ABSTRACT

EFFECTS OF PSEUDOSONIC AND ELECTROACOUSTIC
WAVES ON ANTENNA RADIATION

BY

Garth Maxam

The purpose of this investigation is to study the
properties of radiating systems immersed in hot lossy
plasma media. Specifically, the dissertation considers
two problems: (1) a spherical antenna coated with a
finite layer of hot lossy plasma, and (2) a cylindrical
antenna immersed in an infinite, hot, lossy plasma.

In the first problem, a spherical antenna, covered
with a layer of plasma described by the linearized hot
electron and ion equations, is studied theoretically. It
is found that in the layer of hot plasma, a pseudosonic
wave, an electroacoustic wave, and an electromagnetic wave
can be excited by the antenna. The effects of these waves
on the radiated power and input admittance of the plasma-
coated antenna are investigated. Significant findings
are the resonances due to the pseudosonic and electro-
acoustic waves and the enhanced radiation phenomenon which

implies that under certain conditions a plasma-coated

Garth Maxam

antenna will radiate more power than the same antenna in
free space.

In the second problem, we study theoretically and
experimentally the input impedance of a cylindrical
antenna immersed in an infinite, hot, lossy plasma. The
theoretical development is based on the linearized hot
electron equations and considers the ions to be motion-
less. An integral equation is develOped for the current
on the antenna surface. A zeroth order current distri-
bution is assumed and a zeroth order input impedance is
derived.

An eXperiment is performed to measure the input
impedance of a cylindrical antenna in a laboratory plasma
and the results are found to be in good qualitative agree-

ment with theoretical results.

EFFECTS OF PSEUDOSONIC AND ELECTROACOUSTIC

WAVES ON ANTENNA RADIATION

BY

r‘ SJ
GarthfiMaxam

A THESIS

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

DOCTOR OF PHILOSOPHY
Department of Electrical Engineering

1971

ACKNOWLEDGMENTS

The author is eternally grateful to his major pro-
fessor, Dr. K. M. Chen, for his guidance and encouragement
throughout the course of this work.

He also wishes to thank the other members of the
guidance committee, Dr. J. Asmussen, Dr. M. M. Gordon,
Dr. G. Kemeny, and Dr. D. P. Nyquist for their time and
interest in this work.

A special note of thanks is extended to the Office
of Education, U.S. Department of Health, Education, and
Welfare, for the National Defense Graduate Fellowship
which allowed the author to complete this study. The
research reported in this dissertation was supported in
part by the National Science Foundation under Grant
GK-2952.

Finally, the author wishes to thank his wife,
Rosemary, for her encouragement, understanding, and

patience throughout his graduate training program.

ii

TABLE OF CONTENTS

LIST OF FIGURES O O O O O O O O O O O 0

Chapter

I.

II.

III.

PART I. PSEUDOSONIC AND ELECTROACOUSTIC

WAVES EXCITED BY A PLASMA-COATED
SPHERICAL ANTENNA

INTRODUCTION AND BASIC EQUATIONS . . . .

1.1 Motivation and Background. . . . .

1.2
1.3

Linearized Hydrodynamic Equations . .
Dielectric Sheath . . . . . . .

LONGITUDINAL WAVES IN A HOT PLASMA . . .

2.1
2.2

2.3
2.4

General Relations . . . . . . .
Differential Equations for the

Electron and Ion Perturbation

Densities . . . . . . . . . .
Uncoupling the Differential Equations
for ne and ni. . . . . . . . .
Physical Interpretation of n1 and n2 .

RADIATION AND INPUT ADMITTANCE OF A
SPHERICAL ANTENNA SURROUNDED BY A FINITE
LAYER OF HOT, LOSSY PLASMA . . . . . .

.1

U) wwww U
o o o o
U'chUN

o
0‘

Statement of the Problem and Method
of Analysis . . . . . . . .
Region I: Dielectric Sheath Region
Region II: Plasma Layer . . . .
Region III: Free Space . . . .
Imposition of Boundary Conditions at
Interfaces. . . . . . . . . .
Radiated Power and Input Admittance .

iii

Page

vi

17

19
23

30

Chapter
IV. NUMERICAL TECHNIQUES AND RESULTS .
4.1 Numerical Techniques . . .

4.2 Numerical Results . . . .
4.3 Conclusions . . . . . .

PART II. RADIATION OF A CYLINDRICAL ANTENNA

IN A COMPRESSIBLE PLASMA INCLUDING THE
EFFECT OF AN ELECTROACOUSTIC WAVE

v 0 INTRODUCTION 0 C O O O O O O

5.1 Historical Development. . .
5.2 Outline of the Investigation.

VI. THEORETICAL DEVELOPMENT OF THE INTEGRAL

EQUATION FOR THE CURRENT ON A LINEAR

ANTENNA IN A HOT LOSSY PLASMA AND THE
ZEROTH ORDER SOLUTION FOR THE CURRENT

DISTRIBUTION AND INPUT IMPEDANCE .

6.1 Geometry and Basic Equations.

6.2 Integral Equation Formulation

6.3 Zeroth Order Current and Input
Impedance . . . . . . .

VII. NUMERICAL AND EXPERIMENTAL RESULTS

7.1 Numerical Techniques
7.2 Numerical Results .
7.3 Experimental Results
7.4 Conclusions . . .

REFERENCES . O O O O O O O O O O
APPENDICES

Appendix

A. Uncoupling the Differential Equations

the Electrons and the Ions . .

B. Some Properties of Legendre Functions

iv

for

Page
75
75

81
90

112

112
115

116

116
120

135
138
138
143
148
152

175

180
194

Appendix Page
C. Method of the Auxiliary Integral . . . . 197
D. The Input Resistance of a Very Thin

Cylindrical Antenna in a Hot Lossless
Plasma . . . . . . . . . . . . 199

LIST OF FIGURES

Figure Page

2.1 Plot of Various Parameters Obtained in
Uncoupling Equations (2.3.1) and (2.3.2).
The Plasma (Oxygen Atoms) is Assumed to be
Hot (Ve/C = 0.01, Te = Ti) and Lossless
(Ye = Yi = 0) . . . . . . . . . . . 22

2.2 Phase Velocity of n1 and n2 in a Hot Lossless
(Ve/C = 0.01, Te = Ti, Ye = Yi = 0.0) Plasma
as a Function of the Plasma Frequency
Squared Over the Source Frequency Squared.
The Plasma is Assumed to Consist of Oxygen
Atoms. . . . . . . . . . . . . . 25

3.1 A Spherical Antenna Covered by a Hot, Lossy
Plasma O O O O O O O O O O O O O 31

4.1 Theoretical Power Radiated by a Spherical
Antenna in a Hot (Ve/C = 0.01) Plasma as
a Function of Plasma Density for Various
Collision Frequencies . . . . . . . . 92

4.2 Theoretical Input Conductance of a Spherical
Antenna in a Hot (Ve/C = 0.01) Plasma as a
Function of Plasma Density for Various
Collision Frequencies . . . . . . . . 93

4.3 Theoretical Input Susceptance of a Spherical
Antenna in a Hot (Ve/C = 0.01) Plasma as a
Function of Plasma Density for Various
Collision Frequencies . . . . . . . . 94

4.4 Theoretical Power Radiated by a Small Spheri-
cal Antenna in a Hot (Ve/C = 0.01), Lossy,
(ye/w = 0.01, yi/w = 0.0000584) Plasma as
a Function of Plasma Density for Various
Thicknesses of the Plasma Layer.

vi

Figure Page

4.5 Theoretical Power Radiated by a Spherical
Antenna in a Hot (Ve/C = 0.01), Lossy
(ye/m = 0.01, y-/w = 0.0000584) Plasma as
a Function of PIasma Density for Various
Thicknesses of the Plasma Layer . . . . . 96

4.6 Theoretical Power Radiated by a Large
Spherical Antenna in a Hot (Ve/C = 0.01),
Lossy (ye/w = 0.01, Yi/w = 0.0000584) Plasma
as a Function of Plasma Density for Various
Thicknesses of the Plasma Layer . . . . . 97

4.7 Theoretical Power Radiated by a Spherical
Antenna in a Hot (Ve/C = 0.01), Lossy
(ye/w = 0.1, yi/m = 0.000584) Plasma as a
Function of Plasma Density. . . . . . . 98

4.8 Comparison of Experimental Values by Lin and
Chen [2,3] with Our Theoretical Radiation of
a Spherical Antenna (a = 2.54 cm) in a Hot
Lossy Plasma Driven at Various Frequencies as
a Function of Plasma Density . . . . . . 99

4.9 Comparison of Experimental Values by Lin and
Chen [2,3] with Our Theoretical Radiation of
a Spherical Antenna (a = 1.27 cm) in a Hot
Lossy Plasma Driven at Various Frequencies
as a Function of Plasma Density . . . . . 100

4.10 Theoretical Power Radiated by a Spherical
Antenna in a Hot (Ve/C = 0.01) Plasma as a
Function of Plasma Density for Various
Electron Collision Frequencies with the Ion
Collision Frequency Set Equal to Zero . . . 101

4.11 Theoretical Power Radiated by a Spherical
Antenna in a Hot (Ve/C = 0.01) Plasma as a
Function of Plasma Density for Various
Collision Frequencies . . . . . . . . 102

4.12 Theoretical Input Conductance of a Spherical
Antenna in a Hot (Ve/C = 0.01) Plasma as a
Function of Plasma Density for Various
Collision Frequencies . . . . . . . . 103

4.13 Theoretical Input Susceptance of a Spherical
Antenna in a Hot (Ve/C = 0.01) Plasma as a
Function of Plasma Density for Various
Collision Frequencies . . . . . . . . 104

vii

Figure Page

4.14 Theoretical Power Radiated by a Spherical
Antenna in a Hot (Ve/C = 0.01), Lossy
(ye/w = 0.01, yi/w = 0.0000584) Plasma
as a Function of Plasma Density for
Various Thicknesses of the Plasma Layer . . 105

4.15 Theoretical Power Radiated by a Spherical
Antenna in a Hot (Ve/C = 0.01), Lossy
(ye/w = 0.01, yi/w = 0.0000584) Plasma as
a Function of Plasma Density for Different
Size Antennas. . . . . . . . . . . 106

4.16 Theoretical Power Radiated by a Spherical
Antenna in a Hot (Ve/C = 0.01) Lossy
(Ye/w = 0.01, yi/w = 0.0000584) Plasma as
a Function of Dielectric Layer Thickness
for Various Plasma Densities. . . . . . 107

4.17 Theoretical Input Conductance of a Spherical
Antenna in a Hot (Ve/C = 0.01) Lossy
(Ye/w = 0.01, y-/w = 0.0000584) Plasma as
a Function of Dielectric Layer Thickness
for Various Plasma Densities. . . . . . 108

4.18 Theoretical Input Susceptance of a Spherical
Antenna in a Hot (Ve/C = 0.01) Lossy
(ye/w = 0.01, yi/w = 0.0000584) Plasma as
a Function of Dielectric Layer Thickness
for Various Plasma Densities. . . . . . 109

4.19 Theoretical Power Radiated by a Small Spheri-
cal Antenna Surrounded by a Thin Layer of
a Hot (Ve/C = 0.01) Plasma as a Function of
Plasma Density for Various Collision
Frequencies . . . . . . . . . . . 110

4.20 Theoretical Power Radiated by a Spherical
Antenna Surrounded by a Thin Layer of a Hot
(Ve/C = 0.01) Plasma as a Function of
Plasma Density for Various Collision
Frequencies . . . . . . . . . . . 111

6.1 A Cylindrical Antenna of Radius a and Half
Length h Immersed in an Unbounded Hot
Lossy Plasma . . . . . . . . . . . 117

6.2 Source and Field Points on or Near the
Surface of the Antenna. . . . . . . . 126

viii

Figure

7.1

7.4

7.8

7.9

7.10

Experimental Setup for the Measurement of
Impedance of a Cylindrical Antenna . . .

Theoretical Input Impedance of a MonOpole
(h/Ao = 0.147, a/lo = 0.008) in a Hot
(Ve/C = 0.01) Lossy Plasma as a Function of
Plasma Density. . . . . . . . . .

Current Distributions on a Dipole with
h/lo = 0.147 and a/Ao = 0.0072 for Various
Values of weZ/w2 and y/w as a Function of
Z/h O I O O O O I O O O O O 0

Theoretical Input Impedance of a MonOpole
(h/lo = 0.192, a/lo = 0.0072) in a Hot
(Ve/C = 0.01) Lossy Plasma as a Function of
Plasma Density. . . . . . . . . .

Current Distributions on a Dipole with
h/Ao = 0.192 and a/Ao = 0.0072 for Various
Values of weZ/w2 and y/w as a Function of
Z/h O O O O O O O O O O O I 0

Theoretical Input Impedance of a Monopole
(h/Ao = 0.251, a/Ao = 0.0064) in a Hot
(Ve/C = 0.01) Lossy Plasma as a Function of
Plasma Density. . . . . . . . . .

Current Distributions on a Dipole with
h/Ao = 0.251 and a/lo = 0.0064 for Various
Values of weZ/w2 and y/w as a Function of
Z/h O O O O I O O O O O O O 0

Theoretical Input Impedance of a Monopole
(h/Ao 0.313, a/lo = 0.008) in a Hot
(V /C 0.01) Lossy Plasma as a Function of
Plgsma Density. . . . . . . . . .

Current Distributions for a Dipole with
h/lo = 0.313 and a/lo = 0.008 for Various
Values of weZ/w2 and y/w as a Function of
Z/h O O O O O O I O I O I O O

Antenna Resistance of a Cylindrical Antenna
(Half Length = 0.25 A , Radius = 0.001 10)
in a Hot (Ve/C = 0.01? Lossless Plasma as a
Function of Plasma Density. . . . . .

ix

Page

150

154

155

156

157

158

159

160

161

162

Figure Page

7.11 Antenna Reactance of a Cylindrical Antenna
(Half Length = 0.25 A , Radius = 0.001 10)
in a Hot (Ve/C = 0.01 Lossless Plasma as
a Function of Plasma Density. . . . . . 163

7.12 Comparison of Theoretical and Experimental
Current Distributions on a MonOpole with
Boh = 1.54 agdw2 a = 0.615 cm for Various
Values of we/ The Driving Frequency
in the Experiment was 1.25 GHZ . . . . . 164

7.13 Input Impedance of a Dipole of Half Length
h/Ao = 0. 25 Normalized to 1009. Normalized
Electron Density (we 22/w ) Values are
Indicated . . . . . . . . . . . . 165

7.14 Input Impedance of a Dipole of Half Length
h/Ao = 0.12 Normalized to 1009. Normalized
Electron Density (we 2/w2) Values are
Indicated . . . . . . . . . . . . 166

7.15 Input Impedance of a Short Dipole Antenna in
a Hot (Ve/C = 0.001) Lossless Plasma as a
Function of Plasma Density . . . . . . 167

7.16 Input Impedance of a Short Dipole Antenna in
a Hot (Ve/C = 0.001) Lossless Plasma as a
Function of Plasma Density . . . . . . 168

7.17 Input Admittance (y = G + jH) of a Dipole
Antenna in a Hot (Ve/C = 0. 001) Lossless
Plasma as a Function of the Plasma Density . 169

7.18 Experimental and Theoretical Input Impedance
of a Monopole (h/lo = 0.117, a/Ao = 0.0064)
in a Hot Lossy Plasma as a Function of
Plasma Density . . . . . . . . . . 170

7.19 Experimental and Theoretical Input Impedance
of a MonOpole (h/lo = 0.132, a/Ao = 0.0072)
in a Hot Lossy Plasma as a Function of
Plasma Density . . . . . . . . . . 170

7.20 Experimental and Theoretical Input Impedance
of a MonOpole (h/Ao = 0.147, a/Ao = 0. 008)
in a Hot Lossy Plasma as a Function of
Plasma Density . . . . . . . . . . 171

Figure Page

7.21 Experimental and Theoretical Input Impedance
of a MonOpole (h/Ao = 0.171, a/A9 = 0.0064)
in a Hot Lossy Plasma as a Funct1on of
Plasma Density . . . . . . . . . . 171

7.22 Experimental and Theoretical Input Impedance
of a MonOpole (h/Ao = 0.192, a/lo = 0.0072)
in a Hot Lossy Plasma as a Function of
Plasma Density . . . . . . . . . . 172

7.23 Experimental and Theoretical Input Impedance
of a MonOpole (h/Ao = 0.213, a/AQ = 0.008)
in a Lossy Hot Plasma as a Funct1on of
Plasma Density . . . . . . . . . . 172

7.24 Experimental and Theoretical Input Impedance
of a Monopole (h/Ao = 0.251, a/Ao = 0.0064)
0.0064) in a Hot Lossy Plasma as a Function
of Plasma Density . . . . . . . . . 173

7.25 Experimental and Theoretical Input Impedance
of a MonOpole (h/Ao = 0.282, a/Ao =
0.0072) in a Hot Lossy Plasma as a Function
of Plasma Density . . . . . . . . . 173

7.26 Experimental and Theoretical Input Impedance
of a Monopole (h/lo = 0.313, a/Ao =
0.008) in a Hot Lossy Plasma as a Function
of Plasma Density . . . . . . . . . 174

xi

PART I

PSEUDOSONIC AND ELECTROACOUSTIC WAVES
EXCITED BY A PLASMA-COATED

SPHERICAL ANTENNA

CHAPTER I

INTRODUCTION AND BASIC EQUATIONS

The research described in this part of the disser-
tation is concerned with the radiation of a spherical
antenna through a concentric layer of a compressible plasma
surrounding the antenna. The antenna is assumed to be
separated from the plasma by a thin sheath region which
is also concentric with the sphere.

In this chapter we motivate the above problem and
give some of the historical background dealing with this
problem. Also, the linearized hydrodynamic equations are

developed and discussed.

1.1 Motivation and Background

The study of an antenna surrounded by a finite layer
of plasma is motivated by two important unsolved problems:
(1) the well—known "blackout" phenomenon which occurs when
a satellite reénters the atmosphere, and (2) the audible
noise generated by power lines when a corona forms on the

conductors of the line.

The conventional approach to solve the blackout
phenomenon is to raise the antenna frequency to a level
above the electron plasma frequency of the surrounding
plasma medium. This approach is usually hampered by the
practical limitation of available high-frequency sources.

In this dissertation it will be shown that under
certain conditions the radiation of a spherical antenna
covered by a concentric spherical layer of plasma can be
enhanced if the antenna frequency is adjusted to be much
lower than the electron plasma frequency.

The phenomenon of enhanced radiation from a small
antenna covered by a cold collisionless plasma layer was
first studied by Messian and Vandenplas [1] in 1967. Lin
[2] and Lin and Chen [3] later studied the same problem
and extended it to include the electroacoustic wave and
collisional losses in the plasma. The electroacoustic wave
consists of a longitudinal compression of the electron
fluid with the ions forming a uniform positive background
necessary for overall charge neutrality.

In this work the same problem is again studied but
this time,effects due to the finite temperature of the
ions are included. It is shown that a psuedosonic wave
may prOpagate in the plasma for antenna frequencies much
less than the electron plasma frequency of the medium.

Pseudosonic waves are longitudinal compression

waves in a plasma which are quite analogous to sound waves

in a gas. The election and ion fluids are constrained to
move very nearly in phase by the requirement that the
plasma remain nearly neutrally charged.

Pseudosonic waves were first predicted theoretically
by Tonks and Langmuir [4] in 1929 and probably first ob-
served experimentally by Revans [5] in 1933. Since 1933
pseudosonic waves have been observed by many other workers
in the area such as Barrett and Little [6] and Alexeff,
Jones, and Lonngren [7].

Cook and Buchanan [8] have shown that a significant
amount of power may be radiated in the pseudosonic wave
into an infinite plasma above a ground plane. The exci-
tation they use is an infinitesimal slot in the ground
plane.

When an antenna on a reéntry vehicle is covered by
a plasma layer and suffers blackout, a possible scheme of
overcoming this problem will be to reduce the antenna fre-
quency to a value which will excite the pseudosonic wave
in plasma. The pseudosonic wave will excite an electro-
magnetic wave at the outer surface of the plasma and, thus,
radio contact with the space vehicle may be maintained.

The second problem stated earlier, that of the audible
noise generated by power lines in the presence of a corona,
is not solved here but the mechanisms discussed may be
those involved in that problem. More needs to be done to

verify this.

The remainder of this chapter is devoted to a dis-
cussion of the basic linearized hydrodynamic equations to
be used later. Chapter II studies the pseudosonic and
the electroacoustic waves in an infinite plasma while
Chapter III applies the results of Chapter II to the
specific problem of a spherical antenna covered by a
spherical layer of compressible plasma. Chapter IV
discusses the techniques used to numerically solve the
problem in Chapter III and discusses some specific

numerical results.

1.2 Linearized Hydrodynamic Equations

 

It is necessary to specify a mathematical model to
describe the antenna and the plasma in order to determine
their interaction. The hydrodynamic model of the plasma
which is used throughout this investigation is presented
in this section. A discussion of the models used for the
spherical antenna is presented in later chapters.

Basically there are two ways of describing a plasma:
a microscopic gas—kinetic treatment using the Boltzmann
equation together with Maxwell's equations of electro-
dynamics: or a macroscopic, hydrodynamic approach using
the momentum transport equations together with Maxwell's
equations. The kinetic theory treatment is generally
much more difficult mathematically and requires serious

physical restrictions be placed on the model to make the

problem tractable. For this reason the hydrodynamic
equations together with Maxwell's equations are used
throughout this investigation. It must be noted that the
hydrodynamic equations do not describe Landau damping
which is included in the more general kinetic theory.
Thus, in following investigation,caution should be exer-
cised when the phase velocity of the waves is nearly equal
to the average thermal velocity of the plasma components
because in this range Landau damping can be significant
[9].

A plasma consists of electrons, ions, and neutral
particles. The neutral particles contribute to the
dynamics of the plasma by collisions with the charged
particles and are considered by including a neutral
particle collision frequency for the electrons and the
ions. Thus, in our investigation, the plasma consists
of two fluids, the electrons and the ions.

The basic equations may be written in such a general
way that both the problem in this part and the problem in
Part II are included as special cases. Gravitational
forces, static electrical and magnetic fields, and macro-
scopic gradients of density and temperature are not in-
cluded in this analysis. The plasma is assumed to be
macroscopically neutral and consists, on the average, of
n electrons per meter3, and of the same number of singly

O

ionized ions.

Let E and H be the time varying electrical and mag-
netic fields and let Ye and Yi represent the average fluid
velocities of the electrons and the ions. The universal
constants are the elementary charge (electron charge: -e):
the electron and ion masses me and mi; the permeability of
free space no: and the permittivity of free space so.

The MKS system of units is used throughout.
The hydrodynamic equations of motion for electrons

and ions are [10]

 

 

_8 . -——e
at Ye + (Ye V)Ye _ meIE + Ye x g]
-1 VP-yv (121)
N m e e ~e ' '
ee
—aV+(V'V)V=—e[E+VxB]
3t ~i ~i ~i m ~ ~ ~
-1 VP-yV (122)
m.N. i i ~i° ' °
11

These equations include a damping term proportional to the
velocities where Ye and Yi are termed the mean electron-
neutral particle collision frequency and the mean ion-
neutral particle collision frequency. Pe and P1 are
scalar pressures for electrons and ions. The gradients

of these pressures are discussed in detail later. The

equations of continuity are

N = 0 (1.2.3)

3%”
m

V ° (NeYe) +

N. = 0 (1.2.4)

0)
wk”
H

V ' (NiYi) +

The Maxwell equations become

3H
V x E = - uo 8? (1.2.5)
S 3E
V x H = g + e(NiYi - NeYe) + 60 3? (1.2.6)

V ° H = 0

where (3S and JS are externally supplied sources and are

related by

5
%fi? + v . J5 = 0 (1.2.7)

Equations (1.2.1) through (1.2.7) are nonlinear and
hence are very difficult to solve exactly. In order to
simplify the equations, a small signal excitation is
assumed. That is, the various field quantities are

assumed to be of the form

§(§,t) = EDC(§) + EAC(§,t) (1.2.8)
Ij(§,t) = IjDChg) + {IAc(§,t) (1.2.9)
Ye(£lt) = YeO(E) + Ye(£rt) (1.2.10)

(U

Yi(r,t) = Yio(5) + vi (r, t) (1.2.11)

Ne(r,t) = Neo(r) + ne (r, t) (1.2.12)

Ni(r,t) = Nio(r) + ni(r,t) (1.2.13)
where E

DC' HDC, Yeo' Yio’ Neo and N10 are the average
electric field intensity, magnetic field intensity,
electron velocity, ion velocity, electron density and ion
density, respectively. These average values may vary with
position but not with time (i.e., they are steady state
values). The small perturbation quantities EAC' HAG,
Ye’ Yi' ne and ni are functions of both position and time.
In this investigation it is assumed that the average
electron and ion densities are equal and do not vary with

position

Neo (5) a no: _ ”10(5’ (1.2.14)

and that the average electron and ion velocities are zero
since allowing Yeo and Yio to be finite introduces no new
physical results but it does seriously complicate the
mathematics [11]. In addition in this investigation
externally applied static fields are not considered and
static electric fields set up in sheath regions are not

considered hence

BBC (5) = EDC (g) = 0. (1.2.15)

In general the ion and the electron fluids can be
considered to act as neutral particle gaseous media with
one main difference. The interactions of particles in an
ion or electron fluid are over much larger distances than
those for neutral particles.

For both fluids we will later be concerned with VP
where P is the pressure of the fluid.

If we are concerned with a static pressure (D.C.
case), the pressure is established by an isothermal pro-
cess. That is, the temperature of the gas is fixed through-

out the volume of interest, then
P = n k T (1.2.17)

where T is the fixed temperature of the fluid, n is the
number density of the fluid and k is Boltzmann's constant.

If an external force disturbs n, such that

n(r,t) = no(r) + nl(r,t) (1.2.18)

and n is a fast function of time such as a high frequency
disturbance, then the temperature of the gas is not fixed
simply due to the fact that there is not enough time for
the exchanging of energies in the gas to keep the tempera-
ture fixed. In this type of problem, the adiabatic law

should be used, that is

Pn-Y = constant (1.2.19)

10

where Y is the ratio of specific heats such that

 

c
y =E£=m+ 2 (1.2.20)
V m

where m is the degrees of freedom of the gas.

For high frequency plasma oscillations, the motion of
the electrons is usually in one direction only, so we can
assume m = 1, so that y = 3.

Now for the case of a small r.f. perturbation, as
in eqn. (1.2.18), the relationship between pressure and
election density is

Pn-Y = PonoaY = constant (1.2.20)

since P Po and n = n0 initially. Then

n'Y
P = P [-—] . (1.2.21)
0110

Remembering that the static pressure is established by an

isothermal process, we have

P = no k T (1.2.22)

Therefore

ll

VP

ll
<
"U
0
r-—-\
5k,
H—d
.4

Y Y
9—] VP + P v[—"-]
n O 0 no

Y Y
n1 III
II 0 O n
O 0

 

 

n1 Y n1 Y-l n1
= 1 + H— kTVno + ano Y 1 + H— V H-
o o 0
Y F' Y'1 _
_ n1 n1 noan n1Vno
— l + —— kTVn + an y 1 +1——
n o o n 2
o o n
_ o
y-l
n1 Y n1 n1
= kT l + —— - Y 1 + -— —— Vn
no no no 0
n1 Y-l
+ ykT l + —— an
0
or Since n1 << no
VP = kTVno + ykTan. (1.2.23)

For the case of a uniform average electron density, n ,

o
Vno = 0, (1.2.24)

so that
VP = ykTVn (1.2.25)

1.

12

For our case of a two fluid gas, we have

VP 3kTeVne(r) (1.2.26)

and

VPi 3kTiVni(r). (1.2.27)

Assuming an e3“)t suppressed time dependence along
with the above assumptions and neglecting products of
small perturbation quantities, the linearized hydrodynamic

equations and Maxwell's equations in a plasma media are

 

 

V x EAC = -quo §AC (1.2.28)
.. S - ..
V x EAC — g + Jweo gAC + enohzi Ye) (1.2.29)
noV ° Ye + jwne = 0 (1.2.30)
noV - Yi + jwni = 0 (1.2.31)
e 3kTe
(3w + ye)ve = - E- E - n m Vne (1.2.32)
e o e
e 3kTi
(3w + Yi)Yi = 57 E - n m. Vni (1.2.33)
1 o 1

Equations (1.2.28) thru (1.2.33) are a complete set of
equations which along with the source continuity equation,
equation (l.2.7),completely describe the fields in a plasma

medium.

13

In a study of plasma media, certain characteristic
parameters appear frequently. It is convenient to make
the following symbolic definitions. we, the electron

plasma frequency or simply the electron frequency is

 

 

defined by
2 noe2
me = m 6 (1.2.34)
e 0
while mi, the ion plasma frequency is
2
m n e
w.2=—-e-w2= ° . (1.2.35)
1 m. e m.e
1 1 0

Another pair of parameters, the thermal velocities, of the

electrons Ve and of the ions Vi are defined by

 

 

2 3kTe
Ve = m (1.2.36)
e
2 3kTi
V. = . (1.2.37)
1 mi

The definitions (1.2.36) and (1.2.37) are debatable, but
other commonly used definitions lead to the same order of
magnitude result as long as the linearized equations are
used [11]. Therefore, these definitions are used in this
investigation. Characteristic lengths in a plasma are
often measured in terms of the Debye lengths, which for

the electrons is

14

D)e = -———5 (1.2.38)

and for the ions is

2 eokTi
(AD)i = -;-;§ . (1.2.39)
0

Physically the Debye length is range of effectiveness of
any electrostatic fields due either to a surface at some

nonzero potential or to a charge within a plasma.

1.3 Dielectric Sheath

When a conducting solid is placed in an otherwise
homogeneous plasma medium, a transition region between the
main body of the plasma and the solid is formed. If the
potential of the object is allowed to float, the object
will acquire a negative potential and the electrical
neutrality of the plasma will be disturbed in the vicinity
of the object. Due to the high thermal velocity of the
electrons with respect to the ions, the object will become
negatively charged so that at equilibrium,equa1 numbers
of electrons and ions will hit the object per unit time.
The potential distribution in the vicinity of the object
causes a perturbation of the number densities of the
electrons and the ions. The electron density in this
transition region is less than the ion density and,thus,

the transition region for such a situation is called an

15

ion sheath region. From an electrical viewpoint, this
sheath behaves as a vacuum sheath, or simply as a
dielectric sheath.

In this investigation, the sheath region will be
considered as an electron depletion layer, or a vacuum
adjacent to the antenna. It is assumed that the outer
boundary of the sheath is rigid to the elections and ions
in the plasma and, thus, it reflects all particles that
come into contact with it. Also, the sheath layer is

taken to be a few Debye lengths in thickness [12].

CHAPTER II

LONGITUDINAL WAVES IN A HOT PLASMA

In this chapter we devote our attention to two
purely longitudinal waves that are excited in an infinite
hot lossy plasma by a source current gs and source charge
density 98. The plasma is considered to be a weakly
ionized gas so that linearized hydrodynamic equations

developed in Chapter I may be used.

2.1 General Relations
The source terms are related by the continuity

equation

S S

v - .3 + jwp = 0 (2.1.1)

From Chapter I the linearized equations of motion for the

electrons are

jwne + noV ° v = 0 (2.1.2)

° — _ ii - ___
(3w + Ye)Ye - m E Vn (2.1.3)

16

17

and the linearized equations of motion for the ions are

jwni + noV - Y1 = 0 (2.1.4)
e Vi2
(jw + Y1)Yi = + a; E - 1i: Vni (2.1.5)

The fields E and H in the plasma satisfy Maxwell's

equations which from Chapter I are
V x E = -jwuoH (2.1.6)

_ s _ .
V x H - g + eno(vi Ye) + jweoE (2.1.7)

2.2 Differential Equations for the Electron
and Ion Perturbation Densities

Rearranging equations (2.1.2) and (2.1.4), we have

. __...J'£
V Ye n ne (2.2.1)

o

. =-i£
V vi no ni. (2.2.2)

Taking the divergence of equation (2.1.7) yields

- o S o — 0 ° 0
0 - V g + eno(V Yi V Ye) + jweoV E (2.2.3)

or using equations (2.2.1) and (2.2.2) and rearranging,

V . g = _i. [? . gs - jwe(ni - neg]. (2.2.4)

(DE
0

18

Taking the divergences of equations (2.1.3) and (2.1.5)

gives
e Ve2 2
~ e o
and
e V12 2
(jw + Yi)V ' Vi = I-n— V ' 1‘1: - -n— V ni. (2.2.6)
1 0

Putting equations (2.2.1) and (2.2.4) into equation (2.2.5)

and multiplying through by - nO/Ve2 and rearranging yields

 

 

2 ]Y m 2
2 m we e e
Mug-:1“?- T]ne+;_2ni
e e
we2 s
=-j 2 v-J (2.2.7)
Ve em ~

 

where we = /Hoe2/meeo is the electron plasma frequency.

Using the equation of continuity for the sources and

setting
2 w 2 y
2 _ me e . e
e
we get
2 2 wez weZ Bi
V ne 4' Be ne 4’ F ni = " F e (2.2.9)

19

-By a similar procedure starting with equations (2.2.2),

(2.2.4), and (2.2.6) we obtain

 

Vn. + 5. n. +—n =——L (2.2.10)
1 1 1 2 e e
V. V.
1 i
where
2 2 y.
2 _ w w . 1
V. w.
1 1
and mi =/£o e 2/mi 80 is the ion plasma frequency.

Equations (2.2.9) and (2.2.10) are two coupled
differential equations for the electron and ion pertur-

bation densities.

2.3 Uncoupling_the Differential Equations

for nn and ni

Multiplying equation (2.2.9) by Ve/we and equation

(2.2.10) by Vi/wi, we obtain

 

 

 

 

 

V V w m V. m S
V2 $2 ne + 8e2 $2 ne + VeV Ki ni = V2 %;
e e e 1 i e
(2.3.1)
and
V V w m. V m S
V2 Ki n1 + 8i2 Hi ni + VeV1 $2 ne = Vi E— '
i i e i e 1. e

 

 

(2.3.2)

20

Equations (2.3.1) and (2.3.2) can be written compactly as

the matrix equation

 

 

 

 

 

 

2 S
v§+§§=%§ (2.3.3)
where
R’e ‘1
we
n =
1 V-
.i ni
_“.’i 1
B 2 wewl
9 vv.
e 1
g:
wew1 B 2
Vevi i
__ _J
F u) “
_ .2
Ve
S =
“’1
‘7.- (2.3.4)
L. 1J

 

 

In Appendix A it is shown that equations (2.3.1) and
(2.3.2) can be uncoupled resulting in two differential
equations which describe two new waves n1, an ion wave
and an electron wave denoted by n2. The differential

equations that describe nl and n2 are

21

 

Vzn + k 2 - s 95 (2 3 5)
1 1 n1 ' 1. e ° °
Vzn + k 2n — s 3§ (2 3 6)
2 2 2 - 2 ea ' °
where
2 w 2w.2
2 _ 1 2 2 2 _ 2 e 1
e 1

 

 

and

 

2w.2
_ 1 2 2 2 2 we 1
e i

and S1 and 82 are defined in Appendix A. nl and n2 are

linear combinations of ne and ni

(.0
- .2
ne — Ve (Tllnl + lenz) (2.3.9)
”1
hi = V_. (T21n1 + T22n2) (2.3.10)

1

where T ,and T11 are given in Appendix A.

22' T12' T21
Equations (2.3.5) thru (2.3.10) provide a complete solution
for the electron perturbation density and the ion pertur-
bation density in an infinite homogeneous plasma.

Figure 2.1 is a plot of the coefficients relating

2 2
ne and n1 to 111 and n2 and of S1 and 52 versus we /w .

1089)
1070

Q

V

10

 

1.0 A J A. L
V V

10" 10"3 10'2 10‘1 1.0

Figure 2.1.

22

10 102 103 1041105 106

(012/002 u

:7 :8
fir“

Plot of various parameters obtained in
uncoupling Equations (2.3.1) and (2.3.2).

The plasma (oxygen

atoms) is assumed to be hot (Ve/C - 0.01, Te 8 T1) and lossless

(Ye - Y1 = 0).

23

2.4 Physical Interpretation
of nl and n2

In order to discuss n1 and n2

necessary to specialize equations (2.1.2) thru (2.1.5) to

in more detail it is

the specific case of a monochromatic plane wave which pro-
pagates in the positive 2 direction in a cartesian coordi-
nate system. The variables describing the wave are ex-

pressed in the form:

ej(wt-kz)

A = A0 (2.4.1)
where A0 is, in general, a complex coefficient.

We use the following linear Operator:

V = _ j k (2.4.2)

where k is the propagation vector in the z direction.

Equations (2.1.2) and (2.1.4) can be written as

n = nokvez/w (2.4.3)
n. = nokviz/w (2.4.4)

and the 2 components of equations (2.1.3) and (2.1.5) are

2
. e . Ve
(3w + ye)vez = - E— E2 + 3 1r— kne (2.4.5)
e o
e Vi2
(jw + yi)viz = ET Ez + 3 7T_ kni. (2.4.6)

1 O

24

Specializing equations (2.4.5) and (2.4.6) to a collision—

 

 

less plasma (Ye = Yi = 0) and solV1ng for vez and viz
yields
. e m
V = _ J __ E (2.4.7)
ez me k2V 2 _ w2 z
e
. e w
v. = J — E o (2.4.8)
12 mi kzviz _ w2 z

The electron and ion average velocities are seen to
be 90° out of phase with the electric field. In addition
the simple theory predicts singularities at k = w/Ve and
k = w/Vi due to the use of the linearized equations.

Two other useful quantities are the phase velocities

of nl and n2

_. U)

Vphl — E: (2.4.9)
: 00

Vph2 — E; (2.4.10)

for a collisionless plasma. Figure 2.2 shows a plot of

V and V versus w 2/w2 for a weakly ionized collision-
phl ph2 e

less hydrogen gas at equilibrium (Te = Ti)“ A study of
Figure 2.2 indicates that n1 propagates at all frequencies

but that n2 propagates only when w > me. It must be noted

that this theory does not include collisionless or Landau

25

 

10

 

108)

ph

10 1

V

 

 

10 10 1.0 10 102 103 1041105 106 107 108
2 2-—*'
wi = w we /w

Figure 2.2. Phase velocity of n1 and n2 in a hot lossless
(Ve/C 3 0.01, Te = Ti, e = Yi ' 0.0) plasma as a function Of the
plasma frequency squared over the source frequency squared. The
plasma is assumed to consist of oxygen atoms.

26

damping wh1ch damps nl when Vphl : V1 and n2 when
Vph2 = Ve‘
Some physical insight into the nature of n and n

l 2
can be obtained by studying these waves in the high and

low frequency limits. Using the parameter

 

2 2
viz _ me k Ve - w
V_— _ - ET 2 2 45 (2.4.11)

and choosing freely from limit forms of the parameters kl,
k2, T11, T12, T21,and T22 calculated 1n Append1x A, th1s

will now be done.

(a) Electron Waves, 112

In the high frequency limit (w2:>we2 >> wiz)
w we2 k
k2 = V_ l - —-2— (2.4.12)
e w
and
V
n .. - J”: n (2 4 13)
2 ” w e ' '
e
v. m w 2
and —13 = - £- -9— . (2.4.14)
v m. 2
e2 1 w

Result (2.4.14) shows that in the electron wave in the high
frequency limit the ions are essentially immobile. This
agrees with equation (2.4.13) which says that in the high

frequency limit, the electron wave n2 consists only of

27

electron oscillations. Equation (2.4.14) also shows that
in the electron wave, the ions and electrons oscillate out

of phase. From equation (2.4.12) it can be seen that Vph ,
2

 

2 2

is always greater than the thermal velocity of the

electrons and that in the very high frequency limit

2

(w >> wez) the phase velocity of n2 tends to Ve'

In the low frequency limit (in fact for all w < we)
k2 is purely imaginary and the electron wave does not

propagate.

(b) Ion Waves, 111

For the ion wave, the phase velocity is always in

the range

Vi < Vphl < VS (2.4.16)

where VS is the low frequency limit of the phase velocity

of the ion wave:

k T + T.
e

= 1
VS mi . (2.4.17)

 

On the other hand, the electron temperature is, in
most cases, equal to or greater than the ion temperature.

Hence VS is much smaller than Ve' giving

28

V << Ve. (2.4.18)

phl

In the high frequency limit

___, .. (2.4.19)

 

 

indicating the electron velocity is much smaller than the
ion velocity. In the high frequency limit n1 consists
mainly of the motion of ions justifying calling n1 an ion
wave.

In the low frequency limit

 

 

 

 

2
Ve
m
2 e 2
m mvi+K% 1
12 2 _ _g 1
vez mi V.2
1
m
2 e 2
Vi+I-II.-Ve-l
1
m
2 2 e 2
m Ve - Vi - ET ve
;__e 1
mi V 2 _ V 2 _ TE'V 2
i i m. e
1
v.2 m
:1-1-3
2 m.
Ve 1

II
[.4

(2.4.20)

29

and
k1 s m/[vi2 + 1%- vezr (2.4.21)
and
n1 : igfine +¥ni]. (2.4.22)
/2 e 1

 

for an equilibrium plasma (Te = Ti)’

From the above equations we can conclude that in the
low frequency limit the ion waves consist of electrons and
ions moving in phase with approximately equal velocities
and the medium remains practically neutral. However, this
is rigorously true only in limit of w-—+ 0. For finite
values of w there exists a slight deviation from neutral-
ity; the ion oscillations are slightly larger than those
of the electrons. Even though this deviation is weak, an
electric field resulting from the space charge produces a
coupling between the aggregate motion of the elections
and ions.

Hereafter, we shall refer to the electron wave in
the high frequency range as the electroacoustic wave and
the ion wave will be called the pseudosonic wave in the
low frequency range. These are the regions of interest

for the two waves and the prOperties discussed above will

be used later in the solution of a specific problem.

CHAPTER III

RADIATION AND INPUT ADMITTANCE OF A SPHERICAL
ANTENNA SURROUNDED BY A FINITE LAYER

OF HOT, LOSSY PLASMA

In this chapter the radiation and input admittance
of a spherical antenna surrounded by a finite layer of a
hot lossy plasma is studied. In addition to the electro-
magnetic wave, two longitudinal waves, an electroacoustic
wave and a pseudosonic wave, may prOpagate in a hot plasma.
These are included in the following analysis. In addition,
a thin dielectric sheath is assumed to surround the spheri-
cal antenna and separates it from the plasma layer.

3.1 Statement of the Problem
and Method of Analysis

 

The geometrical configuration is shown in Figure 3.1
using a spherical coordinate system (r, 0, 0). A Spherical
antenna of radius a is centered at the origin and is covered
by a thin dielectric sheath of outer radius b. The permit-
tivity of the sheath is ed and the permeability is taken
as the free space permeability, no. The sheath is covered

by spherical layer of hot lossy plasma which has an outer

30

31

 

 

Z
A
9
\/(r'e ’¢)
I
/
I
/
b I C
4!!::;=‘ :=:y
'49
II III

Region I: dielectric coating
(110,60)
Region II: hot lossy plasma
Region III: free space
(00,20)

Figure 3.1. A spherical antenna covered by
a hot, lossy plasma.

32

radius of C. The plasma is assumed to be a weakly ionized
gas so that the linearized hydrodynamic equations apply.
It can be regarded as consisting of two fluids, the ions
and the electrons with the neutral particles being taken
into account by assuming finite collision frequencies be-
tween the ions and the neutral particles and between the
electrons and the neutral particles.

As an idealized approximation the sheath is con-
sidered to be a lossless coating which is perfectly rigid
to the inward radial flow of the ions and the electrons.
It is also necessary to impose a boundary condition on the
outward flow of the ions and electrons at the outer sur-
face of the plasma layer. To make the problem tractable
it is assumed that the outer boundary of the plasma is
rigid to the outward radial flow of ions and electrons.
Without these assumptions a solution to this problem would
be very difficult.

The spherical antenna is perfectly conducting except
for a narrow equatorial gap between n/2 - 0 j 0 : n/Z + 01.
Across the gap the antenna is driven by a constant voltage
generator with a voltage, V, and an angular frequency, w.
The total space excluding the antenna is divided into
three regions. Region I is the dielectric coating,

Region II is the hot, lossy plasma layer and Region III

is an infinite free space region.

W}

33

We aim to solve for the fields in all three regions
and the ion and electron densities in Region II. The
solutions contain nine arbitrary constants. These con-
stants can be evaluated by matching the tangential electric
field in the dielectric region to that on the antenna, by
matching the tangential electric and magnetic fields across
the boundaries at r = b and r = c,and by requiring that the
radial velocities of the ions and the electrons go to zero
at r = b and r = c as discussed earlier. This procedure
gives us a complete solution to our rather idealized
problem.

In this study rationalized MKS units are used.
Rotational symmetry and an infinitesimal driving gap are
assumed. Furthermore, exp(jwt) time dependence is assumed
for the generator and all the fields.

3.2 Region I: Dielectric
Sheath Region
The basic equations which govern Region I (dielectric

layer, a i r i b) are Maxwell's equations

V x E (g) - jwuo 21(5) (3.2.1)

V x H (r) = jwed E1(r) (3.2.2)

where E1 and H are the electric and magnetic fields, no
is the permeability of free space, and Ed is the permit-
tivity of the dielectric medium. The suppressed time

dependence is exp(jwt).

34

From the symmetry of the antenna it can be seen that
there is no variation in the 0 direction and that the mag-
netic field has only a 0 component. Thus equations (3.2.1)

and (3.2.2) can easily be reduced to three scalar equations

 

such as
BE

3 1r _ _ .

§;(rEle) - 39 - jwuorHl¢ (3.2.3)
-—1.——- -3— (sineH ) = jwe E (3 2 4)
r s1n0 30 10 d 1r ' '
-JL (rH ) = jwe rE (3 2 5)
3r 1¢ d 16' ° '

Differentiating equations (3.2.4) and (3.2.5) and substi-
tuting them into equation (3.2.3) leads to a partial differ-

ential equation

 

l 2 l 2 . 2 _
__7 (rHl¢) + :7 §8[sin9 55(s1n0 rH1¢I] + 8d (rHl¢) — 0
(3.2.6)

where B 2 = wzu e . To solve equation (3.2.6), we use the
d o d

method of the separation of variables. Since 81¢ is inde—

pendent of 0 we can assume

rH = R(r)®(8) (3.2.7)

1¢

31".

fig

5135

Lege

35

where R is a function of r alone and ®is a function of
0 only. The substitution of equation (3.2.7) into equation

(3.2.6) leads to

2 2
r d R 2 2 l. d 1 d .
R dr2 ® d0[:51n0 (19 :l

(3.2.8)

where n(n+1) is the separation constant. Equation (3.2.8)

generates two ordinary differential equations

d l d .
EVE—fr)? ‘5(®Sln9):l + n(n+1) @= 0 (3-2-9)
2 2
€r-g—g + der2 - n(n+1) = 0. (3.2-10)
dr

Let us consider equation (3.2.9) first. Making the

substitutions,

u = cose , V1 - u2 = sine , é%-= - ll - u é%,
equation (3.2.9) can be reduced to
2
(l-u2)§-—@-2ug-@+ n(n+1) -—-—L—®=O
2 du 2
du l - u
(3.2.11)

Equation (3.2.11) is a special case of the associated

Legendre's equation,

36

d2 d m2
(1-x ) __X.- 2x _X.+ n(n+1) _ y = o
2 dx 2
dx 1 - x

(3.2.12)

which has a solution, y = an(x),which is called an associ-
ated Legendre function of the first kind of order n and
degree m. h

In order to have finite solutions on the interval
-1 :Ix 3’1 the parameter n must be zero or a positive
integer and m must take on only values -n, -(n-l),...,0,
...,n-1,n, i.e., n :_|m|.

Thus a solution to equation (3.2.11) is

®= Pnl (u) = Pnl(cos9) (3.2.13)

where n must be a positive integer.

Note that only one solution for this second order
differential equation (3.2.11) has been considered. The
other solution diverges on the 0 = 0 and 0 = 180° axes
and so it must be excluded from the solution.

Some other properties of the associated and ordinary
Legendre functions that will be useful to us in later
developments and in the numerical calculations are tabu-
lated in Appendix B.

We must now solve the other differential equation,

equation (3.2.10). With the substitution

R = R ‘3 (3.2.14)

37

equation (3.2.10) becomes

d R dR
1 1 1 2 (M8) _

Equation (3.2.15) is a form of Bessel's equation which has

a solution

(2)

R n+3

(Bar) + Bn H(l)(8dr) (3.2.16)

= An H n+8

1

. (1)
where An and Bn are arb1trary constants and Hn+8(8dr)
(2)

and Hn+k(8dr) are Hankel functions of the first and
second kinds with order n+8, which represent radially
inward and outward traveling waves respectively.
Combining equations (3.2.7), (3.2.13), (3.2.14» and

(3.2.16) we have

1 (2) (l)
1 Pn(cose)[Aan+k(Bdr) + Ban+%(Bdr{].

(3.2.17)

1
H = —-
l¢ /? n

"P18

The r and 0 components of the electric field can now be
found using equations (3.2.4) and (3.2.5).

Substituting equation (3.2.17) into equation (3.2.4)
and using the identities (B-7) and (B-8) from Appendix B

yields

38

co

_ ' (2)
E1r _ __l_§7§ £1 n(n+l)Pn(cose)E\n Hn+8(8dr)

wedr n-
+ B 11mm n] (3 2 18)
n n+8 d ' ‘ °

To derive E19, we need two differentiation formulas

for Hankel functions

5 H(1) _, n+k (l) (1)
dx Mn+g( X) - - x Hn+k(x) + Hn -3‘H’ (3.2.19)
§LH HAiL‘X’ = - an Hfif3<x) + H‘2;(x) (3.2.20)

The substitution of equation (3.2.17) into equation (3.2.6)

and using equations (3.2.19) and (3.2.20) leads to

' l (2) _ (2)
E16 = - ———1§7§- E1 Pn(cose) An[§ Hn+3(8dr) 8d r Hn -H(Bdril

(1) r n-
Ed

+ Bn[§ Héi;(8dr) _ 8dr H(1;(Bdr{] . (3-2-21)

The solutions for the fields in Region I can thus

be summarized as follows:

Hn+1:

"P18

1 Pi(cos€)[§n H(2)(Bdr) + Bn H(1;(Bdr{]

ihl“

10 _

n

(3.2.22)

39

co

._ ' (2)
Elr — ——1V2- El n(n+l)Pn(COSG)E\n Hn+k(8dr)

r n—
(08d

(1)
+ Bn Hn +15(8dr):] (3.2.23)

' °° (2) _ (2)
E16 - —J-37§ :1 P n(cose){n Elfin+;§(8dr) 8dr Hn_;§(8dr)]

wedr n
(1) (1)
+ Bn[§ Hn+%(8dr) - 8dr Hn_%(8dnj} (3.2.24)
le = H16 = E1¢ = 0 (3.2.25)

3.3 Region II: Plasma Layer

 

In Region II (plasma layer, by: r 3.9)! the plasma
medium is considered to be a two component, ion-electron
fluid. That is,both the ions and the electrons are allowed
to be mobile. The plasma is also assumed to be a weakly
ionized gas having average number densities of no elec-
trons and n0 singularly ionized atoms which are assumed
to be constant in the plasma layer. The deviation of the
electron density from the mean density, no, is denoted
by ne and the average induced velocity of the electrons is
denoted by Ye' The collision frequency of the electrons
with the neutral particles is denoted by Ye“ Similiarly
defined quantities for the ions are given by n., v. and 71'

1 ~1
Electron-electron, electron-ion and ion-ion collisions are

40

assumed to be negligible in a weakly ionized gas and thus
they are ignored.

In its unperturbed state the plasma is assumed to be_
homogeneous and neutrally charged and the perturbation of
the plasma is assumed to be sufficiently small, i.e.,
ne << no and ni << no, so that the linearized hydrodynamic
equations discussed in Chapter I apply. No static electric
or magnetic fields are present.

For a harmonic time dependence of exp(jwt) the basic

equations in the plasma layer are Maxwell's equations

V x E (f) = - jwuo g (5) (3.3.1)
V x H (5) = - enolye(§) - 31(5)] + jweo §2(§) (3.3.2)
7 - g (5) = - g: [ne(r) - ni(r)] (3.3.3)
V ° H2(r) = 0 (3.3.4)

and the linearized continuity and force equations for

electrons
no[V - ve(r)] + jwne(r) = 0 (3.3.5)
e Ve2
(Ye + 3w) ye(§) = - fi;-§2(g) - 7:; Vne(§) (3.3.6)

and the linearized continuity and force equations for

the ions

41

\

nOIV ° Yi(r)] + jwni(r) = 0 (3.3.7)
v.2

”1 + jw)Yi(§) = m—e; 132(5) - 31i- V.ni(§) (3.3.8)
1 0

where -e and me are the electronic charge and mass of the
electrons, e and mi are the electronic charge and mass of
the ions, “0 and so are the permeability and permittivity
of free space, and Ve and Vi are the thermal velocities
of the electrons and the ions,respectively.

It should be noted that the last terms on the right
hand side of equations (3.3.6) and (3.3.8) represent the
force due to a pressure gradient and the definitions given
for Ve and Vi in Chapter I are valid under the assumption
of an adiabatic pressure variation and a one-dimensional
compression.

In our formulation there are fourteen scalar un—

knowns, E2, H2, n , n., v , and vi. We will determine

e l ~e

H2, he, and ni f1rst and then calculate E2, Ye' and Yi'

It has been shown in Chapter II that ne and ni are

solutions to a pair of coupled differential equations

2 2 weZ we2 S
Vn +3 n+——n.=-—E— (3.3.9)
e e e V 2 1 'V e
e e
2 2 wiZ wi2 S
V n. + 8. n. + ——— n = ——— E— (3.3.10)
1 1 1 V 2 e V 2 (a

42

 

 

where
2 w 2 y
2 _ w _ e _ . e
Be - -—§'l -1r 3 —5] (3.3.11)
V w
e
w.2 y.
2 _ (32 1 - —£— - j —i (3.3.12)
B. _— 2 w
1 V 2 w
i
S

and p is the imposed source charge density. In Appendix A
equations (3.3.9) and (3.3.10) are algebraically uncoupled

to obtain solutions for ne and ni

 

 

 

 

w
= .2
ne Ve(T11nl + T12n2) (3.3.13)
“1

hi = T(T21n1 + T22n2) (3.3.14)
where

T11 = l -_ (3.3.15)

V 2V 2
l e i 2
1 + 4 2 ZIBe2 B 2 A ]
w w. 1 o
e 1
2 2
lVV Be - Bi _Ao
T - — _
21 2 w w.
2 2
V V. 2
1+1 3 lIBz-BZ-A]
4 w 2w 2 e 1 o
e i

(3.3.16)

43

 

 

 

 

 

 

T = 1 (3.3.17)
12
v 2v 2 2
l e i 2 2
1 +3—2—‘2lee ‘ 31 “‘01
w w.
e 1.
2 2
_ l Vev1 Be - 8i + Ao
T22 ' 2 w w
e 1 J// v 2V 2
l e 1 2 2 2
l + [B - 8 + A l
4 we2w12 e 1 0
(3.3.18)
and
_ 2 _ 2 2 w 2w.2

Ao ‘ //)Be Bi ) + 4 -53—$§- (3.3.19)

v v

e 1

and n1 and n2 are solutions to the differential equations

2 2 Bi
V n1 + kl nl = S1 e (3.3.20)
v2 + k 2n = 5 BE (3 3 21)
n2 2 2 2 e ° °
where
2 _ 1 2 2
kl — 2(Be + Bi + A0) (3.3.22)
2 _ 1 2 2 _
wi -we
81 = V; T12 + ‘7'; T22 (3.3.24)

44

(1) (A)
e 1

V— T21 + ‘T- T11] . (3.3.25)
8 1

 

In a sourceless region like Region II under consideration,

equations (3.3.20) and (3.3.21) become

+ k 2n = 0 (3.3.26)

2
V n1 1 1

V n + k n = 0 (3.3.27)

Both equations (3.3.26) and (3.3.27) are of the

form

Vzn + kzn = 0 (3.3.23)

which will now be solved by the method of the separation
of variables.

Due to the rotational symmetry, (no ¢ dependence),
the Laplacian of the scalar field n can be expressed in

spherical coordinates as

2 1. a[ 2 an] 1 a [ a ]
r2 Br I r2 sine 36 6
Since
a 2 an _ 32
51,-[r 'a—r-J - r 327(1'11) (3.3.30)

and using equation (3.3.29), equation (3.3.28) can be

reduced to a partial differential equation

an;

5..
”TN

45

3 l 3 . a 2
jun) + —— [sine —(rn)] + k (rn) = 0
8r r2 sine 55 39
(3.3.31)
Since n is independent of o, we can assume
rn = R(r)®(e) (3.3.32)

where R is a function r alone and®is a function of. 6
only. The substitution of equation (3.3.32) into equation

(3.3.31) leads to

2 d2

dr

5!!

E.
R

N

2 _ _ 1 l d . __
+ k r - @5135 “(Sine % — 2(2'1'1) (3.3.33)

where 2 is an integer and £(£+l) is the separation con-
stant. Equation (3.3.33) generates two ordinary differ-

ential equations,

1 d . _
m a—e-(s1n6 3% + £(£+1)®— 0 (303-34)
and
2 dzR 2 2
r ——5 + k r R - £(£+l) R = 0. (3.3.35)
dr

Let us consider equation (3.3.34) first. Making

the substitutions

u = C089: 1 - “2 = Sine, 3% = ' 1 ‘ “2 fi§ r

eq

EqL
Leg

sol

whe)
one
has
axes

cal

as e<

where

equat

ThEIEfI

46

equation (3.3.34) and be reduced to

(l - uz) ig- 2u §+ £(£+l) @= 0. (3.3.36)
u

Equation (3.3.36) is the standard form of the ordinary
Legendre's equation. This equation has the standard

solution

®= P£(u) = P£(cos6) (3.3.37)

where 2 is zero or a positive integer. Note that only
one solution for this second-order differential equation
has been considered. The other solution diverges on the
axes and so it can be excluded from the solution on physi-
cal grounds.

Since equation (3.3.35) is in exactly the same form

as equation (3.2.10), its solution can be written as
_ (2) (1)
R — /E[%£H£+a(kr) + D£H£+B(kr{] (3.3.38)

where C2 and Dz are arbitrary constants. Combining

equations (3.3.32), (3.3.37) and (3.3.38), we have

1 °° (2) (1)
n = 7% 2,-5.0 P£(cose) [EILHRA-lsmr) + D£H£+%(kr):|.

(3.3.39)

Therefore nl and n2 can be written as follows:

47

1 °° l (2) (1)
:11 = —; 2:0 P£(cose) C1£H£+k(klr) + D12H£+k(klr{]
(3.3.40)
_ _1_ °° (2) (1)
n2 ‘ ,—r 2:0 P2(°°39) Ezznua‘kzr’ + Dzznumkzri

(3.3.41)

nl and n2 are the perturbations due to the pseudosonic and
the electroacoustic waves respectively. Substitution of
equations (3.3.40) and (3.3.41) into equations (3.3.13)
and (3.3.14) yields explicit representations of ne and ni.
We must now determine the magnetic field g in
Region II. Taking the curl of equations (3.3.2), (3.3.6)

and (3.3.8), we obtain

 

 

V x V x g2 = - eno(V x Ye -V x vi) + jweoV x g
(3.3.42)

va=- e . VxE (3343)

~e m (y + 3w) ~2 ' '
e e
and

va.= 3. VxE. (3.3.44)

~1 mi(Yi + 3w) ~2

The substitution of equations (3.3.43), (3.3.44), and

(3.3.1) into equation (3.3.42) giVes

48

 

 

V x V x H = wzu e l + . wez T7 + . wiz . H .
~2 0 o Jw(ve + 3w) levi + 3w) ~2
(3.3.45)
Using the vector identity of
v x v x g2 = V(V - 52) - Vzgz (3.3.46)

and equation (3.3.4), equation (3.3.45) can be reduced

to a homogeneous wave equation
2 2 _
(V + ke )gz - 0 (3.3.47)

where ke is the complex propagation constant of the

electromagnetic wave in a two fluid plasma given by

2 _ 2
ke _ w “05 (3.3.48)

where E is the equivalent complex permittivity defined by

 

 

 

 

wez (1)12
g = e l + . . + . .
o Jw(Ye + 3w) levi + 35)
T
_ e 1 _ weZ _ wiz _ J u’ezYe
o w2 + y i w2 + y 2 w(w2 + yez)
u) . 27 .
+ 1 1 2 . (3.3.49)

 

w(w2 + y. )

49

From the symmetry of the antenna it can be seen that
there is no variation in the ¢ direction, i.e., 3/3¢ = 0
and the magnetic field has only a ¢ component. Thus the
Laplacian of the vector magnetic field in spherical coordi-

nates takes the form

2 _ A 2 2
v 52 9(VVH2¢ ;§ csc 6H2¢)
3H ‘ 3H
1 8 2 29 l 3 . 2
= ¢ -7-—— r + -— Sine
r J

 

where 3 is the unit vector in the ¢ direction. Using

~

equations (3.3.30), (3.3.50) and the following identity

 

3H
1 __a_ . 2g; _ 2 _ _g_ 1 3_ .
sine 36(51n9 as J C“ 6H24> " aeE-Tne' e (“wring

2
3 H ER
_ -——%i + cot6 a: - csczeH2 (3.3-51)
39 ¢

in equation (3.3.47) leads to a partial differential

equation,
32 (rH ) + 1 JL--+l—-JL sinerH +-k 2(rH ) = 0
8r: 2¢ r2 39 s1n6 36 2¢ e 2¢

(3.3.52)

50

which is in exactly the same form as equation (3.2.6).

Thus the solution to equation (3.3.52) can be written as

__1_ °° 1 (2)
H2¢(r,6) - /§ nil Pn(cos6)[:En Hn+8(ker)
+ Fn Hriizi(ker):l (3.3.53)

where En and Fn are arbitrary constants.

g2, ne,and ni have been determined explicitly and
are expressed in equations (3.3.13), (3.3.14), (3.3.40),
(3.3.41), and (3.3.53). We now must express g2, Ye' and
vi in terms of these known quantities.

~

From equations (3.3.6) and (3.3.8) we easily get

 

 

 

 

2
9 V9 v (3 3 54)
V = - r E - I n o o
~e me(ye+3w) ~2 no(ye+jw) e
e Vi2
V. = r E - T VII-o (303.55)
~1 miTYi+Jw) ~2 no(yi+jw) 1

The substitution of equations (3.3.54) and (3.3.55) into

equation (3.3.2) leads to

2 2
1 e V e V.

e 1
= +—— - + .
E2 Jwfi Vx§2 jw£(ye+jw) vne 3w£(yi+jw) vn1

(3.3.56)

where E is given in equation

51

(3.3.49). The substitution

of equation (3.3.56) into equations (3.3.54) and (3.3.55)

 

 

 

 

 

 

 

 

 

 

 

yields
2 2
v = - e . .1 VxH - - eoVe . 1 - mi
-e me(ye+3w) ng ~2 Eno(ye+3w) w2+YiZ
. 2 . 2 2
JYiwi v Jwe e°vi v (3 3 57)
- n + . u n. o o
w(w2+YiZ) e no£w(yi+3w)(ye+3w) 1
2 2
v = e —l—»VxH - eoVi 1 - we
-1 mi(yi+3w)3w§ ~2 Eno(yi+3w) wZ+YeZ
. 2 .1 . 2 2
Jyewe v Jwi €°Ve v (3 3 58)
- n. + . . n . . .
w<w2+ye2ij 1 nogw(yi+3w)(ye+jw) e

 

Under rotational symmetry the two vector differential

operators in spherical

_ A l
V x g2 — E r sine
and
_ A an A l 3
Vn — 5 3r + g r

coordinates can be expressed as

3

8r

.3.

36(sine H

) (3.3.59)

1CD)

1
2¢) E' (r32¢

“ (3.3.60)

A A I I O I
where r and 6 are unit vectors in the r and 6 directions

respectively.

we obtain

Combining equations (3.3.56) to (3.3.60),

52

. 2
1 1 a 3 9 Va ane

 

 

 

 

 

 

 

 

 

 

 

 

 

E:2r g J'wE; r sine 5—6—(Slne HM) + w; (yea-f») 3r
e Vi2 ani
_ 3 6377;1367'337 (3.3.61)
2
_ 1 1 a . 3 Va 1 ane
E - - v—— - ——(rH ) + 3 . — ———
26 JwE r 8r 2¢ w§(ye+jw) r 36
e v.2 3n.
— j 1 l ——1 (3 3 62)
w§(yi+jw) r 66 ‘ °
_ _ e l 1 .§_ .
ver - me(ye+jw) ng r sine 36(31n6 H2¢)
_ m
2 2 2
_ ‘60 Ve . 1 _ wi - j yiwi ane
Eno(ye+3w) w2+Y12 w(w2+yiz) 3r
wezeoVi2 ani
+ j T . (3.3.63)
nOEw (Yi-i-jm (Ye-i-jw) 3r
.. e ,_1 1 2.
ve6 _ me(ye+jw) JwE r 3r(rH2¢)
2 2 y w 2
_ 86 Va 1 _ “i _ j i i 1 ane
Eho(ye+3w) w2+Yiz w(w2+Yi2) r 36
2
2 V.
w e 1 an.
3 ° 1 1 (3.3.64)

+ j noEw(yi+jw)(Ye+jw) E 36

53

 

 

 

 

 

 

 

 

 

_ e l 1 3L .
vir - mi(yi+jw) SwE r sine 86mme H2¢)
_ _
2 2 2
_ E:o Vi 1 _ we _ j we Ye ani
Eno(yi+jw) wi-I-yei w(w§+ye2) 3r
2 2
+ j H ::(€°+Y:)T +‘w) 2:? (3'3'65)
0 Y1 J Ye J
__ e 111
via _ mi(yi+jw) wa r 3r‘rH2¢)
r' "T
E V 2 w 2 w 2
_ o i l _ e _ j e Ye 1 i
Eno(yi+3w) w2+Ye2 w(w2+Ye2) r 36
2 2
w. s V 3n
+ j 1 ° 8 1'7F2’° (3.3.66)

 

no€w(yi+jw)(Ye+jw)

Using equations (3.3.13), (3.3.14), (3.3.40), and (3.3.41)

to express ne and ni explicitly, we obtain

e l m H(2)
n = -—--—- Z P (cose) T C (k 1r)
e ve /E n=0 11 In Hn+3,
H(1) (2)
+ D1n Wn+s(k ‘1 T12 C2n Hn+k(k2 1’
H(1)
+ D2n Hn+k(k2 r) (3.3.67)

 

54

w on
_ 1 .1 H(2)
“i ‘ 'T'7g n20 Pn(°°3°){%21[%1n“n+a(klr’
(1) (2)
+ Dln Hn+151k1 1"] + T22E2n H114,1,(1‘2 1’)

(1)
+ D2n Hn+k(k2 ré]} (3.3.68)

where T11, T12, T21, and T22 are given by equations (3.3.15)
thru (3.3.18). Using equations (3.2.19), (3.2.20), (3.3.67),

and (3.3.68) we can get

323 - - 22- :1 E P (cose) (n+l)H(2)(kl r)
3r " Ve r372 n=0 n T11 C1n n+2

k lrflézi(klr{] + T11 Dln[En+l)H(l;(k1 r)

7
- k lrué1;(kl r) + 112 c2n (n+1)néi;(k2 r)
_J

-—q

k 2rH(2;(k2 r) + 112 D2n[}n+1)H(1;(k2 r)

*

_ k 2rH(l;(k2r£1} (3.3.69)

 

and

55

°° (2)
nio Pn(cose) T21 Cln[}n+l)Hn+k(klr)

u

l
<|E
P rm

1
. 7,3 2
- klrnéfimlril + T21 1311r1|:(n+1)nr(h{35 (klr)

1.35;; 0.1”] <2)

- k T22 C2n[}n+l)fln+k(k2r)

+

- kzrfléfimzri' + T22 DZnIFn+1)Hr(1}_;(k2r)

(1)
kern_k(k2ri]

From the definition of the associated Legendre functions

(3.3.70)

and the ordinary Legendre functions we can derive the

identity

1 __ d
Pn (cose) - 35 Pn (cosG). (3.3.71)

Using equations (3.3.67), (3.3.68), and (3.3.71) we have

1 (2)
Pn (cose) Tll[é1an+k(klr)

(1) ' (2)
+ Dlan+15(klr):| + TIZEZan-ngc2r)

+ D a”) (kzr):| (3.3.72)

56
and

3h. w m
1 1 (2)

-— 2 P (cose) T C H (k r)

i r n=1 n 21|:ln n+1: 1

Q)

mra

<’l
P.

(1) (2)
+ Dian+g‘k1r{J + T22[§2nfin+g(k2r’

—

(1)
+ DZan+%(k2r) (3.3.73)

 

because P:(cose) = 0. Equations (3.3.61) thru (3.3.66)

can now be expressed explicitly. The first terms on the
right hand sides of equations (3.3.61) thru (3.3.66) can be
obtained using the same procedures as those used in Section
3.2 and the other terms can be derived using equations
(3.3.69), (3.3.70), (3.3.72), and (3.3.73). Since we are

interested in E26, v , and vir for later development, only

er
equations (3.3.62), (3.3.63), and (3.3.65) are expressed
more explicitly as follows:

_ _ ' w 1 (2) _ (2)
E26 - ;E_%7§ Z Pn(cos6) En[}an+3(ker) kean-k(ker{]

r n=1

+

(l) (1)
En [111111445 (ker) - kean_1§(ker):l

m V e
_ e e (2) (1)
72:35—[%11C1nnn+8(klr) + T1101an+g‘k1r’

+

(2) (l)
T1202nun+a‘k2r’ + TlZDZan+k(k2r{] +

57

w.V.e

i i (2) (1)
7:135'T21C1nnn+3(k1r) + T21D1an+g(k1r)

(2) (1)
+ T22C2nfin+8(k2r) + T22D2an+k(k2r)]} (3.3.74)

_ e l 1
Ver - me(?e+jw) ng r375

 

°° (2)
n:1n(n+1)Pn(c°se)[%an+k(ker)

 

Z

Pn(cose)

O

(1) 6o 1
+ Fan+k(keri] + no€(ye+jw) r3/2 n

X

(2) (2)
{gln[}n+1)nn+k(klr) - kern-g(klr{]

(1) _ (l)
D1n [(n+1) Hn-I-lg (klr) kern_;5 (kl-fl} {”eVeTllE

+

 

2
_ “i _ j Yiwi _ j we 91Vi T
w2+Yi§ w(w2+YiZ) w(yi+3w) 2

+

{%Zn[}n+l)fléi;(k2r) - kzrfié3;(k2r{]

DZDEMM}; (1.2., - 1.2.1.311; (kzrflleveTuE

w 2 w 2 w 2w V
i _ . Y1 i _ . e i i T
2 2 J 2‘ 2) J w(yi+3w) 22

w(w +yi

+

 

 

(3.3.75)

3.4

spa

58

_ _ e 1 (2)
vir - mi (Y1+jw) ng -—7§-n 2 ln(n+l)P n(cose)|:I-E:an+;5(ker)

(1) °°
Fan+15(ke r) +W —317—n 2 cup (0089)

 

+
x Cln[En+1)H(i;(kl r) - k er(2)(k gr:]
+ o [:(n+l)H(1)(kl r) - k an
n n+3 1 in iT21[
2 2

_ we _ j we Ye _ j wizw eV e

w +ye w(w2+ye2) wIY (34'3“)ST
+ C n[3n+l)Héi;(k2 r) - k 2rH(2;)5(k2 r{]n
+

D n[(n+l)Hr(li;(k2 r) - keré£;(k2r)] “iViTzzE

2 2 2

 

we we Ye w eeV
-———--j -j—-(———TT
w2+ye2 w<w2+Ye2) w Ye+jw 12

(3.3.76)

3.4 Region III: Free Space
The basic equations which govern Region III (free

space, r >, c) are Maxwell's equations
V x §3(r) = -3wuoH3(r) (3.4.1)

V x H3(r) = jweog (5). (3.4.2)

59

Since Region III is unbounded, no reflected or inward
traveling wave exists in this region. Following an
analysis similar to that in Section 3.2, the solutions

to Maxwell's equations in this region can be written as

.. _1_ °° 1
H3¢ ‘ J— H: Gn Pn(cose) Héi;(80r) (3.4.3)
E3r = ——-i:7§ E Gn n(n+1) Pn(cose) Héi£(80 r)
weor n—l
(3.4.4)
E36 - - ___1377. g G P1 (cose) “(3;(8 r)
weor n=l n n n
(2)
- Bor Hn_;§(80r)] (3.4.5)
and
H = H = E = 0 (3.4.6)

where Gn is an arbitrary constant, n is a positive

integer, and so = w/uoeo.

3.5 Imposition of Boundary
Conditions at Interfaces

In order to determine the arbitrary constants An,

B , C

n ’ D

, C , D , En’ Fn' and Gn' the boundary con-

ln ln 2n 2n
ditions at r = a, r = b, and r = c are applied.
The voltage across the gap of the spherical antenna

is given by

60

g + e “
v = Ele(a,6)ad6 == fEle(a,6)ad6 (3.5.1)
- 9 0

because Ele is zero on the surface of the conducting
sphere except at the gap n/2 - 91‘: 6 i “/2 + 61. Since
the ordinary Legendre functions form a complete set of
orthogonal functions, any function f(x) on the interval
-1‘: x‘: 1 can be expanded in terms of them. Thus the

electric field on the surface of the sphere can be repre-

sented as

00

Ele(a,6) = E an:(cose) (3.5.2)
n—l
where
n
= 2n+l .[ .
bn WT E16(a,6)1’]r"(c036)81n6d6. 3.5.3)
0

If the gap between the two halves of the sphere is

assumed to be small, then

~

1 1
Pn(cose) ~ Pn(0)

(3.5.4)

s1n6 ” 1 61 is small.

Combining equations (3.5.1), (3.5.3), and (3.5.4), we have

(2n+1) Pi‘0’ v
bn = “§E(n+1) a ° (3°5'5’

 

61

From equation (3.2.21)

00

_ _ ' l (2)
Ele(a,6) - -——1§7§) £1 Pn(cose) An[n Hn+g(8da)

_ Bdanrgzgmdafl + BnE 1.335%.)
- Bda Hé£;(8da{] . (3.5.6)

Combining equations (3.5.2), (3.5.5), and (3.5.6) we have

M11 An + M12 Bn = s v (3.5.7)
where
_ (2) _ (2)
M11 ’ “ Hn+k(8da) Baa Hn-k(8da)
_ (1) _ (1)
M12 - n Hn+k(8da) Bda Hn-%(Bda) (3.5.8)
_ . k 1 2n+l
S - Wda Pn(o) m -
The Mij’ i,j = 1,2,...,8,9 refer to position in a matrix

to be set up later.
The continuity of the tangential components of g
and H at the dielectric-plasma interface (r=b) requires

that

E16(b,6) = 329(b,6) (3.5.9)

Hl¢

(b,

62

6) = H2¢(b,9). (3.5.10)

Using equations (3.2.21) and (3.3.74), equation (3.5.9)

gives

where

M21 n

21

22

23

M24

A

+ M B + M C + M

22 n 23 In D

C

24 1n + M25 2n

+ M26D2n + M27En + M28Fn = 0 (3.5.11)

_ (2) _ (2)
-Eil} Hn+k(8db) Bdb an -;’(Bdb):]

d

53E”! H’E‘l‘zfmdb) ' Bab “fibridbfl
(5%; T11 " gill-319,3 T21] “(135* b)
eEEZ'iT T11 ' gig-5 1.211313%“ b)

(3.5.12)

m V w.V.
e e _ 1 1 (2)
elE‘Tj'E T12 y‘i'+"j'w"2 T 2])!!! +1: (kzb)

w V m.V. '
e e _ 1 1H(l)
9E7}: "’12 7337; T22] “2w,“ 1’)

(2) H(2)
-E Hn+35(keb) - kebH n- -e35(k bi]

63

_ _ (l) _ (1)
M28 " |} Hn-I-ku‘eb) keb Hn-k(keb)] °

From equations (3.2.22) and (3.3.53), equation (3.5.10)

can be expressed as

M31An + 143an + M37En + M38Fn = 0 (3.5.13)

where
M = H”) (B b)
31 Hn+k d

_ n(l)

M32 ‘ Hn+k(8db)

(3.5.14)

_ (2)

M37 ’ Mn+5(k b)
_ (1)

M38 - Mn+k(k b)'

The continuity of the tangential components of the
E and H fields at the plasma-free space interface (r=c)

leads to the boundary conditions

E e(c,e) = E36(c,6) (3.5.15)
H2¢(c,9) = H3¢(c,6). (3.5.16)

Using equations (3.3.74) and (3.4.5), equation (3.5.15)

gives

64

 

 

M43cm + “4491:: + M45cm: + M46D2n + ”473:;
M‘an + M49Gn = 0 (3.5.17)
where
r _
m V w.V.
_ e e _ 1 1 H(2)
M43 " e —"ye+3w T11 __F-yi-t-jw T21“ n+15(k1 C)
[E V w.V.
_ e e _ 1 1 H(1)
M44 ‘ e _F'yenw T11 ___Fyi+jw T21H n+15‘k1‘”
T _
m V w.V.
- e e _ 1 1 (2)
M45 - e ye+3w T12 yi+jw T22 Hn+k(k2 C)
I; V w.V.
_ e e _ 1 1 H(1)
M46 - e W T12 W T22 [in-”5% C) (3.5.18)
__ _J
_ H(2) _ (2)
M47 - ”[9 +15(ke c) k ec Hn -k(ke c)
(l) (1) .1
M48 = -[§ Hn+k(kec) - kec Hn -k(ke c)

 

n+
so 8

M49 =.£L[E H(2)(Boc) - 80c HA3;(BOC{].

From equations (3.3.53) and (3.4.3), equation (3.5.16)

can be expressed as

M57En + M58Fn + M59Gn = 0 (3.5.19)

65

where
_. (2)
M57 ' Hn+k(kec)
_ (1)
_ (2)
M59 ' Hn+;,(Bo°"

In the present analysis, it is assumed that the
normal components of the mean electron and ion velocities
vanish at the interfaces at r=b and r=c. These rigid

boundary conditions require that

ver(b,6) = 0 (3.5.21)
ver(c,6) = 0 (3.5.22)
vir(b,9) = 0 (3.5.23)
vir(c,6) = 0. (3.5.24)

Using equation (3.3.75), equations (3.5.21), and (3.5.22)

give

M63C1n + M549m + M55C2n + M66D2n + M67En

F = 0 (3.5.25)

+‘Msa n

and

66

M73cm * ”7491:: + M75C2n + M76D2n + M77En

+ M78Fn = 0 (3.5.26)

where

. (2) (2)
63 - Je[}n+l)fin+k(kl b) - R 1b Hn -k(kl b{][§eVeTll l

 

M _-
2 2 2
_ “’1 _ j Yiwi 3. we ”1V1 T
T": ‘7—7 " . '
u1471 w(w +Yi ) w(y1+jd) 21

3
I

64 — jel:(n+l)Hrg‘_3§(k1 b) - k 1me (k 10:] [av e 11

 

 

 

 

2 2
mi. . y.w. j wez iiV T
w2+yi5 w(w2+in) “(7“ $) 21
(3.5.27)
M65 = jeEn+1)n(2}!(k2 b) - ksz(2) (k 2b:H:eVe T12[l
wiz Yi mi 2 we 2w. 1Vi
_ _ j j T
w2+Yi2 w(:2:yi2 ) Jw(Yi+jw) 22
_ - (l) _ (1)
M66 — jel:(n+l)Hn_._;fi(k2 b) k 2an 1‘(k 2b] Eev e T112[
2 2
_ “’1 _ j 71‘”: _j “’e 2"” 11V
$2:;;2' 5732:??? w(y.1+jw) T22
w 2
M67 - n(n+1) £- Héiékeb)

67

= n(n+1) -—— H(1)(ke b)

M68 w Hn+1:

and the expressions for M i = 3,4,5,6,7,8 can be ob-

7i'

tained by replacing b with c in the corresponding M6i'

Using equation (3.3.76), equations (3.5.23), and (3.5.24)

can be represented as

M83C1n + M84D1n + Masczn + M86D2n + M87En
+ M88Fn = 0 (3.5.23)
and
M93C1n + M949111 + M95C2n + M96D2n + M97En
+ M98Fn = 0 (3.5.29)
where

83 - jeI:(n+1)HI:i;(klb) - klbnm (k1b]l:iV'T2 1[1

 

M _
w 2 w 2y i 2w eve
_ ___—e;— ..j e e _. j T
wZ+Ye2 w(w2+ye2) w(ye+jw)
- - (l) _ (1)
M84 - 3e[:(n-!-1)l-ln+;fi(kl b) k 1an 15(k 1b] [:ivll i T21[
wez w ”27 i 2w eeV
— -————— - j —1-—— - j _1_—wa
w2+y2 w(w +ye w Y +3”

e

68

85 — je|:(n+l)Hr§i;(k2 b) - kzbnm (k 213)] [1V1T22[1

3
l

 

 

2 2
we w HZY j mi w eeV T
— — j .—
w +76:i w(w +Ye 2) 3w(Y e+jw) 12
_ . (1) _ (1)
M86 — je[(n-%-1)Hn_*_z~fi(k2 b) k 2an 35(k2b)] [1V i T122[
2 2
_ we _ j we Ye - jwie 2w eeV T
w2+YeZ w(w2+Ye2) w(y e+jw) 12
(3.5.30)
w.2
M87 = -n(n+l) : Héi;(ke b)
w.2
M88 = -n(n+1) 7%— Hrfigfime b)

and the expressions M91, i = 3,4,5,6,7,8 can be obtained

by replacing b with c in the correSponding M8i' It should
be noted that the summation of the second term on the

right hand side of equations (3.3.75) and (3.3.76) can be
changed from E to ; because the n=0 term makes no
contribution Egothe Ezries. Thus equations (3.3.75) and
(3.3.76) can be written as single summations and the

rigid boundary conditions may be imposed to yield equations
(3.5.25), (3.5.26), (3.5.28), and (3.5.29).

By imposing the above boundary conditions, we obtain

nine algebraic equations for the nine unknown coefficients.

69

For convenience these equations may be written as a single

matrix equation:

 

 

_Gn J

 

)—

SV

0

 

where [m is the matrix

M=

FM

M

M

E431”

 

[is

11

21

31

M

M

M

0

12

22

32

0

M23

0

M43

 

(3.5.31)
0 o _
M28 0
M38 0
M48 M49
M53 M59
M68 0
M78 0
M88 0
M98 OJ

(3.5.32)

70

From equation (3.5.31) we obtain a solution for the

arbitrary constants as

 

 

where [Mij]

sfl

 

 

L0

(3.3.33)

is the matrix inverse of [Mi.].

Equations (3.2.22) thru (3.2.25),

(3.3.61), (3.3.62),

(3.3.67),

(3.3.68),

3
(3.3.53),

(3.4.3) thru

(3.4.6), and (3.5.33) completely determine the fields in

Regions I, II, III as functions of r and 6.

Using the first result from Appendix B it is seen

that S = 0 for even n.

Thus, from (3.5.33), the arbitrary

constants are all zero when n is even.

This means that

the summations for the fields in all three regions may be

changed from

n

IIMS

l

to

co

2
n=1

. The fields in Region III are

n odd

71

of most interest to us and so

here as follows:

__1_°° 1
H3¢ — /F nil Gn Pn(cos
n odd

E = ———l§7— 2 G
3r weor 2 n=1 n
n odd
E = - ———l—7— Z G
36 weor3 2 n=1 n
n odd
(2)
8or Hn-%(Bor{]
and
“3: = “39 = H3¢ = 0

3.6 Radiated Power and Input
Admittance

Two important quantitie

they will be reproduced

(2)

6) Hn+%(80r) (3.5.34)
n(n+1) P (cose) H(2)(B r)
n n+k 0
(3.5.35)
1 (2)

Pn(cos6)[E Hn+%(80r)
(3.5.36)
(3.5.37)

5 that we will use exten-

sively in the next chapter on numerical results are the

power radiated by the spheric
surrounding plasma layer and

spherical antenna. These qua

a1 antenna together with its
the input admittance to the

ntities are derived below

for the specific problem which we are studying.

72

The power radiated from a large sphere is defined

P = lim ReE P - as] (3.6.1)
r+0° S

where Pe is the complex Poynting vector defined by

~

as

(3.6.2)

ll
NIH
It!)

3:
2::
a-

and d5 is a vector quantity, which points in the outward
radial direction, associated with a differential area on
the spherical surface. The integration is over a large

closed spherical surface.

Thus the radiated power can be written as

1 2n n A 2
P = lim 2Xf Jf Re[E x H* - r] r sine d6d¢.
r+00 o o ” ~ ~
(3.6.3)
In Region III in our problem
* o =
E3 x E3 r E36 H3¢
= - Z Z G G * Pm (cosB)P} (cose)

m n
we r m=l n=l

m odd n odd

[ new a - n] (33,.”

(3.6.4)

73

where a superscripted * denotes complex conjugation. In
the far zone (r+w), the asymptotic expansion for large

argument for the second of Hankel function is

lim Hofz’uc) = E"; e'j(""’°‘"’;‘"). (3.6.5)

x+ 00

Thus, neglecting terms of the order l/r3 and higher we

get
lim E3 x H3 - f = %’——l—§ ; ; GmGn P;(cosa)
r+w ” ~ we r m=l n=1
m odd n odd
1 J(11'1-2-1‘1)‘IT
X Pn(cose) e . (3.3.6)

Therefore we obtain

00

P = ’1' -l— :30 Z Re[G G *expl:j(m-n)1] Izwalwose)
" weo m=l n=1 m n 2 o o m
m odd n odd

x Plr'l(cose)sin6d6d¢]

co an

1
2 . n l 1
= — )3 Z Re|:G G *expEMm-n) J I P (x)? (x)dx].
”so mgl n=1 m n 7 -l m n
m odd n odd

(3.6.7)

74

Using result (B-3) from Appendix B, the radiated power

becomes
2 m 2 2n(n+1)
p =— 2: |c| (3.6.8)
weo n=1 n 2n+ .
n odd

The input admittance is defined [13] as

Y = 2na sin9[H¢] (3.6.9)
r=a
6=n/2

which for our problem is

Y = 2n/E

"M8

(2) (1)
n l Pi(0) Aan+k(Bda) + Bn Hn+g(8da{]

(3.6.10)

It is noted that because of the assumption of a delta
function driver only the real part of the series for Y
converges. Infeld [14] has suggested that the imaginary
part be calculated at some angle slightly different from
6 = 90°. He suggests using the angle that a real physical
gap would make with the 6 = 90° axis. In our case this

n

means we must evaluate the susceptance at 6 = 2 - 61.

This is the procedure used in Chapter IV.

CHAPTER IV

NUMERICAL TECHNIQUES AND RESULTS

4.1 Numerical Techniques
In order to complete the solution for the fields in
Regions I, II, and III of Figure 3.1, it is necessary to

solve the matrix equation (3.5.31), i.e.,

PAn q r—SVT
Bn 0
cln o
Dln o

[M] c2n = 0 (4.1.1)

DZn 0
Bn 0
PD 0

_Gn_i Lo_4

 

 

 

 

for the arbitrary constants An, Bn' C1 , Dln' C2n' D

n 2n'

En, Fn’ and Gn where the elements of M and S are given

in equations (3.5.8), (3.5.12), (3.5.14), (3.5.18),

75

76

(3.5.20), (3.5.27), and (3.5.30) for n = l,3,5,...,~.
This is accomplished using the technique of Gaussian
elimination [1?].

Gaussian elimination is a technique to solve a

matrix equation of the form

PX = Q (4.1.2)

where P is a given square matrix and Q is a given column
matrix and X is the unknown column matrix to be deter-
mined. The technique is based on a theorem which states
that P may be factored into a dot product of a lower

triangularized matrix L and an upper triangularized matrix

up 1060'
p = LU (4.1.3)
where
F' .1
Ll’l o . . o o
Ll’z L2'2 . . O 0 O
L = I I I I I I I (4.1.4)
Ln-1,1 Ln-1,2 ' ° ° Ln-1,n-1 °
Ln,1 Ln'z . . . Ln'n_1 Ln'n

 

 

77

 

and
1‘ 7
ulpl 111,2 . . o ul'n-l ul,n
0 u . . . u u
2,2 2,11-1 2,11
U = . . . . (4.1.5)
0 O ' ‘ ' un-l,n-l n-1,n
_O 0 I O O o un'n .

 

Defining a new unknown column matrix

y = UX (4.1.6)
equation (4.1.2) may be written as

Ly = Q. (4.1.7)

Equation (4.1.7) is a set of equations that may be solved
simply by back substitution. Once y is determined X may
readily be determined from equation (4.1.6) by a similar
procedure.

In theory the Gaussian elimination technique will
give an exact solution to the set of equations (4.1.2),
but in practice the solution must be obtained by use of
a computer. This leads to errors due to the fact that
the computer carries only a finite number of significant
figures. Errors are obtained whenever the individual

terms in equation (4.1.2) that are to be added together

78

differ by more than m orders of magnitude where m is the
number of significant figures carried by the computer
being used.

In our work we found that the solution for the Gn s
to be very accurate in all cases judged on comparison of
our results for spherical antennas in free space with those
of Ramo, Whinnery, and Van Duzer [13]. The solutions for
the An 8 and Bn 8 used in the calculation of the input
admittance were found to be accurate only for antennas
which are of the order of 0.1 wavelengths in radius. For
smaller antennas surrounded by a lossless plasma, the
input conductance calculated from equation (3.6.10)
differed from the conductance calculated from the power
radiated, equation (3.6.8), which is known to be correct
in the limit of the plasma density going to zero. This
was probably due to numerical difficulties because the
matrix M was nearly singular for small antennas. For
larger antennas, we were unable to keep enough terms in
the series (3.6.10) to obtain a reasonable result. For
an antenna radius of 0.1 wavelengths it was found that
only the first term of the series (3.6.10) was needed to
obtain five significant figure agreement with the con-
ductance calculated from the power radiated for a loss-
less plasma layer. The power radiated was calculated
retaining the first five terms, n = l,3,5,7,9, in the
infinite series in all our calculations. The susceptances

for the graphs to be described later were calculated

79

keeping the first three terms, n = 1,3,5, which for a
spherical antenna in free space give a result 27% less
than the result given in the above reference [13].

As a conclusion we can say that the results given
for the power radiated and the input conductance should
be very accurate and the results for the input susceptance
are accurate to within an order of magnitude for the
cases plotted.

The Hankel functions required in the matrix M are

calculated for n = 0,1 using the formulas [15]

n

(1) _ 25'.-n-l -1 jz (n+k)! _ . -k
Hn+k(z) - 1T 3 2 e kio ETTE:ETT ( 2J2)

(4.1.8)

(2) _ z .n+l -l -jz (n+k)! . -k
Hn+g‘2) ' /‘7F 3 z e z ETTHIETT"232)
(4.1.9)

and higher order Hankel functions, i.e., n = 2,3,... are
calculated from the results for n = 0 and n = l for a
given complex argument 2 = x + jy by a recurrence

relation [15]

1

fn+1(z) = (2n+1)z' fn(z) - fn_1(z) (4.1.10)

where fn can be Jn722 Héi;(z) or Jw/2z Héi;(z). The

above formulas gave very good results for all values of

80

purely real or purely imaginary arguments that could be
checked with Abramowitz and Stegun [15] for order up to
n = 11.

The associated Legendre functions are calculated

using [16]
l .
Pl (cose) = - Sine (4.1.11)
P21(cose) = - 3 sine cose (4.1.12)

and the recurrence relation

l'lP1

_ 1
n+l(cose) - (2n+l) cose Pn(cose)

- (n+1) Pi_l(cose). (4.1.13)

These formulas gave very good numerical results.

All the numerical calculations were carried out on
the CDC 6500 computer using single precision arithmetic
(fifteen significant figures) except the calculation of
the Hankel functions where double precision (twenty-nine
significant figures) arithmetic was used.

The radiated power and the input admittance of a
spherical antenna surrounded by a concentric layer of
hot lossy plasma have been numerically calculated as a
function of the antenna radius and the plasma parameters.
In a realistic situation, the presence of the plasma
sheath is taken into account by the adoption of a con-

centric dielectric layer which separates the plasma from

81

the metallic surface of the antenna. This adoption may
also be used to account for an actual dielectric coating
of the antenna. The thickness of a usual plasma sheath
may be of the order of a few Debye lengths. In the
present numerical calculation, the sheath is assumed to
be an electron—free region extending from r = a to r = b.
A convenient parameter to describe the thickness of the
sheath is the dimensionless quantity 5 defined by

b - a = (ve//§me) s. It is to be noted that (ve//§me)

is of the order of a Debye length in the plasma and thus,
5 may be regarded as the "Debye thickness" of the sheath
[12]. The permittivity of the sheath is assumed to be the

same as that of free space, i.e., ed = 80.

4.2 Numerical Results

 

The results of the numerical calculations for various
parameters for a spherical antenna surrounded by a finite
layer of a hot lossy plasma are given in Figures 4.1 thru
4.20. The calculations were performed assuming an oxygen

938 plasma so that m1 = 2.57 x 10’20

kilograms. The
electrons and the ions were assumed to be in thermal
equilibrium so that Te = Ti and the average thermal
velocity of the electrons was assumed to be 0.01 times

the speed of light. The antenna was assumed to be driven
by a one-volt time varying source. Except where otherwise

noted the sheath was assumed to be about one Debye length

thick, i.e., b = a + Ve//3we and the ratio of the

82

ion-neutral particle collision frequency to the electron-
neutral particle collision frequency was taken equal to
the ratio of the ion thermal velocity over the electron

thermal velocity, i.e.,

V.
_ _i = ,/_2
Yi - Ye V Ye mi (4.2.1)

Unless otherwise noted we shall state only the electron-
neutral particle collision frequency with the ion-neutral
particle collision frequency being specified by equation
(4.2.1). The values of the susceptance that are plotted
in this chapter are calculated by matching the magnetic
field to the current on the spherical antenna at 61 = 5°
or 6 = 85°. This procedure was suggested by Infeld [14].

It is to be noted that for 1° 1 6 1 7° essentially the

1
same results are obtained.

Figures 4.1, 4.2, and 4.3 plot the radiated power,
the input conductance, and the input susceptance, re-
spectively of a spherical antenna of radius 0.11, where A
is the free space electromagnetic wavelength, surrounded
by a layer of hot plasma 0.031 thick as a function of the
plasma density, i.e., wez/wz. The running parameter in
each figure is the electron-neutral particle collision
frequency. The range of weZ/w2 considered in these
figures is from 0.0 to 2.6 which corresponds to a high

frequency or a low plasma density region. It should be

noted that all plots over this range are actually

83

independent of the ion-neutral particle collision fre-
quency because for high frequencies the ions are essenti-
ally immobile. The vertical scales in Figure 4.1 are

10 log (P/Po) where P is the power radiated by the
'spherical antenna surrounded by a plasma layer as a
function of (1162/11)2 and P0 is the power radiated by the
same antenna without a plasma layer. In Figure 4.2 the
vertical scales are 10 log (G/GO) where G is the input
conductance as a function of (1162/41)2 and Go is the con-
ductance of a spherical antenna in free space. Figure 4.3
is a series of plots of the input susceptance in mhos of

a spherical antenna as a function of wez/wz. A study of
these figures indicates that the inclusion of the electro-
acoustic wave in the theory gives rise to effects in all

three figures for (1162/11)2 < 1.0 and ye/w = 0.0 in the form

of troughs and peaks very close together whenever

A
_ I = _S
C b — N W N 2 (4.2.2)
where k2 is the electroacoustic prOpagation constant and

Re is the electroacoustic wavelength. The trough and peak
pairs are labeled with the appropriate N in the plots.
Physically, this says that the electroacoustic wave has

a large effect whenever the parameters of the plasma layer

are such that the electroacoustic wave may set up a stand-

ing wave of length N le/z in the plasma layer. For other

84

points the inward and outward traveling electroacoustic
waves are out of phase and thus the total fields due to
the waves are small. When losses are introduced into the
plasma, the effects due to the electroacoustic wave are
smaller because the standing wave pattern set up in the
plasma layer will attenuate as one nears the outer sur-
face. For ye/w = 0.01, effects due to standing waves of
integer order in length are lost but standing waves of
half integer order in length still have an effect. In the
case of ye/w = 0.1 all effects due to the electroacoustic
wave are damped out. It is observed that the regions on
these plots that cannot be related to the electroacoustic
parameters are largely unaffected by the varying collision
frequencies and therefore we assert that these results are
due mainly to the electromagnetic wave. The effect of the
plasma on the electromagnetic wave is to reduce the radi-
ated power, the input conductance, and input susceptance as
the antenna driving frequency, w, is reduced to the
neighborhood of the plasma frequency. After the plasma
frequency exceeds the antenna frequency the radiated power
and the input conductance build up to a value larger than
the corresponding free space value. This phenomenon has
been called enhanced radiation [1, 2, 3]. The input sus-
ceptance for wez/w2 > 1.0 shows some odd effects which we
cannot attribute to any physical phenomena and may well be

due to numerical problems. No further attempt was made to

85

find the source of the irregularities. It should be noted
that the curves for the input conductance, which can also
be considered as plots of the relative power radiated by
the spherical antenna by itself, are always greater than
or equal to the power radiated by the spherical antenna
together with the surrounding plasma layer, the difference
being the power absorbed by the lossy plasma due to
collision effects. The losses are large only when the
power radiated is affected by the electroacoustic wave.
Figures 4.4, 4.5, and 4.6 show the power radiated
from progressively larger spherical antennas surrounded
by hot, lossy (ye/w = 0.01) plasmas of varying thicknesses
as a function of weZ/wz. In these plots we again consider

2 i 2.6. By

the high frequency region, i.e., 0 i wez/w
studying the three figures for one thickness of the plasma
layer it is evident that the effects due to the electro-
acoustic wave become relatively weaker for the larger
antennas. Looking at the plots for varying plasma layer
thicknesses, particularly the a = 0.011 case, we can see
that as the plasma layer becomes thicker the effect of the
electroacoustic wave is observed in more regions and for
large values of N. Also looking at any one figure we can
see that the thickness of the plasma layer affects the
power radiated due to the electromagnetic wave. Particu-
larly for wez/w2 > 1.0 it can be seen that the thicker

plasma layer decreases the power radiated due to the

electromagnetic wave.

86

In Figure 4.7 the radiated power from a spherical
antenna of radius 0.11 surrounded by a hot, lossy
(ye/w = 0.01) plasma of thickness 0.031 is plotted as a
function of wez/mz. The region of enhanced radiation is
seen to extend from wez/w2 = 1.3 to about wez/w2 = 40.0
for the parameters chosen. In this figure the effects
due to the electroacoustic wave and to the pseudosonic
wave which will soon be discussed are ignored.

Figures 4.8 and 4.9 are comparisons of our theory with
some experimental results obtained by Lin [2] and Lin and
Chen [3]. Power radiated is plotted as a function of plasma
density for two different size antennas. The outer radius
of the plasma layer is assumed to be nearly constant at
about 7 cm. The electron-neutral particle collision fre-
quency is assumed to be 0.12 GHz in our theory and the
running parameter in each figure is the antenna driving
frequency. Good agreement between the theoretical and
experimental results is observed.

Figure 4.10 shows three curves for the power radiated
by a spherical antenna of radius 0.11 surrounded by a hot
plasma of thickness 0.031 as a function of weZ/wz. The
range of wez/wz, 28400 3 (1162/41)2 3 30400, corresponds to
a high density plasma or a low frequency antenna source.
The running parameter in the figure is the electron-
neutral particle collision frequency with the ion-neutral

particle collision frequency set equal to zero in this

87

figure only. The peaks observed in these plots are due to
the excitation of a pseudosonic wave in the plasma as

verified by the fact that peaks occur whenever

c-b=N§E—Tz-k—IT=N232E (4.2.3)
where k1 is the pseudosonic propagation constant and 1p
is the pseudosonic wavelength. It is observed that in-
creasing the electron-neutral particle collision frequency
decreases the maximum amplitude of the peaks indicating
that the electrons contribute significantly to the propa-
gation of the pseudosonic wave. Figure 4.11 differs from
Figure 4.10 only in that the ion-neutral particle collision
is now determined by equation (4.2.1). Comparison of
Figures 4.10 and 4.11 shows that the ions in the plasma
contribute more significantly than the electrons to the
propagation of the pseudosonic wave.

Figures 4.11, 4.12, and 4.13 are identical to
Figures 4.1, 4.2, and 4.3 except that the first mentioned
figures are plots of the power radiated, the input con-
ductance, and the input susceptance for low antenna fre-
quency or high plasma density, i.e., 28400 :.wez/w2 E 30400.
The main point to note is that effects due to the inclusion
of the pseudosonic wave in the theory are observed in all
three figures for values of weZ/w2 that satisfy equation

(4.2.3). Losses in the plasma layer decrease the effect

88

of the pseudosonic wave on the plotted quantities. The
physical interpretation here is that the pseudosonic wave
has an effect on the quantities considered whenever the
parameters of the plasma layer are such that a pseudosonic
standing wave of a half integer pseudosonic wavelength in
length may be set up in the plasma layer, i.e., in the
vicinity where equation (4.2.3) holds. At other points
the pseudosonic standing wave is either seriously attenu-
ated or cannot be efficiently excited. Note that the
values of the relative radiated power are always less than
or equal to the values of the relative input conductance for
corresponding loss terms, the difference being the amount
of power absorbed by the plasma layer. The input sus-
ceptance plots in Figure 4.13 show effects that cannot
obviously be related to the electromagnetic wave or the
pseudosonic wave, i.e., the effects at weZ/w2 3 28550,
29900,... . No attempt was made to interpret the effect.
Figure 4.14 is a series of plots of the power radi-
ated by a spherical antenna of radius 0.11, surrounded by
a hot lossy (ye/w = 0.01) plasma of varying thicknesses
as a function of wez/wz. Comparing the upper and center
plots, we notice that there are more peaks due to the
pseudosonic wave for the thicker layer but these peaks
are smaller in amplitude. This trend is continued in the
lower plot. Here the effects due to the pseudosonic wave

are so small that they cannot be seen on the scale used.

89

The radiated power from spherical antennas of vary-
ing radiuses surrounded by a hot lossy (Ye/w = 0.01)
plasma which is 0.031 thick as a function of weZ/w2 in
the low frequency region is shown in a series of plots
in Figure 4.15. The only effect to be noted here is that
the relative radiated power is greater for the smaller

antennas.

In Figures 4.16, 4.17, and 4.18 we show the radi-

ated power, the input conductance, and the input sus-

ceptance of a spherical antenna of radius 0.11 surrounded

by a hot lossy (ye/w = 0.01), plasma of thickness 0.031
plotted as a function of the dielectric sheath thickness
in Debye lengths. Plots are given for (1182/11)2 = 0.31,
0.85, and 1.5 in the high frequency region and for

w 2/(1)2 = 29655, 29700, and 29900 in the low frequency

region. w 2/w2 = 0.31 and 29655 represent points where

e

the electroacoustic and the pseudosonic waves, respec-
tively, contribute significantly to the quantities con-
sidered. The other values of wez/w2 plotted are those
where the radiated power is due mostly to the electro-
magnetic wave. These figures indicate that the sheath
thickness has very little effect on the power radiated
and the input conductance for the system that we are
considering. For the input susceptance the high fre-
quency curves also show very little change due to the

varying thickness of the dielectric sheath. The low

frequency plots for the input susceptance do show a

90

considerable variance with changing sheath thickness. No
further interpretation of these results will be attempted
at this time.

In Figure 4.19 the radiated power from a very small
antenna is plotted versus wez/wZ. The antenna is assumed
to be of radius 0.000671 and the plasma layer thickness
is 0.000041. The relative radiated power is given for
four collision factors, i.e., ye/w = 0.0, 0.01, 0.1,
and 10.0. This case is of interest if we are Operating
an antenna five meters in radius which is surrounded by a
plasma layer, 20 cm thick, with an electron density
equivalent to fe z 0.3 GHz' In this case, if we operate
the antenna at a frequency f = 6 x 105, Figure 4.19 pre-
dicts that the power radiated will be much greater than
if the same antenna is operated without the plasma layer
around it for all but the highest collisional losses. The
plasma layer in the above described circumstances is
approximately one-half of a pseudosonic wavelength thick.

Figure 4.20 is for the same situation as Figure
4.19 except the antenna is much larger. The result is

that the plasma layer affects the radiated power very

little except where equation (4.2.3) holds for N = l.

4.3 Conclusions
From the discussion of the numerical results we can

draw four conclusions:

91

The propagation of a pseudosonic wave through

a plasma layer covering a spherical antenna can,
under proper conditions, strongly affect the
radiated power, the input conductance, and the

input susceptance of the antenna.

The propagation of an electroacoustic wave
through a plasma layer covering a spherical
antenna can, under proper conditions, strongly
affect the radiated power, the input con-
ductance, and the input susceptance of the

antenna.

As has been discussed by others [1, 2, 3], the
prOpagation of the electromagnetic wave through
the plasma layer can, under the prOper con-
ditions, strongly affect the radiated power of
the antenna and, in addition, as we have shown,

the input conductance and susceptance.

The thickness of the dielectric sheath has
little effect on the radiated power and input
conductance of a spherical antenna surrounded

by a layer of hot lossy plasma.

Antenna Radiated Power—"-

 

 

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.2 ’5“).

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Figure 4.1. Theoretical power radiated by a
spherical antenna in a hot (V /C = 0.01) plasma as
a function of plasma density for various collision
frequencies.

O

10 Log (G/G )—*-

 

 

 

 

 

 

 

 

 

 

 

 

v
8 §=0.1 b=a+ e c=b+0.031
: Jim
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Figure 4.2. Theoretical input conductance of a
spherical antenna in a hot (Ve/C = 0.01) plasma as a
function of plasma density for various collision
frequencies.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

v
% = 0.1 b = a + e c = b + 0.031
/3me
N=5 N=4 N=3
+0.02II ' Ye = 0 o
if I 7::
8 N=2l
2 +0.01IL , 71
v \ _ = 000 ‘—
N41 (0
g 0'0 = o " 2‘0 4
II I w.2/“’2'“
3‘-0.01 v I
U
m I
{73
-0.02 v ' n
Y
T+0.02.. l _E.= 0,01
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l
7' 0.0 . .
m ' 210 “7‘
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(5 e
4) —0001
Q.
0)
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8 -0 02
m .
0.02 4 I Ye _
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8 I
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G)
o 000 . ' 3 ‘ ‘
c 1 10 1‘
:3 1141.24“
8‘-0.01 u ,
O
m I
3
-0002 8’ l

 

Figure 4.3. Theoretical input susceptance of a
spherical antenna in a hot (Ve/C = 0.01) plasma as a
function of plasma density for various collision fre-
quencies.

 

 

 

 

 

 

 

 

V
c % = 0.01 b = a + e
0 /3w
"-l e
‘5 (db)
:6 +10.
m
u
g" ‘ 2:0 2 2*
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m ‘10‘
0
H
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. +10 = 0.25
g X I
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20.0 2 T
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Figure 4.4. Theoretical power radiated by a small
spherical antenna in a hot (Ve/C = 0.01), lossy (ye/w = 0.01,
yi/w = 0.0000584) plasma as a function of plasma density for
various thicknesses of the plasma layer.

Antenna Radiated Power __4»

Radiation

if

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0

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n.
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96 V

 

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1

  
   
  

 

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4.
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N
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Figure 4.5. Theoretical power radiated by a
spherical antenna in a hot (Ve/C = 0.01), lossy
(Ye/w = 0.01, Yi/w = 0.0000584) plasma as a function
of plasma density for various thicknesses of the

plasma layer.

Antenna Radiated Power ——>

e Radiation

i

e Radiation

i

i

Free-spa
I
I-'
o

 

Free-spa
I
I-'
o

I
N
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+23
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Radiation
CV

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N
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w
o

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Free-spac
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Figure 4.6.
large spherical antenna in a hot (Ve/C

97

 

1p

4

 

Theoretical power radiated by a
0.01), lossy

(Ye/w = 0.01, yi/w = 0.0000584) plasma as a function
of plasma density for various thicknesses of the plasma

layer.

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Figure 4.10.
spherical antenna in a hot (Ve/C = 0.01) plasma as
a function of plasma density for various electron

collision frequencies with the ion collision frequency
set equal to zero.

 

  

Theoretical power radiated by a

Antenna Radiated Power——)-

(db)

Free-space Radiation
I + +
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O O O C

I
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(db)

+
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810

Figure 4.11.
spherical antenna in a hot (Ve/C

I 29500

30000

N=806

F‘-___—"_"~__.pI"_"~x___./"“-~__./f‘—“\___,

Theoretical power radiated by a
0.01) plasma as a

function of plasma density for various collision

frequencies.

 

N=806

10 Log (G/Go)-——+-

 

 

 

 

 

 

 

 

 

 

 

 

v
§ = 0.1 b = a + /_° c = b + 0.031
30)
e
m N=809 N=808 N=807 N=806
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8 T; = M
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I:
8
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8 28500 29000 f 29500 30000 __+.
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l l
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m -40(L

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3 Y1 — 0 000584
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8
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8 28500 29000 f 29500 30000
9 _ 2 2‘
$¢_20‘_ wl - m we /w
0)
(D
H
in '40-“-

 

Figure 4.12. Theoretical input conductance of a
spherical antenna in a hot (Ve/C = 0.01) plasma as a
function of plasma density for various collision
frequencies.

Susceptance (mhos ) —> Susceptance (mhos.)—+-

Susceptance (mhos) -—>

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

% = 0.1 = a +
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+20+ IN=3 w
+10IP
0 it 00 25000
V\ 062/02
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40+- l 1
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+10"
0 : 44 L
22500 29 0 30
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00
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» 7; = 0.000584
0 28500 29000 29500 30000 082/02
“Illl" “IIIIP" “IIII" ‘“IIIIl"
'10" w
‘20+-

 

Figure 4.13.
spherical antenna in a hot (Ve/C

Theoretical input susceptance of a

= 0.01) plasma as a
function of plasma density for various collision
frequencies.

Antenna Radiated Power-——*'

105

 

 

 

 

 

 

a Ve
I = 0.1 b = a +

(db) /§we
g: +40-1L
O
:3 c = b + 0.03A
(6 +201-
:8.
g m 3902—»
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g N=808 N=807
Q:
m
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0)
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In

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-—+- 0 if %7 0 :
8 28500 29000 f’ 29500 30000
to
3 -200 N=2699 ”i ‘ “ N=2686
0
(D
H
m _4OWMWW/

(db)

 

g +40A
:3 c = b + 0.25A
(6 +20.-
?8 2 2
to w /w -—I>-
0.1 e
-> (3 f *1 : :

28500 29000 1‘ 29500 30000

0). = U.)
1

 

Free-space
l
N
o.
%

 

spherical antenna in a hot (V

(Ye/w

Figure 4.14.

= 0001' Yi/w =

Theoretical power radiated by a
/C = 0.01), lossy
0.0000582) plasma as a function

of plasma density for various thicknesses of the plasma

layer.

Antenna Radiated Power-—'>

 

 

 

 

 

 

      

 

 

 

Free-space
l
N
o

    

Figure 4.15.
spherical antenna in a hot (Ve/C = 0.01), lossy (ye/w =
0.01, yi/w = 0.0000584) plasma as a function of plasma
density for different size antennas.

N=809

v
b = a + e c = b + 0.033
/§w
e
(db)
+40 a _
g X — 0.01
.2
3 +20 N=809 N=808 N=807 N=806
, /\
£2
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Q)
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+40 0 3 = 0.1
c A
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g. N=809 N=80 2_ 806
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8
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m . e
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28500 29000 f 29500 30000
N=808 2 N=807 N=806

Theoretical power radiated by a

Antenna Radiated Power _—‘*

Antenna Radiated Power—>

107

 

 

 

 

a Ve
T = 0.1 b = a + S C = b + 0.03%
we

(db)
G+lO+L
.9.
u ______——————""'—_
3,3 2 2 —/
p we /w = 1.5
g s -—9-
T 0 5 a as 12 715
0
§' Inez/02 — 0.31—_/
l 2 2
:_10II~

 

 

(db)

 

 

2 2

 

 

 

 

we /w = 2970
'30-”-
w 2/02 _ 29900—\
e ____~__,..a—~""‘
f S —-)-
3 ' 0 i2 1?

Figure 4.16. Theoretical power radiated by a
spherical antenna in a hot (Ve/C = 0.01) lossy (Ye/w =
0.01, yi/w = 0.0000584) plasma as a function of dielectric
layer thickness for various plasma densities.

10 Log (G/Go) —->-

10 Log(G/GO) ——>

108

 

 

 

 

 

 

 

 

 

 

v
§=0.1 b=a+ es c=b+0.03l
/§w
e
8(de
. 2 2
g+10* we /w - ”X #
*3 2 2
a w /w = 0.31
e
g /—
ELa-o : ii ? i i
8 3 6 9 12 .15.
(U S
8‘ w 2/w — 0 85
g e -_\\\‘L
)4
0.. -10 0
062/02 = 29655
G)
8 ..
m'+20
4.)
8
'2 w 2/02 = 29700
0 e
U
m +10 .. //
U
m #fi/
3. \— Inez/02 = 29900
<1:
G)
H s——+-
[u
-—"’ 0 5 4. ‘r : 1L
3 6 9 12 15

spherical antenna in a hot (V

Figure 4.17. Theoretical input conductance of a
C = 0.01) lossy

/
(Ye/w = 0.01, Yi/w = 0.0000583) plasma as a function

of

dielectric layer thickness for various plasma

densities.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

% = 0.1 b = a + e s c + b + 0.03A
+0.03” /§we
+0.020
weZ/w = 0.31——\\\
‘6 we /w — 0.85-——\
g o : . . -
~, 3 6 9 12 S__15
§-00014b
m 2 2
.p w /w =l.5—-.
3-0.02.1» e
O
U)
:3
m—0.03" *\\\\-\‘\\\\\\
062/02 = 29700
2/02 = 29900 w 2/02 = 29900
+10v 6

J
T J
a; s——+-
U)
o o #5 % <‘__~__‘: fi~
g 3 5 9 -::;__~;M‘_
" r‘ 2 2 _
m we /w - 29655
2 \
B
m -10¢ w 2/02 — 29700
8 k 7 e ‘
U)
:3
U]

 

 

Figure 4.18. Theoretical input susceptance of

a spherical antenna in a hot (Ve/C = 0.01) lossy
(ye/w = 0.01, yi/w = 0.0000584) plasma as a function
of dielectric layer thickness for various plasma
densities.

Antenna Radiated Power—_y.

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

% = 0.00067 b = a + e ' C X b = 0.00004

,(db) /§we
T3 +60% f
m [+400 Ye/w = 0,0 N=l 8
(D
0

+2 0 4
§ 0 yi/w = 0.00 weZ/w2—>
I-+- 0w‘==t# : - : : 1
8 10° 101 10 10 1 4 105 10 7 108
H -200
In

-40)
,(db)
'3 +60" Ye/w = 0.01
“ -+400
0

. = . 584

8 .+200 Yl/w 0 0000
8‘
3 10° 101 10 10 1
u -20<>
[LI

-407
, (db)4

+60
2 ) ye/w = 0.1

+40"
m _-
3 .+200 yi/w - 0.000584
8‘
I—r'O 3 ¢
8 10° 101
a 1—200

410+
, (db) =
'3 +60(* ye/w 10.0
m
8 +401 yi/w = 0.0584
(i +20‘
?-r-o :
h‘ -20

-40‘r

 

Figure 4.19. Theoretical power radiated by a
small Spherical antenna surrounded by a thin layer of
a hot (Ve/C = 0.01) plasma as a function of plasma
density for various collision frequencies.

Antenna Radiated Power-——+-

111

 

 

 

 

 

 

 

 

 

a _ __ C - b =
7 — 0.1 b — a + “X’- 0.00004

. /3w N=l

(db)
2 +40"
“ y /w = 0.0
0.) +201 e 2 2
O
a Yi/w = 0.0 we /w ,
m j l A A A L A
5+ 0 j '1 f2 13 -4 '6 r8
3 1.0 10 10 10 10 10 10 10 0
m —20"

-4qu
6(db)

+ v- : .
g 40( Ye/w 0 01 N 1
8 +200 71/0 = 0.0000584 2
g. 0) /U) -—-D-
m A e_A
I-b- G % + i # f v .
3 1.0 101 102 103 104 1 5 106 10 08
E -20"

-40«
-<db) = 1
*5 “10+ ye/w 0.

. = . 84

8 +20“ Yl/w 0 0005
m 2
g, =1 we /w -—->-
m L j._.
I-u- o : c t 4 5 q t .
3 1.0 101 102 103 104 105 106 10 08
“ -20
‘H 1-

~40?
'(db)
'0
‘2 +40% Ye/w = 1.0
8 +204 Yi/w = 0.0584 2 2
O. we /w -—-p-
U)
l—F- G 3 4. : : : % ¢ :——-0-
8 1.0 101 102 103 104 105 106 107 108
H
m -20"

-40¢

 

Figure 4.20.

Theoretical power radiated by a

spherical antenna surrounded by a thin layer of a hot
(V /C = 0.01) plasma as a function of plasma density

f0?

various collision frequencies.

PART II

RADIATION OF A CYLINDRICAL ANTENNA IN A

COMPRESSIBLE PLASMA INCLUDING THE

EFFECT OF AN ELECTROACOUSTIC

WAVE

CHAPTER V

INTRODUCTION

Advances in space technology in the last few years
have led to an increased utilization of antennas, Operated
in a plasma medium, as ionospheric probes to determine the
state of the plasma, and for communication purposes. Thus
it is important to be able to predict the effect that a
plasma medium will have on the electrical properties of
an antenna. In this part of the dissertation the electri-
cal properties of a cylindrical antenna immersed in a hot

lossy plasma of infinite extent are studied.

5.1 Historical Development

There is an abundance of literature dealing with the
effect of a plasma upon the operating characteristics of a
linear antenna. Varying assumptions have been made in order
to simplify the problem. These assumptions typically in-
volve one or more of the following; neglect the sheath
region entirely [18-32], neglect the temperature effects
of the electrons and ions [28,29], neglect the ions [12,
18, 20-42], neglect losses [18-22, 24-27, 30-32, 34-37,

39, 41], assume a sheath profile [40, 41], replace the

112

113

sheath region by an electron free sheath region [22, 32,
12, 34-38], assume a short filamental antenna with either
a sinusoidal or triangular current distribution [l9-21,
23, 25, 26, 33].

Prior to 1961, the theoretical considerations of the
influence of a plasma on the characteristics of a linear
antenna neglected collisions and the temperature of the
plasma. Thus the plasma was regarded as a cold, nonlossy
medium, which is equivalent to regarding the plasma medium
as a lossless dielectric.

In 1961, King, Harrison, and Denton [28] solved the
problem of a short, linear antenna immersed in a cold,
lossy plasma. In the same month, Hessel and Shmoys [18],
presented their paper dealing with the behavior of a
Hertzian dipole operated in a warm, lossless plasma. Their
results indicated, for the proper frequency range and
acoustic velocity, a large acoustic wave off the ends of
the antenna, in addition to the usual electromagnetic wave
off the sides of the antenna.

In 1963, Whale [43] observed experimentally a larger
real part of the input impedance of a short antenna used
as an ionospheric probe than predicted by using the cold,
lossless plasma theory. He attributed this to the electro-
acoustic wave.

Chen [23], in 1964, studied the problem of a thin

cylindrical antenna of finite length with a sinusoidal

114

current distribution in a hot plasma. Balmain [33]
treated the problem of an electrically short antenna with
a triangular current distribution immersed in a hot
plasma. Both papers gave the antenna resistance only for
we/w < l where we and w are the plasma and antenna fre-
quencies. Later, Kuehl [25, 26] studied the same problem,
but solved the Boltzmann equation instead of using the
simpler linearized hydrodynamic equations. An interesting
result of his work is the existence of an antenna re-
sistance for we/w > 1. The antenna reactance was not
determined in these papers. Meltz, Freyheit and C. D.
Lustig [44] investigated an infinite cylindrical antenna
covered by a set of coaxial plasma layers, based on a
variational formulation. They were able to deduce both
the antenna resistance and the antenna reactance for a
wide range of we/w.

Compared to the wealth of theoretical papers pro-
duced, only a few reports have presented experimental re-
sults dealing with the electrical properties of linear
antennas in plasmas. Some have measured the impedance
of short dipoles in the ionosphere [43, 45, 46] and in a
laboratory plasma [47]. More recently, impedance measure-
ments of relatively long antennas in laboratory DC impulse
discharges [48, 49, 52] and in an RF discharge [50] have
been reported. Also, measurements of the current distri-
bution on relatively long monOpoles in sustained laboratory

plasmas has been reported [51, 59].

115

5.2 Outline of the Investigation

To the best of our knowledge, no theoretical paper
which accurately determines the complete input impedance
of a cylindrical antenna of finite length in a hot lossy
plasma has been published.

In Chapter VI, an integral equation for the current
on a linear antenna in a hot lossy plasma is formulated.
The antenna is assumed to support a two-dimensional sur-
face current so that antennas of diameters comparable to
the electroacoustic wavelength may be considered. An
assumed form for the current distribution is introduced
and then knowing the impressed voltage, the zeroth order
input impedance is derived. The sheath is not considered.

In Chapter VII, numerical solutions for the zeroth
order input impedance and current distribution are calcu—
lated and compared to experimental results. These
solutions are also compared to theoretical and experi-

mental results obtained by other workers.

CHAPTER VI

THEORETICAL DEVELOPMENT OF THE INTEGRAL EQUATION
FOR THE CURRENT ON A LINEAR ANTENNA IN A HOT
LOSSY PLASMA AND THE ZEROTH ORDER SOLUTION

FOR THE CURRENT DISTRIBUTION AND

INPUT IMPEDANCE

The objective of this chapter is to derive an
integral equation for the current distribution on a gap-
excited linear antenna immersed in an infinite, isotrOpic,
homogeneous, compressible plasma. The known function in
the integral equation will be the tangential electric
field intensity on the surface of the antenna. This
function is known from the boundary condition on the
tangential electric field on the surface of a perfect
conductor and the idealization of the excitation (an
assumed constant electric field in the gap). A zeroth
order current distribution is proposed and a zeroth order

input impedance is obtained.

6.1 Geometry and Basic Equations
The geometry of the linear cylindrical antenna is

shown in Figure 6.1. The antenna is taken to have a

116

117

 

Figure 6.1 . A cylindrical antenna of radius a and
half length h immersed in an unbounded hot lossy plasma.

118

half length h and radius a. It is assumed to lie along
the z axis of a cylindrical coordinate system and to be
excited at its center (2 = 0) by a harmonic voltage V with
angular frequency w. The gap of width 2A in the cylinder
at z = O is assumed to be very small so that 2A-+0,
corresponding to a point (or, what is termed, a slice)
generator with rotational symmetry. The antenna is
assumed to be constructed of a perfectly conducting screen
or mesh-like material, with the spacing of the conductors
being much less than the smallest characteristic dimensions
of the system, so that the surface is penetrable to
electrons and ions. This assumption eliminates the need
to consider the formation of a sheath and to impose a
boundary condition on the particle velocities at the sur-
face of the antenna making the problem tractable.

The antenna is immersed in an unbounded weakly
ionized gas which consists of equal numbers of free
electrons and singly ionized positive atoms and a much
larger number of neutral atoms. It is assumed that the
generator frequency w is sufficiently high to neglect ion
motion so that the ions act as a uniform background of
positive charge.

The basic equations that govern this system (source
and plasma) are Maxwell's equations and the linearized

hydrodynamic equations (see Chapter I),

119

V x §(r) = -jwuo§(§)

V x §(r) = gs(r) - enoy(§) + jweo§(§)
EOIV ° E({)] = 05(5) - en(§)

V - H(r) = 0

nOIV ° y(§)] + jwn(§) = 0

(jw + Y)y(§)

II
I
Elm
I!!!
J?
I
'0
<
:3
H

v - 95(5) + jwos(§) = o

where

electric field

magnetic field

permeability of free space

permittivity of free space

charge on the electron

mass of the electron

equilibrium electron density
perturbed electron density

source current density

source charge density

average perturbed electron density

electron-neutral particle collision
frequency

(6.1.1)

(6.1.2)

(6.1.3)

(6.1.4)

(6.1.5)

(6.1.6)

(6.1.7)

120

v = )/§£E-= average thermal velocity of the
o m
electrons
where k is Boltzmann's constant and T is the average

temperature of the electrons. The suppressed time

jwt

dependence is assumed to be of the form e and

rationalized MKS units have been used throughout.

6.2 Integral Equation Formulation

The curl of equation (6.1.2) is
V x V x H = V x J - enO V x v + jwt»:o V x E. (6.2.1)

Using the vector identity V x V x R = V(V-R) - V23,

equation (6.2.1) becomes

V(V°H) — VZH = V x J8 - enOV x v + jweoV x E. (6.2.2)

Substituting the curl of equation (6.1.6) and equations

(6.1.1) and (6.1.4) into equation (6.2.2) gives after

 

simplification,
w 2 w 2y
V2H + wzeopo l - 2e 2 - j -—27——§— H = - V x gs
~ w +y w(w +y ) ”

(6.2.3)

where we2 = ezno/meo. If we define a complex permittivity

as

2 2
we . “’e 7
5:601‘7—7’3—77, (6.2.4)
w +7 w(w +y )

equation (6.2.3) can be written as an inhomogeneous wave

equation of the form

VZH + keZH = - V x gs (6.2.5)
where

kez = 02005. (6.2.6)

Consider the equation

(v2 + ke2)§(§) = - €S(E)° (6.2.7)
It can be shown (by straight-forward, but tedious,
vector manipulations) that

(72 + kez) v x g = — v x gs. (6.2.8)

Comparing equation (6.2.8) with equation (6.2.5) and
recalling that equation (6.1.4) holds for all space, we

can say that H must be the curl of some vector field A or

H = V x A. (6.2.9)

~

The solution for the inhomogeneous wave equation,
equation (6.2.7), may be obtained using standard pro-

cedures [53, S4, 55]. Thus

122

 

A(r) = 4" f JS(r')Ge(r,r')dV' (6.2.10)
.... V1 .. -1.
where
-jke|5-§'|
G (r,r') = 6 (6.2.11)
9"” IE‘E'I

and r is a vector from the origin to the field point,
r' is a vector from the origin to the source point, and

~

the integration is over all source points. Finally

3 (r')Ge(r,r')dv'. (6.2.12)

Taking the divergence of equation (6.1.6) and

solving for V - E yields

V
v - E = - m(jw+y)v - v - 3 -9— V2n. (6.2.13)
~ 6 ~ mno

Substituting equation (6.2.13) into equation (6.1.3) and

using equation (6.1.5) yields upon rearrangement

 

 

2 w2 we2 I we2 S
V n(r) + -—§ 1 ‘ .7? - w n(E) = ' 20 (E)
v w ev
o 0
(6.2.14)
or
w 2
Vzn + kpzn = - e2 pS (6.2.15)

123

where
2
2 w
2.4» __e_--1
kp -;71 (DZ 3 (1) . (6.2.16)
0

Equation (6.2.15) is an inhomogeneous wave equation for
the average perturned electron density n(r) which has a

standard solution

 

2
(A)
n(r) = -—E——§ f pS(r')G (r,r')dV' (6.2.17)
~ 4nev V ~ p ~ ~
0
where
-Jk lr-r'l

e p ” ~
G (r,r') = (6.2.18)
P“ lr-r'l

where r and 5' and the range of integration are the same
as in equation (6.2.10). Using the continuity equation
for the sources, equation (6.1.7), equation (6.2.17) can
be written as

3062

n(r) = —————5— IV' - 85(5')Gp(§,§')dv' (6.2.19)

~ 4nev w
o

where V' - 95(5') is the divergence of gs with respect to
the source coordinates.

The electric field E(r) at any non-source point can
be derived by eliminating y from equations (6.1.2) and

(6.1.6) and rearranging to give

124

6V2
0

w€(w-jy) Vn. (6.2.20)

 

E=-.-];—VXH+
~ ng ~

V x H can be calculated by taking the curl of

equation (6.2.9)

x v x A = V(V°A) - 72A. (6.2.21)

~

<1
X
ICE
ll

<

Equation (6.2.7) in a source free region implies that

-V A = k A (6.2.22)

so equation (6.2.21) becomes

7 x H = V(V°A) + kez A. (6.2.23)

~

Using this result, equation (6.2.20) can be rewritten as

2
ev

_ _ _1_ . 2 0
E w€[V(V 1}) + ke 1}] +mVn (6.2.24)

where the vector field A(r), the perturbed electron

density n(r), and the complex permittivity 5 are given

by equations (6.2.10), (6.2.17), and (6.2.4), respectively.
The current density on the surface of the cylindri-

cal antenna can be represented as

J (r) = 28(r-a)12(z)/2na (6.2.25)

where 2 is the unit vector in the z direction and 6(x) is

the Dirac delta function. Very near the antenna surface

125

the tangential component of E(r) can then be written as

Ez(z)§ where

2 2
. a A (z) ev
Z 2 o 3n(z)
= - + + -————
Ez(z) wE -—;;7—— ke Az(z) w€(w-jy) 32
(6.2.26)
where using equation (6.2.25)
1 h l " e-JkeR
= — ' __ ___—.— I I
Az(z,¢) 4" i Iz(z ) 20a f R ad¢ dz
- -0
(6.2.27)
and
jwez h 312(2') 1 w e'JkpR
MM) - 2 f 27' 27a ’ T381" d2"
41revo w -h -N
(6.2.28)

where R is the distance between the source point r‘ and
the field point r. From Figure 6.2 it can be seen that R

can be expressed as

R = If - r'l = {2a2[l-cos(¢-¢')] + (2-2')2}%
(6.2.29)
where the primed coordinates are the source points on the
antenna surface and the unprimed coordinates are the field

points very near the surface of the antenna. Since the

antenna is rotationally symmetric, A(r) and n(r) cannot

126

 

 

 

 

 

 

Figure 6.2. Source and field points on or near
the surface of the antenna.

127

depend on 0, so we can arbitrarily set 0 equal to zero.

Therefore,

R = [2a2(l-cos¢')2 + (z-z')2]%

[(2a sin 9%)2 + (z—z')2]l‘ . (6.2.30)

On the conducting surface of the antenna, the
electric field is zero and in the gap at z = 0, E2 = -V/2A
or, in general, on the cylindrical surface, from

-h i z i h and at r = a we can say that

Ez(z) = -V0(z). (6.2.31)

Combining equations (6.2.26) and (6.2.31) gives an

integral differential equation of the form

 

 

32Az(z) 2 evo2 3n(z)
_v6(z) = - 3% ——;;§—— + ke Az(z) + wE(w-jy) 32
(6.2.32)
or
2
—————a AZ(Z) + k 2A (z) - f(Z) (6 7- 33)
3 2 e z — ‘ °
2
where
2
f _ - . evo 3n(z)
(Z) - -J(L)€VO(Z) " J w-jY 32 (6.2.34)

128

which is valid on the surface of the antenna. The
solution to the inhomogeneous differential equation,

equation (6.2.33) is given by

P

C
142(2) = AZ (2) + AZ (2) (6.2.35)

where Azc(z) in the complementary function and AZP(z) is
the particular integral. The complementary function is
the solution to the homogeneous differential equation and

is
A c(z) = B sin k z + C cos k 2. (6.2.36)
2 e e

From symmetry requirements, i.e., 12(2) = Iz(-z), B must
equal to zero. By the method of the variation of param-
eters, the particular integral for an equation of the form

of equation (6.2.33) is given by

A P(z) = 1L [2 f(z') sin k (z-z')dz' (6 2 37)
z ke 0 e ° °

Thus

z
A (z) = l; f f(z') sin k (z-z')dz' + C cos k z
z ke 0 e e

(6.2.38)

where f(z) is given by equation (6.2.34) and C is an
arbitrary constant yet to be calculated.

Writing equation (6.2.37) out gives

129

Z
AZ (2) = — j “Li-‘5 I 6(2')sin ke(z-z')dz'
e 0

2
ev z
_ - 0 8n(2') - ..- .
J kezw-JY) é azT— Sln ke(2 2 )dz

. V .
= -j 8%; Sln ke |z|

ev 2 z ,
- j o f EEIETL sin k (z-z')dz'.
ke(w-jy) 0 32 e

(6.2.39)

The integral on the right—hand side of equation (6.2.39)

can be integrated by parts to give

2

f
0

3n(z')

. _ | I = I ° _ '
.__§ET— Sln ke(z 2 )dz n(z )31n ke(z Z )

z
I _ I I
+ Re 3 n(z )cos ke(z 2 )dz

2
— _ ° I _ I I
- n(0)31n kez + Re 6 n(z )cos ke(z 2 )dz .

(6.2.40)

I I
But if we define R' = [z 2 + (2a sin $—)2]k

wez h 312(2') 1 " e-jkpR'
. _ ' '
n(0) J I 32, 20a f ‘——§T—- ad¢ dz

“'1T

 

= 0 (6.2.41)

130

since the integrand is an odd function over the interval

integrated. Thus

P _ _ . w V .
AZ (2) - J 3%; Sln ke |z|
2
evo z
- j 5:3? 3 n(z')cos ke(z-z’)dz' (6.2.42)

Substituting n(z) from equation (6.2.19) into the integral

on the right-hand side of equation (6.2.42) yields

2
f n(29cos k (z-z')dz'
0 e
we2 2 1'! 312(2")
= j __T f COS ke(z-z') f T
41revo w 0 -h
X GP'(z',z")dz"dz' (6.2.43)
where
l " e-jkpR
I I II _ _ II
where

0| 2 l:
R" = [(z'-z")2 + (2a sin %—)] .

Noting that

131

11 312(2") z"=h
f TGP'(Z'IZ")dZ' = Iz(zII)GPI(zI'zII)
-h z"=-h
h a
_ II I I II II
_fi Iz(z ) 33w GP (2 ,2 )dz
h a
_ _ II I I II II
based on the assumption that Iz(z"=:h) = 0. Equation

(6.2.43) can be rewritten as

2
f n(z')cos k (z—z')dz'
0 e

2

we 2 h
= — j ——————— f cos ke(z-z') f 12(2")
47IevO w 0 “h

x'E‘“ GP'(z',z")dz"dz'

w h z

. e
= - 3 -—————— f I (2") f cos k (z-z')
4wev02w -h z 0 e

x g—w-GP'(Z',Z")dz'dz" (6.2.46)

The interchange of the order of integration of equation
(6.2.46) is legal because Iz(z"), cos ke(z-z'), and
GP'(z',z") are continuous functions of z' and z" in the

range of integration. Since

132

3 g u a I II
537 GP (z',z ) = - 557 GP (z',z ) (6.2-47)

equation (6.2.46) can be expressed as

2
f n(z')cos k (z—z')dz'
0 e

2

 

we h z
= 3 f 12(2") f cos ke(z-z')
41revo w -h 0
X a U U H II 3
32. GP (2 ,2 )dz dz
we2 h z
= j I 12(2') f cos ke(z-z")
4nev w -h 0
o
337 GP'(z",z')dz'dz". (6.2-48)

Then Az(z) can be written as

= _ . w§V .
Az(z) 3 2ke Sln ke |z| + C cos kez

w 2 h

e I I I I
+ 4flw(w-jy) _g 12(2 )K (2.2 )dz (6.2.49)

 

where

z
I I _. _ II a I II I II
K (z,z ) - f cos ke(z z ) 53"GP (z ,2 )dz (6.2.50)

0

Combining equations (6.2.27) and (6.2.49) and re-

arranging yields an integral equation for 12(2) as follows

133

h

f I (z')K(z,z')dz' = - j 9E! sin k Izl
-h 2 2k e
e
+ C cos kez (6.2.51)
where
2
I._._1_ I I_ e I I
K(z,z ) — 4" Ge (2,2 ) 5T5:§7T K (z,z ) (6.2.52)
where
1 -n e-jkeR
I I____ I
e (z,z ) - 2n i R d¢ (6.2.53)

where R = [(z-z')2 + (2a sin ¢'/2)2]% and K'(z,z') is
given by equation (6.2.50).

Now we must solve for the arbitrary constant C.
Evaluating equation (6.2.51) at z = h and solving for C
yields

h

C = sec k eh I Iz(z')K(h,z')dz'
-h

+ j 2kg tan ke h

(6.2.54)

Substituting equation (6.2.54) back into equation
(6.2.51) and rearranging gives us the final form for a
Hallén type integral equation for the current 12(2) as

follows:

134

h
f I (z')[cos k h K(z,z') - cos k z K(h,z')]dz'
-h z e e
= j g—E‘l sin ke(h-IZI) (6.2.55)
e
where
2
. _ l . . we 2 ..
K(Z,Z ) — EGG (2,2 ) - mil.) COS ke(z-z)
x 3 G l(z" zl)dz"
az" P '
2
_ 1 I I we k II
_ 3? Ge (z,z ) - 575:3;7 cos e(z-z )
z"=z
X G [(zli'zl)
P z"=0
z
__ ° _ II I II I II
ke é Sln ke(z 2 )GP (2 ,2 )dz
or

- cos kez GP'(0,z')

z
_ ' _ II II I II
ke g Sln ke(z z )GP(z ,2 )dz .

(6.2.56)

135

6.3 Zeroth Order Current and
Input Impedance

 

 

The results of Section 6.2 can be summarized as
follows: the integral equation for the antenna current is
h

—£ Iz(z')[cos keh K(z,z') - cos kez K(h,z')]dz'

= j Y§£ sin ke(h-|2l) (6.3.1)
e

N

where

2
w
e

I _ 1 I I _ I I
K(z,2 ) — fiGe (2,2) W613 (2:2)

2
_ I I _ ° _ II
cos kez GP (0,2 ) ke 6 Sin ke(z 2 )
x GP'(2",z')dz€] (6.3.2)
where
1 n e-JkeR
I I _. __ I
-jk R
G'(z z')—--1—fTre p 616' (634)
P I _ TI -11, R o o

where

. k
R = [32-2') + (2a sin %r)€] (6.3.5)

136

 

w 2 w 2y
2 2 e

k = w u e l - - 3 (6.3.6)

e o o w2+Y2 w(w2+y )

2
2 w

k2=—9—1--£—- . (6.3.7)

p v 2 w2 w

o

The right-hand side of equation (6.3.1) varies with
sin ke(h-|2|) so a reasonable zeroth order current distri-
bution on the surface of the antenna which also varies with
wave number k8 is
I sin ke(h-I2I) -h{: 2 i h

Iz(z) = (6.3.8)
0 otherwise
It is noted that equation (6.3.8) for the assumed current
distribution satisfies the required boundary conditions,
i.e., Iz(ih) = O and 12(2) = Iz(-z).

If the zeroth order current distribution, equation
(6.3.8), is substituted into equation (6.3.1), the
following result is obtained

h

° _ U I
10-; sin ke(h I2 I)[cos keh K(z,z )

- cos kez K(h,z')]dz'

= j ‘-’—}“:-§-sin ke(h-I2I). (6.3.9)
3

N

137

Evaluating equation (6.3.9) at 2 = 0 and solving for ID

yields

sin k h

I=j e

0 2k h

I sin ke(h—|2'|)[cos keh K(0,2') - K(h,z')]dz'

(6.3.10)

so now equation (6.3.8) for the assumed current distri-
bution is completely specified.
The input impedance of an antenna is defined as the

driving point voltage divided by the driving point current,

 

i.e.,
_ V
Therefore
2 = V
in I sin k h
0 e
no 2 h
_ _ ' ° _ I
- 3 if ——-——-- f Sln ke(h Iz I)[cos keh

sin2 k h -h
e
x K(0,2') - K(h,z')]dz' (6.3.12)

Equation (6.3.12) is the result that we require and will

be solved numerically in Chapter VII.

CHAPTER VII

NUMERICAL AND EXPERIMENTAL RESULTS

In this chapter, numerical solutions to equations
(6.3.12) and (6.3.8) for the input impedance and current
distribution of a cylindrical antenna immersed in a hot,
lossy plasma of infinite extent are displayed for various
plasma and antenna parameters. The numerical results are
compared to values obtained for a cylindrical monopole
immersed in DC laboratory plasma. Also, the numerical
results are compared to experiments performed by Graf and
Jassby [48] and with the theoretical results of Lin and
Mei [56], and Wunsch [58] for very small antennas. It is
also shown that this theory under appropriate approxi-
mations agrees with that of Chen (23) who calculated the

input resistance by a Poynting vector method.

7.1 Numerical Techniques

 

The equation to be numerically solved is from

Chapter VI

138

139

/“° 2 h l |
z. =-j ——-——-—f sink(h-z')
1“ 5 sin2 keh -h e

x [cos keh K(0,z') - K(h,z')]dz' (7.1.1)

where

. _ l . . _ we .
K(Z,2 ) - 4—1rGe (2,2) W GP(Z,2)

- ' I
cos kez GP (0.2 )

z
... ' _ II I II I II
ke é Sln ke(z 2 )GP (2 ,2 )d2:]

(7.1.2)

where

I I e I
Ge (2,2 ) Zn f '—————— d¢ (7.1.3)

(7.1.4)

ll
:1
S
(D
w
0.:
'9

GP'(2,2')

and where no, a, ke, h, we, w, y, and R are defined in
Chapter VI. Equation (7.1.1) can be separated into five

integrals each of which must be integrated numerically.

These integrals are

h
I = f sin ke(h-|z'|)Ge'(0,2')dz' (7.1.5)

140

h

12 = -fi sin ke(h-|2'I)Ge'(h,2')dz' (7.1.6)
h

I3 = _g Sin ke(h-|2 I)GP (h,z )dz (7.1.7)
h

14 = _g sin ke(h-|2'I)GP'(0,2')d2' (7.1.8)

and

h h
I5 = -£ sin ke(h—|z'l)[g sin ke(h-z")

X GP'(2",2')d2{]dz'. (7.1.9)

Numerical integration is accomplished using a Simpson's
rule formulation which insures that the numerical result
approximates the true value by successive iteration until
the difference between two succeeding results is within
some prescribed limits. The limit in all cases except
one to be noted later is taken to be 5%.

It was found by actually performing the numerical
integration on the computer that the integral, equation
(7.1.3), could very accurately be approximated by

-jkeR'

Ge'(z,z') = E—§7—— (7.1.10)

where

R' = [(z-z')2 + a2]Li (7.1.11)

141

for the plasma and antenna parameters of interest. Using

equation (7.1.10), integrals 11’ and I reduce to single

2
integrations of a continuous function over the interval
considered. These integrals are thus evaluated directly
using the Simpson's rule technique.

The last three integrals, I3, 14, and I are not

5
handled as simply because the integration in equation
(7.1.4) must be retained in all three cases.

By a change of variables,y = h - z', I becomes

3
1 2h fl e-JkpR

13 = -- f sin k (h-Ih—y|)f ——..— dcb'dy (7.1.12)
n 0 e 0 R

where

R"

[E2 + (2a sin %%)€]k. (7.1.13)

I3 in the form given in equation (7.1.12) is easily inte-
grated using a nested Simpson's rule technique where the
inner integral over ¢' is integrated for each value of y

required in the outer integration. It is found numerically

that I3 is always four or five orders of magnitude smaller

 

than I4.
I4 can be rewritten as
_ijlll
I-th'kh'fflep d'd'
4 - F 0 Sln e( -z ) 0 Runs ¢ 2

(7.1.14)

142

where

R"' = [2'2 + (2a sin QZL)2]%. (7.1.15)

Note that at z' = 0 and ¢' = 0 the integrand is singular.
I4 can be evaluated by the method of the auxiliary
integral described in Appendix C. Using this method we

can write I4 as

_ 4 . h . -1 2a . -1 h
I4 - fl Sln keh[%- Sinh TT»+ Sinh —{]

 

a 2a
_jk Rlll '
2 h 6 sin ke(h-2')e p - sin k h cos $—
+Ff f RIII e 2 d¢'d2'
0 0
(7.1.16)
where the second integral is well behaved at 2' = 0 and
¢' = 0. Thus, the integral I4 can be easily be evaluated

on the computer.

The integral I is the most difficult to evaluate

5
on the computer because it involves a triple integral that
requires a lot of computer time. The procedures for evalu-
ating I5 are similar to those used in evaluating the first
four integrals, namely the method of the auxiliary integral
is used to remove the singularity that occurs in the
integral over 2" and the Simpson's rule integrations are
nested to obtain the required result. For IS the con-

vergence limit is required to be only 10% in order to

save computer time.

143

All numerical calculations were carried out on the

CDC 6500 computer.

7.2 Numerical Results

Except where noted, the electron velocity, Ve' is
assumed to be one one-hundredth of the speed of light, c,
and the antenna dimensions are measured in terms of the
free space electromagnetic wavelength Ao' where 10 =
2flc/w and w is the angular driving frequency. All calcu-
lations have been made for a one volt gap voltage.

The input impedance and current distribution of a
cylindrical dipole antenna as expressed in equations
(6.3.12) and (6.3.8) have been numerically calculated as
a function of the antenna dimensions and plasma parameters.
The theoretical results on the input impedances of cylindri-
cal monopole antennas of various lengths and diameters are
taken to be Zin/2 and are graphically shown in Figures
7.2, 7.4, 7.6, and 7.8. The input impedance is plotted

2 with y/w as the running parameter.

as a function of wez/w
The value of wez/w2 is directly proportional to the
plasma density when the antenna frequency is kept con-
stant and y/w is the ratio between electron collision
frequency and the antenna frequency. The current distri-
butions given by equation (6.3.8) are plotted in Figures

7.3, 7.5, 7.7, and 7.9 as a function of position along

the antenna for various values of weZ/w2 and y/w. The

144

phase, ¢, of each current distribution which is very

nearly constant along the length of the antennas con-

sidered is also given in these figures.

In the figures depicting the antenna impedances,

the solid lines represent the antenna input resistances

while the dashed lines stand for the antenna input

reactances. From these figures, the effects of the

collision frequency on the antenna input impedance can

be summarized as follows:

1.

For low plasma density (weZ/w2 < 0.4), the
antenna input resistance remains nearly con-
stant while the input reactance becomes slightly
more negative. There is little effect due to

the varying collision frequencies.

For 0.4 < wez/w2 < 0.8, the input resistance
increases monotonically as the plasma density
is increased. The antenna reactance decreases
at a faster rate than in case (1) as the plasma
density is increased. Over this range a larger
collision frequency causes a larger input
resistance and makes the input reactance less

negative.

In the range of 0.8 < wez/w2 < 1.2, there are
sharp peaks in the antenna resistance and a
change from capacitive to inductive for the

antenna reactance when the plasma frequency

145

approaches the antenna driving frequency. The
maximum value of the antenna resistance is
reduced considerably by larger collision fre-

quencies.

4. For wez/w2 > 1.2, both antenna resistance
and reactance decrease as the plasma density

is increased.

The significant findings are that: (l) the peaking
of the antenna input resistance at w ~ we, and (2) the
change in sign of the reactance at w ~ we.

The main observation to be noted from a study of the
antenna current distributions is that the amplitude of the
current is larger for greater collision frequencies, this
effect being more evident in the vicinity of w ~ we.

This result can also be determined from the impedance
plots. For weZ/wz = 0.6 and 1.2 the magnitude of the
input current depends mainly on the magnitude of the
reactance which is smaller for larger values of the
collision frequency. At wez/w2 = 0.95 the magnitude of
the input current is determined largely by the magnitude
of the input resistance which is smaller for larger
values of the collision frequency.

In Appendix D it is shown that the antenna input
resistance from equation (6.3.12) under the assumptions
of a line current flowing down the center of a very thin

(a+0) antenna immersed in a hot lossless plasma reduces

146

to precisely the result obtained by Chen [23] using a

poynting vector method. Further, the resistance under
the above limitations may be broken into a part denoted
by Re due to the excitation of an electromagnetic wave

in the plasma medium and a second part, call it R due

PI
to the excitation of an electroacoustic wave in the

plasma. Re in our theory is derived from the integrals

I and I of Section 7.1 while R arises from integrals

1 2 P
I3, 14, and IS. Figure 7.10 is a plot of Re' RP' and
Rin = Re + RP evaluated using equation (6.3.12) for a

one-dimensional current distribution. In addition to

equation (7.1.10), it is assumed that

-ij
e p

GP'(Z,Z') = ___-fi— (7.2.1)

where
1
R = [(2-2')2 + a2]1, (7.2.2)

for an antenna of half length h = 0.25).O and radius

a = 0.001).o immersed in a hot lossless plasma. Also
plotted are resistances calculated using Chen's results.
The agreement between the two theories is almost exact.

, which

Figure 7.11 is a plot of Xe, X and xin = Xe + X

PI
are defined analogously to Re' R

P

P' and Rin’ calculated
using our theory for the same parameters used in Figure
7.10. It is noted that Chen was unable to calculate

reactances using his poynting vector method.

147

Figure 7.12 is a comparison of current distributions
calculated using our theory and those measured by Judson,
Chen, and Lundquist [51] in a finite DC laboratory plasma
for an antenna of half length, h = 5.9 cm and radius,

a = 0.6l5 cm, driven with a frequency of 1.25 GHZ. The
collision frequency in the theory is assumed to be

y/w = 0.12. The agreement between our theory and their
experiment is good.

Figures 7.13 and 7.14 are comparisons of our theo-
retical input impedances with experimental values measured
by Graf and Jossby [48] for two different size cylindrical
antennas immersed in a hot lossy (y/w = 0.2) plasma. Our
theory is found to give much better agreement with their
experimental values than the cold lossy plasma theory
that Graf and Jossby used.

In Figures 7.15 and 7.16 we compare our theory to
that of Lin and Mei [36] which is limited to very short
antennas on the order of an electroacoustic wavelength
long. For these two figures the plasma is considered to
be hot (Ve/C = 0.001) and lossless. In Figure 7.15 our
theory predicts impedances very close to the values calcu-
lated by Lin and Mei for an antenna of half length

h = Ae/4 and radius a = Ae/75 where

Ve wez -%
Ae = 2n 73-1 - —7? . (7.2.3)

148

Also plotted is the input impedance of an antenna of
dimensions h = lea/4 and a = lea/75 where Aeo = 2nVe/w
which do not vary with plasma density. This is a more
physical case to consider because the actual dimensions
of a real antenna do not change as wez/w2 is varied.
Figure 7.16 is a comparison of the two theories for an
antenna of dimensions h = 3.84).D and a = 0.2041D, where
AD is on the order of a Debye length. In both figures
excellent agreement between the two different theories
is observed. It is noted that Lin and Mei's theory is
restricted to an extremely short antenna while our theory
can be used to calculate the input impedance and current
distribution of longer antennas with practical dimensions.
The reason is that our theory is based on a much simpler
formulation.

Figure 7.17 compares the input admittance of a
dipole of half length h = 9fl/2, Ve/I/3we and radius
a a h/60 calculated by Wunsch [58] using Balmain's theory
[33] with the input admittance calculated using our theory.
The plasma is considered to be hot (Ve/C = 0.001) and

lossless. Fair agreement between the two theories is

observed.

7.3 Experimental Results
The radiation of a cylindrical antenna in a plasma
medium has been studied theoretically by many researchers.

As mentioned before, only a few workers have attempted to

149

measure the properties of antennas in plasmas experi-
mentally. Because of the availability of a large volume
of a stable, high density plasma in our laboratory [60],
we have performed an experiment that measured the input
impedances of cylindrical antennas in a plasma medium.

The schematic diagram of the experimental setup for
the antenna impedance is shown in Figure 7.1. The plasma
tube is made of an open-end Pyrex bell jar with dimensions
14 inches in diameter and 18 inches in length. The upper
end of the tube is a circular metal plate used as an
anode in the excitation of the plasma and as a ground
plane for a cylindrical monopole antenna feeding through
the center of the plate. The lower end of the tube is
the cathode which consists of a pool of mercury contained
in a metal dish. A floating metallic ring is placed at
the center of the mercury pool to fix the moving hot spots
of the mercury arc discharge. An ignition circuit is
installed in the mercury pool for the purpose of starting
the plasma. A DC power supply circuit is connected be-
tween the anode and the cathode of the tube. Under normal
Operating conditions the discharge currents range from 0
to 120 amperes which corresponds to a range of from DC to
3GHZ for the plasma frequency, we/Zn. The vacuum pumping
system consists of a mechanical pump and a mercury dif-
fusion pump. The tube is continuously pumped during the

experiment and the pressure in the tube is maintained at

150

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151

about 10-3 mm Hg. The antenna input impedance is mea-

sured by using the standard SWR method.

The experimental results for the input impedance of
various size cylindrical antennas are shown in Figures
7.18 to 7.26. In each figure we have also plotted theo-
retical results calculated from equation (6.3.12) for the
input impedance for the corresponding size antenna and for
a hot (Ve/C = 0.01), lossy (y/w = 0.12) plasma. The
antennas actually used in the experiments were 2.2 cm,
3.2 cm, and 4.7 cm in length and 0.12 cm in radius. The
driving frequencies were l.GGHZ, 1.8GHz, and 2.0GHz for
each antenna size yielding experimental results for
antennas of nine different electrical lengths. In each
figure the solid lines and the circular points are the
theoretical and experimental resistances respectively,
while the dashed lines and square points are the theo-
retical and experimental reactances, respectively.

A study of Figures 7.18 to 7.26 yields the follow-

ing observations:

1. For low plasma density (weZ/w2 < 0.6) the
theoretical and experimental resistances are
nearly constant and in good agreement. The
experimental reactance tends to become more
negative faster than the theoretical values

as the plasma density is increased.

152

2. For 0.6 < wez/w2 < 1.0 the antenna resistances
increases monotonically and reaches a peak at
w ~ we for both the experimental and theoretical
curves and the reactances reach a large negative
value and then increase in value until the

reactances are nearly zero at w ~ we.

3. In the range 1.0 < wez/w2 s 1.6 the antenna
resistances both experimentally and theoreti-
cally decrease monotonically with the theoretical
values decreasing at a faster rate than the
experimental values. The antenna reactances are
inductive in this range and reach a maximum and
then begin to decrease as the plasma density

is increased.

In general good qualitative agreement between theory
and experiment is observed with the resistances reaching
a maximum at w ~ we and the reactances changing from

capacitative to inductive at w ~ we.

7.4 Conclusions

A theory has been developed and an experiment per-
formed to evaluate the input impedance of a cylindrical
antenna immersed in a hot lossy plasma. Good qualitative
agreement between theory and experiment was observed.
To the best of our knowledge our theory is the first that

is able to predict the input resistance for wez/w2 > 1.0

153

and the input reactance over the entire range (0 < wez/

w2 < 2.6) for an antenna on the order of a free space
wavelength in length immersed in a hot lossy plasma.

The theoretical effect of collisional losses on the
input impedances has been demonstrated.

Further, it has been shown that our theory is com-
patible with that of Chen [23] who used a poynting vector
method and with the theories of Lin and Mei [56] and Wunsch
[58] whose solutions were limited to antennas on the order
of an electroacoustic wavelength in length. It has been
shown that our theory is in good agreement with the
experimental results of Graf and Jassby [48] and it has
been demonstrated that the form of our assumed current
distribution, equation (6.3.8) is in good agreement with

experimentally measured current distributions of Judson,

Chen, and Lundquist [52].

154

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Rel. Amp. of Ant. Cur. Rel. Amp. of Ant. Cur

Rel. Amp. of Ant. Cur.

155

 

 

 

 

 

 

1.01
008‘
0'6) y/w = 0.1
¢ = 75°
004Jb
0.20
0.0 : f : : ..z/h
0.0 0.2 0.4 0.6 0.8 1.0
1.04\
0.84.Y/w:2é1' y/w = 0.2 w 2
___. 0
¢ 15 —-‘-‘3—2- = 0.95
0.6? y/w = 0. w
I) = 2 °
‘LY/w = 0..
0.4 ¢ = 34°
002'”-
: I z/h
0.8 1.0
2
we
—2—= 1.2
w
0.0 : : : : a z/h
0.0 0.2 0.4 0.6 0.8 1.0

Figure 7.3. Current distributions on a dipole with
h/Ao = 0.147 and a/lo==0.0072 for various values of wez/w2
and y/w as a function of z/h.

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Rel. Amp. of Ant. Cur. Rel. Amp. of Ant. Cur.

Rel. Amp. of Ant. Cur.

157

 

 

 

 

 

 

1.04 ¢ = 62° w 2
y/w = 0.2 e _ 0 6
w2 - O
0.84. ¢ = 67°
y/w = 0.15
006 0
y/w = 0.05

0 44- ¢ = 79°
0.21»
0.0 s: : .4 : r. z/h

0.0 0.2 0.4 0.6 0.8 1.0
1.0-

\ y/w = 0.2
_ O

o 307/” = 0'1 ¢ - ll 2

' ¢ = 13° we

———-= 0.95
_ 2
0064byém=-19; w
‘Y/w = 0.!

0.4b ¢ = 32°
002‘b
0.0 ¢ : : : z/h

OJO 0.2 0.4 0.6 0.8 1.0
1.0
008‘
0.6“
0.4‘L
0.2i"
0.0 i 5' i 3 z/h

0.0 0.2 0.4 0.6 0.8 1.0

Figure 7.5. Current distributions on a dipole wi h
h/Ao = 0.192 and a/Ao = 0.0072 for various values of we /w2
and y/w as a function of z/h.

158

.>uflmcmu mammaa mo cofluoazm m mm mammam %mmoa Ado.o
Avooo.o u o«\m .HmN.o u 0<\£V wHomo:0E n no moawpwmafi undue Havauwuomna

um.

o\m>0 no: a i

 

 

 

 

 

 

. ooml
w
All N3\m 3
0.; 0 H N04 0 W
m
\ \‘— .r
\ \L: oom+
”n/HHH//II\,, \ 1. fix ---!
/ /./. , \ . ca
/// // / ~_ .m
/ / \ ‘ . oov+
/ /
/ ./ . ‘\
/ /
/ /
/ /( K 0.0.0 3}
x

// mH.o 3\> I oow+

/ 2 oa.o 3\>

I .
___ 1 00 0 a)

Rel. Amp. of Ant. Cur. Rel. Amp. of Ant. Cur.

Rel. Amp. of Ant. Cur.

159

 

 

 

   
 

 

¢ = 50°
1.04 Y/” = 0'2
we2
¢ = 55° ’7? = 0.6
0.8““ /w - 0.15 w
.-y/n = 0.05
0.6 ¢ = 72°
0.44- _
0.2--
0.0 . 1 A; %
0.0 0:2 0:4 0.6 0.8 1.. z/h
1.04
¢ = 3° (.0 2
y/w = 0.2 e =
008‘” /w = 0.1 —w2 0.95
¢ = 3°
0'6“’Y/w = 0.1
u - 15°
0-‘1""y/w = 0.05
¢ = 30°
0.2-(L
0.0 :0 I 04* I
0.0 0.2 0.4 0.6 0.8 1.0 z/h
100‘)
0 = -50°
0.84\\\\\\\\\\ /w = 0.2 w 2
e _
\\\\\‘ ¢ = _590 7:5-- 1.2
0.61; ” /w = 0.15
y/w = 0.1
0.4.; ¢ = -68°
y/w = 0.05
002‘”. ¢ = _‘780
0,0 3 : 4 :
0.0 0.2 0.4 0.6 0.8 1.0
Figure 7.7. Current distributions on a dipole with
h/Ao==0.251 and a/Ao==0.0064 for various values of we /w2

and y/w as a function of z/h.

160

.muwmcwp mamwam mo cowuocsm 6 mm mammam ammoa Aao.o u U\m>v #0: m 20
Amoo.o u o«\m .mam.o u 0K\£V maomocoe n no mocwpmmafi usmcw Hmowumuomse .m.h musmwm

 

 

 

 

 

 

 

 

._ /
._ , 0~.0 u s\>
__
._ 0 00m-
__
.__
0
.All a. 3 _
N \N . ).
m.H v.H N.H . o m.
o v) . m
S
III, (I! l\
\\\\\\\\\\\\I\III III:i
\
\\
\\~ : com
I \c l
/ I, l 1.0ov
1’ /
/ / ~\
/ /
/ / _ .
x //
/
// 00.0 n 3\> 000+
/, \ me.0 u 3x»
/ oa.o u 3\>
/ .4 no.0 u 3\>

Rel. Amp. of Ant. Cur. Rel. Amp. of Ant. Cur.

Rel. Amp. of Ant. Cur.

161

 

 

1.0 . ¢ = 27°
y/w = 0.1 2
4 = 18° we _ 0 6
0.8 n l\\\\\ Y/w = 0.15 mi‘
0.6 u Y/w : gégs
¢ ' y/w = 0.2
= 10°
0.4 0 ¢
0.2 0
: c ; $0 . z/h
0.0 0 2 0.4 0.6 0.8 1.0
¢ = —6° 0.) 2
0.8 .y/w = o. /w = 0'2 -%§-= 0.95
¢ = 2° (1)
0'6 " w = 0.1
u = 13°
0.4 ‘P /w - 0005
II = +300
002 Q

 

 

 

 

 

Figure 7.9. Current distributions for a dipole with
h/lo==0.313 and a/Ao==0.008 for various values of weZ/w2 and

y/w as a function of z/h.

162

.muamsmc

o wannam mo cowuucdm 0 mm MEmMHQ mmmammoH Ado.o u U\m>v #0: m 20 A04 Hoo.o n mswpmu
. K mm.o u sumcma mamsv wccmusm Havanpcfiamo 6 mo mocmunwwmu mcawucd .oa.n wusmflm

 

 

3\ 03 mua qu ~.H o.H m.o v.0 v.0 N.o o.o
m m . x . z . . . 0 0 \o\1
m w w
m m m
\
\.\\\\ AVOOH
.\\\)/
\\\ as
a \ .m
e \\ 00m
nr
m \\
wosmumflmmu mccwucm no a \
usmcomfioo owumsoomouuomam u m \
wonmumflmmu mcsmucm mo 0 \<\ l.oom m
unmaomEoo owumammsouuowam u m \ F
\
wuawumfimmu msamucm Hmuoe u cam \
\
iroov
mm + 0m u new \
.ponuwa uouom> eh
ocwunaom Eoum muasmmm o o 1
L ..oom
:0 L
m ..... 1
mm _
IIIII :
mm “ ..oom
_

 

assoc

numcma «away wasmusm Hmownpcflamo 0 mo mosmuowwu unsound

w .muflmsmp mammam mo cofiuocsm
0 mm mammam mmmammoH Ado.o u 0\ >0 #0: m :0 A04 aoo.o u msfiomn .00 m~.o u

.Ha.h wusmflm

 

 

 

 

3\ ms mHH «NH NHH o.H who mflo ¢.o Nmo o.oombml
q I I u q . a
N N _
oooml
\N oooml
\. m
\ x 0000.
3
”m
IIIIIIIIII||..'I_III\
0
3
ll! .uommu
(libs/I m .usm mo .QEOU a
II x x oauwsoowouuomam u x : oooa
.uommu .usw mo 0
.mEoo owumcmweouuowam u x
as = 0000
wonmuommu wscwucm Hmuoe n .x
0x + 0x 1 sex
CA : ooom
.N.I lllll
Q I'll AV
wx.||| 0m00
x

 

 

finance

Relative amplitude of antenna current

Phase, degrees

164

+ + r Exp. of Judson,
. . . Chen and Lund-
quist

I
‘-

  
  

Our Theory

 

z/h

 

 

 

 

 

I
\I
U1

5

 

Figure 7.12. Comparison of theoretical and eXperi-
mental current distributions on a monopols with son = 1.54
and a = 0.615 cm for various values of we /w2. The driving
frequency in the experiment was 1.25 GHZ.

165

 

 

 

Inductive
___ Graf and Jossby's 4.0 '\
Cold Lossy Plasma 3
Theory 3.0 \
---- Exp. Results of 2.0% 2 0
Graf and Jossby ‘ '
‘$.3
Our Theory ‘/ \ 1.5
/ 2 \ 1.4” \
/ we \ \
/ . __i— = ' Q 1.
/ w 2.-
1.0/ 0 1 \ 1.1
/ I 1.05
1 1 1009 [1.8 0.9;
I ‘ I '
I ' '
h/A = 0.25 I 3 :
a/A = 0.0156 \ '
f = 9.2 Ghz \ 0.2
y/w = 0.2 \
\l.25
\
\.
\
\
Capacitive

 

Figure 7.13. Input impedance of a dipole of half
length h/Ao==0225 normalized to 1009. Normalized electron
density (wez/w ) values are indicated.

166

Inductive

_____ Exp. Results of
Graf and Jassby

Our Theory

 

._____ Graf and Jassby's
Cold Lossy Plasma

 

 

Theory
IOOQ
h/A = 0.12
a/A = 0.0156
f = 9.2 Ghz
y/w = 0.2
w 2 f
532-: 0.0
w
Capacitive wez 0,1
7: 000
w

 

Figure 7.14. Input impedance of a dipole of half
length h/lo§=0.12 normalized to 1000. Normalized electron
density (we /w2) values are indicated.

ohms

167

 

 

 

 

 

 

 

 

 

-X
“71117: x “\~
1051’ .\‘"\\ 7“\‘\
\\
_x \\\
‘\\ j,
/
\
.z/ \
‘,/" \ I
//
[ji’
/’ R
//
1040 //’ I
/’/ v/-§—-—_~\\\
///|,,/””’ \\\\
/
/ \
x I
/
h = Ae/4
Our Theory I
1034.. a = Ae/75
h = 18/4
“--- Lin & Mei I
a = 18/75
h = lac/4
—“—‘ Our Theory '
a = A /75
e0
1024I
I
30 ' 0:2 0:4 0:6 018 170
w 2/w2'-—*>

e

Figure 7.15. Input impedance of a short dipole
antenna in a hot (V /C = 0.001) lossless plasma as a
function of plasma density.

ohms

168

 

 

..I ll

 

     

 

 

 

3.. 3.84)
10 Our Theory D
a = 0.2041D
h = 3.841D
————— Lin & Mei
a = 0.204).D
102‘:
3° 0.0 012 014 016 018 110

wez/w2-——v

Figure 7.16. Input impedance of a short dipole
antenna in a hot (Ve/C = 0.001) lossless plasma as a
function of plasma density.

Conductance in micromhos

12.0

10.0

8.0

Figure 7.17.

 

 

169

 

 

 

1p 7’ 50
x”’
./
.—- //
/ \\
// \\\\H ///
+ “\ "+40
/ -7“\
/ _ / ‘\
// .\‘“\
m / N
I, G /
.. I // (3 .+30
I / H
/
,’ /
/
I /
1b " // 1b4'20
' /
: // h = 2fl_ Ve
l / 2 w /3
l / e
.. /
/ a = h/60 9+10
I
/
/ ————— Our Method
l/
I ---—- Balmain's e.m.f. "0
V Method
l
I
1 4% is I -10
1.0 1.5 2.0 2.5 3.0
2 2 —-"
w /we

Input admittance (y

G + jH) of a

dipole antenna in a hot (Ve/C = 0.001) lossless plasma
as a function of the plasma density.

Susceptance in micromhos

170

 

 

 

 

f
+1000 0 I
(Ohms) R. o o o '
Experimental in
+800 v xin . I I :
. Rin . I
+500 0 Theoretical . ____ l
in
I
+400 '
I
I
+200 I
I
0 1t' 1.2 1.4

--__ h

-400 “‘~~‘
9 . . 7“n\ I'
I “\ ll
“\. ’I

 

Figure 7.18. Experimental and theoretical input impedance of
a monopole (h/Ao==0.ll7, a/Xo:=0.0064) in a hot lossy plasma as a
function of plasma density.

 

 

+10000 .
(ohms) . 1 R1 0 o . '
+8000Experimenta X n . . . l
in I
R ___.

+6000Theoretical xin -___ :

in
l
+400. '
I
l

+20

U
0
I
1‘0

 

\
‘~
-‘-

 

\
\\

Figure 7.19. Experimental and theoretical input impedance of
a monopole (h/Ao 00.132, a/Ao -0.0072) in a hot lossy plasma as a
function of plasma density.

 

 

171

 

 

+1000<I
(011108) . Rin . . . '
+800‘LExper1mental xin' . . |
R. ___. '
+6004>Theoretical x1" -_-_ 1
in I .
I I '\\
+400 0 y/w = 0.12 'q \\ ' .
| I o \\\\‘ -
I Q ‘~“
+200 q» |,
J o
O
o . A L L A 41
I. |.4 1.6 01.3 MO 172 111 1.6
-200 “ \ Il e
I \\ I]
\ I I
“400 . \\ I '

Figure 7.20. Experimental and theoretical input impedance of
a monopole (h/Ao 80.147, a/Ao =0.008) in a hot lossy plasma as a

function of plasma density.

    
  
  

 

   

 

 

 

 

 

+1000 I I
(ohms) EXperimental gin ' ' ' '
+800 0 in I I I l
l
. R. I
+600 0Theoretical xin ___-
1n l
l
+4004) ' ~\\ .
/ 0 12 1 \‘\“~ .
+200 0- Y m . I” ~.
I.
0 . h, - .
_ 0.8 1150 l'.2 126
10“ I] w /w —-o-
-200 0 .\\\\ I, I e
. \\ I l
\
-400 » \\ ’ I
I \ I

Figdre 7.21. Experimental and theoretical input impedance of
a monopole (h/Io =- 0.171, a/lo = 0.0064) in a hot lossy plasma as a

function of plasma density.

172

+1000
(ohms) Rin o o o l
+800‘rExperimental X. . . .
in
+600"Theoretica1 Xln ___" .

    

 

 

+400‘
+200‘
05 . , - .
II ;‘~H“. . /| w 2/w2-—+-
“’200 " - \\ I ' e
g \
\\ . l I
/
“400 " I . .\\/ |

Figure 7.22. Experimental and theoretical input impedance of
a monopole (h/Ao = 0.192, a/Ao =0.0072) in a hot lossy plasma as a
function of plasma density.

    
  
  
 
 
  

 

+1000‘L
(ohms) Rin‘ ' ‘ '
+8oo‘LExperimental X. g g a 0
in
R. O
+sooquheoretical xln..-..
in .

+400 0

+2004

 

lh‘k'u.

 

 

 

0 .
“I-IIM'“2;4 0 6 0.8 ho 1:2 1.4 11?
I ‘~\
-2000 g \\ I, ' e
\
' \\a/ '
-4oo‘~ _ I

Figure 7.23. Experimental and theoretical input impedance of
a monopole (h/Ao =- 0.213, a/Ao 80.008) in a lossy hot plasma as a
function of plasma density.

I- i mwn
‘

1'73

   
    

 

Experimental gin. ° '
+1000 II» in I I I I 0
(ohm) I
. R -——— |
+8000Theoretical xin --_‘ l
in
I
+6004 I /
l/
'l I \s
+400< i
O . h t
' a \\ I l we /w —"
-200 .. ' \\ / I
C \ |
I

 

-400"

Figure 7.24. Experimental and theoretical input impedance
of a monopole (h/lo I=0.251, a/Ao = 0.0064) in a hot lossy plasma
as a function of plasma density.

  
  

   

 

 

Experimental xin '
+1000 .. in ' ' ' '
(ohms) . I
R. —— I
Theoretical in
+800 T , Xin ———- I
I
+600 -» ' \
U \“s‘ I
0 \~‘
' ~
+400‘ J
I
.1 f I a
‘~~0.6 (.0 1.2 .
I \ 2 —.-
' ' \\ / I we /w
’200 ‘* \~\./ I
. I
-4001’ ' l

 

Figure 7.25. Experimental and theoretical input impedance
of a monOpole (h/k: -0.282, a/lo- 0.0072) in a hot lossy plasma
as a function of plasma density.

174

+10004*

(ohms) Experimental in .
+800 “ 1“ ' ' ' °

+600 0Theoretical

+400

+200

 

-200

 

 

-400 + - '

Figure 7.26. Experimental and Theoretical input impedance
of a monopole (h/Ao =0.3l3, a/Ao 80.008) in a hot lossy plasma
as a function of plasma density.

REFERENCES

 

 

3,:

10.

REFERENCES

A. M. Messiaen and P. E. Vandenplas, "Theory and
Experiments of the Enhanced Radiation from a

Plasma-Coated Antenna," Electronics Letters, Vol. 5
3, 26, (1967). i

C. C. Lin, "Radiation of Spherical and Cylindrical

 

Antennas in Imcompressible and Compressible V}
Plasmas," Ph.D. Thesis, Michigan State University, '
(1969).

C. C. Lin and K. M. Chen, "Effect of Electroacoustic
Wave on the Radiation of a Plasma Coated Spherical
Antenna," IEEE Trans GAP, Vol. AP-18, No. 6, 831,
(1970).

 

L. Tonks and I. Langmuir, "Oscillations in Ionized
Gases," Physical Review, Vol. 33, 195, (1929).

 

R. W. Revans, "The Transmission of Waves Through an
Ionized Gas," Physical Review, Vol. 44, 798, (1933).

 

P. J. Barret and P. F. Little, "Externally Excited
Waves in Low-Pressure Plasma Columns," hysical
Review Letters, Vol. 14, 356, (1965).

I. Alexeff, W. D. Jones, and K. Lonngren, "Excitation
of Pseudowaves in a Plasma Via a Grid," Physical
Review Letters, Vol. 21, 878, (1968).

K. R. Cook and R. B. Buchanan, "Radiation Charac-
teristics of a Slotted Ground Plane into a Two
Fluid Compressible Plasma," Proc. of the Conf. on
Environmental Effects on Antenna Performance, 122,
(1969).

 

T. H. Stix, Theory of Plasma Waves, McGraw-Hill,
Chapter 7, (1962).

 

R. A. Tanenbaum, Plasma Physics, McGraw-Hill, 169,
(1967).

 

175

ll.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

176

L. Oster, "Linearized Theory of Plasma Oscillations,"
Reviews of Modern Physics, Vol. 32, 141, (1960).

 

J. R. Wait, "Theory of a Slotted-Sphere Antenna
Immersed in a Compressible Plasma, Part II," Radio
Sci. J. Res. NBS 68D, No. 10, 1137, (1964b).

Ramo, Whinnery, and Van Duzer, Fields and Waves in
Communications Electronics, John Wiley and Sons,
Inc., 701, (1965). 'V“

 

L. Infeld, "The Influence of the Width of the Gap
Upon the Theory of Antennas," Quart. Appl. Math,
Vol. 5, 113, (1947).

 

M. Abramowitz and I. A. Stigun, Handbook of
Mathematical Functions, Dover, Chapter 10, (1968).

 

 

M. Abramowitz and I. A. Stigun, Handbook of
Mathematical Functions, Dover, Chapter 8, (1968).

G. Forsythe and C. B. Moles, Computer Solution of
Linear Algebraic Systems, Prentice-Hall, Chapters
9 and 17, (1967).

 

A. Hessel and J. Shmoys, "Excitation of Plasma Waves
by a Dipole in a Homogeneous Isotropic Plasma,"
Proc. of the Symposium on Electromagnetics and Fluid
Dynamics of Gaseous Plasmas, Interséience, 173,
(1961).

 

S. R. Seshadri, "Radiation from Electromagnetic
Sources in a Plasma," IEEE PGAP, Vol. 13, 79, (1965).

 

Marshall H. Cohen, "Radiation in a Plasma, III,
Metal Boundaries," Phys. Rev., Vol. 126, 398, (1962).

 

James R. Wait, "On Radiation of Electromagnetic and
Electroacoustic Waves in a Plasma," Appl. Sci. Res.,
Sect. B, Vol. 11, 423, (1965).

 

James R. Wait, "On Radiation of Electromagnetic and
Electroacoustic Waves in a Plasma, Part II," Appl.
Sci. Res., Sect. B, Vol. 12, 130, (1965).

 

Kun-Mu Chen, "Interaction of a Radiating Source with
a Plasma," Proc. IEE, Vol. 11, 1668, (1964).

K. R. Cook and B. C. Edgar, "Current Distribution and
Impedance of a Cylindrical Antenna in an Istropic
Compressible Plasma," Radio Science, Vol. 1, 13,
(1966).

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

177

H. H. Kuehl, "Resistance of a Short Antenna in a Warm
Plasma," Radio Science, Vol. 1, 971, (1966).

 

H. H. Kuehl, "Computations of the Resistance of a
Short Antenna in a Warm Plasma," Radio Science, Vol.
2, 73, (1967).

James R. Wait, "Theories of Prolate Spheroidal
Antennas," Radio Science, Vol. 1, 475, (1966).

R. W. P. King, C. W. Harrison, Jr., and D. H. Denton,
Jr., "The Electrically Short Antenna as a Probe for
Measuring Free Electron Densities and Collision
Frequencies in an Ionized Region," Journ. Of Res. of
E§§' Sect. D., Radio Prop., Vol. 65, 371, (1961).

R. W. P. King, "Dipoles in Dissipative Media,"
Electromagnetic Waves, edited by R. E. Langer,
University of Wisconsin Press, 199, (1961).

 

James R. Wait, "Theory of a Slotted-Sphere Antenna
Immersed in a Compressible Plasma, Part 1," Radio
Science, Vol. 68D, 1127, (1964).

A. D. Wunsch, "Current Distribution on a Dipole
Antenna in a Warm Plasma," Electronics Letters,
Vol. 3, 320, (1967).

S. R. Seshadri, "Propagation Coefficient for the
Current Distribution Along a Cylindrical Antenna
Immersed in a Warm Plasma," Proc. IEE, Vol. 112, 877,
(1965).

 

K. G. Balmain, "Impedance of a Short Dipole in a
Compressible Plasma," Radio Science, Vol. 690, 559,
(1965).

James R. Wait and Kenneth P. Spies, "Theory of a
Slotted-Sphere Antenna Immersed in a Compressible
Plasma, Part III," Radio Science, Vol. 1, 21, (1966).

J. Carlin and R. Mittia, "Acoustic Waves and Their
Effects on Antenna Impedance," Canadian Journ. of
Phys., Vol. 45, 1251, (1967).

J. W. Carlin and R. Mittia, "Terminal Admittance of
a Thin Biconical Antenna in an IsotrOpic Compressible
Plasma," Electronics Letters, Vol. 2, 199, (1966).

S. R. Seshadri, "Radiation in a Warm Plasma from an
Electric Dipole with a Cylindrical Column of Insul-
ation," IEEE PGAP, Vol. 13, 613, (1965).

 

38.

39.

40.

41.

42.

43.

44.

45.

46.

47.

48.

178

James R. Wait, "Radiation from Sources Immersed in
Compressible Plasma Media," Canadian Journ. of Phys.,
Vol. 42, 1760, (1964).

 

A. Hessel, N. Marcuvitz, and J. Shmoys, "Scattering
and Guided Waves at an Interfare Between Air and a
Compressible Plasma," IRE PGAP, Vol. 10, 48, (1962).

Edmund K. Miller, "The Admittance of the Infinite
Cylindrical Antenna in a Lossy, Isotropic, Compressi-
ble Plasma," Univ. of Mich., Col. of Eng. Rept.
05627—10-8, (1967).

 

Ronal W. Larson, "A Study of an Inhomogeneously
Sheath Spherical-Dipole Antenna in a Compressible
Plasma," Univ. of Mich., Radiation Lab. Rept.
7000-25-T, (1966).

 

S. Chen and C. T. Tai, "The Input Impedance of Thin
Biconical Antennas Immersed in a Dissipative or an
Ionized Medium," Ohio State Univ., Antenna Lab.
Tech. Rept. 1021-14, (1962).

 

 

H. A. Whale, "The Excitation of Electroacoustic
Waves by Antennas in the Ionosphere," J. Geophys.
Res., Vol. 68, 415, (1963).

 

G. Meltz, P. J. Freyheit, and C. D. Lustig, "Admittance
of a Plasma-Covered Cylindrical Antenna," Radio Science,
Vol. 2, No. 2, 203, (1966).

 

J. A. Kane, J. E. Jackson, H. A. Whale, "RF Impedance
Probe Measurements of Ionospheric Electron Den-
sities," J. Res. NBS (Radio PrOp.), Vol. 66D, 641,
(1962).

 

R. G. Stone, R. R. Weber, and J. K. Alexander,
"Measurement of Antenna Impedance in the Ionosphere—
I. Observing Frequency Below the Electron Gyro-
frequency," Planet. Space Sci., Vol. 14, 631, (1966).

 

K. G. Balmain, "The Impedance of a Short Dipole
Antenna in a Magnetoplasma," IEEE Trans. Antennas
and PrOpagation, Vol. AP-12, 605, (1964).

 

 

K. A; Graf and D. L. Jassby, "Measurements of Dipole
Antenna Impedance in an Isotropic Laboratory Plasma,"
IEEE Trans. Antennas and Propagation, Vol. AP-15,
681, (1967).

 

49.

50.

51.

52.

53.

54.

55.

56.

57.

58.

59.

60.

179

D. L. Jassby, "Experimental Impedance of a Quarter-
Wave Monopole in an IsotrOpic Plasma," IEEE Trans.
Antennas and Propagation, Vol. AP-16, 282, (1968).

 

 

C. Ancona, "Antenna Impedance Measurements and
Sheath Effects in an R.F. Generated Plasma," IEEE
Trans. Antennas and Propagation, Vol. AP-l9, 237,
(19711 .

 

H. Judson, K. M. Chen, R. Lundquist, "Measurement
of the Current Distribution on Monopoles in a Large
Volume of Hot Plasma," Electronics Letters, Vol. 4,
289, (1968).

 

K. M. Chen, H. Judson, and C. C. Lin, "Experimental
Study of an Electroacoustic Wave Excited by an
Antenna in a Hot Plasma," Proc. IEEE, Vol. 55, No. 9,
1656, (1967).

C. C. Johnson, Field and Wave Electrodynamics,
McGraw-Hill, Chapter 1, (1965).

J. D. Jackson, Classical Electrodynamics, John Wiley
and Sons, Chapters 6 and 9, (1962).

J. A. Stratton, Electromagnetic Theory, McGraw-Hill,
Chapter 8, (1941Y.

 

S. H. Lin and K. K. Mei, "Numerical Solution of
Dipole Radiation in a Compressible Plasma," IEEE
Trans. Antennas and Propagation, Vol. AP-16, 23 ,
(19687.

 

F. E. Hohn, Elementary Matrix Algebra, Macmillan,
Chapter 8, (1964).

 

A. David Wunsch, "The Finite Tubular Antenna in a Warm
Plasma," Radio Science, Vol. 1, 901, (1968).

H. Judson and K. M. Chen, "Measurement of Antenna
Current Distribution in a Hot Plasma," IEEE Pro-
ceedings, Vol. 56, 753, (1968).

H. Judson and K. M. Chen, "Construction and Operating
Characteristics of a Large-Volume Mercury Arc

Plasma Tube," IEEE Trans. Antennas and PrOpagation,
Vol. AP-16, 144, (1968).

APPENDICES

APPENDIX A

UNCOUPLING THE DIFFERENTIAL EQUATIONS

FOR THE ELECTRONS AND THE IONS

___._ _ -.-___ ......“

 

APPENDIX A
UNCOUPLING THE DIFFERENTIAL EQUATIONS

FOR THE ELECTRONS AND THE IONS

From Chapter II the differential equations to be

considered are

 

 

 

 

 

 

 

2 Ve 2 Ve wewi Vi we S
V -— n + B —— n + —— n. = — —— 9—
w e e w e V V. w. 1 \I e
e e e 1 i e
(A-1)
and
V V. w m V m. S
V2 Si n + 812 5i n1 + VeV1 $2 ne = Vi %?
11 i eie i
(A-2)
where
2
2 w
2 w e Ye
3.2-71‘7”? (A3)
V
e
2 w 2
82_m1_i_.1_i_ (A-4)
i - _—2 3 w '
Vi w

180

181

The objective of this section is to uncouple
equations (A-1) and (A-2) and obtain two independent linear
differential equations for the variables nl and n2 which
are linear combinations of ne and ni. This can be accom—
plished by using eigenvalue techniques. The uncoupled
equations, the relationship between the variables ne and
ni, and the variables n1 and n2, and the high and low fre—

quency limits of all pertinent coefficients will be

developed and presented in this section.

 

Ir- 2‘

Equations (A—1) and (A-2) can be written compactly

as a matrix equation

 

 

 

 

S
Vzn + 8n = S %; (A-S)
where
”ve '-
$— ne
e
n = ,
~ Vi
Fni
.1 _J
B 2 wewi
e VeVl
§= ,
wewi B 2
_Yevi 1 _

 

 

and

182

<I

§ = .. 2A-.)

.1
v.
t. '1-

 

 

Define a new vector N such that

(A-7)

1::
ll
NH
22
22
ll
'5 5'
'N #4.

where T is specified to be an orthonormal transformation.
8

 

The substitution of equation (A-7) into equation (A-S)

gives
2 s
v T N + B T N = s 9— . (A-8)

At this point it is assumed that the determinant of T,

denoted by det T, is not zero (this is verified later).

1

Then the inverse of T, denoted by T— , is defined by

T’1 = T'1 (A-9)

22 t-J
ll
22 H

N

where I is a 2 x 2 unit matrix. Premultiplying equation

(A-8) by T"1 gives

5
T 1 v2 N + '1 = l s %; . (A-10)

22.-3
22h]
22m
226

22
222-]

Assuming that T is independent of space coordinates, it

follows that

T"1 v2 T N = T"1 T v2 N = VZN . (A-ll)

183

Therefore equation (A—10) becomes

5
va + 1 3 N = l s %; . (A-12)

RH
220-]
222-3

Equation (A-12) reduces to two uncoupled differential
equations for nl and n2 if g is an orthonormal trans-
formation that transforms B into a diagonalized form.
The procedure associated with determining T is the
subject of the next few paragraphs.

The eigenvalues of B are solutions to character-

~
~

istic equation of B

22

A - (B + 312” + BeZB' - —2——-2- = o (A-13)

or

 

(A-14)

The problem now is to find the transformation T such that

~

(A-16)

u
um
and
II
(O
»

From the theory of matrix algebra

 

184

 

ll 12
T- _ E1 T21
~ T21 T22 “
where T1 and T2 are the eigenvectors associated with Al
and A2, respectively. T1 and T2 are solutions to the F“*
matrix equations
1
(8 - XII) T = 0 (A-16)
and
2
(g - AZI) T = 0 . (A-l7)

Equations (A-16) and (A-l7) reduce to one relationship

1 and T2,

each for the two unknowns of the eigenvectors T
respectively. In order to completely specify the trans-

formation we choose to normalize the eigenvectors, i.e.,

i_ 2 2_ ._ _
T T — T1i + TZi — 1, 1 — 1,2 . (A 18)

In what follows a commonly reoccurring term will be

Specified by A0 where

2 2
_ / 2 2 2 we “’1 _
A0 - (Bi - 8e ) + 4 —2-‘-’-§ . (A 19)

 

The matrix equation (A-16) reduces to the single

relationship

_ _ l e i 2 _ 2 _

 

11 (A-ZO)

185

which together with equation (A-18) defines T11 and T

 

 

 

 

 

 

 

 

21
as follows
T = l (A-21)
11
V 2V 2
1 + l e i 8 2 _ B 2 _ A 2
4 w 2w 2 e 1 o
e i
2 2
T _-lVV Be-Bi -A0
21 2 w w. '
V 2V 2
l e i 2 2 2
1+2? 22|:Be-81_A(3:|
w w
e 1
(A-22)
Similarly, equations (A—l7) and (A-18) define T12 and
T22 as follows
T12 = 1 (A-23)
2 2
V V. 2 2 2
l e 1
1+? ZZEe-B1+Ac;l
w w.
e 1
2 2
_ 1 V V1 Be - 8i + Ao
T -— .
22 2 w m.
V 2V 2
l e 1 2 2 2
1*? 22|:Be-Bl +Ac;l
w w.
e 1
(A-24)

For this development it is necessary that det T # 0
as assumed earlier. We are now able to calculate det T

as follows:

 

186

det T

T = T11 T

- T12 T12 = -1 (A-ZS)

22

Therefore T exists and can be written as

-T T

22 12

(A-26)

Re
I

T21 "T11

ne and ni can now be written in terms of nl and n2

and visa versa as follows

0)

_ _E _
ne — Ve (Tllnl + lenz) (A 27)
”i
ni = v: (T21nl + T22n2) (IX-'28)
and
Ve Vi
n1 = ’ 6"T22ne + ET leni (A’Zg)
e 1
Ve Vi
n2 = a; T21ne - “T; Tllni (IX-'30)

Equation (A-12) can now be written as two linear uncoupled

differential equations

N
+
w
M
s
u
m
or},

V n (A-31)

2 2 S
v n + k s %; (A-32)

"—“7

 

187

where
2 - 1 2 2 _
k1 - §[Be + Bi + A0], (A 33)
2 _ 1 2 2 _ _
k2 " 2[Be + 81 A0]: (A 34)
:3
Te Ti r
S]. = + T22 ‘7— + T12 ‘7.— , (A‘35)
e l 1
and
we (Di
S2 = " T21 V; " T11 x7: “’36)

 

Equations (A-27), (A-28), (A-31), (A-32), (A-33),
(A-34), (A-35), and (A-36) are a tractable set of equations
whose solutions specify the electron and ion perturbations
in an infinite homogeneous plasma due to charge density
source pS immersed in the plasma. The conclusion is that
two independent particle waves are able to propagate in

an infinite plasma.

Let us look at the forms of T11, T12, T21, T22,
klz, and k22 in the high frequency and in the low fre-
quency limits. For simplicity, let the ion- and electron-

neutral particle collision frequencies equal zero, i.e.,

Ye=Yi=0-

(a) High Frequency Limit (2»2 > we2 >> wiz)
2
2 w 2
2 w e 2 w
e V 2 (”2 1 v2

e l

188

In this limit

 

 

 

 

(A-37)

 

(A-38)

 

 

Hence

T11:o

189

R
I
H

21

T12 z

T22 2

From equations (A-29) and (A-30)

 

V i I
n1 2 ET “1 (“'39) :
1 ‘1
ve ..
n2 3 - a; he (A-40)

So in the high frequency limit n is wave consisting of

1

ion motion only and propagates with a phase velocity Vi
which is the thermal velocity of the ions. Similarly, n2
is an electron wave which propagates with the thermal

velocity of the electrons.

(b) Low Frequency Limit (to2 << wiz << wez)
2 _ l 2 _ 2 2 _ l 2 _ 2
Be " V 2“” we) Bi "' V 2(0) (”1)
e i
2 2 2 _ 4 _ 2 2 2
(Bl - 8e ) - Be ZBe 81 + 81
= —lE[w - 2w2w 2 + we4] - 3 2[§4
V V V.

1
2 2 2 w 2 w 2 + w 2 w 2
_ we ml 2 e e 1 1
- V 2 - V 2 - 2w V 4 - V 2V 2 + V 4
e 1 e e 1 i 111
+214 1 _ 1 2
4 4
V V.
e 1

 

 

V "_,
‘5
‘ F'V‘BA'L...

where the first term on the right hand side is of the
zeroth order in w, the second term is of the 2nd order

in w, and the third term is of 4th order in w.

Noting that we2 >> wiz and dropping 4th order terms in w
yields
w 2 w 2 2 w 2 2 w 2
2 2 2 e i 2 e e i
(e.-e)T—-—— -2w —-—+—
1 e V 2 V 2 V 4 V 2V 2 V 4
e i e e i 1
Therefore

(I

191

 

 

2 2 me
Te - TiTe + Ti 1T-
w 2v 4 _ w 2V 2V 2 + w 2v 4 i
e i e e i i e _ 1 ---
2472 2 2 2 - 2 2
(we Vi + mi Ve ) mi (Te + Ti)

| A

where the underlined term is small and therefore it is
drOpped in the remaining calculations. By definition
(112/mi2 << 1, so using the binomial expansion and keeping

only the first two terms

 

 

 

 

2 2 2 2 2 4 2 2 2 2 4
A z we V + wi Ve 1 _ wz w Vi we ngl + mi Ve +
o 2 2 2 2 2 '°°
V6 V1 (we Vi + w12Ve )
Then
2 2 2 2
2 w 2 w. .
kzélL—e+w __l._+(:§_+w_§l_
l 2 2 V 2 V 2 V 2 V 2 V 2
e e i i e 1
2 4 2 2 2 2 4
- w2 m V1 - e Ve V. + wl Ve
2 2 2 2 27I2
Ve V1 we Vi + 1 V
or
2 2 2 2 2 2 2 2
2 w m. V V. + we vizve + we Ve Vi
k1 = 2 2 2 2 2

 

192

or dropping the small underlined term

 

2 2
2 .. w we _ 2 2 “’1 2
k1 2 2 2 2 ' “’ Vi + “2' Ve ”‘41)
m V. + m. V w
e 1 1 e e
So the phase velocity of the nl wave is "“

 

<

..a

ll
'<\
....

N

4.
8'8

P
N N

<
(D

N

 

 

 

 

 

ph
e
3k(T. + T’)
= /// l e (A-42)
m.
1
2 2 2 2
1.221 .93.-“_’;e_+_<2.__“.’_1_-‘fs__“_’_1_
2 2 V 2 V 2 V. v.2 V 2 V 2
e e 1 1 e 1
W W
2 m 2V 4 - m 2V 2V 2 + w.2V.2V 4
+ w 1 e e 1 1 e
VeZVi2 wezv 2 + w.2Ve2
2 2
k2 - :e—Tuwi (A-43)
2 x V 2 V 2
e i

where the underlined terms are small and again they are

drOpped. k2 is purely imaginary and hence the n

will not propagate in the low frequency limit.

wave
2

 

2
2 2 “’1
Be - B1 +Aoz 2 7
V.
1
w 2
2 2 e
Be " Bi "Aoz " 2 ‘7—2-
e
T _1_Vew1
11 ~ [2- weVl
T12 = ./1_
2
l
T g —"
21 /§
T~_1Efi
22 f:- we Vi

In an equilibrium plasma (Te = Ti)

1 V: Vi
r11 = -— —— n + —— n. (A-44)
V V.
5‘”. e “’1 1

 

From equations (A-4l), (A-43), (A-44), and (A-45)
we see that in the low frequency limit only the n1 wave

exists and it consists of both electron and ion motion.

APPENDIX B

SOME PROPERTIES OF LEGENDRE FUNCTIONS

 

ll .-: x'u. '

APPENDIX B

SOME PROPERTIES OF LEGENDRE FUNCTIONS

Some properties of the associated and ordinary

Legendre functions that are useful in this thesis are

 

listed below.

(1) Pi (cosB) is zero at 6 = % if n is even.
(2) P: (cose) is maximum at 9 = % if n is odd and the

value of this maximum is given by

/ 1‘03“» 1)
1,5,9,...

for n

2h!”

I‘(§-+ 1:)

1 -- _
an) -( (B 1)

1“(§+ 1)

—r';——— 3,7,11,000
P(§ + 8)

for n

2M”

 

or

I'(-+1)

[P:(0)]2 = for n odd (B-2)

are

N948)

where F(x) is the Gamma function with argument x.

194

195

(3) The associated Legendre functions have orthogonality

properties,

r
0 for n # m
+1 1 1
3E Pn(x) Pm(x)dx =< (B-3)
3%%2%£L for n = m.

 

K

(4) A recurrence formula for the ordinary Legendre

functions is

agi- Pn+l(X) - X ad?" Pn(X) "' (n+l)Pn(X) = 0 (3‘4)

and a relationship between associated Legendre functions

and ordinary Legendre functions is

m/2 de (x)
m m 2 n
Pn (x) = (-l) (l-x ) ____—37 ° (B-5)
dx
For m = 1, equation (B-S) becomes
1 __ d _
Pn (cose) - 35 Pn (cose). (B 6)

Combining equations (B-4) and (B-6) we obtain

1 1 1 _
EIK§[E°SBPn(c°se) - Pn+l(cosei] — (n+1)Pn(cose).

(B-7)

196

(5) A differentiation formula for the associated

Legendre functions is

d1 _1 1 _ 1
55 pn(cosei] - EIE§[§ Pn+l(cose) (n+l)cosePn(cosei].

(B-8)

 

APPENDIX C

METHOD OF THE AUXILIARY INTEGRAL

APPENDIX C

METHOD OF THE AUXILIARY INTEGRAL [56] T”

In the numerical solution of input impedance Z in

Chapter 7, singular integrals of the form

 

 

n h e-ij
I = j] cos¢ dzdcb (C-1)
0 R
-w 0
where
R = [z2 + 4a2 sin2 5%]15 (C-Z)

are often encountered. Integrals of this form can be
handled using the method of the auxiliary integral.

Consider the result from an integration table

In h cos%
11‘ f “Tr—am

-n 0
2a . -l 11
= 4|:2% sinh l T + Sinh Ta] (C-3)

197

198

which is called the auxiliary integral. Equation (C-l)

can be rewritten as

 

Io = 11 + (Io - 11) = I]. + 12 (CI-4)
where
n h e-JkR cos¢ - cos %
12 = I f R dzdd) (C-S)
-n 0

The integrand of 12 is nonsingular since

e-JkR cos¢ - cos %

R20
Therefore the numerical integration of I2, hence IO, can
readily be carried out on the computer. Other singular
integrals encountered in Chapters VI and VII are handled

in the same manner.

APPENDIX D

THE INPUT RESISTANCE OF A VERY THIN
CYLINDRICAL ANTENNA IN A HOT

LOSSLESS PLASMA

 

1
‘ “‘31:-“

APPENDIX D

THE INPUT RESISTANCE OF A VERY THIN
CYLINDRICAL ANTENNA IN A HOT

LOSSLESS PLASMA

In this appendix, we consider the solution for the
zeroth order input resistance to a cylindrical antenna
immersed in an infinite plasma. It is shown that, under
appropriate assumptions, this resistance is the same as
that derived by Chen [23] using a poynting vector method.
The resistance can, under these assumptions, be broken
into a component due to the electromagnetic wave and a
component due to the electroacoustic wave.

From Chapter VI

h
_ _ ' _9. 2 f ' _ I
Rin - Real 3 a —-2-— 8111 Real '2 I)

Sin keh —h

x [cos keh K(0,z') - K(h,z')]dz' (D-l)

where

199

200

2

0.)
K(2.2') = 417 e(2:2') - m [Guam

- cos kezGP(O,z')

z
- ke ‘4 sin ke(z-z")GP(z",z')d{j} (D-2)

where

 

 

 

n .
1 exP[‘3k R] '
Ge(z.z') = 5;] R e a- 64» (Ia-3)
fl ex [-’k R ]
cp'<z.z') =§gf p; 9‘3ch (1)-4)
-n a
where
T
Ra = /Qz-z')2+ (2a sin #7): (D-S)
2 2
k 2 = wzu e l - —22—— - j we Y (D-6)
e o o w2+y2 w(w§+yi)
kz-wzl-ie-i-'l (1)-7)
P - 5 2 3 w '
Vo w

An assumption that we will make is that the antenna
is very thin so that a line current is assumed to flow
along the axis and the radius of the antenna, a, goes to
zero. A second assumption is that the plasma is lossless,

i.e., Y = 0.

201

With these assumptions, equations (D-2) through

(D-7) become

1 (”e
K(2.2') = I? e(Z.Z') - -7r[§P(z.2')
w

- cos kezGP(O,z')

z
- ke j; sin ke(z-z")GP(z",z')dzi] (D—8)

exp[-jkeRa]

 

 

 

I _ .—
Ge(2.z ) - R (D 9)
a
exp(-jk R )
G (2.2') = P a
P R (D-lO)
a
Ra = [2 - z'I (D-ll)
2
2 we
ke = w “060 l - —7?- (D-12)
w
2 (92 “63
k]? = —7 1 - '—2- , (ID-13)
V w
o
and the input resistance, Rin' may be written as
h
__ __2 2 f ' _ c I n
Rin — J g -jf1r———- s1n ke(h Iz |)T(z )dz
Sin keh -h
(D-l4)

where

202

 

 

 

 

1 sin Re 2' sin ke(h-z')
I .. _ ..
T(z ) - 4n cos keh zrii h_zu
(be? sin kP(h-z') sin RF 2'
+ 13?. h-z' - cos keh 2'

h sin kP(z"-z')
" ke‘fo sin ke (11'2") W d2" .
(D-15)

Let us rewirte Rin in terms of five separate

integrals and then examine each of the integrals separately.

 

 

 

 

 

 

With 7 = 0
Ho _ ' uoleo _ 120w
73 _ _ 2 , (D-lS)
2 2 2
/1 we/w /1 we /w
so
we2 k l
Rin = 60 l - —1f -T_§_—__' - cos keh I1 + 12
w Sin keh
2
we
- (”2 E23 - cos keh I4 - ke IS] (D-16)
where
h sin Re 2'
I1 = 1’ sin ke(h-|z'|) iT** dz' (D-l7)

-h

203

 

 

h sin ke(h-z')
_ ' _ l ' ..
12 - 2L Sln ke(h I2 I) h-z' dz (D 18)
I11 sin kP(h-z')
= ' _ v I _
I3 _h Sin ke(h |z I) h_z. dz (D 19)
h sin sz'
I4 = sin k (h - Iz'l) ——.—— dz' (1)-20)
-h e 2

h h
I = 1’ sin k (h Iz'I)j’ sin k (h-z")
-h e o 8

sin kP(z"-z')

x dz"dz'. (D-Zl)
lz"-z'$

First, let us look at 11' Using trigometric
identities and making some changes of integration vari-

ables, Il becomes

I1 = Sln keh Si(2ker) - cos keh Cin (Zkeh) (D—22)

where the Si and Cin functions are defined as

 

R .
_ Sln x _
O
and
c (R) =fR 1 ’ °°S x dx (1)-24)
in '

204

Similarly, it can be shown that
...1.- _
I - 2 Sln Zkeh[Si(4keh) Si(2keh{]

l
+ 5(1 + cos 2keh) Cin(2keh)

- % cos Zkeh Cin(4keh),
13 = % sin 2kehEsi(2kPh) - Si(kPh):l
I4 = 2 sin keh Si(kPh),
and
15 = Si(kPh)[: - i: sin Zkeh + h cos 2ken]

1 .
+ Si(2kph) 2k; Sln Zkeh h cos Zkeé]
where we have assumed that kP >> ke and kP >> 1.

Therefore

NIH

+ [1 + cos Zkeh] Cin(2keh)

+ 1

3 sin Zkeh[Si(4krh) - 28i(2keh)]

(D-ZS)

(D-26)

(D-27)

(D-28)

cos Zkeh Cin(4keh)

205

2
w
e l . .
+ 2F? 5 Sin Zkeh[Si(2kPh) - Si(kPhi]
. . 1 .
+ Sln 2keh Si(kPh) + keSi(kPh) h - k; s1n Zkeh

l .
+ h cos Zkeé] + keSi(2kPh)[%E; Sln Zkerx

h cos 2keh:| (D-29)

For the system we are discussing, i.e., antennas of

the order of wuoeoh = 1, kPh is a very large number. By

definition
_ 1 _
so we can reasonably use the approximation
~ w
Si(kph) 5 (D 31)

With this approximation, equation (D-29) becomes

-%
w
1
R. =3o[--ij -cos2th.(4kh)
1n 2 sin: keh e 1n e

+ 2[E + cos Zkeh] Cin(2keh) + Sin 2keh

w
_ .2. 1 -
X [Si(4keh) 28i(2keh£] + “,2 2 [sin 2keh

+ Zkeh] . (D-32)

206

Comparing equation (D-32) with the results of Chen
[23] we can identify an electromagnetic component of the

input resistance of a cylindrical dipole antenna as

2 '3:
w
- - _e 1 ..
Re - 30 l 2 __T— COS Zkeh Cin(4keh)
w Sln keh

+ 2E. + cos ZkelEl Cin(2keh) + $111 Zkeh

x EiMkeh) - 2 sicken] (D-33)

and a plasma component of the input resistance as

2'1:

(A) w
- —e— - i 1 .
RP - 15w 2 1 2 -T—2_—_—'2keh + Sln Zkegl'
w w Sin keh

(D-34)

Equations (D-33) and (D-34) are exactly the form of the

corresponding radiation resistances derived by Chen [23].

        

   

WNW

604

mm

  

145 2

W

 

M1

3

m;

129

   

11111111711