ABSTRACT
RADIATION OF SPHERICAL AND CY LINDRICAL ANTENNAS
IN INCOMPRESSIBLE AND COMPRESSIBLE PLASMAS
BY

Cheng- Chi Lin

The interaction of an antenna with a plasma becomes one of
the most interesting and important topics in science and engineering,
since the communication between a space vehicle and a ground sta-
tion has become involved with the ionosphere or an ionized gas. The
performance of an antenna in an ionized gas or plasma is entirely
different from its performance in free space. The purposes of this
study are to investigate theoretically and experimentally the radiation
of a spherical antenna and a cylindrical antenna when immersed in a
plasma and to detect the existence of an electroacoustic or a longi-
tudinal plasma wave excited by the antenna.

_ In the theoretical analysis, spherical and cylindrical dipole
antennas are used as the radiating sources. The surrounding plasma
is assumed to be a weakly ionized gas type and is treated either as
a lossy, cold (incompressible) plasma or as a lossy, hot (compres-
sible) plasma. Two rather different physical models and two different
sets of basic equations are adapted for these two kinds of plasmas.

When the antennas is immersed in a lossy, cold plasma, the plasma

CHENG - CHI LIN

is characterized as a lossy medium with equivalent permittivity and
conductivity. For this case, Maxwell's equations are adequate to
treat the problem. When the antenna is placed in a lossy, hot plasma,
the plasma is regarded as a one-component electron fluid with the
motion of positive ions neglected. The basic equations for this case
are Maxwell's equations and the linearized moment equations which
are derived from the Boltzmann equation assuming that the perturba-
tion of the plasma due to the source being small.

The spherical antenna imbedded in a lossy, cold plasma of
finite extent is studied first and then extended to the case of lossy,
hot plasma of finite extent. These two problems are solved directly
from the basic equations mentioned above and the radiated field in
the far-zone of antenna is obtained explicitly as a function of the
antenna dimensions and the plasma parameters.

The cylindrical dipole antenna immersed in a lossy, cold
plasma of infinite extent is examined next. King-Middleton's theory
and King's modified method are employed to determine the approxi-
mate current distribution on the antenna and after that the input im-
pedance of the antenna is determined as a function of the antenna
dimensions and the plasma parameters. This problem is not ex-
tended to the hot-plasma case because of its complexity in the

mathematical development. Finally, an existing theory of a cylin-

drical dipole antenna immersed in a lossless, hot plasma is briefly

CHENG-CHI LIN

reviewed. Extensive numerical results were obtained and compared
with the experimental results.

In order to conduct an extensive and accurate experimental
study on the interaction of an antenna with a hot plasma, a great deal
of time and effort was exerted to produce a large volume of stable,
high-density plasma. In our experiment, the hot plasma was pro-
vided by a mercury arc discharge which was created in three dif-
ferent plasma tubes. A novel method of placing a spot fixer in the
mercury pool was used to stabilize the plasma. The spherical and
cylindrical antennas were used as the radiating sources and they
were fed through a large ground plane. The radiation fields from
both antennas and the input impedance of the cylindrical antenna are
measured as a function of the plasma parameters. The hetrodyne
receiving system was used for the antenna radiation measurement
while the standard SWR method was adopted for the antenna impe-
dance measurement.

It is shown both theoretically and eXperimentally that the
radiation from a plasrna-coated spherical antenna can be enhanced
if the antenna is operated at a frequency much lower than the plasma
frequency and the dimensions of the antenna and the plasma layer
are apprOpriately chosen. This phenomenon may prove useful for

overcoming the blackout problem suffered by a reentry vehicle, or

CHE NG - CHI LIN

may offer a novel method of low-loss tuning of a small antenna.
The experimental results on the input impedance and the
radiation pattern of the cylindrical antenna tend to indicate that in
addition to the usual electromagnetic wave an electroacoustic or a
longitudinal plasma wave can be excited by the antenna in the hot

plasma.

RADIATION OF SPHERICAL AND CY LINDRICAL ANTENNAS
IN INCOMPRESSIBLE AND COMPRESSIBLE PLASMAS

BY

Cheng- Chi Lin

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering and Systems Science

1969

6’6// 7s/
3 -/ iv} 0

To my par ents

Mr. 8: Mrs. Chia-Chung Lin

ii

ACKNOWLEDGMENTS

The author is sincerely grateful to his major professor
Dr. K. M. Chen for his guidance and encouragement in the course
of this research.

He also wishes to thank the members of his guidance com-
mittee, Dr. G. Kemeny, Dr. B. Ho, Dr. J. Asmussen Jr. and
Dr. C. Y. Lo, for reading the dissertation and valuable suggestions.
The help extended by Mr. J. W. Hoffman of the Division of Engineer-
ing Research is also appreciated.

Finally, the author wishes to express a special thank to
his wife, Lee-Ching, for her encouragement, understanding and
patience.

The research reported in this dissertation was supported by

the National Science Foundation under Grant GK-lOZ6.

TABLE OF CONTENTS

Page
ACKNOWLEDGMENTS...................iii
LISTOFFIGURES.....................Vii
INTRODUCTION......................l

RADIATION FROM A SPHERICAL ANTENNA IMBEDDED
IN A LOSSY, COLD PLASMA OF FINITE EXTENT . . . 6

0‘

Introduction............. ...... ..
Geometry and Statement of the Problem . . . . . . 8
Solutions to Maxwell's Equations in the
PlasmaRegion....................10
2.4 Solutions to Maxwell's Equations in the Free-
SpaceRegion.....................l7
Z. 5 Boundary Conditions for Calculating the
RadiationFields...................18
2.6 NumericalResults..................Zl

NPN
(”NH

RADIATION FROM A SPHERICAL ANTENNA DviBEDDED
IN A LOSSY, HOT PLASMA OF FINITE EXTENT . . . . 33

3.1Introduction.....................33
3.2 Geometry and Statement of the Problem . . . . . . 34
3. 3 Fields in Dielectric Region (Plasma-Sheath
Region).......................36
3.4 Fields in Hot-Plasma Region . . . . . . . . . . . . 37
3.5 Fields in Free-Space Region . . . . . . . . . . . . 47
3.6 Matching of Boundary Conditions at Interfaces . . . 48
3.7 Radiated Field in Free-Space Region. . . . . . . . 54
3.8 NumericalResults..................56

iv

EXPERIIVIENTAL INVESTIGATION ON THE RADIATION
FROM A SPHERICAL ANTENNA IN A HOT PLASMA . . .

4.1Introduction......................
4.2 ExPerimentaISetup..................
4.3 Experimental Results and Comparison with
Theories.......................
4.4 Discussion.....................'.

RADIATION OF A CYLINDRICAL ANTENNA IMMERSED
IN A LOSSY, COLD PLASMA OF INFINITE EXTENT. . .

. Introduction. . . . . . . ...... . . ...... .
. Geometry and Statement of the Problem. . . . . . .
O Basic Equtions O O O O O O O O O O O O O ..... O

BoundaryConditions. . . . . . . . . . . . . . . . .
Integral Equation for the Current . . . . . . . . . .
Approximate Solution of the Integral Equation. . . .
Input Impedance of the Cylindrical Antenna . . . . .
NumericalResults..................

mmmpmmmm
mdo‘mhwwn—

EXPERIMENTAL INVESTIGATION ON THE RADIATION

OF A CYLINDRICAL ANTENNA IN A HOT PLASMA . . .
6. l IntroduCfi-ono O OOOOOO O O O O O O O O O O O O O O
6.2 Experimental Setups . . . . . . . ..... . . . . .

6. 2.1 Construction of a Large Volume of
Stable, High-Density Plasma . . . . . . . .
6. 2. 2 Experimental Setup for the Measurement
of Antenna Impedance in a Hot Plasma . . .
6. 2. 3 Experimental Setup for the Measurement
of Antenna Radiation Through a Hot Plasma.
3 Review of a Lossless, Hot-Plasma Theory . . . . .
4 Experimental Results and Comparison with
Theories.......................
6. 4.1 Comparison of EXperiment With a Lossless,
Hot-Plasma Theory on Antenna Input
Impedance........ ..........
6. 4. 2 Comparison of Experiment With a Lossy,
Cold-Plasma Theory on Antenna Input
Impedance..................

Page

77

77
78

81
82
88
88
90
92
95
96
99
106
108
115

115
116

116
117
120
123
127

128

130

Page

6.4. 3 Experimental Results on Antenna Radiation
FieldaO O O O O O O O O O O O O O O O O O O O O 132
6.5 Discussion..... ..... ............l33

REFERENCES. . . . .............. . . . . . .146

2.6

2.7

LIST OF FIGURES
Page

A spherical antenna covered by a lossy, cold
plasma layerO O O O O O O O O O O O O O O O O O O O O O 9

Theoretical radiation of a spherical antenna

(a = 0.635 cm) in a cold plasma (ll/211’ = o. 03 GHz)

driven at various frequencies as a function of
plasmadensity.............. ....... 36

Theoretical radiation of a spherical antenna

(a = 0.635 cm) in a cold plasma (ll/2n = 0. 003 GHz)

driven at various frequencies as a function of
plasmadensity..................... 27

Theoretical radiation of a spherical antenna

(a a 0.635 cm) in a cold plasma (v/er = 0.12 GHz)

driven at various frequencies as a function of
plasmadensity..................... 28

Effect of the electron collision frequency on the
enhancement of antenna radiation. . . . . . . . . . . 29

Effect of the thickness (d) of plasma layer on the
enhancement of antenna radiation. . . . . . . . . . . 30

Effect of the spherical antenna size on the enhance-
ment of antenna radiation with a fixed thickness of
plasmalayer(d=5.08cm).............. 31

Effect of the spherical antenna size on the enhance-
ment of antenna radiation with a fixed radius of
p1asmalayer(b=7.62cm).............. 32

A spherical antenna covered by a lossy, hot
pla.ma layerO O O O O O O O O O O O O O O O O O O O O O 35

Figure

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

3.10

Theoretical radiation of a spherical antenna

(a = Z. 54 cm) in a lossy, hot plasma (v/ 211' = 0.12GHz,
v /c = 0. 01) driven at various frequencies as a
fSnction of plasma density. . . . . . . . . . . . . . .

Theoretical radiation of a spherical antenna

(a = 2. 54 cm) in a lossy, hot plasma (v/ 21: = 0.12GHz,
v /c = 0. 01) driven at various frequencies as a
fgncti’onofplasmadensity. . . . . . . . . . . . . . .

Theoretical radiation of a spherical antenna

(a = 2. 54 cm) in a lossy, hot plasma (v/ 211': 0.12 GHz,
v /c = 0. 01) driven at various frequencies as a
fli’nctgon of plasma density. . . . . . . . . . . . . . .

Theoretical radiation of a spherical antenna

(a = 1. 27 cm) in a lossy, hot plasma (v/ 217: 0.12 GHz,
v /c = 0. 01) driven at various frequencies as a
fenctgon of plasma density. . . . . . . . . . . . . . .

Theoretical radiation of a spherical antenna

(a = 1. 27 cm) in a lossy, hot plasma (v/ 211': 0. 12 GHz,
v /c = 0. 01) driven at various frequencies as a
f8ncti’on of plasma density. . . . . . . . . . . . . . .

Theoretical radiation of a spherical antenna

(a = 1. 27 cm) in a lossy, hot plasma (v/ 21:: 0.12 GHz,
v /c = 0. 01) driven at various frequencies as a
func onofplasmadensity. . . . . . . . . . . . . . .

Theoretical radiation of a spherical antenna

(a = Z. 54 cm) in a lossy, hot plasma (v/ Zn: 0. 03 GHz,
v /co = 0. 01) driven at 0.4 GHz as a function of
pasmadensity... ..... ..

Theoretical radiation of a spherical antenna

(a = 3. 81 cm) in a lossy, hot plasma (v/ 211': o. 03 GHz,
v /c = o. 01) driven at 0.4 GHz as a function of
paamadenaity....................

Theoretical radiation of a spherical antenna

(a = 5. 08 cm) in a lossy, hot plasma (v/ 21:: 0. 03 GHz,
v /c = 0.01) driven at 0.4 GHz as a function of
pasmadensity....................

Page

63

64

65

66

67

68

69

70

71

Figure

3.11

3.12

3.13

3.14

3.15

4.1

4.2

4.3

4.4

4.5

Theoretical radiation of a spherical antenna

(a = 6. 35 cm) in a lossy, hot plasma (u/ an: o. 03 GHz,
v /co = 0.01) driven at 0.4 GHz as a function of
pasmadensity.....................

Theoretical radiation of a spherical antenna

(a = 2. 54 cm) in a lossy, hot plasma (v/ 211': 0. 003 GHz,
v /c = 0.01) driven at 0.4 GHz as a function of
pasmadensity.....................

Theoretical radiation of a spherical antenna

(a = 3. 81 cm) in a lossy, hot plasma (v/ 211': 0. 003 GHz,
v /co = 0. 01) driven at 0.4 GHz as a function of
pasmadensity.....................

Theoretical radiation of a spherical antenna

(a = 5. 08 cm) in a lossy, hot plasma (ll/211': 0. 003GHz,
v /co = 0.01) driven at 0.4 GHz as a function of
pasmadensity.....................

Theoretical radiation of a spherical antenna

(a = 6. 35 cm) in a lossy, hot plasma (v/ 211’: 0. 003GHz,
v /co = O. 01) driven at 0.4 GHz as a function of
pasmadensity.....................

Experimental setup for the radiation measurement
ofsphericalantenna. . . . . . . . . . . . . . . . . .

A hemi-spherical plasma tube (3-inch radius)
under operation inside of a microwave anechoic
Chamber O O O O O O O O O O O O O O O O O O O O O O O O

Experimental setup outside of a microwave
anechoicchamber...................

Experimental and theoretical radiation of a spherical
antenna (a = 2. 54 cm) in a plasma driven at various
frequencies as a function of plasma density . . . . .

Experimental and theoretical radiation of a spherical
antenna (a = 1. 27 cm) in a plasma driven at various
frequencies as a function of plasma density . . . . .

Page

72

73

74

75

76

79

80

80

84

85

Figure Page

4.6 Experimental and theoretical radiation of a spherical
antenna (a = 2. 54 cm) in a hot plasma driven at various
frequencies as a function of plasma density. . . . . . 86

4. 7 Experimental and theoretical radiation of a spherical
antenna (a = l. 27 cm) in a hot plasma driven at various
frequencies as a function of plasma density. . . . . . 87

5. l A cylindrical antenna immersed in an infinite,
108 BY and Cald Plasma O O O O O O O O O O O O O O O O O 91

5. 2 Theoretical input impedance of a monopole
(h/x = 0. 313, a/x = 0. 008) in a cold plasma as
afunctionofplasmai’density. . . . . . . . . . . . . . 111

5. 3 Theoretical input impedance of a monopole
(h/x = O. 282, a/x = 0. 0072) in a cold plasma as
afunctionof plasmgdensity. . . . . . . . . . . . . . 112

5.4 Theoretical input impedance of a monopole
(ll/X = 0. 213, a/ x0 = 0. 008) in a cold plasma as
afunction of plasma density. . . . . . . . . . . . . . 113

5. 5 Theoretical input impedance of a monopole
(h/X = 0.192, a/X = 0. 0072) in a cold plasma
as acfunction of plagma density . . . . . . . . . . . . 114

6. 1 Experimental setup for the impedance measure-
ment of cylindrical antenna . . . . . . . . . . . . . . 118

6. 2 A cylindrical plasma tube (14-inch diameter by
18-inch length) under operation and the surrounding
experimentalsetup.................. 119

6. 3 Experimental setup for the radiation measure-
ment of cylindrical antenna . . . . . . . . . . . . . . 121

6. 4 A cylindrical plasma tube (6-inch diameter by lZ-inch
length) under Operation inside of a microwave ane-
Choic chamber O O O O O 0 O O O O O O O O O O O O O O O 122

Figure

6.6

6.7

6.8

6.9

6.10

6.11

6.12

6.13

6.14

6.15

Experimental and theoretical input impedance
of a monopole (h/x : 0. 313, a/ X0 = 0.008)
in a hot plasma as a function of plasma density. .

Experimental and theoretical input impedance
of a monopole 01/ X0 = 0.282, a/x = 0.0072)
in a hot plasma as a function of plasma density.

Experimental and theoretical input impedance
of a monopole (h/x = 0. 213, a/X : 0.008)
in a hot plasma as 2 function of plasma density. .

Experimental and theoretical input impedance
of a monopole (h/x = 0.192, a/x = 0.0072)

in a hot plasma as a function of plasma density. . .

Experimental and theoretical input impedance
of a monopole (h/x = 0.313, a/X = 0.008)
in a plasma as a function of plasma density. . . .

EXperimental and theoretical input impedance

of a monopole (h/X = 0.282, a/k = 0.0072)

O O O O

in a plasma as a function of plasma density. . . .

Experimental and theoretical input impedance
of a monopole (h/X : 0. 213, a/ x0 = 0.008)
in a plasma as a function of plasma density. . . .

Experimental and theoretical input impedance of
a monopole (h/X = 0.192, a/x = 0. 0072) in a
plasma as a func‘fion of plasma%ensity . . . . . .

Experimental radiation patterns of a monopole
(h/x = 0.6, a/x = 0.008) surrounded by a hot

Page

O O 135

. . 136

. . 137

e 138

. . 139

O O 140

. . 141

O O 142

plasrgla of various densities. . . . . . . . . . . . . . 143

Experimental radiation patterns of a monopole
(h/X = 0.66, a/x = 0. 008) surrounded by a hot

plasma of various ensities. . . . . . . . . . . . . . 144

Experimental radiation patterns of a monopole
01/). = 0.233, a/X = O. 008) surrounded by a hot
plasma of various ensities. . . . . . . . . . . .

xi

O O 145

CHAPTER 1

IN TR OD UC T ION

The research described in this dissertation deals with the
radiation of spherical and cylindrical antennas imbedded in incom-
pressible and compressible plasmas. The first part of the disser-
tation studies the radiation from a spherical antenna when it is
covered by a finite layer of plasma. This study is motivated by
a newly discovered phenomenon that the antenna radiation can be
enhanced if the antenna frequency is lower than the plasma frequency
of the coating plasma layer. The second part of the dissertation
investigates the interaction of a cylindrical antenna with a plasma.
One of the main objectives of this investigation is to detect the
excitation of an electroacoustic wave by an antenna and the effect
of this wave on the characteristics of the antenna. Additional intro-
ductions on these studies are given below.

It is well known that when an antenna is covered by a layer
of plasma with a plasma frequency higher than the antenna frequency,
the antenna radiation is reduced drastically. The conventional
approach to overcome this blackout phenomenon is to raise the

antenna frequency to exceed the plasma frequency of the plasma

volume. This approach is usually hampered by the practical
limitation of available high-frequency sources. In the first part of
this dissertation, a new phenomenon on the enhanced radiation from
a plasma-coated spherical antenna is studied. It is shown both
theoretically and experimentally that the radiation from a spherical
antenna covered by a spherical layer of plasma can be enhanced if
the antenna frequency is adjusted to be much lower than the plasma
frequency of the plasma layer and the dimensions of the antenna and
the plasma layer are properly chosen.

The phenomenon of the enhanced radiation from a small
antenna covered by a plasma layer was reported first by Messiaen
and Vandenplas‘l) in 1967. These authors also predicted a series
of resonance peaks on the antenna radiation and a significant effect

(2' 3) studied

of the plasma sheath on the radiation. Chen and Lin
the same phenomenon on a cylindrical antenna of various lengths
covered by a finite volume of lossy, hot plasma. Instead of find-

ing a series of resonance peaks, they observed a strong enhancement
on the antenna radiation over a wide band of antenna frequencies which
are much lower than the plasma frequency of the plasma volume. They
also found a negligible effect on the antenna radiation due to the plasma
sheath or the DC potential of the antenna. Although the phenomena of
enhanced radiation observed by Messiaen and Vandenplas and by Chen
and Lin are similar, the detailed results are different. The effects

of the dimensions of the antenna and the plasma layer on the pheno-

menon of enhanced radiation have not been investigated before. In

this study the phenomenon of enhanced radiation is examined more
carefully by conducting both theoretical and eXperimental studies on
a plasma-coated spherical antenna. The spherical geometry is
adopted to make the theoretical study tractable. In the experiment
an imaged hemisphere on a ground plane was used as an antenna and
a mercury arc discharge was used as the coating plasma. In the
theoretical models, the spherical antenna is first assumed to be
covered by a lossy, cold plasma layer with the plasma sheath on
the antenna surface ignored. The study is then extended to a lossy,
hot plasma layer with the plasma sheath included. The theoretical
results based on these models are in satisfactory agreement with
the experimental observation.

When an antenna on a reentry vehicle is covered by a plasma
layer and suffers blackout, a possible scheme of overcoming this
problem will be to reduce the antenna frequency to a value which is
much lower than the plasma frequency and in the range for the en-
hanced radiation. Another potential application of this phenomenon
is the low-loss tuning of a small antenna. Theoretically, the plasma
can be made lossless so that this tuning scheme may prove to be more
effective than any conventional impedance tuning.

The second part of the dissertation investigates the interaction
of a cylindrical antenna with a plasma. When an electromagnetic
radiating source is immersed in a hot plasma, an electroacoustic

or a longitudinal plasma wave may be excited in addition to the usual

electromagnetic wave. Numerous theoretical papers have been

(4)

published on the subject. Cohen was the first one to show the

possible excitation of an electroacoustic wave by a radiating source

(5) (6) studied the

in a hot plasma. Hessel and Shamoys and Fejer
. . (7) . .

electroacoustic wave exc1ted by a small source. Chen investi-

gated the effect of an erectroacoustic wave on the radiation of a

(3)

cylindrical antenna. Wait examined the electroacoustic wave
excited by a slotted-sphere antenna. There are many other theore-
tical papers which are not mentioned here.

In contrast with the abundance of theoretical papers, extremely
few experimental studies have been published. The relevant experi-
mental studies are the observation of electroacoustic wave by
Whale”) in a rocket flight, Schmitt'suo) observation of Tonks-
Dattner's resonance excited by an antenna, and the experiments on
a cylindrical antenna in a hot plasma conducted by Jassby and

2
(11) and by Chen, Jackson and Lin‘1 ' 13). Nevertheless.

Bachyski
to our best knowledge, no extensive experimental study has been
conducted to study the electroacoustic wave excited by an antenna
in a hot plasma.

In this part of study an extensive experimental investigation
was conducted to: (I) detect the existence of an electroacoustic
wave in a hot plasma, and (2) study the effect of this electroacoustic

wave on the circuit and the radiation properties of an antenna. Two

approaches: (1) to measure the antenna input impedance as a

function of the plasma parameters, and (2) to measure the antenna
radiation field as a function of the plasma parameters, have been used
to detect the electroacoustic wave. In our experiment, the hot plasma
was provided by a mercury arc discharge which was created in two
plasma tubes. The cylindrical monopole antennas were used as the
radiating sources.

A simple theory on a cylindrical antenna immersed in a lossy,
cold plasma of infinite extent is developed. The antenna input impe-
dance is obtained as a function of the antenna dimensions and plasma
parameters. The significant finding of this theoretical study is the
observation of a peak resistance and a change of reactance from
capactiviet to inductive when the antenna frequency approaches to
the plasma frequency. The comparison between this theory and
the experiment indicates that the effect of the electron collision
frequency on the antenna input impedance is significant in the neigh-
borhood of plasma frequency.

The experimental results are also compared with another set
of theoretical results obtained by Chen”). The agreements on the
antenna input resistance and the observation of peak radiation near
the axial direction of the antenna tend to indicate the existence of an

electroacoustic wave excited by the antenna and its significant effect

on the characteristics of the antenna.

CHAPTER 2

RADIATION FROM A SPHERICAL ANTENNA IMBEDDED
IN A LOSSY, COLD PLASMA OF FINITE EXTENT

2. 1 Introduction

The radiation of a spherical antenna when surrounded by a
layer of lossy, cold plasma is studied in this chapter. This study
is motivated by a newly discovered phenomenon which indicates
that the radiation from an antenna can be enhanced if the plasma
frequency of the plasma is considerably higher than the antenna
frequency and the dimensions of the antenna and the plasma layer
are appropriately chosen.

When an antenna is covered by a layer of plasma with a
plasma frequency higher than the antenna frequency, the antenna
radiation is reduced drastically, a phenomenon known as the black-
out. The conventional approach to overcome this blackout phenome-
non is to raise the antenna frequency to exceed the plasma frequency
of the plasma. This approach is usually hampered by the practical
limitation of available high frequency sources. The new phenomenon

on the enhanced radiation from a plasma- coated antenna to be

studied in this chapter may serve as a solution to the blackout
problem encountered by a reentry antenna.

The phenomenon of the enhanced radiation from a small
antenna covered by a plasma layer was reported first by Messiaen
and Vandenplas(l) in 1967. These authors also predicted a series
of resonance peaks on the antenna radiation and a significant effect

)

of the plasma sheath on the radiation. Chen and Lin(z’ 3 have investi-
gated the same phenomenon on a cylindrical antenna of various
lengths covered by a finite volume of lossy, hot plasma. Instead

of finding a series of resonance peaks, they observed a strong en-
hancement on the antenna radiation over a wide band of antenna
frequencies which are much lower than the plasma frequency of

the plasma volume. They also found a negligible effect on the
antenna radiation due to the plasma sheath or DC potential of the
antenna. Although the phenomena of enhanced radiation observed
by Messiaen and Vandenplas and Chen and Lin are similar, the
detailed results are different. Neither the effects of the dimensions
of antenna nor the size of the plasma layer on the phenomenon of
enhanced radiation have been investigated before. This unusual
phenomenon of enhanced radiation to be studied in this chapter

may be attributed to the coating of antenna with a dielectric of

negative permittivity which is the simple model of an overdense

plasma.

In our study, the spherical geometry is adopted for the sake
of making the theoretical study tractable. In this theoretical model,
the spherical antenna is assumed to be covered by an uniform, lossy
and cold plasma and the plasma sheath on the antenna surface is
ignored.

Based on the finding of our study in this chapter, it appears
feasible that when an antenna on a reentry vehicle is covered by a
plasma layer and suffers a blackout, a possible scheme of over-
coming this problem will be to reduce the antenna frequency to a
value which is much lower than the plasma frequency and in the
range for the enhanced radiation. Another potential application of
this phenomenon is the low-loss tuning of a small antenna. Since
only a lossless plasma with a negative permittivity is needed for
this purpose, this tuning scheme may prove to be more efficient

than any conventional impedance tuning.

2. 2 Geometry and Statement of the Problem

The geometry of the problem is shown in Fig. 2. l. A
spherical antenna of radius a is covered by a spherical layer of
uniform lossy plasma with a thickness of b-a. The antenna sur-
face is perfectly conducting except for a narrow equatorial gap

between w/Z - 91 5 0 _<_ 1r/Z + 9 Across this gap a voltage of

l O
amplitude V and frequency a) is applied. The spherical coordinates

(7, 0, 4’) are adopted and the rotational symmetry is assumed. The

Z

I e
/
\fir. 6. 4n

  

 

 

..-... y

11

Region 1: cold plasma ((10, E, 0' )

Region II: free space ((10, 60)

Fig. 2. l A spherical antenna covered by a lossy,
cold plasma layer.

10

plasma is assumed to be a weakly ionized gas and canzbe considered

 

a)
as a lossy medium with a permittivity of E = 60(1 - JZLZ)’ a conducti-
noeZ v 00W
vity of 0' = 2 2 , and a permeability of p = "'0 where a) and
e a) + u

(up are the antenna and plasma frequencies, e and me are the charge
and mass of electrons. no is the density of plasma, V is the collision
frequency of electrons with neutral particles, and 60 and “o are the
permittivity and permeability of free space. The total space exclud-
ing the antenna is divided into two regions. Region I is the plasma
layer and the rest of the free space is Region II.

The assumption of infinitesimfl driving gap is made to simpli-
fy the problem because only the radiated fields of the antenna are to
be sought in this study. If the input impedance of the antenna is
also to be determined, the assumption of a finite driving gap is
needed to avoid the divergence of some series appeared in the mathe-

matical expr e s sion s .

2. 3 Solutions to Maxwell's Equations in the Plasma Region

The basic equations which govern the system are Maxwell's
equations. Maxwell's equations in Region I (plasma layer, a E r E b)
are

* *

V : .’
x El Jaw-0H1 (Z. 1)
inil a mail (2.2)

where E and H are the electric and magnetic fields and § is the

complex permittivity given by

ll

§=€(l-jf-€-). (2.3)

The suppressed time dependence is exp(joot).

From the symmetry of the antenna it can be seen that there
is no variation in the 4> direction and the magnetic field has only a
4) component. Thus, Eqs. (2.1) and (2. 2) can be easily reduced to

three scalar equations such as

.9. E 3’51: _ . H z 4
3r (r 19) " 89 - - J60 “or l¢ ( 0 )
l . .
m 58(8meHlo) = 1°“; E1.- “-5)
8
--a-;(rH1¢) = ngrEle . (2.6)

Differentiating Eqs. (2. 5) and (Z. 6), and substituting them into
Eq. (2.4) leads to a partial differential equation,

2

 

8 1 8 1 8 , 2 _
:2- (rchb) + -Z' ‘33 [m '55 (81n9rH1¢)]+ k (rHl¢) - 0
r
(2. 7)
where k is the complex prOpagation constant given by
k2 = mzuog . (2.8)
If we write
k = 8 - jo. , (2.9)
8 and a. can be expressed as
i l
2 2 4 ’- 3
50 a) 20) up
Sufi 1-—L—Z 2+ l--—LZ + (2.10)

2
(DH! w+v w(w+V)

12

 

I 1
Z Z '2' E
50 (0 200 mp4
a, = :2? -l+—£—Z 2+ 1-—-E—Z 2+ 2 Z Z (2011)
(1) HI u) +v w (w +1!)
with

: 'J . 2. 2
BO 0) H060 ( l )

To solve Eq. (2. 7), we use the method of the separation of variables.

Since H is independent of 4), we can assume

1¢

rHl¢ = R(r)@(9) (2.13)

where R is a function of r alone and 9 is a function of 0 only. The

substitution of Eq. (2.13) in Eq. (2. 7) leads to

Z Z
r d R z 2 1 d [ l d .
Z
R dr _ 6 d6 sine d6 (2.14)

where n is any integer. Equation (2. 14) generates two ordinary

diff erential equations,

 

d l d . ..

d—e- [m “56' (6811199 '1' n(n+l)9 - 0 (2015)
Z d2

E... R + 1(er - n(n+l) = 0 . (2016)
R drz

Let us consider Eq. (2. 15) first. Making the substitutions,

u: cose, ’Jl-u =sin9, d =- l-ug- .

:18 du

Eq. (2. 15) can be reduced to

ug—f+[n(n+l)- 12] 9: 0. (2.17)

 

——-2

du

Z dze
(l'u) Z

13

Equation (2. 17) is a special form of the associated Legendre's

 

equation,
2 2
d
(l - xz)—Z - 2x91 + [n(n+l) - m ]y = 0. (Z. 18)
2 dx 2
dx 1 -x

The solution to Eq. (2.18) is
m
Y - Pn (X)

and this function is called an associated Legendre function of the
first kind of order n and degree m. These functions are actually

related to the ordinary Legendre functions Pn(x) by the relation,

9- de (x)
Pm(x) = (-l)m(l-xz)2 ——’-‘-—- . (2.19)
n dxm

In order to have finite solutions on the interval -1 E x E 1
the parameter n must be zero or a positive integer and that the
integer m can take on only the values -n, -(n-1), . . ., 0, . . ., (n-l),
n, i.e., n_>_ 1m).

Thus a solution to Eq. (2. 17) can be obtained as
e = Pl(u) = Pl(cos s) (2, 20)
n n

where 11 must be a positive integer and n z 1.

Note that only one solution for this second-order differential
equation (2. 17) has been considered. The other solution becomes
infinite on the axis, and so it should be excluded from this problem

since the axis is included in the geometry.

14

Other properties of the associated and ordinary Legendre
functions that will be useful to us in the later development and

numerical calculation are listed as follows:

1. P (cos 9) is zero at e = "/2 if n is even.

51—051—

2. P (cos 9) is maximum at e = "/2 if n is odd, and

the value of this maximum is given by

N
"1
Nlb
+

forn= 1,5,9...

"1
NI:
4.
Nl" 0-0

(2.21)

N
2
NH:
+

forn= 3,7,11000

H
NH!
4.
N'O—i H

or n
1 2 4 I‘(E+l) 2
[P (08 = — -——-— for n : odd (2.22)
n 1r n 1
I12 +2)

where I‘(x) is the Gamma function with argument x.

3. The associated Legendre functions have orthogonality

propertie 8,

+1 1 1 0 for n ¥ 1
S Pn(u) P! (u) du = (Z. 23)
-1 Zn(n+1)

2n+1 for n = l .

4. A recurrence formula for the ordinary Legendre

functions is

dPn+1 (u) dPn(u)

du 11 du

 

- (n+1) Pn(u) = 0, (2.24)

15

and from (2.19)

d
P;(cos 9) = 35- Pn(cos 9) . (2. 25)

Combining Eqs. (2. 24) and (2. 25) we obtain an expression,

 

sine [€089 Pikos 9) - Pi+l(Cose)] -.- (n+l)Pn(cos6). (2.26)

5. The differentiation formula is

d l l 1
de- [pn(cos 9)] = silne [n Pn+l(C°8 9) - (n+l)cosGPn(cosG)] .

(Z. 27)

 

Going back to the differential equation (2. 16), we make the substi-

tution,

R = 3} . (2.28)
1.2

Equation (2. 16) then becomes

 

 

dle 1 de 2 (n+-;-)2
- - ._____ = 2 2
Z + 1' dr + k 2 R1 0 . ( . 9)
dr r

This is a form of Bessel's equation,

2 2

_1d .1 L” + (k2- _“ )y = o. (2.30)
2 x dx 2

dx x

The solution to Eq. (2.29) can then be expressed as

v

R = A H“) (kr) + B H(1

2.
l n n+§ n n (kt) ( 31)

NT...

2
where A and B are arbitrary constants, H(l)1(kr) and H‘ :(kr)
n n n+5 n+5

are the Hankel functions of the first and second kinds with order

l6

n+%, which represent the radially inward and outward traveling
waves respectively. Combining Eqs. (2. l3), (2, 20), (2. 28) and

(2. 31) we have

H - J- 020 P1(cosB) A Hm (kr) +13 Hm (kr) (2 32)
1¢_~/—rn:l n nn+§ nn-i—g: ° '

The other two components of the electric field can be determined
easily from Eqs. (2. 5) and (2. 6).
Substituting Eq. (2. 32) into Eq. (2. 5) and using Eqs. (2.27)

and (2. 26) we obtain

(1)
_ ' (2) (1)
Elr _ _177 z n(n+l)Pn(cosO)[Aan+%(kr)+Ban+%(kr)].

(D a 1' 11:1
(2. 33)
To derive E19, two differentiation formulas of the Hankel functions
5L (1) _ n+1 (1) (1)
dx Hug: (X) - - 4x Hn+%(x) + Hn_%(x) (2. 34)
d (2) n+1 (2) (2)
... ___ _ a 2, 3
dx Hn+_;_(X) x Hn+%(x) + Hn-%(x) ( 5)

are needed.
The substitution of Eq. (2. 32) into Eq. (2.6) and the utili-

zation of Eqs. (2. 34) and (2. 35) lead to

A
N

. (n 1
E16: 3 2 Z Pn(cosO) An[nI-I

) (kr) - kr Ha) (kr)]
mgr n=1

A
11";

~12.

n

1 1
+ Bn [nHr(1+)%(kr) - kr H;_)%(kr)] . (2. 36)

The solutions to Maxwell's equations in this region under rotational

symmetry can thus be summarized as

17

(D

_ .1. (2 ) (1)
Hl¢(r,6) - 'fr r:11:n (cos 9) An Hn +%(kr) + Ban+%(kr)] (2,32)
°° (2 ) (1)
(r,9) — 3 2 2‘. n(n«l-1)Pn (cos 6)[A H +%(kr)+Bn Hn +2}er
mgr n=1
(2.33)
-' °° 1 (2) (2)
le(r,9): —LIW_2' Z: Pn(cos 6){A nn+3[nH (kr)- er n_%(krfl
mgr n=1
+ Bn nH(l) (kr) .. kr H( )(kr) (2 36)
an+% n-i '
and
E14): lee H19: 0. (2.37)

2.4 Solutions to Maxwell's Equations in the Free-Space Region

Maxwell's equations in Region 11 (free space, r 2 b) are

v = -' 7
x EZ 3(1)}:on (2. 38)
V -o = _ ->

Since Region II is unbounded, no reflected or inward traveling wave
exists in this region. Following the same analysis as in Sec. 2. 3,

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

H 0 - 0; C P1 0 Ha) 2 40
“(a ) - (fr M n n(cos ) “yuan (. )
°° (2)
Ezr(r,0) = 3 2 2 Cnn(n+l)Pn(cose)Hn+%(fior) (2.41)
(0601' n=1
_. oo 12)
Eze(r, 6) = 3 2 2 CnPn(cos 0)[nH an+%(flor)- 80 anz _;(p or]
wear n=1

(2. 42)

18

where Cn is an arbitrary constant, 11 is a positive integer, and 80

is the wave number of free space given by Eq. (2.12).

2. 5 Boundary Conditions for Calculating the Radiation Fields

 

In order to determine the constants An’ Bn and Cu, the
boundary conditions at r = a and r = b are applied.
The voltage applied across the gap is given by

2 1 1r
E16(a' 9)a d9 = S; Ele(a,6)ad6 , (2.43)

because E19 is zero on the surface of the conducting sphere except

at the gap between; - 91 _<_ 6 5% + 91. Since the Legendre polynomials
form a complete set of orthogonal functions, any function f(x) on the
interval -1 f x f 1 can be expanded in terms of them. The electric
field on the surface of the sphere can thus be expanded in a series

of Legendre polynomials as

a)

1 o.
Ele(a, 9) .. 2 ann(cosO) (2.44)
n=1
where
2n+1 " 1
: —— C 0 2’
bn 2110”“ So Ele(a, 6)Pn(cos 0) 81119 d6 ( 45)

If the gap between the two halves of the sphere is assumed to be

small,
P111(cos6) as Piw) 123- algegg+sl
(2.46)
sine Rs 1 9 is small .

1

19

Combining Eqs. (2.43), (2.45) and (2.46), we have

(2n+l) P:(0)V
bn -.- 2n(n+1) a (2.47)

 

From Eq. (2, 36) we obtain

00
_ 1' 1 (2) (2)
Ele(a, 9) _ mg 3 2 Z Pn(cos 9){An[an+%(ka) - kaHn_%(ka)]

a n=1

(1) (l)
+ Bn [an+%(ka) - ka Hn_%(ka)]} . (2.48)
The combination of Eqs. (2.44), (2.47) and (2.48) gives

FA+FB=FV (2.49)
1n n

2 3

where

nH(2) (ka) - ka Ha) (ka)

rr)
n

l n+§ n-%
_ (l) (1)
F2 — an+%(ka) - kaHn-%(ka)

2n+l

. 1
F : 3005 Jet Pn(0)m .

The continuity of the tangential components of E and H fields at the

plasma-free space interface (rzb) leads to

E16(b’e) = Eze(b.9) (2.50)

Hl¢(b,0) HZ¢(b,9) . (2.51)

Using Eqs. (2. 36), and (2.42), Eq. (2.50) gives

0 = o (2. 52)

FA+F
n 6n

4 Bn"F

5

where

(kb) - kb H“
n-

F = nH(Z)

)
. 4 n %(kb)

.612.

20

HO)

(1)
5 an+%(kb) - kan %(kb)

F6 = é—[nH H”) (a b) - a 10H"?; (a In].

n+§o

From Eqs. (2. 32) and (2.40), Eq. (2. 51) can be expressed as

- = 2. 3
F7An+F8Bn F9Cn 0 ( 5)

wher e

7 n+§
_ (1)
F8 - Hn+%(kb)
_ (2)
F9 - Hn+%(pob)'

The notations Fi' i = l, . . . , 9, are used just for convenience. Thus,
the constants An, Bn and Cn can be solved from Eqs. (2. 49), (2. 52)

and (2. 53) as

 

 

 

VF 36F(F F F )
_ 8 5 9
An _ A (2. 54)
VF 341:"(F -F F )
_ 9 6 7
B11 — A (2. 55)
VF 34F(F F F )
8 5 7
= 2
Cu A ( .56)
where
A = F 1(F6F 8- F94F)+F2(F 9- F6F7) . (2.57)

Up to this point the E and H fields in Regions I and II are completely

determined as functions of r and 6.

21

The quantity of main interest in this study is the radiated

field in the far zone of the antenna. Thus, the field E at (rzR,

29

8:900) can be expressed as

, oo F(FF-FF)
o _ -v 3 4 8 5 7 1 (2
E2961, 90 )- —L-§7-2 E A 1311(0) [an
wER

n=1
(n=odd)

v

(l3 R)

 

.51”...
o

(2)
- (30R Hm

mom] . (z. 58)

NIH

It can also be rearranged as

 

n 2
E (R 900)_ 4§~faV :3) 2n+1 “2“) (F4Fs' F5F7’
. _ __ _____.
20 116 R372 n=1 2n(n+1) F(2+'l") A
o 2 2
(n=odd)

, (Z)
[n Hn+%(poR) - (30R Hn_%(floR)] .
(2. 59)

2. 6 Numerical Results

 

The radiated power from the antenna which is proportional to
the square of E62(R' 90°) as expressed in Eq. (2. 59) has been numeri-
cally calculated as a function of the antenna and plasma parameters.
Figure 2. 2 shows the radiated power from a spherical antenna of
0.635 cm radius driven at various frequencies as a function of the
plasma density of the plasma layer. The radius of the spherical
plasma layer is 7. 62 cm and the electron collision frequency is
assumed to be 0. 03 GHz. The distance between the radiating and
receiving antennas is 0. 7 m. In this figure the radiated power at

each driven frequency is normalized to its free-space radiated power.

22

At all antenna driven frequencies we invariably observe that the
antenna radiation is reduced as the plasma density is increased

and it reaches the cut-off point as the plasma frequency is increased
to the neighborhood of the antenna frequency. After the plasma
frequency exceeds the antenna frequency, the antenna radiation
starts to build up somewhat for higher antenna frequency cases

(e. 3.. 1. 8 ~ 1. 4 GHz). This trend becomes more outstanding for
lower frequency cases. For example, when the antenna frequency

is 0. 8 GHz, the antenna radiation builds up to the level of fr ee-space
radiation after going through the cut-off. If the antenna frequency

is further decreased to 0. 4 or 0. 3 GHz, the antenna radiation can
build up to a level about 20 db higher than the fr ee- space radiation
after passing the cut-off. The phenomenon of this enhanced radia-
tion is the most interesting finding of this study. Physically it means
that if an antenna is operated at a frequency much lower than the
plasma frequency of the plasma layer, its radiation will recover
from the cut-off and then be enhanced greatly over the free- space
radiation level . This phenomenon can be applied directly to solve
the blackout phenomenon by simply scaling down the driving frequency
of the vehicle antenna when its radiation is cut off by a surrounding
plasma layer. The phenomenon of this enhanced radiation probably
can be attributed to the tuning effect of the plasma layer on the
antenna input impedance. When the antenna frequency is lower than

the plasma frequency, the equivalent permittivity of the plasma is

23

negative and the plasma layer may act as an inductor to tune out the
large capacitive reactance of the input impedance of a small antenna.
This over simplified eXplanation is by no means adequate as we can
see from more results in other figures.

Figure 2. 3 shows the similar phenomenon of Fig. 2. 2 for the
case of lower electron collision frequency (0. 003 GHz). Since the
loss of the plasma is lower, the antenna radiation is cut off more
drastically and the radiation is enhanced greater for lower antenna
frequency cases. Figure 2.4 also shows the similar phenomenon of
Fig. 2. 2 but for a higher electron collision frequency (0. 12 GHz).
This figure indicates that a higher loss in plasma makes the cut-
off phenomenon less outstanding and the enhancement of radiation
slightly lower.

To investigate the effect of the electron collision frequency
on the phenomenon of enhanced radiation, the same antenna (0. 635 cm
radius) with the same plasma layer (7. 62 cm radius) driven at O. 3
GHz is considered. The antenna radiation is plotted as a function of
pr/wz for various v/w in Fig. 2. 5. It is evident that for a lower
collision frequency higher enhancement of radiation is obtained at
a lower plasma density. If the collision frequency is increased, a
lower enhancement of radiation is obtained at a higher plasma density
and over a wide range of (spa/m2.

The effect of the thickness of plasma layer on the phenomenon

of enhanced radiation is shown in Fig. 2.6. The same antenna

 

-~a..n-

um

ligh- 33.1.5.
L .1

.1.‘ x (d V

 

24

(0. 635 cm radius) is driven at 0.4 GHz and the electron collision
frequency is assumed to be 0. 03 GHz. The antenna radiation is
plotted as a function of pr/mz for various values of plasma layer
thickness. When the thickness of the plasma layer is the same as
the antenna radius (0.635 cm), an enhancement of 15 db is obtained.
A maximum enhancement of about 23 db is obtained when the thick-
ness of the plasma layer is about 1 3/4 inches. As the thickness

is further increased, the enhancement of radiation decreases. When
the thickness of the plasma layer approaches to infinity, the antenna
radiation remains zero after passing the cut-off point. This point
is expected since the phenomenon of enhanced radiation does not
occur if the antenna is placed in a plasma of infinite extent.

The effect of the antenna size on the phenomenon of enhanced
radiation is indicated in Fig. 2. 7. The antennas of various radii are
assumed to be covered by a plasma layer of Z-inch thickness and
driven at 0. 4 GHz. The electron collision frequency is assumed to
be 0. 03 GHz. The antenna radiation is plotted as a function of
(spa/m2. In this figure it is observed that the phenomenon of enhanced
radiation becomes less significant if the antenna size is increased.
This indicates that for a large antenna, the phenomenon of enhanced

radiation may not be observed.

With a fixed radius of 3 inches for the spherical plasma layer,
the effect of the antenna size or the thickness of plasma layer on the

phenomenon of enhanced radiation is shown in Fig. 2. 8. The antennas

25

of various radii are driven at 0.4 GHz. The electron collision fre-
quency is assumed to be 0. 03 GHz. The antenna radiation is plotted
as a function of (cpl/(1)2. In this figure it is also observed that the
phenomenon of enhanced radiation become less significant if the
antenna size is increased and at the same time the thickness of the
plasma layer is decreased. No resonance peaks are observed even
when the thickness of the plasma layer is very thin.

It should be noted that in the numerical calculation only the
first five terms, i. e., n = 1, 3, . . ., 9, on the right-hand side of
Eq. (2. 59) are summed up for that series. The numerical results
indicate that the n=1 term is the dominant term. All the numerical

calculations are made by using a CDC 3600 computer.

26

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CHAPTER 3

RADIATION FROM A SPHERICAL ANTENNA IMBEDDED
IN A LOSSY, HOT PLASMA OF FINITE EXTENT

3. 1 Introduction

 

In Chapter 2, a theory for a spherical antenna imbedded in
a lossy, cold plasma of finite extent was developed. In this chapter,
the surrounding plasma medium is allowed to have the temperature
effect and the excitation of an electroacoustic wave within this medium.
Another modification to the earlier analysis is to assume the existence
of a dielectric coating surrounding the spherical antenna. As an
idealized approximation the plasma sheath is regarded as a lossless
dielectric coating which is perfectly rigid to the electroacoustic
wave. While such a model is highly idealized, it does permit an
analysis to be carried out in a relatively tractable manner.

A number of related investigations of the stated problem have

(8)

been published recently. Wait has studied theoretically a slotted-

sphere antenna immersed in a lossy, hot plasma of infinite extent.

(

Messiaen and Vandenplas l ) have investigated theoretically and

experimentally on a spherical antenna imbedded in a lossless, cold

33

34

plasma of finite extent. Both papers assumed the existence of a
plasma sheath surrounding the sphere. Since the plasma layer
surrounding the antenna of a reentry vehicle and the laboratory-
produced plasma are hot, lossy and finite in nature, all these
characteristics of the plasma including the plasma sheath are con-
sidered in the present study. Because the theoretical model used in
the present study is somewhat different from the models used by
previous investigators, quite different results are obtained.

Based on this theoretical analysis, a similar enhancement
phenomenon on the antenna radiation as discussed in Chapter 1 has
also been observed. In addition to this enhancement phenomenon,

a series of resonance peaks on the antenna radiation has been found
when the plasma layer is thin and the collision frequency of the
plasma is low compared with the driving frequency. Also the res-
onance peaks and the enhancement phenomenon are effected by the
thickness of plasma sheath. These two phenomena are carefully
examined and extensive numerical results on the antenna radiation
are obtained as a function of the antenna dimensions and plasma

parameters .

3. 2 Geometry and Statement of the Problem
The geomdrical configuration is shown in Fig. 3. 1 using
a spherical coordinate system (r, 6, No A spherical antenna of

radius a is covered by a rigid dielectric sheath whose outer surface

35

Z

I
-e\s,
I (1’, 9, ¢)

 

 

 

III

Region I: dielectric coating ('10, 6d)
Region II: hot plasma
Region III: free space ('40, 60)

Fig. 3. l A spherical antenna covered by a lossy,
hot plasma layer.

36

is at r = b. The permittivity of this sheath is Ed and the permeability
is no which is taken to be the same as that of the free space. Over
the sheath, there is a spherical layer of uniform, los sy and hot plasma
with a thickness of (c-b). The plasma is assumed to be a weakly
ionized gas type and it can be regarded as a one-component electron
fluid.

The spherical antenna is perfectly conducting except for a

_<_ 9 _<_ w/Z + 9 Across this

narrow equatorial gap between 1r/Z - 9 1'

1
gap the antenna is driven by a constant voltage generator with a
voltage of V and an angular frequency of a). The total space exclud-
ing the antenna is divided into three regions. Region I is the dielec-
tric coating (plasma sheath), Region II is the hot plasma layer and
the rest of the free space is Region III.

In this study rationalized MKS units are used. The rotational
symmetry and the infinitesimal driving gap are assumed. Further-

more, the time dependences for the radiating source and all the

fields are assumed to be exp (jmt).

3. 3 Fields in Dielectric RegiorLLPlasma-Sheath Region)
The basic equations which govern Region I (dielectric layer,

a _<_ r 5 b) are Maxwell's equations,

—p

a — -. 3.
a — i 3. Z

37

where E and i; are the electric and magnetic fields, so is the

permeability of free space, and E is the permittivity of the di-

d

electric medium. Following the procedures in Sec. 2. 3, it is easy

to obtain the solutions to Maxwell's equations in this region as

_l__ °° (2 ) (I)
H1¢(r,9) = «fr nil P M(cos€)[A Hn +%(fldr r)+13n Hn +%(Bdr)]
(3. 3)
' °° (2 ) (1)
E1r(r,9)=;€—J;—37§ :Inmuwn (cosB)[An Hn +%(fldr)+Bn Hn +§(sdr)]
d (3.4)

_. oo 2
Emma): —3372 2: Pi(cose){An[nH( ’,.(sdr) sdrH(z _)_;(sdr)]

mEdr n=1
+ Bn[anl +g-(sdr) -sdrH‘ _’%(sd rm
(3. 5)
and
E = H = H = 0 (3-6)

1d) 1r 16
where An and B11 are arbitrary constants, n is a positive integer,
H“) xiii-(pdr) and HS :)_1_2(Bdr) are the Hankel functions of the first and
second kinds with. order “i3: Pn(cos9) is the ordinary Legendre
function, Pi(cose) is the associated Legendre function of the first
kind of order n and degree 1, and 5d is the wave number in this

dielectric medium given by pd: (ml flozd.

3.4 Fields in Hot-Plasma Region
In Region II (plasma layer, b 5 r _<_ c), the plasma medium is

regarded as a one-component electron fluid. That is, the ions are

38

neglected in the equation of motion, yet their presence is required
to neutralize electrically the plasma. The plasma is also assumed
to be a weakly ionized gas having an average number density of
electrons no which is regarded as constant in the plasma region.
The density deviation of the electrons from the mean is denoted as
nl and their mean induced velocity is 27. The collision frequency of
electrons with neutral particles of the gas is v.

In its unperturbed state the plasma is assumed to be homo-
geneous and neutral, and perturbation of the plasma due to the
source is sufficiently small that the linearized equations are appli-
cable. No static electric or magnetic field is present.

For harmonic time dependence of expOmt), the basic equa-

tions which govern this region are Maxwell's equations,

-. *

Vx E2 = '30)“on (3. 7)
v -O = - -' . -O
x HZ enov +Jw€o E2 (3. 8)
_. en1
V e E2 : - —€— (3. 9)
o
v - HZ = o (3. 10)
and the linearized continuity and force equations,
v . -. . : 3o
no( v)+anl O ( ll)
2
-9 e -9 V0
- = - ... - —— v 3. Z
(V+Jw)v m Ez no nl ( l )

where e and m are the magnitudes of charge and mass of electrons,

 

39

p0 and 60 are the permeability and permittivity of free space, and

v0 is the r. m. s. velocity of electrons given by

O m

where k is Boltzmann's constant and T is the electron temperature.
It should be noted that the last term in Eq. (3. 12) represents
the force due to pressure gradient, and Eq. (3.13) is valid on the
assumption of adiabatic pressure variation and one-dimensional
compression““. The above set of equations was used by Chen‘ 7 ),
but in here the source is not included in this region.
In our formulation of the problem, there are four unknown

fields E I; and :7. We will determine I; and n first and

2' 2' n1 2 1
then calculate E2 and 3. Taking curl of Eqs. (3. 8) and (3. 12), we
obtain two expressions as
VxVtz= -enOva+Jw€o‘7sz (3.14)
Vx '5 - -—‘-’—— Vx E (3 15)
’ ' m(v+jw) 2 °

The substitution of Eqs. (3. 15) and (3. 7) in Eq. (3.14) gives

 

Z
a)
.. z _.

v v = —P——— 3. 6
x xHZ wPo€o[l+jm(V+jw)]H2 2 ( l)

2 n e

where a) is the plasma frequency defined as a) = o . Using
p p mEO

a vector identity of

VxVxfi = VW'fi2)- vz'fi

Z (3.17)

2

40

and Eq. (3. 10), Eq. (3. 16) can be reduced to a homogeneous wave
equation,

(vz+kez)'fi = o (3.18)

where k8 is the complex prOpagation constant of the electromagnetic

wave given by

Z 2
ke : a) pofi (3.19)

where 5 is the equivalent complex permittivity defined as

 

 

Z Z , Z
a) (s 30) u
§=EOI+—£‘——.(v+.) =60 (1’2 2). Z Z '
3‘” J“) m w (0(a) +v )
(3.20)
Ifwe write
ke: fle - Jae ' (3.21)
fie and 0.6 can be expressed as
l .1.
50 (1) 2 2w Z 224 z 2
pe=n731‘z 2+1'zz+22 z (3'22)
a) H: w HI (0 (w +1!)
2 Z 4 A A
50 (0 Zoo cop z 2
38:31; ..1+——L—-2 2+ ],--—E—Z 2+ 2 Z Z .(3.Z3)
0) 4'” 0) +9 on (0) +V)

fie and ac are the wave number and attenuation constant of the
electromagnetic wave in this plasma medium, and so is the wave
number in free space defined as Bo = (ox/1F; .

From the symmetry of the antenna it can be seen that there

is no variation in the 4; direction, i. e., 3:: = O and the magnetic

41

field has only a 4: component. Thus the Laplacian of the vector
)

magnetic field in Spherical coordinates can be expressed(15 as

_. A 2 2
VZHZ=¢(VH --l—csc6H)

24) r2 24)
A l a 281-12 1 8H
=¢[7'5;(’ a. H. assume)
r rsinG
_ 29H 324
- Z csc 24>] (. )

A
where 4) is the unit vector in 4) direction. Two identities can be

obtained through differentiation. They are

8H 2

 

 

a 2 2 a
'8?" .5152) = r—E (rH2¢) (3.25)
8r
8H
1 a . 2 a 1 a ,
sine 36 (“me To? ) ‘ C" “12¢: 7'56 [ sinG W (”neHN”

‘3sz 3Hz 2

: -—72 + cotO 35$ - CSC 9H2¢ (3o 26)
89

The substitution of Eqs. (3.24), (3. 25) and (3. 26) in Eq. (3.18)

leads to a partial differential equation,

 

Z
8 l 3 l 3 . Z
:3 "sz + :2: at a... as «max-Hz.” “Haw

(3. 27)
which is exactly in the same form as Eq. (2. 7). The solution to

Eq. (3. 27) can then be obtained as

42

H (r 9)- ‘1— 0230 Pl(cose)[C Ha) (k r) +D H”) (k r)] (3 28)
2¢ ' - «fr n=1 n n n e n n e ° °

~12.
NT...

The magnetic field is thus determined and n1 is the next

quantity to be determined.

Taking the divergence of Eq. (3. 12), we have

 

2
v
(v+jw)V-; = -—e—v-E -—°—V2n . (3.29)
m 2 n l
0
From the continuity equation (3.11), it gives
-9 jun-11
V ' V : - , (3. 30)
n
0

Substituting Eqs. (3. 30) and (3. 9) into Eq. (3. 29), we obtain another

homogeneous wave equation ,

(V2 + kPZ)n1 = 0 (3.31)

where kp is the complex prOpagation constant of the plasma or

electroacoustic wave expressed by

2 l 2 Z ,
k : —-2- [(0) -0.) )-J(DV]. (3. 32)
P v P
o
If we write
k = - °a (3. 33)
9 ‘5’9 J p '

fl and up can be expressed as

43

ll
1 z z 2 22 2235
51)-727(0) -u)p +[((n -mp) +ml’]} (3-34)
0
33
l 2 2 2 22 22
op-Uzzf-w +wp +[(w-wp) +wV] } . (3-35)

5p and up are the wave number and attenuation constant of the
electroacoustic wave in this plasma medium. Due to rotational
symmetry, the Laplacian of the scalar field nl in spherical

coordinates can be expressed as

an 3n
1 3 Z l l 3 . l
V2111: 7 '5;- (r -a-1-.-) + 'T— 55 (81119 736-) , (3. 36)
r r sine

Using Eqs. (3. 36) and (3.25), Eq. (3. 31) can be reduced to a partial

differential equation,

8 2 (ml 2 sine 39
r 1'

 

[sine 58-6- (rnl)] + kpz(rnl) = O , (3. 37)

Equation (3. 37) is similar to Eq. (2. 7) but not identical in form. To
solve Eq. (3. 37), we employ the method of the separation of variables.

Since n1 is independent of 4), we can assume

rnl = R(r)9(9) (3. 38)

where R is a function of r alone and 8 is a function of 9 only. The
substitution of Eq. (3. 38) in Eq. (3. 37) leads to

53.21.”er _1_
Rer p ‘ “seine d0

1 _d_. (sine Eli—5'6 ) = n(n+l) (3.39)

 

 

44

where n is any integer and n(n+l) is the separation constant. Equa-

tion (3. 39) generates two ordinary differential equations,

 

 

l d . <19
sine d3 (31119 35 ) +n(n+l)® - 0 (3'40)
2 Z
_r__ d R + k Zrz - n(n+l) = O (3.41)
R 2 p '
dr

Let us consider Eq. (3.40) first. Making the substitutions,

u 2 C089 , ‘Jl-u = 51119 , i = - ‘Jl-u ad;

d9

Eq. (3.40) can be reduced to

Z
Z
(l-u )i—g— - Zu @- +n(n+l)@ = 0. (3.42)
du du

Equation (3.42) is a standard form of the ordinary Legendre's

equation. A solution to this equation can be obtained as
9 = Pn(u) = Pn(cose) (3.43)

where n must be a positive integer and n 3 0. Note that only one
solution for this second-order differential equation (3.42) has been
considered. The other solution becomes infinite on the axis, and
so it should be excluded from this problem since the axis is included
in the geometry.

Since Eq. (3.41) is exactly in the same form as Eq. (2.16),

its solution can be easily written as

A
V
V

(k r) +F H‘1

2
R=~/i-[EH
p nn

n n (k r)] (3. 44)

P

J...
«St.

 

45

where En and Fu are arbitrary constants. Combining Eqs. (3. 38),

(3. 43) and (3.44) we have

n = l .330 p (cose)[1~: Hm (k r)+F H“ (k r)] (3 45)
l ano n n n+§7 p n n+-; p ° '

V

Now I-IZ and n1 have been determined explicitly from Eqs. (3. 28)

and (3.45). We now aim to express IE

2 and I? in terms of these

two known quantities. From Eq. (3. 12) we have

-o -e _, V02
V = W E2 ' m “I “'4‘”

The substitution of Eq. (3. 46) in Eq. (3. 8) leads to

2
"f: _

l -> jevo
2. E VxH2+

 

w§(v+jw) vn1 (3‘47)

where 5 is given in Eq. (3. 20).

The substitution of Eq. (3.47)
in Eq. (3.46) gives

2
- -e l -> v0 60
v = . . x H - ——-—.— Vn . 3.48
moms) we V 2 no€(v+3w) 1 ‘ )
Under rotational symmetry two vector differential operations in

spherical coordinates can be expressed as

.. A
VXsz 1'

 

a , *1 8
sin6_ .8—9- (SmeH )- 9; '5; (13H )

2(1) (3.49)

(3. 50)

A
where rand 6 are unit vectors in r and 9 directions. Combining

Eqs. (3.47) to (3. 50), we obtain

46

 

 

 

 

. Z 8
E - 1 1 3— 'eH Jev° n1 351
2:”wa rsinO 86 (am 24)) + m€(v+jw) 3r ( ° )
Z
3
E - '1 13—(rH )+ Jev° 1:1 (3 52)
29- jg); r 3r 24> (DEW-Um) r 86 °
2
V ’ -e l l 3- (sinGH ) V0 60 an]
r' m(v+js) jog rsinG 36 24> noi(v+jw) 8r
(3.53)
2
v———" llfist-H). v°e° if: (354)
6— m(V+jm) is: r 8r 24> noE(v+jw) r 36 ' °

Using Eqs. (2. 34), (Z. 35) and (3. 45) we obtain

an on

1 _ -1 (Z) (2)
E.— - r72 :oPn(cosG)IEn[ (n+l)Hn+%(kpr) - kpl' Hn-%(kpr)]
(1) (l)
+ Fn[(n+l)Hn+_z,_(kpr) - kpr Hn_%(kpr)] }.
(3. 55)
Using Eqs. (2. 25) and (3.45) we have
3n 00
l _ _L 1 (2) (1)
W .. «fr 2 Pn(cose) [Ean+%(kpr) + Fan+%(kpr)] (3.56)

n: 1
because for n=0, P:(cose) : 0. It is noted that the first terms on
the right-hand side of Eqs. (3. 51) to (3. 54) can be obtained easily
following the same procedures in Sec. 2. 3 while the second terms on
the right-hand side can be obtained by using Eqs. (3. 55) and (3. 56).
Since we are interested in E and v1' for the later development,

29
only Eqs. (3. 52) and (3. 53) are expressed more explicitly as follows:

47

oo
-' l 2 Z
Eze(r,6) = —JVZ Z} Pn(cose){Cn[ nI-Ifxéfi r)-k rH( ) (ker)]

1
e e n--
mgr n=1 2

(l) (l)
+ Dn[an+%(ker) - ker Hn_%

(ker)] }

. z
Jev 00
+ ° [ z P;(cose)[E H‘2 (k r)+F H”)

3 z 1
m£(u+jm)r n=1 n “+7 P n n

V

 

(kpr )1

(3. 57)

NT.

(D

. (2)
_ -e —%— Z n(n+l)P (cosB)[C H 1(k r)
(Irma)- m<u'+jw) 036:3 7‘ n=1 “ n “+3 e

(1)
+ Dan+% (ker)]

vo 6 °° (2)
+ ° 3R z Pn(cose){En[(n+l)H
11050) +10)!‘ n=0

 

n P

J.

(1)

n

(k r) - k rHu)

+F (n+1 H
n[ ) p p n-%

(kpr )1 }

J.

(3.58)

3. 5 Fields in Free-Siace Region
The basic equations which govern Region 111 (free space,

r 2. c) are Maxwell's equations,

Vx§3 = -ju)poH3 (3.59)
v x H3 = 50360153 (3.60)

Since Region III is unbounded, no reflected or inward traveling wave
exists in this region. Following the same analysis as in Sec. 2. 3,

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

2
(k r) - kpr H;_)%(kpr)]

48

(I)

_ _1_ 1 (2)
H3¢(r,9) - «fr :1 GnPn(cose)Hn+%(Bor) (3.61)
2
E3r(r,6)- - —)—§7—“:21Gnn(n+l)Pn(cosa)H:l+)%(80r) (3.62)

E39(r,9)= —'L7—n: 21 an P n(cosGXnHa) +lmo r)-(3o rH(Z) %((30 r)]
(3.63)
and
E34): H3r= H39: 0 (3.64)

where Gn is an arbitrary constant and n is a positive integer.

3.6 Matching of Boundary Conditions at Interfaces

 

In order to determine the arbitrary constants An' Bn’ Cn’

D , E , F and G , the boundary conditions at r = a, r = b and
n n n n
r = c are applied.
Following the same procedures of Sec. 2. 5 in matching the

boundary condition on the antenna surface (r = a), we can obtain

easily an expression as

MlAn + MZBn = M3V (3.65)
where
- (2) HQ)
Ml - an+%(sda) - sdaH was a)
M2 - nH H‘ n’mda) - sda H‘ M’ys a)

. l Zn+l
M3 - JOQEd «fa in) Zn(n+l)

 

49

The continuity of the tangential components of If and H fields at the

dielectric-plasma interface (r = b) is
Ele(b,9) = E26(b'9) (3.66)

(b,9) . (3.67)

Hl¢(b,9) H24>

Using Eqs. (3. 5) and (3. 57), Eq. (3.66) gives

M4An + IMEB]n + M6Cn + M7Dn + M8En + M9Fn = o (3.68)
where
- E (Z) (Z)

 

 

_ Q (1) (l)
_ (2) (2)
M6 - - [an+%(keb) - kean_%(keb)]
- (l) (1)
M7 _ - [an+%(keb) - keb Hn_%(keb)]
CV 2
_ 0 (2)
M8 v +jm Hn-t-i-(kpb)
CV Z
M = ° Hm (k b)

From Eqs. (3. 3) and (3. 28), Eq. (3.67) can be expressed as

= 3.6
M109.n + Mlan + Mlzcn + M13Dn 0 ( 9)

where

M = Hm

10 n+§ (fidb)

50

_ (
_ (2)
M12 - - Hn+%(keb)
_ (1)
M13 ' ' Hn+§(keb) °

The continuity of the tangential components of E and H fields at the

5

plasma-free space interface (r = c) leads to

 

 

E29(c,6) = Emma) (3.70)
HZ¢(C'9) : H3¢(c,8) (3.71)
Using Eqs. (3. 57) and (3.63), Eq. (3.70) gives
: 2
M14Cn + MISDn + Mlen + M171?n + MIBGn O (3.7 )
where
M = -[ nH(Z) (ke c) - ke cH(Z) (k eC)]
14 n+§ -%_-
_ (l) (1)
M15 - -[ an+%(kec) - kecHn_%(kec)]
e 2.
v
_ o (2)
M16 v+jw Hn+%(kpc)
ev 2
_ o (1)
M17 ’ wry.) Hn+%(kpc)
_ _§_ (2)
M - [an +l(po C)-POCHn 1(90 C”
18 '2
60
From Eqs. (3.28) and (3.61), Eq. (3.71) can be expressed as

= 3. 3
M19Cn + MZODn + M216n 0 ( 7 )

51

where
M19 - H112 (kec)
M20 = Hifléfiec)
M21: HiIZé-(fioc) °

In the present analysis, it is assumed that the normal com-
ponent of the mean electron velocity vanishes at the interfaces at

r = b and r = c. These rigid boundary conditions require that

vr(b,9) _ 0 (3.74)
and
vr(c,9) = O . (3.75)
Using Eq. (3.58), Eq. (3.74) gives
Macn + M23Dn + M24En + M251?n = 0 (3.76)
where
(L)
= _2 (2)
M22 n(n+l) a) Hn+%(keb)
‘22 (1)
M23 = n(n+l) w Hn+;j(keb)
_ . Z (Z) (2)
M24 — Jev [(n+l)Hn+%(kpb) - kprn_z1(kpb)]
_ . (l ) (1 )
M — Jevoz [(n+1)Hn1(kpb)-kanl(kpb.)]
25 p n-E
Similarly from Eqs. (3. 58) and (3. 75) we obtain
: 3.
M26Cn + M27Dn + M28En + M29Fn 0 ( 77)

52

(D
- _E (2)
M26 - n(n+l) m Hn+~§ (kec)
‘32 <1)
M27 = n(n+l) w Hn+%(kec)
_ . Z (Z) (2)
M28 - Jevo [ (n+l)Hn+%(kpc) - kpc Hn_%(kpc)]
_ . 2 (I) (1)
M29_ Jevo [(n+1)Hn+%(kpc)-kchn_%(kpc)].

It should be noted that the summation of the first term on the right-
00 on

hand side of Eq. (3. 58) can be changed from Z to 2 because the
n=1 n=o

n = 0 term makes no contribution to the series. Thus we can equate

the two series in Eq. (3. 58) to yield Eqs. (3. 76) and (3. 77) by using

the rigid boundary conditions imposed by Eqs. (3. 74) and (3. 75).

Furthermore, the notations for the coefficients Mi' i = 1, Z, . . ., 28,

29, are used merely for convenience.

By matching the boundary conditions on the interfaces, we
end up with seven algebraic equations for seven unknowns. For

convenience, the matrix representation for these equations is used.

Combining these equations, we can form a matrix equation as

[M]

 

 

 

M

00000000

V

_

 

(3. 78)

 

53

where [M] is the matrix which can be expressed as

 

1 M2 0 o o o o
4 M5 M6 M7 M8 M9 0
10 M11 M1.2 M13 0 0 O
= 3o
[M] o 0 M14 M15 M16 M17 18 ( 79)
° 0 M19 M20 0 ° 21
° 0 M22 M23 M24 M2.5 °
° 0 M26 M27 M28 M2.9 0
From Eq. (3. 78) we obtain the arbitrary constants as
- A q rM vq
n 3
B o
n
C 0
n -1
Dn = [M] O (3.80)
E o
n
F o
n
G o
n

 

 

 

 

where [M] -1 is the inverse matrix of [M] .

Up to this point all the fields in Regions 1, II, and III are
completely determined as functions of r and 9.

The determination of the arbitrary constants An’ Bn’ . . ., Gn
can be done from Eq. (3. 80) by getting the inverse matrix [ M] -1
through ordinary determinant operation or numerical calculation using

a computer.

54

3.7 Radiated Field in Free-Space Region

 

Since our main interest in this study is the radiated field
in the free-space region, we determine the arbitrary constant Gn
only. In order to determine Gn’ the ordinary determinant method
is employed. From the seven algebraic equations in the last sec-

tion we can form

[>|[>
v—N

(3.81)

where Al and A represent for two determinants as follows:

 

 

2
M1 M2 0 o o o 0
M4 M5 M6 M7 M8 M9 0
M10 11 M12. M13 0 0 0
A1 = o 0 M14 M15 M16 M17 M18
0 0 M19 M20 0 0 M21
0 0 M22 M.23 M24 M25 0
0 0 M26 M27 M23 M29 0
M1 M2 0 o o o M v
M4 M5 M6 M7 M8 M9 0
M10 M11 M12 M13 0 0 0
A2 = o 0 M14 M15 M16 M17 0
o 0 M19 M20 0 o o
0 0 M22 M23 M24 M2.5 0
° 0 M26 M27 M28 M29 0

 

 

55

After some steps of determinant operation, 61 and A2 can be ex-

pressed as

: - 2
Al M21(MZM10 M1M11)A4 (3.8 )
92‘ VM31‘419‘1‘451‘410 ' M4M11)‘33 ”'83)
where
M M
14 20
A3‘ ‘M15' M19 "M24M 29' MZSMZB)
M M
22 20
' ‘M23' M19 "M161“ 29 M17M28)
M M
26 20
+‘Mz7' M19 )‘M16M 25 'M17M24)

 

 

A _ M Mlz‘MlMs' M4M2) A13‘M1Ms'M4Mz)A
4’6MM-MM 5'7'MM MM 6

111 102 111 102

+MA-MA

8 7 9 8
and
A-(M MHMM MM) 16(M MM)
5' 15' M21 24 M29' 25 28 ' 23M 29' 25 27
1M7‘M23 28' M24M27)
M19M18
A6= ‘M14' M21 )‘M24M 29' M25Mzs)'M1"‘16‘1‘422 29' M25M26)
17‘MzzM 28' M24M26)
M19M18
A7= ‘M14' M )‘M23M 29" M25 M27HM17‘M22M27' M23M26)

Zl

56

M M
20 18
"M15' M21 "M22M29'M25M26)
A=(M -J—HMM MM)+M (M MM)
8 14 M21 23M 28' 24 27 16 22M 27' 23 26
M M
20 18
"M15' M21 )‘MzzM 28' M24M26)

Combining Eqs. (3. 81), (3. 82) and (3.83) we have

C _ v M31"‘19‘1‘45h’110 M4M11) :3 (3 84)
11 M21""‘21‘410' M1M11) A4 °

From Eqs. (3.63) and (3.84) the field E

o
39at(r—R, 6- 90 )canbe
expressed as

 

o v °° M3M19‘M5M1o' M4M11) A3
Es‘R'W)= 32 z M(MM-MM) Z—in)
ODEOR n=1 21 2 10 1 11 4
(n=odd)

[nH‘i’ 1(so R) - s RH" _1(s om].

 

(3.85)
Equation (3. 85) can be rearranged as
00 n 2
(R, 90°) fd—ff z .211— :2. M19(M5M10'M4M11)
E39 = 2 n=12n(n+1) n 1921\4(M-M M )
«60R (n-odd) I‘(z+ 2) M21 10 l 11
A
. _2 (Z) (2)
A4 [nHo+1(soR) - so RHo _’1<s om] . (3.86)

3. 8 Numerical Results

The radiated power from the antenna in the broadside direction

which is proportional to the square of E93 (R, 90°) as expressed in

57

Eq. (3. 86) has been numerically calculated as a function of the
antenna dimensions and plasma parameters. In a realistic situation,

the presence of the plasma sheath on the antenna surface is taken into

(8)

account by the adoption of a concentric dielectric layer which

separates the plasma from the metallic surface of the antenna. This
adoption can also account for an actual dielectric coating. For a

usual plasma sheath, its thickness may be of the order of several Debye
lengths. In the present numerical calculation, the plasma sheath is
regarded as an electron-free region extending from r = a to r = b.

A convenient parameter to describe the thickness of the sheath is
v

o
the dimensionless quantity 8 defined by b - a = (NI-3w )s. It is to be

V
O

noted that («[3111 ) is of the order of a Debye length in the plasma and,

thus, 8 may be regarded as the ”Debye thickness" of the sheath‘8 ).

 

 

The permittivity of the sheath can then be assumed to be the same
as that of the free space, i. e., 6d = 60. Furthermore, in the numeri-
cal analysis only the first five terms, i. e., n = l, 3, . . . 9, on the
right-hand side of Eq. (3. 86) are summed up for that series and for
the large arguments, z > 10, the asymptotic forms of the Hankel
functions of the first and second kinds have been used. The numeri-
cal calculation was made by using a CDC 3600 computer .

Figures 3. 2 to 3.4 show the radiated power from a spherical
antenna of radius 2. 54 cm driven at various frequencies as a func-

tion of the plasma density of the plasma layer with the Debye

thickness s as the running parameter. The radius of the spherical

58

plasma layer is 7 cm (or the thickness of plasma layer is 4.46 cm
minus the thickness of plasma sheath) and the distance between the
radiating and receiving antennas is O. 915 m. The electron collision
frequency is assumed to be O. 12 GHz and the ratio of the r. m. s.
electron velocity to the velocity of light in free space, vo/co, is
assumed to be 0. 01. In these figures the radiated power at each
driven frequency is normalized to its free-space radiated power.

At all driven frequencies we invariably observe that the antenna
radiation is reduced as the plasma density is increased and it reaches
the cut-off point as the plasma frequency is increased to the neighbor-
hood of the antenna frequency. After the plasma frequency exceeds
the antenna frequency, the antenna radiation starts to build up some-
what for higher antenna frequency cases of 0. 8 GHz and l. 2 GHz.
This trend becomes more outstanding for lower frequency cases of
0.4 GHz and 0. 3 GHz. At 0.4 GHz and 0. 3 GHz, the antenna radiation
can build up to a level 15 db to 21 db higher than the free-space
radiation after passing the cut-off. Physically it means that if an
antenna is Operated at a frequency much lower than the plasma fre-
quency of the plasma layer, its radiation will recover from the cut-
off and then be enhanced greatly over the free-space radiation level.
This phenomenon of enhanced radiation is similar to that discussed
in Chapter 1 especially when the thickness of the plasma sheath is

in the order of one Debye length, i. e., s = l. The effect of the

plasma sheath on the antenna radiation can be observed from these

59

figures. When the thickness of the plasma sheath is increased the
phenomenon of enhanced radiation after passing the cut-off is more
outstanding for both higher and lower frequency cases.

The set of Figs. 3. 5 to 3. 6 shows the similar phenomenon
as the set of Figs. 3. Z to 3.4 but for the case of a smaller antenna
(1. 27 cm radius). The radius of the plasma layer is again 7 cm.
For a smaller antenna, we observe that the antenna radiation is
enhanced greater for both higher and lower frequency cases. A
strong enhancement of 15 db over free-space radiation for the case
of O. 8 GHz in Fig. 3. 7 is rather interesting since it tends to indi-
cate that the phenomenon of enhanced radiation can occur even for
a higher frequency case if a suitable value is as signed to the plasma
parameters.

The effects of the electron collision frequency, the antenna
size, the thickness of the plasma layer and the thickness of the
plasma sheath on the phenomenon of enhanced radiation can be
observed from two sets of figures, Figs. 3.8 to 3.11 and Figs. 3. 12
to 3. 15. In Figs. 3. 8 to 3. 11 the radiated powers from spherical
antennas of various radii, Z. 54 cm, 3. 81 cm, 5. 08 cm and 6. 38 cm,
driven at O. 4 GHz are shown as functions of the plasma density of
the plasma layer with the Debye thickness 8 as the running para-
meter. The radius of the spherical plasma layer is fixed as 7.62. cm
and the distance between the radiating and receiving antennas is

0. 7 m. The electron collision frequency is assumed to be 0. 03 GHz

60

and the ratio of the r.m. s. electron velocity to the velocity of light
in free space, vo/ co, is assumed to be 0. 01. In these figures the
radiated power is normalized to its free-space radiated power. For
FigS. 3.12 to 3.15 the same set of parameters is used except the
electron collision frequency is changed from O. 03 GHz to O. 003 GHz.
The main points of observation on the enhanced radiation from the
antenna can be summarized as follows:

(1) A higher loss in plasma makes the cut-off phenomenon less
outstanding and the enhancement of radiation slightly lower .

(2) For fixed radius of the spherical plasma layer the
phenomenon of enhanced radiation becomes less significant as the
antenna size is increased and the thickness of plasma layer is reduced.

(3) For fixed antenna size the increase of the thickness of
the plasma sheath which is equivalent to the decrease of the thickness
of the plasma layer makes the enhanced radiation more outstanding.

(4) Besides the above observations on the general behaviors
of the enhanced radiation which are consistent with the phenomena
observed in Chapter 1, an interesting finding is made on a series of
resonance peaks which occur when the thickness of the plasma layer
is thin and the electron collision frequency is low compared with
the driving frequency. In Figs. 3. 8 and 3. 9, the antenna sizes are
small with radii 2. 54 cm and 3. 81 cm and the plasma layers are
thick, no resonance peaks are observed. When the antenna size is

increased to S. 08 cm radius and the plasma layer becomes rather thin

61

in Fig. 3.10 the resonance peaks start to show up at (spa/(1)2 = 1. 3.
If the antenna size is further increased to 6. 35 cm radius, and the
plasma layer is further shrunk, more resonance peaks occur at
pr/wz = 0. 7, l. 3 and l. 5. This indicates that if the thickness

of the plasma layer is made thin the resonance will occur. Physi-
cally, it may be attributed to the electroacoustic resonance in this
thin plasma layer. Since the wave length of the electroacoustic wave
is rather small, it can set up the standing wave only when the plasma
layer is thin. A large plasma layer will make the electroacoustic
wave to set up an attenuating traveling wave instead. since the
plasma is lossy in nature. In Figs. 3.12 to 3.15, the plasma
medium is made less lossy (v/ 211 = O. 003 GHz). The resonance
peaks start to show up for the smaller antenna and thicker plasma
layer cases and more resonance peaks occur for the larger antenna
and thinner plasma layer cases. This might be due to the fact that
since the plasma medium is now less lossy, the electroacoustic
wave will suffer less antenuation and even for a larger plasma

layer it is still possible to set up the standing wave for the resonance
within this layer. There are two other possibilities for producing
the resonance peaks. They are: (l) the cavity resonance due to

the electromagnetic wave in the spherical plasma layer, and (2.)

the cavity resonance due to the electromagnetic wave in the plasma
sheath region. It is unfortunate that in the present analysis there is

no way to identify those resonance peaks with those three different

62

causes. From Fig. 2. 8 we found that under the same conditions but
a different theory, no resonance peak was observed even when the
thickness of the plasma layer was made very thin. This fact tends
to indicate that those resonance peaks observed in Figs. 3.10 to
3.15 are due to an electroacoustic wave in the plasma layer or an

electromagnetic wave in the plasma sheath region.

63

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(db)

+20<

antenna radiated

"—'" power

free- space _. 0

+104.

 

c=7.62cm R=O.7m

 

radiation

-104

-ZO+

(db)
+204
+10<

free-space
radiation

-10‘

.201
(db)

+201

+104

,free-space
radiation

.—.O

-10‘

.20.

V

T

 

Fig. 3. 8

N4>

1r-

 

 

ouL

<r

 

 

T—

10

m w
p/

Theoretical radiation of a spherical antenna (a = Z. 54 cm)
in a lossy, hot plasma (v/ 211' = 0. 03 61-12, v /c = 0. 01)
driven at 0.4 GHz as a function of plasma doensi’ty.

 

antenna radiated

power

1

free- space
radiation

free-space _.

radiation

fr ee- space
radiation

 

70

 

 

 

 

 

 

 

 

v
b-a: J50) s c=7.6Zcm R=O.7m
(db) _ P
+20 1* s = 1
+10%
2 2
—e : 4 4 ¢ #— m /u)
4 8 10 P
'10?
.20v
(db)
"’204b 8 = 5
+100
2 2
0 + t + 4r # a) /u)
2 4 6 8 l P
-104»
-201.
(db)
+ZO<> a = 10
+10?
2 2
0 t 4. ¢ 4 t 0) /03
4 6 8 10 p
-lO<+
.20?
Fig. 3. 9 Theoretical radiation of a spherical antenna (a = 3. 81 cm)

in a lossy, hot plasma (v/ 211' = o. 03 GHz, v /c = o. 01)
driven at 0.4 GHz as a function of plasma dgnsi‘ty.

“U
0
JJ
d
0!"
'U
N
a...
d
an
50
.93
c 0
d O.

free-space
radiation

free-space
radiation

free-space
radiation

....0

(db)
+20»

+100

-104»

-204
(db)

«20‘r

+10%

-104»

 

-ZO1r
(db)

+201}

+10 4

 

 

1L

N”
.3.

 

294.1
n 77.8

71

c = 7.62 cm

R=O.7m

 

 

 

N4
“lb

 

298. 8

Ow»

 

 

-10 4b

 

-20.

—f

Fig. 3.10

 

ow

Theoretical radiation of a spherical antenna (a = 5. 08 cm)

in a lossy, hot plasma (ll/211' = 0.03 GHz, v /c = o. 01)
driven at 0.4 GHz as a function of plasma ens1ty.

72

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

'3 v0
3 b-a: (30) -s c=7.62cm R=O.7m
'3 P
a (db) 305.4
E ” 20 t
o
3 3 + .. 8 = 1
5 8
1 +100
free-space Z Z
. ‘ e * 1L 0)
radiation 0 2 4 6 B 10 P A”
-104»
-zol 305.9
100. 3
(db) ‘ n 90. 2
+204, 8 = 5
+100
free-space z/ 2
. . ...... - a * = t u) (0
radiation 2 4 6 8 10 p
-10..
'20? 306.1
(db) ”97. 7
+204; I
s = 10
+10“
free-space L z/ z
d’ation —’ 0 i 1L : 4 ¢ (0 00
ta 1 ‘\ I z 4 6 8 10
-10..
-ZO.L
Fig. 3. ll Theoretical radiation of a spherical antenna (a = 6. 35 cm)

in a lossy, hot plasma (1)/er = 0.03 61-12, v /c = 0.01)
driven at 0. 4 GHz as a function of plasma density.

antenna radiated

power

I

free-space
radiation

free-space
radiation

fr ee-space
radiation

.—.0

«I...

(db)

+20 ._

+104

73

 

R=O.7m

c = 7.62 cm

b-a:=

 

 

-104

-ZOfi
(db)

+ZO<

+10‘

 

t
-39. 8
124.8

 

-10.

 

-20l
(db)
+304

+204

not

1

 

 

8:10

 

-10*

 

-204

Fig. 3.12

2
-..-.- w /w
10

 

 

 

Theoretical radiation of a spherical antenna (a = Z. 54 cm)
in a lossy, hot plasma (v/ Z1r = 0. 003 GHz, v /c = 0. 01)
driven at 0.4 GHz as a function of plasma density.

'U
0
G
d
all.
'0
d
h
:8
3 3
‘10
do.

1

free- space_. 0

radiation

free-space
radiation

free-space“ 0

radiation

(db)
+20L

+104

74

 

b-a: «[300 -s c=7.6Zcm R=O.7m

61.6

 

_10.L
-20?
(db)

+204

+10‘h

 

 

 

 

306' Z 304. 0

 

-10..

.20‘
(db)
+20-»

 

+104

 

 

‘37s 9

 

_104.

-20 b-

 

-36.Z

Fig. 3. 13 Theoretical radiation of a spherical antenna
(a = 3.81 cm) in a lossy, hot plasma (v/ Z1r = 0. 003 GHz,
v /c = 0. 01) driven at 0.4 GHz as a function of plasma
dgnsi y.

 

hulwhonux to‘

\~
. “M‘s?“V

75

 

 

 

 

 

 

 

 

 

 

 

'o
0
U
d b- = ' = 7.62 R 0.7
8 a «[300 I C cm m
a
H
g .. (db)
0 o
a 3 +20 ‘
t: O
a n.
I +10 4 _ a = 1
Z
free-spacL 0 . 3 ¢ 0) /00
radiation 6 8 10 P
~10 i
-204
(db)
+20 <
8 = 5
+10 .
free-space . . L_ m Z/wz
radiation -’ 6 8 Ito P
-10 4
-203
(db)
+20 <
s = 10
+10. 4
free-space Z 2
radiation“ 0 3 3 7‘, g {0 mp /w

-10‘

 

 

T

 

-Zm

Fig. 3.14 Theoretical radiation of a spherical antenna (a = 5. 08 cm)
in a lossy, hot plasma (v/zu = 0.003 GHz, v /<:o = 0.01)
driven at 0.4 GHz as a function of plasma density.

76

 

 

 

 

 

 

 

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radiation 2 8 10 p

-10

 

-20“

 

 

Fig. 3. 15 Theoretical radiation of a spherical antenna (a = 6. 35 cm)

in a lossy, hot plasma (ll/Zn = 0.003 GHz, v /c = 0.01)
driven at 0.4 GHz as a function of plasma density.

CHAPTER 4

EXPERIMENTAL INVESTIGATION ON THE RADIATION
FROM A SPHERICAL ANTENNA IN A HOT PLASMA

4. 1 Introduction

 

The radiation from a spherical antenna imbedded in a finite,
spherical plasma layer has been studied theoretically in Chapters 2
and 3. In order to confirm the theoretical results, an eXperimental
investigation on the subject has been conducted.

The main purpose of this experimental investigation is to
detect the strong enhancement on the antenna radiation over a wide
band of antenna frequencies which are much lower than the plasma
frequency of the plasma volume. This phenomenon has been pre-
dicted theoretically and it has also been confirmed in this experi-
mental study.

In our experiment, the hot plasma was provided by a mercury
arc discharge which was created in a hemi-spherical pyrex tube.
Hemi- spherical antennas were used as the radiating sources and a

large metal plane was used as an image plane.

77

78

4. 2 Experimental Setup

 

The experimental setup for the antenna radiation measurement
is schematically shown in Fig. 4.1. The plasma tube is made of a
hemi- spherical pyrex glass tube with a radius of 3 inches. A small
mercury pool with a floating spot fixer is located at one end of the
tube. The open end of the tube is sucked to the ground plane when
the tube is being pumped. With this arrangement the ground plane
acts as the anode and the mercury pool as the cathode of the plasma
tube and the discharge is maintained by a DC voltage. The‘ discharge
current can be varied from zero to about 8 amperes, corresponding

to a plasma density of about 2 x 1017(m3)-l. The electron temper-

ature and the pressure of the plasma are about 20, 0000K and 3 x 10"3
mm Hg. respectively. A spherical monopole antenna is fed into the
center of the plasma tube through the ground plane and is driven by
a RF signal. The antenna is DC blocked from the rest of the system
to insure a floating potential for the antenna. The radiation of the
antenna through the plasma is measured by a fixed receiving antenna
on the ground plane. The output of the receiving antenna is connected
to a hetrodyne receiving system. The distance between the radiating
and receiving antennas is O. 915 m. The radiating antenna with the
plasma tube and the receiving antenna are all enclosed in a micro-
wave anechoic chamber. Figure 4. 2 shows the photograph of this
plasma tube under operation inside of the microwave anechoic
chamber. The photograph of the experimental setup outside of the

microwave anechoic chamber is shown in Fig. 4. 3.

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operation inside of a microwave anechoic chamber.

 

 

Fig. 4. 3 Experimental setup outside of a microwave anechoic
chamber.

81

4. 3 Experimental Results and Comparison with Theories

 

The experimental results of the radiation from two spherical
antennas of radii Z. 54 cm and l. 27 cm driven by various frequencies
are plotted in comparison with the corresponding theoretical results
in Figs. 4.4 to 4. 7. At each driven frequency the antenna radiation
is measured as a function of the plasma density and the radiated
power is normalized to the value when no plasma is present (free-
space radiation). The antenna radiation is measured at R = 0. 915 m
and in the broadside direction of the radiating antenna.

In Figs. 4.4 and 4. 5. the corresponding theoretical results
are calculated from E29(R = 0. 915m, 90°) in Eq. (2. 59) under the
assumption of a cold plasma and an electron collision frequency of
0.12 GHz. In Figs. 4.6 and 4. 7, the corresponding theoretical
results are calculated from E39(R = 0. 915m, 90°) in Eq. (3. 86)
under the assumptions of a hot plasma, v/ Zn = 0.12 GHz, vo/Co =
0.01 and s = 1. Figures 4.4 and 4. 5 are based on the lossy, cold-
plasma theory studied in Chapter 2 while Figs. 4.6 and 4.7 are
has ed on the lossy, hot-plasma theory developed in Chapter 3. The
theoretical value of the radiated power is also normalized to the free-
space radiation. The comparison of two theories (lines) with the
experiment (dots) indicates a very close agreement. The agreement
between the experiment and the lossy, hot-plasma theory is somewhat
better than the agreement between the experiment and the lossy,

cold-plasma theory. It should be noted that only four cases are

82

considered in Figs. 4.6 and 4. 7 because the numerical result ob-
tained from the lossy, hot-plasma theory, require extensive com-
puting time.

In the experiment, the antenna radiation was observed to be
enhanced about 15 db over the free-space radiation when the antenna
frequency was 0. 3 or 0. 4 GHz and the plasma frequency was at least
twice higher than the antenna frequency. It was also observed in the
experiment that when the antenna frequency was higher than 0. 8 GHz
no enhancement above the free-space radiation could be obtained after
passing the cut-off point. All these phenomena are well predicted by
the theory. Also in the experiment, the DC potential of the antenna
was varied betweeni 20 volts to see the effect of the plasma sheath
on the phenomenon of enhanced radiation. Except for a slight effect
due to the bias circuit, negligible effect by the plasma sheath had
been observed. Furthermore, no resonance peaks was found in the

experiment.

4. 4 Discus sion

 

In this study, the phenomenon of enhanced radiation from an
antenna coated by a layer of plasma is confirmed theoretically and
experimentally. It appears feasible to apply this phenomenon in
overcoming the blackout problem of a reentry antenna or in providing
a low-loss tuning for a small antenna.

There remains two facts which should be pointed out. The

first is the appropriate size of plasma layer for a possible

83

enhancement in radiation. In our experiments and most of numerical
examples, the size of plasma layer was made bigger than the antenna
size for structural convenience. This does not mean that a large
volume of plasma is always needed for a possible enhancement in
radiation. In fact, it has been shown in Fig. 2. 6 that even for a plasma
layer with a thickness equal to the antenna radius can lead to a 15 db
enhancement. Furthermore, in Figs. 3. 10 to 3.15 we observe
resonance peaks when the thickness of the plasma layer is very thin.
The second is that antennas on a space vehicle are usually
matched under free-space condition and in our study this fact was
not taken into account. This is not very important because a more
important fact is that even a matched antenna when covered by a
plasma will suffer a blackout and after that the antenna radiation can
not be recovered by any tuning or impedance matching. Thus, the
method discussed in this paper may offer a possible solution to the

blackout problem.

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CHAPT ER 5

RADIATION OF A CYLINDRICAL ANTENNA IMMERSED
IN A LOSSY, COLD PLASMA OF INFINITE EXTENT

5. 1 Introduction

 

A cylindrical antenna is one of the mo st commonly used
radiators. When it is used on a space vehicle, it is often operated
in a plasma region. The electrical properties of a cylindrical
antenna in a plasma medium thus become important in view of
practical and academic reasons. In this chapter, the electrical
properties of a cylindrical dipole antenna immersed in a lossy,
cold plasma of infinite extent is studied.

Many workers have theoretically investigated the electrical
characteristics of a cylindrical antenna immersed in a hot plasma
of infinite extent. Chen( 7) studied a thin cylindrical antenna of
finite length with a sinusoidally distributed current in a hot plasma.

(16)

Balmain treated the problem of an electrically short antenna with
a triangular current distribution immersed in a hot plasma. Their

results gave the antenna resistance only valid for cop/w < l where

88

89

(0p and m are the plasma and antenna frequencies. Kueh1(l7'18 '19 )

studied the same problem, but solved the Boltzmann equation instead
of using the simpler hydrodynamic plasma equations. An interesting
result of his work is the existence of the antenna resistance for

cop/(o > 1. Since the Poynting-vector and induced-emf methods which
require a certain prescribed current distribution have been adopted
to calculate the impedance of the antenna in those papers, the antenna
reactance was not determined. Meltz, Freyheit and Lustigao) in-
vestigated 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 cop/(s. There are many other theoretical papers
which are not mentioned here.

To our best knowledge, there is no theoretical paper which
accurately determines the complete impedance of a cylindrical
antenna of finite length immersed in a hot plasma of finite or infinite
extent. This problem is intractable both mathematically and physi-
cally. In this study a simpler cold-plasma model is chosen to make
the analysis tractable.

(21)

Employing the King-Middleton theory and King's modified
22

method( ), the complete input impedance of a cylindrical antenna is

determined as a function of antenna dimensions and plasma para-

meters. The effect of the collision frequency on the antenna impedance

is carefully examined.

9O

5. 2 Geometry and Statement of the Problem

The geometry of the problem is shown in Fig. 3. l. A cylin-
drical dipole antenna of radius a and length 2b is center-driven at
z = O by an idealized delta function generator with a voltage of V and
an angular frequency of m. The antenna surface is assumed to be
perfectly conducting except at the small gap 26 at the center of the
antenna. The antenna is immersed in an infinite, homogeneous and
cold plasma. The plasma is assumed to be a weakly ionized gas and

can be characterized as a lossy medium with a permittivity of

 

w n e
E = E (l -—E— z),a conductivity of 0' = o v , and a per-
0 2 m 2. Z
a) +1: 6 w +V

meability of p = ”o where cop is the plasma frequency, no is the
density of plasma, u is the collision frequency of electrons with
neutral particles, e and me are the charge and mass of electrons,
and Go and .10 are the permittivity and permeability of free space.
In order to employ the quasi-one-dimensional theory, the

following dimensional restrictions on the antenna are made:

h>>a

(5.1)
fla, : -Z—-:—a— (<1

where 8 is the wave number and x is the wave length in this medium.
Based on these thin-wire assumptions, the current can be con-
sidered(23) to be concentrated along the axis of the antenna when
calculating the field or vector potential at a point in the medium.
Even in the vicinity of the antenna surface, the error caused by

this approximation is insignificant.

91

I (z) I
z

 

 

I (z) I plasma
2 (Hour. 6)

perfect __
conductor /'

(cr=cx)) d ______ z:-h

 

 

Fig. 5. l A cylindrical antenna immersed in an infinite,
lossy and cold plasma.

92

In this study rationalized MKS units are used. Cylindrical
coordinates (r, 9, z) are adopted and the rotational symmetry is
assumed. The time dependences for the radiating source and all

the fields are assumed to be exp(jm t).

5. 3 Basic Equations

 

The basic equations which govern the system are Maxwell's
equations. For harmonic time dependence of expfimt), Maxwell's
equations in this infinite, homogeneous and lossy medium can be

written as

8
-’ p
VOE :— 5.2
g ( )
Vx-E = -j0)-é (5-3)
Vx—é = p. 3.8 +10}: 6E (5.4)
O O
v-‘fi = o (5.5)

where E and .15 are electric and magnetic fields, i is the complex

permittivity given by
6 = 6(1 4%). (5.6)

s
and-3 and p3 are the volume densities of the source current and

charge which are related by the continuity equation,

*8

v-J +jwps= o. (5.7)
It is convenient for this case to solve the Maxwell's equa-
tions by introducing the vector potential A and the scalar potential

cb. From Eqs. (5. 5) and (s. 2) '15 and '2': can thus be defined in terms

 

93

of these potential functions as

‘13 = VxX (5.8)

-vq; -jw-A. (5.9)

PH
u

The substitution of Eqs. (5. 8) and (5.9) into Eqs. (5.4) and (5. 3)

leads to two inhomogeneous wave equations,

2-9 -0 ->
v 11+sz = -)LOJ8 (5.10)
2 2 9
V¢+k¢=-%— . (5.11)
subject to the Lorentz condition of
(71.15.. = o (5.12)

where k is the complex propagation constant given by

2 2
k = a) not . (5.13)

If we write
k: fi-ja, (5.14)

[5 and u can be expressed as

 

 

2 2 4 l l

90 w 2a) (up 3 2
(0+9 (n+1! (n(wHI)

z z 1.1.

50 a) 241) m4 J3 3

.=— -1+—£—+ 1-—£—+ P (5.16)

42 z 2 2 2 2 2 2
m+v w+v meow)

where 8 is the wave number in this lossy medium, a. is the attenua-

tion constant, and $0 is the wave number in free space defined as

B =wN/u6 . (5.17)

 

94

Equations (5.10) and (5. 11) are also called Helmholtz's equations.

The general solutions to Eqs. (5. 10) and (5. 11) are given by

 

 

 

 

-* -’ P -'D ~> ..ij —> -> -ij
A(1') 2 :0; [‘S‘V.J8(rl)eR dV'+‘g'K8(rl)eR dSI]
(5.18)
-o l 3 "’ -ij
¢(r)=m[&fp(r')e:R dV'+ +vs'n (1")‘3 ds’
(5.19)

where I? and n8 are the surface densities of the source current and

charge, R is the distance between the observation point and the source
. -ij o ' .

pomt, and e /41rR 18 the Green s function.

For the case of a thin-wire linear antenna we have

3"(?')dv' +E8(?')ds'= 212(2')dz' (5.20)
p°(?')dv' + n°('r")ds' = q(z')dz', (5.21)

since the antenna current flows primarily along the axial or z-
direction.
Using Eqs. (5. 20) and (5. 21), Eqs. (5.18) and (5.19) can be

reduced to

 

A(r)::4_1r z 12°(z') dz' (5.22)
511-): 4+, Elam-:— (5.23)

where I2 and q are the source current and charge densities per unit
length along the antenna.

Using the Lorentz condition Eq. (5.12), Eq. (5. 9) becomes

 

95

‘5': = -¥%V(V.X)-jm2, (5.24)
k

Thus E and E fields are expressed in terms of X only. If we can
determine A from Eq. (5. 22), all the fields are then obtained from

Eqs. (5.8) and (5.24).

5. 4 BoundarLConditions

 

(24) that the boundary con-

It is shown by King and Harrison
ditions for an antenna immersed in a lossy medium are the same as
for an antenna in free space. The King-Middleton's method is then
adopted to solve the present problem.

Due to the symmetry about the origin and the end condition of

the antenna, a pair of boundary conditions on the antenna current can

be expressed as

I (z) : I (-z)
z z (5.25)
12(1-1'1) = 0

From the boundary condition that the tangential component of
electric field should be continuous at the surface of the antenna, it

follows
2% = a-) = Ei (r = a”) (5.26)
Z Z

where E;(r = a+) is the induced electric field just outside of the
antenna surface at r = a+, which is maintained by the current and
charge on the antenna, and E:(r = a') is the electric field just inside

the antenna surface at r = a-.

96

5. 5 Integral Equation for the Current

 

Since the cylindrical antenna is assumed to be constructed
of a perfect conductor, its internal impedance per unit length 21
is equal to zero and the electric field inside the conductor surface

at r = a- vanishes except at the small gap, i. e.,

a 0 for -h:z:-6and6:z:h
EZ(Z)= (5.37)

-2‘—:5- for -6<z<6

where V is the applied voltage and 26 is the gap width.
The applied voltage across the gap at z = 0 is
6 a
V: -( E (z)dz . (5.28)
Z
“-6
In the limit of a slice generator where 26 -> 0, the tangential electric
field on the antenna surface can be expressed as
lim 123(2) = - v (5 (z) (5.29)
26*0 z o
where 6 (z) is the Diracdelta function.

Since the current flows only in the axial direction, from

Eq. (5. 22) the vector potential has only the z-component, i. e. ,

X = zAz. (5.30)

From Eqs. (5. 30) and (5. 24) we obtain the induced electric field

on the antenna surface as

97

 

k2 azz

i - 87‘ 2
122(2) = - E[ + k J Az(z) . (5.31)
In order to satisfy the boundary condition given by Eq. (5. 26),
Eqs. (5.29) and (5. 31) are equated to yield a second-order inhomo-

geneous differential equation for the vector potential at the antenna

surface as

 

2 Z
8 2 _ jk
[322 + k] Az(z) .. - -0)_ V06 (z) . (5. 32)

The complementary solution of Eq. (5. 32) is obtained easily as

c _ jig ,
Az(z) _ - (1) (clcoskz +czsinkz) (5.33)

where c1 and c2 are arbitrary constants, and the parameter

(- i5 ) is added merely for convenience.

Since the particular integral for an equation of the form
d
-—2 + bzy - f(x)

is given by

l x
Yp(x) = b S f(s)sinb(x-s)ds ,
o

the particular solution of Eq. (5. 32) can be verified to be

112(2) = ”1245.) v0 sink|z|. (5.34)

The general solution to Eq. (5. 32) is then

98

c P
Az(z) Az(z) + Az(z)

sinkz +31 sinkIzI].

, i}:
(1) [clcoskz+c2 2
(5.35)

From Eq. (5. 22) we obtain the vector potential on the antenna sur-

face as

p.
A(Z):
Z

filo

h
" Sh Iz(z')Ka(z,z')dz' (5.36)

where Ka(z, z') is the kernel defined as

-ij
R

e

 

Ka(z, z') =

and

 

R = [(z-z') +32
By the surnmetry of the antenna current, Iz(z) = Iz(-z), it can be
shown from Eq. (5. 36) that the vector potential is also symmetric
about the origin, i. e., Az(z) : Az(-z). It is obvious that the arbi-

trary constant c2 should be equal to zero and Eq. (5. 35) reduces to

112(2) = -35 [clcos kz Jr; sinklzl]. (5.37)

By letting z = h in Eq. (5. 37), c1 can be determined as

1
c1 " cos kh

 

[11:3 Az(h) -321 sinkh] . (5.38)

Using Eqs. (5. 36) and (5. 37) we can write an equation such as,

99

411' h

= - $[01K08 kz - coskh) +§ (sink)zl - sinkh)]. (5.39)
o

The substitution of Eq. (5. 38) in Eq. (5. 39) leads to

 

h
'4 k V .
Shlz(z')Kd(z. z')dz' = 011L020)! kh ['2' 31111411 - ‘2‘)
- IE9- Az(h)(coskz - cos kh)] (5.40)

where Kd(z, z') is the difference kernel defined as

Kd(z. 2') = K301. 2') - Kaah 2')

 

and

 

Rh = J(h-z') +az .

Equation (5. 40) is an integral equation for the antenna current
which is valid for -h 5 z 5 h and convenient for the further develop-

ment.

5. 6 Approximate Solution of the Inggral Equation

The current distribution on the antenna can be determined
quite accurately by solving the integral equation (5. 40) approximately
following King's modified method .(ZZ). In this method the antenna

current is assumed to be proportional to the vector potential dif-

ference (the difference between the vector potential at a point on

100

the antenna and that at the end of antenna). In other words, it is
assumed that the ratio of the vector potential difference to the
antenna current is relatively constant along the antenna. Since
Az(z) - Az(h) vanishes at z = i h, it is consistent with the end con-
dition of Iz(z = ih) = 0.
By the peaking property of the kernels
Ka(z, 2') ~ 6 (z-z')

Kan). 2') ~ 6 (h-z')

and from Eq. (5. 39) it follows that

411'
)1; [Az(z) - Az(h)] ~ Iz(z) - Iz(h).

Since 1201) = 0 as imposed in Eq. (5. 25), then

12(2) ~ Az(z) - Az(h)
and the antenna current can be assumed to have the form

12(2) = A lc(z) + B 18(2) (5.41)
where A and B are arbitrary constants and

Ic(z) = cos kz - coskh

18(2) sink(h - (2|) .

Note that Eq. (5.41) satisfies the boundary conditions given by

Eq. (5.25).

101

Since the difference kernel is a complex function, it can be separated

into real and imaginary parts as

 

 

 

l _ I ' 1
Kd(z,z) — Kdr(z' z ) +jKdi(z, z) (5.42)
where
e'aR e-
I .. - _—
Kdr(z’ z ) _ R cos BR Rh cos fiRh
-aR e-aRh
' - - s o
Kdi(z,z) .. R sinfiR + R), sinflRh

Substituting Eqs. (5.41) and (5.42) into Eq. (5.40) we have

h

 

ShlAIc(z') + B13(z')][Kd1—(z’ z') + j Kdi(z, z')]dzl
.. j4flk [X sin k(h- Izl) - E A (h)(C°3 kz ' COS kh)]
' muocoskh Z k z

(5.43)
Since Kdr(z' z') becomes very large when 2' is near 2, it
follows that the principal contribution to the part of the integral that

has Kdr(z' z') as kernel comes from elements of current near 2': 2.

On the other hand, K 1 remains very small when 2' is near 2. This

d'
suggests that the contribution to the part of the integral that has
Kdi(z' 2') as kernel comes from all the elements of current along
the antenna. Due to this peaking property of kernel Kdr(z' z')

and non peaking property of kernel Kdi(z’ 2'), various integrals on

the left hand side of Eq. (5.43) can be equated to the functions on

the right hand side of Eq. (5. 43) in the following manner:

102

S (cos kz' - cos kh)Kdr(z, z')dz' ~ (cos kz - cos kh) (5.44a)
(cos kz' - cos kh)Kdi(z’ z')dz' ~ (cos kz - cos kh) (5.44b)
"-h
h
( sink(h - (z'))K (z,z')dz'~sin(3 (h - (2|) (5.44c)
V-h dr 0
(vb sink(h - ‘2")Kdi(z, z')dz' ~ (cos kz - cos kh) (5.44d)
“-h

where Eqs. (5. 44a) and (5.44c) are based on the characteristics of
kernel Kdr(z’ z'), and Eqs. (5.44b) and (5.44d) are justified from
numerical calculation. Indeed, it can be shown numerically that
the integrals on the left side of Eqs. (5.44b) and (5.44d) are roughly
proportional to the shifted cosine function (cos kz - cos kh).

These properties suggest that Eq. (5.43) can be split into two
parts by equating the corresponding terms on the right and left hand

sides of this equation as follows:

 

 

h
SJAIC (z')Kd (z,z z') + j BIB(z')Kdi(z, z')]dzl : ——Po<34:8kh AZIhHCOB kz
- cos kh) (5.45)
511131 (z')Kd1'Z(z'z')dz'zwiZLflSZkh sink(h - (z‘) (5.46)
"-h 0
Equations (5.45) and (5.46) can be rearranged as
S: I (:(z') I8(z') 4"
I ' I I- __
[A— I: (z ) Kd(z, z ) +jBIc(z) Kdi(z' 2 )]dz — pocoskh Az(h)

(5.47)

103

h I (z')

3
Sh B 18(2) Kdr(z, z')dz' =

_jQanfl/
m pocos kh

 

 

(5. 48)
The current Iz(z) can be expressed in terms of a reference
current Iz(zo) and a distribution function f(z) that is unknown. Let
12(2) 2 Iz(zo)f(z) . (5.49)

From Eq. (5.49) we have

 

 

 

I (z') = I (z )f(z') . (5.50)
z z 0
Then
- f(Z') ._
Iz(z') - 12(2) f(z) - 12(2) 8(2. 2')
and
Iz(z')
g(z.z') = Hz) (5.51)
Z
It has been shown by Kingu‘l) that a function \|J(z) can be defined as
It (2)
11(2) = 31 z = 41+ 7(2) (5.52)
o Iz(2)

where 41 is the constant part of \.IJ(z) and is called as the expansion

parameter. 1|)(z) is roughly constant in the central part of the antenna

but increases rapidly at the ends of the antenna. Usually 7(2) is
chosen to be close to zero at the point of maximum current. It can

be shown that 7(2) remains very small over the central part of the

antenna and has a large value only at the ends of the antenna. There-

fore, we can let

 

104

 

\P = M2 ) = -- (5.53)

where 21 is the point of maximum current on the antenna. It has

been commonly assumed that z = 0 when h S X/4 and 21: h - X/4

1
when h > X/4. Using Eqs. (5. 52), (5. 36) and (5. 51) we can express

(Hz) in another form as

M2) = fhg(2.2')K(z,z')dz' (5.54)
".11

where K(z, 2') stands for an arbitrary kernel. Thus from Eqs.
(5. 53) and (5. 54) we obtain an expression for the expansion para-
meter,

) = ((21) = f[g(z.z')1<(z.z')] dz'. (5.55)
-h 2:

Z!

Using Eq. (5. 55), Eqs. (5.47) and (5.48) can be reduced to

 

. 411'
A 4'61 +13 1.2 - 3:33:55 Az‘h’ (5'56)
_ iZka
B"Pd3 - wuocos kh (5° 57)
where
4’61 = 5J1 [gd1(zo Z')Kd(Z, z')] (12'
-h zzz

l

q’dz : (h [gdz(zs z')Kdi(zs z')] dz.
u_h Z:

Z

1

 

105

4’63 = 6[gd3(z'z')Kd1-(z'z')] dz'

 

 

 

 

”-11 2:21
and
( z!) _ Ic(z') _ cos kz'- coskh
gdl 2' - Ic(z) - cos kz - coskh
( z') - 18(2') _ 31111411 " 121)
g<12 7" ’ Ic(z) ‘ cos kz - coskh
18(2') sink(h - (z'))

 

gd3(z'z') 1 (z') ' sink(h- Izj) .

From Eq. (5. 36) we have

31 Az(h) : rlz(z')Ka(h,z')dz' (5.58)

o c

The substitution of Eq. (5.41) into Eq. (5. 58) gives

ilA(h)=AT+BT (5.58)
p. z c s
0
where
h
'rc = 5 Ic(z')Ka(h,z')dz'
-h
'1“B = S:IB(2')Ka(h,z')dz' .

Substituting Eq. (5. 58) into Eq. (5. 56), an expression is obtained
as

A = B T01) (5.59)

106

where

 

From Eq. (5. 57) we obtain

j21rkV

B
41d300 pocos kh

(5.60)

 

The combination of Eqs. (5.41), (5. 59) and (5.60) leads to the

current distribution along the cylindrical antenna as

J 2ka

¢d3w Hocoskh 1313150“ 12)) 1‘ T(h)(cos kz - cos kh)]

(5.61)

 

1(2):
2

Equation (5. 61) is the final expression for the antenna current.

5. 7 Input Impedance of the gylindrical Antenna

 

The input impedance of the cylindrical dipole antenna is
defined as

v .
Zin- 12(z=0) ' Rin+JXin

where Rin is the input resistance and Xin is the input reactance of
the antenna.

From Eq. (5.61), this impedance can be obtained as

¢d3w pocoskh

Zin = j21rk[ sinkh + T(h)(l - cos kh)] (5° 62)

 

where the symbols can be expressed more explicitly as follows:

 

107

TB- jnlldzcoskh

updlcos kh - Tc

 

T0?) =

 

2‘2

h .
T = ( (cos kz'-cos kh) W dz'
c ”-h (h-z') +a
. Z Z
Ts = (h sink(h- (z'))?L-Lk—MrE—l dz'
v-h (h-z') + a

exp(-jk)/(zl-z')2-I:z)

 

 

 

 

l
'4‘ = - 5 (cos kz' - cos kh)[
d1 coskz1 coskh -h {‘21- 21) + a

. 2 2
_ ex - k‘lh-z') +a ]d2,
701-2 ')é+ .2

 

 

 

-1 exp(- (1"(21 -z') 2+a2

2 1
4(d2 = coskz -coskh 8ink(h- Iz'| —T_'_z__7——sinfl'[(zl-z') +3
1 -h (21 -z) +a

- W W sinfl'IOi-z')z+ az]dz'
(h-z') +a

J 2 2'
l h . exp(-o. (zl-z) +a ) r . 2
5‘ s1nk(h-‘z'l) fl_ cosfi (zl-z )+a

 

 

 

 

 

 

 

¢d3 = amkm-‘zln Jul-z.)
Z 2
.. E$6.0—J0211u-é')7—+-a—zcos BJ(h-z')2+a2]dz'
(h-z') +a
zl = 0 when 115%
X X
21: h-Z when h>z.

108

5. 8 Numerical Results

 

The input impedance of a cylindrical dipole antenna as ex-
pressed in Eq. (5. 62) has been numerically calculated as a function
of the antenna dimensions and plasma parameters. All the integrals
given in Eq. (5. 62) are numerically evaluated by the Simpson's rule
using a CDC 3600 computer. The theoretical results on the input
impedances ,of cylindrical monopole antennas of various lengths are
then calculated from zin/Z where zin is given by Eq. (5. 62), and
graphically shown in Figs. 5. Z to 5. 5. The input impedance is plotted
as a function of (spa/(02 with v/m as the running parameter. The value
of (0132/002 is directly prOportional to the plasma density when the
antenna frequency is kept constant, and ”/00 is the ratio between the
collision frequency and the antenna frequency. The antenna is assumed
to be driven at the frequencies of l. 8 and 2. 0 GHz.

In Figs. 5. Z to 5. 5, the solid lines represent the antenna
input resistances while the dotted lines stand for the antenna input
reactances. Observing from these figures, the effects of the collision
frequency on the antenna input impedance can be summarized as

follows:

(I) For low plasma density (mpz/(oz < 0.4) and low collision
frequency (ll/00 < 0. 01), as the plasma density is increased
the antenna resistance decreases and the antenna reactance

becomes more negative in such a way that the antenna behaves

(Z)

(3)

(4)

(5)

109

progressively shorter electrically. The collision fre-

quency has almost no effect on the antenna impedance.

For low plasma density (gopz/mZ < 0.4) and high collision
frequency (ll/(o > 0. 01), the antenna still behaves progressively
shorter electrically as the plasma density is increased. The
collision frequency has small effect on the antenna resistance

but still no significant effect on the antenna reactance.

2
For 0.4 < (cpl/(o < 0. 85 and V/a) < 0.01, the antenna behaves

as the same in case (1).

For 0.4 < mpz/wz < O. 85 and v/(o > O. 01, as the plasma den-
sity is increased the antenna resistance tends to increase
monotonically and the antenna reactance behaves in an opposite
way. The effect of the collision frequency on the antenna
impedance becomes more obvious over this range. The in-
crease of the collision frequency causes the increase of the
antenna resistance and makes the antenna reactance less
negative. This implies that over this range there is more
energy transferred from the electromagnetic wave to the

electron gas of plasma.

In the range of 0. 85 < pr/wz < 1. 15, there are sharp peaks
of antenna resistance and a sharp change from capacitive to
inductive for the antenna reactance when the plasma frequency

approaches the antenna frequency. Over this range, there is

110

tremendous energy transferred from the electromagnetic wave

to the electron gas of plasma.

(6) When mpZ/wz > 1.15, both antenna resistance and reactance

decrease rapidly as the plasma density is increased.

The significant findings of this study are: (1) the peaking of
antenna resistance at (o ~mp due to the collision,and (2) the change

of sign for the antenna reactance at a) ~wp.

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CHAPTER 6

EXPERIMENTAL INVESTIGATION ON THE RADIATION

 

 

OF A CYLINDRICAL ANTENNA IN A HOT PLASMA F“
. !
6. 1 Introduction 5
i
The radiation of a cylindrical antenna in a hot plasma has
{I
been studied theoretically by many researchers as mentioned before. a";

In contrast with the abundance of theoretical papers, extremely few

experimental studies have been published. The relevant experimental

(9)

studies were the observation of electroacoustic wave by Whale
in a rocket flight, Schmitt'sum observation of Tanks-Dattner's

resonance excited by an antenna, and the experiments on the antenna

in a hot plasma conducted by Jassby and Bachyskiu”

Jackson and Linuz' 13). Nevertheless, to our best knowledge, no

and by Chen,

extensive experimental study has been conducted to study the electro-
acoustic wave excited by an antenna in a hot plasma.

The purposes of this experimental investigation are to
(I) detect the existence of an electroacoustic wave excited by an
antenna in a hot plasma, and (2) study the effect of this electro-

acoustic wave on the circuit and the radiation properties of an

115

116

antenna. Two approaches, (1) to measure the antenna input impedance
as a function of plasma parameters, and (Z) to measure the antenna
radiation field as a function of plasma parameters, have been used to
detect the electroacoustic wave.

In order to conduct an accurate experiment on the interaction of
an antenna with a hot plasma, a great deal of time and effort was exerted
to produce a large volume of stable, high-density plasma. In our
experiment, the hot plasma was the mercury arc discharge which

was created in two plasma tubes. The cylindrical monopole antennas

 

were used as the radiating source.

6. 2 Experimental Setups

 

Two experimental setups have been used in the present investi-
gation. One setup consisting of a large plasma tube with the dimensions
of l4-inch diameter by 18-inch length was designed primarily for the
antenna impedance measurement. The other setup using a smaller
plasma tube of 6-inch diameter by lZ-inch length was designed for the
purpose of the antenna radiation measurement. The construction of
the plasma tubes and the details of these two setups are described
below.

6. Z. 1 Construction of a Large Volume of Stable,
High-Densiy Plasma

 

 

To conduct an accurate experiment on the interaction of an

electric source with a plasma at a convenient frequency range, a

117

large volume of stable, high-density plasma is required. Conventional
laboratory-produced plasma are either a gas afterglow discharge or a
mercury glow discharge. A gas afterglow discharge is not a continuous
plasma and it is usually of low density. A mercury glow discharge can
only give a small volume of high-density plasma. In the present investi-
gation, a mercury arc discharge was employed. A mercury arc dis-
charge can give a large volume of high-density plasma. However, it

is inherently unstable due to its moving hot spots on the mercury pool.

A novel method of placing a spot fixer on the mercury pool of the plasma
tube was used to stabilize the plasma. Using this device, a stable plasma
with the plasma frequency of l to 5 GHz can be obtained and the plasma

tube can be operated continuously for many hours.

6. Z. 2 Experimental Setup for the Measurement of
Antenna Impedance in a Hot Plasma

 

The schematic diagram of the experimental setup for the
antenna impedance measurement is shown in Fig. 6.1. The plasma
tube is made of an open-end pyrex bell jar with the dimensions of
l4-inch diameter and 18-inch length. The upper end of the tube
is the anode with a cylindrical monopole antenna feeding through its
center. The lower end of the tube is the cathode which consists of
a mercury pool. A floating metallic ring is placed at the middle of
the mercury pool to fix the moving hot spots of the mercury arc.
An ignition circuit is indalled in the mercury pool for the purpose

of starting the plasma. Between the anode and the cathode a DC

 

 

118

 

 

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120

power supply circuit is connected. Under the normal operation

the discharge current can run from 0 to 120 amperes. The pumping
system consists of a mechanical pump and a mercury diffusion pump.
The tube is continuously pumped during the experiment and the pressure
of the plasma is kept around 10-3mm Hg. The antenna input impedance

was measured by employing the standard SWR method. A photograph m

of this plasma tube under operation is shown in Fig. 6. Z.

6. 2. 3 Egrimental Setup for the Measurement of I
Antenna Radiation Through a Hot Plasma

 

 

 

The experimental setup for the antenna radiation measure-
ment is shown schematically in Fig. 6. 3. The plasma tube is made
of a pyrex glass tube with the dimensions of 6-inch diameter and
lZ-inch length. A small mercury pool with a floating spot fixer is
located at one end of the tube. The open end of the tube is sucked
to the ground plane when the tube is being pumped. The pressure of
the plasma is kept around 10"3 mmHg. With this arrangement the
ground plane acts as the anode and the mercury pool acts as the
cathode of the plasma tube. A DC power supply circuit is connected
between the anode and the cathode. Under the normal operation
the discharge current can run from 0 to 10 amperes. A cylindrical
monopole antenna is fed through the ground plane and into the center
of the plasma tube. The radiation of the antenna through the plasma
is measured by a movable receiving antenna which is connected to a

hetrodyne receiving system. The distance between the radiating and

121

Hetrodyne
Receiver

Local
R . F. Gen.

 

 

'Mereury Mech.
Diff. Pump Pump

Plasma
J I -
Microwave } _J i=5
- ... I
Absorber Rec.Ant. l-o' db Gen. (1.7-4.0 GC)

’7 vvvvvv

VV

 

 

} Slotted
Line Sq. Wave
} - Gen. (1000 o)
!A A A A A. A A
Anechoic Chamber - - SWR Meter
(6' x 8' x 6') - Detector

D.C. Rawer Supply

Fig. 6. 3 EXperimental setup for the radiation measurement
of cylindrical antenna.

 

122

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123

receiving antennas is 0. 7 m. The plasma tube, the radiating and the
receiving antennas are all enclosed in an anechoic chamber. Figure
6.4 shows the photograph of this plasma tube under operation inside

of the anechoic chamber. The experimental setup outside of the ane-

choic chamber is the same as that shown in Fig. 4. 3. F"

6. 3 Review of a Lossless, Hot-Plasma Theory

In order to provide another set of theoretical results for com-

(7) " '

parison with the experimental data, Chen's analysis on the inter-

4...-

 

action of a radiating source with a plasma is briefly outlined here.

In Chen's analysis, the radiating source is a cylindrical
antenna with a length of lb and a radius of a. The antenna is center-
driven by a delta-function generator and immersed in a lossless, hot
plasma of infinite extent. The current and charge distributions of

the antenna are assumed to be

 

T“ = Imsin[ke(h- whim; (6.1)
2 7 '
qs: :jJP'oEou -o)p /(o )Imcos[ke(h- |z|)] ert (6.2)

where 1m is the maximum current on the antenna, 0) and mp are the
antenna and plasma frequencies, p0 and 60 are the permeability and
permittivity in free space, and ke is the prOpagation constant of the

electromagnetic wave given by

124

k zéJw-mz (6.3)

where c is the velocity of light in free space. In Eqs. (6. l) and
(6. Z) the distributions of antenna current and charge are assumed
to be entirely controlled by the electromagnetic mode. This approxi-
mation is necessary since the effect of the electroacoustic wave on
the antenna current and charge is still not well known.
Based on Eqs. (6.1) and (6. 2), the power density of the electro-

magnetic wave can be derived to be

151 Z cos(k hcosB)-cos(k h) I‘
I: e e ]r (6.4)

m

p :2
e val-a) /w r2
p o

where ro is the distance between the observation point and the center

 

sin 6

A
of antenna, 6 is the polar angle and r is the unit radial vector in the
spherical coordinates.
The total power radiated as an electromagnetic wave can be

calculated by integrating 3e over a large sphere. The result is

151 2
m

P = {- cos (2k h)Cin(4l< h)
e Jl-wp /w e e

+ 2[1 + cos(2 keh)] Cin(2keh)
+ sin(Z keh)[Si(4 keh) - 251(2 keh)] } (6.5)

where Si(x) and Cin(x) are the sine and cosine integrals. The electro-

magnetic component of the antenna input resistance is obtained by

125
. . . . Z . .
d1v1d1ng Pe With l/Z Io where Io 18 the antenna input current and
related to I by I = I sin(k h). This gives
m o m e

30
R = 2—-===7- {- C08 (2 keh)Cin(4 keh)

sin2(keh)\/l-wp /w
+ Z[l + cos(Z keh)] Cin(2 keh)
+ sin(Z keh)[Si(4 keh) - 251(2 keh)] } (6.6)

which is valid for a) > cop.

The power density of the electroacoustic wave can be obtained

 

 

as
2 Z
3 w I
-’ - 1.5. L 2. m 1
pp- 1r v 2 Z I 2 2
o a) r l-o) /w
0 P
2
cosB[cos (k hcosB) - cos(k h)
p e *
r (6.7)

2 2
1 - (c/vo) cos 9

where V0 is r.m. s. velocity of electrons and k is the propagation
P

constant of the electroacoustic wave given by

k = 3- sz-m z. (6.8)

P o P
The total power radiated as an electroacoustic wave can be

obtained by integrating 3p over a large sphere. This gives

126

Z 2.
(D
151r (DP /

P:—
2 Z 2
p («up /w

under the condition of c >> v0 and a) > cop. The electroacoustic com-

 

 

[ 2 keh + sin(2keh)] I; (6. 9)

ponent of the antenna input resistance can be determined by dividing

Pp with 1/2 102. The result is

22
(1)/(n

R z—l—E’l—J-=-E_—2_—_—Z-[Zkh+sin(2k h)]. (6.10)
p . 2 e 8
sin (keh) l-(op /m

Equation (6. 10) is valid if c >> v0 and a) > (op.

The total input resistance of the cylindrical dipole antenna is
R. = R + R . (6.11)
in e p

For a monopole antenna the input resistance is Rin/Z° Re and R
were numerically calculated to compare the experimental antenna
resistance.

The antenna input reactance should also consist of both
electromagnetic and electroacoustic components. The elctromagnetic
component can be theoretically determined from King-Middleton's

)
theory(25

once the electrical dimensions of antenna, keh and kea,
are known. The electroacoustic component of the antenna reactance
is neglected in this study due to the lack of knowledge on the effect of
the electroacoustic wave on the reactive power of an antenna. Due
to this reason only the electromagnetic component of the antenna

reactance was used in comparison with the experimental results

in a later section.

127

The radiation field of an antenna in a hot plasma consists of
the fields of electromagnetic and electroacoustic radiations. The
radiation pattern of the electromagnetic wave is completely deter-
mined by the factor keh as eXpressed in Eq. (6.4). The electro-
magnetic radiation has a characteristic of zero radiation in the axial
direction (9 = 0°) of antenna. The radiation pattern of the electro-
acoustic wave is determined by Eq. (6. 7). The special nature of
the electroacoustic radiation is to have peak radiation in the direction
near the axial direction of antenna. If an antenna is surrounded by
a finite plasma, the electroacoustic wave may convert part of its
energy to create an electromagnetic wave at the plasma discontinuity.
This electromagnetic wave may, in turn, be detected outside of the

plasma region.

6.4 Experimental Results and Comparison with Theories

Since the purposes of this investigation are to detect the
excitation of an electroacoustic wave by an antenna and to examine
the effect of this electroacoustic wave on the circuit and the radiation
properties of an antenna, the antenna input impedance and the antenna
radiation field are measured as functions of the plasma density. The
eXperimental results on the antenna input impedance and the antenna
radiation field are carefully studied and compared with two theories.
The two theories are: (l) a lossless, hot-plasma theory reviewed
in the preceding section, and (2) a lossy, cold-plasma theory developed

in Chapter 5 .

128
6. 4.1 Comparison of ExPeriment With a Lossless,
Hot-Plasma Theory on Antenna Input Impedance

The experimental results on the input imPedances of cylindri-
cal monopole antennas of various lengths are graphically shown in
Figs. 6. 5 to 6. 8. The input impedance of the antenna is plotted as
a function of mpZ/mz where mp is the average local plasma frequency
in the antenna vicinity and a) is the antenna frequency. The average
local plasma density is used here because the plasma density in the
antenna vicinity is not uniform due to the plasma sheath on the antenna
surface. Furthermore, it was found that the circuit property of an
antenna is primarily controlled by the plasma condition in the antenna
vicinity. Experimentally this average local plasma frequency was
found to be about 20% lower than the maximum plasma frequency in
the plasma tube. The value of wpz/mz is directly proportional to
the plasma density when the antenna frequency is kept constant through-
out the eXperiment. The antenna input impedance was measured in
the large plasma tube (14-inch diameter by 18-inch length) and at
the frequencies of 1-8 and 2. 0 GHz.

In Figs. 6. 5 to 6. 8, the solid line with solid dots is the measured
antenna input resistance and the dotted line with solid dots is the
measured antenna input reactance. The theoretical values are repre-
sented by circled dots. The eXperimental results on the antenna input

impedance can be summarized as follows:

129

(1) For the case of low plasma density or mpz/wz < 0.6, as
the plasma density is increased the antenna resistance decreases
and the antenna reactance becomes more negative. This phenomenon
indicates that the antenna behaves progressively shorter electrically
and also implies that no electroacoustic wave or only an electro-
magnetic wave is excited over this range. Theoretical impedance
was calculated over this range by assuming the existence of electro-
magnetic mode only. The agreement between experiment and theory
is very good. The assumption of no electroacoustic wave over this
range may be feasible since a longitudinal plasma wave will suffer a
tremendous Landau damping in this range.

(2) For the range of 0.6 <mPZ/mz < l. 0, the antenna resistance
monotonically increases and finally reaches a peak at mp: (o. The
antenna reactance goes to a large negative value and then sharply
changes its sign at cap: a). The large antenna resistance over this
range may be identified as being due to the excitation of an electro-
acoustic wave. The theoretical resistance of the monopole over this
range was calculated from Rin= % (Re+ RP) where Re and RP are given
by Eqs. (6. 6) and (6. 10). Excellent agreement between theoretical
and experimental antenna resistances tends to confirm the excitation
of an electroacoustic wave over this range. For the antenna reactance
the large deviation between theory and eXperiment over this range is
probably due to the fact that the electroacoustic component of antenna

reactance was not taken into account. The assumption of the existence

 

 

 

{.A y .5.
l

flab-i— .

130

of an electroacoustic wave over this range is justified since the Landau
damping is small when mp is close to m.

(3) For the C388 of (spa/mZ > 1. 0, both the electromagnetic
and the electroacoustic waves are essentially cut off. If no loss
mechanism is present in the plasma, the antenna resistance should
drop sharply to zero. Experimental results, however, show that the
antenna resistance decreases rather gradually in this range. The
possible loss mechanisms are the finite collision loss of the mercury
plasma and the possible excitation of Tanks-Dattner's resonance on
the antenna surface and on the tube wall.

The weakness of this lossless, hot-plasma theory is its in-
adequacy of providing a theoretical prediction for the range of
m < mp.

6.4. 2 Comparison of Experiment With a LossyL Cold-
Plasma Theory on Antenna Input Impedance

In Sec. 6.4.1 it is indicated that the experimental antenna
input impedance can only be compared with a lossless, hot-plasma
theory for the range of mpZ/mZ < 1. In order to have a comparison
of experiment and theory for a wide range of mpZ/mz, the same
eXperimental results of the antenna input impedance are compared
with the theoretical results obtained from a lossy, cold-plasma
theory developed in Chapter 5. These comparisons are shown in
Figs. 6. 9 to 6. 12. The effect of the collision frequency on the

antenna input impedance has been carefully studied in Chapter 5.

 

131

The theoretical input impedance of a monopole was calculated from
Zin/z where zin is given by Eq. (5. 62). The ratio between the
collision frequency and the antenna frequency, V/m, was assumed to
be 0. 15 in the numerical calculation.

In Figs. 6. 9 to 6.12, the solid lines with solid dots are the

measured input resistance and ractance. The theoretical values

are represented by the dotted lines. The comparison between

experiment and theory can be summarized as follows:

 

L...

(1) For the case of low plasma density or mpZ/m2 < 0.4, as
the plasma density is increased both eXperimental and theoretical
results indicate that the antenna behaves gradually shorter electri-
cally.

(2) For the range of 0.4 < mpZ/mz < 0.85, the antenna
resistance starts to increase and the antenna reactance continues
to be more capactivie as predicted by the theory.

(3) For the range of 0. 85 < mPZ/mz < 1.15, a sharp peak
of antenna resistance and a change of antenna reactance from
capacitive to inductive at mp: m have been observed both experi-
mentally and theoretically.

(4) For the case of mpg/mZ > 1. 15, as the plasma density
is increased the antenna resistance and reactance decrease
gradually.

The agreement between the experimental results and the

lossy, cold-plasma theory appears to be quite satisfactory over a

132

2 2
very wide range of mp /m . However, there is still a critical

weakness in this lossy, cold-plasma theory. Because the temperature

effect and the electroacoustic mode are completely ignored.

6. 4. 3 Experimental Results on Antenna Radiation Fields

 

The radiation field pattern of a cylindrical monopole antenna
placed in the middle of the smaller plasma tube (6-inch diameter by
lZ-inch length) was measured under various plasma densities. The

measured radiation patterns for three monopoles of different lengths

are shown in Figs. 6.13 to 6. 15. For each antenna the change in the

radiation pattern was observed as the plasma density was varied.
The main points of the observation can be summarized as follows:
2
(1) For the low plasma density case or mp /m2 < 0. 6,
the radiation field is primarily electromagnetic, since no radiation
is observed in the axial direction of the antenna and the measured
pattern resembles with that of antenna with the electrical length of
k h radiated in free space.
e
2 Z . .
(2) For the case of 0.6 <mp /m < l. 0, a peak of radiation
starts to show in the axial direction of antenna. This radiation may
be identified as due to the electroacoustic wave, since the radiation
of an electroacoustic wave .in the axial direction of antenna will
create an electromagnetic wave at the plasma discontinuity which
exists at the tube wall.
2 Z . . .
(3) For the case of mp /m > 1. 0. the radiation peak in the

axial direction of antenna disappears and the measured pattern is

Ft"!

 

133

again mainly of electromagnetic nature. The reason why only the
electroacoustic wave is cut off in this range is that the plasma
dimension is large in terms of electroacoustic wavelength but rather
small compared with an electromagnetic wavelength. Thus an
electromagnetic wave can still be detected outside of the plasma
. . . Ir”

tube after suffering a relatively smaller attenuation.

To compare these eXperimental results more accurately, a

theory dealing with the radiation from a cylindrical antenna through

 

a finite volume of plasma is needed. Such a theory is unfortunately

intractable.

6. 5 Discus sion

 

Extensive experimental study has been conducted to investigate
the interaction of a cylindrical antenna with a hot plasma. Based on
the comparison with the lossless, hot-plasma theory, the experimental
results tend to indicate that an electroacoustic wave can be excited
when the average local plasma frequency in the antenna vicinity is
near the antenna frequency (0. 6 < mpZ/mZ < l. 0). No electroacoustic
wave can be excited if the plasma frequency is sufficiently lower
than the antenna frequency (pr/wz < 0.6). Based on the comparison
with the lossy, cold-plasma theory, it appears that experimental
results agree quite satisfactorily with the theoretical results if a
suitable collision frequency is assigned to the plasma. The effects
of the electroacoustic wave and the collision in the plasma on the

antenna input impedance are quite simlar. While the effect of an

134

electroacoustic wave on the radiation field is much more important
than that due to the collision in the plasma.

Since the laboratory-produced plasma is hot, lossy and finite,
the two theories used to compare with the eXperiment seem to be
inadequate. If an accurate comparison is needed for the eXperi-
mental results, a more complete theory of a plasma-imbedded
cylindrical antenna which takes into account of the plasma temperature,
collision and dimensions should be developed. It is unfortunate that

such a theory has been proved to be quite intractable.

 

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(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

REFERENCES

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K. M. Chen and C. C. Lin, "Enhanced Radiation from a
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147

H. Whale, "The Excitation of Electroacoustic Wave by Antenna
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H. Schmitt, "Plasma Resonances Excited by Antennas, " Proc.
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D. L. Jassby and M. P. Bachynski, "Measurement of the

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F. 'flt‘lmu_ _—_
11...“: ‘

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L. Spitzer, "Physics of Fully-Ionized Gases, " Interscience,
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H. H. Kuehl, "The Kinetic Theory of Waves in a Warm Plasma
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H. H. Kuehl, "Resistance of a Short Antenna in a Warm Plasma, "
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.....JI

. 7..

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(Z3)

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Second Edition, Dover Publication, Inc., 1963.

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J. Res. NBS, Vol. 64D, No. 4, pp. 365-380, 1960. j‘
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