagation constants in x, y, and 7. by the continuity equation,

133+ k5 + k3 = k2, (1.1)

where k,- = 52' —— jar, and i = 2:,y, z.

Knowing the attenuation coefficient (a) and the phase coefficient (fl) , it is possible
to characterize the operational frequencies of a leaky wave antenna along with some
of the radiation behavior; this idea will be clarified later. In air, It is real and can be
written in terms of the wavelength /\ by k = 3 = 27r/A, and in terms of the speed
of light c by k = w/c. The phase constant fig can also be written in terms of the
surface wavelength AZ by 52 = 27r/Az and in terms of the surface phase velocity oz

by fig = w/vz. Therefore,

—=—=—. (m)

Traveling wave antennas are classified according to (1.2). When this ratio is equal or
greater than one, such structures are known as slow wave or surface wave antennas
because their phase velocity is equal to or less than the speed of light. In a surface wave
antenna, a wave propagates along an interface between two media. The radiation only
occurs at discontinuities and nonuniformities. Bagby and Nyquist [27] have identified
this behavior in microstrip lines as the surface wave regime. When the terms in (1.2)
are less than 1, the structures are known as fast-wave or leaky-wave antennas because
their phase velocity is greater than the speed of light. In these antennas a leaky wave
travels and loses energy because of radiation. The fields decay along the structure in

the direction of the traveling wave and increase in others. Since the wave loses energy

as it propagates along a plane interface, (1.1) becomes,

(3:: — jax)2 + (fiz — 2&2)? = 1:2. (1.3)

Equating imaginary parts,

axfix + 01252 = 0, (1-4)

where (1;; and 5,2 are both positive (attenuation and phase constant in the direction
of propagation); [3:5 is also positive (leakage away from the surface). Consequently,
a3; must be negative. This means that the leaky-wave field increases away from the
antenna’s surface. Since the cross section outside the waveguide is unbounded, the
leaky wave must increase transversely to infinite which is a violation of the radiation
condition. Leaky waves are modal solutions that are improper mathematically be-
cause these waves increase in the transverse direction, in contrast to bound waves,
which are prOper and decrease transversely. As a result, the leaky wave must be
mathematically improper, and it corresponds to a complex pole on the improper Rie-
mann sheet of the longitudinal wave number plane. Practical leaky wave antennas
are finite in extent. Therefore, leaky waves never reaches infinity. Hines et a1. [4]
made measurements and found that near the waveguide the field increases but some
distance away it vanishes very rapidly. As it is shown on Figure 1.5, the field inten-
sity decreases exponentially along 2 however following the dashed line in a: direction
the field increases vertically away. Thus this improper behavior happens only in the

wedge-shape region.

10

 

 

 

/;/

Power

Figure 1.5. Leakage from a closed wave guide opened at the tOp (after [3]).

11

1.3 Operation modes of leaky-wave antennas

In the study of any leaky wave antenna, it is fundamental to know its complete prop-
agation mode spectrum. If this mode spectrum is known, the total electromagnetic
field of the antenna can be expanded in terms of its modes. The complete modal
spectrum of this structure is composed of a continuum of orthogonal radiation modes
and a finite number of discrete bound modes [23]. A bound mode does not radi-
ate, which means that the electromagnetic fields are confined to the proximity of the
guided region and no power flows transverse to the guide. An example of this mode
is the E H00 mode for microstrip transmission lines. On the other hand, in radiation
mode the electromagnetic fields are not confined to the guiding region and the power
flows in the transverse region of the guide. In addition to those modes, a leaky wave
mode is a discrete, not confined mode and its field distribution increases exponen-
tially toward infinity. A leaky-wave mode is not part of the proper spectrum; rather
it is used to construct the total field by the method of the steepest descent, which
is an asymptotic technique but not a modal decomposition. Figure 1.6 shows the
three frequency regions associated with the propagation regimes in a microstrip. The
reactive, leaky, surface and bound regime. In the first regime, 03 is large causing the
microstrip behaves as a reactive load. Above f = fc energy begins to propagate along
the microstrip as 5;; grows larger than (12, the field losses energy due to radiation.
The fields in the transverse section of the microstrip increases (Figure 1.5) because
less energy leaks per unit length as the wave travels along the structure. Above the

frequency at which fiZ/ko = 1, ax increases and the leaky wave starts to attenuate.

12

This region is known as the surface regime. Finally, the bound follows the surface
regime and it is the recommended one for microstrip line operation because the fields

are confined inside the transmission line.

 

 

 

 

 

    

 

 

 

 

 

15 l r l n l

4.. Reactive ]( Leaky Surface .. — "
.5 l , v ' Bound
2 : , r
‘5 : , ’
O ' r
O 1 — l I _
C [ ’
: .
(D | [ Z 0
8' i I
E, l I
g 0.5 ~ : ' —
E ' '
‘5 I
2 K ,

~ ~ 1

‘ . ._ l
0 l i l I I
2 4 / 6 3 10 12 14.
fC Frequency (GHz)

Figure 1.6. Typical dispersion characteristics of the hybrid leaky-wave EH 10 mode
(after [24]).

13

1.4 Radiation characteristics of a TLW antenna

It is very revealing to find the radiation pattern of the TLW antenna assuming that the
aperture fields are traveling waves. The radiation field of the TLW antenna may be
found by two different methods. The first uses the Fourier transform of the aperture
field and the stationary phase method. The second method is using the equivalence
principle and the vector potential. In this section, the second method is used. In
order to solve for the radiation of the TLW antenna, it is assumed that the tangential
components of the electric field over the aperture are known. The equivalent model
[28] that will produce the same electric and magnetic fields radiated by the aperture
is shown in Figure 1.7. For convenience, the TLW antenna shown on Figure 1.7 (a)
is a 900 rotated version of Figure 1.4 in order to have the antenna’s aperture on the
x — y plane. Figure 1.7 (0) corresponds to a magnetic current radiating in free space.
If the equivalent currents are known, the vector magnetic potential may be found

using

 

. « «I
Fm = -— ‘ fig/M (F’)e3k""‘ ds’, (1.5)

The magnetic current is given by

M = —22 x 33E e—jkfl. 1.6
0

Therefore, (1.5) becomes

14

 

0 .‘
floocoooococo.’

 

 

 

 

 

 

 

J
h
A :l f.-
I 3
' 3
l o
' 3
' :
lx :
L l M g 2M
= s
I I
. :
I 0
' 3
I I
' 3
w _ __ :m.-.
T .~
PEC J ;
f‘v (“v .03..
(a) (b) (C)

Figure 1.7. Equivalence models for the trough leaky wave antenna mounted on an
infinite flat electric ground plane.

15

_' E —jk7‘
Fm— — sci—‘18

27r T

 

. . A .4/
fsle—kar'r ds’, (1.7)

where 7“ = risinflcosqb + fisinOsingb + 20080, F, = :r’cic + 3,1,3), and the prime and
unprime coordinates denote source and observation points respectively.

For the aperture shown on Figure 1.7, it is found that

 

F('F')— _ stifle—T jkr 112/2 /L/26—j( j(—kx ksinOCos 48:13, e—jky’ sin63in $dx’dy'.

 

u / 2 L / 2

(1.8)

After the integration over the aperture is performed, (1.8) becomes
F (‘3— i2eE0 e 1’" sin [(kx — k sin6 cos d1) L/2] sin [kw/2 sin 68in (1)] (1 9)

T 7r 1" (kg; — k sin 6 cos (1‘)) 1: sin 0 sin ¢ ° '
The far-zone radiation field of an aperture antenna is given by [29]
a k a

E~ ]—-r xF (1 (1.10)

Hence, the resulting 6 and (15 components of the electric field are given by

EB:

 

 

2jkE0 e—jkr sin [(kx -— ksindcos <15) L/2] sin [kw/2 sinfisin d1] , cf)
(kg: -— ksindcos <15) ksinfisinqfi n ’

71' T

(1.11)

16

and

 

_ 2jkE0 6—3.1" {sin [(kz — ksin6cos a5) L/2] sin [kw/2 sin6sin (1)]

E
d) (kg: — ksin6cos d1) ksin6sin¢

} cos 6 cos 915.
7r r

(1.12)
In the E—plane (p = 7r / 2, the 6 component corresponds to a sine function centered at

6 = O. For this antenna this pattern is not of significant interest.

In the H-plane d) = O, (1.12) becomes

51>

 

_ 2jkE0 e‘jkr {sin [(kx — ksin 9) L/2] } €086 (1 13)

7T T (kg; ‘— ksin 0)

where kg: = fl»; — jam.
The radiation intensity in a given direction is defined as “the power radiated from an
antenna per unit solid angle” [30]. The radiation intensity is related to the far—zone

electric field of an antenna by [31]

mm) = % [|E9 (11¢)? + IE¢(6,¢)|2] (1.14)

where 77 is the intrinsic impedance of the medium.

The normalized radiation pattern is found using

17

 

: lEg(6,¢)|2

U9 (0, 41>) max [IE0 (64912] ,
IE¢ (6, 99,2
max “13¢ (6, ¢)t2]

 

U¢ (6, (15) = (1.15)
Using the TRM, it is possible to compute kx, (please see Chapter 2). The wavenumber
kg; depends on frequency, antenna dimensions, and material inside the trough. As
a function of frequency, the leaky wave antenna main lobe steers from broadside to
endfire. The radiation pattern for TLW antenna is shown on Figure 1.8. This pattern
corresponds to a single forward traveling wave. In reality, a finite length antenna has
backward traveling waves that are reflected from the antenna termination.

In (1.13), the maximum of the radiation pattern occurs when (kg; — ksin 6) = 0, so

that the angle of maximum radiation (6m) is determinated to be

Q”. m sin am. (1.16)
k0

(1.16) is a fundamental relation in leaky-wave antennas [9].

Other fundamental relations in leaky wave antennas involve the effective aper-
ture length (L) and the beamwidth (A6). The effective aperture length of a leaky
wave antenna is related to am. If am is small, the effective aperture length is large,
since attenuation as the wave is guided along the length of the antenna is gradual.

Accordingly, the far field radiation pattern has a narrow beamwidth and larger di-

18

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 r l r I r r I ’i‘ r l
G ' ‘
‘5 r' ",v ' ‘ ..,
I, n ,
I " I
'10 '4' ‘ "
. I \
"‘ 1
g -1 5 .’ ‘ -
g 1 '1, \
39 -2011 ‘> . l '. I
'8 \ 2‘
.5 1‘
a -25 ‘2‘
E F-GGH '-‘
0 II I I I I I - Z 2‘
Z .
- - - F=7GHz -_1.
-— F=8GHz ?
:I
l l 1 .

l l

-40 -20 0 20 40 60 80
9(degrees)

 

 

Figure 1.8. H-plane amplitude pattern for a forward traveling wave distribution
aperture mounted on an infinite ground plane.

19

rectivity. On the other hand, if 013; is large, the effective aperture is short and the
field radiation pattern has a wide beamwidth and consequently less directivity. The

relationship between the antenna length (L) and the beamwidth is given by

(\0
A6 z —. 1.17
L cos 6m ( )
The antenna length is usually selected for a given value of ax such that
P_(L_l z el-47r(ax/ k0)(L/ 10)] (1.18)

P (0)

where P (L) is the power remaining in the leaky mode at the antenna termination
and P (0) is the power input.

Therefore, the percentage of power radiated is given by

%Power radiated = 100 [1 — e[_47r(a$/k0)(L/’\0)]] (1.19)

Most leaky wave antennas are designed to radiate 90% or more of the input power.
The remaining 10% is absorbed by a matched load. For a 90% radiation power, the

antenna length (1.19) becomes

 

(1.20)

Since 013; is a function of frequency, the radiation efficiency changes as the beam
is frequency scanned. This 90% criterion is usually applied to the middle of the

operational bandwidth.

20

The characterization of the attenuation and phase coefficients allows for an effective
design procedure. Note that these relations do not give sufficient information for
determining either the feeding or termination requirements of the antenna; for this,

full-wave methods are useful [17], [18], [19].

21

CHAPTER 2

ANALITICAL SOLUTION FOR THE TLW ANTENNA

2.1 Transverse resonance method

The TRM [21] allows rapid computation of the complex propagation constant that
is fundamental for the analysis of leaky wave antennas; it was developed extensively
by Lee [7]. In the TRM, the cross section of the TLW antenna (Figure 2.1 (a))
is represented as a transmision line (Figure 2.1 (b)) operating at resonance. The

boundary conditions at the antenna’s dielectric-air interface are

(E’ m — 15:th x :1: = 0, (2.1)

5: x (13’1“mt — Him) ] = f, (2.2)

where E cast (I? cart) and E int (If int) are the electric (magnetic) external and in—
ternal fields respectivelly.
Assuming a source free interface, a TRM relation is obtained from the continuity of

the tangential components of the electric and magnetic field at :1: = 0. This is given

by

22

E53315 (:1: = 0+,y,z) = Eff/m (:1: = 0_,y,z),
ES” (:1: = 0+,y,z) 2 E27“: (:1: = 0_,y,z) ,
H5” (:1: = 0+,y,z) 2 H37“ (:1: = 0_,y,z) ,

Hg“ (:1: = 0+,y,z) 2 H37“ (:1: =2 0_,y,z).

(2.3)

(2.6)

The directional wave impedance is defined as the ratio between the electric field

component and the corresponding orthogonal magnetic field component. The wave

impedance in the positive :1: direction is

+ _ E5” (2: = 0.11.2) _ E55“ (:1: = 0,y,z)
x —— — _" .
HS” (117 = 0.11.3) Hi?“ (I = 0.31.2)

 

 

Similarly, the wave impedance in the negative :1: direction is written as

313M (53 = 01 y, 2) __ Egnt (:1: = 0, y, z)

Z_=— . —— . .
.1: Hgnt (:1: = 0, y, z) Hént (x = 0,y,z)

 

 

Substituting (2.4) and (2.5) into (2.7) gives

Egnt (:1: = 0, y, z)

z; -_—. — . .
Hf,“ (36 = 0.11.2)

 

23

(2.7)

(2.9)

The right hand side of (2.9) is equal to —Z; given by (2.8). Thus, the TRM relation

is obtained

25! + Z" = 0. (2.10)

It is customary to write (2.10) using the notation given in Figure 2.2

2(:::) + z (1:) = 0, (2.11)

where the impedance looking to the left is denoted by (2 (11:) and the impedance look-
ing to the right is denoted by 201:). Note that in Figure 2.2, the shorting wall is
explicitly shown and the transverse wavenumber kg; depends on the material inside
the trough. Furthermore, the Open edge impedance is unknown but assumed to be
complex-valued. In Figure 2.2, [1 denotes the distance from the shorting wall. Equa-
tion 2.11 can be re—cast in terms of transverse reflection coefficients. The reflection

(—

coefficient looking to the left, F (2:), (i.e., toward the shorting wall) is written as

«— x =?(x)~zo
r() _—_‘z_(x)+zo' (2.12)

where Z0 is the characteristic impedance of the equivalent transmission line.
_)
Likewise, the reflection coefficient looking to the right, 1‘ (2:), (i.e., toward the open

edge of the antenna) is given by

(2.13)

 

IN
5'
\/

 

 

i
I
N

(b)
y

Figure 2.1. Transmission line model of the transverse cross—section of the TLW an-
tenna.

25

 

 

 

Figure 2.2. Transmission line used to illustrate the TRM.

26

Solving equations (2.12) and (2.13) for (2 (1:) and Z) (:1:) respectively gives

‘2— (1') = 20%, (2.14)

(2.15)

Substituting (2.14) and (2.15) into (2.11) yields the desired TRM relation in terms

of the reflection coefficients

F (1:) - F (2) = 1. (2.16)

The unknown transverse impedance, Zt, (Figure 2.2) is characterized by the reflection
coefficient and the load to the left side as a short circuit. This reflection coefficient is
calculated using the approximation for thin substrates provided in [32]. The reflection

coefficient looking to the left at :1: 2 —h is given by

_V—(:l:=—h.)_ZL—ZO
_V+(:1:=—h) _ZL+Z0'

 

(2.17)

where Z L = 0 for a short circuited terminated line. Thus, F (:1: = —h) = —1.
From the theory of traveling waves on lossless transmission lines (Figure 2.3), the

Voltage wave amplitudes at a: = —h are related to those at :1: = 0 by the expression

27

’5
5"
v

 

 

_
fo=-h) -](-x=0)
V'(X=-h) v(x=0)
x=..h X=0

Figure 2.3. Shifting reference planes in the transmission line.

V— (a: = 0) = V“ (:1: = —h) e‘jkxh, (2.18)

v+ (:1: = 0) = W (a: = —h) ejkxh. (2.19)

The reflection coefficient looking to the left at :1: = 0 is defined by the expression

_V"(x==0)

_. m. (2.20)

Therefore, dividing (2.18) by (2.19), and using (2.20) and (2.17), the desired reflection

coefficient is obtained

28

F (:1: = 0) = —e“j2kxh. (2.21)

According to [7], the reflection coefficient looking to the right is found by

15(1: = 0) = m, (2.22)

where X is a complex expression that incorporates the effect of the Open radiating
edge. It is given in [32].
Substituting (2.21) and (2.22) into (2.16), the following transcendental equation is

obtained:

e-jkaher = -1, (2.23)

where, after recognizing —1 = eijmr, n = 1,3, 5, ..., (2.23) can be recast as

e—J'Zkrvhejx _—_ 613m,” = 1, 3, 5... (2.24)

The left-hand side of (2.24) is a complex value function in the form

10(2) = 62 (2.25)

where z is a complex variable, 2 = rejg = :1: + jy.
Since this function is single-valued [13], it is not necessary to consider any Riemann

sheet. A further explanation of this is given on Appendix A.1. Therefore, the trascen-

29

dental equation becomes

x — 2kxh = :l:mr,n = 1, 3, 5, (2.26)

For the hybrid leaky—wave EH 10 mode shown in Figure 1.3, n = 1, and so (2.26) may

be re—written as

x — 213511 + 71 = 0, (2.27)

where X is given on Appendix B.

The propagation constants inside and outside the trough are related by

1:2 = ark-8 = k}, + 13, for 2: < 0, (2.28)

kg = 1181. + k2, k0 = (.2 50,110, for 2: > 0. (2.29)

Therefore, the axial propagation constant kz is related to the transverse propagation

constant kg; by

kz = idle?) — kg. (2.30)

The square root in (2.30) is a double-valued function (appendix A2). The positive
and negative signs correspond to two branch cuts. Only one of these branches is

the correct to represent outgoing waves vanishing at infinity. The complex plane

30

representation of (2.30) is shown on Figure 2.4. Singular points occur along the real

axis at the branch points kg; 2 i130.

he
Q
N

 

 

“k0

 

Figure 2.4. Complex plane representation of kg = :1:” kg — 1:323.

As an example, it is assumed a TLW antenna with w = 0.0787cm, h = 0.75cm and
Er = 2.33. Using TRM the propagation constant is computed and shown on Figure
2.5. Same as microstrip leaky wave antennas [33], the operational band for TLW
antennas is approximately defined from the frequencies in which az/ko = flz/ko and
62/190 = 1. This regime is known as the leaky or fast region because the phase

velocity is faster than the speed of light, i.e. up > c. This is the spectrum of interest

31

in leaky wave antennas design because it is favorable for radiation. The radiation
takes place along the structure at some angle 6. The surface or slow wave region is
located between fiZ/ko > 1 and 63/190 < 1/6r- This region takes this name because

up < c and radiation may take place at discontinuities or at the termination of the

 

 

 

 

 

 

 

 

 

 

 

 

 

structure.
1 , are '
E I ’
Q) I
E —Oz/k0
8 - - I B ”(0 fl
0 2
C
.9
‘5 _
U)
(U
Q.
2
D.
U —I
(D
.5
'16
E
0
Z .—
0 1 1 1 1 1 ~r=_.i_.__1
5 5.5 6 6.5 7 7.5 8 8.5 9

Frequency (GHz)

Figure 2.5. Propagation constant for a TLW antenna filled with RT/Duroid 5880.

32

One of the advantages of the TLW antenna with respect to microstrip leaky wave
antennas is that it does not need a dielectric to support the conductor strip over
the ground plane. Therefore, an appealing design is a TLW antenna filled with air.
The parameters for this design are 11) = 0.0787cm, h = 1cm and 5r = 1.0. Figure
2.6 shows a comparison of the propagation constants of the TLW antenna filled with
RT/Duroid 5880 and that one filled with air. From these figure, it is clear that the
Operational band for the air-filled TLW antenna is much larger than the one for the
RT/Duroid 5880-filled TLW antenna.

One disadvantange of the TRM is that it does not provide sufficient information for
determining either the feeding or termination requirements of the antenna; the FE—BI

method accounts for these important features [17].

2.2 TLW antenna driving point impedance

Knowing the propagation constant in a TLW antenna, it is possible to estimate its
driving point impedance. The approach that is followed is similar to [34].

For the dielectric-filled TLW antenna shown on Figure 2.1(a), it is necessary to con-
sider hybrid modes in order to satisfy boundary conditions between the interface of
air and dielectric. However, it is possible to compute the driving point impedance for
the first higher order mode using the mode T111320.

The field components for the T111320 mode are given by [35]

33

 

 

 

 

 

 

 

1.2” "-u"." .1
E ’v"'
.2 ,x
9 a
% 1 “““““““““ ;" -----------------------------------
O I
0 ”
g 0-8' l’ .—.o"""‘—'-:1
(U ’ “."’
3 I ,w"
8' ’ .v" _
a“. 0.6- 1 ".v —qz/k0(£r-2.33) -
U ’I ’1 /k -233
E (I, \’3 --IBZ 0(£r_ . )
(U '. 2
g IIIIII aZ/k0(£r_10)
z I-I- :

Bz/k0(£r 1.0) _

 

 

 

Frequency (GHz)

Figure 2.6. Propagation constants for a TLW antenna filled with RT/Duroid 5880
and a TLW antenna filled with air.

34

 

 

82 .1 0' .1
E,__ 1 1., HF:&
02,115 827031 11 82
2
1 8y 2
E — — —— — k H — 0
y we (ayz + ) W y
E — 1 82% H — 182113) (2 31)
Z 302115 Byaz Z _ ,u 82: ’ '
where the scalar potential function 16y has the form
16y 2 Csin [kx (2: + h)] (cos kyy) e—jkzz (2.32)

Where C is a constant.

In order to satisfy boundary conditions at y = iw/ 2 and at 2: = —h (Figure 2.1(a)),

kg = 27m/w. For the first higher order mode 71 = 0, hence (2.32) becomes

111;) = 031M111. (:1: + 11)] e—jkzz. (2.33)

Thus according to (2.31), the field component Ey and H3; are

. 1 2 . _'],;
E = — —-k C k h J ZZ 2.34
3) 302,118 sm] 23(33+ )]e , ( )

and

35

Hg; = 3'sz sin [km (2: + h)] e‘jkzz. (2.35)

In Figure 2.1(a), the voltage between the parallel plates at y = —1/2 and y = 1 / 2 is

given by

 

”(U/2 k2 .
v = — / Eya'y = u” C sin [1,; (2: + 11)] mm. (2.30)
—w/2 was

It can be assumed a current density J; flowing in the plate located at y = 1 / 2, the

electric current has the form

0 s
12/ hJS-idx, (2.37)

but is = —@ x H, hence (2.37) becomes

0

Substituting (2.35) into (2.38) and performing the integral, it is found that

I = M (1 — cos kxh) e-jkzz. (2.39)
#192:

Using Ohm’s law, (2.36), and (2.39), the first higher order mode driving point

iIIIIDEClance of the infinitely long TLW antenna is given by

 

wk$k2 sin [km (2: + h)]
Zd -

__ 2.40
p wekz (1 — cos kxh) ( )

36

In order to assess the effectiveness of this method, the driving point impedance is
computed using (2.40). The input parameters are: 10 = 0.0787cm, h = 0.75cm,
and £1" = 2.33. The wave number [172; is computed using TRM and kg; is obtained
from the relation k3; = m, where k 2 k0 = ”ME—0‘ The driving point
impedance result is shown on Figure 2.7 and its corresponding VSWR assuming a
509 termination, is shown on Figure 2.8. In addition, the same computations are
performed for the air—filled TLW antenna (111 = 0.0787cm, h 2 1.00m, and 57: = 1.0).

These results are shown in Figure 2.9 and Figure 2.10, respectively.

37

 

60 . . .

 

 

 

 

50

A
O

 

Impedance (0)
00
O

 

 

 

20“ _
\
\
\
S
10- ~ '
\
\‘~
0 I l---------___-
6 65 7 75 8

Frequency (GHz)

Figure 2.7. Real and imaginary part of the driving point impedance for an infinitely
long TLW antenna (RT/Duroid 5880—filled).

38

 

 

 

 

 

1 1 1 '

6 6.5 7 7.5 8
Frequency (GHz)

Figure 2.8. VSWR for an infinitely long TLW antenna (RT/Duroid 5880-filled).

39

 

 

—de
- - 'Xdp

 

 

 

 

 

 

 

 

 

§
8 —1
g \
8 20 - \ ‘
g, \
_ _ \ ~

15 \

s
10 - ‘ ‘
~
\~ ~
5 _ ~ u n - - - - - _
0 1 1 1 1 1
6 7 8 9 10 11 12
Frequency (GHz)

I‘fiigilre 2.9. Real and imaginary part of the driving point impedance for an infinitely
long TLW antenna (air-filled).

40

VSWR

 

 

 

1 l

 

l l

 

 

6 7 8

Figure 2.10. VSWR for an infinitely long TLW antenna (air-filled).

9 10
Frequency (GHz)

41

11

12

CHAPTER 3

NUMERICAL SOLUTION FOR THE TLW ANTENNA

3.1 Finite element boundary integral method

The finite element—boundary integral (FE—BI) method combines the finite element and
boundary integral formulations in a hybrid technique that uses the best features of

the finite element method (geometric and material flexibility) and a boundary integral
(minimal extent mesh closure condition). This hybrid technique is very attractive for
modeling three-dimensional cavity backed apertures if the aperture lies in a metallic
plane since the order of the resulting linear system is minimal. In this work, the cavity

is discretized using tetrahedral elements and the aperture of the cavity is dicretized
using triangular elements such that the tetrahedral faces lying in the aperture are
congruent with the triangles, thus ensuring proper continuity of fields from the interior
to exterior regions. The general geometry is illustrated in Figure 3.1, it is assumed
for this work that the cavity is filled with a homogeneous material having a relative
per mitivity 57' and a relative permeability pr. In general, the FE—BI method can
r eavCiily accommodate inhomogeneous cavity filling materials. The FEBI method in
ele(3131‘0magnetics has been around 30+ years and considered in many books (e.g, [17],

[22] 3 etc).

42

Infinite ground plane

 

 

Figure 3.1. Geometry of a three-dimensional cavity-backed aperture in an infinite
ground plane.

3-2 Finite element formulation

The fields Eint and Hint in the interior of the cavity shown on Figure 3.1 obey

Maxwell equations

V x Hint = jweEmt + f (3.1)

V x Eint = -jwpI-fint. (3.2)

Equations (3.1) and (3.2) are the well-known Ampére—Maxwell’s law and Faraday’s
l
aw a respectively. 5 is the absolute permittivity, 11 the absolute permeability inside the

Q ~ .. . . . . . . .
avlt)’, and J is an ex01tation Within the cav1ty. These two equations may be combined

43

together to yield a single second-order vector equation in terms of the electric field.
The advantage of using the electric field over the magnetic field formulation is that
the boundary conditions on a perfect electric conductor surface are easily satisfied
as will be shown later in this chapter. The FE—BI method is used to numerically
approximate the solution of the vector wave equation, in particular a linear system
of equations is obtained via Galerkin’s method. Taking the curl of (3.2) and using

(3.1), the following vector wave equation is obtained

V x V x Emt — wzpeEmt = —jwpf (3.3)

Equation (3.3) is usually expressed in a more convenient form using the relation

between the absolute and relative properties of the material, this is

e
= _ 3.4
57' 50 ( )

Where 50 = 8.854 x 10‘12 farad per meter,
11

11 =— 3.5
r #0 ( )

Where #0 = 47r X 10_7 henry per meter.

Therefore, the vector wave equation (3.3) becomes

1 —o' -o' —o
;—v x v x Emt — kgermt = —jk0Z0J, (3.6)
T.

w
here [so = w p.050 is the free-space wave number, Z0 = «110/50 is the intrisic

44

impedance of free-space, and er and 117» are the relative permitivity and permeability
respectively inside the cavity.

A weighted residual [17] is formed taking a inner product between the vector wave
equation (3.6) and a vector subdomain basis function (W1) The objective is to
minimize the difference between the approximated solution and the physical reality.

The inner product [17] over the domain 52 of two vector functions is defined as

where Q denotes a volume, surface or contour.
Taking the inner product of (3.6) and a vector subdomain basis function, W2", an

integro—differential equation is obtained.

1

__ 7.. ~1nt_ 2 “int =_.. ~,_~
M VW, {vaxE 10er ]dV )1.on [V W, JdV (3.8)

Where V denotes the volume of the cavity on Figure 3.1.
The right hand side of (3.8) corresponds to the interior excitation sources and can be

expressed as

f2?“ = —jkOZO [V 17,-de (3.9)

Substituting (3.9) into (3.8), the electric formulation becomes

45

 

1
WV

— 1717,- - {v x v x Em — 135713377“) dV = fimt.

(3.10)

Equation (3.10) contains second-order derivatives of the unknown electric field. To

realize a symmetric operator —in the spirit of reciprocity— one of the derivatives is

transferred from the unknown electric field onto the vector basis function using the

first vector Green’s Theorem [36]

/V[(VxFl)-(V><F2)—F1-VxVxF2]dV=
fgn- (E1 XVXE2)dS.
Using (3.11) into (3.10), the FE—BI equation becomes,

i v] (\7 x 117,) - (v x 1312"“) )dv

_k857~ [V {Wi . Eint}dv

7% S {n (W, x v x 13W) ]dS = ffnt

W
here S is the closed contour bounding the volume V.

46

(3.11)

(3.12)

The cross product between the normal vector and the magnetic field is necessary to
enforce boundary conditions. The electric field Emt is related to Hint by Faraday’s

law. Substituting (3.2) into (3.12), the functional becomes

Z1; V { (v x 147,-) - (v x Eint) ]dV
-135. [V {1717, - Emt}dV

4116030)]? {n (W,- x 13”“) )ds = ff”? (3.13)

Using the vector identity

fiéxéz—B' 4x6, (3.14)
(3- 1 3) can be written as
711; V{ (V x W1) - (V x Eint) }dV
-13... [V (W,- - E'W]dv
~jk0Z0 f9 { “2.. (n x 13““) )ds = ffnt. (3.15)

47

Equation (3.15) is known as the weak form of the wave equation. One important
characteristic of this equation is that the electric field and the testing function have
the same number of derivatives, if Galerkin’s method is used, a symetric linear system

of equations is obtained for isotropic materials.

3.3 Boundary integral formulation

For the exterior region (2: > 0 in Figure 3.1), the external fields E and If also obey

Maxwell’s equations

v x 11633156) = jwsos‘m (r) + fed”t (1") (3.16)

v x 6'6“ (6) = —jwn01‘ie$t (f). (3.17)

Taking the curl of (3.16) and using (3.17), a vector wave equation in terms of the

magnetic field is obtained:

v x v x He“ (1:) — 131168110) = v x .76“ (r). (3.18)
In order to find the radiated field, the dyadic Green function Ea is used [36]. Ce is
a solution of the dyadic differential equation
v x v x fie (F, 1'”) -— 1358 (6,64) = 76 (F— 7“”), (3.19)
where

48

h~1II

= 2:1: + 1717 + 22 (3.20)

is the Idem factor and 6 (F — 1‘") denotes the Dirac delta function.
In order to find the integral solution for (3.18) the second vector—dyadic Green’s

theorem is used:

/V[P‘.V><Vx5— (vaxfi)-5]dV=
—fgfz-[PxVx5+(VxP)x5]dS, (3.21)
letting I3 = fiext (F) and 5 2 66 (F, 7",) on (3.21) yields the expression
/V{ [11‘3“(6) -V x v x 66 (716’) — (v x v x 131mm) .66 (F,F’)] }dV =

_ fig {6 [962(6) x v x 6., (r, 1‘”) + (v x 131mm) x "56 (F.F’)] )4322)

Substituting the double curl terms V x V x Ce and V x V x fiext from (3.19) and

(3.18) into (3.22), the above equation can be converted to

49

[V {Hm (f) . [1368 (1337:”) + 76 (F— F')] —
[18116951102 + v x J8“ (16)] fie (3376’) )dv =
—]fSn- {131mm x v x 660116") +

(v x 13185“ (7’)) x 66 (F, F’) )ds. (3.23)

The terms multiplying kg in the volume integral of (3.23) cancel each other, and

fv {11mm 75 (F— F’) )dv = [V {131mm 6 (F- f") )dv = H83" (1")
(3.24)

thus,

50

(V x fiert (117) x Ce (17,64) }dS. (3.25)

Using the dyadic identity

6-(bx8)=5-(6x6)=(6x5)-a (3.26)

on the surface integral of (3.25), this whole expression becomes

He“ (6’) — /v{ [v x .76” (6)] fie (6, 6’) }dV =
—][g {Hm (6) . [n x v x a; (6,6’)] +

[n x v x 1‘16“ (6)] .66 (6, 6’) )43. (3.27)

The volume integral of (3.27) can be split in two terms applying the dyadic identity

51

V-(6x6)=(Vin)-5—Ei-Vx5 (3.28)

this is,

/v{ [v x 2“?“ (6)] fie (6, 6’) )41/ =

[V (‘7' [fen (F) x 56 (6,6’)] + .76” (6) - v x 56 (6, 6’) }dV. (3.29)

The dyadic divergence theorem and (3.26) are applied to the first volume integral in

the right hand side of (3.29) to obtain the following expression:

[9 { [n x f8" (6)] .68 (6, 6’)} d5. (3.30)

Substituting (3.30) into (3.29) yields the expression

52

/V{ [V x Jmt (6)] fie (6, 6’) )dv =
/S{[n x jg“ (6)] 56 (6,6’)}ds+

/V{f€1’t(6)-V x a, (6,6’) )dv.

In view of (3.31), the integral equation of (3.27) becomes

Hm (6’) — fv {fat (6) - V x 66 (6, 6’) ]dv =

+ [6. x V x 1‘16“ (6)] fie (6, 6’) )ds.

(3.31)

(3.32)

Collecting terms on the right hand side of equation (3.32), this can be written as

53

—1‘1 x [fen (6) — V x 1‘16“ (6)] .66 (6, 6’) )43. (3.33)

In view of the Ampere-Maxwell’s law (3.1), the above equation can be written in the

form

fie“ (6’) — fv {fat (6) . V x 68 (6, 6’) )dv =

—j[9 {Hm (6) . [n x V x a, (6, 6’)]

+6450 [6. x 155mm] 66 (6, 6’) )ds. (3.34)

For the cavity—backed aperture case (Figure 3.1) the volume V is bounded interiorly
by the entire xy—plane and exteriorly by a hemispherical surface Soc at infinity. If

Hext (7‘) satisfies the radiation condition, this is,

54

lim 6 [V x 131m (6) + )106x 131mm] = 0, (3.35)

77—100
and if Ce is chosen to satisfy the same condition, the portion of the surface integral

over the hemispherical is equal to zero; only the contribution from xy-plane needs to

be considered. Therefore, equation (3.34) can be written as

He” (6’) — [V {fat (6) - V x 56 (6, 6’) }dV =

— [I y—plane {Hm (6) - [n x V x a (6, 6’)]

+ 31.150 [n x 15mm] 56 (6, 6’) )ds. (3.36)

The tangential electric field f1. x Eext (f) vanishes at the ground plane except at the
cavity aperture. In addition, He is chosen to satisfy the dyadic Neumman boundary

condition on the xy-plane; namely,

6 x V x 062 = 0 (3.37)

where 562 is the electric dyadic Green function of the second kind. Therefore, (3.36)

reduces to

55

He“ (6’) — [v {fat (6) . V x 662 (6, 6’) )dv =

4.150 [SJ [n x 153mm] 6122 (6, 6’) )ds. (3.38)

It is convenient to cast (3.38) in the standard form using 7" as the position vector for
a field point and f" as that for a point inside a source. Interchanging the primed and

unprimed coordinate systems, the integral expression (3.38) becomes

11mm = [V [fact (6’) -V’ x 662 (6’ ,6) )dv’

_ng0 [SJ [6’ x 136“ (6’)] 5192 (6’ ,6) )ds’. (3.39)

In view of the dyadic identity

17-5: (E)T . 6', (3.40)

where (b) is the transpose of the dyadic 5; the integral equation (3.39) becomes

56

661(6) = [V { [V’ x 6,, (6', 6))?" . .66 (6') )m

-.1..),a{[s,, (>11 ()1 16- «41>

In addition, using the symmetrical relationships of the dyadic Green functions [36];

namely,

[V’ x 5132 (6’ ,6)]T = V x 681 (6, 6’) , (3.42)

16.21601266266), <32

the integral equation (3.41) can be written in the form

E16“ (6) = /V{ [V x Eel (6, 6’)] fie“ (6’) )dv’

Q

...,) 1:266)- 1161116 (3.44)
a

The dyadic electric Green function of first kind (Gel) is given by

57

where

 

6‘30 (F’Fl) = G ‘ '2‘?) Go (F, T"") , (3.46)
_. ..I e—jkOiF_F”
GO (T’T ) = 4,,r lf'_ F’l , (3.47)

and f1”: x’i + y’g — z’é is the image position of f" = x’i + y’f/ + z’é.

Equation (3.45) is known as the half-space electric dyadic Green function of the first
kind and satisfies (3.19), the radiation condition (3.35), and the Neumman boundary
condition (3.37).

Substituting (3.45) into (3.44) yields

19.58202?) - [ ()1 w «w

The first integral on the right hand side of (3.48) is the field radiated by jext (Fl)

58

in free space, this field is denoted as H'ino The second integral is the field reflected
on the ground plane and it is denoted as H‘ref. These two fields are known and
are maintained by j (77’). The third integral is the secondary field radiated by the
aperture, it is denoted as 1-7 sec and needs to be determined. Therefore, equation

(3.48) can be written as

H‘ext = Hinc+ Href + H‘sec. (3.49)

For radiation problems, H'inc = 0 and therefore gref = 0; however these fields are
retained in this analysis for completeness sake.
Since the intrinsic admitance of free space is given by Y0 = WED/#0: it is convenient

to write the secondary field as

H580 = —jk0Y0 [S {562 (F,F’) - [71’ x E8“ (F’)] }ds’. (3.50)

a

The Green function (062) evaluated on the plane of the conductor is twice the free

space Green function

(3.51)

 

 

where R = F—

 

_./|.
Letting z to approach to zero an taking the tangential component of (3.49) the desired

magnetic field integral equation is obtained

59

fixfie$t(fl= [fixfimc+fixfiref]

M) {levee (..,)].[.,.E~exe ()1 )e eee

Using (3.26) the argument on the surface integral of equation (3.52) can be rewritten

38

7‘2 x Hart“) 2 [ft x Him—kn x firef]

—jk0Y0 [s { [a x 562 (F,F’) x 71’] Em (F’) }ds’. (3.53)
a

With the idea of relating the tangential components of the external magnetic field
and the internal magnetic field the inner product between (3.53) and W2. is carried

out.

60

fsa {if/,- [n x 1?“th }ds =
[S {57, [5 x Henna x firef] }dS
a

—jk0Y0 / [5 {Vi/,- [5x562 (715”) xri’] .E'ext (73’) )ds’ds. (3.54)
a

The exterior excitation field is composed by ginc and firef this is,

ff” 2 fsa {17:7,- - [n x Hm + 5 x firef] )ds. (3.55)

Replacing (3.55) in (3.54), the boundary integral expression becomes

[80 {147,- - [n x Heft (75)] )ds +

jkOYO / [S {157,- [a x 582 (5', F') x 5'] .1158“ (5") }dS’dS = ff“. (3.56)
a

The boundary conditions at the aperture of the cavity, this is at z = 0, are given by

2 x (Hm — Him) 0 = 0, (3.57)
z:

61

2 x (Be-”Ct — Em) 0 = 0. (3.58)
2:

In order to couple (3.15) and (3.56) the continuity condition of the tangential compo-

nent of the magnetic field (3.57) is enforced at the aperture; hence the FE-BI equation

takes the form

5 .{me'a-(we'mwiev
_k857~/V {Ix-Vi .Eint}dv

—1.~3//SG{W,- - [a x 562 (7175’) x 5’] .1581“ (5") }dS’dS
= ff” + jkozoffxt. (3.59)

In order to enforce the continuity condition of the tangential component of the elec-
tric field (3.58), the interior and exterior electric fields are expanded with the same

subdomain-vector-basis function this is,

N
“int _ “ext __ . " .
E _ E _ ZIEJW], (3.60)
J:

where N is the total number of edge-unknowns in the element.

62

In this procedure, Galerkin’s method is utilized, this is, the expansion basis function
(Wj) are the same as the testing basis function (VI/i).

When the field expansion (3.60) is substituted into (3.59) elemental matrix is obtained

§E,{#iT/V{(VXW,).(VXW,)}W

J=1

—JZEJ- {kg/[Sa{Wz-- (ixGegxz’)-Wj}d3’dS}

= fgnt + jkOZOffxt. (3.61)

Assuming that the cavity is discretized using tetrahedrons, and the tetrahedron’s
edges follow a global numeration; a linear system of equations can be obtained from
(3.61). This linear system of equations is summarized in a matrix-vector notation

given by

[A] {E} = {'0}, (3-62)

where {E} is the unknown coefficient vector, {b} is the excitation vector, and [A] is

a matrix which is composed of two parts:

63

AFE APE AB1 0

[A] = bb 1’" + bb (3.63)
AFE ARE 0 0
”lb 22

The first submatrix is the finite element (FE) portion and its entries are given by

31ij = “—1? v {(v x m) - (v x Wj)}dV — lager/V (W,- - Wj} dV, (3.64)

and the second is the boundary integral (BI) submatrix with the generic term

A5]! = 463/190 (W,- - (2 x 562 x z") -Wj} ds’ds. (3.65)

3.4 Antenna feed and load

In order to compute the antenna input impedance and radiation pattern, it is assumed
that there is not any external source. Therefore, ffxt in (3.61) is equal to zero. In

addition, assuming an infinitesimally thin current filament in the 3) direction for this

work, flint (3.9) becomes

fgnt = — 93020101,, (3.66)

where lz- is the length of the edge where the feed is located.
In a similar way, an impedance load of Z L [Q] can be modeled as an infinitely thin
pin. Assuming a pin of length l and coincident with the ith edge, the conductivity

must be a = l / Z L but j 2 0E; therefore, the load contribution to the global matrix

64

A is given by [22]

k z: k 212
a-z- 2] —-0—— L02 2W }:rd dy= J—Oi‘: (3.67)
”a ZL

Notice that (3.67) contributes to the main diagonal of the matrix at the location of

the unknown associated with the load position.

3.5 FE matrix entries using tetrahedrons

Tetrahedrons have the versatility to conform to many shapes. In addition, their
corresponding vector basis functions satisfies the divergence-free condition which is
sufficient to avoid fictitious charge within the element. The convention for the local
node and edge numbering for each tetrahedron in this dissertation is shown in Figure

3.2.

Each of the six edges is represented by a vector basis function that has the form [17]

We = (Li1VLi2 — Li2VLz'1) lie (368)

where l,- is the length of edge 13, (2' = 1, 2, ...6), and Li] and L12 are the scalar-basis

functions of nodes 1 and 2 associated to edge 2'. Lil is defined as

1

Lil : 6V6 (a

fl+bflx+c €1y+df1z). (3.69)

L32 has a similar expression to (3.69). The gradient of the node-basis function Lila is

constant and normal to edge 2'. The vector-basis function W, has important properties

 

 

 

Figure 3.2. Tetrahedron element and local edge-node numeration.

66

as well. It has a constant tangential component along the edge 2' and linear normal
components along all the edges (CT/ LN). The divergency of W, is zero and its curl is
not zero. In addition, vector-basis function W, satisfies the continuity of the tangential
field across the edges of each element.

The constants a5, bf, CE, and (if in (3.69) are found as follow: Let (be be a first-order-

polynomial function. Within the tetrahedron, (be is approximated by

qbe (:c, y, z) = a6 + best + cey + dez, (3.70)

where ae, be, CB, and d8 are unknown coefficients.
Enforcing (3.70) at the four tetrahedron nodes the following system of equations is

obtained

65? (:1:, y, z) = ae + bexl + ceyl + dezl

(b5 (:1:, y, z) = (16 + b61132 + ceyg + dezg

(1)3 (1:, y, z) 2 ae + b81133 + cey3 + dez3

(15265, 31,2) : a6 + b65134 + 66314 + (16134 (3.71)

Each unknown coefficient is expressed in terms of the global coordinates :1:, y, and z

67

f the four nodes
0

1
“e = 6V6
1

be"6ve
1

Ce_'6ve

 

 

 

6
35

((53

1'2

5‘73

5’34

$1

5’32

$3

$4

'91
1‘12
313

"J4

3’1
312
y3

314

8
35

4’3

21

z2

23

Z4

31

Z2

23

Z4

31

Z2

z3

Z4

 

 

 

=6Ve

:6Ve(b

=6—ve (

68

66)
6 a¢4
-+a§6§-+a§634— 4
(01¢1

+”4‘154)
§6e+b§6§+b363

C e¢4l
+ C2452 + C3¢3 + 4
C1¢1

«172)

(3.73)

@174)

1 1'1 yl (15613

1133231263

_6V6 e 2676
113 y3 $3

((15633 + d§6§ + d§6§ + 63163) (3.75)

 

 

1:134 y4 62

where V6 is the volume of the tetrahedron which is calculated as

1x1 yl zl

11132 y2 2:2

<1
(D
II

(3.76)

OBIH

1 3:3 313 23

1 :54 M Z4

 

 

or expanding the determinant, equation (3.76) takes the form

v6 = if (x1 — e4) [<42 — 44) (e3 — 24> — (313 — .44) (22 — 261+

(.111 - 314) [(22 - 24) (I133 — $4) - (Z3 — 24) ($2 - 134)] +

(21 — 24) [($52 — e4) (:43 — 44> — (e3 — e4) (42 — 44>] } (3.77)

From (3.72)—(3.75), the coefficients ae,be,ce, and d6 are expressed in terms of the

global coordinates of the four tetrahedron’s nodes as

69

ai = $3 (21325 - 732:?) + 315 (451851 - 22335) + 2:5 ($3.93 - 95851)

05 = 8‘1 (145133 - yfizei) + 1!? (zeim‘é - 23351) + 3? ($5193 - 93:03)

a3 = mi (9523 - 9&3) + vi (2583 - zfix‘é) + 2i (233%? - 9582)

051 = 8i (.7525 - 752%) + 7‘1 (2516 - 25133) + 2‘1 ($53.75 - 173175)

bi” = (.7225 - 9525) + .715 (25 - 25%)) + ZS (95 - vii)
be = (31522 - 31225) + 7‘1 (ZS - 251) + zi (212 - 95)
e

be = (31225 — 11523) + 91 (3:1 — 25) + 213(95— 31:)

bi = (21525 - 9%) + 2113085 — ZS) + Zi (3:3, - 95)

70

(3.78)

(3.79)

(3.80)

(3.81)

(3.82)

(3.83)

(3.84)

(3.85)

df = (95.55, - $3751) + $5 (:93 - 313)) + 95 ($3 — xi)

0’28 = (5639461 — $393) + 1i (313 - 93) + vi (1351 — 1‘3)

d3 = (7513481 - $53451”) + If (951’ - 25) + 9i (135 - 251)

df = ($5313 - 3,555) + mi (315 - 215') + 71363 - 85)

(3.86)

(3.87)

(3.88)

(3.89)

(3.90)

(3.91)

(3.92)

(3.93)

Continuing with the procedure followed in [22], (3.64) is divided in two elemental

matrices given by

71

1 .. _.
8 _ _ 7. . .
Em. _ W 96 (v x 14,) (v x W]) dV, (3.94)
and
e — "' . . _. .
FW- _5,. [Re W, WJdV (3.95)
where
AF-E =Efj — kOFfj. (3.96)

Using the vector basis function (3.68) the curl term becomes

 

[53(Cz'1dz'2 0726131) + 9(d31b12 - bildz'2) + 5’ (b31012 - 011162)] ,
(3.97)

as before the sub indexes 2'1 and 22 correspond to the node 1 and 2 of edge 2' respec—

tively. This local conection is given on Table 3.1 as shown in Figure 3.2.

Since (3.97) is constant, it can come out of the integrand on (3.94), therefore, this

expression becomes

72

Table 3.1. Tetrahedron edge—node local conection.

 

Edge N1 N2
1

 

 

 

 

ibNWrbCJDM

 

0101pr
barb-MHr—tv—I

 

Ee.=fle_‘”f_[(celde2_delce2)(c; 1;.“ (15165.91.
1,] [1.7“ (til/(3)4 Z Z 7, 'l

e e 6 C C 6' e e e e e e C

In order to evaluate (3.95), (3.68) is expanded as

—O

l- A , - .
Wz' = 5&5 {(Lz'ibfz — Li2b51) 5‘? + (1361652 — ”29161)?! + (Lildfz — Li2df1) Z}-
(3.99)

If equation (3.99) is replaced in (3.95), the next expression is obtained

73

Erl'l'
9.: 23 . <3 _ . e . e: _ . 9
172,] (6Ve)2{./Qe (Lzlbz2 Lz2bzl)(L]1b]2 Lj2b]1)dv+

 

[fie (LilCSZ — Luca) (lecsg _ L12C§1)dV +

e e e e
ffie (Lad,-2 — 6,262.1) (lede — ngdj1)dV}. (3.100)
Each one of the integrands on (3.100) are evaluated using the following general formula

obtained from [38]

klllmln!
(k+l+m+n+3)!'

 

[9 (146927 <Lg>m<L4>ndv = eve (3.101)

3.6 BI matrix entries using triangles

Before equation (3.65) can be discretized, some vector and dyadic manipualtion need

be carried out [39]. Substituting (3.51) into (3.65) yields

4 = -J'k0R _.
AzBlzkgf/ Wi- 2x ”:7; e—— x2 -Wj dS’dS.
"7 Sa 55 k0 27TH
(3.102)

Equation (3.102) is split in two parts given in the next two expressions

74

 

%/ / {[W( ‘ x7 ) W](e——jk0R ds’ds (3103)
zx xz - - , .
21, 5a 5' z J R

and

 

H6 —1 W * vv (MGR * W ds’ds 3104
z,j—§;-[9a[98 z‘ ZX R X“ - j ’ (. )

where A231: 1631;]. + 1163]..
Replacing (3.20) into (3.103) and evaluating the first vector-dyad cross product, I f j

takes the form

 

1‘? . = — f / [W - ((1)6 — 63)) x 2) - W] e dS’dS. (3.105)
2,] 27r 5a $5 2 .7 R

If the dyad-vector cross product is evaluated on (3.105), this becomes

—- ”k R
I”: g/Sa [95,“ W Wj)(e JRO )}dS’dS. (3.106)

Equation (3.104) has a third order singularity that can be reduced if the derivatives

 

are transferred to the basis function. With this in mind, it is found that

—o A e_jk0R A _‘ €_jkOR
W,- . z x VV R = — (z x W,) -VV R , (3.107)

75

 

 

thus, equation (3.104) is written as

 

e ‘_ —— A ' o . A . I
Hij— 2W [Sa./Sé{(zxwz) vv( R ) (szJ)}deS, (3.103)

and further, since

 

 

vv e—jkOR vv’ e—jkOR 3109

equation (3.108) becomes

 

— "k R
8 __l_/ A #7. ' I 6 J O . A d. I
Haj—27F 5a 53 (szz) W R (szJ) d8 d5. (3.110)

The unprime gradient can be taken out of the integrand with respect to prime coor-

dinates, therefore (3.110) gives

 

H§j=§1;IgG{(2vai)-vl[gév’ (B‘JEOR).(2xw/j)ds']}ds,

from the vector identity,

fl-Vw=V. (will) —wV-/l (3.112)

76

the inner integral of (3.111) is recasted as

 

 

 

(331V,
[5, ((53:01?) V’. (2 x Wj)} dS’. (3.113)

Applying the divergence theorem to the first integral in the right hand side of (3.113)

 

yields

 

 

 

jg»! { ($3201?) (2 x Wj) m} dl’ (3.114)

where C’ bounds SQ, i.e. the perimeter of the triangle element and the unit vector
1% is tangential to SQ. The closed contour integral vanishes because of the properties
of basis function used to expand the fields. The proof of this statement is given in

Appendix 01. Therefore (3.113) becomes

77

 

 

[3, {(e_J:OR) v’. (2 x Wj)}d3'. (3.115)

Thus, using (3.115) and reorganizing some terms, equation (3.111) takes the form

 

1 ~ e_jk0R ~
e __ A ,’, . I. A . I
Hij_2W[9a[gé{(zxiv,) v( R )v (szJ)}deS. (3.116)

Using similar arguments to obtain (3.113) and the divergence theorem, equation

(3.116) is written as

e—jkoR

 

ng=%[ga/Sa(v.(mew/(2x33)(

) } dS’dS (3.117)

whose kernel singularity is two orders lower than equation (3.104) kernel.

For the exterior basis function, it is assumed a triangular mesh covering the aper-
ture. Let 2 x E (:1:, , y, ) = —A-f, where E (50’, y’) is the electric field in the aperture;
using Rao, Wilton, and Glisson (RWG) basis function [40], the magnetic surface

current on the aperture A? is expanded as

78

M = Z M,f,‘- (F) (3.118)
2'

where

1' ~+ +
2A:— pz. 1' in T2
f; (7“) = 27112352.— 1‘1an (3.119)
2'
0 otherwise

and [2' is the length of 3th edge that is shared by triangles T 1+ and T 27. A3: is the
area of TZ-i; the vector 5:" is directed away from the free vertex of T2.+ and [1' f is
directed towards the free vertex of T”; as it is shown on Figure 3.3. It is important
to notice that each basis function is associated with an interior edge, shared by two
triangles, everywhere else, this basis function vanishes.

RWG basis function is attractive to model surface current within triangular regions
because it only has a component that is normal to the 3th edge (Figure 3.3) and there
are not normal components to the remaining edges. In addition, the component of

current normal to the 3th

is constant and continuous across the edge, i.e., the normal
component of flit along the edge 3 is just the height of triangle Tii which is 214ii / ln.
This factor normalizes f; (7") and its flux density normal to the edge 2' is one. Current
continuity is thus preserved and all the edges of T271" and T2:— are free of line charges.
Using Gauss’ divergence theorem, it can be proved that the basis function (3.119)

are free of charge accumulation on each triangle. The detailed proof can be found on

Appendix C2.

79

 

X

Figure 3.3. Two triangles sharing an edge and their geometrical parameters.

80

Equating (3.118) with the unknown electric field coeficients (Ei)

2 x ZE,W,- = —§:M,-f;, (3.120)
1' 2'

and noticing that 57:): = i (F — r72), the RWG basis function becomes

 

W1: ‘54?" [(93 — $1) 3? — (II — 311') :8], (3-121)
i
and further
_, —l-s-
V- 2xW- = H. 3.122
( z) 7% ( )
Substituting (3.121) into (3.106) yields
—l-l-s-$- ‘jkOR
e_zyz] _./_. _.I__3 I
— m/t, {10 ...,) (. .,) + <21 y» (y 01—. }ds
3 .7
(3.123)
In addition, after using (3.122), equation (3.117) takes the form
613-3sz e‘jSOR I
Hij: 2::AiAj fi/Sa [3,6 d3 (15' (3.124)
where
+1 Fin T:—
—1 Fin Ti—

81

and

+1 F in Tle
s- = 9 (3.126)
—1 7" in T.-
J
3.6.1 Numerical integration of BI equations
The integrals in equations (3.123) and (3.124) do not have closed form and must be
evaluated numerically. The most efficient way to evaluate them is using the Gaussian

quadrature rule for triangles. The normalized area coordinates [38] are defined as

follows

A1
Aq

A_2
Aq

A_3

C1: Ag

(2: (3.127)

C3=

where C,- (2' = 1, ...3) are known as simplex triangular coordinates and satisfy the
following constraint (1 + C2 + C3 = 1. The areas A1, A2, A3 and Ag are shown on

Figure 3.4.

The transformation from Cartesian to normalized area coordinates is written as

172' = (1721 + C273 + (37-33 (3.128)

th

where 7”z is a position vector from the origin to the z' vertex of triangle T9 as it is
shown on Figure 3.4.

Using Gaussian quadrature rule [42], the surface integral over the triangle T9 of a

function C(f’) is given by

82

 

 

 

X

Figure 3.4. Local coordinates for triangle T q.

83

1 1—4
[Tq G (’2) d5 = ”(I/O [0 2 G [(1771 + C2772 + (3773] d61d<2

n

z Aq Z WiG [(1571 + (562 + (5773] . (3.129)

721

In this, n is the number of integration points over the triangle, (3- are the triangular
coordinates, and W,- are integration weights. These parameters are given in Table
3.2 where (11 = 0.0597158717, B1 = 0.4701420641,072 = 0.7974279853, and 52 =
01012865073.
For the case of (3.123) and (3.124), the integrals are evaluated over a source triangle
(Tp) and observation triangle (Tq) instead of edge-by—edge. After computing the
integrals for each triangle, the edge contribution are calculated and accumulated in
the appropriated BI matrix position. This is a full matrix and because of (3.123) and
(3.124) form, it is symmetric. Therefore, it is only necessary to compute the upper

(or lower) triangle of the matrix.
3.6.2 BI self term evaluation

When the observation and the source triangle coalesce, the singularities in the kernels
of (3.123) and (3.124) must be isolated and evaluated analytically. Equations (3.123)
and (3.124) may be rewritten in a more convenient form expressing global as local

coordinates using the expression

84

Table 3.2. Triangular coordinates and integration weights [38].

 

 

 

Number of points (i, C2" (5 W7
1 1/3,1/3,1/3 1
1/3,1/3,1/3 -27/48
4 0.6,0.2,0.2 25/48
0.2,0.6,0.2
0.2,0.2,0.6

 

1/3,1/3,1/3 02250000000

01, 31, 61 01323941527
31, 01, 51
7 51.51.01

012, Hz, 52 0.1259391805
[32, 02, 52
32, fig, 02

 

 

 

 

 

1w» (aw-x.) + w» (xx—01 = (~72) - (77-6) =5:- 6-

 

(3.130)
Substituting (3.130) into (3.123) yields
—l l- -s s +Je- jkOR
IP’FI— ‘9 ——"—j/Tp [p 7+ ,6. ’d .131
2,] 877APAq qu d5 8 (3 )

The inner integral of (3.131) can be written as

85

 

/ fideSl=f fi‘l’e—JkOR_1dS’+
Tq J R Tq .7 R

 

—o, —0
P —P / .. 4 1 I
dS — - —dS .132
/Tq R +(p pJ)/TqR , (3 )

where 13', p”, and [93 are the projections of the position vectors 7", 7"", and 73- respec-
tively as it is shown on Figure 3.5.
The first integral in the right hand side of (3.132) is bounded and can be numerically

evaluated. The last two integrals can be evaluated analytically using [41] (Appendix

D). The term containing uniform source distribution, i.e.

1 I
—dS , 3.133
/T'q R ( )

it is evaluated using Appendix D.1.

In addition, the term containing linearly varying source distribution, i.e.

 

../ _ _.
qup R pdS’, (3.134)

it is evaluated using Appendix D2. The same process is repeated for the outer integral
of (3.132).

The scalar potential in equation (3.124) may be rewritten as

86

   

'1
w

 

 

Figure 3.5. Projections of F, F’, and Fj onto the plain that contains Tq.

87

[,1 s s e—JkO R
J 3 J I
Hf’ qu :2—I—77ApAq/Tp /Tq{-——— }dS dS. (3.135)

The inner integral of (3.135) can be expressed as

 

e—JkOR , e—jkOR _ 1 I 1 I
“...”? [a M R ds./Tq{,}ds (3.136)

The first term of the right hand side of (3.136) is bounded and can be evaluated
numerically, the second term can be evaluated analytically as shown in Appendix

D.1.

3.7 FE—BI program structure

In the previous sections it was outlined the formulation of the FE (3.98)-(3.100) and BI
(3.132)- (3.135) elemental equations. In this work, these equations are implemented in
a F E—BI program written in Fortran [44]. FE—BI programs have common modules [45],
[46]. These modules are the pre-processing, meshing, pre-assembly, matrix assembly,

matrix solution, and post-processing.
3.7 .1 Pre-processing

In this module information about the problem to be solved is collected. This in-
cludes: geometry specification, electric properties of material, type of excitation, and
frequency. In this project, the geometry is an open cavity filled with a uniform di-

electric and the excitation is an infinitely thin coaxial probe.

88

3.7.2 Meshing

Since the TLW antenna has a simple geometry, the meshing starts with the specifica-
tion of a surface triangular mesh. This mesh is generated using SkyMesh [51]. Figure
3.6 shows different apertures that are investigated in this work. The rectangular, the
bow-tie, and the circular are considered in order to design the most efficient TLW
antenna. Of course, these surface meshes are not to scale because the TLW antenna
considered in this work is several wavelengths long (i.e., 7A) and the width is in the
order of 0.1/\. Note that because of the FE—BI formulation in this work, the open
aperture is discretized but the infinite ground plane does not need to be considered.
The surface mesh is extruded to obtain a volumetric mesh. This extrusion process is
3 follows: the number layers and their thickenss needs to be specified. The surface
nodes given in the surface mesh are replicated and displaced by the thickness of each
layer. This creates prisms in each layer. Then, edges are formed for each prism. Fi-
nally, tetrahedrons are created from the prisms [52]. The information that is obtained
from the meshing process is organized in form of arrays. The T etNodes array con-
tains the global nodes forming each tetrahedron. The GobalNodes array contains the
global node location on Cartesian coordinate system. The EdgesNodes array contains
the nodes that form each edge and associated with it is its corresponding unknown
number. The TetEdges array identifies the global edges that are associated with each

tetrahedron. The Tm'Edges array contains the edges that form the aperture triangles.

89

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(b)

 

 

 

A:
$3!

 

. I"
\ . ~
I ‘\
..

 

 

 

 

 

 

\
)r' / (\1
”"J‘

\
,//
/

M

 

 

 

 

 

 

 

 

 

 

<1
/

 

(C)

Figure 3.6. Different configurations of surfacemesh: (a)Rectangular, (b)Bow—tie,
(c)Circular.

90

3.7 .3 Pre-assembly of FE—BI matrices for the TLW antenna

Pre—assembly is used to determine the topology of the FE—BI matrix before computing
its actual entries. The FE part of the matrix is sparse and the BI part is dense. As
mentioned before, the cavity is discretized with tetrahedra and the aperture with
triangles. Each one of the edges that form these elements is an unknown if it is not
on the metallic boundary of the cavity. In other words, that edge must belong to
two different elements. The dimensions of the global matrix depends on the number
of unknowns, for a typical TLW antenna with dimensions 7A x 0.07/\ x 0.4/\ (where
A = 3.33cm) and discretized using 28, 000 tetrahedrons, the number of unknowns is
22, 700. When the number of unknowns is high, it is not practical to store directly the
FE matrix and the BI matrix. For instance, for this T LW antenna, the number of non-
zero elements in the FE matrix is 222, 982, which means that there are 515, 067.020
(99.95%) entries that are zero. Therefore, it is important to use a storage scheme such
as compressed sparse row (CSR) method [17] in order to avoid storing all those zeros.
In this scheme, the values of nonzero elements of the sparse matrix are stored in a
vector (values) and their corresponding column indices in another vector (colIndesr).
The dimension of each one of these vectors is the number of nonzero elements. Also,
there is another vector (numberElemeow) with the number of nonzero elements
per row. The dimension of this vector is the number of unknowns. In addition, since
the BI matrix is symmetric, it is only necessary to compute the elements in the upper

triangle of the matrix. Thus, the storage saving can be significant.

91

3.7 .4 Matrix assembly

In this process the matrix entries are computed and stored in an orderly fashion using
the CSR technique. The matrix is assembled adding the contribution due to each edge

pair of interaction for each tetrahedron.

3.7.5 Matrix solution of FE-BI linear system using biconjugate gradient

(BiCG) method with preconditioning

In order to solve the FE—BI linear system the BiCG method with preconditioning is
used [17]. This method is easy to implement and present fast convergence for the
TLW antennas simulated in this dissertation. The preconditioner used is diagonal
due the simplicity of computing its inverse. The implemented pseudocode in this

dissertation is given in [17] and repeated here:

92

the unknown solution vector.

3.7 .6 Post-processing

Initialization:

x is given

r=b—Ax; p=r; tmp=r~r
Repeat until (resd S to!)
U)q=Ap

(2) a=tmp/(q-p)

(3) x=x+ap

(4) r=r-aq

(5) q=inv(C)><r

<6) resd=flr—-F*‘I

(7) fi=(r-Q)/tmp

(8) tmpzflxtmp

(9) p=q+fip

EndRepeat

where Ax and Ap denotes a complex matrix vector product, C is the preconditioning
matrix, q, p, x and r are complex vectors; (1,6, tmp are complex scalars; resd, tol

are real scalars. An initial guess is given at the beginning of the algorithm and x is

Once the field coefficients are found from solving the linear system (3.62), different
parameters can be computed such as the antenna’s input impedance, reflection coef-

ficient, voltage standing wave ratio (VSWR), and radiation pattern. In addition, the

93

TLW antenna’s propagation constants can be calculated from these coefficients.

3.7.6.1 Input Impedance, reflection coefficient, and VSWR

The input impedance of an infinitely thin probe feet with a current IO and located

on the ith edge is given by

El-
z- = ——-’¢ 2.
2n IO

(3.137)

The reflection coefficient (F) and the VSWR (assuming a 50!) reference impedance)

can be found using

 

P_Zm—50
Zin+50
and
1+|I‘|
W = .
VS R l—[F]

3.7.6.2 Radiation pattern

The far-zone radiated field of an aperture antenna is given by [29]

E: _'1im {33 x Fm}

'r—+oo 5

where 7" = isin6003q5+ fisinBsin¢+ 20086.

The magnetic vector potential F (1") can be found from

94

(3.138)

(3.139)

(3.140)

 

 

4“ (r) = fie—W f9, ('3' X E (5)) e‘jkf'F'dS’ (3.141)

T

where 71' = 2 and ”F" 2 33,5: + y'g’).

Using the expansion of the electric field in each triangle

3
E=ZQi mm

where the electric field coefficients E,- are now known and the vector basis function

is given on (3.121). Therefore (3.141) becomes

—56 —jkr3

 

130":

E E- ,f (2’ x viii) e—jkf'FIdS/. (3.143)

Performing the cross product 2' x W- and substituting this result into 3.143 , it
2

is found that

# —jkr3
F (r: _e

 

. « ~/
2;... WK . _.,..+) (, 444W
(3.144)
The surface integral in (3.144) is computed using Gaussian quadrature (3.129) and

summing the contribution of each triangle in the aperture.

Hence, using (3.144), the electric field (3.140) takes the form

95

E = — {—iFy cosO + 9F; c086 + 2 (Pg sin6cos¢ — F3; sin6’sin (3)} (3.145)

 

 

s
where
_e—jkr (2'5
Fa:(r= 2 215.2711qu (IE—177;)6—jkrrds,
Tq=12=1
F _e—jk7‘ — —jkrr dS’ 3146
y(7‘= 221322,]: (31 yze (-)

Tq—12=1

where N is the total number of triangles in the aperture.
Expressing (3.145) in spherical coordinates system, the 6 and 45 components are,

respectively,

E9 2 Ex cos6cos¢+ Eycosflsingb — E2 sin0 and

E¢ 2 ——E1; sin 45 + By cos (I). (3.147)

The radiation intensity U (6, (25) is defined in (1.14) and it is repeated here:

U(9,¢) = % [IE4(6,4>)|2 + |E¢(0,¢)[2]. (3.148)

96

The total radiated power is obtained by integrating the radiation intensity given by

(3.148) over the entire solid angle of 47r. Thus

2
Fwd: £2 U(6,qb)d§2= [0 7r /O7rU(6,¢)sin6d0dq> (3.149)

Numerically, the total radiated power may be found using

7r 27r M N ,
Prad 2 (IV) (Tl?) jzzlzzzl U (631¢j)3m63 (3.150)

where the 0 and ()5 coordinates were divided in N and M divisions respectively.

The absoulute gain of an antenna in a given direction is defined as “the ratio of
intensity, in a given direction, to the radiation intensity that would be obtained if the
power accepted by the antenna were radiated isotropically. The radiation intensity
corresponding to the isotropically radiated power is equal to the power accepted
(input) by the antenna divided by 47r” [30]. In equation form this can be expressed

as

 

gain :2 47rU (6’ <15) (3.151)
Pin
where the Pin is given by
P,” = IO - {I0}* ~Real {Z,,,} (3.152)

the Operators {}* and Real {} denote complex conjugate and real part, respectively.

The antenna efficiency is defined as the ratio between the radiated power and

97

input power:

P
efficiency 2 M (3.153)
-3

The power absorbed by a load impedance may be computed using:

IEWd
.P = @130
RL 23L

 

where E, is the electric field coefficient at 3th edge and this edge is coincident with
the location of the lumped load. l, is the length of the ith edge and R L is the real

part of the load impedance.
3.7.6.3 Propagation constants

The propagation constants for the TLW antenna are determined using the method of
least squares [43]. Given the data {(231,341) , (2:2,y2) , . . . (xn,yn)}, the error associ-

ated toy=ax+bis

N
e (a, b) = Z [yn — (mm + b)]2 . (3.155)

n=1

The values (a, b) that minimize the error are such that

Be Be _

g _ , 53 _ 0, (3.156)

Differentiating e (a, b) yields

98

N

8
‘8“: = “2 Z lyn — (Gin + bllicn
7221
Be N
5522 2:1[yn—(axn+b)].
7),:

Using (3.156), it is found that

N N N
a:x%+bzmn= Zinyn
11:1 71:1 13:1

N N
a: xn+bN= 23m.
n=1 n21

Solving for the slope in (3.158) this parameter becomes

N 271:1 xnyn '— 25:11:77. 2712;13111
2
N23,,_ 1 3571.— (25:1 3371)

and the independent term is given by

a—

 

b: 2
N 2371117” (2712]:11’71)

Zyjzv—fl 3917. EN —_-1 3m — Z712]:1 5571 2521113713173

(3.157)

(3.153)

(3.159)

(3.150)

In order to compute the propagation constants, it is assumed a TLW antenna

with the aperture at z = 0 and the length along 3:. The electric field in the y

direction is obtained from the values of the expansion coefficients on each edge

99

in the y direction. For each edge along the aperture there is a Ey, this is,

(3:1, Eyl) , (11:2, Egg) ,... (:rn, Eyn). Since

e“a$k0$ej5$k0$ = By, (3.161)
therefore,
kofix 2 angle (Ey) . (3.163)

Using (3.159) ax and 53; are estimated as

NZn_ ll‘nln[Eyn]2- Z3=nN11L'7123n—1lnlEynl

 

 

—k0(11:= (3.164)
2
and
19051:: N2”: 1 am [angle (Eyn)]2 - —2117Y=1 :1:”, 271221 [angle (Eyn)] (3 165)
2 .
where
angle (Eyn) = arctan Im (Eyn) (3.166)

 

Real (Eyn) '

100

CHAPTER 4

ANTENNA DESIGN AND RESULTS

4.1 Design of the RT / Duroid 5880-filled TLW antenna

In order to find the optimum dimensions for the TLW antenna filled with RT / Duroid
5880 the TRM method is used. This method allows to specify h and 212 (Figure 2.1 (a)).
Figure 4.1 shows the effect of three different values of h in the propagation constants
while 11) is kept constant. If the antenna bandwidth is defined as the frequencies
between 02 /k0 = 62/190 and 6 z / k0 = 1, the antenna bandwidth for these three values
is shown on Table 4.1. The greatest bandwidth value happens when h = 0.75cm and

w = 0.0787cm.

Table 4.1. RT/Duroid 5880-filled TLW antenna banwidth for three different values
of h.

 

 

w[cm] h[cm] Bandwidth[GHz]
0.0787 0.75 2.2
0.0787 0.80 1.9
0.0787 0.85 1.8

 

 

 

 

 

Similarly keeping h constant and varying w the behavior of the propagation con-
stant is shown on Figure 4.2. The antenna bandwidth for these three values is shown
on Table 4.2. Again, the greatest bandwidth value happens when h = 0.75cm and

w = 0.0787cm.

101

 

1.6 I I I I

 

 

 

 

 

 

 

— az/k0(w =0.0787cm)
“ 1-4 - - I :-
é — leko(w 0.0787cm)
% h-0.85 cm
0 1.2 "
0
c , u- "
.9 ' ' ' O ‘
*5 1L h=0.80cm , 1" ,.r
O)
to
O.
9 ..
n.
'D
a:
.5 _
E
E
o
z _
0 l J l l ...._ 7
4 4.5 5 5.5 6 6.5 7 7.5 8

Frequency (GHz)

Figure 4.1. Effect of the variation of h in the propagation constants for the RT/Duroid
5880-filled TLW antenna.

102

_s
03

 

 

 

 

 

 

 

—az/k0(h =0.756m) 1
E1.4- _
.g w=0 23 cm .. .. . Bz/ko(h =0.750m)
E .
31.2 _.
0 .‘c‘-’
g 1 \‘ "v" “"o“
8) a!" v" ’7‘;
g ’r’ ”v ”’¢
5 08 " ’ I ~
0. 1.... ,
'8 ” ,’ w=0.15cm
N I I
'7'; 0.6» -
I

E ’ ’
o 04 --"' f I@\
Z ' ’ \ z w=0.0787 cm

0'27 a"\\ _

0 I l \‘k

 

A
4 4.5 5 5.5 6 6.5 7 7.5 8
Frequency (GHz)

Figure 4.2. Effect of the variation of w in the propagation constants for the RT / Duroid
5880-filled TLW antenna.

103

Table 4.2. RT/Duroid 5880-filled TLW antenna banwidth for three different values
of w.

 

 

w[cm] h[cm] Bandwidth[GHz]
0.0787 0.75 2.2
0.150 0.75 1.5
0.230 0.75 0.9

 

 

 

 

 

Since the TRM assumes an infinitely long antenna, in practice it is necessary to
specify its length L. For this, the 90% radiated power formula, equation (1.20), is
used. It is assumed that the antenna is designed for a frequency of 7 Ghz. At this
frequency az/kO a: 0.03, which means that the length is L = 25 cm. Finally, the ter-
minations must be specified. There are three options considered herein: rectangular,
bow-tie and circular termination (Figure 3.6). As a start point, a TLW antenna with
circular shape at both ends [47] is considered (Figure 4.3). Note that this design does
not require the addition of significant complexity. The design parameters are given

on Table 4.3.

Table 4.3. RT/Duroid 5880-filled TLW antenna design parameters.

 

L[cm] w[cm] h[cm] h’[cm] r[cm] er Load[fl]
25.0 0.0787 0.75 0.60 0.3 2.33 50.0

 

 

 

 

 

 

 

 

 

 

104

Infinite ground plane

 

 

 

 

 

 

 

(a)

i

a__.f
l

['V

M

Coax
probe [1’ gr

 

<—->
w
(b)

Figure 4.3. TLW antenna with terminations and design parameters. (a)Front view.
(b)Side view.

105

4.2 Code validation

Using the formulation given in Chapter 3, a FE-BI code is implemented in Fortran
90 [44]. This code is named ctets. As an initial test of the code, it is simulated the
TLW antenna designed in the previous section. In order to validate the implemented
code, the same design is simulated using a code provided by Dr. Kempel (tet). The
simulation results of real and imaginary part of the input impedance are shown on
Figure 4.4 and Figure 4.5 respectively. It is interesting to notice the oscillations
around F = 7.5 GHz in the real and imaginary parts of the input impedance. This
means that a backward wave traveling in Opposite direction (--z) starts to impact
significantly the antenna performance above this frequency.

The front-to—back ratio is defined as the difference between the forward and the
backward lobe on a decibel basis. In leaky wave antennas this ratio is important
because it is desired to optimize the radiation in the forward direction (i.e., 0° <
0 < 90°) . This optimization can be achieved using a termination load to reduce the
reflection wave at the antenna end. In addition, a further improvement is achieved
designing circular shaped baluns for the TLW antenna (Figure 4.3). The radiation
pattern at F = 8.0 GHz is shown on Figure 4.6. The load and baluns partially reduce
the reflected wave (front-to—back ratio of approximately 20 dB).

In general, the simulations shown on Figure 4.4, Figure 4.5, and Figure 4.6 validate
the results provided by the code developed in this work (ctets). The input impedance
and the radition pattern results obtained using tet and ctets agree excellently.

The VSWR for the RT/Duroid 5880—filled TLW antenna is shown on Figure 4.7.

106

 

[f

Defining the impedance bandwidth for a VSWR less than 2 as the percentage of the
frequency difl'erence (upper minus lower) over the center frequency of the bandwidth

[48]. The simulated bandwidth for the RT / Duroid 5880-filled T LW antenna is 15.71%.

 

 

 

 

 

 

 

 

 

65 l I I
- 0 -Rin(ctets)
. 1 — Rin(tet)
60 - ' 3
9 55 - -
8 .
1:
S
.‘L’ 3
8
c: 50 ~ -
45 - —
40 1 l l
6 6.5 7 7.5 8

Frequency (GHz)

Figure 4.4. Real part of the input impedance of the RT/Duroid 5880-filled TLW
antenna simulated using ctets and tet.

107

 

80 j I I

 

 

 

 

- O -Xin(ctets)
70 _ —x in(tet)
60 -
50 ~

Reactance (Q)
.5
O

00
O

 

20-

 

 

 

10 l l J
6 6.5 7 7.5 8

Frequency (GHz)

Figure 4.5. Imaginary part of the input impedance of the RT / Duroid 5880-filled TLW
antenna simulated using ctets and tet.

108

 

 

 

- O -ctets(F=8GHz)
tet(F=BGHz) .. -

 

 

 

 

    
    
 

 

Power gain (dB)

 

 

1 1 1 ' 1 1

-20 0 20 40 60 80
9 (degrees)

Figure 4.6. Radiation pattern at F=8.0 GHz of . the RT/Duroid 5880-filled TLW
antenna simulated using ctets and tet.

109

VSWR

 

 

 

l l l

 

 

6 6.5 7 7.5

Frequency (GHz)

Figure 4.7. VSWR of the RT/Duroid 5880-filled TLW antenna.

110

Other important parameters that can be studied to evaluate the behavior of this
antenna are the tangential electric field in the aperture of the trough (Ey) and the
propagation constant (kg = 62 — jag). Ey is computed from the values of the expan-
sion coefficients on each edge (in the y direction) along the aperture. Its magnitude is
shown on Figure 4.8. At F = 6.0 GHz, it is evident the exponential decay of the field
magnitude ([Eyl) with the distance. Oscillations of the field are significant at F = 8.0
GHz (Figure 4.8). The phase at three different frequencies (F = 6.0, F 2 7.0, and
F = 8.0) is shown on Figure 4.9, Figure 4.10, and Figure 4.11, respectively.

The propagation constant is computed using FE—BI and the least square method
described on subsection 3.7.6.3, in particular equations (3.159) and (3.160) are used
to compute the slope and the independent term respectively. Also, this information
is shown on Figure 4.12, Figure 4.13 and the ratio of these two on Figure 4.14. The
propagation constant results are compared with the ones of the TRM (Figure 4.15).
Even though TRM is an approximation, the agreement with FE—BI is reasonable.

In addition, it is possible to check the accuracy of equation (1.16) given in Chapter 1.
From Figure 4.15 flz/ko = 0.92 at F = 8.0 GHz, using (1.16), the maximum radiation

is at 0 = 660 which agrees the maximum value of the beam shown on Figure 4.6.

4.3 Air-filled TLW antenna

In Chapter 2, using the TRM it was proved that, in theory, the air-filled TLW antenna
have an infinite operation bandwidth (Figure 2.6). That is, fiz/ko never reaches a
value equal to unity. This is the reason to design and fabricate the air-filled TLW

antenna in this dissertation. The design procedure is similar to that given in section

111

 

|Ey|(Normalized)

 

 

1--- F=6GHZ
0.9 Q) I\ —F=7GHZ

‘ ’\ --- =
0.8 __ ’\ F BGHz

I

 

 

 

0.7 .
0.6

I

0.5

I

0.4

I

I

0.3

l

0.2

0.1

 

 

 

"-"‘-'-'-_l_-_L-_LI_-.1

-10 -5 0 5 10

Probe Position (cm)

Figure 4.8. Magnitude of Ey of the RT/Duroid 5880-filled TLW antenna.

112

 

200 I I

150 /
I
100‘

50*

Angle Ey (deg)

-150~

 

 

-200 1 L

I

 

 

—F=GG HZ

 

 

 

l

 

5

Probe Position (cm)

Figure 4.9. Phase of Ey at 6 GHz of the RT/Duroid 5880-filled TLW antenna.

113

10

 

 

 

200 r I . ,
— F=7GHz

 

 

150

100‘

50*

-501

Angle Ey (deg)

-1005

-150+

 

 

 

 

 

 

l l I l

-5 0 5 10

Probe Position (cm)

 

-200 . 7

Figure 4.10. Phase of Ey at 7 GHz of the RT/Duroid 5880-filled TLW antenna.

114

 

200 l l I r
— F=BGHz

 

 

150 ‘-

100 "

50 1:

Angle Ey (deg)

-100 1-

 

 

 

 

 

 

 

-150

 

 

l l l l

-5 0 5 10

Probe Position (cm)

-200 -

 

 

Figure 4.11. Phase of Ey at 8 GHz of the RT/Duroid 5880—filled TLW antenna.

115

 

—‘L
...
..
_.

 

o
(D
1
q
-
1:

 

.0

on

g
-

 

 

.0 .0
O) \l
9

.1

L

.0

(n

I
l

1-{‘

Normalized Slope Term
.0
h
i

p
on
I

.0
N

0.1

 

 

 

0 1
6 6.5 7 7.5 8

 

Frequency (GHz)

Figure 4.12. S10pe term computed via FE—BI and the least square method.

116

 

 

 

 

 

 

 

 

 

 

 

"0.02 I 1 1
1\
-0.04:
_a lk
g o o
- < I I I I [k
1;:- 0.06 ’ 1| BO 0
(u I
”a; I I "
8 ’ ‘ l’ 1
0 -0.08" ‘ , ‘
'O ‘ I
.g ‘I ’ ‘
To | ,
E -0.1~ t I ’ ’ ‘
1 I
I
912 V
-0.14 1 1 1
6 6.5 7 7.5
Frequency (GHz)

Figure 4.13. Constant term computed via FE—BI and the least square method.

117

0 T T l

—--~-‘-—-~--—------------J

 

   

 

 

 

 

 

 

 

-2.5. / -
_3 _ I I IBolfiz
'3.5 l l l
6 6.5 7 7.5 8
Frequency (GHz)

Figure 4.14. Ratio constant to slope term computed via FEBI and least square
method.

118

 

 

 

 

 

 

 

 

 

 

 

1 l l l
0.9_ “(ff—1":
.. 33"“
g 0.8 _ ...?" :1 -
é /’
m 0.7 " \’ _ —
8 ’ ‘ ”‘ ’ GZIKOCI'RM)
C - ’ ' - ’ I I I -
.2 0 6 ’. I pZ/koqRM)
g " 3’ mm-a/k (FE-Bl)
§O.5 7"!” Z 0 4
5: 0.4 _ , -----BZ/k0(FE-Bl) _
'O
Q)
.5
{—5 _
E
O
z _
0 1 1 H” . ' ' "'m—lfi’m'
6 6 5 7 7 5 , 8
Frequency (GHz)

Figure 4.15. Propagation constant of the RT / Duroid 5880-filled TLW antenna com-
puted using FE—BI and TRM.

119

4.1. It starts using the TRM to compute the propagation constants for different
values of h and 7.0. Figure 4.16 shows the effect of three different values of h in the
pr0pagation constants while 112 is kept constant. Similarly, Figure 4.17 shows the effect
in these parameters for different values of 111 while h is kept constant. From these two
figures the greatest bandwidth value happens for w = 0.2361 cm and h = 1.4 cm.
The next step in the TLW antenna design is to specify the antenna length using
the 90% radiated power formula, equation (1.20). It is assumed that the antenna is
designed for a frequency of 12 Ghz. At this frequency az/ko x 0.018 (Figure 4.17),
which means that the length is L = 25 cm. The design parameters are summarized

on Table 4.4

Table 4.4. Air-filled TLW antenna design parameters.

 

L[cm] w[cm] h[cm] h’[cm] er Load[Q]
25.0 0.2361 1.4 1.2 1.0 50.0

 

 

 

 

 

 

 

 

 

In order to optimize this design several terminations are considered. The rect-
angular, the bow-tie and the circular termination are simulated using F E—BI. Figure
4.18 shows these terminations; the points a —- b and a’ — b, denote the position for
the coaxial feed and load respectively. The rectangular termination has six degrees of
freedom (DF). (L, 11}, ET, p, h and h’). The bow-tie has seven DF (L, w, er, p, 0, h and
h’). The circular termination has six DF (L, 11), ET, r, h and h’). The FE—BI simulation

results have shown that the circular termination has the best response in terms of

120

 

 

 

 

 

 

1_4 ' 1 l 1 r
... —aZ/k0(w =O.2361cm)
C =
{:3 12 - - - -BZ/k0(w 0.2361cm)
a:
a:
o
o 1 _ _
.5 h=1.4 cm 1
E ‘ I - —-‘—”-
O) ‘ ‘ I ‘ I -
(U " I i I "' .1
8 v ' a " ‘ ‘-
0': ’ a ’ I “
3 K _
% h=1.2 cm
E
0 _
z

 

 

 

 

 

4 5 6 7 3 9 1011 12
Frequency (GHz)

Figure 4.16. Effect of the variation of h in the propagation constants for the air-filled
TLW antenna.

121

 

 

 

 

 

 

 

 

 

 

 

 

1 l I I I I I I
I II- - = = =-
09 h a! " = = =—‘—-a-¢‘ " - -1
'8 0.8 - w=0.2361 cm ' v‘ ’ v ’ a
E ,',I,I —a/k (h=14cm)1
8 0 7 E’ ’ ’ I ’ Z O .
' I
O I ’ I --- B/k (h=1.4cm)
c I I’ ’ z 0
1% 0.6 I ’ _
o: ’ R:
8 0 5 I, I 1 -
119 ' I , w=0.1574 cm
‘ I
8 0.4 A I .
.u (1 I
'5 I
E 0.3 r K -
E .
0-2 3’ w=0.0787 cm _
I
0.1 - ’ I § _
' ~ H
O l L 1 «—--— ‘:_ — Mum“—
Frequency (GHz)

Figure 4.17. Effect of the variation of w in the propagation constants for the air-filled
TLW antenna.

122

VSWR (Figure 4.19) and front to back ratio (Figure 4.20) of all three terminations.
The antennas with the rectangular and bow-tie terminations have a front to back
ratio of z 15 dB whereas the circular termination case has one of z 20 dB. The
circular termination parameters are shown on Table 4.5.

The input impedance bandwidth for the circular case is 40% (Figure 4.19) and
the front to back ratio is 20 dB (Figure 4.20). The input impedance for the circular
case is shown on Figure 4.21. Similarly to the RT/Duroid 5880 filled case, this input
impedance is dispersive along operational bandwidth. The behavior of the tangential
component of the electric field (Ey) along the antenna aperture is shown on Figure
4.22 through Figure 4.25. It is clear the exponential behavior of this field at low
frequencies whereas at high frequencies (9-12 GHz) standing waves start to appear
along the aperture. The propagation constants computed using TRM and FE—BI are
shown on Figure 4.26. The agreement between these two methods is reasonable.

The efficiency of the T LW antenna with circular termination is shown on Figure
4.27. The maximum efficiency happens at 12 GHz and is about 75%. The ratio
Pin / PR L for the TLW antenna with circular termination is shown on Figure 4.28,
where PR L is computed using (3.154). The maximum power absorbed by the load

impedance is 3.7% at 12 GHz.

Table 4.5. Air-filled TLW antenna circular termination design parameters.

 

L[cm] w[cm] h[cm] h’[cm] Err r[cm] Load[§2]
25.0 0.2361 1.4 1.2 1.0 0.3 50.0

 

 

 

 

 

 

 

 

 

 

123

 

1,.

 

 

 

 

 

 

 

'fi L 1|: 9 a"
I a er W a’: I
b (a) b’
p ,.
L )IL/\ 4‘0
D er w a’fiuu
b (b) b’
. L

 

 

b

(C)

 

 

 

Coax
probe 11’

1

Figure 4.18. Different terminations: (a)Rectangular (b)Bow-tie (c)Circular (d)Side

view.

L
I"
l
l‘
l .I
i],

£r

 

 

(:9
w

(d)

124

<-——='—-——>

 

 

 

 

 

VSWR
01

 

 

 

0 1111111
4 5 67 8 9101112

Frequency (GHz)

 

Figure 4.19. VSWR for the air filled TLW antenna with different terminations.

125

 

 

 

Power gain (dB)

 

 

 

 

 

 

 

 

 

 

l- - 1 l
-80 -60 -40 -20 0 20 40 60 80
9 (degrees)

Figure 4.20. Radiation pattern at 12 GHz for the air filled TLW antenna with different
terminations.

126

 

80 I T T l l

 

 

 

 

 

 

 

 

 

- - -Xin ..
a; _
a)
0
C
(B
'O
a: .
0. '5
E \
~
~ Q
5 - ‘
s _ ~ _ ~
1 l L l l
10 6 7 8 9 10 11 12
Frequency (GHz)

Figure 4.21. Input impedance of the air-filled TLW antenna with circular termination.

127

 

1 l I l f

 

. - . - -F=6GHz
0.9 1 —F=9GHz

I

- - -F=1ZGHz

 

 

0.8 '-

 

 

0.7' -

l

0.6

0.5 '

0.4 '

|Ey|(Normalized)

l

0.3

0.2 ‘

0.1 *

 

 

 

'.~..“LI_-.me.mml

1 I
0 -1o -5 o 5 10
Probe Position (cm)

Figure 4.22. Magnitude of Ey of the air-filled TLW antenna with circular termination.

128

 

200 T l I I

 

—F=6GHZ
150 - -

 

 

1003 -

 

50- -

 

Angle Ey (deg)

-50 i- _

 

-1oo_ —

 

 

-150~ .

 

 

 

-200 l 1 1

—
—

 

Probe Position (cm)

Figure 4.23. Phase of Ey at 6 GHz of the air-filled TLW antenna with circular
termination. ‘

 

129

 

200 I l I I
— F=9GHZ

 

150-

100'

 

50-

Angle Ey (deg)

-100 ‘

 

 

 

 

 

 

 

-15O

 

-5 0 5 1 0.

Probe Position (cm)

-200 -

 

Figure 4.24. Phase of Ey at 9 GHz of the air-filled TLW antenna with circular
termination.

130

 

200 ' ' ' —F=1ZGHz

1 M M-

100 " ‘

 

 

50 " ‘

 

 

 

 

 

 

 

Angle Ey (deg)
0
1‘1

 

 

 

 

-1oo"

 

 

 

 

 

 

 

 

 

 

 

 

-150

 

-200

 

Probe Position (cm)

Figure 4.25. Phase of Ey at 12 GHz of the RT/Duroid 5880-filled TLW antenna with
circular termination.

131

 

 

 

 

_az/k0(TRM) _
- - - BZ/kO(TRM) _
m-maZ/k0(FE-Bl)
----:BZ/k0(FE-Bl)

 

 

 

Normalized Propagation Coefficient

 

 

 

 

 

Frequency (GHz)

Figure 4.26. Propagation constant of the air filled TLW antenna with circular termi-
nation.

132

80

70-

Efficiency (%)
on A 01 a)
O O O O

N
O

10

 

 

 

 

l l l g l

 

7 8 9 10 1 1 12
Frequency (GHz)

Figure 4.27. Efficiency of the TLW antenna with circular termination.

133

 

we. (%>

 

 

 

 

4 5 6 7 8 9 10 1 1 12
Frequency (GHz)

Figure 4.28. Ratio PR L to Pin of the TLW antenna with circular termination.

134

CHAPTER 5

CONCLUSIONS AND FUTURE WORK

The contribution of this dissertation lies in the balun used to feed and in concert
with the load to terminate a trough leaky-wave antenna. The main feature of this
technique is its simplicity because this balun does not require the addition of signif-
icant complexity. The propagation constants are fundamental for the analysis and
design of leaky-wave antennas. In this work, these constants were computed using
the transverse resonance method in order to have an initial idea of the behavior of the
input impedance and radiation pattern of the antenna. Since this method assumes
an axially-infinite structure, the finite element boundary integral method was used to
design a finite-length antenna. For this design several balun shapes were considered
e.g., rectangular, bow-tie and circular. It was found that the circular termination
was the most suitable in terms of VSWR and front to back ratio. The finite element
boundary integral method was shown to be able to extract the propagation constants
in order to validate the transverse resonance approach.

A trough leaky wave antenna filled with RT/Duroid 5880 was designed. Its
impedance bandwidth was approximately 15.71% and its front to back ratio of 20
dB at 8.0 GHz. In addition, a trough leaky wave antenna filled with air was designed.
The impedance bandwidth was approximately 40% and the front to back ratio was

20 dB at 12.0GHz.

In these two designs, a forward traveling-wave was obtained as it is shown on

135

 

Figure 4.8 and Figure 4.22, respectively. At high frequencies, standing waves started
to appear on the antenna aperture. Standing waves are the result of a backward
traveling-wave from the termination of the antenna. An optimum design requires
minimizing backward traveling—waves. Since the magnitude of the oscillations in
the air-filled case are less strong than in the duroid—filled case; it is evident that less
energy is reflected from the load when the trough is filled with air. Reducing standing
waves in a traveling-waves antenna is equivalent to Optimize its antenna operational
bandwidth. As it was expected, the operational bandwidth for the air-filled trough
leaky-wave antennas is much larger that the one for the duroid-filled trough leaky-
wave antenna.

Tfough leaky-wave antennas and microstrip leaky-wave antennas are in general
easy to feed. One of the advantages of the through leaky-wave antenna with respect
to microstrip leaky-wave antennas is that it does not need a dielectric to support the
conductor strip over the ground plane. This feature adds flexibility to the antenna
design process.

At the time of this dissertation defense a trough leaky wave antenna prototype was
in fabrication process. Unfortunately, experimental data was not available for this
defense. As a future work, the simulated data will be compared with the experimental
data in order to validate this design process and assess the feasibility of the trough
leaky-wave antenna. In addition, new feeding techniques will be investigated in order

to design a more efficient trough leaky-wave antenna.

136

APPENDICES

137

APPENDIX A

REVIEW OF SOME COMPLEX VARIABLE FUNCTIONS

A.1 The exponential function 10(2) = 62

In this appendix it is further discussed the properties of the function 112(2) = 62,

considered in chapter 2 (section 2.1). The exponential is a single-valued function of
its argument. Here, the complex variable 2: is defined as 2 = reje = :1: + jy, without
loss of generality it is assumed 'r = 1. Consider any point 20 in the complex 2 plane
and any path from 2:0 through the plane back to 20, a single value function is such
that its value changes continuously along the path, returning to its original value at
20. Figure A.1 (a) shows the circular path in the 2 plane and Figure A.1 (b) shows
the corresponding path in the w plane. If the start point is 20 = 1 and the path
followed is counter clock wise along the unit circle, this path is closed in the .2 plane
(Figure A.1 (a)) as well as the path in the 111 plane (Figure A.1 (b)). For instance, let
Z1 2 (r, 6) and z2 = (r, 6 + 27r). It is clear that 21 and 22 represent the same point

in the 2 plane. Substituting these two points into 10 (2) yields the expressions

(A.1)

w (22) = 622 = ere] (A2)

138

 

Since ej (9+27r)

is periodic with period 27r, (A.1) and (A2) represent the same point
in the 211 plane. Therefore, the same value of 62 is obtained for different circuits e.g.,

0, (9 + 27r, 0 + 47r, etc.

A.2 The square root function w (z) = fl

The square root is a double-valued function of its argument; this function was also
considered in chapter 2 (section 2.1). Consider the definition of the complex variable
2 = rejg (assuming 1‘ = 1) and the closed path shown on Figure A2 (a), starting
at r = 1, 6 = 0. Figure A.2 (b) shows the corresponding path in the w plane, this
path is not a closed one. After making a complete circle around the origin in the 2
plane the point 10 = —1 is obtained instead of w = 1. In order to arrive to w = 1, it
is necessary to make a complete circle one more time (27r _<_ 6’ < 4n) in the 2 plane.
But this new circle is not in the same sheet as the first one. Note that in this way it
is avoided to encircle the origin. For the case 11) (z) = fl two sheets are enough to
characterize the values of w (z) in a single-valued manner. Each one of these sheets
are known as Riemann sheets [49]. The top sheet is cut along the positive real axis
and joined the bottom sheet as shown on Figure A2 (a). This cut is known as branch
cut and z = 0 is known as branch point. The branch cut is chosen arbitrary but the
branch point is a true singularity. It is usually convenient to take the branch cut

along the positive or negative real axis.

139

he

 

 

 

2 plane
r
9 zo=l
-1
-i
(a)
w plane
j-r
W(Z 0)
.j -_

 

(b)

Figure A.1. (a) A circular contour in the 2 plane. (b) The mapping of the 2 plane
into 112 plane by the function 62".

140

 

 

2 plane
Top sheet

 
  

A Branch
.......... cut

 

 

 

 

Branch Bottom
point sheet
(a)

w plane
Top sheet
-1\ 1
Bottom \ / /
sheet \ ~31 ’

 

(b)

Figure A2. (a) A circular contour in the 2 plane. (b) The mapping of the 2 plane
into to plane by the function fi.

141

APPENDIX B

THIN-SUBSTRATE APPROXIMATION FOR MICROSTRIP PATCH

ANTENNAS

Kuester et a1. [32] investigated oblique incidence of a TEM wave in a dielectric-filled

parallel-plate wave guide. This approximation is used to evaluate X on (2.27) which

is given by
1 k2 "' kg 2
X = 2tan_ ———V tanhA — fe (km) (31)
kg:
where

 

71' 57‘

A:wl/k2—k‘%{l_€T[ln(jw\fi€8—k2+k%)+7‘1]

+262 (-55) — 2Q (5,.) } (13.2)

 

- 2_ 2 2 _
—2kxw ln(]w\/k0 k +k$)+’y 1

ET

 

 

fe (k1?) = + 2Q (—55) — 1n (2n) (3.3)

’7 = 0577215665

142

Er—I

 

 

6 =
E Err-+1
Air—1

6 :
H Hr+l

 

Q(—6s) = i [5" _ 1]mln (m)

1
m=1 E'r‘l'

Q(5u)=0

(13.7)

For er 2 2.33, equation (B.6) converges very rapidly, (i.e., m > 10) as it is shown on

Figure B1

143

0.12

 

0.1-

0.08 -

 

 

 

 

—
_

 

 

Figure B.1. Convergence for equation (B.6), (57~ = 2.33).

144

15

APPENDIX C

PROOF OF SOME PROPERTIES OF RWG BASIS FUNCTION

C.1 Proof of expression (3.114)

In this appendix it is proven expression (3.114) given in section 3.6. In addition, it is
shown that its right hand side is zero. In order to be able to use the Gauss divergency
theorem, the argument of the left hand side of (3.114) must have a continuous partial
derivative on S, and C. If the triangles (Figure C.1) Ti+ and T 17 have continous
derivatives across their common edge lc, they do not have any edge on the boundary.
Therefore, Gauss divergency theorem can be used directly on the whole domain, this

is,

 

 

 

 

 

 

 

1 l2
e_-7k0R . —. . e—JkOR - _. -
+A_( R )(ZXWJ) mdl+/l_( R (szj)-mdl (Cl)
1 2
From expression (3.120),
i i

145

assuming that the expansion coefficients E,- = Mi, therefore

2 x i: —f;- (n (0.3)

where f7: (7") are RWG basis function given on expression (3.119). Substituting (C.3)

into equation (C.1), this becomes

 

VI
3!
2 [1+

1

+/l_

1

 

e—jkoR _. I e—jkoR _‘ A
( R )f.(o]ds =fC ——R— fMfl-mdl
—jk0R z- -jk0R ,
e z+fi+-fizdl+/ 6 l3 57".de
R 2A2. 2 I; R 2.41;t z

e—jkOR l- _g_ .
_( R >2.cf,-pz' 'de ((3.4)
2

 

 

 

 

The resulting current interpolation for two triangles is shown on Figure C.1. This
basis function has no component normal to the upper or lower edges of either triangle
but only to the common edge. The direction of current is normal to the unit vector
fit, the four integrals of (C4) turn out to be zero.

On the other hand, if there is a boundary edge on either T2.+ or T27, the Gauss

divergency theorem should be used on each triangle, this is,

146

 

e-jkoR - e-J‘koR ..
[337’- (T)fi(fl]d8’=fc( R )fz-(fl-Thdl

e—J'koR z,- + e_jk0R [-
: *. .mdl+/ z *1de
fit-“l R R

The integral over T:— can be found as

 

 

 

 

 

 

—'k R
T; R 2.4; . 1+

e-J'koR z,- ~+ , e—J'koR 1,- ,+ A
+/l;( R 2.4:” -mdl+/lc R 2A?“ .de

In a similar way, the integral over T; is computed as

 

 

 

 

 

 

 

e—jkOR lz' _.
finmazf
A%‘( R )2A; 1 z—

e—jkOR l’i _
)5. -7hdl
1 ( R 2A.- 2
+/
l

e_jk0R lz- e-jkoR [-
“.— -er+/ -—‘—-5.— ~n‘1dl
2—( R >24,”2 16 R 2A2.- Z

 

 

(0.5)

(07)

Since the current direction is normal to m the inte rals over 1+, l+, l_, and l— are
g 1 2 1 2

cero on (C6) and (C7). Substituting (C6) and (C7) into (C.5) yields,

147

 

 

e—jkOR _. . e_jk0R li _,+ ..
féj( R )fz.(f).mdl——/lc( R 2Ag'pi -mdl+

e_jk0R l’i _,_ ,.
[lc( R )2A27-pi -mdl (G8)

The last two integrals on the right hand side of (G8) have the same magnitude but

 

opposite Sign. Therefore, (C.1) or (C8) becomes

 

e‘jkOR _. ..
ij R ) Mr) .de = 0 Q.E.D (cg)

C.2 RWG basis function are free of charge accumulation over its support

In this section, the Gauss divergence theorem is used to prove that RWG basis function
are free of charge accumulation over its support.

From the current continuity condition for magnetic charges:

V ' M (7") = -ijm (7") (010)

where pm is the charge magnetic density on a triangle patch. The net charge on the

support is given by

cm: [Spmmdrt (0.11)

Substituting (C.10) into (C.11) yields

148

 

 

_]_ _.
cm W S (*1 < )

Using (3.118), equation (C.12) becomes

_1 _. _,
Qm= E/SV';Mifi (6)617:

“2i Mi , 'f 7. f:
———jw [Sv f,(')d (0.13)

At this point, it is possible to use the result obtained on appendix C2, in specific

(C9) to write

_—Z'M‘ r. ~_
Qm— j; z[S'V7-_fz(f')dr—

1%; f.- (F) .de = o ((3.14)

which means that the net charge over the triangles shown on Figure Cl is zero.

149

 

 

 

Figure 01. Current interpolation between two triangles.

150

APPENDIX D

ANALYTICAL COMPUTATION OF INTEGRALS WITH

SINGULARITIES

In this appendix it is discussed the computation of integrals with singularities encoun-
tered in 3.6.2. The derivation of these integrals is found on [41]. These singularities

are of two type: uniform source distribution and linearly varying source distribution.

D.1 Integrals with uniform source distribution

These integrals contain proportional terms to

1 I
—dS D.1
/Tq R ( )

where Tq is a triangle element. The analytical computation for integrals of this type

 

 

 

 

is given by
+ +
1 .. R. +l.
/ —ds’ =ZP9.a, szn-L—L—lcu
N R . z z R.— +1.—
2 Z Z
10.01?r 10.01.—
X tan— 22 2 —tan_1 22 7’ (D2)
0 + 0 -
(12,.) +|d|Ri (12,.) +|d|Ri
where

151

 

 

 

 

d = a. (F— Ff) (13.3)

These vectors and scalars quantities are shown on Figure D1. 1'” is the position

152

vector from the origin to a source point on the triangular patch. I" is the position
vector from the origin to an observation point. p” and [f are the projections of 17’

and 7" respectively onto the plane of the patch. 7" .

it denotes the position vectors from

the origin to the endpoints If. [3i is the projection of the position vector F3: onto
the patch plane. d is the height of the observation point above the patch surface. P29
is the perpendicular distance of the projected observation point in the plane of the
patch to the ith—edge of the patch. The unit vector i,- is tangent to the ith-edge and

points in the direction of increasing length. The unit vector 12,; is the outward normal

vector to the ith—edge.

D.2 Integrals with linearly varying source distribution

These integrals contain proportional terms to

 

../__ ..
/Tq p R pdS' (D4)

The analytical computation for integrals of this type is given by

 

P P [_l A. 02 z z +,_ ——
[W R dS—2§.:u,[(Rz) an_+l_+Rz.z, Ritz. (13.5)
Z

2' i

The vectors and scalars quantities given on (D5) are defined on (D3) and Figure

D1.

153

 

 

 

 

Figure D.1. Geometrical parameters associated with the evaluation of potential inte-
grals over the triangle Tq (after [50]).

154

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155

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