Investigation of Radiation Problem For Two Separated Mediums
Investigation of Radiation Problem For Two Separated Mediums
Investigation of Radiation Problem For Two Separated Mediums
w w w . a j e r . o r g
Page 92
American Journal of Engineering Research (AJER)
e-ISSN : 2320-0847 p-ISSN : 2320-0936
Volume-03, Issue-06, pp-92-100
www.ajer.org
Research Paper Open Access
Investigation of Radiation Problem for two Separated Mediums
Yahia Zakaria
1
, Burak Gonultas
2
1
Systems and Information Department, Engineering Research Division, National Research Centre
El-Behoose Street, p.c. 12622, Cairo, Egypt
Yahia.zk82@yahoo.com
2
Electrical and Electronics Engineering Department, Nigde University
Fertek Build, p.c. 51310, Nigde, Turkey
brk_gnlts@hotmail.com
Abstract: The Hertzian potentials are used to determine the electric and magnetic field components. Moreover,
it is possible for the radiation to start with the dipole to be observed at much greater distances than would be
possible if the waves were generated in an infinite homogeneous medium. In this paper we present the problem
of communication of aradiation in a conducting medium. The problem is analyzed in terms of a dipole radiation
in a homogenous medium separated by a plane boundary from a dielectric half space. Expressions for the
Hertzian potentials of the dipole is reduced to integrals which was obtained by Sommerfeld equations multiplied
by an exponential depth attenuation factor. The analysis is described for both magnetic and electric, vertical and
horizontal dipolesFinally, accurate numerical analyses are derived to illustrate the above statements.
Keywords: - Sommerfeld radiation problem, Dipole radiation, Hertzian potentials
I. INTRODUCTION
The so - called Sommerfeld radiation problem is a well known problem in the field of propagation
of electromagnetic (EM) waves for obvious applications in the area of wireless telecommunications [1], [2].
Furthermore, Sommerfeld expanded his original work to take into account vertical and horizontal, electric and
magnetie dipoles above a plane earth. In 1909, Sommerfeld stated the existence of a surface wave in the
radiation of a vertical Hertzian dipole over a plane earth [1]. The solution of the boundary value problem was
based on the evaluation of Fourier-Bessel integrals which were the solutions of the wave equation. In 1953,
Wait discussed an insulated magnetic dipole in a conducting medium, showing that the fields are independent of
the characteristics of the insulation for an antenna diameter much less than the radiation wavelength in the
conducting medium [4]. Analyzed magnetic dipole solution of a semi-infinite medium including special cases of
frequency, antenna depth, and separation between antennas was discussed [5]. It can be noticed from [6], [7]
that a horizontal electric dipole in a conducting half space was carried out by a mathematical analysis. Also, the
exponential increase of the attenuation with depth was experimentally verified. In [8], [9] the engineering
application of the above problem with obvious application to wireless telecommunications was discussed and
provided approximate solutions to the above problem, which are represented by rather long algebraic
expressions.
II. PREFACE OF RADIATION PROBLEM
The geometry of the problem is given in Figure 1. It is assumed that the dipole is oriented either horizontally in
x-direction or vertically in the z-direction. It can be noticed that the important direction for transmitting dipole
radiation is directly toward the surface of the sea because of the mode of communication. Furthermore,
magnetic and electric fields have been calculated for both the vertical and the horizontal dipoles.
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Figure 1. Geometry of the radiation problem considered in this paper
The theory of a conducting half space and electromagnetic radiation in a conducting medium are discussed in
[9,10].
III. ANALYSIS OF RADIATION PROBLEM
3.1 The two Hertzian functions for the two mediums have to meet the following conditions, [1]:
1
+
1
2
1
= 0 , > 0
(1a)
2
+
2
2
2
= 0 , < 0
1
=
2
,
1
1
2
=
1
2
2
= 0
(1b)
1
= 0, > 0, = , = +
2
= 0, < 0, = , = (1c)
where
r = cylinder radius , R = distance of the point from the transmitter which lies on the origin
(z = 0 , r = 0). As in Figure 2.
Figure 2. Dipole position
Then, Sommerfeld obtained his well-known solution in the form of the following integral, [2]:
1
= 4
1
2
+
2
2
1
2
(
0
)
3
1
2
1
2
+
2
2
2
2
2
0
> 0 (2)
1.2 The Maxwell equations for a conducting medium are given by, [2], [3]:
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2
= (3)
where
H is the magnetic field and E is the electric field.
Furthermore, by taking into account a plane wave travelling through the positive z-direction we get the
following equations
=
0
1+ /
, =
/4
2
(4)
where
n is the unit vector in the direction of propagation
=
2
The Hertzian potentials and
(the Green's function for a dipole source) for the electric and magnetic dipoles
respectively, in the infinite, conducting medium are, [2], [11]:
=
1
,
1
=
4
(5)
=
1
,
1
4
(6)
where
mm
is the magnetic moment = NIS
I is the current.
N is the number of turns in the magnetic loop.
S is the loop area vector.
em
is the electric moment.
R is the distance from the dipole to the observation point.
In this paper, the Hertzian potentials are used for both the electric and magnetic dipoles. It can be easily found
that the electric and magnetic field vectors are functions of the Hertzian potential as follows, [5], [12]:
For the Electric dipole:
1
= ,
1
= * +
1
2
(7)
For the Magnetic dipole:
1
=
+
1
2
,
1
=
(8)
where
1
=
Moreover, the electric and magnetic fields in an infinite, nonconducting medium are as the following equations:
For the Electric dipole:
2
=
0
,
2
= * +
2
2
(9)
For the Magnetic dipole:
2
=
+
2
2
,
2
=
(10)
where
k
2
= 2/
0
, is the wave number
0
is free space wavelength
Therefore, the Hertzian potentials in this case for the electric and magnetic dipole respectively, are as the
following equations, [4]:
2
=
2
,
2
=
4
0
(11)
2
=
2
,
2
4
(12)
Then, by taking into account the modification for the case of a source in a conducting half space separated by a
plane boundary from a nonconducting half space. Therefore, the source (transmitting dipole) is located in the
conducting half space at coordinate position (0, 0, z
t
) from the boundary. Similarly, the point of observation
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(receiving dipole) is located at coordinate position (r, , z
r
), as shown in Figure 1. Moreover, the Hertzian
potentials for the conducting and nonconducting half space must satisfy the wave equation (condition A below),
the radiation condition at infinity (condition B) and the radiation condition near the source (condition C) below
[1], [13]:
For condition A:
1
+
1
2
1
= 0 ,
2
+
1
2
= 0,
2
+
2
2
2
= 0 ,
2
+
2
2
= 0,
1
= 0 ,
= 0 ,
2
= 0 ,
= 0 ,
)
2
For condition C:
1
=
=
0
Moreover, the electric and magnetic fields must satisfy the boundary conditions at the surface of the sea.
Therefore, the boundary conditions for the components of the Hertzian potentials and their derivatives are as the
following described equations, [3]:
=
2
-jg
1
=
2
where
g =
0
=
1
2
2
2
,-jg is an approximation for the complex index of reflection.
By this way, we have to describe that the Sommerfeld has shown that only
z
component of the Hertzian
potential is required to describe the fields of a vertical dipole [2].
3.3 Related to the integral calculations of Hertzian potential, the Hertzian potential can be obtained in integral
form for the four basic dipole configurations: vertical and horizontal, electric and magnetic dipoles. The method
used here for obtaining the potential integrals was first used by Sommerfeld [1,3]. Thus, a general Hertzian
potential for each of the four dipole configurations is obtained satisfying conditions (A) through (C). So, the
Hertzian potentials for the observation point in the sea are presented in integral form for the four basic dipole
configurations as follows, [3], [8]:
For vertical dipole:
1
= +
1 ,
= +
1
(13)
For horizontal dipole:
1
= +
1 ,
= +
1
(14)
where
L =
2
,
1
=
2
+ (
)
2
,
2
=
2
+ (
)
2
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1
= 2
0
3
0
,
1
= 2
0
4 +
0
(15)
1
= 2 ( 4)
0
( +)
(16)
0
where
G =
2
2
2
, F =
2
1
2
It should be noticed that the function L has two parts: exp (jk
1
R
1
/R
1
) which represents the primary source at
the position (0, 0, z
t
), and exp (jk
1
R
2
/R
2
) which represents a secondary source at the image position (0,
0,z
t
). Moreover, at the ease of the horizontal electric and the vertical magnetic antennas, the secondary source
represents the image of the primary source, but in the case of the vertical electric and the horizontal magnetic
antennas, the secondary source represents an image dipole of the opposite polarity. As shown in Figure 1, the
primary source radiates over the direct path R
1
, and the secondary source radiates over the reflected path R
2
.
Consequently, the integral in every case represents the major contribution to the Hertzian potentials if r
>>(z
r
+z
t
). By considering (-jg-1) = -jg as g >> 1 for the frequencies and conductivity in this paper, it
can be shown that for z
r
> 0 , z
t
> 0 , [9], [11]:
1
=
1
2
2
+
1
2
2
+
1
1
2
(17)
1
+jg
1
=
(18)
Also, the integrals I
c2
, I
b2
and I
a2
are interdependent. So, it can be shown that for z
t
> 0 , z
r
< 0
2
=
1
2
2
(19)
=
2
2
20
Furthermore, by taking into account calculating the fields in the sea and at the surface of the sea to concentrate
on the integrals I
b1
and I
a1
. Moreover, all the fields which will be discussed in this paper can be expressed in
terms of integrals I
b1
and I
a1
. Then, by replacing I
b1
and I
a1
with the two new integrals I
a
and I
b
as
follows, [2], [13]:
4
()
2
=
(21)
6
()
3 +
=
(22)
where
=
=
2
0
, c = speed of light ,
0
= wave length
=
c
=
2
0
, =
, z =
M =
=
2
1 , L =
=
2
+
The factors and are used to express r and z in terms of free space wavelength divided by 2. So, the electric
and magnetic field components in the sea as functions of the integrals I
a
and I
b
are as follows, [11]:
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For the Vertical dipole at Electric dipole, [2]:
2
(23)
(24)
(25)
For the Vertical dipole at Magnetic dipole, [6]:
2
(26)
(27)
(28)
For the Horizontal dipole at Electric dipole, [10]:
2
sin
1
(29)
2
cos
(30)
2
sin
(31)
= 2
3
cos
2
2
+
(32)
= 5
3
sin
1
(33)
=
2
3
3
cos
(34)
For the Horizontal dipole at Magnetic dipole, [12]:
2
sin
1
(35)
= 3
2
3
(36)
(37)
3
cos
2
+
(38)
3
sin
1
(39)
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= 5
3
cos
(40)
It can be mentioned from the above equations that the electric and magnetic fields in the air at the boundary can
be calculated from the fields in the sea and the boundary conditions. Moreover, the fields are expressed through
the boundary conditions as follows, [13]:
2
=
1
,
1
=
2
as i = r, ,
1
=
2
as i = r,, (41)
Related to the calculations of the integrals, asymptotic expansions for I
a
and I
b
can be determined by the
method of critical points [12]. In the case of which the source and point of observation both lie on the boundary,
it will be possible to reduce the integrals I
a
and I
b
to those obtained by Sommerfeld [3], [11]. The coordinates of
branch points 4 and 2 are given by = 1 jx/2 and = 1 +jx/2, respectively. where x is a very small
value associated with the conductivity of air. Moreover, the poles occur when (L-gjM) = 0, exactly when, [15]:
2
= 1
2
1+
2
42
where g 1 for the conductivity and frequencies considered in this paper. It can be easily found that to
evaluate the integrals by contour methods it is convenient to write these integrals in forms such that the path of
integration lies along the entire real axis. Therefore, it can be done by the conversion of the Bessel functions of
the first kind into Hankel functions of the first kind to become as follows, [10], [13]:
0
2
=
(43)
0
5 +
=
(44)
whereM is a pure imaginary part.
Morefore, it is possible to close a contour by tottering a semicircle, whose radius is unbounded from the positive
real axis to the negative real axis through the upper half plane. Then, by taking into account a highly conducting
medium, we can observe that the contribution to the integral along branch line 1 is negligible. Furthermore, in
order to integrate I
a
and I
b
with respect to the conductivity and frequency range considered in this case, it can
be written such as the following equations, [3]:
2
(1 )/2
0
1
()
2
=
(45)
2
(1 )/2
0
1
()
5 +
=
(46)
where
=
Then, as a conclusion in the case examined here, it can be easily realized that I
a
and I
b
in case of 0<< l , T=1
can be written as:
= 5j
4
2
T (47)
= 8
7
( 3)
3
(48)
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IV. PHYSICAL INTERPRETATION OF THE DERIVED EXPRESSIONS OF
A DIPOLE RADIATION PROBLEM CONSIDERED IN THIS PAPER
1. It is deduced from our analysis that the path of electromagnetic energy between the transmitting and
receiving dipoles in the conducting medium is as the following : (a) propagation from the transmitter
directly to the surface, (b) propagation along the surface of the medium allowing refraction of the energy
back into the homogeneous medium, (c) propagation descending into the inhomogeneous medium to the
receiver.
2. From equations (43) (48), we can realize that the energy traveling directly through the sea between the
transmitter and receiver is neglected and the ratio of the magnitude of the direct wave through the sea over
the surface wave is of the order of e
z