Table Of ContentCOPYRIGHTED
by
RAYMOND HORACE DuHMEL
19E2
UNIVERSITY OF ILLINOIS
THE GRADUATE COLLEGE
September 13, 1951
I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY
SUPERVISION BY Uaynoacl Horace DuHamel
ENTITLED Anl.pnna Pal.l.nrn Synthesis
BE ACCEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR
THE DEGREE OF Doc Lor of Philosophy
Recommendation concurred inf
M k%sfj~ciu~
Committee
OC<A«JL^
on
?. QLr-
Final Examination-]-
(P Vy^; ^XZi~-o-^v^
t Required for doctor's degree but not for master's.
M440
ANTENNA PATTERN SYNTHESIS
BY
RAYMOND HORACE DuHAMEL
B.S., University of Illinois, 1947
M.S., University of Illinois, 1948
THESIS
SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OP PHILOSOPHY IN ELECTRICAL ENGINEERING
IN THE GRADUATE COLLEGE OF THE
UNIVERSITY OF ILLINOIS. 1051
UUUANA. ILLINOIS
iii
TABLE OF CONTENTS
Page
Aokn owl 0dgment v
I. Introduction • 1
II, Linear Arrays 4
2.1 Polynomial Formulation of the Linear Array . . .. 4
2.2 Fourier Series Formulation of the Linear Array . . 17
2.3 Optimum Current Distribution for a Broadside Array 21
2.4 Optimum Current Distribution for an Endfire Array 29
2.5 Synthesis of the Pattern which Gives Maximum
Directivity 32
2.6 Other Methods of Synthesis 40
2.7 Summary of the Synthesis Methods 44
III. Plane Apertures 46
3.1 Applioation of the Fourier Transform to the Plane
Aperture . 46
3.2 A Summation Pattern Synthesis Method . . . . . .. 54
3.3 A Polynomial Formulation 66
3<>4 Summary „ 62
' IV. Circular Arrays 63
4.1 Pattern Synthesis 63
4.2 Optimum Current Distribution 71
4.3 Examples 83
4.4 Maximum Directivity 94
4.6 Approximation to the Continuous Current Sheet . .. 96
4.6 Summary 101
Page
Elliptical Arrays 106
6.1 Coordinate System and Wave Functions 106
5.2 Radiation Patterns of Antenna Elements 109
5.3 Pattern Synthesis 112
5.4 Beam-co-phasal Current Excitation 118
5.5 Summary 120
Arrays on Simple Closed Surfaces 121
6.1 Pattern Synthesis for the Spherical Array . . .. 122
6.2 The Tchebycheff Pencil-Beam Pattern 125
6.3 Summary 127
Conclusion 129
Appendices 131
Bibliography 144
Vita o 148
•
ACKNOWLEDGMENT
The author is indebted to his advisor, Dr. E. C. Jordan, for his
helpful guldanoe and enlightening diaoussions during the course of this work,
Robert Jaokman performed the calculations for Chapter IV,
1
I INTRODUCTION
The radiation pattern of an antenna is a graphical representation of
the radiation of energy from the antenna as a function of direotion. The radi
ation may be expressed in terms of the eleotrio field strength or in terms of
power per unit angle. For these two oases the radiation pattern is termed as
a field strength pattern and a power pattern respectively. The power pattern
is proportional to the square of the field strength pattern. In the following,
only field strength patterns will be considered unless specified otherwise,,
It is a straightforward task to calculate the radiation pattern of
an antenna system if the distribution of the currents on the system is known.
The inverse problem of determining the current distribution required to produce
a specified radiation pattern (commonly referred to as pattern synthesis) is
much more difficult. The primary objective of this researoh has been to for
mulate and develop new methods of pattern synthesis. The secondary objective
has been to oollect the most important material on the synthesis problem, in
tegrate it, and present it in a logioal manner.
The difficulties connected with antenna pattern synthesis may be
illustrated by the following example. Suppose we are given N separate antenna
elements located in a certain region and a prescribed radiation pattern F(qp,6)
(<p and 9 are the spherical coordinates). The actual pattern of the Bystam,
E(<p,fi), is a linear funotion of the N different antenna ourrents. Let the our-
rent in each element be proportional to I . Then if we set E(<p,6) • F(<p,8)
for N different directions we obtain N simultaneous equations from which the K "
coefficients I may be determined. If these currents are then produced on the
antennas, the resulting pattern will approximate F(?,6) in the sense that
E(<j>,0) is equal to F(<p,Q) at N points. (In this simplified example we have
2
overlooked the difficulties introduced by the orientation of the antennas and
the polarization of the radiation.) If N were large it would be quite tedious
to solve the N simultaneous in order to determine the required antenna cur
rents. This labor may be avoided if it is possible to express the radiation
pattern of an antenna system by an orthogonal set of functions, the coeffioiants
of the funotions being simply related to the antenna currents. If this is the
case, then the antenna ourrentB may be solved for in a simple direct manner.
Until recently, this orthogonalization teohnique had been applied
only to an antenna system of the simplest geometry, the linear array. This
teohnique will be applied to other types of antenna systems in the text. It
is also possible to simplify the synthesis problem by other methods of attack,
the most important being included in the text.
A desired radiation pattern is usually specified as a function of <p
for a constant value 6 (spherical coordinate system). In general, only the
magnitude of the pattern is prescribed* the phase of the pattern being arbitrary.
Obviously, the solution is then not unique. Sometimes, only general properties
of the pattern may be demanded. Examples of suoh synthesis problems are that
the pattern should have maximum directivity (gain) or that the side lobe level
should be a minimum for a given beam width. All of these problems will be
f
disoussed in the following pages.
Chapters II and III present, for the most part, the general theory of
pattern synthesis for the linear array and the plane aperture respeotively A
0
baslo understanding of the material in these chapters is. quite helpful to the -
reader since much of the theory may be applied to other types of antenna sys
tems. The synthesis methods for a circular and an elliptical array are
It is quite important to realise that the methods of pattern synthesis for an
antenna system aiay also be applied to an aooustioal radiating system and vice
versa.
3
developed in chapters IV and V* Finally, a general technique of pattern syn
thesis for an array of radiators placed on a closed surface is discussed in
chapter VI. The original work of the author is represented by chapters IV,
V, VI and a portion of chapter IIo
4
II LINEAR ARRAYS
Linear arrays are used quite frequently for both radio and acous
tical applications. They may be used to produce either a narrow beam
pattern or a prescribed radiation pattern. An attempt will be made to pre
sent the more important aspects of pattern synthesis for linear arrays.
Since linear arrays have been covered extensively in the literature most of
the material in this chapter has been drawn from other sources.
A linear array is a series of radiators, with points, similarily
situated on the radiators, colinear, for which the individual radiatorB are
geometrically identical, similarily oriented, and energised at similarily
situated points. The radiators differ only in the relative magnitudes and
phases of the current excitations. These properties insure that the radia
tion patterns of the radiators are not only the same but are also similarily
oriented. It follows then that the radiation pattern of the array is the
product of the element pattern and the "space factor." The space factor
of an array 1B defined as radiation pattern of a similar array of isotropic
or non-directive radiating elements.
Only equispaced linear arrays will be considered in the following.
First we will study the polynomial formulation of the linear array in which
the space factor is represented by a oomplex polynomial. Many useful charac
teristics of an array may be derived by using this technique. Next we will
consider a direct method of pattern synthesis in which the space factor is
written as a Fourier series. Then this synthesis method will be used to
obtain the optimum ourrent distribution for both a broadside and an endfire
array. Finally a technique for designing a linear array so that the beam
will have maximum directivity will ba disoussed.
(47)
2* •"•• Poly11 omi&l Formulation of the Linear Array. —Schelkunoff
