Spherical antenna array

166 Pages·2016·5.01 MB·English

by  Narong Yoothanom

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Doctoral Dissertations Student Theses and Dissertations
1969
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Narong Yoothanom
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Yoothanom, Narong, "Spherical antenna array" (1969). Doctoral Dissertations. 2271.
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SPHERICAL ANTENNA ARRAY
by S<' "
j 9
NARONG YOOTHANOM / L_/. .2J
A DISSERTATION
Presented to the Faculty of the Graduate School of the
UNIVERSITY OF MISSOURI -- ROLLA
In Partial Fulfillment of the Requirement for the Degree of
DOCTOR OF PHILOSOPHY
in
ELECTRICAL ENGINEERING
Rolla, Missouri
1969
ii
ABSTRACT
The far field radiation pattern of an isotropic
spherical antenna array with two types of element
distributions, a latitude-longitude distribution and an
approximately equally-spaced distribution, has been
analyzed using Poisson's Sum Formula. Beamwidth expres
sions of both element distributions are found in various
planes. All numerical results are given only for the case
of 21 rings.
iii
ACKNOWLEDGEMENTS
The author would like to acknowledge with sincere
respect the guidance and invaluable comments and discussions
provided for him by Professor Gabriel G. Skitek throughout
his graduate study at the University of Missouri - Rolla.
He would also like to acknowledge both the
Anandhamahidol Foundation under the patronage of the King
of Thailand and the University of Missouri - Rolla for the
financial support throughout his graduate work.
Much of the appreciation goes to the Department of
Computer Science for granting a project number and the
extensive use of the IBM/360 computer in the department.
Thanks go to Muriel Johnson for typing the manuscript.
iv
TABLE OF CONTENTS
Page
ABSTRACT ii
ACKNOWLEDGEMENTS iii
LIST OF SYMBOLS vii
LIST OF FIGURES
X
LIST OF TABLES
xiii
CHAPTER I. INTRODUCTION TO SPHERICAL ANTENNA
ARRAY PROBLEM 1
A. Introduction 1
B. Definition of a Spherical Antenna
Array 2
c. Review of Literature 4
D. Scope of This Dissertation 5
CHAPTER II. ELEMENT DISTRIBUTION 8
A. The Latitude-Longitude Distribution 8
1. Spacing 8
2. Element Density 13
B. APPROXIMATELY EQUALLY-SPACED ARRAY 14
1. Element Distribution 14
CHAPTER III. FAR FIELD PATTERN 25
A. Formulation of the Far Field Pattern 25
B. Simplification of the Far Field
Pattern 31
v
CHAPTER IV. ARRAY ANALYSIS 34
A. The Application of the Poisson's
Sum Formula 34
B. The Latitude-Longitude Distribution
Array 42
1. Pattern Series and Its
Coefficients 45
2. The Coefficients of the Pattern
Harmonics 53
C. Beamwidth 55
1. Beamwidth in the Elevation Angle
Plane 56
2. Beamwidth in the Plane of
Azimuth Angle 61
D. The Approximately Equally-Spaced
Spherical Array 63
1. Pattern Series and Its
Coefficients 65
2. Beamwidth 69
a. Beamwidth in the Elevation
Angle Plane 69
b. Beamwidth in Azimuth Angle
Plane 71
CHAPTER V. NUMERICAL RESULTS AND DISCUSSIONS 73
A. Method of Computation 73
B. The Pattern Harmonics 77
vi
C. Far Field Pattern Analysis 77
D. Beamwidth 81
E. The Scanning Property of the
Spherical Array 100
CHAPTER VI. CONCLUSIONS AND RECOMMENDATIONS 107
A. Conclusions 108
B. Recomrnendat~ons 110
BIBLIOGRAPHY 113
APPENDIX A EQUALLY-SPACED SPHERICAL ARRAYS 116
APPENDIX B EVALUATION OF THE ARGUMENTS IN
EQUATION (4-45) 122
APPENDIX C EVALUATION OF THE COEFFICIENT A (MH) 124
m
APPENDIX D THE CLOSED FORM OF THE PATTERN
COEFFICIENT c 126
1
APPENDIX E POLARIZABILITY OF A SPHERICAL ANTENNA
ARRAY 128
APPENDIX F COHPUTER PROGRAMS 133
VITA 151
vii
LIST OF SYMBOLS
a radius of the sphere
th "h f h . .
A s welg t o t e s -polnt Gausslan Quadrature
s max
integration formula
unit vectors in Cartesian coordinates
axial ratio of an elliptical polarization ellipse
BW beamwidth of the LL distribution
BWA bearnwidth of the AES distribution
sth pattern harmonic coefficient
ring spacing
th
antenna spacing in the rn ring
dm
d equator spacing
eq
dx differential of x
unit vectors in spherical coordinates
electric far field pattern
EA electric far field pattern of the AES distribution
f(n) nth term of n-summation
f(8m) resultant of n-surnrnation
g(e,cp,m} mth term of m-surnrnation
sth pattern harmonic
sth pattern harmonic of the AES distribution
mnth antenna current amplitude
j
Jrn(x) mth order Bessel function of argument x
k wave propagation constant
M total number of rings less one
viii
unit vector in the direction of the moth antenna
N total number of antennas in each ring in the
LL distribution
number of antennas in the equator
number of antennas in the mth ring
the integer nearest to Nm
mth order Le gend re po 1 ynom1· a 1 o f argumen t x
r radius of the mth ring, measured from the polar
m
,
axis
R unit vector in the direction of the observation
point
R distance from the observation point to the origin
of the spherical coordinates
s distance from the nth element to the reference
n
point in a linear array
zeros of the Legendre polynomial of order smax
x,y,z Cartesian coordinates
3.14159
1T
e,cp,r spherical coordinates
elevation angle of the roth ring
elevation angle of the rnainbearn
elevation angle of the beamwidth
azimuth angle of the n~h antenna
azimuth angle of the rnainbeam
azimuth angle of the bearnwidth
wavelength
element density
ix
arbitrary constant
0 percentage spacing deviation
th h .
mn antenna current p aslng
~mn
angle subtended by the observation point and
~mn
the mnth antenna at the reference point
E positive number less than unity