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This is to certify that the
thesis entitled

CONTROL OF THE MAIN BEAM OF A HALF-WIDTH
MICROSTRIP LEAKY-WAVE ANTENNA BY EDGE LOADING
USING THE TRANSVERSE RESONANCE METHOD

presented by

MICHAEL LANRE ARCHBOLD

has been accepted towards fulfillment
of the requirements for the

MS. degree in ELECTRICAL ENGINEERING

 

 

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CONTROL OF THE MAIN BEAM OF A HALF-WIDTH
MICROSTRIP LEAKY—WAVE ANTENNA BY EDGE
LOADING USING THE TRANSVERSE RESONANCE
METHOD

By

Michael Lanré Archbold

A THESIS

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

MASTER OF SCIENCE
Electrical Engineering

2008

Copyright by
Michael Lanré Archbold
2008

ABSTRACT

CONTROL OF THE MAIN BEAM OF A HALF-WIDTH
MICROSTRIP LEAKY—WAVE ANTENNA BY EDGE LOADING
USING THE TRANSVERSE RESONANCE METHOD

By
Michael Lanré Archbold

As the need to conserve fuel is ever increasing, developing low-profile, lightweight,
broadband antennas is undoubtedly important to the success of the United States Air
Force. The Half—Width Microstrip Leaky-Wave Antenna (HMLWA) addresses these
important issues along with other beneficial characteristics such as being inexpensive
and conformable (say to the fuselage of an aircraft) all while being half the width of its
full-width counterpart, the Full-Width Microstrip Leaky-Wave Antenna (FMLWA).

It is well known that the main beam angle of a leaky-wave antenna scans with
frequency [1] and that the angle can be controlled by reactively loading the radiating
edge [2H7]. This work will optimize the reactive loading (lumped capacitors) of a
current HMLWA antenna design allowing for fixed-frequency beam steering over the
operational frequency range. The Transverse Resonance Method (TRM) is used to
analyze the HMLWA antenna for various edge loading configurations. Full-width and
half-width antennas are simulated on both finite and infinite ground planes in an
attempt to verify the claim that the HMLWA antenna reasonably approximates its
full-width counterpart. Furthermore, the radiation pattern of the HMLWA antenna is
determined analytically using the Radiating Aperture Method (RAM) and compared
to the patterns of simulated antennas. Finally, the proposed antenna is fabricated

and measured, and the results are presented.

For Luis and Leslie

iv

ACKNOWLEDGMENTS

First and foremost, I would like to thank God. This work would not have been
possible without His grace and mercy.

To my parents, Luis and Leslie Archbold, whom have always placed the highest
value on education. Their relentless motivation fueled my desire to continue learning,
for that I can never repay them. To my loving fiancé Devin Williams, for her patience
throughout this exhausting process. To my sisters, Rose and Josette, you all have set
an impeccable example, one that I could only hope to ever achieve.

Sincere gratitude for Drs. Percy Pierre and Barbara O’Kelly. Thank you for
providing me with the opportunity to attend graduate school. To my advisor, Dr.
Ed Rothwell, thank you for teaching me how to fish, providing infinite meals, rather
than giving me a single meal. To my surrogate advisor, Dr. Leo Kempel thank you
for providing a priceless opportunity.

To Ms. Jackie Toussaint-Barker of AFRL, thank you for investing in my future.
Being an MLP scholar has opened many doors. To my extended AFRL family: Ms.
Nivia Colon-Diaz, Dr. Stephen Schneider, Mr. John McCann, Mr. Kyle Zeller, Dr.
Dan Janning, and Mr. Matt Longbrake, thank you for welcoming me with open arms
and for your assistance. A very special thanks to Dr. Brad Kramer. I appreciate you
taking the time to answer my many questions and for providing guidance when I was
“spinning my wheels”. And last but not least, I would like to thank Mr. Thomas
Goodwin for his exceptional editorial skills that were pushed to the limit when revising

this work. With you all I feel that I have made life-long friends.

TABLE OF CONTENTS

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

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

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

CHAPTER 1

Introduction .....................................
1.1 Background ................................
1.2 Research Overview ............................

CHAPTER 2

Leaky-Wave Antennas ...............................

2.1 Dispersion Characteristics of the Hybrid Leaky-Wave EH1 Mode . . .

2.2 Half-Width Approximation of a Full-Width Microstrip Leaky-Wave
Antenna ..................................
2.2.1 Half-Width Approximation of a Full-Width Microstrip Leaky-

Wave Antenna on an Infinite Substrate-Ground Plane .....
2.2.2 Half-Width Approximation of a Full-Width Microstrip Leaky-
Wave Antenna on a Finite Substrate-Ground Plane ......

CHAPTER 3
Transverse Resonance Method ...........................
3.1 Transmission-line Theory .........................

3.2 Terminated Lossless Transmission Line .................
3.2.1 Short-Circuited Line .......................

3.3 Titansverse Resonance Relation .....................
3.4 TRM method applied to an unloaded HMLWA antenna ........
3.5 Free-edge Admittance ...........................
3.6 TRM method applied to a reactively loaded HMLWA antenna . . . .
3.7 TRM method Implementation ......................
3.8 Fixed Frequency Beam Steering .....................

CHAPTER 4

Analytical Solution .................................
4.1 Cavity Model of HMLWA antenna ....................
4.2 The Radiating Aperture Model .....................

vi

viii

ix

10

11

12

14
14

2O
23

24
28
34
38
46
48

53
53

CHAPTER 5

Antenna Design ................................... 63
5.1 Measurements ............................... 65
5.1.1 Return Loss ............................ 65
5.1.2 Unloaded HMLWA ........................ 65
5.1.3 Reactively Loaded HMLWA ................... 67
5.1.4 Antenna Measurement Calibration ............... 71
5.2 Reactively Loaded HMLWA Antenna Pattern Measurements ..... 73
5.2.1 Radiation Effeciency ....................... 77
CHAPTER 6
Conclusions ..................................... 79
6.1 Future Work ................................ 80
APPENDIX A
Construction of solutions from potentials ..................... 82
APPENDIX B
Radiation patterns of FMLWA and HMLWA antennas on infinite and finite
ground planes .................................... 93
B.1 HMLWA vs. FMLWA simulated on an infinite ground plane ..... 93
B.2 HMLWA vs. FMLWA simulated on a finite ground plane ....... 93
APPENDIX C
Unloaded HMLWA antenna characteristics .................... 103
APPENDIX D
0.10 pF parallel-loaded HMLWA antenna characteristics ............ 109
APPENDIX E

Measured Radiation Patterns of the Capacitively Loaded HMLWA Antenna . 116

BIBLIOGRAPHY ................................. 126

vii

Table 2.1

Table 2.2

Table 3.1

Table 3.2

Table 3.3

Table 3.4

Table 3.5

Table 3.6

Table 3.7

Table 3.8

Table 3.9

Table 3.10

Table 4.1

Table 4.2

LIST OF TABLES

Radiation Pattern Gain Difference of HMLWA vs. FMLWA An-
tennas on an Infinite Substrate-Ground Plane Layer ........

Radiation Pattern Gain Difference of HMLWA vs. FMLWA An-
tennas on a Finite Substrate—Ground Plane Layer .........

Lower cutoff frequency, upper cutoff frequency, and bandwidth of
various capacitive loads. .......................

Capacitive limits used to compute fixed main beam vs. frequency
tabulated values ...........................

Capacitive Load required to fix Main Beam at 10° over C—band
Capacitive Load required to fix Main Beam at 20° over C-band
Capacitive Load required to fix Main Beam at 30° over C—band
Capacitive Load required to fix Main Beam at 40° over C—band
Capacitive Load required to fix Main Beam at 50° over C-band
Capacitive Load required to fix Main Beam at 60° over C-band
Capacitive Load required to fix Main Beam at 70° over C-band
Capacitive Load required to fix Main Beam at 80° over C-band

Main beam direction of unloaded HMLWA antenna determined
analytically using RAM method ...................

Main beam direction of 0.10 pF edge-loaded HMLWA antenna de-
termined analytically using RAM method .............

viii

12

13

46

49

49

50

50

50

51

51

51

62

Figure 1.1

Figure 2.1

Figure 2.2

Figure 2.3

Figure 3.1

Figure 3.2
Figure 3.3
Figure 3.4

Figure 3.5

Figure 3.6
Figure 3.7

Figure 3.8

Figure 3.9

LIST OF FIGURES

Electric field lines on an open microstrip line operated in the dom-
inant (a) and first higher—order (b) modes. (After [13]) .......

Typical dispersion characteristics of the hybrid leaky-wave EH1
mode. Yellow, green, magenta, and cyan correspond to the reac-
tive, leaky, surface, and bound regimes, respectively (After [15]). .

Radiation Patterns of HMLWA antenna vs. FMLWA antenna at
7 GHz mounted atop an infinite substrate-ground plane layer. (10
dB radial spacing)

Radiation Patterns of HMLWA antenna vs. FMLWA antenna at 7
GHz mounted atop a finite substrate—ground plane layer. (10 dB
radial spacing)

General transmission line shown in (a). ’Ifansmission line decom-
posed into differential sections Ax depicted in (b). And equivalent
circuit of each differential section (0) (After [23]). .........

Tfansmission Line terminated in an arbitrary load impedance Z L-

Short-circuited transmission line.

TRM method applied to an unloaded HMLWA antenna.

Dispersion curves of the hybrid leaky-wave EH1 mode. Operational
region: 5889-8115 GHz. BW=31.80% ................

Convergence of Q(——6€) vs. m for £722.33. .............
TRM method applied to an edge—loaded HMLWA antenna .....

Dispersion curves of the hybrid leaky-wave EH1 mode of a 0.10 pF
loaded HMLWA antenna. Operational region: 3928-4869 GHz.
BW = 21.39%.

Phase coefficients for various capacitive loads ranging from 0.02 to

0.10 pF. ................................

ix

16

21

34

37

39

45

Figure 3.10

Figure 3.11

Figure 3.12
Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 5.1
Figure 5.2

Figure 5.3

Figure 5.4

Figure 5.5

Figure 5.6

Figure 5.7

Leakage coefficients for various capacitive loads ranging from 0.02
to 0.10 pF. .............................. 45

340 mm edge-loaded HMLWA antennas. Loads spaced every 1.5
mm. Depicts how large capacitance value shorts the radiating edge. 47

Main beam direction vs. Capacitance for various Frequencies. . . 52
Cavity Model of a Microstrip Antenna ............... 54
Magnitude of the Electric field within the aperture of the HMLWA

antenna ................................. 56
HMLWA antenna geometry with radiating aperture ......... 57

Analytical solution to far fields maintained by equivalent magnetic
current on the surface of the radiating aperture of the unloaded
HMLWA antenna. .......................... 61

Analytical solution to far fields maintained by equivalent magnetic
current on the surface of the radiating aperture with 0.10 pF edge
loading configuration. ........................ 61

Unloaded 340 mm HMLWA antenna with through—substrate feed 64
Loaded 340 mm HMLWA antenna with edge-launch feed ..... 64

Exploded view of loaded HMLWA antenna showing 0.10 pF 0603
multilayer capacitors ......................... 65

Return loss of an unloaded terminated HMLWA antenna ..... 66

Return loss of terminated and non-terminated unloaded HMLWA
antennas. ............................... 68

(a) PHASEFLEX test assembly. (b) Pasternack 50 Q termination 68

VSWR of terminated and non—terminated unloaded HMLWA an-
tennas .................................. 69

Figure 5.8

Figure 5.9

Figure 5.10

Figure 5.11

Figure 5.12

Figure 5.13

Figure 5.14

Figure 5.15

Figure B.l

Figure B.2

Figure B.3

Figure B.4

Figure B.5

Figure B.6

Figure B.7

Simulated vs. Measured [311] Return Loss of Reactively Loaded
HMLWA Antenna. .......................... 70

Simulated vs. Measured [5'22] Return Loss of Reactively Loaded

HMLWA Antenna. .......................... 71
(a) NSI-2000 workstation (b) Agilent E8362B PNA ........ 72
Reactively loaded HMLWA antenna in Near Field Anechoic Chamber 73
(a) 3—dimensional Spherical Pattern and (b) 2-dimensional Contour

Plot at 7.00 GHz ........................... 75
Simulated vs Measured HMLWA Gain at 7.00 GHz ........ 76
Unloaded, Simulated Reactively Loaded, and Measured Reactively

Loaded HMLWA Antenna Main Beam Direction Comparison . . 77
Simulated vs Measured Radiation Efficiency ............ 78

Radiation Patterns of HMLWA antenna vs. FMLWA antenna at
6G'Hz mounted atop an infinite substrate-ground plane layer. . . 93

Radiation Patterns of HMLWA antenna vs. FMLWA antenna at
6.25GHz mounted atop an infinite substrate—ground plane layer. . 94

Radiation Patterns of HMLWA antenna vs. FMLWA antenna at
6.5GHz mounted atop an infinite substrate-ground plane layer. . 94

Radiation Patterns of HMLWA antenna vs. FMLWA antenna at
6.75GHz mounted atop an infinite substrate-ground plane layer. . 95

Radiation Patterns of HMLWA antenna vs. FMLWA antenna at 7
GHz mounted atop an infinite substrate-ground plane layer. . . . 95

Radiation Patterns of HMLWA antenna vs. FMLWA antenna at
7.25GHz mounted atop an infinite substrate—ground plane layer. . 96

Radiation Patterns of HMLWA antenna vs. FMLWA antenna at
7.5GHz mounted atop an infinite substrate—ground plane layer. . 96

Figure B.8 Radiation Patterns of HMLWA antenna vs. FMLWA antenna at
7.75GHz mounted atop an infinite substrate—ground plane layer. . 97

Figure B.9 Radiation Patterns of HMLWA antenna vs. FMLWA antenna at
8GHz mounted atop an infinite substrate—ground plane layer. . . 97

Figure B.10 Radiation Patterns of HMLWA antenna vs. FMLWA antenna at
6GHz mounted atop a finite substrate—ground plane layer. . . . . 98

Figure B.11 Radiation Patterns of HMLWA antenna vs. FMLWA antenna at
6.25GHz mounted atop a finite substrate-ground plane layer. . . 98

Figure B.12 Radiation Patterns of HMLWA antenna vs. FMLWA antenna at
6.5GHz mounted atop a finite substrate-ground plane layer. . . . 99

Figure 313 Radiation Patterns of HMLWA antenna vs. FMLWA antenna at
6.75GHz mounted atop a finite substrate-ground plane layer. . . 99

Figure B.14 Radiation Patterns of HMLWA antenna vs. FMLWA antenna at
70112 mounted atop a finite substrate-ground plane layer. . . . . 100

Figure B.15 Radiation Patterns of HMLWA antenna vs. FMLWA antenna at
7.25GHz mounted atop a finite substrate-ground plane layer. . . 100

Figure B.16 Radiation Patterns of HMLWA antenna vs. FMLWA antenna at
7.5GHz mounted atop a finite substrateground plane layer. . . . 101

Figure 8.17 Radiation Patterns of HMLWA antenna vs. FMLWA antenna at
7.75GHz mounted atop a finite substrate-ground plane layer. . . 101

Figure B.18 Radiation Patterns of HMLWA antenna vs. FMLWA antenna at

8GHz mounted atop a finite substrate—ground plane layer. . . . . 102
Figure C.1 Free-edge resistance of the unloaded HMLWA antenna. ...... 103
Figure C.2 Free—edge reactance of the unloaded HMLWA antenna. ...... 104
Figure C.3 Re(x) of the unloaded HMLWA antenna ............... 105
Figure C.4 I m(x) of the unloaded HMLWA antenna. ............. 105

Figure C.5 [x] of the unloaded HMLWA antenna. ............... 106

Figure C.6 Ang(x) of the unloaded HMLWA antenna .............. 106
Figure C.7 Re(kT) of the unloaded HMLWA antenna. ............. 107
Figure G8 I m(kT) of the unloaded HMLWA antenna .............. 107
Figure C.9 Re(kz) of the unloaded HMLWA antenna. ............. 108
Figure C.10 I m(kz) of the unloaded HMLWA antenna. ............. 108
Figure D.1 Free—edge resistance of the 0.10 pF loaded HMLWA antenna. . . . 110
Figure D.2 Free-edge reactance of the 0.10 pF loaded HMLWA antenna. . . . 110

Figure D.3 Radiating—edge resistance of the 0.10 pF loaded HMLWA antenna. 111

Figure D.4 Radiating—edge reactance of the 0.10 pF loaded HMLWA antenna. 111

Figure D.5 Re(x) of the 0.10 pF loaded HMLWA antenna. .......... 112
Figure D.6 I m(x) of the 0.10pF loaded HMLWA antenna ............ 112
Figure D.7 [x] of the 0.10 pF loaded HMLWA antenna. ............ 113
Figure D.8 Ang(X) of the 0.10 pF loaded HMLWA antenna ........... 113
Figure D.9 Re(kT) of the 0.10 pF loaded HMLWA antenna ........... 114
Figure D.10 I m(kT) of the 0.10 pF loaded HMLWA antenna ........... 114
Figure D.11 Re(kz) of the 0.10 pF loaded HMLWA antenna. .......... 115
Figure D.12 I m(kz) of the 0.10 pF loaded HMLWA antenna ........... 115

Figure E.1 Measured Radiation Pattern of HMLWA antenna at 4.00 GHz . . 116

Figure E.2 Measured Radiation Pattern of HMLWA antenna at 4.25 GHz . . 117

xiii

Figure E.3 Measured Radiation Pattern of HMLWA antenna at 4.50 GHZ . .
Figure E.4 Measured Radiation Pattern of HMLWA antenna at 4.75 GHZ . .
Figure E.5 Measured Radiation Pattern of HMLWA antenna at 5.00 GHZ . .
Figure E.6 Measured Radiation Pattern of HMLWA antenna at 5.25 GHZ . .
Figure E.7 Measured Radiation Pattern of HMLWA antenna at 5.50 GHZ . .
Figure E8 Measured Radiation Pattern of HMLWA antenna at 5.75 GHZ . .
Figure E.9 Measured Radiation Pattern of HMLWA antenna at 6.00 GHZ . .
Figure E.10 Measured Radiation Pattern of HMLWA antenna at 6.25 GHZ . .
Figure E.11 Measured Radiation Pattern of HMLWA antenna at 6.50 GHZ . .
Figure B.12 Measured Radiation Pattern of HMLWA antenna at 6.75 GHZ . .
Figure E.13 Measured Radiation Pattern of HMLWA antenna at 7.00 GHZ . .
Figure E.14 Measured Radiation Pattern of HMLWA antenna at 7.25 GHZ . .
Figure B.15 Measured Radiation Pattern of HMLWA antenna at 7.50 GHZ . .
Figure E.16 Measured Radiation Pattern of HMLWA antenna at 7.75 GHZ . .

Figure E.17 Measured Radiation Pattern of HMLWA antenna at 8.00 GHz . .

xiv

117

118

118

119

119

120

120

121

121

122

122

123

123

124

124

 

Images in this thesis are presented in color.

 

XV

KEY TO SYMBOLS AND ABBREVIATIONS

AFRL: Air Force Research Laboratory

AUT: Antenna Under Test

EIA: Electronic Industries Alliance

EM: Electromagnetic

FBR: Front-to—Back Ratio

FMLWA: Full-Width Microstrip Leaky-Wave Antenna

HMLWA: Half-Width Microstrip Leaky-Wave Antenna

KCL: Kirchoff’s Current Law

KVL: Kirchoff’s Voltage Law

MLWA: Microstrip Leaky-Wave Antenna

NSI: Nearfield Systems Incorporated

OEWG: Open-ended Waveguide

PEC: Perfect Electric Conductor

PMC: Perfect Magnetic Conductor

PUL: Per Unit Length

RAM: Radiating Aperture Method

xvi

RASCAL: Radiation and Scattering Compact Antenna Laboratory

ROI: Region of Interest

RY: Sensors Directorate

SGH: Standard Gain Horn

SMA: SubMiniature Version A

TEM: 'Ilansverse Electromagnetic

TE: Transverse Electric

TM: Transverse Magnetic

TRM: Transverse Resonance Method

TWA: Ttaveling Wave Antenna

VNA: Vector Network Analyzer

VSWR: Voltage Standing Wave Ratio

WRT: With Respect To

xvii

CHAPTER 1

INTRODUCTION

Microstrip Leaky-Wave Antennas (MLWA) are traveling-wave printed antennas
wherein the microstrip is responsible for radiation. Due to their inherently small
size, low-profile, and conformability, they are well suited for aerospace applications
where all of the abovementioned constraints are addressed. As the name suggests,
a Half-Width Microstrip Leaky-Wave Antenna (HMLWA) is one half the width of a
full-width antenna whereby a perfect electric conducting (PEC) septum connects the
microstrip to the ground plane, ridding the need of half of the antenna.

As one side of the HMLWA antenna is open to free space, it is well known that the
edge reflection coefficient affects the propagation characteristics of the antenna. With
knowledge of the admittance looking into the free edge available in the literature [8],
modifying this edge reflection coefficient by reactively loading the free edge alters the
propagation characteristics of the antenna. In this thesis, the main beam direction
of a HMLWA antenna is steered at fixed frequencies by placing lumped multilayer
capacitors (MLC) in shunt with the free edge of the microstrip and the ground plane.

As waves of the hybrid leaky-wave EH1 mode propagate along the conducting
strip, radiation occurs because the hybrid leaky—wave mode is loosely bound to the
microstrip. In order to control the direction of the energy one must study the re-
lationship between the guided wave phase coeflicient, fl, and frequency, w. (Note
that the term “coefficient” is used rather than the more frequently encountered term
“constant” to emphasize the frequency dependence of the phase.) The guided wave
phase coefficient is dependent on the reactive loading at the free edge (see Figure
3.9); and since the main beam direction depends on the phase coefficient, it follows

that the main beam direction can be controlled at fixed frequencies.

 

The Transverse Resonance Method (TRM) is used to determine the complex prop—
agation coefficient of the wave. It simplifies the analysis by employing transmission
line theory rather than complex field theory. Since the TRM method is an approxima-
tion, a full-wave solution using commercially available software (CST MICROWAVE

STUDIO® and FEKO Suite 5.3) is determined prior to antenna fabrication.

1 . 1 Background

In 1976, Helmut Ermert presented a paper on the dominant and higher-order modes
on microstrip lines in which he investigated mode propagation on covered microstrip
lines [9]. In a subsequent publication, [10] Ermert investigated the dominant and the
first two higher-order modes on both covered and open microstrip lines. He concluded
that radiation was not feasible because in the “radiation region” the wave number
was purely real, resulting in only a phase coefficient, and no leakage coefficient.

A few years later in 1979, Wolfgang Menzel, a German scientist, presented a
traveling-wave microstrip antenna that took advantage of the first higher-order EH1
mode [11]. Menzel’s antenna was a full-width microstrip antenna that consisted of
several transverse slots (with respect to the longitudinal guiding axis) nearest to
the feed. The transverse slots served the purpose of suppressing the non-radiating,
dominant EH0 microstrip mode. Although Menzel’s transverse slot technique was
inefficient at suppressing all of the dominant-mode radiation, hence mode purity was
not entirely assured, his antenna was indeed successful, and 65% of the power leaked
away in the form of space waves [12].

More than two decades later, engineers at the Air Force Research Laboratory
(AFRL) developed a more efficient way to suppress the non-radiating EH0 mode
while simultaneously exciting the radiating first higher-order EH1 mode. When a
microstrip is operated in the first higher-order mode, a phase—reversal appears along

the centerline of the structure resulting in a null in the Electric field. Figure 1.1

 

 

 

 

(b)

Figure 1.1. Electric field lines on an open microstrip line operated in the dominant
(a) and first higher—order (b) modes. (After [13]).

shows the cross section of a Full-Width Microstrip Leaky-Wave Antenna (FMLWA)
depicting the Electric field lines of the dominant (EH0) and first higher-order (EH1)
modes. The voltage is equal to the product of the Electric field and the height of
the substrate. Since the height of the substrate is nonzero, the field is identically zero
at the origin. AF RL engineers realized that the introduction of a perfectly electric
conducting (PEC) septum, or shorting wall, bifurcating the conducting strip does
not perturb the fields because it is subject to the boundary condition that tangential
electric fields vanish along its surface. Both simulations and measurements support
the supposition. Initially, this PEC septum was fabricated using shorting pins, then
as a slit covered with copper tape, and finally in its current form, copper plated
through holes, or vias. The PEC septum also simplifies the feeding network because
it inherently excites the radiating EH1 mode.

AFRL engineers later realized that if the conducting strip is truncated to half its
size, resulting in the HMLWA antenna [14], the radiation pattern is nearly the same

as that of the full-width version. (See simulated results in Appendix B.) In this

I

thesis, the main beam direction of a HMLWA antenna is steered at fixed frequencies
by reactively loading the radiating edge.

Although the literature contains several papers on fixed—frequency beam steering of
leaky-wave antennas, this author has not seen this accomplished (or attempted) with
a reactively loaded HMLWA antenna. In [2], Noujeim uses the transverse resonance
and finite-difference time—domain methods for beam steering of a full—width leaky-
wave antenna loaded with a reactive sheet. Here, the conducting strip is bisected
about the centerline, and a reactive sheet connects the two halves. Augustin et al.
describes a reactively loaded slot line leaky-wave antenna in [3], and in a subsequent
publication scans the main beam using varactors [4]. And finally, in [5] a full-width
leaky-wave antenna is reactively loaded on both sides of the conducting strip, resulting
in a tilt in the main beam direction with respect to the unloaded counterpart. More

reactively loaded antenna designs are considered in [6] and [7].

1.2 Research Overview

In order to understand leaky-wave antennas, it is necessary to understand modes, par-
ticularly the first higher-order radiating mode, on microstrip lines. Chapter 2 presents
typical dispersion curves of the first higher-order hybrid leaky-wave EH1 mode. This
chapter also contains the details explaining how the HMLWA antenna reasonably ap-
proximates its full-width counterpart, the FMLWA antenna. In Chapter 3, the TRM
method, the mathematical model used in this work, is presented. Here, the TRM
is initially applied to an unloaded HMLWA antenna, then to the reactively loaded
HMLWA antenna. The transverse resonance relation is derived, and the mathematics
of the method is thoroughly explained. Chapter 3 concludes with tabulated values
of capacitive loads required to fix the main-beam direction of the HMLWA antenna
at various angles over the C-band (4—8 GHZ) of the electromagnetic (EM) spectrum.

In Chapter 4 the Radiating Aperture Method (RAM) is used to compute the radi-

ation fields of the HMLWA antenna. The HMLWA antenna design and measured
data is presented in Chapter 5, along with a detailed description of the apparatus
and associated settings during data collection. Finally, Appendices A-E provide the
details for construction of solutions from potentials, simulated radiation patterns of
HMLWA and FMLWA antennas, characteristics of both unloaded and 0.10 pF loaded
HMLWA antennas, and measured data of the 0.10 pF loaded HMLWA antenna over

the C-band of the EM spectrum.

CHAPTER 2

LEAKY-WAVE ANTENNAS

According to [1], a leaky-wave antenna is an antenna that leaks power all along its
length via some mechanism. A characteristic of this antenna is the frequency depen-
dence of the main beam direction. This frequency dependence results in the main
beam direction scanning over the entire operational range (beginning near broadside
and terminating near end fire) of the particular leaky-wave antenna. Although this
frequency scanning characteristic makes leaky-wave antennas attractive for certain
applications, fixing the main beam at particular frequencies would allow more ap-
plications for the antenna whereby its other attractive attributes could be exploited
(e.g., ease of fabrication, conformability and reasonable cost).

Since the phase coefficient, 5, depends on frequency, and the main beam direction
depends on B, it follows that the main beam direction is frequency dependent. The
problem lies in keeping the main beam direction fixed at a given angle, resulting in
a frequency invariant main beam direction over the operational leaky region. The
operational leaky-wave region is defined in Section 2.1.

The complex propagation coefficient is defined as

kg = Br — jag, (21)

where the real and imaginary parts, fl and a, are known as the phase and leakage
coefficients, respectively, with C = :13, y, z. The phase coefficient for a wave propagating
through a medium is related to the phase velocity of the wave, up, by the following

GXpression

a = —. (2.2)

Further, for a transmission line, which a microstrip is classified as, the phase veloc-
ity of the wave is inversely proportional to the per unit length (PUL) capacitance,

inductance, conductance, and resistance of the line, dictated by

w

 

 

 

V = . 2.3
p Im(\f(R’ +ij’)(GI +ij’)) ( )

In the case of a lossless line, equation 2.3 reduces to
1 (2.4)

Up = W.

The primes on the parameters in equations (2.3) and (2.4) denote the PUL depen-
dence. Substituting either expression ((2.3) or (2.4)) for the phase velocity into (2.2)
reveals a directly proportional relationship between the phase coefficient and the PUL
transmission line parameters. This relationship describes the premise of this thesis:
reactively loading the HMLWA antenna, by placing capacitors along the radiating
edge, decreases the phase velocity of the propagating wave, ultimately changing the
phase coefficient of the propagating wave.

The dominant microstrip mode is a quasi-transverse electromagnetic (TEM) mode
in which both the electric and magnetic fields have a component in the direction of
propagation, and is designated EH0. When the EH0 mode propagates, the electric
field exhibits even symmetry about the center of the structure and the electromagnetic
fields are bound to the structure; hence, the dominant mode does not radiate (for
electrically thin substrates with h << A). Since the HMLWA antenna is a microstrip
leaky-wave antenna, the first higher-order mode must be excited for EM fields to
decouple and radiate away from the structure [15]. The first higher-order mode, the
hybrid leaky—wave EH1 mode, exhibits peculiar phenomena above cutoff. It is the

tOpic of many papers, viz. [16]-[20], and is explained in Section 2.1.

2.1 Dispersion Characteristics of the Hybrid Leaky-Wave EH1 Mode

 

4 276 8 10 12 14 16 18
f Frequency (GHz)

Figure 2.1. Typical dispersion characteristics of the hybrid leaky—wave EH1 mode.
Yellow, green, magenta, and cyan correspond to the reactive, leaky, surface, and
bound regimes, respectively (After [15]).

The operational region of the HMLWA antenna considered in this thesis lies in the
C—band of the EM spectrum. The operational or leaky region is one of four frequency
regions associated with propagation of higher-order modes on microstrip lines [15]. It
is preceded by the reactive regime and succeeded by the surface and bound regimes

Figure 2.1 depicts typical dispersion characteristics of the first higher-order mode,
along with the four regions explicitly labeled. The figure depicts the phase and leakage
coefficients, both normalized with respect to the free-space wave number, kg, in blue

and red, respectively. The first of the four regions is the only region below cutoff and

is one of the two regions in the so called fast region. The fast region is the region where
the wave velocity is faster than the speed of light, i.e. up > c, and the axial phase
coefficient, fly, is less than the free—space wave number, k0. Known as the reactive
region, the region shaded in yellow is named accordingly because here attenuation is
high and the microstrip line resembles a reactive load [15]. The region shaded in green
is the leaky region. It is the first of two regions that together constitute the radiation
region: the region that identifies where space and surface waves exist [16]. The leaky
region is the operational region of the HMLWA antenna and the region of interest
because it is the region where power leaks, or radiates, away from the structure at
some angle 6 (the value of which changes with frequency), in the form of a space wave
[16]. It is the second and final region in the so called fast region, and it is bound on
the lower side (at cutoff, fc) where the phase and. leakage coefficients are equal and
bound on the upper side where the phase coefficient is equal to the free-space wave
number. The main beam direction is near broadside at lower frequencies and scans
toward near end fire at higher frequencies.

The two regions in the slow region, where up < c, are the surface and bound
regions. In the surface region, one of the two transverse leakage coefficients, say az,
dominates and attenuates the leaky wave. It is the region shown in magenta from

0 = 1:0 to B = k3, where k, is the wave number of the surface wave and is given by

SB 2

Just as in the leaky region, in the surface region power radiates away from the struc—
ture at some angle t/J, except here the leakage is in the plane of the structure (the XY
plane) rather than in the plane normal to the structure (the YZ plane). Therefore
the waves are bound to the structure as opposed to radiating away from it. The sur-

face region is the second region for which radiation can occur. The fourth and final

region is the bound region. It is the region shaded in cyan in Figure 2.1. Here, the
remaining transverse leakage coefficient, ax, dominates and attenuates the surface
wave. The bound region begins when 5 2 kg, and ends when the next higher-order

mode propagates.

2.2 Half-Width Approximation of a Full-Width Microstrip Leaky-Wave

Antenna

Although the HMLWA antenna approximates the behavior of the FMLWA antenna,
the behavior is not identical. One of its principal advantages, as mentioned in [21], is
that the feeding network is simplified. Since the half-width configuration inherently
excites the hybrid leaky-wave EH1 mode with the added benefit of suppressing the
dominant EH0 mode, there is no need for an external 180° Hybrid as required for
the FMLWA antenna. Sections 2.2.1 and 2.2.2 present simulated radiation patterns
of HMLWA and FMLWA antennas on both infinite and finite ground planes.

For the simulations, the antenna geometries consist of conducting strip widths
of 15 and 7.5 mm for the full and half-width antennas, respectively. The length
of the conducting strip is 340 mm for both geometries. Optimal feed placement is
determined from numerous simulations and is provided by antenna engineers at AFRL
to be 4.8025 mm in either direction from the centerline of the full-width antenna
(i.e., if the antenna is centered at the origin, to excite the EH1 mode — requiring
a 180° phase shift — feeds are placed at :I: 4.8025 mm from the origin) and 4.8025
m away from the PEC conducting septum of the half-width antenna. Both the
F MLWA and HMLWA antennas are terminated in 50 Q loads placed at the end of
the conducting strip, 340 mm from the feed. The load is placed at the centerline
of the FMLWA antenna, and 4.8025 m away from the PEG conducting septum of
the HMLWA antenna. The dielectric substrate is modeled as Rogers RT/duroid®

5870, characterized by a relative permittivity of 6122.33 and a dissipation factor of

10

 

--— HMLWA
-—-- FMLWA

 

 

 

 

 

 

120°

 

 

Figure 2.2. Radiation Patterns of HMLWA antenna vs. FMLWA antenna at 7 GHz
mounted atop an infinite substrate-ground plane layer. (10 dB radial spacing)

tan 6:0.0012.

2.2.1 Half-Width Approximation of a Full-Width Microstrip Leaky-Wave

Antenna on an Infinite Substrate-Ground Plane

The radiation pattern shown in Figure 2.2, was computed using a F EKO Suite 5.3
simulation of the antennas. Radiation patterns covering the operational range of
the antennas (6—8 GHz), computed every 250 MHz, are given in Appendix B. In
both full-width and half-width geometries, the conducting strip is married to an

infinite substrate-ground plane layer via FEKO’s multilayer planar Green’s function

A .

option. The geometries are meshed using triangles with an edge length of Jifig-q,
c

where Ami.” — ————8 G Hz’ the smallest wavelength of the C-band.

Figure 3.1-Figure B.9 show that when the antennas are mounted atop an infinite

11

substrate-ground plane layer, the elevation plane (YZ) patterns are quite similar. Al-
though there is noticeable attenuation (in some cases exceeding 4.5 dB, see Table 2.1),
in the patterns of the HMLWA antenna over the entire leaky region, the main beam

direction does not vary more than 2° from the FMLWA antenna for all frequencies.

Table 2.1. Radiation Pattern Gain Difference of HMLWA vs. FMLWA Antennas on
an Infinite Substrate-Ground Plane Layer

 

 

 

Frequency (GHZ) Gain Difference (dB)
6.00 4.30
6.25 2.54
6.50 2.89
6.75 3.12
7.00 3.62
7.25 3.67
7.50 4.39
7.75 4.27
8.00 4.76

 

 

 

 

2.2.2 Half-Width Approximation of a Full-Width Microstrip Leaky-Wave

Antenna on a Finite Substrate-Ground Plane

More accurate results are obtained when the HMLWA antenna approximates the
FMLWA antenna on the practical case of a finite substrate-ground plane layer. CST
MICROWAVE STUDIO is used to compute the radiation patterns of this case, and
the radiation pattern of the antennas at 7.00 GHz is shown in Figure 2.3. Here, as
opposed to Figure 2.2, the HMLWA antenna pattern more reasonably approximates
the FMLWA antenna pattern with less than a 0.8 dB difference (see Table 2.2) in the
gain and a 2° angular variation in the main beam direction. Appendix B gives the

radiation patters over the operational range of the antennas (6—8 GHz) at every 250

MHz.

12

 

— HMLWA -3o° . 30°

 

 

 

 

 

 

 

180°

Figure 2.3. Radiation Patterns of HMLWA antenna vs. FMLWA antenna at 7 GHz
mounted atop a finite substrate-ground plane layer. (10 dB radial spacing)

Table 2.2. Radiation Pattern Gain Difference of HMLWA vs. FMLWA Antennas on
a Finite Substrate-Ground Plane Layer

 

 

Frequency (GHz) Gain Difference (dB)
6.00 1.24
6.25 0.96
6.50 1.07
6.75 0.96
7.00 0.78
7.25 0.73
7.50 1.00
7.75 0.26
8.00 0.70

 

 

 

 

13

CHAPTER 3

TRANSVERSE RESONANCE METHOD

The transverse resonance method is a technique used to determine the complex
propagation coefficient of many practical composite waveguide structures as well as
traveling-wave antenna systems [22]. The method uses the behavior of the wave trans-
verse to the guiding direction to determine the characteristics of the wave along the
guiding axis. With the TRM method, the cross section of the structure is represented
as a transmission line operating at resonance. The resonances of the transverse trans-
mission line model provide a transcendental equation that, when coupled with the
constraint equation, yields the propagation coefficient 7. The method is accurate pro-
vided that the transverse resonance relation, a result of the continuity of tangential
fields, is maintained (see Section 3.3).

It is important to note that the TRM method is an approximation and only pro-
vides the complex propagation coefficient of the structure to which it is applied. More
exhaustive techniques, which invoke a full-wave solution, are required to determine
other design parameters including field distributions and wave impedances. However,
a reasonable approximation of the radiation fields are determined using the radiating
aperture method (see Section 4.2) based on the propagation coefficient found using
the TRM method. Before considering the details of the TRM method, a brief review

of relevant transmission-line theory will be explained.

3.1 Transmission-line Theory

The analysis in this section is restricted to the distributed parameter approach, fore-
going the usual circuit theory (lumped—element) technique, because the physical di-
mensions of the HMLWA antenna are not small when compared to a wavelength. This

criterion is based on the rule of thumb that transmission-line effects, such as the phase

14

shift associated with time delay and the reflection due to load mismatch, must be con-
sidered when the broadest dimension of the structure, I, is, at least approximately,

one one-hundredth of a wavelength, represented mathematically as [23]

l> _, (3.1)

To facilitate the understanding of the distributed parameter model, the concepts
used in its derivation will be reviewed. Composed of per unit length inductance,
resistance, conductance, and capacitance, the model represents the physics of the
transmission line using distributed electrical components. To account for the com-
bined resistance, inductance, and capacitance of the conductors, the model includes
PUL parameters R’ (fl/m), LI (H/m), and C' (F/m), respectively. If the medium
separating the conductors is not free space, its effects must be accounted for with
the inclusion of an additional parameter, conductance. Denoted by G’ (S/m), the
PUL conductance describes the resistance of the dielectric medium separating the
two conductors. The parameters are denoted with primes to show the explicit PUL
dependence. If the length of the transmission line is I, then the net resistance of the
line is R’l (Q). Total inductance, capacitance, and conductance are determined in
the same fashion.

A transmission line can be decomposed into several cascaded differential sections,
each of width Ax, provided that each section is electrically small. Each section
is modeled by an equivalent circuit using the PUL parameters previously addressed.
An arbitrary length of transmission line and its equivalent circuit are shown in Figure
3.1.

Propagating waves on a transmission line obey the wave equation. The wave
equation is derived by applying Kirchoff’s voltage and current laws to the electrically

small section of the transmission line depicted in Figure 3.1. Kirchoff’s voltage law

15

O

 

 

 

 

 

3
C 3
(a) General two wire transmission line
C AX C Ax 3 Ax 3
N N+1
C t C 3
(b) Differential sections
R'Ax L'Ax R'Ax L'Ax R'Ax L'Ax
G'Ax C'Ax G'Ax CA): CA): CA):
c ‘ c 1 r 1 :2
g. Ax .+ Ax 4 Ax——-’I

 

 

 

(c) Each section is represented by an equivalent circuit

Figure 3.1. General transmission line shown in (a). Transmission line decomposed

into differential sections Ax depicted in (b). And equivalent circuit of each differential
section (c) (After [23]).

(KVL) states that the sum of the voltages in any loop must vanish. Kirchoff’s current
law (KCL) states that the sum of current into any node must vanish. Application of

KVL to the outer loop in Figure 3.1 yields

—v(x, t) + R’A$i(.r, t) + L'Ax-a—z-(gjtl + v(:c + Ax, t) = 0. (3.2)
Rearranging terms leads to
21(1: + Art, t) — v(x, t) = —R’A:rz'(:r, t) -— L'AxQZ—(gtfl. (3.3)

16

Dividing equation (3.3) by Ax yields

 

 

 

 

 

22(23 + Art, t) —— v(r, t) _ —R’A:rz'(:c, t) _ L'Aa; 82(27, t)
A2: _ A11: A2: at
= —R’z'(:c, t) —- MW), (34)
8t
and in the limit as Ax -—> 0 equation (3.4) becomes
6v(:c,t) _ , ,(92'(:1:, t)
6:1: — R’z(r, t) L 8t . (3.5)

Equation (3.5) is a first-order partial differential equation relating the voltage and
current. It is one of two time-domain transmission-line, or telegrapher, equations.
The second equation is derived by applying KCL to node N +1 in Figure 3.1 and
following the steps used to derive equations (3.3) through (3.5) above. The second

telegrapher’s equation is given as

62'.(:r, t)
8:1:

= —G’v(a:, t) _ dang, t). (3.6)

 

 

To simplify the analysis and remain consistent with the notation expressed in this
thesis, the telegrapher’s equations will be analyzed in the frequency domain using
phasor notation. The phasors are defined with a cosine reference, and conversion

from a quantity in phasor form to its time-varying counterpart is accomplished by
«406,31, 2, t) = R6{A(x,y, Zlejwt}, C”)

where A(:r,y,z,t) represents a time-varying quantity, and A(a:,y, 2:) represents the

phasor. Equations (3.5) and (3.6) can now be recast in their equivalent steady-state

17

 

phasor form as

 

(1:55;) = —Ir1(:c)—ij'1($)
= —(R’+ij’)I(a:). (3.8)
and
61—55;) = —G'V(a:)—J'wC'V(~”‘)
= —(G’+J’wC’)V(‘”)’ (3.9)
respectively.

Equations (3.8) and (3.9) are coupled first-order ordinary differential equations. It
is necessary to decouple these equations in order to solve for the voltage and current

along the transmission line. Differentiating equation (3.8) with respect to x leads to

 

 

 

 

d2V(r) , , dI(:r)
(11:2 — “([2, +JLUL )“r'c-l—x—q (3.10)
and substituting equation (3.9) yields
d2V(:r) , , (11(1)
drz — —(R’+]wL) d3:
= (R’ + ij’)(G’ + ij’)V(r)
= 72(w)V(a:), (3.11)
where 7(a)), the complex propagation coefficient, is defined as
7(a)) E \/(R’ +ij’)(G’ +ij’). (3.12)

The real part of the complex propagation coefficient, the leakage or attenuation coef-

18

 

ficient, is denoted by a, and the imaginary part, the phase coefficient, is denoted by
6.
The current wave equation is derived by differentiating equation (3.9) and substi-

tuting equation (3.8) into that result giving

321(3.)

(12:2 = 72(w)1(:1:). (3.13)

 

Equations (3.11) and (3.13), are linear, second-order, homogeneous, ordinary dif-
ferential equations. Candidate solutions include exponentials, sines, cosines, or a
linear combination of them. The solutions chosen should complement the physics
of the problem. It is for this reason that exponentials are chosen as the solution of
the wave equations, for they represent traveling waves, while sines and cosines rep—
resent standing waves. The traveling-wave voltage and current solutions of the wave

equations are

V(:r:) = V0+e‘7~’” + Vo—e’” (3.14)

and

[(23) = 136‘” + Iget’” (3.15)

respectively. The characteristic impedance of the line can be determined by differen-

tiating equation (3.14)

 

dV(:r) dVo+e—7x + dVo—el‘”
d3: dz dx
= ——7Vo+e—7‘” + yVo-elx

= —v(V.+e“"‘”—V.‘e”>, (3.16)

19

and upon substitution into equation (3.8) leads to

—')(V0+e_lx — Vge‘fl) = —(R’ + ij')I(a:)
(R1 +ij’)

(V0+e—'7:r _ Vo_87x) ’7

I(:r:), (3.17)

(R’ + ij’)

where the quantity is known as the characteristic impedance, denoted

by Z0. The characteristic impedance may also be derived by differentiating equation
7

(3.15), and substituting the result into equation (3.9) leading to W. Hence
the characteristic impedance is defined as:
Zo E (R, +ij’)
7,
Z S —. 3.18
0 (G’ + ij’) ( )

The complex propagation coefficient and characteristic impedance, defined in (3.12)
and (3.18), respectively, fully characterize any transmission line. Using Equation

(3.17) the current on the line can be recast in terms of characteristic impedance as

= ___:L__ + ~73- - WI
[(33) (R[+ijl)(V08 V06 )
1
= ill/JET” " V58”)
o

W ‘-
= 2203“”: — Egg-e”. (3.19)

3.2 Terminated Lossless Transmission Line

The voltage and current traveling-wave solutions given in (3.14) and (3.19) contain
two unknowns, the amplitudes of the incident and reflected waves, V: and V0— ,
respectively. These unknowns are determined by exciting the transmission line with

a source or generator located at x=——l and terminating the line in an arbitrary load

20

characterized by impedance Z L located at x=0. A transmission line terminated in

an arbitrary load impedance is shown in Figure 3.2. The load impedance can be

 

IL
] +
Z0 2L \1

 

 

 

 

|< >|

=4 x=0

Figure 3.2. Transmission Line terminated in an arbitrary load impedance Z L-

described using Ohm’s law, V=IZ, as

 

 

Z _ V(a: = 0) _ Y]:
L — 1(21 = 0) — IL
_ Vie—7° + V5137O
"1"(V0+3—70 “ V5370)
Zb
_ V0+ + V;
_ 1 _
'Z-0(Vo+ - V0 )
+ _
= zw-gilés (3.20)
o "“ 0
Solving Equation (3.20) for V; gives
- Z — Z .
V0 2 V0+§iilj§ (3.21)
and so
VO— ZL - Z0 .
_ = _——._— = , ‘ .22

21

The ratio of the reflected to incident voltage amplitudes, FL, is known as the voltage
reflection coefficient. The voltage and current traveling-wave solutions given in (3.14)

and (3.19) can now be expressed in terms of the voltage reflection coefficient and are

given by:
V(a:) = V0+(e‘7x + I‘Len) (3.23a)
v+
[(113) = —ZQ—(e—7x - I‘Lew). (3.23b)
0

The impedance at any point on the transmission line is given by the ratio of the
voltage to current. The input impedance is the impedance seen looking into the
transmission line, toward the load, at a distance x=—-l from the load. The input

impedance in its most general form is given by

 

. _ V(:c = --—l)
2‘" — I(:1: = —r)
_ V0+(e—7(—l) + FL€7(-I))
_. V0+

7(6“7(‘-l) — I‘Le7(“l))

0

_ (671+ PUB—71)

_ 2009”” _ new). (3.24)

 

22

Using the reflection coeflicient from (3.22) in (3.24) yields: '

 

w + (Z )6“)
Z- = Z L+Z°
m 0( 71_ (§::§:)e-ryl)

 

ZL(e'll + 8"”) + Zo(e7l — 6‘71
OZ0(e7l + 6—71) + ZL(e7l — 6‘71
2Z L cosh yl + 220 sinh '7!
0 2Z0 cosh 'yl + 2Z L sinh yl
ZL + Z0 tanh 'yl

= Z . 3.25
OZ0+ZLtanhcyl ( )

 

 

 

The conducting septum of the HMLWA antenna is analogous to a special case of
loaded transmission lines: the short-circuited loaded line. This special case will be

discussed in the following section.
3.2.1 Short-Circuited Line

A short-circuited transmission line is characterized by a load impedance of Z L20, and
is depicted in Figure 3.3. Therefore, the voltage reflection coefficient is FL=—1, gov-
erned by Equation (3.22). The voltage and current on a short-circuited transmission

line are given by substituting I‘L=—1 into equations (3.23a) and (3.23b) resulting in

 

 

 

 

 

IL
_ M
+
Z0 ZL= 0 \(f 0
m 3
l‘ a»
x=4 x=0

Figure 3.3. Short—circuited transmission line.

23

 

WI) = VON?” - 67$)

and

-12: ~72: 7:5
I(:r)-— Z (6 +6 )

which become

V(:c) = —V0+(e7x —- 6771')
= —2V0+ sinh 7:1:
and
+
[(23) = KZQ—(eTW + 87x)
0

v+
= 2—ZQO- cosh yr.

(3.26)

(3.27)

(3.28)

(3.29)

The input impedance of a short—circuited transmission line is determined by sub-

stituting Z L=0 into Equation (3.25)

V(:1: = —l)
22” [(23 = -l)
0) + Zo tanh 'yl
_ Zo

 

Zo + (0) tanh ”)1
= 20 tanh '71.

where symmetry of the sinusoidal functions was invoked.

3.3 Transverse Resonance Relation

(3.30)

The TRM method is valid as long as the transverse resonance relation is main-

tained. This relation can be viewed as a boundary condition invoking continuity

of impedances across a mathematical boundary. It states that the input impedance

24

 

to the left of the boundary is equal and opposite the impedance to the right of the

boundary. This is mathematically represented as
- +
Z3, + Z3, = 0. (3.31)

Equation (3.31) is derived by invoking boundary conditions on the tangential electric
and magnetic fields at the boundary x=——22£ (see Figure 3.4). The tangential boundary

conditions that must be obeyed are:

a x (13+ — 13—) = 417 (3.32a)

a x (7% — f1“) = J (3321.)

‘0

where fl = E is oriented from left to right, i.e., originating in the negative region and
pointing toward the positive region. Since the region within the HMLWA antenna is

source free, the tangential boundary conditions reduce to

ax(7§’+ “’-) 0 (3.33.)
as

x (3+ — fi‘) = 0, (3.33b)

resulting in the following continuous components:

E; = E; (3.34a)
E: = E; (3.34b)
H; = H; (3.34c)
H: = H;. (3.34d)

25

The directional wave impedance is obtained by dividing the electric field component
by the corresponding orthogonal magnetic field component [22]. It is governed by the
following expression

Ei EJ'

_ F? = (3.35)

k
z "172"

where it = i x} abides by the cyclic nature of the cross product. The wave impedance
in the positive x-direction is

+ +

 

 

 

23+ = g;— = —%, (3.36)
while that in the negative x-direction is given by
Zl“ = E; = —f;. (3.37)
Substituting (3.34a) and (3.34d) into (3.36) gives
Z“6+ = in (3.38)
The right hand side of (3.38) is equal to —Z$- given by (3.37), and leads to
217+ = -—Z“"', (3.39)

the transverse resonance relation.

An alternative form of the transverse resonance relation may be derived by deter-
mining the voltage reflection coeflicient at the boundary x=—%. Substituting Z3;
and Z3: into equation (3.22) gives

an.
32 _ 2m 20

m _ — (3.40)
Zita +20

26

and
+
17
$4»- _ Zln -' Z0

in _ —§,~T'_v
z," + 20

(3.41)

w
the reflection coefficients to the left and right of the boundary x=—§, respectively.

Solving equations (3.40) and (3.41) for Z}; and Z? gives

Tl

 

 

 

_ 1 ff

Z33". = Z0 + :1

1 + P?“
2.3: - 20 "i-

1 _ Pin

The sum of (3.42a) and (3.42b)
- 1 I‘?’ 1 1‘59”"
Z3, +ng =ZO———+ m +2 + m =0,

_. o ,+
l—I‘fn l—an

(3.42a)

(3.42b)

(3.43)

invokes the transverse resonance relation, equation (3.31). Fhrther algebraic manip-

ulation leads to:

1 + rev“ 1 + r99"
20 2n + Z 2n = 0

_. 0 +
l—I‘fn 1"an

— +
1+an 1+an

 

 

= 0

1—I“,?’,,’ 14‘3”;

_ + + 7
(1+I‘fn)(1—I‘fn)+(l+rfn)(1‘Pi’nn) ___ 0

 

— +
(1..an )(1_Firn)
(1+quIn)(1_Fffz)+(1+rffn)(1-an) = 0

-— +
2—2rfnrg’n = 0

:1:- 33+ _
nun. — L

a more usable form of the transverse resonance relation.

27

(3.44)

3.4 TRM method applied to an unloaded HMLWA antenna

Before the TRM method is applied to the edge-loaded HMLWA antenna, the simpler

unloaded case will be presented. Once the unloaded case is explained in detail, the

 

 

 

20 2:0 {=0

v

Figure 3.4. TRM method applied to an unloaded HMLWA antenna.

edge-loaded case is determined by expounding upon this simpler, but non-trivial
model.

It is evident from the transverse resonance relation (equation (3.44)) that an
a priori knowledge of the reflection coefficients on both sides of the mathematical
boundary is required. The reflection coeflicient looking to the right is that of a short-
circuited transmission line transformed to the mathematical boundary (placed at the

radiating edge of the HMLWA antenna) x=—%. The reflection coefficient due to a

28

shorted transmission line is negative one, given by:

Fshort : ZL ‘T 20
L ZL-I-Zo
0—Z0
0+Zo
—ZO
Zo

= —1, (3.45)

 

 

where the impedance of a short-circuited load is zero. But the input reflection co-
w 10

efficient at the mathematical boundary x=—-—2-, a distance -2— from the short, must

11)

account for this displacement. The input reflection coeflicient at the boundary x=-———2—

can be computed by determining the ratio of the magnitude of the reflected voltage

wave to that of the incident wave, i.e.

V0~67T$
I‘(a:) Vie—7T3 (3.46)
which reduces to
V0_€jkT‘B
F($) — V+e“ij"3 (3.47)

for waves propagating and attenuating in the x—direction. Titansverse wave number

[CT is the wave number transverse to the direction of propagation, and is defined as

4T = a; - jag. (3.48)

where (=x, z. Moving the exponential term in the denominator to the numerator

yields
V0. eijTiL‘

W (3.49)
O

F(3:)

The voltage reflection coefficient at the load follows upon substituting x=0 into equa-

29

tion (3.49),

NO) +
0
V—
1‘ 2 __0 , 3.50
L V0+ ( )

Substituting x=——§ into equation (3.49) yrelds the input reflection coefficrent at the

reference plane given by

 

j2kr(-—)
+ w V_8 2
an( E) : 0 V0+
—' —jkT'LU
= V0 6 + (3.51)
Vo

Recall equation (3.50) which states that the reflection coefficient at the load is the

ratio of the reflected to incident waves. Substituting this into equation (3.51) gives

, + V_e_jkTU)
pg" (2:) = _0T/T__
0
= rLe—jkrw. (3.52)

Substituting the reflection coefficient of a short—circuited transmission line from equa—
tion (3.45) yields
r3: (3:) = —e-i’rrw, (3.53)

the input reflection coefficient looking to the right of the mathematical boundary,
x=--t-;—. Here, kT is the transverse wave number given by kT = W-

The reflection coefficient looking to the left is not as trivial. The discontinuity
presented by the abrupt end of the conducting strip, while the substrate and ground
plane continue, introduces a significant amount of reflection for waves propagating

in the substrate. This phenomenon is thoroughly explained in [8] where the authors

30

give the reflection coefficient at this interface. Using their notation, the reflection

coefficient looking to the left is written as
_ . k“
I?" (:13) = 6”“ 1), (3.54)

where XUCT) is a complex function of kT defined in equation (3.70) of Section 3.5.
Substituting (3.53) and (3.54) into (3.44) enforces the transverse resonance rela-

tion and leads to the following transcendental equation
—e—jkrw x er<kT) = 1. (3.55)

Figure 3.4 depicts application of the TRM method to the unloaded HMLWA antenna.

Multiplying both sides of (3.55) by negative one leads to
e‘jk’r’w x erlkT) = —1, (3.56)
and realizing that —1 = eijmr for odd 71 gives

e—jkTw x er(kT) = eijmr. (3.57)

Dividing both sides by eijmr gives

e-jk'l‘w X er(kT)

 

eijmr
eTjkTw x er(kT) x syn” = 1

ej(x(kr)-kTwEmT) = 1,

(3.58)

31

Taking the natural logarithm leads to
ln(ej(X(kT)‘kTan”)) = 0. (3.59)

The term within the parenthesis of equation (3.59) is a complex quantity. It can be
recast in the general exponential form of z = we”, where r = |ele<kT)"kTw:F”")| = 1
and p = arg(ej(X(kT)—kqu:mr)) = x(kT) — kTw 2]: mt. This complex quantity is
muti-valued, and its value depends on which Riemann sheet, or branch, it lies on.
There are an infinite number of sheets which are traversed by increasing multiples
of 27rm where m, an integer, isolates one sheet where the multi-valued quantity is

single-valued. Therefore, the natural logarithm of a complex quantity is given by

ln(z) = lnr+lnejp
= lnr+jp+j27rm

= lnr + j(p + 27rm). (3.60)
Applying this definition to (3.59) leads to

ln(1) +j(x(kT) — [@212 $ mr) + j27rm = 0
j(x(kT) -— kTw $ n7r + 27rm) = 0

X(kT) — kTw + 7r(2m IF n) = 0, (3.61)

but since (2m 2}: 71) results in an integer, conveniently allowing this integer to be 71
yields
XUCT) - kTw -I- mr = 0. (3.62)

32

For the hybrid leaky-wave EH1 mode, 71 = 1 [21] and (3.62) becomes

x(kT) — kTw + 7r 2 0. (3.63)

Equation (3.63) contains one unknown, the transverse wave number 1971, which is
determined by finding the root of the function. Once kT is known, the axial wave

number ky can be determined by solving the constraint equation,

1.3. + 3,3 = k2. (3.64)
Solving for ky yields
k; = k2 - kg.
3,, = k2 - 1%

kg = ‘/€rk3—kr§s, (3.65)

where k0 is the free-space wavenumber given by k0 = cud/1060. The normalized

leakage and phase coefficients are easily extracted from ky as

01(0)) = Im{7]:%} (3.66)
and
6(0)) = Re{%}, (3.67)

respectively. Dispersion curves of the leaky EH1 mode on an unloaded HMLWA

antenna are depicted in Figure 3.5.

33

 

 

 

 

 

 

 

 

 

 

Figure 3.5. Dispersion curves of the hybrid leaky-wave EH1 mode. Operational
region: 5889-8115 GHz. BW=31.80%.

3.5 Free-edge Admittance

Chang and Kuester investigated oblique incidence of a TEM wave incident upon the
open end of a parallel-plate waveguide with a truncated upper plate [8]. This can be
used as a simple model for the open edge of the HMLWA antenna. They employed a
Wiener—Hopf technique to determine the reflection coefficient at the open edge of the

structure. They used this reflection coefficient to define the free-edge admittance as

\ffZ
YFE(kT) = —-,-—‘5’-'- tan 335—712. (3.68)
.7770 2

34

 

The reciprocal of the admittance results in the free-edge impedance,

j k
ZFEUCT) = -7-7\/;£; COt 31-21;), (3.69)

where fir=1 for a non-magnetic substrate. Figure C.1 depicts the resistance of the
free-edge impedance while Figure C.2 depicts the reactance. An alternative to using
the rigorous Wiener-Hopf technique to determine the free-edge impedance is given in
[2]. Here, a y-polarized Gaussian pulse is excited at one end of the microstrip (between
the conducting strip and the ground plane), and the impedance at the opposite end,
the free-edge, is determined by taking the ratio of the Fourier transforms of the y-
polarized electric to z-polarized magnetic fields.

The kaT) term within the argument of the cotangent function is given by

k2 _ k2
X(kT) = 2tanT1 —————VT tanhA(kT) — fe(kT), (3.70)

T

where A(kT) and fe(kT) are given in [33], based on results reported by Chang and

Kuester, and are repeated here:

 

 

 

 

A(kT)= [ e. (ln (Jh\/k0—k +kT)+7—1)+2Q(—6.)— (3.71)
—2Q(6M)]i
and
_ 1 'h k2-k2+k2 + ~1
fe(kT)= QkTh n(] ‘/0 T) 7 +2Q(—5.)—1n(27r) . (3.72)

7T 61-

Here 7, Euler’s constant, is defined as 7:0.577215665, and the other constants are

35

defined as:
er —— 1

 

 

 

6C = fr + 1 (3.73)
_ Mr -1
6“ _ Hr+1
= 0 (3.74)
Q(_‘6€) = Z [- :r l [] ln(m) (3.75a)
m=1 r
Q(—6,.) = 0|)..z (3.75b)

Figure 3.6 depicts that for €r=2.33, Q(—6¢) converges with at least m=15 terms in

the series. A(kT) and fe(kT) are reexpressed as

 

AlkT) =

 

h 32—16%. 1_€T
[ 6 (ln(jh\/k(2)—-k2+k%)+7—1)+
00 :m (3.76)
61'"
+22 [—€r+1] ln(m)],

m=1

 

and

 

 

+2

 

—2kTh ln(jh\/kg—k2+k%)+7—1 00 Cr—1 '
felkT) = ‘-
1

1r (7.

.. 1867.)]

(3.77)

upon substituting equation (3.75) into equations (3.71) and (3.72), respectively.

Finally, an expression for XIle is derived showing its explicit dependence on the

36

0.12

 

 

 

.0
o
N

......................................................................

 

 

 

 

0 1 0 20 30 40 50
m

Figure 3.6. Convergence of Q(—-6€) vs. m for €722.33.

transverse wave number, kT. Substituting equations (3.76) and (3.77) into (3.70)

yields

(ln(jhkr) + 7')+

metre? h(1—e) 32-11:?
kaT) = 2tan_1(——0——T-tanh[ r O T

[CT 7r€r

2L_M i [_E[m1n(m)[)+ (3.78)
m 1

+
7r €r+1

 

 

ér-I-l

21.51 We 0°
+ T [r10 ins/+2:
7r er =

[fr — 1] m ln(m) —1n(2s)] ,

m 1

the final expression for x(kT), as derived by Chang and Kuester, where 7’ = 7 -— 1.
Equations (3.68) through (3.78) are valid for electrically thin substrates, where

kocrh << 1.

37

3.6 TRM method applied to a reactively loaded HMLWA antenna

The impedance to the left of the boundary, given by

 

— Z + jZotan(k$:r')
Z?’ ’ = 2 RE 3.79
m (:r ) 0 Z0 + jZREtan(k1-:r’) ’ ( )
is derived in Section 3.2. The impedance to the right of the boundary
2,3,:(x') = jZotan(kxa:’), (3.80)

derived in Section 3.2.1, is the input impedance of a short—circuited transmission line.
The radiating-edge impedance denoted by Z RE in equation (3.79), is the parallel
combination of the free-edge impedance and the impedance of the lumped element

placed in shunt with the radiating edge, given by

ZLoad X ZFEUCT)
Z — I 3.81
RE ZLoad + ZFElkT) ( )

 

Substituting Z F E from equation (3.69) yields

3312 Got x(kr)

Z

. k .
.7770 cot x( T)

«a? 2

 

ZRE = (3.82)

ZLoad +

 

The following analysis employs equation (3.44) because it is less laborious to manip-

ulate reflection coefficients than input impedances. The reflection coefficient looking
'w .

to the right, that of a short circuited load transformed to the boundary x=—§, IS

derived in Section 3.4 and is repeated here:

r3: (:8) = -—-e_jkTw. (3.83)

38

 

The characteristic impedance is required to determine the reflection coefficient
looking to the left, evident from equation (3.22). The wave impedance of a TEM
plane wave propagating through a dielectric, characterized by a relative permittivity

of 5,», is used for the characteristic impedance [2] and is given by

 

 

 

 

 

 

 

 

“L

(b) TRM applied to loaded HMLWA
rx’

ZRE If; in

 

Figure 3.7. TRM method applied to an edge-loaded HMLWA antenna.

’70
Z = , 3.84
. «a ( )

 

where no, the intrinsic impedance of free space, has a value of ‘ [w z 1207r Q.
60
Post application of equation (3.22), the reflection coefficient looking to the left of

the boundary x=-—% is given as

s- _ ZREfkT) - Z0

39

and substituting equations (3.82) and (3.84) yields

 

 

 

 

 

 

 

 

jllo XUCT)
Z x —— cot
Z -l- 217—0 cot X(kT) V67
.— «a 2
in (IE) — . (3.86)
.7770 x(k:r)
ZLoa X cot
\fl? 2 + 770
3'77 X(kT) 6
ZLoa + 'Jégr‘ COI. 2 fl

Substituting (3.83) and (3.86) into (3.44) enforces the transverse resonance relation

leading to the following transcendental equation

 

 

 

 

 

 

 

 

 

 

F .7770 X(kT) -
Z x —— cot ——-—
Load (Fr 2 — no
' k
ZLoad + 37:.» Got X(2T) x/Gr
V. r x (—e“J’W) = 1. (3.87)
Z .7770 X(kT)
Load X t ——
2 + 770
mo X(kT) \/€r
Z t
_ Load + ,—6T 00 2 1

Figure 3.7 depicts application of the TRM method to the edge-loaded HMLWA

antenna. Multiplying both sides of (3.87) by negative one yields

 

 

 

 

 

 

 

 

 

and realizing that —1 = eijmr for odd n gives

40

 

 

’ jno XUCT)
Z x —— cot
' k
ZLoad + 3:}?- cot X(2T) “67
. r x eflkw = —1, (3.88)
ZL d x 1770 cot x(k.r)
0a \fl: 2 770
Mo XUCT) V3?
Z t
- Load + x/Er: co 2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

oa. X\/—CO 2 __ 770
ka c
2W” cot ><<2 ) r.
L—V/g-C X e—jkT'w : eijmr.
ZLmdx jno cot WW)
2 + 770
inc X(kT) «E?
Z cot ——
Load + \/_TC 2 -
Multiplying both sides by e+jkTw gives
— .1770 X(le q
Z x —— cot _—
' k
ZLoad + 17-72 cot ——X( T) «a:
\/€_r‘ 2 = ej(in7r+krw)
jno X(k:r) ’
Z x cot ——
jno X(kT) J3?
Z + ——- cot ——
L Load «a: 2 .

 

 

and taking the natural logarithm of both sides leads to

In

 

 

 

 

 

 

ZLd x1200 t_(_XkT)
oad X\/_7‘C 2 _ 770
_J_'___no x(k:r) i/e'i?
ZLoad +-—- \/€_TC CO-——t 2
X_J'__no x(kr)
Zoad cot _—
Load X\/_C 2 + 770
Z +11th x(kr) \/'e7

41

 

+j27rm = j(:tn7r + kTw),

(3.89)

(3.90)

(3.91)

post application of (3.60). Rearranging terms gives

 

 

 

ln

 

1n

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Z Load x L}; cot 9—4?)- 770
Z Load + 2% cot 24.31.). ‘fl;
ZLoad x :7; cot X02611) 720
zLoad + #63; t 3%?) V5
and grouping common terms leads to
Z Load x 3:37: t £32122 — 770
Z Load + 3:; cot 31:2 J3;
ZLoad x :2; cot XUST) + 770
Z Load + 227—“; cot 3%.?) fl
Further simplification leads to
Z Load x 320: t 210261) .— 770
ZLoad + 43% cot X0571) J6?
ZLoad x 5:7 cot MST) + 770
ZLoad + 1% t 95%!) fl

+j27rm — j(in7r + kTw) = 0,

+ j(27rm IF mr — kTw) = 0.

 

+j(7r(2m IF n) — kTw) = 0

 

(3.92)

(3.93)

(3.94)

and since (2m 4: n) results in an integer, conveniently allowing this integer to be n

42

yields

' It
.7770 Got X( T)

Z x
L0G \/— 2 _ 770
J90 cot XlkT) \fl;

 

 

 

 

 

Z

Load + «E; 2
.7770 cot x(k:r)
\/€_7“ 2 770
.7770 cot X(kT) J;
«a? 2

ln

 

+ j(7m — kTw) = 0. (3.95)

 

 

ZLoad ><

 

 

ZLoad +

 

 

This reduces to

 

j 710 X(kT)
Z x cot ——
Load {—67‘ 2 770

 

 

J'no x(k:r) «€77

zLoad + — cot _—

./e7 2

 

 

ln . +'7r—kw=0 3.96
ZL d x 2710 (MM/CT) ]( T ) ( )
oa fl 2 770

 

 

£73 cot X(kT) «5
W 2

ZLoad +

 

 

since n=1 for the hybrid leaky-wave EH1 mode [21].

Equation (3.96) contains one unknown, the transverse wave number kT, which is
determined by finding the root of the function. Once kT is known, the axial wave
number ky can be determined by solving the constraint equation (3.64). Dispersion
curves of the hybrid leaky-wave EH1 mode of a 0.10 pF loaded HMLWA antenna are
depicted in Figure 3.8.

Comparing the dispersion curves of Figure 3.5 and Figure 3.8, there are several
notable differences, particularly with the operational leaky region and the bandwidth.
When loaded, the operational leaky region of the HMLWA antenna decreases. This
behavior is expected because loading the antenna with the lumped capacitors essen-
tially slows the propagating wave down. These results are also consistent with those
reported in [2] and [5], where the authors state that capacitive loading shifts the oper-

ational region down in frequency, while inductive loading shifts the operational region

43

 

 

 

 

 

 

 

 

0 y j i i j

4 4.5 5 5.5 6 6.5 7 7.5 8

 

Figure 3.8. Dispersion curves of the hybrid leaky-wave EH1 mode of a 0.10 pF loaded
HMLWA antenna. Operational region: 3928—4869 GHZ. BW = 21.39%.

to a higher frequency band. The bandwidth, defined as the difference between the
exit and entrance of the operational region, i.e., f (,6 = 1:0) — f (a = fl), experiences a
significant decrease from 2.26 to 0.95 GHZ, 58%.

Figure 3.9 and Figure 3.10 depict the phase and leakage coefl‘icients for various
capacitive loads ranging from 0.02 to 0.10 pF. Figure 3.9 shows that when the value
of capacitance placed in shunt with the radiating edge of the HMLWA antenna is
increased, the microstrip appears to be wider causing the cutoff frequency to shift
downward. Table 3.1 lists the lower and upper cutoff frequencies of the EH1 mode,
i.e., ch = f (a = fl) and ch = f (6 = k0), respectively, and the bandwidth for each

loading configuration depicted in Figure 3.9 and Figure 3.10.

44

 

1:
.u g r y 1 r 1 g r

 

 

No Load

l llll

 

 

 

.......

 

 

 

i a ; . . . g i i
5.5 s 6.5 7 7.5 a 8.5
Frequency (GHz)

Figure 3.9. Phase coefficients for various capacitive loads ranging from 0.02 to 0.10
pF.

 

l. ! 1 T v y r v v

 

No Load
: ——.02pF
— .04pF
— .06pF
— .08pF

_ —.1pF

 

 

 

 

 

 

 

 

5.5 e 5.5 7 7.5 I; 3.5
Frequency (GHz)

Figure 3.10. Leakage coefficients for various capacitive loads ranging from 0.02 to
0.10 pF.

45

Table 3.1. Lower cutoff frequency, upper cutoff frequency, and bandwidth of various
capacitive loads.

 

 

Capacitance (pF ) fc L (G H z) ch (GHZ) BW
0.02 5.29 7.03 28.16%
0.04 4.83 6.25 25.59%
0.06 4.47 5.67 23.76%
0.08 4.17 5.23 22.39%
0.10 3.93 4.87 21.39%

 

 

 

 

 

 

3.7 TRM method Implementation

The transverse resonance method is implemented in FORTRAN. The FORTRAN
code uses the secant method to determine the root (transverse wave number kT) of
equation (3.96). The axial wave number, Icy, follows upon substituting the transverse
wave number into (3.64), the constraint equation. The TRM method models a 3-
dimensional structure as a 2—dimensional transverse transmission line infinite in extent
[21]. Therefore, when a loading configuration is simulated, the method inherently
assumes that it is placed all along the radiating edge of the HMLWA antenna, a
physically unrealizable configuration when using lumped elements. Hence, the loading

configurations are actually per unit length.

To ensure that the radiating edge of the HMLWA antenna is not short— or open-
circuited, the impedance regime is determined prior to running the code. The

impedance of a capacitor

ZC = —, (3.97)

is inversely proportional to the value of capacitance. The numerator in (3.97), the

Caapacitive reactance, is defined below:

—-1

The inversely proportional nature of (3.98) allows for a reduction in the test pool of

46

 

 
   
  

~~~~~

 

ldeal Wine L j ;
_4 _. . , . . O 10 nF Capacitor ........... .............. ....... . .....

 

 

 

 

 

|S11| (dB)

-14p ........... ..........

-16 .............. ........... g ............ ,5

 

 

 

-18 g . g .
6

Frequency (GHZ)

Figure 3.11. 340 mm edge-loaded HMLWA antennas. Loads spaced every 1.5 mm.
Depicts how large capacitance value shorts the radiating edge.

candidate capacitors. Capacitors in the nano—Farad range and larger are ignored since
the operational frequency, in the gigahertz range, essentially neutralizes their effect.
For example, computing the capacitive reactance of a 10 nF capacitor using (3.98)
y ields a capacitive reactance of XC=0-002 Q at 8.00 GHZ. This miniscule impedance
effectively shorts the radiating edge transforming the antenna into a waveguide or
CaVity. Figure 3.11 illustrates the agreement between the input reflection coefficients
0f the 10 nF edge—loaded HMLWA and the ideal wire-loaded HMLWA antennas. It
Shows that edge loading with 10 nF capacitors is equivalent to shorting the radiating

edge of the antenna.

47

3.8 Fixed Frequency Beam Steering

The purpose of this thesis is to show fixed frequency beam steering of a HMLWA
antenna. Although edge loading in the TRM method implies a PUL configuration, a
physically unrealizable result, the simulations provide a reasonable amount of insight
into the solution, without requiring a full-wave solution. Table 3.3 through Table
3.10 provide the PUL capacitive loading required to fix the main-beam direction of
the HMLWA antenna at the specified angle throughout the C-band. The negative
entries in the tables represent non-Foster capacitances. This is implemented in the
TRM code by changing the sign of the reactive portion of the complex load, Z Load-

Table 3.2 gives the numerical values used to determine the fixed main—beam di-

rection tabulated results reported in Table 3.3 through Table 3.10.

Table 3.2. Capacitive limits used to compute fixed main beam vs. frequency tabulated
values

 

 

Frequency (GHZ) Lower Capacitive Limit (pF) Upper Capacitive Limit (pF)
4.00 0.000001 1.00
4.50 0.000001 1.00
5.00 0.000001 1.00
5.50 0.000001 1.00
6.00 0.000001 0.10
6.50 0.000001 0.10
7.00 0.000001 0.10
7.50 0.000001 0.10
8.00 0.000001 0.01

 

 

 

 

 

All loads consist of lumped capacitors placed in shunt with the radiating edge of
the HMLWA antenna. It is assumed that the lumped elements are placed infinitely
along the radiating edge with an infinitesimal spacing between consecutive elements.
This assumption is reasonable because the TRM method is a 2-dimensional continuous

model of the 3-dimensional HMLWA antenna. The angle is measured off broadside,

48

normal to the plane of the antenna. Figure 3.12 is a pictorial representation of the

tabulated data reported in Table 3.3 through Table 3.10.

Table 3.3. Capacitive Load required to fix Main Beam at 10° over C—band

 

 

 

 

 

 

Frequency (GHZ) Angle Lumped Capacitor (pF ) Percent Error WRT 10°
4.00 10.55395 9.404792946473237 x10’2 5.54%
4.50 10.10443 5.752970685342671 x 10-2 1.04%
5.00 10.58057 3.151672636318159x10"2 5.81%
5.50 10.04023 1.100649174587294 X 10"2 0.40%
6.00 10.01043 -4.851474737368685 x10—3 0.10%
6.50 10.04327 -1.775805702851426 X 10.2 0.43%
7.00 10.00862 -2.876366983491746x10"2 0.09%
7.50 10.00392 ~3.846861930965483 x10-2 0.04%
8.00 10.05291 .4.742318609304652x10-2 0.53%

 

Table 3.4. Capacitive Load required to fix Main Beam at 20° over C—band

 

 

 

 

 

 

Frequency (GHZ) Angle Lumped Capacitor (pF) Percent Error WRT 20°
4.00 20.49186 0.100551174587294 2.46%
4.50 20.59925 6.403295197598799x10—2 2.99%
5.00 20.75909 3.751972186093047x10-2 3.79%
5.50 20.16161 1.700948724362181x10-2 0.81%
6.00 20.00144 1.051514757378689x10-3 0.01%
6.50 20.02314 —1.190507203601801x10-2 0.12%
7.00 20.02268 .2.291068484242121><10-2 0.11%
7.50 20.05478 -3.2615634317158:~37x10-2 0.27%
8.00 20.08793 -4.157020110055027x10’2 0.44%

 

49

 

 

Table 3.5. Capacitive Load required to fix Main Beam at 30° over C-band

 

 

 

Frequency (GHZ) Angle Lumped Capacitor (pF) Percent Error WRT 30°
4.00 30.39386 0.109055418209105 1.31%
4.50 30.49723 7.153669634817408x10-2 1.66%
5.00 30.21305 4.402296698349175x10—2 0.71%
5.50 30.33996 2.351273236618309x10‘2 1.13%
6.00 30.04200 7.154505252626313x10-3 0.14%
6.50 30.04345 -5.952036018009005x10"3 0.14%
7.00 30.07277 -1.695764882441220x10‘2 0.24%
7.50 30.02271 —2.661257278639320x10—2 0.08%
8.00 30.07005 -3.531701200600300x10"2 0.23%

 

 

 

 

Table 3.6; Capacitive Load required to fix Main Beam at 40° over C-band

 

 

 

Frequency (GHZ) Angle Lumped Capacitor (pF) Percent Error WRT 40°
4.00 40.26788 0.120060909954977 0.67%
4.50 40.29020 8.104143921960981x10-2 0.73%
5.00 40.02176 5.252721060530265x10—2 0.05%
5.50 40.30084 3.151672636318159x10‘2 0.75%
6.00 40.03358 1.470820660330165x10-2 0.09%
6.50 40.03638 1.351661830915458x10-3 0.09%
7.00 40.05529 —9.753974987493747x10-3 0.14%
7.50 40.00808 .1.935887343671836><10-2 0.02%
8.00 40.03607 .2.786321060530265x10-2 0.09%

 

 

 

 

Table 3.7. Capacitive Load required to fix Main Beam at 50° over C—band

 

 

 

Frequency (GHZ) Angle Lumped Capacitor (pF) Percent Error WRT 50°
4.00 50.19959 0.133067400200100 0.39%
4.50 50.03055 9.204693096548273x10—2 0.06%
5.00 50.45237 6.303245272636318x10-2 0.90%
5.50 50.06800 4.052121960980491x10-2 0.14%
6.00 50.00917 2.331242271135568x10—2 0.02%
6.50 50.03428 9.605706353176589x10-3 0.07%
7.00 50.04904 -1.699867433716859x10“3 0.09%
7.50 50.05514 -1.130476588294147x10”2 0.11%
8.00 50.00431 -1.975907753876939x10"2 0.01%

 

 

 

 

50

 

 

 

Table 3.8. Capacitive Load required to fix Main Beam at 60° over C—band

 

 

 

Frequency (GHZ) Angle Lumped Capacitor (pF) Percent Error WRT 60°
4.00 60.31728 0.147074389694847 0.53%
4.50 60.13477 0.104052921960980 0.22%
5.00 60.44472 7.353769484742370x10’2 0.74%
5.50 60.00945 5.002596248124062x10-2 0.02%
6.00 60.03025 3.221678589294648x10-2 0.05%
6.50 60.00767 1.800982441220611x10-2 0.01%
7.00 60.00542 6.404137568784393x10"3 0.01%
7.50 60.06108 -3.300683841920961x10"3 0.10%
8.00 60.00631 —1.175499549774888><10-2 0.01%

 

 

 

 

Table 3.9. Capacitive Load required to fix Main Beam at 70° over C—band

 

 

 

Frequency (GHZ) Angle Lumped Capacitor (pF) Percent Error WRT 70°
4.00 70.16352 0.159580630315158 0.23%
4.50 70.27648 0.1 15058413706853 0.39%
5.00 70.43237 8.304243771885943 X 10‘2 0.62%
5.50 70.44967 5.903045572786392 X 10—2 0.64%
6.00 70.03647 4.022070785392697x10’2 0.05%
6.50 70.03898 2.556352576288145 x10—2 0.06%
7.00 70.06608 1.365769184592296 X 10’2 0.09%
7.50 70.04286 3.702813906953477 X 10"3 0.06%
8.00 70.02250 .4.801449224612306x10-3 0.03%

 

 

 

 

Table 3.10. Capacitive Load required to fix Main Beam at 80° over C-band

 

 

 

Frequency (GHZ) Angle Lumped Capacitor (pF) Percent Error WRT 80°
4.00 80.68323 0.169085373186593 0.85%
4.50 80.70762 0.123062407703852 0.88%
5.00 80.93887 9.004593246623310x10-2 1.17%
5.50 80.26050 6.503345122561280x10-2 0.33%
6.00 80.11734 4.597352676338170x10-2 0.15%
6.50 80.02818 3.091614857428715x10‘2 0.04%
7.00 80.08994 1.876019209604803x10‘2 0.11%
7.50 80.10358 8.655240620310155x10“3 0.13%
8.00 80.00903 3.601400700350175x10‘5 0.01%

 

 

 

 

51

 

 

 

 

90 .' l I r

 

Main Beam Direction 9

 

 

 

 

 

 

 

o i i 1
-0.05 0 0.05 0.1 0.15 0.2
Capacitance (pF)

Figure 3.12. Main beam direction vs. Capacitance for various Frequencies.

52

CHAPTER 4

ANALYTICAL SOLUTION

There are numerous analytical techniques used to determine the field structure within
a microstrip antenna including the transmission—line model, the cavity model, and the
integral equation method [24]. Of these, the cavity model is appealing because it uses
the aperture fields within the dielectric substrate in a fairly straightforward fashion
[24]. If the radiation pattern is desired, a particularly simple, mathematically precise,
analysis can be used to approximate the pattern of an open-circuited microstrip line
[32]. The radiating aperture model is precise if the aperture field distributions are
known exactly, but the model yields a fairly accurate pattern even if the aperture fields
are approximated. In Section 4.1, the cavity model is used to derive the appropriate
boundary conditions that the EM fields must obey at the radiating edge (aperture)
of the HMLWA antenna. In Section 4.2 the EM field in the half space 2 > 0 of the
HMLWA antenna is determined which must obey the boundary conditions discussed

in Section 4.1.

4.1 Cavity Model of HMLWA antenna

Although patch antennas are resonant standing-wave strucutures, we borrow the anal-
ysis method used to derive the EM fields at the conducting-strip edge air interface of
a patch to study the HMLWA antenna. This approach has been dubbed the leaky-
cavity model and is presented here. Elliott [26] mentions that the electromagnetic
pr0perties of a patch (microstrip) antenna can be deduced by viewing it as a leaky
caVity, provided that the substrate is electrically thin, i.e. h << A. This leaky-
caV'ity analysis provides good conceptual knowledge of how fields radiate away from
microstrip antennas. The radiation zone fields are computed using the aperture fields

Within the slots along the patch edges.

53

If the dielectric substrate and the PEC ground plane of the microstrip antenna are
truncated to the dimensions of the patch, the antenna can be viewed as a dielectric

loaded cavity (Figure 4.1). The cavity is bound by perfect electric conducting walls

 

PMC Wall
Figure 4.1. Cavity Model of a Microstrip Antenna

on the top and bottom, representative of the conducting strip and the ground plane,
and bound by pefect magnetic conducting (PMC) walls on the sides The PMC walls
are included because when the patch is energized, charge distribution is established
on the upper and lower surfaces of the conducting strip and on the surface of the
ground plane [28]. Grouped like charge under the conducting strip moves to the top
of the strip because like charges repel each other. This movement creates two electric
current densities, 7t, and 7b: for current densities on the top and bottom of the
conducting strip, respectively. For most microstrips the substrate height-to—width
ratio is small, and the opposing charges on the bottom of the strip and top of the
ground plane attract; and little, if any, charge moves to the top of the conducting
strip. Ideally, as the height-to—width ratio approaches zero, no charge would move
to the top of the conducting strip, hence i=0, and therefore no tangential magnetic
field exists in the planes forming the edges of the patch. The absence of tangential

magnetic fields along the edges of the patch are represented in the cavity model by

54

the inclusion of PMC walls along each edge. The PMC walls ensure that tangential

magnetic fields vanish along the side walls by the tangential PMC boundary condition

-->

ii x H=O.

4.2 The Radiating Aperture Model

In order to use the radiating aperture model, several a priori assumptions are made.
The fields in the aperture are assumed constant with a propagating and attenuating

behavior along the length of the structure where
132’ = 215306-731. (4.1)

Although Equation (4.1) is reasonable for electrically thin substrates, simulation from
CST Microwave Studio verified that the assumption is appropriate for the thicker
substrate used for the HMLWA antenna. Figure 4.2 depicts _E-" at two different heights
(220.40 mm and z=0.55 mm) within the aperture. The fields are approximately equal
at both cuts throughout the length of the aperture. As expected, the field attenuates
as the wave propagates down the length of the antenna. If reflections are considered

from load mismatch, Equation (4.1) becomes
E = 3E0(1 + Ute—73’. (4.2)

For this analysis, Equation (4.1) will be used for the aperture fields because the
antenna is sufficiently long (L=340 mm) that 90% of the energy radiates away before
reaching the load. Any remaining energy is absorbed by the 50 9 load. The HMLWA
antenna has a half-length £21=170 mm, conducting strip half-width of 33:380 mm,
and dielectric substrate half-height of g=0.3935 mm. The antenna is centered at

L L
the origin such that the feed is located at yz—E, the load is located at y=—2—, the

55

 

 

 

 

 

 

 

 

 

 

 

1 V I f I I I 1
\ 1 i 3 ? — =.4 mm

20.9}. ....... ........ ........ E2(2 0 ).
.9 ; g g g o Ez(z=.55 mm)

0.8 ........ .. .. ....... . ........ _ _ . , q
o : ' ' ; : - .
t : . ‘. : : : :
‘50]? .............. 2. . ....... I ....... . ....... :.. ..... .,
ulj : a! t.“ 2 ; 2

06.....- ....... j ..... ...... .. ..... j ...... ..; ....... ..,
g : 2 ; . " , 2 2
:05 ................ 5 ....... : ........ g ...... .. .. .5 ....... q
0 : s 3 2 ' i 3
20.4.. ............... ....... ....... ...... .. .. ....... -
E03. ....... ....... .
.6 . . . . . . .
€02; .....................................................................
o
z ............................................................. -

-5o 0 so 100 150 200 250 300 350 400

MM)

Figure 4.2. Magnitude of the Electric field within the aperture of the HMLWA an-
tenna.

PEC conducting ground plane (of negligible thickness) is located at 22—; the PEC
conducting strip is located at zzg, the PEC conducting septum is located at x=z-:-,
and the aperture is located at x=—~:%. The antenna geometry is depicted in Figure
4.3.

As discussed in Section 1.1, inclusion of the PEC bifurcating wall slightly perturbs
the fields of the EH1 mode with a noticeable, but negligible, effect on the far-field

patterns. The far zone electric field maintained by equivalent magnetic surface current

is derived in equation (A.52) of Appendix A. The magnetic surface current on the

56

w/2 -* ‘-

 

 

 

Figure 4.3. HMLWA antenna geometry with radiating aperture.

t..—
._> :1 ‘—

surface bounding the aperture of the HMLWA antenna is given as

117%?) = —2sx(2E0e~W)

— 2Eoe_’7y§]

= 2Eoe"j’°vygj, (4.3)

a result of the equivalent magentic surface current, A7 = 73. x E, and 7 = jky. For a

wave propagating and attenuating in the y-direction, kg = (,B—ja) and (4.3) becomes
MG?) = 2Eoe<riflw>yg (4.4)

The far—zone electric field follows upon substituing (4.4) into (A52) and is given by

—jk
E m 2—jkr 6 e JI( (2Eoe( Jig—0‘ ”@637"? 5($'—(—w

6 T

 

% E_0_jke___ 1" ”(Hid —jfi— —)’ayejk7"7‘ 7” (W —(—Z—}))17dv'- (4.5)

The factor of 2 is included because the magnetic surface current images into the PEC

ground plane. Equation (4. 5) becomes

 

 

7T

—-) N onk €_jkr A (—j[3—a)y’ 7671?, I U) A I
E ~ 7‘ r x IJIe e] 6(x — (—Z))ydv (4.6)

57

after employing a singular model since the magnetic current lies in the y—z plane
. . . . w .
Within the radiating aperture located at x=-——4—. Once a spherical-to-rectangular

transformation is performed on the 7" unit vector, (4.6) becomes

 

 

. _ 'k:
E g E09158 J T72 X III (e(—jfl—a)y’ejk(:i:sin6cos¢+3jsinOsin¢+2cos6)-7’
1.
V

 

 

7r
(4.7)
-6(:1:’ + 33M) dv',
which reduces to
E" R: Ejrjke J 7'1: X III (ejy’(ksin93in ¢-fl+ja)ejk(:r’sinflcos¢+z’cosa)
r
V (4.8)

-5(:I:' + %)§)) dv'

since 7" = z'i+y’7)+z’2. Since the solution is independent of the order of integration,

it is appropriate to evaluate the integral over a:’ first leading to

 

 

 

 

E) R1 onke .7ka X Qe’JkZSlIIeCOS¢/ / Bil/((3311108in¢-fl+ja)eij’COSOdy’ dz"
7r T 2’ y;
(4.9)
Since the substrate is electrically thin (kh << 1), the integral over 2’ is
A
2 ' I
(2sz “039 dz’ x h, (4.10)
~13
and (4.9) becomes
—-> onke_jkr —jk-1£sinOCos¢ - , k" 0 . fl . ) I
E z 7r 7‘ 1“ x Qhe 4 [6]“ 5‘“ 511103. +30“ dy. (4.11)
y!

58

Rearranging terms leads to

 

 

 

w
. —jk(r+—sin6cos¢)
‘E" R: onkhe 4 f Xfl/ ejy’(ksin63in¢—fl+ja)dyl
77 T yr
'k( +w ' o a)
. —_7 1' —sm cos L 2
as onkhe 4 7" X 3}] / ejy’(k8in65in¢—fi+ja)dy’. (4.12)
W 7‘ —L/2
The integral in (4.12) is of the form
L
L/2 . Sin(p§)
/ 634’” (K = 2 (4.13)
—L/2 p

where p = (k sinflsinq’) — 3 + ja) and C = y’. Therefore, (4.12) becomes

. w ‘. . L . . .
E? N onkh e-Jk(7‘+ 4 511190089407: x A2sm (50$: smfismd) -- 3 +300)
~ 7r 7" y (ksinBsin¢>—[3+ja)

 

 

(4.14)

which becomes

 

E 'khL _jk(r+-:%sin9cosd>) L
E? z 0.77r e r sinc (EU: Sinfisin ¢-,3+j01)) (93 COSQSin ¢—é COS (b) ’

(4.15)
after completing rectangular-to—spherical transformation on unit vector y and taking

the cross product. In the elevation (YZ-plane), (b = 12r_ and :21- S 0 S 32: reducing
(4.15) to
onkhLe—jk’"

7T

—__)
Em

 

cos Osinc (éJ—(k sin 0 — fl + ja)) (f). (4.16)

The normalized radiation intensity is determined to be

 

2

>2]...

(4.17)

 

 

 

7T

—’ _ 2 .4 _.
lE(9,¢-2)max [¢E(g,¢=g).E*(0,¢=

NJ|=I

2

7

9:6,.MainBeam

= cos2 6

 

sinc(-§—l(k sin 6 -- B + ja))

 

from equation (4.16). It is important to note that the far—field analytical solution
derived with the RAM method is an approximation. Albeit a reasonably accurate
approximation considering it uses the leakage and phase coefficients from yet another
approximation technique: the TRM method. The TRM method assumes that the
structure is infinite in extent. It is a reasonable assumption since the structure is
axially invariant and agrees well with full—wave solutions such as the Finite-Difference
Time-Domain method; see [15]. Figure 4.4 and Figure 4.5 depict the normalized far-
field gain patterns of the unloaded and 0.10 pF loaded HMLWA antennas, respectively,
Computed using the RAM method. The figures are limited to the frequencies that lie
in the operational leaky region of each antennna, i.e. 6 to 8 GHZ for the unloaded
antenna (see Figure 3.5) and 4 to 5 GHz for the 0.10 pF loaded antenna (see Figure
3.8). Tabulated main beam angles for the unloaded and 0.10 pF loaded HMLWA

antennas are given in Table 4.1 and Table 4.2, respectively.

60

 

 

 

 

 

Normalized Gain (dB)

 

 

 

 

Figure 4.4. Analytical solution to far fields maintained by equivalent magnetic current
on the surface of the radiating aperture of the unloaded HMLWA antenna.

 

 

 

 

 

Normalized Gain (dB)

 

 

 

 

Figure 4.5. Analytical solution to far fields maintained by equivalent magnetic current
on the surface of the radiating aperture with 0.10 pF edge loading configuration.

61

 

la)

 

ldl

 

Table 4.1. Main beam direction of unloaded HMLWA antenna determined analytically
using RAM method

 

 

Frequency (GHZ) Main Beam
6.00 19°
6.25 29°
6.50 38°
6.75 45°
7.00 51°
7.25 57°
7.50 63°
7.75 67°

 

 

 

 

Table 4.2. Main beam direction of 0.10 pF edge—loaded HMLWA antenna determined
analytically using RAM method

 

 

Frequency (GHZ) Main Beam
4.00 21°
4.25 40°
4.50 55°
4.75 65°
5.00 74°

 

 

 

 

62

CHAPTER 5

ANTENNA DESIGN

The unloaded and reactively loaded HMLWA antennas are depicted in Figure 5.1
and Figure 5.2, respectively. The antennas are printed on Rogers RT / duroid 5870
high frequency laminate characterized by a relative permittivity of er=2.33, standard
thickness of 31 mils, and a dissipation factor of tan 6:0.0012. The antennas consist
of a conducting strip 340 mm long by 7.6 mm wide (7A and 0.15A, respectively at 6
GHZ) sitting atop the grounded dielectric substrate. The antennas are terminated in
Pasternack’s SubMiniature version A (SMA) 50 Q terminations, product identification
number PE6081. The bifurcating wall is constructed of 440 vias spaced every m, or
0.381 mm. The vias have a radius of 0.1905 mm. The unloaded HMLWA antenna is
fed via an SMA coaxial connector with the outer conductor connected to the ground
plane and the inner conductor terminated on the conducting strip (passing through
the substrate). The reactively loaded HMLWA antenna is edge-launch fed via a
matched 50 Q microstrip line. The microstrip line is 2.2860 mm wide, and is designed
in Microwave Office® 2007 to ensure that it is matched at 50 52. Twenty American
Technical Ceramics 6008 series multilayer capacitors (EIA case size 0603) are placed
in shunt with the radiating edge by mounting one terminal on the conducting strip and
the other terminal, 0.6 mm away, on copper pads (3.7860 mm x 2.6430 mm) with
a center-to—center spacing of 16.1905 mm. The pads are connected to the ground
plane by vias. Each pad consists of four vias to ensure contact with the ground

plane. Figure 5.3 shows an exploded view of the mounting pad and the associated

dimensions.

63

 

 

Figure 5.1. Unloaded 340 mm HMLWA antenna with through-substrate feed

 

Figure 5.2. Loaded 340 mm HMLWA antenna with edge-launch feed

64

 

Figure 5.3. Exploded View of loaded HMLWA antenna showing 0.10 pF 0603 multi-
layer capacitors

5. 1 Measurements

The Sensors Directorate (RY) of the Air Force Research Laboratory at Wright—
Patterson Air Force Base provided materials and facilities for the fabrication and
measurement of the reactively loaded HMLWA antenna. Sections 5.1.1 through 5.2.1
provide measured data and detailed descriptions of the laboratory apparatus (and
associated settings) used during the data collection.

5.1.1 Return Loss

The return loss measurement of the antennas is motivated by the bandwidth definition
given in [3] and [4]. Here, the authors define the bandwidth as a percentage over the
—10 dB resonating band.

5.1.2 Unloaded HMLWA

As an example, Figure 5.4 shows the return loss, or I811], of the unloaded HMLWA
antenna. 811 is defined as the reflection coefficient seen at port 1 when port 2 is

terminated in a matched load [27]. The bandwidth, expressed as a percentage around

65

the center frequency, is governed by

fU- fL
BW— —
fL + (fU__-_-____ 2 fL)

where fU and f L are the frequencies of the upper and lower -10 dB crossing points.

x 100, (5.1)

Therefore, the bandwidth of the unloaded HMLWA is 23.19% for an fU=8.2314 GHZ
and an fL=6.5206 GHZ.

 

 

 

 

 

o ._ T T I
_5 ..................................................................
A -10 ............ g ............ g ............ .......... 1..
m . . I .
B :
<0 -15 .................................... g ................ _.
-20 ..................................... .j ...............................
_25 1 l L 1
3 4 5 6 7 e 9
Frequency (GHz)

Figure 5.4. Return loss of an unloaded terminated HMLWA antenna

The return loss data is measured using an HP8510C Vector Network Analyzer
(VNA) powered on an hour prior to data collection, ensuring accurate measurements.

The HP8510C VNA is used in conjunction with the HP83631A 8360 Series Synthe-

66

sized Sweeper and the 8515A S-Parameter Test Set. Two concatenated 96 inch long
GORE PHASEFLEX test assemblies (cables) with an SMA pin connector connects
the VNA to the antenna under test (AUT). 192 inches of cable is used to ensure the
AUT can be placed in the compact antenna chamber, as to avoid unintentional scat-
tering/reflection from nearby objects in the laboratory. To ensure accurate results,
prior to connecting the VNA to the AUT, the HP8510C VNA must be calibrated for
S11 one-port measurements. The pre—calibration process commences by setting the
VNA to the factory preset state, and setting the start and stop frequencies to 3 and
9 GHZ, respectively. Next, the sweep time is changed from the factory preset state of
160 to 100 ms/ Sweep. The scale (located on the FUNCTION block of the VNA) is
changed to 10 dB / division with a reference value of 0 dB, and a reference line position
set at 5 divisions. Calibration begins by placing an open termination at the opposite
end of the 96 inch long GORE PHASEFLEX test assembly followed by a short and
a broadband load. Figure 5.5 compares the return loss of both the terminated and
non-terminated unloaded HMLWA antennas.

Note that the 50 {2 load at the end of the antenna absorbs a significant amount
of the higher frequency reflections. Inclusion of the 50 52 load also increases the
bandwidth from 16.03% to 23.19%. Figure 5.6 shows the PHASEFLEX test assembly
and the Pasternack 50 Q termination. Since the voltage standing wave ratio (VSWR)
is defined as

 

VSWR = (5.2)

the VSWR of both the terminated and non-terminated HMLWA antennas is easily

computed and shown in Figure 5.7.
5.1.3 Reactively Loaded HMLWA

L
Since the HMLWA antenna is not symmetric about the XZ plane (i.e. y: 5), both

ISHI and |822| return loss data are measured. All settings on the HP8510C VNA

67

 

 

. l

     

‘ —Terminated in 500 .
: . —Unterminated . -----

 

s,,(dB)

 

 

 

6.5

l is
Frequency (GHz)

Figure 5.5. Return loss of terminated and non-terminated unloaded HMLWA anten-
has,

   

(a)
Figure 5.6. (a) PHASEFLEX test assembly. (b) Pasternack 50 Q termination

68

 

 

 

 

 

 

 

 

2.5

2...

1.5--
(I
5
>1. ....... ' ._

—Terminated in 50:2 §
. —Unterminated
"6 6.5 a 3.5

7 7.5
Frequency (GHz)

Figure 5.7. VSWR of terminated and non—terminated unloaded HMLWA antennas.

remain unchanged from the |811| measurement of the unloaded HMLWA antenna,
Calibration excepted. lslll is the reflection coefficient seen at port 1, the port on
the left hand side of the picture in Figure 5.2. Figure 5.8 shows the magnitude
of S11 for both the reactively loaded simulated and measured HMLWA antennas.
The [511] results show the bandwidth of the simulated antenna exceeds that of the
measured antenna by 25%. The variation could be due to the microstrip feed line or an
improper choice of the termination. Another potential culprit could be the modeling
of the capacitors. The capacitors are modeled as lossless ideal lumped elements. The
Capacitor data sheet does not contain equivalent series resistance (ESR) data in the
C~band. It does depict an inversely proportional relationship between capacitance
and ESR. Therefore, the inferred ESR in the C—band is 0.10 0. Altering the HMLWA

a-Iltenna simulations to include lossy capacitors is discussed in Section 6.1.

69

 

 

' —— Measured
— Simulated

 

 

 

 

 

IS"! ((13)

 

 

 

 

45 s sis is sis 7
Frequency (GHZ)

 

7.5

8

Figure 5.8. Simulated vs. Measured I311] Return Loss of Reactively Loaded HMLWA
Antenna.

70

 

\‘r

When port 1 is terminated in a 50 (2 load and port 2 is excited, |822| return loss
data is measured. Figure 5.9 reveals simulated results that more accurately predict
the measurements. Here, the bandwidth of the simulated HMLWA antenna is 24.75%

while that of the measured antenna is 22.18%, an 11.6% difference.

 

' —— Measured
—- Simulated

 

 

 

 

 

 

 

 

 

a ........... ........ . .
3 \ z * 1,»
7:. l” i l l/
(_D_ -15 .................................................. ‘ . . H "
-20 ....................................................................
'25 4.5 5 5.5 é sis i 7T5 8
Frequency (GHz)

Figure 5.9. Simulated vs. Measured ISggI Return Loss of Reactively Loaded HMLWA
Antenna.

5.1.4 Antenna Measurement Calibration

The reactively loaded HMLWA antenna is measured in the N earfield Systems Incor-
porated (NSI) spherical near-field measurement system. The near-field measurement
system is capable of measuring planar, cylindrical, and spherical near-field antenna
patterns. The system consists of a Windows based workstation, Agilent E8362B PNA

Network Analyzer, and an anechoic chamber complete with mobile scanner and sled.

71

The workstation and PNA are shown in Figure 5.10 while Figure 5.11 shows the

scanner and sled.

   

(b)
Figure 5.10. (a) NSI-2000 workstation (b) Agilent E8362B PNA

The NSI system is controlled by the NSI2000 Professional edition software, version
4.0527. Before the antenna is characterized, the system must be calibrated to ensure
accurate measurements. The reactively loaded HMLWA antenna is measured over
the C-band of the EM spectrum. Calibration must be done in two steps since the
antenna is measured over the C-band (4—8 GHz), which requires calibration with two
standard gain horns (SGH).

The C—band is discretized into two portions: the lower portion, 3.95-5.85 GHZ, and

the upper portion, 5.85-8.2 GHz. These portions require the use of two sets of SGHS

72

 

Figure 5.11. Reactively loaded HMLWA antenna in Near Field Anechoic Chamber

and open-ended waveguides (OEWG). The WR187 SGH—OEWG pair is appropriate
for the lower portion, while the WR137 pair is appropriate for the upper portion. Data
collection from the lower portion of the band commences with calibration requiring the
WR187 SGH and the WR187 open—ended waveguide (OEWG). Once a full spherical
scan is taken (a measurement time of approximately 1 hour), the data is stored
as a reference for the AUT scan. After removing the WR187 SGH, the AUT is

characterized, and the results stored. The process is repeated for the upper portion.

5.2 Reactively Loaded HMLWA Antenna Pattern Measurements

After the data is collected, the far-field is extrapolated from the near—field and the
NSI2000 software allows the user to view or export the data. The data is displayed
in a variety of ways; the 3—dimensional far-field pattern and 2—dimensional contour

figures reveal a significant amount of information. Figure 5.12 shows snapshots of

73

the 3-dimensional far-field pattern and the 2—dimensional far-field contour plot at 7
GHZ. Although the 3-dimensional far—field pattern is aesthetically pleasing it does
not display all the details (i.e., directivity, beamwidth, etc.) that the 2-dimensional
contour plot displays. Figure 5.12b reveals the main beam direction (approximately
70°) and insight into the more intimate details of the pattern. Here, it is clear that the
main beam has a concave shape covering nearly i35° in the horizontal plane. (Note:
The coordinate system used by NSI is shifted by 90°. Therefore, the terms elevation
and azimuth are opposite that of the standard convention, i.e., elevation corresponds
to the horizontal plane, while azimuth refers to the normal elevation plane.)

Since the goal of this thesis is to show fixed frequency beam steering of the main
beam direction, a 2-dimensional elevation cut in the YZ plane is used to determine
beam direction. Figure 5.13 depicts gain as a function of elevation angle, 0. Here,
6 represents the main beam direction measured off broadside. The figure also shows
good agreement with simulation. There is a significant amount of disagreement near
-60°, but the measurements outperform the simulations with a front-to—back ratio
(FBR) of 13.45 dB as opposed to the simulated FBR of 10.09 dB. Measured patterns
covering the C—band of the EM spectrum, every 250 MHz, are given in Appendix E.
Analysis of Figure E.1 through Figure E.17 show that when the hybrid leaky-wave
EH1 mode begins to propagate, near 5.75 GHZ, the main beam begins near broadside
and scans toward near end fire as frequency increases.

The main beam direction vs. frequency for the unloaded, simulated reactively
loaded, and measured reactively loaded HMLWA antennas is shown in Figure 5.14.
Here, the green curve represents the simulated unloaded data, while the blue and
red curves represent the simulated and measured reactively loaded HMLWA anten-
nas. There is a 15.3% average shift in the main beam direction of the unloaded and
reactively loaded simulated antennas across the operational leaky-wave region. The

unloaded and reactively loaded simulations show that the minimum expected shift in

74

{I .‘5‘ \c 111‘-

 

(a)

40 00—

    

Elevation (deg)
2

  
 

. |
1 1»
l/l/ fl ~§~
\

t

 
 
   

' .
l \\
I -
‘9
/ ‘
- I 7 rl 777777 l H l—7 177 V ,1, F'VIF 7 I
-‘5U 00 400.00 750.00 U DU 5f] 00 IUUDU ISU 00

Azimuth (deg)
(b)

3-di1nensional Spherical gattern and (b) 2-din1ensional Contour Plot
5

I
L

Figure 5.12. (a)
at 7.00 GHZ

 

 

 

 

 

 

 

 

20 T. f f ', Y r ' T '
‘ ‘ ‘ ‘ ' ‘ -—Simulated
. . _ . . . ---Measuned
. : . . _ . . g .
: :\
: 3 I 3‘
0p ...................................... . . .. ...... ". ..
. f l -
c 3 : I l,,l"" il '
m 1 I ll ' ' o't '
'0 . .l . I | .
v_10 ........... 3.’..' '. 'C'I'lg": ............ 7|] . ..
'g i‘ ‘ ii, I l ' 3 ' i 5 "l“
(D ‘A' r: ' lg'l‘l
"I ‘ I ['II
_20 ..... ‘.' ........................................................ ‘1
‘I {II
l
-30 ................................................................. 1....)
_40 i i i i i i i i j
-80 -60 -40 -20 0 20 40 60 80
Elevation Angle 6

Figure 5.13. Simulated vs Measured HMLWA Gain at 7.00 GHZ

the main beam direction is 14° and occurs at 6.50, 7.50, and 7.75 GHZ. The maximum
expected shift occurs at 6.25 GHz, where the main beam is shifted by 18°.

There is 100% agreement (0% error) between the measured and simulated re-
actively loaded HMLWA antenna data between 7.25-7.75 GHz. Unfortuantely, the
simulations do not accurately predict the measurements over the entire operational
region. The percent error between the simulated and measured results is goverened

by the following expression:

lSimulated -— M easuredl
Measured

 

PercentEr'ror = x 100%. (5.3)

The maximum percent error between simulation and measurement is 16.2%, which

occurs at 6.25 GHz. Here, the main beam direction (31°) is 6° off from the actual

76

observed (measured) direction of 37°.

 

 

 

 

 

 

 

 

 

w I 1 I
—o— Reactively Loaded Simulated Z
80,. +Un|oaded Simulated '
—-o-— Reactively Loaded Measured
+TRMSimulation
CD 70 .
c E
.123 60" .................
u... 50 .................. : ...... 1...... .............................. -
D /. -
gww .. ................ ,
d)
m
-: 30... ......................................................... .,
CO
2 20 ................................................................ .
10... ..............................................................
O 1 i 1
6 65 7 75 8

Frequency (GHz)

Figure 5.14. Unloaded, Simulated Reactively Loaded, and Measured Reactively
Loaded HMLWA Antenna Main Beam Direction Comparison

5.2.1 Radiation Effeciency

Radiation efficiency, according to [28], is the ratio of the total power radiated by the
antenna to the total power accepted by the antenna at its input terminals during
radiation. It is frequently expressed as

Gain

RadiationE f ficiency = Dir at 't .
e 2’02 31

 

(5.4)

Figure 5.15 shows broadband simulated and measured radiation efficiencies. The sig—

nificant variation between the simulated and measured results is mostly attributed to

77

 

‘00 T f f Y 7 ' '
: : : : : —Simulated
90,. ........ ........ ......... ......... ...... _-—Measured

 

l

 

 

 

 

80
70 ......... ........ ......... ........ ....... ........ -

60 ......... g ......... f ....... g ......... ........ ....... 1 ........

Radiation Effeciency
8

 

 

 

40
30
20.. ........ ......... I ........ ........ ........ _
. . .1 , . .
I i ,' I i i Z
10). ........ : ......... : ...... 0': ......... ........ 1. ...... ..: ........ -
: t" : : : : :
0 --1~-"l i i 1 i 1
4 4.5 5 5.5 6 6.5 7 7.5 8

Frequency (GHZ)

Figure 5.15. Simulated vs Measured Radiation Efficiency

the lossless capacitors used in the model. The simulated results are questionable re-
porting 50% effeciency before the hybrid leaky-wave EH1 mode propagates. (Leakage
begins near 5.75 GHZ, see Figure 5.8 and Figure E.8.) Further, the simulated curve
shows a peak effeciency of 90% at 6.00 GHZ, while the measured result is slightly less
than half (48%) at this frequency.

The measured radiation efficiency data is consistent with the predictions. As the
EH1 mode begins to propagate (near 5.75 GHZ), the efficiency reaches its maximum
at 62%, and slowly decreases reaching a minimum of 24% at 8.00 GHZ. Something
pecuilar occurs betwen 5.90—6.50 GHz where the effeciency drops below 50%, in the
middle of the operational range. This behavior is unexplained now, but will be

investigated in the future.

78

CHAPTER 6

CONCLUSIONS

In this thesis, several properties of half-width microstrip leaky-wave antennas are in-
vestigated. A brief history of leaky-wave antennas is discussed in Chapter 2. In this
chapter, the half-width approximation of the full-width microstrip leaky-wave antenna
results reported by AF RL researchers are verified. It is found that the half-width mi-
crostrip leaky-wave antenna accurately approximates its full—width counterpart when
the antennas are simulated on a practical, finite ground plane-substrate layer. How-
ever, the approximation does not hold when the antennas are simulated on an infinite
ground plane-substrate layer.

In Chapter 3, a preliminary technique is used prior to invoking a full-wave solution
to save time and resources. This technique, the transverse resonance method, is used
to extract the axial complex propagation coefficient from the transverse complex
propagation coefficient. The results of the method are given in Section 3.8, which
show the main beam direction can be fixed near end fire (at 80°) over the entire
C-band, provided that a per unit length edge loading configuration is used.

An analytical solution to the far-zone fields radiated by the HMLWA antenna is
described in Chapter 4. The solution makes use of the radiating aperture method, and
assumes a well matched load. Therefore, the solution does not account for reflections.

The antenna design, fabrication materials, and measurements are described in
Chapter 5. Here, measured data from the operational region of the HMLWA antenna
shows the average shift in the main beam direction (with respect to the simulated
unloaded HMLWA antenna) is nearly 17%. The largest shift in the main beam di-
rection is 24° at 6.25 GHZ, while the smallest shift, 14°, occurs at three frequencies,

7.00, 7.50, and 7.75 GHZ. Comparisons between measurment and simulation show

79

100% agreeement over a 500 MHz band (7.25 - 7.75 GHZ).

The measured results show that the main beam direction of the reactively loaded
HMLWA antenna is indeed controlled at various fixed frequencies. This is mentioned
as the goal of the work in Chapter 1, and the results show that it was accomplished

(see Chapter 5).

6. 1 Future Work

There is an interest in having the main beam direction fixed over the entire operational
leaky band. Since the main beam direction is dependent on frequency, fixing the main
beam direction over a broad frequency band would require varying the impedance
(by changing the edge loading capacitance) at the radiating edge of the HMLWA
antenna. Based on the results of the TRM method, which requires a per unit length
edge loading, this could be accomplished by placing a reactive sheet at the radiating
edge of the HMLWA antenna.

Another alternative to loading the radiating edge with a reactive sheet is to main-
tain the lumped capacitor edge loading technique used in this thesis. In the proposed
design, the capacitance value of each lumped element would vary, and would no longer
be constant. After having a discussion with my friend and colleague, Mr. Raoul
Ouedraogo, he pointed out that the optimal capacitance values could be determined
through the use of a genetic algorithm. This would require a well written fitness
function. Once the optimal capacitance values are determined, a full-wave solution
will be computed using CST Microwave Studio. The antenna could be fabricated
and characterized contingent upon the results of that simulation. Mr. Ouedraogo
and I will investigate this optimization technique and we will report our findings in

a subsequent publication.

80

APPENDICES

81

APPENDIX A

CONSTRUCTION OF SOLUTIONS FROM POTENTIALS

In this appendix, construction of solutions from potentials is presented. This discus-
sion will parallel that presented in [22] and [29]. Assuming an ej‘“t time dependence,
Maxwell’s equations maintained by electric and magnetic sources in an isotropic, ho-
mogeneous region of space characterized by permittivity e and permeability p are

given by:

l
l

.—}
V x E = —_7pr — M (A.1a)
V x Ff = jade—E + 7 (A.1b)
—-)
V- D = p (A.1c)
V ' B : p7” (A.1d)
In source-free regions, Maxwell’s curl equations reduce to:
—-> —>
V x E = ~jw/1H (A.2a)
—-> —+
V x H = jweE. (A.2b)

Equations (A.2a) and (A.2b) are F araday’s and Ampere’s laws, respectively. In order
to fully express a vector field, both its curl and divergence must be specified; therefore,
it is necessary to define the divergence of If and E. Using the constitutive parameter
relating the magnetic flux density, E, to the magnetic field intensity, If, i.e., E =

nff, Faraday’s law becomes

V x '13 = —wa. (A3)

82

Taking the divergence of equation (A3) and utilizing the following vector derivative

identity

—+

—-) —+
V°(aB)=aV-B+B-Va (A.4)

yields,

V-(Vx E) = V-(—wa)
--+ ——)
= —jwV-B+B-V(—jw)

——+
= —jwV. B (A5)

where V(—jw) = 0 because the product of —jw is a spatially independent constant
whose gradient vanishes. Equation (A5) is further simplified by realizing that the

divergence of the curl of any vector field vanishes, i.e.,
V-(VX B)=0; (A-6)
therefore, equation (A5) becomes
_,
V - B = 0, (A.7)

which is Gauss’ law for the nonexistence of magnetic monopoles. Gauss’ electric law

is derived in a similar fashion. Using the constitutive parameter relating the electric
e —-’ e e e —-) n H H e e

flux denSIty, D, to the electric field intenSIty, E, 1.e., D 2 6E , and substituting the

latter into Ampere’s law, equation (A.2b), and post application of vector derivative

identities (AA) and (A6), yields Gauss’ electric law,

_.
v. D = 0. (A8)

83

By virtue of vector derivative identity (A.6), any vector whose divergence vanishes
——u)
must be the curl of some other vector. Hence B of equation (A.7) is the curl of some

-—+
other vector, say A,

-—> —-)
VxA=B

2 pH. (A9)

—-> -—->
Since this equation shows that H is the field due to vector field A, it is re—expressed

as

“HA = v x 11"; (A.10)

—-) —->
where the the subscript A on H reflects the explicit dependence of H on A . Vector
field A is known as the magnetic vector potential. It is a mathematical tool which
simplifies the construction of EM solutions [22]. Substituting this result into Faraday’s

law, equation (A.3), leads to

—-> -—)
V x EA = —jw(V x A). (A.11)
Rearranging terms leads to
—-) ——>
VX(EA+jw/l)=0, (A.12)

—) —-+
where the following vector identity was used since both E and A have continuous

first and second partial derivatives
—) —-) —+ —-)
Vx(A+B)=VxA+VxB. (A.13)

Just as a divergenceless vector is the curl of some other vector, similarly, a curl—free

vector is the gradient of some scalar, defined through the following vector derivative

84

identity
V x (Va) = 0. (A.14)

Thus, the quantity within the parenthesis in equation (A.12) can be represented as

the gradient of some scalar; i.e.,
—) —>
EA +ij = —V(I>e, (A.15)

where <I>e is known as the electric scalar potential. Rearranging terms leads to an

expression for E A as a function of the magnetic vector and electric scalar potentials,
—r ——)
EA = -—-(ij + V<I>e). (A.16)
Taking the curl of equation (A.10) and utilizing the vector derivative identity
—> -—-> —->
Vx(aB)=anB—BxVa (A.17)
yields

Vx(Vx7f) = Vx(nHA)
——> —+
= #VXHA-HAXVOI)

W x “In, (A.18)

because the gradient of a scalar vanishes. Rearranging terms leads to an expression

—__’
for V x H A as a function of the magnetic vector potential,

—-> 1 ——>
VXHA=EVXVXA. (A.19)

Substituting equations (A.16) and (A.19) into Ampere’s law maintained by electric

85

-—’
current density J , equation (A.1b), results in

1 ——) , ——> —§
EV x V x A = jwe(—ij —V<I>e) + J

= —j2w257i -— jweV<I>e + 7, (A20)
and distributing the u throughout yields

VxVxA = —]wueA—jwpeV<I>e+/1J
2 —') . “—i
= w ueA —]wu6V<I>e+uJ

— [€274 — ' V<I> '7
— game e + n J , (A.21)
where k is the wavenumber given by k = wfliE. Equation (A.21) becomes I
V(V-A)—V A=k A-jwpeV<I>e+)uJ (A.22)
post application of the vector identity
V x v x 71’ = V(V - 71’) - v22, (A.23)

where V2 represents the vector Laplacian. The curl of the magnetic vector potential
is given in equation (A.9), but the divergence of this quantity, necessary to uniquely
specify A, has yet to be defined. The divergence of A is judiciously chosen to simplify
equation (A.22) as
_-’ .
V- A = —que<l>e, (A.24)

which is also known as the Lorentz condition. Substituting the Lorentz condition into

86

equation (A22), and applying the vector derivative identity
V(ab) = aVb + bVa (A25)

yields,

—-+

—jwp€V(<I)e) + (DeV(—jw,ue) — v2 A = 1:234? —— jwueVCDe + #7,

or
<I>eV(—jw,u€) — V22 = k2}? + [17, (A26)
which reduces to
v2? + 1374’ = —u7, (A27)

the inhomogeneous complex magnetic vector potential wave equation. The electric

—)
field due to the magnetic vector potential, E A: is given by

 

V(V - 74’) (A28)

upon taking the gradient of the Lorentz gauge (equation (A.24)), and substituting
the result into equation (A.16).
In a similar fashion, the magnetic field intensity, if, can be written in terms of
—*
an electric vector potential, F. From Gauss’ electric law, equation (AB), and vector
—.)
derivative identity (A.6), D must be the curl of some other vector, say a vector field
F, where
--) ——)
—V x F = D. (A29)

Therefore,

-—) —)
CEF = -V X F (ABC)

87

——) —->

upon invoking the constitutive relationship D = 6E. The magnetic field due to the
—+

electric vector potential, H F, results upon substituting equation (A30) into Ampere’s

law, equation (A.2b), and invoking vector derivative identity (A.14), and is given by
——§ ~—>
HF = -(ij + V(Pm), (A31)

Where (Pm is known as the magnetic scalar potential. Taking the curl of equation
(A30) and equating it to Faraday’s law maintained by magnetic current density A7,

equation (A.1a), leads to

—1 —-> _ —> —)
TVXVX Fz—quHF—M, (A32)
which becomes
—1 -—-> . . —-) —)
—€—V X V x F = —Jw,u(—]wF — V<I>m) — M, (A33)

upon substituting T? F from equation (A31). Applying vector derivative identity

(A23) and defining the divergence of F) as
V - f3 = —jw,ue<I>m, (A34)

leads to

V2?" + k2? 2 ---J4’, (A35)

the inhomogeneous complex electric vector potential wave equation. The magnetic

—-+
field due to the electric vector potential, H F, is given by

V(V - 13) (A36)

 

88

upon taking the gradient of equation (A34), and substituting the result into equation
(A31).
The total EM fields are obtained by the superposition of the individual fields due

-—-> —§
to A and F [22], i.e.,

——> —+ ——+
E = EA-l-EF

 

 

—-> 1 ——> 1 —+
= -ij+, V(V°A)--VXF
game e
—jw ——> —-> 1 —->
= —k2—(k2A+V(V-A))—-6-V>< F (A37)
and
—.)
H = HA+HF
_—_ 1VXA—jw?+ .1 V(V F)
# Jwflf
1 ° —->
= Zv A—%(AZF+V(V.F)) (A38)
These become
—) —' —-> 1
E=-7:7w-(VXVXA)—;VX? (A39)
and
fiZiVXj—%(VXVXF) (A40)

for all field points external to the source system. Solutions of equations (A27) and
(A35) can be found using the method of separation of variables. For problems in-

volving radiation, appropriate solutions of equations (A27) and (A35) are

74%?) = u (H 7(?”)G(T~’|7”) dv’ (A.41)
V

89

and

 

7177’) = 6 (fl 117(7”)G(7’|7~*’) dv’, (A42)
V
respectively. Here C(r I? '), the 3— dimensional free-space Green’s function, is
.4 .4, 10(7) 7‘")
G( T l T ) _ 47',
(flew—m A 43
_ 47T|7+—?’|. ( ' )

Substituting equation (A43) into equations (A.41) and (A42) yields

 

 

—'k|7’—'F”|
—’ —> _ _E; "'7 —->I 3 J I
A(r)—47TIJIJ(T) W—‘F’I dv (A44)
and
-‘k|7’-—'F”|
“* —) _ _€_ ‘3 —9/ 6 J I
F(r)-4flfJfM(r) |?__F,,l dv, (A.45)

respectively. Equations (A44) and (A.45) are appropriate solutions to equations

(A27) and (A35) because 1,!)(7’I—F") satisfies the homogeneous scalar wave equation
V.—> r #r ’.

The total fields are re—expressed as

T5: :I—g—Zw(V x V x fig/IJ(7")e|7]".]€|_—’_-1‘_>,l?ll dv’)
— EV x 47? Hflf M(‘?’)G(?’|7”)dv’
V

 

and
—§ _'—)[|

= _vX gm m 141:”: 4,, d.

_. 1"l

_ 11:3(VxVx—fflm m’el;k|_:,,l dv’)

I

 

 

90

upon substituting (A44) and (A45) into equations (A39) and (A40), respectively.

These equations reduce to

\

 

 

kl? 7”|
-7 jwu 7))"er
E: 47Tkz'x7xV/xfflJrf’ )Ir—H’l do

and

 

= .._v x (ff—7 J ->I) ‘3 7“" LII" .17

)IT-

when the field is maintained only by 7, and
E=LfoflMu “*(’) G(—*|’?’)dv

and

 

 

Hfiivmmm 115/17’

when maintained only by 117.

(A46)

(A47)

(A48)

(A49)

The Fraunhofer or far-field region is that region that lies beyond the radiating

near-field (Fresnel) where the radiation pattern is unchanging with distance [30].

Equations (A46) through (A49) can be simplified by approximating the kernels of

the integrals in the far zone provided that 7" << 7'. The simplifications result in:

_’
E

22

 

‘30}“3 jkr Tug] J( —>I )eJkr-V r dv

“jWAT

itow“

22

91

(A50)

and

 

2

—’ :__jk~ 4“ e -37“. —>I k I
H ~ x— J( )e3 T r dv
,. lll<
:J—kf‘ x 2 (A51)
11

22

when the field is maintained only by 7, and

 

22

E" J'_k. Mie:krlllM (-—>I)ejkr-’I7idv

E

22

3%? x F" (A.52)

and

mi
22

 

 

itof

_::e 8":k7" [III M(?’)ejkf'?’dvl]
V

H
—jUJFT (A.53)

22

when maintained only by A7. KT and ET of equations (A50) and (A53) are the
transverse components of the magnetic and electric vector potentials. They uphold
the Sommerfeld radiation condition which states that fields far from sources must

behave as waves TEM to the r—direction [31].

92

APPENDIX B

RADIATION PATTERNS OF FMLWA AND HMLWA ANTENNAS
ON INFINITE AND FINITE GROUND PLANES

The following radiation patterns verifies the claim that the HMLWA antenna reason-
ably approximates the FMLWA antenna. The antennas are simulated on both infinite

and finite ground planes using FEKO and CST MICROWAVE STUDIO, respectively.

There is a 10 dB radial spacing (i.e., 10 dB per division) for all patterns.

B.1 HMLWA vs. FMLWA simulated on an infinite ground plane

 

-——- HMLWA
— FMLWA

 

 

 

 

 

Figure 8.1. Radiation Patterns of HMLWA antenna vs. FMLWA antenna at 6GHz
mounted atop an infinite substrate-ground plane layer.

B.2 HMLWA vs. FMLWA simulated on a finite ground plane

93

 

 

 

 

 

 

 

 

Figure B2. Radiation Patterns of HMLWA antenna vs. FMLWA antenna at 6.25GHz
mounted atop an infinite substrate-ground plane layer.

 

—— HMLWA
--- FMLWA

 

 

 

 

 

Figure B3. Radiation Patterns of HMLWA antenna vs. FMLWA antenna at 6.5GHz
mounted atop an infinite substrate-ground plane layer.

94

 

—— HMLWA -30° 30°
-——FMLWA

 

 

 

 

 

Figure B4. Radiation Patterns of HMLWA antenna vs. FMLWA antenna at 6.75GHz
mounted atop an infinite substrate-ground plane layer.

 

 

 

 

 

9
00
-—- HMLWA -3o° __ _ 30°
-— FMLWA .. . " 7‘
~60°/.' 60°
-90° F- 4 90°
420°: 120°

 

 

Figure B.5. Radiation Patterns of HMLWA antenna vs. FMLWA antenna at 7 GHZ
mounted atop an infinite substrate-ground plane layer.

95

 

— HMLWA —30° 30°
—— FMLWA 7 “

 

 

 

 

 

 

Figure 8.6. Radiation Patterns of HMLWA antenna vs. FMLWA antenna at 725G'Hz
mounted atop an infinite substrate-ground plane layer.

 

— HMLWA —30° 30°
—— FMLWA ,, ~-

 

 

 

 

 

 

Figure B.7. Radiation Patterns of HMLWA antenna vs. FMLWA antenna at 7.5GHz
mounted atop an infinite substrate-ground plane layer.

96

 

— HMLWA -3o°
— FMLWA

 

 

 

 

 

Figure 83. Radiation Patterns of HMLWA antenna vs. FMLWA antenna at 7.75GHz
mounted atop an infinite substrate-ground plane layer.

 

—— HMLWA -30° 30°
—— FMLWA ‘

 

 

 

 

 

Figure B.9. Radiation Patterns of HMLWA antenna vs. FMLWA antenna at 8GH2
mounted atop an infinite substrate-ground plane layer.

97

 

—— HMLWA
-— FMLWA

 

 

 

 

 

 

Figure B.10. Radiation Patterns of HMLWA antenna vs. FMLWA antenna at 60112
mounted atop a finite substrateground plane layer.

 

 

 

 

 

 

 

— HMLWA
— FMLWA
410°, 60°
A“ \x
1" 2')
-90° 9 7) 90°
1 Ir
-1 20& . 1 20°

 

 

Figure B.11. Radiation Patterns of HMLWA antenna vs. FMLWA antenna at
6.25GHz mounted atop a finite substrate-ground plane layer.

98

 

—- HMLWA -3o° /__ 30°
--— FMLWA ,-

 

 

 

 

 

 

 

Figure B.12. Radiation Patterns of HMLWA antenna vs. FMLWA antenna at 6.50 H z
mounted atop a finite substrate-ground plane layer.

 

 

 

 

 

 

 

-— HMLWA
—— FMLWA

-60°/ 60°

-90° 5—7 1~~ + 90°
420‘ . \ -. , 120°
\ \\\ r ” i‘x‘ ./ I
‘ ‘ I [Dr ,../
-150° F 150°
180°

Figure B.13. Radiation Patterns of HMLWA antenna vs. FMLWA antenna at
6.75GHz mounted atop a finite substrate-ground plane layer.

99

 

— HMLWA -30° 30°
— FMLWA ~ ’ ~

 

 

 

 

 

 

Figure B.14. Radiation Patterns of HMLWA antenna vs. FMLWA antenna at 7GHz
mounted atop a finite substrate-ground plane layer.

 

 

 

 

 

 

 

Figure B.15. Radiation Patterns of HMLWA antenna vs. FMLWA antenna at
7.25GHz mounted atop a finite substrate-ground plane layer.

100

 

-— HMLWA -30° _ 30°

 

 

 

 

 

 

Figure B.16. Radiation Patterns of HMLWA antenna vs. FMLWA antenna at 7.5GHz
mounted atop a finite substrate-ground plane layer.

 

—— HMLWA -30° 30°

 

 

 

 

 

 

Figure B.17. Radiation Patterns of HMLWA antenna vs. FMLWA antenna at
7.75GHz mounted atop a finite substrate-ground plane layer.

101

 

 

 

 

 

 

 

Figure B.18. Radiation Patterns of HMLWA antenna vs. FMLWA antenna at 8GH2
mounted atop a finite substrate-ground plane layer.

102

APPENDIX C

UNLOADED HMLWA ANTENNA CHARACTERISTICS

In this appendix, characteristics of the unloaded HMLWA antenna including the free-
edge impedance, transverse and axial wave numbers and other characteristics are
presented.

Figure Cl and Figure C2 depict the resistance and reactance at the free-edge
of the antenna. It implements the free-edge impedance described by equation (3.69)
of Section 3.5. Figure C.3 through Figure C.6 show the real, imaginary, magnitude,
and phase of x, the phase of the free-edge impedance (equation (3.78) of Section 3.5).
Finally, Figure C.7 through Figure C.10 show the real and imaginary parts of the

transverse and axial wavenumbers, [CT and kg, respectively.

 

400 r I ._
350 -'

300 7

Re( ZFE )
N
O
C?

150-

100'

 

 

 

 

504 5 6 7 8
Frequency (GHz)

Figure C.1. Free-edge resistance of the unloaded HMLWA antenna.

103

 

-1000 7 t
-1500

—2ooo~u

"“(ZFE)

-2500 r

 

 

 

 

—30004
Frequency (GHz)

Figure C2. F ree—edge reactance of the unloaded HMLWA antenna.

104

Re(x)

|m(x)

 

-O.15

 

 

 

 

4 5 6 7 8
Frequency (GHz)

Figure C.3. Re(x) of the unloaded HMLWA antenna.

 

0.1 ) 4
0.08 7
0.06 7
0.04 7

0.02 ~

 

 

 

 

Frequency (GHz)

Figure C.4. I m(X) of the unloaded HMLWA antenna.

105

0.4

 

 

 

 

E
0.25 ~
4 5 6 7
Frequency (GHZ)

Figure C.5. lxl of the unloaded HMLWA antenna.

3.25

 

3.2 I ......

3.15--~--

3.15 A»

3.05 ..

An9( x )

2.95 7'

2.9 -

 

 

2.85

 

 

Frequency (GHz)

Figure C.6. Ang(x) of the unloaded HMLWA antenna.

106

 

 

196 r r T
194
192-1

1907-

Re( kT)

1887'-

 

 

 

 

1 84 L 3. A
4 5 6 7 8
Frequency (GHz)

Figure C.7. Re(kT) of the unloaded HMLWA antenna.

 

 

 

 

 

Frequency (GHZ)

Figure C.8. I m(kT) of the unloaded HMLWA antenna.

107

 

200

Re(kz)

50-

 

 

 

 

 

Frequency (GHZ)

Figure C.9. Re(kz) of the unloaded HMLWA antenna.

 

140
120
100 -

80" ,

lm(kz)

607-

20_.

 

 

 

 

 

Frequency (GHz)

Figure C.10. I m(kz) of the unloaded HMLWA antenna.

108

APPENDIX D

0.10 PF PARALLEL-LOADED HMLWA ANTENNA
CHARACTERISTICS

In this appendix, characteristics of the 0.10 pF loaded HMLWA antenna including
the free-edge impedance, transverse and axial wave numbers and other characteristics
are presented.

Figure D1 and Figure D2 depict the resistance and reactance at the free-edge
of the antenna. It implements the free-edge impedance described by equation (3.69)
of Section 3.5. Figure D3 and Figure D4 show the resistance and reactance of the
parallel-combination of the 0.10 pF lumped capacitor and the free-edge impedance at
the radiating edge of the HMLWA antenna, respectively. Figure D.5 through Figure
D.8 show the real, imaginary, magnitude, and phase of x, the phase of the free-edge
impedance (equation (3.78) of Section 3.5). Finally, Figure D.9 through Figure D.12
show the real and imaginary parts of the transverse and axial wavenumbers, kT and

102, respectively.

109

 

4000 w

2000>» ------- ~ §.w ,Hg. 2 w ...... .

 

Re( ZFE )

-2000 7-

 

-4000 7

 

 

 

 

-60004
Frequency (GHz)

Figure D.1. Free-edge resistance of the 0.10 pF loaded HMLWA antenna.

x10

 

lm( ZFE )

 

 

 

 

 

 

l
N
I-

Frequency (GHz)

Figure D2. Free—edge reactance of the 0.10 pF loaded HMLWA antenna.

110

 

 

 

lRe(ZRE)
)2)
O

 

 

 

 

 

4 5 6 7 8
Frequency (GHz)

Figure D3. Radiating-edge resistance of the 0.10 pF loaded HMLWA antenna.

 

 

 

 

 

-200 .
-250- i
’31 i
hf ;
.5 :
-300..H. .H.H.? “Mg.
-350 i
4 5 6 7 8
Frequency (GHz)

Figure D4. Radiating-edge reactance of the 0.10 pF loaded HMLWA antenna.

111

 

0.5 .r

..........

0.4» ~

Re(x)

 

 

 

 

Frequency (GHz)

Figure D.5. Re(x) of the 0.10 pF loaded HMLWA antenna.

 

 

 

 

 

 

 

 

' 4 5 6
Frequency (GHZ)

Figure D.6. I m(X) of the 0.10pF loaded HMLWA antenna.

112

0.7

 

le

 

 

 

 

0 1 1 1
4 5 6 7 8
Frequency (GHz)

Figure D.7. Ix| of the 0.10 pF loaded HMLWA antenna.

 

 

 

 

 

 

 

 

 

 

 

 

3 —l ...... 1
2r. ............ .. J
A 1... .............. _.
x i
‘5 0 ............ .............
5 3
_1.. ........... ........
-2“ .....
-3. ........... . ,,,,,,,,,,, .
-4 .
4 5 6 7 8

Frequency (GHZ)

Figure D.8. Ang(x) of the 0.10 pF loaded HMLWA antenna.

113

 

125
120? , f w ,
1155

1107

Re( k1.)

1oo~~~

 

 

 

4 5 6 7 8
Frequency (GHz)

Figure D.9. R6(kT) of the 0.10 pF loaded HMLWA antenna.

 

lm( kT)

 

 

 

 

 

-0.5 I i

4 5 6 7 8
Frequency (GHZ)

Figure D.10. I m(kT) of the 0.10 pF loaded HMLWA antenna.

114

 

 

 

 

 

250 f f I
200» w
A 150—w
J‘
E
100-~---~-
50
0 .f g
4 5 6 7 8
Frequency (GHz)

Figure D.11. Re(kz) of the 0.10 pF loaded HMLWA antenna.

 

|m(kz)

 

 

 

 

 

Frequency (GHZ)

Figure D.12. Im(kz) of the 0.10 pF loaded HMLWA antenna.

115

APPENDIX E

MEASURED RADIATION PATTERNS OF THE CAPACITIVELY
LOADED HMLWA ANTENNA

The patterns shown in Figure E.1 through Figure E.17 were measured in the NSI
spherical near-field measurement system. The author would like to express his sincere

gratitude to team RASCAL of AFRL for providing the resources required to obtain

the patterns.

20

 

107

O
I

Gain (dBi)

 

 

 

 

~30 ~60 ~40 ~20 0 20 40 so so
Elevation Angle 0

Figure E.1. Measured Radiation Pattern of HMLWA antenna at 4.00 GHz

116

an
‘V I I T F Y V Y T 1

 

Gain (dBi)

 

 

 

 

40 60 80

‘80 -60 —40

20 o 20
Elevation Angle 9

Figure E2. Measured Radiation Pattern of HMLWA antenna at 4.25 GHZ

 

Gain (dBi)
I

 

 

 

L J i

 

-00 -62) -41) 4'0 6) 20
Elevation Angle 0

Figure E3. Measured Radiation Pattern of HMLWA antenna at 4.50 GHz

117

an
‘U r fl l r l T I y

 

Gain (dBl)

 

 

 

L

 

-80 -60 -40 4O 60 80

— o o 20
Elevation Angle 0

Figure E4. Measured Radiation Pattern of HMLWA antenna at 4.75 GHz

 

Gain (dBi)

 

 

 

L 1 4:

40 60 80

 

"" -30 -60 —20 o 20
Elevation Angle 0

Figure E.5. Measured Radiation Pattern of HMLWA antenna at 5.00 GHz

118

 

20 W I ' T I T j .7 ‘7

107

Gain (dBi)

 

 

 

4' 1 j i

 

_40 _L '1 i i i
-80 -60 -40 -20 0 20 40 60 8O

Elevation Angle 9

Figure E6. Measured Radiation Pattern of HMLWA antenna at 525 GHz

 

20 T Ti I T 7 I I I I

Gain (dBl)

 

 

1 i i l j '1 L 1
~00 ~00 ~40 ~20 0 20 40 so so
Elevation Angle 0

L

 

 

Figure E.7. Measured Radiation Pattern of HMLWA antenna at 5.50 GHz

119

 

23

 

Gain (dBi)

 

 

 

 

w-80-60-40-20 o 20 40 so so
Elevation Angle 6

Figure E.8. Measured Radiation Pattern of HMLWA antenna at 5.75 GHZ

2i

 

Gain (dBl)

 

 

 

a L 1 I 4

 

-80 -60 -40 -20

0 20 40 so so
Elevation Angle 0

Figure E.9. Measured Radiation Pattern of HMLWA antenna at 6.00 GHZ

120

 

Gain (dBl)

 

 

 

-80 -60 ~40 ~20 0 20
Elevation Angle e

 

40 60 80

Figure E.8. Measured Radiation Pattern of HMLWA antenna at 5.75 GHZ

 

Gain (dBi)

 

-30 ..

 

 

 

7" -80 —60 ~40 ~20 o 20 4o 60 so
Elevation Angle 6

Figure E.9. Measured Radiation Pattern of HMLWA antenna at 6.00 GHZ

120

on
LV

 

Gain (dBl)

 

 

 

s
3

~20 o 20 40 so so
Elevation Angle 6

Figure E.10. Measured Radiation Pattern of HMLWA antenna at 6.25 GHz

 

Gain (dBi)

 

 

 

 

 

-80 -60 -40 40 60 80

-2'0 0 2‘0
Elevation Angle 9

Figure E.11. Measured Radiation Pattern of HMLWA antenna at 6.50 GHZ

121

 

Gain (dBi)

 

 

 

 

—80 -60 ~40 ~20 o 20 40 so so
Elevation Angle 0

Figure B.12. Measured Radiation Pattern of HMLWA antenna at 6.7 5 GHz

 

Gain (dBi)

 

 

 

 

~20 6 2'0 40
Elevation Angle 6

Figure E.13. Measured Radiation Pattern of HMLWA antenna at 7.00 GHZ

122

 

Gain (dBl)

 

 

 

 

-80 -60 -40 -20 0 20
Elevation Angle 0

Figure E.14. Measured Radiation Pattern of HMLWA antenna at 725 GHZ

 

Gain (dBi)

 

 

 

an
w

 

-80 -60 -40 40 60 80

-22) 6) 2‘0
Elevation Angle 0

Figure E.15. Measured Radiation Pattern of HMLWA antenna at 7.50 GHZ

123

an
Lu

 

 

 

 

 

~80 ~60 ~40 ~20 0 20 4o
Elevation Angle 0

Figure E.16. Measured Radiation Pattern of HMLWA antenna at 7.75 GHZ

 

Gain (dBl)

 

 

 

 

~80 ~60 ~40

~20 o 20 4o 80 60
Elevation Angle 0

Figure E.17. Measured Radiation Pattern of HMLWA antenna at 8.00 GHZ

124

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125

 

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[2]

[4]

[5i

[7]

[8]

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129

   

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