Clemson UniversityTigerPrints

All Theses Theses

5-2007

AN EFFICIENT ANALYSIS OF VERTICALDIPOLE ANTENNAS ABOVE A LOSSY HALF-SPACEYongfeng HuangClemson University, yongfeh@clemson.edu

Follow this and additional works at: https://tigerprints.clemson.edu/all_theses

Part of the Electrical and Computer Engineering Commons

This Thesis is brought to you for free and open access by the Theses at TigerPrints. It has been accepted for inclusion in All Theses by an authorizedadministrator of TigerPrints. For more information, please contact kokeefe@clemson.edu.

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

REFERENCES

[1] R.S. Elliott, Antenna Theory and Design, Prentice-Hall Inc., Englewood Cliffs,

New Jersey 07632, 1981. [2] H.C. Pocklington, “Electrical Oscillations in Wire,” Cambridge Phil. Soc. Proc.,

Vol. 9, 1897, pp. 324-332. [3] E. Hallen, “Theoretical Investigations into transmitting and receiving qualities of

antennas,” Nova Acta Regiae Soc. Sci. Upsaliensis, January 1938, pp. 1-44. [4] C.M. Butler, “Integral Equation Method,” ECE 839 Lecture Notes, Clemson

University, Clemson, SC 29634-0915. [5] J.R. Wait, Electromagnetic Waves in Stratified Media, New York: Pergamon,

1970. [6] R.M. Shubair and Y.L. Chow, “A closed-form solution of vertical dipole antennas

above a dielectric half-space,” IEEE Transactions on Antennas and Propagation, Vol. 41, No. 12, pp. 1737 – 1741, December 1993.

[7] A. Sommerfeld, Partial Differential Equations, New York: Academic Press, 1949. [8] L.B. Felson and N. Marcuvitz, Radiation and Scattering of Waves, Prentice Hall

Inc., Englewood Cliffs, New Jersey, 1973. [9] K. Sarabandi, M.D. Casciato, and I-S. Koh, “Efficient Calculation of the fields of

a dipole radiating above an impedance surface,” IEEE Transactions on Antennas and Propagation, Vol. 50, No. 9, pp. 1222 – 1235, September 2002.

[10] Ismo V. Lindell, Esko Alanen, “Exact Image Theory for Sommerfeld Half-space

Problems, Part I: Vertical Magnetic Dipole,” IEEE Transactions on Antennas and Propagation, Vol. AP-32, No. 2, pp. 126-133February 1984.

[11] Ismo V. Lindell, Esko Alanen, “Exact Image Theory for Sommerfeld Half-space

Problems, Part II: Vertical Electric Dipole,” IEEE Transactions on Antennas and Propagation, Vol. AP-32, No. 10, pp. 1027-1032 October 1984.

[12] Ismo V. Lindell, Esko Alanen, “Exact Image Theory for Sommerfeld Half-space

Problems, Part III: Vertical Electric Dipole,” IEEE Transactions on Antennas and Propagation, Vol. AP-32, No. 8, pp. 841-847 August 1984.

35

[13] R.F. Harrington, Field Computation by Moment Methods, Robert E. Krieger Publishing Company, Malabar, Florida, 1982.

[14] R.F. Harrington, Time-Harmonic Electromagnetic Fields, IEEE Press, John Wiley

and Sons, Inc., 2001. [15] R.W.P. King, The Theory of Line Antennas, Cambridge, Mass: Harvard

University, 1956. [16] Warren L. Stutzman, Antenna Theory and Design, John Wiley and Sons, Inc.,

1981. [17] Constantine A. Balanis, Antenna Theory Analysis and Design, John Willy &

Sons, Inc., 1997.

36