Antennas in Inhomogeneous Media

302 Pages·1969·5.57 MB·English

by  Janis Galejs

Antennas in
Inhomogeneous Media
BY
JANIS GALEJS
Senior Scientist, Applied Research Laboratory,
Sylvania Electronic Systems, Waltham, Massachusetts, U.S.A.
4?
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Copyright© 1969
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First edition 1969
Library of Congress Catalog Card No. 68-21384
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08 013276 6
LIST OF TABLES
NUMBER PAGE
2.1 Impedance of Half-wave and Full-wave Antennas 17
7.1 Slot Conductance Computed by the Hallen's Method (G^,
Variational Method (G ) and from a Complementary
v
Dipole(G ). 95
c
8.1 Comparison of Slot Admittances 116
9.1 Self-impedance of Insulated Buried Antennas 166
12.1 Low Frequency Antenna Resistance 268
ix
ACKNOWLEDGMENTS
THIS monograph summarizes the results of the author's recent work at the
Applied Research Laboratory of Sylvania Electronic Systems. The author
is particularly indebted for support to Dr. J. E. Storer, Director of the
Laboratory. Among his colleagues, Drs. R. V. Row, S. R. Seshadri, and
R. M. Wundt have contributed with a number of discussions. Appreciation
is expressed to Mr. W. Rotman of the Air Force Cambridge Research
Laboratory, and to Dr. A. Shostak of the Office of Naval Research, for
their encouragement in the investigation of a number of topics.
The author acknowledges several helpful discussions with Prof. R. W. P.
King of Harvard University, and valuable comments by Dr. J. R. Wait,
Editor of the Series on Electromagnetic Waves.
xi
LIST OF PRINCIPAL SYMBOLS
A, B amplitude of trial functions or a constant
a, b, c radii
a ,AnMw,A(u, v) amplitude coefficients of TE modes
nm
b , B , B™ B(u, v) amplitude coefficients of TM modes
nm n q9
B = lm(Y) susceptance
B magnetic induction
# static magnetic induction
0
C capacitance; a constant; Euler's constant (0.5772 ...)
Ci(z) cosine integral
Cin(z) = C + logz-Ci(z)
C(z) combination of sine and cosine integrals defined by (2.47)
Cs, Ct defined following (11.77)
c free space velocity of electromagnetic waves
d thickness
da area element
dv volume element
D denominator or abbreviations defined by (8.17) or (12.18)
@ differential operator defined by (12.21)
E electric field
E(z) combinations of sine and cosine integrals defined by (2.45)
e = 2.71828 ... base of the natural logarithm
e particle charge; denotes "electric" as a subscript
F , F functions in impedance formulations
A B
F(R), G(R), F(u, v), F(w) functions in impedance formulations
F (z), G (z) functions used in Section 2.2
n n
fA(z)>fii(z) trial functions for antenna current or aperture fields
/ frequency in c/s
© a dyadic function
G scalar function; conductance
G, G radiation conductance, loss conductance
r L
gA> SB functions in impedance formulations
H magnetic field
//(m)(jt) Hankel function of kind m and order n
n
H(k), H(k ), H(u, v), H(n, m) functions in impedance formulations
n
hj metric coefficients
/ electric current
xiii
XIV LIST OF PRINCIPAL SYMBOLS
Idl electric dipole moment (ampere meters)
K> IA> h abbreviations of integral expressions
i unit vector in x direction
x
i electric current density (amperes/meter2)
e
i = IJI relative current amplitudes in Section 2.4
n 0
J , J4, J J electric surface current density (amperes/meter)
se X) z
J magnetic surface current density (volts/meter)
sm
J (x) Bessel function of first kind and order n
n
K constants in Sections 2.3 and 7.12
y
Kdl magnetic dipole moment (volts/meter)
K coefficients in Section 2.4
n
K (x) modified Bessel functions of second kind and order n
n
k wave number
k wave number of free space
0
k , k wave numbers of electromagnetic and plasma waves in Chapter 11
e p
k , k wave numbers of trial functions
A B
k\ = kle — p2 in Section 12.2
3
k shape factor defined by (4.48)
a
/ length
log logarithm of base e
m particle mass; denotes "magnetic" as a subscript
N number of turns in Chapter 4; particle density in Chapters 11 and 12;
numerator
N normalizing factor for radial functions
n
n unit vector in a normal direction
P power
P radiated power
r
P power loss
L
P power carried by surface wave
s
p radiation power factor in Chapter 4; scalar pressure in Chapter 11;
denotes "plasma" as a subscript in Chapter 11
P functions in Sections 11.3 and 12.2
Q functions in Chapters 8 and 11
R resistance
R, R radiation resistance and loss resistance
r L
R resistance associated with a sinusoidal current distribution
s
R reflection coefficient
R , R reflection coefficients of TE and TM modes
a b
R reflection coefficient of scalar pressure in Chapter 11
c
R (p) radial function comprised of a combination of Bessel functions
n
r = RIR in Section 4.2
r L
S Poynting's vector
Si(z) sine integral
LIST OF PRINCIPAL SYMBOLS XV
S(z) combination of sine and cosine integrals defined by (2.46)
S parameters for computing loop impedance in Chapter 10
n
tan 8 = crl(o)€) loss tangent
r
U, W functions in Chapter 11
u acoustic velocity in Chapter 11
u(x) unit step function
V voltage
V velocity in Chapter 11
v phase velocity
ph
w width
X = lm(Z) reactance
X reactance associated with sinusoidal current distribution
s
X amplitude ratio of E in the two modes in Section 12.2
zj
Y= G + iB admittance
Y characteristic admittance
0
y = Y/Y normalized admittance
s 0
y shunt admittance per unit length
s
Z = R + iX impedance
Z impedance computed with sinusoidal current distribution; impedance
s
with symmetrical antenna excitation
Z impedance with antisymmetrical antenna excitation
a
Z characteristic impedance
0
z series impedance per unit length
s
x y, z cartesian coordinates
y
u, v, w transform variables
p, (f>, z circular cylindrical coordinates
r, 6, $ spherical coordinates
a attenuation constant
a coefficients in Section 2.4
n
j8 phase constant; radial wave number in Section 12.1
j8 defined following (7.2)
c
j3 , (3 parameters characterizing the cross-sectional field variation in
x y
rectangular waveguides.
y = — a + ip = ik propagation constant
y parameters of impedance formulations
NM
A function in a variational impedance formulation defined by (2.40) or
(7.18); perturbation of a real wave number in Section 11.1.2; abbrevi-
ation defined by (11.52) in Section 11.2
8 skindepth; function defined by (2.16)
8Z variation of Z
8(x) delta function
8 Kronecker delta (S = 1 if / =j\ 8 = 0 if / ^ j)
y y tf
€ complex permittivity
XVI LIST OF PRINCIPAL SYMBOLS
€o = (1/3 677) X 10"9 Farad/meter permittivity of free space
e real part of permittivity
r
€1, € , € elements of the dyadic permittivity defined by (12.3)
2 3
e = 1 if m = 0, e = 2 if m ¥^ 0
m m
e = vv/2 half-width
£ transform variable
17 efficiency
A(x) remainder term defined by (6.15) or (7.33)
\ eigenvalues
n
k Debye length in Chapter 11
D
X = lirlk wavelength
H permeability
fx = 4TT x 10~7 Henry/meter—permeability of free space
{)
v collision frequency
Il , Il Hertz potential (vector) of the electric and magnetic type
e m
p = V(u2 + v2), where u and v are transform variables in Section 12.2
cr conductivity in mho/meter
<£ scalar function used for deriving TM modes and defined by (1.20)
^ scalar function used for deriving TE modes and defined by (1.19)
11 = 2 log (211 a) or 2 log (411 e) expansion parameter
fl = (ojwp normalized cyclotron frequency
r
Q = (*>J<*>p normalized upper hybrid resonance frequency
u
£l (x) Lommel-Weber function in Section 2.4
n
ay = lirf angular frequency in radians per second
<x) cyclotron frequency, defined following (12.6)
c
cop plasma frequency, defined following (12.6)
o) = V(a)2 + (ol) upper hybrid resonance frequency
u
CHAPTER 1
INTRODUCTION
1.1. Scope and Limitations of the Treatment
Most antenna texts emphasize antenna performance in the classical free
space environment, but a number of important topics involving dielectric
loading, buried antennas, antennas in a reentry environment or in the iono-
sphere require a more specialized treatment. This monograph presents an
introduction to methods on analyzing antennas in such inhomogeneous
media. This involves complex geometrical configurations in which strictly
assumed antenna current or field distributions are not justified and which
generally require formulations in terms of integral equations. Variational
formulations are selected for the treatment of most configurations. Such
formulations appear to provide a straightforward mechanism for determining
approximations to antenna current or field distributions, even when the
current distributions can be expected to vary between relatively wide limits
as shown for cavity-backed slot antennas or for linear antennas in the
presence of dielectric antenna loading. The zero or first order iterative
solutions of integral equations for the antenna current distributions are
shown to be inadequate in several examples, whereas the variational solution
appears to offer an accuracy comparable to higher order iterative solutions
in geometries calculated in the past. However, very long antennas are not
treated in this monograph.
The developments are devoted almost exclusively to calculation of
antenna impedances; antenna radiation patterns not being considered. Most
of the antenna geometries can be described in rectangular or circularly
cylindrical coordinates; spherical geometries being mentioned only briefly,
and conical antennas omitted altogether. Emphasis is on dielectric media
but also included are discussions of antennas in cold (incompressible) and
warm (compressible) isotropic plasma and in cold anisotropic plasma. The
equations describing wave propagation in ferrite media are quite similar
to the equations characterizing magnetoionic media [Epstein, 1956].
However, ferrite media are not specifically treated in this monograph.
It is assumed that the reader is familiar with basic electromagnetic field
theory and antenna fundamentals. No attempt is made to duplicate the fun-
damental theoretical developments available in most modern texts on
electromagnetic theory. This applies in particular to the introductory material
l
2 ANTENNAS IN INHOMOGENEOUS MEDIA Ch. 1
of this chapter, where the usual field equations and boundary conditions, the
Hertz vector and the properties of elementary electric and magnetic sources
are defined by summarizing the results found in more detailed developments
of Stratton [1941]. The decomposition of fields in transverse electric (TE)
and transverse magnetic (TM) components is discussed following Collin
[I960]; and antenna impedances and admittances are defined following
Harrington [1961] and King [1956]. However, similar fundamental
information is also available in several other references listed at the end of
this chapter [Jordan, 1950; King, 1953; Schelkunoff, 1943; Smythe, 1950;
Wait, 1959].
In Chapter 2, past work on thin-wire antennas in free space is summarized.
The topics discussed are the EMF method for linear antennas of Carter
[1932], the integral equation formulation and the iteration method of
Hallen [1938]. The variational formulation of Storer [1952] is used
extensively in subsequent developments for antennas in more complex
environments. For loops in free space the Fourier series solution [Storer,
1956] is shown to give the same impedance as a variational formulation if
identical trial functions of the antenna current are used.
In Chapter 3 Booker's concept of complementary antennas [1946] is
derived from the properties of elementary electric and magnetic sources.
Examples are given that consider radiation from rectangular waveguides
[Lewin, 1951] and coaxial lines [Levine and Papas, 1951] terminated in
an infinite flange.
Dielectric loading of small antennas is discussed in Chapter 4. It is shown
that small loops are less sensitive to the presence of lossy material in the
vicinity of the antenna than are dipole-type antennas.
The transmission line theory of insulated antennas in a lossy medium is
reviewed in Chapter 5, and the self and mutual impedance of wires are
calculated. The transmission line theory has also been used to calculate the
surface excitation from a buried veritcal wire.
The essential characteristics of thin insulated wires in the interface
of two media are briefly explained in Chapter 6. Slots in a dielectric inter-
face are discussed using the Hallen's iterative method for narrow slots,
but the integral equation has been solved using Fourier transforms for
wider slots.
Cavity-backed slot antennas are discussed in Chapter 7, and the vari-
ational solution is shown to be superior to simple iterative solutions for
rectangular slots. Annular slots backed by cylindrical and coaxial cavities
are also discussed.
Slot antennas covered with a stratified dielectric or an isotropic plasma
are treated in Chapter 8. More general solutions show that the impedance of
the rectangular waveguide can be satisfactorily computed by assuming the
principal waveguide mode for the aperture field distribution and by neglecting