TRAVELING WAVE ANTENNAS WITH
IMPEDANCE LGADING

Thesis far the Degree of Ph. D.
MICH'a‘GAN STATE umvmsw‘r
Dennis P. Nyquést
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THESIS

  
      

  

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Michigan Scam
Univcmity

This is to certify that the

thesis entitled

TRAVELING WAVE ANTENNAS
WITH IMPEDANCE LOADING

presented by

Dennis P. Nyquist

has been accepted towards fulfillment
of the requirements for

Ph.D. degree inElec. Engr.

 

 

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Major professor

‘1 Date November 11. 1966

0-169

TRAVELING WAVE ANTENNAS

WITH IMPEDANCE LOADING
BY

Dennis P ‘ Nyqui s t

AN ABSTRACT

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

DOC TOR OF PHILOSOPHY

Department of Electrical Engineering

1966

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ABSTRACT
TRAVELING WAVE ANTENNAS
WITH IMPEDANCE LOADING

by Dennis P. Nyquist

The circuit and radiation characteristics associated with thin-
wire traveling wave antennas are desirable for certain applications.
In this thesis, specific consideration is given to two members of the
class of thin-wire antennas: (l) the linear antenna and (2) the
circular 100p antenna. An impedance loading technique is utilized
in either case to modify the usual standing wave distribution of
antenna current. It is indicated that a traveling wave of current may
be excited on such thin-wire antennas through the use of an Optimum
impedance loading.

It is well known that conventional thin-wire antennas support
an essentially standing wave distribution of current. Such antennas
are highly frequency sensitive, in that their input impedance is a
strong function of the excitation frequency. As a consequence of
this sensitivity, antennas of this type are ordinarily used only at
a single frequency or over a very narrow band of frequencies.

A traveling wave antenna supports a distribution of current
which is essentially an outward traveling wave. In contrast to its
standing wave counterpart, the input impedance of a traveling wave
antenna is relatively broadband as a function of frequency. Further-
more, the radiation fields of such an antenna are considerably

modified as compared with those of a corresponding conventional

a:tenna. The.“
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DENNIS P. NYQUIST

antenna. These radiation patterns are in general characterized by
a wider beamwidth for electrically small antennas, and improved
directivity with a notable absence of minor lobes for large antennas.

It is the object of this research to realize a high efficiency
traveling wave antenna and to evaluate its corresponding circuit and
radiation characteristics. An impedance loading technique is
utilized whereby the antenna is doubly loaded with a pair of idential
lumped impedances. The position and impedance of an optimum
loading to yield an outward traveling wave of antenna current are
determined in terms of the antenna dimensions and its frequency of
excitation. It is indicated that a purely non-dissipative optimum
loading may be utilized if its position is properly chosen. Through
the use of such a loading, a traveling wave antenna may be realized
while maintaining the high efficiency characteristic of a conventional
unloaded antenna.

A theoretical analysis is made of the impedance loaded linear
and loop antenna configurations. It is the object of this analysis to:

(1) determine approximately the distribution of antenna current as

a function of its dimensions, the impedance and position of the loading,
and the frequency of excitation; (2) determine the optimum loading to
yield an outward traveling wave of current; (3) investigate the possibility
of utilizing a purely non-dissipative loading; and (4) calculate the
corresponding input impedance and radiation fields. Particular
emphasis is placed upon the use of a non-dissipative optimum loading,
since the high efficiency associated with such a loading is of fundamental

interest.

 

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DENNIS P. NYQUIST

It is verified experimentally that a traveling wave of current
may indeed be obtained on a linear antenna through the use of a properly
located, purely reactive loading. The experimentally measured values
of the parameters for such an optimum loading are demonstrated to
compare favorably with those which were determined theoretically.

The frequency dependence of the input impedance to a linear antenna
utilizing an optimum non-dissipative loading is evaluated experimentally.

Perhaps the most significant result of this research is the
conclusion that a traveling wave antenna may be realized through the
use of a purely non-dissipative loading. Explicit expressions for the
reactance and position of such an optimum loading are presented. It
is indicated that a purely reactive loading can be optimum only at a
single frequency, a resistive component being required at all other
frequencies. Approximate expressions for the input impedance and
radiation fields of the traveling wave antenna are given in terms of
its dimensions and the frequency of excitation. Numerical examples
are presented for antennas of specific dimensions to illustrate these

theoretical results.

TRAVELING WAVE ANTENNAS

WITH IMP EDANC E LOADING

BY

Dennis P. Nyqui st

A THESIS

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

DOC TOR OF PHILOSOPHY

Department of Electrical Engineering

1966

The a;
chairman Dr.
the preparati;
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ACKNOWLEDGMENT

The author wishes to express his indebtedness to committee
chairman Dr. K. M. Chen for his guidance and encouragement in
the preparation of this thesis. He also wishes to thank Mr. J. W.
Hoffman of the Division of Engineering Research for support in the
form of a graduate fellowship during the conduct of this research.
The research reported in this thesis was supported in part by the

Air Force Cambridge Research Laboratories under contract

AF19(628)-5372.

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TABLE OF CONTENTS

Acknowledgment ..... . . . . ......... . .......
ListofFigures ............ ......
PART I. Traveling Wave Linear Antenna with
Optimum Impedance Loading
1 Introduction. . . . . . . . . . . .............
l. l. IntrOduction O O O O O O O O O O O O O O O O O O O O
l. 2. Definition of a Traveling Wave Linear Antenna . .
l. 3. Important Characteristics of a Traveling
WaveDipole . . . . . . . . ............
1. ,4. Previous Research on the Traveling Wave
Linear Antenna . . ...............
l. 5. Object of the Present Research . . ........
l. 6. Outline for Investigation of Traveling Wave
Linear Antenna with Optimum Impedance
Loading C O O O O O O C O O O O O O O O O .....
2 Approximate Distribution of Current on a Doubly
LoadedLinearAntenna. . . . . . . . . . . . . . . . . .
2.1. Geometry of the Doubly Loaded Linear Antenna .
Z. 2. Dimensions of Interest for a One-Dimensional
Theory. 0 o o o o o ooooo o o o o o o o o o o o
2. 3. Formulation of an Inegral Equation for the
Distribution of Cylinder Current - . . - . - . . .
2. 4. Approximate Solution for the Distribution of
Current on the Doubly Loaded Cylinder. . . . . .
2. 5 Input Impedance of the Doubly Loaded
Linear Antenna. O O O O O O O O O O O O O O O O O O
3 Optimum Loading for a Traveling Wave Distribution

ofAntennaCurrent.............. ..... .
3.1. Physical Interpretation of the Distribution of
Current ona Doubly Loaded Dipole - - . . . . . .
3. 2 Optimum Loading Impedance for a Traveling
Wave Distribution of Cylinder Current . . . . . .
3.3. Purely Resistive Optimum Loading . . . . . . . .
3.4. Purely Reactive Optimum Loading . . . . . . . .
3. 5. The Distribution of Current and Input
Impedance Corresponding to an Optimum
Loading.......................
3. 6. Calculation of the Expansion Parameter

\II(z).........................

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PAR T II.

TABLE OF CONTENTS (continued)

Page

Radiation Characteristics of a Traveling Wave
LinearAntenna..................... 72
4.1. Distribution of Cylinder Current for

Calculation of Radiation Fields . . . ..... . 72
4. 2. Radiation Fields of the Traveling Wave

LinearAntenna.... 75
4. 3. Comparison of Radiation Patterns for

Traveling Wave and Standing Wave

LinearAntennas................. 81

Experimental Study of Traveling Wave Antenna
with Non-Dissipative Loading . . . . . . . . . . . . . 87

5.1. Object of the Experimental Investigation . . . . 87
5. 2. Description of the Experimental Arrangement . 9O
5. 3. Traveling Wave Distribution of Current on

Monopole with Purely Non-Dissipative Loading. 99
5. 4. Effects of Variations in Loading Parameters

and Frequency Upon the Traveling Wave

Distribution of Current . . . . . . . . . . . . . 103
5. 5. Input Impedance of a Traveling Wave Linear

Antenna with Non-Dissipative Loading . . . . . 109

Traveling Wave LOOp Antenna with Optimum

Impedance Loading
Introduction. . . . ................. . . 114
6.1. Introduction . ...... . ...... . . . . . 114

. 2. Definition of a Traveling Wave Loop Antenna . . 114
3 Important Characteristics of a Traveling
WaveLoopAntenna 116
Previous Research on the Traveling Wave
LoopAntenna.......... .... ..... 118
Object of the Present Research . . . . . . . . . 120
Outline for Theoretical Investigation of a
Traveling Wave Loop Antenna with Optimum
Impedance Loading . . ...... . . . . . . . 121

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Distribution of Current on a Doubly Loaded
CircularLOOpAntenna................. 122

7.1. Geometry of the Doubly Loaded Circular

LoopAntenna............. ...... 122
7. 2. Dimensions of Interest for a One-

Dimensional Theory. . . . . . . . . . . . . . . 124
7.3. A Rigorous Fourier Series Solution for the

Distribution of Current on a Doubly Loaded

Loop; Its Failure to Yield the Parameters

of an Optimum Loading . . . . . ........ 125
7. 4. Approximate Distribution of Current on a

Doubly Loaded Loop Antenna . . . . . . . . . . 135
7. 5. Input Impedance of a Doubly Loaded Loop

Antenna..................... 144

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TABLE OF CONTENTS (continued)
Page

8 Optimum Loading for a Traveling Wave
Distribution of Loop Current. . . . ...... . . . 145

8.1. Physical Interpretation of the Distribution

of Current on a Doubly Loaded Loop Antenna . 145
8. 2. Optimum Loading Impedance for a Traveling

Wave Distribution of Loop Current . . . . . . 146
8.3. Purely Reactive Optimum Loading. . . . . . . 148
8. 4. The Distribution of Current and Input

Impedance Corresponding to an Optimum

Loading. ........ . ..... . . . . . . 162
8. 5. Calculation of the Expansion Parameters
‘I’i(9), ‘I’q(9), and {1(9) ........... . . 169

9 Radiation Characteristics of a Traveling Wave
LoopAntenna................ ..... 182

9.1. Distribution of Loop Current for Calculation
ofRadiationFields............... 182
9. 2. Radiation Fields of the Traveling Wave
LoopAntenna.................. 184
9. 3. Comparison of Radiation Patterns for
Traveling Wave and Standing Wave Loop

Antennas . . . . . . . . . . . . . . . . . . . . 194
References ....... O O O O O O O O O O O O O O O O O O O O 201
Appendix A. Electromagnetic Potentials in Antenna Theory. 203

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3.7.

3.8.

3.9.

3.10.

3.11.

LIST OF FIGURES

Page

Geometry of Impedance Loaded Dipole . . . . . . . .

Resistance and Position of Optimum Dissipative
Loading as a Function of the Antenna Electrical
Length 0 O O ..... O O O O ..... C I O O O O O 0

Optimum Impedance (purely resistive at 600 mhz)
of Loading with Fixed Position as a Function of the
Antenna Electrical Length . . . . . . . . . . . . . .

Reactance and Position of Optimum Non-Dissipative
Loading as a Function of the Antenna Electrical
Length (h = 50 cm 2 X0 at 600 mhz) ..... . . . . .

Reactance and Position of Optimum Non-Dissipative
Loading as a Function of the Antenna Electrical
Length (h = 100 cm : 2X0 at 600 mhz) ........

Optimum Impedance (purely reactive at 600 mhz) of
Loading with Fixed Position as a Function of the
Antenna Electric Length (h = 50 cm) . . . . . . . . .

Optimum Impedance (purely reactive at 600 mhz) of
Loading with Fixed Position as a Function of the
Antenna Electrical Length (h = 100 cm) . . . . . . .

Amplitude and Phase of Antenna Current Corresponding
to an Optimum Resistance Loading [ RL] 0 as a Function
of Position Along Antenna. . . . . . . . . . . . . . .

Amplitude and Phase of Antenna Current Corresponding
to an Optimum Non-Dissipative Loading as a Function
PositionAlongAntenna . . . . . . . . . . . . . . . .

Input Impedance of Antenna with Optimum Resistance
Loading as a Function of its Electrical Length
(h : 31. 25 cm) 0 O C O O O O I O O O O O O O O C O I O 0

Input Impedance of Antenna with Optimum Reactance
Loading as a Function of its Electrical Length
(h = 100 cm) 0 O O O O O O O O O O O O O O I O O I O O 0

Input Impedance as a Function of Antenna Electrical

Length for Constant Resistance Loading of Fixed
Position (optimum at 600 mhz) ....... . . . . .

Vi

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38

39

42

43

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52

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56

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Figure Page

3.12. Input Impedance as a Function of Frequency for
Antenna with Loading Consisting of Reactive
Component of Optimum Impedance . . . . . . . . . . 61

3.13. Comparison of Approximate and Exact Values of
Expansions Parameters ‘I’(z) as a Function of
Electrical Position Along Antenna. . . . . . . . . . . 67

3.14. Comparison of Approximate and Corrected Values
of Expansion Parameter ‘1’ as a Function of
Antenna Electrical Length. . . . . . . . . . . . . . . 70

4.1. Geometry for Calculation of Radiation Zone
Electromagnetic Fields of a Traveling Wave

LinearAntenna..................... 73

4. 2. Radiation Patterns of Traveling and Standing Wave
Antennaswith80h=27r................. 82

4. 3. Radiation Patterns of Traveling and Standing Wave
Antennaswithfioh=71r/Z ............... 83

4. 4. Radiation Patterns of Traveling and Standing Wave

Antennas with Bob: 41T. . . . . . . . . . . ...... 84
5.1. Experimental Arrangement . . . . . . . . . . . . . . 91
5. 2. Structure of Model Monopole Antenna . . ...... . 92

5.3. Structure of L00p Type Current Probe . . . . . . . . 92

5. 4. Experimental Current Distribution of Antenna with
h = a ; Unloaded and with Optimum Reactance
o
Loadlng O O I O O O O O O O O O O O O O O O I O O O O O O 101

5. 5. Experimental Current Distribution of Antenna with

h = 2X0; Unloaded and with Optimum Reactance

Loading . C O O I O O O C O O O O O O O O O ....... 102
5. 6. Effect (experimental) of Variations in Monopole End

Length (h-d) upon its Traveling Wave Distribution
0f current 0 O O O O O O O O O O O O O O O O O O O O O O 105

5. 7. Effect (experimental) of Variation in Loading

Reactance on the Traveling Wave Distribution of
MonopoleCurrent................... 106

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LIST OF FIGURES (continued)

Figure Page
5. 8. Effect (experimental) of Variations in Excitation
Frequency on the Traveling Wave Distribution of
MonopoleCurrent................... 108
5. 9. Experimental Input Impedance as a Function of
Frequency for Loaded and Unloaded Monopole
Antennas O O O O O O O O O O O O O O O ........ O 111

7.1. Geometry of the Doubly Impedance Loaded

LoopAntenna..................... 123
8.1. Real and Imaginary Parts of the Normalized Complex

Wave Number as a Function of the Electrical Loop

Circumference O O O O O O O O O O O O O I O O O O O O 151
8. 2. Position of Optimum Non-Dissipative Loading as a

Function of the Electrical Loop Circumference. . . . 156
8. 3. Reactance of Optimum Non-Dissipative Loading as a

Function of the Electrical Loop Circumference. . . . 157
8. 4. Impedance of Optimum Loading (purely reactive for

[3 b = 2. 5) with Fixed Position as a Function of

Electrical Loop Circumference 80b. . . . . . . . . . 159

8. 5. Amplitude and Phase of Current Along Loop
Corresponding to an Optimum Non-Dissipative
Loading. . O O O O O O O O O O O O O O C C O O O O C O O 166

8. 6. Input Impedance of Loop with Optimum Loading as a

Function of the Electric Loop Circumference . . . . 168
8. 7. Current Expansion Parameter ‘I’i(9) as a Function

of Position Along the Loop . . . . . . . . . ..... 175
8. 8. Charge Expansion Parameter \I' (9) as a Function

of Position Along the Loop . . q . . . . . . . . . . . 176

8. 9. Current Expansion Parameter ‘111 as a Function of
the Electrical Loop Circumference . . . . . . . . . . 179

8.10. Charge Expansion Parameter ‘I’ as a Function of
the Electrical Loop Circumfere ce . . . . . . . . . . 180

8.11. Expansion Parameter ‘II as a Function of the
Electrical Loop Circumference . . . . . . . . . . . . 181

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LIST OF FIGURES (continued)

Loop Geometry for Calculation of Radiation
ZoneFields................. ..... 186

Radiation Pattern in Plane of Loop (6 :900) as a
Functionof¢forBob=l............... 195

Radiation Pattern in Plane of Loop (9 2900) as a
Functionof¢forfiob=1.5 196

Radiation Pattern in Plane of Loop (9 2900) as a
Functionof¢forf30b22.5 ............. 197

Radiation Pattern in Plane of Loop (0 2900) as a
Functionof¢forfiob=4.0 198

ix

PART I

TRAVELING WAVE LINEAR ANTENNA
WITH OPTIMUM IMPEDANCE LOADING

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CHAPTER I

INTRODUCTION

1. 1. Introduction

It is the object of the first part of this research to realize
a high efficiency traveling wave linear antenna and to evaluate its
corresponding circuit and radiation characteristics. An impedance
loading technique is utilized in which the cylindrical dipole antenna
is doubly loaded with a pair of identical lumped impedances. A
theoretical investigation of this configuration is carried out to
determine approximately the distribution of current on the cylinder
as a function of its dimensions, the excitation frequency, and the
impedance and position of the double loading. The optimum loading
impedance to yield a traveling wave distribution of current over most
of the cylinder is determined from this result. Particular emphasis
is placed upon the possibility of utilizing a purely non-dissipative
loading, since this would provide the means of realizing a high
efficiency traveling wave linear antenna. It is verified experimentally
that a traveling wave of current can indeed be excited on a linear
antenna through the use of a properly positioned purely reactive loading.
The input impedance and radiation fields of a traveling wave linear
dipole having such a non-dissipative optimum loading are evaluated
as a function of the cylinder dimensions and the frequency of
excitation.
l. 2. Definition of a Traveling Wave Linear Antenna

A traveling wave linear antenna is defined as a linear antenna

which supports a traveling wave distribution of current. The traveling

wave of current ;

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wave of current is excited by a voltage generator at. the center of the
linear dipole, and travels outward toward its ends. While the
amplitude of the current wave decays as it advances outward from the
excitation point, since it continuously radiates energy into space, its
phase is essentially a linear function of position along the antenna. In
this research, a restriction is made to the class of long thin-wire
anteims such that an expedient but approximate one-dimensional theory
may be utilized with good accuracy. In this one-dimensional approx-
imation, the distribution of dipole current is assumed to flow parallel
to the axis of the thin cylinder comprising the linear antenna.

The characteristics of thinuwire center fed linear antennas
have been studied extensively. Historically, Hallenl developed the
first mathematical theory describing the circuit properties of a linear
antenna. The result of this theory was an integral equation for the
axial distribution of current on the linear dipole. Hallen was, however,
unable to find a simple closed form solution to this integral equation.
Some time later, KingZ obtained an approximate closed form solution
to Hallen's integral equation through the use of an iterative technique.
This solution indicated that the distribution of current on a conventional
linear antenna consists essentially of a standing current wave. It was
found that these approximate theoretical results agreed very well with
corresponding experimental measurements of the antenna current.

Since it has been observed that the current distribution on an
ordinary linear antenna is essentially a standing wave, then evidently
some modification of its structure is necessary in order that it might

support a traveling wave of current. In the present research, an

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impedance loading technique is utilized to modify the distribution of
antenna current. This method consists of doubly loading the dipole
with a pair of identical impedances. When the loading is optimum,
that is, when its impedance and position are properly chosen, the
antenna may be made to support the desired traveling wave distri-
bution of current along the majority of its length.

A linear antenna is completely characterized by its distribution
of current. The dipole is fully described by its circuit and radiation
characteristics, which are readily determined from its current
distribution. From a knowledge of the current at its excitation point,
the input impedance of a linear antenna may be immediately calculated.
Similarly, the radiation pattern of the dipole is determined in a straight-
forward manner in terms of its distribution of current. Since the
circuit and radiation characteristics of a linear antenna are determined
by its current distribution, then it might be expected that these
characteristics should differ greatly for distributions corresponding,
respectively, to standing and traveling current waves.

1 . 3. Important Characteristics of 3. Traveling Wave Dipole

A conventional linear dipole antenna is highly frequency sensitive,
in that its input impedance is a strong function of the excitation frequency.
This frequency dependence is a direct consequence of the standing wave
distribution of antenna current. As the frequency of excitation is varied,
the maxima and minima of the standing wave of current shift in position
along the dipole. With the excitation potential fixed therefore, the
current at the driving point of the dipole, and hence its input impedance,

varies strongly Ravith changes in the excitation frequency. As a consequence

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of this frequency sensitivity, a conventional linear antenna is ordinarily
used only at a single frequency, or over a very narrow band of
frequencies.

In contrast to (the standing wave dipole, a traveling wave
antenna has an input impedance characteristic which is relatively
independent of frequency. This broadband character is a consequence
of the traveling wave distribution of dipole current. Since the
amplitude of the traveling wave of current is essentially constant along
the antenna, except for the smooth decay due to radiation, then a
variation in the excitation frequency does not result in a rapid change
in the current at the driving point. The input impedance of a traveling
wave dipole is therefore a relatively weak function of frequency. It
is this broadband character which is the most important property of
a traveling wave linear antenna.

The radiation pattern of a conventional standing wave dipole is
characterized by a single major lobe when the antenna is electrically
short. As the electrical length is increased, this single lobe splits
to form a new major lobe in conjunction with a minor lobe structure.
The beamwidth of the major lobe decreases as the antenna length
increases, which would result in increasingly improved directivity
(high gain) if it were not for the presence of the minor lobe structure.
These minor lobes, however, have an amplitude which is a large
fraction of that of the major lobe. Consequently, the directional
characteristics of a long linear antenna are not desirable for most

applications.

The rad;

0' ‘\-§ 1“": '-
fileCu.‘.e CoA. 8‘

. - ‘\ ‘
€.t\IIiCdl.}' Sr". ‘
'N‘" ‘ A -

sclerr. haA’lI‘fl a

 

'5?
Hi}! h
. U
e Isa?
“l—L‘f: t
3%

The radiation characteristics of a traveling wave linear antenna
are quite different from those of its standing wave counterpart. An
electrically short traveling wave dipole is characterized by a radiation
pattern having a single major lobe with a very wide beamwidth. As
the electrical length of the antenna is increased, the beamwidth of
this single lobe continually decreases. A minor lobe does not appear
in the pattern until the antenna length is much greater than that of the
comparable standing wave antenna. An electrically long traveling wave
antenna may thus be utilized to realize an improved directivity, this
improvement being a consequence of the relatively narrow major lobe
beamwidth which may be obtained without the appearance of a minor
lobe structure. When the traveling wave antenna is sufficiently long
that minor lobes finally do appear, their amplitude is lower than that
of the initial minor lobe structure associated with the standing wave
counterpart antenna.

The modified radiation pattern characteristic of a traveling
wave linear antenna may be desirable for certain purposes. In
particular, the wide beamwidth of a short dipole and the absence of
minor lobes associated with an electrically long antenna may be
useful for some applications.

1. 4. Previous Research on the Traveling Wave Linear Antenna

It has been established that a traveling wave linear antenna
may be realized through the use of a resistance loading technique.
Research has been reported on methods which utilize both lumped and
distributed purely resistive loadings. A traveling wave distribution

of current may be excited on a linear antenna having a purely dissipative

Icac'ing throng}:
Altshuie
named on a lit

:: sptzmum hint:
O I

162:? .

 

 

 

 

loading through the application of either of these techniques.

Altshuler3 proposed that a traveling wave of current could be
obtained on a linear antenna by doubly loading the dipole with a pair
of optimum lumped resistances placed a quarter wavelength from its
ends. This technique was motivated by an analogy between the
standing wave distribution of current on a linear antenna and the
standing wave of current on a section of lossless transmission line
having an equal length and terminated in an open-circuit. Such a
section of line may be matched by placing a resistance equal to its
characteristic resistance in series with the line a quarter wavelength
from the open-circuited end. The distribution of current on the
matched line is a traveling current wave, except on the quarter wave-
length section at the end where the standing wave persists. It was
reasoned by analogy therefore, that, by placing an optimum resistance
loading a quarter wavelength from the ends of a linear antenna, a
traveling wave distribution of current might be excited on all but its
' end quarter wavelength. That such a lumped resistance loading
technique could indeed yield the desired traveling wave of current was
verified experimentally, where a slowly decaying traveling current
wave was found to exist on the antenna between its excitation point
and the position of the loading.

It was found further than an approximately traveling wave
distribution of current was maintained for a range of frequencies
about that where the resistance and position of the loading were
optimum. Consequently, a relatively broadband input impedance

characteristic was measured for a dipole having a fixed resistance

Significantly ITK
cite loading xx.
cpsle current r
the input intpec
characteristic c
riiatic-n Chara.

experimentally :

 

 

‘Sad‘u‘
.qntage
:EC‘M-u
.““QUe
a S h
a"?
‘. E

loading of fixed position. When the excitation frequency was varied
significantly from its center value, where the resistance and position
of the loading were optimum, it was found that the distribution of
dipole current reverted back to an essentially standing wave. Further,
the input impedance again displayed the strong frequency dependence
characteristic of a conventional unloaded antenna. The expected
radiation characteristics of a traveling wave antenna were measured
experimentally for such a resistance loaded dipole.

It has been demonstrated by Wu and King4 that a traveling
wave dipole may be realized by constructing it of a dissipative
conductor whose resistance varies in a prescribed manner with
position along the antenna. Theoretically, such a loading will yield
a rapidly decaying traveling wave distribution of current along the
entire dipole. This distributed loading technique has the advantage
that the optimum resistance of the loading is essentially independent
of the excitation frequency, so that the traveling wave of current may
consequently be maintained over a wide band of frequencies. The
broadband input impedance characteristic and modified radiation
pattern generally associated with traveling wave antennas were
demonstrated to characterize this particular dipole structure.

Although each of the resitance loading techniques discussed in
the preceding paragraphs may be utilized to obtain a traveling wave
distribution of current on a linear antenna, they share a common
disadvantage. Traveling wave antennas realized through these
techniques have a very low efficiency due to the great amount of power

dissipated in the resistive loadings. In either case, the efficiency is of

A"
'I

the order of in"
bat. techniqu
1.5. Object of

11 has b(
5;: obtaining a :

:terna are V8?

ass :neffmiem
It is the

V . ‘
n- ~Lu

’ .‘i' '
.O‘:Aq e‘ I ‘
‘ “Cief'lC ‘\. I.

 

re." ‘
«n.121r10 ‘
b a 1mm

A
.e experienced a

will b “
e e.:ectire

san-.r.g wax e rii
In th.s r
per.ectly C’
Lily loaded ~
“7411116 excna‘
C"Wing a loam
‘L
we J '
“9111mm La

the order of 50% or less, ‘which severely limits the practical value
of both techniques.
1. 5. Object of the Present Research

It has been indicated that the resistance loading techniques
for obtaining a traveling wave distribution of current on a linear
antenna are very inefficient. Although these schemes may be
applied to yield the desired traveling wave of current, 50% or more
of the power supplied by the source is lost in joule heating of the
dissipative loading, rather than being radiated into space. Such a
gross inefficiency is intolerable in the majority of applications.

It is the object of the present research therefore to realize a
high efficiency traveling wave antenna through a new technique
utilizing a lumped non-dissipative impedance loading. Since the
loading is to be non-dissipative with this method, no power loss will
be experienced and essentially all the power supplied by the source
will be effectively radiated. The efficiency of such a traveling wave
linear antenna will thus be comparable to that of a conventional
standing wave dipole.

In this investigation, the antenna is assumed to consist of a
thin perfectly conducting cylinder which is excited at its center and
doubly loaded with a pair of identical impedances placed symmetrically
about the excitation point. There are two degrees of freedom in
choosing a loading with such a configuration; its impedance and position.
The optimum loading to yield a traveling wave distribution of current
on the cylinder is to be determined. In particular, the possibility of
utilizing a properly positioned purely reactive optimum loading is to

be investigated.

. . i-
3.16. radiation C»

.. - .5 -
Extinction u. k

   

1.6. Outline 5-
with Optir

 

*.... 60.11111..le 1.“.
“.;A1‘ h “ vx
m...<1..\, all GD“)
r! ' 1 ,. ’ ,. .
‘f“e ‘5 “nCE¢ -C

1.: .l'ns Of tL-V
the paramete S
Lstr.but:on .0
:fctziizing a r

- H"
CITIESandU‘l"

Through this reactive loading technique, the desirable circuit
and radiation characteristics associated with a traveling wave
distribution of current may be obtained without the introduction of
dissipative elements. A traveling wave linear antenna is thus
realized while retaining the high efficiency of a conventional standing
wave dipole.

l. 6. Outline for Investigation of Traveling Wave Linear Antenna
with Optimum Impedance Loading

The present investigation of a traveling wave linear antenna
with optimum impedance loading is broken into two distinct parts.
Initially, an approximate theoretical study of the doubly loaded
dipole is undertaken to determine the parameters of an optimum
loading which will yield a traveling wave distribution of cylinder
current. At a later point, an experimental study is made to verify
these theoretical results. It is demonstrated in particular that the
theoretically predicted optimum impedance loading will indeed yield
a traveling wave of current on a linear antenna.

It is the purpose of the theoretical analysis to: (1) determine
approximately the distribution of current on the doubly loaded cylinder
as a function of its dimensions, the excitation frequency, and the
impedance and position of the loading; (2) obtain from this result
(in terms of the cylinder dimensions and its frequency of excitation)
the parameters of an optimum double loading to yield a traveling wave
distribution of current on the cylinder; (3) investigate the possibility
of utilizing a purely non-dissipative optimum loading; (4) calculate the
corresponding circuit and radiation characteristics of a linear antenna

utilizing such an optimum impedance loading.

An expe :-
x'erifx'that, at a;
current may in

antrely reactz‘.

ups: the distrzl.
mace cf the in:

wave antenna \x' -

 

 

 

10

An experimental arrangement is to be assembled in order to
verify that, at a given frequency, a traveling wave distribution of
current may indeed be excited on a linear antenna through the use of
a purely reactive optimum loading. Further, the circuit characteristics
of such a traveling wave dipole are to be studied experimentally. The
effects of variations in the excitation frequency and loading parameters
upon the distribution of antenna current are evaluated. A study is also
made of the frequency dependence of the input impedance to the traveling
wave antenna with purely non-dissipative loading. These experimental
results are compared with similar ones for a corresponding conventional

standing wave linear antenna.

APPROX.
Dc

3.1. Geometry
The geort.

take: to be as in"

‘ V
“van

...c-.ar cylinde r
:ezzer by a barn

p:tenti“l V "
C

o -
:centical lumne d

-
\

center. Witt
..eec:m in chess

JT-pedance s a re

2mm

er at the e'

 

CHAPTER 2

APPROXIMATE DISTRIBUTION OF CURRENT ON A
DOUBLY LOADED LINEAR ANTENNA

2. 1. Geometry of the Doubly Loaded Linear Antenna

The geometry of the doubly impedance loaded linear dipole is
taken to be as indicated in Figure 2. l. A thin perfectly conducting
circular cylinder of length 2h and diameter 2a is excited at its
center by a harmonic voltage source of angular frequency w and
potential VO . The cylinder is symmetrically loaded with a pair of
identical lumped impedances ZL at a distance d on either side of
its center. With such a configuration, there are two degrees of
freedom in choosing a loading; its impedance and position.

In this research, both the source of excitation and the loading
impedances are idealized to be point elements. The gap in the
cylinder at the excitation point z = 0 is assumed to be centered about
that point and to have a length of 2.6 . Similarly, the gaps at the
loading impedances at z = :I: d are assumed to have a length of 26
and to be centered about those points. The point element assumption
then corresponds to letting 6 tend to zero as a limit. This mathe-
matical approximation is equivalent to the physical requirement that
the linear dimensions of the excitation and loading elements be
negligibly small compared with the length of the cylinder itself.

2. 2. Dimensions of Interest for a One -Dimensiona1 Theory
It is assumed that the linear antenna consists of a long thin

cylinder whose half-length is very much greater than its radius,

where the latter is taken to be a small fraction of the wavelength.

ll

12

 

12(2)

 

 

-- *Za

 

 

Figure 2.1. Geometry of Impedance Loaded Dipole

Under these Cir

351;: its axis .

recited on the c

(U

wzzcn :IC-ws DE: 2‘

which are two:

'1‘. y
01"",

t..-s BBQ?“
can

.5 as ~ ‘

J
‘-
.::'Cx ‘

as
c1:
1 a “‘er
:55.

n

a... ‘ .
as. .s, tne cur

 

earl”?
‘J‘led

. I"
it”;

N. ate

3 {L
A‘e

in. s

'k‘d

 

13

Under these circumstances, and due to the symmetry of the cylinder
about its axis, it may be assumed that the distribution of current
excited on the cylinder by the source at its center is one-dimensional.
That is, the current is assumed to have only a z-component Iz(z)
which flows parallel to the cylinder axis. The dimenionsal restrictions
which are implicit in such a one ~dimensiona1 theory are thus

h > > a

(2.1)

[30a < < l
where (30 : ZTr/XO is the free space wave number which corresponds
to the free space wavelength )‘o .

Conditions (2.1) are also sufficient to validate the usual
approximation technique utilized in the study of thin wire antennas.
With this technique, the vector potential at the antenna surface is
calculated as the contour integral of the total antenna current, which
is assumed to flow along its axis. In reality, the current flows
throughout the cross section of the cylinder, and is actually most
concentrated along its surface due to the skin effect phenomena. The
vector potential at the antenna surface should in general, therefore,
be calculated as a volume integral of the current density on the
cylinder. However, it has been indicated by Hallen1 and King2 that,
when conditions (2.1) are satisfied, the error introduced by the above
mentioned approximation is negligible. This approximation technique
facilitates the solution for the distribution of antenna current, which

would otherwise be very much more complicated.

13. Forrnulat:
C\'ll!‘.d€ I' (-

The b03r*

smsfied,then t

 

..becne~din:en
.zfier t

L .‘ .
“€38 CII'C

.:.‘:
v.51 '1‘. ‘
e cynnce r v.

l4
2. 3. Formulation of an Integral Equation for the Distribution of
Cylinder Current
The boundary condition on the electric field at the antenna

surface is
(sx'fi) = o (2.2)

where fl is a unit outward normal vector at a point on the surface and
E the electric field at the same point. This condition requires that

the tangential component of electrical field be continuous across the
surface of the cylinder. Since conditions (2.1) are assumed to be
satisfied, then the distribution of current on the cylinder may be taken
to be one -dimensional, i. e. , to have only an axial component Iz(z) .
Under these circumstances, the tangential electrical field at the surface

of the cylinder will have only a z-component and condition (2. 2) becomes

E:(z) = Eiz(z) (2.3)

where E:(z) is the field just within the surface of the cylinder at
r = a- and E:(z) is the field at r =a+ just outside its surface.

Since the cylinder comprising the linear antenna is taken to
be perfectly conducting, then the applied field E:(z) may be non-
vanishing only in the gaps at z = 0, id . Thus E:(z) may be
expressed as

I z 1(d)
—£—?—-—— for -d-6<z<-d+6

N
0'!

V0

3.
Ez(z) = i ‘ 26

for -6< z< 6 (2.4)

 

z L12m)
K "—2 8—“

 

for -d-6< z<d+6

where 1 (d) iS
z
zer: in accords~

the spare: r y c

-Ln 0 0-1 “
...t‘ .O.dl \‘Cltai‘

l
‘4 ‘ avwlt .
- fa“. conszr

«sere 6(2) AS ‘
Tne Indy
H enenna, dc

“2e 9‘ .
.letlnje
:i’cm ‘
A\". it :
«S D083 L
:‘r-‘tfi-F‘
“«d
is and :

 

15

E:(z) = 0 for all other point on - h 5 z: h (2. 5)

where Iz(d) is the current at the loading impedance and 26 tends to
zero in accordance with the point element assumption. In result (2. 4),

the symmetry of the distribution of current has been utilized as
Iz(-z) 2 12(2) (2. 6)

The total voltage drop along the cylinder must be given by

h
-S‘h E:(z) dz = VO - 2 ZL Iz(d) (2-7)

A result consistent with equations (2. 6) and (2. 7) is
E:(z) = - v0 5(2) + zL Iz(d)[6(z-d) + 6(z+d)] (2. 8)

where 6(z) is the Diract delta function.
The induced electric field E1z(z) just outside the surface of
the antenna, due to the current and charge on the cylinder, may be

calculated from the vector and scalar potentials5 (see Appendix A)

A, 4) as
° 87?
1
gm — - (WMZ - (57) (2.9)
Z
Since the time variation is assumed to be harmonic of the form ert ,

then it is possible to make the replacement 5%- -* jw, where the
potentials and fields are then understood to be complex valued. There

is thus obtained

E:(z) = -(vc)>)z -ijz (2.10)

The vector a I:

which may be

vat: Which 9c 1..

 

.- Sati‘:
5.}. eh, .
5A.L‘ :
lie
C‘I‘“nder .
2‘ sz' v
e ut‘C't‘ I"
3
dz“ ’
‘T-ls r44:
‘4”:ere
‘ri.
:PF‘NI
".‘JiErv

16

where Az(z) and ¢(z) are the potentials at the surface of the cylinder.
The vector and scalar potentials are related by the Lorentz condition,
which may be expressed in the form

4. =l‘Bzv-X (2.11)
F5

O

with which equation (2. 10) becomes

Ei(z) = - i”— [ WV- lib]z .. ijz (2.12)

Since the distribution of cylinder current is axial, then the vector

potential will have only a z-component Az(z) as well, 5 and equation

(2. 12) gives

1 '0.) z .
Ez(z) = - 32 2 - JUJAZ (2.13)

(3 8z

 

If results (2.8) and (2.13) are substituted into condition (2. 3),
to satisfy the boundary condition on the electric field at the surface of
the cylinder, then a second order inhomogeneous differential equation

for the vector potential at the antenna surface is obtained as

 

82 Z jBoZ
$2— + 50 Az(z) = w ( -VO6(z) + ZLIz(d)[5(z-d)+5(z+d)]}

(2.14)
This differential equation must be satisfied for - h E z _<_ h. A

complementary solution of equation (2.14) is well known to be

112(2) = c1 eJBOZ + cze‘lfioz - h < z < h (2.15)

 

surface is (Ara
4A (2) :
Z
.31”? ‘ . ‘1‘
is ‘us LAle ehc‘
e:-~

315% ,
.Lt.\,n (2.17

(D

3 C1, Wk:

A (2')

 

17

where C1 and C2 are arbitrary complex constants. The particular

solution is determined as

 

-jsolzl _ M[e-jfiolz-dl ,e-jaolzml]

2v
0

Alz)(z) = e

-h_<_th (2.16)

which is readily verified by direct substitution into differential equation
(2.14). In this last result, v0 = 119/130 is the velocity of propagation in
free space. The complete solution for the vector potential at the antenna

surface is obtained by the superposition of results (2.15) and (2.16) as

 

. . V .
Az(z) = (31.3130Z + Ca e'JF3oz + 2 V: e-JBOI 2|

[e'jfioiz‘di + e'jBolz'l'dl] _ h < z< h
(2.17)

Since the distribution of current on the cylinder is symmetric
about the excitation point, then the vector potential at its surface

exhibits a similar symmetry2 such that

A (-z) = A (z) (2.18)

Z Z

Solution (2.17) may be made to satisfy this boundary condition only if

C2 = C1 , which yields the simplified result

. . V .
A (z) : c eJfiOZ +c e'JFOZJ. _2. e-Jfiolzl
z 1 l 2vO
Z I d . .
_ L z() [e—molz-dl +e'JBOIZ’FdI] _h< M h
ZVO _ _.

(2.19)

(‘ shtz-uid be not

been obtains d s

afmctional cie;

rte-re p, 15:;
o ..

.c ,'_
3- ...e Green's g

 

 

 

18

It should be noted that a solution in terms of complex exponentials has
been obtained since a traveling wave distribution of current having such
a functional dependence is to be sought eventually.

As indicated in section 2. 2, the vector potential at the cylinder
surface may be written as the Helmholtz integral over an assumed
axial distribution of current (see Appendix A) as

p. h
Az(z) = 1,95 Iz(z')K(z.z')dz' -h_<_ 25h (2.20)

where (.10 is the permeability of free space and the kernel K(z, z')

is the Green's function

 

-j(30~/(z--z')2 + a2
K(z, z') = e (2.21)

~/(z-z')2 + a2

The factor R = ~l(z-z')2 + z‘2 in the Green's function (2. 21) represents

 

 

the Euclidean distance between an element of current on the cylinder
axis at z' and an observation point on its surface at z .

If the two expressions (2.19) and (2. 20) for the vector potential
Az(z) at the antenna surface are equated, there is obtained for - h 5 z

_<_ h the result

 

p. h . . V .
—9' I (Z') K(Z. Z') dz' = C 831302 + C e'Jfioz + O e'JBol zl
417 h z 1 1 2V
"’ 0
Z I (d) . .
.. __ZI"v—z_ [e-JBOI Z-dl +e'JfiO|z+dl] (2.22)
o

This expression is an integral equation for the distribution of current

19

12(2) on the doubly loaded linear antenna. The cylinder dimensions

h, a as well as the impedance and position Z d of the double loading

L’
appear as parameters in the equation. Two as yet undetermined constants
C1 and Iz(d) appear on the right hand side of the integral equation,
and must be evaluated through the application of a pair of subsidiary
conditions.

Since the antenna structure terminates at z = h, then the
cylinder current at that point must vanish. Further, the distribution
of cylinder current must be continuous, with the result that the
condition Iz(z=d) = Iz(d) must hold at the location of the loading. The

solution Iz(z) of integral equation (2. 22) must therefore satisfy the

subsidiary conditions

II
C

I ( =h)
Z z (2.23)

Iz(z=d) Iz(d)

These conditions are sufficient to facilitate evaluation of the constants

C and I (d).
z

1

It is to be noted that, in the special case where Z = 0,

L

. . . . . 2
equation (2. 22) reduces to a variation of Hallen's integral equation
for the distribution of current on a conventional linear dipole. This

is to be expected, since when Z = O the structure of the doubly

L

loaded cylinder reduces to that of. an ordinary linear antenna.

’IAI1 I.‘/\
1". “UR (
.

3: CLLTI'E.iI 1.. ..

> .3

'F ”-9; )
C‘s-.‘C.1_“(~I 2
1‘1?" Va ..

a. L..e r‘L:.t .-

“62‘. 1.“ ..
. H

.JQr‘e- .
d C}“r.r:(

‘ a‘mple I-Lh
t: .. -
163Q1t(3 )
9 g‘
3 -.,

20
2. 4. Approximate Solution for the Distribution of Current on the
Doubly Loaded Cylinder

It has been indicated that, for the special case where ZL = 0,
equation (2. 22) reduces to Hallen's integral equation for the distribution
of current on a conventional linear antenna. Hallen's equation has
been solved approximately by King. 2 A relatively simple closed form
expression for the distribution of cylinder current may be obtained
through what has become known as the King -Middleton iterative
technique. The application of this technique to obtain a solution to
equation (2. 22) is, however, impractical since: (1) the constant Iz(d)
on the right hand side of the integral equation depends upon the as yet
undetermined distribution of current; (2) the additional terms occurring
on the right in equation (2. 22) for ZL )6 0 are equivalent to a pair of
shifted sources, which complicate such a solution to the extent that it
becomes very unwieldy.

An integral equation similar to result (2. 22) has been encountered
by Chen6 in the investigation of electromagnetic scattering from a doubly
loaded cylinder. Through an approximate technique, a rather complex
solution for the distribution of cylinder current was obtained in terms
of simple functions. This integral equation was essentially identical
to result (2. 22), with the term involving the excitation potential VO
omitted. The inclusion of this term in equation (2. 22) makes Chen's
approximation technique too intractable to provide a useful solution.

A more approximate technique than those mentioned in the
preceding paragraphs has been reported by Wu and King4 and later by

Chen in conjunction w1th determining the distribution of current on an

'maeéance l:

wxhccrresp

II 1:16 Dear-(12‘ .

Zievectcr pg?

I'r]

I'Ier. by

Where K( 2, Z,

 

15a ;‘ h

tu‘lCtI(Jn (
I'eC‘r y-

-d

‘ pot€111:
C"
C ‘ .

‘uqtlon

:e‘ |

 

21

impedance loaded cylinder. In the latter case, an excellent agreement
with corresponding experimental results was observed. This approxi-
mation technique consists essentially of assuming that the ratio of
vector potential at a point on the antenna surface to the current at the
same point is constant along the cylinder.

The motivation for this approximate method is a consequence
of the peaking property of the kernel K(z, z'). It has been found that

the vector potential at a point z on the surface of the cylinder is

 

 

 

given by
“0 h
A(z) =—— I(z')K(z,z')dz' -h<z<h
z 417 -h z — —
(2.20)
where K(z, z') is the Green's function
e-j130~/(z..z')Z + a2
K(z,z') = (2.21)
~/(z-z')2 + a2

The peaking property of this kernal is exploited to formulate an
approximate solution for the distribution of current on the doubly
impedance loaded cylinder.

Since a < < h by the thin-wire assumption, 1. e. , condition
(2.1), then K(z, 2') has a very sharp peak at z' : z when considered
as a function of z' on - h E z‘ _<_ h. Hence the contribution to the
vector potential Az(z) at each point on - h j z 5 h, as calculated
from equation (2. 20), is due primarily to current elements in a small

neighborhood about the point z' = z . Because the distribution of current

Iz(z) is continuous, then the current Iz(z') for z' = z makes the major

contribution ‘
I: is expectec
“hould be ess
-h_<_ z E h .

The p:
identified app

:15 I'Eplacena‘

91:1 - .

“‘Stant dimer

 

 

22

contribution to the vector potential at a point z on the cylinder surface.
It is expected from this argument therefore, that the ratio Az(z)/Iz(z)
should be essentially constant at each point along the cylinder, or for
- h 5 z _<_ h .

The preceding conclusion is obtained directly if K(z, z') is
identified approximately with the Dirac delta function 6(z-z') . Making
this replacement in equation (2. 20), there is obtained

A212)

12(2)

 

(.1
74.? -115th (2.24)

which is equivalent to the remarks of the last paragraph. An essentially

constant dimensionless quantity ‘II(z) is thus defined by

Az(2) H

0
12(2) = 71-.”- \Il(z) - h: 2: h (2.25)

 

and is designated as the "expansion parameter. " It has been indicated
by King, 2’ 4 in agreement with the above arguments, that ‘Il(z) is
essentially independent of 2 except near z = :1: h, and is determined
primarily by the antenna dimensions. It is thus asserted that ‘I'(z) = \II
is indeed a constant depending only upon the antenna dimensions. The
validity of this assumption will be discussed more fully in section 3. 6
of the following chapter.

According to the approximation discussed above, the distribution
of current on the cylinder is related to the vector potential at its surface
as

Iz(z) = $713; Az(z)

O

Y
,‘
u.

9

' l

. l
‘
41.

C1

attention (2 )-.
o h I

inexpressio:
has been obta.
aztenna curre-
frsn‘. equati:

,7“
..

1(2)
2

Z :
) 12-2)

R651-

\A

l:

and I (d)
2

 

23

An expression for the vector potential at the surface of the cylinder
has been obtained, however, as equation (2.19). The distribution of
antenna current on - h E z E h is therefore obtained approximately
from equations (2. 26) and (2.19) as

ZTrV

 

 

12(2) : 51‘? C1 ejfioz + “41:1, Cl e‘jfioz + 27172 e‘jfiolzl
O O 0
2w 43 lz-dl as lz+dl
' to? ZL Iz(d) [ e O + e 0 ] (2.27)

where {,0 = «ML 760 = 120w is the 1ntr1n31c impedance of free space.
It is to be noted that this expression satisfies the symmetry condition
Iz(z) = Iz(-z) .

Result (2. 27) contains the two as yet undetermined constants
C1 and Iz(d) . These constants may be evaluated by applying to

solution (2. 27) the boundary conditions (2. 23), which are given by

ll
0

Iz(z=h)
(2.23)

I (d)

Z

Iz(z=d)

The straightforward application of these conditions to expression (2. 27)
yields finally the approximate distribution of cylinder current on
- h E z E h as

Vn’ D D

 

Iz(z) : CO‘I’ __1 e'jfioz + T31 ejfioz + Ze‘jfiolzl
o 2 2
Z D . . .
_—_—L __1 cos 5 d + e'JBod e'JBolz'dl + e'JBolz+d|
30T 2 o

(2.28)

upon the exc

impedance a:

An 3;);

me doubly loa

«terms Of its
ind ; -
~POS.ticn 0‘

(v

 

 

24

In this result, the factors T, D1, and D are constants depending

2

upon the excitation frequency, the cylinder dimensions, and the

impedance and position of the loading as

 

 

Z .
_. L 'Jzflod
T — ‘I’+—6—O (1+e ) (2.29)
Z
L -‘ d
D1 = 30,1. e Jfio cos pod —1 (2.30)
jp h ZL 2
o -
D2 e cos [30h 30T cos (30d (2.31)

An approximate expression for the distribution of current on
the doubly loaded cylinder has been obtained in equation (2. 28). This
distribution completely characterizes the linear antenna, and is given
in terms of its dimensions, the excitation frequency, the impedance
and position of the double loading, and the as yet undetermined expansion
parameter. The optimum loading to yield a traveling wave distribution
of current on the cylinder will be obtained from this result in the
following chapter. This traveling wave current distribution will then be
utilized to calculate the value of the expansion parameter ‘I’ .
2. 5. I Input Impedance of the Doubly Loaded Linear Antenna

The input impedance to the doubly loaded linear antenna is

defined by

V

_ o
Zin — —-————Iz(z:0) (2.32)

This expression is readily evaluated by using result (2. 28) for the
approximate distribution of current on the doubly loaded cylinder,

and is found to be

 

pro); 13‘. "

Li.

a
r.

A
l\~o

25

D Z D

_ _1 _L as d _1 "350d
Zin — 60‘1’ 1+ D2 - 3OT e 0 D2 cos (30d + e

(2.33)

An approximate relation for the input impedance to the doubly loaded
linear antenna is thus obtained in terms of its dimensions, the frequency

of excitation, and the impedance and position of the loading.

the cons. V .V

-v-~- a ‘-
i215 Svlutlk

....'.’ g . .

:u...L.enL 5‘
"rtw; o ' ,
55-..C.€C U

'2‘:‘: ‘1
“A has
e L“.

 

 

 

CHAPTER 3

OPTIMUM LOADING FOR A TRAVELING WAVE
DISTRIBUTION OF ANTENNA CURRENT

3.1. Physical Interpretation of the Distribution of Current on a
Doubly Loaded Dipole

An approximate expression for the distribution of current on
the doubly loaded linear antenna has been developed as equation (2. 28).
This solution is valid on - h E z E h, but since Iz(-z) = Iz(z) it is
sufficient to consider only the current on O E z E h. If attention is
restricted to the regions 0 E z E d and d _<_ z _<_ h, then the various
terms in expression (2. 28) may be combined to yield a pair of results
which are more physically meaningful.

The distribution of cylinder current for O E z i d is obtained

from the general result (2. 28) as

 

 

V TT D Z . D . .
IZ(Z) = Z231, 2 + B; - 30.1.. e JBO (T5; COS Bod + e JBO ) e 350
D Z . D . .
_1 _ L flfiod _1 160d Jfioz
+ D2 3OTe (D2 cos Bod+e e

(3.1)

while that on d E z E h becomes

V 11' D Z D . .
_ o 1 L l 160d -JBoZ
Iz(z)— C—J‘F 2+DZ ~15Tcos Bod (D2 cos 80d+e ) e

1 )st (3.2)

26

5:22: expre

.__4
H
)4
In

CLLI'I'EDE ma

r

 

27

A physical interpretation of the current distribution is readily obtained
from expressions (3.1) and (3. 2).

It is observed from equation (3.1) that on O E z E d the total
current may be considered as a superposition of a pair of outward and
inward traveling current waves. The first term in this result represents
an outward traveling wave of current which is excited by the source at
z = 0. At z = d this current wave is partially reflected and partially
transmitted. The second term of equation (3.1) represents an inward
traveling wave of current which results from the reflection of the
outward traveling wave by the impedance discontinuity at z = d.

Similarly, equation (3. 2) indicates that the distribution of cylinder
current on d E z _<_ h is composed of a pair of Oppositely directed
traveling current waves. The first term of this expression represents
an outward traveling wave of current which is excited at z = d by the
transmitted portion of the outward wave on O E z E d. This current
wave is reflected by the structural discontinuity in the cylinder at
z = h. The second term of equation (3. 2) represents the inward traveling
wave of current which results from the reflection of the outward wave at
z = h.

It is indicated therefore, that inIgeneral both outward and inward
traveling waves of current are supported on each of the two regions
0 E2 E d and d E z E h. The superposition of the oppositely directed
traveling waves results in a standing wave distribution of current along
either region. Thus in the usual case a standing wave of current is

supported along the entire cylinder.

3.2. Optiniu:
Cylinde

1132a:

I).

Lcaded cylin-

t

current over

”If It: OX'e r

 

28

3. 2. Optimum Loading Impedance for a Traveling Wave Distribution of
Cylinder Current

It has been indicated in the preceding section that the doubly
loaded cylinder generally supports a standing wave distribution of
current over its entire length. The possibility that this distribution
might be modified, through the selection of an optimum impedance
loading, to yield a purely outward traveling wave distribution of
current over most of the cylinder is now to be investigated.

It is evident physically that no choice of the loading will give
a purely outward traveling current wave on d E z E h, since the
inward traveling wave on that region is due to a reflection from the
unalterable structural discontinuity at z = h. However, it is reason-
able to suspect on physical grounds that if the loading is properly
chosen the inward traveling wave on O E z E d might be eliminated,
leaving only the desired outward traveling wave of current over that
region. Since the inward traveling wave on O E z E d is actually
reflected from the impedance discontinuity at z : d, it is expected
that such an optimum loading should exist.

The optimum loading impedance to yield an outward traveling
wave distribution of cylinder current on O E z E d may be obtained
from expression (3.1). This situation evidently requires that the
inward traveling current wave on that region should vanish. Realization
of this condition is accomplished by equating the coefficient of the second
term in equation (3.1), which corresponds to the amplitude of the inward

traveling wave, to zero as

‘ .
_..
“-~er C (:‘n C T r

' o . 9—. v
kcs‘ ‘rV-U {L‘E
[4
I'L
h en c
“ ‘u'le 7(-
.‘ a

 

 

 

Z9

 

D Z . D .
Di - 334T euJfi0d Di cos [30d + e-Jp0d> = O (3.3)
2 2

Using the defining relations (2.29), (2.30), and (2. 31) for T, D1, and
D2 , respectively, this equation may be solved for the Optimum loading
impedance, designated as [ ZL] O , to yield
eJ’Bod

. (3. 4)
cos [30d - eJBOul'd) cos Boh

 

[z = 30w

L]O

After considerable straightforward manipulation, result (3. 4) may be

cast into the simpler form

[Z = 30‘I’[l + j cot 80(li-d)] (3.5)

L] 0
When the loading impedance is given by this relation, the cylinder
current on O E z E (1 becomes the desired purely outward traveling
wave, while that on d E z _<_ h remains the usual standing wave.

Expression (3. 5) gives the optimum loading impedance in terms
of its position, the cylinder dimensions, and the frequency of excitation.
For a given set of antenna dimensions, this optimum impedance is a
function only of its position d and the frequency w . At this point
the loading location is completely arbitrary, and may be freely
specified in order that the corresponding impedance may satisfy
certain prescribed conditions.

It has been indicated that the optimum loading impedance [ ZL] 0
depends only upon its position d and the excitation frequency (0 once
the cylinder dimensions h and a have been specified. This leads one

to suspect that, at least at a single frequency, it should be possible to

choose an 0?
either pure‘i‘f

will be c ans:

CT. 1: terms

 

3O

choose an optimum position for the loading such that [ ZL] 0 will be
either purely resistive or purely reactive. These two special cases
will be considered individually in the following two sections.

Since ‘11 is in general a complex number, it may be written
as \II = u + jv . With this designation, the Optimum loading impedance

of expression (3. 5) becomes

[Z : 30(u + jv)[l + j cot (30(h-d)] (3.6)

L10

or, in terms of real and imaginary parts

2 3O {[ u - v cot (30(h-d)]

+ j[ v + u cot pom-(1)] } (3.7)

An explicit expression for the expansion parameter \I’ is to be
developed in section 3. 6.
3. 3. Purely Resistive Optimum Loading

In order to obtain the condition for a purely resistive optimum
loading, it is only necessary to equate the reactive component of the

optimum loading impedance given by expression (3. 7) to zero as
v + u cot (30(h-d) = O (3. 8)

This result yields the necessary position of an optimum purely resistive
loading as

h—d l -1 u

 

“—1

while its resistance becomes

lZEC‘I'GILC

. ’lm"I“
L-Jkgo‘.“ .-
.

Irequencv j

1“
h J
‘ ..(
““A‘\Gt€‘d ‘ V‘
— A -
.2 k t, -
*Vr I:

 

a t

.4 L416 ‘9‘.'
A. i"

N

f».
‘K, V

lr"...

< .

 

31

[RL]O = 3o[u -v cot 50(h'd1]
2
= 30(u+%—) (3.10)

Theoretically then, a linear antenna which is doubly loaded with an

optimum resistance [R as given by result (3.10), whose position

L] o
d satisfies equation (3. 9), will support a purely outward traveling
wave of current on O E z E d.

It will be indicated in section 3. 6 that ‘I’ = u + jv is a relatively
weak function of the excitation frequency. Thus it is observed from

expressions (3. 9) and (3.10) that (h—d)/>\ o and [R are essentially

L] 0
frequency independent.

The parameters of an optimum purely resistive loading are
indicated in Figure (3.1) as a function of the antenna electrical length
h/X o for the typical case of an antenna having a half-length of h = 31. 25 cm
and a diameter of 2a : O. 25 inches. These dimensions correspond to
a half-length of h 2 0. 625 X 0 at an excitation frequency of 600 mhz. In
obtaining these numerical results from expressions (3. 9) and (3.10) use
was made of equation (3. 37), which will be developed in section 3. 6.
This equation gives the expansion parameter \I' in terms of the antenna
dimensions h and a, and the excitation frequency w. It is noted
from the figure that both (h-d)/>\ o and [RL] 0 are essentially constant
over a wide range of frequencies. .Thus while the optimum resistance
of such a loading is almost constant, its necessary position is a strong
function of the frequency of excitation. This is evident since d then
depends directly upon X o = Vo/f’ where f is the frequency and v0 the

C
D

velocity of propagation in free space.

 

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33

The preceding theoretical results correspond to the research
reported by Altshuler. 3 As indicated earlier, Altshuler carried out an
experimental investigation of a linear antenna doubly loaded with an
optimum resistance. This experiment indicated that the value of the
optimum loading resistance was essentially constant, i. e. , for a
given set of antenna dimensions it was a weak function of the excitation
frequency. It was found further that such a purely resistive loading
should be placed approximately a quarter wavelength from the antenna
ends in order to obtain a traveling wave distribution of current.

For an antenna having the same dimensions (h = 31. 25 cm,
2a = 0. 25 inch) as those which were used in obtaining the theoretical
results indicated in Figure 3.1, Altshuler determined the parameters
of an optimum purely resistive loading experimentally as: [ RL] 0 =
240 ohms; (h—d) = 0. 25 X o' The excitation frequency utilized in the
experiment was 600 mhz (X o = 50 cm). A corresponding set of
theoretical values are obtained from equations (3.10) and (3. 9),
respectively, as: [RL] 0 = 220 ohms; (h-d) = 0.17 X 0' An excellent
quantitative agreement is thus observed between the present theory
and the available experimental results.

It has been indicated that the necessary position of a purely
resistive optimum loading is a strong function of the excitation
frequency. In a practical physical situation, .however, the location
of the loading must be fixed at some point along the antenna. Hence
an optimum purely resistive loading is possible only at a single
frequency. If the position of an optimum loading is chosen such that

its impedance is purely resistive at a given frequency, then at any

steer freque
reactive con
It is
:ptimum loa
becomes pur

ice dimensic

The c
a: this 10cati
electrical 19,
it is nOtEd th.‘
53: the frequ

Optimum 1r»- h

 

 

34

other frequency an optimum impedance must have both resistive and
reactive components.

It is interesting to consider the frequency dependence of an
optimum loading whose fixed position is so chosen that its impedance
becomes purely resistive at a given frequency. An antenna having
the dimensions h = 31. 25 cm and 2a = O. 25 inch will again be
considered. If the optimum loading is placed such that (h-d) = 0. 171.0,
then its impedance becomes purely resistive with [RL] 0 = 220 ohms
at a frequency where h = O. 6251\0 (600 mhz).

The optimum impedance of a loading having its position fixed
at this location is indicated in Figure 3. 2 as a function of the antenna
electrical length. This result is obtained from equation (3. 5), and
it is noted that the reactive component of the impedance vanishes only
for the frequency where h = 0. 625 x0 . At any other frequency, the
optimum impedance must have a reactive component in order to yield
a purely outward traveling wave of current.

It is noted from the figure, however, that for small frequency
deviations about the value where the reactive component of the optimum
impedance vanishes, the resistive component of this impedance is
essentially constant. Furthermore, for such frequencies, the reactive
component of the optimum loading impedance is much smaller than its
resistive component. Thus a constant purely resistive loading of
fixed position may be utilized to realize an approximately traveling
wave distribution of antenna current for a band of excitation frequencies.

As the frequency deviates further from its value corresponding to an

35

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36

optimum loading, the current distribution slowly reverts back to an
essentially standing wave. These are exactly the results which were
obtained by Altshuler in his experimental study.
3. 4. Purely Reactive Optimum Loading

The condition for a purely non-dissipative optimum loading is
obtained by equating the resistive component of the optimum impedance

given by expression (3. 7) to zero as

u-vcot (30(h-d) 2‘ O (3.11)

This result requires that the necessary position of an optimum purely

non-dissipative loading be given by

 

x _ 21? tan ‘3’ (3.12)

With this condition satisfied, the corresponding optimum loading reactance

becomes
[x = 3O[v+ucotBO(h-d)]

2
30(v+3V—) (3.13)

L10

It is indicated theoretically therefore, that a linear antenna which is
doubly loaded with an optimum non-dissipative impedance, whose
reactance [XL] 0 is given by expression (3.13) and whose position d
satisfies equation (3.12), will support a purely outward traveling wave
of current on 0 E z E d.

In section 3. 6 it will be indicated that ‘I’ = u + jv is relatively

independent of the excitation frequency, and depends primarily upon the

anenna c
indicate,

fiequencx

antenna 6‘ 1‘

ii13L T?

 

37

antenna dimensions h and a. Expressions (3.12) and (3.13) therefore

indicate, respectively, that (h—d)/>\O and [XL] are essentially

0
frequency independent.

Numerical values for the parameters of an optimum non-
dissipative loading will be obtained for a pair of antennas having the
following dimensions:

(1) h = 50 cm, 2a : 0.25 inch; h : k0 at a frequency of

f = 600 mhz.

(2) h = 100 cm, 2a : 0.25 inch; h : 2X0 at a frequency of

600 mhz.
The optimum loading reactance [XL] 0 and its necessary location
(h-d)/)\O are indicated in Figures 3. 3 and 3. 4 as a function of the
antenna electrical length h/kO for antennas (l) and (2), respectively.
These numerical results were obtained from expressions (3.12) and
(3.13). The value of the expansion parameter \I' was calculated from
equation (3. 37) of section 3. 6, which gives 11 in terms of the antenna
dimensions h and a and the excitation frequency w .

It is observed from the figures that, for either antenna, both
(h-d)/>\O and [XL] 0 are essentially constant for a wide band of
frequencies. Hence while the optimum reactance of such a non-
dissipative loading is rather constant, its necessary position is a
strong function of the excitation frequency. As for the case of a purely
resistive loading, this is evident since the loading position d depends
directly upon the wavelength X0 = vO/f, where f is the frequency at

which the dipole is excited.

2. . C:

 

38

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40

A comparison of Figures 3. 3 and 3. 4 indicates that these
numerical results are nearly identical for antennas (l) and (2). The
frequency dependence of the optimum loading reactance [XL] 0 and
its necessary position (h-d)/)\O are almost the same in either case,
even though the half-length of one antenna is twice that of the other.
In particular, at a frequency of 600 mhz the parameters of an

optimum non-dissipative loading for these two antennas are

(l) h

)‘o; [XL] 0

2x0; [ XL] 0 = - 363 ohms, (h-d) = 0.417 x0.

= - 366 ohms, (h-d) = O. 418 X0 .

(2) h

This result is a consequence of the fact (as will be discussed in
section 3. 6) that the expansion parameter ‘I’ is essentially indepen-
dent of the antenna half-length whenever this dimension is of the order
of a wavelength or greater. A similar situation is not observed when
the cylinder diameter 2a is varied, however, since ‘I’ is a strong
function of the cylinder diameter for all values of its half-length h.

It has been indicated that, for a given set of cylinder dimensions,
the necessary position of an optimum non-dissipative loading is strongly
dependent upon the frequency of excitation. Physically, however, a
practical arrangement of such a doubly loaded linear antenna requires
that the position of the loading be fixed at some point along the cylinder.
With this restriction, an optimum purely reactive loading may therefore
be realized only at a single frequency. If the location of such an optimum
impedance loading is so chosen that it becomes purely reactive at a
given frequency, then at any other frequency it must consist of both

resistive and reactive components.

 

 

41

Since the position of the impedance must be fixed, it is of
interest to consider the frequency dependence of an optimum loading
whose location is so chosen that its impedance becomes purely
reactive at a given frequency. In order to obtain some specific
numerical results, the two antennas considered previously will
again be utilized. The loading positions are chosen according to
equation (3. 12) such that the corresponding optimum impedances be-
come purely reactive at a frequency of 600 mhz (X0 = 50 cm). The

specific antenna dimensions and loading locations are then as follows:

(1) h = 50 cm, 2a = 0.25 inch, d = 29.1 cm; at 600 mhz:
h = 1.0, (h-d) = 0.418 10, [XL] 0 = - 366 ohms.

(2) h = 100 cm, 2a = O. 25 inch, (1 = 79.2 cm; at 600 mhz:
h = 210, (h-d) = 0.417 ho, [XL] 0 = — 363 ohms.

The optimum impedance of a loading with fixed position is
indicated as a function of the antenna electrical length in Figures 3. 5
and 3. 6 for the configurations corresponding, respectively, to antennas
(l) and (2) above. Since the optimum impedance is no longer purely
reactive at every excitation frequency, these results were obtained
from the general expression (3. 5), with the dimensions h and d
fixed at the appropriate values. It is noted that in either case the
resistive component of the optimum impedance vanishes only at the
frequency where h 2 X0 or h = 210, respectively (600 mhz). At
any other frequency, an optimum loading impedance must have a
resistive as well as a reactive component in order to yield a traveling

wave distribution of antenna current.

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The following important characteristics are observed from

Figures 3.

5 and 3. 6 for the case of an optimum impedance loading of

fixed position:

(i) In general, an optimum loading impedance requires both

(ii)

(iii)

(1V)

resistive and reactive components. The resistive
component vanishes only at a single frequency.

Both the resistive and reactive components of the optimum
impedance are strong functions of frequency. There is a
frequency just above that where the resistive impedance
component vanishes at which both impedance components
tend to infinity.

The sign of both impedance components changes within a
relatively narrow frequency range. An optimum impedance
thus requires a negative resistance component (active
element) at certain frequencies, and its reactive component
varies from capacitive to inductive as the frequency is
increased.

The reactive component of the optimum impedance has a

negative slope as a function of frequency.

It is therefore evident that synthesis of an exact optimum impedance,

to yield a purely outward traveling wave distribution of current at

every frequency, is out of the question. Although an Esaki diode might

be utilized to obtain a given negative resistance component, there is

no practical; means of realizing a variable negative resistance having

a frequency dependence of the type indicated.

Since the high efficiency associated with a non-dissipative

ts erzectix'z

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45

loading is of fundamental interest, one is led to consider, when
dealing with a loading of fixed position, a loading consisting of the
reactive component of the optimum impedance. Such a purely non-
dissipative loading is optimum only at a single given frequency, and
its effectiveness diminishes as the frequency of excitation is varied
from this value. That is, the distribution of cylinder current on

O E z E (1 will consist of a purely outward traveling wave at an
excitation frequency corresponding to the given center frequency, but
will gradually revert back to an essentially standing wave as the
frequency of excitation deviates from this value.

For reasonably small excursions about the center frequency,
the current distribution corresponding to such a non-dissipative loading
will remain an essentially outward traveling wave. This is evident
since, referring to Figures 3. 5 and 3. 6, for such small frequency
variations the resistive component of the optimum impedance is
small compared with its reactive component. For large deviations
in the frequency of excitation, however, the magnitude of the resistance
component becomes comparable with that of the reactive component,
and the effectiveness of a purely non-dissipative loading will be
necessarily reduced.

An approximately traveling wave distribution of current may
therefore be maintained on a linear antenna over a band of excitation
frequencies through the use of a non-dissipative loading of fixed
position. The frequency dependence of this purely reactive loading
must match that of the reactive component of the optimum impedance

of fixed position. The circuit and radiation characteristics of an

antenna W

9353102, 2
is to reali.
Due to the

the use cf

U!

 

 

 

46

antenna with such a non -dissipative loading will thus be typical of
those for an ideal traveling wave antenna over this range of frequencies.

In order to realize an effective purely non-dissipative loading,
it is necessary that its reactance depend upon frequency in the same
way as the reactive component of the optimum impedance of fixed
position, as indicated in Figures 3. 5 and 3. 6. The problem therefore
is to realize such a frequency dependent reactance characteristic.

Due to the negative slope of this reactance as a function of frequency,
the use of an ordinary lumped capacitance would be relatively ineffective.
A similar difficulty is encountered with a short circuited transmission
line section, or a coaxial cavity. Such a frequency dependent reactance
is thus not realizable with simple circuit elements, and presents a
difficult synthesis problem. The consideration of this problem is
beyond the scope of the present research.

In order to verify the preceding theoretical results, an
experimental investigation of a linear antenna having a purely non-
dissipative loading is made in Chapter 5. In particular, it is
demonstrated that a traveling wave distribution of current may indeed
be excited on a linear antenna through the use of a prOperly positioned
optimum purely reactive loading.

Expressions (3.12) and (3.13) for the position and reactance of
an optimum non-dissipative double loading are perhaps the most
significant results of this research. Given the antenna dimensions and
its excitation frequency, the parameters of an optimum non-dissipative
loading to yield a traveling wave distribution of cylinder current are

readily calculated from these simple expressions. Through the use of

SSCh a 108d:
re:ai..s the l
earthy of n:

Altshuler3 1

:1: 111:5 case

Yge
‘ycc‘ ._.
* '9 Q;
La'fih
iii‘g blifi}.
3,3) 1%. ,
AAJlrIS

 

47

such a loading, a traveling wave linear antenna may be realized which
retains the high efficiency of a conventional unloaded dipole. It is
worthy of note that the simple transmission line analogy utilized by
Altshuler3 in the study Of a resistance loaded dipole fails completely
in this case. A lossless transmission line section terminated in an
open circuit cannot be matched with a purely reactive series loading
regardless of its position. It is thus indicated that there is no real
amlogy between a linear antenna and an open circuited section Of
transmission line.

3. 5. The Distribution Of Current and Input Impedance Corresponding
to an Optimum Loading

A general expression for the Optimum loading impedance to
yield a traveling wave distribution of cylinder current on O E z E d
was determined as equation (3. 5). Whenever the impedance is given
by this expression, whether it consists of a purely resistive or
reactive Optimum loading Of proper position or one of arbitrary position
having both resistive and reactive impedance components, then condition

(3. 3) holds, i. e. ,

 

Z . D .
Di -— 301.1. e-Ji30d (TD—1 cos 130d + e-Jflod) = 0 (3-3)
2

The distribution Of cylinder current on O E z E (1 therefore becomes

from equation (3.1)

12(2) = T)? e‘Jfioz (3.14)

while by r:

These exp:

WM.
...st be 3a

one result

 

48

while by result (3. 2) that on d E z E h is given by

V Tl' Z D . .
12(2) : 2?th 2 - ——__I:'— (.1 C08 god + e-Jfiod) e'JBoz

3OT 132
D .

+ .1 63502 (3.15)
D2

These expressions represent, respectively, the outward traveling
current wave which has been realized on O E z E d and the standing
wave which remains on d E z E h .

It is desired to simplify result (3.15) in order to indicate more
clearly the distribution of current on d E z Eh . This equation may

be written in the general form
2V 71'

12(2) = 2:91? [Ae-jfioz + BejBOz] (3.16)

where A and B are complex constants depending upon the antenna
dimensions and the loading parameters. The direct evaluation Of A
and B is very tedious, so an alternate method will be used. Since

the cylinder current is continuous at z : d, then the condition
1( ~d') — 1( -d+) (3 17)
z z— _ z z— .
must be satisfied, along with the Obvious requirement that
Iz(z=h) = 0 (3.18)

The result Of equation (3.14) may be utilized to Obtain

_ 2V07r "B d
Iz(z=d) = W e 3 0 (3.19)

O

evaluate

.1expres;

susstitute

IL
(A
N
(A
1

iii
CEz<h
‘“°ecance

 

 

49

Applying the two boundary conditions (3.17) and (3.18) to expression
(3.16) in conjunction with result (3.19), the constants Ai and B are

evaluated in a straightforward manner as

A z 1 (3.20)

1 _ e'j250(h-d)

 

_ e'jzfioh
1 _ e'JZpo(h'd)

 

(3.21)

If expressions (3.20) and (3. 21) for A and B, respectively, are
substituted into equation (3.16), the distribution Of current on
d E z E h is obtained as

I j4VO1r e—jpoh
z‘z) ‘ LOW ,_e-12130(h-d)

 

sin 80(h-z) (3.22)

It is Observed from result (3. 22) that the cylinder current on
d E z Eh has a sinusoidal distribution. Thus although an Optimum
impedance loading yields an outward traveling wave of current on
0 E z E d, the distribution on d E z Eh remains a purely standing
wave. This standing wave distribution is in fact identical with the
zeroth-order distribution Of current on a conventional unloaded linear
antenna.

Since the antenna current is symmetric about its excitation
point, then Iz(-z) = Iz(z). In summary then, the distribution of
cylinder current corresponding to an optimum impedance loading may

be expressed on - h E z E h as

'

Iz(z) = afie-Jfiolzl -dE zE d (3.23)

k1

previous

iimensio

 

 

 

50

jVO e-jfioh .
Iz(z) = 3M, 1 - e-jZBo(h-d) sm (soul-Isl) (3.24)

 

_h< z<-d, d:Z:h

The distribution of current corresponding to an Optimum impedance
loading will be illustrated by considering a pair of antennas having the
purely resistive and purely reactive Optimum loadings discussed
previously in sections 3. 3 and 3. 4. Specifically, the antenna

dimensions and Optimum loading parameters are taken as:

(l) h = 31.25 cm, 2a : 0.25 inch, d = 22.9 cm, [RL] 0 = 220 ohms;
purely resistive optimum loading at 600 mhz where:
h=0.625k,(h-d)=0.l7>. .

O O
(2) h = 100 cm, 2a = 0.25 inch, d : 79. 2 cm, [XL] 0 = - 363 ohms;

purely reactive Optimum loading at 600 mhz where :
h = 2110, (h-d) = O. 417 he .
In Figures 3. 7 and 3. 8 are indicated the amplitude and phase

Of current (Obtained from expressions (3. 23) and (3. 24)) for antennas
(l) and (2), respectively, as a function Of position along the cylinder.
It is noted that in either case the amplitude Of the current on 0 E z E d
is constant while its phase is linear, corresponding to a traveling wave
Of current along that region. On (1 E z E h, however, both figures
indicate a sinusoidal standing wave with constant phase. The appearance
of the standing wave on this region is quite different for the two cases.
In Figure 3. 7, corresponding to the optimum resistive loading, the
length (h-d) is only 0.17 he so the standing wave exists only on a
region whose length is less than a quarter free space wavelength. For

the Optimum reactive loading of Figure 3. 8, however, (h-d) = O. 417 RC

51

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53

and nearly a full half wavelength of the standing wave is present.
The input impedance to a traveling wave linear antenna is
obtained from equation (3. 23) as

V
0

1n Iz(z=0)

60 ‘II (3.25)

It is to be noted that this expression is valid only when the impedance
and position of the double loading are chosen to be optimum, i. e. ,
equation (3. 5) is satisfied. Under any other circumstances, when
ZL is not [ ZL] 0’ the general input impedance expression (2. 33)

must be utilized as
-l

 

Z . D .
_ _1. L 'JB d __l_ ’Jfiod
Zin—60\II 1+D -30Te 0 (D2 cosfiod+d

(2.33)

This general result reduces to equation (3. 25) when the loading
impedance is Optimum.

In order to demonstrate the input impedance characteristic of
a linear antenna having an optimum impedance loading, the two dipoles
having purely resistive and purely reactive loadings will again be
considered. In this instance, however, the loadings are taken to be
optimum at every frequency, that is, to have the impedance and
position given by Figures 3.1 and 3. 4, respectively. The specific
antenna dimensions and loadings are thus:

(1) h = 31. 25 cm, 2a = O. 25 inch; Optimum resistance loading

as in Figure 3.1.

in indicath’“

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54

(2) h = 100 cm, 2a : O. 25 inch; optimum reactance loading

as in Figure 3. 4.

An indication of the frequency dependence of the input impedance for
these two configurations is given, respectively, in Figures 3. 9 and
3.10. These numerical results were calculated from expression
(3. 25) using the approximate value of ‘1’ given by equation (3. 37) of
section 3. 6. It is noted that the input impedance of a linear antenna
which supports a traveling wave of current at every frequency is
essentially independent of the frequency of excitation. The
approximate theory developed in this research therefore predicts
an input impedance characteristic which is in general agreement
with the well known circuit properties of a traveling wave linear
antenna.

It was indicated in sections 3. 3 and 3. 4 that a purely resistive
or purely reactive loading of fixed position can be optimum only at a
single frequency. Consideration was thus directed to loadings consisting
of either the resistive or reactive components of the optimum impedance,
with their locations selected in such a way that they became optimum at
a given frequency. Since such loadings are optimum, and hence yield
a purely outward traveling current wave on O _<_ z 5 d, only at a
single frequency, then the frequency dependence of the corresponding
input impedance must be calculated from expression (2. 33).

In Figure 3. 2 of section 3. 3, the optimum loading impedance
was indicated as a function of antenna electrical length for a dipole
having dimensions h = 31. 25 cm, 2a : O. 25 inch, and for which the

loading position was fixed at d = 22. 9 cm. The reactive component

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57

of the optimum impedance was found tO vanish at a frequency Of 600 mhz,
where h = 0.625 x0 and (h-d) = 0.17 X0. A resistance loading Of

[RL] 0 = 220 ohms is Optimum at that frequency. The input impedance
corresponding to a constant resistance loading Of RL 2 220 Ohms
positioned at d = 22. 9 cm is now to be considered. Since the loading

is Optimum for h = O. 625 X0 , then it is approximately Optimum for

a range Of frequencies centered about this point. The input impedance
is therefore expected to be relatively broadband about the frequency
where h = O. 625 RC.

The input impedance to a linear antenna having such a constant
resistance loading Of fixed position is indicated as a function of its
electrical length in Figure 3.11. Both the theoretical result calculated
from expression (2. 33) and Altshuler's experimental result3 are
presented. It is Observed that in either case the input impedance is
relatively constant for a range of frequencies about that where
h = O. 625 X0. This broadband character is quite in contrast to the
frequency dependent impedance of a conventional unloaded linear
dipole, which is a very strong function Of the frequency of excitation.

It is noted from the figure that the theoretical and experimental
values for the resistive component Of input impedance are in close
agreement. There is, however, considerable error present for the
corresponding reactive components. While the two reactance curves
display basically the same functional dependence, they are displaced
from one another along the impedance axis. There are several probable

sources for this error as follows:

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59

(i) The approximate technique utilized to Obtain the distribution
Of current on the doubly loaded cylinder.
(ii) The approximate nature of the expression utilized for the
expansion parameter ‘1! (to be discussed in section 3. 6).
(iii) There is some question as to whether end-effect correction

factors were applied to the experimental results.

It is felt that the major contribution to the error is due to point (ii)

above. As will be indicated in the following section, the approximate

expression utilized to calculate \I’ is accurate only for the case of a

long antenna. It is necessary that the length d Of the cylinder on

which the traveling wave of current exists be considerably greater than

the length (h-d) which supports a standing wave distribution. In the

present case, the antenna length is relatively short, and the accuracy

of the approximate expression for ‘I’ is consequently questionable.

This particular cylinder length was chosen initially only to correspond

with the dimensions used by Altshuler in his experimental investigation.
An antenna having the dimenions h = 100 cm and 2a 2 O. 25 inch,

with a loading impedance fixed at the position d = 79. 2 cm was considered

in section 3. 4. The corresponding optimum loading impedance was

indicated in Figure 3. 6 as a function Of the antenna electrical length.

With this particular choice of the loading position, the Optimum

impedance becomes purely reactive at a frequency Of 600 mhz where

h = 2X0 and (h-d) = 0. 417 X0 . The input impedance Of an antenna

with similar dimensions having a non-dissipative loading consisting

of the reactive component of the above Optimum impedance is now to

be considered. Since this loading is optimum for h : 2X0, then the

distribL‘
:ravelir‘.
corresp:

broadbar

calculatec

is noted I}

F)
' O

Yrequen

i5 Cptintun

 

 

60

distribution Of cylinder current on 0 E z E d will approximate a
traveling wave for a range Of frequencies about this point. The
corresponding input impedance is therefore expected to be relatively
broadband about the frequency where h 2 2x0 .

In Figure 3.12 is indicated the input impedance Of a linear
antenna having such a non-dissipative loading Of fixed position, as a
function of its electrical length. This theoretical result was
calculated from the general input impedance expression (2. 33). It
is noted that the input impedance is relatively constant for a range
of frequencies about that where h : 2X0, at which point the loading
is optimum. This broadbanding effect is not as pronounced, however,
as for the case Of a purely resistive loading. Since a purely resistive
loading is more nearly Optimum over a band Of frequencies, as
indicated by Figures 3. 2 and 3. 6, such a difference is to be expected.

From Figure 3.12 it is Observed that the input impedance of the
doubly reactance loaded dipole is reasonably constant for h/lxO between
the values Of 1. 75 and 2. 25. Thus a broadbanding effect is evident for
a half-length variation Of 0. 5 wavelengths, or for a variation of a full
wavelength in the total antenna length. This is a notable improvement
over the situation which exists with a conventional unloaded dipole,
where the antenna input impedance is a much stronger function of its
electrical length. It is evident therefore that the desirable circuit
properties Of a traveling wave linear antenna may be realized through
the use of a purely non-dissipative loading. In contrast to the resistance
loading method, this technique maintains the antenna efficiency at the

same high level as that of a conventional linear dipole.

I”

X
I.

 

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It should be remarked that if a non-dissipative loading Of fixed
position is desired, then the reactive component Of [ZL] O , as
considered above,is not the optimum reactance. Although it is not
evident from the present theory, there may exist a more suitable
non-dissipative loading for the purpose. Further, if some dissipation
is allowed, a more effective broadbanding Of the input impedance than
is indicated in Figure 3.12 should be obtainable.

3. 6. Calculation of the Expansion Parameter \Il(z)

The expansion parameter ‘I/(z) has been defined in Chapter 2

by equation (2. 25), and may be written in the form

Az<z)

12(2)

 

99(2) 2 iii -h< z< h (3.26)
no - —

where Az(z) is the vector potential at a point on the antenna surface
and Iz(z) is the corresponding cylinder current. It has been indicated

that this vector potential may be expressed as
h -
A (z) : —— f Iz(z')K(z,z') dz' (2.20)

where K(z, z') is the Green's function

 

e--jf30~[(z-z')2 + a2
K(z,z') = (2.21)

2 2
z-
J( z') +a

 

Using result (2. 20) in equation (3. 26), the expansion parameter becomes

h
1 l I 1
‘I’(Z) = T;(—Z—)— S111 IZ(Z ) K(z, Z ) dz (3.27)

traveling
it is thi s
paraniete

independe

actually (jg

is not
[2L

 

63

Since the current distribution of fundamental interest is the
traveling wave corresponding to an optimum impedance loading, then
it is this distribution which will be used to evaluate the expansion
parameter. It has been indicated by King3 that ‘I’(z) is relatively
independent of the distribution of cylinder current, and depends
primarily upon its dimensions. Hence no great error will be made
by using the value of ‘I’(z) corresponding to a traveling wave
distribution when carrying out calculations where the cylinder current
actually departs moderately from a traveling wave, i. e. , when ZL
is not [ ZL] O .

In the preceding section, it was found that the distribution of
cylinder current on O E z 5 h corresponding to an optimum impedance

loading may be expressed as

 

ZVOTr "B z

12(z)=—Z——O‘IJ eJO ogzgd (3.14)
2V TT . .

12(2) 2 LOO‘II [A e-Jfioz + B eJBOZ] d E z _<_ h

(3.16)

where the complex constants A and B are given by equations (3. 20)
and (3. 21), respectively. It is clear from equation (3. 27) for the
expansion parameter, that, since the cylinder current appears both
in the integrand and in the denominator, the constant multipliers of
the current expression may be dropped. Furthermore, since Iz(-z) :
12(2), then for the calculation of \Il(z) the distribution of cylinder

current may be taken as

q

obtained

 

 

 

64

e-jfiolzl _d< z

Iz(z) : _ E d
. . (3.28)

12(2) = A5350,” +BeJBOlz' -11: zfi-d,

2 h

With the current distribution of equations (3. 28), there is

obtained the result

1‘ 'd :3 IN (5 W
S Iz(z')K(z, 2') dz' = S [A e J 0 + B eJ 0 ]K(z, z') dz'
-h -h

d ~13 lz'l
+5 e J 0 K(z,z') dz' (3.29)
-d

h . , . .
+54 [Ae'JBOIZl + BeJfiolZ l] K(z,z') dz'
d

which may be written as

h h :3 (z') (5 lz'l
S Iz(z') K(z, 2') dz' = S [Ae J 0 + BeJ 0 ] K(z, z') dz'
-h -h

d _. l2"
+S e 3‘30 K(z,z') dz' (3.30)
-d

d . .
-S [Ae-Jfiolz'l +BeJfiolz'l] K(z,z')dz'
-d

Result (3. 30) may finally be expressed in the form

h
S‘ Iz(z')K(z, z') dz' = (A+B)Ca(h, z) - j(A-B) Sa(h, 2)
-h
+(1-A-B) Ca(d, z) - j(1 +B-A) Sa(d, z) (3. 31)

where Ca(h, z) and Sa(h, z) represent the integrals

were nun
expansic-r

If

then ‘I/(z)

 

65

h
Ca(h,z) : 5 cos Boz' K(z,z') dz'
-h
(3.32)

h
Sa(h,z) = 5 sin Bolz'l K(z,z') dz'
h

These integrals occur frequently in the theory of linear antennas, and
were numerically machine calculated to facilitate evaluation of the
expansion parameter ‘Il(z) .

If equations (3. 2.8) and (3. 31) are used in expression (3. 27),

then ‘I’(z) is immediately obtained as

M2) : (A+B)Ca(h, z) - j(A-B)Sa1h,z) +(1-A—B)Ca(d,z) - j(l+B-A)Sa(d, z)
e'jfioz

0: z: d (3.33)

_ (A+B)Ca(h,z.) Lj(A-B)Sa(h, z) + (1 -A-B)Ca(d, z) - 1(1H3-A)Sa(d, 2)
M2) _. .
Ae JfioZ + B eJfiOz

dfth (3.34)

It .is noted from these results that the expansion parameter depends
upon the position d of the optimum impedance loading as well as the
antenna dimensions h and a. This poses a new problem, since the
loading position was previously determined in terms of \I' , but now ‘I’
is found to depend in a complicated way upon the location d of the
loading- Furthermore, it was postulated that the expansion parameter
‘Il(z) = \I’ was indeed a constant, while expressions (3. 33) and (3. 34)
indicate that \Il(z) is generally a function of position along the antenna.
Some further approximations are evidently in order, and it must be

demonstrated that ‘I’(z) is essentially independent of z .

It is
expanSion pa
distribution
having an op
cylinder sup
with that on
be made in c
current to e.
compared \vi
|

in result (3 ..

expression

In elm:
L...S apprc

err» -
-...enSIOns

66

It is a well known result in linear antenna theory that the
expansion parameter ‘Il(z) is a relatively weak function of the
distribution of cylinder current. 2 In the present case of an antenna
having an optimum impedance loading therefore, if the portion of the
cylinder supporting a standing wave of current is short compared
with that on which a traveling wave exists, then no great error will
be made in calculating \Il(z) by assuming the traveling wave of
current to exist over the entire cylinder. That is, if (h-d) is small
compared with d, then it should be a valid approximation to let d = h
in result (3. 33). Making this substitution yields the greatly simplified

expression

‘I’(z) i Ca(h, :L-jbgzsaw’ z) 0 f z _<_ h (3. 35)
In this approximate result, ‘Il(z) depends only upon the antenna
dimensions h and a. The methods of sections 3.3 and 3. 4 may there-
fore be applied to directly evaluate the necessary position of an optimum
loading.

An indication of the error in approximation (3. 35) is now to be
obtained. The special case of an antenna having a half-length of h = 100 cm
and a diameter of 2a = O. 25 inch will be considered. It is assumed
that an optimum impedance loading is positioned at d = 79. 2 cm such
that a traveling wave distribution of current exists on O E z E d.

This configuration corresponds to the antenna with non-dissipative
loading discussed in conjunction with Figures 3. 8 and 3.12 in the

preceding section. In Figure 3.13 a comparison is indicated between

67

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o
A\u daemons mcofim coflwmoa 33.3603

o.N om; . co; om.o 00.9

EU om
~35 coo

use coo an o .N
£2: mm .o

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Eu 2: u a

 

0A:

 

 

 

 

(2)15 Jaiaurexed noisu'edxa ssaIuotsuaunp

the EliaCt

parts Of ‘

correspon'
for both it:
results is

would be e:
beyond with
of the appr<

justified on

 

68

the exact result of expression (3. 33) and the approximate one of equation
(3. 35). The figure indicates the dependence of the real and imaginary
parts of \II(z) upon position along the cylinder for O f z _<_ d.

It is observed from Figure 3.13 that there is a very close
correspondence between the approximate and exact values of \II(z)
for both its real and imaginary parts. The deviation between the two
results is small near z = O and becomes progressively larger (as
would be expected by physical reasoning) as the position of the loading,
beyond which a standing wave of current exists, is approached. Use
of the approximate expression (3. 35) for \I'(z) is therefore quite well

justified provided that:

(i) The ratio d/(h-d) is relatively large (in the preceding case
this ratio had the value 3. 81).

(ii) The expression is not applied for values of 2 close to d.

A second very important observation from Figure 3.13 is that
the expansion parameter is indeed nearly independent of position along
the cylinder. This fact justifies the original approximating assumption
of section 2. 4, in which ‘Il(z) was taken to be a constant of value ‘I’.
Since ‘I’(z) is well represented by its value at the excitation point
z = O, and since this is the point at which the input impedance is
defined, then it will be taken that the constant value of ‘1’ is given by
‘11 = \I’(0) . The exact and approximate expressions for ‘I’ then

become from equations (3. 33) and (3. 35), respectively,

mate ya
of the cy
position
each freq
section 3,
“'33 Obtair

deviation )3

 

69

\II = (A+B) Ca(h, 0) — j (A-B) Sa(h, 0)
+ (l-A-B) Ca(d, 0) - j(1+B-A) Sa(d, 0) (3.36)
\II é Ca(h, 0) - jSa(h, 0) (3.37)

Again, result (3. 37) will be a satisfactory approximation whenever
(h-d) is small compared with d.

A comparison between the exact and approximate values of ‘11
given by expressions (3. 36) and (3. 37), respectively, is indicated in
Figure 3.14. The antenna is again taken to have a half-length of
h = 100 cm and a diameter of 2a : 0. 25 inch. A plot of the approxi-
mate value of ‘11 given by equation (3. 37) is presented as a function
of the cylinder's electrical length. From this value of ‘I’ , the
position d of a purely reactive optimum loading was calculated for
each frequency (see Figure 3. 4) according to the method indicated in
section 3. 4. Using these results for d, a corrected result for ‘I’
was obtained from expression (3. 36), as indicated in the figure. The
deviation between the approximate and corrected values of \II is
negligible for its real part, but becomes appreciable for the imaginary
part in the case of a short antenna.

In all of the preceding numerical calculations, the value of \I' was
obtained from the approximate expression (3. 37). The error inherent
in this approximation is small whenever the antenna half-length is of
the order of a wavelength or greater, and the ratio of d to (h-d) is
sufficiently large. In the case of the resistance loaded antenna of the
preceding section, the half-length was relatively short (always less than

a wavelength), and the accuracy of equation (3. 37) is accordingly doubtful.

7.

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EU 2:

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approxiniati
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connection \

71

As was noted in connection with Figure 3.14, the error due to the
approximation occurs primarily in the imaginary part of ‘I’. It is
felt that this inaccuracy in ‘II is primarily responsible for the
deviation between theoretical and experimental values for the

reactive component of input impedance which was observed in

connection with Figure 3.11.

teristics c
distributio
the doubly
that cox-reg
in Chapter
the trax-eli]
are to be C
CondlthD (:
ElEment on
Figure 4. 1‘
the Obserxw
by many Wa
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partlcular,

d

 

lstribufim

 

 

CHAPTER 4

RADIATION CHARACTERISTICS OF A
TRAVELING WAVE LINEAR ANTENNA

4.1. Distribution of Cylinder Current for Calculation of
Radiation Fields

It was indicated in the introduction that the radiation charac-
teristics of a linear antenna are completely characterized by its
distribution of current. The approximate current distribution on
the doubly loaded cylinder was determined in Chapter 2, while
that corresponding to an optimum impedance loading was established
in Chapter 3. In the present chapter, the electromagnetic fields of
the traveling wave linear antenna at the radiation zone (or far zone)
are to be calculated. These radiation fields are defined by the
condition (30R > > 1 , where R is the distance from a current
element on the cylinder to an observation point P as indicated in
Figure 4.1. This condition is equivalent to the requirement that
the observation point P be separated from every point of the dipole
by many wavelengths. In order to determine these electromagnetic
fields, the distribution of cylinder current corresponding to an
otpimum impedance loading will be utilized. Since it is, in
particular, the radiation fields which are to be determined, this
distribution will be further approximated to simplify the calculations.

In section 3. 5 of the preceding chapter, the current distribution
on the doubly loaded cylinder corresponding to an optimum impedance

loading was found as

72

Figui

 

 

Figure 4.1.

73

 

 

 

 

 

 

 

 

zz-h

Geometry for Calculation of Radiation Zone

Electromagnetic Fields of .a Traveling Wave
Linear Antenna.

'1)

These ex;
current ox
- h < z < -
ofa linear
9
“'1
current.
Ofcurrent
eXiStS’ the:
traveling W
35 reasonab

made in cal

ecNation (3.

 

74

V

Iz(z) = 36% e'jfiolzl -d:z:d (3.23)
jVO e'jfioh

 

Iz(z) — 30‘11 1-e-J°Zl30(h-<TY sin (30(h-lzl) (3.24)

These expressions represent, respectively, a traveling wave of cylinder
current over the region - d E z i d and a standing wave on the regions
- h _<_ z 5 - d and d _<_ z _<_ h. It is well known that the radiation fields
of a linear antenna are not a strong function of its distribution of
current. 2’ 5 If the regions of the dipole supporting a standing wave

of current are short compared with the one on which a traveling wave
exists, then it is a reasonable approximation to assume that the
traveling wave is present over the entire cylinder. That is, if (h-d)
is reasonably small compared with d, then no great error will be
made in calculating the radiation zone electromagnetic fields if
equation (3. 23) is assumed to be valid for - h _<_ z 5 h.

From the results of sections 3. 3 and 3. 4, the range of antenna
lengths for which the above approximation is applicable may be deduced.
Consider an antenna with a purely resistive optimum loading having
the dimensions utilized in conjunction with Figure 3. 1. It is observed
from that figure that the ratio d/(h-d) is greater than 2. 0 whenever
the half-length h of the cylinder is of the order of O. 5 wavelengths
or greater. Similarly, for the antenna with the non-dissipative loading
considered in Figure 3. 4, it is found that d/(h-d) is of the order of
2. O or greater whenever the half-length h is greater than a wave-

length. Hence, for either type of loading, there exists a range of

cyhnder

is a reas
will be 11'
linear an
4.2. Ra.

T
time liar:
(See Appg

0: I'GSPe

In the par
are 8i\'en

1ntegréils

“,here,

 

Cylinder g
is the d1".

l
12(2) the

 

75

cylinder lengths for which the distribution of current

V

Iz(z) = 60x13 e‘jfiolZl -h:z:h (4.1)

 

is a reasonable approximation. It is this current distribution which
will be utilized to calculate the radiation fields of a traveling wave
linear antenna.
4. 2. Radiation Fields of the Traveling Wave Linear Antenna

The electric and magnetic fields at a point in space due to a
time harmonic current-charge distribution are given quite generally
(see Appendix A) in terms of the vector and scalar potentials A and

4), respectively, as

_\
E

- V43 - ij
(4.2)

_I

VxA

U31.
n

In the particular case of a linear antenna, the electromagnetic potentials

are given in terms of the charge-current distribution by the Helmholtz

 

integrals
-‘ ”0 ha eujBOR
_ __ I __ l
A — 4w 5. zIz(z) R dz (4.3)
-h
h -j[3 R
_. 1 I e O 3
¢ — 4“ j q(z) R dz (4.4)
o -h

where, as indicated in Figure 4.1, g is a unit vector parallel to the
antenna axis and R is the distance between a source point on the
cylinder at z = z' and the point of observation at P . Further, q(z)
is the distribution of charge per unit length along the cylinder and

Iz(z) the corresponding distribution of current. Actually the charge

 

distributic

through th

determine
The gener;
Obserx-atig
Y- 1 I
it Has bee

antenna n: -

Where Er
TESpecti‘,€
to equatic,
1' Which 1
of the ant.

Lfi
(new)

by “luau.

 

 

76

distribution is not independent, but is related to the cylinder current

through the equation of continuity as

q(z) = g— ,3; 12(2) (4.5)

By substituting the electromagnetic potentials given by equations
(4. 3) and (4. 4), in conjunction with relation (4. 5), into expressions
(4. 2), the electromagnetic fields E and B at any point in space are
determined in terms of the distribution of cylinder current Iz(z) .
The general expressions for these fields are quite complex, but for
observation points in the radiation zone they are greatly simplified.

It has been indicated by King5 that the radiation fields of a linear

antenna may be expressed as

'B'ro‘) = ”30”,.er (4.6)
EH?) = vo[i§’r(r) x 3] (4.7)

where Erfi’) and Bra!) are the radiation electric and magnetic fields,
respectively, An?) is the radiation zone vector potential corresponding
to equation (4. 3), and r is a unit vector in the direction of the vector
'1'" which locates the observation point P with respect to the center
of the antenna.

Let the point P be specified by the spherical coordinates
(r, 9 , (b) as indicated in Figure 4.1. The electromagnetic fields given
by equations (4. 6) and (4. 7) then become

firm

- ifio[ 9x MiG-“)1 = no A3?) sin 9 $

- ijAgm 3 (4.8)

lhis the ra
another as

mat finall‘

Where

In
the in

tegn

Since in t}

 

be dropped.

EXpanSiOn

 

L
Tile value

 

77

m

AH

3}
II

A
— jBOvOAg (“1") (:3 x 3)

- ijg(‘r’)3 (4.9)

'llm. the radiation electric and magnetic fields are transverse to one
another as well as to the direction ’1“ of prOpagation. These fields

may finally be expressed in the form

Erfi‘) = -ijN-r‘) (4.10)
9 9
r _I l r __I
B¢(r) — -—v Ee(r) (4.11)
o
where
1‘ .s I‘ .4 .
Ae(r) — - Az(r) $111 9
H h ‘35 R
_ . o , e o ,
_ .. sin 9 TV S—h Iz(z ) ——————v R dz (4.12)

In order to carry out the integration indicated in equation (4.12),
it is necessary to obtain an expression for the distance R in terms of
the integration variable z' . With reference to Figure 4.1, R may
be expressed by the law of cosines as

R_[2 2 ]1/2

r + z' - 2rz' cos 9

r[l - 2(z'/r) cos 9 + (z'/r)2] l/Z

Since in the radiation zone z' < < r, then the term in (z'/r)2 may
be dropped. Retaining only the two leading terms of a binomial

expansion then gives
R 1- r-z‘c‘ose (4.13)

The value for R in expression (4.12) will thus be taken approximately

as

and makin.

A

This resu’.
are essem
are modif:
which depe

In :
Of the tray
it is Only I,
the apprO);
ME‘king th:

POtentia1 b I

r
A9

ine inteor

 

78

as
R i r ........... . amplitude term
(4.14)
R : r - z' cos 9 ...... phase term
and making this substitution gives
1* ‘JB r h ' 1
r _ _ _g e 0 . , jfioz cos 9 ,
A6 (1*) — 411' —-—---r 3m 95‘ Iz(z ) e dz (4.15)

-h

This result indicates that the radiation zone electromagnetic fields
are essentially outward traveling spherically diverging waves which
are modified by a function of the polar angle 9 , the exact form of
which depends upon the distribution of cylinder current.

In order to determine the radiation zone electromagnetic fields
of the traveling wave linear antenna with optimum impedance loading,
it is only necessary to evaluate the integral in equation (4.15) using
the approximate distribution of cylinder current given by equation (4.1).
Making this substitution, the 9 -component of radiation zone vector

potential bec ome s

V -' r h . .
(3") ‘ HO 0 e J50 sinGS‘ e—JBOIZ'I eroz' COS 9 dz'

A ' 2407M]?— _h

r
9
(4.16)

The integral in this expression may be rewritten in the form

h . , . , h . ,
S. e-jfiolz ' eJfloz COS 9 dz' : ZS e-JfioZ cos(BOz' cos 9) dz'
-h o

(4.17)

and a straightforward integration yields the result

It is c onvt

C(aoh,

in terms <

'7»
0: an Ong

W‘th the Q

 

Symmetré

L
“e "ariat

 

henCe des

 

79

h . .
5' e'Jpol Z" eJBoz' C03 9 dz'
-h
: —-——?L—Z—— {e-JBOh[jcos([30hcos 9) - cos9 sin((30h COS 911-1}
Bosin 9
(4.18)

It is convenient to define a quantity C(Boh, 9) as

C(fioh, 9): siln 9 (e-jfiohfj cos (80h cos 9) - cos 9 sin (80h cos 9)] -j}

 

 

(4.19)
in terms of which the vector potential of equation (4. 16) becomes
P- V “1(3 r
r .1 o o e 0
z _ 9 .
Aem goq, for coach, ) <4 20)

If the last result is used in conjunction with expressions (4.10)
and (4.11), the radiation electric and magnetic fields, respectively,

of the traveling wave linear antenna are obtained as

 

 

iv 413 r

Egrr’) = W0 E—r—O— C(poh, e) (4.21)
jv -j(3 r

Bio-‘1 = V (I? e O coach. 9) (4.22)
0

These expressions give the electromagnetic fields of the traveling wave
antenna as a function of its electrical length and the coordinates (r, 9)

of an observation point P in the radiation zone. There is no variation
with the ¢ -coordinate since the fields of a linear antenna are azirnuthically
symmetric. For fixed values of Bob, the function C(fioh, 9) describes
the variation of the radiation fields with the polar angle 9 , and is

hence designated as the "polar pattern factor. "

80

It is noted that C(fioh, 9) is a complex number, and the
radiation fields therefore vary in phase as well as amplitude with the
polar angle 9 . Physically however, it is the amplitude of the fields
at each point in the radiation zone which is of primary interest.
Consideration will henceforth be restricted, therefore, to the modulus
of the polar pattern factor. After considerable calculation, (C(fioh, 9 )l

is obtained as

 

(C(fioh, 9)] = [cosZUBOh cos 9) + c0529 sinzflioh cos 9)

sin 9
- 2 cos (30h cos (80h cos 9) +1
. . 1/2
- 2 $111 80h cos 9 8111(501'1 COS 9 )1
(4.23)

This expression gives the relative amplitude of the radiation zone
electromagnetic fields as a function of the polar angle 9 and the
electrical length of the traveling wave antenna.

Considering the complexity of expression (4. 23), it is clear
why the approximation discussed in the preceding section was necessary.
If, instead of approximating the distribution of cylinder current as a
traveling wave over its entire length - h _<_ z _<_ h, account were
taken of the standing wave on the regions - h E z j - d and d _<_ z E h,
then the position d of the loading impedance would appear in the
expression for the polar pattern. The result would be a further
complication of this expression, to the extent that it would become
quite intractable for calculating even specific radiation patterns. On

the other hand, the error associated with the approximate expression

81

(4. 23) will be small, and its relative simplicity renders it useful for
making practical calculations.

4. 3. Comparison of Radiation Patterns for Traveling Wave and
Standing Wave Linear Antennas

As was mentioned earlier, it is quite well known that the
radiation characteristics of traveling wave and standing wave linear
antennas differ considerably. The radiation patterns of a conventional
dipole and a traveling wave dipole utilizing an optimum lumped
resistance loading were measured experimentally and compared in the
research reported by Altshuler. 3 Further, Wu and King4 determined
theoretically the polar pattern of a traveling wave antenna having a
distributed resistance loading. These investigations have disclosed
that the radiation patterns of traveling wave antennas are in general
characterized by a wider major lobe beamwidth and an absence of
minor lobes, as compared with an ordinary standing wave dipole. It
is to be expected, therefore, that the radiation patterns resulting
from the present theory should disply essentially the same character.

The approximate expression (4. 23) may be utilized to evaluate
numerically the radiation patterns of traveling wave linear antennas
having various electrical lengths. These polar patterns are obtained
by plotting IGmOh, 9 )l as a function of 9 in polar coordinates, with
the appr0priate constant values of 80h as parameter. Typical
patterns are indicated in Figures 4. 2, 4. 3, and 4. 4 for 80h values
of 21r, 717/2, and 417, respectively. The corresponding respective
half-lengths are h 2 k0, h =1. 75 X0, and h = 2RD. In each case,
the radiation patterns of the corresponding standing wave dipoles,

given by lFmoh, 9 )l are included for comparison.

0

82

   

  

lcmoh. 9)!

traveling wave

 

 

 

    

Irmoh, 9)!

standing wave

 

Figure 4. 2. Radiation Patterns of Traveling and Standing Wave
Antennas with Bob = 2n .

83

       
   
 

1093.0) !

traveling
wave

 

 

 

!F((30h. 6H

standing wave

 

 

Figure 4. 3. Radiation Patterns of Traveling and Standing Wave
Antennas with (3011 : 777/2 .

84

   
 

! Gmoh. 9)!

  
 

traveling wave

 

 

 

     
 

l I l
; F1001‘1,0),

standing wa ye

 

Figure 4. 4. Radiation Patterns of Traveling and Standing Wave
Antennas with (30h = 477 .

85

An inspection of the figures reveals that a traveling wave linear

antenna possesses the following radiation characteristics:

(i) The polar pattern of a traveling wave antenna has a major
lobe beamwidth which is wider than the one of its standing
wave counterpart.

(ii) A minor lobe does not appear in the pattern of a traveling
wave antenna until its length is much greater than that of
the corresponding standing wave dipole. In particular,
the first appearance of a minor lobe in the traveling wave
antenna pattern occurs for a half-length of h 2 2X0 . The
first minor lobe occurs for h = O. 75 to in the case of a
standing wave dipole.

(iii) The radiation pattern of a traveling wave antenna is much
less dependent upon the cylinder's electrical length
(frequency of excitation) than is the one of a standing
wave antenna.

In these figures, the radiation patterns of the traveling wave and standing
wave antennas are not normalized to the same absolute values of field
intensity, but are merely independent relative field strength patterns.
Thus while it appears from the figures that the traveling wave antenna
radiates more power than its standing wave counterpart, this is only
apparent. Actually, in the case of a traveling wave antenna with non-
dissipative loading, the efficiencies of the two antennas are equal
(essentially 100%) and the power radiated by each will be the same.

From the above remarks, it is noted that the radiation patterns

calculated from the approximate theoretical expression (4. 23) display

86

the characteristics generally associated with those of traveling wave
antennas. Whether or not these characteristics represent any
particular advantage will depend, of course, upon the intended
application of the antenna. The relatively wide beamwidth associated
with a short traveling wave antenna, and the absence of minor lobes in
the radiation pattern of a long antenna should be useful in certain
applications.

It should be emphasized that the radiation characteristics
calculated in section 4. 2 correspond to the traveling wave distribution
of antenna current associated with an optimum impedance loading.

As the loading deviates from its optimum value, the traveling wave

of current gradually reverts back to an essentially standing wave.
Under these circumstances, the corresponding radiation patterns
would again be characterized by the narrower beamwidth and presence

of multiple minor lobes associated with a conventional unloaded dipole.

CHAPTER 5

EXPERIMENTAL STUDY OF TRAVELING WAVE
ANTENNA WITH NON-DISSIPATIVE LOADING

5.1. Object of the Experimental Investigation

An approximate expression for the distribution of current on a
linear antenna consisting of a doubly impedance loaded cylinder was
obtained theoretically in Chapter 2. This result gave the current
distribution on the cylinder as a function of its dimensions, the
excitation frequency, and the impedance and position of the double
loading. In Chapter 3 the optimum loading impedance to yield a
traveling wave of current over most of the antenna was determined
from its current distribution. This optimum impedance was
expressed in terms of the cylinder dimensions, the frequency of
excitation, and the position of the loading. It was found that a pure
resistance or a pure reactance could constitute an optimum impedance
loading if their locations were properly selected. The results of this
theory indicate therefore, that a traveling wave distribution of current
may be excited on a linear antenna having either a purely resistive
or a purely non-dissipative loading of proper position.

As was indicated earlier, the observation that a traveling wave
linear antenna may be realized through the use of a properly located
lumped resistance loading is not original with the present research.
An experimental study of such a resistance loaded dipole was made
by Altshuler. 3 In order to determine the necessary position of the

resistance loading, he relied upon a transmission line analogy.

87

88

It is well known that an analogy exists between a linear antenna

and a section of lossless transmission line terminated in an open
circuit, since: (1) the current at the end of either is zero; and (2)
both support an essentially standing wave distribution of current.
The section of transmission line may be matched by placing a series
resistance equal to the characteristic resistance of the line a quarter
wavelength from its open-circuited end. A traveling wave of current
will then exist on all but the end quarter wavelength of the line.

Through this analogy, Altshuler proposed that a traveling
wave linear antenna might be realized by placing an optimum
resistance loading a quarter wavelength from the cylinder ends. He
verified experimentally that this was indeed possible, and determined
the optimum value of the loading resistance. The distribution of
current on the antenna with optimum resistance loading was found
experimentally to consist of a traveling wave between the excitation
point and the position of the loading, and a standing wave between the
loading and the cylinder ends.

The theoretical prediction that a traveling wave current
distribution may be obtained on a linear antenna through the use of
a purely resistive loading is thus found to be in agreement with
Altshuler's experimental results. Specific numerical comparisons
between the theoretical and experimental values for the parameters
of an optimum loading were made in section 3. 3 of the preceding
Chapter. The close correspondence between these results gives an
indication of the validity of the approximate theory presented in

Chapter 2.

89

Although it has been predicted theoretically that a traveling
wave distribution of current may be exicted on a linear antenna
through the use of a properly located lumped non-dissipative loading,
there are no existing experimental results to verify this assertion.
In fact the transmission line analogy just considered appears to
indicate that such a technique must fail to yield a traveling wave of
current, since an Open-circuited section of lossless transmission
line cannot be matched with a purely reactive loading regardless of
its position. The need for an experimental study of such a linear
antenna utilizing a non-dissipative loading is thus apparent. In
particular, the experiment should verify that a traveling wave
antenna may indeed be realized through the use of a lumped purely
reactive loading.

In accordance with the above remarks, an experimental
investigation of a linear antenna utilizing a purely non-dissipative
loading was conducted. The object of the experiment was threefold:
(l) verify that a traveling wave of current may be excited on a dipole
having a purely reactive loading; (2) study the effect of variations in
the loading parameters and excitation frequency upon the distribution
of antenna current; (3) determine the frequency dependence of the
input impedance to a traveling wave antenna utilizing a particular
reactance loading which is optimum only at a single frequency.

In order to facilitate comparison of these experimental results
with the theory presented in Chapter 3, the dimensions of the model
antennas were taken to be the same as those considered in the numerical

results of section 3. 4. The theoretical and experimental values for the

90

position of an optimum non-dissipative loading and the input impedance
to the corresponding traveling wave antennas may therefore be directly
compared.

5. 2. Description of the Experimental Arrangement

The experimental arrangement is indicated in Figure 5.1. An
anechoic chamber was constructed with an aluminum image plane on
one wall and with R. F. absorber covering the remaining walls. The
monopole antenna consisted of an extension of the movable centerwire
of the exciting coaxial line, and emerged from the center of the image
plane. A purely reactive loading impedance was obtained through the
use of an adjustable coaxial cavity structure in the end section of the
antenna, as indicated in Figure 5. 2. The length (h-d) of the end
section was made adjustable so that the effective position as well as
the reactance of the loading could be varied. A movable current probe
of the type indicated in Figure 5. 3 was employed to sense the amplitude
of the antenna current. A slotted section in the exciting transmission
line was provided to facilitate measurement of the antenna input
impedance.

The anechoic chamber consisted of a cubical wooden structure
having dimensions of 6. 0 feet on each side. Since an excitation frequency
of 600 mhz (X0 = 50 cm) was used in the experiment, the chamber
dimensions were of the order of 4. O wavelengths. An aluminimum
ground plane having a thickness of 0. 125 inches completely covered
the chamber wall through which the antenna projected. The remaining
walls of the chamber were covered with radio frequency absorbing

material, which was effective, at the frequency utilized, in reducing

6 ft.

 

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Figure 5.1. Experimental Arrangement.

 

92

 

 

 

 

adjustable short-circuited image :
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(cavity) ‘
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exciting :
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Figure 5. 2. Structure of Model Monopole Antenna.

solder joint to
center conductor

  
 
  
   
 

0. 030 in.

solid wire

solder joint

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line pro ortional

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Figure 5. 3. Structure of Loop Type Current Probe.

93

the reflected power by at least 20 db. It is felt that any interaction

of the reflected fields with the radiating antenna was entirely
negligible. Access was provided to the chamber through a door
opposite the ground plane, this being the region of minimum radiation.
Only the antenna itself was interior to the chamber, with all excitation
and monitoring equipment situated on the outside.

A coaxial transmission line was used to excite the antenna,
and its outer conductor terminated in an electrical contact with the
aluminum ground plane at its center. The outer conductor of this
line consisted of a brass tube with an outside diameter of l. 0 inches
and an inside diameter of O. 875 inches, while the inner conductor
was a brass tube having an outside diameter of O. 25 inches. Several
thin dielectric wafers were used to support the centerwire and
maintain it concentric with the outer tube, except for which the
coaxial line was air filled. The corresponding characteristic
resistance of the line was therefore calculated as RC = 75 ohms.

At the end of the coaxial line opposite the ground plane, and
approximately 1. 5 wavelengths (k0 = 50 cm at 600 mhz) from it, an
adjustable short-circuit was provided between the inner and outer
conductors. The coaxial line was fed through a tee-section located
about 0. 5 wavelengths from its short-circuited end. At the tee-section,
the joint between the two center conductors was effected through the
use of a coupling collar about the centerwire of the line exciting the
antenna. This arrangement allowed the centerwire of the exciting
line, which becomes the antenna after emerging through the ground

Plane, to remain freely movable. The short-circuited tuning stub

94

was utilized to tune out the reactive component of the input impedance
to the coaxial exciting system, as viewed from its driving point at the
tee-section. This arrangement allowed a reasonably good impedance
match to be realized between the coaxial exciting system and the
transmission line driving it at the tee-section.

In the region between the ground plane and the tee-section, the
outer conductor of the coaxial line was slotted axially to allow the
insertion of a charge probe. A movable carraige was constructed to
support a conventional probe in the slot, and thus facilitate monitoring
the III-field of the line at each point along its length. This arrangement
was utilized to make the standing wave ratio measurements necessary
in the experimental evaluation of the antenna input impedance.

The monopole antenna itself consisted of an extension of the
center conductor from the exciting coaxial line, and projected into
the anechoic chamber through the center of the ground plane. Since
this monopole was imaged into the highly conducting aluminimum
ground plane, its distribution of current was exactly that of a
corresponding dipole. The monopole is therefore equivalent to its
dipole counterpart, except that it is driven by an effective voltage
of VO/Z , where V0 is the excitation potential of the corresponding
dipole. Measured values of the monopole input impedance thus
correspond to one half those of the equivalent dipole.

A detailed drawing of the impedance loaded monopole antenna
is given in Figure 5. 2. The section of the monopole for 0 _<_ z 5 (1
consisted of the extended centerwire from the coaxial line, while

that for d j z E h was a separately constructed segment. Since

95

the center conductor of the coaxial line was free to move along its
axis, then the total length of the antenna was readily adjustable.

The portion of the monopole for 0 _f z E d, i. e. , the center conductor
of the coaxial line, was constructed of a brass tube having an outside
diameter of 0. 25 inches. This section of tubing was slotted axially to
allow a small current probe to project through its surface. The
current probe consisted of a small loop, and was supported in the
slot by a plastic guide which prevented it from making contact with
the antenna. With this arrangement, it was possible to measure the
relative amplitude of axial current at each point 0 E z _<_ d along

the monopole.

Figure 5. 3 indicates the detailed construction of the current
probe. The probe consisted of a loop having a diameter of approxi-
mately 0. 25 inches, and was fabricated from a section of Microcoax
coaxial line. This coaxial cable had a solid copper outer sheath with
a diameter of O. 030 inches, and its characteristic resistance was
50 ohms. At its end, this section of Microcoax line was bent into a
semicircle to form half of the current loop. The second half of the
loop was formed from a semi -circular arc of solid wire having the
same diameter as the Microcoax, and was soldered to the coaxial
section at the base of the loop. At the point where the two halves
of the loop met a small gap was left, and the center conductor of the
Microcoax line was soldered to the solid wire loop segment.

The circumferential magnetic field near the antenna surface
links the open surface subtended by the plane loop and induces an e. m. f.

3around its circumference. Since the loop is essentially perfectly

96

conducting except at the small gap, then this induced voltage appears
across the gap and excites a wave on the Microcoax line. At each
point along the surface of the monpole, the magnetic B—field is
essentially proportional to the axial current at the same point. The
amplitude of the voltage wave excited on the Microc oax line by the
loop probe is therefore proportional to the amplitude of the axial
monopole current.

A length of thin flexible 50 ohm coaxial line was joined to the
Microcoax comprising the loop, and was passed through the center
of the tube forming the monOpole, as well as the center conductor
of its exciting line, to the monitoring instruments outside the
chamber. Since the loop was free to move in the axial slot of the
monopole, its position could be varied over 0 5 z E d by simply
feeding this line in or out of the center conductor of the coaxial line
driving the antenna. Dimensional calibrations were placed on the
flexible line such that the position of the loop along the monopole
could be determined accurately. With this arrangement, it was
possible to measure the amplitude of the current distribution on
the monopole by operations completely exterior to the chamber.

A pair of concentric telescoping brass tubes comprised the
end section of the monopole having length (h-d), as indicated in
Figure 5. 2. The inner tube had an outside diameter of 0. 25 inches
and an inside diameter of 0.188 inches, while the outer tube made
a tight electrical contact with the inner one, had a wall thickness
0f 0. 015 inches, and was fitted with a hemispherical end cap. This
telescoping arrangement allowed the end length (h-d) of the monopole

to be adjusted to any desired value.

97

Within the inside tube of the concentric end section was
constructed a short-circuited section of coaxial transmission line.
The details are again indicated in Figure 5. 2. A center conductor
having a diameter of 0. 063 inches was used for this coaxial section
and, since the line was air filled, its characteristic resistance was
calculated as RC = 66 ohms. A short-circuiting disk terminated
the coaxial section, while a dielectric wafer supported its center-
wire at the excited end. The center conductor and shorting disk
were threaded to facilitate adjusting the length I of the short-
circuited section.

The input impedance to the short-circuited coaxial line

section is given by the well known expression

Zin = j Rc tan (301 (5.1)

where Rc is its characteristic resistance and (30 the free space
wave number. It is clear from Figure 5. 2 that this input impedance
is excatly the impedance which loads the mon0pole antenna at z = (1.
Hence, in accordance with the definitions made in Chapter 2, the

loading impedance appearing at z = d is

zL = J RC tan (3012 (5.2)

This impedance is purely reactive, and may be made to have any value
between plus and minus infinity by proper adjustment of the length I
of the short-circuited coaxial section.

The experimental arrangement just described provided the
means for realizing a monopole antenna having a purely non-dissipative

loading. With this apparatus, the total length h of the monopole, the

98

effective position d of the impedance loading, and its reactance XL
were all readily adjustable. Further, means were available for
monitoring the distribution of current along 0 _<_ z _<_ d on the
monopole, and for measurement of its input impedance.

Excitation of the entire experimental setup was provided at the
input arm of the tee-section in the coaxial line driving the antenna.
A 50 ohm coaxial transmission system was used to deliver energy
from the R. F. oscillator (General Radio 1209-B) through a 50 ohm
slotted section (General Radio 874-LBA) to the input of the tee-section.
The antenna tuning stub was then adjusted to minimize the S. W. R.
measured on this slotted section, and consequently result in maximum
power being transferred to the monopole.

In order to avoid the necessity for complicated measuring
apparatus, the 600 mhz signal from the R. F. oscillator was
amplitude modulated at l khz by a sinusoidal signal from an audio
oscillator (General Radio ZOO-C), as indicated in Figure 5.1. The
R. F. signals from the current probe and the charge probes of the
two slotted sections were then amplitude detected by coaxial detectors
(General Radio 874-VQL) terminated in the 50 ohm characteristic
resistance common to all three transmission systems. Bolometer
detectors (Narda N-6lOB) were utilized to assure accurate square
law detection. These detector outputs were coupled to standard S.W. R.
indicators (General Radio 415-B), which provided the necessary bolo-
meter bias currents and which were calibrated for square law detection.
The desired values of standing wave ratio or relative amplitude of

monopole current could then be read directly from the meter scales.

99

5. 3. Traveling Wave Distribution of Current on MonOpole with
Purely Non-Dissipative Loading

It was the primary object of the experimental study to verify
the theoretical prediction that a traveling wave of current could be
excited on a linear monopole antenna through the use of a properly
positioned non-dissipative loading. The experimental arrangement
described in the preceding section was ideally suited to this purpose
since:

(i) It provided an impedance loaded monOpole whose total

length h was freely variable.

(ii) The loading impedance was purely non-dissipative, and its
reactance could be varied to obtain any desired value from
- 0° to +co ohms.

(iii) The position of the loading d and the length (h-d) of the
end section of the monopole were readily adjustable.

(iv) Means were provided to measure the distribution of
monopole current on 0 E z E d for each combination of
antenna length and loading parameters.

For this portion of the experiment, the antenna was excited at a
constant frequency of 600 mhz corresponding to a wavelength of
)‘o = 50 cm.

Two different monopole antennas, having lengths of h 2 X0 =
50 cm and h 2 2X0 = 100 cm, were studied experimentally. These
monopoles correspond to the dipoles considered in section 3. 4, where
the reactance and position of an optimum non-dissipative loading were
given in Figures 3. 3 and 3. 4, respectively, as a function of the

antenna electrical length. In either case, the end length (h-d) was

100

first set to its theoretically optimum value. The optimum loading
reactance was then determined by observing the distribution of
current on the monopole as the length 1 of the coaxial cavity was
varied. Through this technique, it was possible to determine the
cavity setting which minimized the current standing wave ratio on

the monopole. Similarly, the end length (h-d) was slightly readjusted
until a traveling wave of current was realized over the region

0 f z _<_ d of the monopole.

The experimentally measured distributions of monopole current
are indicated in Figures 5. 4 and 5. 5. In theSe figures, the traveling
wave of current obtained with the purely non-dissipative loading is
compared with the standing wave which exists on a conventional
unloaded monopole. It is noted that, in either case, a decaying
traveling wave of current was obtained over the region 0 _<_ z E (1
through the use of a properly located purely reactive loading. The
current over the region d E z E h of the monopole with optimum
loading remains a standing wave. On this region the current was
not measured, but represents the theoretical sinusoidal standing
wave distribution. These results provide direct experimental
verification of the theory presented in Chapter 3.

A comparison is made between the theoretically and
experimentally determined values of the Optimum monopole end length
(h-d) in Table 5.1. It is noted that the agreement between these two
sets of results is quite close, in that the experimental values are only
of the order of 9% smaller than the theoretical ones. Further, the

theoretical prediction that the optimum end length should be essentially

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independent of the total antenna length is found to be fully verified by

the experimental measurements .

Table 5.1. Comparison of Optimum MonOpole End Lengths

 

Optimum monopole end length (h-d)

 

total length h

 

 

 

of monopole theoretical experimental
h = X0 = 50 cm (h-d) = 0. 418kO : 2.0. 9 cm (h-d) = 0.378)\O =18. 9 cm
h = 2x0 = 100 cm (h-d) = O. 417kO = 20. 8 cm (h-d) = 0.378kO = 18.9 cm

 

 

 

 

 

No attempt was made to measure the loading reactance provided
by the coaxial cavity, since it would be virtually impossible to account
for the stray capacitances which would necessarily influence the cavity
adjustment. At the frequency of 600 mhz utilized in the experiment,
the theoretically optimum loading reactance of approximately - 365 ohms
corresponds to a capacitance of less than one picofarad. It therefore
becomes evident that any stray capacitances, which are effectively
paralleled with the input to the cavity across the antenna gap at z = d,
would have a definite but unpredictable effect upon the necessary
optimum length I of the coaxial cavity.

5. 4. Effects of Variations in Loading Parameters and Frequency
Upon the Traveling Wave Distribution of Current

In the preceding section, the parameters of an optimum non-
dissipative loading to yield a decaying traveling wave distribution of
current on a linear monOpole antenna were determined experimentally.
It was found in particular that, for a monopole having a length of

2X0 = 100 cm at an excitation frequency of 600 mhz, the optimum

104

reactance corresponded to a cavity length Of I = 12. 2 cm while its
necessary position was d = 81.1 cm. The sensitivity of the traveling
wave distribution of monopole current on O _<_ z E d to variations in
these loading parameters and the frequency Of excitation is to be
evaluated experimentally in the present section.

The first effect to be considered was that due to variations
in the loading position (1, or equivalently the end length (h-d) of
the monopole. An optimum non -dissipative loading requires an end
length of (h-d) = 18. 9 cm. Figure 5.6 indicates a comparison
between the traveling wave of current obtained with this loading
position and the distribution of current which was measured
corresponding to end lengths of (h-d) = 17. 9 cm and (h-d) : l9. 9 cm.
It is noted that due to the l. 0 cm variation from the optimum end
length (about 5%) the traveling wave distribution of monopole current
is altered substantially, and begins to revert back to a standing
current wave. This indicates that the current distribution on the
monOpOle is a strong function of the loading position, in accordance
with the theroetical observations of Chapter 3.

An Optimum loading reactance corresponding to a coaxial
cavity length of I = 12. 2. cm was determined experimentally for the
monopole having a total length Of h : 2X0 : 100 cm. If the cavity
length is changed, then the position but not the reactance of the non-
dissipative loading is Optimum, and the current distribution will
deviate from the traveling wave corresponding to an optimum loading.
In Figure 5. 7 a comparison is indicated between the traveling wave Of

monopole current associated with an Optimum loading and the measured

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distribution Of current obtained when the coaxial cavity length was set
to values of I = 11. 7 cm and 1 : l3. 0 cm. It is again noted that the
traveling wave distribution of monopole current is substantially altered
by such variations, and that it deviates toward the standing wave
distribution characteristic of a conventional dipole. The dependence
of the monopole current distribution upon the reactance of the non-
dissipative loading is thus Observed to be very pronounced.

It has been indicated, both in theory and by experiment, that
a purely reactive loading of fixed position may be optimum only at a
single frequency. At any other frequency, an Optimum loading
impedance must have a resistive component as well as a reactive
one. In section 3. 4, consideration was given to a purely non-dissipative
loading consisting at each frequency of the reactive component Of the
corresponding optimum impedance. It was observed that, with such
a loading, an approximately traveling wave distribution of current
could be maintained for a band of frequencies about the one where
the loading became optimum.

The monopole antenna having a length of h : 2X0 = 100 cm
at an excitation frequency of 600 mhz was found experimentally to
require the optimum loading parameters: d = 81.1 cm, I = 12. 2 cm.
A decaying traveling wave of current was measured on 0 E z E d
for an antenna having such a loading. Figure 5. 8 indicates a.
comparison between this traveling wave of current and the
distribution of monopole current measured when the excitation
frequency was varied to 560 mhz and 640 mhz, with the loading

parameters fixed at the above indicated values. Evidently the

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109

frequency dependence of the antenna current distribution is quite
strong. There are two reasons for this result: (1) there was no
resistive impedance component present in the loading; (2) the fre-
quency dependence of the coaxial cavity reactance does not at all
match that of the reactive component of an Optimum impedance
loading. The distribution of monOpole current therefore reverts
rather rapidly back to an essentially standing wave as the excitation
frequency deviates from 600 mhz. This situation could be greatly
improved upon if a more apprOpriate Optimum loading reactance
were realized.

5. 5. Input Impedance of a Traveling Wave Linear Antenna with
Non-Dissipative Loading

It is of particular interest to consider the frequency dependence
of the input impedance to a traveling wave linear antenna utilizing a
purely non-dissipative optimum loading. According to the theory of
section 3. 5, this input impedance should be relatively broadband about
the frequency at which the purely reactive loading becomes Optimum.
The extent of this broadbanding will depend, of course, upon just how
well the frequency dependence of the loading reactance matches that
of the reactive component of an optimum loading impedance.

The monopole having a length of h : ZXO : 100 cm at an
excitation frequency of 600 mhz was again utilized for this part Of
the experimental study. A purely reactive coaxial cavity loading
having the parameters d = 81.1 cm, I : 12.2 cm was utilized. Such
a loading is optimum at a frequency of 600 mhz. With the loading

parameters held constant at the indicated values,the input impedance

110

to the monopole was measured for excitation frequencies between
480 and 720 mhz. Since the input impedance of the monopole
antenna is effectively the impedance which terminates its exciting
coaxial line, then this impedance was readily determined by
conventional S.W. R. measurements8 on the slotted section of that
line.

The experimentally measured input impedance to the reactance
loaded dipole is indicated as a function of frequency in Figure 5. 9,
where it is compared with that of an unloaded antenna Of the same
dimensions. Since the input impedance to a monopole antenna is
one half that of its dipole counterpart, the values given in the figure
are just twice the experimentally measured ones. It is noted that
the input impedance of the reactance loaded antenna is somewhat
broadband about the frequency Of 600 mhz, where the loading becomes
optimum, as compared with the impedance of the conventional dipole.
It must be emphasized that the frequency dependence of the coaxial
cavity reactance does not at all match that of the reactive component
Of an optimum loading impedance. If a more appropriate non-dissipative
loading could be realized, the above indicated broadbanding would be
greatly increased.

A direct comparison may be made between the experimentally
measured value of input impedance to a traveling wave antenna and
the one calculated from the theory of section 3. 5. At an excitation
frequency of 600 mhz where the purely reactive loading becomes
Optimum, i. e. , yields a traveling wave distribution Of antenna current,

the corresponding values of input impedance are: (l) theoretical,

lll

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112

Zin = 316 - j 184 Ohms; (2) experimental, Zin = 264 -j156 Ohms. The
error in the theoretical result is approximately 15%, which is well
within the range to be expected for such an approximate theory. It is
felt that the major contribution to this error is due to the approximate

nature of expression (3. 37), which was used to evalute the expansion

parameter ‘1’ .

PART II

TRAVELING WAVE LOOP ANTENNA
WITH OPTIMUM IMPEDANCE LOADING

113

CHAPTER 6

INTRODUCTION

6 . 1 . Introduction

In the second part Of this research, attention is to be directed

t o the realization of a traveling wave loop atenna and to the evaluation
of its corresponding circuit and radiation characteristics. An imped-
ance loading technique is utilized whereby the circular loop antenna
5. s doubly loaded with a pair of identical lumped impedances. A
th eoretical study of this configuration is made to determine approxi-
mately the distribution of current on the loop as a function of its

dimensions, the excitation frequency, and the impedance and position

of the double loading. From this result the optimum loading impedance

to yield a traveling wave of current over the major extent of the loop

of. :- cumference is determined. In particular, the possibility of utilizing

a pu rely non-dissipative optimum loading is thoroughly investigated.
Finally, the input impedance and radiation fields of a traveling wave
loop antenna having a non -dissipative Optimum loading are evaluated
as a function of its dimensions and the frequency of excitation.

6- 2 - Definition of a Traveling Wave Loop Antenna

A traveling wave loop antenna will henceforth be considered
as a. C ircular loop antenna which supports a traveling wave distribution
0f c111‘rent. The traveling wave of current is excited by a voltage
gene, rator at a point on the loop, and advances outward along its
Circumference toward a point 180 degrees removed from the point

of -
exCItation. While the phase of the current is essentially a linear

114

115

function of position along the loop circumference, the amplitude of the

traveling wave decays as it advances outward from the excitation point,

since it constantly radiates energy into space. A restriction to the

class of thin-wire loop antennas is made in this research, such that
an expedient one -dimensional theory will be approximately valid. In
this one -dimensional approximation, the distribution Of loop current
i s assumed to flow parallel to the axis of the thin wire constituting

the loop.

The theory of circular loop antennas remains relatively un-

de veloped when compared with the extensive research which has been

reported concerning the linear antenna. A mathematical theory

de scribing the circuit properties of a circular loop antenna was first

developed by Hallen, 1 at the same time at which he formulated the

cor responding theory for a linear antenna. The result of this theory

wa. s an integral equation for the distribution of loop current. Hallen

solved this integral equation through a Fourier series technique, but
POinted out that the series did not converge when the loop diameter
became an appreciable fraction of the wavelength. Some eighteen
year 8 later, Storer9 found that, by a more careful evaluation of the
Fouri er coefficients, the Fourier series for the distribution of loop
curr ent could be made to converge. This modified theory yielded
mime rical results which were in good agreement with corresponding
exPe rimental measurements of the loop current, and indicated that
the CUrrent distribution on a loop antenna consists of an essentially

standing wave.

116

Since it is well known that a conventional loop antenna supports
an essentially standing wave of current, then evidently some modification

of its structure is necessary in order that it might support a traveling

wave current distribution. In the present research, an impedance

1 oading technique is utilized to obtain the desired traveling current

wave on the circular loop. This method consists of doubly loading the

1 cap with a pair Of identical lumped impedances placed symmetrically

with respect to its excitation point. When the loading is Optimum,

that is, when its impedance and position are pr0perly selected, a
traveling wave distribution of current may be excited on the loop over
m cat of its circumference.

A loop antenna is completely characterized by its distribution

of current. The loop is fully described by its circuit and radiation

cha racteristics, which are readily determined in terms Of the current

di 3 t ribution. If the current at its excitation point is known, then the

input impedance to the circular loop may be immediately calculated.
Furthermore the radiation pattern of the loop is determined in a
strai ghtforward manner in terms of its distribution of current. Since
the c i rcuit and radiation characteristics Of a loop antenna are determined
by it 8 current distribution, then it might be expected that these
Characteristics should differ greatly for distributions corresponding,
respe ctively, to the standing and traveling current waves.
6’ 3 ° Important Characteristics of a Traveling Wave Loop Antenna

A conventional circular loop antenna is very frequency sensitive,

th ~ . . . .
at 1 S 3 its input impedance depends strongly upon the exc1tation frequency.

Th’
18 frequency dependence is a direct consequence of the standing wave

11?

distribution of antenna current. As the frequency of excitation is
varied, the maxima and minima of the standing wave of current shift
in position along the circumference of the loop. With the excitation
potential fixed therefore, the current at the driving point of the 100p,
and hence its input impedance, varies strongly with changes in the
frequency of excitation. As a result of this frequency sensitivity,

a conventional loop antenna is ordinarily used only at a single
frequency or over a very narrow band of frequencies.

A traveling wave antenna, in contrast to its standing wave
counterpart, has an input impedance which is relatively frequency
independent. This broadband character is a consequence of the
traveling wave distribution of current. Since the amplitude of the
traveling wave of current is essentially constant along the circum-
ference of the loop, except for the smooth decay due to radiation,

a variation in the excitation frequency does not result in a rapid
change in the current at the driving point. The input impedance of
a traveling wave loop antenna is therefore a relatively weak function
of frequency. It is this broadband character which is the most
important property of a traveling wave loop antenna.

The radiation pattern in the plane of a relatively small
conventional loop antenna consists of two broad lobes separated in
Space by 180 degrees. As the diameter of this standing wave loop
is increased, both the shape and the spatial orientation of the pair
of lobes in its radiation pattern undergo radical variations, although
they remain oppositely directed in space. Finally, as the loop circum-
ference is further increased, the pair of lobes split to form several

narrower lobes of equal but smaller amplitude.

118

On the other hand, the radiation pattern of a traveling wave
loop antenna of small dimensions is essentially unidirectional, having
a very broad major lobe in one spatial direction and a narrow minor
lobe of relatively small amplitude in the opposite direction. As the
electrical diameter of the traveling wave loop is increased, the
narrow lobe grows in amplitude while the broad lobe shifts in its
spacial orientation, decreases in amplitude, and finally splits to form
a minor lobe structure of relatively small amplitude. An electrically
large traveling wave loop antenna is thus characterized by a single
narrow major lobe which is accompanied by a relatively low minor lobe
structure. The major lobe of this pattern is spatially oriented in a
direction 180 degrees removed from the excitation point of the lOOp,
i. e. , in the direction of the traveling wave of current.

It is indicated therefore that the radiation characteristics of a
traveling wave loop antenna in no way resemble those of its standing
wave counterpart. The modified radiation pattern characteristic of
the traveling wave loop may be desirable for certain purposes. In
particular, the broad unidirectional pattern of a small loop or the
directive pattern of an electrically large loop may be useful for
certain applications.

6. 4. Previous Research on the Traveling Wave Loop Antenna

It was mentioned earlier that little research has been directed
to the study of circular loop antennas, and that the theory of such
antennas remains relatively undeveloped. A formulation for the
distribution of current and circuit characteristics of such a loop was

presented by Storer. 9 This theory expresses the current distribution

119

on the loop antenna in the form of a Fourier series, a result which is
very intractable for practical calculations. No theory has been
developed which yields a closed form solution for the distribution of
loop current in terms of simple functions. Further, no complete
formulation for the radiation field of a circular loop antenna in terms
of its distribution of current is available. Only the special case of an
electrically small loop, on which the current is assumed to be constant
along its circumference, has been considered in any detail. It is
therefore perhaps not surprising that no specific consideration has
been given to the realization of a traveling wave circular loop antenna.
A circular loop antenna multiloaded with lumped resistors has
been considered by Iizuka. 10 In this theory, use was made of Storer's
technique and the principle of linear superposition to obtain a Fourier
series solution for the distribution of current on the multiloaded loop.
It was noted that, when the loop was loaded with a single positive
resistance of approximately 100 ohms at a point 180 degrees removed
fr om the excitation point, its input impedance became relatively broad-
band as a function of frequency. A study of the corresponding amplitude
and phase of the current distribution on the loop, which was presented
by Iizuka, indicates that it approximates a traveling wave distribution.
Thus, while Iizuka's research was not directed toward realizing a
traveling wave loop antenna, his results appear to indicate that such
a traveling wave of current may be obtained through the use of a lumped
impedance loading technique, and that the corresponding input impedance

will exhibit a broadband character.

120

It will be demonstrated in Chapter 7, through a formulation
similar to Iizuka's, that the Fourier series solution for the distribution
of loop current cannot yield an explicit expression for the optimum
loading impedance to excite a traveling wave of current. A more
approximate closed form solution in terms of simple functions must be
resorted to in order to determine the Optimum impedance in terms of
the dimensions of the loop and its frequency of excitation.

6. 5. Object of the Present Research

It is the object of the present research to realize a traveling
wave loop antenna through the use of a lumped impedance loading
technique. In this investigation, the loop antenna is assumed to
consist of a thin perfectly conducting circular cylinder bent into the
form of a circular loop. The 100p is excited at an origin point and
doubly loaded with a pair of identical impedances which are placed
symmetrically along its circumference with respect to the origin.
With such a configuration, there are two degrees of freedom in
choosing a loading; its impedance and position. The Optimum loading
to yield a traveling wave distribution of current on the loop is to be
determined. In particular, the possibility of utilizing a properly
located purely reactive optimum loading is to be investigated.

Through the use of such a reactance loading technique, the
desirable circuit and radiation characteristics associated with a
traveling wave distribution of current may be obtained while avoiding
the introduction of dissipative elements. A traveling wave loop antenna
is therefore realized while maintaining the high efficiency of a

conventional standing wave loop.

121
6. 6. Outline for Theoretical Investigation of a Traveling Wave Loop
Antenna with Optimum Impedance Loading

The present research concerning a traveling wave circular loop
antenna with Optimum impedance loading is restricted to a theoretical
investigation. It will be the primary goal of this theory to determine
the parameters of an optimum loading to yield a purely outward
traveling wave of current over the circumference of the loop. No
experimental investigation will be conducted in conjunction with this
portion of the research, since the theoretical results will be found to
be very similar to those obtained for the case of an impedance loaded
linear antenna, which were thoroughly verified by an experimental
study.

It is the purpose of the theoretical analysis to: (1) determine
approximately the distribution of current on the doubly loaded loop
as a function of its dimensions, the excitation frequency, and the
impedance and position of the loading; (2) obtain from this result
(in terms of the loop dimensions and its frequency of excitation) the
parameters of an optimum double loading to yield a traveling wave
distribution of current along the circumference of the loop; (3) investi-
gate the possibility of utilizing a purely non -dissipative optimum
loading; (4) calculate the circuit and radiation characteristics of a

circular loop antenna utilizing such an optimum impedance loading.

CHAPTER 7

DISTRIBUTION OF CURRENT ON A DOUBLY
LOADED CIRCULAR LOOP ANTENNA

7.1. Geometry of the Doubly Loaded Circular Loop Antenna

The geometry of the doubly impedance loaded circular IOOp
antenna is taken to be as indicated in Figure 7. 1. A thin perfectly
conducting circular cylinder of diameter 2a is bent into the form of
a circular loop having an outside diameter of 2b. A system of polar
coordinates (r, 9) is established with its origin at the center of the
plane loop. The loop antenna is excited at 9 = O by a harmonic
.voltage source of frequency w and potential VO , and is symmet-
rically loaded at 9 : :teo with a pair of identical lumped impedances
ZL . With such a configuration, there are two degrees of freedom
in choosing a loading; its impedance and position.

In this research, both the source of excitation and the loading
impedances are idealized to be point elements. The gap in the loop
at its excitation point 9 = O is assumed to be centered about that
point and to have a length of 2b69 . Similarly, the gaps at the loading
impedances at 9 = :I: 90 are taken to have a length of 2b69 and to be
centered about those points. The point element assumption then
corresponds to letting 69 tend to zero as a limit. This mathematical
approximation corresponds to the physical requirement that the linear

dimensions of the excitation and loading elements be negligibly small

compared with the circumference of the 100p itself.

122

123

 

 

2b 2a

 

 

 

 

Figure 7. 1. Geometry of the Doubly Impedance Loaded
Loop Antenna.

124

7. 2. Dimensions of Interest for a One-Dimensional Theory

It is assumed that the circular loop antenna consists of a very
thin cylinder, whose radius is much smaller than the loop diameter
and at the same time is a small fraction of the wavelength. Under
these circumstances, it may be assumed that a one-dimensional
distribution of current is excited on the thin loop by its source at
9 = 0 . That is, the current is taken to have only a 9 -component
I9 (9) which flows parallel to the cylinder axis at each point along
the circumference of the loop. The dimensional restrictions which

validate this one-dimensional theory are thus

a < < b
(7. 1)
Boa < < l
where BO 2 Zir/XO is the free space wave number corresponding to
the free space wavelength )‘o .

Conditions (7.1) are also sufficient to validate the usual
approximation technique utilized in the study of thin-wire antennas.
With this technique, the vector potential at the antenna surface is
calculated as a contour inegral over the total antenna current, which
is assumed to be confined to flow along the axis of the thin wire. In
reality, the current flows throughout the cross section of the wire,
and is actually most concentrated at its surface due to the skin effect
phenomenon. The vector potential at the antenna surface should in
general, therefore, be calculated as a volume integral over the
current density on the thin wire. It has been indicated by Hallen, 1

however, that when conditions (7.1) are satisfied the error introduced

125

by the above mentioned approximation is negligible. This approximation
facilitates the solution for the distribution of current on the thin wire
loop antenna, which would otherwise be very much more difficult.
7. 3. A Rigorous Fourier Series Solution for the Distribution of
Current on a Doubly Loaded Loop; Its Failure to Yield the
Parameters of an Optimum Loading

The boundary condition on the electric field at the surface of

the loop is given by
(n x E) = O (7. 2)

where ’fi is a unit outward normal vector at a point on the surface and
E the electric field at the same point. This condition requires that
the tangential component of electric field be continuous across the
surface of the cylinder forming the circular loop. Since conditions
(7.1) are assumed to be satisified, the distribution of loop current
may be taken to be one-dimensional, i. e. , to have only a 9 -component
Ie (9 ) . Under these circumstances, the electric field at the loop
surface will have only a 9 -component and an r-component. The
tangential component of electric field at the surface of the loop is

therefore Ee (9 ) , and condition (7. 2) becomes

153(9) = Eye) (7.3)

where Eg(9) is the field just within the loop surface and 1223(9) is
that just outside its surface.

Since the cylinder comprising the loop is taken to be perfectly
conducting, then the applied field Eg(9) may be non-vanishing only

in the gaps at 9 : 0,:t00. Thus Eg(9) may be expressed as

126

 

r
Z I (9)
L9 0
2b69 for 00 69 9 90 + 60
a V0
Ee(9)‘< -3138? for -69<9< 69 (7.4)
Z I (9)
L9 0
.. <
—T—Zb6 for 90 69 < 9 90 + 69
K
r
6<9<9 -6G
— — o
Eg(9)=0 for ( eo+59595_-eo-59 (7,5)
-9 +69<9<-69
L o — —

 

where I6 (90) is the current flowing at the loading impedances and
269 tends to zero in accordance with point element assumption. In
result (7. 4), the symmetry of the distribution of current has been

utilized as
Ie(-9):Ie(9) (7.6)

The total voltage drop along the circumference of the loop must be

given by

TI
a
_5‘ Ee(e)bde = vO — 2 ZLIG(90) (7.7)

-7r
A result consistent with equations (7. 6) and (7. 7) is

123(9) 2 .11)— {- V06(6) + ZLIG(90)[ 6(6 -90) + 6(9 +eo)]} (7.8)

where 6(9) is the Dirac delta function.

127

The induced electric field Elem) just outside the surface of
the loop, due to the current and charge on the loop, may be calculated

from the vector and scalar potentialss(see Appendix A) A, d) as

Em) = 4%), (95%),, (7.9)

. . . . . . out
Since the time variation is assumed to be harmonic of the form eJ ,

it is possible to make the replacement Eaf - jw , where the potentials
and fields are then understood to be complex valued. There is then

obtained

ng): -(V¢)e -ij (7.10)

0

where A6 and (b are the potentials at the surface of the loop.
The vector and scalar potentials at the surface of the loop may
be expressed in terms of its distribution of current and charge by the

Helmholtz integral (see Figure 7.1) as

 

“o 1* , . e-jBoR '
Ae(9) = 31:5: 19(9 )cos(8-G )———R—— bde (7.11)
1 1T eu‘j‘30R
¢(6) : 4Tr€ 5‘ q(e') —-R—— bd9' (7.12)
o —17

In these expressions, HO and 60 are the permeability and permittivity
Of free space, respectively, q(9) is the charge per unit length
distribution along the loop, and R is the Euclidean distance between
an observation point on the loop surface at 9 and a source point on

its axis at 9 ' . The distribution of charge on the loop is related to

its current distribution by the equation of continuity as

v. T(e)+jwq(e) = o (7.13)

128

and since the current is one-dimensional this becomes

 

_ .L l 2.
r=b
_ .L .3.
_ wb 89 Ie(0) (7.14)
With this result, the scalar potential may be expressed as
'jfioR
(7.15)

11'
j 8 , e
¢(6 ) 2 411-60“) 5-1T 8 9| 19(6 ) —'_R'_- (19 1

Referring to Figure 7.1, the Euclidean distance R may be

calculated from the law of cosines to give

[b2 + (b-a)2 - Zb(b-a) cos (9 -9')]1/2
(7.16)

R
b[4(1-a/b) sinz (9.222;) + aZ/b2]1/Z

In the thin wire approximation a/b may be dropped, but aZ/b must
_ l
9 0 may also be very small. An approxi-

be retained since )
mate expression for R which is valid for a < < b is thus

92.9 ') + az/bz]1/.z

 

sin2 (

(7.17)

 

R ‘= b I 4 sin2(
Equation (7.15) for the scalar potential (He) may be integrated

by parts to obtain
9 I211'

. -so
¢(9) = —J——41T€ w [19(6') —-————“’ R :l
O 9'2-7T
(7.18)

TI' -jfiR
a e 0
) 19(99561‘7— de'

The integrated term vanishes since, by equations (7. 5) and (7.16),
=-w).

respectively, it is found that 19(4) = I9 (it) and R(9 '=TT) = R(9'

129

Further, it is noted from expression (7.17) that

8R 8R

sew-‘55

and equation (7.18) therefore becomes

. 1r -jf3 R
¢(e) : Erie—“)5 19(9') 5%- i—fi—O— d9' (7.19)
0 -TT

Relations (7.11) and (7.19) may be substituted into expression
(7.10) to obtain the induced electric field just outside the surface of the

loop as

. . -' R
1 - .1— 2... __J___ Tr I _a_ e J50 [
139(9) ‘ ' b as [zineow 5fl16(9) as R ‘19

jwpob 17 e-jBOR
-717— 16(9')cos (9-9') —-—§—-— d9' (7.20)

Upon combining terms and rearranging slightly, equation (7. 20)

may be expressed in the form

1 e jgo W 9! e 9' e!
E9( )- - m - 19( )K( : )d (7-21)
In this result, {.0 = p. 60 is the intrinsic impedance of free space

and the kernel K(G , 6 ') is defined by

 

 

f 1 32 e-ifioRI
K(9,9 ): Lfiob COS (9 -9 )+ 50b 392 T— (7.22)
where
R1 = B}: = [4 sin2 (92‘9') + aZ/b2]1/2 (7.23)

130

If expressions (7. 8) and (7. 21) for 133(9) and Elem), respectively,
are used in equation (7. 3), an integral equation for the distribution of
loop current is Obtained as
jéo w
*5 19(9')K(9,9')d9' = V06(9) (7.24)
TI'

4~rr

— zLIe(eO)[ 5(e-eo) + 5(e+eo)]

The unknown distribution of loop current 19(9) appears in the integrand
on the left side of this integral equation, while its value 19(90) at the
location of the loading impedances appears on the right hand side in a
term analogous to the excitation function V06(9 ) . This equation is
parameterized by the 100p diemsnions a, b which occur in the kernel
K(9 , 9 ') and by the impedance and position ZL’ Go of the double

loading. In the special case where Z = 0 , result (7. 24) reduces to

L
the integral equation obtained by Storer9 for the current on a conventional
circular loop antenna.

A rigorous Fourier series solution has been Obtained by Storer9
for the distribution of current on a conventional loop antenna. Integral
equation (7. 24) for the current distribution on a doubly impedance loaded
loop is very similar to the one obtained by Storer for an ordinary loop,
the only difference being the additional shifted Dirac delta function
terms on its right hand side. It is a consequence of this similarity
that Storer's Fourier series technique may be applied, almost without
modification, to obtain a rigorous solution to integral equation (7. 24).

The application of this method to obtain an exact solution for the

distribution of loop current will be sketched in the development to follow.

131

It is noted that R1(9 , 9') = R1(9 -9 ') is continuously differentiable

and that Rl(9 -9 ') > 0 for -TI' E (9 -9 ') 5 1T . The function

e-jBObRue -9 ')

f(6 .9!) = R1“; .9!)

 

(7.25)

is therefore bounded and continuous with a continuous first derivative for
- TI' 5 (9 -9 ') < 1r , and is periodic with f(-TT) = f(TI’) . This function may

thus be expanded in a Fourier series as

 

-.-jBobR1<e -9 '> .. me -e .,
, = Z k e
R1(f-9 )

n: .00

(7.26)

n

which converges uniformly to f(9 -0 ') on - 11' _<_ (9 -9 ') _<_ 7r . The

Fourier coefficients of this expansion are given by

 

1 Swr e-jBObR1(¢)

e‘jn¢ d4) (7.27)
R1(¢)

kn = ‘2';
Expansion (7. 26) may be substituted into equation (7. 22) to obtain
a new expression for the kernel K(G, 9 ') . The Fourier series (7. 26)
may be differentiated term by term, and K(9 , 9 ') becomes after some

straightforwa rd manipulation

 

 

' _ I
K(9,6'): :2 new“9 9) (7.28)
n:-00
where
kn+1 -kn-l nzkn
an : (30b 2 - (30b (7.29)

Equation (7. 28) expresses the Fourier series expansion of K(9 -9 ') on
- 1r 5 (9 -9 ') E 77 with coefficients specified by relation (7. 29). The
Fourier series of an absolutely integrable function may be integrated

term by term, thus substituting expansion (7. 28) for K(G , 9 ') into

132

the left hand side of integral equation (7. 24) and interchanging the order

of summation and integration gives

jg oo 77 .
7 (9-9') 7
Z772 n-Zoo (1113:.“ 19(9 )eJn d9
= VO 5(9) - ZLIG(eo) [6(8 -60) + 6(6+60)] (7.30)

Since the loop current is bounded and continuously differentiable
(by physical necessity) with Ie ( -77) = 10 (77) , then it may be expanded in

a Fourier series as

(I)
I 9 = Z I 7.31
9H mm n e ( )

which converges uniformly to 19(9) on - 77 5 9 _<_ 77 . If expansion
(7.31) is substituted into equation (7. 30) and the order of summation
and integration interchanged (by the uniform convergence), there is
obtained

jg co 00 TT . .
_2 Z) a Z I 5‘ ejn(0 -0!)er97 de' :

411. n: _CD p: .0) p -TI‘

= V06(0) - zLie(eo) [ 6(9 -90) + 5(e+eo)] (7.32)

However, by the orthogonality of the set {eJn<1> }

77 . . 7 .
S, eJne ej(p-n)9 d9' = 277 eJne 6!}: (7.33)

-1T

where 6: is the Kroneker 6-function, and result (7.32) becomes

-22. )3 a 111.3119 = 7705(9) - zLIG<eO)I6(e-eo> +6<9+90n (7.34)

133

-‘ 9
If equation (7. 34) is first multiplied by e Jm and then integrated
with respect to 9 on - 77 _<_ 9 j 77 there is obtained, after interchanging
the order of summation and integration on the left hand side
jgo °° 1‘ jnG -jm0
T 2 O. I e e d6
11: _oo n n -77

(7.35)

77 .
_ -jm0
_ S {V06(0) - zL19(eo)[ 6(9 -90) + 6(6+90)] } e de

-1T

Making use of the orthogonality of the set {ejmb} and carrying out
the straightforward integration on the right of expression (7. 35) results
in

m

jQOTr Z a I 6n = VO - Z ZLI9(90) cos (meo) (7.36)

n: _CD n n

From equation (7. 36), a relation between the Fourier coefficients In

and the coefficients on is immediately Obtained as

_ 1 7
In _ j77§ a [v0 - 2 ZLIG(90) cos (1190)] (7.37)
o n
It is a simple matter to complete the solution for the distribution
of loop current by writing out the Fourier series for 19(9 ), with

coefficients given by result (7. 37), as

00 V - 2 Z I (0 )cos(n9 ) .
z 0 L9 0 ° eJne (7.38)
:-CI) an

 

 

10(8) : j77§

The coefficients on are given by equation (7. 29), and have been
evaluated in detail by Storer. 9 Expression (7. 38) indicates that the
distribution of loop current depends upon the impedance and position

ZL and 90, respectively, of the double loading as well as upon the

134

loop dimensions (which are implicit in on). In the special case where
ZL = O , this solution becomes identical with the one obtained by Storer.
The constant 19 (90) in solution (7. 38) for the distribution of

loop current remains as yet undetermined. Since the loop current is

continuous, however, the condition
19(0200) : 19(90) (7.39)

must be satisfied. With this condition, equation (7. 38) may be solved
for 19 (90) in terms of the loop dimensions and the impedance and

position of the loading as

 

 

 

 

 

. -l
9 .
9 o j77 o jwéo n=-°° an n:_co an
(7.40)
If H(ZL,90) is defined as
'ne -1
ZZL ZZL co eJ 0cos(n90) 00 ejneo
H(ZL, 60) = T 1+ T Z 2
J“ O I" O n=-°° an n:-00 an
(7.41)

then the final solution for the distribution of loop current becomes

 

V co 1 - H(Z ,9 )cos(n9 ) .
19(9) = ——£’-— 2 L 0 0 (am9 (7.42)
N o n:-°° C1n

It is observed from solution (7. 42) that the coefficients of the
Fourier series for the distribution of loop current depend in a very
complicated way upon the impedance and position of the double loading.

Furthermore, it has been indicated by Storer9 that this is a slowly

135

converging Fourier series. The possibility of obtaining an approximate
solution by truncating the series and retaining only a few of the leading
terms is thus precluded.

Expression (7. 42) does, however, represent a formal solution
for the distribution of loop current. For a given set of loop dimensions,
excitation frequency, and loading parameters, the series could be
machine summed using Storer's values for the coefficients on . It is
evident, however, that there is no way of determining from this
solution an explicit expression for the parameters of an optimum
loading to yield a traveling wave distribution of loop current. A trial
and error process would therefore be required and, since there are
two degrees of freedom in choosing a loading for each set of loop
dimensions and each excitation frequency, this process would appear
to be highly impractical.

The failure of the formal Fourier series solution to provide
an explicit expression for the optimum loading impedance severely
limits its practical usefulness. A more approximate technique is
therefore required to obtain a simple closed form solution for the
loop current, from which an expression for the parameters of an
optimum loading may be determined. Such an approximate theory
is the subject of the following section, where explicit expressions
for the impedance and position of an optimum loading are obtained.

7. 4. Approximate Distribution of Current on a Doubly Loaded
Loop Antenna

In this section, an approximate theory for the distribution of
current on a doubly impedance loaded loop antenna is presented.
Quite in contrast to the rigorous Fourier series solution obtained

in the preceding section, a closed form result is obtained in

136

terms of simple functions. From this approximate solution, explicit
expressions may be obtained for the parameters of an optimum loading
to yield a traveling wave distribution of current.

It was found in the preceding section that the electric field at

the surface Of the loop must satisfy the boundary condition
a .
129(9) = 7229(9) (7.3)

where E3(9) is the field just within the loop surface and E19(9) is
that just outside its surface. An expression for the applied field

E3(9) was determined as
123(6) = %{ - V06(6) + zL19(eO)[ 6(6 -90) + 6(6+60)]} (7.8)

where 6(9) is the Dirac 6-function. The induced electric field E219 (9)
at the surface of the antenna, due to the current and charge on the

loop, was written as

Eye) = 4%), -ij9 (7.10)

where A9 (9) and ¢(9) are the time harmonic vector and scalar
potentials, respectively, at the antenna surface.

The vector and scalar potentials at the surface of the loop
may be expressed in terms of its distributions of current and charge

by the Helmholtz integrals5 (see Appendix A) as

 

p. 77

739(9) = 3,3 559(9) cos (e-e') K(e.9')bd9' (7.43)
1 1T

¢(e) = 4M 5 q(O')K(G,6')bd9' (7.44)

0 -1T

137

where K(9 , 9 ') is the Green's function

e'jpoR(e 9 e I)

K(999') : R(ejel)

 

(7.45)

In the above expressions, 19(9) is the distribution of total loop current,
q(9) is its charge per unit length distribution, and R(9 , 9') is the
Euclidean distance between a source element on the axis of the loop

at 9 ' and an observation point on its surface at 9 . An approximate

expression for R(9 , 9') was developed in the preceding section as
_ l
R(9, e .) = b[ 4 3171‘2 (979—) + aZ/bz']1/2 (7.17)

which is valid in the thin-wire approximation where a < < b .

The peaking property of the kernel K(9 , 9 ') is exploited to
formulate an approximate theory for the impedance loaded loop antenna.
Since (a/b)2 < < 1 by the thin-wire assumption, then K(9, 9 ') has a
very sharp peak at 9' = 9 when considered as a function of 9' on
- 77 E 9 ' _<_ 77 . The contribution to the vector and scalar potentials
Ae(9) and 43(9), respectively, at each point on - 77 f 9 5 77, as
calculated from equations (7. 43) and (7. 44), is therefore due primarily
to source elements in a small neighborhood about the point 9' = 9 .
Because the source distributions are continuous, then the current and
charge 19(9') and q(9 ') , respectively, for 9' : 9 make the major
contributions to the vector and scalar potentials at a point 9 on the
loop surface. It is therefore expected from this argument that the
ratios A9 (9 )/I9 (9) and (M9 )/q(9) should be essentially constant

at each point along the loop circumference, or for - 77 _<_ 9 5 77 .

138

A pair of essentially constant dimensionless quantities are
defined by

Aew)

:11 __
19(9)

{11(9) : -77:9:77 (7.46)

_ (#9
7111(9) _ 4460 7119—1 -4593” (7.47)

and are designated as the "current expansion parameter" and "charge
expansion parameter, ” respectively. In accordance with the foregoing
argument, the functions \I’i(9) and \I’q(9) should be essentially
independent of 9 , and be determined primarily by the loop dimensions.
Furthermore, these functions will not depend strongly upon the source
distributions I6 (9) and q(9 ) . It is therefore asserted that \I’i(9) =
\IIi and ‘Ilq(9) = ‘I’q are indeed constants depending only upon the loop
dimensions. The validity of this assumption will be discussed more
fully in section 8. 5 of the next chapter.

According to the approximations described in the last paragraph,
the vector and scalar potentials at the loop surface are related to the

corresponding current and charge distributions as

p'o \Pi
191.9(9) = T 19(9) (7.48)
‘1!
7(6) = fif— q(9) (7.49)
O

The distribution of charge is, however, not independent, but related to

the current distribution through the equation of continuity as

139

v-i‘

q(9)
r=b

C31? 3%— 19(9) (7.50)

With this last result, expression (7. 49) becomes
J" a

we): 4765:3055 19(6) (7.51)

From equation (7.10), -the induced electric field E19(9) at the

surface r = b of the loop may be expressed as

I .x
'.,.l

Ego) -(V¢)e -ij9
=.. - % 3% - ije (7.52)

If expressions (7. 48) and (7. 51) are used for A9(9) and ¢(9),

respectively, result (7. 52) becomes

i e '3‘1’9 ()2 2 2‘I’i
E ( ) = —— + [3 b I (9) (7.53)
9 2 2 o T 9
1;, 4TT€Owb 89 q

2 2
where the definition (30 = to H060 has been used. A complex wave

number (3 is defined as

(s = (30 W1 (7.54)

in terms of which equation (7. 53) may be expressed as

° 'j‘I’ 32 2 2
121(9) = q + (3 b 1 (e) (7.55)
9 417(50wa (:an 9

 

140

If results (7. 8) and (7. 55) are substituted into equation (7. 3),
in order to satisfy the boundary condition on the electric field at the
surface of the loop, a second order inhomogeneous differential

equation for the distribution of loop current on - «_<_ 9 5 1r is

obtained as
Z Z
3 Z 2 _ '4Trb
7+pb 19(e)_fi {-V06(9)+
89 0 1

+ ZLIG(6)[ 5(9 -90) + 5(e+eo)]} (7.56)
A complementary solution to this equation is well known to be
13(9) = cl erbe + c2 8-]pr -n _<_ e 5 71' (7.57)

where c1 and c2 are arbitrary complex constants. The particular
solution is determined as
217V . .
p _ o wblel 2w -Jsble-eol
19(9)- W e 'W ZLIe<90>[e

O O

+e'j5b'9+90'] -1T:9_‘_<_1T (7.58)

which is readily verified by direct substitution into differential equation

(7. 56). In this last result, L0 = VHO7€O is the intrinsic impedance

of free space, and the "expansion parameter” ‘11 has been defined as
\I’ : «I W. \I’ (7. 59)

The complete solution for the distribution of loop current is obtained

by superposition of results (7. 57) and (7. 58) as

(7. 60)
It should be noted that a solution in terms of complex exponentials
has been obtained since a traveling wave distribution of current having
such a functional dependence is to be sought eventually.

Solution (7. 60) contains the three as yet undetermined constants
c1, c2, and 19(90) . There are three physical boundary conditions
which facilitate the evaluation of these constants. Due to the symmetry
of the loop, its distribution of current must be symmetric and its
charge distribution antisymmetric about the point of excitation at

9 = 0 . These conditions may be expressed mathematically as
Ie(-9) 2 19(9) (7.61)
<1(-9) = - q(9) (7-62)
Since the distribution of charge must be continuous at 9 = 1r , then

condition (7. 62) may be satisfied at that point only if q(Tr) = O . However,

q(9) is related to 19(9) through the equation of continuity as

q(9): “—33 3%- 19(9) (7.50)

The condition q(Tr) = 0 thus implies that

9
'5'6‘ 19(9)

ll
O

(7.63)
9:17

142

A third condition is obtained from the continuity of the distribution of

current at 9 = 90 as
Ie(9:90) : 19(90) (7.64)

Solution (7. 60) is consequently subject to the set of three boundary

c onditions

Ie('e) : 19(6)

3
57,- 19(9) 2 o (7.65)
9:1.

16(9 :9 O) : 16(90)
The first of conditions (7. 65) may be satisifed by the distribution

of loop current (7. 60) only if c2 = c1 , which yields the simplified result

ZTTV

'be 6'pr o sable!
I (9) =c eJfi +c e + e
9 1 1 COW
13% Z 1 (9 ,[e-ijIG-eol .e-jsblewol]
O

- 11' E 9 _<_ 1T (7. 66)
A straightforward but tedious application of the last two of boundary

conditions (7. 65) to result (7. 66) yields finally the approximate

distribution of current on the loop as

O

N

V 17 P . P . .
° [_1. eJBbe +__l e-Jflbe + 2 e-JBblel

Z P . .
- 591% (F1 cos [3b90 + e-Jfibec) (e-Jfible -90l _
Z

+ e-jfible +960] -1. 5 e 5 1. (7.67)

143

In this result, the factors T, Pl , and P2 are constants depending
upon the excitation frequency, the loop dimensions, and the impedance

and position of the double loading as

 

Z .
T = ‘I’+ 737]):- (1+ e-328b90) (7.68)
ZL —jpbe
P1 = 1 - 30.1. e Ocos (3b9O (7.69)
'[3bn‘ ZL 2
P2 = jeJ sin fibw + 367i.- cos 9b90 (7.70)

An approximate expression for the distribution of current on
the doubly loaded loop antenna has been obtained in equation (7. 67).
This distribution completely characterizes the loop antenna, and is
given in terms of its dimensions, the excitation frequency, the
impedance and position of the double loading, and the as yet
undetermined expansion parameter. The optimum loading to yield
a traveling wave distribution of current on the loop will be obtained
from this result in the next chapter. This traveling wave current
distribution will then be used to calculate the value of the expansion
parameter ‘1'.

The similarity between the distribution of current (7. 67) on an
impedance loaded loop antenna and the expression (2. 28) for that on a
doubly loaded linear antenna is quite apparent. If the constants (30,
D1 , and D2 in the result for the linear antenna are replaced by 8b,
P

and P2 , the corresponding expression for the loop antenna is

1’

obtained. In the case of a loop antenna, however, (3 is a complex

number, and the expression for P is quite different from its counter-

2

part D2 . The functional depedence of the current upon position along

the antenna is otherwise identical in either case.

144

7. 5. Input Impedance of a Doubly Loaded Loop Antenna

The input impedance to the loop antenna is defined by

v
zin- = TTGEE") (7.71)

9

This impedance is readily evaluated by using result (7. 67) for the
approximate distribution of current on the doubly loaded loop, and

is found to be

-I
P Z . P
_ __1_ L -J[3b9 __l_ -jpbeo
Zin—6O\II 1+P2-30T e 0 (Pacosfib9o+e

 

(7.72)

An approximate relation for the input impedance of a doubly impedance
loaded loop antenna is thus obtained in terms of its dimensions, the
frequency of excitation, and the impedance and position of the loading.
Again the similarity between this result and the corresponding

expression (2. 33) for a linear antenna is immediately apparent.

CHAPTER 8

OPTIMUM LOADING FOR A TRAVELING WAVE
DISTRIBUTION OF LOOP CURRENT

8.1. Physical Interpretation of the Distribution of Current on a
Doubly Loaded Loop Antenna

An approximate expression for the distribution of current on a
doubly impedance loaded loop antenna has been developed as equation
(7.67). This solution is valid on -1r _<_ 9 _<_1r, but since Ie(-9) = 19(9)
it is sufficient to consider only the current on O E 9 5 TT . If attention
is restricted to the regions 0 E 9 _<_ 90 and 90: 9 _<_ 11' , then the
various terms in expression (7. 67) may be combined to yield a pair

of results which are more physically meaningful.

The distribution of loop current on O 5 9 _<_ 90 is obtained

 

from the general result (7. 67) as
V TT P Z . P . .
_ _2_ _1 L -Jfib _l -Jfib9 -Jf3b9
o 2 2
P Z . P
_l _._1_:. -Jfib90 __1_ -j[3b90 jpbe
+ P2 -3OT e (P2 cos Bb90+e e (8.1)

VTr P z P
_ o 1 L 1 -ij9 -J'F5b9
19(9) ‘ {7'13 [2+P '15": C05 F3bee (T72 COS pbeo+e 0)]e

O

_l. jfibe
+ P e (8.2)

A physical interpretation of the current distribution is readily obtained

from expressions (8.1) and (8. 2).

145

146

It is noted from equation (8. 1) that the total loop current on
O E 9 S 90 may be considered as a superposition of a pair of outward
and inward traveling current waves. The first term in this expression
represents an outward traveling wave of current which is excited by
the source at 9 = O . At 9 : 90 this current wave is partially
reflected and partially transmitted. The second term of equation
(8.1) represents an inward traveling wave of current which results
from the reflection of the outward traveling wave by the impedance
discontinuity at ' 9 = 90

Similarly, equation (8. 2) indicates that the distribution of
loop current on 90 5 9 f 7r is composed of a pair of oppositely
directed traveling current waves. The first term of this expression
represents an outward traveling wave of current which is excited by
the potential difference at 9 = 90. The second term of equation (8. 2)
represents the inward traveling wave of current which results from
the reflection by ZL at 9 = - 9O.

It is indicated by the above results therefore, that in general
both outward and inward traveling waves of current are supported on
each of the two antenna regions 0 _<_ 9 _<_ 90 and 90 E 9 5 1r . The
superposition of these Oppositely directed traveling waves results
in a standing wave distribution of current along either region. Thus
in the usual case a standing wave of current is supported along the
entire circumference of the loop.

8. 2. Optimum loading Impedance for a Traveling Wave Distribution
of Loop Current

It was indicated in the preceding section that the doubly loaded

loop generally supports a standing wave distribution of current over

147

its entire circumference. The possibility that this distribution might
be modified, through the selection of an Optimum impedance loading,
to yield a purely outward traveling wave of current over most of the
loop is now to be investigated.

It is evident physically that no choice of the loading will give
a purely outward traveling current wave on 90 _<_ 9 5 1r , since the
current should be maximum at 9 = 7r subject to the boundary condition
at that point. However, it is reasonable to suspect that if the loading
is properly chosen the inward traveling wave on 0 _<_ 9 E 90 might be
eliminated, leaving only the desired outward traveling wave of current
over that region. Since this inward traveling wave is actually reflected
from the impedance dicsontinuity at 9 = 90, it is expected that such
an optimum loading should exist.

The optimum loading impedance to yield a purely outward
traveling wave of loop current on O i 9 E 90 may be obtained from
expression (8.1L). This condition evidently requires that the inward
traveling current wave on that region should vanish. Realization of
this condition is accomplished by equating the coefficient of the second
term in equation (8.1), which represents the amplitude of the inward

traveling wave, to zero as

ZL -jpb90 fl
30T 8 P2

 

cos pheo + e‘jfibeo = o (8.3)

"UI'U
NH

Using the defining relations (7.68), (7. 69). and (7.70) for T , P1,

and P2 , respectively, this equation may be solved for the optimum

loading impedance, designated as [ZL] , to yield
0

148

eJ7me.

: 3” ejfibhr J?

(8.4)

 

cos fib90 + j 0) sin [Sbrr

After some lengthy but straighforward manipulation, result (8. 4) may

be put into the simpler form

[2 = 3o\Ir[1 - j tan [31)(1T -90)] (8.5)

L]o

When the loading impedance is given by this relation, the loop current
on O _<_ 9 5 90 becomes the desired purely outward traveling wave,
while that on 90 _<_ 9 5 17 remains the usual standing wave

Expression (8. 5) gives the optimum loading impedance in terms
of the excitation frequency, the loop dimensions, and the position of
the loading. For a given set of antenna dimensions a and b , this
optimum impedance is a function only of the frequency w and its
position 90 . At this point the loading location is completely arbitrary,
and may be freely specified in order that the corresponding impedance
may satisfy certain prescribed conditions.
8. 3. Purely Reactive Optimum Loading

A purely non-dissipative optimum loading is of particular
interest, since such a loading would permit the realization of a high
efficiency traveling wave 100p antenna. It has been indicated that the
optimum loading impedance [ ZL] 0 depends only upon its position
90 and the excitation frequency 6) once the loop dimensions a and
b have been specified. This leads one to suspect that, at least at a
single frequency, it should be possible to choose an optimum position
for the loading such that [ ZL] 0 will become purely reactive.

Since ‘11 and [3 are in general complex numbers, then the

149

following designations will be established

‘1’ = M+ jN (8.6)
mow-90) = x+jy (8.7)
tan (3b(7r -90) = u + jv (8. 8)

With these definitions, expression (8. 5) for the optimum loading

impedance becomes

[2 3o (M+jN)[1 - j<u + M]

L]o
3o {[M(l+v) + Nu] + j[N(1+v) - Mu] } (8.9)

The condition for a non-dissipative loading is obtained by equating

the real part of equation (8. 9) to zero, which gives
u _ M
— — - N (8.10)

With this result, the corresponding optimum loading reactance becomes

2
[x = 30(l+v) <N+&) (8.11)

L]o N

It is indicated therefore that a loop antenna which is doubly loaded
with an optimum non-dissipative impedance, whose reactance [ XL] 0
is given by expression (8.11) and whose position 90 is so selected
that equation (8.10) is satisfied, will support an outward traveling
wave of current on 0 _<_ 9 _<_ 90.

The necessary position of an Optimum non-dissipative loading
may be obtained from relation(8. 10). It is first necessary to express
u, v in equation (8. 8) in terms of x, y from definitions (8.7) as

sinin)
cos (2x) + cosh (2y)

 

(8.12)
sinh (2y)
cos (2x) + cosh (2y)

 

150

In terms of these results, condition (8.10) for a purely reactive

optimum loading becomes

sin (2x) _ M
COS (2x) + cosh (2y) + sinh (2y) 7 " N (8-13)

From this expression, it is desired to determine (Tl’ - 90) where by

definition 8b(7r-90) = x + jy, and M/N and 8b are known quantities.
Evidently some approximations are in order to facilitate the solution

of this trancendental equation.

According to definition (7. 54) for 8 , it is possible to express

0/00 as
111.
_ 1
1 T17 (8.14)

The current and charge expansion parameters ‘I’i and ‘Ilq, respectively,
corresponding to a traveling wave distribution of current, will be
evaluated in section 8. 5. These results may be utilized to calculate

the ratio 8/8O , as expressed by equation (8.14). Figure 8.1 indicates

a plot of 8/80 as a function of the electrical loop circumference 80b

for a loop having dimensions specified by a/b = 0. 0423 or n. = 2 ln
(ZTrb/a) = 10 . It is noted that the real part of 8/80 is always of the
order of unity, while its imaginary part is always less than approximately
0. 05 in absolute value.

In accordance with the remarks of the last paragraph, it would
appear that for a zeroth-order approximation the imaginary part of 8
may be dropped entirely and the value of 8 taken as 8 = 80 . Under
these circumstances, the position of an optimum pure reactive loading

is determined immediately by equating the real part of equation (8. 5)

151

(Cd/d) }o 119d Aleuifiemi

.oucouomguuwo mood amofiuuuofim 23 .«o cofiugh .m an

 

yonfifiz 0.283 xanoU pouSdEqu 05 mo atom .anfiwmfiu pad 30% 4 .w 6.5th
me .o: r .
o
o. u oocouofitdoumo moofi 3373001»
or. mg as. no. o.~ m.V\\ ouzlllho
oo.o 4 q q . \ q q 1 Odd
\" \
1 / \
/ \
No .0 1 / 1 o.
/ \
/ \
/ \ o

oo .o

mod

O10.-

 

    

A638;

3 u Ens: E N u c

 

ea .0

mo .0

co;

(08/8) 30 incl 129i

152

to zero as
_ -l M
80b(1r -9O) — tan (vfi) (8.15)

The ratio M/N is always negative and greater than 3. 0 in absolute
value for a loop with n = 10 and for O. 25 5 80b 5 4. 0 . Thus
8Ob(7r -90) is always of the order of 17/2 for a loop of such
dimensions.

Using the result 8Ob(Tr -90) i Tr/Z of the zeroth-order
approximate solution, a more accurate formulation may be obtained
from equation (8.13). Since the maximum value of Im(8/8O) is of

the order of O. 05, then

[y] = [ Im (awn-80)] x é (005)12— = 0. 08

max ma

and the maximum value of 2y is of the order of 0.16. The hyperbolic

functions in expression (8.13) have the power series expansions

cosh(2y) : 1+QZ-X)— +
(2 )3
sinh(2y) 2 (2y) + —3—)’—— +

and since it has been found that 2y < 0.16 , then a very good approxi-
mation is obtained by retaining only the leading term in each expansion.

With this approximation, condition (8.13) becomes

sin (2x)
1 + cos (2x) + 2y

 

-_M
‘ N

which may be written in the form

sin x cos x

 

2|:

(8.16)

2
cos x+y

153

Result (8.16) may be further approximated by noting that

x = Re[ (WW-90)] = (3068.00) = w/2
Power series expansions of sin x and cos x are then made as
. l 2
Sinxzcos(1r/2-x)=1 - 7 (TT/Z-X) + (8.17)
. l 3
cosx= sm(1r/2-x)=(1r/2-x) -6-(17/2-x) +... (8.18)
and since x E Tr/Z only the first few terms of each expansion need

be retained. Substituting expansions (8.17) and (8.18) into equation

(8.16), and dropping all terms in (Tr/2-x)3 and higher powers gives

(I/ng) : _ LIE/[I- (8.19)
(Tr/Z-x) +Y

If the constants 81, 82, <1) , and Q are defined as

Re (13b)

pl
(32 Immb)
¢ (v-00)

Q = -M/N

(8.20)

then

x = Rembw-eon = 01¢
(8.21)
y -- lambs-90)) = (32¢

and condition (8.19) becomes

5
Z 2 1 1r 1
(l) + __2 + — _ _ (p + _ _—

 

154

Since 81, 82, and Q are known constants, then quadratic equation
(8.22) may be solved directly for <1) = (1T -90) to obtain the necessary
position of an optimum non-dissipative loading.

In order to verify the accuracy of result (8.22), it was solved
to obtain 4) = (17 -90) for several values of the electrical loop
circumference 80b. The values of x and y were then calculated
according to equations (8. 21 ). These results were then substituted
back into equation (8.13), which was found to be satisfied identically.
Hence the approximate solution (8. 22) was verified to be essentially
100% accurate. It should be noted that only the smaller of the two‘
roots to equation (8.22) was considered, since it is desired to
minimize the length of the region 90 E 9 3 “IT on which a standing
current wave exists.

To summarize, the position 4) : (Tr-90) of an optimum purely
non-dissipative loading may be calculated from equation (8. 22) in
terms of the loop dimensions and its frequency of excitation if
80b (1r ~90) = Tr/2 . The corresponding optimum reactance of the
loading is then given by expression (8.11), which may be written

appr oximately a s

2 .
[XL]O = 30 (1+——X-Z——)(N+-Mfi—) (8.23)
COSX

When a non-dissipative loading having these parameters is utilized,

then the corresponding distribution of loop current on 0 _<_ 9 _<_ 9 0

becomes a purely outward traveling wave.
In order to obtain some specific numerical results, a loop

antenna specified by the dimensions a/b = 0. 0423, or n = 2 ln(21rb/a) = 10

155

will be considered. The position (17 -90) and reactance [XL] 0 of

an optimum non-dissipative loading for such a loop are indicated in

Figures 8. 2 and 8. 3, respectively, as a function of the electrical

loop circumference 80b, as 80b varies from 0. 25 to 4. 0. These

results were calculated from expressions (8.22) and (8.23 ) using the

relations for ‘Pi, ‘I’q, and \I' which will be presented in section

8. 5. Since 80 = (1)/vO where V0 is the velocity of propagation in

free space, then 80b = cob/v0 is a linear function of the angular

frequency w of excitation. It is noted from Figure 8. 2 that (Tr -90)

is a strong function of frequency, and that in fact 8Ob(7r -90) is

essentially frequency independent. Further, the nearly constant

value of 8Ob(1r -90) is of the order of TT/Z , which justifies the

approximations leading to solution (8.22). Figure 8. 3 indicates

that the optimum loading reactance [ XL] 0 is a strong function of

frequency for relatively small loops, but settles down to a value of

approximately - 500 ohms when 80b is greater than 1. 5 .
Expressions (8.22) and (8.23 ) for the position and reactance

of an optimum non-dissipative double loading are perhaps the most

significant results of the present research. Given the loop dimensions

and its excitation frequency, the parameters of an optimum non-

dissipative loading are readily calculated from these simple expressions.

By utilizing such a loading, a traveling wave antenna may be realized

which retains the high efficiency of a conventional loop. Since the

theory leading to results (8. 22) and (8. 23) is very similar to that of

the linear antenna of Part I, which was successfully verified

156

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experimentally, it was felt that no experimental study of the reactance
loaded loop antenna was warranted.

It has been indicated that, for a given set of loop dimensions,
the necessary position of an optimum non-dissipative loading is
strongly dependent upon the excitation frequency. Physically, however,
a practical arrangement of such a doubly loaded loop requires that the
position of the loading be fixed at some point along its circumference.
With this restriction, an optimum purely reactive loading may there-
fore be realized only at a single frequency. If the position of such an
optimum impedance loading is so chosen that it becomes purely
reactive at a given frequency, then at any other frequency it must
have both resistive and reactive components.

Since the position of the impedance must be fixed, it is of
interest to consider the frequency dependence of an optimum loading
whose location is so chosen that its impedance becomes purely
reactive at a predetermined frequency. In order to obtain specific
numerical results, the loop specified by n = 10 will again be con-
sidered. A loading position of (1T ~90) = 27.10 is chosen so that the
optimum impedance becomes purely reactive when 80b = 2. 5 (see
Figure 8. 2). Figure 8. 4 indicates the Optimum impedance Of such a
fixed loading as a function of the antenna's electrical circumference
80b. These results were calculated from expression (8. 9) for [ ZL] o’
with the values of u, v taken from equations (8.12). It is noted
from the figure that the resistive component Of the optimum impedance
vanishes only for 80b 2 2. 5. At any other frequency, an optimum

loading must consist of both resistive and reactive components in order

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160

to yield a traveling wave distribution of loop current.

From Figure 8. 4 the following important characteristics are

observed for the case of an Optimum impedance loading of fixed

position:

(i)

(ii)

(iii)

(iV)

In general an Optimum loading impedance requires both
resistive. and reactive components.

Both the resistive and reactive components of the
optimum impedance are strong functions of frequency.
There is a particular frequency where the resistive
component takes on a very large negative value.

The sign of both impedance components changes within
a relatively narrow frequency range. An optimum
impedance thus requires a negative resistance component
(active element) at certain frequencies, and its reactive
component changes from capacitive to inductive as the
frequency is increased.

The reactive component of the Optimum impedance has a

negative slope as a function of frequency.

It is thus evident that synthesis of an exact Optimum loading impedance,

to yield a purely outward traveling wave distribution of loop current

at every frequency, is out of the question. Although an Esaki diode

might be utilized to yield a given negative resistance component,

there is no practical means of realizing a frequency variable negative

resistance characteristic of the type indicated.

Since the high efficiency associated with a non-dissipative

loading is of fundamental interest, one is led to consider, when dealing

161

with a loading of fixed position, one consisting of the reactive component
of the Optimum impedance. Such a purely non-dissipative loading is
optimum only at a given center frequency, and its effectiveness
diminishes as the frequency of excitation is varied from this value.

That is, the distribution Of loop current on 0 _<_ 9 _<_ 90 will be an
outward traveling wave at an excitation frequency corresponding to the
chosen optimum center frequency, but will gradually revert back to

an essentially standing wave as the frequency of excitation deviates
from this value.

For relatively small excursions about the center frequency,the
current distribution corresponding to such a non-dissipative loading
will remain an essentially outward traveling wave. Referring back
to Figure 8. 4, this is evident since for such small frequency variations
the resistive component Of the Optimum impedance is small compared
with its reactive component. For large deviations in the frequency of
excitation, however, the magnitude Of the resistive component becomes
comparable with that of the reactive component, and the effectiveness
of a purely non-dissipative loading will be necessarily reduced.

An approximately traveling wave distribution of loop current
may therefore be maintained over a band of excitation frequencies
through the use of a purely reactive loading of fixed position, provided
that its reactance is a proper function of frequency. The circuit and
radiation characteristics of a loop with such a non-dissipative loading
Will thus be typical of those Of an ideal traveling wave loop over this

range of fr equencie s .

162

In order to obtain an effective non-dissipative loading therefore,
it is necessary that the frequency dependence Of its reactance match
that of the reactive component of the optimum impedance, as indicated
in Figure 8. 4. The consideration of this synthesis problem is beyond
the scope of the present research.

8. 4. The Distribution of Current and Input Impedance Corresponding
to an Optimum Loading

A general expression for the optimum loading impedance to
yield a traveling wave distribution of loop current on 0 E 9 E 9O
was determined as expression (8. 5). Whenever the impedance is

given by this equation, then condition (8. 3) holds, 1. e. ,

P Z . P .
fi'l' " 30% e-JBbeo (51 cos (3b90 + e-Jfibe 0) = 0 (8-3)
2 2

Hence the distribution of current on 0 E 9 _<_ 90 becomes from
equation (8. 1)
2 V Tr .
_ 0 -J8b9

0

while by result (8. 2) that on 90 _<_ 9 E it may be expressed as

 

V TT Z . P . .
0 e 0 - L JBbBo __1_ -J8b90 -J8b9
Ie( ) W Z 30T e P2 cos 8b9O + e e
P .
+ 15—1 eJfibe (8.25)
2

These expressions represent, respectively, the outward traveling wave
which has been realized on 0 E 9 5 90 and the standing wave which

remains on 90 5 9 _<_ 17 .

163

It is desired to simplify result (8. 25) in order to indicate
more clearly the distribution of loop current on 9") E 9 _<_ 1T . This
equation may be written in the general form
2 V 17 . .
- 9 8b9
9 z 0 36b J .
Ie( ) W [Ae +Be ] (8 26)

0

where A and B are complex constants depending upon the antenna
dimensii‘gns and the loading parameters. The direct evaluation of A
and B is very tedious, so an alternate method will be utilized. Since

the loop current is continuous at 9 = 90, then the condition

- +
19(0seo) : 1 0:00) (8.27)

9(

must be satisfied. Further, it was found in the preceding chapter

that the antisymmetry Of the distribution of charge implies that

.3.

99 19(9) : 0 (8.28)

9:7r

The result of equation (8.24) may be utilized to obtain
2 V 11' .
_ - _ O -j8b9
19(9_90) — W e (8.29)

O
Applying the two boundary conditions (8.27) and (8. 28) in conjunction
with result (8.29) to expression (8.26 ), the constants A and B are

evaluated in a straightforward manner as

1
A s 1 + e-J'ZfibW-éo) (8.30)

 

e-j28b1r .,
l + e-jzfib("-B 0)

 

(8.31)

164

If expression (8. 30) and (8.31) for A and B , respectively, are
substituted into equation (8.26 ), the distribution of loop current on

90 _<_ 9 E 11’ is Obtained as

 

4 Von e-j8b7r
19(9) = W 1 + e-j29b(Tr-9O) cos 8b(17-9) (8.32)

It is Observed from result (8.32) that the loop current on
90 E 9 E 17 has a cosinusoidal distribution. Thus although the
Optimum impedance loading yields an outward traveling wave of
current on 0 :9 E 90 , the distribution on 90 E 9 _<_ 1r remains a
pure standing wave. This standing wave distribution is in fact
identical with the zeroth-order distribution of current on a conventional
loop atenna.

Since the loop current is symmetric about the excitation point,
then I9 (-9) = Ie (9 ). In summary then, the distribution of loop current
corresponding to an Optimum impedance loading may be expressed on

-7rf_9:1r as

 

V .
_ 0 -JF3b lel
19(0) _ 601, e -0050500 (8.33)
V e-j8b1'r

 

19(9): 0 cos (awn-lei) (8.34)

30\Ir 1 + e.3‘2816(n.00)

-11-<e<..e’9 <9<1T

In order to illustrate the distribution of loop current corresponding
to an Optimum impedance loading, as given by expressions (8.33) and
(8.34), the loop having dimensions specified by n = 10 will again be

considered. If it is taken that 80b 2 2. 5 , then the parameters of an

165

optimum non-dissipative loading are found from Figures 8. 2 and 8.3

O

to be: (Tr-9O) : 27.1 , [X z - 421 ohms. Figure 8. 5 indicates

L] o
the amplitude and phase of the distribution of loop current corresponding
to such a loading, as a function of the angular position 9 along the
circumference of the loop. These results were Obtained from equations
(8. 33) and (8.34), where the very small imaginary part of 8 was
neglected. It is noted that the amplitude of the current on 0 _<_ 9 _<_ 90
is constant while its phase is linear, corresponding to a traveling wave
along that region. On 90 E 9 5 TT , however, the current is represented
by a sinusoidal standing wave of constant phase.

The input impedance to the traveling wave 100p antenna is

Obtained from equation (8. 33) as

V

__ O
Zin ‘ ‘16”(050)

60 ‘I’ (8.35)

It is to be noted that this expression is valid only when the position and
impedance of the double loading are chosen to be Optimum, i. e. ,
equation (8. 5) is satisfied. Under any other circumstances, when

ZL is not [ ZL] O, the general input impedance expression (7. 72)

must be utilized as

-1
P Z . P .
_ 1 L ~JBb9 I -Jfibeo
Zin — 60‘1’ 1 + —Pz -————30T e 0 -—P2 cos 8b90 + e

(7.72)

This result reduces to equation (8. 35) when the loading is optimum.

166

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Figure 8. 6 indicates the input impedance of a loop antenna
with n. = 10 as a function Of its electrical circumference 80b .
These results were obtained from expression (8. 35), and it is
assumed that the loading is Optimum at every frequency. It is
noted from the figure that the input impedance of a loop antenna
which supports a traveling wave of current at each frequency is
essentially independent Of the excitation frequency.

In section 8. 3 it was indicated that a non-dissipative loading
of fixed position can be Optimum only at a single frequency. Special
consideration was thus given to a loading consisting of the reactive
component of the Optimum impedance. The position Of the loading
was selected in such a way that it became Optimum at a given
frequency. Since such a loading is optimum, and hence yields a
purely outward traveling wave of loop current on 0 f 9 _E 90, only
at a single frequency, then the frequency dependence of the
corresponding input impedance must be calculated from expression
(7. 72). As indicated in Figure 8. 4, for a loop with n. = 10, an
Optimum impedance loading located such that (1r -90) = 27.10 becomes
purely reactive when 80b = 2. 5 . The input impedance to a loop
antenna having a loading consisting of the reactive component of the
optimum impedance given in Figure 8. 4 must therefore be calculated
from expression (7.72) for all frequencies except that where 80b = 2. 5 .

In the corresponding case for a linear antenna, similar
calculations were made from expression (2. 33) and the results presented

in Figure 3.12. Since the input impedance expressions (2.33) and (7. 72)

168

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169

are essentially identical, and resulted from similar approximate
theories for the distribution of current on linear and loop antennas,
respectively, then the input impedance Of the two antenna types will
exhibit a very similar frequency dependence. Thus the calculations
outlined in the preceding paragraph will not be carried out for the
case of a loop antenna. By analogy to Figure 3.12, however, the
frequency dependence of the input impedance to a loop antenna having
such a purely reactive loading, which becomes Optimum for 80b = 2. 5,
will be relatively broadband about the frequency where 80b 2 2. 5.
8. 5. Calculation of the Expansion Parameters ‘I’i(9), \I’q(9), and \I’(9)
The expansion parameters \I'i(9), \I'q(9) , and ‘I’(9) were

defined in section 7. 4 as

 

0 477 A9(9)
‘I’i( )2 I": W (7.46)

_ ., 1121
‘I/q(9) - 4 60 q(9) (7.47)
111(0) 2 «Ii/1(0) 001(6) (7.59)

where A9 (9) and (M9) are the vector and scalar potentials at the
surface of the 100p, and 19(9) and q(9) are the corresponding
distributions of current and charge. It has been indicated that the

potentials may be expressed as

 

u 77
119(0): 47?.) 19(0') cos (8 -0!) K(9,9') bd9' (7.43)
11'
6(0) = 41115 S q(9 1) K(0,8 1) bd9' (7.44)
O .-

170

where K(9 , 9 ') in the Green's function

e'jPoR(e 7 9 ')

 

K(9,9') : R(erg') (7.45)
and the Euclidean distance R(9 , 9 ') is given approximately by
_ I
R(9, 0') = b [ 4 sinz (92e ) + aZ/bz]1/2 (7.17)

 

Using expressiors(7. 43) and (7. 44) for the potentials, the current and
charge expansion parameters \Ili(9) and ‘I’q(9) , respectively, are

obtained from their definitions (7. 46) and (7. 47) as

\II.(0) e 1

TI'
1 3679—) S4. 16(6')cos (919') K(e,9')bd6' (8.36)

w
\qu(0) = 335— 3“ q(9') K(9,9') bd9' (8.37)
-77

Since the current distribution Of fundamental interest is the
traveling wave corresponding to an Optimum impedance loading, then
it is this distribution which will be used to evaluate the expansion
parameters. By analOgy to the linear antenna, it is expected that
{11(9) and ‘I’q(9) will be relatively independent Of the distribution
of loop current, and depend primarily upon its dimensions. Hence
no great error will be made by using the values of \Ifi(9) and ‘Ilq(9)
corresponding to a traveling wave distribution when the loop current
actually departs moderately from a traveling wave: i. e. , when ZL
is not [ ZL] O .

In the preceding section, it was found that the distribution of

loop current on - 77 E 9 _<_ 77 corresponding to an Optimum loading

171

may be expressed as

 

e — V0 -j8b[9| 8 <0<0 8
IeH-me -O——o (33)
I (9) V0 67% BM lel) (834)
= . COS 1T- .
9 30111 1+e-325b(TI'-60)
-77 < 9 < -9 , 9 < 9 < 77

It is necessary to know the position 90 of the loading impedance before
results (8.33) and (8.34) may be utilized for the calculation of ‘I’i(9)
and ‘Ilq(9) . This presents, some difficulty, since the loading position
was previously determined in terms of ‘II = Nf—‘If-l‘l’q and 8 = Bofiy‘i’é ,
but now it is found necessary to known the location 90 of the loading

in order to evaluate ‘I‘i(9) and ‘I’q(9) . Some further approximations

are consequently in order to allow the explicit evaluation of the
expansion parameters.

It is a well known result in linear antenna theory that the
expansion parameter ‘I’(z) associated with such an atenna is a weak
function of the distribution of cylinder current. 2 This fact was
demonstrated in section 3. 5 and was indicated by Figure 3.13. It
was found that no great error was introduced by neglecting the standing
wave of current on d f z 5 h and assuming a traveling wave to exist
on 0 f z E h when calculating ‘Il(z) . Such an approximation was
found to yield reasonably accurate results whenever the length d of
the antenna supporting a traveling wave of current was sufficiently
greater than the length (h-d) which supports a standing wave distri-

bution. It was indicated that,‘ whenever the ratio d/(h-d) is Of the

order Of three or greater, sufficiently accurate results may be Obtained

 

172

through this approximation technique.

A close analogy may be drawn between thin-wire traveling wave
linear and loop antennas. Not only are the current distributions
corresponding to an Optimum impedance loading essentially identical
for the two antenna types, but a very close correspondence between
the theoretical developments Of Chapters 2 and 7 exists as well. It
is thus to be expected that an approximation technique analogous to
the one described in the preceding paragraph will be valid for the
calculation Of the loop expansion parameters \I’i(9) and ‘I’q(9) .

According to the above arguments, if (77 ~90) is reasonably
small compared with 90, no great error will be made in the
calculation of {11(9) and ‘I’q(9) if the traveling wave distribution
Of current is assumed to exist over the entire loop, i. e. , for
- 77 E 9 5 77 . As indicated in Figure 8. 2, the necessary position
(77 -90) of an optimum non-dissipative loading is always less than 500
when 80b is greater than 1. 5. Thus, for a loop with n = 10, the
above indicated approximation should be valid whenever the electrical
loop circumference is of the order 80b = l. 5 or greater. It was
indicated in Figure 8.1 that the imaginary part of the complex wave
number 8 is always very small, while its real part is essentially
equal to 80 . The further approximating assumption that 8 is real
and has the value 8 = 80 will thus be made to facilitate the calculation
of ‘I’i(9) and ‘I’q(9 ).

In accordance with the approximations outlined in the preceding
paragraph, the distribution of loop current to be utilized in calculating

the expansion parameters will be taken as

173

V

-. b 9
19(6) : 600‘? e J60! I -1T:9 in (8.38)

 

The corresponding approximate distribution of charge is related to

the loop current through the equation of continuity as

q(9) = a}? 5% 19(0) (7.50)
21T€OVO ‘Ifi blel
=T—eJo sgn(e) ”150517 (8.39)
q

where the signum function sgn(9) is defined by

lfor9>0

sgn (9) = O for 9

II
C

(8.40)

-1for9<0

Using the distributions of loop current and charge given by results
(8.38) and (8.39), the expansion parameters \I’i(9) and \I’q(9) are

obtained from expressions (8.36 ) and (8.37 ), respectively, as

. 1T . '
1111(0) = eJBObeS e'3‘3oble I cos(9 -9 ')K(9,9 ')bd9' (8.41)
Jr 0 E 9 E 77
. 1T . '
“11(9) = 6960109) e'moble 'sgnw ') K(9.e ')bde' (8.42)
-77
O _<_ 9 _<_ 77

The results may be written in the form
\Ir(0)=e3130b9[ci (fib9)-jsi (8760)] 0<9<77 (843)
1 a,b O ’ a,b o ’ — — '

)I/qw) = ejfiobe [ cibmob, e) - j sibmob, 0)] o 5 0 5 77 (8.44)

174

. . i i q
where the quantlties Ca, b((30b, 9 ), Sa, bmob’ 9 ), Ca, bmob’ 9 ), and

q .
Sa, b(fiob, 9) are defined by

. 11'
Cal, b([30b, 9) = 5:“ cos flobe' cos (9 -9 ') K(G, O ') bd9' (8. 45)
. Tr
8;,b((30b, e) = Swan poble'l cos(9 -e ') K(e,e ') bd9' (8.46)
TI'
C2, b((30b, 9) =S:1T sgn(e ') cos fiobe' K(9 , 9 ') bde' (8.47)
1'!”
Sibmob, e) = 5:” sgn(e ') sin 80ble'l K(e,e ') bd9' (8.48)

The Green's function K(9 , 9 ') is given by equation (7. 45). Integrals
(8. 45) through (8. 48) were numerically machine calculated for the

' following values of the parameters a/b, 50b, and 9 :

a/b
(30b

0 = 00 through 1800 for each 80b

0.0423, or n = 21n(21rb/a) = 10

O. 25 through 4. 0

With these numerical results, the values of ‘I’i(9) and ‘I’q(9) are
readily calculated from expressions (8.43) and (8.44), respectively.
The dependence of \Ili(9) and ‘I’q(9) upon the angular position
9 along the loop circumference is indicated in Figures 8. 7 and 8. 8,
respectively, for the case of Bob : 2 . These results were calculated
from expressions (8. 43) and (8.44 ), and correspond to a loop having
n. = 10 and an electrical circumference of [30b 2 2.. 0 . It will be
recalled that in the approximate theory of Chapter 7 it was asserted
that ‘I’i(9) and \I’q(9) were indeed essentially constant. A study of

Figure 8.7 reveals that {11(9), is relatively independent of 9 for

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The variation of \I’q(9) as depicted by Figure 8. 8 is
more pronounced, but its value is relatively constant for 300 f 9 E 1300.
In either case, the rapid variation for large values of 9 (near 1800)
is very likely due to the approximate nature of expressions (8.43) and
(8.44), where the standing wave of current on 1450 _<_ 9 E 1800 was
neglected. The strong variation in ‘qu(9) for small arguments may
attributed to the discontinuity in both the scalar potential ¢(9 ) and
the charge distribution q(9) at the excitation point 9 = O .

It has been indicated by King2 that in the case of a linear
antenna the greatest accuracy is obtained by evaluating the expansion
parameter ‘P(z) at a point of maximum antenna current. By the
analogy discussed previously, this criterion should also be valid for
a loop antenna. Thus ‘Ili should be evaluated at a point of maximum
loop current and \Ilq at a point of maximum charge. Due to the
traveling wave of current on O E 9 _<_ 90 , the amplitudes of both
19(9) and q(9) are constant for O _<_ 9 f 90, while q(9) is
discontinuous with q(O) = O at 9 = O . It would thus appear that
\I’i(9) and ‘Ilq(9) might be evaluated at any point where the traveling
wave exists, except near the discontinuity at 9 = 0 or the standing
wave at 9 : 90, to yield the constant values ‘Pi and ‘I'q. Since both
‘I’i(9) and ‘Ilq(9) are well represented by their values at = TT/Z ,

it is taken that

75‘
ll

\I’.(9 : TT/Z)
1 1 (8.50)

\I/qw = n/2)

*6
ll

178

These values of ‘I’i and ‘I’q were utilized to obtain all the preceding
numerical results of this chapter.

Figures 8. 9, 8.10, and 8.11 indicate the variation of ‘Pi, ‘qu,
and \P = Vii-11': , respectively, as a function of the electrical loop
circumference (30b . These results were obtained from expressions
(8.43) and (8.44) in conjunction with conditions (8. 50) for a loop with
n = 10 . The numerical values given by these figures may be used in
conjunction with the theory of sections 8. 3 and 8. 4 to determine the
parameters of an optimum non-dissipative loading and the corresponding

input impedance to the traveling wave loop antenna.

 

179

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CHAPTER 9

RADIATION CHARACTERISTICS OF A
TRAVELING WAVE LOOP ANTENNA

9. 1. Distribution of L00p Current for Calculation of Radiation Fields
It was indicated in the introduction that the radiation charac-
teristics of a loop antenna are completely characterized by its
distribution of current. The approximate current distribution on
the doubly loaded loop was determined in Chapter 7, and that
corresponding to an optimum impedance loading was established in
Chapter 8. In the present chapter, the radiation zone fields of the
traveling wave loop antenna are to be calculated. These fields are
defined by the condition [30R > > 1 , where R is the distance from
a current element on the 100p to an observation point P. This
condition is equivalent to the requirement that the point of observation
P should be separated from every point of the loop by many wave-
lengths. To determine these fields, the distribution of current
corresponding to an optimum loading will be utilized. Since it is,
in particular, the radiation zone fields which are to be determined,
this distribution will be further approximated to simplify the
calculations.
In section 8. 4 of the preceding chapter, the current distribution
on the doubly loaded loop corresponding to an optimum impedance

loading was found as

 

_ o arable!
19(6) _ 60g, e -0059 590 (8.33)

182

183

 

8 V0 e-jfibw la] 4
e
-Tr < 9 <-9 , 9 < 9 < 11'

These expressions represent a traveling wave of loop current over

the region - 90 _<_ 9 _<_ 90 and a standing wave on the regions - 1r _<_ 9 _<_-9O
and 60 5 6 5 17. It is well known that the radiation zone fields of an
antenna are not a strong function of its distribution of current. If the
regions of the loop supporting a standing wave of current are short
compared with the one on which a traveling wave exists, it is reasonable
to assume that the traveling wave is supported over the entire loop.

That is, if (Tr ~90) is reasonably small compared with 90, then no

great error will be made by assuming equation (8. 33) t9 hold on

- 1r _<_ 9 _<_ 17 when calculating the radiation fields of the loop.

From the results of section 8. 3, the range of loop sizes for
which the above outlined approximation is applicable may be deduced.
Consider a loop with n. = 10 and having a purely non-dissipative
optimum loading. It is noted from Figure 8. 2 that the ratio 90/(1r-90)
is greater than 2. 0 whenever the electrical loop circumference is of
the order of (30b : l. 25 or greater. Further, it was indicated in
Figure 8.1 that the complex wave number, (3 is essentially equal to
the free space wave number 80 . It is therefore a valid approximation
to take (3 = (30 in calculating the radiation fields. Hence there exists

a range of loop sizes for which the distribution of current

 

v .
19(9) = 6:41 e'JBOblel —«n-< e <n (9.1)

184

is a reasonable approximation. It is this current distribution which
will be utilized to calculate the radiation fields of a traveling wave
loop antenna.
9. 2. Radiation Fields of the Traveling Wave Loop Antenna

It was found that a general formulation for the radiation zone
fields of a circular loop antenna is not available in the existing
literature. The case which is invariably treated is that of an
electrically small loop for which the distribution of current is assumed
constant along its circumference. No theory could be found which
formulated the radiation fields of a circular loop in terms of a general
current distribution. This is a rather surprising fact in view of the
relative popularity of loop type antennas. It is first necessary there-
fore, to calculate the radiation zone fields of a loop of arbitrary size
in terms of a general distribution of current. The fields corresponding
to standing and traveling waves of current may then be obtained from
this general result.

The electric and magnetic fields at a point in space due to some
localized time harmonic current-charge distribution are given quite
generally in terms of the vector and scalar potentials A and <1) ,

respectively, as

ML

-V¢ -jw_A

_, (9.2)
V XA

UdL
II

It is possible to express the potentials in terms of the Helmholtz
—}
integrals over the volume density of current JG") and the volume

density of charge p5") as

185

 

_I I"'0 _.| _: e-jBOR '
A(r) = 74-1? v J(r ) R dV
(9. 3)
-' R

"’ _ 1 "l .2192..— I

¢(r) - 4“ S p(r) R dv
o v
where“ R = I? - :‘I is the distance between a source point located

_I
by r' and an observation point at 3:, and v the region of non-
vanishing source densities. By substituting the potentials (9. 3) into
expressions (9. 2), it has been shown by King5 that the radiation zone

electromagnetic fields become

 

' its ,. -* —) -jpoR
E14?) : #5 [R P(ifi) - J—(vi-L ] S-T— dV' (9. 4)
O V O
_, _, 39 P. .3 -j(30R
Br(r) : - 4: O S, R X .11?) E—R-— dVI (9. 5)
where
.1 _n A '33 _}‘I
R = 'r ' r'l ' R = _s _l
l r - I"!

and v0 is the velocity of prOpagation in free space, with [30 the free
space wave number.

Figure 9. 1 illustrates the geometry of interest for calculating
the radiation fields ofa circular loop antenna. The loop is taken to lie
in the x-y plane of a system of rectangular coordinates (x, y, z) and
to be centered about the origin 0 at (O, O, O) . A position vector 3“
locates the observation point P(r, 9 , 4)) , having the spherical
coordinates (r, 9 , 4)) , with respect to the origin 0 . The vector '11"
locates a source element along the loop having the spherical

_l
coordinates (b, Tr/Z, 4)‘). The loop current 1(3) thus has onlya

186

 

 

 

 

P(r.9.<I>)
I
I
l
I
I
I
I
I
I
I
I
‘9 I
r I
I
b I
R I
2a 0 § “’3,
\ I
\ I
I
Vo \\ |
\ I
“c \ |
.I ' \
‘ I¢(¢) \ :
4" "—~ \ I
I
l
\I

Figure 9.1. Loop Geometry for Calculation of Radiation

Zone Fields.

 

H

I.

4"
mg“ ..

187

(I) -component and is a function of (I) ' , with the result that

 

 

   

 

_, A
15") = ¢'I¢(¢') (9.
It is assumed that the loop is excited at the point (I)' = O .
Due to the assumption of a thin-wire loop where (30a < < l ,
it may be taken that
—) _I _I _I A
JII“) dV' = I(r') bd¢' = <I>' I¢(<I>') bd<I>' (9.
ME") dv' = q(I-"I bd¢' = q(¢') WV (9.
where I¢(<I)) and q(<I)) are the current and charge per unit length,
respectively, along the loop. Further, q(<I)) is related to 143(4))
through the equation of continuity as
q<¢> = J— 1 I (4) (9
ob Bo (b °
so that finally
_I _\ A
J(r') dv' 2' <I)'I¢(<I)') bd¢' (9.
p5“) dv' = i—a— 1(4')d¢' (9
w 8<I>' <1) '
If results (9.10) are substituted into expressions (9. 4) and (9. 5), the
radiation zone electromagnetic fields become
_. -J'I30R
r4 — RJ. _3_ I §___ I
E (r) _ Z31:: [3,): R0) 9'4) 143(4)) R d4)
143””) e'jfioR
_ __._____ I
R b dd) (9.
_I jIS I” W A A 'jflOR
r —I _ o o , l e
B(r)— - 411' SRX¢ I¢(¢)—__R bdd)‘ (9.

6)

7)

8)

9)

10a)

10b)

11)

12)

 

188

Before proceeding further, it is necessary to evaluate the
Euclidean distance R between a source element at 7r" and the
observation point P at 71“. With reference to Figure 9.1, an

application of the law of cosines gives

2 Z Z

R r +b -2rbsin9cos(¢-<I>')

r2[l + (b/r)2 - 2(b/r) sin9 cos (<I)-<I>')] (9.13)
Since the radiation zone is characterized by r > > b, the term in
(b/r)2 may be dropped, leaving
. . , 1/2
R = r[l -2(b/r)sm9 cos (<I)-<I> )] (9.14)

Using the two leading terms of a binomial series expansion gives

finally
R 5 r-bsin9cos(<I)-<I)') (9.15)

It will thus be taken that, for observation points P in the radiation

zone, the distance R is given approximately by

r ................ amplitude terms

R z (9.16)

r - b sin 9 cos (<I) -<I)') . . . phase terms

A
Further, the unit vector R may be approximated in the radiation

zone as
A . A
R = r (9.17)

If approximations (9.16) and (9.17) are substituted into equations

(9.11) and (9.12), the radiation fields may be expressed as

 

Iv,

nun '

 

_‘ jI3 -j(3 r 1T . . . ,
Era.) : o e 0 5' ,1} i; _8_ I¢(¢')6JBObSInG cos(<I) —<I) )dd)'

360 e-J‘fior I" 14M“) 3;, ejpobsin9c08(¢-¢')bd¢t

 

4Tr€ r _ v0
(9.18)
_, jg IJ' 'jB r Tr A ° ' _ I
Bra") = ‘ 4: o e i- 0 I I¢<¢')I?x¢') elfiobsmecos‘d’ ‘I’Ibdct'
-11'
(9.19)

The integration in the first term of result (9. 18) may be carried out
by parts, and the integrated term is found to vanish. Making use of

the fact that
sin 9 sin (c) -¢') = r - 8' (9.20)

the remaining term may be combined with the second integral in

equation (9.18) to give

mob e-jsor

41r€v r
00

 

Era!) =

I [?<’r‘-$') - 9'] I¢(¢')

-1T

ejBObsin9 cos(¢-<I>') d¢' (9.21)

If the vector identity

1?- 9')? «’5' (9.22)

A
9x6x¢9=<
is utilized, then result (9. 21) may be rewritten, and the radiation

zone fields bec ome

 

‘jfi r I’- b 17 A
e r0 f“ I (9x ¢')I¢(<I>')
-TI'

ejfiobSine COSICI’ “‘I”) d¢I (9. 23)

190

e-Jfior Hob
r 41r

 

‘IT
I (9 x$')1¢(¢'>

-1T

firm = - jao

ejBObsin9c08(¢'¢') d¢' (9.24)

From these expressions, it is observed that
-'r —sr A A
E (E) = vO[B (r) x r] (9.25)

and ETC?) and BIG) are orthogonal to one another as well as to the
direction of propagation ’1}, as is typical of radiation fields.

A
Since the unit vector <I)‘ may be expressed as

A A II A
¢' = sin9 sin(<I)-<I)') r+cos 9 sin(<I)-<I)') 9 +cos(<I>-<I>') <1)

 

(9.26)
then
A A A
I1"XCI)' = -cos(<I)-¢')9 +cos9 sin(<I)-<I)')<I) (9.27)
The radiation zone vector potential Ara.) is given by
H 'jB r IT ° - _ I
Krag) : _g e O 5' 9'1 ((1),) eJflobsm9 cos(¢ ¢)bd¢'
4w r -TT (I)
[.1 b -j(3 r 1T A
= 4: e : S. [sin9 sin(<I)-<I)')’1"+cos9 sin (<I)-<I)')9
-Tr

+ cos(¢ _¢I) $] I¢(¢I)e.II30b sin9 COS (‘I’ '¢')d¢t

(9.28)

Using result (9. 27) in expressions (9. 23) and (9. 24) and comparing

with equation (9. 28) shows immediately that

 

In:

191

Err?) = jo?x[ saga-)3 +1435) 8] (9.29)
fir... r A r A
B (r) = jBOIA¢(?) 9 -A9(?) 4)] (9.30)

If the cross product in result (9. 29) is expanded, the radiation fields

may finally be expressed as

 

_l _| A A
Er(r) -- -jw[A;I'r‘)¢ mad-)9] --
A (9. 31)
A .J A .3
Br(r) = -j00[A;G'I <I> -A§,(r> 6]
The components of vector potential are obtained from equation (9. 28) ,J
as

 

II b ~16 1' Tr
A25?) = 4: e r0 S I¢(¢') cos (9-9')
W

ejBOb sin 9 C08 (¢’¢') dd): (9.32)

A p. b -j[3 r 1T
Ash) = 4: e r° cos 954.199” sin (9-9')

 

ejBOb sin 9 cos (<I>-<I>') d¢' (9.33)

Expressions (9. 31) give the radiation fields ER?) and Brfif)
of a loop antenna, at an observation point P(T’) in the radiation zone,
in terms of its distribution of current. These fields depend in general
upon each of the spherical coordinates (r, 9 , 9) of the point P('r') .

A special case which is commonly considered is that of the radiation

fields in the plane of the loop, specified by 9 = TT/Z . For this case,

A; (_r.) = O by equation (9.33) while Air?) becomes

192

II b -J'I3
r _ o e
A4965 — 4n

 

r 11’ - I
O 5 14““) cos (4) -¢') erobcosM) '4) )d<I>'

-1T

1‘

(9.34)

The radiation fields in the plane of the loop are therefore given by

_a _. d A
firm = - )9 Aim 9
(9. 35)
_a _. . A
Br(r) = 380 Aim 9
It was indicated in section (9. 1) that the traveling wave
distribution of current corresponding to an optimum impedance
loading may be approximated as
V .
_ O -J50b I4) I _ < <
I¢(¢) — m e TT_¢_TT (9.36)

Further, the zeroth-order standing wave current distribution on a

conventional loop antenna may be written as
= - .. <
I¢(¢) Im cos (30b(TT I¢I) TT_ (I) :11" (9.37)

Using distributions (9. 36) and (9. 37) in equation (9. 34), the vector
potential in the plane of the loop, corresponding to the traveling and

standing waves of current, respectively, becomes

Vobp.O e-jfior

IA;(‘r’)]T = 240.9 1. GT(I30b.¢) (9.38)

 

I I’ll "jfior
r A m 0 e
[A¢(r)] S = 4.".

 

osmob, 9) (9.39)

1'

where

9149099) =I Cos<9-9')ejfiowasW'I'I‘I‘I'”49' (9.40)

-TT

 

193

11’ . ‘ '
(35(501319) = S. cos {30b(1T-I4>' I)cos(4)-¢')eJI30bCOS(‘I’ “I5 I d4)‘

-1T

(9. 41)
With result (9. 38), the electric field in the plane of the traveling

wave loop may be obtained from expression (9. 35) as
jV p b 'jpor
r .~. _ o o e
[E¢(I‘)]T - 'T r GT(flOb’¢) (9. 42)
while that corresponding to the standing wave of current is determined

from equations (9.35) and (9. 39) as

)1 99b -J'I3 r
[Egc—r‘IIS = - ‘21, ° 6 r° GS(I30b.<b) (9.43)

 

The functions GT(BOb, 4)) and Gsmob, 4)) are defined as the polar
pattern factors of the traveling wave and standing wave loop antennas,
respectively. These expressions describe the variation of the
amplitude and phase of the radiation fields as a function of the electrical
loop circumference (30b and the azimuth angle 4) of spherical
coordinates. For fixed values of Bob , these factors become functions
of 4) alone, and describe the radiation patterns of the fields in the
plane of the loop.

It is noted that GT((30b, 4)) and GS(BOb, 4)) are in general (
complex numbers. Hence the radiation fields vary in phase as well
as amplitude with the‘angle 4) . Physically, however, it is the
amplitude of the fields at each point in the radiation zone which is
of primary interest. Consideration will henceforth be restricted,
therefore, to the modulus IGT(80b, 4)) I and IGS([30b, 4))I of the

polar pattern factors. These expressions give the relative amplitudes

194

of the radiation fields as a function of the angle 4) and the electrical
circumference of the traveling or standing wave loop antennas.

9. 3. Comparison of Radiation Patterns for Traveling Wave and
Standing Wave Loop Antennas

The functions GT(BOb, 4)) and GS(BOb, 4)) , which describe
the radiation patterns of traveling and standing wave loop antennas, bl

respectively, were determined in the preceding section as

 

TT _ ' '
GT(I)Ob.¢I =§ .05”.an eJBobICosW-d) )-I9 I] dd), (9.40)
GSIBOb. ‘I’I : g“ cos gob(fi_|¢1|) cos (4) -<I>') ejfiobcos(¢ -43!) d4)' J

-11'
(9.41)
These integrals cannot be evaluated in closed form to yield a result
in terms of simple functions. It was therefore necessary to determine
the values of GT([30b, 4)) and GS(BOb, 4)) by numerical machine
calculation. Specific numerical results were obtained for values of
Bob between 0. 25 and 4. O. For each value of Bob, the angle 4) was
allowed to take values of O - l80 degrees.
The radiation patterns corresponding to the above numerical
results are obtained by plotting IGT((30b, 4))I and IGS(BOb, 4))I as
a function of 4) in polar coordinates, with the appropriate values of
[Bob as parameter. Typical patterns are indicated in Figures 9. 2
through 9. 5 for Bob values of l. O, 1. 5, 2. 5, and 4. 0, respectively.
In each case, the traveling wave and standing wave patterns are
plotted in the same figure to facilitate comparison. Each pattern is
actually symmetric about the 4) = 0 axis, but only half of every

pattern is shown to avoid obscuring the figures.

195

 

 

IGT(1.O.¢>H

 

 

 

 

Figure 9. 2. Radiation Pattern in Plane of Loop (6 =900) as a
Function of 4) for Bob =1 .

196

 

 

 

IGSU. 5.¢)|

  
 

IGT(1.5.4>)I

  

 

Figure 9. 3. Radiation Pattern in Plane of Loop (9: 90 o) as a
Function of 4) for Bo b =1. 5.

 

 

1' a. "fll! 1—A “-

197

 

 

 

   

 

IGS<2.5.¢)!

IcT(2.5.¢)l

 

 

Figure 9. 4. Radiation Pattern in Plane of Loop (9 :900) as a
Function of 4) for [Bob = 2. 5 .

198

 

 

 

 

 

IGS<4.0. ml

1
IGT(4.0,4>)1

 

 

Figure 9. 5. Radiation Pattern in Plane of Loop (9 =9OO) as a
Function of 4) for Bob : 4. 0 .

199

An inspection of Figures 9. 2 through 9. 5 reveals the following
radiation characteristics for traveling and standing wave loop antennas:
(i) The radiation pattern in the plane of a conventional loop
antenna consists of two broad lobes separated in space by

180°.

On the other hand, the pattern of a small traveling
wave loop is essentially unidirectional, having a very broad
major lobe in one spatial direction and a narrow minor lobe
of relatively small amplitude in the opposite direction.

(ii) As the diameter of the standing wave 100p is increased,
both the shape and the spatial orientation of the pair of
lobes in its radiation pattern undergo radical variations,
although they remain oppositely directed in space. Finally,
as the loop size is further increased, the pair of lobes
split to form several narrower lobes of smaller but equal
amplitude. As the electrical diameter of the traveling wave
loop is increased, the narrow minor lobe grows in amplitude
while the broad major lobe shifts in its spatial orientation,
decreases in amplitude, and finally splits to form a minor
lobe structure of relatively small amplitude.

(iii) An electrically large traveling wave loop antenna is
characterized by a single narrow major lobe accompanied
by a minor lobe structure. As the diameter of the loop
increases, the amplitude of the major lobe steadily
increases with respect to that of the minor lobe structure.
The major lobe of this pattern is spatially oriented in a
direction 1800 removed from the excitation point of the loop,

i. e. , in the direction of the traveling wave of current.

 

‘ku;m

200

It is indicated by the above remarks that the radiation characteristics
of a traveling wave loop antenna in no way resemble those of its
standing wave counterpart.

Whether or not the radiation characteristics of a traveling
wave loop offer any particular advantage will depend of course upon
the intended application of the antenna. The modified radiation pattern '1
which characterizes the traveling wave loop may, however, be desirable I l

for some purposes. In particular, the broad unidirectional pattern of

 

a small loop or the narrow directive pattern of an electrically large i
loop should be useful for certain applications. v
It should be emphasized that the radiation characteristics

determined in the preceding section correspond to the traveling wave
distribution of current associated with an Optimum impedance loading.
As the loading deviates from its optimum value, the traveling wave of
current gradually reverts back to an essentially standing wave. Under
these circumstances, the corresponding radiation patterns would again

become similar to those characteristic of a conventional loop antenna.

R EF ER ENC ES

1. Hallen, E. , "Theoretical Investigations into the Transmitting and
Receiving Qualities of Antennae, " Nova Acta Regiae Societatis
Scientiarum Upsaliensis, Uppsala, Sweeden, Series IV, Vol. II,
No. 4, Nov. 1938, pp. 1-44.

2. King, R. W. P., The Theory of Linear Antennas, Harvard Uni-

 

versity Press, Cambridge, Massachusetts, 1956.
3. Altshuler, E. E. , "The Traveling Wave Linear Antenna, " IRE

Trans. on Antennas and Propagation, July, 1961, pp. 324-329.

 

a;

4. Wu, T. T. and R. W. P. King, "The Cylindrical Antenna with
Nonreflecting Resistive Loading, " IEEE Trans. on Antennas and
Propagation, Vol. AP-13, No. 3, May, 1965, pp. 369-373.

5. King, R. W. P., Fundamental Electromagnetic Theory, Second

 

Edition, Dover, New York, 1963.

6. Chen, K. M. , "Minimization of Back Scattering of a Cylinder by
Double Loading, " IEEE Trans. on Antennas and Propagation,
Vol. AP-l3, No. 2, March, 1965, pp. 262-270.

7. Chen, K. M. , "Minimization of End-Fire Radar Echo of a Long
Thin Body by Impedance Loading, " IEEE Trans. on Antennas
and Propagation, Vol. AP-l 4, No. 2, May, 1966, pp. 318-323.

8. Stewart, J. L., Circuit Analysis of Transmission Lines, John

 

Wiley and Sons, New York, 1958.
9. Storer, J. E. , "Impedance of Thin-Wire Loop Antennas, " Trans.
AIEE, Vol. 75 (Communications and Electronics), Nov. , 1956,

pp. 606 -6l9.

201

i.

10.

11.

202

Iizuka, K. , "The Circular Loop Antenna Multiloaded with Positive

and Negative Resistors, " IEEE Trans. on Antennas and Propagation,

Vol. AP-l3, No. 1, Jan., 1965, pp. 7-20.

Jackson, J. D. , Classical Electrodynamics, John Wiley and Sons,

New York, 1962.

 

C

APPENDIX A

ELECTROMAGNETIC POTENTIALS
IN ANTENNA THEORY

It is convenient to formulate the theory of thin-wire antennas in terms
of the electromagnetic scalar and vector potentials. Such a formulation
is expedient since the distributions of antenna current and charge are
more closely related to these potential functions than to the electric
and magnetic fields themselves. A brief survey of the set of electro-
magnetic potentials useful in the study of thin-wire antennas is presented
here.

It will be assumed that the antenna is immersed in an infinite free
space region characterized by a permittivity 60 and a permeability

“'0' Maxwell's equations for such a free space region may be expressed

11
as
V-E = E"— (A.1)
O
-s 3???
VXE = -'5-E- (A.Z)
BE
_3 A
VxB = “OJ +11on3? (A.3)
_J
V-B = o (A.4)

_1 _1 .1"
where E is the electric field, B the magnetic field, and J and p
are the volume densities of current and charge, respectively. The
basic problem in the theory of thin-wire antennas is to determine the
distributions of antenna current and charge as solutions to equations
(A. 1) through (A. 4). Such a solution is greatly facilitated by the

introduction of a set of electromagnetic potentials.

203

204

Equation (A. 4) implies that
E“ = v xA (A. 5)

_l
where A is the vector potential. If expression (A. 5) is used in

equation (A. 2), there is obtained

which implies

where 4) is the scalar potential. The electromangetic fields may

therefore be expressed in terms of the vector and scalar potentials as

-: 32
E = 'W - 75?
(A. 6)
_I c—I
B = V xA
If equations (A. 6) are substituted into equations (A. 1) and (A. 3), a
pair of equations for the potentials are obtained as
2 8 . -‘ _ E.
V o + at V A _ _ e
o
z-s 3221' -s 34> —'
V A 'Hoeo —-2- -V(V° A+uoeo W) = -uOJ
8t
Since V - 75: is as yet unspecified, it may be taken that
. "' 34> _
V A+HO€O—a-E — O (A.7)

. . . . Lorentz condition

A
and the differential equations for 4) and A become

205

2
V 4’ - ”060—: = -
(A. 8)

2-' -4
VA-HOEO—7—-|J. J

In the special case where the sources are time harmonic of

the form ,2

the results of the preceding paragraph become J

 

.1
E

- V4) - ij
(A. 9)

DUI
ll
<1
)4
M.

0 (A. 10)

 

° 4> = o (A.11)

. Lorentz c ondition

jwt

The time factor e is implied in these results, and the fields

and potentials are undertstood to be complex valued. Further, the
free space wave number has been defined by £502 = wzuoeo . It is

possible to integrate the inhomogeneous wave equations (A. 8) through

the use of a Green's function technique11 to obtain

_,' e‘jBoR
5. P(r) -—-§— dv
V

¢<‘r‘) --

 

41m (A. 12)
O

206

e_______"J:oR
= -—Sv J(r' dv' (A.l3)

where R = I? - 3‘"! is the distance between an observation point
located by I" and a source point at 'r“ .
For the case of a thin-wire antenna, solutions (A. 12) and (A. 13)

may be integrated over the cross section of the wire to obtain

_, _, 49 R
Mr) 4,360 SCH q(r') 3+ dz'

T '
411' OCH d!

A
where q(?) is the charge per unit length and I('r.) the total current on

 

XG’)

the wire forming the contour c . It has been indicated by Hallenl and

King ’ that the potentials at the surface of a thin-wire antenna may be

.4
calculated by assuming q('r‘) and HE") to be concentrated along the
axis of the wire. In such a case, R = I'i'! - 33" is the distance between
an observation point on the wire surface at T“ and a source point on

o a J'
its ax1s at r .