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4563-39-T
INTERACTION OF A RADIATING SOURCE
WITH A PLASMA
EFFECT OF AN ELECTROACOUSTIC WAVE
KUN-MU CHEN
July 1963
Radar Laboratory
ift a Seoe a*" 7eS"
THE UNIVERSITY OF MICHIGAN
Ann Arbor, Michigan
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NOTICES
Sponsorship. The work reported herein was conducted by the Institute of
Science and Technology for the U. S. Air Force, under Contract AF 33(616)8365. Contracts and grants to The University of Michigan for the support of
sponsored research by the Institute of Science and Technology are administered
through the Office of the Vice- President for Research.
Acknowledgments. The author is grateful to Mr. Ralph E. Hiatt, Professor
C-M Chu, and Dr. R. F. Goodrich for reading the manuscript.
Distribution. Initial distribution is indicated at the end of this document.
DDC Availability. Qualified requesters may obtain copies of this document from:
Defense Documentation Center
Cameron Station
Alexandria, Virginia
Final Disposition. After this document has served its purpose, it may be
destroyed. Please do not return it to the Institute of Science and Technology.
ii
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CONTENTS
Notices..................................
List of Figures..............................
Abstract.................................
1. Introduction.............................
2. Basic Equations...........................
3. Separation of the Field into EM and P Modes...........
4. Potential Equations for EM and P Modes...............
5. Radiation Resistance and Fields of a Hertzian Dipole in a Lossless
Plasm a....................................ii...... 1.....2.... 7....2....4.. * * 5 5. * * * 7
6. Radiation Resistance and Fields of a Cylindrical Dipole Antenna in a
Lossless Plasma.............................. 12
7. A Hertzian Dipole in a Lossy Plasma.................... 23
Appendix. Evaluation of the Integral in Equation 82.............. 32
References.................................... 35
Distribution List................................. 36
iii
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FIGURES
1. A Hertzian Dipole................................ 7
2. Radiation Resistance of a Hertzian Dipole in a Plasma.............. 10
3. A Cylindrical Dipole Antenna........................ 13
4. Radiation Resistance of a Half-Wave Dipole (h = X0/4) in a Plasma
vs. 2/X2................................. 19
5. Radiation Resistance of a Short Dipole (h = X /10) in a Plasma
vs..wp2/w 2................................. 20
6. Radiation Resistance of a Dipole Antenna vs. Antenna Length at
wp2/p 2 = 0.5............................. 21
7. Radiation Resistance of a Dipole Antenna vs. Antenna Length at
wp2/W2 = 0.8................................ 22
8. A Hertzian Dipole in a Lossy Plasma Surrounded by an Imaginary
Sphere of Radius r1............................ 23
9. EM Component of Radiation Resistance of a Short Dipole (d2 = AO/10rT)
in a Plasma as a Function of Plasma Collision Frequency, v/co.... 28
10. Paths for Contour Integrations....................... 33
iv
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INTERACTION OF A RADIATING SOURCE
WITH A PLASMA
Effect of an Electroacoustic Wave
ABSTRACT
When a radiating source is immersed in a homogeneous plasma of infinite extent, an electroacoustic wave may be excited in addition to the usual electromagnetic
wave. The electroacoustic wave becomes a longitudinal plasma wave in the far zone
of the source.
The case of a Hertzian dipole in a lossless plasma is considered first. The fields
of both the EM and plasma modes excited by the dipole are explicitly obtained, and the
EM and plasma components of the radiation resistance of a Hertzian dipole are determined. The case of a cylindrical dipole antenna in a lossless plasma is investigated next. The far zone fields of both the EM and plasma modes are explicitly obtained, and the EM and plasma components of the radiation resistance are then derived
as functions of the antenna dimension and the plasma parameters. Finally, the case
of a Hertzian dipQle in a slightly lossy plasma is studied; the effect of collisions in a
plasma on the radiation resistance of the immersed antenna is discussed.
1
INTRODUCTION
The interaction of an antenna with a plasma is one of the most interesting and important
topics in physics and engineering, since communication between a satellite and the ground involves the ionosphere. The performance of an antenna in an ionized gas or in a plasma is entirely different from the performance in the case of free space. The behavior of an antenna in
a plasma medium has been widely investigated, but in most cases a plasma has been treated as
/ 2 \ 2
a dissipative medium with E = - = and p =- - where v m
2\ 2 z / u m 2 2 p
+ v eX +v
w, and v are the plasma frequency, the antenna frequency, and the plasma collision frequency,
respectively, and nO is the ambient electron density. This treatment has completely ignored a
plasma wave or an electroacoustic wave which may be excited by an antenna in a plasma.
Cohen has shown in a series papers [1, 2, 3] that an electric source immersed in a homogeneous plasma of infinite extent can excite an electroacoustic wave in addition to the usual
electromagnetic wave. In the absence of a static electric or magnetic field, Cohen [1] and Field
[4] have shown that the electroacoustic wave and the electromagnetic wave are uncoupled and
can be separated into two independent modes. The electromagnetic wave becomes an ordinary
transverse electromagnetic wave, and the electroacoustic wave becomes a longitudinal plasma
1
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wave at great distances from the source. Furthermore, Whale [5] has observed the existence
of an electroacoustic wave in a rocket flight in the ionosphere. These studies tend to predict
that the radiation resistance of an antenna can be seriously affected by this electroacoustic wave.
The purpose of the present study is to investigate specifically the interaction of a plasma
with a dipole antenna and the effect of an electroacoustic wave on the radiation of a dipole antenna. We shall concentrate on tracing the effects of an electroacoustic wave on the input resistance and the fields of a dipole antenna in a plasma. The case of a Hertzian dipole in a lossless plasma is considered first. The fields of both the EM and the plasma mode excited by the
dipole are explicitly obtained. The radiation resistance of a Hertzian dipole is determined as
the sum of the EM and plasma components, where the former is due to the excitation of an EM
wave and the latter is due to the excitation of an electroacoustic wave. The case of a cylindrical
dipole antenna in a lossless plasma is investigated next. The far zone fields of both the EM and
plasma modes are explicitly obtained, and the EM and plasma components of the radiation resistance are derived as functions of the antenna dimension and the plasma parameters. Finally,
the case of a Hertzian dipole in a slightly lossy plasma is studied; the effect of collisions in a
plasma on the radiation resistance of the immersed antenna is discussed.
In this study MKS rationalized units are used and the time variation for the radiating source
is assumed to be ejw. The plasma involved is assumed to be a weakly ionized gas type.
2
BASIC EQUATIONS
-0-s s
We assume that a radiating source with current, J, and charge, p, is immersed in a
homogeneous plasma of infinite extent. No static electric or magnetic field is present. The
current and charge of the radiating source are related by the equation of continuity as
~S aps
v.J+- - =o (1)
The plasma is assumed to be a weakly ionized gas having an ambient electron density, n0. The
collision frequency of electrons with neutral particles of the gas is v.
In its unperturbed state the plasma is assumed to be homogeneous and neutral and the perturbation of the plasma due to the source is assumed to be small, so that the linearized equations
are applicable. This assumption may be poor in the very vicinity of the source, where the field
can be very strong and the perturbation in the plasma may not be small.
The basic equations which govern this system are Maxwell's equations,
VXE = -/0 at (2)
2
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V XH J=s- en V + aE ()
0- a (3)
oat.E =pS/0 - en1/e0 (4)
V* H =0 (5)
and the linearized Euler equations,
an1
n(V V) + = 0O (6)
nome + n0m vV = -n eE - m v Vn (7)
0 e 0 e 0 eG 1
In the above notation, n1 is the deviation of the electron density from its ambient density, n0.
m and e are the mass and charge of an electron. V is the mean induced velocity of electrons.
e
v0 is the rms velocity of electrons and can be expressed as
v0= ym (8)
where k is the Boltzmann constant and T is the electron temperature.
The power relation, which can be governed by Poynting's theorem as shown by Field [4], is
V (E e n ) + at E2 + 1 M H2 + n20
1 2 22 2 -+ n Om vo (n1/no) =-E J (9)
time as eW, the basic equations become
V x E = -jW /oH (10)
V x H = 7s - enOV + jweo (11)
V. E = pS/e0 en1/E (12)
V+H =0 (13)
and
n(V * ) +jwn1 =0 (14)
3
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(jw + v) = - e/m E - v02/n0Vn (15)
3
SEPARATION OF THE FIELD INTO EM AND P MODES
In our formulation of the problem, there are four unknowns, E, H, V, and ni, and two given
s
quantities, J and p. In the absence of a static electric or a static magnetic field and under
the constant temperature, these fields can be separated into two groups when the perturbation
in a plasma due to the electric source is small. The first group consists of three vector
fields, E e' H and V, and is called the electromagnetic (EM) mode. The second group consists
of two vector fields, E, V, and a scalar field, n1, and is called the plasma (P) mode. The EM
mode is the usual electromagnetic mode with an electric and a magnetic field but no charge
accumulation; this mode gives a transverse electromagnetic wave in the far zone of the source.
The P mode has the charge accumulation and an electric field but no magnetic field; this mode
gives a longitudinal plasma wave at great distances from the source. Physically, the separations
of the fields into two modes means that the electromagnetic wave and the plasma wave excited
by the radiating source are uncoupled.
After the separation of the field the following relations hold:
E = E + E (16)
e p
V = V + V (17)
e p
VXE =0 (18)
Equation 18 holds because there is no magnetic field in the P mode.
The substitution of Equations 16 to 18 in the basic Equations 10 to 15 gives the following
two groups of equations.
For the EM mode (Ee, H' Ve):
V x E =-jwo/ H (19)
e o
V X H = J - en 0V + jW 0Ee (20)
V. H= 0 (21)
(jc + V) = e (22)
e m e
e
One more equation for V * E can be obtained by using Equations 20, 22, and 1. The result is
V E = P /e0 (23)
4
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where
/ 2 \ 2
- } - 2 2/ -( + 2) (24)
o2 + V CO(c + v )
2
2 n0e
2 =- (25)
P e0me
For the P mode (Ep, V, n1):
V X E = 0 (26)
jOEp - enV ~ (27)
2
v
(jwo +)V e E- V n1 (28)
p m p n 1
These vector equations may be solved for some particular cases, but a more general approach
is to find some potential functions which are derived from those equations.
4
POTENTIAL EQUATIONS FOR EM AND P MODES
For the EM mode a vector potential can be derived conventionally. For the P mode, n
is found to act as a scalar potential.
For the EM mode:
Let us assume that
xH = V x a (29)
= -0 - jcA (30)
e
An equation for the vector potential, A, can be obtained from Equation 20 as
(V + e2) A = -O'Js (31)
where
2e =2 C )O (32)
and Equation 31 is subject to a condition of
V * A + jcOE00Q =0 (33)
5
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/ is the conventional scalar potential of the EM field.
Thus from Equation 31, A can be determined if J is given. After A is obtained, all fields,
E, H, and V, can be completely determined.
Physically, this mode is an EM wave set up in a dissipative medium with a complex dielectric constant, e = e0t. This EM wave propagates with a complex propagation constant of
co V//c0 in the medium.
For the P mode:
From Equations 27 and 28 we have
2
ev0
E 21Vn (34)
p -p -J(> 1
The substitution of Equations 23 and 34 in Equation 12 leads to
2
(v2 + p2)n 2 (35)
ev0
where
2 12 2\ 2 2o (36)
Thus, if the charge source, p, is given, n1 can be obtained by solving Equation 35. After n
is determined, E and V can be obtained from Equations 34 and 27. It is noted at this point
p p
that the deviation of the electron density from its ambient density, n1, in this case acts as a
scalar potential for Ep
Physically, n1 behaves like an acoustic wave and propagates roughly with the rms velocity
of electrons, v0. We call this an electroacoustic wave because it is excited by the electric
charge, p.
Up to this point, we are theoretically able to solve for all fields which are excited by a
-s 5
radiating source (J, p ) in a plasma medium. It is interesting to observe that the EM wave
S
is excited by both J and p, but the plasma wave or the electroacoustic wave is excited solely
s
by p. The electroacoustic wave affects the radiatio n antenna to a great extent,
as we shall see in the following sections.
6
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5
RADIATION RESISTANCE AND FIELDS OF A HERTZIAN DIPOLE IN A
LOSSLESS PLASMA
The first and simplest case to be investigated is a Hertzian dipole in a lossless plasma.
We shall attempt to find the EM and plasma waves which are excited by a Hertzian dipole in a
lossless plasma, and, after the fields are determined, to obtain the dipole's radiation resistance.
A Hertzian dipole is an oscillating electric dipole with a uniform current in a short wire,
df, and the concentration of charges at the two ends, as shown in Figure 1. The current and
charge of a Hertzian dipole can be symbolically represented as
=s I df 65(x)5(y)5(z) ejt z (37)
p = I [6(z - 2 dk) - 5(z + 2 df)] 6(x)5(y) e (38)
where 6 is the delta function.
s
The EM and plasma waves excited by J and p are analyzed separately below.
qs
+++
di i
qS
FIGURE 1. HERTZIAN DIPOLE
5.1. THE EM MODE (Ee, H' Ve)
We assume that the plasma is lossless and v = 0. With J as expressed in Equation 37,
the vector potential A can be found from Equation 31 to be
- ~0 e e
AT -id r
-jf er
/10 e e Jot A
r47 Idk ej ct z (39)
where r = x +y +z = the distance between the observation point and the source point. In
spherical coordinates, A can be expressed as
7
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j(cot-e r)
- 0 /10A A
A - Id - (cos 0 r - sin 0 0) (40)
4-I r
The scalar potential, c, can be determined from Equations 34 and 40 to be
<= J V A
j(wt-O r) e
-jId2 e (41)
= 4 jd e J-+ Cos 0 (41)
47Te0w ( )o
From Equations 30, 40, and 41, the electric field E can be determined as
E = -V( - jcwA
e
-jldf 1 j(ot-oe r) A
= rc (1+ jK) cos 0 e r
-j+d_ l?3e2\\ j(cot-oer) A
-jildf 1 ie fe2 A
+4jdO1 +1 e - ~ sin 0 e 0 (42)
The magnetic field can be obtained from Equations 29 and 40 as
=1
H =-V x A
/10
4T (2 - e sin 0 e e (43)
r
Equations 42 and 43 give the complete expressions for the electric and magnetic fields of the
EM mode, which is excited by a Hertzian dipole in a plasma. Ee has a radiation term (1/r),
2 3 a aito n nidien
an induction term (1/r ), and an electrostatic term (1/r ). H has a radiation and an induction
term. Both E and H fields are exponentially attenuating when if has an imaginary component,
e e
if the plasma is assumed to be lossy. In this section,3 is assumed to be a real number.
To determine the EM component of radiation resistance of a Hertzian dipole, we use a
Poynting vector method. This method, without modification, fails if he is assumed to be a
complex number. (This is discussed in Section 7.)
The radiation terms of E and H are
e
2
- r -jldf /e j(cot-e r) A
E r -Id - sin 0 e e 0 (44)
e 47T e0w r
8
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-r jIdf Pe sn co3 e A
4H = esin 0 e h (45)
It is assumed above that
W co
e 2 = c~e 4 1 - P and ^ = 1- p (46)
2 / CO
as the result of v = 0.
The Poynting vector of the EM wave is
- 1 (Er x-r*
e = (E XH
Idi e.2 A (47)
327r e co r
0
The total power radiated as an EM wave in a lossless plasma by a Hertzian dipole is obtained
by integrating the Poynting vector over a large sphere. That is,
r27T rT 2 - A
P = d d Or sin 0 p * r
I df2 3
127re 0 co e
= 407r2(2 ) 1 - 2/2 2 (48)
The corresponding radiation resistance can be obtained by dividing P by 1/2 I, which gives
e
Re = 807T (21 - aohms (49)
where X0 is the wavelength in free space. Equation 49 gives the expression for the EM component of the radiation resistance of a Hertzian dipole in a lossless plasma. In free space or
when w = 0, Equation 49 reduces to a well known value for the radiation resistance of a
Hertzian dipole. The behavior of R as a function of w /c2) is shown graphically in Figure 2.
e p
9
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0.8- O 1
E
E 0
o 0.6 —X\\0 3
|o 0
0 0.4 \0.2 '
0.2 —.\1 -
0 0.2 0.4 0.6 0.8 1.0
^ /u 2
FIGURE 2. RADIATION RESISTANCE OF A HERTZIAN DIPOLE IN A PLASMA.
R = EM component of radiation resistance; R = plasma component of radiation
resistance.
5.2. THE PLASMA MODE (E, Vp, nl)
sp p
If the charge source, p, is expressed as in Equation 38, the deviation of the electron
density from its ambient density, nl, can be determined as
2 r -jpr
n=41 2 Jer p dv
ev0
2
_j2d~ C~p /~ ~) j(cot-o r)
- jdJp 1 cos 0 e (50)
evd r
It is noted that n1 has been transformed from a rectangular coordinate form to a spherical
coordinate form.
From Equations 34 and 50 the electric field fo the plasma mode, Ep, is evaluated as
10
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2 /2 \ j(cot-j r)
E= P - i cos r + + sin 0 e P (51)
P 4 2 2 2 r r3r2
O47re W - co) r r r
Equation 51 shows that Ep has a radiation term (1/r) in the radial direction.
The induced velocity of electrons, Vp, in a plasma is easily obtained from Equations 27
and 51 as
-c_ Idj If 2A3 j(/3\ Ij w t-/3 r)
2 2
v + 2 - cos 0 r+ + ( sin 0 e (52)
47ren() - r r / r r
V also has a radiation term in the radial direction.
P
From a Poynting theorem in a plasma, as expressed in Equation 9, it can be shown that
there is a net flow of power from the source. The power density flowing out of the source as
a form of plasma wave is
2 r- r
Pp = mevO n1 Vp (53)
where n1 and V are the radiation terms of n1 and V fields. Since
p P
nd Cp P cos 80 e (54)
nl 4mw 2 r
ev0
and
(P p ) Av 2 2 2 r
47Ten0 - 2) r
The time average value of p can be found to be
21 2 A
(p) av = c2 p3 ~P cos 0 r (56)
32ir cv0 P r
The total power radiated by a Hertzian dipole as a form of a plasma wave is then
P = d d(r sin 0 (p ) av * r
P
11
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2
2p 23 2
24 E3
00
p= 2G2 (^) (c)V~ 1 - p 2/2 h 2I (57)
T he corresponding radiation resistance is obtained by dividing P by 1/2. This gives
2 ldk~ 2 c 0 coP 22
dipole in a lossless plasma. For a typical case of ionosphere, R can be much greater than
p
R if w is not very small compared to w. This may cause a serious change of the input
impedance of a Hertzian dipole in the ionosphere. The behavior of R as a function of o /7w
P P
is shown graphically in Figure 2.
5.3. INPUT RESISTANCE OF A HERTZIAN DIPOLE IN A LOSSLESS PLASMA
The input resistance of a Hertzian dipole in a lossless plasma is the sum of the EM and
plasma components of the radiation resistance of a Hertzian dipole. That is,
R =R + R (59)
0 e p
The total input power to a Hertzian dipole is 1/2 I R0, out of which 1/2 I R is radiated as an
2 0' e
electromagnetic wave and 1/2 I R is radiated as a plasma wave.
It is important to note that R appears as a part of input resistance of a Hertzian dipole
only when an electroacoustic wave is excited by the dipole and propagates away from it. If a
radiating source is surrounded by a plasma of limited size, an electroacoustic wave may be
excited but a part of this wave may be reflected at the plasma boundary so that R in that case
can be considerably smaller than the value expressed in Equation 58.
6
RADIATION RESISTANCE AND FIELDS OF A CYLINDRICAL DIPOLE
ANTENNA IN A LOSSLESS PLASMA
The second case to be studied is that of a cylindrical dipole antenna in a lossless plasma.
This type of antenna has a great practical importance and also can be analyzed quite accurately.
12
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Our objective is to find the far zone fields set up by a cylindrical dipole antenna in a lossless
plasma. The near zone fields are not considered in this study for the sake of simplicity. The
radiation resistance of a cylindrical dipole antenna can be determined after the far zone fields
are obtained.
A dipole antenna as shown in Figure 3 can be assumed to have the following current and
charge distribution.
h + /
FIGURE 3. A CYLINDRICAL DIPOLE
ANTENNA
= I sin je(h - Izi) ej z (60)
~p = ~f0 I cos pe(h - Izl) et (61)
/ X 2\
where 3e = (wjo/0M and t = (1 - )2/. In this section we also assume that v = 0. These
distributions of current and charge along the antenna are approximate but are quite sufficient
for our study (see Reference 6).
With the current and the charge distribution specified in Equations 60 and 61, the far zone
fields for the EM mode and plasma mode can be obtained separately.
6.1. THE EM MODE (Ee, H, V )
The vector potential A maintained by the antenna current I is
-- ~0 ~Isi e~zWjcot e-JderA
A = I sin (h - Iz 1ee dz z (62)
where r = I[ - zl is the distance between an observation point in space and a point on the
antenna, and r0 is the distance from the center of the antenna to the observation point in space.
To avoid further complication in analysis, we evaluate only the far zone field of A. At the
far zone of the cylindrical dipole antenna,
13
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r= r - z cos 0 (63)
and A is simplified to
j(cot-3 r0)
/ e01 e 0 h j z cos 0
A 7Tr f sin he(h - Izl)e dz z
0 h
j(cot-e r0)
Om 0 e e cos (P3h cos 0) - cos (e h) A
= -- 2^ -. -- * --- 2 — ~ z (64)
2er0 _ sin2 0
If A is expressed in terms of spherical coordinates, A can be represented as
0
A = -A sin 0 (65)
0 Z
The electric field in the far zone of the antenna is directly related to A as follows:
0
r A
E r = -jcoA0 =jco sin 0 A 0 (66)
e 0 z
where the superscript r stands for the radiation field and subscript e stands for the EM mode.
The far zone magnetic field can be obtained from Maxwell equations as
Hr v r A
H -E (67)
CO
/ co
where {0 = 1207r ohms and V= 1
~0 ~ ~ 2
The Poynting vector of the EM wave is
e =2 e
2 os h cos s0)- cos (eh)12
30^V~~r 2f~ e r (68)
The total power radiated as an EM wave in a lossless plasma is obtained by integrating the
Poynting vector over a large sphere; that is,
r27T C7 2 - -
P = dO d o dr0 sin 0 P. r
e u e
730 2 7 [cos (e h cos 0) - cos (3eh) ]2
I 2 -- - - -- - dO (69).m r n sin 0
14
</P>
<P><PB REF="00000019.tif" SEQ="00000019" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="788" N="00000019">
Institute of Science and Technology
The University of Michigan
The integral in Equation 69 has been evaluated [7], and the final form of P can be written as
e
P ~ = gIV 2 { - cos (2j3eh) Cin (4/3 eh) + 2 [1 + cos (21eh)] Cin (2e3 h)
e m e e e e
+ sin (213eh) [Si (4/3eh) - 2Si (2/3 eh)]} (70)
The electromagnetic component of the radiation resistance of a cylindrical dipole antenna is
2
obtained by dividing P by 1/2 I. This gives
e m
m 30
R m= -cos (21eh) Cin (43 h) + 2 [1 + cos (2/3 h)] Cin (2/3eh)
e 'A 2 / 2 I - e e e e
e - ( h /h
+ sin (2/3eh) [Si (4Peh) - 2Si (23 eh)]} (71)
where 3 = co p e 1 - cw /aw. It is noted that R m symbolizes the radiation resistance
pe
referred to the maximum antenna current.
6.2. THE PLASMA MODE (Ep V, n1)
Following a procedure similar to that of the previous section, we can determine the deviation of the electron density from its ambient density, n1, as, 2 h -jP r
n - P2 pdz (72)
n 47T ev02 h
where 3p = (2 - p) 2/v0 (because in this section v is also assumed to be zero) and v0 = rms
velocity of electrons.
Again, for simplicity, only nI at the far zone of the antenna is determined. With Equation
63, Equation 72 leads to
_jp. 2 E00 ' Im j(Wt-0pr0) r jo pz cos 0
n =2 e I cos f (h - z)e dz
41 ev r0 e
0 j3 z cos 06
Ji h 3e( ) dzj (73)
- cos 3 (h + z)e p dz (73)
That is,
-w JE0 I pr p cos 0[cos (3 h cos 0) - (/3h)]
P mv e PP (74)
0 t 4.evWt2 r1 r J7s2 2 2
1 4mev 2 r0 /3 cos -03 )
0 p e
15
</P>
<P><PB REF="00000020.tif" SEQ="00000020" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="649" N="00000020">
Institute of Science and Technology The University of Michigan
Since 2
ev
E -2 Vn (75)
p co- E
the electric field of the plasma mode, E is determined as
p
wop 2 I 5I 3 f1 co [csGh c os 0 [c0) (h cos ) - cos (e3h)]
0Im jC op +_e Ah1 A
E = 2 L- 27;-2 2
'P 4T- LrO rl Lcos 03-e
3 sin 0 (32 cos2 0 + 2) [cos (/3 h cos 0) - cos (eh)]
r 2 (i32 cos2 0 2
h 3pcos 0 sin ( h cos 0) ((wt-r) ()
The radiation term of E which is significant in the far zone of the antenna is
j- 2 I j(t Cos) 0 [cos (h cos 0) -cos (3 h)l
p (fI 2__e 2 e2 (hc cos )_ c oS (78)
4v 2en O 0 2 cos 0 -
The radiation term of the induced velocity of electrons, V Equa, can be determined simtipon 74.
n e 0 p
Tp mefo ofr -co 7eh (78)
00 sbtutio of e0 p eostio
The radiation term of n wis exactly the same as Equation 74.
- 2 r r (79)
The substitution of Equations 74 and 78 in Equation 79 gives
16
</P>
<P><PB REF="00000021.tif" SEQ="00000021" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="737" N="00000021">
Institute of Science and Technology
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C ( 3 1 Os 0 [scos (3 h cos 0) - cos3 h) 2
pJ d si0 (1) -.
) 15 U) H7 ) tm iO 6i 0co [cos p cos r)- cs(ph) ] (80)
p ay 32w2 kv 0 \w 4 w/w2 rl L 1-c02/v0 cos 6 J
In the evaluation of Equation 80, the relation
=p _ 1 2 (81)
2 2 2 2 2
e 0 0 0 0 -
is used.
The total power radiated by a cylindrical dipole antenna as a form of plasma wave is then
27T c7T 2
P = dO d0r sin 0 ( * r
0P av
c 3, 2j \ I 2 7T sin 0 cos 2 0[cos (3 h cos 0) - cos h)
152J (v) a0 1- 2 2 2 2 pA d (82)
2 Woi 2/2 ( 02_0 cos 0
The integral in Equation 82 can be evaluated by a contour integration for the case of c 0/v0 ~ 1
in the appendix. The final expression for P is
/ 2\
p =15u ^ 1 I 2 (23 h + sin 23 h) (83)
p 8 \2 / /- 22 m e e
for c0/v0O ~ 1. The plasma component of radiation resistance referred to the maximum antenna current is obtained by dividing P by 1/2 I
p m
R m i 5-L.1-(20 h + sin 230 h) ohms (84)
p 4 \ 2 / _~p/2 e e
2 2
for c0/v0 >> 1, wherej3 = ~0t0 V I1 wp '
6.3. INPUT RESISTANCE OF A CYLINDRICAL DIPOLE ANTENNA IN A LOSSLESS PLASMA
The total radiation resistance of a cylindrical dipole antenna referred to the maximum antenna current is
17
</P>
<P><PB REF="00000022.tif" SEQ="00000022" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="833" N="00000022">
Institute of Science and Technology
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Rm =R m +R m (85)
e p
Since the power relation of
1 2 Rm 1 2
I R = RIo
2m 2 0 0
with I0 as the input antenna current and R0 as the input antenna resistance is satisfied, the
input resistance of a dipole antenna in a plasma is obtained as
(I) (Rem m) (86)
For short antenna (Im/Io) 1/sin t h.
m and m 22
R and R p as functions of wp /w are shown graphically for the cases of h = X 0/4 and
h = /10 in Figures 4 and 5. Rem and Rm as functions of h/A are shown graphically for the
2 22
cases of wC /w =0.5 and D /w = 0.8 in Figures 6 and 7.
p p
18
</P>
<P><PB REF="00000023.tif" SEQ="00000023" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="602" N="00000023">
3 -(A
C
CD
0
C-,
A
n
(D
Q
0
0
CD
0
cO
0
H
3
t M
0
St'
0
til
W?
- Electromagnetic Component of Radiation Resistance, Re
o_ ro 0 0 oa
o o o.o o o O
I 0
3 D3
Cl)
'F
N
\
6a1
"I,.%
eK
~0
C m
I3 3
I U
m
- Plasma Component of Radiation Resistance, Rp
0
v
-I4
CD
CD
U,
0
0
-.
1,0
</P>
<P><PB REF="00000024.tif" SEQ="00000024" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="730" N="00000024">
Institute of Science and Technology
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Rm
e
(ohms)
Rm
p
(ohms)
E ~t
G3
0
Us
0
0
0.c
0
cr
4 -0
0
E
0
o0
0
0
C
0
0
b\O.
E a
0.
C
0
c
0
0
0
0
c'4 -0
1)I
C
0.
E
0
0
E
to
{3
FL
2 /A( 2
FIGURE 5. RADIATION RESISTANCE OF A SHORT DIPOLE (h = X /10)
IN A PLASMA VS. C) 2/co)2
p
20
</P>
<P><PB REF="00000025.tif" SEQ="00000025" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="308" N="00000025">
I
C
(D
0
Rp"
Re o0
el
(ohms) A~~~~~~~~~~~~~~~(ohms) ~~~(ohms) CD
525- '- I I!, 4 }- 175 "
E~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
/ 6-:
o / 5 /,CD
450 - 0.5 150
a) (~~~0
M" /. -. 0 c
/> io^ \)
37'54~~~~~~~- ~ / \-125 -
O5 e^^ \
0)~~~~~~~~~~~~~~~~~~~~~
Os300'- "\ / \ / \ "'~~&
1225-~ ~~~ 4-""1\\ 7, 30 0- nO)
cr~
300 — ~~~~~~~~~~~~~~~~~~~~ —I00o
0 / 0
(. / ^ 0.
a. 75".~7"2.2 ^^ / E
225- /
C~~~~~~~~~~~~~~~
E 0
0 E.
i ~~~~~/ o,
0h. 0
= 15o_ -5
~~~~~/
75 —3-5
'I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(
Q
3
0.25 0.50 0.75 1.0 1.25 1.50 1.75 2.0 2.25 2.5
-~ h/X
0~~~~~~~~~~~~~~~~~~
o ~ ~..~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-.
2 2
FIGURE 6. RADIATION RESISTANCE OF A DIPOLE ANTENNA VS. ANTENNA LENGTH AT CO Aw 0.5-A
"" 75-' / - 25
o~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
</P>
<P><PB REF="00000026.tif" SEQ="00000026" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="515" N="00000026">
c
m
CD
0
Rn m
R e R m o
(ohms) (ohms) '
EGv 700o- — i I.1 t t -280 28
I aE
/ /
— I
600- /w2 = 0.8 / 240
p~~~~~~~~~
00~ ~ ~ ~ ~ ~ ~~~~~~",,,~,/ 240
n.'.~ 600 -... D~ o
0 m o
500 — R / 200c
r e p1
o* /.1
/ \ ~ ~~ ~ ~~~~~~~/ 4-.
0.
0 400"+\ - -160
CL
I0)^ / 0
o..
E 300 — / 120
0 /
200 80
o /E
E
00
4- LI..
g100 / "40
s ~ ~ ~~~~~~~~~~~~~~~~~/ ^ ~CT CD
0.25 0.50 0.75 1.0 1.25 1.50 1.75 2.0 2.25 25,
— ~ h/X '
0
0
2 2 -FIGURE 7. RADIATION RESISTANCE OF A DIPOLE ANTENNA VS. ANTENNA LENGTH AT w /0 = 0.8
p:
(Q
0
</P>
<P><PB REF="00000027.tif" SEQ="00000027" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="883" N="00000027">
Institute of Science and Technology
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7.A HERTZIAN DIPOLE IN A LOSSY PLASMA
In the previous sections on the interaction of an antenna with a plasma, the plasma has
been assumed to be lossless. It is the purpose of this section to observe the effect of loss in
plasma on the radiation of an antenna. In restricting our study to a weakly ionized gas type of
plasma, we assume that the loss in a plasma is merely due to the collision between electrons
and neutral particles. We will find the radiation resistance of a Hertzian dipole as a function
of the collision frequency of the plasma, among other parameters. Actually, when the medium
is lossy, the radiated power at the far zone of an antenna is zero, but the power radiated by an
antenna into the medium is finite. The power radiated by an antenna in a lossy medium can be
obtained approximately by integrating the power radiating from a small sphere which surrounds
the antenna. The radiation resistance can then be determined after the radiated power is obtained.
The geometry of the problem is shown in Figure 8. A Hertzian dipole is immersed in a
lossy plasma and surrounded by an imaginary sphere of radius ri.
FIGURE 8. A HERTZIAN DIPOLE IN A LOSSY
PLASMA SURROUNDED BY AN IMAGINARY
SPHERE OF RADIUS r
The current and the charge on the Hertzian dipole are assumed to be
= I d2 6(x)6(y)6(z)ei z
(87)
23
</P>
<P><PB REF="00000028.tif" SEQ="00000028" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="725" N="00000028">
Institute of Science and Technology
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p 1[( 2 d ) - z + d) 6(x)6(y)ejwt
(88)
The EM and the plasma mode excited by J and p are analyzed separately as follows.
7.1. THE EM MODE
The propagation constant of the EM mode, fe, in this case is complex and can be expressed
as
- 2 2= o 2
2 \
p2 2
co + v
2 ~
c p
j (2 2)
wL(w? +v^)j
(89)
If we write
e = e - ja
e e e
(90)
we have
2 - a2 =
e e E00(1
2 \
w\
P.
2 2
w + v
(91)
(92)
( 2\
2e e 0 20 2
co + v
From Equations 91 and 92, /3 and a can be expressed as
-13e0 2= _2 4 - 1/2
_ ^^ P + P P 2 p
e -.22 2 2 2 2 2 2
+ 2
(93)
car=~
eV=Vo E0 2+
w + v
2w 2
1 - -- P
2 2
W +1)
4-1/2
+
2(w2 + v2
w (w + v )
(94)
The electric field of the EM mode, E, set up by a Hertzian dipole in a lossy plasma can
be found to be
be found to be
- -jId (i1 j(wt-er)
E = ( + j cos 0 e r
e 2-feO WI r r/
-jI / - 22 ( ) A
47TEW 3I-2- "r sin0 e
0 E r r
(95)
24
</P>
<P><PB REF="00000029.tif" SEQ="00000029" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="852" N="00000029">
Institute of Science and Technology
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The magnetic field of the EM mode, H, set up by a Hertzian dipole in a lossy plasma is
_jI _ d 1 e j(wt-0r) A
H =4 (*j- 2 --- 06e sn (96)
r
The radial component of the complex Poynting vector is given by
(Pe)r 2 (Ee OH *) (97)
and the real part of (Pe )r is obtained from Equations 95 and 96 as
2 2 2 2 22 2 2
Idf 3a 1 2a ^ +a 3 +a 1 -2a r
_, /- e e e e e \e e/ e
Real (e)r 16 20 3 -2 5 -- + 3 2a -2sin e (98)
167 ~ / r r r e r
The total power radiated by the dipole into the plasma is the sum of the power radiated outward from the sphere of radius rI and the power dissipated within the sphere. Symbolically,
27T w71 r
' P = d dO sin 0 r Real (e)r+ [W]0 1 (99)
e Pe 0
The first term of the right-hand side (RHS) of Equation 99 can be evaluated, but the second
term, the power dissipated within the sphere of radius r1, cannot be evaluated unless the exact
shape of the dipole and the exact expression of its near zone field are known. The relative
values of the first and the second terms of Equation 99 depend on the conductivity of the medium.
King [8] has found the following results.
The following observations are based on a short dipole of,feh = 0.3, and 3 r = 1, radian
-3 eel
sphere. (1) If a /j < 10, 90% or more of the power supplied to the antenna is transferred
e ebeyond the radian sphere; in other words, the first term dominates the RHS of Equation 99 in
this case. (2) If a /je < 10, only about 8% of the power is dissipated outside and 92% of the
e e
power is used to heat the medium inside the radian sphere; that is to say, the second term
dominates the RHS of Equation 99 in this case. (3) When a /fe 1, virtually all of the power
e e
is dissipated as heat within the radian sphere.
Because of this difficulty, only the case of a slightly lossy plasma is studied. We shall try
to predict the effect of the collision in a plasma on the radiation of the dipole from the first
term of the RHS of Equation 99. Let us write the first term of the RHS of Equation 99, or the
power radiated beyond the sphere of radius r1, as
2 7T 7T 2
Pie = X d(P dO sin 0 r1 Real (P ) (100)
e 0V f0 er
25
</P>
<P><PB REF="00000030.tif" SEQ="00000030" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="764" N="00000030">
Institute of Science and Technology
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By using Equation 98, P' can be determined as
p I Pe ee 2 ~e 2 2\ 1 2 21 e
' 1E 2= - + r 2+ k(e2+ + e a+a)ee (101)
e 12+, 2W2 A12 3 r 2 e e ae e e
0 01
Equation 101 gives the expression for the power radiated outward from the sphere of radius r1.
It is interesting to show that this amount of power is actually dissipated as heat in the medium
outside the sphere of radius r1. To prove this, we will first consider the power dissipated in
the unit volume of the medium. If we assume that the conductivity of the medium is a and the
electric field in the medium is E, as given in Equation 95, the power dissipated in the unit
volume of the medium is
w = uE 2 (102)
e
With Equation 95, the above equation leads to
+ 22 + ) + O s2 a 2e (103)
h 2 t wr di i t med o e t e oe e c 2
or[We =J'-dridJ+ +2 sin n Ee
e 3-T2 22 d2 | 4|2 2 + 2)r
+ —~2 - r
+ 2 e + e cos 2 e e e (103)
The total power dissipated in the medium outside the sphere of radius r can be obtained as
[W ] oo= dr d6\ do r sin 6 oE
e r1 ^ ^0 O e
2 2O
wr d 2 + - + 2, 2 2
247E0 [W r1 r 1
1( 2 2\2 - Vrl
+~ ~ + a e (104)
a e e
e
In a plasma of weakly ionized gas type, a can be expressed as
or = 2, 2 E (105)
(A + V
26
</P>
<P><PB REF="00000031.tif" SEQ="00000031" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="842" N="00000031">
Institute of Science and Technology
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It is now very easy to prove that Equations 101 and 104 are identical.
The EM component of the radiation resistance of a Hertzian dipole in a lossy plasma is
defined as
P 2 r12 r1 1
eR =2 =2 f ( Pe + [We]j = - [W + [We] (106)
e 1/2 I2 I2e -0 I rl e
r
Since the power, [W ], dissipated within the sphere of radius r1 cannot be evaluated, we
0
evaluate the first term of the RHS of Equation 106 only and write R as
R = R' + R (107)
e e eh
where R' is that part of the radiation resistance which is responsible for radiating an EM wave
e
out of the sphere of radius rl, and R h is that part of the radiation resistance which takes into
account the power dissipated as heat within the sphere of radius r1.
With Equation 101 br 104, R' can be determined as
R'= 80, (A-d_ VT) 2 +ee 2- eer 1 (108)
where
2w 2 4
+ - (109):=/ 1 2 2 — 2 2 (109)
W + v (a 4 + v )
2 4 ae /e 2
L +-e + (110)
+ r / r 2r1
Pe and a are expressed in Equations 93 and 94.
Rte, expressed in Equation 108, reduces to the free space value of the radiation resistance
of a Hertzian dipole when ae is zero. In general, R e is a function of a /fe and r1, a fact which
implies that the value of R' is dependent on the size of the sphere on which the integration of
the Poynting vector is performed. R', therefore, can not give a very accurate and unique estimation of the EM component of the radiation resistance except for very small a /3 or a slightly
lossy plasma. R' is plotted as a function of v/w in Figure 9 for the case of a Hertzian dipole
e
27
</P>
<P><PB REF="00000032.tif" SEQ="00000032" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="854" N="00000032">
Institute of Science and Technology
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Ohm
0.7ja)
X 0.6
~ < J ) 2 //2 0.
=0.5 -
z. /
z /
o 0.4 -
0
0 0.3 1
0.2- / 2 =1
0
o 0.1' (er = 1)
0.01.02.03.04.05.06.07.08.09 0.1
FIGURE 9. EM COMPONENT OF RADIATION RESISTANCE OF A SHORT
DIPOLE (dk = X /107r) IN A PLASMA AS A FUNCTION OF PLASMA COLLECTION FREQUENCY, v/W
with df = X0/107r and 3 r = 1. R' is seen to increase with v/w, and this effect becomes very
significant when w is close to w. From this behavior, we may conclude that the collision in a
p
plasma tends to increase the radiation resistance of an antenna.
It is noted that the Poynting vector method used in this section is not adequate to find the
radiation resistance of an antenna in a lossy plasma. King [8] has obtained the EM component
of the radiation resistance in a lossy medium by using an integral equation method. In his resuits, R becomes much greater than R' when v/w is greater than 10. This tends to show
e -2 e
that for v/w greater than 10, Rh can be much greater than R' e; we may therefore be forced
to conclude that the collision in a plasma may greatly increase the radiation resistance of an
antenna.
28
</P>
<P><PB REF="00000033.tif" SEQ="00000033" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="754" N="00000033">
Institute of Science and Technology
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Institute of Science and Technology The University of Michigan
7.2. THE PLASMA MODE
The propagation constant of the plasma mode, /3, in this case is complex and can be expressed as follows:
= —2[ - )- jv (111)
If we write
/ p = p - ja (112)
we have
2 2 1_2 2\
/32 - 2 ( — ) (113)
v0
2p a = v (114)
pp
v0
From Equations 113 and 114 we get
f l / 2 7 — -1/2
1 221
1 2 2 + 2 2 22
I-1_ 2 2 J/ 2 2\" 22,^,.
a c v o0 + + (2 2) + w2v (116)
The deviation of the electron density from its ambient density, n1, caused by a Hertzian
dipole in a lossy plasma can be found to be
Idn Wp / j (eJt-(pr)
1I= 47u 2r 2 -F-J cos 0 e (117)
ev r
and the radial component of induced velocity of the electrons of the plasma mode, (Vp)r, in
this case is
w 2 Idf 2 2 fi j(t-fpr)
(Vp)r = 2 -- cos e (118)
47ren foCi - ji r r
The power density flowing from the dipole as a form of an electroacoustic wave is
29
</P>
<P><PB REF="00000034.tif" SEQ="00000034" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="825" N="00000034">
Institute of Science and Technology
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Real (Pp) Real n 1 (V)
212d22 + 2+ 2a\
32T2CwEp K2 2 2)2 j p p r2
/ 2 2O[2 2 2 2 2 -2 r
+ ( p + p + 2 P ) cos 0 e p (119)
r r r
The total power flowing from a sphere of radius r1 as a form of an electroacoustic wave is
r27T rl 2
dP' = d d sin 0 r Real (p ) (120)
r JQ JQ P
The total power transferred from the Hertzian dipole into the lossy plasma in a form of an
electroacoustic wave is then the sum of P'p and the power dissipated within the sphere of radius
r1 by the electroacoustic wave, [W ]0. Symbolically,
r
p = ' + [Wp]0 (121)
p p
Again, we are only able to evaluate P', for the same reason as in the EM mode case.
The plasma component of the radiation resistance of a Hertzian dipole in a lossy plasma is
Rp 2 P'+ [W]0 } (122)
Let us define
R = R' + R (123)
p p ph
with R' as that part of the radiation resistance which is responsible for radiating a plasma
p
wave out of the sphere of radius rl, and Rh as that part of the radiation resistance which is
responsible for the power dissipated as heat within the sphere of radius r. We can then express R' as follows:
p
2 c p -2ar
R'P =e40 7 1(- w 2 f(a/3, r) e p1 (124)
where
30
</P>
<P><PB REF="00000035.tif" SEQ="00000035" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="875" N="00000035">
Institute of Science and Technology
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4x / 1 \8x
+ + 2)2 3 3 3)J (125)
and
x = a1p/ip
R' expressed in Equation 124 reduces to the free space value when a is zero. To observe
p p
the effect of the collision in a plasma on R'p, Equations 124 and 125 are carefully studied. Let
us consider the case of a Hertzian dipole with d2 = X 0/10T. To have a sphere of reasonable size
surround the dipole, we let 3ern = 1, consistent with the EM mode case. If r1 is set to be 1/3e,
ap r1 and p3r1 become very large, even for a very small v/w. On the other hand, f(a p/ p, rl)
remains quite close to unity. This implies that R' is very small, compared to its free space
value, when a is not zero. This also implies that Rh is much larger than R' when a is not
p.ph p p
zero. We then conclude that the major part of the power of the plasma mode is dissipated in
the very vicinity of the antenna if the plasma is lossy. Unfortunately, we cannot observe the
effect of the collision in a plasma on the change of R in Equation 124. However, the effect of
the collision in a plasma on R may not be significant as in the case of the EM mode since in
the plasma mode case the major part of the power is dissipated in the very vicinity of the
antenna. It may also be fair to state that R does not change very rapidly with i/wo, as does
p
R.
e
Since the total radiation resistance of an antenna is the sum of R and R, the effect of the
e p
collision in a plasma on the total radiation resistance may be very significant. Therefore, when
v//w is not equal to or smaller than 10 3, the results of R and R for the case of lossless plasma
e p
cannot be applied to the lossy plasma case. We can only say that when v/Co is larger than 10,
both R and R increase with v/w, and R increases very rapidly with v/'w when w is very close
e p e
to wo.
p
In a wide range of communications involving the ionosphere, v/w is smaller than 10 and
w is not close to co. For this case R can be approximately calculated from Equation 108, and
R may be approximated with fair accuracy by the free space value of R.
P P
31
</P>
<P><PB REF="00000036.tif" SEQ="00000036" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="735" N="00000036">
Institute of Science and Technology
The University of Michigan
Appendix A
EVALUATION OF THE INTEGRAL IN EQUATION 82
The integral is
Jo sin 0 cos 0 [cos 3h cos 0cos )- cos ((eh)]
(1 - 2c2/v0 2 2 2dO
If we let c0/v0 cos 0 = x and 3eh = a, then I becomes
I = 2( x [cos (x)- cos (a)]2 d
For the case of c >> 1 I can be represented(1 - x2)2
For the case of c0/v0 ~ 1, I can be represented as
(126)
(127)
( 3
I = 2 f-~-l x [cos (ax) - cos (c<)] 2
VW ^0 (1 - x2)2
(128)
The integral
x1 [cos (ax) - cos ()]2dx
a1 n(1 - x2)2
can be evaluated by contour integration as follows:
= 1 {22x2 cos (a)x) 2 c os ( ax)
I h 2 fr dx -i -Eu2a dx
T-e 0 (1 - x9 i E (1 - xp
The first integral in Equation 129 can be expressed as
1 f cos (a) x
0 -9. (1 - x2)2
(129)
oc 2 2
x2 cos (ax)
09 (1 - x2)2
dx = f(z)dz ff2(z)dz + f (z) dz
l-lfoo f-oo -coc
(130)
where
2 2iaz
f(z) = z e2
(1 - )2 (1 + )2
f2() = 2z2
(1 - z) (1 + z)
32
</P>
<P><PB REF="00000037.tif" SEQ="00000037" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="803" N="00000037">
Institute of Science and Technology
The University of Michigan
2 -2icaz
z e
f (z) z e
3 ( - z )2 (1 + z)2
By integrating along the closed paths as indicated in Figure 10, we can obtain Equation 130 as
ro 2 2
X Cos (x) -27T i
- (1 - 2)2 dx= 4 {Res [f3(z), z = -1] + Res [f (z), z = 1]
= - -cos 2a - 7 sin 2a (131)
2c 4
z-Plane
Path for fl, f2 f4, and f5
Path for f3 and f
FIGURE 10. PATHS FOR CONTOUR INTEGRATIONS
The second integral in Equation 129 can be expressed as
cos (a) x cos (ax)d cos (a) f (z)dz + f (z)dzj
(1 - 2 2 2 oo J
(132)
where
33
</P>
<P><PB REF="00000038.tif" SEQ="00000038" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="766" N="00000038">
Institute of Science and Technology The University of Michigan
2 icaz
z e
f4(z ) 2 2
(4( -z ) (1 +z (1 z)
2 -iaz
f5(z) = -z e
5 2 2
5 (1 -z)2 (1 + z)2
By integrating along the closed paths as indicated in Figure 10, we can express Equation 132 as
f cos (c) x2 cos (X)dx = -7ri cos (a) {Res [f5(z), z = -1] + Res [f5(z), z =1]}
-cc (1-x)
= -cos a- cos.c + - sin (133)
The third integral in Equation 129 can be expressed as
00Cc2 2 2 0
cos (c vx dx= cos (a) f6(z)dz = 0 (134)
cc (1 - x ) 2
where
2
f62z) = 2 2
6 (1 -z)2 (1 + z)2
Equation 134 is found to be zero if the integration is performed along the closed path indicated
in Figure 10.
By summing up Equations 131, 133, and 134, we obtain the following result:
3
I =c0 ( --- — - cos 2a o_ c i+T sin a + cos
= (0 (V) (+ 74sin 2cv)
(135)
= c0~ [4 (2eh + sin 2e h)] (135)
Vo/ 4 e
34
</P>
<P><PB REF="00000039.tif" SEQ="00000039" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="896" N="00000039">
Institute of Science and Technology
The University of Michigan
REFERENCES
1. M. Cohen, "Radiation in Plasma, I. Cerenkov Effect," Physical Review, 1 August
1961, Vol. 123, No. 3, pp. 711-721.
2. M. Cohen, "Radiation in a Plasma, II. Equivalent Sources," Physical Review,
15 April 1962, Vol. 126, No. 2, ppo 389-397.
3. M. Cohen, "Radiation in a Plasma, III. Metal Boundaries," Physical Review,
15 April 1962, Vol. 126, No. 2, pp. 398-404.
4. G. Field, "Radiation by a Plasma Oscillation," Astrophysical J., November 1956,
Vol. 124, No. 3, pp. 555-570.
5. H. Whale, "The Excitation of Electroacoustic Wave by Antennas in the Ionosphere,"
J. Geophys. Res., 15 January 1963, Vol. 68, No. 2, pp. 415-422.
6. R. King and C. Harrison, "Half-Wave Cylindrical Antenna in a Dissipative
Medium," J. of Research of NBS-D Radio Propagation, July-August 1960, Vol.
64D, No. 4.
7. R. King, Theory of Linear Antennas, Harvard University Press, 1956, ppo 560 -561.
8. R. King, Dipoles in Dissipative Media, Tech. Report No. 336, Cruft Laboratory,
Harvard University, 1 February 1961.
35
</P>
<P><PB REF="00000040.tif" SEQ="00000040" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="889" N="00000040">
Institute of Science and Technology
The University of Michigan
DISTRIBUTION LIST
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Aeronautical Systems Division
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1 repro.)
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AF Unit Post Office
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Mr. S. J. Grossman
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U. S. Army Combat Surveillance Agency
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Arlington 1, Virginia
ATTN: Mr. Leonard Plotkin
36
</P>
<P><PB REF="00000041.tif" SEQ="00000041" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="891" N="00000041">
Institute of Science 'and Technology
The University of Michigan
Copy No. Addressee
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P. 0. Box 618
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46 Massachusetts Institute of Technology
Electronics Systems Laboratory
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ATTN: Mr. Laurence Swain, Jr.
37
</P>
<P><PB REF="00000042.tif" SEQ="00000042" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="897" N="00000042">
+
+
+
AD
Div. 25/4
Inst. of Science and Technology, U. of Mich., Ann Arbor
INTERACTION OF A RADIATING SOURCE WITH A
PLASMA: Effect of an Electroacoustic Wave by Kun-Mu
Chen. July 63. 35 p., incl. illus., 8 refs.
(Report No. 4563-39-T)
(Contract AF 33(616)-8365) Unclassified report
When a radiating source is immersed in a homogeneous
plasma of infinite extent, an electroacoustic wave may be
excited in addition to the usual electromagnetic wave.
The electroacoustic wave becomes a longitudinal plasma
wave in the far zone of the source.
The case of a Hertzian dipole in a lossless plasma is
considered first. The fields of both the EM and plasma
modes excited by the dipole are explicitly obtained, and
the EM and plasma components of the radiation resistance of a Hertzian dipole are determined. The case of a
cylindrical dipole antenna in a lossless plasma
(over)
UNCLASSIFIED
I. Chen, Kun-Mu
II. U. S. Air Force
III. Contract AF 33(616)8365
Defense Documentation Center
Cameron Station
Alexandria, Virginia
UNCLASSIFIED
AD
Inst. of Science and Technology, U. of Mich., Ann Arbor
INTERACTION OF A RADIATING SOURCE WITH A
PLASMA: Effect of an Electroacoustic Wave by Kun-Mu
Chen. July 63. 35 p., incl. illus., 8 refs.
(Report No. 4563-39-T)
(Contract AF 33(616)-8365) Unclassified report
When a radiating source is immersed in a homogeneous
plasma of infinite extent, an electroacoustic wave may be
excited in addition to the usual electromagnetic wave.
The electroacoustic wave becomes a longitudinal plasma
wave in the far zone of the source.
The case of a Hertzian dipole in a lossless plasma is
considered first. The fields of both the EM and plasma
modes excited by the dipole are explicitly obtained, and
the EM and plasma components of the radiation registance of a Hertzian dipole are determined. The case of a
cylindrical dipole antenna in a lossless plasma
(over)
Div. 25/4
UNCLASSIFIED
I. Chen, Kun-Mu
II. U. S. Air Force
III. Contract AF 33(616)8365
Defense Documentation Center
Cameron Station
Alexandria, Virginia
UNCLASSIFIED
+
+
+
AD
Div. 25/4
Inst. of Science and Technology, U. of Mich., Ann Arbor
INTERACTION OF A RADIATING SOURCE WITH A
PLASMA: Effect of an Electroacoustic Wave by Kun-Mu
Chen. July 63. 35 p., incl. illus., 8 refs.
(Report No. 4563-39-T)
(Contract AF 33(616)-8365) Unclassified report
When a radiating source is immersed in a homogeneous
plasma of infinite extent, an electroacoustic wave may be
excited in addition to the usual electromagnetic wave.
The electroacoustic wave becomes a longitudinal plasma
wave in the far zone of the source.
The case of a Hertzian dipole in a lossless plasma is
considered first. The fields of both the EM and plasma
modes excited by the dipole are explicitly obtained, and
the EM and plasma components of the radiation resistance of a Hertzian dipole are determined. The case of a
cylindrical dipole antenna in a lossless plasma
(over)
UNCLASSIFIED
I. Chen, Kun-Mu
II. U. S. Air Force
III. Contract AF 33(616)8365
Defense Documentation Center
Cameron Station
Alexandria, Virginia
UNCLASSIFIED
AD
Inst. of Science and Technology, U. of Mich., Ann Arbor
INTERACTION OF A RADIATING SOURCE WITH A
PLASMA: Effect of an Electroacoustic Wave by Kun-Mu
Chen. July 63. 35 p., incl. illus., 8 refs.
(Report No. 4563-39-T)
(Contract AF 33(616)-8365) Unclassified report
When a radiating source is immersed in a homogeneous
plasma of infinite extent, an electroacoustic wave may be
excited in addition to the usual electromagnetic wave.
The electroacoustic wave becomes a longitudinal plasma
wave in the far zone of the source.
The case of a Hertzian dipole in a lossless plasma is
considered first. The fields of both the EM and plasma
modes excited by the dipole are explicitly obtained, and
the EM and plasma components of the radiation resistance of a Hertzian dipole are determined. The case of a
cylindrical dipole antenna in a lossless plasma
(over)
Div. 25/4
UNCLASSIFIED
I. Chen, Kun-Mu
II. U. S. Air Force
III. Contract AF 33(616)8365
Defense Documentation Center
Cameron Station
Alexandria, Virginia
UNCLASSIFIED
+
+
+
</P>
<P><PB REF="00000043.tif" SEQ="00000043" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="897" N="00000043">
AD
is investigated next. The far zone fields of both the EM
and plasma modes are explicitly obtained, and the EM
and plasma components of the radiation resistance are
then derived as functions of the antenna dimension and
the plasma parameters. Finally, the case of a Hertzian
dipole in a slightly lossy plasma is studied; the effect of
collisions in a plasma on the radiation resistance of the
immersed antenna is discussed.
UNCLASSIFIED AD
DESCRIPTORS
Electromagnetic waves
Plasma physics
is investigated next. The far zone fields of both the EM
and plasma modes are explicitly obtained, and the EM
and plasma components of the radiation resistance are
then derived as functions of the antenna dimension and
the plasma parameters. Finally, the case of a Hertzian
dipole in a slightly lossy plasma is studied; the effect of
collisions in a plasma on the radiation resistance of the
immersed antenna is discussed.
UNCLASSIFIED
DESCRIPTORS
Electromagnetic waves
Plasma physics
UNCLASSIFIED
UNCLASSIFIED
+
AD
is investigated next. The far zone fields of both the EM
and plasma modes are explicitly obtained, and the EM
and plasma components of the radiation resistance are
then derived as functions of the antenna dimension and
the plasma parameters. Finally, the case of a Hertzian
dipole in a slightly lossy plasma is studied; the effect of
collisions in a plasma on the radiation resistance of the
immersed antenna is discussed.
UNCLASSIFIED
DESCRIPTORS
Electromagnetic waves
Plasma physics
AD
is investigated next. The far zone fields of both the EM
and plasma modes are explicitly obtained, and the EM
and plasma components of the radiation resistance are
then derived as functions of the antenna dimension and
the plasma parameters. Finally, the case of a Hertzian
dipole in a slightly lossy plasma is studied; the effect of
collisions in a plasma on the radiation resistance of the
immersed antenna is discussed.
UNCLASSIFIED
DESCRIPTORS
Electromagnetic waves
Plasma physics
UNCLASSIFIED
UNCLASSIFIED
</P>
<P><PB REF="00000044.tif" SEQ="00000044" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="897" N="00000044">
+
AD
Div. 25/4
Inst. of Science and Technology, U. of Mich., Ann Arbor
INTERACTION OF A RADIATING SOURCE WITH A
PLASMA: Effect of an Electroacoustic Wave by Kun-Mu
Chen. July 63. 35 p., incl. illus., 8 refs.
(Report No. 4563-39-T)
(Contract AF 33(616)-8365) Unclassified report
When a radiating source is immersed in a homogeneous
plasma of infinite extent, an electroacoustic wave may be
excited in addition to the usual electromagnetic wave.
The electroacoustic wave becomes a longitudinal plasma
wave in the far zone of the source.
The case of a Hertzian dipole in a lossless plasma is
considered first. The fields of both the EM and plasma
modes excited by the dipole are explicitly obtained, and
the EM and plasma components of the radiation registance of a Hertzian dipole are determined. The case of a
cylindrical dipole antenna in a lossless plasma
(over)
UNCLASSIFIED
I. Chen, Kun-Mu
II. U. S. Air Force
III. Contract AF 33(616)8365
Defense Documentation Center
Cameron Station
Alexandria, Virginia
UNCLASSIFIED
AD
Div. 25/4
Inst. of Science and Technology, U. of Mich., Ann Arbor
INTERACTION OF A RADIATING SOURCE WITH A
PLASMA: Effect of an Electroacoustic Wave by Kun-Mu
Chen. July 63. 35 p., incl. illus., 8 refs.
(Report No. 4563-39-T)
(Contract AF 33(616)-8365) Unclassified report
When a radiating source is immersed in a homogeneous
plasma of infinite extent, an electroacoustic wave may be
excited in addition to the usual electromagnetic wave.
The electroacoustic wave becomes a longitudinal plasma
wave in the far zone of the source.
The case of a Hertzian dipole in a lossless plasma is
considered first. The fields of both the EM and plasma
modes excited by the dipole are explicitly obtained, and
the EM and plasma components of the radiation resistance of a Hertzian dipole are determined. The case of a
cylindrical dipole antenna in a lossless plasma
(over)
UNCLASSIFIED
I. Chen, Kun-Mu
II. U. S. Air Force
III. Contract AF 33(616)8365
Defense Documentation Center
Cameron Station
Alexandria, Virginia
UNCLASSIFIED
+
+
+
AD
Div. 25/4
Inst. of Science and Technology, U. of Mich., Ann Arbor
INTERACTION OF A RADIATING SOURCE WITH A
PLASMA: Effect of an Electroacoustic Wave by Kun-Mu
Chen. July 63. 35 p., incl. illus., 8 refs.
(Report No. 4563-39-T)
(Contract AF 33(616)-8365) Unclassified report
When a radiating source is immersed in a homogeneous
plasma of infinite extent, an electroacoustic wave may be
excited in addition to the usual electromagnetic wave.
The electroacoustic wave becomes a longitudinal plasma
wave in the far zone of the source.
The case of a Hertzian dipole in a lossless plasma is
considered first. The fields of both the EM and plasma
modes excited by the dipole are explicitly obtained, and
the EM and plasma components of the radiation resistance of a Hertzian dipole are determined. The case of a
cylindrical dipole antenna in a lossless plasma
(over)
UNCLASSIFIED
I. Chen, Kun-Mu
II. U. S. Air Force
III. Contract AF 33(616)8365
Defense Documentation Center
Cameron Station
Alexandria, Virginia
UNCLASSIFIED
AD
Div. 25/4
Inst. of Science and Technology, U. of Mich., Ann Arbor
INTERACTION OF A RADIATING SOURCE WITH A
PLASMA: Effect of an Electroacoustic Wave by Kun-Mu
Chen. July 63. 35 p., incl. illus., 8 refs.
(Report No. 4563-39-T)
(Contract AF 33(616)-8365) Unclassified report
When a radiating source is immersed in a homogeneous
plasma of infinite extent, an electroacoustic wave may be
excited in addition to the usual electromagnetic wave.
The electroacoustic wave becomes a longitudinal plasma
wave in the far zone of the source.
The case of a Hertzian dipole in a lossless plasma is
considered first. The fields of both the EM and plasma
modes excited by the dipole are explicitly obtained, and
the EM and plasma components of the radiation resistance of a Hertzian dipole are determined. The case of a
cylindrical dipole antenna in a lossless plasma
(over)
UNCLASSIFIED
I. Chen, Kun-Mu
II. U. S. Air Force
III. Contract AF 33(616)8365
Defense Documentation Center
Cameron Station
Alexandria, Virginia
UNCLASSIFIED
+
+
+
</P>
<P><PB REF="00000045.tif" SEQ="00000045" RES="600dpi" FMT="TIFF6.0" FTR="UNSPEC" CNF="898" N="00000045">
AD
is investigated next. The far zone fields of both the EM
and plasma modes are explicitly obtained, and the EM
and plasma components of the radiation resistance are
then derived as functions of the antenna dimension and
the plasma parameters. Finally, the case of a Hertzian
dipole in a slightly lossy plasma is studied; the effect of
collisions in a plasma on the radiation resistance of the
immersed antenna is discussed.
UNCLASSIFIED AD
DESCRIPTORS
Electromagnetic waves
Plasma physics
is investigated next. The far zone fields of both the EM
and plasma modes are explicitly obtained, and the EM
and plasma components of the radiation resistance are
then derived as functions of the antenna dimension and
the plasma parameters. Finally, the case of a Hertzian
dipole in a slightly lossy plasma is studied; the effect of
collisions in a plasma on the radiation resistance of the
immersed antenna is discussed.
UNCLASSIFIED
DESCRIPTORS
Electromagnetic waves
Plasma physics
UNCLASSIFIED
UNCLASSIFIED
+
AD
is investigated next. The far zone fields of both the EM
and plasma modes are explicitly obtained, and the EM
and plasma components of the radiation resistance are
then derived as functions of the antenna dimension and
the plasma parameters. Finally, the case of a Hertzian
dipole in a slightly lossy plasma is studied; the effect of
collisions in a plasma on the radiation resistance of the
immersed antenna is discussed.
UNC LASSIFIED
DESCRIPTORS
Electromagnetic waves
Plasma physics
AD
is investigated next. The far zone fields of both the EM
and plasma modes are explicitly obtained, and the EM
and plasma components of the radiation resistance are
then derived as functions of the antenna dimension and
the plasma parameters. Finally, the case of a Hertzian
dipole in a slightly lossy plasma is studied; the effect of
collisions in a plasma on the radiation resistance of the
immersed antenna is discussed.
UNCLASSIFIED
DESCRIPTORS
Electromagnetic waves
Plasma physics
UNCLASSIFIED
UNCLASSIFIED
</P>
</DIV1>
</BODY>
</TEXT>