Application of Method of Moments to thin wire antennas

Application of Method of Moments to thin wire

antennas

Ivan Tim Oloya

Thesis submitted in partial fulfilment of the requirements for a

Bachelor’s degree

In

Electrical and Electronic engineering

University of Mauritius

Faculty of Engineering

April 2014

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Table of Contents

Application of Method of Moments to thin wire antennas ................................................... 1

Table of Contents ................................................................................................................... i

List of Figures...................................................................................................................... iii

Acknowledgements ............................................................................................................. iv

Project Declaration form ...................................................................................................... v

Abstract .............................................................................................................................. vi

List of symbols and abbreviations ...................................................................................... vii

1 Introduction ..................................................................................................................... 1

1.1 Problem definition ................................................................................................... 1

1.2 Aim and objectives .................................................................................................. 2

1.3 Assumptions ............................................................................................................ 2

1.4 Thesis Organisation ................................................................................................. 2

2 Literature review ............................................................................................................. 4

2.1 Computation in electromagnetism ........................................................................... 4

2.2 Types of Antennas.................................................................................................... 4

2.2.1 The Dipole Antenna ............................................................................................ 4

2.2.2 The Loop Antenna .............................................................................................. 5

2.2.3 The Gray Hoverman antenna .............................................................................. 6

2.3 The Method of Moments ......................................................................................... 6

2.4 The Integral Equations ............................................................................................. 7

3 Method of Moments ........................................................................................................ 8

3.1 Terminology ............................................................................................................. 8

3.2 Maxwell’s equations ................................................................................................ 8

3.3 Solving the matrix equation ................................................................................... 10

3.4 Basis functions ....................................................................................................... 11

3.4.1 Subdomain functions ........................................................................................ 12

3.4.2 Entire domain basis functions........................................................................... 16

3.5 Source modelling ................................................................................................... 17

3.5.1 Delta-gap .......................................................................................................... 17

3.5.2 Magnetic frill generator .................................................................................... 18

4 Application to thin wire antennas .................................................................................. 20

4.1 Charge Distribution ............................................................................................... 20

4.1.1 The Straight Dipole .......................................................................................... 20

4.1.2 The Gray Hoverman antenna ............................................................................ 24

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4.2 Current Distribution ............................................................................................... 27

4.2.1 The Straight Dipole .......................................................................................... 27

4.2.2 Simplification of Pocklington’s Equation for Arbitrary shaped antennas ........ 37

5 Results and Discussions ................................................................................................ 46

5.1 Charge distribution on a straight wire.................................................................... 46

5.2 Charge distribution on a bent wire antenna ........................................................... 48

5.3 Charge distribution on the Gray Hoverman antenna ............................................. 49

5.4 Pocklington’s equation solution for current distribution on a dipole .................... 52

5.5 Hallen’s integral solution for current distribution on a dipole ............................... 54

5.6 Current distribution on the gray hoverman antenna. ............................................. 55

5.7 Current Distribution on a circular Loop Antenna .................................................. 57

6 Conclusion ..................................................................................................................... 60

6.1 Achievements and results ...................................................................................... 60

6.2 Limitations and practical considerations ............................................................... 60

6.2.1 Effects of antenna radius .................................................................................. 60

6.2.2 Choice of basis function ................................................................................... 60

6.2.3 Speed of solutions ............................................................................................. 61

6.3 Further work .......................................................................................................... 61

REFERENCES ................................................................................................................... 62

APPENDICES .................................................................................................................... 63

Appendix 1 ..................................................................................................................... 63

Progress Log ............................................................................................................... 63

Appendix 2 ..................................................................................................................... 64

MATLAB Programs For Charge Distribution ............................................................ 64

Appendix 3 ..................................................................................................................... 73

MATLAB Programs for Current Distribution ............................................................ 73

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List of Figures

Figure 3.1 Piecewise constant subdomain functions .................................................. 12

Figure 3.2 Piecewise linear subdomain functions ...................................................... 14

Figure 3.3 Piecewise sinusoids subdomain functions ................................................ 15

Figure3.4 Truncated cosines subdomain functions .................................................... 16

Figure 3.5 Cylindrical dipole, its segmentation and gap modelling .......................... 17

Figure 4.1 Electric potential V at a point P at a distance r from a charge Q .............. 20

Figure 4.2 Straight wire of constant potential and its segmentation .......................... 21

Figure 4.3 Geometry of a bent wire ........................................................................... 24

Figure 4.4 A uniform plane wave obliquely incident on a conducting wire .............. 28

Figure 4.5 Dipole segmentation and its equivalent current ........................................ 30

Figure 4.6 Wire antenna with N segments ................................................................. 35

Figure 4.7 Relations among the vectors of an arbitrary shaped wire ......................... 38

Figure 4.8 Unit tangential vectors for the thin wire ................................................... 40

Figure 5.1 Charge distribution on a straight dipole; Length=1m, nm=5 ................... 47

Figure 5.2 Charge distribution on a straight dipole; Length=1m, nm=20 ................. 47

Figure 5.3 Charge distribution on a bent wire ............................................................ 48

Figure 5.4 Original gray hoverman antenna structure ............................................... 49

Figure 5.5 Current distribution on the original Gray Hoverman antenna structure ... 50

Figure 5.6 Charge distribution on a thin straight dipole; L=1m, nm=80 ................... 51

Figure 5.7 Charge distribution on a Gray Hoverman antenna; L=1m, nm=80 .......... 51

Figure 5.8 Current distribution on a thin dipole; L= 0.47𝜆, nm=5 ............................ 53

Figure 5.9 Current on a thin dipole with more segments. L= 0.47𝜆, nm=13 ............. 54

Figure 5.10 Current distribution on a linear half wavelength dipole antenna ............ 55

Figure 5.11 Current distribution on a half wavelength Gray Hoverman antenna ...... 57

Figure 5.12 Circular loop antenna and its associated vectors .................................... 57

Figure 5.13 circular loop antenna and its associated vectors ..................................... 59

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Acknowledgements

I would like to thank my supervisor, Dr. Yasdeo Bissessur for his great expertise and en-

couragement throughout the working on my project. Without his guidance, motivation and

support I would not have been able to complete this project. Am also very grateful that he

was always available and made time whenever I had an issue with the project

I would also like to thank my fellow classmates for their support and encouragement, and

single out Mr William Joubert who offered me great advice especially in times when I was

stuck and couldn’t find a way through.

Finally, I would like to extend my gratitude to my family who have been of great assis-

tance to me during the entire length of my project, keeping me focused and making me

believe, I have the will to complete this project.

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Project Declaration form

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Abstract

Before the invention of computers; analysis, testing and design of complex structures

were a major problem. However, over the last two decades, high speed and reliable comput-

ers have since been built which are capable of solving some of the most sophisticated

problems hence greatly simplifying such tasks.

In order to analyse antenna structures, one of the most reliable and common methods de-

veloped has been the Moments method. This involves dividing the wire into a number of

segments and obtaining matrices than can be used to obtain various parameters such as the

charge distributions, current distributions, impedances to mention but a few

In this thesis, the thin wire antenna will be analysed by applying the method of moments

and using techniques such as the famous Hallen’s integral and Pocklington’s integral equa-

tions. The pulse basis function will also be introduced; all programs and codes have been

written using MATLAB software.

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List of symbols and abbreviations

HF High Frequency

VHF Very High Frequency

UHF Ultra High Frequency

RF Radio Frequency

MM Moments method

IE Integral Equation

k Wave number

휀 Electrical permittivity

𝜌 Charge density

nm Number of segments the antenna is divided into

𝜂 Intrinsic impedance

𝜇 Electrical permeability

𝜆 Wavelength of antenna

f Frequency of operation

c Speed of light

FMM Fast Multiple Method

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1 Introduction

1.1 Problem definition

The most commonly used antenna for Television reception is the Yagi-Uda array. The original

design and working principles of this antenna were first published in Japanese by Professor S.

Uda of Tohoku Imperial University in Japan. However, in a later but more circulated and read

article, one of Professor Uda’s colleagues H. Yagi described the operation of the same antenna

in English. This antenna is very common due to versatility to operate the HF, VHF and UHF

frequency spectrums. The Yagi-Uda array is made up of a number of linear dipole elements.

One is energized directly by a feed transmission line while the others act as parasitic radiators

whose currents are induced by mutual coupling, the most common feed element being the

folded dipole. This radiator is accomplished by having the parasitic elements in the forward

beam acting as directors while those in the rear act as reflectors [1]. However, one major dis-

advantage of the Yagi-Uda array is that it is only suitable for short range receptions.

For long range receptions however, the performance of the Yagi-Uda array has not been as

good. As a result, recent studies have been focusing on a better solution for long distance re-

ceptions. One better substitute is the Gray Hoverman antenna that was designed by Mr. Doyt

Hoverman in 1959 which he further improved in 1964.

Recent analysis and studies have shown that the Gray Hoverman exhibits better capture of

scatter from deep fringe signals and is also easier to aim for such long distances. Moreover, the

fact that it provides an almost constant and larger positive gain over a much wider frequency

range compared to the Yagi-Uda array has increased its popularity especially among do-it-

yourselfers since it’s easy to build.

One of the most important parameters in antenna analysis and design is the impedance. The

impedance of an antenna can be defined as the ratio of the voltage to the current at the input to

the antenna. The real part of the impedance represents the power radiated or absorbed by the

antenna whereas the imaginary part represents the power stored in the near field of the anten-

na. An antenna is therefore said to be resonant if it has no imaginary part [2].

Various methods can be used to determine an antenna’s impedance. Some of these methods

include; the boundary-value method, the poynting vector method, the transmission line meth-

od and the integral equation method.

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1.2 Aim and objectives

The aim of this thesis is to apply the Method of Moments technique as a solution to solve an-

tenna parameters.

Objectives

To find a solution and compare charge distribution on a dipole antenna and the

Gray Hoverman antenna.

To obtain the current distribution on a dipole antenna as well as other antennas of

arbitrary shapes.

To calculate the self-impedance of a dipole antenna as well as other antennas of ar-

bitrary shapes.

1.3 Assumptions

For this thesis, two main assumptions have been made. Firstly, that the antenna is thin, that is

the radius of the antenna is very small compared to the wavelength at which it is operating.

(𝑎 ≪ 𝜆).

The second assumption is that the wire from which the antenna is made is perfectly conduct-

ing. Copper is a very good conductor and hence provides very good accuracy.

1.4 Thesis Organisation

Chapter 2 of this thesis gives a bigger picture of the project. First, a brief summary of how

growth in technology has aided in solving electromagnetic field problems. Next, various an-

tenna types are discussed. Later, the Method of moments is introduced and finally, various

integral equations used in solving electromagnetic field problems are introduced.

In chapter 3, the Method of Moments is discussed in details. The solution of Green’s functions

for free space is derived from Maxwell’s equation. The basis functions used to segment a thin

wire antenna are explained. Finally, source modelling techniques are discussed.

The concepts of chapter 3 are then extended to chapter 4, and applied to straight wire an-

tennas. Pocklington’s equation of chapter 2 is simplified to cater for thin arbitrary wire

configurations.

In chapter 5, a solution for three antenna parameters is presented. These three parameters

are; the charge distribution on a thin wire antenna, the current distribution on a thin wire an-

tenna and finally, the self-impedance on a thin wire antenna is calculated from the current

distribution graphs.

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In chapter 6, conclusions which includes; achievements, limitations, and further work has

been presented.

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2 Literature review

2.1 Computation in electromagnetism

Before the invention of digital computers, the analysis, modelling and design of structures

and machines was to a large extent experimental. When computer languages such as C, C++,

FORTRAN where developed, it enabled people to start using them in solving problems in

electromagnetism that where not easy to solve experimentally. This work starts from the phys-

ical structure of the system being designed that is, the geometry and electrical properties of the

structure. Most electromagnetic field problems usually do not have an analytical solution and

hence often a numerical approach is crucial to solving them. Down the years a number of

techniques have been developed which has made these studies much easier. This has been

helped by development of modern computers and properly developed techniques.

The disadvantage of solving these problems experimentally was due to the fact that it was

becoming very costly, further still it required a large amount of manual labour. The main rea-

son behind the computation of algorithms is the ability to visualize and simulate structures and

systems before physically building them. By doing this, an engineer can optimize the system,

determine faults and errors and derive suitable solutions with minimal expenses [3].

2.2 Types of Antennas

In this thesis, three different antenna types are discussed. These are; the dipole antenna, the

loop antenna and the Gray Hoverman antenna.

2.2.1 The Dipole Antenna

The Dipole antenna is the most widely used and simplest antenna. It consists of a conductive

wire rod which is half the length of the maximum wavelength that the antenna has to generate.

This wire rod is split in the middle, and the two sections on either side are separated by an in-

sulator. The two sections are connected to a coaxial cable at the end closest to the middle of

the antenna. In dipole antennas, radio frequency voltages are applied at the centre between the

two conductors.

The Radio Frequency (RF) current in dipole antennas is maximum at its centre and minimum

at its ends. Dipole antennas are often used in rabbit-ear television antennas but can also be

used as driven elements in other types of antennas. Dipole antennas are of three main times;

the ideal half-wavelength dipole, folded dipole, and the Hertzian dipole.

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2.2.2 The Loop Antenna

The loop antenna is a very versatile yet simple and inexpensive type of antenna. Loop anten-

nas take many different forms such as rectangular, square, triangular ellipse, circular, as well

as other different configurations. Due to its simplicity in analysis, design and setting up, the

loop antenna is one of the most popular antennas and has received the widest attention. A

small loop antenna (circular or square) is equivalent to an infinitesimal magnetic dipole with

its axis perpendicular to the plane of the loop. This in other words means that, fields radiated

by an electrically small circular or square loop are of the same mathematical form as those

radiated by an infinitesimal magnetic dipole.

Loop antennas are divided into two broad categories, electrically small loop antennas and

electrically large loop antennas. Electrically small antennas refer to those whose overall length

is usually less than about one-tenth of a wavelength. Electrically large loop antennas on the

other hand are those whose circumference is about a free-space wavelength. There are three

main frequency bands in which loop antenna applications are applied and these include the HF

(3-30 MHz), VHF (30-300 MHz), and UHF (300-3000 MHz). However, when loop antennas

are used as field probes, they find applications in the microwave frequency range as well. This

is the main reason behind the loop antenna’s popularity.

Electrically small loop antennas are however rarely employed for transmission in radio

communication. This is mainly due to the fact that they have small circumferences hence their

radiation resistances are usually smaller than their loss resistance, making them poor radiators.

However, they may be employed as receiving antennas for example in portable radios and

pagers where antenna efficiency is not of much importance as the signal-to-noise ratio. They

can also be used as probes for field measurements and as directional antennas for radiowave

navigation. As the overall length of the loop increases and its circumference starts to approach

one free-space wavelength, the maximum of the pattern shifts from the plane of the loop to the

axis of the loop which is perpendicular to its plane.

By increasing the perimeter and/or the number of turns of the loop, its radiation resistance

can be increased, and made comparable to the characteristic impedance of practical transmis-

sion lines. The radiation resistance can also be increased by inserting, within its circumference

or perimeter, a ferrite core of very high permeability which will raise the magnetic field inten-

sity and as a result, the radiation resistance. This results in the so-called ferrite loop.

Electrically large loop antennas are used primarily in directional arrays, which include heli-

cal antennas, Yagi-Uda arrays and quad arrays. For such applications, the maximum radiation

is directed toward the axis of the loop forming an end-fire antenna. Therefore, to obtain such

directional pattern characteristics, the circumference of the loop should be about one free-

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space wavelength. The overall directional properties are enhanced by proper phasing between

turns [1].

2.2.3 The Gray Hoverman antenna

The Gray Hoverman antenna was first created by Mr. Doyt R. Hoverman, during a time

when computer modelling programs did not exist. His work would have been entirely created

and improved by applying field testing techniques, trail and error, and with a great amount of

calculation without the benefit of electronic devices.

The first patent of the Gray Hoverman antenna was released on 25-june-1959. It consisted

entirely only of the driven array. This array was composed of two subsections, each subsection

was 56 inches long with 8 segments of 7 inches.

The second patent released was much more improved. It consisted of a similar driven array

as the original design, however it also had additional 4 rod reflectors both full wavelength and

co-linear half wavelength.

Commercial television signals are telecast in two broad frequency spectra referred to as

“very high frequency” often abbreviated as VHF and “ultra high frequency” abbreviated as

UHF. Due to the extensive separation between the UHF and VHF spectra, it is almost impos-

sible to use a receiving antenna designed for one frequency spectrum in another frequency

spectrum. In other words an antenna designed to operate in UHF spectrum cannot operate in

the VHF spectrum. To adapt receiver antennas to operate in both UHF and VHF stations, often

two separate antennas are used; one for VHF reception and the other for UHF.

The Gray Hoverman antenna consists of two antenna subsections assembled in substantial

parallelism and in a common plane; each subsection is bent into V-shaped elements forming

eight zigzag segments. According to Mr. Doyt Hoverman, this antenna is capable of operating

channels 2 to 7 in the VHF spectrum and channels 14 to 35 in the UHF spectrum. However,

recent analysis and modelling results have not found any positive net gain for the VHF chan-

nels [4].

2.3 The Method of Moments

The Method of Moments (MM) technique was introduced by Roger F. Harrington in his

1967 seminar paper, “Matrix Methods for Field Problems”. In 1970, the Method of Moments

was implemented by Poggio and Burke at Lawrence Livermore National Labs establishing this

technique as a mainstay in the design of wire antenna arrays [5]. Since then, fast performing

high capacity computers have been built which have aided in solving problems in electromag-

netism. The MM is a technique that can be used to solve electromagnetic boundary or volume

integral equations in the frequency domain. Since the electromagnetic sources are the quanti-

- 7 -

ties of interest, the MM is very useful when finding solutions to radiation and scattering prob-

lems [3].

2.4 The Integral Equations

The Integral Equation (IE) Method of Moments is a technique commonly used to obtain the

current distribution and impedance of an antenna. In the late 1960’s, this technique was ex-

tended to solve electromagnetic field problems. There are two commonly applied integral

equation methods used to solve electromagnetic field problems;

Pocklington’s integral equation was derived by Pocklington in 1897 [6]. In 1956, Erik Hallen

derived the famous Hallen’s integral equation to give an exact treatment of current antenna

wave reflection at the end of the tube shaped cylindrical antenna. However, his work on the

project dates back to 1938. Both these integral equations enabled them to prove that on thin

wire antennas, the current distribution is approximately sinusoidal and propagates with nearly

the speed of light.

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3 Method of Moments

In this chapter, the Moments Method is discussed. A detailed explanation on how to apply MM

has been discussed. Firstly, a brief description is given for the MM. Next, Maxwell’s equation

from which the electromagnetic field equations are used to solve various antenna parameters

are presented. Matrix solutions for solving MM are then discussed followed by various basis

functions and a brief explanation on how they may be applied to antennas. Finally, the two

most commonly applied source modelling techniques are presented.

3.1 Terminology

The method of moments is a step by step process used for reducing an integral equation of

the form

∫𝐼(𝑧′)𝐾(𝑧, 𝑧′)𝑑𝑧′ = −𝐸𝑖(𝑧) 3.1

into, a system of simultaneous linear algebraic equations, in terms of the unknown current

𝐼(𝑧′) by applying point-matching technique [6]. This unknown current is multiplied by an ap-

propriate function 𝐾(𝑧, 𝑧′) which is referred to as a kernel. For fields in a vacuum, the kernel

for the Lorentz potential is the free-space Green’s function. Certain equations are also imposed

based on boundary conditions. Boundary conditions present a relationship between the tangen-

tial and normal components of field vectors at a surface of discontinuity. If we consider the

surface of a perfectly conducting body, the tangential component of the electric field vanishes.

In antenna analysis, the method of moments can be applied alongside techniques such as

Gaussian elimination, LU factorization as well as other methods used in solving simultaneous

equations and matrices. The method of moments is especially useful when simulating struc-

tures of arbitrary configuration such as; the inverted v-dipole, the Gray Hoverman antenna,

and circular loop antennas due to the fact that it is very versatile.

3.2 Maxwell’s equations

The first step in obtaining the current distribution on any antenna, is to derive the appropriate

integral equation. By use of maxwell’s equations, these integral equations can be easily

derived:

𝐻𝐴 =1

𝜇∇ × A

3.2

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𝐸𝐴 = −∇∅𝑒 − jωA

3.3

∇ × 𝐻𝐴 = 𝐽 + jωε𝐸𝐴 3.4

∇ × 𝐸𝐹 = −𝑀 − jωμ𝐻𝐹 3.5

∇. 𝐸 =𝜌𝑒

𝜀 3.6

∇.𝐻 =𝜌𝑚

μ 3.7

∇. 𝐴 = −𝑗𝜔휀𝜇∅𝑒 3.8

Where: H = Magnetic field intensity

E= Electric field intensity

𝜌𝑒 = electric charge density

𝜌𝑚 = magnetic charge density

J = electric current density

M = magnetic current density

∇2𝐴 + 𝐾2𝐴 = −𝜇𝐽 3.9

By combining Eq. (3.3) and Eq. (3.8), we are able to obtain the scattered electric field as:

𝐸𝑠 = −∇∅𝑒 − 𝑗𝜔𝐴 = −𝑗𝜔𝐴 − 𝑗1

𝜔𝜇𝜀∇(∇. 𝐴) 3.10

By solving the vector equation:

𝐴𝑟 = 𝜇

4𝜋∬ 𝐽𝑠(𝑟′)𝐺(𝑅)𝑑𝑠′𝑠

3.11

Where G(R) is the free space green’s function defined by

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𝐺(𝑅) =𝑒−𝑗𝑘𝑅

4𝜋𝑅

3.12a

𝑅 = √(𝑥 − 𝑥′) + (𝑦 − 𝑦′) + (𝑧 − 𝑧′) 3.12b

k is the wave vector and is given by, 𝑘 = 𝜔√휀𝜇 [1]

3.3 Solving the matrix equation

The basic form of any equation being solved by the Moments Method is,

𝐹(𝑔) = ℎ 3.13

Where: F = known linear operator

𝑔 = response function

h = known excitation function

By applying the method of moments technique, the unknown response function, 𝑔 can be

expanded as a linear combination of N terms and written as:

𝑔(𝑧′) ≅ 𝑎1𝑔1(𝑧′) + 𝑎2𝑔2(𝑧

′) + ⋯+ 𝑎𝑁𝑔𝑁(𝑧′) = ∑ 𝑎𝑛𝑔𝑛(𝑧

′)𝑁𝑛=1 3.14

Where: 𝑎𝑛 = unknown constant

𝑔𝑛(𝑧′) = basis or expansion function

By substituting Eq. (3.14) into Eq. (3.13) and applying the linearity of the f operator:

∑ 𝑎𝑛𝐹(𝑔𝑛)𝑁𝑛=1 = ℎ 3.15

Eq. (3.15) can be evaluated by applying boundary condition at N different points using

point-matching techniques (collocation),doing this reduces Eq. (3.15) to:

∑ 𝐼𝑛𝐹(𝑔𝑛)𝑁𝑛=1 = ℎ𝑚 m=1,2,3….N 3.16

In matrix form, Eq. (3.16) can be expressed as:

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[𝑍𝑚𝑛][𝐼𝑛] = [𝑉𝑚] 3.17

Where: [𝑍𝑚𝑛] = 𝐹(𝑔𝑛)

[𝐼𝑛] = 𝑎𝑛

[𝑉𝑚] = ℎ𝑚

Now, by applying matrix inversion techniques, the current distribution can be obtained [1]:

[𝐼𝑛] = [𝑍𝑚𝑛]−1[𝑉𝑚] 3.18

3.4 Basis functions

The choice of a basis function is very crucial in solving any numerical problem in antenna

design. The basis function chosen should resemble and represent the anticipated unknown

function. It’s not wise to choose basis functions with smoother properties than the unknown

being represented.

Basis functions are subdivided into two broad categories. The first category is that of the

subdomain basis functions. Subdomain basis functions are functions that are nonzero only

over a part of the domain of the function 𝑔(𝑥′) ; its domain being the surface of the structure.

The second category is that of the entire domain basis functions which as the name suggests is

nonzero over the entire domain of the function.

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Figure 3.1 Piecewise constant subdomain functions (source: Figure 8.8 in [1])

3.4.1 Subdomain functions

Subdomain basis functions are the most commonly used of the two categories in antenna

analysis and design. This is due to the fact that they may be used without having prior

knowledge of the nature of the function that they must represent.

By subdividing the struction into N nonoverlapping segments as illustrated in Fig. 3.1(a),

the subdomain basis functions can be easily computed. In the above illustration, the segments

are assumed to be collinear and of equal length, however, this may not necessarily be the case

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3.4.1.1 Piecewise constant

The simplest and most commonly applied basis function is the piecewise constant. It is

sometimes referred to as the “pulse” function. This piecewise constant is demonstrated in

figure 3.1(a) and is defined by:

𝑔𝑛(𝑥′) = {

1 𝑥𝑛−1′ ≤ 𝑥′ ≤ 𝑥𝑛

′

0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒 3.19

Once the associated unknown coefficients have been determined, a staircase representation

of unknown function will be produced as demonstrated by figures 3.1(b) and 3.1(c).

3.4.1.2 piecewise linear

The piecewise linear sometimes referred to as the “triangular” functions is another common

basis function. Shown in figure 3.2(a), its defined by:

𝑔𝑛(𝑥′) =

{

𝑥′ − 𝑥𝑛−1

′

𝑥𝑛′ − 𝑥𝑛−1

′ 𝑥𝑛−1′ ≤ 𝑥′ ≤ 𝑥𝑛

′

𝑥𝑛+1′ − 𝑥′

𝑥𝑛+1′ − 𝑥𝑛

′ 𝑥𝑛′ ≤ 𝑥′ ≤ 𝑥𝑛+1

′

0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒

3.20

Unlike the piecewise constant which covers only one segment, the piecewise linear covers

two segments, and overlaps adjacent segments. The advantage of the piecewise linear is that a

smoother representation is obtained as shown in figure 3.2(c), however, there is increased

computational complexity.

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Figure 3.2 Piecewise linear subdomain functions (source: Figure 8.9 in [1])

3.4.1.3 piecewise sinusoidal

Some integral operators may be evaluated without numerical integration when their

integrands are multiplied by a sin (𝑘𝑥′) or cos(𝑘𝑥′) function where x’ is the integration

variable. During such scenarios, it would be more appropriate to use basis functions such as

the piecewise sinusoidal and truncated cosine. The piecewise sinusoidal basis function is

defined by:

𝑔𝑛(𝑥′) =

{

sin[𝑘(𝑥′ − 𝑥𝑛−1

′ )]

sin[𝑘(𝑥𝑛′ − 𝑥𝑛−1

′ )]𝑥𝑛−1′ ≤ 𝑥′ ≤ 𝑥𝑛

′

sin[𝑘(𝑥𝑛+1′ − 𝑥′)]

sin[𝑘(𝑥𝑛+1′ − 𝑥𝑛

′ )]𝑥𝑛′ ≤ 𝑥′ ≤ 𝑥𝑛+1

′


3.21

- 15 -

Figure 3.3 Piecewise sinusoids subdomain functions (source: Figure. 8.10 in [1]).

3.4.1.4 Truncated cosine

The truncated cosine basis function is defined by:

𝑔𝑛(𝑥′) = {

cos [𝑘 (𝑥′ −𝑥𝑛′ − 𝑥𝑛−1

′

2)] 𝑥𝑛−1

′ ≤ 𝑥′ ≤ 𝑥𝑛′


3.22

- 16 -

Figure 3.4 Truncated cosines subdomain functions (source: Figure. 8.11 in [1]).

The piecewise sinusoidal is very similar to the piecewise linear basis function. The only

difference being that instead of being triangular, they are curved, however, in both, there is

overlap between adjacent segments.

3.4.2 Entire domain basis functions.

Unlike the subdomain basis functions, in the entire domain basis functions, no

segmentation is used. The commonest entire domain basis set is the sinusoidal functions,

defined by:

𝑔𝑛(𝑥′) = cos [

(2𝑛 − 1)𝜋𝑥′

𝑙] −

𝑙

2≤ 𝑥′ ≤

𝑙

2 3.23

- 17 -

This function would be very useful in obtaining the current distribution on a wire dipole,

which is known to have a primarily sinusoidal distribution. The main advantage of the entire

domain basis functions over the subdomain basis functions lies in the fact that it has reduced

complexity. However, when modelling arbitrary and complicated unknown functions, the

entire domain basis function would be very inefficient and impossible to use.

Representing a function by entire domain basis function is similar to the fourier series

expansion of arbitrary functions.

3.5 Source modelling

There are two methods that have been often used to model the source. The simplest is the

delta-gap excitation and the second being the magnetic frill generator sometimes referred to as

the equivalent magnetic ring current.

Figure 3.5 Cylindrical dipole, its segmentation and gap modelling (source: Fig. 8.7 in

[1]).

3.5.1 Delta-gap

The delta-gap source modelling is the most widely used of the two, however it is also the least

accurate, especially when calculating impedances. However, for thin wire antennas, it provides

considerable accuracy. By using the delta-gap, it is assumed that the excitation voltage at the

- 18 -

feed terminals is of a constant, 𝑉𝑖 value and zeros elsewhere. Therefore the incident electric

field 𝐸𝑧𝑖(𝜌 = 𝑎), 0 ≤ ∅ ≤ 2𝜋,−

𝑙

2≤ 𝑧 ≤ +𝑙/2 is also a constant,(

𝑉𝑖∆⁄ where ∆ is the gap

width) over the feed gap and zero elsewhere, hence the name delta gap. For the delta gap

model, of Fig. 3.5(a), the feed gap ∆ is replaced by a narrow band of strips of equivalent

magnetic current density of:

𝑀𝑔 = −�̂� × 𝐸𝑖 = −�̂�𝑝 × �̂�𝑧

𝑉𝑖

∆ = �̂�∅

𝑉𝑖

∆ −

∆

2≤ 𝑧 ≤

∆

2 3.24

3.5.2 Magnetic frill generator

The magnetic frill generator was introduced to calculate the near as well as far-zone fields

from coaxial apertures. In this model, the feed gap is replaced with a circumferentially

directed magnetic current density that exists over an annular apertue with inner radius a, which

is usually chosen to be the radius of the wire, and an outer radius b, as shown in Fig. 3.5(b).

Since the dipole is usually fed by transmission lines, the outer radius b of the equivalent

annular aperture of the magnetic frill generator is found using the expression for the

characteristic impedance of the transmission line.

Over the annular aperture of the magnetic frill generator, the electric field is represented by the

TEM mode field distribution of a coaxial transmission line given by:

𝐸𝑓 = �̂�𝜌𝑉𝑖

2𝜌′ ln(𝑏/𝑎) 𝑎 ≤ 𝜌′ ≤ 𝑏

3.25

Therefore the corresponding equivalent magnetic current density Mf for the magnetic frill

generaor used to represent the aperture is equal to:

𝑀𝑓 = −2�̂� × 𝐸𝑓 = −2�̂�𝑧 × �̂�𝜌𝐸𝜌 = −�̂�∅𝑉𝑖

𝜌′ ln (𝑏𝑎) 𝑎 ≤ 𝜌′ ≤ 𝑏 3.26

The fields generated by the magnetic frill generator of Eq. (3.26) on the surface of the wire are

found by using:

𝐸𝑧𝑖 (𝜌 = 𝑎, 0 ≤ ∅ ≤ 2𝜋,−

𝑙

2≤ 𝑧 ≤ +

𝑙

2) ≅ −𝑉𝑖 (

𝑘(𝑏2−𝑎2)𝑒−𝑗𝑘𝑅𝑜

8 ln(𝑏 𝑎⁄ )𝑅𝑜2 {2 [

1

𝑘𝑅𝑜+ 3.27

- 19 -

𝑗 (1 −𝑏2−𝑎2

2𝑅𝑜2 )] +

𝑎2

𝑅𝑜[(

1

𝑘𝑅𝑜+ 𝑗 (1 −

𝑏2+𝑎2

2𝑅𝑜)) (−𝑗𝑘 −

2

𝑅𝑜) + (−

1

𝑘𝑅𝑜2 + 𝑗

𝑏2+𝑎2

𝑅𝑜3 )]})

Where

𝑅𝑜 = √𝑧2 + 𝑎2 3.28

The fields generated on the surface of the wire computed using Eq. (3.27) can be

approximated by those found along the axis (𝜌 = 0). Doing this leads to a simpler expression

of the form [1]:

𝐸𝑧𝑖 (𝜌 = 0,−

𝑙

2≤ 𝑧 ≤

𝑙

2) = −

𝑉𝑖

2 ln(𝑏 𝑎⁄ )[𝑒−𝑗𝑘𝑅1

𝑅1−𝑒−𝑗𝑘𝑅2

𝑅2] 3.29

Where

𝑅1 = √𝑧2 + 𝑎2

𝑅2 = √𝑧2 + 𝑏2

- 20 -

4 Application to thin wire antennas

In this chapter, mathematical formulations for calculating the needed antenna parameters have

been derived. Firstly, an equation for charge distribution on a straight wire is derived. The so-

lution is then further extended to arbitrary wire shapes. Finally, the equations to solve for

current distribution on arbitrary wire shapes are presented.

4.1 Charge Distribution

Charge density refers to the measure of electric charge per unit volume of space, in one, two or

three dimensions. Linear surface charge density, 𝜌 is the amount of electric charge per unit

length

4.1.1 The Straight Dipole

4.1.1.1 The Electric Field Integral Equation

In order to get a clear understanding of how the Moments Method (MM) is applied, con-

sider the electrostatic charge distribution on a wire.

4.1.1.2 Electrostatic Charge Distribution

Figure 4.1 Electric potential V at a point P at a distance r from a charge Q (source:

Figure 9-19 in [2]).

The electric potential, V at a point P, due to a charge Q a distance r apart is given by

𝑉 =𝑄

4𝜋휀𝑟

4.1

Where Q is the charge in coulombs

r is distance between charge Q and point P in meters

- 21 -

ε is permittivity of the medium in F m-1

The potential at some observation point, P for a line charge, ρ over the length, Ɩ is given by

𝑉(𝑟) =1

4𝜋𝜀∫

𝜌(𝑟′)

𝑅𝑑𝑥

𝑙

0 4.2

Where: ρ(r’)= charge per unit length of the line, C m-1

r’(x’,y’,z’) denotes the source coodinates,

r(x,y,z) denotes the observation point coodinates , and R is the distance

between a source and observation point. In general,

𝑅(𝑟, 𝑟′) = ǀ𝑟 - 𝑟′ǀ = √(𝑥 − 𝑥′)2 + (𝑦 − 𝑦′)2 + (𝑧 − 𝑧′)2 4.3

Consider a straight wire of length, Ɩ with a constant potential, V along its entire length, lying

along the y-axis as shown in the Fig. 4.2. The wire is of radius a.

Figure 4.2 Straight wire of constant potential and its segmentation (source: Fig 8.1 in

[1]).

- 22 -

In order to avoid singularity, the observation point and source point cannot be placed on the

same axis. It is usually more convinient to place the observation points along the wire axis and

the source points along the surface of the wire.

Since the wire is placed along the y-axis, Eq. (4.3) becomes:

𝑅(𝑟, 𝑟′) = 𝑅(𝑦, 𝑦′) = ǀy - y’ǀ = √(𝑦 − 𝑦′)2 + 𝑎2 4.4

If the potential along the wire is 1V, Eq. (4.2) can be simplified to obtain

1 =1

4𝜋휀∫

𝜌(𝑦′)

√(𝑦 − 𝑦′)2 + 𝑎2𝑑𝑦′

𝑙

0

4.5

The above Eq. (4.5) can then be used to solve for the unknown charge density, 𝜌(𝑦′) by

reducing it into a set of linear algebraic equations.

This can be done by approximating the unknown charge distribution 𝜌(𝑦′) by enlarging N

known terms with a constant but unknown coefficients, the N known terms represent the

number of segments the wire is divided into. Each segment being of length, ∆=𝑙

𝑁.

Therefore charge distribution can be represented as:

𝜌(𝑦′) = ∑𝑎𝑛𝑔𝑛(𝑦′)

𝑛

𝑛−1

4.6

Where: 𝑎𝑛 is the unknown charge density

𝑔𝑛(𝑦′) is the basis function.

Various forms of basis functions exist, however, to simplify the equation, subdomain

piecewise constant shall be applied which is defined by

𝑔𝑛(𝑦′) = {

0 𝑦′ < (𝑛 − 1)∆

1 (𝑛 − 1)∆≤ 𝑦′ ≤ 𝑛∆

0 𝑦′ > 𝑛∆

4.7

Eq. (4.5) can be further simplified to the form

4πε = ∫1

𝑅(𝑦,𝑦′)∑ 𝑎𝑛𝑔𝑛(𝑦

′)𝑛𝑛−1

𝑙

0 4.8

- 23 -

The sum and integral functions in Eq. (4.8) can now be interchanged to obtain:

4πε = ∑ 𝑎𝑛𝑁𝑛=1 ∫

𝑔𝑛(𝑦′)

√(𝑦−𝑦′)2+𝑎2

𝑙

0𝑑𝑦′ 4.9

Replacing y in Eq. (4.9) by an observation point, 𝑦𝑚 and expanding the summation:

4𝜋휀=𝑎1 ∫𝑔1(𝑦

′)

√(𝑦𝑚−𝑦′)2+𝑎2

∆

0𝑑𝑦′+𝑎2 ∫

𝑔2(𝑦′)

√(𝑦𝑚−𝑦′)2+𝑎2

2∆

∆𝑑𝑦′+….+

𝑎𝑁 ∫𝑔𝑁(𝑦

′)

√(𝑦𝑚−𝑦′)2+𝑎2

𝑙

(𝑁−1)∆𝑑𝑦′

4.10

Therefore for N segments, N linear equations will be obtained hence expanding Eq. (4.10)

to:

4𝜋휀 = 𝑎1∫𝑔1(𝑦

′)

√(𝑦1 − 𝑦′)2 + 𝑎2

∆

0

𝑑𝑦′ + 𝑎2∫𝑔2(𝑦

′)

√(𝑦1 − 𝑦′)2 + 𝑎2

2∆

∆

𝑑𝑦′ +⋯+ 𝑎𝑁∫𝑔𝑁(𝑦

′)

√(𝑦1 − 𝑦′)2 + 𝑎2

𝑙

(𝑁−1)∆

𝑑𝑦′

4𝜋휀 = 𝑎1∫𝑔1(𝑦

′)

√(𝑦2 − 𝑦′)2 + 𝑎2

∆

0

𝑑𝑦′ + 𝑎2∫𝑔2(𝑦

′)

√(𝑦2 − 𝑦′)2 + 𝑎2

2∆

∆

𝑑𝑦′ +⋯+ 𝑎𝑁∫𝑔𝑁(𝑦

′)

√(𝑦2 − 𝑦′)2 + 𝑎2

𝑙

(𝑁−1)∆

𝑑𝑦′

⋮

4𝜋휀 = 𝑎1∫𝑔1(𝑦

′)

√(𝑦𝑁 − 𝑦′)2 + 𝑎2

∆

0

𝑑𝑦′ + 𝑎2∫𝑔2(𝑦

′)

√(𝑦𝑁 − 𝑦′)2 + 𝑎2

2∆

∆

𝑑𝑦′ + ⋯+ 𝑎𝑁∫𝑔𝑁(𝑦

′)

√(𝑦𝑁 − 𝑦′)2 + 𝑎2

𝑙

(𝑁−1)∆

𝑑𝑦′

4.11

In matrix form, Eq. (4.11) may be rewritten as:

[𝑣𝑚] = [𝑧𝑚𝑛][𝑎𝑛] 4.12

Each Zmn term in the matrix can be obtained from the integral equation:

𝑧𝑚𝑛 = ∫𝑔𝑛(𝑦

′)

√(𝑦𝑚−𝑦′)2+𝑎2

𝑙

0 4.13

By applying the subdomain piecewise constant to Eq. (4.13), a simplified equation for 𝑧𝑚𝑛

matrix is obtained:

𝑧𝑚𝑛 = ∫1

√(𝑦𝑚−𝑦′)2+𝑎2

𝑛∆

(𝑛−1)∆ 4.14

- 24 -

For a constant potential, 𝑣𝑜 = 1V along the length of the wire:

𝑣𝑚= [4𝜋휀𝑣𝑜]= 4𝜋휀

4.15

𝑎𝑁 is the charge distribution and can be obtained by numerous matrix inversion

techniques.

[𝑎𝑁] = [𝑧𝑚𝑛]−1[𝑣𝑚] 4.16

Eq. (4.14) can be readily solved by use of a digital computer.

4.1.2 The Gray Hoverman antenna

Assume the wire has a bend of angle 𝛼, which remains on the yz-plane as shown in Fig. 4.3

The mathematical formulation of this type of design is quite similar to that of the straight

wire.

Figure 4.3 Geometry of a bent wire (source: Figure 8.3 in [1]).

In fact equation (4.4) still applies to a bent wire as well, however the distance between the

observation point and the source point does not reduce to an expression as simple as |𝑦 − 𝑦′|,

for cases when the source point and observation point are located on different segments. For

this kind of situation, the distance may be expressed as:

- 25 -

𝑅 = √(𝑦 − 𝑦′)2 + (𝑧 − 𝑧′)2 4.17

And as a result of this, the integral equation can be divided into two parts which is given

by:

𝑍𝑚𝑛 = ∫𝜌𝑛(𝑙1

′ )

𝑅𝑑𝑙1′

𝑙1

0

+∫𝜌𝑛(𝑙2

′ )

𝑅𝑑𝑙2′

𝑙2

0

4.18

Where l1 and l2 are the length of each segment and are measured along the corresponding

wire segment from their leftmost end. Eq. (4.18) can be therefore be expressed as [1]:

𝑍𝑚𝑛 = ∫𝜌𝑛(𝑙1

′ )

√(𝑦𝑚 − 𝑦′)2 + 𝑎2

𝑑𝑙1′

𝑙1

0

+∫𝜌𝑛(𝑙2

′ )

√(𝑦𝑚 − 𝑦′)2 + (𝑧𝑚 − 𝑧′)2

𝑑𝑙2′

𝑙2

0

4.19

Since ym is a constant, the integral equations depends on the source 𝑧′ points and 𝑦′. From

integral calculus, the integral of a function f(y), can be assumed to be the sum of the areas

under rectangular strips each having a height equal to the mean of f(y) over the strip which

can be mathematically represented as:

∫ 𝑓(𝑦′)𝑑𝑦′ ≈ 𝑓(𝑦1)∆𝑦′ + 𝑓(𝑦2)∆𝑦

′ +⋯ . .+𝑓(𝑦𝑁)∆𝑦′

𝑙

0

4.20

From equation (4.2), assuming a potential of 1V is applied across the antenna length:

4𝜋휀0 = ∫𝜌𝑛(𝑙1

′ )

√(𝑦𝑚 − 𝑦′)2 + 𝑎2

𝑑𝑙1′

𝑙1

0

4.21

Since we are dealing with thin wire antenna models, the radius 𝛼 is much shorter than

wavelength of the antenna and can be assumd to be zero, by doing this, Eq. (4.21) can be

reduced to:

4𝜋휀0 = ∫𝜌𝑛(𝑙1

′ )

√(𝑦𝑚 − 𝑦′)2𝑑𝑙1′

𝑙1

0

4.22

Applying Eq. (4.20) to Eq. (4.22), the expression is simplified to:

- 26 -

4𝜋휀0 ≅𝜌1∆

|𝑦𝑚 − 𝑦1′ |+

𝜌2∆

|𝑦𝑚 − 𝑦2′ |+ ⋯+

𝜌𝑁∆

|𝑦𝑚 − 𝑦𝑁′ |

4.23

This wire has been divided into N uniform equally spaced segments. Since ym represents

observation points, it should be considered for each of N wire segments. Hence Eq. (4.23) can

be expanded as:

4𝜋휀0 ≅𝜌1∆

|𝑦1 − 𝑦1′ |+

𝜌2∆

|𝑦1 − 𝑦2′ |+ ⋯+

𝜌𝑁∆

|𝑦1 − 𝑦𝑁′ |

4𝜋휀0 ≅𝜌1∆

|𝑦2 − 𝑦1′ |+

𝜌2∆

|𝑦2 − 𝑦2′ |+ ⋯+

𝜌𝑁∆

|𝑦2 − 𝑦𝑁′ |

⋮

4𝜋휀0 ≅𝜌1∆

|𝑦𝑁 − 𝑦1′ |+

𝜌2∆

|𝑦𝑁 − 𝑦2′ |+ ⋯+

𝜌𝑁∆

|𝑦𝑁 − 𝑦𝑁′ |

4.24

For convinience, the observation points are considered to be along the wire axis, with

source points along the wire surface. However, both source and observation points are located

at the midpoint of a segment. This presents a problem of singularity of self terms, that is in

cases when the source point is equal to the observation point (𝑦𝑚 = 𝑦′).

In order to avoid this, the self terms which lie along the matrix diagonals, need to be solved

separately by considering the radius 𝛼. Given the fact that the wire is highly conducting, this

implies that a uniform potential exists throughout the wire’s length. The self terms can

therefore be interpreted as a potential at the center of a uniform tube of an approximate charge

density 𝜌𝑠. Mathematically, this is represented as:

𝜙(𝑡𝑢𝑏𝑒 𝑐𝑒𝑛𝑡𝑒𝑟) =1

4𝜋휀𝑜∫ ∫

𝜌𝑠𝑎𝑑𝜙𝑑𝑦′

√𝑎2 + 𝑦′2

Δ 2⁄

−Δ 2⁄

2𝜋

0

4.25

On solving, Eq. (4.25) reduces to:

𝜙(𝑡𝑢𝑏𝑒 𝑐𝑒𝑛𝑡𝑒𝑟) =2𝜌𝑙4𝜋휀𝑜

ln(Δ 𝑎⁄ ) 4.25a

Where 𝜌𝑙 = 2𝜋𝑎𝜌𝑠.

Thus the diagonal self term becomes:

𝑍𝑛𝑛 = 2 ln(Δ 𝑎⁄ ) 4.26

- 27 -

For cases when the source and observation point are not located on the same segment,

(𝑚 ≠ 𝑛), if the source and observation points exist on the same wire subsection [7]:

𝑍𝑚𝑛 =∆

|𝑦𝑚 − 𝑦𝑁′ |

4.27a

However, if the source and observation points are located on different segments of the

wire:

𝑍𝑚𝑛 =∆

√(𝑦𝑚 − 𝑦𝑁′ )2 + (𝑧𝑚 − 𝑧𝑁

′ )2 4.27b

Where ym is the observation point which is located at the midpoint of the segment,and can

can be represented by:

𝑦𝑚 = (𝑚 − 0.5) × 𝑑𝑒𝑙𝑡𝑎 4.28

Delta is the length of each segment, and can be calculated as:

𝑑𝑒𝑙𝑡𝑎 =𝑡𝑜𝑡𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑤𝑖𝑟𝑒

𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑒𝑔𝑚𝑒𝑛𝑡𝑠 4.29

4.2 Current Distribution

The purpose of the integral equation method is to find the unknown current density which

forms part of the integrand. Once the current distribution is obtained, the impedance can be

readily calculated.

4.2.1 The Straight Dipole

4.2.1.1 Pocklington’s Integral Equation

When an incident electric field, 𝐸𝑖(𝑟) is directed on a conducting wire as observed in Fig.

4.4, it induces a linear current density, 𝐽𝑆 which reradiates producing a scattered electric field,

𝐸𝑠(𝑟). Therefore the total electric field, 𝐸𝑡(𝑟) is given by:

- 28 -

𝐸𝑡(𝑟) = 𝐸𝑖(𝑟) + 𝐸𝑠(𝑟) 4.30

Where 𝐸𝑡(𝑟) = Total electric field

𝐸𝑖(𝑟) = incident electric field

𝐸𝑠(𝑟) = scattered electric field

Figure 4.4 A uniform plane wave obliquely incident on a conducting wire (source:

Figure 8.5 in [1]).

If the wire is a perfect electric conductor and the observation point is moved to the surface of

the wire(𝑟 = 𝑟𝑠), the total tangential electric field vanishes.

𝐸𝑧𝑡(𝑟 = 𝑟𝑠) = 𝐸𝑧

𝑖(𝑟 = 𝑟𝑠) + 𝐸𝑧𝑠(𝑟 = 𝑟𝑠) = 0 4.31

Therefore,

𝐸𝑧𝑖(𝑟 = 𝑟𝑠) = - 𝐸𝑧

𝑠(𝑟 = 𝑟𝑠) 4.32

𝐸𝑠(𝑟) can be obtained from Eq. (3.10), therefore,

𝐸𝑠(𝑟) = −𝑗𝜔𝐴 − 𝑗1

𝜔𝜇𝜀∇(∇. 𝐴) 4.33

- 29 -

𝐸𝑠(𝑟) = −𝑗1

𝜔𝜇휀[𝑘2𝐴 + ∇(∇. 𝐴)]

4.33a

However, for observations at the wire surface, only the z component of Eq. (4.33) may be

considered since the dipole is lying along the z-axis, which can be written as:

𝐸𝑧𝑠(𝑟)=−𝑗

1

𝜔𝜇𝜀[𝑘2𝐴𝑧 +

𝜕2𝐴𝑧

𝜕𝑍2]

4.34

From Eq. (3.11) and neglecting edge effects:

𝐴𝑧 = 𝜇

4𝜋∬ 𝐽𝑧𝐺(𝑅)𝑑𝑠′𝑠

= 𝜇

4𝜋∫ ∫ 𝐽𝑧𝐺(𝑅)𝑎𝑑∅𝑑𝑧′

2𝜋

0

+𝑙2⁄

−𝑙2⁄

4.35

For a thin wire, the current density 𝐽𝑧 is not a function of the azimuthal angle ∅, and can

therefore be written as:

2𝜋𝑎𝐽𝑧 = 𝐼𝑧(𝑧′) 4.36

- 30 -

Figure 4.5 Dipole segmentation and its equivalent current (source: Figure 8.6 in [1]).

Where 𝐼𝑧(𝑧′) is the equivalent filament line-source current located a radial distance 𝜌 = 𝑎

from the z-axis, as shown in Fig. 4.5

Expressing Eq. (4.36) as a function of 𝐼𝑧(𝑧′) and replacing in Eq. (4.35):

𝐴𝑧 = 𝜇

4𝜋∫ [

1

2𝜋𝑎∫ 𝐼𝑧(𝑧

′)𝐺(𝑅)𝑎𝑑∅2𝜋

0] 𝑑𝑧′

+𝑙2⁄

−𝑙2⁄

4.37

𝑅 = √(𝑥 − 𝑥′) + (𝑦 − 𝑦′) + (𝑧 − 𝑧′)

𝑅 = √𝜌2 + 𝑎2 − 2𝜌𝑎 cos(𝜙 − 𝜙′) + (𝑧 − 𝑧′)2

Where: 𝜌 = radial distance to the observation point

a = radius of the wire

if 𝜙 = 0, and observation points are taken on the wire surface, 𝜌 = a, Eq. (4.37) reduces to:

- 31 -

𝐴𝑍 = 𝜇 ∫ 𝐼𝑧(𝑧′)

+𝑙2⁄

−𝑙2⁄

(1

2𝜋∫

𝑒−𝑗𝑘𝑅

4𝜋𝑅

2𝜋

0𝑑𝜙′)𝑑𝑧′

4.38

𝐴𝑧=𝜇∫ 𝐼𝑧(𝑧′)𝐺(𝑧, 𝑧′)𝑑𝑧′

+𝑙2⁄

−𝑙2⁄

4.38a

𝐺(𝑧, 𝑧′) =1

2𝜋∫

𝑒−𝑗𝑘𝑅

4𝜋𝑅

2𝜋

0

𝑑𝜙′ 4.38b

Where:

𝑅(𝜌 = 𝑎) = √4𝑎2𝑠𝑖𝑛2 (∅′

2) + (𝑧 − 𝑧′)2

4.38c

For a thin wire, the radius is very small compared to the wavelength of operation of the

antenna (𝑎 ≪ 𝜆), the free space Green’s function reduces to:

𝐺(𝑧, 𝑧′) = 𝐺(𝑅) = 𝑒−𝑗𝑘𝑅

4𝜋𝑅

4.39

Replacing Eq. (4.38) in Eq. (4.34) and solving for the scattered electric field with the wire

along the z-axis:

𝐸𝑧𝑠(𝜌 = 𝑎) = −𝑗

1

𝜔𝜀(𝑘2 +

𝜕2

𝜕𝑧2) ∫ 𝐼𝑍(𝑧′)𝐺(𝑧, 𝑧′)𝑑𝑧′

+𝑙2⁄

−𝑙2⁄

4.40

Replacing Eq. (4.40) in Eq. (4.31):

−𝑗1

𝜔𝜀(𝑘2 +

𝜕2

𝜕𝑧2) ∫ 𝐼𝑍(𝑧′)𝐺(𝑧, 𝑧′)𝑑𝑧′

+𝑙2⁄

−𝑙2⁄

= −𝐸𝑍𝑖 (𝜌 = 𝑎)

4.41

Finally, by interchanging the integration with differentiation in Eq. (4.41):

𝑗

𝜔𝜀∫ 𝐼𝑧(𝑧′)[(

𝜕2

𝜕𝑧2+ 𝑘2)𝐺(𝑧, 𝑧′)]𝑑𝑧′

+𝑙2⁄

−𝑙2⁄

= 𝐸𝑧𝑖

4.42

- 32 -

Eq. (4.42) is referred to as Pocklington’s integral equation. This equation can be used to

obtain the equivalent filamentory line-source current, and therefore the current density on the

wire.

In order to further simplify the expression, the brackets in Eq. (4.42) can be opened. Doing

this results in [1]:

𝑗

𝜔𝜀∫ 𝐼𝑧(𝑧′)[

𝜕2

𝜕𝑧2𝐺(𝑧, 𝑧′) + 𝑘2𝐺(𝑧, 𝑧′)]𝑑𝑧′

+𝑙2⁄

−𝑙2⁄

= 𝐸𝑧𝑖 4.43

In order to apply Method of Moments to Eq. (4.43) to find a solution for current

distribution on a cylindrical center fed dipole, a suitable basis function used to expand the

unknown current should be chosen. However, since the current distribution on a half

wavelength dipole antenna is known to be sinusoidal, the piecewise sinusoidal basis function

can be used since it reduces the complexity of the computation. The piecewise sinusoidal basis

function is given in Eq. (3.21) that is:

𝐼𝑧(𝑧′) =∑𝐼𝑛

𝑛

{

𝑠𝑖𝑛𝑘(𝑧′ − 𝑧𝑛−1)

𝑠𝑖𝑛𝑘(𝑧𝑛 − 𝑧𝑛−1), 𝑧𝑛−1 ≤ 𝑧

′ < 𝑧𝑛

𝑠𝑖𝑛𝑘(𝑧𝑛+1 − 𝑧′)

𝑠𝑖𝑛𝑘(𝑧𝑛+1 − 𝑧′), 𝑧𝑛 ≤ 𝑧

′ < 𝑧𝑛+1

0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒 }

4.44

The delta gap source model is used to approximate the incident electric field, 𝐸𝑧𝑖

By applying integration by parts to the first term of Eq. (4.43) , the following substitutions

can be made.

𝜕

𝜕𝑙𝐺(𝑅) = −

𝜕

𝜕𝑙′𝐺(𝑅) and

𝜕2

𝜕𝑙2𝐺(𝑅) =

𝜕2

𝜕𝑙′2𝐺(𝑅) with 𝑅 = √(𝑙 − 𝑙′)2 + 𝑎2

On applying the above substitutions, a simpler equation is now obtained in the form

𝑗

𝜔휀𝑜∫ [

𝜕2

𝜕𝑧′2𝐼(𝑧′) + 𝑘2𝐼(𝑧′)] 𝐺(𝑅)𝑑𝑧′

𝐿 2⁄

−𝐿 2⁄

= 𝐸𝑧𝑖 , 𝑅 = √(𝑧 − 𝑧′)2 + 𝑎2 4.45

By applying the piecewise sinusoidal basis function, the pulse weighting function and the

delta gap source model:

- 33 -

∑𝐼𝑛 ∫ ∫𝑘

sin𝑘∆[𝛿(𝑧′ − 𝑧𝑛+1) + 𝛿(𝑧

′ − 𝑧𝑛−1)

𝑧𝑛+∆+

𝑧𝑛−∆+

𝑧𝑚+∆2

𝑧𝑚−∆2

𝑛

− 2𝛿(𝑧′ − 𝑧𝑛) cos𝑘∆]𝐺(𝑅)𝑑𝑧′𝑑𝑧 = ∫

𝑉𝑚𝑠∆𝛿(𝑧 − 𝑧𝑚𝑠)𝑑𝑧

𝑧𝑚+∆2

𝑧𝑚−∆2

4.46

Where: ∆ = length of one segment

𝑧𝑛 = location of the nth observation point

𝑧𝑚𝑠 = location of mth source point

𝑉𝑚𝑠 = voltage of mth source

∆+ = end points to be considered in the interval

By integrating the delta function on the left-hand side of Eq. (4.46), a final equation is

obtained [8]:

∑𝐼𝑛𝑗

𝜔휀𝑜

𝑘

sin𝑘∆∫ 𝐺(𝑅𝑛+1) + 𝐺(𝑅𝑛−1) − 2𝐺(𝑅𝑛) cos 𝑘∆𝑑𝑧

𝑧𝑚−∆2

𝑧𝑚−∆2

𝑛

= ∫𝑉𝑚𝑠∆𝛿(𝑧 − 𝑧𝑚𝑠)𝑑𝑧

𝑧𝑚+∆2

𝑧𝑚−∆2

4.47

Where

𝑅𝑛 = √(𝑧 − 𝑧𝑛)2 + 𝑎2

4.2.1.2 Hallen’s Integral Equation

Referring to Fig. 4.4, if the length of the cylinder is assumed to be much larger than radius

(𝑎 ≪ 𝑙) and also that the wavelength is much larger than the radius (𝑎 ≪ 𝜆) , the effects of the

end faces of the cylinder can be neglected. This implies that the boundary conditions for a wire

with infinite conductivity are those of the vanishing total tangential electric fields on the wire

surface and the vanishing currents at the cylinder’s end.

The equation obtained will be:

- 34 -

𝐸𝑧𝑡 = −𝑗𝜔𝐴𝑧 − 𝑗

1

𝜔𝜇휀

𝜕2𝐴𝑧𝜕𝑧2

= −𝑗1

𝜔𝜇휀[𝜕2𝐴𝑧𝜕𝑧2

+ 𝜔2𝜇휀𝐴𝑧] 4.48

Since total tangential electric field is vanishing on the cylinder’s surface:

𝜕2𝐴𝑧𝜕𝑧2

+ 𝜔2𝜇휀𝐴𝑧 = 0 4.49

Because of symmetry of the current density on the cylinder, 𝐽𝑧(𝑧′) = 𝐽𝑧(−𝑧

′) , the

potential 𝐴𝑧 will also be symmetrical , 𝐴𝑧(𝑧′) = 𝐴𝑧(−𝑧

′). The solution of Eq. (4.49) can be

written as:

𝐴𝑧(𝑧) = −𝑗√𝜇휀[𝐵1 cos(𝑘𝑧) + 𝐶1 sin(𝑘|𝑧|)] 4.50

Where 𝐵1 and 𝐶1 are constants. For a current carrying wire, the potential can also be given

by:

∫ 𝐼𝑧(𝑧′)𝑒−𝑗𝑘𝑅

4𝜋𝑅𝑑𝑧′

+𝑙 2⁄

−𝑙 2⁄

= −𝑗√휀

𝜇[𝐵1 cos(𝑘𝑧) + 𝐶1 sin(𝑘|𝑧|)] 4.51

Eq. (4.51) is referred to as hallen’s integral for a perfectly conducting wire. By applying a

voltage at the input terminals of the wire, the constant 𝐶1 is found to be 𝐶1 = 𝑉𝑖

2. The constant

𝐵1 can be obtained from the boundary conditions that require current to vanish at the wire’s

endpoints.

Let 𝜂 = √𝜇

𝜀 so that Eq. (4.51) becomes:

∫ 𝐼𝑧(𝑧′)𝑒−𝑗𝑘𝑅

4𝜋𝑅𝑑𝑧′

+𝑙 2⁄

−𝑙 2⁄

= −𝑗

𝜂[𝐵1 cos(𝑘𝑧) + 𝐶1 sin(𝑘|𝑧|)] 4.52

If the wire is of length, l and divided into N uniform segments, the length of each segment,

Δ = 𝑙

𝑁 . By applying a pulse basis function and point matching shown in Fig. 4.6, with

observation points taken to be at the midpoint of the subinterval and source points taken at the

leftmost end of the subinterval [9]

- 35 -

𝑍𝑛 = (𝑛−1)𝑙

𝑁 and 𝑍𝑚 =

(𝑚−1)𝑙

2𝑁 , m = 1,2,….., N + 1 4.53

Where 𝑍𝑛 denotes the source points on the subinterval and 𝑍𝑚 denotes the observation

point of the subinterval

Figure 4.6 Wire antenna with N segments (source: Figure 2.6 in [10]).

By applying the Moments method solution, the current distribution 𝐼(𝑧) can be approximated

as:

𝐼(𝑧) ≈ ∑ 𝐼𝑛

𝑁

𝑛=1

𝑃𝑛(𝑧) 4.54

𝑃𝑛(𝑧) is the basis function which can be defined by:

𝑃𝑛(𝑧) = {1 , 𝑧𝑛 ∈ ∆𝑧 0 , 𝑧𝑛 ∈ ∆𝑧

4.55

- 36 -

Substituting (4.54) back into hallen’s integral Eq. (3.46):

∑𝐼𝑛∫ 𝐺(𝑧𝑚, 𝑧′)𝑃𝑛(𝑧

′)𝑑𝑧′+𝑙 2⁄

−𝑙 2⁄

= −𝑗

𝜇𝐵 cos(𝑘𝑧𝑚) −

𝑗

2𝜇𝐵 sin(𝑘|𝑧𝑚|)

𝑁

𝑛=1

4.56

A simpler expression can be obtained by dividing the wire in N+1 uniform segments to

obtain:

∑𝐼𝑛∫ 𝐺(𝑧𝑚, 𝑧′)𝑑𝑧′ = −

𝑗


𝑗

2𝜇sin(𝑘|𝑧𝑚|)

𝑧𝑛+1

𝑧𝑛

,

𝑁

𝑛=1

𝑚 = 1,2,… ,𝑁 + 1

4.57

In matrix form, Eq. (4.57) can be written as:

[𝑍𝑚𝑛][𝐼𝑛] = [𝑉𝑚] 4.58

Where 𝑍𝑚𝑛 = ∫ 𝐺(𝑧𝑚, 𝑧′)𝑑𝑧′

𝑧𝑛+1𝑧𝑛

𝑉𝑚 = −𝑗


𝑗

2𝜇sin(𝑘|𝑧𝑚|)

Since the value of B is not known, boundary conditions are applied at the end of the wire.

This results into a matrix of the form:

[

[

𝐴11 𝐴12𝐴21 𝐴22

⋯𝐴1,𝑁−1 𝑑1𝑁𝐴2,𝑁−1 𝑑2𝑁

⋮ ⋱ ⋮𝐴𝑁1 𝐴𝑁2 ⋯ 𝐴𝑁,𝑁−1 𝑑𝑁,𝑁

]

]

[ 𝐼1𝐼2...

𝐼𝑁−1𝐵 ]

=

[ 𝑏1𝑏2...

𝑏𝑁−1𝑏𝑁 ]

4.59

Where 𝑑𝑚 = 𝑗

𝜂cos(𝑘𝑧𝑚) , 𝑏𝑚 =

𝑗

2𝜂sin(𝑘|𝑧𝑚|)

By solving the matrix in Eq. (4.59), current distribution on a thin wire dipole antenna can

be obtained [10]:

- 37 -

4.2.2 Simplification of Pocklington’s Equation for Arbitrary shaped antennas

By now, it should be obvious, that by knowing the current distribution on any conductor, we

can be able to understand its electromagnetic behaviour. By simplifying Pocklington’s integro-

differential equation into an integral one, we are not only able to save time but also obtain

solutions for antennas of other shapes. A simplified solution is also obtained.

4.2.2.1 The Electric Field Integral Equation

The relationship between the electric field and current distribution on the conductor is

described by the electric field integral equation. Applying the concept of the magnitude

potential vector and electric potential scalar to Maxwell’s equations, this equation can be

derived and obtained as:

𝐸𝑡(𝑟) = −𝑗𝜂

𝑘∫ 𝐽(𝑟′)[𝑘2 + ∇2]

𝑒−𝑗𝑘|𝑟−𝑟′|

4𝜋|𝑟 − 𝑟′|𝑑𝑣′

𝑣′ 4.60

Where: 𝐸𝑡(𝑟) is the total electric field vector measured at the observation point

𝐽(𝑟′) is the current distribution located at the source point 𝑟′

[𝑘2 + ∇2] is the Helmholtz’s operator

𝑒−𝑗𝑘|𝑟−𝑟

′|

4𝜋|𝑟−𝑟′| is the Green’s function for free space

k is the wave number which is given by; 𝑘 = 𝜔√𝜇𝑜휀𝑜

𝜂 is the intrinsic impedance for free space given by; 𝜂 = √𝜇𝑜 휀𝑜⁄

4.2.2.2 Pocklington’s equation for arbitrary shaped thin wires

By applying boundary condition over the arbitrary shaped thin wire, Pocklington’s

equation can be derived from Eq. (4.60). The boundary condition states that the total electric

field vanishes over the surface of any perfectly conducting wire. Since antennas are usually

made from almost perfect conductors such as copper, this conditions provides reliable

accuracy. If the surface of the wire is represented by 𝑟 = 𝑟𝑠;

𝐸𝑡(𝑟𝑠) = 𝐸𝑖(𝑟𝑠) + 𝐸

𝑠(𝑟𝑠) = 0 4.61

Due to skin effect, current is located over the wire’s surface for high frequencies. As a

result, the more the frequency is increased, the better the current will be confined in an

infinitesimal layer that cover the wire. The skin depth is given by:

- 38 -

𝛿 = √2 𝜔𝜇𝜎⁄ 4.62

Where 𝜎 is the electrical conductivity of the metal from which the antenna was made.

Combining Eq. (4.60) and Eq. (4.61), and assuming that current is located only on the surface

of the wire, the impressed electric field on the wire’s surface shown in Fig. 4.7 can be obtained

as:

𝐸𝑠(𝑟 = 𝑟𝑠) = −𝑗𝜂

𝑘∫ ∫ 𝐽𝑠(𝑟

′ = 𝑟𝑠)[𝑘2 + ∇2]

𝑒−𝑗𝑘|𝑟−𝑟′|

4𝜋|𝑟 − 𝑟′|𝑎𝑑𝜑′𝑑𝑠′

𝜑𝑠

4.63

Where 𝜑′ is the azimuthal angle across the section of the wire. Using thin-wire

approximation, if the wire has a radius, a, then the current distribution can be expressed as:

2𝜋𝑎𝐽𝑠 = 𝐼𝑠(𝑠′)

𝐽𝑠 =𝐼𝑠(𝑠′)

2𝜋𝑎

4.64

Figure 4.7 Relations among the vectors of an arbitrary shaped wire (source: Fig.1 in

[11]).

Due to azimuthal independence, the current can be assumed to be confined in a very thin

current filament 𝐼𝑠(𝑠′) over the surface of the wire.

If the observation points are placed along the axis of the wire, the axis of the wire is

represented by the following vectorial equation:

- 39 -

𝑟(𝑠) = 𝑥(𝑠)𝑖 + 𝑦(𝑠)𝑗 + 𝑧(𝑠)𝑘 4.65

The current filament that lies along the wire surface is represented by the vectorial

equation:

𝑟′(𝑠′) = 𝑟(𝑠′) + 𝑎𝑛(𝑠′) 4.66

Where 𝑛(𝑠′) is the unit vector normal for the wire’s axis. This implies that the curve which

represents the current filament is a parallel curve to that which represents the wire’s axis.

Pocklington’s equation can now be obtained by replacing Eq. (4.64) into Eq. (4.63):

𝐸𝑠𝐼 = −

𝑗

𝜔휀∫ 𝐼𝑠(𝑠

′)[𝑘2𝑠. 𝑠′ +𝜕2

𝜕𝑠𝜕𝑠′]𝑒−𝑗𝑘|𝑟−𝑟

′|

4𝜋|𝑟 − 𝑟′|𝑑𝑠′

𝑠′

4.67

Where: 𝐸𝑠𝐼 is the tangential impressed electric field.

Equation. (4.67) is the general Pocklington’s equation applicable to wires of any possible

geometry. The geometry of wire is expressed by the dot product 𝑠. 𝑠′,

Where: s(s) is the unit tangential vector for the wire’s axis

s’(s’) is the unit tangential vector for the parallel curve representing the

current filament as shown in Fig. 4.8. These vectors can be calculated from:

𝑠(𝑠) =𝑑𝑥(𝑠)

𝑑𝑠𝑖 +

𝑑𝑦(𝑠)

𝑑𝑠𝑗 +

𝑑𝑧(𝑠)

𝑑𝑠𝑘

𝑠′(𝑠′) =𝑑𝑥′(𝑠′)

𝑑𝑠′𝑖 +

𝑑𝑦′(𝑠′)

𝑑𝑠′+𝑑𝑧′(𝑠′)

𝑑𝑠′

4.68

- 40 -

Figure 4.8 Unit tangential vectors for the thin wire (source: Fig. 2 in [11]).

The geometry is also expressed as the magnitude of the difference between the vectors |r-r’|

and is therefore given by:

𝑅 = |𝑅| = |𝑟 − 𝑟′| = √[𝑥(𝑠) − 𝑥′(𝑠′)]2 + [𝑦(𝑠) − 𝑦′(𝑠′)]2 + [𝑧(𝑠) − 𝑧′(𝑠′)]2 4.69

After obtaining Eq. (4.69), the remaining task involves only finding the vectors

representing the parallel curve and the axis curve for the wire under consideration.

4.2.2.3 The MM solution to Pocklington’s equation for arbitrary shaped wires

By now, we know that the MM is simply one of many numerical techniques used to

transform an operational equation into a matrix equation. The objective is to get the current

distribution Is(s’) of the wire.

By expressing the unknown function in terms of a linear combination of linearly

independent functions 𝑖𝑛(𝑠′), the current distribution can be obtained as:

𝐼𝑠(𝑠′) = ∑ 𝑐𝑛𝑖𝑛(𝑠′)

𝑁

𝑛=1

4.70

Where: 𝑐𝑛 are the unknown coefficients that must be determined

𝑁 is the number of basis functions.

Substituting Eq. (4.70) into Eq. (4.67), an equation with unknowns is obtained as:

- 41 -

𝐸𝑠𝐼 = −

𝑗

𝜔휀∑ 𝑐𝑛 ∫ 𝑖𝑛(𝑠

′) [𝑘2𝑠. 𝑠′ +𝜕2


′|

4𝜋|𝑟 − 𝑟′|𝑑𝑠′

𝑠′

𝑁

𝑛=1

4.71

To obtain a more consistent equation system, it is necessary to find N linear independent

equations, which can be obtained by taking the inner product of Eq. (4.71) with another set of

linearly independent functions 𝑤𝑚(𝑠) referred to as weighting functions:

⟨𝑤𝑚, 𝐸𝑠𝐼⟩ = −

𝑗

𝜔휀∑ 𝑐𝑛 ⟨𝑤𝑚, ∫ 𝑖𝑛(𝑠

′) [𝑘2𝑠. 𝑠′ +𝜕2


′|

4𝜋|𝑟 − 𝑟′|𝑑𝑠′

𝑠′

⟩

𝑁

𝑛=1

𝑚

= 1,2,… . , 𝑁

4.72

Eq. (4.72) can be rewritten in the form:

∫ 𝑤𝑚𝐸𝑠𝐼𝑑𝑠 = −

𝑗

𝜔휀∑ 𝑐𝑛∫ 𝑤𝑚 ∫ 𝑖𝑛(𝑠

′)

𝑠′𝑠

𝑁

𝑛=1𝑠

[𝑘2𝑠. 𝑠′ +𝜕2


′|

4𝜋|𝑟 − 𝑟′|𝑑𝑠′𝑑𝑠 𝑚

= 1,2, . . , 𝑁

4.73

Eq. (4.73) may now be written in matrix form as:

[

𝑧11 𝑧12 ⋯ 𝑧1𝑁𝑧21 𝑧22 … 𝑧2𝑁⋮𝑧𝑁1

⋮𝑧𝑁2

⋮⋯

⋮𝑧𝑁𝑁

] [

𝑐1𝑐2⋮𝑐𝑁

] = [

𝑣1𝑣2⋮𝑣𝑁

]

[𝑧𝑚𝑛][𝑐𝑛] = [𝑣𝑚]

4.74

Where the 𝑧𝑚𝑛 elements represent the impedance matrix and can be easily calculated from:

𝑧𝑚𝑛 = −𝑗

𝜔휀∫ 𝑤𝑚 ∫ 𝑖𝑛(𝑠

′) [𝑘2𝑠. 𝑠′ +𝜕2


′|

4𝜋|𝑟 − 𝑟′|𝑑𝑠′𝑑𝑠

𝑠′𝑠

4.75

𝑣𝑚 elements represent the voltage matrix and can be obtained from:

𝑣𝑚 = ∫ 𝑤𝑚𝐸𝑠𝐼𝑑𝑠

𝑠

4.76

- 42 -

𝑐𝑛 represent the current matrix.

The current matrix can then be obtained through various matrix inversion techniques as:

𝑐𝑛 = [𝑧𝑚𝑛]−1(𝑣𝑚) 4.77

4.2.2.4 Point Matching Simplification

Elements of the matrix in Eq. (4.74) could be quite difficult to obtain since it involves

double integration over the Helmholtz’s operator and green’s function. This is because the

calculation time and programming requirements increases as the number of segments is

increased. By applying point matching technique, Eq. (4.75) can be simplified. Point matching

uses the Dirac’s delta function as a weighting function. This employs the following properties:

∫ 𝛿(𝑠 − 𝑠𝑚)𝑑𝑠 = {1 𝑖𝑓 𝑠𝑚 ∈ ∆𝑠0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒

∆𝑠

∫ 𝑓(𝑠)𝛿(𝑠 − 𝑠𝑚)𝑑𝑠 = {𝑓(𝑠𝑚) 𝑖𝑓 𝑠𝑚 ∈ ∆𝑠0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒

∆𝑠

4.78

Replacing Eq. (4.78) into Eq. (4.75) and Eq. (4.76):


𝜔휀∫ 𝑖𝑛(𝑠

′) [𝑘2𝑠. 𝑠′ +𝜕2


′|

4𝜋|𝑟 − 𝑟′|𝑑𝑠′|𝑠 = 𝑠𝑚

𝑠′

𝑣𝑚 = 𝐸𝑠𝐼(𝑠 = 𝑠𝑚)

4.79

Applying the Dirac’s delta implies that boundary electromagnetic conditions have been

applied only at discrete points on the wire’s structure, exactly where the Dirac’s functions

have their roots 𝑠𝑚. It is therefore important to choose the location of the roots. It is usually

convinient to choose the roots to be at the axis of the wire.

Eq. (4.79) can be further simplified by choosing a pulse basis function which is defined by:

𝑖𝑛(𝑠′) = {

1 𝑖𝑓 (𝑛 − 1)∆𝑠′ ≤ 𝑠′ < 𝑛∆𝑠′

0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒 4.80

The pulse function divides the wire’s structure into N segments of length ∆𝑠′, producing a

stair representation of the structure current. Usually the whole structure is equally divided such

that each segment is ∆𝑠′ = 𝐿 𝑁⁄ , however, the length of each segment can be freely chosen as

per the person’s demands. Replacing Eq. (4.80) into Eq. (4.79):

- 43 -


𝜔휀∫ [𝑘2𝑠. 𝑠′ +

𝜕2


′|

4𝜋|𝑟 − 𝑟′|𝑑𝑠′

𝑛∆𝑠′

(𝑛−1)∆𝑠′

| 𝑠 = 𝑠𝑚 4.81

4.2.2.5 Kernel simplification

It can be noted that in Eq. (4.81), that the integral operator deals with an iterated

differential operator. It is possible to develop a derivative operation to prevent the software

making such operations numerical each time for the impedance element. This is possible by

using Eq. (4.69) to find the iterated derivative for Green’s function:

𝜕2

𝜕𝑠𝜕𝑠′𝑒−𝑗𝑘𝑅

𝑅= −

𝜕

𝜕𝑠′[(1 + 𝑗𝑘𝑅)𝑒−𝑗𝑘𝑅

𝜕𝑅𝜕𝑠

𝑅2]

= 𝑒−𝑗𝑘𝑅(𝜕𝑅𝜕𝑠) (𝜕𝑅𝜕𝑠′

) [2 − 𝑘2𝑅2 + 2𝑗𝑘𝑅] − 𝑅(1 + 𝑗𝑘𝑅)𝜕2𝑅𝜕𝑠𝜕𝑠′

𝑅3

4.82

The value of the multiple derivatives of R that appears in Eq. (4.82) can be found as:

𝜕𝑅

𝜕𝑠′=1

2𝑅

𝜕

𝜕𝑠′{[𝑥(𝑠) − 𝑥′(𝑠)′]2 + [𝑦(𝑠) − 𝑦′(𝑠)′]2 + [𝑧(𝑠) − 𝑧′(𝑠)′]2} = −

1

𝑅𝑅 ∙ 𝑠′

𝜕𝑅

𝜕𝑠=1

2𝑅

𝜕

𝜕𝑠′{[𝑥(𝑠) − 𝑥′(𝑠)′]2 + [𝑦(𝑠) − 𝑦′(𝑠)′]2 + [𝑧(𝑠) − 𝑧′(𝑠)′]2} =

1

𝑅𝑅 ∙ 𝑠′

𝜕2𝑅

𝜕𝑠𝜕𝑠′= −

1

𝑅{𝜕𝑥′(𝑠′)

𝜕𝑠′𝜕𝑥(𝑠)

𝜕𝑠+𝜕𝑦′(𝑠′)

𝜕𝑠′𝜕𝑦(𝑠)

𝜕𝑠+𝜕𝑧′(𝑠′)

𝜕𝑠′𝜕𝑦(𝑠)

𝜕𝑠} + 𝑅 ∙ 𝑠′

1

𝑅2𝜕𝑅

𝜕𝑠

= −1

𝑅𝑠 ∙ 𝑠′ +

1

𝑅3(𝑅 ∙ 𝑠)(𝑅 ∙ 𝑠′)

4.83

Substituting Eq. (4.83) back into Eq. (4.82):

𝜕2

𝜕𝑠𝜕𝑠′𝑒−𝑗𝑘𝑅

𝑅= 𝑒−𝑗𝑘𝑅

𝑅2(1 + 𝑗𝑘𝑅)𝑠 ∙ 𝑠′ − (𝑅 ∙ 𝑠′)(𝑅 ∙ 𝑠)(3 − 𝑘2𝑅2 + 3𝑗𝑘𝑅)

𝑅5 4.84

Replacing Eq. (4.84) into Eq. 4.81), a simplified kernel of Pocklington’s equation is

obtained:

[𝑘2𝑠 ∙ 𝑠′ +𝜕2

𝜕𝑠𝜕𝑠′]𝑒−𝑗𝑘𝑅

4𝜋𝑅= 4.85

- 44 -

[𝑅2(1 + 𝑗𝑘𝑅 − 𝑘2𝑅2)𝑠 ∙ 𝑠′ − (3 + 3𝑗𝑘𝑅 − 𝑘2𝑅2)(𝑅 ∙ 𝑠)(𝑅 ∙ 𝑠′)]𝑒−𝑗𝑘𝑅

4𝜋𝑅5

The elements of the impedance matrix can now be expressed for:


4𝜋𝜔𝜀∫ 𝑅2(1 + 𝑗𝑘𝑅 − 𝑘2𝑅2)𝑠 ∙ 𝑠′

𝑒−𝑗𝑘𝑅

𝑅5𝑑𝑠′

𝑛∆𝑠′

(𝑛−1)∆𝑠′|𝑠=𝑠𝑚

+

𝑗

4𝜋𝜔𝜀∫ (3 + 3𝑗𝑘𝑅 − 𝑘2𝑅2)(𝑅 ∙ 𝑠)(𝑅 ∙ 𝑠′)

𝑒−𝑗𝑘𝑅

𝑅5𝑑𝑠′

𝑛∆𝑠′

(𝑛−1)∆𝑠′|𝑠=𝑠𝑚

4.86

4.2.2.6 Source modelling

The most used source used in modelling is the delta gap generator. Mathematically, this is

described as:

𝐸𝑆𝐼(𝑠) = {

𝑉 ∆𝑠⁄ , 𝑖𝑓 𝑠 ∈ ∆𝑠𝑚0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒

4.87

Where V is the value of the fasor of the source connected to the mth segment of the

structure. In practice, the delta gap generator is a very idealized generator that could not be

obtained, however, its use produces results with good accuracy while simplifying the solution.

Since the antenna is using only one source connected to the mth segment, the voltage matrix

can be defined as:

(𝑣) =

[ 𝑣1𝑣2⋮𝑣𝑚⋮𝑣𝑁]

=

[ 00⋮

𝑣 ∆𝑠⁄⋮0 ]

4.88

Usually, the value of v is taken as 1, however, other values can be used.

By applying Kirchoff’s voltage law. If the impedance Z is connected to the mth segment, the

potential difference in its ends can be calculated from:

𝑉𝑚 = 𝑣𝑚 − 𝑐𝑚𝑍 4.89

Where 𝑐𝑚 is the current on the segment. Substitution Eq. (4.89) in Eq. (4.74):

- 45 -

[ 𝑧11 𝑧12 … 𝑧1𝑚 … 𝑧1𝑁𝑧21 𝑧22 … 𝑧2𝑚 … 𝑧2𝑁⋮𝑧𝑚1⋮𝑧𝑁1

⋮𝑧𝑚2⋮𝑧𝑁2

⋱…⋱…

⋮𝑧𝑚𝑚⋮

𝑧𝑁𝑚

⋱…⋱…

⋮𝑧𝑚𝑁⋮𝑧𝑁𝑁 ]

[ 𝑐1𝑐2⋮𝑐𝑚⋮𝑐𝑁 ]

=

[

𝑣1𝑣2⋮

𝑣𝑚 − 𝑐𝑚𝑧⋮𝑣𝑁 ]

4.90

By using the definition of the matrix product, the mth segment of the voltage matrix can be

expressed as:

𝑐1𝑧𝑚1 +⋯+ 𝑐𝑚𝑧𝑚𝑚 +⋯+ 𝑐𝑁𝑧𝑁1 = 𝑣𝑚 − 𝑐𝑚𝑧

𝑐1𝑧𝑚1 +⋯+ 𝑐𝑚(𝑧𝑚𝑚 + 𝑧) +⋯+ 𝑐𝑁𝑧𝑁1 = 𝑣𝑚 4.91

hence the impedance matrix can be expanded as:

[ 𝑧11 𝑧12 … 𝑧1𝑚 … 𝑧1𝑁𝑧21 𝑧22 … 𝑧2𝑚 … 𝑧2𝑁⋮𝑧𝑚1⋮𝑧𝑁1

⋮𝑧𝑚2⋮𝑧𝑁2

⋱…⋱…

⋮𝑧𝑚𝑚 + 𝑧

⋮𝑧𝑁𝑚

⋱…⋱…

⋮𝑧𝑚𝑁⋮𝑧𝑁𝑁 ]

[ 𝑐1𝑐2⋮𝑐𝑚⋮𝑐𝑁 ]

=

[ 𝑣1𝑣2⋮𝑣𝑚⋮𝑣𝑁 ]

4.92

It can therefore be concluded that connecting an impedance to the mth segment is the same

as adding its value to the corresponding zmm matrix element [11].

- 46 -

5 Results and Discussions

In this chapter, worked examples have been developed for the equations obtained in chapter 4.

All the programs for the solutions have been solved using MATLAB software and can be

found in the appendix section.

5.1 Charge distribution on a straight wire

To illustrate the solution for Eq. (4.14) and Eq. (4.16), consider a wire of length, 1m and

radius, a=0.001m having a constant potential of 1V along its length. By applying Eq. (4.14)

and dividing the wire into nm=5 uniform segments, the impedance matrix is obtained as [1]:

[ 10.5967 1.0986 0.5108 0.3365 0.25131.0986 10.5967 1.0986 0.5108 0.33650.5108 1.0986 10.5967 1.0986 0.5108 0.3365 0.5108 1.0986 10.5967 1.09860.2513 0.3365 0.5108 1.0986 10.5967]

[ 𝑎1𝑎2𝑎3𝑎4𝑎5]

=

[ 4𝜋휀𝑜4𝜋휀𝑜4𝜋휀𝑜4𝜋휀𝑜4𝜋휀𝑜]

By solving the above matrix, the amplitude coefficients of the charge distribution are

obtained as:

a1 = 8.790pC/m

a2 = 8.071pC/m

a3 = 7.954pC/m

a4 = 8.071pC/m

a5 = 8.790pC/m

However, the use of such few segments produces a rigid graph shown in Fig. 5.1. In order

to obtain much better results, more segments can be used. Employing 20 segments gives more

accurate readings and a smoother graph as shown by Fig. 5.2

- 47 -

Figure 5.1 Charge distribution on a straight dipole; Length=1m, nm=5.

Figure 5.2 Charge distribution on a straight dipole; Length=1m, nm=20.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 17.9

8

8.1

8.2

8.3

8.4

8.5

8.6

8.7

8.8x 10

-12

length(m)

charg

e d

ensity (

C/m

)

Charge distribution on a straight wire L=1m, v=1 volt

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 17.5

8

8.5

9

9.5

10

10.5x 10

-12

length(m)

charg

e d

ensity (

C/m

)


- 48 -

5.2 Charge distribution on a bent wire antenna

In order to illustrate the results of Eq. (4.26) and Eq. (4.27), consider a 1m long wire, with a

90o bend at its midpoint, subdivided into 20 equally spaced segments. If the wire has a radius,

a=0.001m and a potential of 1V is applied across its entire length, the charge distribution

across the length of the wire obtained is shown in Fig. 5.3.

It can be observed from Fig. 5.3 that the charge is relatively more concentrated near the

ends of the wire than was the case of the straight wire in Fig. 5.2. It can also be observed that

the overall density, and therefore capacitance, on the structure has been decreased [1].

Figure 5.3 Charge distribution on a bent wire.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 17.5

8

8.5

9

9.5

10

10.5x 10

-12

length(m)

charg

e d

ensity (

C/m

)

Charge distribution on a bent wire L=1m, v=1 volt

- 49 -

5.3 Charge distribution on the Gray Hoverman antenna

Figure 5.4 Original gray hoverman antenna structure (source: Fig. 1 in [4]).

Having analysed the bent wire, the same approach can be applied to the gray hoverman

antenna. A picture of the original hoverman designed in 1958 by D. R. Hoverman is shown in

Fig. 5.4. On close observation, one can tell that the structure of the gray hoverman is similar to

that of the bent wire, however the gray hoverman is a bit more complex as compared to the

bent wire antena since each subsection is made up of 8 segments instead of only 2 on either

side. However, the calculations follow similar steps to the bent wire structure.

To obtain the charge distribution on a gray hoverman antenna subsection, consider a wire,

1m in length with a radius of a=0.005m, divided into 8 parts of equal length. Suppose each

part is divided into nm=10 uniform segments of equal length. By solving the impedance

matrix in Eq. (4.26) and Eq. (4.27), and applying inversion techniques to obtain the charge

distribution, the charge distribution on the hoverman antenna subsection is obtained as shown

in Fig. 5.5.

- 50 -

Figure 5.5 Current distribution on the original Gray Hoverman antenna subsection.

From Fig. 5.5, it can be observed that the charge density is highest at the ends of the

structure and significantly more concentrated than for the simple bent wire structure.

The hoverman antenna design can be converted into a dipole antenna by simply altering the

angles into a straight wire structure.

In order to illustrate this, consider the charge distribution on the thin wire dipole of Fig.

5.2. Suppose the number of subsections is further increased to 80, the charge distribution

obtained will be almost perfectly smooth. This is shown in Fig. 5.6. By applying the necessary

angles required to straighten the Gray Hoverman antenna design, the charge distribution

obtained is represented in Fig. 5.7 which is very similar to that of the thin dipole antenna.

However, it can be observed that the hoverman has pulses that are evenly spaced. This is

simply a representation that the hoverman design has got eight partts with seven bends in

between. In fact by analysing Fig. 5.7, it is possible to tell how many bends the antenna is

made of. The even spacing between the bends is an indication that the antenna segments are of

equal length.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 16

7

8

9

10

11

12

13x 10

-12

length(m)

charg

e d

ensity (

C/m

)

Charge distribution on a gray hoverman L=1m, v=1 volt

- 51 -

Figure 5.6 Charge distribution on a thin straight dipole; L=1m, nm=80.

Figure 5.7 Charge distribution on a Gray Hoverman antenna; L=1m, nm=80.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.7

0.8

0.9

1

1.1

1.2

1.3x 10

-11

length(m)

charg

e d

ensity (

C/m

)


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.7

0.8

0.9

1

1.1

1.2

1.3x 10

-11

length(m)

charg

e d

ensity (

C/m

)

Charge distribution on a gray hoverman L=1m, v=1 volt

- 52 -

5.4 Pocklington’s equation solution for current distribution on a dipole

To illustrate the solution for Eq. (4.47), consider a dipole antenna with a length of 0.47𝜆,

radius of 0.005𝜆, divided into 6 segments of equal length with a source having a voltage of 1V.

Since we are applying the delta-gap source model, it is always more convinient to choose an

odd number of unknowns (or even number of segments). The number of unknowns will

always be nm-1, where nm is the number of segments. In this particular case we have 5

unknowns. However, if an even number of unknowns is chosen, to obtain the desired output,

half of the desired source voltage can be placed across each of the two segments located at the

center.

On applying the given parameters to Eq. (4.47), the impedance matrix is obtained as:

𝑍𝑚𝑛 =

[ 488.49∠ − 89.42°256.56∠88.93°32.29∠82.08°9.19∠64.72°4.47∠43.18°

256.56∠88.93°488.49∠ − 89.42°256.56∠88.93°32.29∠82.08°9.19∠64.72°

32.29∠82.08°256.56∠88.93°488.49∠ − 89.42°256.56∠88.93°32.29∠82.08°

9.19∠64.72°32.29∠82.08°256.56∠88.93°488.49∠ − 89.42°256.56∠88.93°

4.47∠43.18°9.19∠64.72°32.29∠82.08°256.56∠88.93°488.49∠ − 89.42°]

And the voltage matrix obtained is given by:

𝑉 =

[ 00

1∠0°

00 ]

By applying matrix inversion, the current vector is obtained as

𝐼 = 10−3

[ 8.1∠ − 1.0°13∠1.9°14.8∠7.5°13∠1.9°

8.1∠ − 1.0°]

The above impedance matrix is observed to be symmetrical. This is due to the fact that the

dipole is straight. The voltage and current vectors are also noted to be symmetrical about the

center segment. This is because the dipole is cenre-fed [8].

- 53 -

Figure 5.8 Current distribution on a thin dipole; L= 0.47𝝀, nm=5.

The input impedance, which can also be referred to as the self-impedance can be obtained

from Fig. 5.8 .

𝑍 =1

𝐼𝑛𝑚𝑎𝑥 5.1

Where: Z = self-impedance of the dipole

𝐼𝑛𝑚𝑎𝑥= peak value of current density.

From the graph of Fig. 5.8, the self-impedance obtained is 66.92-j8.8 ohms. However, for a

dipole antenna that is perfectly resonant, the self-impedance should be approximately 72 + 𝑗0

which is varying from the value obtained from the graph. This is as a result of very few

unknowns used which in this case was 5. By increasing the number of unknowns to 13, an

input impedance of 72 - 𝑗0.65 is obtained which is much closer to the expected value. As

smoother graph is also obtained shown in Fig. 5.9. The thin wire approximation has one

limitation however. As the length of the segment approaches the radius of the wire, the input

impedance starts to break down. In case the number of unknowns was raised to 101, an input

impedance of 80+𝑗0.74 is obtained which is above the expected value. This breakdown is as a

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.008

0.009

0.01

0.011

0.012

0.013

0.014

0.015

length (m)

curr

ent

density (

A/m

)

Current distribution on a dipole antenna nm=5 f=300 MHz

- 54 -

result of numeric and modelling constraints on the solution algorithms. The choice of the

expansion function could also be another factor.

Figure 5.9 Current on a thin dipole with more segments. L= 0.47𝝀, nm=13.

5.5 Hallen’s integral solution for current distribution on a dipole

To illustrate the solution for the matrix in Eq. (4.59), consider a half-wavelength dipole

operating at a frequency of 1.5x108 Hz, divided into nm=40 uniform segments with a radius,

a=𝑙𝑒𝑛𝑔𝑡ℎ

518. A current distribution shown in Fig. 5.10 is obtained [10].

Note that the green curve denotes the real part of the current distribution whereas the red

curve demonstrates the imaginary part. The absolute current distribution is represented by the

blue curve.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.54

5

6

7

8

9

10

11

12

13

14x 10

-3

length (m)

curr

ent

density (

A/m

)Current distribution on a dipole antenna nm=13 f=300 MHz

- 55 -

Figure 5.10 Current distribution on a linear half wavelength dipole antenna.

It can be observed that the current distribution is sinusoidal as expected in dipole antennas.

The only variation occurs at the endpoints of dipole which according to boundary conditions,

must be at zero.

5.6 Current distribution on the gray hoverman antenna.

The first step in obtaining the self-impedance of the Gray Hoverman antenna is to find the

equation describing its axis. If the antenna is lying in the Z-Y plane, then its vectorial equation

and its unit tangential vector are given by:

𝑟(𝑠) = (𝑠 cos 𝑎1 + s cos𝑎2 + 𝑠 cos𝑎3 +…+ 𝑠 cos𝑎15 + 𝑠 cos 𝑎16)𝑖 +

(𝑠 sin 𝑎1 + 𝑠 sin𝑎2 +𝑠 sin 𝑎3 +…+ 𝑠 sin𝑎15 +𝑠 sin 𝑎16)𝑗 5.2

𝑠(𝑠) = (cos 𝑎1 + cos 𝑎2+cos𝑎3 +⋯+ cos 𝑎15 + cos 𝑎16)𝑖 + (sin𝑎1 +

sin𝑎2 +sin𝑎3 +…+ sin𝑎15 +sin𝑎16)𝑗 5.3

Now, all that is needed is to define the parallel curve describing the current filament lying on

the surface of the wire. Hence, the equations are given by:

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

Dipole Length (m)

I(z),

Curr

ent

density (

A/m

)

Current distribution over dipole antenna f=150 MHz

- 56 -

𝑟′(𝑠′) = (𝑠 cos 𝑎1 + s cos 𝑎2 + 𝑠 cos𝑎3 +…+ 𝑠 cos𝑎15 + 𝑠 cos 𝑎16)𝑖 +

(𝑠 sin 𝑎1 + 𝑠 sin𝑎2 +𝑠 sin 𝑎3 +…+ 𝑠 sin𝑎15 +𝑠 sin 𝑎16)𝑗 + 𝑎𝑘 5.4

𝑠′(𝑠′) = (cos 𝑎1 + cos 𝑎2+cos𝑎3 +⋯+ cos 𝑎15 + cos 𝑎16)𝑖 + (sin𝑎1 +

sin𝑎2 +sin𝑎3 +…+ sin𝑎15 +sin𝑎16)𝑗 5.5

Where 𝑎1, 𝑎2, 𝑎3, … , 𝑎16 are the angles the wire segments make with the Y-Z plane, a is the

distance between the axis of the wire and its current filament. In other words, a is the wire

radius.

From the vectorial equation of the current filament 𝑟′(𝑠′) obtained in Eq. (5.4), it is easy to

note that the Gray Hoverman antenna is 3 dimensional.

Since, the antenna is being fed at its centre, the voltage matrix is given by:

(𝑣𝑚) =

[ 0⋮0

𝑣 ∆𝑠⁄0⋮0 ]

5.6

The original Gray Hoverman antenna was composed of 16 segments, with 8 segments on each

of the two subsections located on either side of the voltage source. Each segment was 7 inches

long hence the entire length of the antenna was 2.8448m long. Also, according to the specifica-

tions, each of the two subsections was a quarter wavelengths at a frequency lying in the VHF

spectrum. As a result, the entire antenna was a half wavelength type of antenna. By using these

specifications, the frequency of operation of the original Gray Hoverman design can be easily

calculated as:

𝑓 =𝑐

𝜆 5.7

Applying the defined specifications to Eq. (4.86) and Eq. (4.88), with v=1 volt, the current

distribution on the original Gray Hoverman design is obtained in Fig. 5.11

- 57 -

Figure 5.11 Current distribution on a half wavelength Gray Hoverman antenna.

The self-impedance of the antenna can now easily be obtained as:

|𝑧𝑖𝑛| =|𝑣|

|𝐼|=

1

0.003285= 304.4Ω 5.8

The feed impedance of the Gray Hoverman antenna is 300Ω, the results obtained are therefore

of very close accuracy.

5.7 Current Distribution on a circular Loop Antenna

Figure 5.12 Circular loop antenna and its associated vectors (source: Fig. 5 in [11]).

-1.5 -1 -0.5 0 0.5 1 1.5-4

-3

-2

-1

0

1

2

3

4

5x 10

-3

Antenna Length (m)

I(z),

Curr

ent

density (

A/m

)

Current distribution on a gray hoverman antenna nm=57 f=300 MHz

- 58 -

In order to obtain the current distribution on a circular loop antenna using Eq. (4.86) and Eq.

(4.88). Firstly, the vectorial equation for the axis and filament curve is obtained. Suppose the

circular loop has a radius A as shown in Figure 5.12.

The vectorial and tangential equations for the axis of the wire are given by:

𝑟(𝑠) = 𝐴 cos (𝑠

𝐴) 𝑖 + 𝐴 sin (

𝑠

𝐴) 𝑗

𝑠(𝑠) == −sin (𝑠

𝐴) 𝑖 + cos (

𝑠

𝐴) 𝑗

5.9

The vectorial and tangential equation for the filament curve are given by:

𝑟′(𝑠′) = 𝐴 cos(𝑠′

𝐴) 𝑖 + 𝐴 sin (

𝑠′

𝐴) 𝑗 + 𝑎𝑘

𝑠′(𝑠′) == −sin(𝑠′

𝐴) 𝑖 + cos (

𝑠′

𝐴)𝑗

5.10

Where a is the distance between the axis of the wire and the filament curve. It can be

clearly observed that the axis curve and the filament curve are of the same length. The circular

loop antenna is end-fire. This implies that the feeding is at its ends, and as a result the voltage

matrix obtained if given by:

(𝑣𝑚) =

[ 𝑣 2∆𝑠⁄00⋮0

𝑣 2∆𝑠⁄ ]

5.11

Consider a circular loop antenna of length 3𝜆 with the following parameters:

𝑓 = 1 𝐺𝐻𝑧,𝑁 = 41 𝑠𝑒𝑔𝑚𝑒𝑛𝑡𝑠, 𝐿 = 2𝜋𝐴 = 3𝜆, 𝑎 = 0.005𝜆, 𝑣 = 1𝑣𝑜𝑙𝑡

By applying Eq. (5.9), Eq. (5.10) and Eq. (5.11) to Eq. (4.86) and Eq. (4.88), the current

distribution graph obtained is shown in Fig. 5.13.

- 59 -

Figure 5.13 circular loop antenna and its associated vectors (source: Fig. 5 in [11]).

Since the antenna is end-fire, to obtain the self impedance, the equation to obtain the

impedance is applied to the value of current density at the ends of the graph:

|𝑧𝑖𝑛| =|𝑣|

|𝐼|=

1

0.0008016= 1247.51Ω 5.12

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-8

-6

-4

-2

0

2

4

6

8

10x 10

-4

length (m)

current density (A

/m

)

Current distribution over a circular loop antenna f=1 GHz

- 60 -

6 Conclusion

6.1 Achievements and results

The purpose of this project was to apply Method of Moments to thin wire antennas of various

shapes. Using the Method of Moments, charge distribution on a straight wire antenna as well

as antennas of arbitrary shapes has been solved. The Method of Moments has also been used

to solve for current distribution and self-impedance on wire antennas of three different config-

urations namely; the straight dipole, the circular loop antenna and also the Gray Hoverman

antenna.

6.2 Limitations and practical considerations

6.2.1 Effects of antenna radius

The solution for Pocklington’s integral equation and Hallen’s integral equation discussed in

this project apply only to thin wire antennas. As such, the best results are obtained when the

wire antenna is of radius, a=0.005𝜆 where 𝜆 is the wavelength at which the desired antenna is

operating. This is mainly due to the fact that Method of Moments involves dividing the wire

into N segments. It is hence appropriate for antennas whose radius is much smaller compared

to the wavelength of operation (𝑎 ≪ 𝜆).

According to [2], the integral equations are sensitive to small changes in distance, r between a

source and an observation point (particularly in the region when 𝑟 = √1.5𝑎 𝑜𝑟 𝑙𝑒𝑠𝑠). The

segment with the smallest r contributes negligible resistance to the antenna’s impedance but a

large negative reactance. For larger r’s however, there is a contribution to the resistance and

the reactance becomes positive which is desired. This is also the main reason why the use of

few segments results in inaccurate results. The solution is hence, to use a larger number of

segments.

6.2.2 Choice of basis function

In the case of centre-fed antennas like the half wavelength dipole, the graph for current distri-

bution should be perfectly sinusoidal with zero current density at the end of the antenna due to

the boundary conditions explained earlier. However, in the graphs obtained in Fig. 5.8, Fig. 5.9

and Fig. 5.10, it can be noticed that this is not the case. This can be attributed to the choice of

the basis function. For the case of solution of current distribution on the dipole antenna using

Hallen’s integral, the piecewise constant basis function was applied. This was to reduce com-

plexity of computation. The piecewise constant is not the most judicious to be taken as a basis

- 61 -

function, however, the results near the feeding point closely match with the assumption of si-

nusoidal current distribution over a wire with 𝑙 =𝜆

2 [10].

6.2.3 Speed of solutions

In using the MM in the thin wire antennas to solve for the unknown current distribution, it can

be noted that even by using fast computers, obtaining a solution for such structures with large

number of segments is complex since a large amount of memory is needed to store the data in

order to be used for other applications.

Take for example, in order to solve the matrix equations obtained in chapter 5. First, the im-

pedance matrix has to be generated. Next the inverse of the matrix is calculated. Finally, the

calculated inverse matrix is multiplied on either sides of the equation to obtain the values of

the unknown charge densities. In case of large number of segments, this is very expensive in

terms of time and memory usage.

This solution to this would be the application of the Fast Multipole Method.

The FMM was first developed by Rokhlin to solve acoustic scattering problems more effi-

ciently [12]. The FMM allows interaction due to a group of particles, to be computed as if

there is a single particle. Applying the FMM allows the matrix equation to be rapidly calculat-

ed in an iterative solver without the need to store any of the matrix elements. This helps to

speed up calculation while also reducing amount of memory consumed.

6.3 Further work

In this project, a solution for charge distribution, current distribution as well as self-impedance

has been obtained. However, further work needs to be done solve for other parameters of the

antennas such as the mutual impedance, near field radiation pattern and far field radiation pat-

terns as well as the gains. The solution for the mutual impedance can be easily obtained from

Pocklington’s equation for thin wires of arbitrary shapes derived in chapter 4. The only chang-

es necessary would be properly defining where the antenna feed voltage is applied. According

to [2], mutual impedance of 2 coupled circuits is defined in circuit theory as the negative of

the ratio of the emf V21 induced in circuit 2 to the current I1 flowing in circuit 1 with circuit 2

open. By properly defining the geometry, orientation and distance between two or more anten-

nas, the mutual impedance matrix can be obtained.

- 62 -

REFERENCES

[1] Constantine A. Balanis (1982), Antenna Theory Analysis and Design, second editions,

John Wiley & Sons, 379-404.

[2] John D. Kraus (1988), Antennas, second edition, McGraw Hill, 359-396.

[3] Walton C. Gibson (2008), The Method Of Moments in Electromagnetics, Chapman &

Hall/CRC, 1-9.

[4] D. R. Hoverman (1959), “Broadband U.H.F – V.H.F Television Antenna”, June, 1-3.

[5] W. D. Rawle, “The Method of Moments: A Numerical Technique for Wire Antenna De-

sign”, 42-46.

[6] Waren L. Stutzman (1981), Thiele, Antenna Theory and Design, John Wiley & Sons, 306-

314.

[7] Leonard L. Tsai and Charles E. Smith, “Moment Methods in Electromagnetics for Under-

graduates” IEEE Trans on Education, vol. E-21, pp. 1-9, February.

[8] Eric D. Caswell, “Analysis of a Helix antenna Using a Moment method approach With

Curved Basis and Testing Functions”, 7-14.

[9] Robert S. Elliot (2003), Antenna Theory and Design, John Wiley & Sons, 287-289.

[10] Yazdan M. Attaei, “Current Distribution on Linear Thin Wire Antenna Application of

MOM and FMM”, February, 18-23.

[11] J. Sosa Pedroza, V. Barrera-Figueroa and J. Lopez-Bonilla (2004), “Simplification of

Pocklington’s integral equation for arbitrary bent thin wires”, August, 69-75.

[12] Nader Engheta, William D. Murphy, Vladimir Rokhlin & Marius S. Vassiliou. “The Fast

Multipole Method (FMM) for Electromagnetic Scattering Problems” IEEE Trans on An-

tennas and Propagation, vol. 40, No. 6, 634.

[13] Tapan K. Sarkar, Antonije R. Djordjevic & Branko M. Kolundzija (2002), Method of

Moments Applied to Antennas, CRC Press LLC, 1-3.

[14] Zbynek Raida and Ivan Hudak (1997), “Analysis of Wire Antennas by Moment Methods”

Radioengineering, vol 6, No. 4, 2-4.

[15] Kamel S. Sultan, Haythem H. Abdullah, and Esmat A. Abdallah (2012), “Method of Mo-

ments Analysis for Antenna Arrays with Optimum Memory and Time Consumption”,

1353.

[16] D. R. Hoverman (1964), Broadband Unidirectional U.H.F – Television, 1-5.

[17] James R. Nagel (2012), “Solution to Static Charge Distribution on a Thin Wire Using the

Method of Moments” April, 1-7.

- 63 -

APPENDICES

Appendix 1

Progress Log

- 64 -

Appendix 2

MATLAB Programs For Charge Distribution

2-1: Straight wire (dipole)

length=input('wire length=');

- 65 -

nm=input('number of subsections='); delta=length/nm; zmn=[]; v(1:nm)=1.11e-10; y=v'; a=0.001; spacing=linspace(0,length,nm); for m=1:nm for n=1:nm ym=delta*(m-0.5); syms x v=-1/sqrt(x.^2 + a.^2); k=inline(v); zmn(m,n)=quadv(k,ym-(n-1)*delta,ym-n*delta); end end an=zmn\y; plot(spacing,an) xlabel('length(m)') ylabel('charge density (C/m)') clear all

2-2: Bent wire

l1=input('length of wire 1='); l2=input('length of wire 2='); nm=input('number of subsections='); alpha=input('enter angle for l2='); tl=l1+l2; delta=tl/nm; v(1:nm)=1.11e-10; y=v'; spacing=linspace(0,tl,nm); zmn=zeros(6); for m=1:nm for n=1:nm if(m<=nm/2 && n<=nm/2) ym=(m-0.5)*delta; zmn(m,n)=quadv(@k,(ym-(n-1)*delta),(ym-n*delta)); elseif(m>nm/2 && n>nm/2) ym=(m-0.5)*delta; zmn(m,n)=quadv(@k,(ym-(n-1)*delta),(ym-n*delta)); elseif(m>nm/2 && n<=nm/2) absym=((nm/2)-n+0.5)*delta + (m-(nm/2)-

0.5)*delta*cosd(alpha); zm=(m-(nm/2)-0.5)*delta*sind(alpha); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(m<=nm/2 && n>nm/2) absym=((nm/2)-m+0.5)*delta + (n-(nm/2)-

0.5)*delta*cosd(alpha); zm=(n-(nm/2)-0.5)*delta*sind(alpha); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); end end end an=zmn\y; plot(spacing,an) clear all

- 66 -

2-3: Gray Hoverman antenna subsection

tl=input('total length of hoverman='); l1=tl/8; l2=l1; l3=l1; l4=l1; l5=l1; l6=l1; l7=l1; l8=l1; nm=input('number of subsections(must be a factor of 8)='); a2=input('enter angle for l2(<90)='); a3=input('enter angle for l3(>90 && <180)='); a4=input('enter angle for l4(<90)='); a5=input('enter angle for l5(>90 && <180)='); a6=input('enter angle for l6(<90)='); a7=input('enter angle for l7(>90 && <180)='); a8=input('enter angle for l8(=180)=');

delta=tl/nm; v(1:nm)=1.11e-10; y=v'; spacing=linspace(0,tl,nm); zmn=zeros(80); for m=1:nm for n=1:nm

if(m<=nm/8 && n>nm/8 && n<=(2*nm/8)) absym=((nm/8)-m+0.5)*delta + (n-(nm/8)-

0.5)*delta*cosd(a2); zm=(n-(nm/8)-0.5)*delta*sind(a2); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(n<=nm/8 && m>nm/8 && m<=(2*nm/8)) absym=((nm/8)-n+0.5)*delta + (m-(nm/8)-

0.5)*delta*cosd(a2); zm=(m-(nm/8)-0.5)*delta*sind(a2); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(m<=nm/8 && n>(2*nm/8) && n<=(3*nm/8)) absym=(nm/8-m+0.5)*delta + l2*cosd(a2) + (n-(2*nm/8)-

0.5)*delta*cosd(a3); zm=l2*sind(a2) + (n-(2*nm/8)-0.5)*delta*sind(a3); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(n<=nm/8 && m>(2*nm/8)&&m<=(3*nm/8)) absym=(nm/8-n+0.5)*delta + l2*cosd(a2) + (m-(2*nm/8)-

0.5)*delta*cosd(a3); zm=l2*sind(a2) + (m-(2*nm/8)-0.5)*delta*sind(a3); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(m<=nm/8 && n>(3*nm/8) && n<=(4*nm/8)) absym=((nm/8)-m+0.5)*delta+l2*cosd(a2)+l3*cosd(a3)+(n-

(3*nm/8)-0.5)*delta*cosd(a4); zm=l2*sind(a2)+l3*sind(a3)+(n-(3*nm/8)-

0.5)*delta*sind(a4); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(n<=nm/8 && m>(3*nm/8) && m<=(4*nm/8)) absym=((nm/8)-n+0.5)*delta+l2*cosd(a2)+l3*cosd(a3)+(m-

(3*nm/8)-0.5)*delta*cosd(a4);

- 67 -

zm=l2*sind(a2)+l3*sind(a3)+(m-(3*nm/8)-

0.5)*delta*sind(a4); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(m<=nm/8 && n>(4*nm/8) && n<=(5*nm/8)) absym=(nm/8-

m+0.5)*delta+l2*cosd(a2)+l3*cosd(a3)+l4*cosd(a4)+(n-(4*nm/8)-

0.5)*delta*cosd(a5); zm=l2*sind(a2)+l3*sind(a3)+l4*sind(a4)+(n-(4*nm/8)-

0.5)*delta*sind(a5); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(n<=nm/8 && m>(4*nm/8)&& m<=(5*nm/8)) absym=(nm/8-

n+0.5)*delta+l2*cosd(a2)+l3*cosd(a3)+l4*cosd(a4)+(m-(4*nm/8)-

0.5)*delta*cosd(a5); zm=l2*sind(a2)+l3*sind(a3)+l4*sind(a4)+(m-(4*nm/8)-

0.5)*delta*sind(a5); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(m<=nm/8 && n>(5*nm/8)&& n<=(6*nm/8)) absym=(nm/8-

m+0.5)*delta+l2*cosd(a2)+l3*cosd(a3)+l4*cosd(a4)+l5*cosd(a5)+(n-

(5*nm/8)-0.5)*delta*cosd(a6); zm=l2*sind(a2)+l3*sind(a3)+l4*sind(a4)+l5*sind(a5)+(n-

(5*nm/8)-0.5)*delta*sind(a6); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(n<=nm/8 && m>(5*nm/8)&& m<=(6*nm/8)) absym=(nm/8-

n+0.5)*delta+l2*cosd(a2)+l3*cosd(a3)+l4*cosd(a4)+l5*cosd(a5)+(m-

(5*nm/8)-0.5)*delta*cosd(a6); zm=l2*sind(a2)+l3*sind(a3)+l4*sind(a4)+l5*sind(a5)+(m-

(5*nm/8)-0.5)*delta*sind(a6); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(m<=nm/8 && n>(6*nm/8)&& n<=(7*nm/8)) absym=(nm/8-

m+0.5)*delta+l2*cosd(a2)+l3*cosd(a3)+l4*cosd(a4)+l5*cosd(a5)+l6*cosd(

a6)+(n-(6*nm/8)-0.5)*delta*cosd(a7);

zm=l2*sind(a2)+l3*sind(a3)+l4*sind(a4)+l5*sind(a5)+l6*sind(a6)+(n-

(6*nm/8)-0.5)*delta*sind(a7); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(n<=nm/8 && m>(6*nm/8)&& m<=(7*nm/8)) absym=(nm/8-

n+0.5)*delta+l2*cosd(a2)+l3*cosd(a3)+l4*cosd(a4)+l5*cosd(a5)+l6*cosd(

a6)+(m-(6*nm/8)-0.5)*delta*cosd(a7);

zm=l2*sind(a2)+l3*sind(a3)+l4*sind(a4)+l5*sind(a5)+l6*sind(a6)+(m-

(6*nm/8)-0.5)*delta*sind(a7); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(m<=nm/8 && n>(7*nm/8)&& n<=(8*nm/8)) absym=(nm/8-

m+0.5)*delta+l2*cosd(a2)+l3*cosd(a3)+l4*cosd(a4)+l5*cosd(a5)+l6*cosd(

a6)+l7*cosd(a7)+(n-(7*nm/8)-0.5)*delta*cosd(a8);

zm=l2*sind(a2)+l3*sind(a3)+l4*sind(a4)+l5*sind(a5)+l6*sind(a6)+l7*sin

d(a7)+(n-(7*nm/8)-0.5)*delta*sind(a8); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(n<=nm/8 && m>(7*nm/8)&& m<=(8*nm/8)) absym=(nm/8-

n+0.5)*delta+l2*cosd(a2)+l3*cosd(a3)+l4*cosd(a4)+l5*cosd(a5)+l6*cosd(

a6)+l7*cosd(a7)+(m-(7*nm/8)-0.5)*delta*cosd(a8);

- 68 -

zm=l2*sind(a2)+l3*sind(a3)+l4*sind(a4)+l5*sind(a5)+l6*sind(a6)+l7*sin

d(a7)+(m-(7*nm/8)-0.5)*delta*sind(a8); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(m>nm/8 && m<=(2*nm/8) && n>(2*nm/8) && n<=(3*nm/8)) absym=((2*nm/8)-m+0.5)*delta*cosd(a2) + (n-(2*nm/8)-

0.5)*delta*cosd(a3); zm=((2*nm/8)-m+0.5)*delta*sind(a2) + (n-(2*nm/8)-

0.5)*delta*sind(a3);%start here zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(n>nm/8 && n<=(2*nm/8) && m>(2*nm/8) && m<=(3*nm/8)) absym=((2*nm/8)-n+0.5)*delta*cosd(a2) + (m-(2*nm/8)-

0.5)*delta*cosd(a3); zm=((2*nm/8)-n+0.5)*delta*sind(a2) + (m-(2*nm/8)-

0.5)*delta*sind(a3); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(m>nm/8 && m<=(2*nm/8) && n>(3*nm/8) && n<=(4*nm/8)) absym=((2*nm/8)-m+0.5)*delta*cosd(a2) + l3*cosd(a3)+ (n-

(3*nm/8)-0.5)*delta*cosd(a4); zm=((2*nm/8)-m+0.5)*delta*sind(a2) + l3*sind(a3) + (n-

(3*nm/8)-0.5)*delta*sind(a4); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(n>nm/8 && n<=(2*nm/8) && m>(3*nm/8) && m<=(4*nm/8)) absym=((2*nm/8)-n+0.5)*delta*cosd(a2) + l3*cosd(a3)+ (m-

(3*nm/8)-0.5)*delta*cosd(a4); zm=((2*nm/8)-n+0.5)*delta*sind(a2) + l3*sind(a3) + (m-

(3*nm/8)-0.5)*delta*sind(a4); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(m>nm/8 && m<=(2*nm/8) && n>(4*nm/8) && n<=(5*nm/8)) absym=((2*nm/8)-m+0.5)*delta*cosd(a2) +

l3*cosd(a3)+l4*cosd(a4)+ (n-(4*nm/8)-0.5)*delta*cosd(a5); zm=((2*nm/8)-m+0.5)*delta*sind(a2) + l3*sind(a3)

+l4*sind(a4)+ (n-(4*nm/8)-0.5)*delta*sind(a5); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(n>nm/8 && n<=(2*nm/8) && m>(4*nm/8) && m<=(5*nm/8)) absym=((2*nm/8)-n+0.5)*delta*cosd(a2) +

l3*cosd(a3)+l4*cosd(a4)+ (m-(4*nm/8)-0.5)*delta*cosd(a5); zm=((2*nm/8)-n+0.5)*delta*sind(a2) + l3*sind(a3)

+l4*sind(a4)+ (m-(4*nm/8)-0.5)*delta*sind(a5); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(m>nm/8 && m<=(2*nm/8) && n>(5*nm/8) && n<=(6*nm/8)) absym=((2*nm/8)-m+0.5)*delta*cosd(a2) +

l3*cosd(a3)+l4*cosd(a4)+l5*cosd(a5)+ (n-(5*nm/8)-0.5)*delta*cosd(a6); zm=((2*nm/8)-m+0.5)*delta*sind(a2) + l3*sind(a3)

+l4*sind(a4)+l5*sind(a5)+ (n-(5*nm/8)-0.5)*delta*sind(a6); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(n>nm/8 && n<=(2*nm/8) && m>(5*nm/8) && m<=(6*nm/8)) absym=((2*nm/8)-n+0.5)*delta*cosd(a2) +

l3*cosd(a3)+l4*cosd(a4)+l5*cosd(a5)+ (m-(5*nm/8)-0.5)*delta*cosd(a6); zm=((2*nm/8)-n+0.5)*delta*sind(a2) + l3*sind(a3)

+l4*sind(a4)+l5*sind(a5)+ (m-(5*nm/8)-0.5)*delta*sind(a6); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(m>nm/8 && m<=(2*nm/8) && n>(6*nm/8) && n<=(7*nm/8)) absym=((2*nm/8)-m+0.5)*delta*cosd(a2) +

l3*cosd(a3)+l4*cosd(a4)+l5*cosd(a5)+l6*cosd(a6)+ (n-(6*nm/8)-

0.5)*delta*cosd(a7); zm=((2*nm/8)-m+0.5)*delta*sind(a2) + l3*sind(a3)

+l4*sind(a4)+l5*sind(a5)+l6*sind(a6)+ (n-(6*nm/8)-

0.5)*delta*sind(a7); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(n>nm/8 && n<=(2*nm/8) && m>(6*nm/8) && m<=(7*nm/8))

- 69 -

absym=((2*nm/8)-n+0.5)*delta*cosd(a2) +

l3*cosd(a3)+l4*cosd(a4)+l5*cosd(a5)+l6*cosd(a6)+ (m-(6*nm/8)-

0.5)*delta*cosd(a7); zm=((2*nm/8)-n+0.5)*delta*sind(a2) + l3*sind(a3)

+l4*sind(a4)+l5*sind(a5)+l6*sind(a6)+ (m-(6*nm/8)-

0.5)*delta*sind(a7); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(m>nm/8 && m<=(2*nm/8) && n>(7*nm/8) && n<=(8*nm/8)) absym=((2*nm/8)-m+0.5)*delta*cosd(a2) +

l3*cosd(a3)+l4*cosd(a4)+l5*cosd(a5)+l6*cosd(a6)+l7*cosd(a7)+ (n-

(7*nm/8)-0.5)*delta*cosd(a8); zm=((2*nm/8)-m+0.5)*delta*sind(a2) + l3*sind(a3)

+l4*sind(a4)+l5*sind(a5)+l6*sind(a6)+l7*sind(a7)+ (n-(7*nm/8)-

0.5)*delta*sind(a8); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(n>nm/8 && n<=(2*nm/8) && m>(7*nm/8) && m<=(8*nm/8)) absym=((2*nm/8)-n+0.5)*delta*cosd(a2) +

l3*cosd(a3)+l4*cosd(a4)+l5*cosd(a5)+l6*cosd(a6)+l7*cosd(a7)+ (m-

(7*nm/8)-0.5)*delta*cosd(a8); zm=((2*nm/8)-n+0.5)*delta*sind(a2) + l3*sind(a3)

+l4*sind(a4)+l5*sind(a5)+l6*sind(a6)+l7*sind(a7)+ (m-(7*nm/8)-

0.5)*delta*sind(a8); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(m>(2*nm/8) && m<=(3*nm/8) && n>(3*nm/8) && n<=(4*nm/8)) absym=((3*nm/8)-m+0.5)*delta*cosd(a3) + (n-(3*nm/8)-

0.5)*delta*cosd(a4); zm=((3*nm/8)-m+0.5)*delta*sind(a3) + (n-(3*nm/8)-

0.5)*delta*sind(a4); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(n>(2*nm/8) && n<=(3*nm/8) && m>(3*nm/8) && m<=(4*nm/8)) absym=((3*nm/8)-n+0.5)*delta*cosd(a3) + (m-(3*nm/8)-

0.5)*delta*cosd(a4); zm=((3*nm/8)-n+0.5)*delta*sind(a3) + (m-(3*nm/8)-

0.5)*delta*sind(a4); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(m>(2*nm/8) && m<=(3*nm/8) && n>(4*nm/8) && n<=(5*nm/8)) absym=((3*nm/8)-m+0.5)*delta*cosd(a3)+ l4*cosd(a4) + (n-

(4*nm/8)-0.5)*delta*cosd(a5); zm=((3*nm/8)-m+0.5)*delta*sind(a3) + l4*sind(a4)+ (n-

(4*nm/8)-0.5)*delta*sind(a5); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(n>(2*nm/8) && n<=(3*nm/8) && m>(4*nm/8) && m<=(5*nm/8)) absym=((3*nm/8)-n+0.5)*delta*cosd(a3)+ l4*cosd(a4) + (m-

(4*nm/8)-0.5)*delta*cosd(a5); zm=((3*nm/8)-n+0.5)*delta*sind(a3) + l4*sind(a4)+ (m-

(4*nm/8)-0.5)*delta*sind(a5); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(m>(2*nm/8) && m<=(3*nm/8) && n>(5*nm/8) && n<=(6*nm/8)) absym=((3*nm/8)-m+0.5)*delta*cosd(a3)+ l4*cosd(a4) +

l5*cosd(a5)+ (n-(5*nm/8)-0.5)*delta*cosd(a6); zm=((3*nm/8)-m+0.5)*delta*sind(a3) +

l4*sind(a4)+l5*sind(a5)+ (n-(5*nm/8)-0.5)*delta*sind(a6); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(n>(2*nm/8) && n<=(3*nm/8) && m>(5*nm/8) && m<=(6*nm/8)) absym=((3*nm/8)-n+0.5)*delta*cosd(a3)+ l4*cosd(a4) +

l5*cosd(a5)+ (m-(5*nm/8)-0.5)*delta*cosd(a6); zm=((3*nm/8)-n+0.5)*delta*sind(a3) +

l4*sind(a4)+l5*sind(a5)+ (m-(5*nm/8)-0.5)*delta*sind(a6); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(m>(2*nm/8) && m<=(3*nm/8) && n>(6*nm/8) && n<=(7*nm/8))

- 70 -

absym=((3*nm/8)-m+0.5)*delta*cosd(a3)+ l4*cosd(a4) +

l5*cosd(a5)+l6*cosd(a6)+ (n-(6*nm/8)-0.5)*delta*cosd(a7); zm=((3*nm/8)-m+0.5)*delta*sind(a3) +

l4*sind(a4)+l5*sind(a5)+l6*sind(a6)+ (n-(6*nm/8)-0.5)*delta*sind(a7); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(n>(2*nm/8) && n<=(3*nm/8) && m>(6*nm/8) &&

m<=(7*nm/8)) absym=((3*nm/8)-n+0.5)*delta*cosd(a3)+ l4*cosd(a4) +

l5*cosd(a5)+l6*cosd(a6)+ (m-(6*nm/8)-0.5)*delta*cosd(a7); zm=((3*nm/8)-n+0.5)*delta*sind(a3) +

l4*sind(a4)+l5*sind(a5)+l6*sind(a6)+ (m-(6*nm/8)-0.5)*delta*sind(a7); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(m>(2*nm/8) && m<=(3*nm/8) && n>(7*nm/8) &&

n<=(8*nm/8)) absym=((3*nm/8)-m+0.5)*delta*cosd(a3)+ l4*cosd(a4) +

l5*cosd(a5)+l6*cosd(a6)+l7*cosd(a7)+ (n-(7*nm/8)-0.5)*delta*cosd(a8); zm=((3*nm/8)-m+0.5)*delta*sind(a3) +

l4*sind(a4)+l5*sind(a5)+l6*sind(a6)+l7*sind(a7)+ (n-(7*nm/8)-

0.5)*delta*sind(a8); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(n>(2*nm/8) && n<=(3*nm/8) && m>(7*nm/8) &&

m<=(8*nm/8)) absym=((3*nm/8)-n+0.5)*delta*cosd(a3)+ l4*cosd(a4) +

l5*cosd(a5)+l6*cosd(a6)+l7*cosd(a7)+ (m-(7*nm/8)-0.5)*delta*cosd(a8); zm=((3*nm/8)-n+0.5)*delta*sind(a3) +

l4*sind(a4)+l5*sind(a5)+l6*sind(a6)+l7*sind(a7)+ (m-(7*nm/8)-

0.5)*delta*sind(a8); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(m>(3*nm/8) && m<=(4*nm/8) && n>(4*nm/8) && n<=(5*nm/8)) absym=((4*nm/8)-m+0.5)*delta*cosd(a4)+(n-(4*nm/8)-

0.5)*delta*cosd(a5); zm=((4*nm/8)-m+0.5)*delta*sind(a4)+(n-(4*nm/8)-

0.5)*delta*sind(a5); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(n>(3*nm/8) && n<=(4*nm/8) && m>(4*nm/8) && m<=(5*nm/8)) absym=((4*nm/8)-n+0.5)*delta*cosd(a4)+(m-(4*nm/8)-

0.5)*delta*cosd(a5); zm=((4*nm/8)-n+0.5)*delta*sind(a4)+(m-(4*nm/8)-

0.5)*delta*sind(a5); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(m>(3*nm/8) && m<=(4*nm/8) && n>(5*nm/8) && n<=(6*nm/8)) absym=((4*nm/8)-m+0.5)*delta*cosd(a4)+l5*cosd(a5)+(n-

(5*nm/8)-0.5)*delta*cosd(a6); zm=((4*nm/8)-m+0.5)*delta*sind(a4)+l5*sind(a5)+(n-

(5*nm/8)-0.5)*delta*sind(a6); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(n>(3*nm/8) && n<=(4*nm/8) && m>(5*nm/8) && m<=(6*nm/8)) absym=((4*nm/8)-n+0.5)*delta*cosd(a4)+l5*cosd(a5)+(m-

(5*nm/8)-0.5)*delta*cosd(a6); zm=((4*nm/8)-n+0.5)*delta*sind(a4)+l5*sind(a5)+(m-

(5*nm/8)-0.5)*delta*sind(a6); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(m>(3*nm/8) && m<=(4*nm/8) && n>(6*nm/8) && n<=(7*nm/8)) absym=((4*nm/8)-

m+0.5)*delta*cosd(a4)+l5*cosd(a5)+l6*cosd(a6)+(n-(6*nm/8)-

0.5)*delta*cosd(a7); zm=((4*nm/8)-

m+0.5)*delta*sind(a4)+l5*sind(a5)+l6*sind(a6)+(n-(6*nm/8)-

0.5)*delta*sind(a7); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(n>(3*nm/8) && n<=(4*nm/8) && m>(6*nm/8) && m<=(7*nm/8))

- 71 -

absym=((4*nm/8)-

n+0.5)*delta*cosd(a4)+l5*cosd(a5)+l6*cosd(a6)+(m-(6*nm/8)-


n+0.5)*delta*sind(a4)+l5*sind(a5)+l6*sind(a6)+(m-(6*nm/8)-

0.5)*delta*sind(a7); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(m>(3*nm/8) && m<=(4*nm/8) && n>(7*nm/8) && n<=(8*nm/8)) absym=((4*nm/8)-

m+0.5)*delta*cosd(a4)+l5*cosd(a5)+l6*cosd(a6)+l7*cosd(a7)+(n-

(7*nm/8)-0.5)*delta*cosd(a8); zm=((4*nm/8)-

m+0.5)*delta*sind(a4)+l5*sind(a5)+l6*sind(a6)+l7*sind(a7)+(n-

(7*nm/8)-0.5)*delta*sind(a8); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(n>(3*nm/8) && n<=(4*nm/8) && m>(7*nm/8) && m<=(8*nm/8)) absym=((4*nm/8)-

n+0.5)*delta*cosd(a4)+l5*cosd(a5)+l6*cosd(a6)+l7*cosd(a7)+(m-

(7*nm/8)-0.5)*delta*cosd(a8); zm=((4*nm/8)-

n+0.5)*delta*sind(a4)+l5*sind(a5)+l6*sind(a6)+l7*sind(a7)+(m-

(7*nm/8)-0.5)*delta*sind(a8); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(m>(4*nm/8) && m<=(5*nm/8) && n>(5*nm/8) && n<=(6*nm/8)) absym=((5*nm/8)-m+0.5)*delta*cosd(a5)+(n-(5*nm/8)-








(6*nm/8)-0.5)*delta*sind(a7); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(m>(4*nm/8) && m<=(5*nm/8) && n>(7*nm/8) && n<=(8*nm/8)) absym=((5*nm/8)-

m+0.5)*delta*cosd(a5)+l6*cosd(a6)+l7*cosd(a7)+(n-(7*nm/8)-


m+0.5)*delta*sind(a5)+l6*sind(a6)+l7*sind(a7)+(n-(7*nm/8)-

0.5)*delta*sind(a8); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(n>(4*nm/8) && n<=(5*nm/8) && m>(7*nm/8) && m<=(8*nm/8)) absym=((5*nm/8)-

n+0.5)*delta*cosd(a5)+l6*cosd(a6)+l7*cosd(a7)+(m-(7*nm/8)-

0.5)*delta*cosd(a8);

- 72 -

zm=((5*nm/8)-

n+0.5)*delta*sind(a5)+l6*sind(a6)+l7*sind(a7)+(m-(7*nm/8)-

0.5)*delta*sind(a8); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(m>(5*nm/8) && m<=(6*nm/8) && n>(6*nm/8) && n<=(7*nm/8)) absym=((6*nm/8)-m+0.5)*delta*cosd(a6)+(n-(6*nm/8)-








(7*nm/8)-0.5)*delta*sind(a8); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); elseif(m>(6*nm/8) && m<=(7*nm/8) && n>(7*nm/8) && n<=(8*nm/8)) absym=((7*nm/8)-m+0.5)*delta*cosd(a7)+(n-(7*nm/8)-




0.5)*delta*sind(a8); zmn(m,n)=delta/(sqrt(absym^2 + zm^2)); else ym=(m-0.5)*delta; zmn(m,n)=quadv(@k,(ym-(n-1)*delta),(ym-n*delta)); end end end an=zmn\y; plot(spacing,an) xlabel('length(m)') ylabel('charge density (C/m)') title('Charge distribution on a gray hoverman L=1m, v=1 volt') clear all

- 73 -

Appendix 3

MATLAB Programs for Current Distribution

3-1: Dipole antenna using Hallen’s integral

clc clear all clc %______________________________Constants___________________________% c=3.0e+08; etha=377; Frequency=1.5e+08; Voltage=1; lamda=c/Frequency; L=lamda/2; a=L/518; k=2*pi/lamda; N=40; delta=L/(N); %__________________________________________________________________% t=linspace (-0.5*L,0.5*L,N); zm=-0.5*L-delta; A=zeros(41); V=zeros(41); P=zeros(40); for m=1:N+1 zm=zm+delta; zn=-0.5*L+delta/2; for n=1:N+1 if (n==N+1) A(m,n)=-1i*cos(k*zm)/(etha); else syms z R=sqrt(a^2+(z-zm)^2); g=inline(exp(-1i*k*R)/(4*pi*R)); G=quad(g,zn,zn+delta/2); A(m,n)=G; end zn=zn+delta; end V(m,1)=-1i*Voltage*sin(k*abs(zm))/(2*etha); end I=(A\V); plot(t,abs(P),t,real(P),'g',t,imag(P),'r') xlabel('Dipole Length (m)') ylabel('I(z),Current density (A/m)') title('Current distribution over dipole antenna f=150 MHz')

3-2: Dipole antenna using Pocklington’s Equation

c=3.0e+08; f=300e+06;

- 74 -

lambda=c/f; l=input('length in lambda='); nm=input('number of subsections(odd number)='); a=input('radius='); delta=l/(nm+1); k=2*pi/lambda; w=2*pi*f; e=8.854e-12; spacing=linspace(0,l,nm); vt=[zeros(1,((nm+1)/2-1)) 1 zeros(1,((nm+1)/2-1))]; v=transpose(vt); zmn=zeros(15); for m=1:nm for n=1:nm zm=m*delta; zn=n*delta;

syms z R1=sqrt((z-zn).^2 + a.^2); R2=sqrt((z-(zn-delta)).^2 + a.^2); R3=sqrt((z-(zn+delta)).^2 + a.^2); fun1=inline(exp(-1i.*k.*R1)./(4.*pi.*R1)); G1=quad(fun1,zm-delta/2,zm+delta/2); fun2=inline(exp(-1i.*k.*R2)./(4.*pi.*R2)); G2=quad(fun2,zm-delta/2,zm+delta/2); fun3=inline(exp(-1i.*k.*R3)./(4.*pi.*R3)); G3=quad(fun3,zm-delta/2,zm+delta/2); G=G2+G3-(2.*G1.*cos(k*delta)); zmn(m,n)=(1i./(w.*e)).*(k./(sin(k.*delta))).*G;

end end k=radtodeg(angle(zmn)); In=zmn\v; mag=abs(In); theta=radtodeg(angle(In)); impedance=1/max(In); plot(spacing,In) xlabel('length (m)') ylabel('current density (A/m)') title('Current distribution over a dipole antenna f=300 MHz') clear all

3-3: Dipole antenna using Pocklington’s Equation for arbitrary shaped antennas

nm=41; c=3.0e+08; f=150e+06; %f=1.0e+09; lambda=c/f; a=0.005*lambda; L=0.47*lambda; uo=1.25663706e-6; eo=8.85418782e-12; w=2*pi*f; k=w*sqrt(eo*uo); delta=L/nm;

- 75 -

vt=[zeros(1,((nm+1)/2-1)) (1/delta) zeros(1,((nm+1)/2-1))]; v=transpose(vt); spacing=linspace(0,L,nm);

zmn=zeros(41); for m=1:nm for n=1:nm s=(m-0.5)*delta; syms sp R=sqrt((0-a)^2 +(s-sp)^2); C=1; D=s-sp; E=s-sp; fun1=inline(R.^2*((1+1i*k*R)-((k.^2)*(R.^2)))*C*((exp(-

1i*k*R))/R.^5)); fun2=inline(((3+3*1i*k*R)-((k.^2)*(R.^2)))*D*E*((exp(-

1i*k*R))/R.^5)); G1=quad(fun1,(n-1)*delta,n*delta); G2=quad(fun2,(n-1)*delta,n*delta); zmn(m,n)=(-1i/(4*pi*w*eo))*G1 + (1i/(4*pi*w*eo))*G2; end end In=zmn\v; impedance=1/(max(In)); plot(spacing,abs(In),spacing,real(In),'g',spacing,imag(In),'r') xlabel('length (m)') ylabel('current density (A/m)') title('Current distribution over a dipole antenna f=150 MHz')

3-4: Circular loop

nm=input('number of subsections='); v=input('excitation voltage='); c=3.0e+08; f=1.0e+09; lambda=c/f; L=3*lambda; a=0.005*lambda; uo=1.25663706e-6; eo=8.85418782e-12; w=2*pi*f; k=w*sqrt(eo*uo); delta=L/nm; vt=[(v/(2*delta)) zeros(1,nm-2) (-v/(2*delta))]; v=transpose(vt); spacing=linspace(0,L,nm); A=(3*lambda)/(2*pi);

zmn=zeros(41); for m=1:nm for n=1:nm

s=(m-0.5)*delta; syms sp R=sqrt((A*cos(s/A)-A*cos(sp/A))^2 + (A*sin(s/A)-A*sin(sp/A))^2

+ a^2); C=(sin(s/A)*sin(sp/A))+(cos(s/A)*cos(sp/A));

- 76 -

D=(-

A*cos(s/A)*sin(s/A))+(A*cos(sp/A)*sin(s/A))+(A*sin(s/A)*cos(s/A))+(-

A*sin(sp/A)*cos(s/A)); E=(-

A*cos(s/A)*sin(sp/A))+(A*cos(sp/A)*sin(sp/A))+(A*sin(s/A)*cos(sp/A))+

(-A*sin(sp/A)*cos(sp/A)); fun1=inline(R.^2*((1+1i*k*R)-((k.^2)*(R.^2)))*C*((exp(-

1i*k*R))/R.^5)); fun2=inline(((3+3*1i*k*R)-((k.^2)*(R.^2)))*D*E*((exp(-

1i*k*R))/R.^5)); G1=quad(fun1,(n-1)*delta,n*delta); G2=quad(fun2,(n-1)*delta,n*delta); zmn(m,n)=(-1i/(4*pi*w*eo))*G1 + (1i/(4*pi*w*eo))*G2; end end zmn; In=zmn\v; plot(spacing,abs(In),spacing,real(In),'g',spacing,imag(In),'r') xlabel('length (m)') ylabel('current density (A/m)')

title('Current distribution over a circular loop antenna f=1 GHz') clear all