The Shape Synthesis of Non-Planar and
Closely-Spaced Electrically-Small
Conducting Surface Antennas
Anas Mohammad Ishaq Alakhras
Thesis submitted in partial fulfillment of the requirements for the
Doctorate in Philosophy degree
in Electrical and Computer Engineering
Ottawa-Carleton Institute for Electrical and Computer Engineering
School of Electrical Engineering and Computer Science
Faculty of Engineering
University of Ottawa
© Anas Alakhras, Ottawa, Canada, 2020
ii
ABSTRACT
Antennas are needed in wireless communications of any kind and are key to the quality of any wireless link,
irrespective of its use. There remains a need for electrically-small antennas that can fit on/in a restricted
physical surface/volume for use in sensors (eg. Internet-of-Things) that need to “talk” to some data centre
via the wireless network. The design issues associated with electrically-small antennas are well-known, and
thus it will be appreciated that it would be advantageous to be able to design the above antennas in a way
that exploits as much of the space allotted to them, as well the proximity of other objects present, to arrive at
functioning antennas. Antennas of conventional shape do not necessarily do this. It might thus be best to
perform a shape synthesis of an antenna that lets the electromagnetics decide on the shape of the final
antenna in order to obtain the performance desired under the geometrical restrictions mentioned. Such shape
synthesis has only been done for planar (2D) conducting surface antennas. In this thesis we develop the
machinery for a computational electromagnetics-based tool capable of performing the shape synthesis of 3D
conducting surface antennas for the first time. It has purposefully been developed in a way that allows it to
capitalize on commercially available software for both the computational electromagnetics (CEM) and the
genetic optimization algorithm (GA). A comprehensive ‘shaping manager’ software tool has been developed
that controls the whole shaping process and communicates with the commercial software, in so doing
implementing a desired shaping prescription. It is a complex tool whose development is based on a
considerable amount of software engineering. The advantage of being able to utilize the commercial codes is
that the work becomes accessible to others, such codes are more powerful and flexible than in-house codes,
and they are also able to export files describing the complicated shaped geometries for fabrication. The
shaping manager utilizes the characteristic mode concept so that one only needs to select the feed point once
the shape synthesis has been completed. Fully 3D electrically-small conducting surface antenna examples
are then successfully shape synthesized subject to various geometry restrictions. The validity of the
modelling process is experimentally validated. The shape synthesis procedure is then extended to apply to
the design of two closely-spaced electrically-small antennas operating at different frequencies, again with
some experimental validation. Such shape synthesis has also not been done before. These procedures can be
extended to the case where the two antennas in question are mounted on some electrically-large platform.
However, if such platforms are electrically very large the full-wave computational burden may become
prohibitive. We show that the characteristic modes for the antennas under shaping in such situations can in
fact be found using hybrid method of moment / geometrical theory of diffraction (MM/GTD) methods, with
only the antenna proper needing to be MM-meshed. An example is provided to demonstrate the computation
of characteristic modes in this way. This possibility for characteristic mode computation has not yet been
explicitly stated elsewhere (and so perhaps not realised), nor example computations provided. We believe
this realisation might widen the scope of application of characteristic mode analysis.
Keywords:
Antenna shape synthesis, characteristic modes, sub-structure, perfect conductors, method of moments,
geometrical theory of diffraction, electrically-small antennas.
iii
Acknowledgements
I thank God for blessing me with the knowledge I have attained and I pray that my work be of benefit for
those who come after me.
I am especially grateful to my supervisor, Professor Derek McNamara, for his support, guidance, advice, and
many suggestions throughout the course of my research, and allowing me to liberally garner review material
for Chapter 2 from the notes of the various courses he presents at the University of Ottawa.
In addition, I would like to thank my PhD Advisory Committee members for their insightful and
constructive advice on my thesis proposal.
Finally, I am very grateful to my family (specially my father) for extending their patience, support and love
to me. Without them, this work would never have come into existence.
iv
TABLE OF CONTENTS
Abstract ii
Acknowledgements iii
Table of Contents iv
List of Figures vii
Glossary xii
List of Tables xii
List of Acronyms xiii
List of Symbols xiv
1. Introduction 1
1.1 Antenna Design 1
1.2 Antenna Synthesis 1
1.3 Summary of the Contributions of the Thesis 3
1.4 Overview of the Thesis 4
2. Review of Key Background Concepts & Techniques 6
2.1 Introduction 6
2.2 Antenna Performance Measures 6
2.2.1 Introductory Comments 6
2.2.2 Antenna Electrical Size Classification 8
2.2.3 Far-Zone Fields & Radiation Patterns 9
2.2.4 Fractional Bandwidth 12
2.2.5 Input Reflection Coefficient of an Isolated Antenna 13
2.2.6 Radiation Efficiency 13
2.2.7 Directivity and Gain 14
2.2.8 Mutual Coupling Between Two Antennas 16
2.2.9 Mean Effective Gain (MEG) 17
2.2.10 Performance Metric of Particular Importance for Electrically Small
Antennas 19
2.3 Antenna Analysis Using Computational Electromagnetics 21
2.3.1 Initial Remarks 21
2.3.2 Electric Field Integral Equation Model 21
2.3.3 Method of Moment (MoM) Solution of an Integral Equation (IE) 23
2.3.4 The Surface Patch Method of Moment Formulation (MM) Solution of an
Integral Equation (IE) 26
2.3.5 Hybrid Method of Moment / Geometrical-Theory-of-Diffraction (MM-
GTD)
Methods 28
2.4 Characteristic Modes 30
2.4.1 Introductory Remarks 30
2.4.2 Conventional Characteristic Modes of Conducting Objects 30
2.4.3 Sub-Structure Characteristic Modes 32
2.4.4 Applications of Characteristic Modes in Antenna Design 36
2.5 A Primer on Optimisation Algorithms 37
2.5.1 Classification of Optimisation Algorithms 37
v
2.5.2 Gradient Based Algorithms 37
2.5.3 Evolutionary Algorithms 37
2.5.4 Further Detail on a Specific Evolutionary Algorithm: The Genetic
Algorithm 38
2.6 Existing Antenna Shape Synthesis Techniques 41
2.6.1 Introduction 41
2.6.2 Existing Shape Synthesis Methods Using Driven Problem Simulation 41
2.6.3 Existing Shape Synthesis Methods Using Characteristic Mode Approaches 42
2.6.4 Computational Considerations 42
2.7 Concluding Remarks 44
3. Construction of a General Shape Synthesis Tool 45
3.1 Preliminary Remarks 45
3.2 Selection of the Computational Electromagnetics Engine 46
3.3 Selection of the Optimization Algorithm 46
3.4 Shaping Manager – Part I 47
3.5 Geometry of Starting Shape, Shaping, and Shaping Constraints 48
3.6 Connecting Expansion Function, and Geometry, Chromosomes 49
3.7 Characteristic Mode Eigenanalysis 53
3.8 Objective Function Definition 54
3.9 Customization of the Shaping Prescription 54
3.10 Feed Point Selection & Triangle Removal After Shaping 60
3.10.1 Feed Point Selection 60
3.10.2 Removal of Islands & Vertex-Only Connections 60
3.11 Supplementary Antenna Performance Computation 61
3.11.1 Quantities 61
3.11.2 Quality Factor (Q) 62
3.11.3 Mean Effective Gain (MEG) 62
3.11.4 Bandwidth-Efficiency Product 63
3.12 Shaping Manager- Part II 63
3.13 Concluding Remarks 65
4. The Shape Synthesis of Non-Planar Conducting Surface Antennas 66
4.1 Preliminary Remarks 66
4.1.1 Outline of this Chapter 66
4.1.2 Some Motivation 66
4.2 Three-Dimensional Shape Synthesis Example #1: Open Cuboid Antenna 69
4.2.1 Preamble 69
4.2.2 Starting Geometry 69
4.2.3 The Objective Function 69
4.2.4 Outcome of the Shape Synthesis Process 71
4.2.5 Computed Performance of the Shaped Antenna 74
4.2.6 Scaling, Fabrication and Measurement 78
4.2.7 Shaping of a Closed Cuboid of the Same Size 83
4.2.8 Shaping of a Closed Conducting Sphere 85
4.3 Three-Dimensional Shape Synthesis Printed Electronics Compatible Antenna 86
4.3.1 Preamble 86
4.3.2 Starting Geometry & Objective Function 87
vi
4.3.3 Outcome of the Shape Synthesis Process 88
4.3.4 Computed Performance of the Shaped Antenna 90
4.4 Three-Dimensional Shape Synthesis Example #3: Cuboid with Battery Inside 94
4.4.1 Starting Geometry 94
4.4.2 Outcome of the Shape Synthesis Process 95
4.4.3 Computed Performance of the Shaped Antenna 95
4.5 Three-Dimensional Shape Synthesis Example #4: Open Cuboid with Pattern
Constraints 98
4.5.1 Preamble 98
4.5.2 Maximum Effective Directivity for a Single-Mode Antenna 98
4.5.3 The Objective Function 100
4.5.4 Outcome of the Shape Synthesis Process 101
4.6 Possible Extensions 103
4.7 Concluding Remarks 104
5. The Shape Synthesis of Closely-Spaced Antennas 106
5.1 Preliminary Remarks 106
5.2 Closely-Spaced Antennas: Starting Geometry 106
5.3 Objective Functions 108
5.4 Shaping Prescription 109
5.5 Shape Synthesis Example: Two Antennas with Broadsides Parallel 111
5.5.1 Starting Geometry & Shaping Constraints 111
5.5.2 Outcome of the Shape Synthesis Process 112
5.5.3 Performance Computation & Measurement of the Shaped Antennas 113
5.6 Possible Extentions 118
5.7 Concluding Remarks 118
6. Sub-Structure Characteristic Mode Computation Utilizing Field-Based
Hybrid Methods 119
6.1 Goals of the Chapter 119
6.2 Proposed Idea 120
6.3 Sub-Structure Characteristic Modes Computed Using the MM/GTD Hybrid
Approach 123
6.4 Conclusions 129
7. General Conclusions and Future Works 130
7.1 Contributions of the Thesis 130
7.2 Future Work 131
8. References 133
9.
Appendix A: Estimation of Measured Radiation Efficiency of the Shaped-Synthesized
Open Cuboid Antenna 140
vii
List of Figures
Figure 2.2-1 Power budget for an antenna without (a), and with (b), a matching
network. (After [MCNA 17])
8
Figure 2.2-2 Spherical coordinate system (After [MCNA17])
12
Figure 2.2-3 Two-port network representation of two coupled antennas (After
[MCNA17])
17
Figure 2.3-1 Scattering of the fields, from impressed sources, from a perfectly
conducting (PEC) object
23
Figure 2.3-2 PEC object (cubic bottomless rectangular box) meshed into triangular
elements
27
Figure 2.3-3 Four of the triangles from the mesh in Figure2.3-2, shown exaggerated
in relative size
28
Figure 2.3-4 PEC object from Figure2.3-2, but now in the presence of an
electrically very large additional PEC object
29
Figure 2.3-5 Perturbation of the values of the MM matrix accounted for using the
GTD
29
Figure 2.4-1 Two objects considered for the sub-structure CM analysis
33
Figure 2.5-1 A demonstration of pixelization of a geometry. Those pixels (or
elements) occupied by conductor are sketched in black; unfilled
elements are left clear
39
Figure 3.6-1 (a). Simplified unaltered geometry consisting of four triangles and four
expansion functions and (b). with triangle#1 removed (as suggested by
the dashed line). Triangle numbering is shown encircled. The red dots
are the vertices of the triangles
52
Figure 3.6-2 (a). MM impedance matrix [Z] corresponding to the unaltered
geometry in Figure 3.6-1(a), and (b). The reduced MM matrix
formation upon removal of expansion function #1 and expansion
function #2
53
Figure 3.9-1 Trial Run
58
Figure 3.9-2 Actual Run
59
Figure 3.10-1 Illustration of a shape-synthesized antenna with highlighted vertex-
only connections
61
Figure 3.11-1 Current density on a centre-fed PEC strip of roughly one-half-
wavelength
62
Figure 3.12-1 Shaping Manager
64
Figure 4.1-1 Applications of the Internet of Things (Adapted from [NEZA19])
67
Figure 4.2-1 Starting non-planar cuboid 3D conductor shape with bottom face
removed, and sides of dimension 30mm d  . Diagram shows tilted
cuboid viewed from below
70
viii
Figure 4.2-2 Progress of the value of the objective function (fitness function) of the
fittest member of the population as a function of generation number in
the “actual run”
72
Figure 4.2-3
Eigenvalues
n
 vs mode index n for the final shaped antenna at
o
f ,
normalised to
1
 and expressed in decibels
72
Figure 4.2-4 Shape-synthesized antenna geometry. The feed point is indicated by
the red bar
73
Figure 4.2-5
Three views, at frequency
o
f , of the current density of the lowest order
CM on the shaped structure in Figure4.2-4 are shown in (a) through
(c). The current distribution on the driven shaped antenna is that in (d),
and is referred to in Section 4.2.5
73
Figure 4.2-6
Modal excitation coefficient (
n
 ) vs. mode index ( n ) at
o
f
75
Figure 4.2-7 Computed input impedance of the shaped antenna
75
Figure 4.2-8 Computed input reflection coefficient of the shaped-synthesized
antenna for reference impedance
o
50 Z  
76
Figure 4.2-9
Normalized radiation pattern cuts at
o
f
. The coordinate system sketch
in (a) shows which pattern cuts are referred to. The pattern cuts are as
follows - (b).
90 ,0 360     
(c).
0 ,0 360     
, (d).
90 ,0 360     
77
Figure 4.2-10 Computed input resistance and reactance of the scaled antenna
78
Figure 4.2-11 Flat-patterned antenna shape scaled by a factor of two for fabrication
purposes
79
Figure 4.2-12 Fabricated scaled antenna
79
Figure 4.2-13 Schematic illustration of the coaxial cable feedline location. The red
line represents a small gap (first depicted in Figure4.2-4), the black
line the coaxial cable outer conductor, and the white line the latter’s
inner conductor
80
Figure 4.2-14 Plot of the computed and measured reflection coefficient of 3-D shape
electrically small antenna for a 50Ω reference impedance
80
Figure 4.2-15 Polar plot of the computed and measured normalized patterns in the
azimuth plane
82
Figure 4.2-16 (a). The starting shape of the closed cuboid, and (b). the resulting
shape synthesized antenna. Islands and vertex-only connections have
not yet been removed in (b)
84
Figure 4.2-17 The objective function value of the fittest shape in each generation,
versus the generation number
84
Figure 4.2-18 (a). Computed input impedance of the antenna shape-synthesized from
85
ix
the closed cuboid starting shape and (b). its computed input reflection
coefficient for reference impedance
o
50 Z  
Figure 4.2-20 The starting shape of the 10mm radius closed sphere is shown on the
left, and the shaped antenna on the right (along with its feed location)
86
Figure 4.3-1 (a).Cuboid-like starting shape with purposefully missing corners, used
in this example, and (b).Possible alternative shape that compensates for
assembly ‘imperfections’. Note that w 2s d  
87
Figure 4.3-2 Two different views of the geometry of the shaped antenna. The left
view shows the feed location as a red line
88
Figure 4.3-3 The cost function of the fittest member in each generation of shapes
versus the generation number
89
Figure 4.3-4
Eigenvalues
n
 vs mode index n for the final shaped antenna at
o
f ,
normalised to
1
 and expressed in deciBels
89
Figure 4.3-5 Eigenvalues of the first three characteristic modes versus frequency
90
Figure 4.3-6
(a) Computed
in ( )
f  of the shaped antenna for a reference
impedance
o
50 Z   , and (b). the associated current distribution on
the fed antenna at 3 GHz
91
Figure 4.3-7
Modal excitation coefficient
n
 versus modal index n, at 3 GHz
91
Figure 4.3-8 Computed input resistance and reactance of the shape-synthesized
antenna
92
Figure 4.3-9
Computed normalized radiation pattern cuts at
o
f . The elevation plane
cuts are (a). 0 ,0 360      and (b). 90 ,0 360      . The
azimuth pattern cut 90 ,0 360      is shown in (c)
93
Figure 4.4-1 Starting shape consisting of an outer open cuboid with an inner closed
cuboid
94
Figure 4.4-2 The shape synthesized antenna is shown to the left, with the feedpoint
identified in red. The same geometry is shown to the right, with the
MM mesh removed and the (unshaped) battery shown in blue
95
Figure 4.4-3
Modal excitation coefficient
n
 versus modal index n, at 2 GHz
96
Figure 4.4-4 Current distribution on the driven shaped structure at 2 GHz
96
Figure 4.4-5 Computed input impedance of the shaped-synthesized antenna
97
Figure 4.4-6 Computed total gain patterns (in dBi). The elevation plane cuts shown
in (a) are for 0   , 45   , 90   and 135   . The azimuth
pattern cut is that in (b)
97
Figure 4.5-1 Computed total gain patterns (in dBi). The azimuth plane cut is shown
in (a), as the solid line, along with the corresponding result from
102
x
Section 4.3 shown as a dashed line
Figure 4.5-2 (a) The geometry of the shaped antenna, and (b). Current distribution
103
Figure 4.7-1
Compendium of bandwidth-efficiency products (
rad
B  ) of numerous
antenna designs
105
Figure 5.2-1 Two closely-spaced starting shapes
107
Figure 5.2-2 Two closely-spaced shape-synthesized antennas
108
Figure 5.4-1 Trial Run - Flowchart#4
110
Figure 5.5-1 Starting conductor shapes used
111
Figure 5.5-2
Value of
B
obj
F at the end of each iteration in Table 5.4-1
112
Figure 5.5-3
The value of
A
obj
F (blue circle) and
B
obj
F (red asterisk), versus the
iteration number
113
Figure 5.5-4 Shaped geometry and feed port locations. The port labelled “Feed#A”
is the feedport for Antenna#A, and similarly for Antenna#B
113
Figure 5.5-5
Computed current distribution (at 1.5GHz
a
f  ) on the complete
radiating structure when excited at the feed port of Antenna#A
115
Figure 5.5-6
Computed current distribution (at 2.0GHz
b
f  ) on the complete
radiating structure when excited at the feed port of Antenna#B
115
Figure 5.5-7 Photograph of the fabricated antenna structure
116
Figure 5.5-8 (a) Computed input impedance of Antenna#A. (b) Computed total
gain
116
Figure 5.5-9 (a) Computed input impedance at the feed port of Antenna#B.
(b) Computed total gain in the azimuth plane (xz-plane) of Antenna#B
117
Figure 5.5-10 Computed and measured input reflection coefficients (referenced to
50) of the two antennas at their respective feed ports
117
Figure 6.2-1 (a) Several PEC bodies grouped together as a single isolated object,
and (b) several PEC bodies grouped as Object A and Object B
122
Figure 6.3-1 PEC Object#A (small plate) and Object#B (arrangement of large
plates) used in this paper. The size of square PEC plate Object#A is
127.5mm per side. Object#A is shown larger than it actually is in
relation to Object#B
124
Figure 6.3-2 Eigenvalues of the 1 st (──), 2 nd (──) and 3 rd (──) sub-structure CMs
obtained using the MM/MM approach to find
per
[ ] Z . Eigenavlues (- -
-) of the first three (two of which are degenerate) conventional CMs of
Object#A in the absence of Object#B
125
Figure 6.3-3 Eigenvalues of the 1 st (blue), 2 nd (red) and 3 rd (black) substructure CMs
computed using the MM/GTD method. Those found using the
MM/MM (exact) approach to find
per
[ ] Z , as in Figure6.3-2, are shown
as dashed lines. Those found using the hybrid MM/GTD approach to
126
xi
find
per
[ ] Z are shown as solid lines
Figure 6.3-4 Ship model from which the Object#B in Figure6.3-1 was extracted.
The sections on the rest of the model are for visualisation reasons only;
they do not represent the mesh needed for computational purposes
using the method of moments. The ship model is courtesy of Altair Inc.
[FEKO]
126
Figure 6.3-5 Eigenvalues (▬▬) of the 1 st sub-structure CM found using the
MM/MM approach, and those (▬ ▬) using the hybrid MM/GTD
approach (with all GTD terms included). Curve (• • •) is that with
double diffraction terms removed, () with corner diffraction
contributions also removed, and (- - -) when the only GTD terms
included are the direct and reflected ray terms
127
Figure 6.3-6 Magnitudes of the coupling coefficients expressed in deciBels between
the first five sub-structure CMs obtained when the hybrid MM/GTD
approach is used to find
per
[ ] Z .
128
Figure A-1 Curve of the nominal gains (dBi) of the horn antenna used as a gain
reference
141
xii
Glossary
Computational Electromagnetics (CEM) Section 1.1
Antenna Shape Synthesis Section 1.2
Electrically Small Antenna Section 2.2.2
Far-Zone Fields & Radiation Patterns Section 2.2.3
Bandwidth Section 2.2.4
Input Reflection Coefficient Section 2.2.5
Radiation Efficiency Section 2.2.6
Directivity and Gain Section 2.2.7
S-Parameters Section 2.2.8
Mean Effective Gain (MEG) Section 2.2.9
Electrically-Small Antenna Performance Measures Section 2.2.10
Electric Field Integral Equation Model Section 2.3.2
Method of Moment (MoM) Solution of an Integral Equation (IE) Section 2.3.3
Expansion Functions Section 2.3.4
Hybrid Moment-Method / Geometrical-Theory-of-Diffraction (MM-GTD) Methods Section 2.3.5
Characteristic Modes Section 2.4
Sub-structure Characteristic Modes Section 2.4.3
Optimization Algorithms Section 2.5
Antenna Shape Synthesis Techniques Section 2.6
List of Tables
Table 4.2-1 Summary of some performance measures of the shaped open cuboid antenna 78
Table 4.3-1 Summary of some performance measures of the shaped printed electronics
compatible shaped
88
Table 4.4-1 Summary of some performance measures of the shaped antenna with a “battery”
enclosed
94
xiii
List of Acronyms
1D One-Dimensional
3D Three-Dimensional
EFIE Electric-Field Integral Equation
IE Integral Equation
MS Modal Significance
TE Transverse Electric
CEM Computational Electromagnetic
CM Characteristic Mode
DMM Direct Matrix Manipulation
ESA Electrically Small Antenna
GA Genetic Algorithm
MIMO Multiple-input Multiple-output
MOM Method of Moments
PEC Perfect Electrical Conductor
RWG Rao-Wilton-Glisson expansion functions
UWB Ultrawideband
VNA Vector Network Analyzer
MED Mean Effective Directivity
MEG Mean Effective Gain
AoA Angle-of-Arrival
PAS Power Angular Spectrum
XPR Cross-Polarisation Ratio
Q Quality Factor
VSWR Voltage Standing Wave Ratio
GTD Geometrical Theory of Diffraction
MM-GTD Moment-Method / Geometrical-Theory-of-Diffraction
ASCII American Standard Code for Information Interchange
xiv
List of Symbols
inc
P Incident Power
refl
P Reflected Power
in
P Accepted Power
rad
P Radiated Power
loss
P Power Losses
f Frequency
L
f Upper Frequency
U
f Lower Frequency
a Spherical Volume Radius
 Wavelength
k Wavenumber
E
Electric Field
H
Magnetic Field
S
Time-Averaged Power Density
r Observation Point
o
 Free-Space Intrinsic Impedance
o
 Permeability
o
 Permittivity
U Radiation Intensity
f  Absolute Bandwidth
o
f Self-Resonant Frequency
in
 Input Reflection Coefficient
in
Z Input Impedance
o
Z Characteristic Impedance
in
R Input Resistance
in
X Input Reactance
rad
R Radiated Power
loss
R Power Dissipated
rad
 Radiation Efficiency
D Directivity
G Gain
realized
G Realized Gain
[ ] Z Impedance Matrix
[ ] S Scattering Matrix
G  Partial Gain

P Angular Power Density Function
xv

P Angular Power Density Function
Q Quality Factor
1
 Eigenvalue of Mode#1
z
Q Quality Factor Estimated from the Frequency Derivative of the Input Impedance
 Conductivity
c
S Surface of the Scatterer
tan
scat
E
Scattered Field
imp
J
Impressed Electric Source Current Densities
imp
M
Impressed Magnetic Current Densities
s
J
Electric Surface Current Density
G Kernel (Green’s Function)
( )
n
f z Expansion Functions
( )
n
z  z-Dependent Function
inc
V
 
 
Incident Voltage
fs
G
Green’s Function In Free Space
n
 CM Eigenvalues
[ ]
n
J CM Eigenvectors (Eigencurrents)
[ ] R Real Part of the Impedance Matrix
[ ] X Imaginary Part of the Impedance Matrix
mn
 Kronecker Delta
n
 Weighting Coefficient
n
E
Eigenfield
MS Modal Significance Factor
sub
n

Sub-Structure CMs
[ ]
Asub
n
J
Sub-Structure Eigencurrents
pop
N
Population Size
obj
F Objective Function
gen
N
Number of Generations
bit
N Number of Bits in a Chromosome
N  Number of Triangles in the Mesh
.out FEKO Output File
.cfm FEKO Geometry File
dof
N Number of Expansion Functions
init
C Initial Chromosome
rad
B  Bandwidth-Efficiency Product
1
CHAPTER 1
Introduction
1.1 ANTENNA DESIGN
Prior to the availability of computational electromagnetics (CEM), antenna analysis had severe limitations.
The quantity that dictates the form of the radiation pattern, namely the current distribution on the antenna,
had to be approximated, and input reflection coefficients at the antenna terminals could not be calculated
accurately. No doubt this hampered design innovation on the antenna front; but then the constraints on the
antenna design (eg. amount of room allocated for the antenna; the bandwidth required of the antenna; and so
on) were often not prohibitive and quite standard antenna shapes were useable and so communications
system development as a whole was not hampered. The availability of CEM changed the way antennas are
designed, and commercial CEM software arrived on the scene in a timely manner just when wireless
communications began to boom, and demands for improved antennas increased, along with constraints on
the real estate they may occupy. CEM analysis now allows us to actually compute the current distributions
on antennas no matter what their shape, and hence also find their port properties more accurately than was
previously the case. We need no longer only use a straight wire dipole or circular loop. During the design
process we can bend it to fit on some device and accurately determine what the performance consequences
are. We can determine what the effect of the chassis (or platform) is, on which the antenna is mounted. And
much more. This ability has moved antenna design into the domain of most RF designers who work on other
RF “circuit” aspects, and creativity has flourished. The rapid expansion of what can be done wirelessly is
proof of this. However, by nature engineers always “push the envelope”, and so antenna topics remain that
are in need of research.
1.2 ANTENNA SYNTHESIS
Antennas are essential in all wireless communications, being key to the quality of any radio link,
irrespective of its use. Transmitting antennas transform electrical signals in a transmitter into an
electromagnetic wave that is radiated into space. Similarly, receiving antennas capture power from incident
electromagnetic waves and deliver the induced electrical signals to a receiver. In most cases the same
2
antenna is used for transmitting and receiving. Depending on the application, an antenna may take one of
several different geometrical forms. Widely-used handbooks [VOLA 07], [BALA 08] discuss conducting
wire geometries, slot radiators, patch radiators, dielectric resonator antennas, and horn radiators, for low
(broad radiation patterns) or medium directivity performance 1 . If high directivity (narrow radiation pattern)
is needed and cannot be achieved using the above-mentioned antenna types, engineers then use a properly
shaped reflector (or reflectarray), or a lens-like arrangement (or transmitarray), with one of the above-
mentioned antennas used as a feed radiator to illuminate the reflector or lens. Or several of the above-
mentioned radiators (in which case they are called radiating elements) are correctly arranged in space to
form an array antenna.
The shape synthesis approach in antenna design is to not select a particular antenna geometry beforehand,
but to “let the electromagnetics tell us what the shape must be”. This is a tall order. We are rightfully 2
warned that the ideal antenna synthesis procedure would allow one to “start with a set of electrical,
mechanical, and system specifications that would lead to a particular antenna together with its specific
geometry and material specification”, but that “this ideal general antenna synthesis method does not exist”.
Nevertheless, progress in antenna shape synthesis has been made 3 . In order to clarify what we mean by
shape synthesis, we can group antenna design optimization approaches as follows:
■ Excitation optimization: This includes the determination of the complex excitations of an array of known
radiating elements required to provide some desired radiation pattern performance, usually called array
pattern synthesis.
■ Size optimization: This is essentially the adjustment of the values of a pre-determined set of feature
dimensions with the feature shapes determined from physical insight and experience. The majority of
antenna design procedures fall into this category; it has been very successful.
■ Shape optimization: In this category the shape of the final antenna structure is an outcome of the synthesis
process, although there is usually some starting structure that is perturbed by the synthesis procedure to
result in the final antenna shape. It is already used for microstrip and wire antennas, with only the conductor
geometry undergoing shaping through the removal or retention of conducting pixels/segments into which the
1 The most important antenna performance indices are reviewed in Section 2.2.
2 W.Stutzman and S.Licul, “Synthesis Methods for Antennas” in C.A.Balanis (Edit.), Modern Antenna Handbook (Wiley, 2008).
3 This is reviewed in Section 2.6.
3
starting shape is divided. It will be reviewed in Section 2.6. This approach is related to topology optimisation
in structural engineering4 (where it has been used for some years), and inverse design in photonics (where
its use appears to be relatively recent). In the antenna field it includes shaping of planar elements for
frequency selective surfaces and reflectarrays.
■ Material optimization: In this technique a rectangular parallelepiped starting shape is divided into voxels,
each of whose material properties can individually be selected (during the shaping process) to be conducting
or consist of dielectric/magnetic material. This resembles the inverse design methods that are coming into
use in the photonics area.
The goal of the shape synthesis methods to be developed in this thesis is not to ‘break existing performance
records’ of antennas designed by more standard approaches. Instead, the aim is to extend, to a wider range of
problems, those methods that begin with some “starting shape” and then through automated shaping arrive at
an antenna that has some desired performance, subject constraints both on the electrical performance and on
the space the final antenna may occupy.
1.3 SUMMARY OF THE CONTRIBUTIONS OF THE THESIS
The principal contributions of this thesis are as follows:
■ The development of a computational tool for the shape synthesis of three-dimensional conducting
antennas. This has previously not been available.
■ Use of the tool for the shape synthesis of 3D conducting electrically small antennas using a “shape-first
feed-next” approach. This is the first time that characteristic mode concepts have been utilized in this way to
shape synthesize 3D conducting surface antennas (with constraints on the surface/volume that may be
occupied), and that directivity-related requirements have been incorporated into such an approach by
expressing mean effective directivity in terms of characteristic mode quantities.
■ Use of the tool for the shape synthesis of closely-spaced antennas that operate at two different frequencies,
using sub-structure characteristic modes to permit a “shape-first feed-next” approach. This sort of shape
synthesis has previously not been reported elsewhere.
4 But the pixelation approach is not used there.
4
■ Demonstration of sub-structure characteristic mode computation using a field-based hybrid MM/GTD
method. This does not yet appear to have been explicitly stated elsewhere (and so perhaps not realised), nor
example computations provided.
1.4 OVERVIEW OF THE THESIS
The research described in this thesis lies in the area of antenna shape synthesis. It deals exclusively with the
shape synthesis of antennas whose principal radiation mechanism is an electric current density on
conducting surfaces 5 .
Chapter 2 assembles the key technical concepts that are needed to conduct the research described in the
thesis, and upon which it builds. This includes the definition of antenna performance measures, as well as
reviews of the method of moments for radiation/scattering from conducting objects, characteristic mode
concepts, and (briefly) the optimization methods available, in particular that used in this thesis. We also
recognize and review what has been achieved by others on the topic of the shape synthesis of conducting
antennas, and clarify some aspects that we believe have not yet been adequately dealt with in the literature.
Chapter 3 describes the development of the computational tool that permits one to perform certain types of
antenna shape syntheses for the first time. This development is thus, in its own right, an important
contribution. A chapter of this sort might be found to be a rather tedious one, but is necessary from the
standpoint of a reader who might want to repeat this work. It will also serve to remove any vagueness as to
exactly how things are done in the remaining chapters, and indeed allow descriptions in the remaining
chapters to be more concise and to the point. Importantly, it establishes a structured way to use the multitude
of parameter settings when using the GA in antenna shaping.
Chapters 4 and 5 describe two classes of shape synthesis problem. Chapter 4 demonstrates the
implementation of the shape synthesis process for fully 3D conducting surface structures. Chapter 5
considers the shape synthesis of conducting surface antennas that operate in each other’s presence, at
different frequencies. Both these classes of shape synthesis problem are being described here for the first
time. Chapter 6 contributes to the theory of sub-structure characteristic modes by demonstrating how these
5 Thus, for example, we do not consider dielectric resonator antennas (DRAs), whose principal radiation mechanism (for non-
hybrid DRAs at least) is electric polarisation current in the dielectric material.
5
can be found using field-based hybrid methods. This would allow characteristic mode based shape synthesis
methods to be used even when the antenna’s platform is electrically too large (for even reasonable high-
performance computing facilities) to be completely modelled using the method of moments.
Finally, some general conclusions are drawn in Chapter 7, and the research reported in the thesis put into
perspective.
6
CHAPTER 2
Review of Key Background Concepts & Techniques
2.1 INTRODUCTION
We are interested in the shape synthesis of antennas that are electrically-small 6 or of intermediate size. In
order to place the work of this thesis in the context of previous works on the topic, and so to demonstrate
that it has made measurable contributions to the field, a review of the state of the art of such shape synthesis
is necessary. This is provided in Section 2.6. Before this can be done with clarity, it is necessary to first
summarize the computational electromagnetics methods that have been used; this is done in Section 2.3. As
mentioned in Chapter 1, computational electromagnetics provides superb tools for antenna analysis. The
source densities and electromagnetic fields from such modelling incorporate all the physics associated with a
particular configuration. The difficulty is to extract and exploit the "hidden information" in the solutions
presently not always used to the full. The characteristic mode analysis of a structure, which is found using
computational electromagnetics modelling, will be used in this thesis as a way of doing this, and so is
reviewed in Section 2.4. The shaping tool to be developed in the thesis needs to use some optimization
algorithms, and so these are briefly put in context in Section 2.5. Section 2.2 defines the essential antenna
performance measures that will be used in the thesis. Section 2.7 concludes the chapter with a summary of
shape synthesis problems for electrically small antennas that have not been adequately dealt with in the
literature, and which form the subject of this thesis.
2.2 ANTENNA PERFORMANCE MEASURES
2.2.1 Introductory Comments
Antennas form the transducers between the electrical signals in the circuitry of transmitters and receivers,
and the electromagnetic waves (in the surrounding space) that are radiated or absorbed, respectively. Any
antenna has a set of terminals (a port) at which the desired signals are input (in the transmitting situation) or
extracted (in the receiving situation). Fig.2.2-1(a) shows a transmission line that connects an antenna to the
6 By which is meant those whose dimensions are a fraction of the free space wavelength at the operating frequency.
7
transmitter 7 . A certain amount of power
inc
P is incident on the antenna terminals. Due to mismatch between
the transmission line and the antenna some of this power
refl
P is reflected back towards the transmitter while
the remainder
in
P is accepted by the antenna (i.e. actually enters the antenna terminals). Thus we have
in inc refl
P P P   (2.2-1)
Of the power (2.2-1), a portion
loss
P is dissipated on the antenna structure due to material losses, and the rest
rad
P is radiated. In other words
rad in loss
P P P   (2.2-2)
The antenna designer’s task, for a given
inc
P , is to maximize
rad
P by minimizing
refl
P and
loss
P . In addition,
the antenna is normally required to radiate most of
rad
P in preferred directions. The electromagnetic wave
transporting
rad
P is in many cases also required to be of a certain polarization. And all the above
requirements must be met at frequencies f over a specific frequency band
L U
f f f   between lower
frequency
L
f and upper frequency
U
f . In order to minimize
refl
P it is often necessary to use a matching
network between the transmission line and the radiator, as shown in Fig.2.2-1(b). The antenna then consists
of the radiator plus the matching network, and
loss
P then consists of losses in the matching network 8 and the
radiator.
7 As is often the case, we will always discuss antennas as if they are transmitting. All of the antennas in this thesis are composed of
isotropic linear materials, and so the performance in the receiving situation will always be the same as in the transmitting (that is,
reciprocity holds).
8 Because the matching network must be considered part of the antenna.
8
Antenna
Terminals
Transmission Line
Reflected Power
Incident Power
Accepted
Power
Radiated Power
Power Loss
Transmitter
Antenna
inc
P
refl
P
in
P
loss
P
rad
P
IN
Z
0
Z
in
I
in
V
(a)
Antenna
Terminals
Transmission Line
Reflected Power
Incident Power
Accepted
Power
Radiated Power
Power Loss
Transmitter
Antenna
inc
P
refl
P
in
P
loss
P
rad
P
Matching
Network
Radiator
(b)
Fig.2.2-1 : Power budget for an antenna without (a), and with (b), a matching network. (After [MCNA 17])
2.2.2 Antenna Electrical Size Classification
As originally defined by Wheeler [WHEE 47], an electrically small antenna is any antenna that fits within
a spherical volume with radius a less than 2   , or as Wheeler stated it, any antenna that can fit within the
radiansphere. This can be stated as
2
a


 (2.2-3)
Recalling that the free space wavenumber 2 / k    , this can be rewritten as the condition 9
9 Some prefer to define an electrically small antenna to be one for which
0.5 ka  , but we will use the more common (2.2-4).
9
1 ka  (2.2-4)
The properties of electrically small antennas are restrictive : narrow bandwidth, low radiation efficiency,
high reactive energies, and small radiation resistance. The design of electrically small antennas centres
around trying to overcome these properties. A half-wavelength dipole has 1.57 ka  , and is not considered to
be electrically small. Further discussion on electrically small antennas will follow in Section 2.2.10 after the
various antenna performance metrics mentioned above have been defined.
2.2.3 Far-Zone Fields & Radiation Patterns
Any electromagnetic field, and hence the electromagnetic field of any antenna can be written (with
respect to the chosen coordinate origin) as
ˆ ˆ
ˆ ( , , ) ( , , ) ( , , ) ( , , )
r
E r E r E r E r r
 
             (2.2-5)
and
ˆ ˆ
ˆ ( , , ) ( , , ) ( , , ) ( , , )
r
H r H r H r H r r
 
             (2.2-6)
where ( , , ) r   specifies an observation point in the three-dimensional space surrounding the antenna. The
pair of angles ( , )   gives the direction of the observation point and r its distance from the origin of the
coordinate system. The first two components in each of the above expressions are the transverse components
of the antenna’s electromagnetic field. The last component in each of (2.2-3) and (2.2-4), namely ( , , )
r
E r  
and ( , , )
r
H r   , are the radial components of the electromagnetic field. The magnetic field ( , , ) H r   can
be found from ( , , ) E r   using Maxwell's equations, and vice versa.
The time-averaged power density at any point ( , , ) r   , at which the antenna fields are ( , , ) E r   and
( , , ) H r   , is given by the (time-averaged) Poynting vector
 
*
( , , ) ½ Re ( , , ) ( , , ) S r E r H r         (2.2-7)
and is measured in Watts/metre 2 . If we think of an imaginary spherical surface
ant
S that encloses the antenna
entirely, as illustrated in Fig.2.2-2, then the time-averaged power (in Watts) passing through surface
ANT
S is
given from Poynting’s theorem as the integral of ( , , ) S r   over the closed surface
ANT
S , namely
10
 
2
2
0 0
ˆ ˆ ( , , ) ( , , ) ( , , ) sin
ANT ANT
rad
S S
P S r dS S r rdS S r r r d d
 
              
   
(2.2-8)
The physical space around an antenna (in fact this is all of space) can be broadly classified as the near-zone 10
in the immediate vicinity of the antenna where the variation of the fields with observation point ( , , ) r r    is
very complicated and sensitive to the geometrical details of the structure, and the far-zone region that is distant
from the antenna. In the far-zone region the fields are found to have the following properties:
▪ The radial components of the fields (namely E r and H r ) are negligible compared to the transverse
components, so that we can assume
ˆ ( , , ) ( , , ) 0
r
E r r E r        (2.2-9)
and
ˆ ( , , ) ( , , ) 0
r
H r r H r        (2.2-10)
▪ The components of the transverse fields take the forms
( , , ) ( , )
jkr
e
E r F
r
 
   

 (2.2-11)
and
( , , ) ( , )
jkr
e
E r F
r
 
   

 (2.2-12)
so that the transverse electric field as a whole is
( , , ) ( , )
jkr
e
E r F
r
   

 (2.2-13)
with
ˆ ˆ
( , ) ( , ) ( , ) F F F
 
          (2.2-14)
▪ In other words, in the far-zone region the fields decay as 1/ r . If we set
r
E and
r
H to zero, then from the
Maxwell curl equations we find that the electric and magnetic fields in the far-zone of an antenna are related
through
1 1
ˆ ˆ ( , , ) ( , , ) ( , )
jkr
o o
e
H r r E r r F
r
     
 

    (2.2-15)
10 Some prefer to divide this near-zone region into the reactive near-zone region, and the radiating near-zone region (also called the
Fresnel region). The far-zone region is also called the Fraunhofer region.
11
with /
o o o
    . The result is that, in the far-zone, expression (2.2-7) becomes
ˆ ( ) ( ) S r, , = S r, , r     (2.2-16)
The radiation intensity U(  ,  ) of an antenna, in a specific direction (,) is defined by the IEEE as the power
radiated from the antenna per unit solid angle in that direction. Radiation intensity is a far-zone quantity, as
emphasised by the fact that it is purposefully denoted as a function of the angular coordinates  and  only.
The relation between the radiation intensity and the far-zone power density is simply
2
( ) ( ) U , = r S r, ,     (2.2-17)
Note that although ( ) S r, ,   is a function of r, in the far-zone its r-dependence is always the 1/r 2 decay, and
hence the radiation intensity U(,) is indeed independent of r, being
2
( , )
( , )
2
o
F
U
 
 

 (2.2-18)
Graphical representations of ( , ) U   are represented in either polar or rectangular form. The radiation intensity
is usually maximum in some direction, say ( o , o ), and the radiation pattern is most often plotted in deciBels in
normalised form, namely
 
2
2
( , ) ( , )
( , ) 10log 20log 20log ( , ) ( , )
( , )
( , )
dB N N
dB
o o
o o
F F
U F F
F
F
   
     
 
 
 
 
   
   
   
   
 
 
(2.2-19)
This is usually called the radiation pattern of the antenna.
12

r
Antenna
ˆ r
Point ( , , ) r  
ˆ

ˆ

Fig.2.2-2 : Spherical coordinate system (After [MCNA17])
2.2.4 Fractional Bandwidth
The bandwidth of an antenna is defined as the range of frequencies within which the performance of the
antenna, with respect to some characteristic, conforms to a specified standard. The bandwidth can be
considered to be the range of frequencies, from lower frequency
L
f to upper frequency
U
f , namely
U L
f f f   on either side of some center frequency
o
f , over which the required antenna characteristics are
within acceptable values. The various characteristics of an antenna will not necessarily have the same
frequency dependence. For instance, the input reflection coefficient might be within acceptable values over a
smaller frequency range than are the sidelobe levels. Thus there is no unique definition for bandwidth. We
can talk about the impedance bandwidth (also referred to as the reflection coefficient bandwidth), the pattern
bandwidth, and so forth. If an antenna satisfies a required set of specifications over the frequency band
L U
f f f    (2.2-20)
the quantity f  is called the absolute bandwidth. Given the above absolute bandwidth, the center-frequency
is
 
1
2 2
o L U L
f
f f f f

    (2.2-21)
All the above quantities are of course measured in Hertz (Hz). The fractional bandwidth of the antenna is
defined as
13
o
f
Fractional Bandwith
f

 (2.2-22)
It is dimensionless and usually expressed as a percentage.
2.2.5 Input Reflection Coefficient of an Isolated Antenna
Consider once more the set-up shown in Fig.2.2-1(a), which shows an antenna of input impedance
in
Z fed by
a transmission line of characteristic impedance
o
Z . The reflection coefficient at the antenna input terminals is
in o
in
in o
Z Z
Z Z

 

(2.2-23)
Denoting the power incident at the antenna terminals by
inc
P , the power actually accepted by the antenna is
 
2
1
in IN inc
P P    (2.2-24)
At any frequency  we have an input impedance
( ) ( ) ( )
in in in
Z R jX      (2.2-25)
Resistance ( )
in
R  models the energy that is input from the source and is not returned to the source. In the case
of the antenna it thus represents two physical effects, namely power that is radiated by the antenna and power
that is dissipated on the antenna structure itself. We can write
( ) ( ) ( )
in rad loss
R R R      (2.2-26)
The portion
rad
R models the radiated power, and ( )
loss
R  the power dissipated (eg. Ohmic loss in conducting
material; dielectric loss in printed circuit substrates, and so on). We have purposefully shown the frequency
dependence explicitly to emphasise the fact that the values of these quantities are not the same at all frequencies
of interest 11 .
If ( ) 0
in o
X   without the need for a matching network we say the antenna is self-resonant at 2
o o
f    . If
an antenna is self-resonant, the matching network can often be dispensed with or need be of a much simpler
design (an impedance transformer).
2.2.6 Radiation Efficiency
The radiation efficiency of an antenna is defined as the ratio of the total power radiated by the antenna to the
power accepted by the antenna from the connected transmission line, namely
11 In what follows we will not always show this frequency dependence explicitly, but it should always be kept in mind.
14
rad
rad
in
P
P
  (2.2-27)
It follows that we can rewrite this as
rad RAD
rad
rad loss RAD LOSS
P R
P P R R
  
 
(2.2-28)
We observe that
rad
 is not affected by
in
 . However, the quantity 

2
rad
1
in
   is sometimes referred to as
the “total radiation efficiency”.
2.2.7 Directivity and Gain
A. Directivity
It is usually desired that an antenna increase the amount of radiation in particular directions and minimise
it in others. A measure of the ability of an antenna to do this is its directivity. The directivity 12 D(,) of an
antenna in direction (,) is officially defined as
Radiation Intensity in Direction ( , )
( , )
Spatially Averaged Radiation Intensity
D
 
   (2.2-29)
Hence it follows that we can express directivity as
( )
( )
rad
4 U ,
D , =
P
  
  (2.2-30)
where
rad
P the total power radiated by the antenna, as described in Section 2.2.1. Directivity is typically
expressed in deciBels, namely
  ( , ) 10log ( , )
dBi
D D      (2.2-31)
It is said to be in dBi, the “i” standing for “isotropic radiator”. An isotropic radiator is a non-physical radiator
that radiates equally well in all directions. If it were physically possible to make such an antenna its directivity
would be 1, and so we imagine any directivity as being measured relative to it.
Due to (2.2-16) and (2.2-17), the expression for the power radiated by an antenna can be shown to be
2
2
rad
0 0
1
( , ) sin
2
o
P U r d d
 
    



(2.2-32)
12 Also called the total directivity.
15
If we write
2
( , )
( , )
2
o
E
U


 
 

 (2.2-33)
and
2
( , )
( , )
2
o
E
U


 
 

 (2.2-34)
we can define associated partial directivities
( )
( )
rad
4 U ,
D , =
P


  
  (2.2-35)
and
( )
( )
rad
4 U ,
D , =
P


  
  (2.2-36)
Due to the way these are defined, we of course have
( ) ( ) ( ) D , = D , D ,
 
       (2.2-37)
and also
 
2
0 0
( , ) ( , ) sin 4 D D d d
 
 
         
 
(2.2-38)
B. Gain
The gain of an antenna is related to the directivity as
rad
( , ) ( , ) G D       (2.2-39)
The realized gain of the antenna is defined as
 
2
rad
( , ) 1 ( , )
realized in
G D         (2.2-40)
Similarly, we have the partial gains
rad
( , ) ( , ) G D
 
      and
rad
( , ) ( , ) G D
 
      , and the associated
realized partial gains of the antenna are defined as 

2
rad
1 ( , )
in
D       and 

2
rad
1 ( , )
in
D       .
16
2.2.8 Mutual Coupling Between Two Antennas
Any two antennas can be regarded as a two-port network, in terms of voltages and current defined at the ports
of the two antennas, as indicated in Fig.2.2-3. If we number the ports as #1 and #2 as shown, these are related
as
1 11 1 12 2
V =Z I Z I  (2.2-41)
and
2 21 1 22 2
V =Z I Z I  (2.2-42)
These apply no matter how the 2-port network is excited. The access transmission lines are only shown for
emphasis. The characteristic impedances
01
Z and
02
Z are assumed to be real. The elements
ij
Z of the
impedance matrix [ ] Z of the two-port are defined by the relations
2
1
11
1
0 I
V
Z
I


1
1
12
2
0 I
V
Z
I


2
2
21
1
0 I
V
Z
I


1
2
22
2
0 I
V
Z
I

 (2.2-43)
The input impedance of antenna#1 in the presence of antenna#2 is
2
(1)
1
in
1
L
Z
V
Z
I
 (2.2-44)
where
2 L
Z is the load placed at the terminals of antenna#2, and similarly
1
(2)
2
in
2
L
Z
V
Z
I
 (2.2-45)
Normally,
1 L
Z and
2 L
Z are the reference impedances. Impedance matrix [ ] Z is related to the scattering
matrix [ ] S of the two-port by
17
11 01 22 02 12 21
11
11 01 22 02 12 21
( )( )
( )( )
Z Z Z Z Z Z
S
Z Z Z Z Z Z
  

  
12 01 02
12
11 01 22 02 12 21
2
( )( )
Z Z Z
S
Z Z Z Z Z Z

  
(2.2-46)
21 01 02
21
11 01 22 02 12 21
2
( )( )
Z Z Z
S
Z Z Z Z Z Z

  
11 01 22 02 12 21
22
11 01 22 02 12 21
( )( )
( )( )
Z Z Z Z Z Z
S
Z Z Z Z Z Z
  

  
Fig.2.2-3 : Two-port network representation of two coupled antennas (After [MCNA17])
2.2.9 Mean Effective Gain (MEG)
The sensors we will refer to in Section 4.1 will, like mobile devices, usually be surrounded by scattering
objects that are physically spread over a wide range of angles around the device. The question that arose in
early wireless systems work was what value of antenna gain was to be used when calculating link budgets in
such circumstances. The concept of a mean effective gain (MEG) was therefore introduced and defined in
[TAGA90] to be
2
0 0
1
( , ) ( , ) ( , ) ( , ) sin
1 1
XPR
MEG G G d d
XPR XPR
 
   
          
 
 
 
 
 
 
P P (2.2-47)
Terms ( , ) G    and ( , ) G    are the partial gains defined in Section 2.2.7. The other terms will be
explained below. Clearly, if we wish to show radiation efficiency explicitly, this can be written as
2
rad
0 0
1
( , ) ( , ) ( , ) ( , ) sin
1 1
XPR
MEG D D d d
XPR XPR
 
   
           
 
 
 
 
 
 
P P (2.2-48)
18
Some authors interpret ( , ) G    and ( , ) G    as the realized partial gains 13 of the antenna, and in this case
we would need to write (2.2-48)
 
2
2
rad
0 0
1
1 ( , ) ( , ) ( , ) ( , ) sin
1 1
in
XPR
MEG D D d d
XPR XPR
 
   
           
 
   
 
 
 
 
P P (2.2-49)
Of course, if the antenna is well matched then the ‘mismatch loss’
2
1
in
  will be almost unity anyway.
The terms in the expressions for the MEG are discussed in [KILD 07], [SAUN 08], [HUAN 08], [DOUG 98]
and [KALL 02]. These can be summarized as follows:
■ XPR is known as the cross-polarisation 14 ratio of the wireless environment (or channel). It is
determined/estimated as follows : The received signal level in two orthogonal polarisations (eg. -  and
-polarisations  ) are measured as the receive probe moves in the environment over random paths, and then
averaged. The ratio (usually of the vertical over the horizontal 15 polarisation levels) is the XPR. This is
determined experimentally. [KALL 02] reports that the value of XPR has been found to range between 0dB
and 9dB, depending on the particular environment (eg. indoor, suburban outdoor, urban, dense urban,
medium-density urban, and so on).
■ ( , )

  P and ( , )

  P are known as the angular power density functions, probability distributions of the
angle-of-arrivals (AoA) of the incident waves, or the power angular spectrum (PAS). They are a description
of the probability that an incident signal 16 will be incident on the antenna (whose MEG is being determined)
from direction ( , )   . They must satisfy the conditions
2
0 0
( , )sin 1 d d
 

     
  P
(2.2-50)
and
2
0 0
( , )sin 1 d d
 

     
  P
(2.2-51)
13 Defined in Section 2.2.7.
14 Unrelated to the antenna whose MEG is being calculated.
15 This will be the
-polarised  over the -polarised  levels with the coordinate systems defined as we do here, with the z-axis
vertical.
16 That is
-polarised  or -polarised  , respectively.
19
These functions are either established through measurement campaigns, or are conjectured from arguments
based on wireless systems experience. Examples given in [KALL 02] include the Gaussian function and
double exponential functions (of which a special case is the so-called Laplacian function [PEDE 97]).
■ The MEG clearly depends on the far-zone patterns of the antenna, the orientation of the antenna, its
polarization, the AoA distributions of the wireless environment, and the XPR of the environment.
■ In an environment where there is equal probability of a wave to be incoming from any direction 17 with any
polarisation, we have ( , ) 1/4

    P , ( , ) 1/ 4

    P and XPR 1  . The integral in (2.2-49) then
evaluates to one-half, and we have
 
2
rad
0.5 1
in
MEG     , or in other words MEG is equal to half the
‘total’ radiation efficiency. The maximum value of MEG is
rad
4  [GLAZ 09].
■ Some [VOLA 07] have defined the mean effective directivity (MED) as
 
2
rad
0.5 1
in
MEG MED     ,
because MED then only depends on the environment and the radiation pattern.
2.2.10 Performance Metric of Particular Importance for Electrically Small Antennas
We first defined the term “electrically small antenna” in Section 2.2.2. Much theoretical and practical work
has been reported on the topic; interest in the subject has increased over the past 15 years because antennas
need to be used on physically small mobile communications devices. The works [YAGH 05] and [SIEV 12]
provide a particularly concise and to-the-point examination of what is important on the subject 18 , and has
been used as a guide in this work.
A. Antenna Quality Factor (Q)
The narrow bandwidth of antennas that are electrically small comes as a surprise when one has only had
experience with antennas that are electrically much larger, but it is fundamental. As [COLL 19] relates : “It
is very disappointing to measure a complete device 19 , contained in a smart, expensively-designed case, that
has an efficiency 20 of only 15%, but this is not unusual”. This narrow bandwidth has resulted in engineers
viewing such antennas as resonators (which they are, albeit ones where the portion of the loss, due r
17 Sometimes (perhaps misleadingly, in the light of definitions used in antenna engineering) referred to as an “isotropic”
environment.
18 [SIEV 12] remains a yardstick in spite of having been published 7 years before the writing of this thesis.
19 Antenna on its host platform.
20 The total radiation efficiency defined in Section 2.2.6.
20
radiation, is desirable). And like resonators their bandwidth can be assessed from the value of their quality
factor (Q).
B. Bounds on the Q of an Antenna
A useful formula for a lower bound for the Q of an antenna is [SIEV 12], when the polarisation is
unconstrained, is
 
3
1 1
2( )
Q
ka
ka
   (2.2-52)
Here k is the free space wavenumber at frequency  , and a the radius of the minimum sphere enclosing the
antenna. If an antenna is a single-mode 21 one in the sense of characteristic modes, then [ETHI 14] the Q at a
self-resonance frequency of the antenna can be found from purely characteristic mode parameters as
 
1 1
cm
2 2
o
o
o o
o
f f
f d d
Q
d df
 
  




  (2.2-53)
C. Approximate Computation of the Quality Factor of an Actual Antenna
Yaghjian and Best [YAGH 05] derived an approximate expression for antenna quality factor that permits
this quantity to be found, at frequency
o
 , for an actual antenna from its input impedance properties as
 
         
 
2 2
/
2
o in o in o in o o
z o
in o
R X X
Q
R
    


   
 (2.2-54)
The expression is valid for single-resonance antennas in their resonant or anti-resonant regimes, and so are
valid for all frequencies. It allows the bandwidth of an antenna to be estimated from the frequency derivative
of the input impedance at the desired frequency. We will always denote the quality factor determined in this
manner by the symbol
z
Q , as has become common practice. The bandwidth for a voltage standing wave
ratio no larger than the value VSWR can be estimated as [YAGH 05]
21 This is defined in Section 2.6.3.
21
1 1 VSWR
B
Q VSWR
  

 
 
(2.2-55)
where of course
1
1
in
in
VSWR
 

 
(2.2-56)
2.3 ANTENNA ANALYSIS USING COMPUTATIONAL ELECTROMAGNETICS
2.3.1 Initial Remarks
In the present thesis there is just one CEM method that will be used, namely the method of moments (MM),
because it enables us both to find the characteristic modes (CMs) of an object, and can be used to model the
driven antenna. The CM concept will be reviewed in Section 2.4. But in order to do that properly we need to
set down set down some ideas on integral equation models for conducting antennas, and their solution using
the MM. Also mentioned in the thesis is the asymptotic method known as the geometrical theory of
diffraction (GTD), and so this will be also be touched on in the present section.
2.3.2 Electric Field Integral Equation Model
The most widely used integral equation in CEM is the electric-field integral equation (EFIE) that models
scattering from a perfectly conducting (PEC) object [PETE97], a situation sketched in Fig.2.3-1. We have an
arbitrarily shaped perfectly conducting (PEC) scatterer, situated in unbounded space of permittivity  ,
permeability  and zero conductivity ( 0   ). The surface of the scatterer is denoted
c
S . Although we
show a single scatterer, there could be more than one, with
c
S the union of the surfaces of the individual
scatterers. The EFIE can be written as
    tan tan
, , , , ,
scat fs inc imp imp fs
s c
E r J E r J M r S    G G (2.3-1)
■ The subscript “tan” on a quantity implies the component of the quantity tangential to the surface
c
S of the
PEC object.
22
■  
imp
J r and  
imp
M r are impressed electric and magnetic source current densities exterior to surface
c
S .
Recall that by impressed is meant that these are unaffected by the electromagnetic fields themselves, and that
if all the impressed sources were turned off there would be no fields whatsoever.
■ In the absence of the scatterer these sources generate fields  
inc
E r and  
inc
H r at all points r in free
space; these are of course the incident fields. Recall that an incident field is by definition that which exists
when the object identified as the scatterer is removed, and that this incident field exists at all points (even
within the scatterer).
■ Because of the incident field there will be an induced electric surface current density
s
J on
c
S . This will
give rise to additional electromagnetic fields, namely the scattered fields  
scat
E r and  
scat
H r .
■ The total field will at all observation points 22 r be the sum of the incident and scattered fields, namely
     
inc scat
E r E r E r   .
■
 
, , ,
inc imp imp fs
E r J M G is the incident electric field, which is a known quantity. It can be found using
the known impressed current densities and the free space Greens function (identified by the superscript “fs”).
■
 
, ,
scat fs
s s
E r J G is the tangential scattered electric field, and is an integral involving the unknown
electric surface current density
s
J on
c
S , and a free space Green’s function.
■ The boundary condition on a PEC object requires
           
ˆ ˆ 0
inc scat
c
n r E r n r E r E r r S       (2.3-2)
and results in the EFIE in (2.3-1)
22 Both inside and outside the PEC scatterer.
23
c
S
ˆ n
( ), ( ) E r H r
imp
J
imp
M
( , )  
( ) 0
( ) 0
E r
H r


PEC
Fig.2.3-1 : Scattering of the fields, from impressed sources, from a perfectly conducting (PEC) object.
2.3.3 Method of Moment (MoM) Solution of an Integral Equation (IE) 23
The method of moment is a numerical method for solving integral equations. The approach is best
appreciated by considering a simple one-dimensional (1D) integral equation and not the 3D one in (2.3-1).
The 1D integral equation can be written as
( ) ( ) ( )
U
L
z
L U
z
f z z,z dz = g z z z z     

G (2.3-3)
The qualifier
L U
z z z   means that the equation (2.3-3) applies only for points in space with z in the interval
L U
z z z   . The function ( ) f z is the unknown 24 . The ( ) g z is a known function 25 . The function ( , )
z z
G , is
called the kernel 26 of the integral equation, is also a known function, of two variables z and
z
. In such integral
equations we need to be able to evaluate ( ) f z and ( ) g z at different z values in the same equation. Thus we
need to use the primed and unprimed notation, with
L U
z z z    also. Note that the integration is with respect to
z
.
23 Portions of this section have been taken, with permission, from [MCNA 16].
24 Such as an induced current density.
25 Related to the incident field.
26 In the case of CEM-related integral equations this kernel will be some appropriate Green’s function.
24
In order to find an approximate solution for ( ) f z we first expand ( ) f z in terms of some selected (and hence
known) expansion functions ( )
n
f z ,
dof
N of them in total 27 , as
1
( ) ( )
N
n n
n
f z a f z

  (2.3-4)
Since ( ) f z is the unknown, the coefficients
n
a ,
dof
1,2,...., n N  are of course unknown at this stage. If we
can find them, then (2.3-4) constitutes an approximation to the solution. If we substitute the series expansion
(2.3-4) into the original integral equation (2.3-3) we get
dof
1
( ) ( , ) ( )
U
L
z N
n n
n
z
a f z z z dz g z

 
   

 
 
 
G (2.3-5)
The linearity of the integral operation next allows us to move the summation from under the integral sign,
giving
1
( )
( ) ( , ) ( )
U
L
z
N
n n
n
z
n
z
a f z z z dz g z


 
 
 
 
 
 
 
G (2.3-6)
Since the ( )
n
f
z
are known expansion functions, the z-dependent function ( )
n
z  can be evaluated 28 for any
value of z .
We next select a second set of
dof
N functions 29
dof
1 2
{ ( ), ( ),...., ( ),...., ( )}
m N
g z g z g z g z . We need to select
them and thus they are known. These are conventionally called the weighting or testing functions. We can then
multiply both sides of (2.3-6) by each of these weighting functions in turn, each time integrating the products on
the left and right hand sides with respect to z (as opposed to with respect to
z
) over the interval
L U
z z z   .
This process yields
27 The subscript in the symbol
dof
N denotes “degrees of freedom”.
28 If not in closed form, then at least numerically.
29 It is called a Galerkin Method of Moment approach if
( ) ( )
n n
g z f z  for all
dof
1,2,...., n N  .
25
dof
1
dof
( ) ( ) ( ) ( )
1,2,.....,
U U
L L
z z N
n n m m
n
z z
a z g z dz g z g z dz
m N

 
   
 
 

  
(2.3-7)
where
dof
1,2,....., m N  indicates that (2.3-7) holds for each m in turn. We can tidy (2.3-7) up to read
( ) ( )
U
L
z
mn n m
z
Z z g z dz  

(2.3-8)
and
( ) ( )
U
L
z
m m
z
b g z g z dz 

(2.3-9)
so that the set of linear equations represented by (2.3-7) can be written in matrix notation as
  
inc
Z I V   
 
(2.3-10)
where the known method of moment matrix is
 
















 
    
    
 
 

NN N N
N
N
Z Z Z
Z Z Z
Z Z Z
Z
2 1
2 22 21
1 12 11
(2.3-11)
The known column matrix (also referred to as the excitation matrix)
1
2
inc
N
b
b
V
b
 
 
 
 
   
 
 

 
 
 
(2.3-12)
while the column matrix
 

















 
N
a
a
a
I
2
1
(2.3-13)
26
is the matrix of unknown coefficients. Solution of the matrix equation (2.3-10) then yields [ ] I , and the
approximate solution is known through (2.3-4).
2.3.4 The Surface Patch Method of Moment Formulation (MoM) Solution of an Integral Equation
(IE)
Consider the PEC object shown in Fig.2.3-2. It has been meshed into triangular elements. A set of
expansion functions ( )
n
J r ,
dof
1,2,...., n N  can be defined over this surface, with each ( )
n
J r associated
with the n-th common edge between all pairs of triangles, as indicated in Fig.2.3-3. The expansion functions
were developed by Rao, Wilton and Glisson [RAO 82], and have thus become known as RWG expansion
functions. They are cleverly defined in sense that continuity of the current density component normal to any
edge is guaranteed, as is physically required. This is true even if the two triangles to which the common edge
belongs do not lie in the same plane. Note that the number of expansion functions
dof
N is not equal to the
number of triangles N  . Any triangle that is connected to three other triangles will have three expansion
functions defined on it, one for each of its three edges that are connector to the other triangles. An isolated
single triangle has no expansion function defined on it (because it has no edge in common with another
triangle) on it, and hence as far as the CEM analysis is concerned does not exist 30 . Details of the formulation
can be found in [RAO 82] and [JOHN 90]. Higher-order versions of the RWG expansion functions have also
been developed [GRAG 97]. These RWG expansion functions can be used [PETE 97] in a MM formulation
to discretize the EFIE in (2.3-1) to a matrix equation 31 of the form (2.3-10) that can be solved for coefficients
n
a ,
dof
1,2,...., n N  (in other words, column vector [ ] I ) and hence the current density
dof
1
( ) ( )
N
s n n
n
J r a J r

  (2.3-14)
This current density can be used to find all the associated electromagnetic fields. In the case of the EFIE the
expression (2.3-8) for the terms in the MM impedance matrix [ ] Z becomes
 
( ) ,
m
fs
mn o m n n
S
Z C J r E r J dS  

,G (2.3-15)
30 More will be said about this in Section 3.10.
31 Column vector [ ] I
is also often written as [ ] J .
27
Quantity
 
,
fs
n n
E r J ,G is 32 the field due to expansion function ( )
n
J r , obtained using the free-space
Green’s function
fs
G ; the integration is over the region
m
S occupied by the m-th expansion function;
o
C is
a known constant. In essence,
mn
Z can be interpreted is the mutual impedance between the m-th and n-th
expansion functions. The impedance matrix [ ] Z will be of size
dof dof
N N  . The number of degrees of
freedom
dof
N must be large enough so that (2.3-14) is an accurate representation of the true ( )
s
J r . This
means that there must be a sufficient number of triangles in the mesh 33 . Another way of stating this is that
dimensions of the sides of the triangles must be small enough in terms of wavelength. Most commercial
codes based on the MM use RWG expansion functions.
Fig.2.3-2 : PEC object (cubic bottomless rectangular box) meshed into triangular elements.
32 Its 1D equivalent is
( )
n
z  in expression (2.3-6).
33 The mesh must be fine (or dense) enough.
28
Fig.2.3-3 : Four of the triangles from the mesh in Fig.2.3-2, shown exaggerated in relative size.
2.3.5 Hybrid Moment-Method / Geometrical-Theory-of-Diffraction (MM-GTD) Methods
If a scattering object is electrically very large, then the field scattered from it can be found in terms of
rays 34 that reflect or diffract from key features of the object’s geometry. The details are beyond the scope of
this review chapter, but can be found in [MCNA 91] and [PAKN 16], and are known as the geometrical
theory of diffraction (GTD). Often the overall electrical size of the PEC object, such as the situation shown
in Fig.2.3-4, is such that
dof
N would need to be too large for feasible computation. In such cases a hybrid
approach can be used, as will next be briefly outlined. The original PEC object is meshed as usual with
dof
N
expansion functions, but the very large additional object is not. The impedance matrix is still of size
dof dof
N N  . However, the presence of the very large object will alter the values of the matrix terms
mn
Z
from what they would be if the very large object were not present. This perturbation of each matrix element
is found using the GTD. The term
 
,
fs
n n
E r J ,G in (2.3-5) is altered by adding the term
 
,
GTD n
E r J to it,
as we try to illustrate in Fig.2.3-5. The pioneering work is described in [EKEL 80], with a subsequent review
34 Ray-based methods are only accurate when the scattering objects are large in terms of wavelength.
29
provided in [MEDG 91]. More recent improvements have been described in [THER 00], [JAKO 10] and
[JAKO 97].
Fig.2.3-4 : “Original” PEC object from Fig.2.3-2, but now in the presence of an electrically very large
additional PEC object.
Fig.2.3-5 : Perturbation of the values of the MM matrix accounted for using the GTD.
30
2.4 CHARACTERISTIC MODES
2.4.1 Introductory Remarks
The theory of characteristic modes was first introduced by Garbacz [GARB 65], but their determination for
conducting objects was reformulated by Harrington and Mautz [HARR 71a] in terms of integral equations. It
was the last-mentioned work that made it possible to compute the characteristic modes (CMs) of an arbitrary
conducting object through the numerical solution of an integral equation using the method of moments
[HARR 71b]. A few years after its introduction, a means of computing the CMs of penetrable objects
[HARR72] [CHAN77] was published. This work is now also described in a recent textbook [CHEN 15b]. In
this thesis we are concerned with antennas composed of conducting material only, and thus the CM
formulations reviewed below are for this case only.
2.4.2 Conventional 35 Characteristic Modes of Conducting Objects
As stated in Section 2.3, the EFIE modelling of a conducting object 36 is most often a driven problem, with
an impressed source (responsible for the incident field) specified. This allows us to solve for the resulting
induced current density on the PEC object using (2.3-1). Characteristic mode analysis on the other hand
involves an eigenvalue problem, independent of any excitation, and provides information dependent on the
scattering object itself. This eigenvalue problem is [HARR 71b]
[ ][ ] [ ][ ]
n n n
X J R J   (2.4-1)
where
n
 and [ ]
n
J are the CM eigenvalues and eigenvectors (eigencurrents), respectively. The matrices [ ] R
and [ ] X are defined via [ ] [ ] [ ] Z R j X   . The [ ] Z is the MM matrix from Section 2.3.4. Note that [ ]
n
J is a
column vector of length
dof
N ; the subscript “n” identifies the mode and not an element in the column
vector. Both [ ] R and [ ] X must be symmetric [HARR 71a] in order to obtain real eigenvalues and
eigenvectors for a lossless (eg. PEC) case, and so a Galerkin MM approach 37 must be used. The eigenvalues
can be shown to be the ratio of the reactive power to the radiated power in the modes. So, an eigenvalue is
equal to zero when the associated mode is at resonance, and equal to infinity (or large number) when the
35 As opposed to the sub-structure characteristic modes to be described in Section 2.4.3, and of late also being referred to as full-
structure characteristic modes [XIAN 19].
36 Which will henceforth, at least as far as CM analyses are concerned, be assumed to be a perfectly conducting (PEC) object.
37 Defined in Section 2.3.3.
31
mode is non-radiating (reactive). The eigenvalues can be negative or positive indicating a capacitive or
inductive mode. The CM eigencurrents are orthogonal in the following sense [HARR 71b]:
*
[ ] [ ][ ] [ ] [ ][ ]
T T
m n m n mn
J R J J R J    (2.4.2)
*
[ ] [ ][ ] [ ] [ ][ ]
T T
m n m n n mn
J X J J X J     (2.4-3)
and
 
*
[ ] [ ][ ] [ ] [ ][ ] 1
T T
m n m n n mn
J Z J J Z J j      (2.4-4)
where
mn
 is the Kronecker delta,
0
1
mn
m n
m n

 
 


(2.4-5)
Note that the currents are normalized so that individual modes radiate a power of 1 Watt. The CM fields in
the far zone (i.e. over the sphere at infinity) are orthogonal in the following sense [HARR 71a]:
0
1
= ( , ) ( , ) sin
mn m n mn
S
C E E d d        



 

(2.4-6)
and
0
1
( , ) ( , ) sin
m n mn
S
H H d d        



 

(2.4-7)
where
0
/
o o
    is the free-space intrinsic impedance, and ( , )
m
E   and ( , )
m
H   are the far-zone
eigenfields (with the distance dependent term /
jkr
e r

suppressed) due to eigencurrent
m
J . The expressions
in (2.4-2) through (2.4-4), (2.4-6), and (2.4-7), are referred as the current and field orthogonality
relationships. The modal weighting coefficient
n
 of the n-th mode, is found as [HARR 71a]
tan
,
1 1
i
i
n
n
n
n n
J E
V
j j

 
 
 
(2.4-8)
32
In (2.4-8) the terms
i
n
V are seen to be as the inner product of the CM eigencurrent with the incident field due
to the impressed source in the absence of the PEC object. It allows one to determine “how much” of each of
the characteristic modes is excited for a given excitation.
Due to the orthogonality properties, any arbitrary surface current
s
J and its fields ( E , H ) can be expressed
as a linear sum of the CM eigencurrents
n
J and eigenfields (
n
E ,
n
H ) of the object, namely
0
( ) ( )
s n n
n
J r J r 


  (2.4-9)
1
( ) ( )
n n
n
E r E r 


  (2.4-10)
and
1
( ) ( )
n n
n
H r H r 


  (2.4-11)
The quantity
1
1 


n
MS
j
(2.4-12)
is called the modal significance factor, the term contained within the modulus signs also occurring in
expression (2.4-8). This quantity will lie between zero and unity; if a CM is resonant (has zero eigenvalue)
then 1 MS  .
2.4.3 Sub-Structure Characteristic Modes
The sub-structure CM idea for conducting objects was introduced in [ETHI 12] and in more general form
in [ALRO 14] and [ALRO 16]. It is next described in the form given in the latter reference.
33
a
S
imp
J
imp
M
0 0
( , )  
PEC
Object A
PEC
Object B
b
S
Fig.2.4-1 : Two objects considered for the sub-structure CM analysis.
Consider the PEC scatterer shown in Fig.2.4-1. We recognize that it is possible to purposefully view the
structure in Fig.2.4-1 as consisting of two portions, Objects#A and #B, with distinct expansion function
subsets located on these two portions. We can then partition the method of moment matrix equation to read
[ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ]
AA AB A A
BA BB B B
Z Z J V
Z Z J V
    

    
    
(2.4-13)
This does not alter the scattering problem in any way. We will call (2.4-13) the primary MM formulation
equation. It is possible to find the conventional CMs of the complete structure as the solutions of the matrix
eigenvalue problem (2.4-1).
We can next define the general sub-structure CM concept. In order to do this we manipulate matrix
equation (2.4-13) into the two equations
 
1
1
AA AB BB BA A AB BB B
[ ] [ ][ ] [ ] [ ] [ ] [ ]
A
Z Z Z Z J V Z Z V


      
    (2.4-14)
and
   
1 1
BB BA BB B B A
J Z Z J Z V
 
          
        (2.4-15)
In (2.4-14), we can write
1
AA AB BB BA sub
[ ] [ ][ ] [ ] [ ] Z Z Z Z Z

  (2.4-16)
34
and obtain the sub-structure CMs of Object#A in the presence of Object#B from eigenvalue problem
[ ][ ] [ ][ ]
Asub sub Asub
sub n n sub n
X J R J   (2.4-17)
with sub-structure eigenvalues
sub
n
 different from those
n
 of the conventional CMs, and
[ ] [ ] [ ]
sub sub sub
Z R j X   . Note that the set of
sub
n
 are not a subset of the set of
n
 .
The [ ]
Asub
n
J are the sub-structure eigencurrents on Object#A in the presence of Object#B. In order to find
the eigenfields due to sub-structure eigencurrents [ ]
Asub
n
J we must first find the “secondary” currents
sec
[ ]
B
n
J
on Object#B. This is a separate driven problem. We must view [ ]
Asub
n
J as an impressed current density (with
Object#A absent) that illuminates Object#B, and use a MM formulation (which we will call the secondary
method of moment formulation) to find
sec
[ ]
B
n
J . The elements of [ ]
BA
Z in the primary formulation are of the
form
 
, ( ) ( )dS
B
ij j i
S
Z E r J r J r  

. Here   , ( )
j
E r J r would be the electric field at r due to the j-th
expansion function ( )
j
J r , which is on Object#A, whereas ( )
i
J r is the i-th weighting function, which is on
Object#B. Now the matrix equation for the secondary problem would be of the form
sec sec
BB
[ ] [V ]
B
n B
Z J   
 
(2.4-18)
We write
sec
[V ]
B
because it is the excitation with
Asub
n
J as the impressed current, and is not the same as in
primary formulation (2.4-13). If we use the same expansion functions on Object#B for both the primary and
secondary problems then
BB
Z  
  is the same in the two problems. The elements of
sec
[V ]
B
in the secondary
formulation will be of the form
 
, ( ) ( )dS
B
inc Asub
n i
S
E r J r J r  

, where   , ( )
inc Asub
n
E r J r would be the
electric field at r due to
Asub
n
J viewed as an impressed current density. Of course, we have
( )
Asub
n
J r described in the form of expansion functions and a set of coefficients. Thus the expression for
sec
[V ]
B
in fact reduces to
35
sec Asub
BA
[V ] [ ]
B n
Z J     

(2.4-19)
If we substitute this into (2.4-18) the latter becomes
ec Asub
BB BA
[ ] [ ]
B
n n
Z J Z J      
   
(2.4-20)
which can be rewritten as
1
Bsec Asub
BB BA
[ ] [ ]
n n
J Z Z J

      
  
(2.4-21)
Thus it turns out that we already have all the matrix information needed for the secondary method of
moment formulation from the primary formulation. Expression (2.4-15) just happens to be the same as (2.4-
21) with
B
[0] V 
 
 
, but has nothing to do with
B
V  
  or
A
V  
  of the primary problem. The sub-structure
CMs do not depend on
B
V  
  or
A
V  
  in any way, as expected because of the fact that CM formulations are
eigenvalue and not driven problems. The sub-structure CM field
n
E is the superposition of the fields due to
Asub
n
J and
sec B
n
J . It is shown in [ALRO 16] that the sub-structure CMs also satisfy orthogonality conditions
on the current densities and fields; the reader is referred to that reference for a catalogue of these properties.
Suppose we knew the modified Green’s function which can be used to determine the field in all of space
for an arbitrary electric current density distribution in the presence of Object#B, and used it to derive an
EFIE for the current density on Object#A. In the associated method of moment formulation the expressions
for the impedance matrix and excitation vector now involve the appropriate modified Green’s function and
so are not the same as when the free-space Green’s function is used. It would provide the same sub-structure
CMs as those described in the paragraphs above. However, such modified Green’s functions are usually not
known, except in idealised cases such as when Object#B is an infinite PEC groundplane, of infinitely long
PEC circular cylinder, and so on. The route described in the paragraphs above effectively uses the modified
Green’s function without explicitly needing it.
36
2.4.4 Applications of Characteristic Modes in Antenna Design
Computational electromagnetics provides superb tools for antenna analysis. The source densities and
electromagnetic fields from such modelling incorporate all the physics associated with a particular
configuration. Several authors have shown how the use of a CM analysis of a structure allows one to exploit
this information to the full, and have used it in systematized antenna design.
Characteristic modes can be used in two ways in antenna design. Most authors use them to “understand”
the antenna design problem being tackled and then use the insight gained to come up with a good design;
this might be called the indirect approach. Another way is more direct, namely the use of CMs in the
automated shaping of the antenna. In this thesis we are interested in the latter shape synthesis, and will
extend the range application of the direct approach, and will review the topic of shape synthesis (even those
that do not use CMs) in much more detail in Section 2.6. Suffice it to say here that a representative cross-
section of the indirect use of CMs in antenna design includes :
■ Tuning and bandwidth enhancement of electrically small antennas [ADAM 11]
■ The excitation of chassis modes as antennas for small terminals [MART 11][WU 13].
■ MIMO antennas [ETHI 08][DENG 16][MART 16][LI 14][MIER 13].
■ Reconfigurable MIMO antennas that utilise chassis modes [KISH 13].
■ Coplanar-fed dielectric resonator antennas [BERN 15].
■ Shipboard antennas [CHEN 15a].
■ Slotted patch antennas [CHEN 12].
■ UWB antennas [ANTO 10].
■ Wearable antenna design [YAN 18]
■ UHF RFID antennas on tin cans [SHAR 19]
The above applications, and many more, have steadily increased interest in characteristic modes, and hence
their accurate computation [CAPE 17][DAI 16].
37
2.5 A PRIMER ON OPTIMISATION ALGORITHMS
2.5.1 Classification of Optimisation Algorithms
An optimisation algorithm is one that tries to find the minimum (or maximum) of some function
obj
F ,
called the objective function, fitness function or cost function, with respect to several independent variables.
These algorithms are, in the numerical mathematics literature, described as performing optimisation without
constraints (unconstrained optimisation) or with constraints (constrained optimisation). In antenna work the
constraints often do not fit into a neat mathematical framework, and so unconstrained optimisation
algorithms are used with the constraints built into the objective functions themselves. This is especially so
for evolutionary algorithms.
2.5.2 Gradient Based Algorithms
Gradient based optimization algorithms are essentially greatly advanced versions of the technique we learn
in beginning calculus courses for finding the location of the minimum (or maximum) value of a given
mathematical function. In such courses we start with a real function ( ) f x of the single independent variable
x and wish to know at what x value (i.e. at what point) the function is a minimum. We recall that this
involves use of the derivative (and possibly second derivative) of the function with respect to x. When the
function to be maximized is dependent on several independent variables the more general gradient
operations are involved instead of a simple derivative. The now-multidimensional point is sought at which
the function (still a single real number) is a minimum. The various algorithms devised to do this differ in the
way we navigate in the multidimensional landscape to reach the desired point where the function is
maximized. Good tutorial accounts of these “gradient based” algorithms remain [BAND 69] and [BAND
88]. The objective functions used must of course be such that their gradients are defined and can be found.
2.5.3 Evolutionary Algorithms
Optimization algorithms are available that do not use gradients [RAHM 12]. This has made them very
flexible indeed. They go by interesting names (that attempt to hint at how they work) such as particle swarm
optimization, differential evolution [OLIV 12][ROCC 11], and genetic algorithms [JOHN 97][MITC 99].
Only the genetic algorithm (GA) will be used in this thesis, and so it will be reviewed immediately below.
Some other evolutionary algorithm could be used in place of the GA in the shape synthesis tool, but this will
38
not be done. Optimization algorithms are available already implemented in commercial software, and their
development is not considered a contribution in this thesis 38 .
2.5.4 Further Detail on a Specific Evolutionary Algorithm : The Genetic Algorithm
A. Aim of Sub-Section
In the shape synthesis tool that is the subject of Chapter 3 we will use the genetic algorithm (GA)
exclusively, and so it will be discussed in some detail. However, this will be done in the context in which it
is used in this thesis, namely in relation to the shaping of conducting surface antennas.
B. Pixelation of a Conducting Surface
Consider a fictitious planar surface that has been divided into a mesh of
bit
N elements 39 , as shown in
Fig.2.5-1. Each element can be filled with conductor or left unfilled 40 . We can associate a 1 or a 0 with each
element, a 1 if it is filled with conductor and a 0 if it is not. We can define a chromosome that consists of a
string of
bit
N genes (or bits), for example {110011000110011............0100} , and map each chromosome bit
to a different element in the meshed geometry. Thus by altering the binary values of each bit a different
conducting geometry can be defined. Another way of thinking of it is that each shape is described by a bar-
code. An objective (or fitness) function
obj
F is defined that is a function of the binary value of each gene
constituting the chromosome. In other words, it can be considered a function whose indpendent variables are
the bits in the chromosome. Thus it is a function of the shape of the conductor. The definition of
obj
F is
naturally problem-specific. The operation of the GA will next be explained in outline, emphasizing those
aspects that will need to be referred to when discussing the development of the new antenna shape synthesis
tool in Chapter 3.
38 They are merely used as a building block in the actual contribution, namely the extension of shaping procedures to include 3D
conducting surfaces (Chapter 4) and closely-spaced antennas (Chapter 5)
39 In the case shown these are rectangular. This will not be so in the shape synthesis tool developed in Chapter 3; reasons will be
given there.
40 Each cell is actually filled completely with conductor or has no conductor at all. The gaps shown in the diagram are for clarity of
explanation only.
39
Fig.2.5-1 : A demonstration of pixelization of a geometry. Those pixels (or elements) occupied by conductor
are sketched in black; unfilled elements are left clear.
C. Outline of the Genetic Algorithm
■ Initialisation
All optimization algorithms have to have a starting point. In the GA this is a set of possible, but far from
final, solutions (called the initial population of chromosomes) that are generated randomly. The number of
members in the population is called the population size
pop
N . The value of
pop
N has to be chosen. The
larger the value of
pop
N the better the chances of the optimization scheme actually allowing
obj
F to reach its
physically possible maximum value. This can be viewed as allowing enough genetic diversity to exist in the
population. There is always a trade-off though. For instance, in antenna work the evaluation of
obj
F for each
population member usually needs a full-wave electromagnetic analysis. This is computationally time-
consuming in the present context, since such an analysis will be needed for each member of each new
population as the GA iterates (that is, for each new generation).
■ Selection
Once a population is created, members of the population must be evaluated according to the fitness
function. As stated above, it takes the conductor geometry of each member, and outputs a numerical value
for the associated
obj
F that allows each member to be ranked relative to the other members. Once the fitness
of all members has been computed, a portion (call it
select pop
N N  for convenience) of the “top-scoring”
members is selected. Many sophisticated methods of selection 41 have been developed (eg. fitness
41 There are seven possibilities in the particular GA whose choice we discuss in Section 3.3.
40
proportionate selection; tournament selection) for deciding which population members will be mated. These
“elite” population members are then mated, and generate offspring that share characteristics with their
parents, and that are subsequently integrated into the population. One can keep the population size
pop
N constant (in which case the lowest fitness members not selected for mating are removed), or a more
complicated variable population size can be used in which some of the lowest fitness members are retained
but simply not allowed to mate. The hope is that a gradual improvement in the fitness of the members of the
population will occur as the GA iterates from generation to generation – in other words that we will find the
chromosome that gives
obj
F its minimum (or maximum) value.
■ Mating Operators
These could be called mating operators, and include the operations of crossover, mutation and migration. As
previously stated, the population members that are selected for mating (crossover) produce offspring that
contain a mixture of the parents’ qualities. Crossover 42 is the process that combines two individuals (parents)
in the population to form a new individual (child) for the next generation. Treatises have been written on
how crossover may be accomplished, the particular GA whose choice we discuss in Section 3.3 allows eight
possibilities. In order to prevent the optimization algorithm from quickly becoming trapped in local extrema,
new genetic material must be introduced into the generation. This is done using mutation and migration.
Mutation 43 changes a small portion of the offspring 44 so that they no longer perfectly reflect subsets of the
parents’ genes. The chance of a particular offspring undergoing a mutation, as well as the severity of the
mutation, is governed by a probability distribution. Genetic diversity can also be increased using a migration
operator (that makes small random changes), whose name at once allows us to grasp its function. Migration
can be forward, backward, or both.
■ Termination
The algorithm must eventually be halted. There are two cases in which this usually occurs: either the
algorithm has reached some specified maximum number of generations (say
gen
N ), or the value of
obj
F is
ound to not improve significantly from one generation to the next (eg.
obj
F changes in value by an amount
less than Tol .
42 In essence the “crossover” point in the chromosome that decides which bits will be altered to produce a new chromosome.
43 It too has been studied extensively by numerical analysts. In the particular GA whose choice we discuss in Section 3.3 there are
seven different ways of achieving mutation that one can decide on.
44 By flipping some of the bits in the chromosome.
41
2.6 EXISTING ANTENNA SHAPE SYNTHESIS TECHNIQUES
2.6.1 Introduction
The classification of the way optimisation is used in antenna design has already been mentioned in Section
1.1. In this thesis we are solely interested in the shape optimisation (or shape synthesis) of antennas that are
electrically small or possibly of intermediate size 45 . Reasons for the practical interest in the latter type of
antenna will be presented in the introductory sections of Chapters 4 and 5.
2.6.2 Existing Shape Synthesis Methods Using Driven Problem Simulation
In order to use optimization algorithms for the shape synthesis of any antenna it is necessary to translate
the design goals into a quantity that, when minimized, provides an antenna that has the performance we
desire. The most obvious way to define an objective function in antenna work would be to use some well-
chosen combination of the input reflection coefficient, the directivity, radiation efficiency, and so on, as
defined in Section 2.2. However, such quantities can only be found (during the shaping process) if the
complete antenna is modelled, as a driven problem. This implies that the type and location of the feed has to
be selected before the shaping process. Most previous work on shape synthesis has adopted this route. Such
shape optimization of low gain microstrip [JOHN 99a][JOHN 99b][THOR 05], and wire antennas [YAMA
09], with only the conductor geometry undergoing shaping through the removal or retention of conducting
pixels/segments into which the starting shape is divided, has allowed miniaturization and bandwidth
increases; the structure planarity is preserved, and so this can be described as two-dimensional shaping.
Fully three-dimensional shaping of thin-wire antennas is performed in [ALTS 02][LIND 99][CHOO 05],
with some restrictions placed on the allowed occupied volume to maintain a sense of practicality. The above
all used the GA as the optimisation method. More recently [MIRH 16] other evolutionary algorithms have
been tried in place of the GA in the pixelisation-based shaping; it works but is not more effective than when
the GA is used. Binary differential evolution has also been used as a replacement for the GA, but has only
been applied [GOUD 17] to relatively simple antenna shaping problems already satisfactorily tackled with
the GA in place.
One alternative to the pixelisation approach used so-called topology optimization (gradient based), based
on methods used in structural engineering, and is discussed in [HASS 14] and [EREN 11]. It allows
45 The shaping of electrically large antennas with high-directivity antennas are not of interest here. This includes the shaping of
large reflector surfaces that form a component of a completely different class of antenna, requiring a separate feed antenna to
illuminate the reflector, which together with it form the actual antenna.
42
extremely fine resolution, as is apparently required in its structural application, but it is not clear whether
such resolution is really significant in antenna problems. Another non-pixelisation route uses gradient based
optimisation to shape antennas whose geometry is defined in terms of splines [GRIF 06][TOIV 10][SALU
18]; the spline parameters are adjusted during the optimisation process. With the exception of [EREN 11],
all the shape synthesis work for conducting surface antennas (as opposed to thin wires) that we have
surveyed, such as that reviewed above, has been applied to planar shapes only.
2.6.3 Existing Shape Synthesis Methods Using Characteristic Mode Approaches
A method for the shape synthesis of single electrically-small antennas, unique in the sense that the feed
location need only be selected after the shaping has been performed, was described in [ETHI 14]. A related
route was used in [YANG 19] for MIMO antennas, and has been termed a shape-first, feed-next approach.
This work applies to electrically small or intermediate-size antennas that are “single-mode” in the sense that
only one dominant CM is sufficient to describe the behaviour of the antenna. If the antenna proper radiates
in the presence of other objects (that need not be electrically small) the approach of [ETHI 14] can still be
used, except that the sub-structure CMs of the antenna proper (with the other objects in place) must be used.
The antenna proper is then of an electrical size that makes it “single mode” in the sense of its sub-structure
CMs [ETHI 12][ALRO 16]. The reason the antenna must be single-mode in a CM sense is that the method
uses CMs to construct the objective functions whose minimization will constitute the shaping process; this is
the key reason it is not necessary to select the feedpoint location before shaping but only after shaping using
information provided by the CMs of the shaped structure(s). The pixelisation of Section 2.5.4 and 2.6.2 is still
used, except that it uses an eigenvalue analysis to evaluate objective functions and not the output of driven
electromagnetic analyses. All the shape-first feed-next approach has been for planar conducting surface
antennas. In this thesis we extend it to fully 3D conducting surface antennas, and to the shaping of two
closely-spaced antennas operating at different frequencies.
2.6.4 Computational Considerations
All of the above-mentioned shape synthesis methods need to utilize full-wave techniques of computational
electromagnetics. If one remembers that during shape optimization such full-wave analysis needs to be done
repeatedly, and possibly at several different frequencies each time, it becomes clear that this process will be
computationally demanding, in spite of ever-improving access to powerful numerical processors. The burden
of repeated full-wave analysis can often be relieved somewhat through the use of neural network modelling
43
or space-mapping. However, these are better suited to size optimization 46 , where the number of optimization
variables is greatly reduced compared to the
bit
N (in the hundreds) of them when using the shape
optimization intended here. If the number of variables is larger than roughly 50, then the use of conventional
neural networks has been reported to become problematic. Also, for the large number of variables being
used here, it becomes difficult to ensure that the coarse model (in space mapping) faithfully “follows” the
fine model with respect to all such variables. As a result, the so-called direct matrix method (DMM) was
developed in [JOHN99a] and [JOHN 99b] to speed up computation. The DMM can be explained as follows:
Suppose we have an EFIE/MM model for any of the conductor shapes that might be constructed out of cells
as indicated in Fig.2.5-1. In the DMM the Z-matrix for the object is constructed for the case when all cells
are filled with conductor (the starting shape), and expansion functions have been put in place over the entire
object. We will call this [ ]
start
Z . In successive iterations of the shaping process different potential shapes,
obtained by removing conductor from some of the cells, are created in each generation. Each such shape will
have its own Z-matrix that is a reduced version of [ ]
start
Z ; in other words the reduced Z-matrix can be
constructed from elements already existing in [ ]
start
Z and so do not need to be re-computed. This saves
computation time. It of course requires careful manipulation of the Z-matrix. It will be used in the new
developments in Chapter 3.
46 Where they have resulted in quite superb design routes.
44
2.7 CONCLUDING REMARKS
We began this chapter by pledging the interest in this thesis work to be that of shape synthesizing antennas
that are electrically-small or of intermediate size. Section 2.2 defined the antenna performance measures that
will be needed in what follows. Sections 2.3 and 2.4 identified the computational electromagnetics
modelling concepts that will be used in the work. The shaping process always has some sort of optimizer at
its core, and so the genetic algorithm, the optimization algorithm of choice in this thesis, was briefly
described in Section 2.5.
Finally, Section 2.6 reviewed existing shape synthesis work on electrically small conducting antennas,
from which a number of conclusions can be drawn. Firstly, no shape synthesis method that does not
constrain the location of the feed-point prior to shaping has been available for fully three-dimensional (3D)
conducting surface antennas 47 . Chapter 4 applies a procedure to do this for the first time 48 . Chapter 3
develops the new computational tool needed to achieve this. Secondly, such shape synthesis methods have
not been applied to the problem of two closely-spaced antennas, operating at different frequencies. The
contribution of Chapter 5 fills this void. Chapter 6 does not undertake the devising of any new synthesis
method, but instead identifies and demonstrates a novel way of finding the sub-structure characteristic
modes of a conducting object in the presence of an electrically very large second object using a hybrid
MM/GTD scheme.
47 As noted in Section 2.6, by calling them thus, we distinguish them from conducting thin-wire antennas.
48 Reasons why this might be of interest in practice are given in Section 4.1.
45
CHAPTER 3
Construction of a General Shape Synthesis Tool
3.1 PRELIMINARY REMARKS
A tool that applies the DMM technique 49 , and incorporates the characteristic mode route to allow a “shape-
first feed-later” approach to be used, for the shape synthesis of 3D conducting surface antennas, has not been
available. The development of such a tool was needed to achieve the goals of this thesis. It is the subject of
this chapter, and will be described in terms of the following essential elements :
■ Section 3.2 - Selection of the CEM Engine
■ Section 3.3 - Selection of the Optimisation Algorithm
■ Section 3.4 - Shaping Manager : Part I
■ Section 3.5 - Geometry of Starting Shape, Shaping, and Shaping Constraints
■ Section 3.6 – Connecting Expansion Function, and Geometry, Chromosomes
■ Section 3.7 – Characteristic Mode Eigenanalysis
■ Section 3.8 - Objective Function Definition
■ Section 3.9 – Customization of the Shaping Prescription
■ Section 3.10 – Feed Point Selection & Triangle Removal After Shaping
■ Section 3.11 – Supplementary Antenna Performance Computation
■ Section 3.12 - Shaping Manager : Part II
■ Section 3.13 – Concluding Remarks
It will become clear as this chapter proceeds that the 3D shape synthesis tool in question consists of far
more than “merely” the optimization algorithm, or simply running the CEM engine.
The nature of the material in this chapter is such that the use of footnotes greatly facilitates explanations,
and hence footnotes will be used freely. Without them the explanations soon become unacceptably
convoluted.
49 The DMM is described in Section 2.6.4.
46
3.2 SELECTION OF THE COMPUTATIONAL ELECTROMAGNETICS ENGINE
Any shape synthesis tool needs a CEM engine that, given a certain conductor geometry, is able to compute
all related current densities and fields on demand. If research of the type described here is ever to be taken to
the point where it might be used in engineering practice, one has to use a commercially available
electromagnetic simulator and optimisation algorithm software. These are not only vastly more powerful
than most “in-house” simulators, but are available to others as well. In order to do this it is necessary to
develop a deeper insight into the operation of such codes than might otherwise be required for conventional
use of the codes. Several commercial software tools are available that use a Method of Moment (MoM)
formulation for full-wave electromagnetic analysis 50 . Access to the details of the MM formulation, such as
the mesh geometry, its association with the expansion functions used, and other “book-keeping” details, as
well as the impedance matrix [Z], is typically restricted in commercial codes for proprietary reasons.
However, access to [Z] is required in order to apply the DMM approach, and to perform sub-structure CM
analyses 51 . Access to the vertex coordinates of the triangles in the mesh, and expansion function numbering,
is needed in order to be able to manipulate the geometry as the shaping proceeds. The commercially
available CEM code [FEKO] was selected because it has the enlightened attitude of giving the user access to
resource files that provide this information 52 . Sub-algorithms and associated scripts were developed 53 to link
each expansion function with the geometrical data on its associated pair of triangles, as will be detailed in
Section 3.8. The selected CEM engine uses a mesh of triangles over the conducting objects, and RWG
expansion functions, as described in Section 2.3.4.
3.3 SELECTION OF THE OPTIMISATION ALGORITHM
Many researchers have developed in-house genetic optimization algorithms. As with the CEM engine
selected, our approach is similar to that in a measurement laboratory where one simply uses a vector network
analyser (VNA) rather than develop such a piece of equipment oneself. So we have selected a particular
“make” of GA, namely that available in the MATLAB optimization toolbox. As is the case of our use of the
commercially available CEM engine (FEKO), utilization of the commercially available GA algorithm (in
50 This work relies on an ability to find the characteristic modes of a structure. The MM is the only one that currently allows this to
be done reliably.
51 Detailed in Section 2.4.3.
52 Resource files with such information is not available from other commercially available CEM codes.
53 This was a considerable and far from straightforward task, and hints at reasons why others might not have attempted 3D
conducting surface antenna shaping. We believe that the reason the shape synthesis of 3D conducting surface antennas has not
been reported in the literature might be that the CEM engines used have typically been in-house codes, with limited geometrical
ability.
47
MATLAB) has allowed the development of a shape synthesis tool that can be accessible to others. It can
benefit from the superior coding practices in such software, as well as future upgrades. The performance of
any GA is always problem-specific, and its degree of success depends on the use of just the correct settings
of the various parameters 54 already mentioned in relation to the GA in Section 2.5.4. The GA in MATLAB
allows one to experiment with these settings. This has been done, and will be discussed for the class of
problems of interest in this thesis in Section 3.9.
3.4 SHAPING MANAGER – PART I
In the remaining sections of this chapter we will describe the steps/processes required to completely
perform the antenna shaping. These are all implemented as MATLAB scripts that collectively form what can
be called the “shaping manager”, which controls the overall shape synthesis procedure. More will be said
about the capabilities of the shaping manager in Section 3.12 once the above actions it controls have been
individually described. In the latter descriptions, when we name the operation (eg. “the conducting surface is
meshed”) we mean the shaping manager instructs FEKO or the GA to do so under its control, or performs
the operation itself.
Although they are technicalities, it is worthwhile naming where the required pieces of data can be found in
FEKO resource files, since it is not mentioned elsewhere in the thesis. The ( , , )
n n n
x y z , 1,2,3 n , of the
vertices of each triangle are available in FEKO resource file *.cfm. The data are written in ASCII format,
and so must be converted to text format (that is, what MATLAB refers to as ‘double format’, which simply
means ‘numbers’) for further processing by the MATLAB-based shaping manager. The FEKO resource file
*.out is a text file containing information on the number of triangles, the triangle numbering and the
placement of the expansion functions onto the mesh. It also becomes populated with true output such as
current densities and fields; in the present context this means after a characteristic mode analysis during
shaping or after a driven analysis once the shaped antenna is known. The shaping manager needs to pick out
the specific information that is needed from this file at various stages in the shaping process. The FEKO file
*.mat contains the impedance matrix 55 . As with the other files mentioned, such *.mat files are opened by a
MATLAB script (that forms part of the shaping manager) that does the data format conversion, and
thereafter assembles the impedance matrix terms into a recognisable “matrix format”.
54 The settings for the population size, selection, crossover, mutation and migration mentioned in Section 2.5.4.
55 As mentioned in Section 3.2, such information is not accessible to users of other commercial codes, at least not at the time of
writing of this thesis.
48
3.5 GEOMETRY OF STARTING SHAPE, SHAPING, AND SHAPING CONSTRAINTS
We imagine a fictitious specification that requires an antenna to be designed subject to the restriction that it
only occupy some surface of arbitrary shape, or collection of sub-surfaces. At the start of the shaping process
these surfaces are considered to be completely occupied by conductor, and referred to as the starting shape.
The shaping procedure synthesizes an antenna by removing portions of conductor from this starting shape.
Once the starting shape and centre-frequency of operation (
o
f ) have been specified, the starting shape is
meshed into N  triangles. We ensure that the mesh is sufficiently dense to guarantee both the accurate
representation of the computed current density and the geometrical resolution desired. The number and
layout of triangles sets the number of edges common to two triangles. There is an expansion function
associated with each such common edge 56 , of which there are
dof
N in number 57 . The number of bits in an
expansion function chromosome 58 (eg. [10110011100…..0100]) that is manipulated by the GA is denoted 59
by bits
bit
N . In other words, each bit indicates the presence or absence of an expansion function and not the
presence or otherwise of a triangle 60 . Thus as the GA performs the shaping it is removing and/or returning
particular expansion functions rather than triangular elements. Once the shaping process is complete the
expansion functions that remain are known. The mapping between each expansion function and its triangle
pair allows one to determine what triangles (that is, pieces of conductor) remain; this mapping is described
in Section 3.6.
The location of the expansion function bit in a chromosome, and the geometrical location of the expansion
function on the actual conducting surface geometry, are roughly 61 related so that, as often as possible,
adjacent bits in an expansion function chromosome imply almost-adjacent geometrical location of these
functions. This allows us to approximately interpret the behaviour of the GA process in physical terms,
which will be useful in the discussion in Section 3.9.
56 Recall from Section 2.3.4 that a tringle can thus have up to three different expansion functions defined on it. In general
dof
N N

 .
57 The shaping manager immediately instructs FEKO to generate the MM matrix
start
[ ] Z of the starting shape. As the shaping
proceeds the value of
dof
N will change from its value for the starting shape, as the geometry (and hence N  ) is altered.
58 That we will show in Section 3.8 can be one-to-one mapped to a geometry chromosome that defines a particular antenna shape.
59 In general
bit dof
N N  , as will be explained in the last paragraph of this section.
60 We will later, in order to facilitate discussion in Section 3.8, also refer to geometry chromosomes in which a 1 or 0 indicates the
presence or absence of a triangle. It will be of length N  and denoted in parentheses viz. {100110101…….001} instead of square
brackets.
61 Because the expansion function chromosomes and geometry chromosomes are not of the same length.
49
There may be certain portions of the starting shape that we insist must remain conductor and cannot
participate in the shaping process. There may be many reasons for wanting this, including ones related to the
fabrication of the shaped antenna, or the way it is to be assembled with associated circuitry, and so on. The
shaping manager can be used to disallow participation of specified portions of the starting shape in the
shaping process. In such a case
bit dof
N N  . The “disallowing” is achieved by keeping the appropriate
locations in the geometry chromosome always set to 1, that is achieved as follows : The starting shape is
built as a union of sub-geometries. This is done in a way that all fixed regions that may not be changed
during shaping can be identified as a collection of specific sub-geometries. Although the complete starting
shape is always described by the chromosome of length
dof
N , those bits associated with the fixed region are
prevented (by the shaping manager) from changing.
Of course, of certain portions of a surface must not contain conductor, one simply does not include such
portions in the starting shape.
3.6 CONNECTING EXPANSION FUNCTION, AND GEOMETRY, CHROMOSOMES
Consider the chromosome 62 {1 1 1 1} that describes the starting object geometry 63 shown in Fig,3.6-1(a).
The fact that each bit is a “1” means that each of the four triangles is “filled” with conductor. There are four
common edges and so there are four expansion functions, each associated with a common edge 64 . The
starting impedance matrix [Z] is therefore a 4 4  matrix. The expansion function chromosome of the starting
geometry is [1 1 1 1].
Now suppose that in the shaping process triangle#1 is removed. The chromosome describing the altered
geometry is thus {1 1 1 0}. With two expansion functions removed the reduced Z-matrix is 2 2  , as shown
in Fig.3.6-1(b). All terms needed for any reduced matrices that might arise are contained in the starting [Z].
62 The position in the geometry chromosome is {Triangle#4 Triangle#3 Triangle#2 Triangle#1}. Observe that we are
purposefully using parentheses {…} to represent this chromosome. Square brackets […] will be used to represent chromosomes
that describe which expansion functions (not triangles) are present. The position in the expansion function chromosome is in the
format [Expansion#4 Expansion#3 Expansion#2 Expansion#1] .
63 Of course, in the actual shape synthesis work, there would be hundreds of triangles used to resolve a geometry. The four-triangle
one being used here
64 As outlined in Section 2.3.4.
50
The procedure just described in words must be implemented using the limited information available in the
workspace files of the CEM engine. In this work we have done so using a binary logic approach, as the next
set of bullets will explain 65 . The reader is once more referred to the geometry in Fig.3.6-1.
■ Each individual triangle (as opposed to expansion function) can be represented by chromosomes:
Triangle #1 {0 0 0 1}
Triangle #2 {0 0 1 0}
Triangle #3 {0 1 0 0}
Triangle #4 {1 0 0 0}
■ The FEKO output file gives the information which indicates the connection between these triangles and
their expansion function indices, in the following format (annotated here for clarity):
This shows that expansion function #1 (for example) is defined on the edge common to triangles #1 and # 4,
and expansion function #2 is defined on the edge common to triangles #1 and #3, and so forth.
■ The TA and TB index ‘vectors’ for the entire starting geometry are converted to binary format:
TA
1  1 0 0 0
1  1 0 0 0
2  0 1 0 0
2  0 1 0 0
TB
4  0 0 0 1
65 This is needed because we use the DMM approach; if not the process would not be as complicated. The reward is that, using the
DMM, the [ ] Z need be computed only once.
51
3  0 0 1 0
4  0 0 0 1
3  0 0 1 0
The above binary vectors are then added (adding in binary format implies an OR logic operation) :
TA • OR • TB
1 0 0 1
1 0 1 0
0 1 0 1
0 1 1 0
The outcome of the OR operation is then flipped :
1 0 0 1
0 1 0 1
1 0 1 0
0 1 1 0
Now suppose we have a chromosome of triangles {1 1 1 0}. This defines a specific conductor geometry. We
wish to identify the corresponding chromosome of expansion functions in square brackets as [x x x x]. In
order to do this, {1 1 1 0} undergoes an AND operation with each element in the array of the flipped TA •
OR • TB quantities on the previous page:
{1 1 1 0} • OR • {1 0 0 1} = {1 0 0 0}
{1 1 1 0} • OR • {1 0 1 0} = {0 1 0 0}
{1 1 1 0} • OR • {0 1 0 1} = {1 0 1 0}
{1 1 1 0} • OR • {0 1 1 1} = {0 1 1 0}
or in other words
1 0 0 0
0 1 0 0
1 0 1 0
0 1 1 0
which, when rotated, becomes
1 0 1 0
0 1 0 1
0 0 1 1
0 0 0 0
52
The decimal addition of the bits in each complete column provides
1 1 2 2
A digit “1” signifies that there is no expansion function at the particular location in the expansion function
chromosome, whereas “2” signifies the opposite 66 . The above digits thus tell us that expansion function
chromosome associated with the triangle function chromosome {1 1 1 0} is [0 0 1 1]. Only expansion
functions #3 and #4 are present. This fits with Fig.3.6-1(b).
The above process in essence does the following : Given the “barcode” for the expansion functions
generate the associated barcode for the conductor layout, so that the conducting shape is known.
#4 Function
#1 Function
#2 Function
4
1
2
3
#3 Function
#4 Function
4
1
2
3
#3 Function
(a) (b)
Fig.3.6-1 : (a). Simplified unaltered geometry consisting of four triangles and four expansion functions and
(b). with triangle#1 removed (as suggested by the dashed line). Triangle numbering is shown encircled. The
red dots are the vertices of the triangles.
66 This is true no matter how many triangles are present.
53
(a) (b)
Fig.3.6-2 : (a). MM impedance matrix [Z] corresponding to the unaltered geometry in Fig.3.6-1(a), and (b).
the reduced MM matrix formation upon removal of expansion function #1 and expansion function #2.
3.7 CHARACTERISTIC MODE EIGENANALYSIS
If we require conventional characteristic modes of some shape, the shaping manager takes the reduced [Z]
of the particular shape, separates it into real and imaginary parts, and solves the matrix eigenvalue problem
of expression (2.4-1) using MATLAB eigenvalue problem routines. If we require sub-structure characteristic
modes of some shape, the shaping manager takes the reduced [Z] of the particular shape, separates it into
real and imaginary parts, and executes the matrix operations (2.4-16), (2.4-17) and (2.4-21) using MATLAB
routines. This eigenanalysis of course supplies the column vectors [ ]
n
J that are the eigencurrents. FEKO
does not support sub-structure CM analysis.
If at any stage 67 the associated eigenfields are needed the modal currents 68 [ ]
n
J are returned to FEKO by
the shaping manager so that it can compute these fields. The eigencurrents [ ]
n
J of any shaped geometry
will not be of the same length as the starting shape, because the starting shape has more expansion functions
on it. In order to be able to export this reduced [ ]
n
J back to FEKO 69 to obtain current distribution plots and
compute the fields due to it (i.e. secondary parameters), a special MATLAB script that is part of the shaping
manager has been written that does the following: It alters the relevant FEKO resource files so that it accepts
an altered geometry, and regenerates the (smaller) new [ ] Z for this altered geometry 70 . It then converts the
[ ]
n
J generated in binary format by MATLAB to a so-called *.str file 71 in ASCII format that can be
67 Albeit usually only once the shaping has been completed.
68 Whether for conventional or sub-structure CMs.
69 That allows us to fully exploit its superior post-processing capabilities.
70 As a double-check, one can compare this [ ]
Z to the associated reduced [ ] Z that had been obtained from applying the DMM.
These should be identical. This was always found to be true.
71 This is FEKO terminology.
54
interpreted by FEKO. Once this has been done FEKO is able to compute secondary quantities based on these
current density values.
When using these column vectors of coefficients for the CM currents it is necessary to instruct FEKO to
“only calculate the scattered part of the field”; this is an option provided by FEKO. The script must also
explicitly request FEKO to read the current column vectors from the created *.str file.
In the case when sub-structure CMs are used, the shaping manager first concatenates [ ]
Asub
n
J and
sec
[ ]
B
n
J ,
as defined in Section 2.4.3, to form a single column vector of coefficients before sending it to FEKO for
secondary parameter computation.
3.8 OBJECTIVE FUNCTION DEFINITION
Objective functions translate the design requirements into mathematical quantities that can be minimized
by the optimization algorithm. Use of an algorithm that is not gradient-based allows one to use objective
functions that are not merely simple classical mathematical function. It can also incorporate some sub-
algorithm that can bias the decisions of the shape optimization algorithm based on known physical
knowledge. At any rate, it is best to customize objective functions to the design targets of a specific shaping
example. They are easily defined as a script in the shaping manager. Objective functions will therefore be
revisited when particular shaping examples are discussed in Chapter 4 and Chapter 5. These can be easily
incorporated into the shaping manager.
3.9 CUSTOMIZATION OF THE SHAPING PRESCRIPTION
We mentioned earlier that the parameter values needed for successful operation of a GA are always
specific to the class of optimization problem. Such necessary details are most often not provided, even
though they are essential if the reader wishes to repeat what has been described. Exhaustive testing of the
shaping process done using a particular case 72 as a vehicle has resulted in the development of a customized
adaptive approach to the application of the GA for antenna shaping. Although the outcome of this case will
only be discussed in Section 4.2, the customized procedure that resulted from the numerical experiments on
it are provided here in an effort to keep the thesis properly organized. It became the route that was followed
in all subsequent shape synthesis examples.
When the GA was used with the default settings for its various parameters the shaping process either
stagnated or, to put things plainly, the GA removed too much conductor too soon in its initial few iterations,
72 Reported as Example #1 in Chapter 4.
55
and then did not seem able to put back conducting pixels in an acceptable number of iterations, when it
“realised its mistake” in later iterations. Adjustment of the GA’s parameters 73 such the crossover, mutation,
migration direction, and so on, did not consistently improve matters. A comprehensive organized set of
numerical experiments was clearly needed. The outcome was a structured procedure that successfully
achieves the shaped antennas desired.
We have found that it is necessary to divide the shaping process into two parts, a trial run 74 and only then
the actual run 75 . These are summarized in the flowcharts Fig. 3.9-1 and Fig. 3.9-2, respectively. They will
next be elaborated on as a series of bullets written in the form of a ‘recipe’, and several other observations.
A. Trial Run – Flowchart#1
▪ Assume that the value of
pop
N has been specified. The GA randomly generates a single beginning
expansion function chromosome (call it [
init
C ]), and from this breeds an initial population of
pop
N possible
antenna shapes 76 .
▪ Allow the GA to cycle through several generations, using default crossover, mutation and migration
settings, until the value of
obj
F has decreased substantially (eg. decreased by a factor of 10 or so). Halt the
GA. Up to this stage the iterations of the GA are conventional.
▪ Select the chromosome with the smallest value of
obj
F . Use this chromosome as the new initial
chromosome
init
C and start the GA afresh. If any chromosome in subsequent populations contains fewer 1’s
than
init
C do not place it in custody 77 . Otherwise place it in custody and using, it as the next
init
C , do a
freshly-started GA run.
▪ Begin the fresh GA runs using this
init
C with a crossover value 78 of 0.2.
▪ After the first iteration, record the value of
obj
F
▪ Change the crossover value to 0.3, and complete another generation. Record the value of
obj
F .
73 These actions can be performed in a number of ways, usually controlled by a particular deterministic or probability function
type that has to be chosen, and numerical values that define the details of the choice.
74 The trial run finds a physically feasible “warm start” for the actual shaping run.
75 The actual run eventually produces the shaped antenna.
76 Even though it actually generates an initial population of
pop
N expansion function chromosomes, we know from Section 3.8 that
these can, when required, be translated into geometry chromosomes that each define a shape. Thus we will simply say that
different antenna shapes are created.
77 By placing a chromosome in custody we mean we record it for later possible use.
78 This means 20% of the bits are kept fixed and 80% altered.
56
▪ Repeat this with increasing crossover values until a generation has been completed with a crossover value 79
of 0.8.
▪ Compare the values of
obj
F of the collection of
init
C chromosomes. Select the
init
C in the set that gave the
smallest
obj
F , and the associated crossover value 80 that allowed it to be spawned. Use this
init
C as the initial
chromosome, and the crossover value, and then run the GA in the conventional way for very many
generations. In other words, move to the next stage of the shaping process, namely the actual run.
B. Actual Run
The flowchart in Fig. 3.9-2 provides the entire list of steps for the actual run. Although it is not absolutely
necessary, this author has done the following once the shaping procedure just described is complete (and
hence
obj
F small): The mesh is refined on the final shape (that is, for the final chromosome). This
chromosome is then used as an initial chromosome
init
C , the GA allowed to run for one more generation,
and the value of
obj
F for the bets population member observed. This was done for all the examples to be
described in Chapters 4 and 5; it never altered the final shape.
C. Further Observations
▪ Both forward and backward migration should be attempted for the 3D antenna shaping problem. This is not
necessarily the case with all physical problems.
▪ There are often rules of thumb stated, which in the present notation takes the form 81
pop dof
N mN  , with
suggested values provided for integer m . Experience in this thesis has made us believe that this should not
be used as a general rule. The selection of
pop
N has to be studied for particular classes of problem. This is
seldom detailed in the literature. The best 82 values of
pop
N for the antenna shaping problem has been
established through supervision of the progress of the GA for several specific cases (when beginning form
79 If the crossover value of larger than 80% too many bits are kept fixed for sufficient progress to take place.
80 In all the shaping experiments carried out the best crossover value has been found to lie between 0.5 and 0.8.
81 Because
dof
N is the expansion function chromosome length, the expansion function chromosome being the ones directly acted
upon by the GA.
82 In the sense that it will actually lead to a shape that has a very small
obj
F .
57
the starting geometry). We use
pop
20 N  in order to arrive as fast as possible at the stage where refined
runs are performed. Only after doing so is the value of
pop
N increased significantly.
▪ We have found that values of Tol (defined in Section 2.5.4) should be set to
12
10  in the actual run. The
GA will stop if the changes to the value of
obj
F of the best member of the population changes by less than
Tol from one generation to the next.
▪ If one has fixed regions on the starting geometry it is observed that the GA takes longer to reduce the value
of
obj
F . This can be understood in the sense that one is interfering in the route the algorithm wishes to
follow. In a sense one is “protecting” possibly weak members of the population. In such cases stagnation can
occur unless the crossover values are adjusted slightly at different stages of the shaping process; this is one
aspect falling within the User Supervision block in flowchart#2.
▪ In the same User Supervision block the mutation rate might need to be adjusted of necessary. In all the
shape synthesis examples completed in this work we have found that mutation rates between 0.3% and 0.5%
are satisfactory.
▪ In all the shaping experiments carried out the best crossover value has been found to lie between 0.5 and
0.8. The reason is clear. If the crossover value is 0.x (stated as x%), then (100-x)% of the bits in the
chromosome are altered in generating new shapes for the next generation. Bits in chromosome locations
close to each other are changed. This implies that expansion functions located geometrically close to each
other are changed. If 0.5 x  we end up with too many shapes in a population that have far too little
conductor, that we know for physical reasons are not going to give us what we need.
▪ In all the applications shown in this thesis the mating scheme employed is uniform crossover and the
selection scheme used is tournament selection. The population size (
pop
N ) was held constant in all trial
runs; although its value was experimented with in some actual runs (in an attempt to save electromagnetic
simulation time) this is not really necessary.
58
Store
Its Associated
Corresponding Crossover Value
Select The Stored
With The Lowest
Randomly Generate Initial Chromosome
q = 1
Breed Chromosomes
(Single New Generation Only)
Compute For Each Chromosome
Crossover = Crossover + 0.1
Is Crossover > 0.8?
q = q+1
Set Crossover = 0.1
Set
pop
20 N 
init
[ ] C
pop
N
obj
F
init q
[ ] [ ] C C 
Select Chromosome with Smallest
Call it
obj
F
q
[ ] C
q
[ ] C
( )
obj
q
F
q
[ ] C
( )
obj
q
F
No
Yes
( ) q
r
C
Fig. 3.9-1 : Trial Run - Flowchart#1
59
Is ?
Shaping Complete
Set Crossover Equal To Value From Flowchart#1
Outcome
Set
Breed Chromosomes
Find Associated For Each
Chromosome
Is ?
Input
Compute
pop
N
Select Chromosome With Largest
Call It And
obj
0 F 
max
[ ] C
No
Yes
pop
1 2 p N
[ ],[ ],...,[ ],...,[ ] C C C C
max

pop pop
2 N N  
(p)

(p)

max init
  
init max
[ ] [ ] C C 
Yes
Evaluate For
obj
F
max
[ ] C
No
User
Supervision
init max
[ ] [ ] C C 
( ) q
r
C
init

init
[ ] C
Fig. 3.9-2 : Actual Run - Flowchart#2
60
3.10 FEED POINT SELECTION & TRIANGLE REMOVAL AFTER SHAPING
3.10.1 Feed Point Selection
At the end of the shaping process the feed point must be selected on the shaped antenna 83 . This is
determined based on the current distribution (in other words the column vector of coefficients) of the
dominant CM on the antenna. In the shaping manager a script therefore simply search for the coefficient
with the maximum value, and the known link (form Section 3.6) between the expansion function index and
the geometry used to determine exactly where the feedpoint should be placed. Visual observation of the
displayed CM current on the shaped antenna is not sufficiently precise to reliably select this location.
3.10.2 Removal of Islands & Vertex-Only Connections
When fabricating the antenna under consideration the synthesized shape that is produced by the shaping
manager will contain triangles that are joined only at their vertices (vertex-only connections), or are not
joined to any other triangles at all (islands). The RWG expansion functions 84 are so-defined that there is no
current crossing such junctions. Such junctions must not be fabricated, since they would allow current flow
through them and hence the fabricated model would not correspond to the electromagnetic model upon
which the shaping was based. Such “vertex only” connections must therefore be separated slightly (“split”)
in the geometry file before it is used as input to the fabrication process, to prevent such current flow.
Furthermore, any single isolated triangles occurring in the shaped geometry must be removed; they will not
have had any current on them (again due to the manner in which the RWG expansion functions are defined)
during the simulations.
83 After which a driven MM analysis is performed using FEKO to determine the antenna properties.
84 Described in Section 2.3.4.
61
Fig.3.10-1 : Illustration of a shape-synthesized antenna with highlighted vertex-only connections.
3.11 SUPPLEMENTARY ANTENNA PERFORMANCE COMPUTATION
3.11.1 Quantities
Definitions of all the antenna performance parameters needed have been provided in Section 2.2. Most of
these are provided directly by the chosen computational engine (FEKO). However, the mean effective gain
(MEG), antenna quality factor (Q), and bandwidth-efficiency product (
rad
B  ) are not, and so scripts in the
shaping manager were written to determine these once the shape synthesis of an antenna has been
completed.
It is often easy to be mislead by colour-coded plots of the current density over a conducting surface of
unconventional shape with which one is not yet familiar. In order to explain what we mean Fig.3.11-1 shows
the current density on a centre-fed strip dipole. The current is high at the location of the source, and at the
edges of the strip in the vicinity the source 85 . The temptation is to assume that the portions of the PEC strip
(the ends in this case) where the current density is low are of little importance to the antenna operation, and
are “not being used”. Of course, we know from experience that these portions (physically connected 86 as
they are to portions where the current density is higher) are essential for proper operation of the antenna,
85 Current density parallel to a PEC edge peaks at the edge; the so-called edge condition.
86 In a structure (not the dipole depicted here) containing small isolated pieces of PEC on which the current density is everywhere
almost zero will indeed not play a significant role in the antenna operation.
62
because this is where most of the net electric charge density resides. This author had to continually recall this
fact when observing currents on the unconventional shapes to be discussed in Chapters 4 and 5.
Fig. 3.11-1 : Current density on a centre-fed PEC strip of roughly one-half-wavelength.
3.11.2 Quality Factor (Q)
The actual quality factor is computed using the expression for Q z given in Section 2.2.10. At any
frequency f the derivatives ( )
in
R f  and ( )
in
X f  can simply be computed via first-order finite differences
using values at frequencies a very small amount lower and higher, than the frequency of interest (eg.
1.5MHz “deltas” on either side of 1 GHz).
3.11.3 Mean Effective Gain (MEG)
The MEG is computed by the shaping manager using expression (2.2-47). The gain, or directivity
functions and radiation efficiency, are extracted from FEKO by the shaping manager. One can define the
angular probability density functions for a specific environment in the shaping manager. It then normalises
them so that (2.2-50) and (2.2-51) are satisfied. The integrals in the definition of the MEG are then
numerically computed the integration over S  . It is found that, for the relatively broad radiation patterns of
the electrically small antenna under discussion here, values of   and   less than 1 allow the MEG
integral to be evaluated accurately as a simple Riemann sum.
63
3.11.4 Bandwidth-Efficiency Product (
rad
B  )
This quantity was defined in Part E of Section 2.2.10. The fractional bandwidth B is found from the actual
in ( )
f  curve, and the radiation efficiency as that computed by FEKO multiplied by a correction factor 87 if
needed.
3.12 SHAPING MANAGER - PART II
The overall operation of the shaping manager is summarized by the flowchart in Fig. 3.12-1. As stated
previously, it is implemented in MATLAB and is thus flexible enough to be built-upon if new requirements
arise. It links the CEM engine and the GA algorithm, makes the changes to the geometry during shaping,
performs the required CM analyses, evaluates the objective functions for each candidate geometry, is able to
enforce symmetry or not during shaping. It is able to disallow shaping over portions of the geometry. Most
of the shaping process is completed without having to continually appeal to FEKO; it is not necessary to re-
compute the MM-matrix several times during each iteration of the GA because the DMM approach is used
by the shaping manager. This is so unless the objective function includes not only CM eigenvalues but also
the far-zone fields 88 of the CMs. The shaping manager is able to interpret FEKO resource files and hence can
exploit all that FEKO is able to provide as far as electromagnetic computation, CAD capabilities for
describing the geometry (which can be quite complex), and even the export of CAD files for fabrication
purposes are concerned. The latter can also be used to designate parts of a complete object as Object#A and
Object#B (or more) if the shape synthesis is to be done 89 using sub-structure CMs.
87 The correction factor will be discussed in Section 4.2 and, as will be discussed there, Appendix A.
88 As explained in Section 4.5.
89 This will be done in Chapter 5.
64
Define Objective Function
Specify Starting Shape + Shaping Constraints
Record : MM Matrix; Mesh Details;Expansion
Function Data; Triangle Geometry Data.
Feed Point Selection
User
Supervision
Controllable
CEM Engine
Mesh Generation
MM Matrix Generation
for Starting Shape
Current Density
Reconstruction
Computation of Field
of Current Density
Conduct Trial Run to Establish
Best GA Settings
(Flowchart#1)
Conduct Actual Shape Synthesis
Runs
(Flowchart#2)
Controllable
GA
Algorithm
Geometry Filtering
Driven Antenna Performance
Computation
Stop
Generate CAD File for
Antenna Fabrication
Data Mining
File Format Translation
ASCII Fornmat - Text
Format Conversion
Shaping Manager
Fig. 3.12-1 : Shaping Manager - Flowchart#3
65
3.13 CONCLUDING REMARKS
Chapter 3 has developed the very complex machinery for a computational electromagnetics-based tool
capable of performing the shape synthesis of conducting antennas. It has been developed in the form of a
software shaping manager that controls the shaping process by exploiting a commercially available CEM
engine and a commercially implemented genetic optimisation algorithm. This is an important step, because it
allows, contrary to pixilation-based shaping implementations used by others, an ability to perform true 3D
shaping of conducting surface antennas here for the first time. The use of a commercially available CEM
engine (FEKO), and a commercially available GA algorithm (in MATLAB) has allowed the development of
a shape synthesis tool that can be accessible to others. It benefits from their superior pre- and post-
processing capabilities, and will so from future upgrades 90 . Another reason for using a commercial code such
as FEKO is that files describing complicated shaped geometries can be exported for fabrication. However, it
will hopefully have been appreciated from the description in this chapter that the shaping manager is
considerably more than merely “running FEKO”.
One might wonder why the GA is used at all, and whether a simple direct search approach should be used.
However, the number of permutations of the bits in a binary word of length m is 2 m . If m is on the order of
800 (which is less than that in the first example in Chapter 4) there will be an impossibly large number of
arrangements to examine. This obviously discourages a direct search, since a full-wave electromagnetic
simulation would be required for each of the population members (that is, antenna shapes) considered. The
GA’s evolutionary approach much reduces the number of shapes that need to be “tested” in reaching a shape
that is optimal according to the selected objective function.
90 We believe that the reason shape synthesis of 3D conducting surface antennas has not been reported in the literature might be
that the CEM engines used have typically been in-house codes, with limited geometrical ability.
66
CHAPTER 4
The Shape Synthesis of Non-Planar Conducting Surface Antennas
4.1 PRELIMINARY REMARKS
4.1.1 Outline of this Chapter
The review in Section 2.6 allowed us to correctly conclude that shape synthesis methods which do not
constrain the location of the feed-point have not yet been available for, or hence applied to, fully three-
dimensional (3D) conducting surface antennas. A full-wave based synthesis tool was therefore developed in
Chapter 3 to enable the antenna designer to do this. In the present chapter several examples of the
application of this synthesis tool are treated, and some experimental validation of the approach discussed.
Chapter 3 has been written in such a way that descriptions of the set-up for these examples can be concisely
laid down here in Chapter 4. The first shaping example, in Section 4.2, starts with an open cuboid shape and
synthesises an antenna from it for self-resonance at a specified frequency. This, as well as the further
examples considered in this chapter, confirms the validity of both the shaping idea for 3D conducting surface
antennas, and the success of its implementation tool described in Chapter 3. The specific antenna (in Section
4.2) is fabricated and its performance measured, to demonstrate that the CEM modelling process is sound.
Sections 4.3 and 4.4 further demonstrate the successful application of the shape synthesis process, with
their examples representative of how the starting shape could be altered (eg. to accommodate possible
fabrication), and then when there is an inclusion (eg. battery) within the starting shape. Section 4.5
demonstrates a case where shaping is done, not only to achieve self-resonance (which it continues to do), but
to simultaneously maximise the mean effective directivity (MED) of the antenna. Section 4.6 briefly
mentions some possible extensions of the work of Chapter 4. The contributions of the work in this chapter
are summarized in Section 4.7. Before proceeding to the shape synthesis examples it will be instructive to
motivate the need for such 3D conducting surface antennas.
4.1.2 Some Motivation
There is much hyperbole associated with new technology, and the Internet of Things (IoT) concept is no
exception. However, even if only a small percentage of this “hype” is eventually realized, it will represent an
67
enormous industry, conservatively 91 estimated at around $300 billion annually. It is recognized [NEZA19]
that the IoT is still in its infancy, and that details of its eventual structure and constituents are yet to be
determined. There are challenges at all levels, involving virtually every discipline within electrical
engineering. Indeed, there is even now an IEEE Internet of Things Journal. One aim of the Internet of Things
is to allow embedded sensors to be connected to some host, so that once connected they can be supervised,
tracked, and monitored via the Internet. What is certain is that there will be sensors on many devices that
communicate with their data centres via the wireless network. Each sensor therefore needs an antenna and
associated RF circuitry, because whenever communications takes place in a wireless, as opposed to a wired,
manner, this is essential. The number of wirelessly connected devices is expected to grow to 50 billion 92 by
2020. The range of possible applications is summarized in Fig.4.1-1.
Fig.4.1-1 : Applications of the Internet of Things (Adapted from [NEZA19])
The type we have in mind in this work is antennas on such things as surveilllance cameras, weather
monitoring stations, parking meters, road cones that alert the public to construction work (‘smart’ cones),
and so forth. In such applications, if one can make the antennas electrically small, then they will also be
acceptably physically small and allow use of the lower frequency ranges not crucial for future very high data
rate communications. In such applications the goal is to have antennas on the surface of some package (eg.
cuboid) capable of having sensor circuitry packaged within the antenna structure. This has resulted in the
terminology “antenna-on-package” (AoP). Although it may be considered technically vague terminology,
the trade literature describes the antenna requirements as providing “near isotropic” radiation patterns so that
the sensor is not over-sensitive to orientation. It also mentions that the use of additive manufacturing
techniques to make and fit, or directly deposit, the antenna on the package would be favourable.
91 The conservative estimation process has consisted of extracting dollar figures from tens of websites, flyers in the trade literature,
and statements by investors hoping to profit hugely, and simply averaging their predictions.
92 This includes mobile phones and tablets. Thus some are even referring to the Internet of Everything (IoE) or Internet of People
(IoP). However, IoT is apparently the term accepted in the engineering community.
68
There are only a limited number of references directly related to electrically-small 3D conducting surface
antennas 93 on a package of the type in mind in the present work. Good representative examples are those
reviewed below:
[KRUE 09] : The antenna consists of a non-conducting cube into which three meander-line dipoles, fed
using T-match configurations, have been folded. The antenna has 0.5169 ka  , has a measured 75%
rad
  ,
and a -10dB bandwidth of 1.3%. The authors refer to this as “smart packaging” because sensor equipment
may be integrated inside the cube’s hollow interior.
[NASS 12] : The antenna consists of a meander-line dipole folded onto a non-conducting cube which
“makes available the cube interior for sensor electronics equipment. Two antenna were designed. The
antenna with 0.55 ka  had a measured 78%
rad
  , and a -10dB bandwidth of 2%. The antenna with
0.45 ka  had a measured 73%
rad
  , and a -10dB bandwidth of 1.5%.
[ADAM 15] : The authors discuss the 3D printing – direct writing of silver nanoparticle ink - of a spherical
dipole onto a non-conducting spherical shell, using 3D printing. The with 0.367 ka  had a measured
57%
rad
  . It should be remembered that silver nanoparticle ink has a much lower conductivity than
conventional printed circuit board copper.
[LIU 16] : The antenna consists of a meandered loop antenna whose plane is located at the (vertical) centre
of the a cubic package. The antenna has 0.12 ka  , a measured 22%
rad
  , and a -10dB bandwidth of
1.1%. Electrically, this antenna is extremely small. It is not self-resonant, and needs a lumped component
matching network.
[SU 18] : The antenna consists of meander-line antennas folded onto a non-conducting rectangular package.
The antenna has 0.78 ka  , and was constructed using additive printing onto a flat surface that is then folded
onto the package and any required connections made using conductive epoxy. The emphasis in the paper is
that the resulting antenna patterns are near-isotropic, and the fact that dimensions on the antenna were
93 None of which can be considered to have been shape synthesized.
69
(manually) altered in trade-off studies to obtain the said behaviour in the presence of a battery inside the
supporting package.
In all of the above instances the design process consisted of using conventional antenna geometries and
folding them to conform to a prescribed host surface. The resulting changes in the antenna behaviour are
accepted as inevitable, and matching networks are sometimes needed. In this chapter we will demonstrate
that, with the shaping tool of Chapter 3 now available, the intended surface of some package can be
specified (with portions designated as a starting shape for the eventual conducting surface antenna), and we
can let the electromagnetics guide the shape synthesis of the antenna so that it is self-resonant at some
frequency of interest.
4.2 THREE-DIMENSIONAL SHAPE SYNTHESIS EXAMPLE #1 : OPEN CUBOID ANTENNA
4.2.1 Preamble
This example was chosen to serve as the test object on which to run computational experiments in an effort
to observe and understand the behaviour of the proposed shaping process, in particular the manner in which
to use the genetic algorithm. An enormously large number of runs were made and the progress (or lack
thereof) observed. The result was the customization that has already been described in Section 3.9. The
results shown here 94 are the outcome when this customization was finally implemented. Furthermore, we
used 1 ka  for this test case. Such an antenna is on the border of the usual “electrically small” designation,
and could be said to be of intermediate electrical size. Such an antenna, susceptible as it is to not being
single-mode 95 , represents a slightly more tricky shaping problem when using the shape-first/feed-next
approach, and thus serves as a good test case. Subsequent examples have much smaller values of ka .
4.2.2 Starting Geometry
The starting shape is the non-planar cuboid with one face removed, as shown in Fig.4.2-1. The centre
frequency is 2
o
f GHz  , at which the wavelength 150mm   , and so the sides 96 are each of electrical
dimension 30mm /5 d    . The radius a of a sphere that just encloses a cuboid of sides d can be shown to
be ( 3/ 2) a d  . Thus we here have / 0.1732 a   and hence 1.09 ka  . The starting shape is densely
94 And indeed for all shape synthesis examples to be given in this thesis.
95 In the sense defined in Section 2.6.3.
96 There is no groundplane. A “half-wavelength” self-resonant dipole operating at the same frequency would have to have a length
0.48 72mm   and have (as noted in Section 2.2.2) 1.57 ka  .
70
meshed into 860 N   triangles. This results in an initial
dof
1023 N  , and implies that the average length of
any side of the triangles in the mesh is 2 mm, which is well below /60  . This very fine mesh was selected
so that we could experiment with the shaping process prescription without having to be concerned about the
accuracy of the electromagnetic modelling. It also provides good geometrical resolution.
We selected the starting shape to be open at the bottom to emphasize that it can in fact be any shape
whatsoever; the implication in this case is that such a starting shape would ensure that the final synthesized
antenna can easily be assembled by pushing it over a package. Furthermore, we envisaged that the intention
is to have the antenna fabricated out of a flat sheet of copper and then folded into a structurally rigid form 97 .
We therefore instructed the shaping manager to disallow the removal of triangles on either side of all edges
in the shaping process. A lower number of optimization variables always makes the shape synthesis process
slightly “tougher”; but we have not been able to quantify this in any way. No symmetry was enforced during
the shaping process.
Fig.4.2-1: Starting non-planar cuboid 3D conductor shape with bottom face removed, and sides of
dimension 30mm d  . Diagram shows tilted cuboid viewed from below.
97 Further comments on how alternative intended fabrication methods might be accommodated by the choice of the starting shape,
or constraints, are made in later sections of this chapter.
71
4.2.3 The Objective Function
The objective function used was simply
1 o
( )
obj
F f   (4.2-1)
with
1 o
( ) f  the magnitude of the CM with the lowest eigenvalue at
o
f . The range of possible values is
0
obj
F    . The reasoning is that, as long as the structure is indeed a single-mode one, a low
1 o
( ) f  value
will signify that the shaped structure is resonant in a CM-sense, and it will then be possible to find a feed
point with respect to which, as an antenna, it will be self-resonant 98 .
4.2.4 Outcome of the Shape Synthesis Process
The shaping procedure flowcharted in Chapter 3 was followed. Fig.4.2-2 reveals that the shaping took
roughly
gen
250 N  generations to converge. The value of
1 o
( ) f  reached was 0.004. Fig.4.2-3 is a plot of
a normalised eigenvalue  
1
10log /
dB
n n
    . It is clear that the lowest eigenvalue is orders of
magnitude lower than that of the other CMs. The synthesized shape is that in Fig.4.2-4, which also indicates
the location of the feed, found using the approach described in Section 3.10.1. The surface current density
distribution of the lowest CM is depicted in Fig.4.2-5 (a) through (c), and shows pictorially why the feed
location turns out to be where it is; we wish to excite this particular CM strongly to ensure self-resonance at
o
f . Note that these are eigencurrents. The current for the driven structure is that in Fig.4.2-5(d); the driven
structure will be discussed in the next section.
98 If the modal Q-factor were to be included as part of the objective function this could allow the bandwidth to be maximised
subject to the given constraints.
72
Fig.4.2-2 : Progress of the value of the objective function (fitness function) of the fittest member of the
population as a function of generation number in the “actual run”.
Fig.4.2-3 : Eigenvalues
n
 vs mode index n for the final shaped antenna at
o
f , normalised to
1
 and
expressed in decibels.
73
Fig.4.2-4 : Shape-synthesized antenna geometry. The feed point is indicated by the red bar.
(a) (b)
(c) (d)
Fig.4.2-5 : Three views, at frequency
o
f , of the current density of the lowest order CM on the shaped
structure in Fig.4.2-4 are shown in (a) through (c). The current distribution on the driven shaped antenna is
that in (d), and is referred to in Section 4.2.5.
74
4.2.5 Computed Performance of the Shaped Antenna
With the feeding location selected, we can proceed to simulate the shaped structure as a driven problem to
obtain the input impedance and reflection coefficient, radiation efficiency and antenna quality factor at
frequency
o
f and hence assess how successful the shape synthesis has been. It is clear from the modal
weighting coefficients shown in Fig.4.2-6 that, with the feed located as shown in Fig.4.2-4, the antenna is
indeed a single-mode structure. Higher-order CMs are all excited at levels around 20dB or more lower than
the dominant lowest one.
The computed input impedance in Fig.4.2-7 confirms that at
o
2GHz f  (or 1.997 GHz to be precise) the
antenna is self-resonant, with
in
0 X  and
in
47.3 R   . As expected, the input reflection coefficient
in
 ,
referenced to 50Ω , is very low in the vicinity of
o
f . This can be seen from Fig.4.2-8. Computed radiation
patterns 99 are shown in Fig.4.2-8. Fig.4.2-8(a) has been included for convenience in recognizing which cuts
are being referred to. The azimuthal pattern cut is that in Fig.4.2-8(b); it is seen to be almost omni-
directional in this plane. Two elevation cuts are shown in Fig.4-2-8 (c) and (d), namely the xz- and y-z-
planes respectively. Such broad patterns are expected of an antenna with 1 ka  and are highly desirable for
sensor applications. Additional computed performance measures will be referred to Section 4.2.6, in relation
to their measured values for the fabricated version of this antenna.
Fig.4.2-7 reveals that there is an ordinary resonance at 1.76 GHz, but that 3
in
R   there, which is very
low. Interestingly, the shaping procedure automatically selected the anti-resonance point with the higher
in
R
value without user intervention. If bandwidth maximization were to be included in the objective function
(see comments later) then it might very well select the ordinary resonance and deliver the low
in
R to us. At
frequencies below 1.76GHz the input resistance (and hence radiation resistance) drops quickly towards zero,
whereas
in
X becomes increasing negative (capacitive).
99 The maximum total gain was computed to be 2.8 dBi.
75
Fig.4.2-6 : Modal excitation coefficient (
n
 ) vs. mode index ( n ) at
o
f .
Fig.4.2-7 : Computed input impedance of the shaped antenna.
76
Fig.4.2-8 : Computed input reflection coefficient of the shaped-synthesized antenna for reference impedance
o
50 Z   .
77
(a) (b)
(c) (d)
Fig.4.2-9 : Normalized radiation pattern cuts at
o
f . The coordinate system sketch in (a) shows which
pattern cuts are referred to. The pattern cuts are as follows - (b). 90 ,0 360      (c).
0 ,0 360      , (d). 90 ,0 360     
78
4.2.6 Scaling, Fabrication and Measurement
In an effort to facilitate fabrication, we increased all linear dimensions of the shape-synthesized antenna by
a factor of two, and hence the behaviour of all performance measures will decrease in frequency by the same
factor 100 . In other words, the centre-frequency of the scaled antenna will be 1 GHz. Just for confirmation, the
computed input impedance is shown in Fig.4.2-10 and is seen to have the same behaviour as that shown in
Fig.4.2-7 scaled down in frequency to be centered around 1 GHz. In the remainder of the present Section
4.2, when we refer to computed values, it is the scaled antenna to which we are referring.
Fig. 4.2-10 : Computed input resistance and reactance of the scaled antenna.
The flat-patterned shape of the antenna, as shown in Fig.4.2-11, was laser-cut out of a sheet of C110
copper of thickness 0.45mm. This copper has a nominal conductivity of
7
5.8782×10 S/m. The flat-patterned
shape was then folded to form the open cuboid shape, and the corners soldered, as shown in Fig.4.2-12.
A coaxial cable was run across the one side of the cuboid antenna, as shown in the schematic diagram in
Fig.4.2-13. The cable follows a current path on the cuboid shape; the outer conductor of the coaxial cable is
soldered to the cuboid all the way to one side of the feedpoint gap; the centre conductor of the coaxial cable
then jumps the gap and is soldered to the other side of the gap. This approximates an infinite balun. In
practice antennas of this type that are used with a sensor package would be fed directly from the enclosed
electronics. The coaxial feed arrangement is simply used here to permit experimental work.
100 Basically, if the dimensions are scaled by a factor x, the ‘frequencies’ are shifted by
1
( ) x  . This is expected from the theorem
of similitude [SINC 48] [POPO 81].
79
Fig. 4.2-11 : Flat-patterned antenna shape scaled by a factor of two for fabrication purposes.
Fig. 4.2-12 : Fabricated scaled antenna.
80
Fig.4.2-13 : Schematic illustration of the coaxial cable feedline location. The red line represents a small gap
(first depicted in Fig.4.2-4), the black line the coaxial cable outer conductor, and the white line the latter’s
inner conductor.
Fig.4.2-14 : Plot of the computed and measured reflection coefficient of 3-D shape electrically small
antenna for a 50Ω reference impedance.
81
The measured
in ( )
f  performance is compared to that computed in Fig.4.2-14. It is immediately clear
that the centre-frequency (as defined by the best reflection coefficient performance) of the fabricated antenna
is shifted up (by 0.8%) with respect to the computed one. Other parameters for the antenna are summarized
in Table 4.2-1, to which we now refer. The measured self-resonance frequency is 995 MHz instead of the
desired 1GHz, which represents a shift down (0.5%) in frequency from that computed. Using as a reference
impedance the actual computed
in
47.3 R   instead of 50 moves the centre-frequencies slightly closer,
but does not make them coincident. There are many different features on this shaped antenna. We have
measured a set of randomly selected features, and calculated the difference between their values on the
computed and fabricated shape. The fabricated shape dimensions differ on average by 2.3% from those of
the computed shape. Thus we might expect the self-resonant frequency of the fabricated antenna (measured
as 995 MHz) to be the just-mentioned percentage lower than the computed value (1000 MHz); it is in fact
only 0.5% lower. Although it has been difficult to quantify, we conjecture that these shifts have been caused
by the frequency-behaviour of the feeding network on the fabricated antenna, a fact supported by the
observation (in Table 4.2-1) that the measured
in
R at self-resonance is in fact 61 and not 47.3. The value
of
z
Q found using (2.2-54) from computed and measured data are also shown in Table 4.2-1. The bandwidth
(for a -9.54dB reflection coefficient) obtained using the
z
Q in expression (2.2-55), and that using the
in ( )
f  plot are in agreement when computed quiantities are concerned. However, there is a significant
difference between the bandwidth gleaned directly from the
in ( )
f  plot and (2.2-55) when the measured
data is used.
Next we turn to the simulated radiation efficiency. We have considered that the antenna is constructed
from copper, with the resulting radiation efficiency computed to be
rad
98.94%   . This is clearly an over-
optimistic value 101 , and confirms the known fact that it is difficult to accurately predict
rad
 . Although the
goal in this research is the establishment of the shape synthesis tool and a demonstration of its use to obtain
self-resonant antennas that conform to some specified 3D surface, we nevertheless wished to obtain a more
realistic estimate of
rad
 . This has been done using the argument and results described in Appendix A. It
results in a more realistic (if pessimistic) estimate of
rad
65%   .
101 If the CEM engine’s capabilities of computing
rad
 were to improve this would automatically become part of the shaping
manager. This supports the argument that a commercially available CEM engine be used as an existing resource in the shaping
manager.
82
Finally, the computed normalized radiation pattern in the azimuth plane shown in Fig.4.2-9(b) is compared
to the measured pattern in Fig.4.2-15. The decrease in the pattern level in the vicinity of the 180  
direction is attributed to the location of the coaxial cable of the antenna test chamber equipment (emerging
from the pedestal holding the antenna-under-test, and connecting to the latter there).
Fig.4.2-15 : Polar plot of the computed and measured normalized patterns in the azimuth plane.
Table 4.2-1 : Summary of some performance measures of the shaped open cuboid antenna
Quantity Intended
Value
Computed
Value
Measured
Value
Frequency of Minimum
in ( )
f 
- 998.5 MHz 1007 MHz
Minimum
in ( )
f 
- -22.7 dB -14.7 dB
Bandwidth (-9.54dB) extracted from
in ( )
f  plot
- 2.8% 1.6%
Self-Resonance Frequency (
in
0 X  ) 1000 MHz 1000 MHz 995 MHz
in
R at Self-Resonance Frequency - 47.3 61
z
Q - 25.82 19.44
Bandwidth (-9.54dB) from
z
Q at
o
f - 2.8% 3.7%
83
4.2.7 Shaping of a Closed Cuboid of the Same Size
It is of interest to briefly consider the difference in the outcome of the shape synthesis if the starting shape
is a cuboid of the same size as that just discussed, but is closed (that is, has its bottom face present) as shown
in Fig.4.2-16(a). The centre frequency is 2 GHz, and up/down geometrical symmetry is enforced. Fig.4.2-17
shows that the shaping procedure takes 150 generations. The shaped geometry is that in Fig.4.2-16(b), which
also shows the feedpoint (located after shaping has been completed) denoted by the red line. The computed
in ( )
Z f and
in ( )
f  are shown in Fig.4.2-18, and is the principal point of interest here. Although an anti-
resonance behaviour occurs at a much lower frequency (and we could use this, with 1 ka  at 1.85 GHz),
with the additional conductor available (the bottom face) the shaping procedure has selected the second
“ordinary-resonance” where the variation of the real and imaginary parts of
in ( )
Z f are more gradual with
respect to frequency. Indeed, at the self-resonance frequency (2 GHz) we have 10.62
z
Q  , which is less
than half the computed value (in Table 4.2-1) when the open cuboid starting shape was used. This predicts a
-9.54dB bandwidth of 6.7%.
We observe that the open cuboid
in ( )
Z f result in Fig.4.2-7 differs in character (not just values) from that
of the present closed cuboid in Fig.4.2-18(b). Dipole-like antennas have a first anti-resonance at a frequency
higher than that at which its ordinary resonance occurs, whereas the opposite is true for loop-like antennas. It
would appear that the different current paths that emerge from the shaping procedure thus also determine
which kind of input impedance behaviour will result, depending on the starting shape and hence the
associated conducting material available to it.
84
(a) (b)
Fig.4.2-16 : (a). The starting shape of the closed cuboid, and (b). the resulting shape synthesized antenna.
Islands and vertex-only connections have not yet been removed in (b).
Fig.4.2-17 : The objective function value of the fittest shape in each generation, versus the generation
number.
85
(a) (b)
Fig.4.2-18 : (a). Computed input impedance of the antenna shape-synthesized from the closed cuboid
starting shape and (b). its computed input reflection coefficient for reference impedance
o
50 Z   .
4.2.8 Shaping of a Closed Conducting Sphere
The conducting sphere has been of interest for a long time, and its analysis harks back to a time when it
was the only finite-sized PEC object that could be treated in a rigorous analytical way [HARR 61], and so as
much information on electromagnetic scattering was gleaned from it in order to understand scattering in
general. Also, the so-called spherical wave modes used in deriving various forms of the Q-bounds mentioned
in Section 2.2.10 are in fact the characteristic modes of a PEC sphere of the same radius as the minimum
sphere [LI 2019]. The spherical dipole antenna has also retained the interest of many [BEST 05]. But of
course these are restricted to a spherical surface. We very briefly consider the spherical starting shape on the
left in Fig.4.2-19, essentially only to dispel any concerns that the shaping tool needs starting shapes to be
rectangular in form. They can literally be anything. The geometry here is such that the radius of the
minimum sphere is obviously that of the actual starting shape itself, and in this case 0.4189 ka  for a 10mm
radius sphere at 2 GHz. The starting shape was densely meshed into 856 N   triangles, which resulted in
an initial
dof
1284 N  . The objective function was again
1 o
( )
obj
F f   . The resulting shape is that on the
right in Fig.4.2-19, obtained in roughly 450 generations of the shape synthesis process. The computed
53.88,
z
Q  and so (2.2-55) predicts a -9.54dB bandwidth of 1.3%.
86
Fig.4.2-19 : The starting shape of the 10mm radius closed sphere is shown on the left, and the shaped
antenna on the right (along with its feed location).
4.3 THREE-DIMENSIONAL SHAPE SYNTHESIS EXAMPLE #2 : PRINTED ELECTRONICS
COMPATIBLE ANTENNA
4.3.1 Preamble
Although printed electronics can be done over 3D surfaces, indications are that cost considerations likely
encourage the use of such printing on a flat surface that is then folded. In such cases the desire is to simply
fold and bond the printed shape onto the package without any need for any sort of conductive bonding (or
soldering) at the edges.
87
4.3.2 Starting Geometry & Objective Function
The starting shape is the non-planar cuboid-like structure shown in Fig.4.3-1(a), with 10mm d  ,
w 6mm h   and 2mm s  . The corners are purposefully missing so that the resulting shape satisfies the
assembly requirements noted in Section 4.3.1. The centre frequency 3 GHz
o
f  , at which the wavelength
100mm   , and so /10 d   . Use of some elementary analytical geometry shows that the radius of the
minimum sphere is
2 2
2 /2 a h d   , and so in this case / 0.0768 a   and hence 0.4826 ka  . The
starting shape is meshed into 644 N   triangles, resulting in an initial 922
dof
N  . The objective function is
the same as that in expression (4.2-1).
Although we do not do so here, the shape synthesis procedure and tool can deal with any starting shape.
For example, if it is determined that quick assembly (folding) of such an antenna would result in a “cubical”
shape with some rounding of the edges, as depicted in Fig.4.2-1(b), the starting shape could be specified to
accommodate this.
Perfect
Corner
Bulging of
Substrate at
Corner
(a) (b)
Fig.4.3-1: (a).Cuboid-like starting shape with purposefully missing corners, used in this example, and
(b).Possible alternative shape that compensates for assembly ‘imperfections’. Note that w 2s d   .
88
4.3.3 Outcome of the Shape Synthesis Process
The shaped antenna geometry is that on the left in Fig.4.3-2. This shape, which with 0.4826 ka  is
electrically much smaller than that in Section 4.2, was achieved 5000 generations, and resulted in
1 o
( ) 0.034.
obj
F f    It is clear from Fig.4.3-3 that the eigenvalues of the higher-order CMs are much
larger than that of the lowest order one. The feedpoint location was found to be as shown on the right in
Fig.4.3-2. In Fig.4.3-5 the first three eigenvalues of the shaped structure are plotted versus frequency,
confirming that
1
 indeed goes through zero at 3 GHz. We will make further use of this information in
Section 4.3.4.
Fig.4.3-2 : Two different views of the geometry of the shaped antenna. The left view shows the feed location
as a red line.
89
Fig.4.3-3 : The cost function of the fittest member in each generation of shapes versus the generation
number.
Fig.4.3-4 : Eigenvalues
n
 vs mode index n for the final shaped antenna at
o
f , normalised to
1
 and
expressed in deciBels.
90
Fig.4.3-5 : Eigenvalues of the first three characteristic modes versus frequency.
4.3.4 Computed Performance of the Shaped Antenna
With a feed located at the location on the shaped antenna as shown in Fig.4.3-2, a driven analysis provides
its performance measures in Fig.4.3-6 through Fig.4.3-8, as well as Table 4.3-1. Some comments are in
order. Firsly, we note from Fig.4/3-7 that the modal excitation coefficient of the lowest CM are more than
70dB more than that of the higher order CMs. This is truly a single-mode antenna. More so than the open
cuboid case for which a corresponding result is that in Fig.4.2-6. The reason is that the open cuboid had ka
roughly double that of the present antenna. Fig.4.3-8 shows the input impedance character is that of utilizing
an ordinary resonance at the desired
o
f , this occuring at a frequency lower than that at which the first anti-
resonance occurs. Table 4.3-1 shows that the -9.54dB bandwidths predicted from
in ( )
f  and from
z
Q are
much the same. Also shown is the the quality factor
cm
Q predicted from the slope of the curve
1 ( )
f  in
Fig.4.3-5, at
o
f , via expression (2.2-53). The value of
cm
Q (predictable from the eigeanalysis of the shaped
antenna before ever specifying the feed location), and
z
Q (predictable from ( )
in
Z f and a driven analysis
once the feed has been specified), are almost identical for such a single-mode antenna. It is not necessary to
use
cm
Q , but showing its equivalence to
z
Q anchors the shaping procedure being used here in the firmly in
the characteristic mode framework. Finally, the computed total gain patterns
91
(a) (b)
Fig.4.3-6 : (a). Computed
in ( )
f  of the shaped antenna for a reference impedance
o
50 Z   , and (b). the
associated current distribution on the fed antenna at 3 GHz.
Fig.4.3-7 : Modal excitation coefficient
n
 versus modal index n, at 3 GHz.
92
Fig.4.3-8 : Computed input resistance and reactance of the shape-synthesized antenna.
Table 4.3-1 : Summary of some performance measures of the shaped printed electronics compatible shaped
antenna
Quantity Intended Value Computed Value
Frequency of Minimum
in ( )
f 
- 3.0002 GHz
Self-Resonance Frequency (
in
0 X  ) 3.0 GHz 3.0002 GHz
in
R at Self-Resonance Frequency - 46 
Maximum Gain at 3 GHz - 1.47 dBi
cm
Q at 3 GHz - 60
z
Q at 3 GHz - 59
rad
 at 3 GHz - 89%
Bandwidth (-9.54dB) extracted from
in ( )
f  plot
- 1.10 %
Bandwidth (-9.54dB) from
z
Q - 1.20 %
93
(a) (b)
(c)
Fig.4.3-9 : Computed normalized radiation pattern cuts at
o
f . The elevation plane cuts are
(a). 0 ,0 360      and (b). 90 ,0 360      . The azimuth pattern cut 90 ,0 360      is
shown in (c).
94
4.4 THREE-DIMENSIONAL SHAPE SYNTHESIS EXAMPLE #3 : CUBOID WITH BATTERY
INSIDE
4.4.1 Starting Geometry
It was noted in Section 4.1.2 that sensor packages often have batteries within them. In this demonstration
of the shaping procedure that has been developed in this thesis we consider the starting shape to be the open
cuboid with one face removed, and a closed smaller cuboid inside it, as depicted in Fig.4.4-1. The inner
closed cuboid represents a battery, and it is constrained from participating in the shaping. The outer open
cuboid is that to be shaped. The starting shape dimensions are 13mm d  , 10mm h and 6.5mm s  .
The centre frequency is 2
o
f GHz  , at which the wavelength 150mm   , and so the sides of the outer
cuboid are each of electrical dimension 0.0867 d   ( /11.5 d   ). The radius a of the minimum sphere is
then ( 3/ 2) 0.0751 a d    , and hence 0.4716 ka  . The starting shape is a densely meshed into 822 N  
triangles, which results in an initial
dof
1219 N  . The objective function is that in expression (4.2-1).
Fig.4.4-1: Starting shape consisting of an outer open cuboid with an inner closed cuboid.
95
4.4.2 Outcome of the Shape Synthesis Process
The shaping process reduced
1 o
( )
obj
F f   to a value 0.0028 within several thousand generations,
resulting in the shape shown to the left in Fig.4.4-2.
Fig.4.4-2 : The shape synthesized antenna is shown to the left, with the feedpoint identified in red. The same
geometry is shown to the right, with the MM mesh removed and the (unshaped) battery shown in blue.
4.4.3 Computed Performance of the Shaped Antenna
As for most electrically small antennas the curve is
in ( )
f  similar to that in Fig.4.3-6(b), with reflection
coefficient
in
30dB    at (in this case) 2 GHz, referenced to 50. The modal weighting coefficients in
Fig.4.4-3 once again reveal this 0.4716 ka  shape synthesized antenna to be a perfectly single-mode one. It
has achieved self-resonance in the presence of the inner cuboid (battery) by accounting for the presence of,
and indeed exploiting, the latter object during shaping, as confirmed by the fact (not unexpected) that
Fig.4.4-4 shows a significant induced current density on the battery. The input impedance character in
Fig.4.4-5 reveals that the shaping process has utilized the first anti-resonance of the structure, there being no
ordinary resonance lower in frequency where 0
in
R  . The total gain patterns are given in Fig.4.4-6 and, like
most electrically small antennas, these are broad (some even call such patterns “close to isotropic”). The
performance is summarized in Table 4.4-1; this information will be used along with that in the earlier tables
for comparison purposes in Fig.4.7-1. Although we mentioned earlier that (and this is not unusual) the
96
computed
rad
 values are optimistic, we note the trend that as we go to an antenna with a smaller ka value
the predicted
rad
 values decrease. This follows the trend, at least, found and expected by antenna
researchers in practice.
Fig.4.4-3 : Modal excitation coefficient
n
 versus modal index n, at 2 GHz.
Fig.4.4-4 : Current distribution on the driven shaped structure at 2 GHz.
97
Fig.4.4-5 : Computed input impedance of the shaped-synthesized antenna.
(a) (b)
Fig.4.4-6 : Computed total gain patterns (in dBi). The elevation plane cuts shown in (a) are for 0   ,
45   , 90   and 135   . The azimuth pattern cut is that in (b).
98
Table 4.4-1 : Summary of some performance measures of the shaped antenna with a “battery” enclosed
Quantity Intended Value Computed Value
Frequency of Minimum
in ( )
f 
- 1.9972 GHz
Minimum
in ( )
f 
- < -30 dB
Self-Resonance Frequency (
in
0 X  ) 2.0 GHz 1.9972 GHz
in
R at Self-Resonance Frequency - 59.1 
Maximum Gain at 2 GHz - 1.13 dBi
z
Q at 2 GHz - 66.18
rad
 at 2 GHz - 92.3 %
Bandwidth (-9.54dB) extracted from
in ( )
f  plot
- 1.03 %
Bandwidth (-9.54dB) from
z
Q - 1.07 %
4.5 THREE-DIMENSIONAL SHAPE SYNTHESIS EXAMPLE #4 : OPEN CUBOID WITH
PATTERN CONSTRAINTS
4.5.1 Preamble
Until now we have used an objective function that strives simply to arrive at an antenna that is self-
resonant. This is often sufficient. The electrical size of the resulting antennas is such that their radiation
patterns are relatively broad almost by default. However, many sensor antennas will need to operate in the
same wireless environment as mobile devices, and so the mean-effective-directivity (MED) might be an
important quantity. This quantity is easily defined (and hence calculated) for a driven antenna, as explained
in Section 2.2.9. However, recall that in the shaping procedure being pursued here the feed location is not
known until the shaping is completed. Fortunately, we have been able to show (in Section 4.5.2) that it is
possible to determine the MED for a single-mode antenna without actually knowing the feed location. Once
this expression has been derived the starting shape form Section 4.3 is re-shaped with MED as an additional
component of the objective function, all other specifications being the same as in Section 4.3.
99
4.5.2 Maximum Effective Directivity for a Single-Mode Antenna
Based on the comments in Section 2.2.9, the mean effective directivity (MED) is
2
0 0
1
( , ) ( , ) ( , ) ( , ) sin
1 1
XPR
MED D D d d
XPR XPR
 
   
          
 
 
 
 
 
 
P P (4.5-1)
If an impressed source is used to drive a radiating structure whose CM fields   ,
n
E   are known, we saw in
Section 2.4 that it is possible to compute the modal weighting coefficients
n
 such that the far-zone radiated
field is
1 1 1
ˆ ˆ
( , ) ( , ) ( , ) ( , )
n n n n n n
n n n
E E E E
 
            
  
  
  
  
(4.5-2)
The orthogonality property of the far-zone fields means that the power radiated by the driven structure is
2
1
rad n
n
P 


  (4.5-3)
The partial radiation intensities are
 
2
1
1
( ) ,
2
n n
n
o
U , = E
 
    




(4.5-4)
and
 
2
1
1
( ) ,
2
n n
n
o
U , = E
 
    




(4.5-5)
The definitions in Section 2.2.7 then allow us to write the partial directivities of the antenna as
 
2
1
2
1
,
( ) 2
( )
n n
n
rad
o
n
n
E
4 U ,
D , =
P



  
   
 









(4.5-6)
and
 
2
1
2
1
,
( )
2
( )
n n
n
rad
o
n
n
E
4 U ,
D , =
P



  
  

 









(4.5-7)
100
The model weighting coefficients
n
 can of course not be found unless we specify how the antenna
structure is excited. In other words, until we specify the feed we cannot find the partial directivities.
However, if we have an electrically small single-mode antenna 102 then
1 m
   for 2 m  . The partial
directivity expressions (4.5-4) and (4.5-5) then become
 
2
1
2
( ) ,
o
D , E
 

   

 (4.5-8)
and
 
2
1
2
( ) ,
o
D , E
 

   

 (4.5-9)
In other words, on the assumption that a single CM will be excited on the antenna structure, we can find the
partial directivities without having to specify how this will be achieved 103 . Thus we can find the MED of
each single-mode candidate during the shape optimisation process. Use of (4.5-6) and (4.5-7) in (4.5-1) gives
   
2
2 2
1 1
0 0
2 1
, ( , ) , ( , ) sin
1 1
o
XPR
MED E E d d
XPR XPR
 
   

          

 
 
 
 
 
 
P P (4.5-10)
4.5.3 The Objective Function
The objective function used is now
o
1 o
( )
( ) 1
4
obj
MED f
F f 

 
  
 
 
(4.5-11)
with
1 o
( ) f  the magnitude of the CM with the lowest eigenvalue at
o
f . We know from Section 2.2.9 that
the maximum value of MED is 4  , and that MED is always positive. Hence maximizing MED is the same
as minimizing the 2 nd term on the right hand side in (4.5-11).
The goal is to demonstrate the utility of being able to include MED in the objective function; we did not
have a specific propagation environment in mind. We therefore used XPR=1 (0 dB) and Gaussian
distributions [DOUG 98]
102 Only one significant CM, which we always label as mode
1 n  .
103 Of course, once the shaping has been completed, we then decide where the structure will be fed.
101
 
2
2
/2 /2
( , )
V V
m
A e
  
 
 
    
 
 P (4.5-12)
and
 
2
2
/2 /2
( , )
V V
m
A e
  
 
 
    
 
 P (4.5-13)
with 0
V H
m m   , and 40
V H
    . This should result in the shaping attempting to achieve self-
resonance and making the azimuth radiation pattern closer to omni-directional. This was indeed the
outcome.
4.5.4 Outcome of the Shape Synthesis Process
Fig.4.5-1 compares the azimuth patterns of the present shaped antenna (with MED included in
obj
F ) and
that from Section 4.3 where it was not included. It is very clear that the present antenna is more omni-
directional, the maximum-to-minimum variation decreasing to 2.8dB from a value of 7.8dB when the MED
was not included in
obj
F in Section 4.3. But there has been a price to pay. The elevation patterns of the
present antenna in Fig.4.5-1(b) show that there is now a dip in the pattern for the 135   elevation plane;
this does not occur for the shaped antenna in Section 4.3. Note, however, that the shaping scheme, being
mindful of the desired MED, has ensured that this dip does not occur in the azimuth plane. It has attempted
to “twist” it out of the region of MED interest. The shape synthesis manager has ensured that we have gotten
what we asked for, as defined by
obj
F . In the interests of completeness, we show the present shaped antenna
geometry
102
Fig.4.5-1 : Computed total gain patterns (in dBi). The azimuth plane cut is shown in (a), as the solid line,
along with the corresponding result from Section 4.3 shown as a dashed line. The plane cuts shown in (b) are
for 0   , 45   , 90   and 135   .
103
Fig.4.5-2 : (a). The geometry of the shaped antenna, and (b). Current distribution on the driven antenna.
4.6 POSSIBLE EXTENSIONS
It is known [COLL 19] that the “platform” on which an antenna that is of small or intermediate electrical
size (with its required broad radiation patterns) can have a significant influence on the antenna performance.
If it is possible to model the platform, plus the antenna proper, in the CEM engine, then the process just
described in this chapter can tale into account the effects of the platform by using the sub-structure CMs of
the antenna proper (or rather, the starting object which is shaped into the antenna) in place of the
conventional CMs that have been used. If such sub-structure CMs are used the shape synthesis procedure is
valid even if the platform (but not the starting object) is electrically large. The main difficulty is that an
electrically large structure is computationally taxing for the MM that we need to use as the underlying CEM
analysis to find the CMs.
In the example of Section 4.4 the “battery” effects were accounted for by simply designating the battery as
a region which could not be altered. An alternative approach would be to use the sub-structure CM concept
with the battery being Object#B, and then shape Object#A using its sub-structure CMs in the shaping
process. One advantage of this is that if the interior of the starting shape (Object#A) is occupied by both
conducting and dielectric material (eg. to more carefully the scattering effects of the embedded electronics),
this could be accommodated in the sub-structure CM concept 104 .
104 Sub-structure characteristic modes will be used in Chapter 5, albeit for slightly different purposes.
104
The computational time to perform a complete shape synthesis described in Sections 4.2 through 4.5 is
large, that the DMM approach has been exploited in the shaping manager to reduce this. A large portion of
the time is spent on the eigenanalyses needed to find the CMs of each candidate shape in the population for
each generation. This could be parallelized.
It is possible to augment
obj
F with additional terms, as was done in Section 4.5, as long as such terms can
be expressed purely in terms of CM quantities. The quality factor term in expression (2.2-53) thus could be
used to augment
obj
F in an effort to squeeze out some more bandwidth. However, as remarked at the end of
Section 4.2.5 this might do what we ask, but decrease
in
R for example. There is a trade-off. As [COLL 19]
reminds us, for instance, “efficiency and bandwidth can be traded, but they cannot be maximised together”.
4.7 CONCLUDING REMARKS
A recent authoritative paper 105 [COLL 19] based on its author’s many years of practical experience
identifies a number aspects related to the design of “small antennas”. In particular, it refers to antennas
whose applications will be in sensors that perform “monitoring, measuring and metering in almost every
area of economic and social activity”, within the framework of the Internet-of-Things. Several statements
relevant to the present work can be extracted from [COLL 19]:
 The electrical smallness of the antenna, and sometimes even the platform, is a challenge.
 Antenna performance is a function of the shape and size of the whole platform.
 Accounting for nearby conducting objects is crucial.
 The industrial design of the platform is often “unfriendly to antennas”.
The examples discussed in this chapter (and pre-emptively summarized in Section 4.1.1 at its beginning)
have shown that shape-synthesis can in principle contribute significantly to the accounting of the matters
bulleted above and, with the availability of the shaping tool, can do so in practice. The success, in practical
terms, of the shape synthesis process, and hence the shaping manager that has been developed, is dependent
on the accuracy of the CEM modelling used. Any improvements in the CEM engine used will seamlessly
improve the usefulness of the shaping process. Nevertheless, some experimental work was done as a “sanity
check”.
105 We mention [COLL 19] as a footnote here for convenience : B.S.Collins, “Practical application of small antennas in hardware
platforms”, IET Microwaves, Antennas & Propagation, Vol.13, No.11, pp.1883-1888, 2019.
105
An insightful comparative study of a very large number of electrically small and intermediate-size
antennas was done in [SIEV 12]. Fig.4.7-1 shows an illuminating graph taken from this reference. In order to
place our shape-synthesized antennas in this context we have shown “windows” within which the
performance of the examples in Section 4.2 through 4.4 lie. They are “frames” instead of “dots” because we
used computed or measured bandwidths with computed radiation efficiency, and radiation efficiency
multiplied by a factor 0.66, as described in Appendix A, whatever combinations were available for a
particular example, and so obtained more than one
rad
B  value for each antenna. The shape-synthesized
antennas clearly have a better performance than some other published antennas, but have a lower
rad
B  than
some others of the same electrical size. The difference between the shaped ones obtained in this chapter is
that we have been able to achieve this performance even though the antenna is constrained to lie precisely
on the surface prescribed. This is not so with any of the antennas whose
rad
B  points are shown on the
graph.
Fig.4.7-1 : Compendium of bandwidth-efficiency products (
rad
B  ) of numerous antenna designs after
[SIEV 12], showing the location of this measure for the antennas shape-synthesized in Sections 4.2 through
4.4 as “red frames”. The vertical axis is the bandwidth-efficiency product and the horizontal axis is ka , with
a the radius if the minimum sphere.
106
CHAPTER 5
The Shape Synthesis of Closely-Spaced Antennas
5.1 PRELIMINARY REMARKS
In Section 2.6.3 we referenced an existing method for the shape synthesis of single electrically-small or
intermediate-size (‘single-mode’) antennas for which the feed location need only be selected after the
shaping has been performed, but noted that its use had been demonstrated for planar conducting antennas
only. Chapter 4 removed this geometrical restriction, and successfully extended the shape synthesis
technique to that of antennas comprised of 3D conducting surfaces, for the first time. It was done using a
new shape synthesis tool whose development was detailed in Chapter 3; such a facility has not been
previously available.
The present chapter extends the shape synthesis work to the case of two closely-spaced antennas that
operate at two different frequencies. The sub-structure characteristic mode concept is exploited in a
completely new way. As in Chapter 4, we are concerned with a synthesis in which the shape of the final
antenna structures is an outcome of the synthesis process, there being some starting shape that is perturbed
by the synthesis procedure to produce the final antenna shapes. In addition, the selection of the feed-point of
each of the two antennas must only need to be done after the shaping.
Precisely what is meant by “two closely spaced antennas” is explained in terms of the specification of the
starting geometry for the synthesis process in Section 5.2. Section 5.3 points out why a simple extension of
what has been done in Chapter 4 by using appropriately altered objective functions, is not necessarily an
acceptable option. A new shaping prescription is then described in Section 5.4, and it is pointed out how this
can be accommodated by the shaping tool developed in Chapter 3. Section 5.5 describes an example of such
closely-spaced-antenna shaping and provides experimental validation. Section 5.6 concludes the chapter.
5.2 CLOSELY-SPACED ANTENNAS : STARTING GEOMETRY
We begin with a starting geometry, which consists of two conducting objects that will be referred to as
Object#A and Object#B, as was done when discussing sub-structure characteristic modes in Section 2.4.3.
This is depicted in Fig.5.2-1. These objects must together now be shaped to provide two antennas. We
require that after shaping there must eventually be a feedpoint on Object#A that that acts as the feedport for
an antenna that operates at frequency
a
f , and a similar feedport on Object#B that acts as the feedport for an
107
antenna operating at
b
f . This is shown schematically in Fig.5.2-2. Without loss of generality we
assume
a b
f f  . When the feedport on Object#A is excited we have an antenna that utilizes both Object#A
and Object#B, and we will call this Antenna#A. Similar comments apply to Antenna#B.
We assume the dimensions of the starting shapes are chosen such that, individually, each separate object is
electrically not too large, at
a
f and
b
f , respectively. But the other object may be electrically large. In other
words, Object#A has a ka value 106 at frequency
a
f that allows it to be a single-mode structure 107 in a sub-
structure CM sense, even though the two objects taken together may not be sufficiently small electrically
that the combined structure is single-mode in a conventional CM sense at
a
f . Similarly, Object#B has a ka
value at frequency
b
f that allows it to be a single-mode structure in a sub-structure CM sense, even though
the two objects taken together may not be sufficiently small electrically that the combined structure is single-
mode in a conventional CM sense at
b
f . Frequencies
a
f and
b
f could be widely separated, as long as the
above conditions apply. The overall starting geometry, which includes both starting objects, is meshed 108
into a set of triangles. The mesh should be sufficiently dense to ensure accuracy of the computational
electromagnetics (and so should be so at the highest frequency of interest) and to provide sufficient
geometrical resolution for the shaping process.
Object#A
Object#B
Fig.5.2-1 : Two closely-spaced starting shapes.
106 We continue to use the symbol a to denote the radius of the minimum sphere surrounding each object separately.
107 Recall that the shape synthesis method works when the object being shaped is “single-mode” in the sense that only one dominant
CM is sufficient to describe the behavior of the antenna.
108 All the steps outlined in this chapter are of course executed by the shaping manager described in Chapter 3.
108
Shaped
Object#A
Shaped
Object#B
Feedport of
Antenna#A
Feedport of
Antenna#B
Fig.5.2-2 : Two closely-spaced shape-synthesized antennas.
5.3 OBJECTIVE FUNCTIONS
Similar to what was done in Chapter 4 when using conventional CMs, except that here we use the
eigenvalues of the sub-structure CMs of Object#A and Object#B 109 respectively, we can define objective
functions for Antenna#A and Antenna#B as
1
( )
A Asub
obj a
F f   (5.3-1)
and
1
( )
B Bsub
obj b
F f   (5.3-2)
We might be tempted to suggest that the closely-spaced antenna synthesis can then be done by simply
defining some composite objective function such as (with w 1 and w 2 being user-selected weights)
1 1 2 1
w ( ) w ( )
Asub Bsub
obj a b
F f f     (5.3-3)
  1 1
,
( ) , ( )
Asub Bsub
obj a b
a b
F Max f f    (5.3-4)
or even
109 Given the terminology introduced in Section 5.2, it is incorrect to talk of the sub-structure CMs of Antenna#A or Antenna#B,
since these antennas will eventually “use” both objects to fulfil their duties.
109
2 2
1 1
1
( ) ( )
2
Asub Bsub
obj a b
F f f     (5.3-5)
The latter objective functions incorporate the dominant eigenvalues of both sub-structure modes, at
frequencies
a
f and
b
f , respectively. We could instruct the shaping manager execute the GA to shape
synthesize both objects simultaneously by minimising (5.3-3), (5.3-4) or (5.3-5). The problem with this
shaping prescription is that the feed locations resulting from the shaping may both lie on one of the two
objects. Section 5.1 has recognized that this is not what is desired in most such two-antenna situations; we
want the respective ports to end up on physically different objects. An alternative shaping prescription is
needed.
5.4 SHAPING PRESCRIPTION
The alternative shaping prescription is best explained by referring to the flowchart in Fig.5.4-1. In the
shaping prescription used in Chapter 4 some trial runs were first performed, and then these were followed by
actual runs that eventually produced the shaped antenna. This was described in Section 3.9. A similar
philosophy is used here.
The first step is to perform a trial run of the GA to shape Object#A using its dominant sub-structure
eigenvalue (at
a
f ) to minimize
1
( )
A Asub
obj a
F f   , without altering Object#B. At the end of this trial run
Object#A will have been altered. Then a trial run of the GA is performed to shape Object#B using its
dominant sub-structure eigenvalue (at
b
f ) by minimizing
1
( )
B Bsub
obj b
F f   , without any shaping of the
(now-altered) Object#A. At the end of this trial run Object#B will have been altered. In other words, both
objects are now different from their starting shapes. As in Chapter 4 these trial runs will have provided
“advice” on various parameter settings for the GA. Actual runs are then executed as follows : An actual run
(consisting of several generations) is carried out to minimise
A
obj
F and hence further shape Object#A without
altering Object#B. An actual run (again consisting of several generations) is then carried out to minimise
B
obj
F and hence further shape Object#B without altering the current shape of Object#A. This pair of actual
runs can be called an iteration, as shown in Fig. 5.4-1. These iterations are then repeated until both of the
fitness functions are satisfactorily minimised.
110
At the end of the shaping process the mesh density on the resulting shapes is refined if necessary, the sub-
structure modes at the respective frequencies computed, and the modal currents on the objects observed.
This allows one to select feedpoints in the manner described in Section 3.10.
TRIAL RUN:
Keep Object#B Fixed
Shape Synthesize Object#A at Frequency
f a Using its Sub-Structure Modes.
No
TRIAL RUN:
Keep Object#A Fixed
Shape Synthesize Object#B at Frequency
f b Using its Sub-Structure Modes.
ACTUAL RUN:
Keep Object#A Fixed
Shape Synthesize Object#B at Frequency
f b Using its Sub-Structure Modes.
Shaping Complete
0?
a
obj
F 
Yes
0?
b
obj
F 
ACTUAL RUN:
Keep Object#B Fixed
Shape Synthesize Object#A at Frequency
f a Using its Sub-Structure Modes.
Single
Iteration
Fig. 5.4-1 : Trial Run - Flowchart#4
111
5.5 SHAPE SYNTHESIS EXAMPLE : TWO ANTENNAS WITH BROADSIDES PARALLEL
5.5.1 Starting Geometry & Shaping Constraints
We use the starting geometry structure in Fig.5.5-1, consisting of two parallel conducting plates (Objects
#A and #B). The goal was to shape synthesize two antennas, one to operate at 1.5
a
f GHz  and the other at
2.0
b
f GHz  . The dimensions were 60mm
a
d  , 40mm
b
d  and 24.5mm. h  We again let the symbol a
represent the radius of the minimum sphere, depicted by the outer (blue) circle in Fig.5.2-1, and assume
a b
d d  . Some analytical geometry then shows that the minimum sphere radius is 2 /
a
a d  as long as
2 ( )
b a b
h d d d   . In this case, therefore, 42.4mm a  , and 1.33 ka  at
a
f , whereas we have 1.78 ka 
at
b
f . It should be remembered that this minimum sphere encloses both objects.
We have enforced symmetry about the xz-plane. This was done to allow one half of the eventual shaped
antenna to be placed above a conducting groundplane. There were no constraints placed on any portions of
the geometry (no “disallowed” regions that may not be shaped). The starting shape was densely meshed into
844 N   triangles, with 592 on Object#A and 252 on Object#B, which resulted in the initial number of
degrees of freedom being
dof
1215 N  .
The shape synthesis procedure outlined in Section 5.4 was used to in essence reduce the eigenvalue of the
dominant sub-structure CM of Object#A (Object#B) in the presence of Object#B (Object#A) at
a
f (
b
f ) to
values close to zero.
Fig.5.5-1 : Starting conductor shapes used.
112
5.5.2 Outcome of the Shape Synthesis Process
Since the shaping prescription is different form that used in Chapter 4, some view of how such a
prescription progresses is in order. For instance, Fig.5.5-2 shows the behaviour of
B
obj
F during the actual run
of the final iteration. Fig.5.5-3 shows the value of
A
obj
F (blue circle) and
B
obj
F (red asterisk), versus the
iteration number, at the end of the first actual run (top graph) and the second actual run (bottom graph) in
each iteration. It is clear that eventually both antennas (that is, as seen in a sub-structure CM sense) become
self-resonant. The shaped antennas, and the resulting feedports, are shown in Fig.5.5-2.
Fig.5.5-2 : Value of
B
obj
F at the end of each iteration in Fig. 5.4-1
113
Fig.5.5-3 : The value of
A
obj
F (blue circle) and
B
obj
F (red asterisk), versus the iteration number, at the end of
the first actual run (top graph) and the second actual run (bottom graph) in each iteration
Fig.5.5-4 : Shaped geometry and feed port locations. The port labelled “Feed#A” is the feedport for
Antenna#A, and similarly for Antenna#B.
5.5.3 Performance Computation & Measurement of the Shaped Antennas
The current distributions on the complete radiating structure when each of the antenna feed ports are excited
in turn, with the other terminated, are shown in Fig.5.5-5 and Fig.5.5-6. It is clear that each antenna relies for
its operation on both Object#A and #B. The antennas were fabricated in the same manner as that for the
114
open cuboid in Section 4.2, and of the same material. An annotated photograph of the fabricated antenna pair
is shown in Fig.5.5-7.
Two issues should be emphasized. Using this shaping prescription, the method of Section 3.10.1 does
indeed ensure the feedports are on different objects, as seen by the feedports of Antenna#A and Antenna#B
indicated in Fig.5.5-4 and Fig.5.5-7. Reference to the computed input impedances in Fig.5.5-8(a) and
Fig.5.5-9(a) confirms that, as far as the theoretical model is concerned, the antennas are self-resonant at the
desired frequencies of 1.5GHz and 2.0GHz, with 43
in
R   for Antenna#A, and 46
in
R   for Antenna#B,
there. Thus the new shaping prescription does successfully achieve the goal set in Section 5.1. We note the
unusual character of the impedances in Fig.5.5.9(a) due to the coupling between the objects forming the
antenna.
Fig.5.5-11 gives the computed and measured
in ( )
f  of the shaped antennas. The computed frequencies
for minimum
in ( )
f  are 1.4725 GHz and 2.0005 GHz, slightly shifted 110 from the prescribed (and
obtained) self-resonance frequencies of 1.5GHz and 2.0GHz. This can be so because
in ( )
f  does not
appear in any objective function, since it would be necessary to select a feed location before shaping if we
wished to do that. In most cases the minimum
in ( )
f  does occur close to the self-resonance frequency,
especially for antennas with smaller ka values. In the present case the ka values of Object#A (taken
individually) is 1.3, and so the resulting structure is not as purely single-mode (in its individual sub-structure
CM sense) as it would have been had it (individually) been electrically smaller. This results in the shift
between the computed self-resonance frequency and frequency of minimum
in ( )
f  for Antenna#A.
The predicted -9.54dB (VSWR = 2) bandwidths of Antenna#A and Antenna#B, extracted from the
in ( )
f  plots, are 6.54% and 2.55%, respectively. The corrresponding measured values are 111 1.42% and
7.3%. Furthermore, the computed and measured frequencies for minimum
in ( )
f  for Antenna#A differ by
some 150 MHz. This is attributed to fabrication errors.
110 Although the computed
in ( )
f  values there are better than -14dB.
111 It is unclear why there is such a large difference between the measured and computed bandwidths.
115
Fig.5.5-5 : Computed current distribution (at 1.5GHz
a
f  ) on the complete radiating structure when excited
at the feed port of Antenna#A.
Fig.5.5-6 : Computed current distribution (at 2.0GHz
b
f  ) on the complete radiating structure when
excited at the feed port of Antenna#B.
116
Fig.5.5-7 : Photograph of the fabricated antenna structure.
(a) (b)
Fig.5.5-8 : (a). Computed input impedance at the feed port of Antenna#A. (b). Computed total gain in the
azimuth plane (xz-plane) of Antenna#A.
117
Fig.5.5-9 : (a). Computed input impedance at the feed port of Antenna#B. (b). Computed total gain in the
azimuth plane (xz-plane) of Antenna#B.
Fig.5.5-10 : Computed and measured input reflection coefficients (referenced to 50) of the two antennas at
their respective feed ports.
118
5.6 POSSIBLE EXTENTIONS
The sub-structure CM formulation in Section 2.4.3 can be extended from one involving two objects to one
involving more than two objects. For example, two closely-spaced antennas on an electrically large platform,
operating at different frequencies, could be shaped. The two objects to be shaped (as has been done earlier in
this chapter) would be Objects #A and #B as before, and the vehicle Object#C that is not altered by shaping.
As long as the ka for Objects#A and Object#B, taken individually, is sufficiently small, these will be single-
mode in a sub-structure sense. And hence the shape synthesis procedure can be applied. Furthermore, one
could define three objects as just suggested, where the third is not an electrically large platform, but a third
antenna, operating at a third frequency, which is also to be shaped. We have not exercised the shaping tool
for such cases because of the computational burden.
5.7 CONCLUDING REMARKS
At the start of Chapter 5 we stated a different two-antenna requirement, and devised a new shaping
prescription to achieve this. The shaping manager developed in Chapter 3 is such that the above shaping
prescription could be incorporated in a relatively routine manner. This is important, as different ones can
arise for different design requirements. It was then demonstrated by an example, and some experimental
backing, that the shaping prescription is able to achieve its goal. It provides self-resonance at the specified
frequencies of the two closely-spaced antennas, with each antenna “leaning” on the other to achieve the
desired performance at its particular operating frequency. It is a “shape-first feed-next” procedure that
ensures that each of the resulting feedports is on a different object. This sort of shape synthesis has
previously not been reported elsewhere.
119
CHAPTER 6
Sub-Structure Characteristic Mode Computation Utilizing Field-
Based Hybrid Methods
6.1 GOALS OF THE CHAPTER
Chapter 3 developed the machinery for a complex computational electromagnetics-based tool capable of
performing the shape synthesis of 3D conducting surface antennas through use of a full-wave method of
moments approach combined with the genetic optimisation algorithm.
Using this tool, Chapter 4 demonstrated for the first time how characteristic mode concepts can be utilized to
shape synthesize 3D conducting surface antennas, using the “shape-first feed-later” approach.
Chapter 5 extended the shape synthesis methods, using sub-structure characteristic modes, to the case of
closely-spaced antennas operating at different frequencies. One begins with two adjacent starting structures
from which two antennas are to be shaped, with the feed locations identified by the process, once shaping is
complete, situated separately on each of the two structures. One of the antennas can operate at a much lower
frequency than the other, since the single-mode requirement of the said shape synthesis method is preserved
through use of the sub-structure instead of conventional modes. We mentioned at the end of Chapter 5 that
the sub-structure approach can relatively easily be extended to the case of two structures (that must be
shaped in the manner stated) are situated in the presence of a third very much (electrically) larger object,
such as a platform of some sort. The question arises as to what one could do if the size of this third object,
for instance, is electrically so large that a MM analysis is not feasible.
Several researchers have, in verbal discussions at engineering conferences, pondered the idea of using
asymptotic methods to find the (sub-structure) characteristic modes of an object that sits on such a platform
that is electrically very large. In this chapter we demonstrate, for the first time, how this can be achieved
through the use of field-based hybrid MM/GTD methods. The fundamental concept is described in Section
6.2. Section 6.3 uses the method in an example. Sections 6.4 concludes the chapter.
In Section 2.4.3 the altered MM matrix for the sub-structure CM eigenvalue problems was denoted by the
symbol
sub
[ ] Z , as has become customary. In the context to be discussed in the present chapter (and only for
120
this chapter) we will alter the notation slightly in order to improve the clarity of the explanations to follow.
Thus some re-stating of facts from Sections 2.4.2 and 2.4.3 will be needed.
6.2 PROPOSED IDEA
Recall from Section 2.3 that if we model the electrically perfectly conducting (PEC) structure in Fig.6.2-
1(a) using an EFIE [KRIS 16] for the electric current density
s
J everywhere on the structure, the MM can be
used [VOLA 12] to discretize the EFIE into a matrix equation involving the MM matrix (generalized
impedance matrix) [ ] Z . If the object is alone in free space, the kernel of the EFIE will be a free space
Green’s function, and we can write, for ease of explanation in the present paper at least, the MM matrix as
[ ] [ ]
fs
Z Z  . The “fs” explicitly recognizes the use of the free-space Green’s function in the EFIE. It is then
possible (see Section 2.4.2) to find the characteristic modes (CMs) of the structure as the solutions of the
matrix eigenvalue problem [ ][ ] [ ][ ]
n n n
X J R J   , with [ ] [ ] [ ] Z R j X   . Quantities
n
 and [ ]
n
J are the
eigenvalue and eigencurrent of the n-th CM, respectively. The far-zone eigenfields can be found for each
eigencurrent [ ]
n
J , and are mutually orthogonal over the sphere at infinity. In Section 2.4.2 we called these
the conventional CMs of the PEC structure.
If we wish to find the CMs for a microstrip patch above an infinite groundplane, for example, an EFIE
must be used that utilizes a modified Green’s function to account for the presence of the infinite groundplane
and substrate [CHEN15a][ALRO 15]. The unknown in the EFIE is then the
s
J on the patch only. The
general situation can be depicted as in Fig.6.2-1(b), where an Object#A (eg. a patch) is in the presence of an
Object#B (eg. infinite groundplane), with the modified Green’s function being that which gives the fields
due to any current density radiating in the presence of Object#B. The EFIE unknown would be the
s
J on
Object#A only. Although [ ] Z would be obtained directly in such a case, without
per
[ ] Z being explicitly
computed separately, for the purposes of comparison with what is to be discussed below, we will write it as
fs per
[ ] [ ] [ ] Z Z Z   (6.2-1)
with the superscript “per” intended to indicate “perturbation”. It is this [ ] Z that would be used in the
eigenvalue problem mentioned earlier. In order to distinguish the resulting CMs from ones found via integral
equation formulations that use free space Green’s functions in their kernel, we explained in Section 2.4.3
121
that these can be called the sub-structure CMs of the patch (say Object#A) in the presence of the infinite
groundplane (say Object#B). Finding conventional CMs is not an option, as they would not make sense due
to the presence of the infinitely large Object#B. Thus, albeit not often explicitly stated, whenever CMs are
used in design work in such circumstances they are in fact sub-structure CMs [ALRO 17].
If the groundplane (of the microstrip patch antenna) spoken of above is of finite extent, which means that
an appropriate modified Green’s function in analytical form that will account for its presence is not
available, sub-structure CMs can still be found. As before (and not only for the patch and groundplane
situation), the EFIE uses the free space Green’s function, implying that the unknown is the
s
J on both
Object#A and Object#B taken as a whole. But in order to be able to find the sub-structure CMs, when
discretizing this EFIE using the MM, distinct expansion function subsets are purposefully located on these
two objects, so that the usual MM matrix can be partitioned as 112
[ ] [ ]
[ ] [ ]
AA AB
BA BB
Z Z
Z Z
 
 
 
(6.2-2)
All block matrices in (6.2-2) are found as part of the usual MM procedure. The sub-structure CMs of
Object#A in the presence of Object#B are then found from the [ ] Z in (1), but now with [ALRO 14][ALRO
16][ALRO 18]
[ ] [ ]
fs
AA
Z Z  (6.2-3)
and
per 1
[ ] [ ][ ] [ ]
AB BB BA
Z Z Z Z

  (6.2-4)
The above route for the sub-structure CM eigenvalue problem effectively uses a discrete form of the
modified Green’s function that accounts for the presence of Object#B, although this Green’s function is not
needed explicitly. We will, for convenience, here 113 refer to this way of finding the sub-structure CMs as the
MM/MM approach; the reason for doing so will become clear later. The reader is referred to [ALRO 16] for
a catalogue of the properties of sub-structure CMs; these are similar to their conventional CM counterparts.
Previously [ALRO 16] the altered MM matrix for the sub-structure CM eigenvalue problems was denoted by
112 Up to this point in the present chapter the notation is that used in Section 2.4, but this will now change, for reasons mentioned
at the end of Section 6.1.
113 That is, here in Chapter 6, but not elsewhere in the thesis.
122
the symbol
sub
[ ] Z . In the context to be discussed in the present paper we believe the notation in (1) improves
the clarity of the explanations to follow. It is also useful to reiterate that, if we were to want the conventional
CMs of the entire structure (namely Object#A and Object#B viewed as a single structure) then the [ ] Z used
in the eigenvalue problem would be the complete matrix in (6.2-2). However, if the antenna proper
(Object#A) were in the presence of an electrically large Object#B, such as a microstrip patch above a large
but finite groundplane, the conventional CMs would lose their usefulness. There would be too many
significant conventional CMs, and it would be difficult to relate these to the characteristics of the antenna
proper [ALRO 18][OUTW 17]. The use of sub-structure CMs allows the insight lost from conventional CMs
in such situations to be gleaned instead from the sub-structure CMs and once more aid design, as discussed
in [ETHI 14,Sect.VIII], [BERN 15] and [XIAN 19], for instance.
Now suppose that Object#B is electrically very large, for instance in the case of an Object#A (antenna
proper) mounted on a platform. Finding the sub-structure CMs of Object#A using the MM/MM approach
may not be computationally feasible if Object#B is electrically too large. Matrix (6.2-2) might be huge even
though [ ] [ ]
fs
AA
Z Z  is not, and finding
per
[ ] Z as per (6.2-4) might be computationally excessive,
especially if needed at several frequencies and if the geometry of Object#A is being repeatedly altered
during some design process. In this paper we wish to point out that such a dilemma can be circumvented if
per
[ ] Z is approximated using a MM/GTD hybrid approach. This fact could be discerned, as we have done,
from the definition of sub-structure CMs. However, this has not yet been explicitly stated anywhere or
example computations provided.
PEC Bodies
Grouped as
an Object
(a)
PEC Bodies
Grouped as
Object B
( ), ( ) E r H r
0 0
( , )  
PEC Bodies
Grouped as
Object A
(b)
Fig.6.2-1. (a) Several PEC bodies grouped together as a single isolated object, and (b) several PEC bodies
grouped as Object A and Object B.
123
6.3 SUB-STRUCTURE CHARACTERISTIC MODES COMPUTED USING THE MM/GTD
HYBRID APPROACH
The field-based MM/GTD method was pioneered by Thiele and co-workers [EKEL 80]. A review of this,
as well as current-based hybrid methods, is provided in [MEDG 91], with later enhancements described in
[THER 00]. Here we make use of the field-based MM/GTD hybrid method as implemented in FEKO [JAKO
10][JAKO 97][FEKO]. In the context of the present paper Object#A is meshed and expansion functions for
the unknown
s
J defined on it as usually done when using the MM. In spite of the looming presence of the
very large Object#B, the free space Green’s function is used to determine the fields of each expansion
function on Object#A over every other one (as if Object#B did not exist), and hence [ ] [ ]
fs
AA
Z Z  is found.
These ‘first-order’ fields are, however, supplemented by finding the contribution of Object#B via the GTD
[EKEL 80]. Each expansion function is in turn considered to be a source and the GTD-determined field
perturbations over every expansion function used to supply a perturbation
per
[ ] Z to the MM matrix.
Expression (6.2-1) then gives the [ ] Z needed in the eigenvalue problem. The reader is referred to the
aforementioned references for details, and information on how the approach is actually implemented in
practice. The above description is intended to elucidate the principle only.
Fig.6.3-1 shows a rectangular PEC plate (Object#A) in the vicinity of another large PEC plate arrangement
(Object#B). For the numerical experiment to follow Object#B was selected to be large, but not so large that
the MM/MM approach to finding the sub-structure CMs was unfeasible 114 , because we wish to compare
MM/MM and MM/GTD results. If we mesh both Object#A and #B and determine the sub-structure CMs of
Object#A with (6.2-4) used to find
per
[ ] Z , in other words the MM/MM approach, the first three sub-
structure CM eigenvalues obtained are those shown by the solid lines in Fig. 6.3-2. The corresponding
eigenvalues of Object#A with Object#B absent (and hence
per
[ ] 0 Z  ), namely the conventional CMs of
Object#A alone, are shown on the same graph as dashed lines. This has been done merely to show that, as
expected, the presence of Object#B has shifted the frequencies at which the CM eigenvalues are zero (the
CM resonance frequencies) lower in frequency, due to its ‘loading’ of Object#A. We show the frequency
range over which the eigenvalues are small, because the CMs with small eigenvalues are those of most
importance in antenna design.
114 We emphasize, however, that this is for the purposes of this numerical experiment only; the utility of the sub-structure CM
computation approach developed in this chapter is precisely that it can be used when Object#B is electrically so large that a
MM/MM route would not be feasible. Indeed, even for the numerical experiment in question, Object#B required 300,000
expansion functions.
124
If we mesh only Object#A, and use the MM/GTD approach to effectively find
per
[ ] Z , the sub-structure CM
eigenvalues are as shown as the solid lines in Fig. 6.3-3. They clearly approximate those found using the
(exact) MM/MM approach well, but without the need to handle the large MM sub-matrices related to the
electrically large Object#B. Observe that as the frequency increases the MM/GTD results approach the exact
(MM/MM) ones increasingly closely, as anticipated when using asymptotic methods in computations. If the
Object#B were not just the portion shown Fig. 6.3-1, but the whole ship in Fig. 6.3-4 from which it was
extracted, then the MM/GTD approach would even more computationally attractive. We here used the
Object#B in Fig. 6.3-1 simply because we wanted to demonstrate that the sub-structure CMs found via the
MM/GTD approach are indeed those obtained using the computationally more demanding MM/MM one.
The particular Object#B used was not so large that use of the MM/MM approach was prohibitive. It would
be so of Object#B were prohibitively large for the complete structure in Fig. 6.3-4, but the MM/GTD
approach would not be.
Fig.6.3-1 PEC Object#A (small plate) and Object#B (arrangement of large plates) used in this paper. The
size of square PEC plate Object#A is 127.5mm per side. Object#A is shown larger than it actually is in
relation to Object#B.
125
Ray tracing rules 115 [PAKN 16] decide which particular direct, reflected, edge-diffracted, corner-diffracted
and double-diffracted contributions are included in effectively finding
per
[ ] Z in the MM/GTD approach.
The manner in which the eigenvalues from the MM/GTD approach move closer to the exact MM/MM ones
as more GTD ray types are included is demonstrated in Fig. 6.3-5. As a further check the coupling integral 116
mn
C [HARR 71a,Eq(22)] between the sub-structure CM far-zone fields found using the MM/GTD method
was computed. The results for the first five modes are shown in Fig. 6.3-6. Theoretically 117 the value for
m n  should be zero, but are here around -40dB or lower, which is expected in actual CM computation.
Fig.6.3-2 Eigenvalues of the 1 st (──), 2 nd (──) and 3 rd (──) sub-structure CMs obtained using the
MM/MM approach to find
per
[ ] Z . Eigenavlues (- - -) of the first three (two of which are degenerate)
conventional CMs of Object#A in the absence of Object#B.
115 Briefly discussed in Section 2.3.5.
116 Defined in Sections 2.4.2 and 2.4.3.
117 But never precisely so numerically.
126
Fig.6.3-3 Eigenvalues of the 1 st (blue), 2 nd (red) and 3 rd (black) substructure CMs computed using the
MM/GTD method. Those found using the MM/MM (exact) approach to find
per
[ ] Z , as in Fig.6.3-2, are
shown as dashed lines. Those found using the hybrid MM/GTD approach to find
per
[ ] Z are shown as solid
lines.
Fig.6.3-4 Ship model from which the Object#B in Fig.6.3-1 was extracted. The sections on the rest of the
model are for visualisation reasons only; they do not represent the mesh needed for computational purposes
using the method of moments. The ship model is courtesy of Altair Inc. [FEKO].
127
Fig.6.3-5 Eigenvalues (▬▬) of the 1 st sub-structure CM found using the MM/MM approach, and those
(▬ ▬) using the hybrid MM/GTD approach (with all GTD terms included). Curve (• • •) is that with double
diffraction terms removed, () with corner diffraction contributions also removed, and (- - -) when the
only GTD terms included are the direct and reflected ray terms.
128
Fig.6.3-6 Magnitudes of the coupling coefficients expressed in deciBels ( 20log
mn
C ) between the first five
sub-structure CMs obtained when the hybrid MM/GTD approach is used to find
per
[ ] Z .
129
6.4 CONCLUSIONS
This chapter began with the puzzle of how to compute sub-structure CMs in a case where the antenna
proper (to be shaped) is small, but the platform is electrically so large that meshing of the complete structure
(radiator plus platform 118 ) for sub-structure CM computation using a MM is computationally unfeasible.
Essential to the solution of this problem was the important recognition, in Section 6.2, that sub-structure
CMs can in fact be computed using field-based hybrid methods. Section 6.3 then confirmed, by application
of this approach to an example, that it is indeed feasible to find the sub-structure CMs for some Object#A in
the presence of Object#B using such an approach. This greatly extends the range of problems to which the
shape synthesis methods, that are the subject of this thesis, can be applied, in a manner that has not yet been
done by others.
118 Object#A (radiator) plus Object#B (platform).
130
CHAPTER 7
General Conclusions
7.1 Contributions of the Thesis
The principal original contributions to antenna shape synthesis which have been presented in this thesis can
be described as follows :
■ At the start of this work only planar (1D) conducting surface antennas had been shape synthesized, either
with the feed specified beforehand or using characteristic mode theory with the feed only needing to be
specified after shaping. Using the work of Chapter 4, fully 3D conducting surface antennas can, and now
have been, shape synthesized using characteristic mode theory with the feed only needing to be specified
after shaping. Importantly, through definition of the starting geometry, the antennas can be constrained to fit
on, and exploit for their operation, some required shape. The starting geometry can also be selected so that
the resulting antenna is favourable as regards fabrication by some chosen method. The presence of nearby
objects, such as an enclosed battery or associated circuitry, can be accounted for in the shape synthesis. In
addition to shaping being done to achieve a self-resonant antenna, additional terms can be added to the
objective function to obtain radiation patterns favourable in an expected wireless environment, as long as
these expectations can be expressed in terms of characteristic mode quantities.
■ In Chapter 5, the characteristic mode based shape synthesis procedure was extended to apply to the design
of two closely-spaced antennas operating at different frequencies.
■ Previously, if sub-structure modes were needed for antenna located in the vicinity of some platform
structure, and sub-structure characteristic modes were needed, the platform had to be small enough
(electrically) for method of moment meshing/modelling of the complete structure (antenna + platform) to be
computationally feasible. With the work of Chapter 6 such sub-structure characteristic modes can now be
computed for platforms of any electrical size as long as geometrical-theory-of-diffraction (GTD) modelling
of the platform, and moment-method (MM) modelling of the antenna proper, is possible. We recognise that
such use of the field-based hybrid MM/GTD approach to sub-structure characteristic modes computation
could be inferred from the basic idea of a sub-structure mode. However, this does not yet appear to have
131
been explicitly stated elsewhere (and so perhaps not realised), nor example computations provided. We
believe this realisation might widen the scope of application of characteristic mode analysis. The use (as
done here) of asymptotic methods as part of the procedure to find useful sets of characteristic modes can be
viewed as one end of a classification scale in terms of which that in [SCHU 75], in which the entire object is
electrically very small, lies at the opposite end.
■ The implementation of the shaping tool itself, in Chapter 3, apart from being needed to validate the
shaping process, is a contribution in its own right. It has purposefully been fully implemented using
commercially available software for the computational electromagnetics modelling and the optimization
algorithm, and so can benefit from improvements therein. It allows the shaping prescription (eg. use of
different objective functions; designation of portions of the starting geometry that may not be altered during
shaping) to be adapted to new requirements that may arise.
7.2 Possible Future Work
The following possible future work might prove fruitful by exploiting more of what shaping tool synthesis
has to offer:
 Apply it to antennas that have to occupy non-cuboid shapes.
 Implement the two different frequency / closely-spaced antenna technique when the two objects are
mounted on an electrically large platform.
132
Preliminary portions of the work have been presented in two conference papers :
A.Alakhras and D.A.McNamara, “The shape optimisation of closely-spaced electrically-small antennas”,
IEEE Int. Antennas & Propagat. Symp. Digest, San Diego, USA, July 2017.
A.Alakhras and D.A.McNamara, “Further observations on the shape synthesis of closely-spaced antennas
using sub-structure characteristic modes”, IEEE Int. Antennas & Propagat. Symp. Digest, Boston, USA, July
2018.
The following are in preparation for publication as journal letters :
A.Alakhras and D.A.McNamara, “Sub-structure characteristic mode computation utilizing field-based
hybrid methods”.
A.Alakhras and D.A.McNamara, “The shape synthesis of three-dimensional electrically-small conducting
surface antennas”.
A.Alakhras and D.A.McNamara, “The shape synthesis of closely-spaced electrically-small conducting
surface antennas operating at different frequencies”.
There remain some issues/cases whose investigation in the future might prove useful. These have already
been outlined in Section 4.6 and Section 5.6.
133
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APPENDIX A
Estimation of Measured Radiation Efficiency of the Shaped-
Synthesized Open Cuboid Antenna of Section 4.2
As noted in Section 4.2.6 the computed radiation efficiency of 98.9%
Computed
rad
  is considered over-
optimistic. It is relatively difficult to perform accurate measurements of the radiation efficiency of an
antenna, and so a method of estimating a more realistic measurement was devised as follows. From Section
2.2.7 we know the partial directivity
q
D , where q denotes either  or  , is related to the partial gain
q
G
though the radiation efficiency. We can write
Measured Measured Measured
q rad q
G D  
and
Computed Computed Computed
q q rad
G D  
It is well-known that, due to the fact that they are found using integrals of the current distribution, the far-
zone fields are relatively insensitive to small details in the predicted antenna current distribution
s
J . The
directivity is computed directly from these far zone fields, as outlined in Section 2.2.7, and so is also reliably
predicted. There are only small changes in the predicted
s
J between those cases where losses are included
in the computation and when it is not. We therefore make the assumption that
Measured Computed
q q
D D  .
If we take any point ( , )
o o
  on the radiation pattern, and compute ( , )
Computed
q o o
G   , followed by a
measurement ( , )
Measured
q o o
G   , then we can use the above two expressions, and the assumption
( , ) ( , )
Measured Computed
q o o q o o
D D      , to write
( , )
( , )
Measured
q o o
Computed Measured
rad rad
Computed
q o o
G
G
 
 
 
 
  
 
 
141
The pyramidal horn antenna used as a gain reference is the H-1734 broadband (0.5 GHz – 6.0 GHz) horn
antenna supplied by Cobham. Although it is not a calibrated standard gain horn antenna, the nominal gains
of this horn are as shown 119 in Fig.A-1. In the case in question, the computed partial ( polarised   ) gain of
the AUT in the selected azimuth direction was 1.2dBi
Computed
G    , with the relative level on the VNA
reading a value -50.8dB. We add the mismatch factor 120 of 0.05dB (determined from the computed
in
 of
-22.7dB), to obtain a computed realised gain of 1.16dBi
Computed
G    . When the AUT was substituted with
the above-mentioned horn antenna, with its maximum pointed in the direction of the source antenna, the
relative level on the VNA was -40.7dB. The difference in received power levels is thus 10.1dB. This implies
the actual measured gain of the AUT is in fact 6.8dBi minus this 10.1dB, or in other words
3.3dBi
Measured
G    . We correct this to a measured realized gain by adding the mismatch factor of 0.3dB
(determined from the measured
in
 of -14.7dB) to obtain a measured realized gain 3.0dBi
Measured
G    . It
then follows that
( 3.0 1.16)/10
10 0.66 0.65
Computed Computed Measured
rad rad rad
  
 
  
This is possibly a pessimistic estimate, but it at least bounds the radiation efficiency as 65% 98.9%
rad
   .
Fig.A-1 : Curve of the nominal gains (dBi) of the horn antenna used as a gain reference.
119 Data accessed from www.cobham.com on 15 August 2019.
120 The factor
2
1
in
  .