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ANALYSIS AND SYNTHESIS OF BROADBAND TRAVELING
WAVE ANTENNAS

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Lanwu Zhao

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ANALYSIS AND SYNTHESIS OF BROADBAND
TRAVELING WAVE ANTENNAS

BY

Lanwu Zhao

A THESIS

Submitted to
Michigan State University
In partial fulfillment of the requirements
For the degree of

MASTER OF SCIENCE
Electrical and Computer Engineering

2004

ABSTRACT

ANALYSIS AND SYNTHESIS OF BROADBAND TRAVELING WAVE
ANTENNAS

By

Lanwu Zhao

GA (Genetic Algorithms) is proving to be useful for solving complex electromagnetic (EM)
problems. This thesis presents one important application to wideband antenna design, a set of
traveling-wave spline wire antennas. These antennas are designed to operate in the frequency
band from 200 MHz to 2 GHz. These antennas are optimized based on the VSWR, the
radiation efficiency and the power gain over the frequency band of interest.

Topics covered include the basic concepts of GA and its application to the spline antenna
design. A brief history of the wideband antenna design is also covered. A set of spline antennas

are analyzed and synthesized. The EM properties are studied and discussed. A time domain
impulse response, hNJX (t) is introduced to described the antenna’s capability for impulse
radiation. Comparisons are made between the loaded dipole, the rhombic antenna and the

spline antenna for a GMC (Gaussian Modulated Cosine) input. Finally, conclusions drawn

from this research are discussed.

ACKNOWLEDGMENTS

I am sincerely grateful to my academic advisor Dr. Edward Rothwell who has provided too
much guidance, counsel and encouragement throughout the course of this research. This

thesis couldn’t be done without him. Thanks to him for being my major advisor and teacher.

I would like to thank Dr. Leo Kempel and Dr. Edward Rothwell for giving me the chance
to be their research assistant; thanks to the National Science Foundation Division of
Undergraduate Education Award DUE—0231312 for providing the financial support that

makes the completion of thesis possible.

Many thanks to Dr John Ross, without whose technical support and guidance the
completion of this thesis would be impossible. Thanks to him for letting me access to his

powerful GA-NEC program to finish this research.

Thanks to Dr. Balasubraniam for serving as committee members in the defense of this

thesis and giving me a deep understanding of computational electromagnetics during my study

in Michigan State University.

I would also like to thank all my friends and colleagues at the EM group here at Michigan
State University. They have been a constant source of friendship and support thought my

many years of study and research.

Most of all, I would like to thank my wife Susan Wang and my daughter Jessie. They are the

constant source of my happiness.

TABLE OF CONTENTS

Chapter 1

INTRODUCTION .................................................................................................................................... 1
Chapter 2

THE APPLICATION OF GENETIC ALGORITHMS .................................................................. 4

2.1 Basic GA Concepts .................................................................................................................... 5

2.1.1 GA operations .................................................................................................................. 7

2.1.1.1 Selection Scheme ................................................................................................... 8

2.1.1.2 Crossover Scheme ................................................................................................. 8

2.1.1.3 Mutation Scheme ................................................................................................... 9

2.1.2 Fitness Functions ........................................................................................................... 10

2.2 GA-NEC ................................................................................................................................... 11

2.2.1 Selection, Crossover and Mutation ............................................................................. 11

2.2.2 Generation Gap ............................................................................................................ 13

2.2.3 Convergence ................................................................................................................... 13

2.2.4 Fitness ............................................................................................................................. 13

2.3 The applications of GA-NEC ............................................................................................... 14

2.3.1 Coding the spline antennas .......................................................................................... 15

2.3.1.1 Coding the symmetric spline antenna .............................................................. 15

2.3.1.2 Coding the asymmetric spline antenna ............................................................ 20

2.3.2 The searching space ....................................................................................................... 22

2.3.3 The external Visual Basic program ............................................................................. 22
Chapter 3

THE OPTIMIZATION RESULTS AND ANALYSIS .................................................................. 25

3.1 Antennas with Symmetric Geometry ................................................................................... 29

3.1.1 Symmetric geometry with open ends ........................................................................ 30

3.1.1.1 Feed impedance of 600 Ohm$ .......................................................................... 30

3.1.1.2 Feed impedance of 300 Ohms .......................................................................... 38

3.1.1.3 Feed impedance of 150 Ohms .......................................................................... 45

3.1.2 Symmetric geometry with closed ends ....................................................................... 52

3.1.2.1 Feed impedance of 600 Ohms .......................................................................... 52

3.1.2.2 Feed impedance of 300 Ohms .......................................................................... 59

3.1.2.3 Feed impedance of 150 Ohms .......................................................................... 66

3.2 Antennas with asymmetric Geometry .................................................................................. 73

3.2.1 Antennas with open ends ............................................................................................. 73

3.2.1.1 Feed impedance of 600 Ohms .......................................................................... 73

1V

3.2.1.2 Feed impedance of 300 Ohms .......................................................................... 80

3.2.1.3 Feed impedance of 150 Ohms .......................................................................... 87

3.2.2 Antennas with closed ends ........................................................................................... 94

3.2.2.1 Feed impedance of 600 Ohm$ .......................................................................... 94

3.2.2.2 Feed impedance of 300 Ohm$ ....................................................................... 101

3.2.2.3 Feed impedance of 150 Ohms ....................................................................... 108

3.3 Conclusions ............................................................................................................................ 1 15
Chapter 4

COMPARISONS ON GMC PULSE RESPONSE ....................................................................... 117

4.1 Dipole ...................................................................................................................................... 1 19

4.2 Loaded dipole ........................................................................................................................ 123

4.3 The rhombic antenna ........................................................................................................... 126

4.4 The spline antenna ................................................................................................................ 132

4.5 Comparison and Conclusion .............................................................................................. 138
Chapter 5

CONCLUSIONS ................................................................................................................................... 140

LIST OF TABLES

Table 3.1 Different study cases ............................................................... 29

Figure 2.1

Figure 2.2

Figure 2.3
Figure 2.4
Figure 3.1
Figure 3.2
Figure 3.3
Figure 3.4
Figure 3.5
Figure 3.6

Figure 3.7
Figure 3.8
Figure 3.9
Figure 3.10

Figure 3.11
Figure 3.12
Figure 3.13
Figure 3.14
Figure 3.15

Figure 3.16

LIST OF FIGURES

Example of the crossover procedure .................................................. 9
The mutation operator randomly changes the elements within the
Chromosome in case of P < P (mutation) ......................................................... 10
An example of antenna with symmetric geometry... .............................................. 16
An example of antenna with asymmetric geometry... 17
Antenna Geometry .............................................................................................. 32
Input Impedance vs. Frequency... . . . 33
VSWR vs. Frequency .......................................................................................... 33
Radiated Power vs. Frequency... . . . . . . . . . .. 34
Efficiency vs. Frequency .................................................................................... 34
Power Gain vs. Frequency ................................................................................. 35
E¢ (xz-plane) and E9 (yz-plane) patterns at 200 MHz.... . . . 36
E¢ (xz-plane) and E9 (yz-plane) patterns at 800 MHz.... 36
E’, (xz-plane) and E9 (yz-plane) patterns at 1400 MHz.... 37
E¢ (xz-plane) and E,9 (yz-plane) patterns at 2000 MHz.... 37
Antenna Geometry ............................................................................................ 39
Input Impedance vs. Frequency... . . . . . . . . . . . . . . . . 40
VSWR vs. Frequency ........................................................................................ 40
Radiated Power vs. Frequency... . . . . . . . . . . . . . . . . .. 41
Efficiency vs. Frequency .................................................................................. 41
Power Gain vs. Frequency... . . . . . . 42

vii

Figure 3.101 Antenna Geometry ....................................................................................... 102
Figure 3.102 Input Impedance vs. Frequency... . . 103
Figure 3.103 VSWR vs. Frequency .................................................................. 103
Figure 3.104 Radiated Power vs. Frequency... . . .. 104
Figure 3.105 Efficiency vs. Frequency... . . . 104
Figure 3.106 Power Gain vs. Frequency... 105
Figure 3.107 E¢ (xz-plane) and E9 (yz-plane) patterns at 200 MHz.... . . . 106
Figure 3.108 E¢ (xz-plane) and E9 (yz-plane) patterns at 800 MHz....106
Figure 3.109 E ¢ (xz-plane) and E9 (yz-plane) patterns at 1400 MHz ........................ 107
Figure 3.110 E¢ (xz-plane) and E0 (yz-plane) patterns at 2000 MHz ........................ 107
Figure 3.111 Antenna Geometry ...................................................................... 109
Figure 3.112 Input Impedance vs. Frequency... . . . . . . . . . . . . . . . . 110
Figure 3.113 VSWR vs. Frequency .................................................................. 110
Figure 3.114 Radiated Power vs. Frequency... 111
Figure 3.115 Efficiency vs. Frequency... . . . . . . 111
Figure 3.116 Power Gain vs. Frequency... . . . 112
Figure 3.117 E“ (xz-plane) and E,9 (yz-plane) patterns at 200 MHz.... . . . . . . . . . . . . 113
Figure 3.118 E¢ (xz-plane) and E0 (yz-plane) patterns at 800 MHz.... . . . . . . . . . . . . 113
Figure 3.119 E¢ (xz-plane) and E0 (yz-plane) patterns at 1400 MHz ........................ 114
Figure 3.120 E¢ (xz-plane) and E9 (yz—plane) patterns at 2000 MHz ........................ 114
Figure 4.1 Time domain version of the frequency GMC window function ................. 120

Figure 4.2
Figure 4.3
Figure 4.4
Figure 4.5
Figure 4.6
Figure 4.7
Figure 4.8
Figure 4.9
Figure 4.10
Figure 4.11
Figure 4.12
Figure 4.13
Figure 4.14
Figure 4.15
Figure 4.16
Figure 4.17
Figure 4.18
Figure 4.19
Figure 4.20
Figure 4.21
Figure 4.22
Figure 4.23

Figure 4.24

Energy in the waveform radiated by the dipole. . . . . . . . . . . . . . .. 121
Waveform radiated in far-zone field at 30 degrees. . . . . . . . . . . . . . .. 121
Waveform radiated in far-zone field at 60 degrees. . . . . . . . . . . . . . .. 122
Waveform radiated in far-zone field at 90 degrees... .. . ... . ... 122
Energy in the waveform radiated by loaded dipole. . . . . . . . . . . . . . .. 124
Waveform radiated in far-zone field at 30 degrees. . . . . . . . . . . . . . .. 124
Waveform radiated in far-zone field at 60 degrees. . . . . . . . . . .. 125
Waveform radiated in far-zone field at 90 degrees. . . . . . . . . . . . . . .. 125
The geometry of the rhombic antenna. 127
Energy in the waveform radiated by the rhombic antenna in y-z plane ........... 128
Waveform radiated in far-zone field at 0 degrees. . . . . . . . . . . . . . .. 128
Waveform radiated in far-zone field at 30 degrees. . . . . . . . . . . . . . .. 129
Waveform radiated in far-zone field at 60 degrees... .. . .. . .. ...... 129
Waveform radiated in far-zone field at 90 degrees. . . . . . . . . . . . . . .. 130
Waveform radiated in far-zone field at 120 degrees. . . . . . . . . . . . . . .. 130
Waveform radiated in far-zone field at 150 degrees. ... 131
Waveform radiated in far-zone field at 180 degrees. . . . . . . . . . . . . . .. 131
The geometry of the spline antenna geometry. . . . . . . . . . . . . . .. 133
Energy in the waveform radiated by the spline antenna in y-z plane. . 134
Waveform radiated in far-zone field at 0 degrees. . . . . . . . . . . . . . .. 134
Waveform radiated in far-zone field at 30 degrees. . . . . . . . . . . . . . .. 135
Waveform radiated in far-zone field at 60 degrees. . . . . . . . . . . . . . .. 135
Waveform radiated in far-zone field at 90 degrees. . . . . . . . . . . . . . .. 136

Figure 4.25
Figure 4.26
Figure 4.27
Figure 4.28

Figure 4.29

Waveform radiated in far—zone field at 120 degrees. . . . . . . . . . . . . . ..
Waveform radiated in far—zone field at 150 degrees. . . . . . . . . . . . . . ..
Waveform radiated in far-zone field at 180 degrees. . . . . . . . . . . . . . ..
Normalized energy radiation patterns (GMC pulse) of the four antennas. .

Radiation efficiency of the four antennas. . . . . . . . . . . . . .

xiv

.136

137

137

139

139

Chapter 1
INTRODUCTION

The imminent widespread commercial development of Ultra—wideband systems has sparked
renewed interest in the subject of Ultra-wideband antennas [1]. Since the radiated power of the
antenna over the whole particular frequency band must meet the requirements regulated by the
FCC, a UWB antenna with specific properties plays a critical role in the overall UWB system.
This thesis introduces a set of broadband traveling wave antennas that are suitable for the
band from 200 MHz to 2 GHz. The properties of the antennas including the radiation pattern,
the gain, the efficiency, the VSWR and the GMC (Gaussian modulated cosine) pulse response

are investigated in this study.

Antenna performance is often the main factor which determines the overall performance of
a UWB system. Many types of antennas with wideband properties had been designed and used
for various UWB radio systems. In 1898, Oliver Lodge disclosed spherical dipoles, square
dipoles, biconical dipoles, and bow-tie dipoles in his patent [2]. During World War Two and
subsequent years additional antennas that had much wider bandwidth were introduced such as
Carter’s biconical antenna and conical monopole [3-4], Schelkunoff’s spherical dipole [5],
Lindenbald’s coaxial horn spherical dipoles [6], and Brillonin’s omni-directional and directional
coaxial horn [7]. Later on, more manufacturable wideband antennas were pioneered, such as
Stohr’s ellipsoidal monopole and ellipsoidal dipole [8], RH. DuI-Iamel and DE. Isbell’ log
periodic arrays [9], Marie’s wideband slot antenna [10], Lalezari et a1 ’5 broadband notch antenna

[11], Thomas et al’r circular element dipole [12], Harmuth’s large current radiator/ magnetic

antenna [13] and Barne’s UWB slot antenna [14-16]. These antennas are some highlights of

UWB antennas in the past century.

For some purposes, the directional and broadband properties of traveling wave antennas are
desirable. The first work on traveling wave dipoles was reported by Altschuler [17] who
inserted a lumped resistor at quarter wavelength from the ends of the antenna to form
traveling wave propagation along the dipole antenna. Soon after there came many reports on
traveling wave antennas. The most famous one was reported by T.T. Wu and R.W.P king in
1964, who studied the cylindrical antenna with variable internal impedance per unit length to
form a pure outward traveling wave along the antenna at a certain frequency [18]. They derived
the distributed impedance loading theoretically to form a nonreflecting traveling wave along
the antenna. Based on this, they also derived the efficiency and the far zone electric field of
their antenna. The efficiency of their antenna was only 50 percent due to impedance
distributed along the antenna. Moreover, since the nonreflecting property only worked at the
center frequency, a very wide bandwidth was not expected. For directional transmission

purposes, high directivity is desired for a traveling wave antenna.

The goal of the research presented here was to develop traveling wave antennas that have
higher directivity, better efficiency and wider bandwidth properties than those reported in the
literature. In this study, a set of spline traveling wave wire antennas were investigated, such as
the ones with close or open end, ones with symmetric and asymmetric geometry, ones that the
wire radius were constant or varying, and ones with or without resistive loading at the end.
With the help of GAs (Genetic Algorithms) in conjunction with NBC (Numerical
Electromagnetic Code), a set of spline antennas were analyzed and synthesized using the GA-

NEC program developed by John Ross [19]. The application of GA-NEC and the associated
2

external programs and the external Visual Basic program that called by GA-NEC are described

in Chapter 2.

The GAS optimized traveling wave antennas with different geometry shapes, such as ones
with geometrical symmetry and asymmetry, ones with constant wire radius and varying wire
radius, and ones with open ends or with lumped resistive loading at the ends. The main
features of these antennas, the far field radiation pattern, the VSWR, the input impedance and
the efficiency are simulated in a free space environment over the frequency band from 200
MHz to ZGHz. The transfer function is examined and the impulse response is studied by

looking at the far-zone energy distribution in time domain. This is covered in Chapter 3.

The comparison with other broadband wire antennas is carried out in Chapter 4. In this
chapter, the GMC pulse response is studied. The GMC pulse energy radiation pattern is
discussed and comparisons are made. The radiation efficiency simulated by NECZ is

compared.

Chapter 5 presents the conclusions as well as some future considerations on the broadband

spline antenna design.

Chapter 2

THE APPLICATION OF GENETIC ALGORITHMS

Genetic algorithms (GAS) are stochastic search procedures modeled on the principle concepts
of natural selection and evolution. With the advantage of computers and powerful
computational techniques, nature’s optimization processes can be applied in the form of
genetic algorithms (GAS) to problems with high-dimension and multi—modal functions. As
optimizers, GAs are effective at solving complex, combinatorial and related problems [20].
Electromagnetic (EM) optimization problems usually involve a large number of parameters.
These problems are expensive to evaluate because the objective functions that arise in
electromagnetics are often nonlinear, inflexible and non-differentiable. GAs are attractive for
EM optimization since they can be easily applied to problems involving non-differentiable
functions and discrete search spaces. It has been proven that GA is very useful for solving
complex electromagnetic (EM) problems [21].

The main features that make GAs attractive to the application of this thesis, the design of
broadband traveling wave antennas, are that the algorithm converges to a global extreme in a
global search, its ability to optimize functions with a large number of parameters and the ability
to handle complex objective functions. GAs can simultaneously operate on the global
information to obtain a global extreme (desired solution) rather than a local extreme. In this
application, the goal is to synthesize broadband traveling wave wire antennas with desired
characteristics in terms of high directivity, good efficiency and lower Voltage Standing Wave

Ratio (V SWR) over a frequency ranging from 200 MHz to 2 GHz. Since these features are
4

highly depended on the shape of the antennas, the radius of the wire and resistive loading of
the antennas, the objective is to optimize the shapes of the antennas, the radius of the wires
and the loading at the ends of the antennas to attain the optimal solutions.

The EM problems can be solved through numerical methods. The Numerical
Electromagnetics Code version 2 (N EC2, available in the public domain) which is based on
the Method of Moments (MOM) is applied to calculate the EM solutions to the traveling wave
antennas. In GAS, by coding the parameters of the antennas into chromosomes, each
individual represents one type of antenna, and the properties [of the antenna are calculated
through the NEC2 analysis. As the fitness-weighted functions guide the population toward to
the optimal solution, NEC2 is called repetitively by the GA to solve the EM problems.

In order to take the advantage of the GA, a program is desired to have the ability to
implement the GA. It must be able to generate the output file for each individual, call outside
functions to perform calculations of the antenna geometry, and weight the results using
fitness-weighted functions. The program GA-NEC is a perfect platform for performing the
GA computation. The GA-NEC program was developed by Dr. John Ross as part of the
antenna modeling and design effort at Delphi Research Labs. It is a general purpose GA based
optimizer for NEC2. Dr. Ross also developed another program named AntennaCAD, which
provides pre and post-processor functions that simplify the construction of the NEC2 input

and output files.

2.1 Basic GA Concepts

In genetic algorithms, a set of possible solutions are generated which evolve towards an

optimal global solution. An optimal solution can be achieved by a fitness-weighted selection

process. The exploration of the solution space is accomplished by crossover and mutation of

the characteristics in the current population.

As powerful optimizers, GAS are effective at solving complex and combinational problems.
They are especially good at finding the goal maxima in a high—dimension, multi-modal function

domain. They differ from conventional algorithms in three aspects:
1. They operate on a population of trial solutions in parallel,

2. Usually, they operate on a coding of the function parameters (a chromosome) rather

than the parameters themselves, and

3. They use the sample operators (selection, crossover and mutation) to search for the

optimal solution in the solution domain.

In the GA procedure, successive populations of trial solutions are called generations. The
generations followed are made up of children produced through the crossover and mutation
operations by the selected parents with the fitness-weighted ranked selection. During the
optimization, a set of individuals (populations) are chosen. They evolve toward the optimal
solution as determined by the fitness function. Therefore, GA optimizers must have the ability

to perform the following six tasks [20]:
1. Encode the solution parameters as genes,
2. Create a string of genes to form a chromosome (Binary-code form),

3. Initialize a starting trial solution,

4. Evaluate and assign fitness values to individuals in the population,

5. Perform reproduction through the fitness-weighted selection of individuals from the

population, and

6. Perform crossover (recombination) and mutation to produce the members of the next

generation.

In a simple GA, an initial population is generated with randomly assigned parameter strings
or chromosomes. Each of these chromosomes represents an individual. The set of individuals
form the first generation. Each of them is assigned a fitness value by evaluating the fitness
function. After the individuals are sorted according to their fitness value, a ranked individual
list is obtained. The ones with the higher fitness get a higher ranking. The reproduction phase
produces a new generation from the current generation. During this process, a pair of
individuals is selected to act as the parents. A new pair of children are generated though the
crossover and mutation operation. These new children are placed in the new generation by
replacing their parents. This process is repeated until there are enough children to fill the new
generation. Again, the individuals in the new generation will be evaluated and assigned a fitness
value in the same manner as before. The individuals in the new generation are ranked in a
sorted order. The reproduction process is repeated and is terminated when the desired fitness

value is met.

2.1.1 GA operations
The GA operates on a coding of parameters instead of operating on the parameters

themselves. The coding is a mapping from parameter space to chromosome space. In a binary

coding, the parameters are presented by finite length of binary strings. The binary strings that
represent all the parameters form a set of binary codes and are referred to as a chromosome.
The binary coded parameter is called a gene. The reproduction phase involves three

operations: selection, crossover and mutation.

2.1.1.1 Selection Scheme

The selection operator generates a new population from the existing one. The individuals in
the new population are selected through a certain criteria. The most commonly used selection
operator is known as the weighted roulette wheel selection [22]. Dunng this selection, a
roulette wheel is divided into certain number of slots to represent the individuals in the current
generation. The area of each Slot is proportional to the value evaluated by the fitness-weighted
function. Therefore, the probability of the individual to be selected is proportional to the value
of the individual itself. Highly fit individuals are more likely to be selected. This procedure is

important because the convergence of the GA is strongly dependent on this choice.

2.1.1.2 Crossover Scheme

The crossover operator accepts two parents and produces two children of the next generation.
It works directly on the chromosomes themselves. The purpose of the crossover operation is
to combine blocks of highly fit sequences and form new sequences with even better fitness
values. The simplest crossover scheme is one-point crossover, which is favored by Holland
[23]. It has become the most common method of hybridizing binary chromosomes. The one-
point crossover provides the simplest way to hybridize binary chromosomes without
disrupting the main structure. This process is controlled by the crossover probability. In one-

point crossover, if the value of a random variable is less than a specified probability, Pcross, a

8

random crossover point is selected to divide the binary set into two parts. The portion of the
chromosomes preceding the selected point is simply copied from the parent 1 to child 1 and
parent 2 to child 2. The portion of the chromosome of parent 1 following the selected point is
copied to child 2 in the corresponding position. Similarly, the remaining portion of the
chromosome of parent 2 is placed in the corresponding position of child 1, as shown in figure
2.1. Otherwise, if the random variable is greater than Pcross, the entire chromosome of parent
1 is copied into child 1 and similarly for parent 2 and child 2. It can be seen that the operation
of one-point crossover precludes the possibility of the first and last bit ever remaining on the
same chromosome. The effect of crossover is to rearrange the genes with the objective of
producing better combinations of genes to get more fit individuals. Typically, crossover
probability between 0.6 and 0.8 has been found to work best in most situations. In this thesis,

the probability of 0.75 has been chosen.

Mir—011 101 W
M M1191
Parents Children

Figure 2.1. Example of the crossover procedure
2.1.1.3 Mutation Scheme
The mutation operator allows the possibility of a slight change in portions of the chromosome
of the child. It tends to prevent the optimization process from being trapped in a local
minimum. Thus, Mutation rate can affect the ability of the GA to locate a solution and gain
more diversity. In mutation, if a random variable is less than the probability of the mutation, a

binary element in the chromosome strings is randomly selected and changed. In most cases,

9

the mutation operator is not very effective to improve the GA’s performance. A high mutation
value often leads to the destruction of valuable information stored in the highly fit sequences.
It usually disrupts the process toward to the optimal solution and interferes with the beneficial
action of the selection and crossover. Basically, the mutation process should occur at low
probability, usually in the range from 0.001-0.01(although in some cases values as high as 0.1
are used [24]). In this thesis a value of 0.001 is used. The mutation operator is illustrated in
figure 2.2 in binary case. In this figure a randomly selected element in the chromosome is

changed to a new value (0 to 1 or 1 to 0).

101101101l::> 101100101

Figure 2.2. The mutation operator randomly changes the elements within the chromosome
in case of P < P (mutation).

2.1.2 Fitness Functions

The fitness function, also known as the objective function or the cost function, is used to
assign a fitness value to each individual in the GA population. This value represents the
goodness of the individual. A desired objective value is specified and stored prior to the
optimization process. The fitness value of every individual is calculated and assigned to the
individual. By comparing the assigned value with the stored one, the goodness of the
individual is obtained. In some complex problems, there may be more than one objective
function. For instance, for an antenna problem, the radiation gain of the antenna is not the
only objective value of interest; also important are the efficiency and voltage standing wave
ratio (V SWR). The final objective function should be a combination of the three sub-

objectives. The weighted value should be applied to each of them according to the importance

10

associated. The desired value assigned to each objective function may vary. It could be
assigned as a minimum, a maximum or in a constrained range. Constraints are implemented by
imposing a large penalty whenever the constraint is violated. A simple method is to define a

function of the multiple objective optimizations:

F = inlCm -—S,,,| (2.1)

Here F is the total raw fitness metric, M is the total number of the objective functions, Wm is
the weight applied to the m“ basic fitness metric, Cm is output value of the mm basic fitness

metric and Sm is specified value of the m”I basic fitness metric.

2.2 GA-NEC

The GA-NEC program was developed by Dr. John Ross as part of the antenna modeling
and design effort at Delphi Research Labs. It is a general purpose GA based optimizer for
NEC2 with multiple functions for antenna design. In GA-NEC, parameters may be specified
to control the selection, the crossover, the mutation operations, the generation gap, the

convergence type and multiple fitness functions.

2.2.1 Selection, Crossover and Mutation

Based on the selection method used, parents are selected in pairs, and thus the number of the
initial individuals in the population must be even. The population size used depends on the
complexity of the problem and the length of the chromosome. Typically, longer chromosomes
require larger population size. The population size can not be too large or too small. For a

large population, the GA moves very slowly towards to the good points, and there are many

11

children in each generation, requiring many simulations. The large population along with its
gene pool diversity overpowers these individuals by sheer numbers, and generally it is a large
hill that will be exploited [25]. If the hill does not contain the best answer, the algorithm will
often be trapped in local optimum. It is thus suboptimal. If the hill contains the best answer, it
will take many simulations to arrive at good answers with such a large population. For a small
population, there is not enough diversity to attain the optimal solution. It may make sense to
solve a problem with several runs of a small generation GA rather than just one run of a large
population GA under the same time cost. For binary chromosome, it is efficient to use a
population of about four or five times of the number of bits in the chromosome. The size of
the population is usually on the order of 20-500. Values around 100 are commonly used. In
this study, there are most likely around ten parameters for the symmetric spline antenna and 13
for the asymmetric ones. The length of chromosome is at the range from 40 to 60. As the
result of the rule, the population for the symmetric ones and asymmetric ones is set to 150 and

200 respectively in this study

The selection strategy determines how the selection operator chooses the parents to
generate the children. The elitist .rtrateg is chosen along with roulette wheel selection. This
strategy ensures that the best solution in each generation is most probably preserved. This will
improve convergence. In the convergence criteria, the maximum number of generations that

will be computed is specified. The number is set to 150 in this study.

The one-point crossover operation is adopted is this algorithm. The effect of the crossover

is to rearrange the genes to produce better combinations of genes, and, as a result to get better

12

fit individuals. It has been found that the probability of crossover on the order of 0.6-0.8

works best in most problems [19]. This value is set to 0.7 in the GA-NEC program.

Mutation rate can affect the GA’s ability of locating a solution to avoid the optimization
process being trapped in local minima. Higher probability of mutation will destroy the valuable

information that is already stored in the highly fit sequences. The value is set to 0.05 in this

study.

2.2.2 Generation Gap

In the GA operation, the fraction of the population, the generation gap, to be selected as
parents highly affects the GA convergence. It specifies the number of individuals involved in
selection, crossover and mutation operations. A generation gap between 0.95 and 1 is adequate

for most problems. It is set to 0.95 in this study.

2.2.3 Convergence

The convergence type is set to none-me max generation: to specify that the run will stop only
when the maximum number of generations has been reached. The convergence variable is set
to rawfitrm: minimum to control the convergence. This is the common option because one is

generally only interested in finding a single solution.

2.2.4 Fitness

Fitness is a single numerical quantity evaluated by a fitness function. It is assigned to each
individual to describe how well an individual meets predefined design objectives and
constraints. For multiple objectives, fitness can be computed based on the outputs of multiple

analyses using a weighted sum. The definition of good fitness functions depends on the

13

problems themselves. For broadband antenna design problems, the three main objectives, the

VSWR, efficiency, and gain, should be taken into consideration. The total fitness of an

individual is the sum of the three weighted ones as shown in equation 1.

In order to design broadband traveling wave antennas with optimal characteristics, three
fitness functions are specified. Each of them should be the summation of fitness values
evaluated at certain frequencies from 200 MHz to 2 GHz. Since the VSWR plays a more

important role, the corresponding weighting value should be considerably higher than that of

efficiency and gain. We set the weighting value of VSWR, WVSWR to 10000, and set wefliciency and

W

gain

to 100 and 1, respectively.

2.3 The applications of GA-NEC

GA has been shown to be a very powerful tool for antenna design. It is very efficient to design
an antenna with special desired characteristics. The wire antenna design with special
characteristics has received considerable attention in the literature. For instance, a circularly
polanzed antenna formed by several segments connected in series has been studied by
Altshuler and Linden [26]. Ultra-wideband wire antennas have been designed by loading the
wires with resonant RLC circuit [27—28]. In this application, the load location and the circuit
elements have been optimized by a GA. The first GA designed Ultra-broadband antenna for

communication was a loaded monopole with height of 1.75 m and frequency range of 30-450

MHz [29].

This study focuses on the topic of Ultra-broadband wire antenna design using a GA. The

goal of this study was to design a wire antenna with loading at one end which had the desire

14

characteristics in terms of VSWR, efficiency, and the directivity over the frequency range 200

MHz to 2 GHz.

2.3.1 Coding the spline antennas

As discussed earlier, GA-NEC can not work on the antenna’s parameters themselves. It must
work on the binary strings or chromosomes that are formed by coding the parameters. Thus,
the parameters that specify the physical information of the antennas must be determined
before the optimization process can begin. The transient field produced by a traveling-wave
wire antenna was studied by Rothwell [30]. It was shown that the traveling-wave time-domain
antenna has the ability to directly radiate wideband waveforms for a certain impulse width. The
optimal wideband properties can be attained by optimizing the shape of the antenna. Two
types of spline antennas are taken into consideration: one with symmetric geometry as shown
in figure 2.3, and the other with asymmetric geometry in figure 2.4. In order to use GA-NEC
to optimize the shape of the antenna, the antenna parameters must be represented by binary

strings.

2.3.1.1 Coding the symmetric spline antenna

There are several parameters that can affect the EM properties of the symmetric spline
antenna. They are the length of the antenna, the radius of the wires, the shape of the curves
and the load at the end of the antenna. In order to have GA-NEC optimize the antenna, the
fitness function should be specified which defines the goal of the optimization process. All
antenna parameters should be represented by binary strings since GA-NEC can’t operate on

the parameters themselves.

15

To obtain the spline shape antenna, an external program is called by GA-NEC. The external
program is written in Visual Basic. Its purpose is to generate smooth “spline” curves by
passing lines through several discrete points. Therefore, the shape of the curve is decided by
the points which the curve should pass through. The optimal Shape of antenna can be
achieved by optimizing the positions of the points. The more points that are chosen, the more
geometrical diversity the antenna has. But, more points means more binary strings are
required. As a result, there will be a long chromosome which will slow down the process to
find the best solution. In this particular study, eight points are selected to form one side of the

curve. The other side is generated automatically by the symmetry of the geometry.

16

Figure 2.3. An example of antenna with symmetric geometry

17

Figure 2.4. An example of antenna with asymmetric geometry

18

Fixing the beginning points, the shape of the antenna can be optimized based on the other
seven points. There are also two other parameters, the radius of the wire and the load at the
ends that should be optimized in order to reach the goal guided by the fitness functions. The
nine parameters, also called the genes, represented in binary strings form one chromosome.
One chromosome stands for an individual. In the real-valued chromosome, there are nine

genes. Its representation is:

[P1P2P3 P4 P5 P6P7 R1L1]
Where the P5 are the position genes, R is the radius gene, and L is the load gene.

In binary representation, GA-NEC will use 5 bits for each position gene, 3 bits for the
radius and 2 for the lumped load. The 40-bit chromosome will be constructed in the following
typical way:

[P11...P15, P21...P25, P31...P35, P41...P45, P51...P55, P61...P65, P71...P75, R11 R12
R13, L11 L12]
Where the P’s indicate the locations of each bit in the binary representation of the point
positions variables, while the R’s indicate the location of the 3 bits that make up the radius
variable, and the L’s indicate the 2 bits compose the third variable, the load. The first binary bit
of each variable is the most significant one for that variable. It has higher weight than any

others in the strings that compose that variable. One possible chromosome can be

[0010001011001001010000100100111010101111]

It represents one possible solution, an individual. For this individual, GA-NEC will translate it
into a set of those parameters and pass them to the external program to generate the spline

shaped antenna. In conjunction with NEC2, the antenna will be simulated and the
19

electromagnetic response of the antenna will be calculated. The results from the NEC2
analysis will be evaluated using the specified fitness functions. The yield value, the fitness
value, is associated with the individual. The individuals in each generation are ranked according

to their fitness values.

Several types of symmetric spline antennas are simulated in this study. They are varying,
constraint and constant wire radius with closed and open end. GA-NEC only optimizes the
point location and the load for a closed end. Hence, the number of binary strings can be
reduced to 37 for open end and 39 for closed end. For the first case, the radius of the wire
varies as a function of separation to keep the constant impedance (600 Ohms in this study).
The second case allows the radius to vary over a specified range and GA-NEC optimizes the
point location and the radius as well The number of binary strings in this case will be 41 for
the one with open end and 43 for the one with closed end. In the third case, the radius is set to
constant along the whole antenna. GA—NEC optimizes the point locations and picks the
optimal radius for the wire. In this case the number of binary strings is 39 for open end one

and 41 for closed end.

2.3.1.2 Coding the asymmetric spline antenna

Similar to the symmetric antenna, the optimal asymmetric spline antenna can be achieved by
optimizing the shape of the curves. But unlike the symmetric one, there are two wires that
must be worked on. As a result, the number of points that need to be optimized will be twice
that of the symmetric antenna. The population in each generation should be larger to have an
efficient GA-NEC optimization. This will increase the simulation time for each run. The GA-

NEC will move very slowly to the optimal solution. On the other hand, a large population can

20

bring more diversity, which may yield better-fitted value points. To speed up the simulation
time and give GA-NEC more diversity, the number of points on each side is reduced to six
instead of seven. Thus there are fourteen parameters that decide the characteristics of the

asymmetric spline antenna. In the real-valued chromosome, there are fourteen genes. Its

representation is:

[P1P2P3P4P5P6P7P8P9P10P11P12R1L1]

As with the symmetric antenna, PS represent the position gene, R the radius gene, and L the

load gene.

The size of the gene or the number of bits used to represent a parameter is important to the
accuracy of the solution and the time needed for GA~NEC to converge. In this case, GA-
NEC uses 5 bits for each position gene, 4 bits for the radius and 2 bits to present the lumped
load. This will bring enough accuracy to the solution and requires less time to converge. Each

chromosome is encoded by 66 binary bits. The 66-bit chromosome can be constructed as:

[P11...P15, P21...P25, P31...P35, P41...P45, P51...P55, P61...P65, P71...P‘75, P81...P85,
P91...P95, P101.. .P105, P111...P115, P121...P125, R1...R4,L1 L2]

One possible chromosome can be represented in binary strings as
[000110100111011100110101111101000000101100001100011000110001010111]

The number of bits in the chromosome is 66, which is bigger than that of the symmetric
antenna, 40. This will enlarge the searching space. It seems that the asymmetric coding will
obtain better solutions than the symmetric one if the population and number of runs are

chosen properly. On the other hand, this will increase the simulation time.

21

2.3.2 The searching space

In order to run GA-NEC, the possible values of the parameters that will be optimized must be
constrained. This will make the search space manageable. It is important that the ranges of the
parameters should be large enough such that the optimal solution is likely to be included. Also,
in GA-NEC, the number of bits that represent the parameters should be carefully chosen.
This number together with the search space gives the possible value from which GA-NEC can
choose. In the binary GA-NEC, the number of bits used to represent a parameter provides an
integral number of different possible values of the parameter. If it is chosen to be small, it is
possible that the optimal solution is not included in the search domain. But on the other hand,
if it is big, as discussed previously, a large population is necessary to have GA-NEC search for
the optimal solution efficiently. In this study, the point position search space is set from 0 to 1
meter. The radius varies from 00005-0008 meters. The lumped load impedance search space

is from 0 to 600 ohms with a step of 200 ohms.

2.3.3 The external Visual Basic program

The purpose of external program is to create the shape of the antenna using spline functions
by working on the information in the input file that is generated by GA-NEC. In the input file,
called splineinp, the information that is needed to create the spline antenna is written in the
proper order such that VB program can read it correctly. The first line of splineinp specifies
the total length of the antenna from the input end to the load end. The minimum separation is
defined in the second line, which determines the minimum separation between the two halves
of the antenna. Setting the proper minimum separation avoids segment separation violations
when the NEC2 program is called. The third line specifies the input impedance at the input

end of the antenna. The fourth line defines the number of segments that the antenna can be
22

divided into along the z—axis, the direction of the antenna. The fifth line specifies the number
of points that the spline will pass through. The positions of the spline points are generated
right after the fifth line. For an asymmetric spline antenna, the number of points is twice that
of the symmetric one. The VB program will generate the antenna geometry data based on the

information that is given in the input file, splineinp.

In the VB program, there is a main function that defines the names of the input file and
output file. Once these two files are specified in the main function block, a sub function is
called to generate the antenna geometry. In the sub function block, the input file is opened and
the parameters are read line by line as input information to create the antenna. The input file
will close when the last line is fetched. After all the information is read from splineinp,

another function is called to create the curve that passes through the specified points.

In order to minimize reflections, it is desirable for the antenna to have constant
transmission line impedance along its length. The separation and the radius of the wire
determine the impedance of the antenna at a certain distance. Once the separation is known,

the radius corresponding to impedance Z0 can be calculated by
Radius = Separation/ (2 * Cosh (ZO/ 120)) (2.2)

Here Cosh is the hyperbolic cosine function and Z0 is the input impedance of the feed twin

lines.

After this process is complete, the information about segments that comprise the antenna is
known. The information that is associated with the segments includes: the positions of the

segments, the coordinates of each end, and the radius of the segments. The VB program then
23

writes this information to an output file, called splineni2, line by line. The information is
formatted in the manner of a GW card that can be recognized by the N EC2 program. In the
output file, each segment is written in its own GW card. All the GW cards for the segments
that make up the spline antenna compose the splineni2 file. All the GW cards in this file are
taken as part of the GA-NEC input file, which is the input of the NEC2 program. As
mentioned earlier, the purpose of the VB program is to generate the geometry of the antenna
and write it into a format which is readable by NEC2. Because the positions of the segments
are unpredictable, it is almost impossible to write the GW cards for a spline antenna in a
conventional manner, as might be done for a rhombic antenna. During the GA-NEC
optimization process, each splineinp and splineni2 file are associated with an individual. The
program is called repeatedly by GA-NEC as the optimization process proceeds. For more

details about this VB program, interested readers are referred to the appendix.

24

Chapter 3
THE OPTIMIZATION RESULTS AND ANALYSIS

The EM properties are mainly controlled by three system factors: the shapes of the antennas,
the radius of the wires, and the load inserted at the end of these antennas. Therefore,
optimization of the antenna properties is the goal of this study. The purpose of this chapter is
to optimize the antenna properties. A set of spline antennas with the optimal wideband
properties were studied in terms of gain along a certain direction, VSWR, and efficiency over a

frequency range from 200 MHz to 2 GHz.

The GA-NEC program is used for optimization in conjunction with NEC2 (open version).
The GA—NEC program performs optimization based on the NEC2 simulation results. In this
study, both symmetric and asymmetric antenna geometry antennas are considered. The
asymmetric geometry requires more binary Strings to represent each individual than the
symmetnc one. As a result, it provides more diversity than the symmetric geometry, but needs
a larger population and a longer run time. Both open ends and closed ends were studied for

each geometric type. In the closed ends case, the lumped terminating load resistant was also

optimized by GA-NEC.

The antennas were divided into 200 segments. Three criteria were used to optimize the radii
of the segments: varying radius, constant radius and varying radius with constraint. The varying
radii were calculated according to equation 2.2 to keep the characteristic impedance of the
two-wire line constant. A constraint radius is to have the radius varying within a constrained

range based on equation 2.2. The constant radius was optimized by GA-NEC with different

25

input irnpedances. The length of each of the antennas studied is 3 meters. An impedance of
600 ohms is chosen for varying radius case, 300 ohms for constraint radius antennas and 150
ohms for constant radius antennas. Table 3.1 shows all the cases that are discussed in this

study.

Term definitions:
VSWR

An antenna’s ability to receive power from a source is determined by the input impedance. For
maximum power transfer, the input impedance should exactly match the output impedance of
the source. For broadband antennas, the input impedance differs from the output impedance

of the source. The complex reflection coefficient at the input of the antenna is

z, ,—z
”inn—+72 (3”
input 0

Here Zim, is the antenna’s complex input impedance and Z0 is the source impedance. The

power reflected is equal to the incident power multiplied by the square of the magnitude of the

complex input reflection coefficient. The net power accepted by the antenna is given by

P = incident (1 - 1“2) (3'2)

input

In terms of the reflection coefficient, the voltage standing wave ratio is given by

1+|r|
1-lFl

 

VSWR = (3.3)

26

The reflection coefficientr , ranges from 0 to 1. The VSWR ranges from 1 to infinity. The

VSWR is useful to describe the input match when the match is not good.

Antenna efliciengl

The antenna efficiency, sometimes called the radiation efficiency, is defined as the ratio of the
power radiated by the antenna to the net power accepted by the antenna from the connected

transmitter. It can be described as:

_ *
Radiated _ Bnput ”radiation (3'4)

Where Pram,“ is the power radiated by the antenna, P

input

is the power accepted from the

source, and ”radiation is the antenna efficiency.

Radiated power

The radiated power is computed as the input power minus the structure and network loss. In

this study, it is assumed that there is no network and the wire is lossless. The structure loss is

ohmic loss in the load.
Radiated = Pinput — Road (35)
Power gain

The power gain of an antenna (in a given direction) is defined as the ratio of the intensity, in a
given direction, to the radiation intensity that would be obtained if the power accepted by the
antenna were radiated isotropically. The radiation intensity corresponding to the isotropically
radiated power is equal to the power accepted (input) by the antenna divided by 47! [31]. The

desired direction refers to along the length of the antenna (z-axis) in this study.
27

GA -NE C parameter setting

In the GA-NEC program, the initial population is set to 150. In some cases, GA—NEC does
not converge after a long run time and therefore, the maximum number of generation is
chosen to control the convergence. The value is set to 100 for all cases. The crossover
probability is set to 0.75. The mutation probability is set to 0.008 and the generation gap is

0.95. The Elitist Strategy is chosen as the selection strategy.

The lengths of the antennas are 3 meters and the antennas are in yz-plane The desired
radiation direction is along the length of the antennas (z-axis). VB program generates the
antennas by passing splines through 8 equally spaced points. 5 bits for symmetric geometry
and 4 bits for asymmetric geomen'y are used to represent each point. The search domain for
each point is from 0 to 1.2 meters. Two bits are used to represent the terminating load at the
ends in the case of closed ends. For constraint or constant radii, the search space is from 0.001
to 0.01 m and is represented by 3 bits. As a result, in the cases of symmetric open ends, there
are 35 bits for varying radii and 38 for constraint/ constant radii. In the cases of symmetric
closed ends, there are 32 and 35 bits in a chromosome for varying radii and
constraint/ constant radii, respectively. In the cases of asymmetric open ends, there are 56 bits
for varying radii and 59 for constraint/ constant radii. In the cases of symmetric closed ends,
there are 50 and 53 bits in a chromosome for varying radii and constraint/ constant radii,
respectively. GA-NEC sweeps the frequency from 200 MHz to 2 GHz with a step of 100
MHz. GA-NEC only evaluates the value at integer of hundreds during the optimization
process. A voltage source of 1 volt is chosen to excite the antennas. The goal of the fitness

function is chosen as “constrain within range”. As discuss in chapter 2, a penalty will be

28

applied if the value is out of the range. In this study, the constraint range of VSWR is set to

range from 1 to 3, 5 to 20 for the power gain and 80 to 100 for efficiency.

Table 3.1 Different study cases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Geometry Ends Impedance Radius type Optimization Bits
type (Ohms) points
Open 600 varyinj 7 35
_ ends 300 constraint 7 38
Symmetric 1 50 constant 7 38
Closed 600 varfl'ng 6 32
300 constraint 6 35
ends
1 50 constant 6 35
Open 600 varyinj 14 56
. ends 300 constraint 14 59
Asymmetric 150 constant 14 59
Closed 600 varying 12 50
300 constraint 12 53
ends
1 50 constant 12 53

 

 

 

 

 

 

 

3.1 Antennas with Symmetric Geometry

The shape of the antenna is generated by spline curves passing through certain points. GA-
NEC2 optimizes the shape of the antenna by optimizing the positions of these points. For the
symmetric antennas, the number of points optimized is just half of that for the asymmetric
antenna. As shown in table 3.1, the number of optimization points is 6 for antennas with
closed ends and 7 for that with open ends. There are 8 points along the antennas equally
spaced along the length of the antennas. One point at one end is fixed in the case of open ends

and two points at the ends are fixed in the case of closed ends.

29

3.1.1 Symmetric geometry with open ends

For symmetric antennas with open ends, the current is zero at the ends. The input power is
radiated when the current travels along the antenna. In this case, the shape of the antenna and
the radius of the wire dominate the EM properties. Several cases with different input
impedance are studied (table 3.1). Optimizing the shape and the radius of the wire is the goal

for this situation.

3.1.1.1 Feed impedance of 600 Ohms

Because of the symmetric geometry, there are only seven parameters that need to be optimized
in GA-NEC. They are the positions of the seven points that the wire passes trough. The radius
of the wire is obtained with equation 2.2. Figure 3.1 shows the geometry of a 3~meter antenna.
The thickness of the line is proportional to the radius of the wire. It varies from 0.00013 to

0.013 meters. Some segments with small radii near the ends cannot be displayed in figure 3.1.

In Figure 3.2, the real part of input impedance is close to 20 (600 Ohms) and the imaginary
part tends to zero as the frequency increases. The VSWR (to Z0) varies from higher that three
at low frequency to just over one at high frequency. Therefore, the radiated power changes in a
large range at low frequency and become stable at high frequency (Figure 3.4). There is no
ohmic loss when the wave travels along the antenna and therefore, the efficiency is 100
percent for the entire frequency range (Figure 3.5). In Figure 3.6, the power gain of the
antenna is 4 dB at 200 MHz and increases to about 9 dB at 2000 MHz. It indicates that the
antenna has higher directivity at higher frequency. As shown in Figure 3.7, the antenna has
lower directivity in both xz-plane and yz-plane at low frequency. When frequency increases,

the side lobes become smaller (Figure 3.8 - 3.10). Figure 3.1 -3.10 indicates that this antenna

30

has high radiation efficiency (100 percent), low VSWR (ranging from 1.01 - 3.3), and good
directivity at higher frequency Furthermore, it will show in chapter 4 that it also has good

GMC pulse response in transmission.

31

Figure 3.1 Antenna Geometry

32

Impedance (Ohms)

2000

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

        

  

 

 

   

 

W T T T -—Iree|
-+- lmeg
1500
1000 a
500i [
l r
00.? if: iiifiii kiwi! RQIW
j.» [margjtfllrftfi‘ly 3
l ’uiijlfi
. 1’
-5oo~ 0 -
’1}
.100 00 400 600 800 1000 1m 1400 16100 18100 2000
Frequency (MHz)
Figure 3.2 Input Impedance vs. Frequency
305 I— f T
3:
2.5
a: V *
3
m 4)
>
2..
1.5_
200 400 600 000 1000 1200 1400 1000 13100 2000

Frequency (MHz)

Figure 3.3 VSWR vs. Frequency

33

Efficiency (96)

Redlated Power (Watts)

 

d d d d
' C . ‘
-b N . a a
fi’ ‘ V V U
3

 

 

 

 

 

 

 

 

 

 

 

 

 

0.4 [

 

 

o I r 1 1 1
300 1000 1200 1400 1600 1800
Frequency (MHz)

400 000 800

Figure 3.4 Radiated Power vs. Frequency

 

100.8 >

100.0 ~

100.4 ~

.5
O
P
N

.b
G

e

99.4 -

99.2 ~

 

 

 

9 r r 1 4 r
300 1000 1200 1400 1600 1800

Frequency (MHz)

400 600 800

Figure 3.5 Efficiency vs. Frequency

34

2000

Power Gain ((18)

 

 

.

a. .

7~ 1

a» -

5. .

4. ~
m

 

 

300 400 600 800 1000 1200 1400 1600 1800 2000
Frequency (MHz)

Figure 3.6 Power Gain vs. Frequency

35

 

 

 

 

 

. nu
mm 9
N“ o «0
armr “w as
11 4|
x v:
_ _
o
u
o mu
“w u :u
L- 1
n
.
u
o
u
D
o
n
B. m e
-
mu ..................................... w
1
mu ...... mu
1
no. 2

 

 

     
      

  

 

 

 

2
H
M
m w
00
nb/ul mm 0 9 00
3 Ti. W " e2!
m an. m
c ...... .U .....
a I m .
m. w .. ......
m
m.
m “\ ..n o
..oo ....... . ....... .o.
9 1
why

330°

\
Ire
e
1‘1

0 O
M w
3 2

Figure 3.7 E¢ (xz-plane) and

E9 (yz—plane) patterns at 800 MHz

Figure 3.8 E¢ (xz—plane) and

36

 

 

 

 

.0
mm
at...
nu.
__
w

II
1!

 

lllllllllllllllllllll

II
a:

o
o
9

-----..L-------

180°

.
o
a
c
-
o
o
J
u
.
.

240°

270°

-plane) patterns at 1400 MHz .

Figure 3.9 E¢ (xz-plane) and E9 (yz

 

 

 

 

 

0
0 o
m m ...
0
an r _ a
2 1 0 u 1
H y n
n
_ m .....
. ...- n -
.r n m.
. 5
3 ..r II
a o
0
000 W ................... 8
1
w W
3 2

it
lllllll

 

E9 (yz-plane) patterns at 2000 MHz

Figure 3.10 E, (xz-plane) and

37

3.1.1.2 Feed impedance of 300 Ohms

As in the 600 ohms case, there are seven points to be optimized. The VB program calculates
the radii of the segments (equation 2.2) and constrains the value in the constraint range if the
calculated results are out of it. Figure 3.11 shows the optimized result. Most of the radii have a

value of 0.01 meter except for a few segments near the feeding point that have radii varying

from 0.00163 to 0.01 meter.

In Figure 3.12, the real part of the input impedance is close to 300 ohms as frequency
increases. The VSWR is in a range of 1.08 - 2.46 (Figure 3.13). More power is radiated at
higher frequency (Figure 3.14). The efficiency is 100 percent since there is no loss (Figure
3.15). Figure 3.16 show that the power gains have several dips at frequency 550, 850 and 1600
MHz. The antenna is not very directive at low frequency (Figure 3.17). Moreover, there are
more side lobes at higher frequency. The directivity is not as good as that of the previous case

as seen in Figure 3.17 — 3.20.

38

Figure 3.11 Antenna Geometry

39

Impedance (Ohms)

VSWR

 

 

 

 

--— reel

 

 

 

 

 

 

 

 

 

 

 

 

 

 

500
400
300 >
200 r 1
100 r e
o- W’
.... . chfW" .
i I tifw "
it It: I? T Q
.200 r: 1".) ‘g V i
I * 3
3o ‘ r r I J_ L r r r
800 400 800 800 1000 1200 1400 1800 1800 2000
Frequency (MI-12)
Figure 3.12 Input Impedance vs. Frequency
2.5 ’ fi' r if I fir
2 0 _
’ 0
1D ’
1.5 r
0 o i ,
I
200 400 800 800 1000 1200 1400 1800 1800 2000

Frequency (MHz)

Figure 3.13 VSWR vs. Frequency

Efficiency (16)

 

d J d
e e e
‘ a O
I T I
1 l

v

Redleted Power (Watts)
’N

d
Jr
1

 

I
0.8 -. 1

 

 

 

oI I I I I I I L I
300 400 600 800 1000 1200 1400 1600 1600 2000
Frequency (”112)

Figure 3.14 Radiated Power vs. Frequency

 

101 r fl . x t 1
100.8 — J
100.8 ~ 1
100.4 ~ 4
1 00.2 ~ 'l
100W
99.8 ~ 4
99.8 F i
99.4 [ a
99.2 r .

 

 

 

9 I I I I I I I I
300 400 600 800 1000 1200 1400 1600 1800 2000
Frequency (MHz)

Figure 3.15 Efficiency vs. Frequency

41

Power Gsln (dB)

 

11

10P

 

 

 

200

400 600 800 1000 1200 1400 1600
Frequency (MHz)

Figure 3.16 Power Gain vs. Frequency

42

1800

2000

 

 

— xz-plene

 

 

30° -- yz-plene

  
    

I,
,r 00°
120°

IIII 00!!
I

.......5.

sls
\\
|

180°

0 |\
l lllll

III
I!
ll

and E0 (yz-plane) patterns at 200 MHz

Figure 3.17 E, (xz-plane)

 

 

 

 

om
mu
Mm.
xv:
_.
_
mu.

 

1 80°

Figure 3.18 E ¢ (xz-plane) and E9 (yz—plane) patterns at 800 MHz

43

 

— xz-plsne
30° .... yz-plsne

      
    
 

 

 

  

 

4-------

-.-~
.0

A’s"--
O

.......
o -.
'o

..---'
l

as ’ . ' . ‘
270° """" ‘ ‘ _ - """" 90°

,p
.........

  

1 50°

180°

Figure 3.19 E ‘ (xz-plane) and E9 (yz-plane) patterns at 1400 MHz

 

 

— xz-plsne
--- yz-plene

 

 

 

240° ‘ ,. _ 120°

-“
~__

_.

Figure 3.20 E ¢ (xz-plane) and E,9 (yz—plane) patterns at 2000 MHz

3.1.1.3 Feed impedance of 150 Ohms
The radii of the segments are constant throughout the antenna. The program optimizes the

value in a range of 0.001 -— 0.01 m. Figure 3.21 shows the optimized antenna shape. The GA-

NEC optimized radius value of segments is 0.0058 m.

The real part of the input impedance is close to 150 ohms as frequency increases while the
imaginary part is close to -50 ohms (Figure 3.22). Figure 3.23 shows the VSWR is in a range of
1.3 - 3.44. The radiated power increases at higher frequency (Figure 3.24). In Figure 3.26, the
power gain has several clips at frequency 300, 550 and 1250 MHz. There is no ohmic loss for
this antenna (Figure 3.25). The directivity is not as good as that of the previous ones as shown
in Figure 3.27 — 3.30. The direction of maximum radiation is around 60 degree instead of 0

degree at 200 MHz (Figure 3.27).

45

Figure 3.21 Antenna Geometry

250

200

150

100

Impedance (Ohms)

 

 

_..— ma]
--e- lmsg _

 

 

 

 

 

 

.2 I I I I_ L 4 I L
5300 400 600 800 1000 1200 1400 1600 1800 2000

VSWR

Frequency (MHz)

Figure 3.22 Input Impedance vs. Frequency

 

3e5 I I I 1 1’ f ‘ r

2.5 l r

 

1.6+ ~

 

 

 

200 400 600 800 1000 1200 1400 1600 1600 2000
Frequency (MHz)

Figure 3.23 VSWR vs. Frequency

47

Efficiency (16)

3.5 1 r

 

2.5L,

 

Radiated Power (Watts)

1.5

 

 

1000 1200 1400 1600 1600
Frequency (MHz)

200 400 800 800

Figure 3.24 Radiated Power vs. Frequency

101 1 1 v . a t r I

2000

 

100.8 -

1 00.8 -

100.4 ~

a
O
P
N

I-

99.8 r

e

99.4 >

99.2 -

 

1

.4

 

 

9 I I L I
200 1000 1200 1400 1600

Frequency (MHz)

400 600 800 1800

Figure 3.25 Efficiency vs. Frequency

48

2000

Power Geln (d8)

 

 

 

 

l

 

 

.300 400 600 800 1000 1200
Frequency (Ml-ls)

1400

1800 1800 2000

Figure 3.26 Power Gain vs. Frequency

49

 

 

 

 

 

.0
"n
h“
as
a”
__
a
w

 

180°

330°

Figure 3.27 E ¢ (xz-plane) and 1i"9 (yz-plane) patterns at 200 MHz

 

 

 

 

O.
mm
as
an.
m...
B
ow
o .......
..o
n

      
         

-5-------

IIIII
I Ill

3299,.

\
0
III
e.\1

O
o
7
2

III

   

'-~-—

180°

E0 (yz-plane) patterns at 800 MHz

Figure 3.28 E, (xz-plane) and

50

 

0° — xz-plsne
330° 30° "'- yz-plene

 

 

 

 

270°

Figure 3.29 E, (xz-plane) and E9 (yz-plane) patterns at 1400 MHz

 

— xz-plsne
--- yz-plane

 

 

 

.
.
2700 ....... ‘f j . ............ 90°
9 . 4 l
I \ ‘
' . § I

~.‘
~—
.....

Figure 3.30 E ¢ (xz-plane) and E9 (yz-plane) patterns at 2000 MHz

51

3.1.2 Symmetric geometry with closed ends

For symmetric antenna with closed ends, the current may not be zero at the end. The input
power is radiated as the current travels along the antenna. Some of the power may be lost in
the load. In this case, the EM properties are affected by three factors: the shape of the antenna,
the radius of the wire, and the load on the antenna. Several cases with different input
impedance are studied (Table 3.1) to optimize the shape, the radii of the composed segments
and the terminating load to the antenna. Since the ends are closed, only 6 points control the

shape of the antenna.

3.1.2.1 Feed impedance of 600 Ohms

The optimized geometry is shown in Figure 3.31. Some segments with small radii near the
ends cannot be displayed. The thickness of the line represents the radii of these segments,
which vary from 0.00013 to 0.01123 m. The load to the end segment is 600 ohms which is

equal to the characteristic impedance of the two—wire line.

Figure 3.32 shows that the real part of the input impedance is around 600 ohms and the
imaginary part is around zero. The VSWR varies from 1 to 2.6 (Figure 3.33). In Figure 3.35,
the efficiency is 100 percent over the frequency band because there is no current passes
through the load. In figure 3.36, the power gain is higher than 2.2 dB over the frequency band
and reaches to 12 dB at around 1600 MHz. The direction of maximum radiation is 90 degree
at 200 MHz (Figure 3.37). It is in the desired direction at higher frequency as seen in figure

3.38 — 3.40.

52

Figure 3.31 Antenna Geometry

53

1400

 

1200 ~

1000 *

 

800

Impedance (Ohms)
‘
8

 

 

 

 

 

 

I
-+- Imeg

 

 

 

 

 

2.8

400

800 800 1000 1200 1400 1800 1800 2000

Frequency (Ml-Ix)

Figure 3.32 Input Impedance vs. Frequency

 

2.61!
2.4 -

2.2 >

 

 

 

 

)

 

 

loo

400

000' 000 1000 1200 1400 1s00 1000 2000

Frequency ("112)

Figure 3.33 VSWR vs. Frequency

54

14

12

10

Redlsted Power (Watts)
es

101

100.8

100.8

100.4

an.
O
P
N

Efflclency (10)

99.4

99.2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

200

400

600 600 1000 1200 1400 1600
Frequency (MHz)

1 800

Figure 3.34 Radiated Power vs. Frequency

2000

 

l.

 

1 00W

 

 

9Boo

400

600 800 1000 1200 1400 1600
Frequency (MHz)

Figure 3.35 Efficiency vs. Frequency

55

1800

2000

Power Gsln (dB)

 

 

 

 

 

200 400 600 800 1000 1200 1400 1600 1600 2000
Frequency (MHz)

Figure 3.36 Power Gain vs. Frequency

56

 

 

 

 

 

o
00 0
mm 9 o
' 0
«.1 w m
22 1
XV.
: .......
o
o 0
w 5
1
a...
m .
.
0 .......................... o
0 L. W
..
0 0
m 40:
3 2

t
I \
ttttttttt

O o
M w
3 2

 

E9 (yz-plane) patterns at 200 MHz

Figure 3.37 E, (xz-plane) and

 

 

 

 

 

0
mm m
“h 0 00
an m ....
xv.
a «0
M Isl.
B 0
min. ................................ n

 

E9 (yz-plane) patterns at 800 MHz

Figure 3.38 E, (xz-plane) and

57

 

 

 

 

0.
mm
an
an
__
w

w

e|\\
III
IIIIII

0
o
7
2

 

180°

-plane) patterns at 1400 MHz

Figure 3.39 E, (xz-plane) and E9 (yz

 

 

 

 

0
60 0
nn 9
.Ieh 0 m0
0 . 2
”w mu :0 n I.
XV: ”
__ ..... .. ......
. .. m
0 n ““04
w . 1

 

a
a
--T

10111. I IIIII IIIIII I IIIIIIIIIIIIIIIIIIII

180°

Ill \
CCCCC
llllll

.
.
.
.
_
c
.
J
.
.
.
.
.
.
.

270°

E9 (yz-plane) patterns at 2000 MHz

Figure 3.40 E, (xz-plane) and

58

3.1.2.2 Feed impedance of 300 Ohms
GA-NEC optimizes the shape of the antenna and constrains the radii of the composed
segments in a specified range of 0.002 — 0.01 m. Figure 3.41 shows the optimized result in this

case. The radii vary from 0.002 — 0.009 m. The optimal load to the end is 600 ohms.

The real part of input impedance becomes stable at 300 ohms and the imaginary part tends
to zero as frequency increases (Figure 3.42). Figure 3.43 shows that the VSWR is from 1.03 to
3.85. There is a VSWR burst around frequency 250 MHz. This is because GA-NEC only
evaluates the value at integer of hundreds during the optimization process. The radiated power
varies at different frequencies (Figure 3.44). Figure 3.45 shows the power radiation efficiency is
100 percent through the study frequency band. The power gain along the length of the
antenna is between 4.88 and 11.62 dB (Figure 3.46). From Figure 3.47 to 3.50, it can be seen
that this antenna become more directive at higher frequency. In figure 3.47, there are three big
side lobes in yz-plane the antenna there is not directive in xz-plane. The side lobes getting

smaller at higher frequency indicate that the antenna is more directive.

59

Figure 3.41 Antenna Geometry

60

1000 r v . f a

Impedance (Ohms)

 

 

I
-— reel
-+- Imeg
600 ~ 4

 

 

 

cool

 

 

1

vii .inrmwymw.

 

 

 

400 600 600 1000 1200 1400 1600 1600 2000
Frequency (III-ls)

Figure 3.42 Input Impedance vs. Frequency

 

3.5 ~

 

 

 

 

300 400 600 600 1000 1200 1400 1600 1600 2000
Frequency (1411:)

Figure 3.43 VSWR vs. Frequency

61

Efficiency (16)

Radiated Power (Watts)
N

3.5

I"
on

.5
0|

0. L I I I L I L I
300 400 600 600 1000 1200 1400 1600 1800 2000

101

100.8

100.8

100.4

a
O
P
N

99.4 -

99.2 —

9 I I I I I L I L
200 400 600 800 1000 1200 1400 1600 1800 2000

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Frequency (MHz)

Figure 3.44 Radiated Power vs. Frequency

 

l. .1

 

 

 

Frequency (MHz)

Figure 3.45 Efficiency vs. Frequency

62

Power Gsln (dB)

 

 

 

 

 

4 I I a I I I L I
200 400 600 800 1000 1 200 1400 1600 1600 2000
Frequency (MHz)

Figure 3.46 Power Gain vs. Frequency

63

 

 

 

 

mm
um um
re
an.
I
w

300°

0
0 II

9“
III

'ca-

IIOIIII

‘~_

120°

240°

         

210°

180°

Figure 3.47 E, (xz-plane) and E9 (yz-plane) patterns at 200 MHz

 

 

 

 

mm
hm hm
WW
ya u:
o
M
B
an
“mvnu
o
nu
ea
ea

Gill

>-------r---

III!

I
I
III
III

  
  

llllllllll

   
 

180°

E9 (yz-plane) patterns at 800 MHz

Figure 3.48 E, (xz-plane) and

64

 

0° -— xz-plene
30° -- yz-plsne

 

    

 

 

 
    
        
    

J-------

.........

 
  

O)
O
o
0
v0---
N
O

........
a
o u‘

9 -~

-----J-_---_-

2700 ......... ’ . - ---.°.°- ‘ ' x ,_ ......... 90°

5
-------

210° ‘ 150°

 

0° — xz-plsne
30° --- yz-plene

 

 

 

........

.........

-----

.---.-
i

l\§

I e

......... . . , ‘ ~ 0
270° ‘ i A El ......... 90

a-‘.
a

......

"C- -o-‘

- -0'
........

180°

Figure 3.50 E, (xz-plane) and E9 (yz-plane) patterns at 2000 MHz

65

3.1.2.3 Feed impedance of 150 Ohms
In this case, the radii of the segments of the antenna are set to be constant. The GA-NEC
optimizes the value in a range of 0.001 — 0.016 m. The optimized geometry is shown in figure

3.51. The segments that made the wires have constant radii of 0.005 m. The load to the ends is

600 ohm$.

Figure 3.52 shows that the real part of the input impedance is around 150 ohms and the
imaginary part is around -50 ohms. The VSWR values are in a range of 1.15 — 2.8, but most of
them are larger than 1.4 (Figure 3.53). The radiated power increases at higher frequency
(Figure 3.54). The efficiency is 100 percent (Figure 3.55). It can be seen that there are three big
dips in the power gain at around 450, 950 and 1450 MHz (Figure 3.56). The worst one is
around 950 MHz at which there is almost no radiation along the antenna. Figure 3.57 shows
that the antenna has low directivity at lower frequency. There is a subtle improvement at

higher frequency.

66

Figure 3.51 Antenna Geometry

67

Impedance (Ohms)

VSWR

400

300

200

100

@300

 

 

 

-+- Imeg

 

 

 

 

 

1000 1200 1400 1600 1600
Frequency (MHz)

400 800 800 2000

Figure 3.52 Input impedance vs. Frequency

 

2.8

2.8

2.4

 

2.2

2

1.8

1.8 ~

1u‘<'

1.2

4D

 

 

 

 

 

loo

1000 1200 1400 1600 1800
Frequency (MHz)

400 800 800 2000

Figure 3.53 VSWR vs. Frequency

68

Efficiency (16)

3.2

 

 

Radiated Power (Watts)
N
N

1.4

 

 

 

 

‘Lioo

101

400

600 600 1000 1200 1400 1600
Frequency (MHz)

1 800

Figure 3.54 Radiated Power vs. Frequency

2000

 

100.8 -

100.8 ~

100.4 -

_s
O
P
N

100W

99.8 F
99.8 -
99.4 F

99J2l

 

 

 

°ioo

400

600 800 1000 1200 1400 1600
Frequency (MHz)

Figure 3.55 Efficiency vs. Frequency

69

1800

2000

Power Geln (d8)

15

 

10»

 

 

 

 

 

0 ~ 1
.5 p o .i
-10 . -
I
.13 I I I I L L I I
00 400 600 800 1000 1200 1400 1600 1 800 2000

Frequency (MHZ)

Figure 3.56 Power Gain vs. Frequency

70

 

— xz-plene

 

 

 

30° -- yz-plsne

 

180°

E9 (yz-plane) patterns at 200 MHz

and

Figure 3.57 E, (xz-plane)

 

 

 

 

mm
h“
as
an.
__
w
a
000

120°

 

180°

E9 (yz-plane) patterns at 800 MHz

and

Figure 3.58 E, (xz-plane)

71

 

0° — xz-plane

 

 

 

 

270° """"

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

Figure 3.59 E, (xz—plane) and E9 (yz-plane) patterns at 1400 MHz

 

— xz-plsne
3300 0‘3 . 300 -- yz-plsne

 

 

 

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

......
-----

o ..

270° ----------- 0 f: j f if ; :37. ----------- 90°

s
h a
m —
-.- --o
-5- .-.-

180°

Figure 3.60 E, (xz-plane) and E0 (yz-plane) patterns at 2000 MHz

72

3.2 Antennas with asymmetric Geometry

As shown in table 3.1, 14 points in open ends cases and 12 in closed ends cases define the
shape of the asymmetric antenna. More binary strings are required than in the symmetric case
to code these parameters in GA-NEC. The length of the chromosome to represent an
individual is almost twice of that in symmetric cases. As discussed in chapter 2, this gives more
diversity, and the program searches the optimal solution in a larger domain. Therefore, a bigger
number of individuals in the initial population and more running time are needed. In this

study, the initial population is set to 200 and the maximum number of generations is 100.

3.2.1 Antennas with open ends

3.2.1.1 Feed impedance of 600 Ohms

The radii of the segments are calculated with equation 2.2. The characteristic impedance of the
two wires is a constant of 600 ohms. Figure 3.61 shows the geometry of the optimized
antenna. The thickness of the line is proportional to the radius of the segments ranging from
0.00013 to 0.01189 m. Some segments with small radii near the ends cannot be displayed in
Figure 3.61.The real part of the input impedance varies around 600 ohms and the imaginary
part varies around 0 ohms (Figure 3.62). The input VSWR is below 2.8 (Figure 3.63) and the
efficiency is 100 percent over the frequency band (Figure 3.65). The radiated power versus
frequency is shown in figure 3.64. In Figure 3.66, the antenna has good power gains. The
direction of the maximum radiation is around 10 degrees away from the desired direction at

800 MHz (Figure 3.68) and 15 degrees away at 2000 MHz (Figure 3.70). The antenna pattern is

73

not symmetric. In Figure 3.68 — 3.69, the direction of the maximum radiation is close to the

desired direction (along the length of the antenna).

Figure 3.61 Antenna Geometry

74

Impedance (Ohms)

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

--—Ireel
-+- Imeg
1000
500l
0 5-54 iiififrfii’WWWW
115*” iii
I
45001
400200 400 600 600 1000 1200 1400 1600 16100 2000
Frequency (MI-ll)
Figure 3.62 Input Impedance vs. Frequency
2.6 f
2.6
2.4.
(I
2.2L
I: 2’0
3
In
”I
1.6- "
1.4~ .
1.2» l
200 400 600 600 1000 1200 1400 16°00 16100 2000

Frequency (MHz)

Figure 3.63 VSWR vs. Frequency

75

Efficiency (16)

1.8

1.8-

1.4>

1.2’

08»
O

Radiated Power (Watts)

0.8

0.4 I

0.300

101

100.8

100.8

100.4

d
O
P
N

100W

99.4

99.2

°ioo

 

 

 

 

400 600 600 1000 1200 1400 1600 1800 2000
Frequency (MHz)

Figure 3.64 Radiated Power vs. Frequency

 

[-

 

.4

J

 

 

400 600 600 1000 1200 1400 1600 1800 2000
Frequency (MHz)

Figure 3.65 Efficiency vs. Frequency

76

Power Gain (d8)

11 Y r w

 

 

 

 

 

4 L I I I I I I I
200 400 600 800 1000 1200 1400 1600 1600 2000
Frequency (MHz)

Figure 3.66 Power Gain vs. Frequency

77

 

— xz-plene

 

 

 

me ~- yz-plsne

 

1 80°

E9 (yz-plane) patterns at 200 MHz

Figure 3.67 E, (xz-plane) and

 

 

 

 

..
mm
as
an.
__
0..

III!

y-.-----

eels

120°

III IIIIIIIIIII

 

--.-

180°

and E9 (yz-plane) patterns at 800 MHz

e 3.68 E, (xz-plane)

F'

78

 

— xz-plene
-- yz-plsne

 

 

 

 

‘ S
a ' ‘ I
l
o r ' “ 0
I ' ‘
\
\‘ ‘
. ~ 4
- e ' .
Q . . T ‘ C

~ \ I r
“ s I I ' .
*~ ‘ \ I . ‘ '
I Q ~ ... __
o \ I I o
270 ‘- - . , - ............. 90

s I , ' '

' ' . I

a' “ I " \
\‘ ,
\ I I
240° ; ., 120°
I“‘.~~ ° ""\‘
210° 150°
180°

 

0° — xz-plene
330° .°°° 30° “" "Wm

 

 

 

300° 80°

.........

a s I \ , ~
' \ , I ‘ ' ~
“ l I
o _ l \ I o
\‘ I 8 \ "
\ ' I ‘ I

\ e . \"

l“~ ' "-\
. \

I ‘. \

o
-
...........

210° : 150°
130°

Figure 3.70 E, (xz-plane) and E9 (yz-plane) patterns at 2000 MHz

79

3.2.1.2 Feed impedance of 300 Ohms

Figure 3.71 shows the optimized antenna shape. The thickness of the wire is proportional to
the radius. Most of the radii of segments of the antenna are 0.005 m except for a few segments
near the feeding point that have radii values varying from 0.00165 to 0.005 m. Figure 3.72
shows the input impedance to the antenna. The real part of the impedance varies around 300
ohms (the desired impedance), and the imaginary part has the mean value about zero. The
input VSWR ranges from around 1 to less than 2.6 (Figure 3.73). Figure 3.74 shows the
radiated power as a function of frequency. The radiation efficiency is 100 percent over the
frequency band (Figure 3.75). In figure 3.76, all power gains along the length of the antenna
are above 4 dB. There are two small dips at around 650 MHz and 1500 MHz. Figure 3.77
indicates that this antenna is more directive that the previous one at low frequency in the
desired direction. But, it is not as good at higher frequency (Figure 3.78 — 80). The direction of

the maximum value is around 10 degree away from the desired one.

80

 

Figure 3.71 Antenna Geometry

81

 

 

 

 

--— real

-+- lmag

 

 

 

 

600 800 1000 1200 1400 1600 1800 2000
Frequency (MHz)

400

 

 

700

 

.255. coca—39:.

 

40200

Figure 3.72 Input Impedance vs. Frequency

2000

 

 

2.‘

 

 

 

o
no
a
1
o
no
6
1
m
141
\I
m
o
-.....m
'1 1y
c
0m
0“
4W0.
F
o
no
a
m
o
10
‘
o
o
Al2

Figure 3.73 VSWR vs. Frequency

82

Efficiency (96)

Radiated Power (Watts)

x10

 

3.50

2.5

 

 

 

 

 

 

J.__

 

 

1000 1200 1400 1600 1800

Frequency (MHz)

400 600 800

Figure 3.74 Radiated Power vs. Frequency

101 Y Y r l I I 1 Y

2000

 

100.8 ~

100.6 1-

100.4 ~

d

O

.0

N
I

100W

99.2 [

 

1

—4

 

 

9 l 1 1 i 1
300 1000 1200 1400 1600 1800

Frequency (MHz)

400 600 800

Figure 3.75 Efficiency vs. Frequency

83

2000

Power Gain (dB)

 

8.5L

8..
1.5»

7.
0.5»
5r
5.5» k. U i.
5

y
4.5

 

l

 

 

200 400 600 800 1000 1200
Frequency (MHz)

1400

1600 1800 2000

Figure 3.76 Power Gain vs. Frequency

84

 

0° — xz-piane
330° 0‘3 300 .... yz-plene

 

 

 

 

-- o-
_______
_— o
........
- ~

.....
s
I 0“

J'"-----l~. ‘: ‘ ~ ,1 ~

I

.....
n
u‘-‘

4:
........
........

 

0° — xz-plene
“3 ..... yz-plene

 

 

 

Figure 3.78 E' (xz—plane) and E9 (yz-plane) patterns at 800 MHz

85

 

— xz-plene
330° Ode 30° ”" yz-plene

 

 

    
 

 

.....
..‘-

-.‘
u
u

-----

.
270° ------------ .. ~ ‘ o
, V
‘ .5 4' .

 

0° — xz-plane
30° --- yz-plene

 

 

   

 

I»

f-----

a’ ‘o

 
 

-—----J------.

270° ....... i """"" - . .‘ . ' VA ' ............. 900

‘-
‘c—

210° 150°

Figure 3.80 E¢ (xz-plane) and E9 (yz-plane) patterns at 2000 MHz

86

3.2.1.3 Feed impedance of 150 Ohms

The range of the radii of all the segments is from 0.0005 to 0.01 m. The optimized value is
0.00864 m. Figure 3.81 shows the shape of the antenna. In Figure 3.82, the input impedance
changes as a function of frequency. The VSWR is in a range of 1.7 — 2.65 (Figure 3.83). In
figure 3.84, more power is radiated as the frequency increases. The radiation efficiency is 100
percent over the frequency range (figure 3.85). All of the power gains of the antenna are above
4.8 dB over the whole frequency band (Figure 3.86). There is a dip at around 750 MHz and a
peak at 1300 MHz. The maximum radiation in the length of the antenna direction is shown in
Figure 3.87. It is 20 degrees away in the xz-plane at 800 MHz (Figure 3.88) and 10 degrees
away in yz-plane (Figure 3.89). At 2 GHz, the maximum radiation is 15 degrees away in the yz-

plane.

87

 

Figure 3.81 Antenna Geometry

88

250

 

200

150-

100-

soh

Impedance (Ohms)

.50.

400 r

 

-1 50

 

qgtyfl'

 

__ mg]
-+- Imag

 

 

 

 

 

"°¥oo

400 600 800 1000 1200 1400 1600 1800 2000
Frequency (MHz)

Figure 3.82 Input Impedance vs. Frequency

 

2.8

2.6

2.4

VSWR

2.2

1.8*

 

 

 

 

1 .300

400 600 800 1000 1200 1400 1600 1800 2000
Frequency (MHz)

Figure 3.83 VSWR vs. Frequency

89

Efficiency (16)

Radiated Power (Watts)

4.5

101

100.8

100.0

100.4

.5
0
.°
N

1 DOW——

e

90.4

90.2

x10

 

3.5 -

 

9300

400

l

600 800 1000 1200 1400 1600
Frequency (MHz)

1 800

Figure 3.84 Radiated Power vs. Frequency

 

2000

 

t

 

-1

 

 

400

600 800 1000 1200 1400 1600
Frequency (MHz)

Figure 3.85 Efficiency vs. Frequency

90

1000

2000

Power Gain (d8)

 

 

 

 

 

300 400 600 800 1000 1200 1400 1600 1800 2000
Frequency (MHz)

Figure 3.86 Power Gain vs. Frequency

91

 

— xz-plane
330° 30° -- yz-plane

 

 

   

 

.....

.....
4- \‘

   
   
   

4_------L--

-
- .
‘a— -.-

§
......
........

1 80°

 

 

 

 

 

Figure 3.88 E, (xz-plane) and E9 (yz-plane) patterns at 800 MHz

92

 

 

 

 

 

. o
m m w .
| I o o
o . 2
x y w
_ m .....
. t- n -
o u 00
w H
a o
0
menu ...................... um

IIIIII
iiiii

o 0
M w
3 2

 

and E9 (yz-plane) patterns at 1400 MHz

-plane)

Figure 3.89 E‘ (xz

 

 

 

 

o

.3 0
mm 9 0
l' o o
9? W 2
11 al
XV.
—.

.

u

.

o

o

3

 

1 00°

...0 v
o ‘
3 2

plane) patterns at 2000 MHz

and E 0 (yz-

-plane)

Figure 3.90 E‘ (x2

93

3.2.2 Antennas with closed ends

3.2.2.1 Feed impedance of 600 Ohms

Figure 3.91 shows the optimized shape of the antenna. The radii of these segments vary from
0.0005 m to 0.01125. Some of the radii are too small to be displayed, especially at the two
ends. The input impedance as a function of frequency is shown in Figure 3.92. It shows that
the input impedance is not as a smooth function as that of the symmetric antenna. The
VSWRs vary from 1.01 to 2.5 and the mean value is around 1.5 (Figure 3.93). It is not a
smooth function of frequency. Figure 3.94 shows the radiated power as a function of
frequency. The power gain is above 3.3 dB over the frequency band (Figure 3.95). It reaches
9.5 dB at 600 MHz and drops to 5 db at around 1150 MHz. As shown in figure 3.97, the
maximum radiation direction is 30 degrees away in the xz-plane at 200 MHz, and about 3
degrees away in the yz-plane at 800 MHz (Figure 3.98). The maximum radiation is in the yz-
plane which is 5 and 10 degrees away from the desired direction as show in figure 3.89 and

figure 3.90, respectively.

94

Figure 3.91 Antenna Geometry

95‘

 

 

--- real

-+- Im an

 

 

 

f

 

1200

1000

 

0
0
dl A q o
2
0
r 0
8
1
M
L 0
1

D 0
v. m
Re.

9:... M
We. .2.
”......
0
1 0
..0..
t H
1.0.0.1. - W
10mm! 8
90mm“!

Mutant”! r W
tmmw... s
”MUN“:
‘uumnnleeo
‘HHIIII'! o

IIUHHWUIIO i o

”wining-i 4
IIIHI "Hi
I..." Ilka-uni
o m m o m .0. 0.
m 4 2 .... 4 m

.255. 3:239...

Frequency (Ml-ix)

Figure 3.92 Input Impedance vs. Frequency

5
2000

 

 

 

1000 1800

1400

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

'
'1
1 Lo
1 1
,l
l
||
0
’ 1
0
‘1 o
1 1 F 1
P P 1 o
1 1
‘ ’ 1
’ ‘
1 1
‘ ’
P 1
1
1 P
‘ 1‘
P 1
-4 lo 0
N r
‘ 0
0'0"“ 4 8
’ ’
lHt 1
0 F
F 1
P P
P1 1
A.
1 M o
P
1 i P
P 3‘ h 1
1 V1 0
1 F
‘
P l 1
‘ ’
1 V
P 1
1 ‘
1

’

1 1 P o
D 1 1 10
‘IF 0

4 ‘
P 1 ‘
’
1 It!
’ .‘

 

 

 

 

 

 

Frequency (MHz)

Figure 3.93 VSWR vs. Frequency

96

Efficiency (96)

1.6

 

1.4r

A
v

A
e
N
A

Radiated Power Matte)
O
b .a

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.300

101

400

600 800 1000 1200 1400 1600
Frequency (Mi-i2)

1000

Figure 3.94 Radiated Power vs. Frequency

 

100.8 -

100.6 ~

100.4 r

.3
O
.0
N
I

99.0 r
99.6 -
99.4 -

99.2 -

 

 

 

9300

400

600 800 1000 1200 1400 1600
Frequency (MHz)

Figure 3.95 Efficiency vs. Frequency

97

1800

2000

10 , . r

 

Power Gain (dB)

 

 

 

 

300 400 600 800 1000 1200 1400 1600 1800 2000
Frequency (MHz)

Figure 3.96 Power Gain vs. Frequency

98

 

— xz-piane
330° 7 30° --- yz-plane

 

       

 

 

....

5.---

...a.

{v
‘i

1

i

. ,—
‘-- -.-

 

 

 

 

   

‘s

.....

-----J---

a
N
O

-- -.
v' ~-
5
o' “
a

— -
n ..

- 0
270° ------- , . ---------- 90

-
e.-
-‘--

--“
.-.

Figure 3.98 E ¢ (xz-plane) and E0 (yz-plane) patterns at 800 MHz

99

 

 

 

 

..
0

mm 00v m 00

mm .. u

I
w. w

 

180°

210°

plane) patterns at 1400 MHz

-plane) and E9 (yz

Figure 3.99 E, (xz

 

 

 

 

..
n"
h“ o
95 0
ll 6
‘y
o
o
3
B
..M
o .......
o
o
3
3

 

120°

IICe

-------L--...

llllllllllllllllllllll

180°

ttttt
00000

o

o
7
2

plane) patterns at 2000 MHz

and E9 (yz

—plane)

Figure 3.100 E¢ (xz

100

3.2.2.2 Feed impedance of 300 Ohms

The optimized shape is shown in figure 3.101. The radii of these segments vary from 0.00163-
0.006 m. The terminating load to the ends is zero (short circuit). The input impedance versus
frequency is shown in figure 3.102. The VSWR varies from 1.05 to 2.5 (Figure 3.103). The
radiated power and efficiency are shown in Figure 3.104 and Figure 3.105, respectively. The
power gain starts from 3.5 dB at 200 MHz and reaches 12.8 dB at 2000 MHz (Figure 3.106). It
indicates that this antenna has good directivity along the length of the antenna. Figure 3.107 —
120 show the radiation pattern in xz-plane and yz-plane. The maximum radiation direction is

not always along the length of the antenna in yz-plane over the frequency band.

101

 

 

Figure 3.101 Antenna Geometry

102

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-— real
500 -+- imag
400
. 300
I
E
4:
g 200»
3
g 100 ,1
5 o ' 1,11 t
1 1 1:1111111W&111th*
11351 1
.2001 12>}?
30 4:11“ i r L i i i i i
. 300 400 600 600 1000 1200 1400 1600 1600 2000
Frequency (MHz)
Figure 3.102 Input Impedance vs. Frequency
2.5 L fl
I i,
2
a: U
3 4
e)
>
1.5 11
. . '
' ' ' i i '
200 400 600 000 1000 1200 1400 1600 1000 2000

Frequency (14111)
Figure 3.103 VSWR vs. Frequency

103

Efficiency (96)

2.0 T . - . f f . .

 

2.4 - 1

2'21 l’ 0“ .

 

 

 

 

Radiated Power (Watts)
0

 

 

 

 

 

 

 

 

 

 

o. 7 L 1 1 1 1 L 1 1
300 400 600 800 1000 1200 1400 1600 1800 2000
Frequency (MHz)

Figure 3.104 Radiated Power vs. Frequency

101 1 7 f r . . ' '

 

100.3 - .
100.01 ~

100.4 ~ -

1- d

100;“

99.0 1 _

.3
0
.°
N

99.6 '- -<
99.4 r 1

99.2 ~ _

 

 

4

9 1 1 1 L 1 1 1
200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency (MHz)

 

Figure 3.105 Efficiency vs. Frequency

104

Power Gain (d8)

13 r—

 

12~

10~

 

1

200 400 600 800 1000 1200 1400 1600
Frequency (11112)

Figure 3.106 Power Gain vs. Frequency

105

1800

 

2000

 

 

 

 

mm
M.“
a?
ll
XV:
_“
u
w
a
..M
0 .......

I'll

300°

II.

'-.-

‘--

120°

llllll

240°

 

1 50°

180°

210°

Figure 3.107 E ¢ (xz-plane) and E9 (yz-plane) patterns at 200 MHz

 

— xz-piane

 

 

 

300 -- yz-plane

 

100°

Figure 3.108 E" (xz-plane) and E9 (yz-plane) patterns at 800 MHz

106

 

OdB — xz-plane
330° 30° VFW”

 

 

 

 

.. I ‘ ‘
i
a I
O \
I
300° ‘ ' ‘ 60°
I
‘ l | I \ ,
~‘ I ‘ ‘p , \ "—

........
‘Q
~ ‘ v
s‘ ‘ , 'a \ ,
~ (I, 4 0 s
I ‘ I ' 1
.91. o
O ............... - .-----
s , _‘ . .. I
0 \x .
o ‘ - . a
1 .

r ‘
’. _-' \ 1 ‘.
\ I -- _-- s ,
u , a
\
O ‘ ° ' O
\‘ I \ "
o ' ‘ I
. ‘ 4
x‘ I -’
I ‘ r \
I ' \

210° ‘ 150°
100°

Figure 3.109 E ¢ (xz—plane) and E9 (yz-plane) patterns at 1400 MHz

 

0° — 111-plane
30° --- yz-plane

 

 

 

900

o
e ’e
e '-

------ ' 120°

 

210° 150°
100°

Figure 3.110 E ‘ (xz-plane) and E9 (yz-plane) patterns at 2000 MHz

107

3.2.2.3 Feed impedance of 150 Ohms

This antenna has the optimized constant radius of 0.007 m and the shape is shown in Figure
3.111. The terminating load to the ends is 300 ohms. The real part of the input impedance
varies around 150 ohm$ and the imaginary part is close to -50 ohms as frequency increases
(Figure 3.112). The VSWR varies from 1.28 to 2.0 and the mean value is around 1.45 (Figure
2.113). The radiated power increases with frequency (Figure 3.14). The antenna efficiency is
from 81.5 percent at 200 MHz to 99.7 percent at 2000 MHz (Figure 3.115). The input power is
loss due to the ohmic loss at the load. The antenna has good power gain along the length
except there are two dips at around 1000 MHz and 1500 MHz. In Figure 3.117, the direction
of maximum radiation is around 30 degrees away from the desired direction in the xz—plane at
200 MHz. It is at around 10 and 15 degrees away in the yz-plane at 800 MHz (Figure 3.118).

The maximum radiation is in yz—planes but not in the desired directions (Figure 3.119-120).

108

 

Figure 3.111 Antenna Geometry

109

 

250

 

200 .

150

100 r

50-

Impedance (Ohms)

 

I

 

 

—— ng'
-+- Imag

 

 

 

 

400 600 800 1000 1200 1400 1600

Frequency (MHz)

1800 2000

Figure 3.112 Input Impedance vs. Frequency

 

1.0 »,

1.7%

 

 

vswn
3

 

1.5

1.4

1.3

 

 

 

 

 

 

 

 

 

1 .200

400 600 800 1000 1200 1400 1600
Frequency (MHz)

Figure 3.113 VSWR vs. Frequency

110

1900 2000

Efficiency (96)

Radiated Power (Watts)

 

1d
a
1
1
q
.1

1° 9’ f"
O h. N .1
' Y 1 I
1 1 1 1

1"
a
'

1

 

P
‘1
r

1

 

1"
N
1

 

N
1

1.0 -

 

 

1 . 1 1 1 1 1 L 1 1
:00 400 600 000 1000 1200 1400 1600 1000 2000
Frequency (MHz)

Figure 3.114 Radiated Power vs. Frequency

100

 

94

92

88.

 

 

 

 

8 1 1 1 1 1 1 1 1
300 400 600 800 1000 1200 1400 1600 1800 2000
Frequency (MHz)

Figure 3.115 Efficiency vs. Frequency

111

 

Power Gain (dB)

10

 

 

 

 

 

400 600 800 1000 1200 1400 1600
Frequency (MHz)

Figure 3.116 Power Gain vs. Frequency

112

1 000

2000

1 u—v ..v— .-—

 

 

 

 

 

 

mm w

hh o ...u

m... w a

I

.... .0.

    

100°

     
   

3.,”

1'

-------J

o
o
7
2

-plane) patterns at 200 MHz

Figure 3.117 E¢ (xz-plane) and E9 (yz

 

 

 

 

.0
n"
hr.“
09
11
way
_.
m
.0.

90°

270°

120°

 

--.-

w

 

100°

-plane) and E0 (yz-plane) patterns at 800 MHz

Figure 3.118 E¢ (xz

113

 

—- xz-plane

 

 

 

 

...........
.o

______
,a

J-------

_._

. 4> ' .
27o ....... , ..... "‘2' .. N f, tn .......... 9°

240° ‘ ,. ' 120°

 

0;“ — xz-plane
330° 30° --- yz-piane

 

 

 

270° ------- W . 1 ------- 90°

.
~‘_

Figure 3.120 E y (xz-plane) and E9 (yz-plane) patterns at 2000 MHz

114

3.3 Conclusions

In this research, twelve different types of antennas are synthesized and discussed. Simulation
results show that all the antennas have 100 percent radiation efficiency except the one shown
in figure 3.115. It is because of the ohmic loss at a 300 ohms terminating load to the ends. The

efficiency starts at 81.5 percent at 200 MHz and go up to 99.7 percent at 2000 MHz.

The VSWR versus frequency curves vary with different shapes and segments radii types.
Generally, the ones with varying segment radius have lower VSWR values at higher frequency.
As shown in figure 3.3, 3.33, 3.63 and 3.93, the VSWR starts at around 3 at low frequency, and
decrease to approximately 1 as the frequency goes higher. In figure 3.13, 3.43, 3.73 and 3.103,
the VSWR starts from around 3 and slowly decreases to around 1.1 while it is 1 as frequency
increases. With the feed impedance of 150 ohms, the VSWR starts at around 3 and drops to
around 1.4 at high frequency as shown in figure 3.23, 3.53, 3.83 and 3.113. The mean value of
VSWR is higher than that with feed impedance of 600 and 300 ohms. On the other hand,
these antennas with symmetric geometry have smoother impedance versus frequency curves
than those of asymmetric geometry. The ones optimized with higher feeding impedance have

better VSWR properties.

For the antennas with the feed impedance of 600 ohms, the real parts of the input
impedance vary around 600 ohms and the imaginary parts vary around zero. For ones with 300
and 150 ohms feed impedance, the real parts vary down to around 300 and 150 ohms,
respectively. The imaginary parts go up close to zero in the 300 ohms case while they approach

to -50 ohms in the 150 ohms case at high frequency.

115

In figure3.4, 3.34 and 3.64, the radiated power vanes around 0.8"'10_3 watts. It starts
around 0.8 "‘10-3 watts but increases around 1.1"‘10'3 because of the reduced input
impedance at higher frequency as shown in figure 3.94. Figure 3.14, 3.44, 3.74 and 3.104 show

the averaged radiated power increases as a function of frequency and the average is 1.7 "'10—3

approximately. The average radiated power increases as frequency goes higher as shown in

figure 3.24, 3.54, 3.84 and 3.114.

Most of the power gains are above 5 dB over the frequency band as shown in figure 3.6,
3.36, 3.46, 3.66, 3.86, 3.96 3.106 and 3.116. There are several minor dips in figure 3.16 and
3.76. Two big dips happen at 300 MHz and 1200 MHz in figure 3.26. Figure 3.56 shows that

there are clips at around 400 and 950 MHz; the one at 950 MHz is especially deep.

All these antennas have good directivity along their length even though the maximum

radiation may not be along the same direction over all frequencies. The maximum radiation is

mostly in the xz—plane (E0) at low frequency and in the yz-plane (E 9) at higher frequency.

116

Chapter 4
COMPARISONS ON GMC PULSE RESPONSE

In this chapter, a normalized antenna impulse response is introduced to study the wideband
properties of the traveling wave antennas in the time domain. A single waveform, h N (t) ,

describes an antenna’s performance in both transmission and reception. It describes the
antenna’s responses to an impulse function in reception or to a step function in transmission.
A number of useful simplifications can be taken in the antenna equations by using this impulse

response.

There are several advantages to using the impulse response to describe the antennas. First of
all, the transmission coefficients between the feed line and the antennas are eliminated.
Secondly, the antenna impedance does not appear in these equations. Thirdly, the normalized

impulse response applies to both transmission and reception equations. Fourthly, the

expressions are made very simple by introducing the normalized impulse response h N (t).

Finally, by writing the equations in this manner, it is possible to tie the theory back to the

measurements that are usually made on transient antennas.

In this study, 11,, (I) cannot be obtained directly because the outputs of GA-NEC are in
frequency domain. Its impulse response form in the frequency domain,h~ ((0), can be
expressed as a function of input voltage and radiated filed at a certain distance. When It” (0))

is available at a certain number of frequency points, a band-limited version of hN (t) can be
obtained with the inverse Fourier transform.

117

The normalized impulse response is valid for both reception and transmission. Only the
response in transmission is studied in this research. The normalized expression for the

radiation field is described in [32] as

 

Erad (t) _ ___1__ a; 1 strc (t)
JZ—o — 27170 hNJ‘X (t) J5: dt (41)

Here Em, (t) is the radiated electric field, Z0 is the characteristic impedance of free space (377
Ohms), r is the distance away from the antenna and is chosen to be 100 meters in this study,
6 is the speed of light in free space, hmrx (t) is the normalized impulse response in
transmission with the units of meters per second, Z c is the characteristic impedance of the
feed cable, * is convolution operator, and Vm (t) is the source voltage. In this equation, the
antenna’s transmission characteristic is described completely by h N,” (t) and there are no

antenna impedance or transmission coefficients involved. The above equation refers by default
to the direction of the maximum radiation, but it is easily extended to multiple angles. By

taking the Fourier transform, the above equation can be expressed in the frequency domain as

72—. E (w)

J2: jme(w)

 

h ”X ((0) = 272m (4.2)

There are two unknown variables in this equation, Em, ((0) and Vm ((1)). Vm (60) is set to 1
volt for simplicity and E rad ((0) can be easily extracted from the NEC2 outputs. h N,TX (t) can

be obtained by taking the inverse Fourier Transform of 11”,“, ((0). The radiated energy of the

118

normalized impulse response can be calculated by integrating the square of h NIX (t) over the

time window. By calculating the energy at multiple circumferential angles the impulse energy

radiation pattern can be plotted.

The frequency band of interest is limited from 200 MHz to 2 GHz. An appropriate
weighting function is needed to reduce the Gibbs oscillation when performing the FFT on
swept frequency data. In this study, a GMC (Gaussian modulated cosine) waveform is used in
order to reduce the oscillation and preserve the bandwidth. The GMC spectrum used in this

research is given by [33]
W. (f) = r(e"’“"’r"” + WWW) (4.3)

Here ft is the center frequency, and T is width of the pulse. In this study, fc= 1 GHz,

2' =0.002. The waveform of the GMC pulse is shown in figure 4.1

The loaded dipole described by Clayborne D. Taylor in [33], an optimized dipole and the
rhombic antenna mentioned in [34] are chosen to compare with the spline antenna. Both the

energy radiation patterns and radiation waveforms at certain angles are studied.

4.1 Dipole
The length of the dipole is optimized in GA-NEC for the GMC pulse radiation (described in
equation 4.3). The searching space for the optimal length is in the range of 0.1 to 0.8 m. The

value to be optimized is the theta component of the electric field at 100 meters away from the

origin alongH = 90° . The weighting function is point-based. “Greater than the value of 0.02”

is chosen in the constraint file. A total of 37 equally spaced points are weighted over the
119

frequency band in the fitness function. The weighting values obey the GMC distribution
described in equation 4.3. The optimal length is 0.1504 m which is half wavelength long at 1

GHz.

Figure 4.2 shows the GMC pulse energy radiation pattern. It is a @-independent pattern and
the maximum radiation is in the 6 = 90° plane. The waveforms of the GMC pulse radiated
alongH = 30° , 0 = 60° and 0 = 90° are shown in figure 4.3-4.5, respectively. The pulse
consists of ringing due to the strong multiple reflections at the ends. The individual reflections
can not be recognized in the time domain because of the short length of the dipole. The first
arrival and the reflections overlap in a time period of around 0.02 us. The width of the pulse

response is considerably wider than the GMC pulse shown in Figure 4.1.

500

 

400 ~

300~

200 ~

100~

 

 

.100 ,.

Ian (0 (mus)

-200 r

.300»

 

 

 

 

0.3 0.35 0.4 0.45 0.5
Tlmc (us)

Figure 4.1 Time domain version of the frequency GMC window function

120

h“ (In/us)

 

Figure 4.2 Energy in the waveform radiated by the dipole

40 1 , .

 

 

 

 

 

 

4 1 1
%.3 0.35 0.4 0.45 0.5
Tlmo (us)

Figure 4.3 Waveform radiated in far-zone field at 30 degrees

121

th (t) (mlus)

-20 r

 

80

60

40~

20»

 

40*

430»

 

 

 

 

‘33.:

0.35 0.4
Time (us)

0.45

Figure 4.4 Waveform radiated in far—zone field at 60 degrees

h“ (t) (mlus)

 

80
60 ~
40 ~

20[

 

 

.20 .—

~40-

-60~

 

 

 

"3.

3 0.35 0.4

Time (us)

0.45

0.5

Figure 4.5 Waveform radiated in far~zone field at 90 degrees

122

 

4.2 Loaded dipole

The impedance loaded dipole described here is based on the ones that described in [33]. The
dipole can be formed to a traveling wave antenna by insert some impedance loads at certain
positions along the antenna. Due to the NEC2 limitation on the number of load cards, 24
recalculated impedance loadings are inserted along the wire instead of 48 as listed in [33]. It is
simulated in GA-NEC and optimized using the same fitness function as the one for dipole.
The searching domain for the loaded dipole length is from 0.1 to 0.8 m. The optimal length of

the loaded dipole is 0.453 m.

As with the dipole, waveforms are plotted at 30, 60 and 90 degree as shown in figures 4.7-
4.9. There are no reflections seen in these figures, which indicates that the current vanishes at
the ends due to the radiation and ohmic loss as the current wave propagates along the wires.

The pulse response is a very good reproduction of the GMC pulse shown in figure 4.1.

However, the amplitude of hN’TX (t) is much smaller than that of the dipole, due to the

resistance of the loading. The normalized energy radiation pattern is obtained over 360 degree

with interval of 2 degrees (figure 4.6). There is no radiation at 0 degrees and it increases
smoothly as Bgoes up. The antenna has maximum radiation along 9 = 90° and vanishes to

zero at 6 = 180° . The half energy beam width is slightly wider than that of the dipole.

123

 

 

Figure 4.6 Energy in the waveform radiated by loaded dipole

 

 

 

 

 

 

10 T I Y
5 r a
A 0
fl
3
E
X
p.
8 .5 L _[
.10 ~ .4
.1 t l 1
3.3 0.35 0.4 0.45 0.5

Time (us)

Figure 4.7 Waveform radiated in far-zone field at 30 degrees

124

 

20 r . 1

 

15~ -

10~ J

 

 

th (t) (mlus)

 

 

 

 

.2 1 1
0.3 0.35 0.4 0.45 0.5
Tlmo (us)

Figure 4.8 Waveform radiated in far-zone field at 60 degrees

 

zo , ,
15» 4
1o- 4
5» .

 

 

th (t) (mlus)
O
l

 

 

 

 

.5 . .
.10 ~ -
.15 ..

'23.; 035 0:4 0.35 0.5

Tlmo (us)

Figure 4.9 Waveform radiated in far-zone field at 90 degrees

125

4.3 The rhombic antenna
Rhombic antennas are traveling-wave long wire antennas. The wires usually have lengths much
greater that one half wavelength long. The rhombic antenna that is simulated here is the one

that is described in [35] shown in figure 4.10. The length of each side is set to 1.8 m which

corresponding to 6 times the wavelength at 1 GHz and the flare angle a = 16° .The radius of
the wire is 0.001 m and the rhombic terminating impedance is set to 600 Ohms as

recommended. The antenna is placed in the y-z plane along +2 axis.

The GMC pulse energy radiation is plotted in the y-z plane as a function of azimuth in
figure 4.11. It shows that the maximum radiation direction is along z-axis. There are two 10 dB
down side lobes located at around 30 degree away from z-axis and one 14 dB down lobe along
—2 axis. The GMC pulse radiation at different azimuth angles is plotted in figure 4.12-4.18.
There are four events observed in figure 4.12. The first event originates when a voltage
impulse is induced at the feeding end. compared to figure 4.1, this first event is close to the
GMC pulse, but not as close as was the loaded dipole. The second event is significantly
smaller, and is due to the current reflecting at the suddenly changing wire direction. The third
event is produced by current reflecting from the wire end. The fourth is observed because of
the second reflection at the end. The significant different between the amplitudes of first event
and the others in figure 4.12 is due to radiation and ohmic loss. The radiation from the two
sides of the antenna is out of time synchronization seen from figures 4.13-4.18. The total
radiation is the superposition of the distributions of the two sides. The event that is caused by

multiple reflections from the ends becomes more apparent as the azimuth angle increases. In

126

 

figure 4.18, the dominated event is caused by the reflection from the ends. Figure 4.12 shows

that at 0 degrees, h N.” (t) has much larger amplitude than for a dipole or a loaded dipole.

Figure 4.10 The geometry of the rhombic antenna

127

 

IF

 

Figure 4.11 Energy in the waveform radiated by the rhombic antenna in y-z plane

 

250 . . ,

200 - -

100 ~ _

 

 

h“ (t) (In/us)
O
4r

 

 

 

 

«50 ~ .
-1oo » J
.150 ,. ..
.200 .- ..
4521.3 0.35 0:4 0.35 0.5

Tlmo (us)

Figure 4.12 Waveform radiated in far-zone field at 0 degrees

128

 

 

so . m

40~

20*

4!

h‘.x (mlus)
o

.20 _

.40»

 

 

 

 

7%.3 0.35 0:4 0.45 0.5
Time (us)

Figure 4.13 Waveform radiated in far-zone field at 30 degrees

 

20 . , -

15»

10~

 

th (t) (mlus)

.10.

 

.15..

 

 

 

.2 1 1 m
3.3 0.35 0.4 0.45 0.5
Tlme (00)

Figure 4.14 Waveform radiated in far-zone field at 60 degrees

129

 

 

 

 

 

 

 

 

1o . f r
8 ».

a i J

E .‘ " t‘

5- ! f N W1].-. ~-

X n

:0-

I.“
"13.3 0.35 01.4 0.35 0.5
Time (us)

 

Figure 4.15 Waveform radiated in far—zone field at 90 degrees

 

 

hTX (t) (m/us)

 

 

 

 

 

3.3 0.35 0.4 0.45 0.5
Tlmo (us)

Figure 4.16 Waveform radiated in far-zone field at 120 degrees

130

 

15

"IX (t) (mlus)

 

 

 

 

 

'13.:

0.35

of.
Time (us)

0.45

Figure 4.17 Waveform radiated in far-zone field at 150 degrees

 

60

40—

20*

th (t) (mlus)

.20 -

40*

 

 

 

 

 

 

“1.3

Figure 4.18 Waveform radiated in far-zone field at 180 degrees

0.35

oia
Time (us)

131

0.45

0.5

 

4.4 The spline antenna

Figure 4.17 shows the geometry of the spline antenna in the y-z plane. The wires have a
constant radius of 0.001 m. The antenna has the same length as the rhombic antenna, 3.46 m.
The optimized quantity is the theta component of the electric field 100 meters away along the
z-axis. GA-NEC uses the same fitness method as was applied to the loaded dipole including
the same fitness file. The optimal impedance load to the ends is 600 ohms.

The GMC pulse energy radiation pattern (figure 4.18 in the y-z plane) reveals that this
antenna is directive and radiates most strongly along the z-axis. Two 11 dB down side lobes
occur at around 45 degrees away from the maximum radiation direction. There is also a 15 dB
down back lobe as with the rhombic antenna. Unlike the rhombic antenna, the second
reflection from the wire ends is negligible. Furthermore, the wire shapes are smooth so there is
no reflection as with rhombic antenna. The GMC pulse radiation at any angle is the
superposition of the contributions from the two wires. Along z-axis, the distributions are in
time synchronization. There are only two events that can be seen from figure 4.21. One is
from the feed and closely reproduces the GMC pulse and the other is the first reflection from
the ends. As shown in figures 4.20-4.25, the distributions from the two wires are out of time

synchronization. At 180 degree, the reflection is the main event of the GMC pulse radiation.

132

 

 

 

Figure 4.19 The geometry of the spline antenna geometry

133

 

   

 
  
 

‘--- --
.3

......

------_-‘-

     

I
"10
I

..........

210° 5 150°
130°

Figure 4.20 Energy in the waveform radiated by the spline antenna in y-z plane

h“ (t) (mlus)

 

300 t I .

200 * a

100* ~

 

 

.100 r- '1

-200 ~ -

 

 

 

 

.30 1 r
(0.3 0.35 0.4 0.45 0.5
Tlmo (us)

Figure 4.21 Waveform radiated in far-zone field at 0 degrees

134

 

30

20*

10*

111.x (mlus)
O

.10.

.20 ..

"3%.

 

 

 

 

 

 

0.35 0.4 0.45 0.5
Tlmo (us)

 

Figure 4.22 Waveform radiated in far-zone field at 30 degrees T...

20

hTx (t) (mitts)

 

 

 

 

 

 

 

 

0.35 0.4 0.45 0.5
Tlmo (us)

Figure 4.23 Waveform radiated in far-zone field at 60 degrees

135

30

 

20*

10*

 

 

h“ (t) (mint)
6
t
r

 

 

 

 

.10 ..
.20 .. 4
.3 L 1 1

$3 0.35 0.4 0.45 0.5

Tlmo (us)

Figure 4.24 Waveform radiated in far-zone field at 90 degrees

15 , r

 

 

th (t) (mlus)

 

 

 

 

.1 l 1 1
0.3 0.35 0.4 0.45 0.5
Tlmo (us)

Figure 4.25 Waveform radiated in far-zone field at 120 degrees

136

 

 

 

20

th (t) (mlus)

 

 

 

 

Figure 4.26 Waveform radiated in far-zone field at 150 degrees

0.35

0:4
Tlmo (us)

0.45

0.5

 

60

40*

20*

 

 

h" (t) (mlus)
O

40*

 

 

 

 

’53.

Figure 4.27 Waveform radiated in far-zone field at 180 degrees

0.35

0:4
Tlmo (us)

137

0.45

0.5

 

 

4.5 Comparison and Conclusion

Figure 4.28 shows the normalized GMC pulse energy radiation pattern of the dipole, the
loaded dipole, the rhombic and the spline antennas. Except for the dipole antenna, the other
three are traveling-wave antennas and have wideband properties. The loaded dipole has a
azimuthally independent radiation pattern. The maximum radiated energy is nearly 20 dB
down to that of the spline antenna and 12 dB down to the dipole. It trades the directivity for
wideband radiation. This restricts the loaded dipole for short range signal transmission and
reception. The rhombic and the spline antennas have directive radiation abilities. The spline
antenna has higher directivity that the rhombic one. There is a 1.6 dB difference along the z-
axis. The half energy beam width is 10 degrees for the spline antenna and 12 for the rhombic
one. As shown in figures 4.12 and 4.21, the difference between the feed radiation and the
reflected radiation is significant. For the spline antenna, there is no reflection from rapidly
changing curves as with the rhombic antenna. It is important to have higher directivity and
preserve the bandwidth of the impulse radiation.

Figure 4.29 shows the radiation efficiency of the four antennas. It indicates that the
radiation efficiency is 100 percent for the dipole and the spline antennas. That means there is
no ohmic loss for those two antennas. The radiation efficiency of the rhombic antenna is in
the range from 16.8 percent at 200 MHz to 84.9 percent at 2 GHz due to the ohmic loss. The
loaded dipole has the even lower radiation efficiency which is in the range from 4.2 to 25.9
percent over the frequency band of interest. Thus, the spline antenna is the choice for the

GMC pulse radiation among the four.

138

 

 

Efficiency (56)

 

 

 

 

 

 

110 T T 1 T T

100.. __ . -,__

80*
70*

 

50*

 

-— dipole ~
— loaded dipole
~ — - mombic 4
—— spline

 

 

20

10

 

 

 

 

800 400 600 800 1000 1200
Frequency (MHz)

1400 1600 1800 2000

Figure 4.29 Radiation efficiency of the four antennas

139

 

 

Chapter 5
CONCLUSIONS

This thesis has introduced a set of spline traveling-wave antennas that are suitable for the band
from 200 MHz to 2 GHz. The EM properties of these antennas such as the input impedance,
the radiation efficiency, the VSWR, the radiated power and the power gain were presented and
the results are discussed. The responses to a GMC (Gaussian modulated cosine) pulse function

were studied and comparisons were made with the loaded dipole and the rhombic antenna.

With the use of GA (Genetic Algorithm), the optimal results were obtained based on the
fitness function specified. There were three aspects specified in the fitness function as a goal
for GA operations, Gain, efficiency and VSWR. Several types of geometries were optimized,
including varying and constant segment radii with symmetric geometry and with asymmetric
geometry. These are optimized with varying input irnpedances. Several runs were made for
each case to avoid the GA-NEC optimization process from becoming trapped in a local

maximum domain.

Simulation results show that all the optimized spline antennas have 100 percent radiation
efficiency except the asymmetnc one with 150 ohms feed line, which is from 81.5 to 99.7
percent over the band of interest. The spline antennas show much better efficiency compared
with the rhombic antenna with efficiency from 16.8 to 84.9 percent and loaded dipole from 4.2
to 25.9 percent. Most of the VSWRs are in the range from 1 to 3 over the frequency band,
except some values exceed 3 at certain low frequencies as shown in figures 3.3 and 3.23. Some
of the optimal antennas also yield high power gains (over around 4 dB) over the frequency
band as shown in figure 3.6 and figure 3.106. Radiation patterns indicate that the spline

antennas are directive along the z-axis.

As a wideband traveling-wave antenna, the spline antenna has better radiation ability for a
certain GMC pulse over the loaded dipole and the rhombic antenna as discussed in chapter 4.
It can radiate 1.6 dB higher energy than the rhombic antenna and more that 20 dB higher than

the loaded dipole along the maximum radiation direction. Also, the spline antenna that was
140

 

discussed in chapter 4 is the optimal one for the GMC pulse radiation described by equation
4.3. The antenna should be redesigned for different GMC pulse radiation by changing the

weighting value in the fitness file as described in chapter 4.

As an optimizer, GA-NEC searches the best fitted individual in a specified domain. A wider
domain and a finer searching step will give GA-NEC more diversity. Thus better results may

be found with more powerful computer than used in these studies.

141

 

APPENDIX

142

 

 

APPENDIX A

VISUAL BASIC CODE TO GENERATE THE SPLINE ANTENNA WITH
SYMMETRIC GEOMETRY

This code is written in Microsoft Visual Basic. It reads the input parameters such as the
position of the points and the radius of the segments to generate the spline shape antenna and
create an output file as a part of the GA—NEC input file to NEC2.

Program: Spline_sy.vbp

Sub Main()

Dim strCommandLineArgs As String
Dim intLocSpace As Integer

Dim strInputFile As String

Dim strOutputFile As String

.. l_' . m“.'1
I-
ll

 

'get the command line arguments :4".

strCommandLineArgs = LTrim(RTrim(Command))

'there are two arguments separated by one or more spaces
‘locate the space
intLocSpace = InStr(1, strCommandLineArgs, " ")

'take apart the string
strInputFile = Left(strCommandLineArgs, intLocSpace)
strOutputFile = Right(strCommandLineArgs, Len(strCommandLineArgs) - intLocSpace)

'call the function used to createt the desired geometry - in this case a splined fit transmission
line

Call SplineTL(strInputFile, strOutputFile)

End Sub

Sub SplineTL(strInputFile As String, strOutputFile As String)
'this subroutine makes a transmission line that is flared

'using a spline fit to several points along the z axis

'segment endpoints

Dim x1 As Single

Dim y1 As Single

Dim 21 As Single

Dim x2 As Single

Dim y2 As Single

Dim 22 As Single

'starting point along z-axis
1 43

Dim zstart As Single

'increment along z-axis
Dim delta As Single

'number of wires
Dim lngWireNumber As Long

'temporary counter
Dim i As Integer

'radius of wires
Dim Radius As Single

'delimiter
Dim strDelimiter As String
strDelimiter = ","

'These variables read from argument file
Dim Length As Single

Dim Separation As Single

Dim 20 As Single

Dim NumberofSegments As Integer
Dim NumberOfTerms As Integer

'open input file and read variables provided by GA-NEC
Open strInputFile For Input As #1

Input #1, Length

Debug.Print "Length = " 8: Length

Input #1, Separation

Debug.Print "Separation = " 8: Separation

Input #1, 20

Debug.Print "Impedance 2 " & 20

Input #1, NumberofSegments

Debug.Print "NumberofSegments = " & NumberofSegments
Input #1, NumberOfTerms

Debug.Print "NumberOfTerms = " 8r NumberOfTerms

'these variables store the y and z coordinates for each point along the transmission line
ReDim z(NumberOfTerms) As Single

ReDim Y(NumberOfTerms) As Single

ReDim y2a(NumberOfTerms) As Single

'This variable defined below
Dim sngelta As Single
sngelta = Length / (NumberOfTerms - 1)
144

 

 

 

 

'counter for segments
Dim intNumSegments As Integer

'read the terms, compute z coordinate for each point along the way
Fori = 1 To NurnberOfTerrns
z(i) = (i - 1) * sngelta
Input #1, Y(i)
Debug.Print z(i), Y(i)
Next i
Close #1

'open output file and start writing GW commands
Open strOutputFile For Output As #1

'initialize spline function
Call Spline(z(), YO, NumberOfTerms, 1E+30, 1E+30, y2a())
delta = Length / NumberofSegments

'make excitation segment
lngWireNumber = 1
intNumSegments = 1

x1 = 0
y1 = Separation / 2#
21 = -1
x2 = 0
y2 = -Separation / 2#
22 = -1

Radius 2 Separation / (2# * Cosh(20 / 120#))

Print #1, "GW " & lngWireNumber & strDelimiter & intNumSegments & strDelimiter & x1
& strDelimiter & y1 & strDelimiter & 21 & strDelimiter & x2 & strDelimiter & y2 &
strDelimiter & 22 8c strDelimiter & Radius

'Make Lead in wires
lngWireNumber = 2
intNumSegments = 50

x1 = 0
y1 = Separation / 2#
21 = -1
x2 = 0
y2 = Separation / 2#
22 = 0

Radius = Separation / (2# * Cosh(20 / 120#))
Print #1, "GW " & lngWireNumber & strDelimiter & intNumSegments & strDelimiter & x1
& strDelimiter & yl 8r. strDelimiter & 21 & strDelimiter & x2 & strDelimiter & y2 &
strDelimiter & 22 8c strDelimiter & Radius

145

 

 

lngWireNumber = 3

intNumSegments = 50

x1 = 0

y1 2 -Separation / 2#

21 = -1

x2 = 0

y2 = -Separation / 2#

22 = 0

Radius 2 Separation / (2# * Cosh(20 / 120#))

Print #1, "GW " & lngWireNumber & strDelimiter & intNumSegments & strDelimiter & x1
& strDelimiter & y1 & strDelimiter & 21 & strDelimiter & x2 & strDelimiter & y2 &
strDelimiter & 22 & strDelimiter & Radius

'make the positive half of the twin wire
lngWireNumber = 99
intNumSegments '-'—' 1

zstart = 0

x1 = 0

x2 = 0

21 = zstart

Call Splint(z(), YO, y2a0, NumberOfTerms, 21, y1)

y1 = Separation / 2#
Fori = 1 To NumberofSegments
lngWireNumber = lngWireNumber + 1
22 = 21 + delta
Call Splint(2(), YO, y2a0, NumberOfTerms, 21, y2)
Ify2 < 0Theny2= 0
y2 = y2 + Separation / 2#
Radius = Abs(2# * y2 / (2# * Cosh(20 / 120#)))

Print #1, "GW " & lngWireNumber & strDelimiter & intNumSegments & strDelimiter & x1
& strDelimiter & y1 & strDelimiter & 21 & strDelimiter 8: x2 & strDelimiter & y2 &
strDelimiter & 22 & strDelimiter & Radius

21 = 22

y1 = y2
Next i

Call Splint(2(), YO, yZaO, NumberOfTerms, 21, y1)

y1 = Separation / 2#

lngWireNumber = lngWireNumber + 1

Print #1, "GW " 8c lngWireNumber 8r strDelimiter 8c intNumSegments & strDelimiter & x1
& strDelimiter & y1 & strDelimiter & 21 & strDelimiter & x2 & strDelimiter & y2 &
strDelimiter & 22 & strDelimiter & Radius

146

 

 

'Make the negative half of the twin wire
zstart = 0
21 = zstart

Call Splint(z(), YO, y2a0, NutnberOfTerms, 21, y1)

y1 2 -Separation / 2#

Fori = 1 To NumberofSegments
lngWireNumber = lngWireNumber + 1
22 = 21 + delta

Call Splint(2(), Y0, y2a(), NumberOfTerms, 21 , y2)

Ify2< 0Theny2 = 0

y2 = -y2 ~ Separation / 2#

Radius = Abs(2# * y2 / (2# * Cosh(20 / 120#)))

Print #1, "GW " & lngWireNumber & strDelimiter & intNumSegments & strDelimiter & x1
& strDelimiter & y1 & strDelimiter & 21 & strDelimiter & x2 & strDelimiter & y2 &
strDelimiter & 22 & strDelimiter 8t Radius

21 = 22

y1 = y2
Nexti

Call Splint(z(), YO, y2a(), NumberOfTerms, 21, y1)

y1 = ~Separation / 2#

lngWireNumber = lngWireNumber + 1

Print #1, "GW " & lngWireNumber 8r. strDelimiter & intNumSegments & strDelimiter & x1
& strDelimiter & y1 & strDelimiter & 21 & strDelimiter & x2 & strDelimiter & y2 &
strDelimiter & 22 & strDelimiter & Radius

Termination segment

lngWireNumber = 1000

intNumSegments = 1

x1 = 0

y1 = Separation / 2#

x2 = 0

y2 = -Separation / 2#

Radius = Abs(Separation / (2# * Cosh(20 / 120#)))

Print #1, "GW " & lngWireNumber & strDelimiter & intNumSegments & strDelimiter & x1
& strDelimiter & yl & strDelimiter & 21 & strDelimiter & x2 & strDelimiter & y2 &
strDelimiter & 22 & strDelimiter & Radius

Close #1

End Sub

Sub Spline(X() As Single, YO As Single, n As Integer, YP1 As Single, YPN As Single, y2() As
Single)
147

 

 

'this routine is translated from the one in Numerical Recipes in F ortran

'Given arrays X and Y of length N containing a tabulated function, i.e. Y(i) = f(X(i))
'with X1 < X2 < XN, and given values YP1 and YPN for the first derivative of the
'interpolating function at points 1 and N, repectively, this routine returns and

'array Y2 of length N which contains the second derivatives of the interpolating
'function at the tabulated points X]. If YP1 and/ or YP2 are equal to 1x10“30 or
'larger, the routine is signalled to set the conesponding boundary condition for

'a natural spline, with zero second derivative on that boundary.

Dim i As Integer

Dim sig As Single

Dim p As Single

Dim qn As Single

Dim un As Single

Dim K As Integer

ReDim U(n)

If (YP1 > 9.9E+29) Then 'The lower boundary condition is set either to be "natural"
3’20) 2 0

U(l) = 0

Else 'or else to have a specified first derivative
y2(1) = -0.5

EUfiif: (31‘?t / (31(2) - XO)» * (Y(Z) - Y(1)) / (XQ) - X0) - YP1)

'this is the decomposition loop of the tridiagonal algorithm.
'Y2 and U are used for temporary storage of the decomposed factors.
Fori = 2 To n - 1
sig = 0(6) -X(1- 1)) / <X<1+ 1) - X<1- 1))
p= sig*y2(i- 1) +2
y2<1) = (s1g-1#) / p
U(i) = (611 * «Ya + 1) -Y<i)) / <X<1 + 1) — X<1)) - (Y(i) -Y<i - 1)) / (Xa) -X<i- 1))) / 046 + 1)
-X(i-1))-s1g*U<i-1))/p
Next i
If YPN > 9.9E+29 Then 'The upper boundary condition is set either to be "natural"
qn ‘2 0
un = 0
Else 'or else to have a specified first derivative
qn = 0.5
un = (311 / <X<n) -X<n- 1))) * (YPN - (Y(n) -Y<n- 1)) / (X01) -X<n - 1)))
End If
y2<n> = (un - qn * U(n - 1)) / (qn * y2(n - 1) + 1)
'this is the backsubstitution loop of the tridiagonal algorithm
ForK=n- 1 T01 Step -1
y2(K) = y2(K) * Y2(K + 1) + U(K)
Next K
148

 

 

End Sub

Sub Splint(XA() As Single, YAO As Single, y2a() As Single, n As Integer, X As Single, Y As
Single)

'this routine is translated from Numerical recipes in fortran

'given the arrays an and ya() of length N, which tabulate a function
'(with the XAi's in order). adn given the array YZAO, which is the
'output from the SPLINE routine, and given a value of X, this routine
'returns a cubic-spline interpolated value Y.

Dim klo As Integer

Dim khi As Integer

Dim K As Integer

Dim a As Single

Dim B As Single

Dim H As Single

klo = 1

khi = n

'we will find the right place in the table by means bisection. This is
'optimal if sequential calls to this routine are at random values of
'x. If sequential calls are in order, and closely spaced, on would
'do better to store previous values of K10 and khi and test if they
'remain appropriate on the next call
Splintl:
If('khi-klo>1)Then

K = (khi + klo) / 2#

If (XA(K) > X) Then

khi = K

Else
klo = K

End If

GoTo Splintl
End If
'K10 and Khi now bracket the input of x
H = XAO‘ihi) - X4010)
'The XA's must be distinct
If H = 0 Then

MsgBox "Warning! Bad value of XA in Splint!"
End If

'Debug.Print "high, low = ", XA(khi), XA(klo), YA(khi), YA (klo)
'cubic spline is now evaluated

a = 01110111) -X) / H

B = (X - XA(klo)) / H

149

 

 

Y=a*YA(klo) +B*YA(khi) + ((a"3#-a)*y2a(klo) + (BA3#-B)*y2a(khi))*(H"2#)
/6#
EndSub

Function Cosh(X) As Double
'Purpose:
'Compute hyberbolic cosine of X

Cosh = (Exp(X) + Exp(—X)) / 2#
End Function

 

 

150

 

BIBLIOGRAPHY

151

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[34] Warren L. Stutzman, “Antenna theory and design” John Wiley & Sons, inc

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