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5-2007
AN EFFICIENT ANALYSIS OF VERTICALDIPOLE ANTENNAS ABOVE A LOSSY HALF-SPACEYongfeng HuangClemson University, [email protected]
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Recommended CitationHuang, Yongfeng, "AN EFFICIENT ANALYSIS OF VERTICAL DIPOLE ANTENNAS ABOVE A LOSSY HALF-SPACE " (2007).All Theses. 147.https://tigerprints.clemson.edu/all_theses/147
AN EFFICIENT ANALYSIS OF VERTICAL DIPOLE ANTENNAS ABOVE A LOSSY HALF-SPACE
A Thesis Presented to
the Graduate School of Clemson University
In Partial Fulfillment of the Requirements for the Degree
Master of Science Electrical Engineering
by Yongfeng Huang
May 2007
Accepted by: Dr. Xiao-Bang Xu, Committee Chair
Dr. Anthony Q. Martin Dr. L. Wilson Pearson
ii
ABSTRACT
The electromagnetic modeling of radiation by vertical dipole antennas above a
lossy half-space has become an important subject due to many applications where these
dipoles are involved. The modeling often encounters Sommerfeld-type integrals that are
normally highly oscillatory with poor convergence. Recently, an efficient computation of
the electric field radiated by an infinitesimal electric dipole above a lossy half-space has
been reported, in which the Sommerfeld-type integrals are reduced to rapidly-converging
integrals. Taking advantage of such efficiently-calculated electric field and using it as the
Green’s function, in this thesis, an electric field integral equation (EFIE) is formulated for
the analysis of a vertical dipole antenna above a lossy half-space. Then, the EFIE is
solved numerically employing the Method of Moments (MoM). Sample numerical results
are presented and discussed for the current distribution as well as the input impedance
and radiation pattern of a vertical thin-wire antenna above a lossy half-space. First, the
EFIE solutions of the current distribution on an antenna in free space are compared with
that obtained using a traditional approach of solving the Pocklington’s equation, and a
good agreement is observed. Then, the current distributions on an antenna above a lossy
half-space of various conductivities are compared with that for the antenna above a PEC
plane. The comparison illustrates that as the conductivity increases, the current
distribution gradually approach to, and finally match that for the antenna above a PEC
plane, as one would expect. Data of the current distributed on an antenna above a lossy
half-space at different heights show that for an antenna close to the media interface
iii
separating two half-spaces, the lower half-space can significantly affect the current
distribution on the antenna and its input impedance. But as the antenna is located farther
apart from the media interface, the influence of the lower half-space would become
weaker and weaker, and eventually negligible. The radiation patterns of an antenna above
a very lossy half-space and that for an antenna at different heights above a lossy half-
space are also presented. They exhibit properties as expected and similar to that
documented in literature for infinitesimal vertical dipoles above a lossy half-space.
iv
ACKNOWLEDGMENTS
I would like to express my deep and sincere gratitude to my advisor, Professor
Xiao-bang Xu. His wide knowledge and logical way have been a great value for me
throughout the development of this thesis. Also, I wish to express my warm and sincere
thanks to Dr. Anthony Martin and Dr. Wilson Pearson for serving on my thesis
committee.
I would like to present this to my father Ben-xiang Huang and my mother Hui-lan
Li as a special gift from their son.
v
vi
TABLE OF CONTENTS
Page
TITLE PAGE............................................................................................. i
ABSTRACT............................................................................................... iii
ACKNOWLEDGEMENTS....................................................................... v
LIST OF FIGURES ................................................................................... ix
LIST OF TABLES..................................................................................... xi
CHAPTER
I. INTRODUCTION ......................................................................... 1
II. FORMULATION OF THE ELECTRIC FIELD INTEGRAL EQUATION (EFIE)............................................. 5 III. NUMERICAL SOLUTION TECHNIQUE, RESULTS, AND DISCUSSIONS............................................ 14 IV. CONCLUSIONS............................................................................ 33
REFERENCES .......................................................................................... 35
vii
viii
LIST OF FIGURES
Figure Page
1. A vertical dipole antenna above a lossy half-space ....................... 6 2. Highly oscillating integrand of the Sommerfeld-type integral
3
( )00( ) zjk z z
vz
kJ k e dk
kρ
ρ ρρ∞ ′+Γ∫ for ' 4,z z+ = 30ρ = ,
and 0.3 0.1n jη = + .................................................................... 8 3. Convergence of the integrand of the semi-infinite integral
0 0 ( )
0 ( )
nk jk Re e dR
η ξ ξ
ξξ
′− −∞
′∫ for ' 4z z+ = , 30ρ = and
0.3 0.1n jη = + ........................................................................... 10 4. Comparison between the current distributions obtained using Pocklington’s equation and that employing EFIE on a thin-wire vertical antenna ( 0.01a λ= , 0.5l λ= ) in free space ........................................... 19 5. Comparison between the current distributions obtained using Pocklington’s equation and that employing EFIE on a thin-wire vertical antenna ( 0.01a λ= , 1l λ= ) in free space ............................................... 19 6. Comparison between the current distributions on a vertical dipole antenna ( 0.01a λ= , 0.5l λ= ) above a lossy half-space ( 1.001rε = , 0.26h λ= , 300f MHz= ) of various conductivities with that on the antenna above a PEC plane .................................................................... 21 7. Comparison between the current distributions on a vertical dipole antenna ( 0.01a λ= , 0.5l λ= ) above a lossy half-space ( 1.001rε = , 0.35h λ= , 300f MHz= ) of various conductivities with that on the antenna above a PEC plane .................................................................... 22
ix
List of Figures (Continued)
Figure Page
8. Current distributions on a vertical dipole antenna ( 0.01a λ= , 0.5l λ= ) at different heights above a half-space with normalized intrinsic impedance 0.3 0.1n jη = + .......................................... 24 9. Current distributions on a vertical dipole antenna ( 0.01a λ= , 0.5l λ= , 0.251h λ= ) above a half- space with various real part of normalized intrinsic impedance ................................................................... 26 10. Current distributions on a vertical dipole antenna ( 0.01a λ= , 0.5l λ= , 0.251h λ= ) above a half- space with various imaginary part of normalized intrinsic impedance ................................................................... 27 11. Radiation patterns of a vertical dipole antenna ( 0.01a λ= , 0.5l λ= ) above a very lossy half-space ( 1.001rε = , 0.26h λ= , 300f MHz= ) of various conductivities compared with that for the antenna above a PEC plane .................................................................... 31 12. Radiation patterns of a vertical dipole antenna ( 0.01a λ= , 0.5l λ= ) at different heights above a half-space with normalized intrinsic impedance 0.3 0.1n jη = + .......................................... 32
x
LIST OF TABLES
Table Page
1. The input impedance of a vertical dipole antenna ( 0.01a λ= , 0.5l λ= ) above a lossy half-space ( 0.3 0.1n jη = + ) with various heights ...................................... 29 2. The input impedance of a vertical dipole antenna ( 0.01a λ= , 0.5l λ= , 0.251h λ= ) above a half-space with various real part of normalized intrinsic impedance ................................................ 29 3. The input impedance of a vertical dipole antenna ( 0.01a λ= , 0.5l λ= , 0.251h λ= ) above a half-space with various imaginary part of normalized intrinsic impedance ................................................ 30
xi
xii
CHAPTER I
INTRODUCTION
Wire antennas are the oldest and perhaps still the most prevalent one of all antennas
forms. For an accurate electromagnetic modeling of wire antennas, the electric current
distributed on the wire must be determined [1]. In the past, Pocklington’s equation [2]
and Hallen’s equations [3] have been formulated by expressing the scattered electric field
in terms of the magnetic vector potential and electric scalar potential, and then solved for
the unknown current distribution on thin-wire antennas [1], [4]. This approach is widely
used for analyzing wire antennas located in an open space where no media interface
presents.
As pointed out in [5] and [6], the electromagnetic modeling of radiation by vertical
dipole antennas above a lossy half-space has become an important subject of research and
development, due to many applications where these dipoles are involved. The modeling
often encounters the Sommerfeld-type integrals [7] that represent the effect of the media
interface. Asymptotic techniques, including the saddle-point method [8], have been
developed to efficiently evaluate the Sommerfeld-type integrals. But these methods are
limited for the evaluation of the Sommerfeld-type integrals involved in the far-zone field
computations, which are not applicable for integral equation formulations that require
information of the near-zone fields. In [6], the input impedance of a vertical dipole
antenna above a dielectric half-space is calculated, employing the induced EMF method
[1]. In the calculation, a complex image Green’s function is used and a sinusoidal current
1
distribution is assumed. However, the complex image Green’s function is only an
approximation of the exact Green’s function, using a series of exponential functions, and
the sinusoidal current assumption may not faithfully represent the actual current
distribution, especially when the antenna is close to the media interface. Recently, an
efficient computation of the electric field radiated by an infinitesimal electric dipole
above a lossy half-space has been reported [9]. The efficient computation is based on an
exact image theory [10]-[12] derived by means of applications of integral transforms and
appropriate identities. In that way, the Sommerfeld-type integrals involved in the
computation of the electric field are reduced to integrals that converge very rapidly, and
the computation time is greatly reduced. Using such efficiently-computed electric field as
the Green’s function will make the application of an integral equation approach practical
for determining the current distribution on the surface of a dipole antenna above a lossy
half-space.
In this thesis, an electric field integral equation (EFIE) is formulated for the
current distribution on a vertical dipole antenna above a lossy half-space, where the
Green’s function is the vertical component of the electric field radiated by an
infinitesimal vertical dipole, which is readily determined making use of the exact image
theory and is presented in [9]. Then, the integral equation formulated is solved
numerically employing the Method of Moments (MoM) [13]. The numerical solution of
such formulated integral equation is expected to be more efficient because the semi-
infinite integral included in the Green’s function converges very rapidly. Finally, based
on knowledge of the current distribution, the antenna characteristics of interest, such as
the input impedance and the radiation pattern, are computed.
2
The outline of the rest of this thesis is as follows. Section II presents the
formulation of the EFIE for the unknown current distribution. In the formulation, we start
with a vertical dipole antenna in free space first, and then extend it to that for the antenna
above a lossy half-space, employing the exact image theory. The EFIE is solved
numerically employing the MoM and the numerical solution procedure is described in
Section III. Then, sample numerical results of the current distribution on a vertical dipole
antenna above a lossy half-space, as well as data of its input impedance and radiation
pattern are presented and discussed. Finally, some conclusions are drawn in Section IV.
3
4
CHAPTER II
FORMULATION OF THE ELECTRIC FIELD INTEGRAL EQUATION (EFIE)
As shown in Fig. 1, a vertical dipole antenna of length l and of a circular cross
section with radius a is located above a planar interface 0z = at a height h measured
from the center of the vertical antenna. The interface separates two semi-infinite
homogeneous spaces. The upper half-space ( is taken to be free space representing
the air characterized by
0z > )
0 0( , , 0)μ ε σ = ; and the lower half-space is assumed to be a lossy
earth with electromagnetic parameters 0 0( , , 0)rμ ε ε σ ≠ . The antenna has a very narrow
feed gap at its center and is fed by a delta-gap source [4] so that the incident electric field
can be expressed by ( )izE V zδ= , where ( )zδ is a delta function. Under the excitation, a
z-directed electric current is induced on the surface of the antenna that can be considered
as a perfect electric conductor (PEC). To determine the unknown current distribution, an
electric field integral equation (EFIE) is formulated in this section. Different from the
Pocklington’s equation and the Hallen’s equation, the Green’s function used in the
integral equations formulation presented in this section is the electric field, rather than the
potentials, radiated by an infinitesimal vertical dipole above a lossy half-space. The EFIE
is formulated by enforcing the boundary condition on a PEC surface, which requires that
the tangential component of the total electric field be zero,
, (1) tan tan 0s iE E+ =
where tansE is the scattered electric field generated by the induced current and
5
tan tan tans fE E E= + d , (2)
in which tanfE is the electric field of the antenna if it were in free space and tan
dE is the
diffracted field due to the existence of the lower half-space. Therefore, in this section, we
present the electric field radiated by an infinitesimal vertical dipole in free space first,
then derive the diffracted field by an infinitesimal vertical dipole above a lossy half-
space, and finally, incorporate the electric fields obtained in the EFIE formulation as the
Green’s function.
0 0( , , 0)ε μ σ =
0 0( , , 0)rε ε μ σ ≠
l
h2l
z=0 plane
z
Fig.1 A vertical dipole antenna above a lossy half-space
A. The electric field radiated by an infinitesimal vertical dipole in free space
The electric field radiated by an infinitesimal z-directed electric dipole of dipole
moment in free space is given in [14] as Il
0 00 0 02 3 2 3
0 0
1 1ˆ( , ) ( ) cos ( )sin , (3)2 4
jk R jk Rfdipole
jIl IlE r r r e eR j R R R j Rη ωμ ηθ θ θ
π ωε π ωε− −′ = + + + +
6
where R is the distance between the source point and the field point, and
2 2( ) ( ) ( 2)R r r x x y y z z′ ′ ′= − = − + − + − ′ , in which ( , , )x y z′ ′ ′ and ( , , )x y z locate the
source and field point, respectively. Also, in equation (3), k0 and 0η are the wavenumber
and the intrinsic impedance of free space, and θ is the angle between vector R and the z
axis, defined by . Then, from equation (3), the z-component of the
electric field is readily found to be
1cos [( ) / ]z z Rθ − ′= −
0
,
20 0 2 2 2 2
0 0 0 0
3 3 1[(1 )cos (1 )]4z dipole
jk Rf e j jE Il jk
R k R k R k R k Rη θ
π
−
= − − − − − . (4)
B. The electric field radiated by an infinitesimal vertical dipole above a lossy half-space
The electric field radiated by an infinitesimal dipole of orientation , where l̂
ˆ ˆ ˆ ˆx y zl l x l y l z= + + , located above a lossy half-space was originally derived by
Sommerfeld and given in [9]
0 02 0 20
2 2( )
1 0 2 22 2 20 0 0
0 0
ˆ( , ) [ ( ( ) cos 2 ( )) ( )sin 2 )]4 2
2[ cos ( ) ( ( ) ( ) cos 2 ) ( )sin 2 ]
2
ˆ [4 2
z
ddipole h x y
z
z jk z zz zv z x y
z
hz
kk IE r r x l J k J k l J kk
k jk k k kl J k l J k J k l J k e dk k k k
kk Iyk
ρρ ρ ρ
ρ ρ kρ ρ ρ ρ ρ
ρ
η ρ φ ρ ρ φπ
φ ρ ρ ρ φ ρ φ
ηπ
∞
′+
⎧′ = Γ − + −⎨
⎩⎫
+Γ + − − ⎬⎭
+ Γ −
∫
2 0 20
2 2( )
1 2 0 22 2 20 0 0
20 0
0 12 200 0
( ) sin 2 ( ( ) ( ) cos 2 )]
2[ sin ( ) ( ) cos 2 ( ( ) ( ) cos 2 )]
2
ˆ [ ( ) ( )(4
z
x y
z jk z zz zv z x y
z
zv z x
z
l J k l J k J k
k jk k k kl J k l J k l J k J k e dkk k k k
k k jk kk Iz l J k J k lk k k
ρ ρ ρ
ρ ρρ ρ ρ ρ ρ
ρ ρ ρρ ρ
ρ φ ρ ρ φ
φ ρ ρ φ ρ ρ φ
η ρ ρπ
∞
′+
∞
⎧− −⎨
⎩⎫
+Γ − + + ⎬⎭
− Γ +
∫
∫ ( )cos sin )] , (5)zjk z zyl e dkρφ φ ′++
where ρ is the radial distance between the observation and source points, φ is the angle
between ρ and the x-axis, and z z′ are the heights of the observation and source point
above the lossy half-space, kρ and are spectral wave numbers and zk 2 20zk k k 2
ρ= − . Also,
7
in equation (5), 0 ( )J ⋅ , , and 1( )J ⋅ 2 ( )J ⋅ are Bessel functions of order 0, 1 and 2, and
are the horizontal and vertical Fresnel reflection coefficients given by
hΓ
vΓ
0 0
0 0
/ / , / /
n z n zh v
n z n z
k k k kk k k k
η ηη η
− − +Γ = Γ =
+ +,
in which ηn is the normalized intrinsic impedance of the lower half-space, defined by
0/ 1/[ /( )]n r j 0η η η ε σ ωε= = − . For a vertical infinitesimal dipole, , , and
, equation (5) reduces to
0xl = 0yl =
1zl =
2( )0 0
, 1 1 02 2 200 0 0
ˆ ˆ ˆ( , ) [ cos ( ) sin ( ) ( ) ]4
zz z jk z zdv dipole v
z
k jk k jk k kk IlE r r J k x J k y J k z e dkk k k k
ρ ρ ρ ρ .ρ ρ ρη φ ρ φ ρ ρπ
∞ ′+′ = Γ + −∫ ρ
(6)
One notes that the semi-infinite integral in (6) is a Sommerfeld-type integral, which does
not have a closed-form analytic result, and is difficult to be evaluated numerically
because of its poor convergence and highly oscillatory nature as shown in Fig. 2.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
kρ
imaginary partreal part
Fig. 2 Highly oscillating integrand of the Sommerfeld-type integral
3
( )00( ) zjk z z
vz
kJ k e dk
kρ
ρ ρρ∞ ′+Γ∫ for ' 4,z z+ = 30ρ = , and 0.3 0.1n jη = +
8
In order to improve the convergence properties of the Sommerfeld-type integral, exact
image theory is employed by the application of integral transforms and appropriate
identities as described in [9], the procedure is outlined below. First, equation (6) is
rewritten to an equation that contains the zero-order Bessel function only as
2 2 2 2
( )0, 02 2 0
0
ˆ ˆ ˆ( , ) [ ( ) ] ( ) ,4
zjk z zdv dipole v
z
kIlE r r x y z J k e dkk x z y z x y k
ρρ ρ
η ρπ
∞ ′+∂ ∂ ∂ ∂′ = − + − + Γ∂ ∂ ∂ ∂ ∂ ∂ ∫ (7)
by using the following identities
2
12 20 0
1cos ( ) ( )zk kj J k J
k k x zρ
ρ 0 kρφ ρ ∂− =
∂ ∂ρ , (8a)
2
12 20 0
1sin ( ) ( )zk kj J k J
k k y zρ
ρ 0 kρφ ρ ∂− =
∂ ∂ρ , (8b)
and
2 2 2
0 02 2 2 20 0
1( ) ( ) ( )k
J k J kk k x y
ρρ ρρ ρ∂ ∂
− = +∂ ∂
. (8c)
Then, the vertical Fresnel reflection coefficient vΓ contained in (7) is expressed in the
form of a Laplace transform as
0( )0 0
1 2 n zk kv nk e dη ξη ξ
∞ − +Γ = − ∫ . (9)
Substituting equation (9) into (7) and then solving the resulting integrals in terms of kρ
analytically by applying
( )00( ) z
jkRjk z z
z
ke j J k e dkR k
ρρ ρρ
′′ ∞ ′+=′′ ∫ , (10)
where ''R is defined by 2 2( ) ( ) ( )2R x x y y z z′′ ′ ′ ′= − + − + + , we arrive at the final form
of as ,dv dipoleE
9
0 00
( )2 2 2 20
, 02 2 00
ˆ ˆ ˆ( , ') [ ( ) ][ 2 ],4 (
n
jk R jk Rkd
v dipole nj Il e e
)E r r x y z k e d
k x z y z x y R R
ξη ξη η ξ
π ξ
′− −∞ −′′∂ ∂ ∂ ∂
= − + − + −′′ ′∂ ∂ ∂ ∂ ∂ ∂ ∫
(11)
in which R′ is defined by 2 2( ) ( ) ( ) ( )2R x x y y z z jξ ξ′ ′ ′ ′= − + − + + + . As illustrated in
Fig. 3, the integrand of the semi-infinite integral on the right-hand side of equation (11)
with the same parameters as those used in Fig.2 decays rapidly as ξ increases, due to
both exponentially decaying factors 0nke η ξ− and 0 ( )jk Re ξ′− . Subsequently, the semi-infinite
integral with such a rapidly decaying integrand will converge quickly, making the
application of an integral equation approach practical, where ,dv dipoleE is used as part of the
Green’s function in the integral equation formulation.
0 1 2 3 4 5 6 7 8-0.035
-0.03
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
kρ
imaginary partreal part
Fig. 3 Convergence of the integrand of the semi-infinite integral 0 0 ( )
0 ( )
nk jk Re e dR
η ξ ξ
ξξ
′− −∞
′∫ for ' 4z z+ = , 30ρ = , and 0.3 0.1n jη = +
10
C. Formulation of EFIE for a vertical dipole antenna above a lossy half-space
Based on knowledge of the electric field of an infinitesimal dipole in free pace and
that above a lossy half-space, derived in the previous two sub-sections, an EFIE is
formulated for the current distributed on the surface of a vertical dipole antenna above a
lossy half-space. The EFIE is formulated for a thin wire antenna, the radius of which is
much less than the wavelength and the length of antenna ( , a a lλ<< << ). The antenna is
assumed to be a PEC, the current on its surface is assumed to be in direction and is
independent of φ due to the azimuthal symmetry of the cylindrical configuration of the
antenna, and the current is supposed to vanish at the two ends of the antenna.
z
The first step of the formulation of the EFIE is to enforce the boundary condition
which requires that the tangential component of the total electric field be zero on a PEC
surface, as shown in equation (1). Under the assumption that the current distributed on
the antenna surface is z-directed and φ -independent, the tangential component of the
electric field should be . Then, equation (1) reduces to zE
, (12) 0s iz zE E+ =
where izE is the z-component of the incident electric field and
( )izE V zδ= , (13)
in which V is a voltage applied across a very narrow feed gap and δ(z) is a delta function.
Also, in equation (12), szE is the total scattered field that contains the free-space
component and a correction term that counts the effect of the lossy half-space as
s fz z
dzE E E= + , (14)
11
where the free-space component fzE can be found as an integral, on the antenna surface,
of the z-component of the electric field generated by an infinitesimal vertical dipole in
free space, given in equation (4),
020 0
, 2 2 2 20 0 0 0
3 3 1[(1 )( ) (1 )]4
jk Rf
z dipolee jk j z z jE Il
R k R k R R k R k Rη
π
− ′−= − − − − − . ( 4′ )
Also, in equation (14), the diffracted field due to the lossy half-space dzE can be obtained
as an integral, on the antenna surface, of the z-component of the electric field generated
by an infinitesimal vertical dipole above the lossy half-space, given by equation (11) as
0 00
( )2 20
, 02 2 00
( )[ 24 (
n
jk R jk Rkd
z dipole nj Il e eE k
k x y R R
ζη ζη ]
)e dη ξ
π ξ
′′ ′− −∞ −∂ ∂= + −
′′ ′∂ ∂ ∫ . (11′ )
Then, the total scattered field by the vertical dipole antenna can be found as the surface
integral of and , shown in equations ( ) and (11,f
z dipoleE ,dz dipoleE 4 ' ′ ), and
0
0 00
20 022 2 2 2
0 0 0 02
( )2 20
02 2 00
3 3 1[(1 )( ) (1 )]4
+ ( )[ 2 ]4 ( )
n
l jk Rhslz zz h
jk R jk Rk
n
jk e j z z jE J aR k R k R R k R k R
j e ek e d dz dk x y R R
π
φ π
ζη ζ
ηπ
η η ξ φπ ξ
−+
′=−′= −
′′ ′− −∞ −
⎧ ′−= − − − −⎨
⎩⎫∂ ∂ ′ ′+ − ⎬′′ ′∂ ∂ ⎭
∫ ∫
∫
−
(15)
in which is the surface current density on the antenna surface. For convenience, we
define a current as
zJ
( ) 2 ( )zI z aJ zπ= . After substituting such defined ( )I z into (15) and
taking the derivatives in the equation, we rewrite equation (15) as
0
20 022 2 2 2
2 0 0
( ) 3 3 1[(1 )( ) (1 )]2 4
l jk Rhslz z h
I z jk e j z z jER k R k R R k R k R
π
φ π
ηπ π
−+
′=−′= −
⎧′ ′−= − − −⎨
⎩∫ ∫
0 0
− −
0 2 2 20 0 0
40
[(3 / 3 ) [( ) ( ) ] 2 ( 1)] 4
jk Rj e R jk k R x x y y R jk Rk R
ηπ
′′− ′′ ′′ ′ ′ ′′ ′′+ − ⋅ − + − − ++
′′0
12
00
2 2 20 0 0 0
40
[(3 / 3 ) [( ) ( ) ] 2 ( 1)]2
n
jk Rknj e R jk k R x x y y R jk Re d
Rη ξη η dz dξ φ
π
′−∞ − ′ ′ ′ ′ ′ ′ ⎫+ − ⋅ − + − − + ′ ′− ⎬′ ⎭∫
(16)
Finally, substituting equations (13) and (16) into (12), the EFIE is formulated as
0
0
0
20 022 2 2 2
2 0 0 0 0
2 2 20 0 0 0
40
00
( ) 3 3 1[(1 )( ) (1 )]2 4
[(3/ 3 ) [( ) ( ) ] 2 ( 1)] 4
2n
l jk Rh
lz h
jk R
kn
I z jk e j z z jR k R k R R k R k R
j e R jk k R x x y y R jk Rk R
j e
π
φ π
η ξ
ηπ π
ηπ
η ηπ
−+
′=−′= −
′′−
∞ −
⎧′ ′−− − − − −⎨
⎩′′ ′′ ′ ′ ′′ ′′+ − ⋅ − + − − +
+′′
−
∫ ∫
∫0 2 2 2
0 0 04
[(3/ 3 ) [( ) ( ) ] 2 ( 1)]
( )
jk Re R jk k R x x y y R jk R d dz dR
V z
ξ φ
δ
′− ⎫′ ′ ′ ′ ′ ′+ − ⋅ − + − − + ′ ′⎬′ ⎭= −
(17)
13
14
CHAPTER III
NUMERICAL SOLUTION TECHNIQUE, RESULTS, AND DISCUSSIONS
A. The numerical solution technique
In this section, the EFIE formulated in the previous section is solved numerically,
employing the MoM, for the unknown current ( )I z distributed on the surface of a
vertical dipole antenna above a lossy half-space. Then, based on knowledge of the current
distribution, the antenna characteristics of interest, such as the input impedance and the
radiation pattern, are computed. Sample numerical results are presented and discussed.
One notes that the integral equation presented in equation (17) contains double
integrals. To eliminate one of the double integrals for an efficient numerical solution of
the integral equation, the reduced kernel approximation [4], [15] for thin wires under the
conditions that a λ<< and a is employed. Realizing that for both the source point
and field point on the antenna surface, , then
using the median value of ( d
l
2)
<<
2 2 2 2( ) ( ) 4 sin ( /x x y y d a φ′ ′− + − = =
d a= ) to approximate it, R, 'R , and ''R can be
approximated by
2( )r2R R z z′≈ = − + a , (18a)
2 (r2)R R a z z jξ′′ ≈ = + + +′ , (18b)
and
2 ( )r2R R a z z′′′′ ′≈ = + + . (18c)
Substituting these approximations into equation (17), the integrands in the integrals of
15
(17) become φ -independent. Subsequently, the integral equation reduces to
0
20 022 2 2 2
2 0 0 0
3 3 1[(1 )( ) (1 )]4
rl jk Rh
l zz hr r r r r
jk e j z z jI0 rR k R k R R k R k R
ηπ
−+
′= −
⎧ ′−− − − − −⎨
⎩∫
0
00
2 20 0 0 0
40
2 20 0 0 0
0 4
[(3 / 3 ) 2 ( 1)]4
[(3/ 3 ) 2 ( 1)] ( ). (19)2
r
rn
jk Rr r r r
r
jk Rkn r r r r
r
j e R jk k R a R jk Rk R
j e R jk k R a R jk Re dR
η ξ
ηπ
η η ξ φ δπ
′′−
′−∞ −
′′ ′′ ′′ ′′+ − ⋅ − ++
′′
⎫′ ′ ′ ′+ − ⋅ − + ⎪ ′ ′− =⎬′ ⎪⎭
∫ dz d V z−
To solve (19), we employ the MoM. First we divide the vertical antenna into N segments,
the length of each is , then approximate the unknown current /l NΔ = ( )I z by means of
a linear combination of pulse functions as
1
( ) ( )N
n nn
I z I=
= ∏∑ z , (20)
where the pulse function is defined by
1, ( , )( ) 2 2
0, otherwise,
n nn
z z zz
Δ Δ⎧ ∈ − +⎪∏ = ⎨⎪⎩
(21)
in which is the median point of the segment. Then, substituting the pulse
expansion in (19) for
nz thn
( )I z and then testing the resulting equation with the pulse function,
the integral equation is converted to a matrix equation as
[ ][ ] [ ]mn n m ,Z I V= (22)
where the forcing function is
(23) / 2
/ 2( ) ( )
h l im z mh l
V E z z+
−= Π∫ dz
and the impedance matrix elements mnZ are
16
020 02 2
2 2 2 20 0 0 02 2
3 3 ( ) 1[(1 )( ) (1 )]4
rjk R
mn z zr r r r r
jk e j m n z jZR k R k R R k R k R
ηπ
Δ Δ −
Δ Δ′=− =−
⎧ ′− Δ −= − − − −⎨
⎩∫ ∫
r
−
0 2 20 0 0
40
[(3 / 3 ) 2 ( 1)]4
rjk Rr r r r
r
j e R jk k R a R jk Rk R
ηπ
′′− ′′ ′′ ′′ ′′+ − ⋅ − ++
′′0
00
2 20 0 0 0
0 4
[(3 / 3 ) 2 ( 1)] , (24)2
rn
jk Rkn r r r r
r
j e R jk k R a R jk Re dR
η ξη η ξπ
′−∞ −⎫′ ′ ′ ′+ − ⋅ − + ⎪ ′− ⎬
′ ⎪⎭∫ dz dz
in which
2 2[( ) )rR a m n z′= + − Δ − , (25a)
2 2(2 ( ) ) ,rR h l m n z a′′ ′= − + + Δ + + (25b)
and
2 2(2 ( ) )rR h l m n z j aξ′ ′= − + + Δ + − + . (25c)
Based on knowledge of the current distribution on the surface of a center-fed antenna, its
input impedance can be readily found by
/ (0)inZ V I= , (26)
where is taken to be 1 volt and V (0)I is the input current at 0z = . Also, the far-zone
radiation pattern of the antenna can be found [16] by
0 0/ 2 cos cos
/ 2( ) sin ( )( ) - / 2 / 2
h l jk z jk zu h l
F I z e e dzθ θθ θ π+ ′ ′−
−′ ′= + Γ∫ θ π< < , (27)
where
2 2
2 2
cos / 1/ sin
cos / 1/ sinn n
n n
2
2
θ η η θ
θ η η
− −Γ =
+ − θ , (28)
for the upper half-space, and
/ 2 cos
/ 2( ) sin ( ) / 2 3 / 2
h ljkR jkzl h l
F e I z Te dzθθ θ π+ ′−
−′ ′= <∫ θ π< , (29)
17
where
2
2coscos 1 sin /n n
T2
θη θ θ
=− − η
. (30)
for the lower half-space. In equations (27) and (29), k0 and k are the wavenumbers in the
upper and lower half-space, respectively.
B. Numerical results and discussion
Sample numerical results of the current ( )I z distributed on the surface of a thin-
wire vertical antenna above a lossy half-space as well as the input impedance and
radiation pattern of the antenna are shown and discussed in this section. First of all, the
EFIE results of the current distribution on a half-wavelength-long thin-wire antenna (a =
0.01λ) located in free-space are compared with that obtained by a traditional approach of
solving the Pocklington’s equation. The comparison is illustrated in Fig. 4, where one
observes that the two sets of data resulting from these two methods fall on top of each
other. The same comparison is also made for a one-wavelength-long thin-wire antenna in
free-space, and is presented in Fig. 5, which shows again that the EFIE results are almost
the same as the Pocklington’s equation solutions.
18
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25-5
0
5
10
I(z) (
mA
)
z
Pocklington Imag. partPocklington Real PartEFIE Imag.partEFIE Real part
Fig. 4 Comparison between the current distributions obtained using Pocklington’s Eq. and that employing EFIE on a thin-wire vertical antenna ( 0.01a λ= , 0.5l λ= ) in free space
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-3
-2
-1
0
1
2
3
4
z
I(z) (
mA
)
Pocklington Imag. partPocklington Real partEFIE Imag. partEFIE Real part
Fig. 5 Comparison between the current distributions obtained using Pocklington’s Eq. and that employing EFIE on a thin wire vertical antenna ( 0.01a λ= , 1l λ= ) in free space
19
Then, data of the current distribution on a half-wavelength-long thin-wire antenna
( 0.01a λ= ) above a lossy half-space are compared with that on the antenna above a PEC
plane. The comparisons are illustrated in Fig. 6 for 0.26h λ= and Fig. 7 for 0.35h λ= .
From these two figures, one observes that as the conductivity of the lower half-space
increases, the current distribution gradually approaches to that for the antenna above a
PEC plane. When the conductivity is taken to be high enough ( ), the two sets of
data match very well.
410σ =
20
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.250
1
2
3
4
5
6
7
z-h
Rea
l par
t of I
(z) (
mA
)
PEC σ=1
σ=50σ=10000
(a) Real part
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
z-h
Imag
inar
y pa
rt of
I(z)
(mA
)
PECσ=1σ=50σ=10000
(b) Imaginary part
Fig. 6 Comparison between the current distributions on a vertical dipole antenna ( 0.01a λ= , 0.5l λ= ) above a lossy half-space ( 1.001rε = , 0.26h λ= , 300f MHz= ) of various conductivities with that on the antenna above a PEC plane.
21
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.250
1
2
3
4
5
6
7
8
9
10
z-h
Rea
l par
t of I
(z) (
mA
)
PEC σ=1
σ=50σ=10000
(a) Real part
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
z-h
Imag
inar
y pa
rt of
I(z)
(mA
)
PECσ=1σ=50σ=10000
(b) Imaginary part
Fig. 7 Comparison between the current distributions on a vertical dipole antenna ( 0.01a λ= , 0.5l λ= ) above a lossy half-space ( 1.001rε = , 0.35h λ= , 300f MHz= ) of various conductivities with that on the antenna above a PEC plane.
22
Fig. 8 depicts the data of the current distributed on the surface of a thin-wire
antenna ( 0.01a λ= , 0.5l λ= ) above a lossy half-space with normalized intrinsic
impedance 0.3 01n jη = + , at different heights. One notes from this figure that when the
antenna is very close to the media interface separating the two half-spaces ( 0.26h λ= ),
its current distribution is significantly different from that obtained for the antenna located
in free space. However, as the height of the antenna increases, the current distribution
data gradually approach to the free-space result. This makes sense because when the
antenna location is moved away from the media interface, the influence of the lower half-
space on the antenna current distribution becomes weaker and weaker. If the antenna is
located high enough above the lossy half-space, then the influence of the lower half-space
on the antenna would be negligible, making its current distribution be about the same as
the free-space results.
23
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0
1
2
3
4
5
6
7
8
9
10
z-h
Rea
l par
t of I
(z) (
mA
)
free-space resulth=0.26λh=0.35λh=0.75λh=2λ
(a) Real part
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
z-h
Imag
inar
y pa
rt of
I(z)
(mA
)
free-space resulth=0.26λh=0.35λh=0.75λh=2λ
(b) Imaginary part
Fig. 8 Current distributions on a vertical dipole antenna ( 0.01a λ= , 0.5l λ= ) at different heights above a half-space with normalized intrinsic impedance 0.3 0.1n jη = +
24
To see how the electromagnetic parameters of the lower half-space can affect the
current distribution on an antenna above it, in Figs. 9 and 10, we present the current
distribution on a thin-wire vertical antenna ( 0.01a λ= , 0.5l λ= ) above the lower half-
space, at a height of 0.251h λ= , with various normalized intrinsic impedance nη . In Fig.
9, the imaginary part of nη is taken to be unchanged, only its real part varies. The data
presented in Fig. 10 are for the case that the real part of nη remains to be a fixed value,
only its imaginary part changes. The data depicted in these two figures show that when
the normalized intrinsic impedance of the lower half-space varies, the current distribution
changes, and they all are significantly different from that for the antenna located in free
space.
25
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.250
1
2
3
4
5
6
7
8
9
z-h
Rea
l par
t of I
(z) (
mA
)
free-space resultηn=0.05+0.1j
ηn=0.12+0.1j
ηn=0.2+0.1j
ηn=0.3+0.1j
(a) Real part
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25-5
-4
-3
-2
-1
0
1
z-h
Imag
inar
y pa
rt of
I(z)
(mA
)
free-space resultηn=0.05+0.1j
ηn=0.12+0.1j
ηn=0.2+0.1j
ηn=0.3+0.1j
(b) Imaginary part
Fig. 9 Current distributions on a vertical dipole antenna ( 0.01a λ= , 0.5l λ= , 0.251h λ= ) above a half-space with various real part of normalized intrinsic impedance.
26
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.250
1
2
3
4
5
6
7
8
9
z-h
Rea
l par
t of I
(z) (
mA
)
free-space resultηn=0.3+0.01j
ηn=0.3+0.1j
ηn=0.3+0.15j
ηn=0.3+0.23j
(a) Real part
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25-5
-4
-3
-2
-1
0
1
z-h
Imag
inar
y pa
rt of
I(z)
(mA
)
free-space resultηn=0.3+0.01j
ηn=0.3+0.1j
ηn=0.3+0.15j
ηn=0.3+0.23j
(b) Imaginary part
Fig. 10 Current distributions on a vertical dipole antenna ( 0.01a λ= , 0.5l λ= , 0.251h λ= ) above a half-space with various imaginary part of normalized intrinsic
impedance.
27
Based on knowledge of the current distribution on a thin-wire antenna above a
lossy half-space, its input impedance and radiation pattern are obtained and presented in
Tables 1 – 3 and Figs. 11 - 12, all for 0.01a λ= and 0.5l λ= . In Table 1 are listed data
of the input impedance of the antenna above a lossy half-space with normalized intrinsic
impedance 0.3 0.1n jη = + , at different heights. One observes that when the antenna is
very close to the media interface separating the two half-spaces, its input impedance is
significantly different from that for the antenna located in free space. As the height of the
antenna increases, its input impedance gradually approaches to the free-space result.
Again, this is what one would expect and is due to the fact that as the antenna is placed
farther and farther apart from the media interface, the influence of the lower half-space
becomes weaker and weaker, and eventually negligible if the antenna is located high
enough above the interface. Tables 2 and 3 present the input impedance for the antenna
above a lossy half-space, at a height of 0.251h λ= , with different normalized intrinsic
impedance nη . Data listed in these two tables show that variation of the electromagnetic
parameters of the lower half-space can significantly change the input impedance of a
vertical antenna above it.
28
Table 1 The input impedance of a vertical dipole antenna ( 0.01a λ= , 0.5l λ= ) above a lossy half-space ( 0.3 0.1n jη = + ) with various heights
( )inZ Ω
Free space result 106.8188 25.1592 j+
2h λ= 106.7627 + 25.2726j
1h λ= 107.3416 + 24.2268j
0.5h λ= 103.0509 + 18.2636j
0.26h λ= 132.4766 + 6.4395j
Table 2 The input impedance of a vertical dipole antenna ( 0.01a λ= , 0.5l λ= , 0.251h λ= ) above a half-space with various real part of normalized intrinsic impedance
( )inZ Ω
Free space result 106.8188 25.1592 j+
0.05 0.1n jη = + 198.8489 - 9.8498ij
0.12 0.1n jη = + 183.6976 - 4.7526j
0.2 0.1n jη = + 168.5355 - 0.4902j
0.3 0.1n jη = + 152.2572 + 3.2005j
29
Table 3 The input impedance of a vertical dipole antenna ( 0.01a λ= , 0.5l λ= , 0.251h λ= ) above a half-space with various imaginary part of normalized intrinsic
impedance ( )inZ Ω
Free space result 106.8188 25.1592 j+
0.3 0.01n jη = + 153.9227 + 17.1085j
0.3 0.1n jη = + 152.2572 + 3.2005j
0.3 0.15n jη = + 150.474 - 4.1298j
0.3 0.23n jη = + 146.5436 - 15.0825j
Radiation patterns of an antenna above a lossy half-space are presented in Figs. 11
and 12. Fig. 11 depicts the radiation patterns of an antenna above a very lossy half-space.
One observes that the magnitude of the field in the upper half-space is larger
corresponding to higher conductivity of the lower half-space, which would result in
stronger reflection. Also, one note that as the conductivity increases, the radiation
patterns gradually approach to that for the antenna above a PEC plane. This pattern is
similar to that presented in [16] for an infinitesimal vertical dipole above a very lossy
half-space. To shows how the height of a vertical antenna above a lossy half-space can
affect the radiation pattern, in Fig. 12, we present the radiation patterns corresponding to
different antenna heights. One observes that for an antenna with larger height h above the
lower half-space, its radiation pattern has more lobes. This phenomenon is similar to that
presented and discussed in [16] and [17] for infinitesimal vertical dipoles above a lossy
half-space. The magnitude of the far-zone field in the lower half-space is supposed to be
very small, and to approach to zero, due to an exponentially decaying factor jkRe− in
30
equation (29), representing the lossy property of the lower half-space. In order to show
the angular distribution of the field in the lower half-space, the field pattern depicted for
that region is enlarged by a factor of 1/ jkRe− . It illustrates that as the antenna is moved
away from the media interface (as its height h increases), the magnitude of the field in the
lower half-space is getting smaller.
0.2 0.4 0.6 0.8 1
330°
150°
300°
120°
270° 90°
240°
60°
210°
30°
180°
0°
σ=10σ=100σ=1000PEC
Fig. 11 Radiation patterns of a vertical dipole antenna ( 0.01a λ= , 0.5l λ= ) above a very lossy half-space ( 1.001rε = , 0.26h λ= , 300f MHz= ) of various conductivities, compared with that for the antenna above a PEC plane
31
0.2 0.4 0.6 0.8 1
330°
150°
300°
120°
270° 90°
240°
60°
210°
30°
180°
0°
h=0.26λh=0.35λh=0.75λh=2λ
Fig. 12 Radiation patterns of a vertical dipole antenna ( 0.01a λ= , 0.5l λ= ) at different heights above a lossy half-space with normalized intrinsic impedance 0.3 0.1n jη = + .
32
CHAPTER IV
CONCLUSIONS
In this thesis, an EFIE is formulated and solved numerically for the analysis of a
vertical thin-wire antenna above a lossy half-space. In the EFIE formulation, the
Sommerfeld-type integrals, which are often encountered in electromagnetic modeling
involving media interfaces, are reduced to semi-infinite integrals that converge rapidly,
making use of the exact image theory. The EFIE solutions of the current distribution on
an antenna in free space have been compared with that obtained using a traditional
approach of solving the Pocklington’s equation, and a good agreement is observed. A
comparison between the current distributions on an antenna above a lossy half-space of
various conductivities with that on the antenna above a PEC plane illustrates that as the
conductivity increases, the current distribution data gradually approach to that for the
antenna above a PEC plane. And when the conductivity is taken to be high enough, the
two sets of data match each other, as one would expect. Data of the current distributed on
an antenna above a lossy half-space at different heights show that for an antenna close to
the media interface separating the two half-spaces, the lower half-space can significantly
affect the current distribution on the antenna and its input impedance. But as the antenna
is located farther apart from the lower half-space, its influence would become weaker and
weaker, and eventually negligible if the antenna is placed high enough above the lower
half-space. The radiation patterns of an antenna above a very lossy half-space of various
conductivities show that the magnitude of the field in the upper half space is larger
33
corresponding to a higher conductivity of the lower half space and the radiation patterns
approach to that for the antenna above a PEC plane as the lower half-space conductivity
increases. The radiation patterns for an antenna above a lossy half-space at different
heights illustrate that as the height increases; the field pattern in the upper half-space has
more lobes. All these properties of the radiation patterns presented are similar to that
documented in literature for infinitesimal dipoles above a lossy half-space.
34
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[11] Ismo V. Lindell, Esko Alanen, “Exact Image Theory for Sommerfeld Half-space
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[12] Ismo V. Lindell, Esko Alanen, “Exact Image Theory for Sommerfeld Half-space
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35
[13] R.F. Harrington, Field Computation by Moment Methods, Robert E. Krieger Publishing Company, Malabar, Florida, 1982.
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36