Analysis and Synthesis of Wire Antennas
ELECTRONIC & ELECTRICAL ENGINEERING RESEARCH STUDIES ANTENNAS SERIES Series Editor: Professor J. R. James The Royal Military College of Science, Shrivenham, Wiltshire, England 1.
Flat Radiating Dipoles and Applications to Arrays G. Dubost
2.
Analysis and Synthesis of Wire Antennas B. D. Popovic, M. B. Dragovic and A. R. Djordjevic
Analysis and Synthesis of Wire Antennas B. D. Popovic, M. B. Dragovic and A. R. Djordjevic Department of Electrical Engineering, University of Belgrade, Yugoslavia
RESEARCH STUDIES PRESS A DIVISION OF JOHN WILEY & SONS LTD Chichester· New York· Brisbane· Toronto· Singapore
RESEARCH STUDIES PRESS Editorial Office: 588 Station Road, Letchworth, Herts. SG6 3BE, England.
Copyright© 1982, by John Wiley & Sons Ltd. All rights reserved. No part of this book may be reproduced by any means, nor transmitted, nor translated into a machine language without the written permission of the publisher. Library of Congress Cataloging in Publication Data: Popovic, Branko D. Analysis and synthesis of wire-antennas. (Research studies on antennas series; 2) Includes index. 1. Antennas (Electronics) I. Dragovic, M. B. II. Djordjevic, A. R. IV. Series TK7871.6.P68 1982 621.38'.0283 ISBN 0 471 90008 7
Ill. Title 82-11078
British Library Cataloguing in Publication Data: Popvic, B. D. Analysis and synthesis of wire~antennas. -(Research studies on antennas series; 2) 1. Antennas (Electronics) I. Title II. Dragovic, M. B. Ill. Djordjevic A. R. IV. Series 621.3841'1 TK656.A6 ISBN 0 471 90008 7 Printed in Great Britain
Editorial Preface
Wire
dipole
and arrays
have
nearly a century yet
a gulf
existed up until recently between the
theory and practice.
This came about because
only be
antennas
applied
to
has
been
extensively
the
the idealised geometries of dipoles
The
advent of computers promised
since, in principle, late
the
numerical methods
any arbitrarily shaped wire
achieved
because
computational
for
exact theories could
while engineering applications embraced a multitude configurations.
analysed
and monopoles
of radiating wire
to
bridge this gulf
enable engineers to calcu-
radiator.
This
techniques
can
has
not always been
introduce
additional
problems which obscure the value of computers as a design tool for engineers.
In contrast, this present monograph presents
an exhaustive computational treatment of sense
and
clearly demonstrates
the
wire
the
reader with
antennas in their widest
excellent results that
can be ob-
tained both by the numerical analysis and the synthesis of such radiating structures.
A particularly
interesting feature is
the
use of al-
most-entire domain polynomial representations of current instead of various sub domain from
the
basis
function representations
gion behaviour of
elsewhere.
Apart
advantages of computational economy it does question the need
for more complicated methods.
some
used
the
and
the careful construction of
many notable
physical appreciation Popovic and
his
The attention paid to the excitation re-
the
aspects
of
authors have
colleagues have
a
the for
high
practical antennas are
book that exhibit the sound their research.
international
Professor
reputation for
their contributions to engineering electromagnetics and this
book
is a
vi culmination of many years lays bare
the
essential
of
research.
de tails
in an
The
text
characteristically
economic yet lucid manner and
will appeal to postgraduates, research scientists
and engineers alike,
establishing beyond doubt that wire antennas can be designed by computer with confidence.
April 1982
J. R. JAMES
Preface
Thin-wire the
antennas,
or
similar
antenna
structures,
were
only antennas used for radio-communication purposes
essentially
from
the dis-
covery of electromagnetic radiation by H. Hertz in 1887 until about the mid-thirties. possible sizes.
the
At
that
design
time,
of
utilization
other antenna
of
higher
types
frequencies
made
of practically acceptable
However, wire antennas have remained in a wide use until today.
Analysis of wire antennas was first based on a sinusoidal approximation
of
current distribution
known to predict it is
fairly
along
the
wires.
This approximation is
accurately the antenna radiation pattern,
but
usually quite insufficient for accurate determination of the an-
tenna impedance.
Although an integral equation for the current distri-
bution along
cylindrical wire dipoles was derived by H. E. Pocklington 1 as early as in 1897, a more accurate current distribution than sinuso2 idal along such dipoles was first obtained by E. Hallen in 1937, who calculated a tion
for
his name. antennas his
few
terms of a series solution
current distribution along R. W. P. King following,
added
to
another integral equa-
cylindrical antennas, which bears
considerably
to
our knowledge of wire
largely, Hallen's basic approach, culminating in 3 in 1956. About a decade later, wide usage
classical monograph
of high-speed digital computers changed radically antenna analysis. aided 1
design
The numbers graph.
the methods of wire-
In addition, it opened the door for recent computer-
(synthesis)
of
such antennas
by means
refer to the List of References
at
of
optimization
the end of the mono-
viii methods.
At
the
present time, with adequate precautions and clear in-
sight into the physical and numerical aspects of the problem, computeraided analysis and synthesis of wire-antenna structures of electrically moderate sizes can be so accurate that experimental verification of the results
thus
obtained
can
tom than of necessity. analysis
and
due, at
least
techniques
Nevertheless, these powerful modern methods for
synthesis
widely accepted
and
almost be regarded more as a matter of cus-
of wire-antenna
recognized
partly, to
underlying
the
the
by
antenna
do not
seem to be
design engineers.
This is
fact that the modern ideas and numerical
analysis and,
wire-antenna structures are for periodical literature.
structures
in particular,
synthesis of
the most part still to be found in the
The present monograph, in which
certain modern
methods for wire-antenna analysis and synthesis are presented concisely and with
the
needs
of
design
engineers
and
university educators
in
mind, is intended to fill this gap to some extent. Essentially,
the monograph
long research activity
represents
to
methods
develop accurate, for
analysis
reached,
because
the theoretical
but
and,
wire-antenna structures. ly
The principal
in
objective
conceptually
in
the
practically found
the
structures.
no
final
and
computationally simple
stage,
synthesis
of
general
all to
cases which were be
considered
in excellent agreement with
the limits
of
experimental error.
of this monograph is to present, in an orderly
the main
Belgrade concerning Almost
over-a-decade
It could be said that the aim has been large-
results were
compact manner,
of
The aim adopted in the very beginning
experimental results, almost within
and
summary
on wire-antenna analysis and design at the Uni-
versity of Belgrade, Yugoslavia. was
a
results
obtained at
the
University of
analysis and synthesis of diverse wire-antenna attempt
and evaluate various methods
was
made
to
present, discuss, compare
for solving the wire-antenna problem pro-
posed by other authors; that would have been a task of exceptional complexity.
However, considerable
care was
exercised
to make
the mono-
graph as self-contained and complete as possible. Although the
some
aspects
of
wire-antenna structures are not treated in
monograph explicitly (e.g.,
general wire-antenna arrays, antennas
ix made
of
circular wires with abrupt change in diameter, or of non-cir-
cular wires,
etc.) ,
most
of
them
can
be
analysed and/ or synthesized
using the simple and accurate theory presented in tively
little
oretical
additional effort.
and experimental
On
result"s
the
design
book with
rela-
other hand, many useful the-
(most often, coupled to each other)
are presented throughout the monograph, as for
the
well
as some practical data
engineers, e.g. , accurate graphs of conductance and suscep-
tance of vertical monopole antennas above conducting ground plane driven
by
coaxial lines of various sizes (Appendix 5) and of una t tenua ted
electric-field intensity of
such antennas versus radiated power, their
thickness and height (Appendix 6).
It is believed, therefore, that the
book might be of equal value to university professors, design engineers and graduate students interested in wire-antenna structures. The monograph is divided into two parts: antenna analysis and antenna synthesis.
The
first
part
is
devoted
to
the numerical determination
of current distribution along various unloaded and loaded wire antennas in a vacuum or
in homogeneous and
general, lossy),
and
media
(in
to the analysis of excitation regions and of wire
junctions and ends. modern
inhomogeneous dielectric
The
computer-aided
second part
design
constitutes
of wire-antenna
an
introduction
to
structures by means
of
optimization methods. Although much of the material, as presented, has not been published, a
substantial
larger years with
or
part of
the
monograph
in various journals this,
the authors
and
wish
in
Institution the
material
Electronic
published
Springer-Verlag
for
in
adapting, authors In
to
a
over the
connection
to express their sincere gratitude to the
the Proceedings IEE of
the
conference proceedings.
Institution of Electrical Engineers published
was written by
smaller extent, articles published by
for
and
and Radio
permission to use the material
in
Electronics Letters,
Engineers
for
permission
The Radio and Electronic Engineer,
permission
to
use
the material
published
to
the
to use and
to
in the
Archiv fur Elektrotechnik. During authors
the had
years a
of work which made
permanent
support
from
this monograph possible, the
Department
the
of Electrical
X
Engineering computer,
of
the
Belgrade
laboratory,
University,
workshop
and
in
the
form of
other facilities.
free use of
A part of
the
program was
also supported by the Serbian Academy of Sciences and Arts
and by
Serbian Research Foundation.
the
the Department participated in tributing greatly In
this
respect
Paunovic,
graph.
a
cheerful
authors
an active
member
several
problems,
in solving permanent
to the
interest
The
in
larger
monograph were
the
the
part
and
stimulating working
particularly
the
and
indebted
atmosphere.
to Dr Dj.
S.
antenna group, for his cooperation
to Professor A. S. Marincic, for his
project
of
faculty members at
project in one way or another, con-
are of
Several
and
in
the
the experimental
obtained by patient
and
progress of this mono-
results
presented in the
reliable work of a number of
the authors' students, and most of the antenna models and special parts of the measuring equipment were expertly made by the staff of the workshop of the Department. The
monograph was written
(B.D.P.)
during
as Visiting Professor
the
stay of
Institute and atmosphere at
creative
Virginia.
VPI & SU and,
particular,
the understanding
Department
of
the authors
The
Blacksburg,
in
of
at Virginia Polytechnic
State University,
Hodge, Head,
one
of
Professors D. B.
H. H. Hull, Assistant Head and I. M. Besieris, all of the Electrical Engineering,
complicated process of writing a
were
of substantial help in the
book with co-authors on the two sides
of the Atlantic. The
authors would
Studies
on Antennas,
also
like
to
thank
the
Professor J. R. James,
Editor for
of
the
his initiative which
resulted in this monograph.
Blacksburg, Virginia, U.S.A., Belgrade, Yugoslavia, April 1982
Research
B. D. P.
M. B. D. A. R. Dj.
Table of Contents
PART I: 1.
ANALYSIS OF WIRE-ANTENNA STRUCTURES
DETERMINATION OF CURRENT DISTRIBUTION IN ARBITRARILY EXCITED WIRE STRUCTURES 1.1.
INTRODUCTION, 3
1.2.
TWO-POTENTIAL EQUATION FOR CURRENT DISTRIBUTION IN ARBITRARY THIN-~VIRE STRUCTURES, 5 1.2.1.
1.3.
SOME EQUATIONS FOR DETERMINING CURRENT DISTRIBUTION IN CYLINDRICAL CONDUCTORS, 13
..
-
.._,
1.3.1~
'---~
-:L~·2)
1.4. 2.
Approximate solution of the two-potential equation, 10
The two-potential and vector-potential equations, 14 Hallen's equation, 16
1.3.3.
Pocklington's equation, 18
1.3.4.
Schelkunoff's equation, 18
CONCLUSIONS, 20
APPROXIMATIONS OF EXCITATION REGIONS 2.1.
INTRODUCTION, 23
2.2.
DELTA-FUNCTION GENERATOR, 25
>~
2.3.
2.2.1.
Solution of Hallen's equation with delta-function generator, 29
APPROXIMATIONS OF COAXIAL-LINE EXCITATION, 34
xii
2. 4.
2. 3 .1.
TEH magnetic-current frill approximation of coaxial-line excitation, 35
2.3.2.
Belt-generator approximation of coaxial-line excitation, 44
2.3.3.
Higher-order approximations of coaxial-line excitation by means of wave modes, 48
APPROXIHATIONS OF TWO-WIRE LINE EXCITATION, 56 2.4.1.
2.5. 3.
CONCLUSIONS, 66
TREATHENT OF WIRE JUNCTIONS AND ENDS 3.1.
INTRODUCTION, 69
3.2.
CONSTRAINTS RESULTING FROH FIRST KIRCHHOFF'S LAW, 70
3.3.
JUNCTION-FIELD CONSTRAINTS, 73
'3.4. 3.5. 4.
A method for measuring admittance of symmetrical antennas by reflection measurements in coaxial line, 63
TREATHENT OF WIRE ENDS, 79 CONCLUSIONS, 90
WIRE ANTENNAS WITH DISTRIBUTED LOADINGS 4.1.
INTRODUCTION, 91
4.2.
EQUATIONS FOR CURRENT DISTRIBUTION ALONG ANTENNAS WITH SERIES DISTRIBUTED LOADINGS, 93
~~4.;~.~Examples
of analysis of antennas with series loadings, 96
~distributed
5.
4.3.
WIRE ANTENNAS WITH DIELECTRIC OR FERRITE COATING, 100
4.4.
CONCLUSIONS, 108
WIRE ANTENNAS WITH CONCENTRATED LOADINGS 5.1.
INTRODUCTION, 109
5.2.
HODIFICATION OF EQUATIONS FOR CURRENT DISTRIBUTION, 110
;:x.S~
Examples of cylindrical antennas with concentrated resistive loadings, 114
xiii
·--·.. 5.3.
0
-·-~
5.5. 6.
/
Examples of cylindrical antennas with concentrated capacitive loadings, 118
NOTES ON MEASUREMENTS OF CONCENTRATED LOADINGS, 129 5.3.1.
Compensation method for measuring lumped reactances, 129
5.3.2.
Measurement of lumped reactances mounted on the antenna by means of a coaxial resonator, 132
WIRE ANTENNAS WITH MIXED LOADINGS, 139 CONCLUSIONS, 144
WIRE ANTENNAS IN LOSSY AND INHOMOGENEOUS MEDIA 6.1.
INTRODUCTION, 145
6.2.
WIRE ANTENNAS IN HOMOGENEOUS LOSSY MEDIA, 146
6.3.
DETERMINATION OF CURRENT DISTRIBUTION ALONG WIRE ANTENNAS ON PLANE INTERFACE BETWEEN TWO HOMOGENEOUS MEDIA, 150
6.4.
WIRE ANTENNAS ABOVE IMPERFECTLY CONDUCTING GROUND, 154
6.5.
CONCLUSIONS, 164
PART II: 7.
5.2-~.
SYNTHESIS OF WIRE-ANTENNA STRUCTURES
GENERAL CONSIDERATIONS OF WIRE-ANTENNA SYNTHESIS 7.1.
INTRODUCTION, 167
7.2.
GENERAL PRINCIPLES OF WIRE-ANTENNA SYNTHESIS, 169
7.3.
7.4.
7.2.1.
Possible optimization functions, 170
7.2.2.
Possible optimization parameters, 172
OUTLINE OF SOME OPTIMIZATION METHODS, 173 7.3.1.
Complete search method, 175
7.3.2.
A gradient method, 175
7.3.3.
The simplex method, 177
CONCLUSIONS, 178
xiv
8.
OPTIMIZATION OF ANTENNA ADMITTANCE 8.1.
INTRODUCTION, 179
8.2.
OPTIMIZATION OF ANTENNA ADMITTANCE BY VARYING DISTRIBUTED ANTENNA LOADINGS, 183 8.2.1.
Some general examples of optimization, 186
8.2.2.
Some remarks on loaded cylindrical antenna optimization, 187
. ---~9 8.3.
SYNTHESIS OF PARALLEL LOADED CYLINDRICAL ANTENNAS WITH MINIMAL COUPLING, 192 8. 3.1.
·-:::-:::.:.-:(9 8.4.
Numerical examples, 188
Outline of the method, 194 Resistive cylindrical antennas with minimal coupling, 197
OPTIMIZATION OF ANTENNA ADMITTANCE BY VARYING CONCENTRATED Optimal broadband capacitively loaded cylindrical antennas, 206 8.4.2.
Limits of VSWR for optimal broadband capacitively loaded cylindrical antennas versus their length, 212
OPTIMIZATION OF ADMITTANCE BY VARYING DISTRIBUTED AND CONCENTRATED LOADINGS, 214 8.6.
8.7. 9.
OPTIMIZATION OF ADMITTANCE BY MODIFICATION OF ANTENNA SHAPE, 220 8.6.1.
Synthesis of broadband folded monopole antenna, 223
8.6.2.
Synthesis of broadband monopole antenna with parasitic elements, 224
8.6.3.
Synthesis of cactus-like antenna matched to feeder at two frequencies, 228
8.6.4.
Synthesis of vertical monopole antenna with susceptance-compensating element, 230
CONCLUSIONS, 232
OPTIMIZATION OF ANTENNA RADIATION PATTERN 9.1.
INTRODUCTION, 235
9.2.
OPTIMIZATION OF RADIATION PATTERN BY VARYING DRIVING VOLTAGES OF ANTENNA-ARRAY ELEMENTS, 239
XV
9. 3.
OPTU1IZATION OF RADIATION PATTERN BY VARYING ANTENNA LOADINGS, 241
9.4.
OPTIMIZATION OF RADIATION PATTERN BY VARYING ANTENNA SHAPE, 243
9.5.
9.4.1.
Synthesis of Uda-Yagi array with one and two directors and two reflectors, 244
9.4.2.
Synthesis of inclined monopole antenna, 248
9.4.3.
Synthesis of Uda-Yagi array with folded monopole as a driven element, 249
9.4.4.
Synthesis of moderately broadband Uda-Yagi array, 252
CONCLUSIONS, 254
APPENDIX 1.
NOTES ON EVALUATION OF INTEGRALS ENCOUNTERED IN ANALYSIS OF WIRE STRUCTURES ASSEMBLED FROM STRAIGHT WIRE SEGMENTS, 257
APPENDIX 2.
NOTES ON HALLEN'S EQUATION, 261
APPENDIX 3.
EVALUATION OF THIN-WIRE ANTENNA RADIATION PATTERN AND INDUCED ELECTROMOTIVE FORCE, 263
APPENDIX 4.
A3.1.
Evaluation of radiation pattern, 263
A3.2.
Evaluation of electromotive force induced in a receiving wire antenna, 266
NOTES ON TEM MAGNETIC-CURRENT FRILL APPROXIMATION OF COAXIAL-LINE EXCITATION, 269 A4.1.
Near-zone field of TEM magnetic-current frill, 269
A4.2.
Radiation field of TEM magnetic-current frill, 271
A4.3.
Determination of antenna admittance from power generated by magnetic-current frill, 272
A4.4.
Antenna admittance correction when boundary conditions are satisfied inadequately, 275
APPENDIX 5.
ADMITTANCE OF MONOPOLE ANTENNAS DRIVEN BY COAXIAL LINE, 279
APPENDIX 6.
FIELD INTENSITY VERSUS RADIATED POWER, HEIGHT AND THICKNESS OF A VERTICAL MONOPOLE ANTENNA ABOVE PERFECTLY CONDUCTING GROUND PLANE, 285
APPENDIX 7.
SIMPLEX OPTIMIZATION PROCEDURE, 287
REFERENCES, 291 INDEX, 301
PART I
Analysis of Wire-Antenna Structures
CHAPTER 1
Determination of Current Distributioni"Arbitrarily Excited Wire Structures
1.1.
INTRODUCTION
This book deals with analysis
and
synthesis of wire-antenna structures
assembled
from arbitrarily interconnected wire segments.
"wire" we
shall
refer
to resistive wire-like structures (e.g., a dielectric a
resistive
i.e., wires of a
layer). the
We
shall
of
the
rod
the
term
covered with
consider only electrically
diameter of which
plane wave
By
to metallic, highly conducting wires, but also
thin wires,
is much smaller than the wavelength
frequency considered propagating in
the
sur-
rounding medium. A wire structure can be ments
in many ways, and
curved.
the
Regions in which
referred to
as
junctions.
along the wires and which
constructed from a given number of wire seg-
the
tively as
segments may in principle be
two
or more segments
are
straight or
connected will be
Junctions, wire ends, concentrated loadings
possible transition regions
along
the segments in
diameter of the wire is changed will be referred "discontinuities".
discontinuities.
In this chapter we
shall not
to
collec-
deal with
They will be treated in detail in the third and fifth
chapters. A wire
structure
the electric
field
can be of
a
excited in many ways. wave
propagating
in
the
If it is excited by surrounding medium
(e.g., the field of an incident plane wave), it behaves as a scatterer. If it is excited at
one or more electrically small regions, it behaves
as a transmitting antenna.
The
term "excitation region"
will
be used
4
Ch.l.
to
designate
small regions
impressed field.
of
Determination of current distribution the antenna structure with
Excitation regions will
any kind of
be treated in more detail in
the next chapter. The definition of differs field"
somewhat
impressed field as
from
the
will be used in this monograph
usual definition.
By
the
term "impressed
shall understand the field due to any type of known sources.
we
For example, it may
be the field
of
an incoming plane wave, the field
due to kno•vn localised electric or magnetic currents, etc. If
a
wire
excitation a
structure
field
scatterer
it
and a
is
used
can be
for
receiving
treated as
a
transmitting antenna.
that virtually all properties of interest impedance and
antenna
combination of the cases of It of
is
well-known, however,
a receiving antenna (emf,
radiation pattern) are known if the antenna transmitting
properties are known. separately.
purposes, regarding the
We shall not therefore
Actually,
st rue tures.
treat
receiving antennas
our main concern will be the transmitting wireSome details of
the theory of receiving antennas
are presented in Appendix 3. All the antenna properties mining has
current
been
There is
As
a
the wire segments of
this monograph is
not
particular method,
and
general wire
to
present
which
structures
adopted
potential equation,
or
for
the structure
number of possibilities how
already mentioned
one
starting point
be deduced once the problem of deter-
distribution along
solved.
that problem.
can
in
and
discuss
appears
to
analysed analysis
briefly
the
the Preface,
the
approach
purpose of
of them.
Instead,
be most suitable, is chosen,
using
is
all
to
the
that
method only.
The
so-called vector-scalar-
two-potential equation, which will
be derived in the next section. When
considering cylindrical wire
cylindrical antennas, are in Since
some
respects more
isolated
frequent
several
integral
are
parallel
available which
coupled cylindrical dipoles
brief survey of
along such antennas will
equations
of
convenient than the two-potential equation.
and parallel
occurrence, a
antennas, or arrays
are
of very
integral equations for current
be presented later in this chapter and
numerical methods for their solution described.
some
Sect.l.2. 1. 2.
Two-potential equation
5
TWO-POTENTIAL EQUATION FOR CURRENT DISTRIBUTION IN ARBITRARY THIN-WIRE STRUCTURES
Let
us
consider an arbitrary wire structure sketched in Fig.l.l, situ-
ated in a linear, isotropic mittivity uum or
£
and
and permeability \1.
homogeneous dielectric medium, of per(Most
often
the medium will be a vac-
air, of parameters equal or very close to
E 0
and \.1
0
For the
• '")
moment we shall assume that all the segments of the structure have constant radius, that they are
~ade
of perfectly conducting wires and that
no concentrated loadings are connected along the segments.
FIG.l.l. Arbitrary wire str~cture in impressed field Ei. (A) wire junction, (B) wire end, (C) antenna terminals.
Let the structure be lectric field
of
situated in a ->-
intensity E. and of
->-
structure, which are
£.
As a reac-
l
tion to Ei, currents and
sity
given time-harmonic impressed eangular frequency w.
charges are induced along the segrnen ts of the
sources of the secondary electric field of inten-
These induced currents and
tial component
of
charges
are
such that the tangen-
the total electric-field intensity vector is zero at
all points of the (perfectly conducting) wire-structure surface:
CE +E.) l tang As
E
can be
structure in
=
o
(1.1)
on the wire surface.
expressed in terms of currents and charges induced in the the
form of
rents and charges are
certain integrals, given below, and
interconnected through
the
as
cur-
continuity equation,
* If not stated otherwise, in all numerical examples presented in the monograph £ and \.1 will be used for permittivity and permeability of 0 0 the medium.
6
Ch.l.
eqn. (l.l) essentially represents
Determination of current distribution an
integral equation for current dis-
tribution along the wire structure. Let us
assume that a curvilinear s-axis, described by a vector func-
tion -;: (s) with respect to a convenient coordinate system, runs along s axis of a perfectly conducting wire segment, of radius a, as shown
the
in Fig.l. 2. tween s
Let
and
1
s
2
the
radius
of
curvature of the s-axis everywhere be-
be much larger than a.
The currents
and
charges in-
FIG.l.2. Curved currentcarrying wire segment and the field point P. Not drawn to scale.
/
s=O
duced
in
segment
this wire segment form an surface
speaking,
a
s.
surface-current density,
component locally parallel
ferential component. usually very problem of
The
infinitesimally thin layer on
small,
to
has,
the
generally
the wire axis and a circum-
In the case of thin wires the latter component is except
at
the
antenna
discontinuities.
As the
discontinuities will be treated in later chapters, we shall
neglect here
the circumferential component.
Also,
away
from the dis-
continuities currents and charges are distributed practically uniformly around the circumference C of the segment local cross-section, for any 4 . ->I(s)-r Q'(s) s. Thus we have approx1mately J =-- - 1 (s) and p = - - - along C, s 2 1ra s s 21fa where I(s) is the segment current intensity, 1 (s) the unit vector lo-
s
cally tangential to the s-axis, ps the surface-charge density and Q'(s) the
segment
charge
per
unit length.
there is no field inside it.
Since the
conductor is perfect,
We can, therefore, imagine that the inte-
Sect.l.2. rior
of
Two-potential equation
7
the segment is filled with any medium.
is a medium with parameters E and usual expressions
for
If we imagine that it
~.
the medium is homogenized, and the 5 the retarded potentials can be applied. Thus,
the electric field ~ due to this segment, at a point P, can be computed as -+
-+
E
(1. 2)
-jwA- grad V,
where (1. 3)
g(r ) dS e
is the magnetic vector-potential,
v
( 1. 4)
is the electric scalar-potential,*
(1. 5)
is Green's function for unbounded homogeneous medium, -+ r
(~ - ~ )
is
s
p
e the
field
- ~c
-+ r- r
-+
(1. 6)
c
exact distance between point
P,
the
segment surface element
dS
~ is the distance between P and the point P'
and the at the
s-axis, and k is
=
(l)~
the
( 1. 7)
phase coefficient.
The integration around
the
segment circum-
ference yields
-+
A
I(s)
1s (s)
G(s) ds
( 1. 8)
* We assume that there are no concentrated charges at the segment ends, i.e., that at both ends the segment current is continuing into adjacent segment, or equals to zero, so that the first Kirchhoff's law is satisfied at these points.
8
Ch.l.
Determination of current distribution
and s2 1
v
Q' (s) G(s) ds ,
f
E
( 1. 9)
s1 where G(s)
Pg(r e )
2!a
( 1. 10)
dl .
c
The integration around the circumference is very time consuming, and it is performed with difficulties when the field point P is at the segment surface (because
the
integrand
is
singular).
integration, G(s) is usually approximated by G(s) "' g(r)
In order
to avoid this
6
,
(1.11)
where r
is
2
2
(r +a )
a
an
surface.
the
average
off
a very
good
s-axis
is
yields exact values
straight segment.
the axis
of
for
and the
the potentials
On the axis of a curved seg-
a straight or curved segment, it represents
approximation provided that the radius of curvature of the
much
error
distance between the field point P
Eqn. (1.11)
s-axis of a
ment, and
The
(1.12)
approximate
segment along
!,;2
larger
than a,
introduced by
using
or that r>>a g(ra)
and
instead of
ka<<1, respectively. G(s)
is
largest
for
points close to the segment surface. According
to
the equation
of
continuity, Q' (s)
can
be expressed in
terms of I(s) as Q' (s) =
i w
dl(s) ds
Combining eqns.(l.2), (1.8), (1.9), (1.11)
(1.13) and
(1.13)
we
find for the
-+
vector E s2 -+
E
-jW\1
grad] g(r ) ds k s a f [I(s) 1.s (s) +~dld(s)
.
( 1. 14)
Sect.l.2.
Two-potential equation
9
Consider now a structure consisting of N arbitrarily interconnected wire segments.
Obviously, only
tric field is responsible be _the
for
the
impressed electric field
(1.14)
and
obtain
that
the concept
axial component of the total elec-
the axial current .in
~
wires.
Let Ei
From eqns. (1.1) and 7 5 extended boundary conditions, • we easily
of
along
the
the wires.
current along the segments satisfies the integra-differen-
tial equation N
-r [ i p •1 sm ~
I m=1
I
(s ) m m
di (s ) m m k2 dsm
+ 1-
i
i •E
•grad]g(r )ds = p a m
-?----i JW]l
'
(1.15)
-+
where
i
is the unit vector at any point at the axis of the pth segp ment of the structure considered (p=1 ,2, ... ,N), locally tangential to the axis.
Eqn. (1.15) is
known as the vector-scalar-potential equation
or two-potential equation.
Note that it was derived
assuming
that the
first Kirchhoff's law was satisfied at all junctions and wire ends. In the case
of
general wire structures, the kernel in eqn. (1.15) has
terms proportional to Other known in
1/ra and
integral equations
the wire-antenna
higher-order equals
a,
to
terms
the
structures in
radius
pseudo-singular,
but
case of eqn.(1.15).
1/r . a the
of has
the
the first derivative of
for
determining
in
the general
Since
ra
1/ra only.
current distribution case
have in addition
can be very small (for r=O it
wire),
the
lowest
order
kernel
in
all
equations
pseudo-singularity
in
is the
Since all the equations can be solved only numeri-
cally, the more \·leakly pseudo-singular is the kernel, the easier it is to evaluate
the
the equation. suitable
integrals
encountered in
the
approximate solution of
Therefore the authors consider eqn. (1.15) to be the most
starting point for
arbitrary shapes.
analysis of wire-antenna
Only in the case
of
of such parallel antennas
can equations
venient kernel, containing
terms
structures of
cylindrical antennas and arrays be derived having a more con-
proportional to
1/ra alone,
as
will
be explained in Section 1.3. In
the
present discussion
tions, such as
the
problem of
proper treatment of junc-
junction A in Fig. 1. 1, and the influence of ends, such
?
10 as
Ch.l. B in Fig.l.l,
were not
Determination of current distribution
included.
These
discontinuities will
be
treated in detail in Chapter 3. There are tion
of
several methods which have been used
eqn. (1.15).
larly simple
and
One
of
can be
used with
computers
size and
speed for analysis of wire-structures
ity, will
be
determining
presented in current
for
numerical solu-
the methods, which appears to be particu-
the
of of
relatively moderate reasonable complex-
following subsection.
Some methods for
distribution along cylindrical wire
antennas will
be described in Section 1.3. 1. 2 .1.
Approximate
ical methods
for
solution of the two-potential equation.*
solving eqn. (1.15) can be classified (a) according to
possible mathematical manipulations with ical
solution
the
the
equation prior to numer-
(for example, further integration
appropriate weighting proach of
functions,
the
of moments ) ,
obtained by various
and
Broadly
methods
sufficiently high order
of
the equation, with
and
speaking, using
the
(b) according
current distribution,
basis functions adopted.
of
in accordance with
9
method
approximation of
Numer-
or
general
to
the type of
so-called expansion
it seems
different
that
basis
ap-
or
the results
functions and
approximations practically do not differ.
However, amount of the computing time and computer storage requirements vary
significantly
from
present time limitations
one on
to
another.
Although at
time
tend
be quite pronounced when
do not
exist for
small
structures, they
considering large structures or when
using optimization procedures for computer-aided antenna synthesis. is therefore useful
to
the
both computer memory requirements and pro-
cessing to
usually almost
procedure
It
adopt a numerical method for solving eqn. (1.15)
which is as fast as possible and has modest memory requirements. In
the
authors'
opinion,
one
of
the
most
suitable methods in this
sense for solving eqn.(l.l5) is the point-matching method with entire or almost-entire-domain polynomial approximation for current distribution.
©
* This subsection is adapted from the material of Reference 8, 1979 Springer-Verlag Berlin-Heidelberg-New York, with kind permission of the publisher.
Sect.l.2.
Two-potential equation
(In
of
terms
the
11
general method of moments, this amounts
to
Dirac's
delta-function used as weighting or testing function, and powers in coordinates
as
basis or expansion functions.)
According to this method,
we divide the wire structure into N convenient segments, and approxi. 10 11 mate current distribution along the segments by power ser1es ' n
I
m
I
(s )
m m
. s
I
ml
i=O
i
( 1. 16)
m= 1, ... ,N,
m
where
nm is the desired degree of polynomial approximation for current
along
the
mth
segment.
We next
substitute
this
approximate current
distribution into integral equation (1.15), and stipulate that the equation be the wire
satisfied at structure
in
a
sufficient number of order
to
obtain a
"matching points" along
determined system of linear
equations in unknown current-distribution parameters Imi" Obviously, structure
the
influences
therefore useful of
distribution of
to
the
matching
points· along
considerably accuracy of have
a
the
the wire
solution.
It is
unique procedure for determining positions
the matching points which guarantees that in most cases an accurate
solution would
be obtained.
the matching points
The authors
should be
adopted
the
following rule:
equidistantly spaced along each antenna
segment, the distance between them being twice the distance between the segment end and the closest matching point.
An example of this rule- is
shown in Fig.1.3. Although, in
this
as
already pointed out
in the Preface, no attempt is made
monograph to compare various existing methods
for
analysis of
FIG. 1. 3. Illustration of the rule adopted for choice of matching points along wires of a wire structure. The segments are numbered 1- 8, and the matching points designated by dots.
12
Ch.l.
thin-wire
antenna
Determination of current distribution
structures, the choice
combined with polynomial
of
approximation of
the point-matching method current warrants
certain
justifications and explanations. The point-matching method for solving eqn.(1.15), with Im(sm) substituted by the expression in eqn.(1.16), requires numerical evaluation of single integrals only. moment method,
Other procedures
corresponding
for
to weighting
solving eqn. (1.15) by the functions
Dirac's delta-function, require integration of is known without
to
be numerically much more complicated and
yielding noticeable
improvements
different from
double integrals, which
in
time-consuming,
accuracy
in
almost all
cases. The polynomial expansion
for
current
is very flexible
and
can suf-
ficiently accurately approximate smooth functions (such as current distribution functions
along
wires of constant cross-section), as well as
some of their derivatives, using only few terms. mating the first derivative
of
current arises
[The need of approxi-
from
explicit existence
of the term di/ds in eqn.(1.15). equations
We shall see in Section 1.3 that some 2 2 the cylindrical antennas involve also d I/ds .] On the
for
other hand,
polynomials
point of view.
are
simple and
When compared with a
fast
from
the
computational
sequence of simple rectangular or
triangular pulses used to approximate current distribution along wire 12 antennas, polynomials require a substantially smaller number of unknown coefficients
to
be determined with only a
complexity of the solving procedure.
moderate increase in
They are, therefore, particularly
suitable for obtaining solutions using modern minicomputers. It should be is independent
emphasized that the polynomial approximation of current of
the basis
set
used (for example, power functions as
in eqn.(1.16), some special polynomials, such as Lagrange interpolation polynomials, practically other.
etc.). coincide,
The
final
since
results
they are
for
the
current
distribution
simply rearrangements
of
each
Therefore nothing seems to be gained except unnecessary complex-
ity if any other polynomial basis functions are used instead of simple power functions given in eqn.(1.16). The
theory
presented above
is valid
for wire
structures assembled
Sect.l.3. from
Equations for cylindrical conductors
curved wire
bent wires general it is
at
case
not
have a
segments,
any
provided
that
13
the
point is much larger than
radius
the
of curvature of
wire radius.
Such a
can be solved numerically with relative ease, but usually
of interest.
Instead, most
computer program for
often
it
is
more convenient to
numerical analysis of wire structures as-
sembled from straight segments only, for the following reasons. ly, that is
the
most frequent case encountered in practice.
curved wire segments of
straight
assembled for
the
from
Secondly,
can always be approximated by a sufficient number
segments.
Finally, numerical analysis
straight wire
general
First-
segments is simpler and faster than that
Therefore we
case.
of wire structures
shall
consider
only
such wire
structures. In Appendix tion
of
(1.15)
several
hints
integrals encountered with
Im(sm)
given
are in
given relating to numerical evaluathe point-matching solution of
in eqn. (1.16)
for
eqn.
the case of straight wire
segments.
1.3.
SOME EQUATIONS FOR DETERMINING CURRENT DISTRIBUTION IN CYLINDRICAL CONDUCTORS
As already mentioned, integral equations for current distribution along cylindrical
antennas
potential equation, section is aimed at
are
at
available which are
least from
the
simpler
than
numerical point of
the
view.
twoThis
presenting some of these equations and at explain-
ing certain numerical procedures for their approximate solution. Consider a
perfectly conducting cylindrical antenna of
length b, situated and
~.
in
radius a and
a homogeneous dielectric medium of
parameters
E:
Let the z-axis of a coordinate system coincide with the antenna
axis,
and
shown
in Fig.l.4.
symmetric
let
the
impressed
coordinates Assume
that
of
the antenna
and z , as 2 the antenna is situated in an axially
time-harmonic electric
ends
field
be
z
1
->-
Ei, of angular fre-
quency w, the z-component of which along the antenna axis is Eiz (generally a
function of z).
a structure we
can use
As already explained in Section l. 2, for such the usual expressions
tials given by eqns.(l.3) and (1.4).
for
the retarded poten-
Applying again the extended bound-
14
Ch.l.
Determination of current distribution
b
~--
2a
I ( z)
t+T=--
z=z1
FIG.l. 4.
Cylindrical antenna.
z=O
ary conditions, we can begin the antenna analysis postulating that along the z-axis between z
1
and z
2 (1.17)
where E
is due to currents and charges induced in the antenna.
z
to
Due
axial symmetry, the antenna surface-currents and charges pro-
duce only axial electric field along the z-axis, which can be expressed in terms of the retarded potentials as
E2
= -jwA2
av az
-
(1.18)
Note, however, that
it
is strictly not
possible
to
consider a cylin-
drical antenna without taking into account the antenna ends. see
Chapter 3 that
in
the influence of
the antenna ends
We shall
in some in-
stances is very pronounced and
should be taken into account (e.g., for
thicker antennas and
antenna lengths),
resonant
instances it can be neglected without Substituting eqn. (1.18) tegral equation ed.
However,
for
the
in expressing A
into
eqn. (1.17)
the
current distribution along
final form and V in
of
while
in
some
other
introducing a significant error. general form of the inthe
antenna is obtain-
the equation depends on further steps
terms of
current
l(z), which results in a
determining
current
distribution.
2
number of
equations
for
Some
of
these equations are derived and briefly discussed below. and z V in eqn.(l.18) are substituted by their expressions in eqns.(1.8) and 1. 3.1.
(1.9), general
The two-potential and vector-potential equations.
the
two-potential equation results.
form
of
If A
Since we already have the
the equation, eqn. (1.15), the special
form valid for
Sect.l.3.
cylindrical antennas is obtained simply i f we
i p =is =iz
15
Equations for cylindrical conductors
i z •grad = a/az.
and
[ I(z')
+ __!_ k
2
di(z') dz'
l] az
let
in the equation N=1,
The result is
g
E.
lZ
(r) dz'
( 1. 19)
jw]l
Note that in eqn.(1.19) r
=
[ (z-z')
is the exact
2 +a 2] h2
( 1. 20)
distance between the source point
and the field point at the z-axis.
at
the antenna surface
Differentiation of the equation ker-
g(r) [given by eqn. (1.5)], with 2 terms proportional to 1/r and 1/r .
nel,
respect
to
z
results
again in
If, on the other hand, \ve first make use of the Lorentz condition for the retarded potentials, -+
divA= -jwc:]l V, and note
i.e.,
_j_ divA
V
(1.21)
WE]l
that in the case considered everywhere A=A -+
r . and
z z
thus along
the z-axis divA=dA /dz, eqn.(1.18) becomes z (1. 22)
Combining eqns. (1.8), (1.11)
and
(1.22), with ra replaced by r, we ob-
tain from eqn.(1.17) z2
I
E.
lZ
I(z') g(r) dz'
( 1. 23)
jW]l
z1 This integra-differential equation, for obvious reasons, might be termed
the
vector-potential equation.
It is not
analysis of cylindrical antennas, because tion
of
the second derivative
equation with respect to z.
of
it
convenient for numerical implies numerical evalua-
the integral on the left side of the
16
Ch.l. 1.3.2.
Hallen's equation.
ing with
Determination of current distribution
Further branching is now possible start-
the vector-potential equation.
One possibility is to rewrite
eqn.(1.23) in a compact form, (k
2
+-
i
dz and
to
2
) F(z)
k
=
determine
2
z
S(z) ,
F(z),
which equals
(1.24)
the
integral
in eqn.(l.23),
in
terms of the source function S(z), which stands for the right-hand side in eqn.(1.23).
Solution of eqn.(1.24) is known to be
z
F(z)
+
c1 cos kz
c2 sin kz
+
k
I
( 1. 25)
S(z') sin k(z-z') dz'.
zo The first
two
terms
on
the right-hand side represent
the homogeneous equation. eqn.(1.24).
The last term is
the
the
solution of
particular solution of
It can be obtained, for example, by the Lagrange method of 13 In Appendix 2 it is proved that F(z) [eqn.(1.25)]
varying constants.
satisfies eqn. (1.24) for an arbitrary constant z . 0 Thus, if we
consider eqn. (1.23)
integral shown, distribution
we
along
obtain
the
as
a
differential equation in
the
following integral equation for current
the antenna (the constants c
have still to 2 be determined, so that the sign in front of them is irrelevant): 1
and
c
z2
I
I(z') g(r) dz'
+
c1 cos kz
+
c2 sin kz
z1 z
j~~ This equation is kernel
I zo
E.
lZ
(z')sink(z-z')dz'
kno~vn
containing
only
as
the Hallen equation.
1/r
terms.
( 1. 26)
2
Note
Therefore, from
that
the
it has the
computational'
point of view, it represents a very convenient equation, which has been widely used
for
analysis
parallel antennas. for
the
here).
of cylindrical antennas
and arrays
of
such
(It is quite simple to generalize Hallen's equation
case of parallel cylindrical antennas, but we
shall not do it
Sect.l.J.
17
Equations for cylindrical conductors
Eqn.(1.26) contains three constants (C , c and z ). As already men2 1 0 z can be adopted at will. The constants c and 1 0 can be determined in two ways. If the point-matching technique is
tioned, the constant
c2
used for
determining I (z),
we
can simply adopt two matching points in
addition to the number of matching points necessary for determining unknown coefficients in
c1
consider
(1.26) must
c2
and
the
finite series approximating the current, and
as additional unknowns.
Alternatively, since eqn.
true for any z on the segment [z ,z ], we can adopt two 1 2 arbitrary values of z on the segment and require that the equation be satisfied
be
for
these
two values
z.
In this manner a system of two
and c is obtained, from which they 2 1 expressed in terms of certain integrals of I (z'). These inte-
linear algebraic equations in can be grals
of
C
can be then introduced into eqn. (1.26).
pression under
the
A more complicated ex-
integral sign is obtained, but
for
numerical solu-
tion of the equation it is irrelevant, except that programming is somewhat more complicated than in the first case.
In the latter case, how-
ever, the order of the system of linear equations when using the pointmatching technique
is
reduced by
two, which in some
instances can be
advantageous. As already pointed out, the end effects are not taken into account in this form of Hallen's equation.
If we consider very thin antennas only
(in terms of wavelength), we can stipulate that I(z )=I(z )=0, although 1 2 this strictly is never the case. For example, if the antenna is in the form of a rod with flat ends, we have small radial currents on the flat caps, going
to
zero only at the center of the caps.
A certain correc-
tion is obtained if the antenna is assumed to be for a/2 longer at each end in the case of flat caps, because such an additional antenna length has
the
mately
same area as the
the antenna flat
ends
and, presumably, approxi-
same charges are localized on the two surfaces.
If the an-
tenna ends are hemispherical, the real length of the antenna, including the caps, can be
approximately taken as the antenna length in Halle'n 1 s
equation, because the area of the hemispherical cap is exactly equal to that
of
the curved part
These corrections are
of
a right cylinder of radius a and length a.
only approximate, however, and
certain instabilities in
the
cannot
alleviate
numerical solution of Hallen's equation,
18
Ch.l.
which tend
to
Determination of current distribution
be fairly pronounced for antennas of approximately reso-
nant lengths.
A much better remedy is
the field
to charges and
due
to
incorporate in the equation
currents at
the antenna ends.
However,
this cannot be done in a simple manner with Hallen's equation. 1. 3. 3.
Pocklington' s equation.
Another possibility for
rearranging
the vector-potential equation, eqn.(l.23), is to introduce the operator in parentheses under
the
integral
sign.
Since in
the
integrand g(r)
alone is a function of z, this becomes
22
f
I(z')
a
2 + 21 - 2 )
(1
k
dz
E.
lZ
g(r) dz'
( 1. 2 7)
jW\1
This equation is known as the Pocklington equation. ed differentiation
1
When the indicat-
is
performed, the kernel of the equation turns out 2 3 to have terms proportional to 1/r, l/r and l/r . As already explain-
ed, this is very inconvenient from the numerical point of view. cally,
this
(1. 2 7)
is
result
follows
proportional
this approach
the
ian dipoles.
The
posite charges
at
to
from the
the
field
fact
that
of
Hertz ian
a
Physi-
the integrand in eqn. dipole.
Thus, in
antenna is implicitly envisaged as a chain of Hertz3 l/r term is a consequence of almost equal, but opthe adjacent ends
of any
two
successive dipoles in
the chain. 1. 3. 4.
Schelkunoff's
equation.
potential equation, eqn. (1.19), for drical antennas. (1.20), 3r/3z
=-
Note
3r/3z'.
that
Finally, current
3g(r) /az
Therefore
=-
the
consider again
the
distribution along
:Jg(r) /az'
because,
two-
cylin-
from
eqn.
second term of the integral can
be integrated by parts, to obtain
_ dl(z') g(r) dz'
I
2
2 + z '=z 1
( 1. 28)
Sect.l.3.
Equations for cylindrical conductors
19
Thus the two-potential equation is transformed into z2
Iz'=z 1 This equation is known as venient
for
the
Schelkunoff equation.
numerical analysis
of
cylindrical
14
E.
lZ
It is very con-
antennas.
side, it has the kernel of the most convenient type possible ing on the
r
as
current
1/r only.
On -
the
one
depend-
On the other side, this equation requires that
distribution be
approximated by a functional series which
is twice differentiable, in order brackets under
( l. 29)
jw)J
to
find the second derivative in the
the integral sign, as well
as the values of di (z') I dz'
at
z'=z and z'=z on the left side of the equation. In that case it 1 2 can be solved numerically as easily as the Hallen equation. In the authors' opinion, Schelkunoff's equation has
not received an appropriate
attention it deserves as the starting point for analysis of cylindrical antennas.
For
this
reason we
shall explain in more detail how it can
be solved using numerical methods. The simplest possibility is bined with polynomial,
to
use the point-matching technique com-
a convenient expansion in which
that described
in
case
for
current.
the process
connection with
However, another possibility is
to
the
of
The expansion
can
be
solution closely parallels
general two-potential equation.
approximate current distribution by
a polynomial series in which the first two terms are replaced by trigo15 16 •
nometric functions
I (z)
c1 cos kz
+
c2 sin kz
+
i
n
I
( l. 30)
Ii z
i=2 If we substitute this expression for current into eqn.(1.29), the first two terms of the expansion vanish under the integral sign, because each makes
the expression
in the brackets
reduces to approximate equation
equal to zero.
Thus eqn. (1.29)
20
Ch.l.
Determination of current distribution
n
I
[z'i + i(i-1) z' (i-2) )g(r) dz' - _i_ z' (i-1) g(r)
k2
i=2
c1
the
points tions
next along in
E.
c2 k cos kz'
+ ksin kz'
As
I
k2
~z
jW\1
step we require that this equation be the antenna, which results
(n+1) unknown
complex
in
parameters
( l. 31)
satisfied at (n-1)
(n-1) complex linear equa-
c1 , c2 ,
I , ... ,In. 2
Two ad-
ditional equations are obtained requiring that ( l. 32)
and
Expansion given by eqn.(1.30) is particularly useful when considering relatively long antennas (longer than about case
the
current distribution is well
tric terms with a
one
wavelength).
approximated by
In this
the trigonome-
corrective polynomial term which is relatively small
everywhere except in
the
vicinity of the antenna discontinuities (like
excitation region and antenna ends). ed that the origin,
z=O,
[This conclusion is valid provid-
is sufficiently
far
from the antenna discon-
tinuities, and if z <0
CONCLUSIONS
In the present chapter analysis
of
an
arbitrary wire
obtaining approximate structures.
integral equation was derived convenient for structures and a
solutions
for
current
However, the problems of
method was explained for distribution along
excitation
region
the
and various
antenna discontinuities were not considered. As a special, but important case, thin cylindrical antennas were analysed in more detail. analysis was
A number of equations which can be used for their
derived.
distribution in all
A numerical procedure
the cases was
that excitation field was function along the antenna.
a
for
determining current
outlined, but it was
continuous
and
again assumed
relatively slowly varying
1.4. The
21
Conclusions next
four
chapters are
which the excitation i.e.,
localized in a
field
devoted
to analysis of wire antennas in
is close to that in
small
region,
the
and thus quasi-discontinuous, and
along which discontinuities exist of various kinds. theory
presented
in this
transmitting case,
chapter will be
In this manner the
completed for analysis of a
wide class of real transmitting and receiving wire-antenna structures.
CHAPTER 2
Approximations of Excitation Regions
2.1.
INTRODUCTION
The approximate integral and integra-differential equations for current distribution along wire structures and the
preceding
chapter are
valid for
cylindrical antennas derived in any impressed electric
field
in
which they might be situated, but simple numerical procedures suggested for
their
approximate solutions
can be applied only
if the impressed
electric field is relatively slowly varying function along the wires of the s.tructure.
However, the impressed field is, often, concentrated in
a relatively small
region of
the antenna structure (the so-called ex-
citation region), in which it varies rapidly along the wires, as in the case of transmitting antennas.
Rapid variations of the impressed field,
and thus also rapid variations
of
the induced current and, especially,
charge in this region, must be treated numerically with additional precautions.
They usually
require
separate, rapidly varying expansion
functions to be used in the excitation region, and specification of appropriate boundary conditions for these expansions. to do field
In order to be able
that, a closer insight into variations of the impressed electric along antennas in real circumstances is needed.
devoted to modelling of transmitting wire
This chapter is
some systems used frequently for excitation of
antennas
and
for connection between
receiving wire
antennas and the receiver. Only rarely a
wire antenna is connected to the terminals of a trans-
mitter or receiver directly.
Usually, this is done by means of a con-
24
Ch.2.
venient transmission line are
used most often
(feeder).
to
Coaxial lines
for that purpose.
distance between conductors spect
Approximations of excitation regions
of
the wavelength of
and
two-wire lines
Almost without exceptions, the
a feeder in practice is
small with re-
An antenna
the wave radiated or received.
connected to one end of a feeder has, therefore, two electrically close terminals, and
can
in the network
theory
be regarded as
a lumped impedance element connected at the feeder end.
In
the
be considered as sense.
a two-terminal (one-port) network
In the transmitting case, an antenna can
receiving case, the antenna
Thevenin's feeder.
(or,
alternatively,
It is well
known that
generator equivalent
to
can
be considered as an equivalent
Norton's) the
generator
to
the
internal impedance of the Thevenin
a receiving antenna is equal
of the same antenna when used for transmitting. the receiving antenna
connected
to
the impedance
Electromotive force of
can be determined in various ways, one
of
which
is based on knowledge of the current distribution along the anten~a in the transmitting case, in addition the electric field 3).
For
this
of
to
orientation and
distribution of
the received electromagnetic wave
reason we
(see Appendix
shall consider transmitting antennas only, as
already mentioned in the introduction to the first chapter. To be able to define an antenna as a two-terminal element, it is necessary to define two electrically close points in the excitation region in which
the
feeder
terminates
and
the antenna
begins.
This can be
done fairly arbitrarily if rough analysis only is needed, but considerable care has to be exercised if more accurate analysis is required. In spite of the assumed electrically small volume occupied by the excitation region, its geometry has considerable influence on the antenna impedance or admittance, although it practically does not influence the antenna radiation pattern.
Therefore precise modelling of
excitation region is necessary
for
the antenna
obtaining accurate theoretical val-
ues of the antenna impedance or admittance. In this chapter gions will
several
theoretical models
be described and
these models examined.
of
actual generating re-
discussed, and the impressed field due to
This chapter, combined with the preceding chap-
ter, represents a basis for accurate numerical analysis
of
cylindrical
Sect.2.2.
Delta-function generator
25
and other wire-antenna structures without junctions and the wires.
It contains
also
certain numerical results
loadings along obtained using
computer programs based on the theory presented so far, as well as comparison of these results with available experimental results.
2.2.
DELTA-FUNCTION GENERATOR
The delta-function generator is the simplest - and the least accurate theoretical model can approximate
of
to
a
the real excitation region certain degree
any
type
of of
wire antennas.
It
excitation, and has
been widely used for analysis of wire antennas. Without ral forms
going of
into historical details,
the delta-function generator
~•e
shall only note that seve-
have
been used.
In Fig.2.1
the generator is assumed in the form of an ideal voltage generator connected across a gap of negligible width along the antenna.
The antenna
admittance is defined in this case as Y=I /V ,where V is the generator g g g voltage, and I its current. As the gap width, d, is assumed to tend to g zero, such a gap introduces an infinite capacitance across the generator terminals.
This makes
the
antenna susceptance singular, which, of
course, in reality is never the case. In
order
to
avoid
in theoretical analysis
the
radial
currents
and
charges in the gap, and to circumvent the gap capacitance, another representation of
the delta-function generator is used, shown in Fig. 2. 2.
This model
be
can
deduced
from
that
shown
in Fig. 2.1 by postulating
FIG.2.1. Ideal voltage generator connected across a narrow gap as approximation of excitation region. The gap width is not drawn to scale.
26
Ch.2.
Approximations of excitation regions
z
that
the
field in the gap
be
(i.e., outside the surfaceS be achieved +
H
be
5
that
in the exterior region
shown in Fig.2.1) be unchanged. This can + 5 Let ES and
by applying the general equivalence theorem.
the total fields on S, due to all the sources (i.e., to the gen-
erator current inside
zero, while
FIG.2.2. Magnetic-current ring as approximation of excitation region.
and
to the antenna currents
and
charges).
The sources
S can be replaced with respect to the domain outside S by equi-
valent surface electric and magnetic current on S given by +
J
+
J
s
(2. 1)
ms
where ~is the outward unit vector on S (see Fig.2.1). valent
sources
medium can
on S, the
total
be homogenized.
field inside
S
With these equi-
is zero,
5
so
that
the
Note that the magnetic surface currents on
Sin Fig.2.1 exist only on the hatched surface shown in Fig.2.2, because the antenna is perfectly conducting. The voltage
across
the magnetic-current ring
can be defined as the
integral of the total electric field along a path lying outside S (Fig. 2.2).
This voltage is the same
as
that in Fig.2.1 (the fields outside
S in the two cases are identical), and is very nearly equal to V , since g is small and thus the field essentially quasi-static. The antenna
a
admittance is defined in
this
antenna electric current
at
is not equal
to
2.1 (they differ
the current for
case as the
ratio
the antenna surface for z=O. I
This current
of the ideal voltage generator in Fig.
g the displacement current in
the antenna admittances as
Y=I/V, where I is the
the
gap).
Therefore
defined for the systems in Figs. 2.1 and 2. 2
Sect.2.2.
27
Delta-function generator
are not equal. If in Fig. 2. 2 we let the \vid th
d
tend to zero, the surface magnetic-
current layer degenerates into an infinitesimally thin magnetic-current ring of surface current Jms(z)=Vo(z), the total current of the ring being Im =V.
As
a
consequence, a we
citance appears across the two sides of the ring, resulting again in an infinite antenna susceptance. In
the
generator
current ring is
the
representation sketched source
in Fig.2.2
the magnetic-+
of
the impressed electric field, Ei.
field can be computed in terms of the electric vector-potential, -+
->
E.
1
Ae ,
as
(2.2)
curl A e
E
This
where
Ae
E
-+
J
J
g(r) dS
IDS
s
(2. 3)
'
S is the surface with magnetic
currents,
-+
r
the
distance between
the
element dS and the field point, and g(r) Green's function, given by eqn. (1.5). under
Combining eqns. (2.2) and (2.3), introducing the "curl" operator the
integral
in eqn. (2. 3)
and noting
that
it operates only on
g(r), it can be easily obtained that ->
E.
1
If
d
JJ S
IDS
1 xgradg(r)dS=-4 1T
J (J dS)x[~l+j3krexp(-jkr)]
is assumed
to
r
be zero, I
JIDS dS.
(2.4) instead of
(2.4)
IDS
S
di=v di should be introduced into eqn. m In that case the impressed electric field E.
1
at the antenna surface exhibits the properties of a delta-function with respect to
the
z-coordinate.
However, in order
boundary conditions, the impressed field along Starting from eqn. (2.4), with
J
IDS
to
the
apply the extended z-axis is required.
dS substituted by Vdi, it is easy to
show that, along the antenna axis,
E.
1Z
(z)
va 2
2
l+jkr exp(-jkr) , 3 r
(2.5)
28
Ch.2.
Approximations of excitation regions
where r = (z
2
+
2
a )
1:2
(2.6)
From eqn. (2 .5) it
can
along
(actually,
the
z-axis
practically confined eral antenna assumed to of
the
be seen that
only
radii.
be
as
to a region the length of which is only sev-
In this
region
kr<< 1 since the antenna radius is
much smaller than the wavelength, and hence the integral
impressed field in eqn. (2.5) along the z-axis in the excitation
region is very closely equal this integral is equal can be
the impressed field rapidly decays 3 l/r in the near zone), and is thus
approximated
Eiz (z) "'V 6 (z),
to
by
to V.
0. 97 V.)
the
Dirac
(For example, from z=-4a
to
z =4a
Therefore in some instances E.
~z
delta-function,
i.e.,
we
can
(z) set
as discussed in the next subsection.
The impressed electric field given by
eqn. (2 .5) can be used when de-
termining current distribution starting from the equations described in Chapter 1.
However, if
the
point-matching method is used with any
of'
these equations, except the Hallen equation,Ja relatively large number of matching points is necessary in the excitation region in order to properly sample
the
impressed field.
Alternatively, the general method of
moments can be applied for this region.
For example, the integral con-
straint (2. 7)
0
can
be applied, where
nity of the ring.
L
is a short path along the z-axis in the vici-
In this way the effect of the generator is taken in-
to account integrally, neglecting the local variations of the impressed field. the
Such an approach yields reasonably stable numerical results for
antenna
in Fig. 2. 2 ring.
admittance
Secondly,
bution cannot vicinity of tance.
for
introduces only any
several reasons. a
weakly
Firstly, the model shown
singular capacitance across
low-order expansion function
adequately represent
the
for
the
current distri-
rapid charge variations in the
the ring, and thus cannot account for the infinite capaci-
Thirdly, the application of the extended boundary conditions in
Sect.2.2.
29
Delta-function generator
a numerical solution virtually eliminates this singularity since, viewed from
the
antenna axis, numerically
it
can
be
hardly distinguished
whether the magnetic-current layer has a finite or zero width d. ly,
the
integral constraint (2. 7) introduces
by neglecting
the
computation of which is
time
local distribution of
still
Final-
further softening,
the electric field.
However,
the integral in eqn. (2. 7) can be done only numerically, consuming.
tional effort more
In addition, with almost
sophisticated approximations
of
the
same
computa-
the excitation re-
gions can be utilized yielding more stable and reliable results for the antenna admittance.
Therefore
the
delta-function
primarily with
the Hallen equation, because it
corporated into
the
can
generator
right-hand side of the equation, without
cial numerical precautions, as will
be
is
used
be very simply inany spe-
explained in the following sub-
section. 2.2.1. Let,
for
Solution of Hallen's equation with simplicity, in
the
delta-function generator be
delta-function generator.
Hallen equation [eqn.(l.26)j z =0 and the 0 that location. The i~pressed electric
at
field, given by eqn. (2. 5), can be right-hand side of eqn.(l.26).
in traduced into the integral on the
This impressed field practically exists
only in the immediate vicinity of
the point z=O, i.e., in the interval
zE(-s,s), where sis of the order of few wire radii.
Thus we have
-s
I
z
I
E.
(z') sin k(z-z') dz' , z
E.
(z') sink(z-z') dz', z>O
lZ
0
E.
lZ
(z') sin k(z-z') dz' I;
0
I
lZ
0
(2. 8)
E. (z') dz' lZ
for
Iz I >>s,
because
very nearly equal
ks<
with eqn.(2.5), we have
Since
the
V/2, according
last to
integral
in eqn. (2. 8) is
the discussion
in
connection
30
Ch.2.
Approximations of excitation regions
z
J
Ei/z') sin k(z-z') dz'
"~sin kJ zJ
(2. 9)
.
0 Ass is very small compared with the antenna length, eqn.(2.9) is valid practically along the whole antenna. would be
exact if
the
It is worth noting that eqn.(2.9)
impressed electric field
along
the z-axis were
taken to be a Dirac's delta-function, i.e., Ei (z) 2
=
(2. 10)
V o(z)
With eqn.(2.9) substituted into eqn.(1.26), the Hallen equation reads: z2
I
I(z') g(r) dz'
+
+
c1 coskz
c2 sinkz =
2~~)1
sinkJzl
z1 (asymmetrically driven antenna at z =0). 0 If
the
antenna is symmetrically driven and
z =-h and z =h. 1 2 So we have
In addition,
c =0 2
its length is
(2 .11)
2h, we have
in this case, because of symmetry.
h
I
I(z') g(r) dz'
+
c
1
coskz =
-h
2 ~~)1
sinkJzJ
(symmetrically driven antenna at z =0). 0
(2. 12)
As already mentioned, eqns.(2.11) and (2.12) have been used extensively for analysis
of
cylindrical antennas.
It is interesting
these equations physically cannot have a solution. derstood
if we have in mind
actly corresponds
to
that
to note that
This is easily un-
the last term in these equations ex-
the impressed electric field given by eqn. (2 .10).
Although eqn.(2.10) can be considered as an approximation of eqn.(2.5), such a singular impressed electric field can neither be obtained physically by any sources located at the antenna surface, nor can be compensated by any physically realisable distribution currents and charges located at that surface. tively
rapid ultimate divergence
creasing order of
the
of
of antenna electric
This might explain rela-
the antenna susceptance with
in-
expansion functions when using eqns. (2.11) and
Sect.2.2. (2.12).
Delta-function generator However,
if
31
low-order expansion functions
are
used, this ap-
proximation yields fairly accurate results for the antenna admittance. It is interesting
to
present some results obtained using
with polynomial approximation sion for z.
for
current.
eqn. (2.12)
Note first that the expan-
current along symmetrical dipoles must be an even function of
For example, we may choose the expression n
I
I(z)
I
m=1
(2 .13)
m
as was done in Reference 10, to take the condition I(±h)=O into account 11 automatically. However, we can also adopt the simple power expansion n
I
I (z)
I
m=O
(2.14)
m
with the constraint
I(±h)=O
added to the point-matching equations.
As
mentioned in Chapter 1, the form of the polynomial approximation is irrelevant, since all
are
rearrangements
of
one another.
The matching
points should be distributed relatively evenly along one arm of the dipole.
The simplest choice is
the points
at
to
take
them equidistantly, with one of
the antenna end (i.e., at z=h).
Once we have determined
the current distribution coefficients Im, the antenna admittance is obtained, for V=1 V, as n
r
y
I(O)
v
L
I
{··:
m
(2.15) 0
As
a
for expansion in eqn. (2.14)
numerical example, Table 2.1 gives theoretical and experimental
antenna admittances lengths
for expansion in eqn.(2.13)
and
from Reference 10
fixed antenna radius.
for
three
antenna
electrical
It is seen that the theoretical re-
sults remain relatively stable with increasing degree
of
approximation
and that they are in good agreement with experimental results. As the next example, Fig. 2. 3 presents experimental current distributions.
two
cases
It is seen that
of
theoretical and
the overall thea-
32
Ch.2.
Approximations of excitation regions
TABLE 2.1. Admittance of cylindrical dipoles of radius a=0.007022 A for increasing degree of polynomial approximation of current, obtained as a solution to eqn.(2.12). (Ref.10)
h/A
Admittance (mS) degree of polynomial approximation 2 4 3
measured
17*
0. 250
9.16-j3.57
9.16-j3.55
8.81- j3.31
8.92- j3.46
0. 375
1.52- jO. 36
1.54-j0.27
1.53- jO.lO
1.58- j0.18
0.500
0.98+j1.54
0. 96 + j 1.58
0.97+j1.76
1.02 + j1.68
* In Reference 17 two sets of results for admittances of isolated monopole antennas driven by a coaxial line are given. The first set corresponds to admittances of antennas as elements terminating the coaxial line. In the second set these values are corrected as proposed by King to take approximately into account the influence on the antenna susceptance of the specific coaxial line used in measurements as contrasted to the delta-function excitation. In this section it was natural to use the latter set of results, but the first set was used with more accurate approximations of coaxial-line excitation (see Table 2. 3). In both cases averaged measured values are shown (computed from a number of experimental results presented in Reference 17).
Z/A
0.4
Z/A
0.7
-4 -3 -2-1 0 (b)
FIG.2.3. Current distribution along isolated dipoles of length (a) 2h=0.750 A and (b) 2h=1.250 A, of radius a=0.007022 \; (1) real part, (2) imaginary part, (3) magnitude. o o o experimentl7, ----- theory, second degree polynomial approximation of current for case (a) and third degree for case (b), obtained as solution to eqn. (2. 12). (Ref .10)
Sect.2.2.
Delta-function generator
33
retical current distribution represents quite a good approximation, in spite of very low degree of polynomial approximation of current and the inaccurate model of the excitation region. Finally, a rough sketch showing dependence of
the antenna admittance
versus the (half-length)/(wavelength) ratio, i.e., versus h/A, presented in Fig.2.4, indicates that solutions of
the Hallen equation with
simple delta-function generator and polynomial approximation of current of quite low degree (second or third) predict fairly accurately the antenna admittance behaviour as
a function of frequency.
(In Appendix 5
more accurate curves are given for admittance of cylindrical antennas valid for a wide range of the ratios h/A and a/A.) It is also shown in Reference
10 that Hallen's equation with
the
delta-function generator with polynomial approximation for current can efficiently be used for analysis of coupled cylindrical dipoles, and in References 18, 19 and 20 some results are given for isolated and coupled asymmetrical dipoles and for coupled symmetrical dipoles of unequal
14
G,B (mS)
12 10
(\
8
I \
\
6
\~
4 2
0
h/ A 0.1
0.2
0.6
-2 -4 FIG. 2 .4. Conductance (G) and susceptance (B) of isolated dipoles of radius a=0.007022 A, for different ratios h/A of dipoles. xx oo experimental, l 7 polynomial approximation of current, second degree for h/;\<0.5, third degree for h/A>0.5, obtained as a solution of eqn.(2.12). (Ref.10)
34
Ch.2.
lengths.
Approximations of excitation regions
Thus, in spite of conceptual difficulties, the delta-function
generator appears
to
be a convenient approximation for fairly accurate
analysis of isolated and coupled cylindrical antennas.
We shall see in
the next sections that much better approximations of excitation regions are available, which yield accurate admittance,
and
and which can be more
reliable values of the antenna
efficiently incorporated
into
the
two-potential equation for arbitrary wire structures.
2.3.
APPROXIMATIONS OF COAXIAL-LINE EXCITATION
A frequently
used
system for
feeding wire antennas is a coaxial line.
Being inherently an asymmetrical structure, it is used mostly
to
feed
asymmetrical antennas, like the monopole-antenna shown in Fig.2.5.
The
antenna is in the form of a simple protrusion of the inner coaxial-line conductor
through
the
conductor terminates.
ground
or
small
at \vhich
In practice, the
of a very good conductor. flat, large
plane,
in
ground
the
outer coaxial-line
surface is usually made
In shape and size it can be diverse: curved, terms
of
the wavelength.
Often, however, at
least as far as the antenna admittance is concerned, the ground surface can be
approximated by an infinite, perfectly conducting ground plane.
This is the only case we shall consider in what follows.
z
z=h 2a E
)J
z=O /
FIG.2.5. Sketch of a monopole antenna driven by coaxial line. Dielectric is assumed to be homogeneous everywhere.
Sect.2.3.
Approximations of coaxial-line excitation
35
In principle, the system shown in Fig.2.5 can be analysed if all currents and charges in the system are taken into account.
These currents
and charges are distributed on the antenna itself, on the ground plane, on the
inner and outer coaxial-line conductors along
of the line,
and
in
Obviously, such an
the
generator
approach would be extremely complicated but, fortu-
nately, it is not necessary. citation of
the whole length
to which the antenna is connected.
It can be avoided if the coaxial-line ex-
the antenna is replaced by
a suitable approximation which
allows the half-space above the ground plane to be treated independently of
the coaxial line and
the
image theory to be applied in order to
reduce the problem to that of a symmetrical dipole driven by an equivalent generator. tion
are
Three such approximations
of
the coaxial-line excita-
presented in the following subsections.
for the case
of
They are all derived
a vertical monopole, of the form shown in Fig.2.5, but
they can also be applied successfully to arbitrary wire-antenna configurations situated above
a
perfectly conducting ground plane and fed in
that plane by one or more coaxial lines. Throughout the following discussion we ing frequency is sufficiently low along
the
coaxial line.
major part of propagating
shall assume that the operat-
so that TEH waves only can propagate
In that case the electromagnetic field in the
the line is given practically as a sum of two TEM waves,
in
opposite directions.
At the line end, however, higher-
order (TE and TM) modes must be taken into account. this, the antenna admittance is most often defined coaxial-line mode,
at
current
In connection with as
the ratio of the
intensity and voltage, corresponding to
the
TEM
the coaxial-line opening (i.e., in the plane z=O in Fig.2.5).
The antenna
admittance
thus
defined uniquely determines
the
TEM-mode
standing-\vave pattern along the coaxial line. 2.3.1.
TEH magnetic-current frill approximation of
citation. results for
At frequencies the
for which,
approximately,
coaxial-line exkb
antenna admittance are obtained if the electromagnetic
field at the coaxial-line opening in Fig.2.5 is approximated by the TEM mode Only. 21,22
Th ere f ore, we can set, f or z= 0 ,
36
Ch.2.
Approximations of excitation regions
v
E (p)
p ln(b/a)
p
a
(2. 16)
I(O)
H<jJTEM(p)
=
2--;;p
where V is the voltage and opening,
and
p
the
I(O)
distance
current intensity at the coaxial-line
of
the point considered from
the
z-axis
(see Fig.2.5). According to the equivalence theorem? the perfectly conducting ground plane
can
be extended to cover the coaxial-line opening, provided that
an annular layer of
surface magnetic currents is placed at
opening, immediately above
the
plane,
magnetic currents are circular, and
as
their
shown
the former
in Fig.2.6(a).
These
density, with respect to the
reference direction shown in Fig.2.6(a), is given by
If the coaxial-line voltage,
V,
is assumed to be known, from the first
of eqns.(2.16) and eqn.(2.17) it follows that Jms<jl(p) is also known.
z
z
z
I
I
->-
->-
2Jms
Jms /
I I
I
I
I 12a I
I
I
FIG. 2.6. Equivalent systems line sketched in Fig.2.5.
z=O
--1I
;+
z=O
I
I
~ (a)
t:rs
I I I I I ->I I 2J I 1 ms
I I I I I I I I
I
I
I
E, ll
E, ll
E' ll
l
(b)
(c)
for monopole-antenna driven by
coaxial
Sect.2.3.
Approximations of coaxial-line excitation
The image theory
can now be
sketched in Fig.2.6(b).
applied to obtain the equivalent system
In that system the magnetic-current density of
the frill is twice that in Fig.2.6(a). source
of
tenna.
37
These magnetic currents are the
the impressed electricfield for
the
symmetrical dipole an-
Finally, we can apply once mote- the- equivalence theorem to re-
move the perfectly conducting antenna, which leaves us with the magnetic-current frill and the induced currents and charges on the former antenna surface, situated tained in which
in
a homogeneous medium.
the usual expressions
for
A system is thus ob-
the retarded potentials can
be applied. The impressed shown
electric
in Fig. 2.6(c),
at
field a
due
to the
TEM magnetic-current frill
point P(x,O,z) lying in the xOz plane (see
Fig.A4.1 in Appendix 4), has the components b -4zV ln(b/a)
Eix(x,z)
1f
II
~ dg(r) d
~
r
d
(2.18)
P
a 0 and
I
b
1f
-4V ln (b/ a)
Eiz (x, z)
(2. 19)
g(r) lp=a d¢ '
0
where g(r)
r is
can be
is
the
the
distance between the source and
Green
function.
Details on
found in Appendix 4. 1.
the field points, and
derivation of these formulas
The integrals in eqns. (2 .18) and (2 .19)
can be evaluated numerically without difficulties, except when the field point P is very close to the magnetic-current frill. The expressions in eqns.(2.18) and (2.19) are important when analysing general wire-antenna structures. antenna, like that nent along
the
When considering a
vertical monopole
in Fig. 2.5, we need to evaluate only the Eiz compo-
z-axis, because E. (O,z)=O.
This expression can be ob2 2 J, tained from eqn.(2.19) noting that now r=(p +z) . It has the form lX
(2.20)
38
Ch.2.
Approximations of excitation regions
where
r
a
(a
=
2
2 !<2
+ z )
and
rb = (b
The equivalent dipole antenna
2
2 Y, + z ) .
shown
(2.21)
in Fig.2.6(b) can next be analysed
by any of the equations presented in Section 1. 3 if
Eiz
is replaced by
the expression in eqn.(2.20). As
a
consequence of
the
annular magnetic-current
frill
excitation,
current derivative has a discontinuity at the frill location.
The val-
ue of the derivative (di/dz)Jz=O+ is uniquely determined by the coaxialline dimensions and below.
This
fact
the voltage at
must be
the
line
opening, as demonstrated
taken into account
in a numerical solution
for stable and accurate results in this case to be obtained. To determine the value of (di/dz) J
, note that at the antenna surz= 0 + face in Fig.2.6(b) the electric-field vector at z=O+ has only the radial component, E
P
, which
is tangential to the magnetic-current frill.
Ac-
cording to the boundary conditions, E (p=a,z=O+) - E (p=a z=O-) = -2J (p=a) . ms¢ p p ' The antenna charge
per
Q' (z)=21f£a E (p=a, z).
unit length is related to the electric field as The first current derivative and this charge are
p
interrelated through
(2.22)
the
continuity equation, i.e., di(z)/dz=-jwQ' (z).
From eqns. (2.16), (2.17) and (2.22) it hence follows that di(z)
dl(z)
~Jz=O+
~Jz=O-
In the case considered
the
. 4Tr£wV - J ln(b/a)
(2.23)
current derivative is an odd function of z.
Therefore we finally have dl (z)
dl(z)
~Jz=O+ where Y
c
~Jz=O-
. 2Tr£wV -J ln(b/a) = -jkYcV '
(2.24)
is the characteristic admittance of the coaxial line.
The conditions
in eqns. (2.23)
or
(2.24)
can easily be incorporated
into any smooth approximation of the antenna current dis-tribution, such as
the
polynomial approximation, resulting in very stable and accurate
Sect.2.3. results
Approximations of coaxial-line excitation
for
current
and
magnetic-current frill. varying along
the
charge
distributions
in
39
the
However, these distributions
vicinity of the are
very rapidly
z-axis in the excitation region, and special precau-
tions must be taken in order to provide a sufficiently accurate approximation
for
these
distributions
near
the
magnetic-current
frill,
as
will be discussed later in this section. When using (see
the magnetic-current frill with
Subsection
parts this
the
1. 3. 4) ,
care
should be
the Schelkunoff equation
exercised when integrating by
integral shown on the left-hand side of eqn. ( 1. 28) .
integral
should be
split
in to
two integrals, one of
Namely,
them being
performed on avoid
the
the interval (z ,0), and the other on (O,z ), in order to 1 2 discontinuity of di/dz at z=O, where the magnetic-current
frill is located.
Thus instead of (1.28) w·e obtain
di(zl) - - g(r) dzl = _ dl(z dz 1 dz 1 ()z 1
2
1
(r)\
)
g
2
+ di(z dz 1
1
z =z
1 )
I O+
(r)
z=O-
1
g
+ J
z=O
0-
+(J
(2. 25)
Eqn.(2.25) should now be introduced into eqn.(l.l9), and Eiz substituted from eqn.(2.20). can be the
The second
interpreted as
first
term on the right-hand side of eqn.(2.25)
the electric field
current derivative
at
due
to the discontinuity in 2 -k /(jwlJ). Since
z=O, multiplied by
the discontinuity is determined by eqn. (2. 23), this field is known and can be treated as a part of the impressed electric field (as defined in Section l.l). eqn. (1.19),
Therefore
where
it
should be moved to the right-hand side of
it exactly
field given in eqn.(2.20).
cancels the first term of
the impressed
Thus we finally obtain the Schelkunoff equa-
tion for a cylindrical antenna driven by a magnetic-current frill: z2
0-
(I
+
I)( I(z
__!_dl(z 2 dz 1 k
1 )
0+ j4rrV
W\1 ln(b/a) g(rb)
1
E.lrz jwlJ
)
()1
g r
2
2
z 1 =z
1
(2. 26)
40
Ch.2. Note that
the
Approximations of excitation regions
resulting impressed field
Eirz
in
eqn. (2. 26) is less
pseudo-singular along the z-axis than the field given by eqn.(2.20). can be shown that
the
It
z-component of this resulting impressed electric
field is singular only
for
(p=b,z=O), i.e., at the outer magnetic-cur-
rent frill edge, while the field
due
to the frill alone, given by eqn.
(2.19), is singular at the antenna surface also, i.e., for (p=a, z=O). Once
the
antenna current distribution
antenna admittance, defined
in
terms
of
is
determined,
the
monopole-
the coaxial-line TEM mode, is
approximately given by* I(O)
y
(2. 2 7)
v
where V is
the
the antenna
current
assumed voltage for
z=O.
the electromagnetic field
at
at
the coaxial-line opening, and I(O)
Although based on
the approximation of
the coaxial-line opening by
the TEM mode
only, eqn.(2.27) yields very accurate results at lower frequencies. higher frequencies more
At
accurate analysis of the electromagnetic field
at the coaxial-line opening is necessary, as demonstrated in Subsection 2.3.3. The monopole
radiation pattern
be determined easily Appendix 3. of
the
The
from
current radiation field.
tails on
to
its
(in
the
frill,
upper half-space)
about
which should
be
This radiation field
the field radiated computation
are
can
also
known current distribution, as shown in
question arises, however,
magnetic-current
proximation
the
from
given
the
radiation
field
added to the antenna-
is, essentially, an ap-
the coaxial-line opening.
in Appendix A4. 2.
It
is
De-
usually
negligibly small, except in cases when very short monopoles are considered (of lengths
of
the order of few outer coaxial-line radii) or when
the coaxial-line circumference
is
length.
radiation
Determination
of
the
a significant fraction field
from
the
of
the wave-
coaxial-line
opening is, therefore, required quite rarely. The
exact
solution
for
the antenna
current
distribution results in
* Note that other approaches for determining the antenna admittance are possible, one of them being presented in Appendix A4.3.
Sect.2.3.
41
Approximations of coaxial-line excitation
zero total electric field in the interior of the monopole (dipole) surface,
and
ever,
an
zero total electric
field
approximate solution does
cept, possibly,
at
some points.
tangential to that surface. not satisfy
these
How-
conditions, ex-
For gaining insight into
the
quality
of a numerical solution, it is useful to compute the axial field due to the
obtained
approximate current distribution and to see how much it 23 24 25 • • As a spec~"f·~c examp le , s hown in Fig . 2 . 7
differs from -E.~z (z). is
the
tation
z-componen t of the total electric field, region,
for
a
point-matching polynomial
(E +E. ) , in the exci-
z
~z
solution of
the two-
potential equation for a quarter-wavelength cylindrical monopole sketched in Fig.2.5.
The monopole radius was
a;O.Ol :\,
the
outer coaxial-
line conductor radius b;2.3a, and the monopole had a hemispherical cap. In
the
analysis the monopole was divided into four segments: the exci-
tation region (6a long), the hemispherical end cap (see Section 3.4), a 3a
long
segment adjacent
0.005JJ
o
1
to
the end cap
~
and
the rest (central, main)
1\ 9
10 11
12 z/ a
mll!J~wru!JJmm--~ -o.oosl!/j';, ~ 1
-0.010
N
,J
-0.015 -0.020
FIG. 2. 7. Real part of the axial c..:Jmponen t of the normalized electric field in the excitation region of a vertical monopole driven by a coaxial 1 ine. Small circles indicate the positions of matching points which are not masked by the graph. Imaginary part is several times smaller in magnitude. (Ref. 25)
42 part
of
the monopole.
required
to
Ch.2.
Approximations of excitation regions
Both the
current and its first derivative were
be continuous functions along
the polynomial approximation were cap, where a
z-axis.
used
to
the
The degrees of
on all the segments except the end
third-degree polynomial was adopted.
were distributed along tions were
4
the
z-axis (i.e.,
the
The rna tching points
extended boundary condi-
formulate the equation).
The total field is shown
normalized with respect to the intensity of the excitation field at the origin, i.e., with respect to Eiz(O). As it can be
seen from Fig. 2. 7,
the
is practically zero at
the matching
excitation region
field
not only at
the
surface (i.e., the
true
this
of
n
is very
small
in
magnitude CIE 1<0.01) n
z-axis, but also everywhere within and on the monopole for
p/a..::_1).
This indicates that
boundary conditions
middle part
total normalized electric field 6 points (where IE I <10- ). In the
the monopole
are the
satisfied
both
to
the extended and
a high degree.
total electric field
in
for an order of magnitude smaller than in the excitation zone. conclusions were
found
distribution along a
to
be valid
for
In the
this case is Similar
any good solution for current
monopole of any dimensions.
However, if
the
ap-
proximation of current distribution is poor (e.g., polynomial degree is inadequate,
or
the matching
points are
improperly distributed) , the
boundary conditions are also poorly satisfied.
This results in signif-
icant error in the antenna admittance and even in the radiation pattern. In Appendix A4. 4 a
method is given for estimating and reducing the er-
ror in the antenna admittance in such a case. For electrically fourth-degree
thicker
antennas
(ka_::_0.01)
polynomial approximation
for
region, a conclusion reached by extensive
it
suffices
current in numerical
trically thinner antennas (i.e. ,
the
z-component of
the the
to
take a
excitation
experiments.
length of the excitation region should be between 3a and of 6a was adopted in the above example).
the
The
lOa (a length
However, in the case of elec-
second derivative of the antenna current charge gradient) in the vicinity of the
excitation region is also varying very rapidly along
the
antenna axis.
In that case a different division of the antenna into segments was found (by numerical experiments) to
be necessary.
A segment several antenna
Sect.2.3.
Approximations of coaxial-line excitation
43
radii long is required at the monopole base in order that the excitation field can be sampled adequately, but one or more adjacent, progressively longer segments should be
adopted
sufficiently accurate approximation of segment.
in addition, in order to provide 2 2 d I/dz further from the first
This is essential for obtaining a good solution with the two-
potential and other equations based directly on the boundary conditions for the electric field. As an illustration, Table 2.2 presents the antenna admittance and the peak
total normalized electric
antenna cap)
for
field along
the
z-axis (excluding the
a relatively thick quarter-wavelength monopole with a
hemispherical cap, versus the polynomial degrees for the excitation region (which was part
of
adopted
the monopole.
to
be
20a/3 long) and for
The antenna was
analysed by
the major, middle the two-potential
equation. As
can be seen
from
Table 2. 2, very stable results for the monopole
admittance, and the peak total normalized electric field strength smaller
than
1% (excluding
the
end cap), are obtained
for
Note, ho\vever, that this remarkable stability would not the end
nl-.::_4 and n .:::_3. 2 be obtained if
effect were not properly included in the analysis.
eral rule, following
from
As
a
gen-
a large number of numerical experiments, the
TABLE 2. 2. A quarter-wavelength monopole admittance (in mS) and peak normalized electric field strength along the monopole axis (in %, excluding the end cap), versus degrees of polynomial approximation n1 (in the excitation region) and n 2 (along the main monopole part); a=0.01 :\, b/a=2.3.
2
3
4
5
2
21.92-j6.90 mS 30 %
19 .14-j6 .56 mS 7. 2 %
18.32-j6.30 mS 0.9 %
18.40-j6.40 mS 0.5 %
3
21.42-j7.36 mS 30 %
18.24-j6.60 mS 7. 2 %
17.70-j6.42 mS 1 %
17.70-j6.50 mS 0.2 %
4
21.64-j7.50 mS 30 %
18.12-j6.54 mS 7.2 %
17. 72-j6 .42 mS 1 %
17. 76-j6.48 mS 0.1 %
5
21. 86- j 7 . 6 0 mS 30 %
18.10-j6.50 mS 7.2 %
17. 74-j6 .56 mS 1 %
17.80-j6.48 mS 0.1 %
44
Ch.2.
density
of
the matching
antenna (away from
the
points
Approximations of excitation regions along
the
main
part of any thin-wire
excitation region, the end-cap regions and pos-
sible other discontinuities) should not be less than 6 points per wavelength in order
to
obtain reasonably accurate results for the antenna
admittance. 2.3.2.
Belt-generator
Essentially,
the
approximation
belt genera tor
model sketched in Fig.2.2.
is
Instead
of
coaxial-line
a mod if ica t ion of
of
assuming that
excitation. the
the
26
genera tor
width, d, of
the generator tends to zero, as in the delta-function generator approximation, it is assumed to be finite. a
belt
The belt generator thus resembles
of finite width >
In contrast to the
approach presented in Subsection 2. 3.1, where approximation of pressed field sources (i.e., the magnetic currents) belt-generator considered.
theory approximation
of
was
the impressed
the im-
sought, in the field
itself is
The main idea is to determine the size (length) of the ex-
citation region and the distribution of the impressed field along it in such a
way
that
the belt generator approximates as nearly as possible
the actual excitation of belt generator modes
near
can
take
into
account
It is worth mentioning that the to
some extent the higher-order
the coaxial-line opening, and not
approximation described cerning
the antenna.
the
in
derivations
only the TEM mode as the
the preceding subsection.
can be found in Reference 26.
The details conHere, only the
main idea will be explained and the final conclusions given, illustrated by some numerical examples. Consider first
the
real coaxial-line excitation of a monopole anten-
na, sketched in Fig. 2. 8 (a).
In the domain bounded by the surface S , 1 parts of the ground plane and the outer coaxial-line conductor, and the coaxial-line cross-section at complex.
For example, in
in addition avoid
the
to
the
the TEH mode,
analysis
of
z=z , the electromagnetic field is very 0 coaxial line a multitude of modes exists as
indicated in the figure.
In order to
the electromagnetic field in the domain, we can
approximate that domain with respect to both the antenna above z=z the coaxial line below z=z in
the
end section of
and 1 by a simple network, extract non-TEM modes
0 the line and take their influence approximately
Sect.2.3.
45
Approximations of coaxial-line excitation
+ TEM
mode only
(b)
(a)
(c)
FIG.2.8. Excitation region of monopole antenna driven by coaxial lin~ (a) actual case, (b) approximation to actual case, (c) circuit-theory representation of case (b). (Ref.26)
into account by appropriately chosen network. that the
Finally, we
TEM mode in the line extends up to the plane z=O.
proximation is shmm in Fig.2.8(b). approximating networks
can assume 27 This ap-
(Hare complicated and more accurate
can, of course, be
used
if desired.)
Thus the
system in Fig.2.8(b) can be approximated with respect to the end of the coaxial line and
the monopole
above
sentation shown in Fig.2.8(c).
z=z by the circuit-theory repre1 The parameters Gel and L were determin-
ed in References 26 and 28. Consider now
the same monopole antenna driven by a hypothetical belt
generator, shmm in Fig. 2. 9 (a). we obtain
the
Following similar reasoning as above,
approximation in Fig. 2. 9 (b)
and
its circuit-theory rep-
resentation, shown in Fig.2.9(c). In order that the two excitation regions shown in Figs.2.8(b) and 2.9 (b)
behave
in
the
same way with respect to the antenna, the two-port
networks in Figs.2.8(c) and 2.9(c) should be identical.
In the approx-
imation adopted this means that Cbg should be equal to Gel"
It is shown
46
Ch.2.
Approximations of excitation regions
(c)
(a)
FIG.2.9. Excitation region of monopole antenna driven by finite-size belt generator; (a) actual case, (b) approximation to actual case, (c) circuit-theory representation of case (b). (Ref.26)
in Reference 26 that a very simple conclusion is reached in this manner under certain conditions.
If
the
impressed field
at
the antenna sur-
face in the excitation region is assumed to be of the form
l(1 +COS 1TZ) E.
lZ
(z)
aa
aa
{0
,
Iz l.::_aa (2. 28)
IzI>aa
,
where V is the voltage across the coaxial-line opening, the coefficient a, determining
the
length aa of the belt generator on the monopole an-
tenna approximately equivalent
to
coaxial-line excitation, is obtained
from the following simple approximate equation: a=
In
2.18 (l-1). a
this manner
the
(2.29) belt generator approximately equivalent to a given
coaxial-line excitation can easily be found, The image theory
can now be applied to obtain the symmetrical system
equivalent to that in Fig.2.9(a). en in eqn. (2.28)
can be
The excitation field of the form giv-
introduced into any equation for current dis-
tribution along symmetrically fed distribution determined current needs
to
dipoles, and the approximate current
in a desired manner.
be adopted in
the
A separate expansion for
belt-generator region, because the
Sect.2.3.
Approximations of coaxial-line excitation
47
current is a rapidly varying function along the antenna in this region. For obvious reasons, the antenna current should z
satisfying
the width
of
the
condition
the
dl(z)/dzJz=O =0.
belt generator is larger
be
an even function of
Since, than
most
frequently,
the antenna diameter,
for application of the extended boundary conditions we can assume that, approximately, the impressed field given in eqn.(2.28) is the same along the antenna axis. In the case of the vector-potential equation, the two-potential equation, Pocklington's
and
Schelkunoff's equation, Eiz(z) given
in eqn.(2.28) is used as it stands.
Substituting the expression in eqn.
(2.28)
into
equation
the right-hand
side
integral of Hallen's equation (1.26),
for symmetrical dipole antenna with z =-h 1 the form
and
z =h that equation takes 2
h
f
l(z') g(r) dz'
+ c 1 cos
kz
kV
- . - Cg(z)
(2. 30)
JW\1
-h where TTZ 2 - 1 - cos- - (P - 2) cos kz aa 2 2 [ P cos k(z-aa) - (P - 2) cos kz], [P
2
J,
O
(2. 31) z>aa
2 K=kaa(P -1) and P=TT/(kaa). As a numerical example, given of
in
Table 2.3 are values of admittances
several monopole antennas analysed
to increasing degree principal antenna with
the
of
part.
using
eqn. (2. 30), corresponding
the polynomial approximation of current on the The results
are
delta-function generator (using
compared with those obtained the
same division of the an-
tenna into segments) and with experimental results. perimental results
is
seen
generator solution exhibits
to the
Agreement with ex-
be very good, while expected increase
the in
delta-function
susceptance with
increased degree of the approximation {compare with Table 2.1). Finally, it is interesting to note
that
the results for the monopole
admittance and overall current distribution obtained with the magneticcurrent frill
and
the belt-generator approximations
of
the excitation
48
Ch.2.
Approximations of excitation regions
TABLE 2. 3. Theoretical and experimental admittance (in mS) of vertical monopoles above ground plane, with a=3.175 mm, b/a=3, at frequency f=663.5 HHz. (Ref.26)
h/A
0.250
0.375
0.500
0.625
n
1
n
2
Theory, Hallen's equation belt-generator approx- delta-functionimation, a=4.36 generator approximation
4
3
18.4-j7.63
4
4
18.1-j7.74
18.4- j4. 74
4
5
17.5-j7.64
4
6
17.4-j7.78
4
3
3.10-jl.08
4
4
3.10-jl.04
4 5
3.08- j0.96
3.08+jl.95
4
3
2.00+j2.74
2.00+j5.64
4
4
1.97+j2.72
4 5
1.97+j2.73
1.97+j5.64
4
3
2.88 + j 7. 98
2.88+j10.9
4
4
2.78+j7.71
4
5
2.80 + j7. 75
4
6
2.84+j7.87
Experimental, coaxial-line excitation, b/a=3 (Ref. 17*) 17.84 - j 7. 50
17.6-j4.75
3. 10
+j 1. 82
3.16-j0.93
2.05 + j2. 78
2. 96 + j 7. 86
1. 80 + j 10.7
*
Please see Table 2.1 for explanation concerning experimental results in Reference 17.
region are very close sufficient accuracy consequence
of
1,
for
almost
all
If kb<<1,
both yield results of
practical applications.
This is a
very similar impressed electric fields due to these two
generator models, than
in all cases.
as
seen
from Fig.2.10.
or if higher accuracy
is
If kb
is not
much smaller
needed, the theory presented
in
the
following subsection may be used. 2. 3. 3. means
Higher-order approximations of coaxial-line excitation by 29 of wave modes. We have seen that, at lower frequencies, the
TEH approximation of both the electric and magnetic field at the coaxial-line opening, given
in
eqns. (2.16),
yields
relatively accurate re-
Sect.2.3.
49
Approximations of coaxial-line excitation
2a E.
v
0.8
1
0.7
z/a
FIG.2.10. Impressed electric field in excitation region, for ka<
sults for the antenna admittance (and, of course, for current distribution
in
general).
netic fields
at
However, if we consider the real electric
the opening, it becomes obvious that a
cannot approximate
the
30 . 1 ower f requencles.
actual
field
F or examp 1 e, at
TEM wave alone
distribution accurately, the
and mag-
even at
outer coaxial-line conductor
wedge, in reality both the radial (E ) and axial (E ) components of the p z -l/3 31 electric field are singular and proportional to (b-p) when p~b. With p=a
the and
TEM approximation, however, EP p=b.
Therefore, in
an
is
regular everywhere between
exact analysis higher-order wave modes
at the coaxial-line opening should be taken into account in addition to the TEM mode, Rigorous solution of
the electromagnetic field
shown in Fig.2.5 is not known.
for
the whole system
As a boundary value problem this system
is very complicated for direct numerical solution. possible, however, which combines
Another approach is
separate solutions
for
the electro-
rnagnetic fields in the upper half-space in Fig.2.5 (containing the monopole antenna), and in the lower half-space (i.e., in the coaxial line),
50
Ch.2.
in
terms
of
determined
a
Approximations of excitation regions
functional series.
from
that the tangential components of the
two
ciple,
half-spaces be this
approach
curacy, provided
The coefficients of
the boundary conditions
equal
at
the series are
the surface z=O, requiring
electric and magnetic fields in
the coaxial-line
opening.
In prin-
can yield results which may have any desired ac-
that
a sufficient number of
pansion is taken into account. be obtained with
the
at
In practice,
terms of
series ex-
very accurate results can
several terms of the series only.
rection of the antenna admittance thus found at the susceptance, but at
the
The principal cor-
lm~er
frequencies is in
higher frequencies a significant correction of
the monopole conductance results
as well.
In this subsection we shall
describe this method briefly and present some numerical results. The currents in the actual axially-symmetrical system shown in Fig.2.5 are only in the z-direction, or are radial (on the ground plane and the antenna end). and
Therefore the magnetic field cannot have the z-component,
TM modes
principal,
alone
can exist
TEM mode.
in
Of all the
the
coaxial line in addition to the
™mn modes, the
axially symmetrical, i.e., do not depend on the the
solution must
TM
modes.
on
be
sought in this case in
™on modes only are
~-coordinate.
the
form
of
Therefore a series of
modes in the coaxial line, travelling in the direction of The TM on the (-z)-axis, are of the well-known form (see, for example, Reference 32): Ezn (p)
-z 0n (knp)
E
(y /k ) n n
pn
(p)
H~n(p)
z
exp(ynz) 1n
-(jwE/y ) E n
(k p) exp(y z) n n pn
) z
n=l,2, ... ,
(2.32)
(p)
where (2.33)
Z. (x) 1n
J. (x) 1
and N. (x)
being
the Bessel and Neumann functions, respectively,
1
of order i, kn, Cn and Dn coefficients to be determined, and
(2. 34)
sect.2.3.
51
Approximations of coaxial-line excitation
the propagation coefficient of
the nth mode
in the coaxial line.
For
determining the coefficients kn' Cn and Dn we can use the boundary conditions Ezn(a)=O and Ezn(b)=O, and normalize th~ amplitudes of the modes in a convenient manner. system of
For example,
three nonlinear equations
we
can .adopt
z
(k a)=l V /m. A 1n n in kn' Cn and Dn' n=l,2, ••• , is
thus obtained, which can be solved using any convenient procedure. an example, given ponding
to
As
in Table 2.4 are values of these coefficients corres-
the above normalization,
for
n=l, ••• ,6,
and for a coaxial
line with b/a=2.3.
TABLE 2.4.
Coefficients k , C (V/m) and Dn (V/m) of TM modes for a n n on coaxial line with b/a=2.3 and n=l, ... ,6. n
1
k a
2.396
4.822
7.242
cn
1. 923
-2.098
0.526
-0.017
1. 770
-3.328
3.294
now represent
the
n
D
n
Let us
2
field strength at
5
6
9.661
12.079
14.496
2.075
-4.176
4.326
-1.234
-2.0ll
4
3
p-component, E (p), p
of
the total electric
the coaxial-line opening as a sum of the E -field of p
the TEH mode and those for higher-order modes, i.e.,
E (p) = E
P
where
E
V, to
be
po
(p)
+
I
w
n=l
n
E
pn
(p) ,
(2.35)
z=O,
(p) is the TEH-mode field, and E (p) is the TM -mode field, po pn on given in eqns.(2.32). Assuming the voltage at the coaxial-line opening, known, the TEH-mode field is given in eqn. (2. 16) , because the
voltage across the coaxial-line opening due to any higher-order mode is zero,
which
can be
proved
from
eqns. (2. 32) .
The coefficients w
n
in
eqn.(2.35) are unknowns to be determined. The equivalent system for a monopole driven by a coaxial line in this case is the same as before, sketched in Fig.2.6(a), except that now the total magnetic current of the frill is given as the sum
'"~
52
Ch.2.
J
-E
,(p) = -E (p) ms"' p
Eqn. (2. 36) ensures that
po
I
(p)
Approximations of excitation regions
w E
n=l
the
n
pn
(p)
(2. 36)
tangential component of the electric field
at the coaxial-line opening in the systems shown in Figs.2.5 and 2.6(a) be equal. We can now analyse the equivalent system shown in Fig.2.6(c) for every term of In(z).
this series separately, yielding partial antenna
currents,
Using the superposition principle we can find the total current
I(z) along the antenna as a sum of these partial currents,
I (z)
w I (z) n n
(2.37)
Of course, all components of the electric and magnetic field due to this total current can
similarly be
represented as a sum of partial compo-
nents (due to partial antenna currents). According
to
the
boundary
conditions,
the magnetic-field vee tor for z=O in 2.6(a,c) must be equal.
I+ 2TijWE
the
the
tangential
component
of
systems shown in Figs. 2. 5 and
On the other hand, from Ampere's law we obtain
1pEZ(p)
(2.38)
dp .
a Hence we
have an alternative, equivalent condition,
that
currents and
the z-components of the electric field in both systems must be equal for z=O.
This condition appears
easier
to
compute
and
because E (p) is z rapid variations with p for z=O
to be more
exhibits more
convenient,
We thus obtain that the coefficients w
than does
n
must be such
that the equation
a
the
In
this
z=O,
(2. 39)
equation, the left-hand side is the z-component
electric field in the coaxial line, given in terms of the field
Sect.2.3.
Approximations of coaxial-line excitation
of the higher-order modes
[see eqns. (2. 32)], Ezio (p)
53
and Ezin (p) the
fields of the antenna current due to excitation by the TEM and higherorder magnetic-current frills, and E f (p) and E f (p) the fields due to z o z n these frills themselves in the system of Fig.2.6(c).--' In order to determine the coefficients wn numerically, we must consider a truncated series instead of
the infinite series in eqn.(2.39).
We can then determine the coefficients of the truncated series approximately in various ways, of which the point-matching method appears to be the simplest.
A suitable choice of the matching points is
, b-a pp = a+ 2.;- (2p-l) , for a
(2.40)
p=l, ... ,m,
series truncated with the mth term inclusive.
In this manner
the points p=a and p=b are avoided, which should be done, since at p=a eqn. (2. 39)
is,
theoretically speaking,
identically satisfied,
and at
p=b the total field is singular. Once the
first m coefficients wn,
n=l, ... ,m,
are determined, the
problem is solved to the desired degree of approximation.
Current dis-
tribution along the monopole is given in eqn. (2. 37), truncated to the first m terms.
This current for z=O should be equal to the coaxial-
line innner-conductor current for z=O, i.e., m
I(O)
I (0) + 0
I
(2.41)
w I (0)
n=l
n
n
where ITEM is the current corresponting to the TEM mode and Ih.lg h er mo d es to all the higher-order modes in the coaxial line. Note that ITEM(O) is not equal to either the total current I(O), or to the current I (0) due 0
to the TEM-frill excitation alone, since currents in the antenna due to higher-mode frills and these frills themselves create an additional TEM magnetic field at z=O. The monopole admittance is obtained as y
which corresponds to
(2.42) the definition of admittance when measured using
54
Ch.2.
Approximations of excitation regions
standard reflection measurements.
To deter~ine the current ITEM(O) from
eqn. (2.41), note
that
ficients are known and
the ™on modes at z=O- are known, since wn -coefthe modes propagate in the (-z)-direction only.
Therefore \ve can easily determine the total current at z=O corresponding to the higher-order modes in the coaxial line as the sum of line grals of
H~n(a)
inte-
around the inner coaxial-line conductor at z=O,
-j2nawE according to eqns. (2.32).
I
z1n(kna) (2. 4 3)
w n=1 n
kn
However, we have adopted the normalization
z (k a)=1 V/m, so that, finally, we obtain for the monopole admittance 1n n computed with m higher-order modes I (0) y
Note
_o_ _ V that
m
I
+l
V n= 1
w [I (O) n n
+. 2nawE] J
(2.44)
kn
the first term on the right of this equation is the monopole
admittance computed with the TEM-frill approximation, according to eqn. (2.27), and the sum represents the correction. As a numerical example, cap, as in Fig.2.5,
consider a monopole with hemispherical end
with a=3 rnm, b=6.9 mm (i.e., b/a=2.3), at frequen-
cies between 1 GHz (corresponding to b=0.023A) and 8 GHz (corresponding to b=0.184 A).
Given in Table 2.5 are
quarter-wavelength monopole at
pole at f=8 GHz, obtained with m=6. coefficient wn,
the ratios E
pn
(a)/E
po
(a) for a
1 GHz (h/A=0.25), and for a 0.8 A monoThis ratio is proportional to the
and indicates how much the total field at the coaxial-
line opening is influenced by a higher-order mode.
It can be seen that
it is almost real and only weakly frequency-dependent, which means that the field at the opening is very nearly quasi-static. Shown in Table 2.6 are values of the corrective term of the monopole admittance, i.e., the sum in eqn.(2.44), for a quarter-wavelength monopole, a=3 mm,
b/a=2. 3 and at a frequency f=1 GHz, versus the number m
of the higher-order modes used for correction. frequency,
the
At this (relatively low)
correction is practically only in the antenna suscep-
tance, for about -0.21 mS.
Obviously, even with m=1 or m=2 the correc-
Sect.2.3.
55
Approximations of coaxial-line excitation
Ratio E (a)/E (a) for b/a=2.3, a=3 mrn and m=6, for f=l TABLE 2.5. pn po GHz and h=75 mm, and for f=8 GHz and h=30 mm. f=8 GHz
f=l GHz
n
-0.1973+ j0.0041 2
-0.2128+j0.0574
0.134 7- j0.0023
0.1411- j0.0312
3
-0.1088 + j0.0017
-0.1149+ j0.0233
4
0.0981- j0.0015
0.1021-j0.0198
5
-0.0907 + jO. 0013
-0.0955 + j0.0181
6
0. 045 3 - j 0. 000 7
0.0471-j0.0088
tion is sufficient for most purposes.
At f=8 GHz, however, the correc-
tion \vas found to be as high as ( -0.5 7 - j 1. 6 7) mS.
It was found by nu-
merical experiments
independent
monopole length.
that
these values
are almost
of
the
In general, numerical experiments have also indicated
that the correction in susceptance is practically proportional quency (or, more precisely, to ka
to
fre-
for b/ a=constant), while the conduc-
tance increases with frequency much faster.
Essentially, this takes in-
to account radiation from the coaxial-line opening. In conclusion, it may be said that the method presented above is the most accurate method available for analysis of antennas driven by a coaxial line.
However, it
able numerical skill.
is
rather complicated and requires consider-
Therefore it
cannot be recommended for monopole
analysis at lower frequencies (such that, approximately, kb<0.1), except if extremely high accuracy of the numerical results is required, but at higher frequencies (if kb
is larger than about
0 .1) it should be used
if accurate and reliable results are desired.
TABLE 2.6. Corrective term, !:::.Y (mS), of monopole-antenna admittance for a quarter-wavelength monopole, a=3 mm, b/a=2.3, at 1 GHz, versus the number m of higher-order modes. m
!:::.Y
m
!:::.Y
1
-0.021- j0.181
4
0.003-j0.214
2
0.012- jO. 229
5
0.002-j0.211
3
-0.001- j0.210
6
0.003- j0.212
56
Ch.2.
2.4.
Approximations of excitation regions
APPROXIMATIONS OF TWO-WIRE LINE EXCITATION
The two-wire-line excitation metrical wire
antennas.
is
used in practically all
cases of sym-
For example, a horizontal wire-antenna struc-
ture above ground, used for short-wave communications, is fed most frequently by a
two-wire line.
It is, therefore, of considerable practi-
cal interest to analyse this type of excitation in greater detail. Shown in Fig.2.11 is connected
to
is
a sketch of a symmetrical dipole antenna, which
a distant generator by means of a two-wire line.
The
TEM wave propagating from the generator along the line is partially reflected from dition
to
the
the
order modes
line end (at which the antenna is connected).
reflected
TEM wave,
is excited in
flection coefficient
of
the
the
a multitude of
line near
its end.
TEM wave depends
on
In ad-
evanescent higherObviously, the re-
the line termination
(i.e., on the antenna properties) and on the excited higher-order modes. Considering
the
antenna as one-port network connected at the line end,
the antenna admittance
is
defined as
the ratio of current and voltage
of the TEM mode at the antenna terminals. The antenna admittance thus defined the transverse dimensions of
depends
to
the two-wire line.
a
certain extent
on
However, if these di-
-___________,:___.JI -.---
-
TEM waves only
towards generator
Sketch of FIG. 2 .11. two-wire line.
-- ~orderm--~Lde
1TEM + high~d __s ______
a symmetrical wire
antenna driven by very long
Sect.2.4. mensions
Approximations of two-wire line excitation are much smaller
most negligible, mately
using
and
than
the wavelength, this dependence is al-
the antenna admittance
belt generator.
tends
that
to
can be
TEM magnetic-current frill or the
The delta-function generator the distance
zero, and
can
d
between
the
be used with
approximation amounts to
wires of
The other two representations
are not physically obvious in this case, but they lutions
of
of
and
the
reasonable accuracy if
the
can
frill
critical parameters, because in
and the
the
yield stable so-
two-\vire line transverse dimen-
the antenna radius are sufficiently small.
magnetic-current
the two-wire line
the same order of accuracy as in
the case of the coaxial-line excitation.
sions
obtained approxi-
any convenient approximation to the real excitation, such
as the delta-function generator, the
assuming
57
The outer radius
belt-generator length are
corresponding case
not
of an electri-
cally thin monopole antenna driven by a coaxial line the antenna admittance is practically independent of the coaxial-line dimensions. However, if the two-wire line transverse dimensions are of
the
order
of 0.001 A or larger, a more precise analysis is necessary. An approach 3 to this problem was presented by King, based on an equivalent L-C network bet\veen the
two-wire
line and
the
idealised antenna.
Although
yielding good agreement with experimental results, this method involves a somewhat artificial idea of the antenna intrinsic admittance.
Another
approach is to treat the wire antenna and the two-wire line sketched in Fig. 2.11
together,
proximation of tem
as
a
unique wire structure,
the real excitation.
thus
involving no ap-
Current distribution in this sys-
can be determined using, for example, the method described in Sec-
tion 1.2.
In order to be able to analyse the system in this manner, it
is necessary
to
introduce certain assumptions concerning
the
two-wire
line. One possibility is to consider the line to be straight and semi-infinite.
We
can
waves
along
then analyse the structure as having two progressive TEM
the two-wire line up to its end, with a corrective term in
the end zone, requiring that boundary conditions for the electric field along the line and the antenna be satisfied.
This procedure, using the
Hallen equation with polynomial approximation for current along the an-
58
Ch.2.
Approximations of excitation regions
tenna and for the corrective term at the line end, was proposed in Reference 33, resulting in accurate antenna admittance. this method
is, perhaps, that
involved in
the
rather
A disadvantage of
complicated double integrals are
analysis, which is inherent to the Hallen equation for
non-cylindrical wire structures.
This deficiency can be removed if the
two-potential equation is used. Another possibility is of relatively small
to
consider the dipole to be driven by a line
length, as in Fig.2.12, and to solve the system as
any other wire-antenna structure.
In that case, however, the generator
must be included into the analysis, which raises the question concerning the influence of the generator approximation on final results.
Another
question in this approach is the length of the two-wire line needed for obtaining accurate results tribution.
for
the antenna admittance and current dis-
Both problems can, fortunately, be
solved
quite simply, as
outlined below. To be
able
to
obtain unique results
have to assume that
the
sufficient length, so
for
the antenna admittance, we
section of the two-wire line in Fig. 2. 12 is of
that
the coupling between
the
generator and the
antenna exists only through the TEM waves along the line.
generator region
On the other
TEM-waves
X
FIG.2.12. Sketch of symmetrical V-dipole antenna connected to generaThe generator is assumed to tor by short segment of two-wire line. be a TEM magnetic-current frill.
Sect.2.4.
Approximations of two-wire line excitation
hand, the shorter numerically.
the
59
section, the easier it is to analyse the problem
By extensive numerical experiments it was
found that the
length of
this section is not very .critical, and that in some cases as
little
0. 25 A is sufficient to approximate accurately the two-wire-
as
line excitation.
This
can
be also verified
from
the example shown in
Fig.2.13, where the admittance of a symmetrical dipole antenna is plotted versus
the
two-wire-line length, for various degrees of
the
poly-
nomial approximation of current along the line. What concerns
the
type of the generator to be used in representation
shown in Fig. 2.12,. is obviously of
little importance, provided that
the line is
Therefore any desired approximation
of
sufficient length.
can be used, even the delta-function generator. To solve for current distribution in the integral antenna-feeder system shown
in
Fig.2.12, we
can,
in principle, approximate current dis-
G,B (mS)
\
14 13
·'x-----G ,,'" ·-·-·---·-·-·-·---::..:..,,',
·-·-·--·--·-
12
B
FIG.2.13. Conductance (G) and susceptance (B) of dipole antenna shown in Fig.2.12, for a=O.Ol A, 8=90°,, h=0.2149/~ and d=0.03086 A, versus the line length (L) for various degrees--(n) of polynomial approximation of current along the line. - • - n=4; - - - n=S; - - - n=6.
60
Ch.2.
Approximations of excitation regions
tribution along the two-wire line by a sum of two TEM waves propagating in opposite directions, with unknown in
the
vicinity of
of
the
TEM regime
in
these
regions.
line it is much more convenient along
the
amplitudes, with additional terms
the line ends in order to approximate perturbation
whole line, because
to the
However, in the case of a short
use simple polynomial approximation line is
then
treated together with
the antenna in a unique way and no significant adaptations of the program (in which polynomial
approximation
segments is assumed) are necessary.
of
Along
current the
along all
antenna
antenna itself, current
expansion needs to be adopted according to the suggestions given in the preceding chapter. at x=O and
x=L.
Naturally, continuity of current must be postulated Hhat concerns
the
order
of
approximation along
the
two-wire line, it was found that it suffices to adopt a 4th-degree polynomial
for
the
line
segment
if
its
length
is of the order of
A./ 4,
a
6th-degree polynomial being, however, sufficient for segments of lengths up to one wavelength (see Fig.2.13). tion is required near
the
If more precise current distribu-
line ends, separate subsegments can be taken
in these regions. Once the current distribution has been determined, the antenna admittance
can
be computed from
any
two
values
of
current in the central
two-wire-line part, using the formula I(x') sin kx" - I(x") sin kx' I(x') coskx"'
y = jYc I(x") coskx'
(2.45)
xl<x',x"<x2
where y
(2.46)
c
is the characteristic admittance
of
the two-wire line.
the antenna admittance can be expressed in terms
of
Alternatively,
current intensity
and its derivative, di(x)/dx, at a single point x, x <x<x , as follows: 1 2 ki (x) cos kx - I' (x) sin kx + ki(x) sin kx '
xl<x<x
y = jYc I' (x) cos kx
Both formulas
for
the antenna admittance
,
I • (x) = did(xx) .
2 can
easily be
(2. 4 7)
obtained from
the well-known expression for current distribution along a transmission
Sect.2.4.
61
Approximations of two-wire line excitation
line, I (x)
where
j Yc V
0
v0
sin kx
+
I
0
is the voltage and I
TEM mode, and noting that Y=I It was
(2.48)
cos kx , the current
0 0
at x=O corresponding to the
/v • 0
found by extensive numerical experiments that eqns. (2.45) and
(2.47) yield nearly
the
same results, provided
that
the points x' and
x" in eqn.(2.45), i.e., the point x in eqn.(2.47), are within the limits of
the
TEM region along the line, and that x' and x" are not extremely
close together.
The
latter
case, obviously, may result in significant
numerical errors when computing Y according to eqn. (2.45). is presented in
Fig. 2.14 for the case of a symmetrical dipole sketched
G,B (mS) 14
r
--,-------
---------
----
--------
G
B --~
0~~--J_
0.1
-1
-2
r
An example
0.2
--
-~
~"
.,---
x x' I' T
........................
__L _ _ L_ _~~---L--~~--~----0.3 0.4 0.5
0.6
0. 7 0.8
0.9
'\
1.0 \
''
\
'\
FIG.2.14. Conductance (G) and susceptance-~B)_of dipole antenna shown in Fig.2.12, for a=O.Ol A, 8=900, h=0.2194 A, d=0.03086 A, 1=0.5 A and n=6, versus positions x' and x entering eqns. (2.45) and (2.47), with x"=L-x'. - - - G and Bfromeqn.(2.45); - - - GandBfrom eqn.(2.47).
62
Ch.2.
in Fig.2.12.
The
formulas (2.45)
Approximations of excitation regions
antenna admittance was
and
(2.47), but taking
the
computed according
to
both
points x' and x" symmetri-
cally, i.e., x"=1-x'. Finally, theoretical and V-dipole
antenna
sketched
experimental results
for
admittance of the
in Fig. 2. 12, for a=3 mm, 8=45 °, d= 12 mm and
1=200 mm, are plotted in Fig.2.15 versus the antenna-arm length, h', at frequency f=883.3 MHz.
Solid lines denote results obtained by the pres-
ent theory, dashed lines (which almost denote results obtained
in
the solid lines)
Reference 33 using Hallen's equation and an
infinitely long two-wire line, while sults, obtained according
coincide with
to
the
points
represent experimental re-
procedure outlined in
the
following
G,B (mS)
24
.-·· h'(mm) 150
160
FIG. 2.15. Conductance (G) and susceptance (B) of V-dipole antenna shown in Fig.2.12, for a=3 mm, 8=45°, d=12 mm and 1=200 mm, at frequency f=883.3 MHz, versus antenna-arm length, h'. present theory; - - - results obtained by Hallen's equation (Ref. 33) ; oo, xx experimental results (Ref.34);- • - theoretical results for V-dipole antenna driven directly by a TEM magnetic-current frill.
Sect.2.4. subsection. sults
63
Approximations of two-wire line excitation
is
Good agreement between
observed.
theoretical
and
experimental re-
For comparison, theoretical results for admittance
of a symmetrical V-dipole antenna are presented if the antenna is driven directly by
a
TEM magnetic-current frill with
sponds to 1=0 in Fig.2.12). cy with
the
b/a=2 (which corre-
These results are in significant discrepan-
results obtained for the two-wire-line excitation, showing
that adequate modelling
of
the antenna excitation
is
indeed necessary
in the present case. 2.4.1.
A method for
measuring admittance of symmetrical antennas by 34 reflection measurements in coaxial line. Direct measurements of input admittance of symmetrical antennas driven by symmetrical two-wire lines is a very difficult task. measurement of which is
Firstly, there is no
impedances
of
comparable with
Another problem is
symmetrical
that
of
the influence
standard equipment for
systems
the
performance
of
the classical coaxial slotted line. of
the antisymmetric wave mode which
can be excited due to small asymmetries in the system caused by instrumentation situated in two-wire line. by asymmetric
The
the
same
electromagnetic field of the antenna and the problem is present when measurements are made
coaxial-line instrumentation using standard balance-to-
unbalance transformers, the accuracy of which is not high and is always influenced by the presence of antisymmetric wave modes. The best way to overcome these difficulties is to perform the measurements on the asymmetrical system shown in Fig.2.16, which is an approximate asymmetric equivalent of tem.
In
this
case
the
image
the considered symmetrical antenna systheory applies, so that at a sufficient
distance from the coaxial-line opening this valent a
to
situation is closely equi-
the initial symmetrical antenna driven by a short segment of
two-wire
line.
The admittance measured at
the reference plane 2-2'
on the asymmetrical system shown in Fig.2.16 is twice the admittance of the symmetrical antenna considered. Measurements
can
connected behind order
to
now be
the
made using a
perfectly
standard slotted coaxial line
conducting plane
(see Fig. 2. 16).
In
preserve high accuracy achieved by slotted-line measurements,
64
Ch.2.
Approximations of excitation regions
FIG.2.16. One half of a symmetrical antenna, together with a segment of a twowire-line conductor, driven by a slotted coaxial line.
-------}., I
2'
'
',
' .......
slotted coaxial line
a special procedure must the reference
plane
be adopted
2-2'.
If we
for
calculating
the
consider only TEH modes
impedance at on
both the
coaxial line and the wire line, the system between reference planes 1-1' and
2-2'
can be
represented by
a two-port network.
The impedance Zi
seen at the reference plane 1-1' corresponding to the antenna terminating impedance Zt at the reference plane 2-2' can be expressed
in
terms
of the Z-parameters of this two-port network as
(2. 49)
The parameters
z11 , z22
coaxial
the wire
two,
line,
but
z12
and line
depend not only on the geometry of the
and
the transition region between
also on the positions of the reference planes 1-1'
We can, therefore, adopt these positions in
and
these 2-2'.
such a way that the trans-
formation in eqn.(2.49) be as simple as possible. One way on
site
variable
to determine
the
Z-parameters accurately is to measure them
by using three known reference impedances. reference
impedance
at
high
frequencies
The most is
a
reliable
movable short
circuit and it was, therefore, used in the present measurements. fix the movable short circuit in the reference plane 2-2' antenna terminal, so minimum along
the
(i.e., at the
Z =0), we can find the position of a voltage t
the slotted coaxial
reference plane 1-1'. at
that
If we
line
and take
this
position as the
If the two-port network is lossless, the voltage
voltage minimum is zero, and therefore the impedance Zi in this
Sect.2.4.
Approximations of two-wire line excitation
reference
plane
that z
is
also
zero.
For this
65
case we have from eqn. (2. 49)
2
z -z =o, and eqn.(2.49) for arbitrary Zt reduces to 11 22 12
z.l
(2.50)
The parameters z line beyond
and z can now be measured by extending the wire 22 11 the reference plane 2-2' and by sliding the short circuit
along the line, as shown in Fig.2.17.
If we slide the short circuit to
x =xz=A/4, we have Zt=Z~=oo. By locating the voltage minimum in the co2 axial line at x =xi , we can compute the input impedance as Zi_=j zc • 1 1 tan kxi , where zc is the characteristic impedance of the slotted co1 axial line. From eqn.(2.50) we have immediately z =Zi_. By fixing the 11 position of the slotted-line probe at x =xi=A/4, we can now slide the 1 short circuit to a new position x =xz which results in a voltage mini2 mum at x =xl, i.e., in Z'i_=oo at the reference plane 1-1' (since current 1 in that plane is then zero, and voltage is maximal). In this case we have, from eqn. (2.50), z
22 =-Z~=-jzc 2
tan kxz,
where zc
2
is
the charac-
teristic impedance of the wire line, which is one half the characteristic impedance of the symmetrical two-wire line obtained if image theory is applied.
Having thus determined the parameters Z
now remove
the
wire
measure
line,
short Zi
circuit and
connect
the
and z , we can 11 22 desired antenna to the
in the coaxial line and, from
eqn. (2.50),
then
easily compute the antenna impedance Zt at the reference plane 2-2'.
FIG.2.17. Measurement of equivalent- network Zparameters.
Z. l
66
Ch.2.
2.5.
Approximations of excitation regions
CONCLUSIONS
Presented in
this
chapter were various approximations
tations of wire-antenna structures. useful
to
For practical
to
applications
know which of these is most suitable in a
Although this was
mentioned when
actual exci-
given
it
is
situation.
describing a specific approximation,
it may be worth summarizing these suggestions at one place. If radiation pattern of
a wire-antenna structure
alone
is of inter-
est, it suffices to use the simplest, delta-function generator approximation.
The overall current distribution obtained with this approxima-
tion is quite accurate everywhere except near the generator itself (see Fig.2.3).
The antenna radiation pattern obtained with this current dis-
tribution is very close to the actual pattern, so tion suffices for practical applications. tion generator approximation tance which are Fig.2.4). tance,
results
that
this approxima-
In addition, the delta-func-
in values
of
the antenna admit-
good indications of the real values (see Table 2.1 and
However, the antenna admittance, and particularly the suscep-
are
substantially dependent
on
the order of
approximation for
current, so that they cannot be considered as reliable. this, the delta-function approximation can into
the
Hallen equation,
and more
In addition to
easily be incorporated only
sophisticated approximations
are
preferred with other types of equations. If accurate value of the admittance of a wire-antenna structure driven by
a coaxial line is desired, the TEM magnetic-current frill or the
belt-generator approximations two approximations tically
the
Table 2. 3).
same
are
of
the excitation
results,
which are very accurate in most to
used.
The
cases (see
note that for obtaining stable and re-
liable values of the antenna admittance using is
be
of the same order of complexity and yield prac-
It is important
real excitation, it
should
essential,
in
any
particular
approximation to the for
thicker antennas
near resonance, to take into account the end effect. The most accurate approximation of the coaxial-line excitation available
is
the magnetic-current
axial-line modes
in addition
frill to
which
represents
the TEH mode.
higher-order co-
The admittance correc-
Sect.2.5.
67
Conclusions
tion term thus obtained (with respect to the TEM magnetic-current frill excitation) is relatively small at lower frequencies, and is practically imaginary (i.e., 2.6),
but
almost
correction
at higher frequencies
the
both susceptance and conductance.
in
susceptance only - see Table
correction becomes substantial in
Thus, at
latively complicated procedure should be
lower
used
frequencies this re-
only
rate theoretical values of admittance are required. cies, however,
when
radiation
from
the
if extremely accuAt higher frequen-
coaxial-line
opening
becomes
noticeable, this method should be used if accurate value of the antenna admittance is desired. The two-wire-line excitation can sidering
the
antenna and a relatively short segment of
integral wire structure. ture
is
be modelled very accurately by con-
considered
most often difficult
to to
Note, however, that in be
strictly
symmetrical.
the
the line as an
theory the struc-
In practice
this
is
achieve, which may result in discrepancies be-
tween theoretical and experimental results.
CHAPTER 3
Treatment of Wire Junctions and Ends
3.1.
INTRODUCTION
When dealing with wire-antenna structures having several interconnected wires, particular attention must be wires
are
paid
in which
the
joined together, in order that a numerical solution for
the
antenna current distribution be accurate.
to
the
zones
Sharp bends
along otherwise
continuous wires fall into the same category, as well as abrupt changes of wire diameter, and nique.
can be treated using essentially
These three classes
of
discontinuities
antenna structure we shall refer to as "junctions". out in
the
the
same tech-
along wires of a wireAs already pointed
previous chapters, wire ends deserve also special attention
if stable and very accurate results for
the
antenna admittance are de-
sired. At wire junctions and ends and in their immediate vicinity the antenna current
and
charge distributions, as well as their fields, exhibit
rapid variations.
->-
For example, the surface-current density vector, Js'
has locally nonaxial components, which pronounced.
On
the
other hand,
the
for
some geometries can be very
axial component in
these
regions
may vary rapidly along the wire length or circumference, or may even be singular.
Obviously, a rigorous treatment of
complex problem.
junctions and ends is a
In principle, it can be solved by accurate approxima-
tions of current and charge distributions on the surface of the antenna junctions and ends and
in their immediate neighbourhood.
Fortunately,
in most practical cases such an involved analysis is unnecessary if dimensions of these discontinuities are
relatively
small when
compared
70
Ch.3.
with
the
lengths of
the wire segments interconnected at
and with the free-space wavelength. of current distribution of
these
Treatment of wire junctions and ends the junction
In these cases the local variations
can often be neglected if the
integral
variations is properly taken into account.
efficiently by postulating some constraints
on
effect
This can be
done
solutions of the integ-
ral equations for the antenna current distribution described in Chapter 1.
Obviously, the manner in which the wire segments are interconnected
in
space
into
(i.e.,
the
the
antenna geometry) must
be
included appropriately
antenna analysis, since this generally has appreciable influ-
ence on the antenna current distribution. The constraints which
include approximately the effects of junctions
and of ends into the mathematical antenna model are established following
the
same general
together in
the
following
treatment of wire ends strating
to
lines
what
of reasoning. two
is
sections.
Therefore
they
In Section 3.4
are treated the
rigorous
presented, showing its complexity and demon-
extent such an analysis can improve accuracy of the
approximate antenna current distribution.
3.2. If
CONSTRAINTS RESULTING FROH FIRST KIRCHHOFF'S LAW
only
thin-wire structures
junctions or fore
the
are
considered, any
wire ends is small compared with
the
linear
dimension of
wavelength.
There-
fields in the domains of the junctions and wire ends are very
nearly quasi-static, so that these discontinuities can be treated using a kind of circuit-theory relations.
Hence, the wire segments should be
regarded as branches, and the wire junctions and ends simply as the end points of
these branches, i.e. , as
the
nodes.
In this way the actual
shape of the junctions and ends is neglected, but
an
information about
their geometry is retained. In
the
formulation
of
some
equations
for
current
distribution in
wire-antenna structures, the equation of continuity is implicitly taken into account by
adopting the expressions for the vector and scalar po35 tentials in the Lorentz gauge. Therefore the exact solution of these
equations would
sat.isfy the equation of
continuity automatically.
In
the derivation of the two-potential equation for arbitrary wire-antenna
Sect.3.2.
71
Constraints from first Kirchhoff's law
structures,
in eqn. ( 1. 15), as well
given
as in
the
derivation of the
special case of this equation for cylindrical antennas, eqn.(1.19), and the Schelkunoff equation, eqn.(1.29),
(1.13), was used explicitly.
the
equation of continuity, eqn.
However, it is assumed, in addition, that
there are no
excess charges at the ends of
any antenna segment.
implies
the
the wire
that
continuity equation
for
This
currents at every
node is forced to be in the form of the first Kirchhoff's law, i.e., n
I
m=1
I
0 ,
m
where n
is
(3. 1)
the
number of
wire segments meeting
reference direction of a current is from representing a
wire-segment
requiring that
the
was obtained from
end,
current at
this
the sum
at
the node, and the
node outward. reduces to
the wire end vanishes.
a basically artificial constraint,
one
For a node term only,
Since eqn. (3.1) it must be
used
with eqns.(1.15), (1.19) and (1.29). In reality, junctions and along
the
the
distribution
of
charges accumulated at
ends differs significantly from
rest of the wire segments.
the
the antenna
charge distribution
Therefore, in a rigorous numeri-
cal solution these two distributions must be modelled separately. tunately,
the
charges accumulated at
the antenna discontinuities are
usually relatively small compared with of
the wire
nodes
can
segments.
often be neglected,
satisfied by were
Therefore,
the total charge along the rest
the
excess charges at
their
field should be
potential and Schelkunoff's equation. a term should be
the
antenna
and eqn. (3.1) is, indeed, approximately
the wire-structure currents.
not neglected,
For-
Note that
if
these charges
incorporated into the two-
Thus, for example, in eqn.(1.15)
added to the left-hand side for every node, being es-
sentially the electric field due to a point charge at the node. Although it when using
is
not necessary
the equations
based
to on
postulate the first Kirchhoff's law the magnetic vector-potential only
(i.e., the magnetic vector-potential, Pocklington' s tions, given in
and Hallen's equa-
Section 1. 3), the use of eqn. ( 3 .1) is always desirable
in approximate numerical
solutions, because
this
improves convergence
72 of
Ch.3.
Treatment of wire junctions and ends
the
results, particularly with continuous approximations of the an11 36 tenna current distribution. • In some cases current distribution can be
approximated in such a way
that eqn. (3.1) be automatically satisfied at the antenna ends, e.g.' by constructing
special
polynomials which vanish at wire ends,
10
or
by
using overlapping functions which are zero at the ends of their sub23 domains. Explicit formulation of eqn. (3.1) can also be avoided by 36 . . rnan1pu . 1 at1ons. . certa1n rna tr1x However, \ve s h a 11 deal here only with simple polynomial approximation of eqn. (3.1)
needs
alway'?
to
current distribution, and therefore
be explicitly formulated
and
added
to
the
point-matching equations. In with
the
theoretical'· analysis the antenna segments frequently coincide
the wire segments of the real antenna.
real antenna in order
to
segment must be obtain accurate
subdivision is
However, in some
subdivided into two solution for
or more subsegments
current distribution.
necessary when current distribution
along one part of
the
tation region, as pointed out
An example is
This
rapidly varying
the
the
other
antenna exci-
in Chapter 2, with the only exception of
the delta-function generator excitation, in which sion is undesirable.
When wire
wavelength, they
be subdivided
can
is
physical antenna segment, while along
part it is a slowly varying function.
cases a
case such a subdivi-
segments are very long in terms of the into
subsegments in order
degrees of the polynomial approximation relatively low. the degree of the polynomial approximation requires
a
to
keep
An increase of higher-order in-
tegration formula for evaluating the integrals encountered in the equation for
the
antenna current distribution,
the programming.
In addition,
shorter subsegments
the
can be more
which
somewhat complicates
lower-degree polynomials defined on
flexible
than
a single higher-degree
polynomial on the whole segment. Obviously, a junction of not represent Therefore, functions
a
both at
two coaxial subsegmen ts of equal radii does
physical discontinuity in the
current
and charge
the
real antenna structure.
distributions
are
continuous
such a node, so that, in addition to eqn. (3.1), the equa-
tion expressing continuity of
the first current derivative at the node
Sect.3.3. may be
added to the point-matching equations.
rule are two subsegments on as
73
Junction-field constraints
described
in Section
the
is
An exception
of
this
two sides of a magnetic-current frill,
2.3.
first current derivative
26
In
that· case
the
discontinuity in the
determined by eqn. (2. 23).
Formulation of
equations for the first current derivative is not necessary except with the Schelkunoff
equation,
but
it usually improves
the
convergence of
the solution. Eqn. (3.1)
cannot be formulated
for
nodes lying in
the
ground plane
for antennas driven by a coaxial line, because it is automatically satisfied in the equivalent symmetrical system. antenna
segment
plane (or
terminating. at
closely
such a
point
Only if there is a single at a
to zero, and this equation can replace eqn.(3.1). tween
the
right
to that angle), current derival'.ive
angle on the
can be set equal
For other angles be-
segment and the plane, or if there are more segments meeting
at that node, junction-field constraints should be used.
3.3.
JUNCTION-FIELD CONSTRAINTS
Consider a wire-antenna structure assembled from Ns segments (including in this number the wire subsegments), and having N. junctions. Let the J total number of unknown current-distribution coefficients be Nc. Hence, the
total
number
of
equations needed is also Nc.
For such an antenna
we can write N. equations according to the first Kirchhoff's law. Thus, J if only equations expressing the first Kirchhoff's law are formulated N. antenna junctions, the number of the point-matching equaJ It is easy to conclude that for a given number tions must be (N -N.). c J of antenna segments, Ns, the number of point-matching equations depends
for
on
the
the
antenna topology.
tributed along This is very
all
the
Therefore the matching points cannot be disantenna segments
in ·the
same
general manner.
inconvenient, because it prevents a unique choice
number and positions of
the
of
the
matching points along all the antenna seg-
ments. In addition hoff's
law
to
this disadvantage, postulating only
to be valid
for
merically stable solutions
the first Kirch-
a junction is not sufficient to obtain nufor
the antenna current distribution if the
74 two-potential
equation
is
Ch.3.
Treatment of wire junctions and ends
used.
This
can
be
explained
as
follows.
Neither the antenna theoretical model, described by eqn.(l.lS), nor the approximation for current distribution, given in eqn.(l.l6), can represent adequately the antenna junctions (as well as ends). boundary conditions in fied, so
that
the
junction regions cannot be
Therefore the properly satis-
by choosing the matching points arbitrarily and/or close
to the antenna nodes might result in numerical instabilities. Both problems can be solved by introducing additional
constraints at
every junction, which we shall refer to briefly as the junction-field 8 constraints. These constraints essentially minimize the influence of an improper current-distribution approximation in junctions
on
the vicinity of
the overall current distribution, by taking
effect of
the approximate current distribution in
to be
same
the
as in a
rigorous solution.
these inaccuracies is confined only vicinity.
to
the
the
the
integral
junction region
In this way the effect of
the
junction and its immediate
At the same time these constraints enable a unique choice of
the matching points along all segments of the wire-antenna structure. The junction-field constraints represent, in a way, the second Kirchhoff's la\v for
the
antenna nodes.
considerations, as follDws.
They
can
be deduced from the power
In any numerical solution for current dis-
tribution in perfectly conducting antenna structures ditions are satisfied only approximately. el the antenna electric ial
total
for
the
etc.
this
total
and
power, the
antenna current
The
boundary con-
Hence in the theoretical mod-
currents are situated in a nonzero locally ax-
electric field,
The larger
the
thus generate a
less
nonzero complex power.
accurate, on average, are the results
distribution,
admittance,
radiation
pattern,
locally axial electric field may be particularly large
near the antenna junctions (as well as ends) because of inadequate modelling of current distribution in these regions. field the
low,
one
vicinity
increase
in
of the
can
try
the
to
better approximate current distribution in
junctions.
number
of
In order to keep this
However,
unknown
this
leads to a significant
current-distribution coefficients.
If the specific surface distribution of currents in the vicinity of the junctions
is
ignored,
as
has
been
done
in
eqn. (1.15), this increase
Sect.J.J.
Junction-field constraints
75
might even fail to give any improvement. The
length of
the
order of
a few radii
Therefore
current
zone of of
junction is relatively small, of the
the thickest wire
intensity
tion does not vary
the
in
significantly
the wires
connected at the junction. interconnected at the junc-
in this zone.
The complex power gen-
erated by the antenna currents in the vicinity of a junction of
n
seg-
ments can hence be approximately evaluated as n
s
z:
m=1 where
(3.2)
U I* m m '
Im is the current intensity at the junction node in the wire la-
belled "m" (Fig.3.1), the asterisk denoting the complex conjugate,
um
(3.3)
f
L
m
is the voltage along the path Lm lying along the mth wire-segment axis (Fig.3.1),
and
(E+E.) ~
is the total electric field.
From eqn.(3.1) and
Fig.3.1 it follows that n
I* 1
(3.4)
z:
m=2
so that eqn.(3.2) can be written in the form
FIG. 3.1. Symbolic representation of a junction of n wires.
2
76
Ch.J.
Treatment of wire junctions and ends
n
L
S
(Urn-
m=2
u1 ) li
(3.5)
(Urn -u ) is the integral of the total electric field 1 along the path L m lying on the axes of the segments denoted by "1" 1 and "m", as shown in Fig. 3.1. The power generated by the antenna curThe
difference
rents
in
the vicinity of
the junctions will
ences are zero for m=2,3, ... ,n,
0 ,
be zero if these differ-
i.e., if
(3.6)
m=2,3, .•• ,n .
Eqns.(3.6) represent the junction-field constraints. quire that
the
a junction oscillates around zero, so but
they
In fact, they re-
total· electric field along the wires in the vicinity of
allow,
in
principle,
that
that
its average value
be
zero,
the electric field may have high
local extrema. By numerical
trial-and-error method it was
approximately optimal along every segment. tion
are
path
length
is
about
found that in most cases three
to
four wire radii
If radii of the wires interconnected at the junc-
different, this path length is determined by
the largest ra-
dius. The integral in eqn. (3.6) can be repeated midpoint points of vided.
rule, i.e., by simply summing
subsegments of
the
equal lengths into which
integrand at midthe path is subdi-
Eqn.(3.6) is thus reduced to summing (averaging) several point-
matching equations. suffices
approximately computed by using the
to
take
It was also found by numerical experiments that it
only
L m if the radii of 1 more than about 2:1.
two integration points
per
the wires interconnected at
one arm of the path a node do not differ
These points are denoted by asterisks in Fig.3.1.
For larger differences betw·een
the
wire radii the number of the integ-
ration points should be correspondingly larger. According
to
eqns. (3. 6) , for a junction of n
(n-1) equations resulting
from
the
wires we can formulate
junction-field constraints.
These
equations, together 'vith that obtained using the first Kirchhoff's law,
Sect.3.3.
Junction-field constraints
form a set of exactly n equations.
77
This is true for the wire ends also
(where n=l), but it should be noted that constraints in eqns.(3.6) need not be to
be
used for the ends. zero,
and
At these points the currents are postulated
therefore complex pmver
of
these currents is usually
negligible. As
a
consequence, for the whole wire structure exactly
equations can be associated with every antenna segment.
two
of these
Hence, if the
degree of the polynomial approximation is nm, i.e., if there are (nm+l) unknown
current-distribution
point-matching equations are
coefficients required
choice for all the antenna segments. points should
be
for
along
that it,
segment,
which
is
a
(nm- 1) unique
For obvious reasons, the matching
distributed relatively uniformly along every segment,
avoiding junctions and ends, at which the current distribution is relatively inaccurately approximated. has
the
Thus, if
the
origin at one segment end, and if hm is
segment the
local
sm-axis
segment length, a
suitable choice of the matching points is s
.1.E_::-_!_ h 2n -2 m
mp
(3. 7)
p=l,2, ... ,(nm-l) .
m
Of course, for a junction of
two
coaxial wire segments of the same ra-
dius at which a magnetic-current frill is situated, instead of eqn. (3. 6) the equation expressing discontinuity
in
the
first
current derivative
should be used [see eqn.(2.23)]. As
an example, shown
in Fig.3.2 is
a
"top-loaded" monopole antenna
driven by a coaxial line, which is modelled by the TEM magnetic-current frill.
Depicted in Fig.3.3 is the total axial electric field along the
axis of segments 1, 2 the
and
impressed electric
frill.
3 of the antenna, normalized with respect to
field
at
the
center
of
the
magnetic-current
The matching points are denoted by circles in Fig.3.3.
ample illustrates clearly practically all tion relating
to
the
total
wire antenna resulting from
the
axial electric field along the
This ex-
conclusions of this secthe wires of a
approximate numerical solution of
the
two-potential equation with polynomial approximation for current, subject
to
relative
the constraints described above. total
If we
note the scale of the
axial electric field, it may be concluded that the cur-
78
Ch.3.
Treatment of wire junctions and ends
FIG.3.2. Sketch of top-loaded monopole antenna driven by coaxial line. All dimensions are in millimeters. ~
Segment no.2
X
rent distribution obtained is quite accurate. best knowledge,
no
available method
boundary conditions with
According to the authors'
results in better satisfaction of
approximately
the
same order of current-dis-
tribution approximation.
Etotal/Eio Seg-j ment no.1
Segment no.3 (or 4)
Segment no.2
I
0.05 s
-0.05
excitation region
junction
FIG. 3. 3. Real part (1) and imaginary part (2) of locally axial electric field along segments of antenna shown in Fig. 3. 2, normalized with respect to impressed electric field at center of magneticcurrent frill, at frequency f=l GHz. Adopted degrees of polynomial approximation of current were n 1=4, n 2 =3 and n 3 =n4=3. End effects were neglected. - - - normal scale; - - - 10 times reduced scale. Small circles denote matching points.
Sect.3.4.
3.4. As it
79
Treatment of wire ends
TREATMENT OF WIRE ENDs can
be concluded
results in a
very
discontinuities. nificant
from
Fig. 3. 3, improper treatment
large total electric field in In many
errors
24 15 16 • •
in
the
of
wire ends
vicinity of these
practical cases this does not introduce sig-
antenna
analysis, because
the
antenna
current
is
forced to be zero at the wire ends and the power generated by this current is relatively small.
Thus, in the case of electrically thin wires,
this large electric field usually has little effect on the antenna current distribution away admittance and points very
from
the
wire ends, as well
as on the antenna
radiation pattern, provided that there are
close to
the
ends.
no matching
However, neglecting the end effect in
the case of electrically thicker resonant antennas can lead to significant numerical
instabilities,
which
cause
the antenna admittance
to
diverge with increasing order of the current distribution approximation in the end region. A rigorous treatment of mation of
the
end effect must include precise approxi-
the surface current and
ends, taking
into
account
the
charge distribution at
exact shape of the ends.
ment is presented in this section for
the
the
antenna
Such a treat-
case of cylindrical antennas
(of circular cross-section) with hemispherical
and with flat ends, be-
cause these are the simplest structures where there are neither circumferential
currents, nor
charges, and
the
circumferential variations
problem is, in fact, that
current distribution.
of
of
currents
The precise treatment of the end effect requires
an exact integral equation for the antenna current distribution. fore we
and
finding one-dimensional
There-
shall first make some necessary comments on the integral equa-
tions for cylindrical antennas. Let
us
consider a hollow, cylindrical, thin-walled antenna, sketched
in Fig. 3. 4(a).
The antenna currents and charges are localised only on
the cylindrical surface, of the
radius
a.
sumed
to
be perfect,
field
at
the antenna surface must be
If the antenna conductor is as-
tangential component zero.
of
the
total
electric
Due to symmetry, there is
no circumferential component of the electric field, and only the axial,
80
Ch.J.
Treatment of wire junctions and ends
z
(a)
z
z
(b)
(c)
FIG. 3.4. Sketch of (a) hollow, thin-walled cylindrical antenna, (b) cylindrical antenna with hemispherical end caps and (c) cylindrical antenna with flat end caps, driven by TE}I magnetic-current frills.
z-component, should be considered. E
z
Thus the boundary conditions yield
0 ,
+E.
~z
(3. 8)
where
1 dl d ] [ I(s) + k 2 d; ~ G(z,s) ds
E (z)
z
(3.9)
is the electric field along the antenna due to the antenna currents and charges,
E.
is the impressed field
~
(due, for example, to the TEM mag-
netic-current frill), and 1T
G(z,s)
21T
I
(3.10)
g(r) d¢
-1T is the so-called exact kernel, where g(r) is Green's function, given in eqn. (1.5).
The distance r
between the source point P' (z=s,¢) and the
field point P(z,¢=0) at the antenna surface is given as r =
[Cz-s) 2
+ 4a 2 (s~n ..P.. ) 2] lv, , 2 0
(3.11)
Sect.3.4.
Treatment of wire ends
where a is the antenna radius. (3. 8),
exact form of
the
81
When eqn.(3.9) is substituted into eqn.
the
two-potential equation
cylindrical antenna
is
complex to
Thus, instead of
the
hollow
this equation contains a 37 double integral with a logarithmic singularity, which might be rather handle.
called reduced kernel
obtained.
for
is
used
However,
using
almost
the
exact kernel, the so-
without exception.
It has
the
form K(z,s)
g(r)
(3.12)
'
where r
[Cz-s)
a
2
1
2
+a]~
(3. 13)
is interpreted as an approximate average distance (see also Section 1. 2). Eqn. (3.12) yielding
should be the
substituted
into
eqn. (3.9)
already cited two-potential equation
poles, eqn.(l.19).
The
reduced
kernel
in
instead for
eqn.(3.12)
of
G(z,s),
cylindrical diis
often inter-
preted as being only an approximation to more complex exact kernel given in eqn.(3.10) in connection with computing the electric field at the antenna surface surface.
due
to the antenna currents and charges located at the
Alternatively, it
is
interpreted as
the
exact kernel if the
currents and charges are assumed to lie along the antenna axis, and the 3 field is computed at the antenna surface. However, both of these approaches are misleading if one wants to find the source of instabilitia~sociated
es
with the reduced kernel.
If we imagine the field sources to be where they are in reality, i.e., at
the
obtain
antenna surface, the
and
exact expression
hollow-tube antenna.
Hmvever,
substitute eqn. (3.12) into
eqn. (3. 9),
we
for the electric field at the axis of the the
electric field
at
the antenna axis
is not equal to zero, because the antenna is hollow and its axis is not enclosed by (3.8)
to
a perfectly
conducting
surface.
Therefore,
forcing
hold along the z-axis for such an antenna means imposing non-
physical conditions, \vhich, naturally, causes instabilities in merical solution. are
eqn.
located
in
the
nu-
This is particularly pronounced when matching points the
immediate vicinity of the antenna ends,
total field differs significantly from zero.
where
the
82
Ch.J.
Treatment of wire junctions and ends
If the antenna is capped, eqn.(3.8) is rigorously satisfied along the z-axis, but the expression for E and
charges
located at
Fig.3.4(b), or this
the
2
must include the field due to currents
the antenna ends (i.e., at
discs in Fig.3.4(c)].
correction eqn. (3.8) expresses
the
the
hemispheres in
It should be noted that with extended boundary
conditions
exactly, as discussed in Section 1.2. If we consider a smooth, rotationally symmetrical antenna, the charge Q'(z) per unit length measured along the antenna axis is related to the axial antenna current, I(z), by the continuity equation (1.13), just as in the case of wires of constant cross-section. field
along
the axis
of
Therefore the electric
such an antenna is essentially given by eqn.
(1.14), and has the form
E (z)
[I(s)
z
+
1 di a ] k 2 ds ~ K(z,s) ds ,
(3.14)
provided that the variable antenna radius, a(s), is used in eqn.(3.13). eqn.(3.8), the exact two-potential 38 equation for the current distribution of a capped antenna is obtained.
When eqn.(3.14) is substituted
This
equation
"ordinary"
solved with almost no more
two-paten tial
bear in mind ently
can be
less
that
into
equation,
eqn. (1. 19) .
problems
However,
than
one
the
should
the extended boundary conditions result in an inher-
stable equation than the exact boundary conditions, because
in the first case the kernel is quasi-singular, and in the second it is 39 As a consequence, the approximate current distribution ob-
singular.
tained using the extended boundary conditions, being sensitive to location
of
the matching
points,
tends
to
oscillate when the spacing be-
tween the matching points becomes smaller than the distance between the matching points and the antenna surface. As
it
can be
effect in serious
the
seen from
analysis
conceptual
new type number of
of
the
above discussion, inclusion of
cylindrical antennas
problem, except
that
of quasisingular integrals (for extra
terms
is needed
does
not
the end
represent a
it requires
computation
the
Unfortunately, a
caps).
in order to approximate
the
of a
antenna
Sect.3.4.
Treatment of wire ends
83
current distribution in the vicinity of the antenna ends. an increase However,
in
the
visible
solution for admittance,
computer
improvements may be
the antenna as
can be
current
seen
This implies
storage requirements and execution time.
from
gained
in
the
stability of the
distribution,
and
the
example.
following
quarter-wavelength monopole antenna, of
radius
particularly
its
Consider a
a=0.007022 A, driven by
a coaxial line with b/a=3, sketched in Fig.2.5.
According
to
the ex-
perimental results presented in Reference 17, the antenna admittance is Y=(l7. 84- j 7 .SO) mS. was first divided into rest of the monopole.
In
the
two
theoretical analysis the monopole antenna
sub segments: the excitation region, and the
The degree of the polynomial approximation in the
excitation region was n , and along the rest of the monopole n . The 2 1 antenna was analysed using the Hallen and the two-potential equations, with
the
mations the
belt-generator and of
the TEM magnetic-current-frill
the coaxial-line excitation,
end effect.
In Table 3.1
the
without
approxi-
taking into
account
antenna admittance is presented for
various degrees n
and n . It can be seen that the results do not seem 1 2 to converge with increasing the polynomial degree n , although for 2 n =4 or 5 the results are very close to the experimental ones. In 2 order to locate the source of these instabilities, a third antenna sub segment, 6a degree
for
long,
was separated at the antenna end.
this segment was n . 3
The polynomial
The antenna admittance was quite in-
TABLE 3.1. Admittance (in mS) of quarter-wavelength monopole of radius 0.007022 A, versus degrees of polynomial approximation for current distribution, without taking end effect into account. n1 n2
*
Hallen's equation* belt generator
Two-potential equation TEM magnetic frill
3
3
18 . 04 - j 7 . 6 1
19. 80 - j 7. 4 9
19.61-j7.33
3
4
17.65-j7.68
18.39-j7.62
18.33-j7.59
4
4
17. 61 - j 7. 66
18.41-j7.58
18.24-j7.48
4
5
1 7 . 35 - j 7 . 7 3
1 7 . 84 - j 7 . 50
17.89- j7.42
4
6
17.10- j7. 74
17.32-j7.64
17. 30 - j 7. 58
These results differ somewhat from those presented in Table 2. 3 due to different arithmetic precision used in computation.
84
Ch.J.
Treatment of wire junctions and ends
sensitive to variations of both n in drastic
and n , while changes of n resulted 1 2 3 variations of the admittance, as can be seen from the sec-
ond column
of
Table 3. 2 (n
is not applicable to this column). This 4 points out to the improper treatment of the end effect as the source of the instabilities.
If the end effect is rigorously analysed taking in-
to account the hemispherical distribution at
the
cap
and approximating the antenna current
cap by a separate polynomial, the degree of which
is n , and still retaining the short subsegment adjacent to the end, 4 with the polynomial degree n , very stable and accurate results for the 3 antenna admittance are obtained, presented in the last column of Table 3. 2.
The
short
subsegment adjacent
to
the hemispherical end improves
very much the stability of the solution, because the antenna current is rapidly varying along it. axis, associated with Fig.3.5.
The total electric field along
this solution, is very
the
antenna
small, as can be seen in
For comparison, the total axial electric field at the antenna
end obtained without taking the end effect into account is even stronger than the amplitude of the excitation field along the z-axis. The antenna analysis which significantly expedited noting tion
in
the vicinity
of
the
antenna length and frequency.
includes treatment that the shape of
of wire ends the
can be
charge distribu-
ends practically does not vary with the This will be demonstrated in the case of
TABLE 3.2. Admittance (in mS) of quarter-wavelength monopole of radius 0.007022 f., versus degrees of polynomial approximation for current distribution, obtained by two-potential equation. n1 n2 n3 4
4
3
Without end effect 17.00-j7.60
n4
With end effect
3
17.74-j7.47
4
17.72-j7.52
4
4
4
15.36- j 7. 78
3
17.58-j7.51
4
17.60-j7.50
4
4
5
16.50-j7.66
3
17.68-j7.48
4
17.68-j7.50
6
14. 90 - j 7. 80
3
17.63-j7.50
4
17.66-j7.52
4
4
Sect.3.4.
Treatment of wire ends
85
Segment no. 2
3
0.015
!4 I 2
0.010 0.005 z/h
0 1.0
-0.005 2
-0.010 -0.015
FIG. 3.5. Real (1) and imaginary (2) parts of total axial electric field along z-axis of quarter-wavelength monopole of radius 0.007022 :X., normalized with respect to impressed electric field due to TEM magnetic-current frill at z=O. This field was computed from the solution of the t>vo-potential equation which included end effect into account. Degrees of polynomial approximation were · n =n =n =4 and 1 2 3 n =3. Small circles denote matching points. 4
a cylindrical antenna with flat ends, like that sketched in Fig.3.4(c), although the procedure
can
also be applied to other shapes of
the
an-
tenna ends. In
the
case of
antennas with flat ends eqn. (3.14) is not valid, be-
cause di/ds is not defined at the end discs.
In this case the electric
field along the antenna axis due to the disc current is zero, as a consequence of
symmetry.
Therefore
its charge, of density ps(p). metrical, its
field
the
total disc field is
due
only to
As this charge distribution is also sym-
along the antenna axis (see Fig. 3.6) can be calcu-
lated from eqns.(1.2) and (1.4) as
Ps(p) dg(r) 2np dp ds
(3. 15)
86
Ch.3.
2
2 !<2
where r= ( s +p )
The
•
charge
Treatment of wire junctions and ends
density p s (p)
can
be approximated by a
polynomial with even degrees only, 2k (_fl_) a
in which case The
last
the
term
in
+ Dr
o( a-p )
(3. 16)
,
integrals in eqn.(3.15) can be evaluated explicitly. eqn. (3.16) represents
a
line
charge,
i.e., a ring,
which approximates the edge effect. In what follows, the treatment of the end effect is included into the Schelkunoff equation, the
first
drical
part
is
long
can be
3.6). gion.
regarded as
that
a
case
the
source
the
discontinuity of
end of the antenna cylin-
of a strong
the last left-hand term in
practically localised only the
In
current derivative, di/ds, at
field, Ee, given by Ee
eqn. (1. 29).
local
eqn.(l.29).
electric The field
in a region the length d of which a-
s-axis is of the order of several antenna diameters (see Fig.
Therefore this zone \vill be referred to as
the
antenna end
re-
The charge distribution along the antenna cylindrical part (which
is proportional to di/ds) is a rapidly varying function in this region, so
that
its
electric
field, together with
the
field
due to the disc
charge, compensates the major part of the field Ee. The charge distributions
on
the disc
and
on the antenna cylindrical
part are proportional to di/ds at the antenna end (i.e., at s=O).
Being
essentially quasi-static distributions, they depend practically only on the antenna end Therefore
they
geometry, and not on the antenna length and frequency. can be
determined once for all antenna lengths and in-
corporated into the antenna current distribution. The
antenna current distribution along
the antenna cylindrical part
can now be represented as a sum of a slowly varying term, given in eqn. (1. 30),
valid
along
the
whole antenna
length, and a
rapidly varying
term,
I (s), defined only in the end region. This total current for 1 s=O must be equal to the disc current for p=a, \vhich corresponds to the
disc charge must also
given by eqn. (3.16). satisfy boundary
The local current distribution I
conditions
at
the
end of
the
(s) 1 end region
Sect.3.4.
Treatment of wire ends
87
E/Ee{O) 1.4
d
1.2
a 0
s
end region
1.0
{a) I {A) I' {A/rad) kd
1.0
0.5
ks
ks
0
2 (b)
3
(c)
FIG. 3.6. (a) Sketch of cylindrical-antenna end region. (b) Current and its first derivative in end region for ka=0.1321 and I' (0)=1 A/rad: It -main part of current, If=dit/d(ks) -main part of the first current derivative, I - total current, I'=dl/d(ks) - the first derivative of total current. (c) Field distribution in end region: Ee - field due to discontinuity of the first current derivative, Er - sum of Ee and fields due to local distributions on antenna cylindrical part and end discs. (Please see text for further explanations.)
(s=d).
Therefore it is taken in the form of a special polynomial, nl
Il (s) =
L
ck (d-s)k
'
(3. 17)
k=2 so
that
r 1 =0 and di/ ds=O for s=d, thus automatically satisfying cur-
rent and charge continuity conditions for s=d.
88
Ch.3.
Treatment of wire junctions and ends
The coefficients Ck and Dk of the local distributions can be computed using
the
point-matching technique and
requiring that the total elec-
tric field, Er' which is the sum of Ee and the fields due to both local distributions, be a continuous and slowly varying function along the s16
axis, and be zero at the antenna end (s=O+).
Fig.3.6(b) gives an example of the antenna current and charge distributions along I~
are
which are terms.
the
cylindrical antenna
in
Local
the
in the end region.
present case approximated only by
current
is superimposed on
total current distribution along the field tions.
part
It and
the main parts of the antenna current and its first derivative,
Ee
the main
the antenna.
two
Given in Fig.3.6(c) are
and the total field corresponding
to
the local distribu-
Such a slowly varying electric field can now be
sated by
a low-degree polynomial in
rent, requiring no more
trigonometric
part, yielding the
easily compen-
the main part of the antenna cur-
terms and matching points than
on
the rest of
the antenna. The method for
analysis of
particularly suitable in
the
cylindrical antennas described above is 15 16 of electrically long antennas. '
case
As an example, Fig. 3. 7 shows conductance
and
susceptance of a monopole
antenna with ka=0.1321, versus its electrical height, kh.
In spite of
the fact that the distance between successive matching points along the main antenna part was as large as 0.27 wavelengths, the theoretical results are seen to be remarkably accurate. The above example is based on the Schelkunoff equation, but, obviously, the analysis of the local charge and current distributions can also be incorporated into antennas. for
local
other
types of integral equations for cylindrical
It should be noted
that
a similar analysis can be performed 16 current distribution in the excitation region, thus reduc-
ing the number of matching points and current distribution coefficients in that region, while retaining excellent mittance.
results
for
the
antenna ad-
-5
0
5
II
1
& "
I
I
" " "
G
I
1\ I
r 1
1\.. .I
A.
I
I
ll.
I
I
I
FIG.3. 7. Conductance (G) and susceptance (B) of cylindrical monopole antenna of radius (in electrical units) ka=0.1321, versus its electrical height, kh. -----theory, oo, ~*experiment. (Ref.l5)
1111 I
II I
10
I
~~
r 1'
15
"
0
201- II
251.. II
301--
G,B (mS)
kh
~
til lb
"'
00
(IJ
0..
::s
lb
~
'"'·
'!;
...,0
rt
::s
11!
rt
Ill
lb
~
"""
!""" ...,
()
90
Ch.3.
3.5.
Treatment of wire junctions and ends
CONCLUSIONS
Presented
in
this chapter were
wire junctions and ends. hoff's
law be
the
approximate
methods
for
treating
This treatment requires that the first Kirch-
postulated
for
every
junction
and
end.
In addition,
junction-field constraints can be written for every junction, implicitly requiring that boundary conditions be junction region.
These
representation of
the
these constraints be comes possible segment, [Note
so
that
ficients
to
that the
satisfied, on average, in the
constraints are, in a
second Kirchhoff's law. satisfied
for
By requiring
sense, a that
all
every junction and wire end it be-
associate two of these equations with every antenna (nm-1)
(nm+1)
for
junction-field
is
matching points
can
be chosen
per
segment.
the number of unknown current-distribution coef-
segment
considered.]
This simple rule is valid
for
all the segments of a wire-antenna structure, \vhich is very convenient, because
it
enables
automatic
choice of
numbers
matching points in the antenna structure.
and positions of the
These constraints
yield
re-
latively stable and sufficientlyaccurate solutions of the two-potential equation for arbitrary wire antennas for most practical purposes. If more
stable and
reliable results
for
the antenna admittance are
desired, in particular for resonant, or nearly-resonant, thicker antenna
s true tures,
it
is
necessary
distribution more precisely.
to model
presented in this chapter only for
tennas.
Very accurate numerical results
and
admittance
are approximately
of
of
antenna surface-current
Such a treatment is very
it was tion
the the
for
antenna current distribu-
cylindrical antennas are
the
same
complicated and
ends of cylindrical anthus
order of accuracy as
obtained, which the
most precise
results that can be obtained by standard admittance measurements. Using
the
technique described so
sible to analyse with
far in
this
monograph, it is pos-
remarkable accuracy any thin-wire antenna struc-
ture consisting of perfectly conducting wire
segments.
It was assumed
in the analysis, however, that there are no loadings along the segments and
that
electric.
the whole structure is situated
in
a homogeneous perfect di-
These conditions are fulfilled frequently, at least approxi-
mately, but in practice loaded wire antennas and wire antennas in inhomogeneous and/or imperfect-dielectric media are
also encountered.
The
problems associated with such cases are analysed in the next chapters.
CHAPTER 4
Wire Antennas with Distributed Loadings
4.1.
INTRODUCTION
Electrical characteristics of an antenna made of a perfectly conducting material depend, size.
In
some
at a
given
cases it
is
frequency, only on convenient,
or
the
antenna shape and
even necessary,
antenna current distribution for given antenna dimensions. done by
loading
the
antenna with
a
convenient
type
to modify
This can be
of loadings.
In
this way some of the
antenna properties (such as admittance, radiation
pattern, power
etc.)
gain,
the rhombic antenna. nected
to
can be improved.
A well-known
example
At its far end from the feeder a resistor ts
decrease reflections from that end.
current distribution
is
changed
from
is
~on
In this way the antenna
a standing wave
to
a travelling
wave, which results in a remarkably broadband antenna in its admittance. Another well-known example is a medium-wave or short-wave vertical monopole antenna with
an inductive coil inserted between two antenna sec-
tions, which improves
to
some extent radiation from the monopole.
nally, a dielectric, usually protective,cover over
Fi-
a wire antenna also
represents a loading along the antenna. The term "loaded wire antenna" is For
example,
it
is
said
frequently
often that
used in a rather wide sense. a
vertical monopole with an
inverted-umbrella-like wire structure at its top is "capacitively loaded" (see also Fig.3.2).
On the other hand, the same term is used for a
vertical monopole consisting of
two segments, the upper, short segment
being connected to the base segment by means of a lumped capacitor.
To
92
Ch.4.
avoid
this
chapter we
rather
loose meaning of
shall use
narrow sense.
Wire antennas with distributed loadings the word, in this and in the next
the expression "loaded wire antenna"
in a more
As loaded wire antennas we shall consider antennas
hav-
ing one or more of the following four general types of loadings: a)
If
current
in a
wire segment
associated with a continuous field which
segments of the antenna can be of (locally) axial electric
is a function of the current intensity at the point of the
considered, \>Te
segment
or
distribution
distributed loading.
say
that
segment is loaded with a series
the
An example
is
an
antenna made
from a resistive
rod. b)
If
the voltage
function of
the
(theoretically) there is
a
segment.
An
between
two
close
points
along
a
segment
is a
current between these two points and remains finite if the
two
points
become
infinitely
close, we
say
that
series concentrated, or lumped loading at that point of the example
of this type of loading is
an
electrically small
coil inserted between two antenna segments. c) to
If a
the
tric
distributed current exists along
the
local segment axis and is a function of
field,
i.e., of
the
segment perpendicular the
local charge density on
local radial electhe
antenna, we say
that the antenna is loaded with a shunt distributed loading.
A loading
of this type is a thin protective dielectric coating over the wire. d)
If
the
segment, but
preceding effect results
is localised
to
a very small part of a
in a finite radial current, we say that a concen-
trated (lumped) shunt loading exists at that point of the antenna.
An
example of this type of loading is a lumped capacitor, coil or resistor connected between two perpendicular segments, far from both ends of one of them, or a
thin
metallic
disc
mounted on the wire perpendicular to
its axis. In principle, the linear,
as
in
loading
network
linear passive loadings.
can be
theory.
We
passive shall
or
active, linear or non-
consider here
only cases of
The present chapter is devoted to analysis of
distributed loadings, and the next to analysis of lumped loadings.
Sect.4.2. 4.2.
Equations for antennas with distributed loadings
93
EQUATIONS FOR CURRENT DISTRIBUTION ALONG ANTENNAS WITH SERIES DISTRIBUTED LOADINGS
Consider a cylindrical antenna with a linear distributed loading along its length, sketched in Fig.4.1. to
an
(It will
be seen that generalization
arbitrary wire-antenna structure is almost straightforward.)
assume
that
We
the loading is a smooth and slowly varying function of the
z-coordinate (which, as earlier, coincides with the antenna axis).
FIG.4.1. Part of cylindrical antenna with surface S enclosing it.
z
Let E be the electric field due to the antenna currents and charges, and E.
the impressed electric field.
l
on
S,
Es=E+Ei.
ilar manner,
The
-+
total
-+ -+
Hs=H+Hi.
total
electric field
magnetic field can be represented in a sim-
From the
tributed loading it follows
Hence, the
that
definition of
the
linear series dis-
the total current through the antenna
cross-section (equal to the sum of conduction and displacement currents through Scs) must be proportional to the z-component of the total electric field at the antenna surface, i.e., ESz(z) = Z' (z) I(z) = Z' (z) •2TiaHS~ (z) Z' (z)
is known as
the
internal impedance per unit length, and is, es-
sentially, defined by eqn. (4.1). antenna
interior
and
(4.1)
can
It is an integral description of the
be determined, at least
in
principle, if we
know the actual antenna composition. In order
to
apply the usual expressions for the retarded potentials,
the medium must be homogenized. This can be done using the equivalence 5 following the same procedure as in Section 2. 2. Thus from
theorem,
94
Ch.4.
Wire antennas with distributed loadings
eqn.(2.l) we obtain the density of the equivalent surface electric current,
!
I(z) 211a
J s (z)
The density
(4.2)
z
of
the equivalent
surface magnetic currents,
-+
1
ms'
is ob-
tained from eqns.(2.1) and (4.1) as
Jms (z)
=
!,. ,
Z' (z) I(z)
(4. 3)
'i'
which means
that
the equivalent magnetic
currents
are
circulating a-
round the surface S. It is interesting the
total
to
the antenna surface. However, now we current,
note that eqn.(4.2) requires, essentially, that
antenna electric current be
of
have
This is identical with the perfect conductor case. to
assume that, in addition, a
density given
antenna surface.
assumed as a surface current on
in eqn.(4.3),
(Of course,
for
exists
surface magnetic
simultaneously on the
Z' =0 we obtain the perfect-conductor
case.) Current distribution I(z) must be such that the surface currents given
in
eqns. (4.2) and (4.3) make
the
z-component of the total electric
field to be zero inside the domain occupied earlier by the antenna. is simplest to require that this be true along
the
It
antenna axis, since
the equations will be almost the same as those for perfectly conducting antenna obtained in Chapter 1, where we used the extended boundary conditions along
the
antenna axis to formulate the integral equations for
the antenna current distribution. The electric field puted from eqn.(2.4).
due
to
the
surface magnetic currents can be com-
For the axial electric field along the z-axis we
thus obtain
Z' (z') I(z') a 2
E mz
where z
1
and z
2
2
l+jkr 3
exp(-jkr) dz' ,
(4.4)
r
are the end-points of the loaded antenna segment, and
Sect.4.2.
Equations for antennas with distributed loadings
95 (4.5)
The field
Em (z)
should now be simply added to the electric field pro-
2
duced by the electric currents and charges in any equation mining current distribution the
extended boundary
along
for
deter-
perfectly conducting antennas.
condition equation analogous
to
Thus
eqn. ( 1. 17) now
reads E (z) z
+
E (z) mz
-E. lZ
(4.6)
(z) .
In the case of the two-potential equation for cylindrical structures, eqn.(1.19), equation, term
the
vector-potential equation, eqn.(1.23), the Pocklington
eqn.(l.27),
-Em/ (jw)l)
tions.
equation E~
the
Schelkunoff
equation,
eqn.(l.29),
the
should be added to the left-hand sides of these equa-
The Hallen
simple manner.
and
equation,
Namely, in
2 (z')
be
can
also be modified
in
a
integral on the right-hand side of this
(z')], and the mz integral containing Em (z') moved to the left-hand side of the equation. ~
should
eqn.(l.26),
the
substituted by
[E. (z')+E lZ
2
Frequently both Z'(z) and I(z) are slowly varying functions along the antenna axis, such that
they are practically constant along a distance
of several wires' radii.
In that case, provided that the wire is elec5 the duality theorem we can consider the
trically
thin,
according to
electric field Em (z) to be approximately dual to the magnetic field on 2
the axis of a very long, uniformly wound solenoid.
Thus, on the anten-
na axis, (4. 7) This equation tion
for
can
easily be
arbitrary wire
incorporated into the two-potential equa-
structures, eqn. (1.15), by merely
adding the
term Z'(s) I (s )/(jw)l) to the left-hand side of eqn.(l.15), where "p" p p p p is the index of the segment at the axis of which the field point is located. Note that the approximate expression (4.7) strictly cannot be applied close to the antenna ends, because there it gives significantly different results
from
the exact eqn. (4 .4).
vere restriction, for
the
Fortunately, this is not
following reasons.
In all
a se-
the equations in
96
Ch. 4.
which Em (z)
enters directly (the two-potential equation, etc.),
2
rule we
do
vlire antennas with distributed loadings as
a
not choose matching points close to the antenna end (if the
point-matching technique is
used
for obtaining the solution).
In Hal-
len's equation expression (4.7) is integrated, so that the end error is smoothed out. ograph were (4.7).
All the results presented below and further in this monobtained using
the
approximation of
Emz (z) given in eqn.
For example, in the case of Hallen's equation the term
z
f
k jw)J
D(z)
z
(4. 8)
Z'(z') I(z') sink(z-z') dz'
0
should be added on the left side of the equation. 4.2.1. ings.
Examples of analysis of antennas with series distributed load-
In practice,
be resistive, made in
the
continuous
inductive
or
series
loading along wire antennas can
capacitive.
Resistive loading
is
usually 40 41 •
form of a thin resistive layer over a dielectric rod.
Commercial surface-layer resistors can also be used to obtain step-like . 42-44 resistive load1ng. Variable inductive distributed loading is obtained able,
if
the antenna is made in
slowly varying pitch.
made easily.
(For example,
the
form of a wire spiral with vari-
Continuous capacitive loading it
cannot be
can be realized if a molten dielectric
is mixed \vith conducting powder, a thin cylinder made from this mixture, and left to solidify.)
However, quasi-distributed
loading
can be ob-
tained by making the antenna in the form of a row of many short metal45-48 lie segments with dielectric discs between two successive segments. This subsection presents some results for antennas with distributed resistive and quasi-distributed capacitive loading. Fig.4. 2 shows
theoretical and
symmetrical dipoles with and
1400 n/m,
The
theoretical
erator and
left
the
side.
admittance of
constant resistive loading, for Z' (z)=700 n/m 49 electrical half-length kh of the dipoles.
results were
obtained
polynomial approximation for
equation given its
versus
experimental values of
in eqn.(2.12)
with
using
the
delta-function gen-
current in solving the Hallen
the additional term in eqn.(4.8) at
Agreement between theoretical and experimental results
Sect.4.2.
Equations for antennas with distributed loadings
97
G,B (mS)
5
4
4
3
3
2
2
G,B (mS)
kh ( rad) o~~--~--L-~--~--~
2.0 2.4 2.8
3.2 3.6
B
kh ( rad)
..
oL-~L-~--~--~--~--~
4.0 4.4
2.0 2.4 2.8 3.2 3.6 4.0 4.4
(a)
{b)
FIG.4.2. Conductance (G) and susceptance (B) of resistive dipoles with constant distributed loading versus their electrical length, kh; a=O. 32 em, h/a=71. 3; (a) Z' (z)=700 r
is seen to be very good. Shown in Fig.4.3 is theoretical and experimental current distribution along dipole antenna
of
radius
a=0.3175 em and half-length h=0.5 m,
with resistive loading approximately given by
z
I (
z
)
-
-
1 -
at f=450 MHz
720
I z I /h
and
I
" m
(4.9)
,
f=900 NHz.
function generator was used.
50
Again Hallen's equation with the deltaAgreement between
rimental results is seen to be quite good. the
loading
like loading.
theoretical and
expe-
It is worth mentioning that
in eqn. (4.9) in the experiments was approximated by step41 Shown in Fig.4.4 is dependence of conductance and sus-
ceptance of this
antenna on frequency.
erties in admittance of this antenna. that there are
Note excellent broadband prop-
From Fig.4. 3
practically no reflections from
it may be concluded
the
dipole ends, \·lhich 51 52 is precisely the reason for this broadband behaviour . • It was found that the dipole radiation pattern also does not change drastically with 50
frequency.
98
Ch.4.
-2
-1
0
Wire antennas with distributed loadings
-2
2
-1
(a)
0
1 (b)
2
3
4
FIG. 4. 3. Current distribution (I) along resistive dipole antennas with a=O. 3175 em and h=O.S m, loaded as given in eqn. (4. 9), at (a) 450 MHz and (b) 900 MHz. (1) real part, (2) imaginary part, (3) magnitude. ----- theory (5th degree polynomial at 450 MHz, 7th degree at 900 MHz); ---experiment (Ref.41, amplitude available only). (Ref. 50)
Fig.4.5 shows a monopole antenna with quasi-distributed capacitive 47 It had altogether 16 rings, of wall thickness d=0.1 mm.
loading.
The diameter
of
the supporting dielectric rod was D=8.16 mm, its rela-
tive permittivity sr=2.25,
3
and
the
total
G,B (mS) 0
0
B
2 X X
X
:...>
X
X
X
X
X
f (MHz)
1
400
monopole length was b=27 em.
500
600
700
800
900
FIG.4.4. Dependence of conductance (G) and susceptance (B) of the antenna described in caption to Fig.4.3 on frequency. ----- 7th-degree polynomial approximation; oo, XX experiment. 4 1 (Ref .SO)
Sect.4.2.
Equations for antennas with distributed loadings
99
FIG.4.5. Schematic representation of capacitively loaded monopole. (Ref. 4 7)
b
The lengths
of
the rings decreased towards
the
monopole end according
to the relationship (4 .10) where k=1,2, ... ,16,
the zk are as shown in Fig.4.5 and h
rily taken as 27.0 ern.
was arbitra-
The rings were set by appropriate gauges to ap-
proximately 8=0.1 rnrn apart.
If averaged, the impedance per unit length
of the monopole can be represented as
Z' (z)
where
[t/crn ,
Xfref(z)
is
the
After
quency fref
(4.11)
reactance per unit length at the reference fre-
some
estimates it is possible
to
write eqn. (4.11)
in the approximate form
Z' (z)
105
-j
[1 Theoretical and
tance.
that
this
(z/h) ] [3 - 0.878(z/h) experimental results
shown in Fig.4.6. Note
2
2
(4.12) ]
for
f
the
monopole admittance are
The two sets of results are in reasonable agreement.
antenna
is
also
considerably broadband
in
its admit-
100
Ch.4.
Wire antennas with distributed loadings
G,B (mS)
14
FIG.4.6. Conductance (G) and susceptance (B) of monopole with quasidistributed capacitive loading as in eqn. ( 4 • 12), versus frequency. ---theoretical; vertical lines indicate the ranges of measured values. (Ref .4 7)
f (GHz) 2~~---L--~--~~--~--~~~
4.3.
2.6
WIRE ANTENNAS WITH DIELECTRIC OR FERRITE COATING
Dielectric-coated wire direct contact desirable.
of
at improving
the
the
used
in many applications where
antenna
to
the
dielectric coating may
antenna to
the wire
plied for
are
the metallic wire with the surrounding medium is un-
coating, in order order of
antennas
53 54 •
For example, the surrounding medium may be corrosive and/or
conductive, and
the
2.2
1.8
1.4
1.0
radiation
be efficient
radius.
such
be protective and/or aimed
properties. as
The thickness of
the
a protection, is usually of
the
Such coatings influence
an extent
antenna analysis.
the
properties of
that bare-antenna theory cannot be apThe influence
of
the coating is even
more pronounced when the coating is made of a ferrite material. There
are
few papers dealing with this interesting topic. 55 and Newman presented a general method for analysis of die-
Richmond
only a
lectric-coated wire antennas of arbitrary shape.
However, their method
is
the
rather
complicated, because
which is solved
by
addition, it uses of
the
it
is
based
on
reaction integral
piecewise-sinusoidal approximation for current. the delta-function
generator model, so
In
that values
antenna susceptance should be considered only as approximate. 56 57 • dealt with cylindrical antennas covered with a
James and Henderson
thick
101
Antennas with dielectric or ferrite coating
Sect.4.3.
layer
of dielectric
or
ferrite material.
Their method is based
on cavity-type modes excited in the coating and the variational method. However, it was applied only to cylindrical structures and, being general, it is too complex when only thin coatings are considered.
Exten-
sive experimental results for dielectric-coated antennas were presented by Lamensdorf.
58
In this section a simple method is proposed assembled
from
arbitrarily located
wire segments, each of with a
thin
which may be
dielectric
these wires
and
or
ferrite
are loaded with
for
analysis of antennas
interconnected
straight thin-
covered, completely or partially, (Fig.4. 7).
coating
distributed loading.
It is
Essentially, of
the shunt
type if the coating is dielectric, and of the series type if it is magnetic.
Therefore, coated
similar
to
antennas
can
be
analysed
using an approach
that presented in the preceding section.
However, we shall
use here a different approach. The method
is
based
on
the
quasi-static approximation of
both
the
electric and magnetic field in the coating in planes transverse to the wire axis. the
case
The of
an
electric
field
is
supposed
to
be radial, as it is in
infinite, uniformly charged metallic cylinder in elec-
sz E
0, )10
s _,_
coating
->1
r
p
p
s,
-+
rp
X
FIG.4. 7. A current-carrying wire segment geneous dielectric or ferrite coating.
covered with a
thin
homo-
102
Ch.4.
trostatics.
If
the
Wire antennas with distributed loadings
free charge per unit length of the cylinder is Q',
then in the dielectric coating
~
E (p)
a
2rrsp '
p
The magnetic
field
is approximately
(4.13)
has
the
only
the H
same as
that
due to an infinite current-carrying
cylindrical conductor of radius a, i.e., H
I
=
Zrrp ,
(4.14)
a
The approximate equations (4.13) a+O, because at is zero.
At
approximate. than
the
and
(4.14)
are
strictly valid for p=
the metallic cylinder surface the axial electric field
points
off
this
surface eqns.(4.13)
and
(4.14) are only
They are valid if the axial electric field is much smaller
radial field
(in the case of eqn. (4 .13)]
and i f the displace-
ment current (i.e., the flux of jwD) through the circle of radius p can be neglected when
compared with the conduction current in the wire
the case of eqn.(4.14)). and (4.14)
give
[in
It was found that approximate equations (4.13)
sufficiently accurate results for coatings up
to one,
or even two wire radii thick, and for relative permittivity and permeability of the coating less than about 10. The influence of
the
coating on the electromagnetic field can be re-
duced to that of the polarization charges and currents, and of the magnetization currents, situated in a vacuum. From eqn.(4.13) it follows that the polarization vector is radial, of intensity P (p)=(s-s )E (p). p
and
their
0
density
is
p
jwP . p
The polarization currents are hence radial The volume
polarization
charges
coating do not exist, because the coating is homogeneous. surface of
the
in
the
At the inner
coating the surface polarization charges exist, of den-
sity per unit length of the wire Q' =-Q' (1-1/E: ). They combine with the pa r free charges on the metallic cylinder, which yields a total
adjacent
Q'+Q' =Q'=Q' Is . At the outer pa a r coating there are also surface polarization charges, of
charge per unit length at surface of
the
p=a equal to
density per unit length Q' =Q'=Q'(1-1/s ). pb b r
Sect.4.3.
103
Antennas with dielectric or ferrite coating
From eqn.(4.14) it follows that the lines of the magnetization vector are circles centered at the (ll/]l -1)H
rents,
In the
conductor axis.
Its intensity is M
coating there exist radial magnetization cur-
the density of which is -ClM
At the
inner surface of the
coating there are axial surface magnetization currents. is Ima=(11r-1)I.
Their intensity
Together with the wire conduction current, they form a
tubular current of total intensity Ia=11ri.
At the outer surface of the
coating there are magnetization surface currents of intensity Imb =Ib = -(llr-1)1. The coating can now be removed, together with the wire segment, and the currents and charges Ia, Ib, vacuum.
Within the
Q~
and
Q~
imagined to be situated in a
cylinder of radius p=a the electric field due to
all these sources must be zero.
Thus the two-potential equation can be
formulated which includes all the charges and currents.
If the extend-
ed boundary conditions are used, the total electric field along the
is computed
segment axis, where the electric field due to the radial po-
larization and magnetization currents is zero.
When the electric field
due to the coated segment considered is computed along the axis of any other antenna segment, the field due to the radial currents can be neglected, because these
currents are uniformly distributed in all direc-
tions in a relatively small domain.
Thus the radial currents can be
disregarded altogether, and the field due to the wire and the coating currents and charges can be attributed only to
two
tubular layers of
and charge Q' a a per unit length, and the outer, with a total current Ib and charge Q' b per unit length. (Of course, this approximation is less accurate when currents and charges:
the
inner, with a total current I
the field is computed in the coating itself.) Following the reasoning of Section 1. 2,
the electric field due
to
these tubular distributions can be approximately computed by introducing approximate average distances, as in eqn. (1.11). wire
Noting that the
current I and charge Q' per unit length are related by the conti-
nuity equation
(1.13), it is not difficult to derive the two-potential
equation for an arbitrary thin-wire structure the segments of which may be covered with dielectric and/or ferrite coatings.
It has the form
104
Ch.4.
N
1
I
{
m=1
+ [
Wire antennas with distributed loadings
-+ -+ [i p ·i smrmm 11 I (s ) + - - m 2
E
-ip •fsm (\1 rm-1)Im(s m) +
k nn
di ( s ) m m ds m
i p •grad )g(r a )
+
di (s ) } 1 (1---} ~s m fp•grad )g(rb) dsm = Erm k2 m 1 -+ -+ i •E j(u\1 p i
2 2 k2 where ra =(r +am) and structures similar
to
eqn. (4.15)
assembled
2 2 k2 rb =(r +bm) • from
(This equation
straight or
curved
is valid for
segments.)
eqn. (1.15) and can be solved in the same manner. is
division of
valid for
(4.15)
It
wire
is very Note that
segments with uniform coatings only, so that
the wire structure into segments must be performed having
that in mind.
G,B (mS)
30
...
/1 , 1
I
25
20 15 10
5
1
I I I I
I
FIG.4.8. Conductance (G) and susceptance (B) of a bare and a dielectric-coated monopole antenna versus the normalized antenna length h/:>.. 0 for :>. 0 =0.5 m, wire radius 3.18 mm, dielectric-layer thickness 3.18 mm and relative permittivity 3.2. - - - theory, with coating; - - - theory,without coating; oo, xx experiment, with coating. 58 (Ref.53)
Antennas with dielectric or ferrite coating
Sect.4.3. As
the
first example of
Fig.4.8 are a
analysis of
coated wire antennas, shown in
theoretical and experimental results for the admittance of
vertical monopole antenna
length, and tions
lOS
as
a
function of
in Fig.4.9 theoretical and
along
the same antenna.
the
normalized antenna
experimental current distribu-
Very good agreement between theoretical
and experimental results is observed.
10
I/V (mA/V)
8 6 Current distribution FIG.4.9. along a die lee tric-coated monopole antenna for h/"A 0 = 0. 25. Other data are the same as for Fig.4.8. (1) real part; (2) imaginary part. theory; oo, xx experiment. 58 (Ref.S3)
4
2 0
-2 -4
Z/"Ao
0.05 0.10 0.15 0.20 0.25
-6 -8 -10
As the next example, shown in Fig.4.10 are theoretical and experimental results for ally with inset in
the
admittance of a cactus-like antenna covered parti-
dielectric the
coating
figure).
(sketch of
the antenna is
Excellent agreement
between
given
in the
theory and experi-
ment is seen again. As a further example, sketched
in
square loop with a ferrite coating.
the
ferrite coating,
ference between these the magnetic
For
E
energy
consequence of
two in
and
is
=1 and
r the present method yields the
cy f=lOO MHz, without
Fig. 4.11
an electrically small
vr =4 '
at the frequen-
loop inductance L =96 nH 0 The difL=l56 nH \vith the coating.
results can be attributed to the increase of
the coating, but
can
also be
considered as a
a series distributed indue tive loading along
the loop.
106
Ch.4.
G,B (mS)
Wire antennas with distributed loadings
22.6
(a)
(b)
FIG.4.10. (a) Conductance (G) and susceptance (B) of cactus-like antenna shown in the inset, versus frequency. All dimensions are in millimeters. -----theoretical, teflon coating (Er=2.1);- - - t h e oretical, without coating; ••, xx experimental, teflon coating; oo, ++ experimental, without coating. (Ref.54) (b) Photograph of the antenna.
The increase
of
inductance per unit length of the conductor due to the
coating is )J-)J 0
1' f
2n
and
the
ln b a
( 4. 16)
total increase of the inductance thus found is 1f=55 nH.
This
agrees well with
the computed increase, 1-1 =50 nH. The inductance of 0 the loop without coating calculated by means of the Neumann integral is
found
to be 10N=91 nH, which
is
also in good agreement
with 1 =96 nH 0
obtained by the present method. As
the
last example, Fig.4.12
shows
the admittance versus frequency
for a vertical monopole sketched in the figure inset of
the
coating.
The
results
for
the
for various types
coating with Er=l
and
)Jr=4 are
Sect.4.3.
£ 0 ' ll
Antennas
wi~h
60 o
I
£, ll
,_
-~-
~ a=3
107
dielectric or ferrite coating
I I I
I I
b=5
FIG. 4. 11. Sketch of a small square loop with a ferrite coating (£r=1, llr=4). All dimensions are in millimeters.
0 1.0
~ f-
compared with
the solution of Hallen's equation
for
a monopole of the
same size with distributed inductive loading obtained according to eqn. (4.16).
It is seen that the results practically coincide.
G,B (mS)
-.\x····:.. :'.)' •
30
.
'
: I
20
>l"!:J
\ ''
/X'
I \
.
;
5
•
•.
!
10 ( GHz)
0 -10
/
FIG. 4.12. Theoretical conductance (G) and susceptance (B) of the monopole antenna shown in the inset, versus frequency. All dimensions are in millimeters. ----- £r=1, lJr=1; - - - £r=4, l.lr=1; • • • £r=1, l.lr=4; - • - £r=4, l.lr=4; x x x results obtained from Hallen's equation and series loading given in eqn. (4.16), for £r=1, l.lr=4. (Ref. 54)
108
Ch.4.
Wire antennas with distributed loadings
As a general conclusion following from all
the above examples, it is
obvious that the dielectric or ferrite coating, even if quite thin, has an appreciable influence on coating (sr#l ,
llr =1)
can
the
antenna properties.
be regarded as
and purely ferrite coating (sr =1 , Both loadings crease
the
have
essentially
shunt
the
same
However, variations
pronounced when
Purely dielectric capacitive loading,
llr#l) as a series inductive loading. effect:
antenna electrical length, shifting
lower frequencies. come more
a
of
compared with
they
its
the antenna those
for
apparently in-
resonances towards admittance be-
the
same
antenna
without coating. The method presented in this section can obviously be applied also to analysis of antennas with lossy coatings, provided that losses are very pronounced, by simply
replacing
the
not
real permittivity s and real
permeability ll by complex values s=s'-js" and lJ=ll'-jll", respectively.
4.4.
CONCLUSIONS
Presented in equations
this
for
chapter were
the methods for extending the integral
determining current distribution in perfectly conducting
wire structures to structures containing wire segments with distributed impedance loadings.
It was
assumed that the loading was a smooth, re-
latively slowly varying function along the segments.
A method was also
presented for analysis of wire antennas with relatively thin dielectric or
ferrite
coating.
It was
pointed out
that, although
thin,
such a
coating can influence the antenna properties considerably. Finally,
it was mentioned
that
a
continuous
frequently approximated by
a sequence
which was
that
given indicated
of
impedance loading is
lumped loadings.
An example
this approach is possible, but no cri-
teria as to when and how such an approximation should be used were presen ted.
The next
chapter, dealing with
loadings along their segments, among tion also, and presents
certain
antennas having
other
concentrated
questions treats that ques-
guidelines concerning approximation of
distributed loading by lumped loadings.
CHAPTER 5
Wire Antennas With Concentrated Loadings
5.1.
INTRODUCTION
Concentrated loadings along wire antennas can be considered as limiting cases
of
distributed loadings when
segment becomes very
small
the length of the loaded part of a
in comparison with
the segment length and
the wavelength, but its total impedance remains finite. in the preceding chapter, in
some
As pointed out
instances only concentrated loadings
can be made easily, in which cases distributed loadings are approximated by appropriate sequence of concentrated loadings.
For example, this
is the case with series capacitive loadings. In practical realizations, concentrated loadings are lower frequencies.
At
higher
frequencies, however, it is difficult to
realize desired loadings accurately, for mainly it
is
difficult
to make
prescribed properties
easy to make at
the
two
reasons.
Firstly,
impedance elements themselves with
at high
frequencies.
Secondly,
some
capacitance
of
the two segments between which a loading is connected represents a progressively important as
the
count.
part
of
frequency increases,
the
total concentrated loading impedance
which
is very
It is therefore virtually impossible
difficult to take into acto
design theoretically a
lumped loading at higher frequencies accurately.
An experimental meth-
od which can help to obtain a desired loading is therefore presented in one section of this chapter. However,
the
difficulties
loading realization, but
also
are
not only encountered with
the lumped
with its theoretical modelling.
Precise
110
Ch. 5.
r'lire antennas with concentrated loadin(Js
modelling is very complex even in the simplest cases, such as that of a capacitive loading in antenna parts. element
form of a narrow gap between two cylindrical
Fortunately, accurate modelling
along
taken into
the
an
antenna
usually is not
account correctly in an
on
either
For this the
required.
reason
poses.
due
lumped impedance If the loading is
the
immediate neighbour-
Therefore this does not have a significant effect
radiation
field
a
integral manner, the error in cur-
rent distribution is localized practically in hood of the loading.
of
properties
of
the
antenna
or
on its admittance.
the delta-function approximation of the loading and of to
the
Only rarely a
loading more
is sufficient for practically all
pur-
sophisticated approximation of lumped load-
ings is needed. This
chapter
is
devoted
to
loadings along their segments. realized in
some
of wire
antennas with
A description how
lumped
such loadings can be
instances is also given, and the effects which can be
expected from introducing discussed.
analysis
A full
the
loadings into
appreciation of
the
antenna structure are
the usefulness
of
lumped loadings
will be possible, however, only in connection with wire-antenna synthesis, a topic treated in the last three chapters of this monograph.
5.2.
MODIFICATIONS OF EQUATIONS FOR CURRENT DISTRIBUTION
Consider a
wire
segment containing an axially symmetrical concentrated
of impedance z , located at z=z , as sketched in Fig.5 .1. In 0 0 order to avoid a detailed analysis of the electromagnetic field in the
loading
interior of
the
surface S enclosing the wire segment, we can apply the
same procedure as in Sections 2.2 and ties of
4.2.
For this we need the densi-
the equivalent electric and magnetic
surface currents, which
are given by
Js (z)
I(z) 21Ta
1 z
(5 .1)
and
(5. 2) where I(z) is
the
total current through the antenna cross-section, and
Sect.5.2.
111
Equations for current distribution
Short segment of FIG.5 .1. wire antenna with a concentrated loading.
ES
is
the total electric field at
the
surface S.
This electric field
depends primarily on
the
actual geometry of the gap at which the load-
ing is connected and
on
the
in any
simple manner.
tely, in
the
loading itself, and it cannot be computed ->-
Therefore ES is most often not known.
case of concentrated loadings we do not
actual distribution of
this
Fortuna-
need to know the
field, because practically only the integ-
ral effect of the loading (i.e., the voltage between its closely spaced terminals)
is
important.
is practically constant
If
for
any point between z
I z -z 1 <
and z 1 2 rent, the circuit-theory approximations apply, so that we have z2
J
E
Sz
(z') dz'
provided
that
(5. 3)
the capacitance between the loading terminals A and B 1 1 z . Note that the surface magnetic currents,
in Fig.5.1 is included in
Jms ,
0
are also proportional to the total current I(z ), so that from the
0
circuit-theory standpoint they can be regarded collectively as an electric-current dependent voltage generator. We
can
now use
any convenient approximation
the constraint in eqn. (5 .3). gap width
for
ESz (z), subject to
The simplest approximation is
tend to zero, which results
in
to
let the
a a-function generator, dis-
cussed in Section 2.2, the voltage of which is V=-Z I(z ). Hence for 0 0 E (z), the electric field along the z-axis due to the magnetic curmz rents, we obtain the same expression as that in eqn. (2.5), only with V
112
Ch.S.
Wire antennas with concentrated loadings
substituted by -z I(z ). This a-function approximation of lumped load0 0 ings can be incorporated into any equation for the antenna current distribution in
the
same manner as
the
a-function generator described in
Section 2. 2.
Thus, with the two-potential
integral
equation the junc-
tion-field constraint
f Etotal z dz
=
0
(5.4)
L
should be applied the
lumped
for
loading.
a short path (along the segment axis) containing [Of
course,
for
arbitrary wire
structures
local s-axis should be used instead of the z-axis in eqn.(5.4) .]
the Natu-
rally, the field E in eqn. (5 .4) includes the field E due to the tota 1 m magnetic-current ring which approximates the lumped loading. With
the
Hallen equation the same approximation of Emz is introduced
as in Subsection 2.2.1, i.e., (5 .5)
Hence,
for
the symmetrical dipole antenna
shown
in Fig.5.2 the Hallen
equation can easily be obtained in the form h
J
I(z') g(r) dz' + - k -
I
2jw)J i=l
zi I(zl.) [sink(z-z.) + sink[z-z.[] + 1
-h
1
z + cl cos kz
k jw)J
J
E. (z') sink(z-z') dz' lZ
(5. 6)
0
FIG.5.2. Symmetrical dipole antenna with delta-function concentrated impedance loadings.
Sect.5.2.
113
Equations for current distribution
with z =0 at the dipole center. The integral in the last term in eqn. 0 (5.6) depends on the choice of the excitation approximation.
It
is
obviously possible
other ways, although in terpart.
to
some
For example, we
approximate
cases
can
a
lumped
loading
this may not have a physical coun-
approximate not the field ESz (i.e., the
equivalent magnetic currents on S), but rather the field Emz z-axis
due
in many
along
the
to these currents, in the same way as with the belt-genera-
tor approximation of
the
antenna excitation (see
Subsection 2.3.2).
can also approximate the concentrated impedance by a TEM magnetic16 described in Subsection 2. 3. 1. In both cases, when
We
current frill,
calculating the electric field along the z-axis, the voltage V in eqns. (2. 28)
and
(2. 20)
(z-zi).
The
the
frill
TEM
should be replaced by -zi I (zi) /2
parameter
a
of
and z replaced by
the belt generator and the ratio b/a
can be chosen fairly arbitrarily, because
significant influence on
the
for
they have no
antenna current distribution when the an-
tenna is electrically thin. As already mentioned in the type of
approximation of
only on
the
lumped loadings has
The overall current distribution and, in particu-
the antenna admittance are almost unaffected by the type of approx-
imation,
provided
that
the
radius of
loading are much smaller than approximation for
special
quired.
from
the wavelength.
and the length of the Therefore the simplest
the numerical standpoint should be
cases when
into
two-potential
straint. tennas
the wire
a more
adopted, except
precise analysis of the loading is re-
The delta-function approximation of
be incorporated the
a pronounced effect
field and current distribution in the immediate neighbour-
hood of the loading. la~
the
introduction to this chapter, a specific
the loadings
can
readily
the analysis of general wire structures based on
equation,
by
introducing
the
junction-field con-
Also, modification of the Hallen equation for cylindrical anto
account
for
lumped loadings is simple and
straightforward.
The effect of the parasitic capacitance introduced between terminals which loading (see
the
loading
is inherent to the delta-function approximation of the
also
Section 2 .2) is practically not
same reasons as explained
pronounced, for the
in Section 2. 2, provided that the degrees of
114
Ch.5.
polynomial approximation of the loading are not high.
Wire antennas with concentrated loadings current
in
Therefore
the wire
the
segments adjacent to
delta-function approximation
of the loadings will predominantly be used throughout this monograph. 5 .2.1.
Examples of cylindrical antennas with concentrated resistive 11 loadings. All the examples presented in this and the following subsection were analysed starting from eqn.(5.6), using the polynomial approximation for
current
combined with
the
point-matching method and
adopting the delta-function approximation for
the
loadings.
This sub-
section is devoted to examples of antennas with resistive, and the next subsection to examples of antennas with capacitive loadings. loadings
are
inconvenient because
of
losses
they
Resistive
introduce, but they
can produce some very useful effects, as already pointed out in Section 4.2.
Capacitive loadings
dependent.
do
not introduce losses, but
are
frequency-
Therefore both types of loadings can be advantageously used
in various situations. As
the
first example of resistive loadings, consider so-called Alt9 Starting from the transmission-line theory, Altshu-
shuler's antenna? ler predicted of a
that
a travelling wave
could
thin cylindrical monopole, if a
be maintained along a part
resistive loading of appropriate
magnitude is inserted "A/4 from the monopole end. clusion experimentally, on a monopole of quency of f=600 MHz. nopole was "A/4
from
radius
He verified this cona=O. 3175 em, at a fre-
The outer radius of the coaxial feeder to the mo-
equal to Sa.
Altshuler found that a resistance of 240 r2 at
the monopole end was the optimal loading, resulting in an es-
sentially travelling current wave from the excitation point to the load, and a
standing wave
from
the load to
exhibited fairly constant admittance monopole
section
the monopole end.
for
from the ground plane
The monopole
practically any length of the to
the load.
This antenna was
analysed starting from eqn.(5.6), with the delta-function approximation of the coaxial-line excitation. Figure 5.3 shows dependence of conductance (G) and susceptance (B) of the
antenna,
versus
the normalized half-length h/"A of
symmetrical dipole, as measured by Altshuler cally.
Agreement
in
the
equivalent
and as obtained theoreti-
conductance is seen to be very satisfactory.
The
Sect.5.2.
115
Equations for current distribution
G,B (mS) FIG.5.3. Conductance (G) and susceptance (B) of the symmetrical Altshuler antenna; a=0.3175 em, f=600 MHz, Z1=240 Q at ±z 1 =±(h-A/4). theoretical; - - - experimental.59 (Ref.11)
theoretical susceptance shape
of
the
is higher
susceptance
This discrepancy is
due
than
curves in
the
the experimental, although the two cases is very much alike.
to poor approximation of the real driving con-
ditions (coaxial line with outer-to-inner diameter ratio as large as 8) by the delta-function generator. by dividing using
the
the monopole second-degree
into
The theoretical results were obtained segments
polynomial
of
lengths less than A/2, and
approximation
for
current
along
every segment. As an example of current distribution, Fig.5.4 shows experimental and theoretical current distribution
for
h/A=5/8.
It is
seen
that agree-
ment between theoretical and experimental curves is excellent. It was interesting to analyse the possibility of approximating a continuous resistive loading by a sequence of lumped loadings.
Therefore
the following two cases were considered theoretically, using the deltafunction generator excitation: (1) the total loading was evenly distributed along the antenna length, and (2) the total loading of equal magnitude was case
of
divided into several lumped loadings.
We shall analyse the
resistive antenna with
constant continuous loading, for which 49 experimental results are available. Consider a 0.3175 em.
resistive dipole
Let
the
total
of
half-length h=O. 226 m and
which is 1400 Q/m, and let the frequency be f=663 MHz, i.e., the
case
of
such a
radius a=
loading along half of the dipole be 317 Q,
continuous
loading theoretical
kh=~.
For
and experimental
116
Ch.5.
Wire antennas with concentrated loadings
z/>. FIG.5.4. Real part (1), imaginary part (2) and magnitude (3) of current along one half of the symmetrical Altshuler antenna; a=0.3175 em, h = 31.25 em, f=600 MHz, 21=240 ~at ±z1=±(h-A/4)=±18.74 em. -----theoretical;--- experimental.59 (Ref.11)
-3
values
3
of
the admittance
are
4
presented in Reference 49.
For conveni-
ence, these are given again in Table 5.1. Let now the total loading be concentrated at n=1, 2, 3 and 4 equidistant points along
the
dipole arms.
Values of G and B corresponding to
TABLE 5.1. Comparison of admittances of dipoles with continuous and lumped equidistant resistive loadings; a=0.3175 em, h=0.226 m, f=663 MHz, total loading along one dipole arm= 317 ~ (i.e., 1400 ~/m). In the case of lumped loadings, piecewise parabolic approximation of current has been adopted. (Ref.11) Type of loading
Admittance (mS)
Single loading at ±h/2
3. 70 + j2 .11
Double loading, at ±n/3 and ±2h/3
2.48+j2.73
Triple loading, at ±h/4, ±2h/4 and ±3h/4
2. 15 + j 2. 70
Quadruple loading, at ±h/5, ±2h/5, ±3h/5 and ±4h/5
2.04 + j2.67
Continuous loading, theoretical, 3rd degree polynomial approximation for current49 40 Continuous loading, experimental
1. 90 + j 1. 91 1.9 +j2.2
Sect.5.2.
Equations for current distribution
these cases are also four
loadings
are
shown in Table 5. 1.
sufficient
to
117
It is seen that as little as
substitute
the
continuous
resistive
loading, at least as far as the dipole conductance is concerned. oretical susceptance loading case, as for
a
the
is,
of
course,
increasing with n
in
(The-
the lumped
same polynomial approximation for current is used
progressively smaller segment of
the delta-function generator.)
This
the antenna in the vicinity of
indicates
that about
trated loadings per wavelength are needed to approximate
10 concenquite
accu-
rately a smooth, slowly varying continuous loading. Compared case
of
in Fig .5 .5 is
the
theoretical current
distribution in the
continuously loaded dipole of electrical half-length kh=11 with
those corresponding to 2 and
4 lumped loadings.
It is evident that as
little as 4 lumped resistive loadings, i.e., 8 loadings per wavelength, can approximate curately.
the
considered continuous
Note that this need not be
resistive
loading quite ac-
the case if the loadings are not
resistive.
z/h
Z/h
2
3 4
(a)
(b)
FIG.5 .5. Theoretical current distribution along resistive dipoles; a=0.3175 em, h=0.226 m, f=663 MHz, total loading along half of the dipole = 317 ~. ----- lumped equidistant loadings, piecewise parabolic approximation of current;--- continuous loading, 3rd-degree polynomial approximation of current. (1) real part, (2) imaginary part, (3) magnitude; (a) two lumped loadings, (b) four lumped loadings. (Ref.ll)
118
Ch.5.
5.2.2.
Wire antennas with concentrated loadings
Examples of cylindrical antennas with concentrated capacitive
loadings.
6
°
Cylindrical antennas with lumped capacitive loadings along
their length are
known to offer interesting possibilities. Broadband 45 46 47 cylindrical antennas • • and cylindrical antennas sustaining a trav61 ,62 elling wave along its major part are examples of such structures. Due
to
a large number of
parameters (positions
loadings), antennas having appropriate choice of
and magnitudes of the
a variety of properties
these
next part of the monograph.
parameters.
can be obtained by
This will be discussed in the
In this subsection we shall present a num-
ber of examples for admittance and current distribution of these useful lossless structures. For
the
theory to be applied successfully to capacitively loaded an-
tennas, it
is necessary
that
the real loadings be good approximations
to delta-function loadings adopted mation of the
the delta-function loading
form of
Such a
a narrow gap between
construction is
made in
the
shown
are mounted.
inder to slide tightly over the capacitive loading
is
between
the
the
capacitance and
is
two
An excellent approxi-
obtained if the loading is in tubular
inFig.S.6(a).
form of a dielectric rod
desired lengths
gap
in analysis.
parts of the antenna.
Briefly, the antenna
is
onto which metallic cylinders of
A narrow longitudinal slot allows a cylthe
dielectric rod.
The desired amount of
obtained by adjusting
cylinders using appropriate gauges.
the width of the gap The
relation between
the gap width, for different
was obtained experimentally by
the method
gap
described in
geometries,
the
following
section. As
the
adjustment of the gap width by using gauges is not quite pre-
cise, another construction of also used.
the antenna, shown
in Fig.S.6(b,c), was
Several elements with precisely determined
ance were made.
(Here, the dielectric rod and
were glued together.)
fixed
capacit-
the metallic cylinders
In addition, a number of brass cylinders of dif-
ferent lengths were prepared, with a screw at one end and with a threaded hole at
the other.
The fixed gap, or gaps, could be
positioned at
the desired location along the antenna by an appropriate combination of the brass cylinders.
Sect.5.2.
Equations for current distribution
119
longitudinal slot (a)
(c)
(b)
FIG.5.6. Two experimental models of the capacitively loaded monopole antenna: (a) model with variable gap widths; (b) model with a fixed gap width. (Ref .60) (c) Photograph of disassembled antenna with two capacitive loadings with fixed gap widths (on the extreme right side).
In order
to verify the theoretical results, experimental setups were
made for measurements of antenna admittance Fig.5.7).
and
radiation pattern (see
These setups were used for obtaining all the experimental re-
sults presented in
this monograph except those which are referenced to
some other source. The impedance measurements were made for monopole antennas mounted on a vertical ground plane. of 2 m side.
The ground plane was a square aluminium sheet
The hole in the plane through which the monopole was pro-
truding was positioned somewhat off center, to eliminate possible resonances.
The frequency
from l. l GHz
to
range
in which measurements were performed was
approximately 2. 6 GHz.
used, the side of
Thus, at
the square ground plane was
the
lowest frequency
longer than seven wave-
lengths. The value referred to
of the apparent monopole admittance, i.e., the
end of
the
the
admittance
coaxial line, \.as measured by the standard
reflection-measurement technique, using the GR 900 (General Radio Corp.)
120
Ch.5.
Wire antennas with concentrated loadings
(a)
{b)
FIG.5.7. (a) Photograph of part of the instrumentation used for measurements of antenna admittance: precision slotted line (General Radio 900-LB), sweep signal generator (Rohde Schwartz SWU BN 4246), frequency counter (Hewlett Packard 5246L with 5254C frequency converter) and galvanometer (Norma 251N) placed behind vertical ground plane. (b) Photograph of part of the equipment used for measurements of radiation pattern: antenna positioner (with a symmetrical dipole and detector), polar plotter, and transmitting log-periodic array.
slotted line.
Although it was
overall
of
error
not possible to determine precisely the
the measuring equipment, a maximum error of about 5%
seemed to be a realistic estimate. For measurements of the radiation pattern, dipole antennas were used. The antenna under investigation was served as and
the
a receiving antenna.
receiving antennas was about 20 m.
antenna terminals, mode.
mounted on a
A small
bably one
of
was
designed
to
the
transmitting
A detector, mounted at the
receive principally
asymmetric component received by the
10 m-high tower, and
The distance between
the
symmetric
the detector was pro-
causes of slight asymmetry of the radiation patterns
presented below. The available
and
considerable reflections were
observed from nearby trees and buildings.
The dipole was mounted hori-
zontally, pole axis
site was not clear,
with a small distance (about 3 em) existing between and
the
vertical axis
These factors probably also
of
the
di-
rotation of the antenna pedestal.
added to the asymmetry of the measured ra-
Sect.5.2.
Equations for current distribution
121
diation patterns. As mentioned above, in the general case a capacitively loaded antenna is a structure with many parameters. tive examples
of
case
of
is
that
these
structures
Therefore, only some representa-
are
presented below.
a monopole loaded with
The simplest
a single loading (or a dipole
with two identical, symmetrically positioned loadings).
A monopole an-
tenna of the form shown in Fig.5.6(b) was made, of length h=15 em (measured from
the
ground plane) and radius a=O. 3 em.
good accuracy, gaps
of
fixed widths were used, and
each loading was measured agreement
between
separately.
experimental and
The belt-generator approximation so
that
accurate
theoretical
to
Therefore,
In order to achieve the the
capacitance of best
theoretical results was
possible expected.
coaxial-line excitation was
values
of
the antenna
used,
susceptance were
also anticipated. Consider
first
a monopole antenna with fixed value of
the
gap capa-
citance and variable position of the gap along the monopole.
An exam-
ple of conductance (G) and susceptance (B) of the monopole against frequency is
shown
in Fig.5.8.
The theoretical results were
using the measured value of the loading, -j197
~
obtained by
at 1 GHz.
G,B (mS) 18
16 14
FIG.5 .8. Conductance (G) and susceptance (B) of a monopole antenna with a single capacitive loading Z1=-j197 ~ at 1 GHz, at a distance z 1 = 6 em from the ground plane. a=O. 3 em, h=15 em. theory; - - - experiment. (Ref.60)
12 10
8 6
4
2 0
-2 -4
f (GHz)
122
Ch.5.
Wire antennas with concentrated loadings
G,B (mS)
10 8
6 4 2 0 -2
- z
1
(em) FIG.5.9. Conductance (G) and susceptance (B) of monopole antenna with a single capacitive loading Z1=-j284 >l at 1 GHz against distance z 1 of the loading from the ground plane; a=0.3 em, h=15 em. (a) f=l.O GHz, (b) f=l.5 GHz. theory; experiment. (Ref.60)
24 20 16 12 8
4
z
1
(em)
0
-4
Fig.5.9 shows the real and imaginary parts of the monopole admittance plotted against
the position of a fixed capacitive loading of
at 1 GHz, at frequencies of 1 GHz and 1.5 GHz.
-j284
Q
Note that the influence
of the loading position is much larger at 1.5 GHz
than at 1 GHz.
This
is because the unloaded antenna at 1 GHz is approximately antiresonant, h=J./2, and at
1.5 GHz is approximately resonant, h=3>./4.
Note, again,
very good agreement of theoretical and experimental results. Fig.5.10 shows a sequence of radiation patterns in the electric-field strength for
a dipole antenna with two equal, symmetrically positioned
loadings
of values -j284 >l, at 1 GHz,
for
loadings.
The frequency was 1.5 GHz.
Note
different positions the
of
the
considerable influence
of the position of the loading on the radiation pattern. Using again
the
construction shown in Fig.5.6(b), monopoles with two
Sect.5.2.
Equations for current distribution
z
z
dB)
(dB)
0
capacitive loadings \vere investigated next. and 274 r2 at
123
1 GHz (measured values).
FIG.5.10. Radiation patterns in the plane containing the dipole axis for a dipole antenna with two loadings Z1= -j284 r2 at 1 GHz, positioned symmetrically at ±z1; f=l.5 GHz, a=0.3 (a) Z1=3 em, h=15 em. (b) (c) em, z1=6 em, (d) z1=12 em. z1=9 em, ---theory; - - - experiment. (Ref.60)
The two loadings were 185 Q
They were positioned at differ-
ent points along the monopole, and frequency dependence of the monopole admittance was measured and calculated. which loading
Fig.5.11 shows an example, in
z1=-j274
r2 at 1GHz was positioned at z =6.0 em, and load1 ing z =-j 185 Q at 1 GHz was positioned at z =10.1 em from the ground 2 2 plane. Note the considerable smoothing out of the curves when compared
with those tions of ever,
for the
a single loading (Fig.5.8). two
By interchanging the posi-
loadings, similar curves were obtained, which, how-
exhibited a
somewhat more
rapid
change of admittance with fre-
quency. To
gain
some
insight
along a monopole of of experiments
and
into
the
influence of
fixed length on
its
the protrusion of
properties, the following set
computations was performed.
the form of 25 rings of length 1 em each. the
the number of loadings
A monopole was made in
(In fact, the first ring was
inner cable conductor through the ground plane.)
The gap capacitance against gap width has been measured and this value used in computations.
124
Ch.5.
Wire antennas with concentrated loadings
G,B (mS) 12 CIO
0 0
FIG.S.11. Conductance (G) and susceptance (B) against frequency of a monopole antenna with loadings Z1=-j274 Q and Z2=-j185 Q, at 1 GHz, positioned at z1=6.9 em and z2=10.1 em from the ground plane; a=0.3 em, h=15 em. -----theory, oo,ee experiment. f ( GHz) (Ref.60)
0
10
8 6
•
4
2 0
1.6 1.8 2.0 2.2 2.4 2.6
-2 •
First, 24 cylinders.
gaps
of approximately 0.1 mm width were
Next,
the
second cylinder, counted from
left the
between the
ground plane,
was pushed to come into contact with the third, the fourth to come into contact with the fifth, etc. were obtained.
Thus 12 gaps of approximately 0.2 mm width
This procedure was
gaps of widths 0.3 mm,
0.4~m,
repeated to obtain 8, 6, 4, 3 and 2
0.6 mm, 0.8 mm and 1.2 mm, respectively.
Note that the first gap was always 1 em from the ground plane. quency dependence
of
the monopole admittance in
was both measured and computed in all theory and
experiment was
found to
trates two representative cases.
be
these
zero, resistance between the
two
range 1.1-2. 6 GHz Agreement between
satisfactory.
Fig.S.12 illus-
One reason for the discrepancy between
theoretical and experimental results could be
was found that
the
cases.
The fre-
the very small, but non-
rings pressed one against
the
other.
It
d.c. resistance betweem two such rings was not zero
and, in addition, that it was sporadic from case to case. An interesting feature of the admittance curves shown in Fig.S.12 can be seen:
in
both cases
the antenna admittance is much
less
frequency
sensitive than in the known case of the unloaded monopole antenna.
Sect.5.2.
Equations for current distribution
125
G,B (mS) 14 12 10 8 6 4 2 0
f
1.4
1.6
1.8
2.0
G,B (mS)
14 12 10
2.2
2.4
2.6
(a) 0
•••• •
8
Conductance (G) FIG.5.12. and susceptance (B) of a monopole antenna with e(GHz) qual equidistant loadings, against frequency. The first loading was 1 em from the ground plane; a=0.3 em, h=25 .4 em. (a) four loadings, 380 rl at (b) eight 1 GHz each. loadings, 280 rl at 1 GHz each. - - - theory; oo, (Ref.60) •• experiment.
6
4 2
f (GHz)
0
1.6
1.4
2.0
1.8
2.2
2.4
2.6
\
(b)
As an bution
illustration, Fig. 5.13 shows for
the theoretical
the monopole considered, for four
current
distri-
and eight loadings, at a
frequency of 2 GHz. Let us
consider now monopoles with a number of
equal
loadings, the
distances between which become progressively smaller towards
the mono-
pole end.
broadband
This construction was
of interest because better
properties could be expected for tapered loadings than for equidistant 45 loadings. We shall restrict our attention to the monopoles shown in Fig.5.6(a), with the positions of the gap centers defined by n
z
n
where
(n-0.5)8
+
I
[3.2-0.2(k-1)]
k=l
o
is the width of the gaps.
em,
n=l,2, ... ,15,
(5. 7)
Ch.5.
126
Wire antennas with concentrated loadings
z/h
1.0
2
-4
0
4
4
(a)
(b)
FIG.S.13. Theoretical current distribution along monopole antennas of Fig.S.12, at f=2 GHz; (1) real part, (2) imaginary part, (3) magnitude. (a) Four loadings. (b) Eight loadings. (Ref .60)
G,B (mS)
G,B {mS)
24 14 20 12 16 10
12
8 8
6 4 4
••
0
••
-4 1.0
1.5
f ( GHz)
2.0 (a)
2.5
2
• • • • •••• f ( GHz)
0
1.0
1.5
2.0
2.5
{b)
FIG.S.14. Conductance (G) and susceptance (B) of monopole antennas with n loadings of -j320 S1 at 1 GHz each, at distances from the ground plane given in eqn. (5. 7); a=O. 3 em. (a) n=1, h=6. 23 em; (b) n=7, h=20.21 em. - - t h e o r y ; oo, •• experiment. (Ref.60)
Sect.5.2.
Equations for current distribution
z
1 .0 z/h
-8 -4 0 4 8 12 16 20 (a)
2
-8 -4 0 4 8 12 16 20
(b)
Two representative examples susceptance
127
of
are
shown in Fig. 5.14 of conductance and
the monopole admittance
Zload=-j320 Q at
1 GHz, and n=1 and 7.
come flatter with
increasing
the
FIG.5.15. Radiation patterns in the plane containing the dipole axis and current distribution along a dipole antenna with arms the same as the monopole described in caption to Fig.5.14(b); (a) f=1.2 GHz, (b) f=l.8 GHz, (c) f=2.4 GHz; (1) real part, (2) imaginary part, (3) magnitude of current. ----- theory; - - - experiment. (Ref. 60)
plotted
against frequency, for
It is seen that the curves be-
number of the cylinders.
the scale for n=1 is smaller than for the other case.) ing that, with only band properties.
(Note that
It is worth not-
seven gaps, the antenna exhibits remarkable broad-
These are not improved considerably with additional
cylinders, at least in the frequency range considered. Finally, Fig. 5 .15
shows
examples of the theoretical
current
distri-
128
Ch.5.
Wire antennas with concentrated loadings
G,B ( mS) termination
8
c::JOCD
h
4
27.32an
h (em)
0 5 10 tG,B (mS)
(a)
G,B (mS)
'~~
8
~
20
t~
·~ 8~ ~·-- ..
-1>-0..
............,.,
15 (b)
4~4
B
0 .___---=----:-':---c'::c--.___h_______,(c_m) 0 5 10 15 20 (c)
I
I
I
10
5
15 (d)
h. ( em)
I
20
FIG.5.16. Conductance (G) and susceptance (B) of cylindrical antennas with nonreflecting capacitive termination, sketched in figure (a), versus the antenna length; (b) f=l.3 GHz, (c) f=l. 7 GHz, (d) f=2.3 GHz. -----theory; oo, •• experiment. (Ref.62)
but ion and of tenna having GHz.
The
gap
the measured and calculated radiation pattern for an anseven gaps,
at
impedance was
frequencies
of
-j320
1 GHz.
at
Q
1.2 GHz, 1.8 GHz
and
2.4
Note that the current
distribution is in the form of a decaying progressive current wave with small standing waves bet\veen the loadings.
The
radiation
patterns at
1.8 GHz and 2.4 GHz also clearly indicate the travelling-wave behaviour of
the
current.
At
1.2 GHz
the
radiation
region of
the
antenna
is
small, and the pattern resembles that of an electrically short antenna. The
reduction
of
the
last figure suggested unloaded monopoles
reflected wave
along
the
antenna seen in the
that a "nonreflecting" capacitive termination of
could
considerably
increase
their otherwise
poor
broadband properties.
To that aim an antenna of the form shown in Fig.
5.16 (a)
its admittance versus the length h
was made
pole and lengths
and
frequency measured. (30-k· 2) nun,
of the mono-
The termination consisted of 15 rings of
k=O, 1, ... , 14 ,
with gaps
of capacitances 1. 2 pF
left between successive rings. The results are shown in Figs.5.16(b)62 (d). It is seen that, indeed, the termination increases to a large extent the monopole broadband properties.
Sect.5.3.
5.3.
Measurements of concentrated loadings
NOTES ON MEASUREMENTS OF CONCENTRATED LOADINGS
For a thin-wire antenna operating in hertz ,
the
antenna
the
range up to few tens of mega-
problem of determining accurate value of
ing impedance
cut.
129
is
is not complicated.
cut, and
the
In
the
the lumped-load-
simplest solution, the wire
loading is connected between the ends
The lumped-loading impedance
can be obtained by
of the
standard bridge
measurements, prior to introducing it into its place along the antenna. Impedance of the capacitor formed by the two wire antenna parts is usually much
larger
than
that
of the required lumped element, and can be
neglected. However, for frequencies of citor
formed by
which is if a
of
two
the
order of 1 GHz and higher, the capa-
sections of an antenna itself may have impedance
the order of
the
required loading impedance.
(Therefore,
capacitive loading is desired, usually an additional capacitance
need not only as
be used at all.) an
integral
measurement will
This impedance should obviously be measured
part
of
the antenna structure.
be difficult
to
However,
be performed in some other, practically realisable, manner. the results are (apart
from
then only approximate.
the accuracy of
to measure
the
tenna, and
to
impedance in realise
a
Obviously,
For the results to be accurate
the measurement itself), it the
such a
carry out, and usually will have to
is
essential
operating frequency range of
construction that is as
close
the
an-
as possible to
the real antenna structure. This section describes
two methods used by the authors for measuring
concentrated reactive loadings mounted method, outlined in
the
along wire antennas.
following subsection, is essentially a compen-
sation method, and is particularly useful a
single
loading, for example
capacitance on
The first
its width.
The
for
in measuring
determining impedance of the
dependence of the gap
second method, outlined
in Subsection
5.3.2, is basically a resonant method, suitable for measuring reactances on an already assembled antenna. 5. 3.1.
Compensation method
most natural
for measuring
lumped
configuration for measurements of
reactances.
60
The
lumped antenna imped-
130
Ch.5.
Wire antennas with concentrated loadings
FIG.5.17. Schematic view (a) and photograph (b) of short-circuited coaxial line for measurement of concentrated loadings. (Ref. 60)
1~
equivalent plane of short circuit
gap
(a)
(b)
ances appeared to be a coaxial line,
with a part of the antenna itself
replacing the inner line conductor.
A 50 Q air coaxial line was there-
fore made,
having
an
inner-conductor
diameter
diameter of the outer conductor of 13.8 rnrn. with sliding contacts was useful length of
an essential part
the coaxial line was
of 6 rnrn, and the inner
A precision movable piston of
the coaxial line.
160 rnrn, and it was
The
possible to
set the movable piston with a precision of ±0.05 rnrn at any point within this
region.
The movable
which is being measured, loading
in
gap between parts.)
coaxial piston, is
shown
in
together with
Fig.5.17.
the
element
For convenience, the
Fig.S.17(a) is shown as a capacitor in the form of a narrow the
two antenna parts (i.e.,
inner
coaxial-line conductor
To be more specific, we shall assume in further considerations
that the loading is a capacitor, but the principle can be extended easily to any
type of lumped reactive loading the size of which is of the
order of the wire radius. To eliminate
possible errors
supporting dielectric wafers
due
to
connection
discontinuities and
on the measuring line, the following mea-
Sect.5.3. suring
131
Measurements of concentrated loadings
procedure
checked without plane of
was
the
the short
adopted.
capacitor circuit
The
shown
short-circuited
line was
first
in the figure, and the equivalent
determined experimentally.
The capacitor
was then introduced into the line, and the piston moved until the equivalent plane of of
the
the
capacitor.
Fig.S.17(a).
short circuit coincided with the center of the gap This
position of
the
piston is designated by 1 in
By means of the GR900 (General Radio Corp.) slotted coaxi-
al line, the location of a minimum along the slotted line was detected. Note
that,
in
this
position of
the piston, the capacitor is not con-
nected in the circuit [Fig.S.17(a)].
Let us designate this position of
the equivalent plane of the short circuit by x , with respect to an ar1 bitrary reference plane. The piston is next moved so that the capacitor is introduced into the circuit.
As a result, the position of
The piston is
the minimum changes noticeably.
then moved until the position of the minimum on the mea-
suring line coincides with the earlier position (corresponding to position
1 of the piston).
pedance, referred to position of
the
This obviously means that the total series im-
plane
1, is again practically zero.
is the 2 equivalent plane of the short circuit, it follows that
the reactance
of
reactance
the short-circuited coaxial line
of
the capacitor is equal
to
If x
the negative value of the of
length Cx -x ). 2 1
The
capacitance is hence obtained as 1
c
(5. 8)
is the characteristic impedance, and k Z c of the short-circuited coaxial line.
where
the phase coefficient
A comparison between the gap capacitance measured by the above method and
the
concept of a delta-function loading may need some _comments at
this point.
As already explained, a rigorous analysis of
this problem
is intricate, but some comments relating to it can be given. The proposed method enables us refer it
to
to measure certain reactance, and to
a particular cross-section of
the
coaxial line.
Provided
that the gap width is small compared with the outer coaxial-line radius,
132
Ch.5.
Wire antennas with concentrated loadings
C (pF)
1.4
6~5mm
1.2
a~
1.0
FIG.5.18. Measured gap capacitance against gap width, for three different dielectric supports; (a) acrylic rod, (b) air (acrylic rod with deep groove), (c) glass tube. (Ref.60)
0.8 c
0 .. 6 0.4 0.2 d (mm)
0
the
0.2 0.4 0.6 0.8 1.0 1.2
outer
(Although eral
conductor does this
not influence
the value of
that
reactance.
could be proved experimentally, it requires making sev-
short-circuited
coaxial
dius, and has not been
lines
of
different outer-conductor ra-
done by the authors.)
Thus this reactance can
formally be ascribed to a delta-function loading, located on the innerconductor surface, at the center of the gap. As an example of application of the method, the capacitance was measured of
the
gap shown
dielectric supports.
in
the
Curve
a,
inset of Fig .5 .18, for three different shown in
the
figure, refers to a solid
acrylic cylinder of 5 rnrn diameter.
Curve b corresponds to an acrylic
cylinder with a deep
the
case of
groove
an air dielectric.
around
tube of outer and inner diameters 5 pacitances were used
to
gap, made
Finally, curve
obtain
and
the
c
to
approximate the
corresponds
to
3 rnrn, respectively.
theoretical
results
a glass
These ca-
presented in
the preceding section. 5.3.2.
lumped reactances mounted on the antenna by 16 means of a coaxial resonator. Using the compensation method described in
the
Measurement of
preceding subsection it is possible to measure isolated concen-
trated loadings.
It
cannot be used,
however,
for measuring a succes-
Sect.5.3.
Measurements of concentrated loadings
133
hollow conductor short circuit
(a)
(b) FIG.5.19. Schematic view (a) and photograph (b) of coaxial resonator used for measuring concentrated loadings along assembled antennas.
sion of loadings already mounted on many cases.
an antenna, which is preferable in
For example, by using gauges
for
fixing a gap width, only
moderate accuracy can be achieved, so that short-circuited coaxial line shown in Fig .5 .17 cannot be rate values.
used for realising loadings of very accu-
For this reason a special resonator was made which enabled
precise measurements of prefixed loadings on an assembled antenna. The measuring the form of tubes with
structure
is
shown
a coaxial resonator with a short
segment between
antenna into these tubes, any of the resonator
to
nator conductor. resonator circuit,
in Fig.5.19.
the
Basically, it
the inner conductor made them missing.
is
in
of two
By introducing the
loadings can be positioned inside
represent a series loading of the inner coaxial-resoTo measure
the
loading
impedance,
into resonance by either varying the or by varying
the frequency.
The
we
can bring the
position of the short
loading
impedance
can be
134
Ch.5.
Wire antennas with concentrated loadings
then computed from the frequency and the measured values of the lengths lA and lB or, if a more precise measurement is desired, of lengths lA'' lA'" lB' and lB" [see Fig.5.19(a)]. In addition
to
bled antennas,
enabling measurements of loading impedances on assem-
the method has
another
two
useful properties.
First,
as a resonant method in \vhich the measured quantity is obtained by measurements of length and ond,
it
enables
considered as
a
that
frequency, it is a very accurate method.
actual parameters be determined of
the
Sec-
loading
two-port network, and thus check whether in reality it
can be regarded, at least approximately, as a pure series element. The
particular
resonator
shown
in Fig. 5. 19 was made
ments on antennas of diameter 6 mm and
7 mm.
The
inner,
for
measure-
tubular con-
ductor of the resonator should differ in diameter as little as possible from
the
them.
antenna diameter, to reduce the step-like transitions between
Therefore it was made to be 7 mm, viz. 8 mm, for measurements of
loadings on antennas having the two mentioned diameters. The outer resonator diameter should be the
stray
sible.
chosen
so
that its effect on
field due to the loadings being measured be as small as pos-
However, it
should not be too large, in order to eliminate the
possibility of existence of higher coaxial-line modes. value of
the
effect of
the
outer
resonator
For the adopted
diameter (28 mm), it was
found that the
outer conductor on the stray field due to the loading is
negligible even if its length is several millimeters, and sonator can be used for frequencies up to about 5 GHz. er details concerning
the
construction of
the
that
the re-
There were oth-
resonator, but we shall
not elaborate them here. To determine
the
impedance of the loading introduced in to
the
reso-
nator, we shall consider the structure shmvn in Fig.5.19(a) as a series connection
of
three t\vo-port networks:
the
two coaxial-line parts
of
the resonator, designated by A and B in Fig. 5. 19 (a), and the measured loading between
them, designated by D.
lle assume that
D is a linear
lossless reciprocal network (as are also the two coaxial-line parts), so that in the general case it is uniquely defined by three parameters, of either Y- or Z-type.
(If the loading is geometrically symmetrical with
Sect.5.3.
Measurements of concentrated loadings
135
(1) y
v
A
B
FIG.S.20. An equivalent 5.19(a).
circuit
for
the
respect to the median plane, two parameters the Y-parameters resonator is ling, and
the
the
ter,
for
the analysis.
suffice.)
generator
We
in Fig.
shall adopt
One possible representation of the
then as that shown in Fig.5.20.
Due to a very weak coup-
can be represented as an ideal voltage generator,
diode used for detecting the resonance as an ideal current me-
i.e.,
the two-port networks A and B can be assumed to be short-
circuited at their ports 1' and 2 1 For a B of
resonator sketched
,
respectively.
given loading, the lengths of
the resonator can always be
the
adjusted so that the system in Fig.
5.19(a), i.e., in Fig.S.20, be in resonance. maximal current intensity in a measurement procedure quency of
the
to
the
is
A resonance is detected by
current meter.
We
can
always follow
ensure that the first, lowest resonant fre-
system be obtained.
and frequency, it
coaxial-line sections A and
By measuring the lengths lA and lB'
possible to calculate
the
admittance of the two-
port network A, at port 1 looking to the left, and that of network B, at port 2
looking
to
the right
circuited coaxial lines of
course,
the inner 5.19(a)
possible line
take
into
lA and lB, account
conductor, the lengths
the
respectively.
(It is,
different diameters of
of which are designated in Fig.
Let these admittances be YA and YB.
Resonance port
lengths
Fig.S.20, as admittances of short-
by lA', lA"' lB' and lB'" if a more accurate representation is
desired.)
at
to
of
in
in
Fig. 5. 20
is
attained if input admittance
to network D
1, looking to the right, equals -YA, and that at port 2, look-
ing to the left, equals -YB.
We thus obtain the equations
136
Ch.5.
Wire antennas with concentrated loadings
(5. 9)
v2 ;v 1
Eliminating
from these equations we get (5. 10)
(Of course, YDll=jBDll,
etc., since all
the
admittances in this equa-
tion are imaginary, the networks being lossless and passive.) This
is
an equation in 2
YDllYD
-YD
three unknowns, YDll' YD and YD (namely 22 12 To be able to determine these parameters of the D-uet-
).
22 12 work, we must
calculate
the
admittances YA and YB for three different
lengths lA of the resonator, in which case eqn.(5.10) results in three linear equations for determining the three unknowns. urement errors, it urements, and
to
is
advisable
obtain
the
some convenient procedure, be
the
best possible
YDll' YD
and YD
22
12
This method was loadings
used
in
perform several sets of such meas-
unknown parameters of
for
the
example by requiring that
the least-square
sense.
D-networks by the
solution
Note, however, that
are functions of frequency. applied
to measure
in examples
antenna synthesis.
to
To decrease meas-
Two
of
cases
several
types
antenna analysis of importance
of
concentrated
and, in particular, of
for
future
reference
are
the experimental values of reactances of variable lumped capacitive and inductive loadings sults of
the
designed
specifically
antenna synthesis (to be
this monograph).
by varying the distance
o,
part
The
the
element.
distance between the accuracy of
about
aid
in verifying the re-
considered
in
the next part of
These two types of loadings are shown in Fig.5.21.
The value of the capacitive loading
of
to
could
be
varied continuously
i.e., by screwing and unscrewing the movable' screw pitch was
only 0. 25 mm, so that the
capacitor electrodes was
0.01 mm.
possible to adjust with
Note that the physical length of the whole
element is thereby not changed, so that the total antenna length is also not changed by adjusting antenna.
Dependence of
1 GHz
the distance
on
one
the
or more
such capacitors on an assembled
element capacitance and of susceptance at
o between
the electrodes
is
shown in Fig.5 .22.
Sect.5.3.
137
Measurements of concentrated loadings
(a)
-
•
(c)
(b)
FIG. 5. 21. Variable lumped antenna loadings: sketch of (a) capacitive and (b) inductive loading; (c) photograph of both loadings and exchangeable inner conductors of the inductive loading.
Be (mS) C (pF)
25
4
20
6 (mm) 0
0.5
1.0
1.5
2.0
2.5
3.0
FIG.5.22. Capacitance (C) and susceptance at 1 GHz (Be) of the capacitive loading shown in Fig.5.2l(a) versus the distance o between the electrodes. For upper curve the readings for C and Be are to be divided by five.
Ch.5.
138 Note
that
it was possible
Wire antennas with concentrated loadings to
realize with
n
loadings ranging from about 40 The value varied in
of
the
steps by changing the diameter of the
sides of
the
be
screw S shown in the
screw), which was used as the inner conductor
two short-circuited coaxial
on both
at 1 GHz.
the inductive loading shown in Fig.5.21(b) could
figure (by replacing of
this capacitor capacitive
n
to about 700
the gap.
The
lines machined inside
gap
itself was
(4 mm), in order to reduce the gap capacitance.
the
element
left relatively wide Dep7ndence of the ele-
ment inductance and of its reactance at 1 GHz on the screw diameter is shown
in
could
be
Fig.5.23. Note that only a relatively limited range of values realized, due to small size of the element.
This induc-
tance was used primarily for compensation of the antenna susceptance by mounting the element in the excitation zone. Finally, above
it was
cannot
found
experimentally
be strictly regarded as
that
pure
the
elements
series elements,
this is true with a relatively high accuracy. of
the
described but
that
For example, in the case
capacitive loading a shunt reactance of about 5 kQ at 1 GHz ex-
isted in addition
to
the series capacitance, but
it
can be
obviously
neglected without significantly impairing accuracy.
\
(n)
L (nH)
40
6 5
30
4 20
\
' "-.....,
....
""'-....,
.....
3
0 ......... ....,P,
..........
_ --.Jl
10
2 0
d (Rill)
1.0
2.0
FIG.5.23. Inductance (L) and reactance at 1 GHz (XL) of the lumped inductive loading sketched in Fig.5.2l(b) versus the diameter of the screw representing the inner conductor of two short-circuited coaxial lines.
Sect.5.4. 5.4.
Wire antennas with mixed loadings
WIRE ANTENNAS WITH MIXED LOADINGS
It was
shown
distributed
in
the
loading
preceding offer
chapter
case
resistive
antennas with
However, they exhibit
example
re-
losses, which in
of the "travelling-wave" resistive antennas are such that ef51 is of the order of 50%. On the other hand, antennas with
ficiency lumped
that
interesting possibilities, for
markable broadband properties. the
139
capacitive
those of
loading exhibited
resistive antennas, but
indicated that might be
antennas with
expected
to
broadband properties
did not have losses.
inferior
to
These examples
combined distributed and lumped loadings
have properties
superior
to
those
of
antennas
with only one type of loadings. To analyse sented
such antennas we
have simply
in Sections 4.2 and 5.2.
cal cylindrical
antennas
to
combine the theory pre-
For example, in the case of symmetri-
the Hallen equation
(5. 6) should be modified
by adding
to the left side the term D(z) given in eqn. (4.8) with z =0. 0 Such an equation could then be solved by any convenient procedure. As
an
example of mixed distributed and lumped loadings we shall con-
sider a cylindrical antenna with distributed resistive and concentrat42 loadings. The first problem to be solved was that of
ed capacitive
designing a convenient model capacitive loadings. ments
to be
Firstly,
a
of
useful model
it had
to
cylindrical antenna with resistive and
The antenna had
be as
for
to satisfy
investigations
simple as
the loadings.
Finally,
of
basic require-
such structures.
possible, for practical reasons.
Secondly, it had to be flexible with respect to bution of
three
the
the
amount and distri-
physical model
had
to
be such
that it can be analysed theoretically with sufficient accuracy. After considering several possibilities, an antenna model which closely satisfied all
the
three
requirements was made in the form of a row
of commercially available high-quality resistors between which air gaps were left. over
a
row of
The resistors were
in
cylindrical ceramic body, these resistors was
the with
form of
a thin resistive layer
silver endings at both ends.
A
placed in a groove made in a styrofoam di-
electric support suspended in
front
of
a vertical ground plane.
The
140
Ch.5.
Wire antennas with concentrated loadings
air gaps between the resistors represented concentrated capacitors, the capacitance Such a
of which
construction
can be varied
easily by varying the gap width.
is very simple and
provides great flexibility in
obtaining the desired distribution and magnitude of the loadings. ther, the structure very nearly satisfies the system by
the
the
Fur-
conditions for analysing
approximate method described in this and the preced-
ing chapters. A monopole antenna of 1=24 mm and
this
form was made using resistors of lengths
radius a=3.S mm.
With regards to resistance, resistors of
resistances SO, 100 and
200 Q were
used (which correspond to continu-
ous resistive loadings of about 2080, 4170 and 8330 Q/m, respectively). 1 According to some known resultsSO,S and to numerical computations performed by
the authors, step-like resistive and lumped capacitive load-
ings were
adopted,
tenna end.
It was
the magnitudes of expected
that
which
increased towards the an-
such a loading should result in good
broadband properties for frequencies between 1 and
2.S GHz.
Three re-
sistive elements, each of length 4.8 em (two connected resistors of the same resistance) were added to a segment which was S .6 em long presented
a
simple
protrusion
coaxial-line
brass
conductor,
The
first
segment of
through of
the
the same
ground
and re-
plane of the inner
diameter as the resistors.
resistive segment was made of two SO Q resistors, the second two
100 Q resistors
and
the third of
two
200 Q resistors.
The air gap between the first and the second segment was 0.3 mm, corresponding approximately to 1.1 pF, the second gap was 0. S mm, corresponding so
to that
O.S6S pF, and the last gap was the
total
length of
1 mm, corresponding to 0.34 pF,
the monopole antenna was
h=20.18 em.
A
photograph of the antenna is shown in Fig.S.24. Using the method described above and adopting the belt-generator mod-
FIG.S.24. Photograph of the resistive-capacitive monopole antenna, of radius a= 3.S mm and length h=20.18 em. (Ref. 42)
141
Sect.5.4.
Wire antennas with mixed loadings
el of
coaxial-line excitation, a program was
the
prepared for analys-
ing symmetrical dipole antennas with arbitrary continuous trated impedance loadings along their lengths.
and
concen-
The antenna admittance,
radiation pattern and current distribution for the case described above were calculated in the frequency range 1.2-2.4 GHz.
The solid lines in
Fig.5.25 show the calculated conductance (G) and susceptance (B) of the antenna model against frequency. To check nopole
the
reproducibility of the antenna, measurements of the mo-
admittance were
performed
several
times.
After every set of
measurements the whole antenna was dismounted and then assembled again. The difference limits of
the
in
the results was
experimental error.
found to
be
practically within the
One typical set of measured results
is presented in Fig.5.25. A very good agreement between measured can be sults
observed. is not
so
Agreement between good
for
and
theoretical values of G
experimental and
the imaginary part of
the
theoretical readmittance.
The
measured and the computed B curves have practically the same shape, but there is a nearly constant difference between them. asons
for
Three possible re-
the discrepancy are (a) the capacitances of the concentrated
capacitive loadings
are not known exactly, (b) the mathematical model
of the antenna is not sufficiently accurate, e.g., the influence of the silver
resistor endings was not taken
into
account, and (c) the belt
generator can also introduce some error.
G,B {mS) 18 FIG.S .25. Conductance (G) and susceptance (B) of the antenna shown in Fig.S .24 against frequency. theory; oo, •• experiment. (Ref.42)
15 12
9
6
3 0
G
~B :-= • • • ••• •••••
1.2
1.6
2.0
2.4
f {GHz)
142
Ch.5.
Wire antennas with concentrated loadings
It is evident that the described resistive-capacitive type of antenna has
good
broadband 'characteristics
in admittance.
For
the average
measured conductance of Gav=8.26 mS in the frequency range from
1.2
to
2.4 GHz, the average VSWR is only 1.38. Measurement
of
current
distribution
in
complicated task, and was not performed.
the
present
case is a very
However, theoretical analysis
of such structures was
proved to predict quite accurately not only the 60 antenna admittance, but also the radiation pattern. In other words, the theoretical current distribution should be very close to the actual distribution.
The antenna efficiency,
n,
can therefore
be
determined·
theoretically with high accuracy as
P.
n = 1nput
- p
Joule along antenna
(5. 11)
P.
1nput
where (5. 12)
Pl.nput =--G- J I(O) J2 ' G2 +B2 and h
-I
P Joule along antenna-
R'(z)ji(z)j
2
(5 .13)
dz,
0 I(z) representing
the
complex r.m. s. value
of
current intensity along
the antenna and R'(z) the antenna resistance per unit length. ciency of 81
and
better
the
84%
over
than
for which, should be
The effi-
antenna described was found in this manner to be between the whole
frequency
range
considered.
This is much
for purely resistive travelling-wave cylindrical antennas, as mentioned,
noted
that
tremely involved (it
the efficiency is
of
the
experimental determination of
order of 50%.
It
efficiency is ex-
can be, for example, based on measured radiation
pattern and gain) and probably less accurate than the theoretical estimate. Finally,
the
theoretical current distributions exhibited a
caying travelling-wave property,
and
the
quaside-
corresponding radiation pat-
Sect.5.4.
143
Wire antennas with mixed loadings
-6 (b)
(a)
FIG. 5. 26. Theoretical current distribution (a) and radiation pattern in electric-field strength (b) of the antenna shown in Fig.5 .24, at f=l.5 GHz; (1) real part, (2) imaginary part and (3) magnitude of current. (Ref.42)
terns had
the expected shapes, typical
tennas.
Examples
pattern,
in
Figs.5.26(a)
the
of
the
current
for
distribution and of
electric-field strength,
and
(b).
travelling-wave linear an-
at
f=1.5 GHz,
The current magnitude had
the are
radiation shown in
practically the same
shape over the whole frequency range considered, and radiation patterns changed slowly similar
to
from
that shown in Fig.5.26(b), only with somewhat more pronoun-
ced radiation at at 2400 MHz.
that of a Hertzian dipole at 1200 MHz to the shape
Thus
approximately 45°, with the
respect
to the ground plane,
antenna exhibits excellent broadband properties
in the radiation pattern also. In a later chapter a similar antenna will to be
optimized.
drical antenna
be considered as
a system
It will be seen that exceptionally broadband cylin-
can be
thus obtained, having very
markably wide frequency range.
small VSWR in a re-
144
Ch.5.
5.5.
CONCLUSIONS
In this with
Wire antennas with concentrated loadings
chapter a method was presented for
concentrated loadings.
in various ways,
the
Although the
simplest,
analysis of wire antennas
loadings
delta-function
can be represented
approximation
of
the
loadings appeared to be of sufficient accuracy. A number of examples of antennas with both
concentrated loadings analysed
theoretically and experimentally indicated satisfactory agreement
between
the two sets of results.
Particular
attention was devoted to
concentrated capacitive loadings, because they are easy to realise accu- , rately and are lossless. capacitive
loadings,
Most examples related to antennas with lumped
and a
section was
devoted to describing several
kinds of lumped capacitive loadings made by the authors and measurement of their
capacitance
nas with combined drical
in operating conditions.
As an example of anten-
continuous and concentrated loadings,
antenna was described and analysed,
indicating
an RC cylin-
some advantages
of the combined type of loading over one type of loading only. Inductive all
series
(although,
chapter lowing.
applies
of
lumped loadings were not considered as examples at course,
the
method for analysis
to that case also).
Such loadings result in
The
reason
essentially
described in this
for this was the fol-
slow-wave structures.
It
is known that slow-wave structures are less efficient radiating systems than fast-wave structures (antennas with our case),
and
that they usually have
efficiency than the latter.
series capacitive loadings in narrower bandwidth and smaller
CHAPTER 6
Wire Antennas in Lossy and Inhomogeneous Media
6.1. In
INTRODUCTION
deriving
previous
the equations
chapters
it was
for
analysis
assumed
homogeneous lossless medium.
that
of wire-antenna structures in the antenna was
situated in a
Although this assumption does approximate
well the real situation frequently, e.g., in all instances in which antenna is situated in
air
an
far from other objects or possibly adjacent
to a large, theoretically perfectly conducting plane surface, in practice antennas
in
conducting, approximately homogeneous media are
encountered.
Examples of
vehicles travelling through the antenna and
such cases the
are numerous:
antennas
ionosphere (neglecting
the anisotropy of
the
also
on space
sheaths around
ionosphere), antennas buried in
the ground or immersed in the sea, or antennas used for measuring properties of
media
(i.e., plasma or the earth's crust).
The next section
of this chapter is devoted to analysis of wire antennas in such circumstances. In reality, antennas
are never operating
in homogeneous media.
For
example, even if an antenna is situated high above the earth's surface, the presence
of
the supporting structure and
certain influence on this kind of
the antenna properties.
the
feeder certainly has
A common
case
in which
inhomogeneity is practically negligible is a large, plane
conducting sheet a coaxial line.
through which a monopole antenna is
fed
by means of
In most instances, inhomogeneity of one kind or another
is always present in the antenna vicinity, and has at least a small influence on the antenna properties.
146
Ch.6.
Analysis
of
the influence of an inhomogeneity is, as a rule, a very
difficult task. the
final
Wire antennas in lossy and inhomogeneous media
It is therefore often neglected even when this affects
results considerably.
homogeneities, however, which is
There are
two
that when a planar wire-antenna structure
interface between
two media.
is
located on
the
One plane
Approximately this situation is obtained
if the antenna is laid on the flat on the surface of the sea.
important cases of in-
can be analysed with high accuracy.
surface
of the earth or is floating
The other is that of antennas located above
real, imperfectly conducting ground, a situation which quent occurrence in practice.
is of very fre-
These two cases will be considered in the
third and fourth sections of this chapter. 63 WIRE ANTENNAS IN HOHOGENEOUS LOSSY HEDIA
6. 2.
Consider a
perfectly conducting wire-antenna 1
homogeneous lossy medium of actly we
the
parameters £=(£ -j£
same approach as in
the
can homogenize the medium and
potentials.
In
the
structure 11
),
situated in a
ll and cr.
Using ex-
case of antennas in lossless media, use the expressions for the retarded
complex formalism
the
losses
the equations (and in the solutions) by simply
are
changing
· lossless case) to equivalent complex permittivity £
eq
incorporated in £
(real in the
,
(6. 1)
£ · = £'-j(E:"+cr/w) eq
(Generally speaking, ll can also be complex, e.g., for a lossy, approximately
linear ferrite material, but this case is rarely encountered in
the antenna practice.)
As
a
consequence,
the
propagation coefficient
becomes complex also, k
=
w~ eq
=
t3- j a ,
(6. 2)
as well as the intrinsic impedance of the medium,
r,;
=
/ll/E:eq
=
r,;r
+
(6. 3)
jr,;i
Thus, the presence of losses in a (homogeneous) medium leaves all the equations formally intact, but k pagation coefficient k solution of
enters
and
all
r,; become complex.
the integrals which
an antenna structure need
to
be
Since the proin a numerical
evaluated numerically, a
Sect.6.2.
Wire antennas in homogeneous lossy media
computer program valid
for
to be modified.
however,
shall not
This,
their evaluation in
elaborate it here.
cal results
for
is
a
147
the
lossless case has 64 relatively easy task and we
Instead, we shall present some theoreti-
certain cases of cylindrical antennas
for which expe-
rimental data are available. Extensive experimental results were presented by admittance mersed
and current distribution along, monopole antennas im65 6 7 in a liquid conducting medium. The monopole antennas ana-
lysed had a radius a=O. 318 em, the
Iizuka and King for
of,
inner
and
represented a
coaxial-line conductor through
radius of the width of
outer
the
simple
protrusion of
ground plane.
The inner
coaxial-line conductor was b=1.112 em, so that the
the equivalent belt
generator, according
to
eqn. (2.29), was
aa=5 .45 a. In
the
experiments, the coaxial line was
sealed at the ground plane
with a lossless dielectric piece of polystyrene, which is not quite the same as the theoretical model that can be represented by the belt generator.
However, it
is not difficult
to
conclude that this cannot in-
troduce significant error in theoretical results. the experiments was kept constant at 0.036 to 8.8.
to 78£
0
For measurement of forms:
(a)
(bare antenna),
114 MHz, and cr/w£ was varied from
The permittivity £=£ 1 -jO of the liquid solution used was
in the range of 69£
two
The frequency in all
as
a
and
0
•
the admittance, the monopole antenna was made in simple protrusion
of
the
inner
cable con due tor
(b) using an antenna partially covered with a die-
lectric cover (made of penton) to protect a small probe used to measure current
distribution
differ considerably.
along
the monopole.
Naturally, thE first
The set
two
admittance
curves
of results, correspond-
ing to the bare antenna, were considered. Figs.6.1(a)-(c) display the theoretical and experimental curves showing conductance (G) and susceptance (B) of cal dipole antenna versus ured in the
the
conducting medium).
using Hallen's equation and a
the
corresponding symmetri-
antenna electrical halflength Sh (measThe theoretical results were obtained single
polynomial (of degree n)
to
ap-
proximate current distribution along the whole monopole-antenna length.
148
Ch.6.
120
Wire antennas in lossy and inhomogeneous media
G,B (mS)
100 80 60 40 20
Bh
0
4
-20 (a)
80 G,B (mS) 60
_____________ _
...._ G
-
40
------
20
Bh
0 (b)
G,B (mS)
--------. -·-·-·-·-·-
160
G
120 80 40
Bh
o~~~-L----~------~----~4--
:. \ .. ,_1
-40
2
B
3
_ ... 'iiiift ';o;;-~-~~~-~-~~~~==---
(c)
FIG. 6 .1. Conductance (G) and susceptance (B) of dipole antennas immersed in a conducting medium; a=O. 318 em, f=114 MHz; (a) E:r=78, cr/wE:=0.036, (b) E:r=77, cr/wE:=1.06, (c) E:r=69, cr/wE:=8.8. -----belt generator, n=5; • - delta-function generator, n=3; - - - experiment. 65 (Ref .6.3)
Sect.6.2.
Wire antennas in homogeneous lossy media
The
shown, with cr/ws=0.036, 1.06 and 8.8, indicate again clearly
cases
the advantage
of
the belt generator (or, equivalently, of the TEM mag-
netic-current frill) cr/ws=l.06,
the
149
over
the
conductance
delta-function
curve
generator.
corresponding
to
Already for
the delta-function
generator is in substantial error, an error that becomes rapidly larger as cr/ws increases further, in spite of the very low-order approximation for current used.
It is interesting to note that, for cr/ws>1, the more
pronounced effect
of
conductance
rather
the
delta-function generator
than on
susceptance, just as
is
the
on
the antenna
converse is true
for cr/ws<1 (which, of course, should be expected). As already mentioned, for measurements ka and King
used
dielectric cover, protecting fortunately, the
bare
of
current distribution Iizu-
a monopole partially covered along
this
the
probe from
arrangement obviously was
monopole antenna, as
about
the
a
close agreement of
length with a
liquid solution.
Un-
radically different
from
1/3 of the monopole circumference
was not in direct contact with the liquid solution. not expect
its
Therefore, we can-
current distribution obtained by the
present theory with experimental results of Iizuka and King. The entire-domain polynomial used to approximate current distribution introduced
some
difficulties.
points
in
the belt-generator region, we
values
of
current for
point admittance;
By
small
the
shown
three,
about
the
we can
expect
true solution in
difficulty in two ways.
spaced
clearly
the
driving-
indicate that
the mutual distance of the re-
larger than the distance between
certain that
rna tching
expect to obtain q.ccurate
in Fig.6.1
However, if
maining matching points is much first
can
closely
z, i.e., accurate value of
results
this reasoning is correct.
choosing
unstable,
region.
oscillatory
the
solutions
It is possible to avoid this
We can use either a higher-order approximation
for current (thus increasing
the number of the rna tching points and de-
creasing the distance between them), or divide the monopole into several subsegments. As an
The first procedure was adopted here.
illustration of
method, Fig. 6.2
current distribution obtained by the present 5 shows experimental distributions 5 (obtained with the
monopole partially covered with a dielectric cover) for Bh=3rr/4 and for
150
Ch.6.
Wire antennas in lossy and inhomogeneous media
sz
sz
az
0 10 20 30
0 10 20 30
(a)
(b)
-10 0 10 20 30 (c)
FIG. 6. 2. Current distribution along dipole antenna in a conducting medium; a=0.318 em, Sh=3rr/4, f=114 MHz; (1) real part, (2) imaginary part, (3) magnitude; (a) £r=78, a/w£=0.036, (b) Er=78, cr/w£=0.35, (c) Er=77, cr/w£=1.06. -----belt generator, n=8;- • -belt generator, n=5; - - - experiment. 65 (Ref.63)
cr/w£=0.036, 0.35 for n=5
and
and n=8.
1.06, together with the theoretical distributions
It is evident that,
tioned oscillatory solution, latory character is lost.
but
that
for for
n=5, we have the aforemenn=8 this undesirable oscil-
Both solutions are
in a
relatively good a-
greement with experiment. Although
the
above examples relate
to
cylindrical antennas only, it
can obviously be expected that the method for analysis of wire antennas described in the analysis
preceding
chapters
of wire antennas of
can
successfully
be
extended to
arbitrary shapes situated in homogeneous
lossy media.
6.3.
DETERMINATION OF CURRENT DISTRIBUTION ALONG WIRE ANTENNAS 68 ON PLANE INTERFACE BETWEEN TWO HOMOGENEOUS MEDIA
As mentioned termining near the
the introduction
to
this chapter, the problem of
circuit and radiation properties of interface between
situations. so-called
in
A wire
two media arises
antenna situated at
de-
antennas located at or in
a number of practical
or above a
island antenna, and an antenna buried in
lossy ground, the the
earth are ex-
Sect.6.3.
Antennas on interface between two media
amples
such structures.
of
ous solution of bution along
151
The problem is a very complex one.
the problem implies
determination of
the antenna, the electromagnetic field
satisfies boundary conditions at
Rigor-
current distri-
produced by which
the antenna surface and at the inter-
face between the two media. The problem of antennas media
located at
the very interface between
two
can be solved approximately, but in a simple manner, by dividing
it into
two
quasi-independent problems:
the
first of determining (ap-
proximately) the current distribution along the antenna, and then determining the electromagnetic field in both media produced by this current distribution.
The
aim of
the
present section is to show that such an
approximate two-step solution is possible, and
to
present a method for
obtaining the first part of the solution, i.e., for determining approximate current distribution along
a cylindrical antenna located at
the
interface between two media. By a
rigorous mathematical
procedure
it has been
found that along
thin, infinite straight wires parallel to the interface between two media, with time-harmonic current injected at a point along their length, there exists an exponential mode of propagation of current along the 69 wire. If the wire is located at a distance of several diameters from the interface, approximately wire
is
the the
situated.
propagation coefficient (in general complex) equals propagation coefficient of
the medium in which the
If the wire is located at the interface, the propa-
gation coefficient is approximately given by
(6.4) where k ).1
1
and
and k
1 ).1
2
are the propagation coefficients of the two media, and 2 their permeabilities.
Consider a
symmetrical dipole
of radius
a
and
length 2h,
center-
driven by a belt generator of convenient width to approximate a desired coaxial-line excitation,
and
situated (for
the moment) in a homogene-
ous, generally lossy medium of complex parameters E,
).1
and cr.
From the
152
Ch.6.
preceding section we rent
I(z)
know that
any
of the integral equations for cur-
along the dipole is formally valid, except that the propaga-
tion coefficient k complex.
Wire antennas in lossy and inhomogeneous media
and
the
intrinsic impedance
of
the medium are now
The (complex) propagation coefficient k is given by (6.5)
where, of course, all the parameters may be complex. Assume now that the antenna is located at the interface between media , \1 =)1 and cr , and 10: , \1 =\1 and cr . Then the propa1 1 0 2 1 2 2 0 gation coefficient along the dipole is approximately given by equation
of parameters
10:
(6.4), i.e.,
2
w E:
)1
e o
(
1- j
(J
e
I WE e )
(6.6)
where E:
By
(6. 7)
e comparing equations (6.4)
considered as
and
permittivity and
Thus, by simply substituting
in
(6. 7)
we
see
conductivity of
that
E and cr can be e e an equivalent medium.
eqn.(6.5) E by Ee and cr by cre' any in-
tegral equation for current distribution along the dipole situated in a homogeneous
medium can be
used
for
approximately determining current
distribution along the dipole situated at the interface between the two media.
It is
valent medium
also the
possible to show that IJith the concept of the equibelt-generator approximation to coaxial-line excita-
tion of the monopole remains valid, but we shall not consider that here. Extensive experimental results mittance axial
and
line
are
and located at
or
close to the interface between air and a
water solution of sodium chloride. above method were cases agreement was (b) display
presented in Reference 70 for ad-
current distribution of monopole antennas driven by a co-
the
compared with
Theoretical results obtained by the those
found to be good.
theoretical
in
Reference 70.
In
all
the
As one example, Figs.6.3(a) and
and experimental curves
for
the monopole
conductance (G) and susceptance (B) versus the ratio cr/wE for the solu-
Sect.6.3.
Antennas on interface between two media
tion,
a
for
quarter-wave
153
and half-wave monopole antenna
measured in the conducting medium).
(wavelength
As the second example, Figs.6.4(a)
and (b) show real and imaginary parts
of
current along a quarter- wave
and a half-wave monopole (wavelength measured in the conducting medium) for o/w£=1.06
and
Er =77.
In all the cases agreement between theoreti-
cal and experimental results can be considered to be good. It is obvious that the above approximate method for rent distribution along used also
determining
cur-
symmetrical cylindrical dipole antennas can be
for approximate analysis
of
arbitrary planar wire antennas
situated at the plane interface between two media. In the end, two remarks
necessary concerning the method just ex69 according to theoretical conclusions 70 (which were confirmed by measurements ), the antenna current distribu-
plained.
First, note
tion varies
are
that,
rapidly when
it is moved into
one
or other medium, i.e.,
FIG.6.3. Conductance (G) and susceptance (B) of monopole ~~~c-J-___ L_ _-L--~----L---~----~ antenna located at the interface of air and lossy medium, versus o/wE of the medium; a=0.318 em, b/a=3.5, f=ll4 MHz, Er varies from 69 to 78. - - - experimental; 7 0 present method, 6th- degree polynomial approximation along (a) quarter-wavelength monopole, (b) half-wavelength monopole (wavelength measured in the conducting medium). (Ref. 68) o/wE o/wE
7
(b)
154
Ch.6.
Wire antennas in lossy and inhomogeneous media
f3Z
f3Z
2
-40
0
40
80
120
0
40
80
(b)
(a)
FIG.6.4. Real (1) and imaginary (2) parts of current distribution along monopole antennas at the interface of air and lossy medium; a=0.318 em, b/a=3.5, f=l14HHz, cr/wE=l.06, Er=77. - - - experimental; 7 0 - - - present theory, 9-th degree polynomial approximation for current; (a) quarter-wavelength monopole, (b) half-wavelength monopole (wavelength measured in the conducting medium). (Ref.68)
when it is not exactly at the interface. even if
bution along it, field
Second, as pointed out above,
an antenna is at the interface and we know the current distri-
due
to
there
this
still
remains
antenna current in
the
problem of determining
the
two media.
the
That problem we
shall not consider here.
6.4. In
WIRE ANTENNAS ABOVE IHPERFECTLY CONDUCTING GROUND the
fectly field
case
of an antenna placed in a homogeneous medium above a per-
conducting due
ground plane, we know from
to currents and charges induced on
that of the antenna image situated in tenna analysis using any
can be performed
suitable method.
for
However,
the
the
image
theory
that
the plane is the same as
homogeneous medium.
The an-
the symmetrical equivalent system no
image theory
is
available if
the antenna
is placed relatively closely above an imperfectly conduct-
ing ground.
Sommerfeld was
problem, as early as in 1909, shall present here
the first author to analyse this complex 71 and he was followed by many others. lve
a brief review of
the basic principles of
Sommer-
Sect.6.4.
155
Antennas above imperfectly conducting ground
feld' s theory, with
slight modifications.
For a
more
detailed treat-
ment the reader may consult, for example, References 3 and 72. In a homogeneous medium, the magnetic vector-potential due to an elemental current source of intensity
J dv
and
the
electric scalar-poten-
tial due to an elemental charge source of intensity pdv are proportional
to
the respective
source
intensity and
to the appropriate Green's
function, -+
_.,.
I
exp(-jk r-r'l) 4111 r- r'
_.,. r defines
As usual, This
function
the source.
can
the observation point,
be
Let us
(6. 8)
I
interpreted as
the
source point.
a spherical wave emanating from
assume, for convenience, that the source is loca t-
ed at the origin of a rectangular coordinate system, so that ;'=0. cording to Sommerfeld's theory,
71
Ac-
this spherical wave can be mathemati-
cally represented as an integral of plane waves, as follows: 11/2+joo 11 exp(-jkr)
f f
,·k11« --=-2
r
0
exp [ -jk (x sin 8 cos ~ + y sin 8 sin ~ +
-11 • sine
-+
-+
-+
I z I cos 8) ]·
d~
(6. 9)
de ,
-+
-+
where r=xi +yi +zi , r=lrl, and 8 and~ can be interpreted as the angles y
X
Z
of a spherical coordinate system determining the propagation vector the elemental the
complex
to (11/2,0), (11/2,+ 00 ) . en.
plane waves.
The first part
of
8-plane can be taken along the real axis, from and
then parallel
to
of
the integration path in
the imaginary axis,
from
the
origin
(TI/2,0) to
Hence, the following interpretation of eqn.(6.9) can be giv-
For z>O, along the first part of the integration path the exponen-
tial term of the integrand is an ordinary transversal plane wave, propagating in the direction determined by the angles e and the elemental
solid angle, and
whole upper half-space. tion path in
the
the
8
sine
d~
d8 is
integration is performed over
the
However, along the second part of the integra-
complex 8-plane the exponential term of the integrand
is a generalized, non-transversal plane wave. clude i f
~.
This is possible to con-
in the integrand is substituted by (11/2+jt).
planes of that wave
are
The equiphase
perpendicular to the xy-plane and propagate in
156 a
Ch.6.
direction
planes
of
Wire antennas in lossy and inhomogeneous media
parallel
to
the
xy-plane
defined by the angle ¢.
The
equal amplitude are perpendicular to the z-axis, and the am-
plitude of the wave
decays exponentially with increasing z.
A similar
interpretation of eqn.(6.9) is possible for z
over
in eqn. (6. 9)
¢
can be expressed in terms
of
the
Bessel function of the first kind and order zero, to obtain n/2+joo exp(-jkr) r
J
-jk
(6.10)
J (kpsin8) exp(-jk!zlcos8)sin8d8, 0
0 where J
(t) is the 0 2 and p=(x +y 2 ) "'2 • Let us
now
Bessel
function of the
first
kind and order zero,
suppose that the elemental source is situated in a vacuum ~
above flat surface of a homogeneous medium of permeability in Fig.6.5.
The complex relative permittivity Er of
we
to
consider
conductivity.
be
as shown
the medium, which
the ground, is assumed to include in E~
the
ground
Each plane wave propagating in a direction characterized
by an angle n-8 > with respect
0,
to
1T /2
is
incident on
the normal
to
the
ground surface at an angle 8
the surface.
This wave
flected from the ground, and partly transmitted into it.
is partly reLet us denote
the angle of the reflected wave by 8r' and the angle of the transmitted wave by 8t (both with respect to the normal to the surface).
z elemental
FIG.6.5. Elemental source above imperfectly conducting ground. P denotes the point of reflection of an elemental wave.
Sect.6. 4.
Antennas above imperfectly conducting ground
15 7
The reflection and transmission coefficients can be obtained from the boundary conditions at the p)ane z=-d: Ei
+
Er
Et
X
X
Ei z
+
Er z
E
Bi
+
Br
Bt
X
X
+
Br z
Bt z
X
X
Bi z where
the
Ei y
Er y
+
Et y
(6.11)
Et r z
(6.12) Bi y
Br y
+
Bt y
(6.13)
(6.14)
superscripts "i", "r" and "t" denote the incident, reflected
and transmitted waves, respectively. In his solution, Sommerfeld introduced at this point the Hertz potential.
Since
all
the preceding
the equations
chapters were
Lorentz potentials, we
shall
for
current
distribution
presented in
derived essentially on the
basis of the
use them instead of
the Hertz potential,
so that -+
E
-+
B
-+
-jwA - grad V,
( 6. 15)
-+
curl A,
(6. 16)
with +
divA = -jwqtV .
(6.17)
The Lorentz scalar-potential V is a continuous function at all points of the system considered.
Thus, according to eqn.(6.17), -+t
div A Let us
first
assume
that
I z=-d-0
( 6. 18)
.
an elemental
current
source j dv=J dv
z
iz
(i.e., a z-directed Hertzian dipole) is placed at the origin (Fig.6.5). In that
case
Ai has
only the z-component.
The boundary conditions in
eqns.(6.11)-(6.14) and (6.18) can be satisfied if both the reflected and transmitted vector-potentials are assumed to have only the z-component. The boundary conditions ing
eqns.(6.15)
and
for
(6.16)
the two potentials are obtained by insertinto boundary
conditions
(6.11)-(6.14).
158
Ch.6.
Eqns. (6 .11)
are
potential has
Wire antennas in lossy and inhomogeneous media
satisfied automatically, because
no tangential component
to
the magnetic vector-
the interface (z=-d) and the
electric scalar-potential is a continuous function.
Since the incident,
reflected and transmitted vector-potentials are continuous functions of coordinates x and
y on the boundary surface, eqns. (6.13) are satisfied
if
At
Ai + Ar z zlz=-d+O
(6. 19)
zlz=-d-0
From eqn.(6.18) it follows that
(6. 20)
Conditions
in eqns.(6.19)
potential will be
and
(6.20)
for
the
total magnetic vector-
satisfied if they are satisfied by all the elemental
plane-wave components of the incident, reflected and transmitted potentials.
The local phase velocity of these waves along the ground surface
must be equal
on
both
sides
of
the
surface, as in the case of plane
waves, because the boundary conditions could not be otherwise satisfied at all time.
Therefore \ve have that Snell's laws of reflection and re-
fraction are valid in this case also, i.e., 8r=e and ksin8 = ktsin8t, where k
and kt
case
for
sin8 = ~sin8t,
(6. 21)
are the propagation coefficients of the vacuum and the
ground, respectively. this
or
Let RAzz be the local reflection coefficient in
elemental
incident
plane-wave
components
of
the total
magnetic vector-potential, and TAzz the local transmission coefficient. From eqn.(6.19) it follows that, locally,
( 6. 22) The partial derivative with respect to reduces
to multiplying that wave
-jk cos e for
the
reflected wave
wave [see eqn.(6.9)].
z
of the elemental plane wave
by jk cos 8 for the incident wave, by and by jkt coset for
the
transmitted
Eqn.(6.20) therefore yields
(6. 23)
Sect.6.4.
Antennas above imperfectly conducting ground
159
from eqns. (6 .22) and (6.23) and noting that kt/k=~,
Eliminating TAzz we obtain
rs-::- cos 8 -
cos 8 t (6.24)
IEr cos 8 +cos 8t or 2 cos 8 1
+
F
F Azz
Azz
iE cos r
t
(6.25)
8 + cos 8
t
The first term in RAzz (i.e., 1) corresponds to the case when the ground conductivity tends to infinity. regarded
to
reflect
the
Therefore the other term, FAzz' can be
influence of
the
finite ground conductivity.
The reflected magnetic vector-potential can thus be expressed as rr/2+joo +
+
]1Jdvg(ir+2di 0 Z Z {
1>-~ 4 "k
J
IT
FA
ZZ
. J (kpsin8)exp[-jk(z+2d)cos8]•
0
0 • sin 8 de}
(z>-d)
.
The electric scalar-potential in this case is proportional to 3Az/3z. Thus
the
reflection coefficient
for
the
electric
scalar-potential is
simply
~z = -RAzz = - 1 + FVz ' This
result
spaced, charge ment.
is obtained for
opposite as
point
(6.27)
FVz = -FAzz · a Hertzian dipole, i.e., for
charges.
However, it
two
is valid for
closely
one point
well, if properly associated with a z-oriented current ele-
Therefore the reflected electric scalar-potential due to an ele-
mental charge pdv is rr/2+joo pdv + + -E- -g(ir+2dizl) { 0
-i; f "k
FVz Jo(kp sin 8)exp[-jk(z+2d) cos e] •
0 • sin 8 d8 }
(z>-d) •
160
Ch.6.
Let us now assume at
the
Wire antennas in lossy and inhomogeneous media
that
an elemental current source
origin, but is now directed along the x-axis.
is placed again In that case the
incident magnetic vector-potential at the point P of reflection in Fig. 6. 5
has
only the x-component.
However, for the boundary conditions to
be satisfied, the reflected and have
two
components.
and z-components.
3
transmitted potentials in general must
A suitable choice
is
to take them to be the x-
The reflection coefficient for the elemental plane-
wave component of Ax is found in the analogous way to be given by cos e - ;;:-cos e r
cos e + ;;:-cos e r
2 cos e
t
+ --------
-· 1
cos e + ;;:-cos e r
t
-1 + FAxx, (6.29)
t
so that the reflected potential is "IT/2+joo )J
0
~f 4
J dv{-g( J-;+2df J) X
FAxxJO(kp sin e)exp[-jk(z+2d) cos
1T
Z
8]·
0 • sine de }
It
can be further shown
wave components of A
that
each of
the
(6.30)
(z>-d) .
reflected elemental plane-
is equal to the corresponding incident plane-wave
2
component of Ax multiplied by 2(1- Er) sin 8
COS
e
COS
cp
(cos e + ;;;-cos e ) (;;;-cos e + cos e ) IE r t r t r 2 sin e cos 8 cos
cj>
(cos
rscr ;;:-cos e r
e-
;;;-cos e ) r
t
cos
cj>
(6. 31)
F Axz •
+ cos e ) t
The z-component of the reflected potential is "!T/2+joo
x jr
k -11 J d v - 0 X 4"!T p
FAxz J l (kp sin e) exP[ -jk(z+2d) cos 8 J sin 8 de
0 (6. 32)
( z>-d) . i
The incident electric scalar-potential is proportional to 3Ax/3x, and the reflected
to
(3Ar/3x + 3Ar/3z).
x
z
From these relationships, the re-
Sect.6.4.
161
Antennas above imperfectly conducting ground
flection coefficient for the electric scalar-potential is found to be _ 1 + --------~2~c~o~s~8~-------
-1 + Fvx .
/~( I~ cos e + cos et) r
(6. 33)
r
As before, this reflection coefficient mental charge,
can be associated with an ele-
instead of with a dipole, and
the
reflected potential
due to the elemental charge pdv is rr/2+joo Vr=pEdv
{
I
-g(i~+2dizl)-~~
0
0
FVxJ (kpsin8)exp(-jk(z+2d)cos8]• 0
• sin 8 dB }
(z>-d) .
(6.34)
Eqns.(6.28) and (6.34) state essentially the same result as obtained in Reference 73 using a different approach. If the elemental current source is arbitrarily oriented, the reflected magnetic vector-potential is obtained by applying eqns.(6.24)-(6.26) and (6.29)-(6.32) on the vertical and horizontal components of the incident potential.
However,
the
reflection coefficient
scalar-potential is rather complex,
and
for
the electric
the reflected potential can be
found more easily from eqn.(6.17). A similar procedure tric vector-potential
can be used for determining the reflected due
to possible magnetic currents above
elecground,
but this will not be needed in the following example. As an example, consider a Fig.6.6.
magnetic-current frill, The
field
respect
symmetrical, horizontal dipole sketched in
The excitation region of the dipole was approximated by a TEM
due
to
to
with b/a=2.3,
according
this frill is negligible at
to
the
Subsection
2.3.1.
ground surface with
the field due to electric current along the dipole.
There-
fore the reflected electric vector-potential due to these magnetic currents was not
taken into account in
tential equation
for
~vhat
follows.
To fom the two-po-
the antenna current distribution, the x-component
of the total electric field along the x-axis is of interest. note,
as
usual,
by x'
the
x-coordinate
of
If we de-
the source point and by x
162
Ch.6.
Wire antennas in lossy and inhomogeneous media
z X
h
FIG.6.6. Horizontal dipole above imperfectly conducting ground.
z=-d
that of
the field point along the antenna, in all the expressions con-
taining x
the
difference (x-x')
appears
3V/3x=-aV/ax', and the expression for
only.
In
that
case gradxV=
grad V can be integrated by parts.
Thus, the two-potential equation for the problem considered has the form h
n/2+joo
J {r(x')[gd-gr-~~
J
-h
FAxxW(x,x',e)de]+
0
n/2+joo 1 -(g dl -- g k2 dx'
d
J
jk 4tt "
r
h
1
Fvx W(x,x' ,e) de] jx'=-h
0 1
= - . - E . (x) JW\lO l.X
'
(6. 35)
where
I +X
+I )
-+ +ai g( xi -x'i X
Z
-+ -x'i -+ +2dl. -rl ) g(lxi X
and
X
Z
1
,
(6. 36) (6. 3 7)
Sect.6.4.
Antennas above imperfectly conducting ground
W(x,x' ,e) = Jo(klx-x'
I sin8)exp[-jk(z+2d)
163 (6.38)
cos e] sine •
A similar transformation is possible for a vertical dipole above imperfectly conducting ground. As a numerical example, shown in Fig. 6. 7 is pole sketched in Fig.6.6
versus
the
the
impedance of the di-
antenna height above ground, for
h=O. 25 A, a=O. 007 A, Er =10-j 1. 8 and the degrees of the polynomial approximation n =4 in the excitation zone (which was taken to be 6a long) and 1 n =4 on the rest of a dipole arm. The results obtained by the present 2 method are compared with available theoretical results computed using also Sommerfeld's theory, but with Hallen's equation, the delta-function approximation of the generator and the second-degree polynomial approx74 imation of current. Excellent agreement between the two sets of results
is
seen.
impedance results
for
For comparison, a perfectly
clearly
indicate
also
conducting that
in
shown
in Fig.6. 7 is the antenna
ground
the
(i.e., for E:"-+oo).
present
case
These
the approximation
R,X (n) 100 80
60
I
I I
40
I I
20 /
0
/
/
/
I
d/A 0.1
0.2
0.3
0.4
0.5
FIG. 6. 7. Resistance (R) and reactance (X) of the antenna sketched in Fig.6.6 versus the height, d, of the antenna above ground; h=0.25 A, a=0.007A, Er=l0-jl.8. -----results obtained by the method described above; o o o results obtained using Hallen's equation; 74 results for perfectly conducting ground.
164
Ch.6.
that
the
Wire antennas in lossy and inhomogeneous media
ground is perfectly conducting is not adequate when computing
the antenna current distribution and its impedance.
Concerning the ra-
diation pattern of the antenna shown in Fig.6.6 in the upper half-space, it can be computed using the above derived reflection coefficients once the antenna
current
distribution has
been determined.
We
shall not,
however, elaborate that here.
6.5.
CONCLUSIONS
Although more
difficult for analysis than wire antennas in homogeneous
perfect dielectrics, we proximately,
three
have
important
inhomogeneous dielectrics: neous lossy medium, terface between
when
two
the
cases
when
of wire antennas
the
antenna
is
to in
analyse, apimperfect and
situated
in a homoge-
a planar antenna is situated at the plane in-
homogeneous (possibly lossy) media and when a wire
antenna is situated above ably
seen that it is possible
real, lossy ground.
The
last
case is prob-
most important from the practical point of view, but also the
most intricate. With
this
chapter we are concluding
antenna analysis. of
interest
only
the
In engineering practice for
antenna design
topics dealing with wiresuch
purposes.
monograph is devoted to that important topic.
an analysis is usually The
next
part of the
PART II
Synthesis of Wire-Antenna Structures
CHAPTER 7
General Considerations of Wire-Antenna Synthesis
7.1.
INTRODUCTION
In the classical approach to designing a wire antenna or a wire-antenna structure, a certain prior knowledge of properties of a antennas,
such as their admittance,
rigidity,
aerodynamic
knowledge and satisfied of
profile, 1
the designer s
radiation
etc. ,
\vas
experience,
pattern,
class of
size, weight,
indispensable.
Using this
an antenna was
sought which
the necessary requirements (including,
the mentioned antenna properties).
wide
possibly,
Usually, a
single
only
some
step of this
kind did not result in an antenna with the desired properties, and was, therefore, followed by several, or even many, similar corrective steps. The trial-and-error method of the antenna design been used for a long time.
example, in designing medium-wave arrays prescribed
described has
This is still a frequently utilised antenna
design procedure, but it may not, obviously, be
with a
just
radiation
pattern
of
of the
the
optimal one.
For
vertical monopole antennas array
in the horizontal
plane, the procedure amounts to choosing one of the theoretical patterns available in antenna engineering handbooks (based on assumption of sinusoidal current distribution along
the
antennas)
and
then fulfilling
the feeding conditions with which, theoretically, such a pattern is obtained.
Experimental matching
of
the array elements
to
their respec-
tive feeders and the final experimental adjustment of the array follows, which is usually far in
the
from
pattern we wanted.
Yagi array
for
a
simple and, actually, may not result exactly As another example,
single frequency
we may design an Uda-
using available data, but no
such
168
Ch.7.
data
General considerations of antenna synthesis
can be found if we wish to design an Uda-Yagi antenna which
operate successfully at more than one frequency. design
in
least
some
that case would
though it
indicate
can
Although experimental
be possible in principle, we would need at
guidelines to make it
two examples
can
clearly
both economical and feasible.
that
the classical
antenna
These
design, al-
be advantageously used in some cases, in many more cases
is inconvenient or even virtually impossible. With
the
advent of high-speed digital computers
to many antenna structures proximately between 1970
changed
and
1975.
radically.
the
design approach
This trend started ap-
Today, many antenna structures can
be designed with such a high accuracy using the new, computer-aided design approach, that frequently an experimental model of the antenna resulting from the design, intended for possible corrections of theoretical results, is more
a matter of tradition and scientific approach than
of necessity. This part
of
the monograph deals with computer-aided design of wire-
antenna structures.
Such a design may imply' diverse procedures, but it
is always based on a sequence of analyses of a more or less defined antenna
structure with systematically
quence is most
often
perturbed parameters.
created automatically by
sometimes it might be
the
This se-
computer, although
advantageous to use the interactive process.
It
is terminated once an antenna is obtained which is considered to satisfy the desired requirements in "the best
possible" manner.
Thus, such
a process creates, or synthesizes, an antenna with properties "as close as possible" to of
the
used
in
the
desired properties.
expressions "the best the
antenna
possible"
to
refer to
briefly as to the antenna synthesis. of
and
computer-aided design
this reason it is usual
al principles
(We shall explain the meaning
synthesis
of
the
"as close as possible" as in
the next section.)
For
computer-aided antenna design
This chapter discusses the gener-
wire-antenna
structures, outlines some
synthesis procedures and points out to possible general difficulties of the synthesis.
More specific synthesis problems are then dealt with in
the next two chapters.
Sect.7.2.
7.2.
169
General principles of wire-antenna synthesis
GENERAL PRINCIPLES OF WIRE-ANTENNA SYNTHESIS
To synthesize
a
ties,
it
is
first
procedure
wire-antenna structure
possible,
in
principle,
having to
certain desired proper-
follow
is that which is aimed at
two
procedures.
determining
The
a unique solu-
tion, or a number of solutions, which satisfy certain conditions exactly
such
(if
least
two
solutions
is
impractical
for
at
reasons, even if solutions
solutions tend
to
require
impractical
tances.
This approach
exist).
quite 77
be very unstable.
do exist. On the one hand, sucli. 75 76 ' On the other hand, they may
physical elements,
e.g.,
negative
induc-
The second procedure is based, essentially, on the engineering common sense:
determine
so close as
the
to
final solution.
guarantee (e.g.,
an antenna
a
the
properties of which are, if possible,
some desired properties that
that
By
the
solution can be accepted
appropriate constraints
the solution obtained be
this
physically
approach can
easily realizable
cylindrical antenna with capacitive loadings only).
It seems
that, indeed, a meaningful synthesis method (in the engineering sense), which will always result in at least some realisable solution, can only be based on approaching a desired property (or a collection ties)
step
by
step,
and not
of
proper-
on requiring that it be reached exactly.
We shall, therefore, follow in this monograph the second procedure, and by the
term "antenna synthesis" we shall essentially imply any process
which,
on
average,
constantly
improves
the
antenna
properties
(with
respect to the desired properties) although, in principle, it may happen that
the
tion. tion"
final result
it
reaches is rather far from the desired solu-
The solution thus reached we shall term "the best possible soluor
"the solution as
close as possible to the desired solution",
although
these expressions are not at all precise, since we shall see
that
final solution often depends to a large extent on the initial
the
conditions. As formulated, the synthesis problem is, basically, a problem of nonlinear optimization.
Therefore we construct a convenient real (usually
positive definite
semidefinite) function
the
or
desired antenna
properties
are
reached
which
has
(or, in
a
some
minimum when cases, when
170
Ch.7.
the antenna properties
General considerations of antenna synthesis
are better
than
those desired).
we shall refer to as the optimization function. erties
This function
Since the antenna prop-
depend on the antenna parameters, the optimization function is,
essentially, a function
of
these parameters.
We next use any conveni-
ent optimization method to minimize the optimization function. Two preceding paragraphs define the main lines which the antenna synthesis must
follow.
However, there
are many additional
problems
and
dilemmas which have to be solved prior to starting a synthesis process. The following subsections deal briefly with some of them. 7.2.1.
Possible
requirements
on
optimization
upon
specific
antennas which are synthesized, it is possible to con-
struct many
optimization
present
specific
any
Depending
functions.
functions.
In
this
subsection we shall not
optimization function - they will
and justified in the next
two chapters.
be introduced
Rather, we shall discuss. some
details relevant to these functions in general. Once we know current distribution in an quantities
of
interest, such as
pattern,
gain,
electric
field,
efficiency (in etc.,
can be
the antenna admittance, its radiation
the
case
of
calculated
these quantities require a knowledge of whole antenna
structure (e.g.,
antenna, all the electrical
the
lossy structures), maximal
relatively
easily.
Some of
current distribution along the
radiation pattern or
efficiency),
and some a knowledge of current intensity at a specific point along the antenna only (e.g., admittance or voltage across a lumped loading). In one
approach of
the
classical antenna synthesis, current distri-
bution was sought which resulted in a desired ever,
the more difficult part of
rent distribution
the
radiation pattern.
How-
problem, that of how such a cur-
can be realized, was not considered.
sical antenna synthesis, the problem of
Also, in clas-
synthesizing an antenna having
admittance close to a desired admittance was often not of interest, because it was
assumed that it is always possible to match an antenna to
the feeder by means of a matching network. The antenna
synthesis
in
the
sense
used in
this monograph allows, )
however, the synthesis of wire antennas having any property (or a collec~)
Sect.7.2.
tion of properties) as close as possible ample,
171
General principles of wire-antenna synthesis
optimization functions
to
a desired value.
For ex-
can be constructed which result in the
following antenna properties: a)
Antenna admittance as close as possible
to a desired admittance,
at a single frequency or at a number of frequencies. b)
Radiation pattern
as possible
to
of an antenna
or
of an antenna array as close
a desired pattern, at a single frequency or at a number
of frequencies. c)
Antenna admittance and
radiation pattern simultaneously as close
as possible to some desired value, viz. some desired shape. d)
Antenna as broadband as possible
in
its admittance and/or radia-
tion pattern. e) In
Coupled antennas with minimal coupling.
the next two chapters possible optimization functions corresponding
to most of these cases will be described and utilized for wire-antenna synthesis. Two additional general remarks might be added at this point. it is very tennas
First,
important to understand that the range of properties of an-
of
given dimensions is inherently limited.
For example, a dis-
tributed resistive loading along a cylindrical antenna of given dimensions
can result
only
in certain range of
radiation pattern when loading are varied.
the
As
amount and
the
antenna admittance and
distribution of
another example, by adding
the
lumped
resistive capacitive
loadings along a cylindrical antenna of given dimensions it can be made broadband with respect to its admittance only to certain extent. essential
that
It is
the optimization function be constructed having in mind
these limitations. Second,
if we wish
properties,
it
to
optimize
simultaneously
two
or more antenna
should be kept in mind that they might be
or larger extent
contradictory.
For example,
to synthesize
cylindrical antenna having a prescribed radiation pattern in containing
the
to a lesser a
loaded
the
plane
antenna axis and simultaneously a prescribed admittance
172 is
Ch.7. often not
pattern and should be
possible,
except
admittance
obtained
General considerations of antenna synthesis
is
if
a large tolerance in
allowed.
in any
An
7.2.2.
radiation
such situations
to
constructing the
optimiz~tion
Possible optimization parameters. antenna in any sense
the
into
possible manner prior
optimization function and running the
ing an
insight
program.
Any parameter characteriz-
can be varied in the optimization process
in order to achieve desired antenna properties.
He shall restrict here
our considerations only to those antenna parameters which
characterize
its dimensions
optimization
and
parameters may
its electrical properties.
characterize
the
Thus,
antenna shape,
distributed loadings, driving voltages of
the
size,
concentrated or
array elements, current dis-
tribution along the antenna, etc. In principle, there is no limitation concerning the number of optimization parameters.
For example,
in
the
case
of
a
loaded cylindrical
antenna we may simultaneously vary its radius, length, positions of the loadings, their kind
and
their magnitudes.
not done for three reasons. meters
have
much more
tion than variations prolongs
the
However, this
pronounced influence on
of
the rest.
optimization
is
usually
First, most often variations of some para-
time
Second,
the
any
considerably.
optimization func-
optimization parameter Therefore
optimization
parameters with only a small influence on the antenna properties should be avoided. meters
Finally, from the practical point of view only those para-
should
be varied
(i.e.,
should be
taken as
the
optimization
parameters) which in practice can be easily adjusted to any value which might be required by the result of the optimization process. Variation
of
some antenna optimization parameters
requires
complete
new solution of the equation for the antenna current distribution. example, if we
change
the
For
antenna length, the antenna diameter or the
positions of the concentrated loadings in the case of a cylindrical antenna, peated,
the
complete process
including
instances,
the
however,
lumped loadings
at
of
solution
for
the equation must be re-
time-consuming integrations
this
is
not necessary.
fixed positions
along
involved.
For example,
a wire antenna,
In some
if we no
have
integra-
tions need to be repeated if only magnitudes (and even the kind) of the
Sect.7.3.
173
Outline of some optimization methods
loadings are varied, and complete solutions for different values of the loadings needed
are
for
obtained with
solving
again
little more computational effort than that
the basic linear system of
equations.
Such
cases should always be considered as a possibility, at least at an initial
stage
of synthesis, because that
puting time.
Several
can
save a large amount of com-
such examples will be presented in the following
chapters.
7.3.
OUTLINE OF SOME OPTIMIZATION METHODS
It is well-known that there is problems of optimization.
no
best optimization method for all the
A variety of optimization methods are at our
disposal for synthesis of wire-antenna structures, but some may be more convenient than the others. about fic
the
case
effectiveness
It is not possible, however, to say much of
an
optimization
by theoretical considerations alone.
method It
seems
~n
a
~eci
that the only
means to choose an optimization method for a problem at hand, like that of synthesis with
few
of
a wire antenna, is to
try how it works, to compare it
other methods, and possibly to choose one of them on some ra-
tional basis.
It
should be
pointed out, however, that if we
need to
perform the optimization relatively rarely, there is usually no need to try
to
find a faster optimization method if
the
one used gives satis-
factory results. An optimization method can be defined as a specific procedure for de-
termining successive points in the parameter space in which the optimization function should be computed in order that a (usually local) minimum be reached.
Only rarely the number of feasible points in the para-
meter space is finite, in which
case we can make a complete search and
find the global minimum. Most often the search is incomplete, and there is no
guarantee
that
the global minimum is reached.
Rather, possibly
only a local minimum can be approached (with some prescribed accuracy). All
the
optimization methods can roughly be divided into two groups:
(l) those which use only the values of the optimization function itself for determining the
gradient
the
next point of the process, and (2) those which use
(i.e., partial derivatives) of
the
optimization function
174
Ch.7.
for
that
purpose.
methods, and
the
General considerations of antenna synthesis
The first second as
are
the
frequently referred to as the direct
gradient methods.
that in the case of antenna synthesis an explicit function of
the
the
(It should be noted
optimization function is not
antenna parameters, and the derivatives of
the optimization function must be computed numerically.) which cannot be classified into these
two
A special case
groups is the random search,
in which successive points are determined by the use of a random-number generator. In our
case
the
values
of
by practical possibilities.
the optimization parameters
These constraints must
the optimization process, i.e., all
the
are
limited
be introduced into
optimization methods
used
for
wire-antenna synthesis are methods with constraints. In any optimization procedure that of
the
choice of
the decision
on
the
there exist
two
additional problems:
starting point for the process, and that of
termination
of
the process.
In order to expedite, or
even to ensure convergence of the optimization procedure, point the
should be located as
closely as
optimization function.
behaviour of
the
This
the
starting
possible to a local minimum of
requires
optimization function,
some which
prior knowledge about can
be
gained if some
systematic data are available (or can be computed) of the properties of the antenna type considered.
If such data are missing, it may be con-
venient to make a random search in the parameter space and to adopt the best point
thus
found
as
the
starting point
for
a
direct search or
gradient optimization procedure. The decision on termination of an optimization procedure depends both on the method used for optimization and on the goal of the antenna synthesis process.
Basically,
if a minimum of
the
sufficient accuracy sion of
procedure
is
(which depends upon our
can be thus based on
the
the
terminated autorna tically
optimization function is presumably located with a requirements).
The deci-
testing increments of the parameters and/or
optimization function in two successive iterations of the optl.-
mization procedure.
In
some
cases, especially
in early stages of the
optimization, the interactive procedure might be desirable, whereby the optimization process
can be
directed
or
terminated according
to
the
Sect.7.3.
175
outline of some optimization methods
designer's judgement. This section is devoted to a very brief presentation of mization methods
the
authors
wire-antenna structures.
Since
given below are intended as referred
to
used and
a
the
found
useful
descriptions of
rough
in
three optisynthesis
of
the three methods
information only, the
reader
is
specialized literature for a more detailed treatment (see,
for example, References 78 and 79). 7.3.1. of
Complete
the
search method.
The
complete
search method
conceptually simplest optimization methods.
consists in determining lar or
the
is one
Essentially, it
optimization function at nodes of a (regu-
irregular) multidimensional
grid
in the space of the optimiza-
tion parameters and searching for the optimal solution among these. A serious disadvantage of of parameters is of
the
the need
for
optimization function.
with concentrated loadings mization
function
meters
are
number
of
at
case of a large number
computation of a large number of values However, in
some
cases (e.g., antennas
fixed positions) evaluation of the opti-
can be greatly expedited if
the
successively varied only one at a time.
optimization paraThus, if the total
parameters varied is small, the number of evaluations of the
optimization function c.p.u.
this method in the
time
can be greatly increased without
required for
the
optimization.
increasing the
Therefore
the
complete
search in these cases can be used efficiently. This method where to start
can also
be used with a coarse grid to
obtain an idea
the optimization process, i.e., from which point in the
multidimensional parameter
space
to
trigger another,
more
convenient
optimization process, which, however, cannot easily locate the position of, possibly, the global optimum. 7. 3. 2.
A gradient method.
the steepest-descent methods. consists
in adopting
a
The gradient methods The most
starting point,
the multidimensional parameter
space
at
are
also known as
rudimentary form of the method determining that
the
direction in
point in which the opti-
mization function decreases most rapidly, adopting a new starting point in
that
direction at
a desired distance
from
the old starting point,
176
Ch.7.
and repeating is found. of
the
the
General considerations of antenna synthesis
process until a minimum of the optimization function
This amounts
to
calculating
optimization function, and
the
then
components of the gradient
decreasing all
the
parameters
by the corresponding gradient component multiplied by a suitable basic step "length".
Let s be the basic step length, F(x , x , ... , xn) the op1 2 timization function (x ,x , ... ,xn are the optimization parameters) and 1 2 0 0 0 P (x ,x , ... ,xn) the starting point in the parameter space. The com0 1 2 ponents of grad F at P are 0
..
0 grad F 1
0 grad F
,
(dF/Clx )
n
0
n
(7. 1)
.
0 As in our case gradk F, we must
k=l,2, ... ,n, cannot be determined analytically, 0 calculate them approximately as (6F/6xk) , k=1,2, ... ,n which
requires
at each point computation of (n+1) values of the optimization
function.
We then determine the next starting point as k=1,2, ... ,n,
and
repeat
the procedure
possibly equal to)
the
until
(7.2)
the new value of
old value.
F
is larger than (or
Presumably, a local minimum of F is
thus obtained. The brief sketch of given here
in order
application to ent
to
point
problems.
in any application of
methods.) evaluations has
our
the well-known basic steepest-descent method was
to
First, of
the
the
out
some obvious difficulties in its
(These difficulties are, actually, pres-
the method and in most
process
requires
a
optimization function.
considered.
Finally,
the components of
the
the
the
other optimization
relatively large Second, the step
be chosen appropriately, which can be
have a certain insight into
~.
done
number of length s
only if we already
behaviour of the optimization function
increments l'lxk for
gradient of
the
determining
numerically
optimization function should be
small, but large enough that the increment l'IF can be computed accurately, a problem the
solution of which usually requires some prior nume-
rical experiments. Many improvements able.
One of
of
this basic
steepest-descent method are
avail-
these is to follow the negative gradient at the starting
Sect.7.3.
point as long as crease
the
the
basic
optimization
step
in
the
function
gradient
is computed only when we
straight
line
the
in
space
of
this
In all the examples
optimization method,
quadratic interpolation based tion at
on
three successive points
This procedure reduces
later chapters determined
by
values of the optimization func-
(which are
the
in
This minimum
this minimum was
the
The
reach a minimum along this
optimization parameters.
can be located in various ways. which use
decreases, and even to in-
successive points along this line.
complete
line.
177
Outline of some optimization methods
number
nonequidistant) along the
of
gradient evaluations con-
siderably, and thus expedites the optimization process. As most
optimization methods,
local optimum, rather
than
all
gradient methods may
of the optimization parameters considered. is
relatively
whether
the
cedure
to
large,
it
is virtually
this
seems
several starting points in cess. all
We the
If the number of parameters
impossible
to
judge in any way
optimum found is the global optimum or not.
check
can
to
be
the
the
following.
that
We adopt at random
the optimum is the global one if
optimizations result in approximately
ever, it should be noted that, from interested in
The only pro-
domain and repeat the optimization pro-
be fairly certain
are usually not
end up in a
in the desired global optimum in the domain
the
the
the
same optimum.
How-
engineering point of view, we
global optimum if
the
solution ob-
tained can be considered satisfactory. 7 .3.3. for
a
The
simplex method.
The
body in multidimensional
tetrahedron in
three
dimensions.
term "simplex" is used as the name
space which is a generalization of the The simplex optimization method com-
putes the values of the optimization function at the vertices of a simplex in a
new,
the
parameter space, and on
presumably
situated.
smaller
the
bases of these values chooses
simplex within which an
optimum should be
The simplex polyhedron rolls itself down towards the optimi-
zation function valley, elongates itself in the direction of the steepest fall of process
is
the
tained which used
for
function, and contracts itself near
repeated until a simplex of suff.iciently
the minimum. small
locates an optimum with a desired accuracy.
The
size is obThe authors
optimization of some antenna structures the simplex optimiza-
178
Ch. 7 ._
tion algorithm as
General considerations of antenna synthesis
proposed by Nelder
and Mead.
80
This algorithm is
briefly described in Appendix 7. It is fairly obvious also might both
the
not, be
initial
that
able
to find
simplex,
and
progresses.
The authors
on
required
average,
the simplex optimization method might, but
did
less
the
global optimum.
the way how
find,
time
however,
to
This depends on
it shrinks as the process that
the
simplex method,
reach an optimum than the gradient
method.
7.4.
CONCLUSIONS
This chapter summarized
the
basic ideas and procedures used by the au-
thors in synthesis of wire-antenna structures.
It was pointed out that
many optimization functions
can be constructed and
parameters
the
those
can
be
parameters
easily.
used
as
should be
that many antenna
optimization parameters, but that only
varied which
can be
realized in practice
Otherwise the antennas synthesized might be purely theoretical
structures. Concerning possible optimization methods, the the authors were
three methods
briefly explained and discussed.
used
by
Except for the com-
plete search method, the others cannot guarantee to find the global optimum of
the
optimization function
parameters considered.
in
However, if
the
the
domain
of the optimization
solution thus obtained is sat-
isfactory from the engineering point of view, we are usually not interested whether a better solution
does
exist.
In addition, from the en-
gineering point of view, a relatively broad, stable optimum is obviously preferred space,
the
Therefore,
to
an
optimum of
the
latter solution being even
if
in
"deep-well"
unstable
the optimization we
in
type
in the parameter
practical realizations.
miss such an optimum, this
can be considered as positive rather than as negative. The next
two
chapters present a relatively large number of
of wire-antenna synthesis,
in which
the
methods
for
examples
antenna analysis
presented earlier and the methods of synthesis outlined in this chapter are combined in the computer-aided antenna design.
CHAPTER 8
Optimization of Antenna Admittance
8.1.
INTRODUCTION
If we
consider an isolated wire antenna of relatively small electrical
length or an array of
such antennas, it is usually easy to roughly es-
timate their radiation pattern. quency
for
which
the
This is possible because, at
any
fre-
antenna is electrically small, radiation pattern
of one element is relatively independent of the antenna size and change in frequency, and classical
array
the
array pattern
theory.
can
analysis.
can
admittance
determined using the
the
antenna size and the ope-
hardly be predicted without precise numerical
It is, therefore, possible
antenna and
then
However, the antenna admittance is a quantity
which is very sensitive to variations in rating frequency and
be
to
vary certain parameters of an
thus to obtain antennas having a broad range and
simultaneously a
of values of
relatively constant radiation pattern
(or approximately known in the case of an array). This
chapter
is
devoted
to
examples
of synthesis
of wire antennas
with admittance as close as possible to a desired admittance by varying various antenna parameters, one or several of them at a time. of
these
parameters
are
distributed
loadings,
Examples
concentrated loadings,
possible lumped loadings in the antenna excitation zone, and the antenna
size
and shape.
It may be desired to optimize
the
antenna admit-
tance at a single frequency, or in a certain frequency range.
The lat-
ter is usually done if a broadband antenna to a lesser or larger extent is desired.
180
Ch.B.
Optimization functions tance
depend
on
these functions
the are
used
in optimizations
desired
antenna
quite simple.
an antenna broadband in
its
zation function
to
appears
Optimization of antenna admittance of wire-antenna admit-
properties.
However,
the
In most
instances
authors found that if
admittance is desired, a specific optimibe
part'icularly convenient.
We shall con-
sider now that optimization function because it requires certain justifications
and
explanations.
The other optimization functions which we
shall use are fairly obvious, and will be introduced as needed. The antenna admittance is optimized in order to match the antenna closely as possible to its feeder. can be
described by various quantities.
reasons
to
be
explained
as
A measure of the level of the match However, for
at
least three
below, the reflection coefficient at
tenna terminals, or a
quantity closely related 16 very convenient optimization function.
to
the
an-
it, appears to be a
Assume that the antenna is connected to a feeder of a real characteristic admittance Y . Let the maximal power which can be transmitted by c the feeder (determined by the maximal admissible electric field in the feeder) in that
if
the
the
case of a matched load be (P
load
) y • It is well-known max Y= c is not matched to the line, the maximal power which
transmitted by the line is smaller than (P
can be
) y • max Y= c
If we
de-
note the reflection coefficient by R, y R
and
y
c c
- y
+
(8.1)
y '
its magnitude by
J RJ,
the maximal power which can be transmitted
by the line when not matched is given by
(P
max Y#Y
c
(P
) max Y-Yc
r
r is the voltage standing-wave ratio.
where
ficient line. used
1 1 +
)
Thus, the reflection coef-
R determines the maximal power which can be transmitted by the (This
at
case.)
(8.2)
its
is, full
of
course,
important
only
in cases when the line is
power-transmitting capacity, which is not always the
Sect.B.l. The
Introduction
second
important parameter depending directly on
coefficient is Provided
181
that
the
efficiency
of
the
reflection
power transfer to a nonmatched load.
the line is lossless, i.e. , that
its
characteristic ad-
mittance is real, the efficiency is given by
n
=
1 -
IRl 2
(8. 3)
.
The third important quantity proportional to I Rl , more precisely to 2 81 I Rl , is intermodulation noise due to feeder mismatch. This is very important
parameter
if
extreme
sensitivity of
a
receiving
system is
desired, for example in radio surveillance systems. For
these
a convenient quantity to be mini2 mized when optimizing the antenna admittance is IRI , the square of the modulus of
reasons
the
it
seemed
that
reflection coefficient of the antenna with respect to a
Note that, for Y~Yc' this is approximately equivalent to 2 requiring that IY-Y 1 be minimal. c
given feeder.
When optimizing the antenna admittance to be broadband, following the above reasoning it is logical to request that the integral
(8.4)
(f -f ) is the frequency range. The quantity Reff 2 1 termed "the effective reflection coefficient". The integral in
be minimal, can be
where
eqn.(8.4) can be evaluated only numerically.
The simplest way of doing
this is to approximate the integral by a finite sum,
(8.5) where nf
is the adopted number of equidistant frequencies in the range
(f ,f ). Note that computation of R(fi) is often time-consuming, be1 2 cause it requires evaluation of the antenna admittance. Therefore nf should be adopted as small as possible. According
to
eqns.(8.1)
and
(8.2), Reff can be made smaller also by
182
Ch.B.
varying Yc, cannot
be
the characteristic admittance of
the
feeder.
Although Yc
varied in a wide range (e.g., for commercial coaxial feeders
between 50 st and formers
Optimization of antenna admittance
to
90 D), it is always possible to
increase this range considerably.
use
broadband trans-
Therefore the admittance
Y
can be taken almost at will, except that it should be real, and can c be considered as another variable for obtaining a better match. This admittance we shall term "the reference admittance" and denote by Ycref" There always exists an optimal smallest range
effective
reference
admittance
reflection coefficient Reff"
considered does
not
differ
from Yc
If Y in the frequency
considerably,
reference admittance can be determined as follows. X -
1
_!_ ln x 2
X+ 1 because
for
first
the
R
1
2
y c y
ln
2
Note first that
terms
(8.6) of
the
Taylor
series
at x=1 of the two
We can thus approximate R in eqn. (8.1) by y
ln
this optimal
x "' 1 ,
two
functions are equal.
which gives the
c [Y[
j
(we assume Y to be real). c equation (8.4), we obtain
21
arg Y
if
y
y
c
(8. 7)
Introducing this approximation for R into
(8. 8)
2 Requiring now that d(Reff)/dYc=O, we get that the minimum of the effective reflection coefficient is obtained if
If
ln Y c
(8.9)
the
integral in eqn. (8.9) is approximated by a sum over nf frequen-
cies, as we reference
approximated R!ff in eqn. (8.5), we obtain that the optimal
admittance
is
given by
[Y[ at the nf frequencies, i.e.,
the
geometric mean of the values of
Sect.8.2.
(Y
183
Optimization by varying distributed loadings
(8.10)
)
cref opt
This optimal value of
the
reference admittance will be used in several
examples of optimization later in this chapter. obtain
the
Of course, in order to
smallest possible effective reflection coefficient, we need
to require also that zero as possible.
the
be approximated by
) can cref opt geometric mean of the antenna conductances, Gi=
the
Re(Yi), i=1,2, ... ,nf" admittances Yi
arguments of Yi' i=1,2, ... ,nf, be as close to
If these arguments are close to zero, (Y
Another approximation
are close
to
can
be obtained when
the
each other replacing the geometric by the
arithmetic mean value.
8.2.
OPTIMIZATION OF ANTENNA ADMITTANCE BY VARYING DISTRIBUTED 76
ANTENNA LOADINGs In
this
dance
section a method is presented for determining continuous impe-
loading
along
a
cylindrical antenna of
given dimensions, which
approximately
results
in
certain desired input characteristics of the
antenna.
essence
of
the method consists in assuming a dependence
of
the
The
impedance loading on the length along
of power series with
the
antenna, in the form
unknown coefficients, and determining these coef-
ficients by minimizing a convenient optimization function involving the antenna admittance.
The
method can easily be extended
to
arbitrary
wire-antenna structures. Although a continuously varying loading along an antenna is relatively difficult to realize, the method is of considerable practical value. On
the
one hand, continuous loading can efficiently be approximated by
either step-like loading,
or by concentrated
hand, the results obtained by a starting point
for
end up in an
tends
loadings.
On
the
other
present simple method can be used as
optimization of
concentrated loadings, which to
the
antennas with a large number of
both to be very time- consuming and
undesired, local optimum if the initial values of the
loadings are not reasonably close to the optimal values. Consider a
thin symmetrical cylindrical dipole of length 2h and ra-
184
Ch.B.
dius and
a
(with h>>a), driven at
angular
frequency w.
Optimization of antenna admittance
the center by a generator of voltage V
Let the dipole be situated in a lossless ho-
mogeneous medium of parameters
£
and 1.1, and let the internal impedance
per unit length along the dipole be Z'(z), which we assume to be an arbitrary,
but
differentiable function
The current distribution I(z)
along
of the
the
coordinate
z
for O
dipole may be then determined
from the integral equation obtained by combining eqns.(2.30) and (4.8), which has the form h
z
I(z') g(r) dz'
J
+
~ J jw\.1
-h
Z' (z') I(z') sin k(z-z') dz'
+ c cos kz 1
0 =
F (z) g
,
(8.11)
F (z) representing the adopted model of the generator driving the dig pole. We can next eliminate the constant c by requiring that the e1 quation be satisfied at z=O.
To solve eqn.(8.11) for I(z), assume that I(z) can be approximated by a series with complex coefficients Ii to be determined, n
I
I(z)
I. f. ( z)
i=1
l
(8.12)
l
with n
I
I. f. (h) l
i=1
(8.13)
0.
l
Let us assume that Z'(z) can also be represented in a similar way as m
L
Z' (z)
(8.14)
ZJ. gJ, (z)
j=1 With eqns.(8.12)-(8.14) th~
the
integral equation (8.11) for current takes
following approximate form:
n
n
I i=1
I. A. (z) + l
l
m
I I i=1 j=1
I.Z.B .. (z) l
J
lJ
D(z) •
(8.15)
Sect.B.2. Assuming
Optimization by varying distributed loadings f (h)~O,
that
using
n
eqn. (8.13)
we
185
can eliminate the coeffi-
cient In' to obtain n-1
n-1
I
I. A~(z) l
i=1
l
+
m Ii ZJ. B~j (z)
L L
D(z) ,
(8.16)
i=1 j=1
where A~ (z) and B ~. (z) are simple expressions involving Al. (z) and An(z), l
l]
namely B .. (z) and B .(z), respectively. lJ n] To determine the coefficients Ii, i=1,2, ... ,(n-1), the point-matching method may be used.
This results in a system of (n-1) linear equations
in (n-1) unknowns Ii, i=1,2, ... ,(n-1), of the form
m
n-1
L i=1
[A~(zk) + /:
Z.B~.(zk)J I.= D(zk),
j=1
Solutions of tions
of
l]
J
(8.17)
We can construct an optimization func-
J
tion which includes mize
it by varying
that
the
the
current-distribution
the
square matrix
values
[A~k]
of and
parameters
Ii, and mini-
the
impedance parameters Z.. Note J the 3-dimensional matrix [B~jk] in
independent of the values of Z. [A~k and B~ "k stand for J l lJ respectively] . Therefore, for a gi veh problem,
eqn.(8.17)
are
A~ ( zk)
B~j ( zk) ,
these matrices
.
the last system of equations can be considered as func-
the m parameters Z..
and
k=1,2, ... ,(n-1)
l
can
be computed once, and their values used in all sub-
sequent solutions of eqns.(8.17). just evaluation of
This is extremely important, because
these matrices requires the larger part of the com-
puting ti~e necessary
for
solving eqns. ( 8. 17).
During an optimization
process these equations are solved many times, and therefore any saving in the computing time necessary for a single step becomes worthwhile. Finally, it is usually necessary to introduce some constraints on the values
of
Z., imposed by J of the loading.
practical possibilities and the desired type
We shall adopt that f.(z) = jz/hjil
and
1
,
( 8. 18)
186
Ch.B.
Optimization of antenna admittance
. 1
g.(z) = lz/hiJJ In
that
case
thus all
(8.19)
the
integrals Ai (zk)
the elements of
B .. (zk) with
can
[A~k]
the
be integrated numerically, and
matrix
determined.
The integrals
the
adopted g.(z) can be evaluated explicitly using are~J J currence formula, and so the elements of the [B~ .k] matrix evaluated ~J
easily. 8.2.1. it
is
ways.
Some general examples of optimization.
possible
to
As already indicated,
optimize an antenna with impedance loading in many
In this subsection we
shall mention as examples a few of these,
for which numerical results will be presented in Subsection 8.2.3. Perhaps
the
conductance
simplest case is that of requiring
or
susceptance of
an
that
the admittance,
antenna of given dimensions and at a
given frequency be as close as possible to a given value. if
we
For example,
wish the admittance of the antenna to be as close as possible to
a desired admittance Y , we construct the optimization process which 0 will determine the loading along the antenna so that the expression (8.20) be minimal.
(V represents the voltage of the generator, and the anten-
na admittance equals
I/V.)
Similarly, if
we
wish the susceptance to
be as close as possible to zero, we may minimize 1Im(I ) 1 Next, we
can
lectrically
I·
require, for example, that the current wave along an e-
long antenna of
given dimensions and at a given frequency
be as close as possible to a decaying travelling wave from some h=h z=h.
{I Here,
p~l.
to
the
I(hl)\ (1-z/h)p(1-h/h) p
II
i=1
I. (z/h)i-11}2dz.(8.21) ~
Of course, instead of the desired amplitude function of the
form (1-z/h)p from
1
A possible expression to be minimized in this case is
it
value
by necessity be
is possible
to
I I(h ) I at z=h 1 broadband
to
use any other that decreases smoothly to zero at z=h.
Such an antenna will 1 a certain extent both in the. admittance
Sect.8.2. and in
187
Optimization by varying distributed loadings
the
radiation pattern, because the above requirement minimizes,
essentially, the waves reflected from the antenna ends. A truly broadband antenna in either admittance
or
in radiation pat-
tern, or in both, can be synthesized by the following optimization proWe first compute the [A~k] and [B~jk] matrices at a number nf of
cess.
frequencies in mized must shape of
the
now
the
desired frequency range.
include values of
the
radiation pattern at
The function
admittance
all of
these
or
to
be opti-
a measure of the
frequencies.
For ex-
ample, we may wish to minimize the expression
F(Z , ... 1
,z
where Ycref
;f , ... ,f ) m 1 nf
(8.22)
is the desired reference admittance in the frequency range
considered. These few cases indicate how the function to be minimized can be constructed, but obviously they do not exhaust the possibilities for optimization of cylindrical antennas with continuous impedance loading.
8.2.2.
Some
remarks
cylindrical antenna range
of
on
of
properties.
loaded
given
cylindrical antenna optimization.
dimensions
For example,
can
it is
have
clear
A
only a restricted
that we cannot have a
cylindrical antenna with a maximum in the radiation pattern in the direction of
the
conductance. whether we are erties) possible
antenna axis,
However,
requiring
which can be to
very
have a
angle 8=45° with
or
an
antenna with a negative value
often we
are
not
an antenna property (or a collection of prop-
closely approximated.
thin
of
able to tell in advance
For example, is it at all
cylindrical antenna with
respect to
the
the main beam at an
z-axis and simultaneously with an ad-
mittance close sible
to
to Y =(4+j0) mS? Unfortunately, it does not seem pos0 answer such questions in any simple manner, but by the method
described above we
can always find
how
closely the desired properties
can be approached. Whichever
initial values
certain solution,
of
the
Z.-parameters we adopt to reach a J no method seems to be available to check whether it
188 is
Ch.B. the
best one.
Fortunately, this
obtained solution satisfies tioned
the
Optimization of antenna admittance is unimportant, in as ~equirements.
desired
in Section 7. 4, a minimum of
far as the
Also,
as men-
the optimization function
at
the
bottom of a deep, narrow well in a multidimensional space does not represent a desirable engineering solution. sity very
sensitive
Therefore ble
of
for
to
a change
of
Such a
solution is by neces-
parameters, i.e., it is unstable.
the present purpose any simple optimization method capa-
locating a
satisfactory minimum of
the
optimization function
suffices. The results in plain or
the
following subsection were
the modified
Subsection 7.3.2.
steepest
In all
the
obtained by either the
descent method, briefly explained
in
cases the gradient was determined by va-
rying the coefficients for I6Z.I=10 ~/m. It was found that this repreJ sented a safe minimal increment of the coefficients to ensure accurate gradient
computation when working with
about
7 decimal digits.
The
step size I6Z.I in Z. was adopted to be 100 ~/m i f not stated otherwise. J J 8.2.3.
Numerical examples.
Although any type of generator represen-
tation can be used, all the numerical examples presented below were obtained with simplest
the
and
delta-function
generator.
sufficiently accurate
This
generator
appeared
to
representation
be
the
for
the
present purpose of checking the method of synthesis. Probably
the
of determining
simplest case the
of
cylindrical antenna synthesis is that
loading along an
antenna of given dimensions which
ensures that, at a given frequency, the antenna admittance as possible is not
to
a desired value.
of much interest, but it is useful
theory with
be
as close
From the practical point of view this
the experimental evidence
and
as
a means of comparing the
for estimating
the
conver-
gence of the optimization process in a simple case. An antenna was
considered with h=ll.44 em and a=0.32 em at
a fre-
quency f=667 MHz.
When the antenna is loaded with a constant resistive 49 loading Z' (z)=700 ~/m, it is found theoretically that the admittance of
the
antenna
convergence
of
is Y =(5.356 + j0.030) mS. 0 the process, the values of
In order all
to
estimate
the
the Z .-parameters were J
Sect.8.2.
Optimization by varying distributed loadings
set to zero,
189
and a search was made by the plain steepest descent 1'1ethod,
for real Zj such that II /v-Y 1_2_0.1 rnS. The values of the Zj-pararneters 1 0 converged very rapidly for all the values of rn considered (rn=l, 2, ... , 6). For example, 683 n/m;
for m=1
after
only 3 iterations it was found that Z' (z)=
for m=2 after 10 iterations the result was
+ 190 (z/h) D/m,
Z' (z) = 616
(8.23)
and for m=6 the optimization after 20 iterations resulted in
+ 160 (z/h) + 67 (z/h) 2 + 35 (z/h) 3 + 21 (z/h) 4 +
Z'(z) = 611
+ A much more important practical problem of
13 (z/h)
5
'1/m. (8.24)
optimization is
to
find
the loading which ensures a travelling current wave along the antenna. As mentioned, in this way an antenna which is to some extent broadband should be obtained automatically. First an antenna was a=O. 3175 em.
considered
of
half-length h=50 ern and
A resistive loading was
pression given
in eqn. (8. 21)
frequency f=450 MHz.
searched
with p=l
It has been
for
such
radius
that the ex-
and
h =0. 25 h, be minimal at a 1 51 show-n both theoretically and ex-
perimentally that if this antenna is loaded with a resistive loading of the form Z'(z)=Z /(1-z/h), where z is a certain real constant, a trav0 0 elling-wave antenna is obtained with I(z) proportional approximately to (1-z/h).
In. this
example comparison was made
between
the results
obtained by the present theory with those obtained theoretically by Wu 41 51 and King and verified experimentally by Shen. It was loading
reasonable should
initial values
to
increase of
the
assume that towards
for
the
a travelling-wave antenna the Arbitrarily,
antenna ends.
zj-pararneters were
adopted
to
the
be z1 =1000 n/m,
z =2000 D/rn, etc. With n=8 and rn=4, using the second mentioned opti2 mization method, after 30 gradient determinations with step size I nZ .I= J 100 '1/m, then 25 gradient determinations with lnZ.I=50 n/m, and finally J 25 gradient determinations with 1nz. 1=2o n;rn, the optimal loading was J found to be Z'(z) = 492- 509 (z/h)
+
2749 (z/h)
2
+
5017 (z/h)
3
'1/m
(8.25)
190
Ch.B.
For
this
Z' (z)
the
than 0. 0062 (mA/V) Fig. 8.1.
The
2
Optimization of antenna admittance
optimization function given in eqn. (8. 21) was less
•m.
Current distribution in
solution
is
quite
this
case is
shown
in
close to that proposed by Wu and King
and obtained experimentally by Sherr (see Fig.4.3). The
two
previous examples can be considered as a proof that the pro-
posed optimization method
yields
available experimental results. trative numerical examples
for
results The
two
which are
in agreement with
following examples are illus-
which, unfortunately, experimental data
do not exist. It
was
interesting to discover whether a travelling-wave antenna can
also be obtained with
Arbitrarily, an an-
tapered capacitive loading.
tenna with h=15 em and a=0.3 em was considered, and the aim was to synthe size
capacitive
a
loading which will
result
in
a travelling cur-
rent wave of the form given in eqn.(8.21), with p=1 and h1=0.25 h, at a frequency f=1600 HHz. current (n=5) (m=4)
were
and
The
fourth-degree polynomial
approximation for
the third-degree polynomial for loading distribution
adopted.
The initial values
at 1 GHz were taken to be -j3000 ~/m.
of all the four parameters Z. J After 17 gradient determinations
z/h
1.0
FIG.8.1. Current distribution along optimal travelling-wave cylindrical dipole antenna with resistive loading given in eqn.(8.25); a=0.3175 em, h=SO em, f=450 HHz, n=8, m=4, (1) real part, (2) imaginary part, ( 3) magnitude. (Ref. 76)
2
-2
0
2
3
Sect.B.2.
Optimization by varying distributed loadings
191
z/h
0
2
4
6
(b)
6 (d) FIG. 8. 2. Current distribution along capaci tively loaded cylindrical dipole antenna with loading given in eqn. (8.26), which results in travelling current wave at 1.6 GHz; a=0.3 ern, h=15 ern, n=S, rn=4; (a) f=l.4 GHz, (b) f=l.6 GHz, (c) f=l.8 GHz, (d) f=2.0 GHz; (1) real part, (2) imaginary part, (3) magnitude. (Ref. 76)
it was
found with
the modified steepest
descent method that the ex2 pression given by eqn.(8.21) reduces to less than 0.006 (rnA/V) •rn, with Z'(z) =- ___i__ [4894+4093(z/h)+3696(z/h) fGHz Current distributions along the antenna at
2
3 +3465(z/h) ] rl/rn.
(8.26)
f=l.4, 1.6, 1.8 and 2.0 GHz
are shown in Fig.8.2, and variation of the antenna conductance and susceptance with
frequency in Fig. 8. 3.
A relatively good broadband
an-
tenna in admittance is obtained, and at 1.4 GHz and 1.6 GHz there seems to be only a small reflected wave from the antenna end. As
the
final example, consider a capacitively loaded antenna of the
same dimensions
as
in
the
preceding example, and let us determine the
capacitive loading which minimizes the expression in eqn.(8.22) at frequencies f =1.4 GHz, f =1.6 GHz and f =1.8 GHz (i.e., nf=3), with Ycref 1 2 3 in eqn.(8.22) given by
192
Ch.B.
Optimization of antenna admittance
G,B (mS)
5 4 3
2 f
(GHz)
FIG.8.3. Dependence of conductance (G) and susceptance (B) of optimal capacitively loaded dipole antenna on frequency; a=O. 3 em, h=15 em, n=S, m=4; ----- optimal travelling-wave antenna at 1.6 GHz with loading as in eqn,(8,26) and - - optimal broadband antenna described in the text with loading as in eqn. (8.28). (Ref. 76)
OL-~--~--~--~--~----
1.0 1.2 1.4
1.6
1.8
2.0
nf
I
ycref By
(8.27)
1 I 1 ( fk) /V . nf k=1
the modified steepest descent method, after 10 gradient searches it
was found that with Z'(z) =
-~[5205+2535(z/h)+2430(z/h) 2 +2530(z/h) 3 ]
'J/m
(8.28)
GHz the expression in eqn.(8.22) is reduced to less than 0.2 (mS) dence
of
real
and
imaginary parts of the antenna admittance
loading is shown in Fig.8.3 in dashed lines.
2
.
Depen-
for
this
We note considerable sim-
ilarity of the G and B curves, those corresponding to the present example being somewhat flatter in the range of optimization (1.4-1.8 GHz), as expected.
8.3.
SYNTHESIS OF PARALLEL LOADED CYLINDRICAL ANTENNAS WITH MINIMAL 82 83 COUPLING •
If we consider an array of wire antennas, there is always certain coupling between
the elements
closer
spacing, the stronger is the coupling.
their
of
the array..
For a pair of
elements, the
The elements can-
Sect.B.J. not
Synthesis of antennas with minimal coupling
be widely separated, however, for
this would
two
reasons.
193 On
the
one side,
increase the dimensions of the array, and on the other side
radiating properties
of
arrays with widely spaced elements most often
are not suitable, because radiation patterns of such arrays have several beams.
Therefore,
coupling between elements
of antenna arrays al-
ways exists in practice. Because
of
the coupling, properties of individual elements are modi-
fied, with respect to
both
the behaviour of the elements as two-termi-
nal networks, and to their radiation characteristics. array theory
the
array analysis
In the classical
coupling is neglected, \vhich results in a very simple
and
synthesis.
It
turns
out, however,
that
coupling
cannot be neglected if a more precise analysis of an array is desired. For example, in the case of large periodic arrays the properties of all array elements tically
the
ditions
of
except
those
same in spite the array all
quite close to
of
cannot be neglected.
array edges are prac-
coupling, because in usual operating con-
such elements are situated in a very similar
"electrical environment", but
erties of
the
in
determining these properties coupling
In the case of smaller or aperiodic
all their elements are
arrays
prop-
different
due to different influence
been made
to reduce coupling between
on them of mutual coupling. It
seems
that
elements of ments.
This
antenna ments
no
attempt
has
an antenna array by a convenient construction of section
arrays
aimed at
consisting of
coupling
loading along
is
can
parallel,
that
in
ele-
the case of
nonstaggered cylindrical ele-
substantially reduced by
appropriate impedance
elements.
This particular distribution of the load-
ing can be determined by an
optimization procedure, in which the load-
ing
along
the
be
demonstrating
the
the antenna is varied until coupling between the elements as
a function
of
quanti ties
in a certain
the
distance between
parallel nonstaggered numerically and
them or of
becomes minimal.
resistive antennas
verified experimentally
method coupling bet.veen distances be tween
range,
the
the
are that
frequency, or of both As
an
example, two
considered.
It is shown
by means of the proposed
antennas can be practically eliminated for
antenna
axes
larger than about
0. 2 .vavelengths
194 and of
Ch.B. for
a frequency
such elements
range wider
can
Optimization of antenna admittance than
2: 1.
Antenna arrays consisting
be driven as if the elements were
the radiation properties
of
the array determined with
isolated, and
relatively high
accuracy according to the simple classical array theory, in which coupling between
the
array elements
is neglected.
The
authors
anticipate
application of such and similar elements with minimal coupling for antenna arrays with a
small
number of elements, for aperiodic arrays in
which the geometry of the array and/or frequency need to be varied, and for space-periodic arrays with elements in nonuniform electrical environments. 8.3.1.
Outline of the method. antennas
and distance
their axes
be
be tween
driven by ideal val tage
quency w and coordinate
system
that
and that
the
b.
two
identical parallel non-
2, of length 2h, radius a (a«h) Let
the
dipoles, for simplicity,
delta-function generators
of
angular fre-
voltages
V and V , respectively. Let the z-axis of a 1 2 coincide with the axis of dipole no.1, and let the
origin coincide with assume
Consider and
staggered cylindrical
the dipole center, as shown in Fig. 8. 4.
Finally,
the dipoles are situated in a homogeneous lossless impedance per
unit
medium
length along the dipoles, Z' ( z) , is a
continuous, differentiable and even function of the coordinate z. The system being linear, current distributions along
z=h
the
dipoles for
z
v,
+
-
v
2
2a
z=-h b
FIG.8.4. Two parallel symmetrical nonstaggered dipoles with continuous impedance loading.
Sect.B.J.
Synthesis of antennas with minimal coupling
arbitrary driving voltages v
1
and v
2
195
are given by
v Ys(z) + v Ym(z) 1 2
(8.29)
V Ym(z) + v Ys(z) 1 2
(8.30)
If v =v =Vs (symmetric excitation), then r (z)=I (z)=Is(z), and i f 2 1 2 1 v =-V =-Va (antisymmetric excitation), then r (z)=-I (z)=-Ia(z). The 2 1 1 2 functions Y (z) and Y (z) can thus be expressed as
s
m
_!_[Is (z)
v
2
+
I a (z)
v
s
[Is (z)
l
I a (z)]
1 2_v _ _ _v_
s Therefore
(8. 31)
a
·
(8.32)
a
current
distribution along the two antennas corresponding to
any qriving voltages v
and v can be calculated if symmetric and anti1 2 symmetric current distributions are known. Note that the self and mutual admittances of
the
dipoles, Ys
and Ym' are simply equal to Ys(O)
and Ym(O), respectively. Symmetric
and antisymmetric current distribution
can easily be equations.
calculated by
If we
solving one
consider the
along
the antennas
of several existing integral
Hall~n-type
equation, it is only neces-
sary to substitute g(r) in eqn.(8.11) by G(p) (z,z')
g(r )
s
+
(-1)p+l g(r )
(8.33)
p=1 '2
m
where rs
=
[Cz-z')
2
2 !:2
+a ]
r
(8.34)
m
in order to obtain the integral equation from which the symmetric, viz. the
antis)~metric
current distribution can be calculated.
The case p=1
corresponds to the symmetric, and p=2 to the antisymmetric driving conditions. Solutions manner
as
of those
these
integral
of eqn. (8.11)
equations in
the
can be obtained in
preceding
section.
the
same
As before,
196
Ch.B.
these solutions
Optimization of antenna admittance
can be considered as
functions of the impedance para-
meters Z., j=1,2, ... ,m. Among these solutions we can search for those J which correspond to the minimal coupling between the antennas. As the mutual admittance Ym
(or Y
12
as is labelled frequently) of the antennas
is given by
(8.35) where n
v
I
I
i=1
~p)
f. (0)
l
the coefficients
(8.36)
p=1,2 '
l
I~p) l
should be
determined in such a way that a conve-
nient real function F(Ym) be minimal. In
this manner the problem of determining a convenient loading along
the antennas is reduced again to the problem of nonlinear optimization. The
optimization
possible
function F(Ym)
can
be formed
in many ways.
Several
forms of the optimization function F(Ym) for the case of mini-~
mization of coupling between the two antennas are the following: a) F (Y ) = IY I, in which case it is required that the magnitude of 1 m m the mutual admittance be as small as possible;
b)
F (Ym , ... ,YmN)=(IYm 1+ ... +1YmNI)/N, in which case it is re1 2 1 quired that the mean value of the magnitudes of mutual admittances in a certain
range
(in
the
frequency domain,
space
domain, or both at the
same time) be as small as possible; c) quired
F /Ym , ... , YrnN) =max( I Ym 1 1, ... , IYmNI), in which case i t is re1 that the mutual admittance of largest magnitude in a certain
range (in frequency domain,
space
domain
or
in both at the same time)
be as small as possible; F (Ym , ... ,YrnN)=maxCIYm/Yoo 1, ... ,1YrnN/Y00 NI), where Yook denotes 1 1 4 the self-admittance of such isolated antenna, in which case it is red)
quired that the mutual admittance of the largest magnitude in a certain range (in frequency domain, space domain
or
with
antenna when
respect
to
the
admittance
of
the
in both at the same time), isolated,
be as
Sect.B.J.
Synthesis of antennas with minimal coupling
197
small as possible. 8.3.2.
Resistive cylindrical antennas with minimal coupling.
As an
example of the method, distribution of the loading was determined along two thin cylindrical resistive antennas, i.e., antennas for which Z'(z)= R'(z)+jO, which ensures that the mutual admittance between the antennas be minimal.
The function F (Yml, ... , YmN) was adopted as the optimiza4 tion function, in which Ymi' i=l,2, ... ,N, were values of the mutual ad-
mittance for different distances between the antennas. the local minimum of
the optimization function
steepest descent was
used.
Current
and
the
For determining
modified method of
resistance distribution func-
tions were adopted to be polynomials, as in the preceding section. The antennas considered were a
frequency
f=l600 MHz.
of dimensions a=O. 3 em and
During
the
optimization process
h=lS em, at two
natural
tendencies were observed: a)
If the loading was assumed in the form (8. 37)
x=l z/hl ,
minimization of coupling led to increasingly larger values of the first term, R • This could be expected, since in that case current intensi0 ties entering the expression (8.35) for mutual admittance tend to zero, so that Y
m
b)
tend also to zero.
If the loading was assumed in the form (8.38)
R' (z) there was
a tendency during
resistance values
the
[ R' (z) <0 J •
optimization process towards negative
This
could also
be
expected,
since
by
active loading it is possible to cancel the influence of one antenna on the other. In order
to
eliminate
these
two tendencies, constraints were intro-
duced into the computer program in values of the loading R'(z). one
side,
it was
requested
that
the
smaller than a certain maximal loading.
loading R' (z)
at
On the
all points be
On the other side, it
was
re-
198 quested
Ch.B. that
the
loading be
Optimization of antenna admittance
passive
for
all
z,
by
stipulating that
R' (z)>O.
Three
out
of
several
results obtained by
the
optimization
process
with different polynomials representing the distribution of the loading are the following (as before, x stands for
Ri (x) R~ (x)
= 2777
+
+
4000 x
RJ (x) = 1515 x
1172 x
2
+
6000 x
+
2147 x
3
3865 x
+ 3
+
2
+
7000 x
4162 x
4
6000 x
3
lz/hl): (8.39)
rl/m,
rl/m, 4
+
(8.40)
8056 x
5
>1/m .
( 8. 41)
Although it is difficult to realize in practice resistive loading varying continuously along the antenna, it can be approximated either by a stepwise function (as was done in the case considered), or by a certain number of lumped loadings.
4 3
2
0
L--~========:L==~~==~~-L~b~/~A-
0.1
0.2
0.3
0.4
0.5
0.6
0.7
FIG. 8. 5. Magnitude of self and mutual admittance of two identical nonstaggered cylindrical dipoles versus normalized distance between their axes; a=0.3 em, h=15 em, f=l.6 GHz; -----optimally loaded antenna as in eqn.(8.39); ---unloaded antenna of the same size.
Sect.8.3.
Synthesis of antennas with minimal coupling
199
Fig.8.5 shows the magnitudes of self and mutual admittances for antennas with
the
loading as in eqn.(8.39), versus the distance between the
antenna axes. of
self
For comparison, shown in
the
figure also are magnitudes
and mutual admittances of unloaded antennas of the same size.
In this case it is found that efficiency of the loaded antennas, defined by eqn. (5 .11), is about the loading
74
per cent.
lhth some other distributions of
this efficiency, which is quite satisfactory for most pur-
poses, can even be improved. Distribution of current along such loaded antennas varies very little if driving conditions of
the
dipoles are changed from symmetric to an-
tisymmetric, and it is almost identical with current distribution which would
exist
Fig. 8.6
along
shows
an
real
isolated antenna (i.e., if b+oo).
and
imaginary parts
of
currents
As an example, corresponding to
symmetric and antisymmetric cases for the optimally loaded antenna with
z/h
z/h
;:::- ....
...........
''
,
... \
I I I
/
I
I
.... ',
\
I
I
I
\
'
/
''
_,/
' ' .......
/
/
"
I (z)
a
v;_!_(rnA)
v v
-4 -3 -2 -1 0
1 2 (a)
3
4
5
-4 -3 -2 -1 0
1
2
3 4
(b)
FIG.8.6. Symmetric and antisymmetric current distribution along optimal resistive antennas with loading as in eqn. (8.39) and along unloaded dipoles of the same size for distance between the dipoles b= 0.4A; (a) real part, (b) imaginary part; -----loaded antennas, - - unloaded antennas.
200
Ch.B.
loading
Optimization of antenna admittance
in eqn. (8.39), for distance bet~veen the dipole axes b=0.4A.
as
For comparison, shown loaded antennas.
also
in
the
The fourth-degree
figure .are
analogous cases for un-
polynomial
approximation
for
cur-
rent along the antennas was used. Sho>vn in Fig. 8. 7 are magnitudes optimally loaded tance
be tween
and
such
self
and mutual
admittances
of
of unloaded antennas versus frequency, for a dis-
the antenna
antennas optimized for tween
of
axes
b=O. 4 A.
Important property of
minimal coupling is
seen
loaded
clearly: coupling be-
antennas is quite small in a wide frequency range, which is
larger than 2:1 in the case shown in the figure. According
to
eqns.(8.30)
(8.32), i f
and
only
dipoleno.1 is driven
and dipole no.2 is short-circuited (V =0), then 2
1 - [r (z)- I (z)], 2 s a
(8.42)
for V =V s =V a. This indicates that magnitude of current in the short1 circuited parasite can be a measure of coupling between the two antennas.
This conclusion was
1\1 ,!Ym!
used
for
checking theoretical results
in
a
{mS)
'""'\I
I
I
I 1
10
I I
I I
I
I I I
I
5
\ \
I
\\
,, s---'""' \!Y I
/
1 II
\
\
\
I
\
I
/
<, ... ___ ..... .,/
'
' . . . . ____ f ./ (GHz) o~~===r~~~~~~~ 1.0
1.2
I
!Ym!
1.4
'
1.6
l.8
2.0
FIG.8.7. Magnitudes of self and mutual admittances versus frequency of two parallel coupled dipoles; a=0.3 em, h=15 em, b=0.4A; ----- dipoles with loading as in eqn.(8.39), unloaded dipoles.
Sect.8.3.
Synthesis of antennas with minimal coupling
201
R' ( kll/m)
FIG.8.8. Synthesized optimal loading and stepwise approximation of the loading. The model was made in the form of a row of resistors. Dimensions are in millimeters. Note a slight difference in size of the experimental model when compared with the theoretical.
z/h 0.5
simple manner. The optimally loaded (8. 39)
was
cylindrical
first approximated
sketched in
Fig. 8. 8,
After concluding
and
that
its
by
antenna
with
loading as
in eqn.
an antenna with stepwise loading, as
properties
then
the antennas with
checked
theoretically.
stepwise loading have proper-
ties which are quite similar to those of antennas with continuous loading,
such antennas
Note
that
were made
the experimental
and
antenna
some
of
their properties measured.
in Fig.8.8
had a radius a;0.35 em
(as compared with a;0.3 ern used for the theoretical model with continuous loading) and length 14.4 em (as compared with 15 em). From Fig. 8.6
and
eqn. (8.42) we see that very small current should be
expected in a parasitic element optimized for minimal mutual This conclusion was verified experimentally in two ways.
coupling.
First, admit-
tance of the optimally loaded monopole antenna was measured in the presence of identical, short-circuited monopole, versus frequency a
wide
range
of are
distances
The
results
the
parasitic element
shown
for
between the antennas (from 2 em to 16 em).
in Fig.8.9. on
and
It is
seen
the admittance of
the
that
the
influence of
driven element is re-
r,
202
5.5 5.0 4.5
Ch.B.
(mS)
Optimization of antenna admittance
G
.J-
~
6.0 5.0 4.0 3.0
f ( GHz)
1.2
1.4
1.8
1.6
2.0
2.2
2.4
2.6
2.8
FIG.8.9. Conductance (G) and susceptance (B) of the optimal antenna shown in Fig.8.8 in the presence of identical short-circuited antenna, versus frequency, for different distances (in the range from 2 range of measured G and em to 16 em) of the parasitic element. B values;----- isolated antenna (measured).
III
markably small.
Second, radiation pattern
of
such a system of dipole
antennas was measured for a number of frequencies and distances between the elements, and compared with loading along
its length.
that
Some of
of
the
a single dipole with the same results are shown in Fig. 8.10.
It is seen that the optimally loaded parasite has quite small influence on
the
radiation
pattern
of
optimally
loaded antenna,
while
this is
known not to be the case at all frequencies if the antennas are unloaded.
In the case of optimally loaded antennas the influence of the par-
asitic element is small
for
practically all distances and a wide range
of frequencies.
8.4.
OPTIMIZATION OF ANTENNA ADMITTANCE BY VARYING CONCENTRATED LOADINGS
16
Wire antennas with concentrated loadings those with distributed easily.
loadings,
because
are much more important than they
can
be
realized
more
We have seen in Chapter 5 that, in addition, with a relatively
small number
of concentrated
loadings we
can
approximate quite accu-
Sect.B.4.
Optimization by varying concentrated loadings
f=1.3 Gl-lz
203
f=1.8 GHz
z
z
b=2. 3 em
b=6.4 em
FIG. 8.10. Measured radiation pattern in electric-field strength of optimally loaded symmetrical equivalent of the antenna shown in Fig. 8.8 with identical short-circuited parasitic element and theoretical pattern of isolated antenna with the same loading, in the plane containing the antenna axes; o o o experiment, ----- theory.
rately a distributed loading. Synthesis of antennas with concentrated loadings, however, is in general
very
lengthy numerical
procedure.
Namely, if we allow the posi-
tions of the loadings to represent optimization parameters
in
addition
to their impedances, the number of optimization parameters becomes relatively large still more
even
for a small number of
the
loadings.
What makes it
time-consuming is the fact that if we move a single loading
along the antenna,
we
essentially have
to
analyse the antenna practi-
Ch.B.
204 cally from that
the
if we
very beginning.
fix
On the other hand, we shall show below
the positions of
pedances, it is possible
to
Optimization of antenna admittance
the
loadings and vary only their im-
analyse the antenna essentially only once,
all subsequent analyses being reduced to a single matrix inversion. in addition, ings at
a
we
have in mind
If,
that by variation of the impedance load-
sufficient number of
fixed locations we
can to some extent
obtain effects analogous to physical displacement of the loadings along the antenna, it is obvious that the general synthesis case for antennas with variable
loadings
except, possibly, in
at variable
some
locations need not be considered,
very special cases.
Therefore in
this
sec-
tion we shall restrict our attention to synthesis of wire antennas with fixed
geometry,
having variable
concentrated loadings at
fixed posi-
tions along the wire structure. Fig.8.11 shows a schematic representation by a single generator of voltage
v
of
such an antenna, driven
and having n concentrated loadings
0 of admittances Y ,Y , •.. ,Yn (at the operating frequency). The antenna 1 2 can be considered as (n+l)-port linear network, so that the following equations can be written for the structure: n
2:
j=O where
Y .. V. 1J J
v
0
I. 1
i=O,l, ... ,n,
is the generator voltage, I
(8.43)
0
its current and
FIG. 8.11. Schematic representation of wire-antenna structure driven by a single generator and loaded by n concentrated loadings.
Sect.B.4.
Optimization by varying concentrated loadings
205 (8.44)
(with respect to the reference directions shown in Fig.8.ll). The self
and mutual admittances of
the
structure, Yii and Yij, res-
pectively, depend only on frequency and the antenna geometry. be computed from eqns. (8.43)
if we
know voltages
They can
Vi
and currents Ii'
i=O,l, ... ,n, for (n+l) independent driving conditions.
For example, we
can short circuit all the antenna ports except that labelled by "j", at which a voltage generator V. is connected. We next solve for the anJ tenna current distribution for these driving conditions. The currents Ii known, matrix
we
can
compute
elements
of
one
column
of
the
admittance
as
Y .. =I. /V., i=O, l, ... , n. Performing this procedure (n+ l) lJ ]_ J times (for j=O,l, ... ,n) we can compute all the elements Y ... Note that lJ changing driving conditions in this case amounts to changes of the an-
tenna excitation only, which enables great savings of the computer time in ·solving the integral equation for current distribution. Once
the
Y-parameters
of
the structure are known, we can substitute
eqns.(8.44) into eqns.(8.43) to obtain n
I j=O
(Y i]"
Here, oij
+ is
(8.45)
0 .. y . ) v .
lJ ]_
J
the Kronecker
that the generator voltage
delta
v
(oii=l
oij=O i f i#j).
and
Assuming
is known, from eqns.(8.45) we can compute
0 V., j=l, ... ,n, at the loading terminals, and, in partiJ generator current r , for any values of the loading admit0 Obviously, in order to compute the antenna input admit-
the voltages cular,
the
tances
Yi.
tance of
the
for
a given
set
of loadings we
integral equation
for
do not have to repeat solutions
the antenna
current
distribution, which
greatly expedites the optimization process. The admittance Y has no influence on the antenna current distribu0 tion and its radiation characteristics, and from this standpoint it can be discarded.
However,
ristics is very tenna
input
its
influence
on
the
antenna
strong because it is connected
terminals.
This admittance
can
in
thus
input characte-
parallel to the anbe very efficiently
206
Ch.B.
used
for
compensation
of
the
Optimization of antenna admittance
antenna
input
susceptance.
Of course,
similar effect can be obtained with a series impedance element. fore, either included of
the
into
parallel or the series compensating element should be
an optimization of
capacitively
which will
loaded
the antenna admittance.
antennas, which are
be considered as
the antenna susceptance tends 16 4 7 60 84 85 86 ues. • • ' ' ' Therefore we inductive
element,
of
In the case
the easiest
to
make and
examples below, without compensating ele-
ment
ing
There-
to have
quite
large positive val-
shall include a
impedance
z
0
,
in all
series compensa t-
the
examples of this
section. 8.4.1. Due
to
Optimal broadband capacitively loaded cylindrical antennas.
87
their simple construction, unloaded cylindrical antennas are of
considerable practical importance.
However, we
know
that their admit-
tance varies
very much with frequency, so that they cannot be used to
operate
wide
in
considered
a to
frequency
stem from
range.
Basically,
this
the fact that they radiate
property can be
only
from discon14 They do
tinuities, such as the antenna ends and the excitation zone. not radiate
along
their length, because
phase
velocity of the antenna
current wave is equal to the velocity of propagation of electromagnetic . b ances 1n . t h e surroun d.1ng me d.1um. 5 '88 d 1stur In order to enhance radiation, it is necessary to produce a fast wave along
the
lumped
antenna.
series
This
can be
capacitive loadings
additional discontinuities along
achieved most easily by along
it.
Such
introducing
loadings
the antenna, which radiate
the antenna ends and excitation region.
represent just
like
An increase in number of these
discontinuities results in increasingly better broadband antenna properties.60•86 As
the
optimization function, square of the effective reflection co-
efficient, R;ff' given
in eqn. (8.5), will be used in all examples, and
the reference admittance given in eqn. (8.10) will
be
adopted in evalu-
ating Reff'
The self and mutual admittances Y .. andY .. are determined 11 1] by solving the Schelkunoff integral equation with combined trigonomet15 ric and polynomial approximation for current. For the optimization
process itself, a modification of the complete search is used.
Namely,
Sect.8.4. taking
Optimization by varying concentrated loadings
two
of
the
loadings as the optimization parameters (the others
being fixed) a complete search is two loadings determined. other
pairs
cycles.
This
made
and the optimal values of these
procedure
is
of loadings, taken cyclically.
variable loadings It was
207
an
optimum
is
repeated
successively for
In the case of five or six
thereby reached typically in 5
to
20
found that, due to the particular analysis method des-
cribed above, this optimization procedure enabled a significant increase of efficiency in determining
the optimal lumped loadings along the an16 . h ot h er opt1m1zat1on . . . tenna, wh en compare d w1t met h o d s. In order
to
further increase efficiency of the optimization process,
the loading z =jX in the excitation zone (which compensates the anten0 0 na reactance) was treated separately. It can be optimized very simply if
use
is made of
the
following approximation, valid for small values
of the reflection coefficient: zai
R.
+ jXOi - zc 2Z
1
where
Zai
(8.46) c
compensating element (both at actance
x
i the reactance of the series 0 frequency fi). Optimization of the re-
is the antenna impedance and
x
is thereby reduced to a linear fit, which expedites the op0 timization process noticeably, since this, in fact, reduces the number
of optimization parameters by one. A check of
the
optimization procedure described above was performed
by comparing theoretical and experimental results. model, an
antenna with 6
Fig.8.12(a,b).
h, 163 mm.
For
used.
ries
concentrated loadings was
adopted, shown in
The radius, a, of the antenna was 3 mm, and its length,
antenna was reactance
As the experimental
the
present purpose, only the monopole version of the
The compensating element in the excitation zone (of
x
at a reference frequency) was made in the form of a se0 inductance shown in Fig. 5. 21 (b), the reactance of which can be
varied
from
12
to
40
n (at 1 GHz).
Five
lumped
capacitive loadings
were positioned at
unequal mutual distances, in order to minimize pos-
sible
They were
form
resonances. shown
in Fig.5.21(a),
of
also made
as variable elements, of the
capacitance ranging from 0.22
to
4 pF,
208
Ch.B.
Optimization of antenna admittance
(a)
(b)
R,X (S<)
1 VSWR
b 100
I
(c) FIG.8.12. (a) Sketch of the experimental antenna; the dimensions are in millimeters. (b) Photograph of the experimental antenna. (c) Resistance (R) and reactance (X) and VS\\'R for the optimal octave antenna with h/a=54. 3 in the range 1-2 GHz, with values of lumped reactances Xi given in Table 8.1. - - - , - • - theory, oo, •• experiment.
Sect.B.4.
209
Optimization by varying concentrated loadings
which corresponds to reactance of about 700 In order
to
check accuracy of
the
theoretically optimized
~
to about 40
~
at 1 GHz.
adopted optimization procedure, a
number
of
antennas were
examined
experimen-
tally.
As an example, Fig.8.12(c) shows resistance, reactance and VSWR
for an antenna optimized in a frequency range from f =1GHz to f =2GHz. 1 2 It is seen that agreement between theoretical and experimental results is very in
good.
It is interesting
the whole range except in
cy limit, in spite of
to note that VSWR is less than 1.25
the
small region near the lower frequen-
the monopole length being only 0.542 wavelengths
at 1 GHz. A systematic analysis formed as
on a
of
optimal
loaded broadband antennas was per-
theoretical model of the same length and loading positions
in Fig. 8. 12 (a) .
In order
thickness on matching,
to analyse
antennas of
the
radii
influence of the antenna
a=3 mm, 1.5 mm and
were considered, which corresponds to the ratio 217.3,
respectively.
For
0. 75 mm
h/a of 54.3, 108.7 and
such antennas mutual
and
self
admittances
were first determined in the frequency range from 0.5 to 3 GHz, in steps of 0.1 GHz.
Synthesis
rious properties was
of
a large number
of broadband antennas of va-
then performed, of which only some characteristic
examples will be presented here. As
the
first example, antennas with
constant
frequency
to
0.0166
at
of
500 HHz.
h/a=54.3 were optimized, with a
f , the effec1 tive reflection coefficient, Reff' decreased quite rapidly (from 0.3463
at 0.5 GHz
range
1.5 GHz).
With increasing
Such a rapid decrease in
the
ref-
lection coefficient is partly due to decrease of the relative frequency band.
In order to obtain a clearer insight into relative antenna prop-
erties,
two
cases with
constant
relative
the octave antennas (i.e., f /f =2), 2 1 tennas (i.e., f /f =1.5). 2 1 In
the
and
bandwidth were
approximately half-octave an-
case of octave antennas, analysis of influence
reflection coefficient was 8.1
shows
the values of
the
reference
performed on over
the
impedance (Z
ere
effective f)
op
t=1/(Y
considered,
20
f)
op
h/a on the
antenna types.
reflection ere
of
Table
coefficient (Reff),
t' labelled for short'as
Ch.8.
210
Optimization of antenna admittance
TABLE 8.1. Optimal octave antennas; f1 and f2 are given in GHz, Zc in Ohms and Xi in Ohms at 1 GHz; rows for the same f 1 and f 2 correspond to h/a=54.3, 108.7 and (in the last three cases) to 217.3, respecThe antenna dimensions are given in Fig.8.12(a). tively. fl
f2
Reff
zc
xo
xl
x2
x3
x4
x5
0.6
1.2
0.2919
91
66
-75
-25
-100
0
0
o. 3505
102
79
-75
-125
-25
0
-25
0.8
1.0
1.2
1.4
Zc' and
1.6
2.0
2.4
2.8
the
0. 1381
83
25
-100
,roo
-150
-100
0
0.2246
117
20
-75
-100
-275
-25
0
0.0966
88
29
-150
-25
-300
-175
0
0.1397
112
34
-200
-25
-400
-175
0
0.2012
135
39
-275
0
-570
-175
0
0.0659
90
17
-150
-175
-175
-325
-175
0.0978
107
18
-250
-150
-275
-400
-150
0. 1382
130
18
-325
-175
-300
-600
-50
0.0315
98
15
-150
-250
-175
-525
-200
0.0614
128
16
-210
-285
-230
-600
-205
0.0920
15 7
13
-300
-300
-325
-700
-350
optimal reactances of the lumped loadings, for a number of
typical examples of optimal antennas with seen
different ratios h/a.
It is
that magnitudes of the loadings do not differ considerably as the
ratio h/a is increased. Fig. 8.13(a) antennas with
shows VSWR versus h/a=54.3.
frequency
for
three
types
of octave
It is seen that the optimization results in a
large decrease of VSWR in the passband (solid lines) when compared with that outside
the
GHz antenna,
the
filter;
is
this
VSWR is most
passband
(dashed
lines).
[Note
VSWR of which is very similar the
often
same
graph
obtained at
as the
means that there it is most difficult
to
the
case of the 1-2
that of a passband
that in Fig.8.12(c)].
The largest
lower to
frequency limit, f , which 1 achieve matching, a result to
be expected. In order to analyse the influence of the relative bandwidth on matching, approximately half-octave antennas were also optimized, as already
Sect.B.4.
VSWR
VSWR
I. Ir·~y
I
i i
I. \ iI \
I \ I I I I
.i I.
i i i i
3.0 2.5
iI
.iiI
i
\\ \\. \
I
I
\.
,,
i
\.\\
.II. I.
\.
2.0
I
I·I. .I \ Ii
i
1.5
1 o.6-0.9
!I.
ii \\0.7-1.4
2.0
211
Optimization by varying concentrated loadings
I
/
i i
/ 1.5
1.0-2.a·' i 1.4-2.8
f
f
( GH z ) 1. o L-~=---~~::::::::::=.::::::::::::::::::I:::::::~(C.::::GH_:.;.:z)1. oL------L-~~=~:::!!:::...__:. 0.5 1.0 1.5 2.0 2.5 0.5 1.5 1.0 2.0 (a)
(b)
FIG. 8.13. VSWR versus frequency for optimal broadband antennas sketched in Fig.8.12(a) with h/a=54.3. (a) Octave antennas. (b) Halfoctave antennas.
mentioned.
Table 8. 2 gives the values of the reference impedance and
optimal reactances of
the lumped loadings for such antennas, and in
Fig.8.13(b) the VSWR for several "half-octave" antennas is plotted versus frequency.
TABLE 8.2. Optimal "half-octave" antennas, i.e., f2/f1=1.5; h/a=54.3, f1 and f2 are given in GHz, Zc in Ohms and Xi in Ohms at 1 GHz. The antenna dimensions are given in Fig.8.12(a). Reff
zc
0.9
0.2872
112
1.2
0.0728
68
1.5
0.0775
89
1.2
1.8
0.0447
1.4
2.1
0.0216
f1
f2
0.6 0.8 1.0
xo
x1
x2
x3
x4
x5
0
0
61
-200
-75
-50
0
0
-75
-125
25
-125
0
0
-50
-400
-25
0
98
21
-150
-75
93
19
-200
-125
-450
-75
-100
-425
-175
-200
212
Ch.B.
8.4.2.
Limits of VSWR for optimal broadband capacitively loaded cy-
lindrical antennas versus fluence
Optimization of antenna admittance
of
their length.
the antenna length and
of cylindrical antennas with
87
In
this
subsection the in-
thickness on reflection coefficient
concentrated capacitive loadings
is
ana-
lysed, and some diagrams are given indicating the antenna extremal possibilities.
The analysis is based on a large number of results for an-
tennas of the form shown in Fig.8.12(a) optimized with respect to their admittance in a prerequired frequency range. Fig.8.14(a) optimal
octave
shows
the effective
antennas, as
reflection
compared with
coefficient,
that
(Reff) , of 1 for unloaded antenna,
3.0 0.5 \
2.
2.5
\ \ \ R \ max
2.0 (Reff)l
1.5 h
0.1 0.3 0.4
1.0 0.5
-A.-
max
1.0
1.5
1.0
f ( GHz)
(a)
1.5 f ( GHz)
(b)
FIG. 8.14. VSHR and reflection coefficient versus lower frequency limit, f 1 (or h/Amax), for antenna sketched in Fig.8.12(a). (a) Effective reflection coefficient for optimally loaded octave antenna, (Reff)l, and for unloaded antenna, (Reff)u, with h/a as parameter. (b) Effective, Reff• and maximal, Rrnax• reflection coefficient for optimally loaded half-octave antenna (solid lines) and octave antenna (dashed line) for h/a=54.3.
Sect.8.4. (Reff)u
213
Optimization by varying concentrated loadings (with
frequency
respect
limit,
corresponding
f
to
,
its reference admittance), versus
with h/a as
1 curves
parameter.
The
obviously represents a measure of
matching obtained by
the
capacitive loading.
the
difference
lower
between
improvement of
It is also
seen
that it
is more
difficult to match thin antennas, whfch is natural, since VSWR
of
antennas when
thin
unloaded
is also
larger
than for thick anten-
nas, as seen from the figure. Fig.8.14(b)
shows
the
dependence of
the effective reflection coef-
ficient, Reff' and
the maximal reflection coefficient, Rmax' for half-
octave antennas on
the
antenna the
lower frequency limit, f
, viz. on the relative 1 For comparison, in dashed lines are also shown
length h/\max
values
of Reff
and
Rmax
for
octave antennas.
It is seen that an
increase in bandwidth by increasing the upper frequency limit increases the reflection coefficient much quency that
limit
the
of
the
range.
At
less than by decreasing the lower frethe
same time, this clearly indicates
antenna length is a critical parameter for broadband matching
of antennas of the type considered. As
the
2 [ (Reff) / (Reff) u] . 1 just as that in Fig. 8.14(b), that significant
final example, Fig. 8. 15
This diagram indicates, improvement
in matching
shows
can only be
the
ratio
obtained when the antenna length
at the lower frequency limit is above about 0.3 or 0.4 wavelengths. To analyse
the
influence on matching possibilities of
capacitive loadings, antennas
were
ber
of
It was
the
final
loadings (4 results
and was
3).
relatively
also optimized with a smaller numconcluded
small
for
upper
which is
the difference in
can be
below 2 GHz,
considered as a
limit of possibilities of the antenna type considered.
At higher frequencies, icant,
that
frequencies
so that the results for 6 loadings given above practical
the number of
however,
undoubtedly
the difference was found to be signif-
due to a large
distance
(in wavelengths)
between adjacent loadings in these cases. The results
presented
above may be very useful for estimating neces-
sary minimal size of capacitively loaded broadband cylindrical antennas which
guarantees
frequency range.
a desired level of the antenna
matching
in a desired
214
Ch.B.
Optimization of antenna admittance
FIG.8.15. The ratio [(Reff)l/ (Reff)u] 2 for octave antennas, versus frequency (or h/Amax), with h/a as parameter.
h a
-=217.3 0.1
h :\max
0 0.5
1.0
1.5 f { GHz)
8.5.
We
OPTIMIZATION OF ADMITTANCE BY VARYING DISTRIBUTED AND 43 CONCENTRATED LOADINGs
have
seen
in
Subsection
can
result
and
radiation pattern
that
distributed
resistive loading
in cylindrical antennas with remarkably constant admittance
(Section 5 .4) uted resistive
in a
wide
frequency range.
We
have
that addition of lumped capacitive loadings loading
can improve
properties, increasing at tion we
4. 2.1
the
shall explain a method
even
same
to
distrib-
further the antenna broadband
time its efficiency.
for
also seen
In this sec-
optimization of antennas with such
combined loadings andpresent some theoretical and experimental results. Note
first
that matching of
broadband antennas
to
their feeders is
a difficult task if classical single-frequency approach matching reason
is
ceptance
the
antenna
by means
of
a
is
reactive matching
that, by Foster's reactance theorem, reactance (B=-1/X) of
a
linear
reactive
network always
adopted for
network. (X)
The
and sus-
increase with
Sect.8.5.
Optimization by varying mixed loadings
frequency,
while
the
with frequency, and
antenna
cannot
susceptance does
in all
cases
be
215
not
generally decrease
compensated by a reactive
compensating network connected between the antenna terminals. The present taining a
section
loaded
is aimed
broadband
at
describing a simple method
cylindrical
antenna
for
ob-
virtually perfectly
matched to its feeder in a relatively wide frequency range, possibly by means of a broadband transformer, using a parallel reactance in the excitation zone.
One way of achieving matching would be to analyse either
experimentally
or
numerically
the
antennas with
the desired kind
frequency behaviour of a
of loading.
All those antennas having
practically constant conductance, and a susceptance which a
certain
frequency
quency range (which has
to a
a
decreases in
can, in principle, be matched
feeder by means
in
that fre-
of a reactive compensating network
to be synthesized) connected between the antenna terminals,
and possibly a not
range
number of
broadband transformer.
Unfortunately,
this
is usually
simple task, particularly at microwave frequencies, because net-
work synthesis
and
its realization at
these frequencies are most often
very intricate. However, by varying susceptance
can
the
antenna loading, frequency dependence of its
be modified substantially.
Therefore, basically oppo-
site procedure is also possible, i.e., first to adopt a convenient simple reactive element with known frequency behaviour, and thesize
the
loading along
the
antenna in order
then
to obtain the antenna
susceptance which compensates the reactive element susceptance. tain insight
into
possibilities offered by variation
along an antenna in
this
the reactive element, but
to syn-
of
A cer-
the loadings
case is of course necessary prior to adopting for
microwave antennas this method is never-
theless more appropriate, because synthesis and realization of a microwave broadband
compensating
will
that
be
shown
reactive network
is
thereby avoided.
It
in this manner loaded cylindrical antennas can be
obtained having excellent broadband
properties
and
quite
small input
susceptance in a relatively wide frequency range. We
have
lindrical
seen that antennas
the
RC-loaded (as well as only C or R loaded) cy-
usually have
positive
susceptance.
In accordance
216
Ch.B.
Optimization of antenna admittance
with
the method just outlined, an inductive coil connected in parallel
with
the
antenna
terminals was
adopted
being the simplest for realization. B
-1/wL
comp.net.
Following
the
and determine
the
compensating network,
Thus, theoretically,
( 8. 4 7)
c
proposed method, the
as
we
next adopt
distributed and
the
antenna dimensions,
concentrated loadings along the an-
tenna which minimize the optimization function
(8.48)
F
In
this
equation Lc
is also considered to be
an
unknown parameter to
be determined from the optimization process, and f are
and f , as before, 1 2 lower and upper frequencies of the frequency range considered.
the
In this manner both the compensating
the
coil
loading along the antenna and the magnitude of
inductance,
antenna admittance, are obtained. can be
used, a
which
result
in
approximately real
Of several optimization methods that
slight modification
of
the simplex method was adopted
and found satisfactory. Of particular interest stant conductance matching
as
in
practice
is
that
the
antenna has as con-
possible, in order that the broadband transformer
can be applied.
Therefore the optimization function given in
eqn.(8.48) can be modified to require also that the antenna conductance be
as
close as
possible
to
the average conductance in
the
frequency
range [f ,f ]. 1 2 Although the method described above can be applied to any type of the loading
along
the antenna, and indeed to
any
type of antenna (not ne-
cessarily cylindrical), in practice it is restricted to cases which can more or less easily be difficult
realized.
At microwave frequencies it is very
to
obtain a pure concentrated or distributed inductive load-
ing, as well
as pure concentrated resistive loading or distributed ca-
pacitive loading. citive loading
On
the
other hand, we
know
that concentrated capa-
can easily be obtained, and that it is not difficult to
Sect.8.5.
Optimization by varying mixed loadings
realize
distributed
a
therefore
adopted
resistive
loading.
similar to that shown
An
in
217
experimental model was
Fig.5.24.
It consisted of
a row of cylindrical resistors with metallic endings between which teflon discs were inserted to obtain lumped capacitors. was
made
length.
of brass, i.e., having practically Contrary
antenna was
to
the model
made as
small ·hole was
zero
The first segment
resistance
described in Section 5. 4,
a self-supporting structure.
To
per
unit
the
present
achieve
this, a
drilled along the axis of all the antenna elements, and
the antenna parts were
held together by means of a stretched nylon fi-
lament passing through the hole.
By a simple device the nylon filament
could be relaxed, the antenna dismounted, and then assembled again from new desired elements. As a specific example, consider a tenna of
cylindrical RC-loaded monopole an-
diameter 2a=O. 7 em, made of four segments (i.e., three lumped
capacitors), driven by a coaxial line of the inner diameter of the outer conductor 2b=1.38 em. considered ment
and
as
The values of the three lumped capacitors were
variables, as well as the length of the first brass seg-
the values of
the continuous resistive loadings of the other
three segments, each of the length 4.8 em.
Available resistance values
of
200
the
1040,
resistive 2080,
segments were
4170 and
50,
100,
and
400 rl,
8330 s:J/m, respectively, which were
i.e., about incorporated
into the optimization process as the only possible resistance values. Numerical optimization was outlined in Section 5.4.
based on
the method of antenna analysis
It resulted in the length of the first, brass
segment equal to 3.11 em, and in resistances of the other three segments of
100,
200
and 400 s:J, respectively.
The optimal capacitances of the
concentrated capacitors were found to be approximately 1.43 pF, 0.65 pF and 0.4 pF, counted Finally,
the
from
the excitation
optimal value
of
the
found to be approximately Lc =15 nH. the
form
of
few
turns of very
zone
small metallic
coaxial-line
opening,
inductance was
thin wire (of diameter 0. 09 mm), wound
contacts were made between
coil
The compensating coil was made in
on a short styrofoam cylinder of radius 4 mm. very
towards the antenna end.
compensating
the
and
antenna
At both ends of the coil the coil was fixed at the
rod and
the outer coaxial-
218
Ch.B.
Optimization of antenna admittance
(b)
(a)
FIG.8.16. (a) Sketch of monopole-antenna excitation zone with compensating element. (b) Photograph of assembled RC-loaded antenna >vith compensating element.
line
conductor
(Fig.8.16).
By a
rough calculation it was
found that
the coil should have approximately 2 turns of the wire, but the accuracy of made
this and
result was
quite
doubtful.
Therefore several
coils were
the optimal one, having approximately 1. 75 turns,
was
deter-
mined experimentally. Shown
in Fig. 8.17
are
(G) and susceptance (B), also
shown
3 turns)
the computed and measured antenna conductance versus
frequency.
For comparison, curves are
for under-compensation (too high value of Lc, approximately
and
over-compensation (too small value of Lc, approximately 1
turn), as well as the measured results without compensation. broadband properties served, comparable a log-periodic
to
dipole
radiation patterns of have
the
of
the
optimally
compensated antenna
Excellent can be ob-
those of a
much more complicated structure like 89 antenna with seven elements. Fig.8.18 shows the antenna at
three frequencies.
The patterns
expected shapes, typical for travelling-wave cylindrical wire
antennas, and are quite stable in a wide frequency range. By comparing theoretical and
experimental
susceptance
curves
it
is
Sect.8.5.
219
Optimization by varying mixed loadings
G,B {mS) G
;a • ~·~·;..!.J'
ooo o
i,.A..
.....__ +
0
B
+
+
+
+
&
:a_
---
+
+
a
+
•• • • 0
-5 -10
0
0
000 0
0
0
0
0
0
0
0
f {GHz)
3
0
0
0
FIG. 8. 17. Conductance (G) , susceptance (B) and cornpensa ted susceptance (Be) of the RC-loaded cylindrical monopole antenna versus frequency; a=3.5 rnrn, h=17.75 ern;----- theoretical;+++ experimental, without compensation; • • • experimental, optimally compensated; o o o experimental, under-compensated; o o o experimental, over-compensated.
obvious
that
the compensating coil susceptance does not vary with fre-
quency exactly element tual
as (-1/wLc).
the measured G-curves
frequency
Also, in all are
the cases with compensating
affected as well.
Fortunately, ac-
behaviour of the coil appears to be more favourable for
the present purpose than the theoretical one.
z
1.1 GHz
z
1.8 GHz
z
2.7 GHz
FIG.8.18. The optimal RC-loaded dipole-antenna radiation pattern in electric-field strength; o o o experimental; -----theoretical.
220
Ch.8.
For convenience, Fig. 8.17
in a
Table
8. 3
different
summarizes
form.
also was
of
conductance G in
be
noted
over
80%
that in
to
to
the
results shown in
the
ratios
and
computed corresponding to the average
theoretically obtained efficiency of
It should the antenna
If a higher antenna effi-
optimization function can, of course, be modi-
include efficiency as
achieve
the
frequency range considered.
the whole frequency range.
ciency is required, fied
some of
The voltage standing-wave
the reflection coefficients were values
Optimization of antenna admittance
a parameter
performances comparable
above, a smaller frequency range
to
than
to
be optimized.
those of
the
However,
antenna described
in the present example should be
adopted.
TABLE 8.3. Theoretical and experimental average (arithmetic mean) parameters of RC-loaded cylindrical monopole antennas.
Frequency range (GHz) Average (reference) admittance (mS)
Without compensation theory experiment
Optimal compensation theory experiment
1.1-2. 7
1.1-2.7
1. 2-2.6
1.1-2.7
11. 04+j6. 84 11. 2 9+j 5 . 25 11.04-j0.52 12. 07-j0.51
Average reflection coefficient (%)
30.0
23.1
6. 74
3.94
Average VSWR
1.86
1.60
1. 14
1. 08
8.6. In
OPTIHIZATION OF ADHITTANCE BY MODIFICATION OF ANTENNA SHAPE the
case
of
antennas of fixed geometry, the
their properties loadings. ried by
in
the
is
to
Although
the
load
size.
only means of varying
them with distributed and/or concentrated
antenna parameters
a relatively wide range, they are antenna
90 91 •
For example,
can
in this manner
be va-
rather limited, essentially
we have
seen
in Subsection 8. 4. 2
that if we wish to make a capacitively loaded cylindrical antenna broadband by loading
it with lumped loadings, the
lower
limit of the fre-
quency band is determined basically by the antenna length. This section
is
devoted
to
synthesis of antenna admittance by modi-
fication of the antenna shape, instead of by varying the loadings along
Optimization by modification of antenna shape
Sect.8.6. it.
221
Although synthesis of antennas with variable both shape and load-
ings is possible in principle, only perfectly conducting unloaded structures will be considered.
This will be done because analysis of a sin-
gle general
be
case
tends to
quite lengthy, so that synthesis of such
structures is rather uneconomical from the computer-time point of view. Conceptually, however, it
is
a relatively
simple matter to synthesize
such general wire-antenna structures. Since to
the
general principles of
their admittance
mention
some
have
already been
of
antennas with respect
explained, here we
shall only
details relevant to the examples presented in the follow-
ing subsections.
These examples
that they will
serve
the
were
realized,
are
fairly simple, but it is believed
purpose of demonstrating usefulness of nume-
rical antenna synthesis in they all
synthesis
the their
case of variable antenna shape, because properties measured and
compared with
theoretically predicted properties. In
all
the
cases,
monopole antennas driven by
a coaxial
a=3 mm and b/a=2. 3 (i.e., Zc =50 f:l) were considered. structure wires was equal cally synthesized about 1 mm. differed
optimal
The
to a, i.e., 3 mm. antennas were
experimental models,
somewhat
from
the
line
with
The radius of all
The lengths of theoreti-
determined with
however,
optimal antennas.
for
accuracy of
practical
In these
cases
reasons the ex-
perimental model was analyzed theoretically, and these results compared with experimental results. For analysis, equation were proximation
used,
to
approximation
either
the Hallen-type
equation or
two-potential
with magnetic-current frill or belt-generator ap-
coaxial-line excitation, and with for
the
current distribution.
piecewise polynomial
In synthesizing broadband an-
tennas, the reference admittance was assumed in the form
G.l
ycref i.e., in
the
nf frequencies all
cases
(8.49)
form of the arithmetic mean of the antenna conductance at in
the
range
considered, if not
(where applicable),
the
modulus
of
stated otherwise. the
In
reflection coeffi-
222
Ch.B.
IRI,
cient,
Optimization of antenna admittance
given in eqn.(8.1), was computed as a function of frequency
and used for forming the optimization function. In all the examples presented in this section the optimization parameters were taken to be the rectangular coordinates of the antenna nodes, because
they are
the
simplest parameters which can define the antenna
geometry in the general case. Concerning
the
optimization method, a combination of essentially two
different techniques was
found to
be most suitable in the majority of
cases.
At the very beginning of synthesis, when almost nothing is known
about
the
function
behaviour,
it
seemed
convenient
to
apply several
steps of random or interactive search in the whole region of mization parameters, in order antenna properties.
to
the
opti-
gain some insight into the realizable
The best point in the parameter space out of these
was then adopted as the starting point, and an optimization method used for
determining
concluded
that
the
local
optimum.
the simplex algorithm,
By extensive comparisons it was 80 with minor modifications, out-
lined in Appendix 7, appeared to be the most suitable in almost all examples.
It was found to be sometimes far superior to other methods ex-
amined, such as
coordinate search, pattern search and some variants of
steepest descent. In order to provide realizability of the antenna, to prevent possible crossings
of
the
antenna marginal the form these
of
for
dimensions,
leads
analysis
conducting plane, transitive
segments
during
certain
simple inequalities.
inequalities
method
large
wire
in
or
such
positive
to
fails are a
an
way that
value whenever
constraints were
to limit the
introduced, in
Since in some cases the violation of impossible
(e.g. , too
optimization and
wire
antenna
structure,
segments penetrating
or the
into
the
short) , these inequalities were made inthe a
optimization function was set to a
constraint was
violated.
Thus
the
simplex was forced to contract back into the admissible region. At
the
beginning,
structure (by an
it
is necessary
educated
guess,
to
specify
a
convenient initial
or based on previous knowledge), and
to specify the desired properties of the final, optimal structure. initial structure is determined by
the
The
number of wire segments and the
Optimization by modification of antenna shape
Sect.B.6. way
they are interconnected.
work, trying
to
optimize
The computer
the
takes
given structure.
223
over the rest of the
Naturally, there is no
guarantee in advance that the proposed structure can fulfil the requirements, nor there is a general method for estimating in advance the characteristics
which can
be
obtained from a structure.
As usual, a good
initial guess can sometimes be essential for obtaining satisfactory results, since the optimization function is often multimodal. 8.6.1.
Synthesis
of
broadband
folded
monopole antenna.
folded monopole antenna sketched in Fig.8.19. h of the monopole and the distance variable.
d
Consider a
Vie assume that the height
between the two monopole arms are
The aim is to synthesize the antenna so that it be optimally
matched (in
the
described sense) to
the
reference admittance given in
eqn.(8.49) between f =1.0 GHz and f =1.2 GHz. 1 2 Since nf=2. of
the
the
frequency range
No random search was
They were
parameters.
quarter-wavelength) after
only 5
computations,
is
and
relatively narrow, it was adopted that
used in this case to obtain initial values adopted
d=20 mm.
to
Using
iterations,
which amounted
an
antenna was
optimal
z
be
the to
h=75 mm
(approximately
two-potential
equation,
12 optimization function
obtained with h=62 mm and
d=21
d
5
Sketch of a folded monoFIG.8.19. pole antenna. Larger numbers indicate nodes, and smaller the segments. The length of the first segment is given in millimeters, a=3 mm and b/a=2.3. X
224
Ch.B.
Optimization of antenna admittance
G,B (mS)
I Rl
8
0.4
7
0.3
6 5
0.2
4 3 2
0.1 f ( GHz)
0
.9
1.1
1.0
1. 2
1.3
FIG.8.20. Conductance (G), susceptance (B) and modulus of the reflection coefficient R with respect to Ycref=6. 1 mS, for the folded monopole antenna in Fig.8.19, versus frequency; a=3 rnrn, h=61.5 rnrn, d=20.4 mm; ----- theory; oo, ~~. •• experiment.
(I I)
rnrn.
Modulus
of
be about 0.19,
the reflection coefficient at with
respect to
The experimental model 20.4 rnrn.
was
the
somewhat
f
and f was found to 1 2 reference admittance Ycref=6.1 mS. different,
with h=61.5 rnrn and d=
Theoretical and experimental conductance and
that antenna,
as well
as
the
modulus
of
the
susceptance of
reflection
coefficient
(with respect to Ycref=6.1 mS), are shown in Fig.8.20. 8.6.2. 92 ments.
Synthesis Already
of
for
broadband monopole antenna with some
time
it has
parasitic ele-
been known that by adding two
parasitic elements at a small distance from and parallel to a cylindrical monopole antenna near resonance a relatively good broadband antenna 93 94 could be obtained. • The synthesis problem of determining the optimal dimensions of such an antenna by an optimization procedure has not, however, ing
an
been
considered.
optimal monopole
The present subsection is aimed at describantenna with
two symmetrical, closely-spaced
parasitic elements with respect to the monopole admittance.
Sect.8.6.
Optimization by modification of antenna shape
225
z
FIG.8.21. Sketch of antenna with two identical parasitic elements. (a) Coaxial-line feed; (b) belt-generator feed of equivalent dipole. (Ref. 92)
(a)
Consider
(b)
the monopole antenna driven by a coaxial line and with two
identical, symmetrically positioned parasitic elements, 8.2l(a).
The
equivalent
dipole antenna with
two
shown
parasitic
in Fig.
elements,
driven by a belt generator, is shown in Fig.8.2l(b). The Hallen-type
simultaneous integral
equations
for
currents
r 1 (z)
r (z) along the driven and the parasitic dipole elements have the
and
2 following form:
h2
hl
I
r (z') G
11
1
(z,z') dz'
+
I
-hl
-h2
hl
h2
r (z') G 2
12
(z,z') dz'
F (z) g (8.50)
f -hl
1 (z') 1
c21 (z,z')
dz'
+
f -h2
I (z') G (z,z 1 ) dz' 2 22
The kernels Gmn(z,z') are known functions, scribing tions was
the
belt-generator excitation.
0.
and F (z) is a function deg This system of integral equa-
approximately solved by assuming current distribution in the
form of polynomials with point-matching method.
On
unknown complex coefficients and applying the the
driven element the current was approxi-
226
Ch.B.
mated by along
two
the
polynomials (one
rest
of
Optimization of antenna admittance
along
the antenna),
the belt generator, and the other
with
constraints
that
values of the
r (h )
polynomials and their first derivatives at z=c be equal, and that =0.
1
1
Along the parasitic elements it '"as adopted simply that (8.51)
because
the
parasites are electrically short.
A higher-order approxi-
mation for current distribution along the parasitic elements was
found
to be unnecessary. Of particular interest tenna with quency
range.
To
frequency range ric mean
in
the present case was
approximately real
value
that
the
aim,
and
to synthesize an an-
constant admittance in a given fre-
first for nf frequencies in
antenna conductances were
the
desired
computed, their geomet-
determined, and that value used as the reference admit-
tance, nf y
ere£
=
[
nc)
1/nf (8.52)
i
.1
J.=1
The moduli IRil, i=1,2, ... ,nf, of the reflection coefficients were then found at
the
nf frequencies
and
the
The
mean value of
rmean which
with
corresponding voltage
=
r~)
[:£
served
the
as
tends to max(r.) l.
the
respect to the reference admittance,
standing-wave
ratios,
ri ,
calculated.
voltage standing-wave ratio was then defined as
1/m (8 53) 0
optimization function, with m=8.
Note that r
mean
when m-+oo.
79 was performed by the pattern search in ) . mean ml.n the plane of the variables d and h , with a =a =a=0.3 em and h =7.5 em 1 2 2 1 kept constant, for n£=3, with fi=[1+(i-1)•0.1] GHz, i=1,2,3. The search Determination of (r
was programmed to terminate when less than 1.6 mm and
in h
d=l.9 em and h =4.7 em, 2
simultaneously the step size in d was
less than 2.5 mm. Optimization resulted ;in 2 with (rmean)min=l.09, with respect to Ycref=
Sect.B.6.
Optimization by modification of antenna shape
21.9 mS.
227
(If a wider frequency range is required, however, VSWR cannot
be kept so low.) The elements of
the
antenna considered are relatively thick with re-
spect to their lengths and distances between them. present case (d/a)"'6, (h/a),25 rections
to
account for
For example, in the
(h /a)"'15. Therefore certain cor2 end and proximity effects were considered
the
and
to be necessary when comparing theoretical and experimental results. Concerning
the end effect,
the simplest correction was
used, by a-
dopting the experimental antenna length to be for a/2 shorter than that of the theoretical antenna (see Subsection 1.3.2). It was more difficult correction.
the driven and phase.
to
decide on the kind of the proximity-effect
Preliminary theoretical results indicated the
Therefore
distance between
parasitic elements the
the
quasi-static conductors
currents in
be approximately opposite in
approximation
of
amounted to taking somewhat larger
to
the
for
the
equivalent
a t\vo-wire line was adopted, which d
in the theoretical model.
In the experimental model, both the driven and the parasitic elements were made into of
the
the
of several cylindrical pieces other.
elements
mounted onto
In in
thin
this steps
of
radius a=3 mm screwed
one
manner it was possible to change the lengths of
llh=0.5 mm.
strips which
could
The
slide
parasitic along
elements were
a radial slot made
in the ground plane. The synthesized antenna was realized and checked experimentally. results
are
shown in Fig.8.22.
experimental results sults are and
also
Good agreement between theoretical and
can be observed.
plotted
for
the
For comparison, theoretical re-
antenna without correction of the end
proximity effects, showing worse
than
The
agreement with experimental data
those with corrected effects, as well
as for the antenna without
parasitic elements. The
radiation
pattern
of
the
antenna was
found
to be practically
identical with that of a half-wave dipole in the .vhole frequency range, as expected.
Thus, a madera tely broadband antenna
tance and the radiation pattern was obtained.
in both
the
admit-
228
Ch.B.
Optimization of antenna admittance
G,B (mS) 28
24
FIG.8.22. Conductance (G) and susceptance (B) of the monopole antenna shown in Fig.8.21(a), with a1=a2= 0.3 ern. ----- experimental, h 1=7.35 ern, h2=4.55 ern, d=2 ern;- - - computed, h1 = 7.5crn, h2=4.7cm, d=l.9 ern;- • -computed, h 1 =7.35 f hz=4.55 ern, d=2 em; {GHz) ern, ••••• computed, h1=7.5 ern, h2=0 (antenna without parasitic elements). (Ref.92)
8
4
-4
·.·.......... ·· .··
-8 -12
-16
Several other similar cases were synthesized theoretically, and checked experimentally, all of them showing the same degree of agreement between theory and experiment. 8.6. 3.
Synthesis of
frequencies. which
is matched
to
close frequencies.
branches.
situations
feeder
two
As
an
at
to
feeder
an antenna
at
two
is needed
or more arbitrary, relatively
optimizing an antenna with
different lengths.
shown in Fig.8.23, branches.
its
antenna matched
practical
There are many possibilities for solving that prob-
lem, for example by ments of
cactus-like
In various
in
the
example,
The
authors
form of we
shall
a
several parasitic ele-
considered also
the antenna
saguaro-cactus with two or three consider
the
structure with
two
Sect.8.6.
Optimization by modification of antenna shape
229
FIG.8.23. Sketch of cactuslike antennas with (a) two branches, and (b) three branches.
X
(a)
(b)
The antenna the
optimization parameters were
two branches,
20 mm.
It was
coaxial-line
with
c
required
feeder
the lengths h and h of 1 2 and d arbitrarily adopted to be equal, c=d=
that
the antenna be optimally matched
to a
of characteristic impedance Zc =50 11 at frequencies
G,B {mS) 30
20- G
FIG.8.24. Conductance (G) and susceptance (B) of the antenna sketched in Fig. 8.23(a), versus frequency; hl=90 mm, h2=144.2 mm, c=21 mm, d=22.5 mm; - - theory; oo, •• experiment.
G
f {GHz)
-10
230
Ch.B.
The
initial
configuration for
Optimization of antenna admittance
simplex optimization was adopted with
h =h =100 mm. After 11 iterations, i.e., 26 computations of the opti1 2 mization function, using the two-potential equation, optimal antenna was
obtained with h =89 mm and h =142 nnn. Modulus of 1 2 both frequencies wa~ found to be 0.22.
the
coefficient at tal model
differed somewhat
from
reflection
The experimen-
the optimal theoretical antenna, its
dimensions being h =90 mm, h =144.2 mm, c=21 mm and d=22.5 nnn. Theo1 2 retical and experimental results for admittance of this antenna versus frequency are shown in Fig.8.24. 8.6.4.
Synthesis
of
vertical As
pole antenna sketched
in Fig.8.25.
image,
represents
the
monopole
compensating element.
The
an open-circuited
horizontal
two-wire
length appropriately, it should be possible
tively far from the it cannot be theory.
compensating
done
with
its
By choosing
its
compensate the antenna
Of
the
be
segment,
at microwave frequencies it is not simple to do that.
other hand, if
can also
line.
to
vertical mono-
susceptance.
the
this
with susceptance-
element,
but
course,
antenna
last example, consider a
conductor
by a lumped reactive
On
is adopted to be rela-
ground plane and short in terms of the wavelength,
accurately designed on the basis of the transmission-line
Therefore it must be considered as an integral part of the an-
tenna.
z
h
FIG.8.25. Sketch of vertical monopole antenna with a horizontal segment added for compensation of the monopole susceptance.
b
d
X
Sect.8.6.
Optimization by modification of antenna shape
231
If we assume that h=A/4 and a=0.01 A, the monopole without compensation
has
an admittance Yo-(l8-j7) mS.
By compensating
the
susceptance
in this case, an antenna is obtained matched almost perfectly to a SO (20 mS) coaxial line. 1 GHz,
Since we
adopted
a=3 mm,
frequency was
u
set to
d was adopted to be 10 mm, and h and b were considered as opti-
mization parameters.
The initial values of
the simplex optimization process was and b=40 mm
(approximately
which, according
to
the -j7 mS monopole tations of
the
the
parameters, with which
started, were h=7S mm (i.e., A/4)
length of
the
line having susceptance
the electrostatic approximation, susceptance).
would
compensate
After 8 iterations, i.e., 18 compu-
optimization function, the optimal antenna was obtained
having h=82 mm and
b=SS mm.
(Note considerable differenc-e between the
optimal and the initial values of hand b.) coefficient of
the
this
The theoretical reflection
antenna, with respect to a SO
u
line, was found to
be only 0.02 (i.e., the VSWR only 1.04).
G,B (mS) 30
25
20 15
•
•
10 B
5 f (GHz)
0 -5
0.9
1. 15
1.1
FIG.8.26. Conductance (G) and susceptance (B) of the antenna shown in Fig.8.2S, with a=3 mm, h=81.S mm, b=S4.2 mm and d=9.S mm; - - - theory; oo, •• experiment.
232
Ch.B.
The experimental model
Optimization of antenna admittance
differed
having h=81.5 mm, b=54. 2 mm and
somewhat
d=9.5 mm.
from
the
optimal antenna,
The theoretical and experi-
mental results for this antenna are shown in Fig.8.26.
8.7.
CONCLUSIONS
In this chapter some methods of synthesi·s of wire antennas with respect to
the
most
antenna admittance
of
were
explained and illustrated by examples,
which were analysed also
experimentally.
sion might be that wire-antenna synthesis with
The general conclu-
respect to their admit-
tance is a reliable and efficient method of designing such antennas. All
the
examples
of
antennas considered were
such that their radi-
ation pattern was
at least approximately known in advance, because the
length and/or
complexity of
the
the
only cylindrical structures were
structures were not large.
synthesized, possibly with additional
elements which do not influence considerably tern
of
Almost
a single cylindrical antenna.
the
basic radiation pat-
Therefore it was not necessary
to consider the radiation pattern of the antenna as unknown. It was can be
shown
that
efficient optimization of wire-antenna admittance
performed by varying distributed and/or
along it, as well
as
concentrated loadings
by varying the antenna shape and size.
bined, general case of optimization, of varying
The com-
simultaneously the an-
tenna loadings and shape, was not considered, because a large number of optimization parameters is cess
quite
timize
lengthy.
an antenna
then
In principle, however, it
in that
reasoning presented
involved, making the optimization pro-
general
case also,
is
quite simple to op-
following
the lines of
in connection with optimization of the three spe-
cial cases. It was
pointed out
at several places in the chapter
tion of wire antennas is in
the
the
the
choice
very little
the
initial values
can be
suggested
optimiza-
choice of optimization
choice of optimization parameters, of
that
a single-valued process in many respects:
choice of optimization function, in
method, in in
not
of
these
concerning
and,
in particular,
parameters.
It seems that
the
best
possible choice of
Sect.8.7.
Conclusions
233
these quantities, and that the only basis on which numerical synthesis can efficiently be thors
hope
knowledge be
of
that
built
this
is a certain amount of experience.
chapter,
summarizing
the
larger
part
The auof their
and experience in synthesis of wire-antenna structures, may
some help in
that
rely in possibilities of
respect both
to
those who are interested me-
this modern approach
to wire-antenna design,
and to those who have been using the numerical synthesis method for antenna design material for
for
some time.
pre sen ted
In particular, the
in Subsection 8. 4. 2 might
estimating potential broadband
properties
authors
serve as a of
feel that the useful
basis
antennas limited to
a given volume. Frequently a wire antenna has to be designed satisfying as closely as possible certain requirements relating to both its admittance and radiation pattern, or, sometimes, to its radiation pattern only. tion
of wire antennas with
Optimiza-
respect to radiation pattern, and combined
optimization of pattern and admittance, is the topic of the next, last chapter of the monograph.
CHAPTER 9
Optimization of Antenna Radiation Pattern
9.1. The
INTRODUCTION radiation
pattern of
an antenna situated
and bounded by the surface inside
aimed
determining current distribution in
pattern.
The
of current
The
S is uniquely determined by distribution of
currents at
S.
other,
classical methods
more
in a homogeneous medium
difficult
of
pattern
synthesis were
S resulting in a desired
problem, how such a distribution
can be obtained, and whether it can be obtained at all, was
not considered.
In contrast
to
this, the method of pattern synthesis
to be outlined below is always associated with a real antenna structure, which is modified until a radiation
pattern
of
the
structure is ob-
tained which is as close as possible to a desired pattern. As
a
rule, modification of
the antenna structure in order to modify
the radiation pattern influences also the antenna admittance.
In fact,
we
know
that admittance is usually much more sensitive
to variations
of
the antenna shape or loading than radiation pattern.
Therefore op-
timization of the antenna raDiation pattern alone is frequently not advantageous. tern and
Instead,
admittance
simultaneous optimization of seems
to
be
a
better
the
radiation pat-
design approach.
For this
reason, the stress in this chapter will be on simultaneous optimization of admittance and pattern, although some attention will also be paid to optimization of the radiation pattern alone. Concerning
possible
optimization functions when
optimizing simulta-
neously radiation pattern and admittance, they may be of the form
236
Ch.9.
F
w F
a a
where
Optimization of antenna radiation pattern
+ wpF p
(9. 1)
F
and F are convenient optimization functions incorporating a p the antenna admittance and radiation pattern alone, respectively, and
w and w are weighting coefficients. The function Fa can be any of a p the functions used in the preceding chapter (or some other convenient function for admittance optimization). ous forms, some form of
the
of which are
optimization
The function F
p briefly described below.
function
F
can also
can be of variOf course, the
be different from that
given in eqn.(9.1). A frequent requirement
on
the radiation pattern is that
gain be maximal possible in a given direction.
the
antenna
The optimization func-
tion in that case could be any function which decreases with increasing antenna directive gain, structures).
(or power gain in the case of lossy antenna
A simple choice of the optimization function could be
Alternatively, it rection be as
gd
can be required that the antenna gain in a given di-:
small
as possible,
which
a certain direction the antenna radiation
amounts to postulating that in pattern has a null.
The op-
timization function in that case might be (9. 3) In
some
engineering applications
we
can require
that
the antenna di-
rective gain in certain directions be equal or larger than a prescribed value, while in other directions be smaller than a given value. antenna properties
are
of interest in a certain
range
If the
of frequencies,
the optimization function can be of the form
(9. 4)
D ..
lJ
where nf is the number of discrete frequencies in the range considered, and nd fied.
is the number of directions in which the antenna gain is speciD .. should be
lJ
the antenna directive
positive functions gain
in
the
which
rapidly tend to zero i f
specified direction
is
better than
Sect.9.1.
237
Introduction
required, and have
large positive
functions
various
can have
values
forms.
otherwise.
Since
the
Of course, these
antenna gain is usually
specified in decibels, one possible choice of the functions D .. is
lJ
(9.5)
D ..
lJ
where
(9.6) when directive gain in decibels Gdij larger than GdOj is required, and t .. ; A (Gd .. -Gd .) 0J lJ lJ
(9.7)
when directive gain Gdij A is
a constant which
In all used,
the
determines
the
examples presented in this
A was
double
smaller than GdOj
arbitrarily
difference
of
set
the
to
is required.
The quantity
steepness of the D .. functions.
lJ
chapter
in which eqn. (9.4) was
be O.l·ln 10, so that
prescribed and
the
attained
t. . represents lJ directive gain
in the direction j, expressed in nepers. The when
final the
example
of
the optimization function
antenna is required
F is for the case p have a specified shape of the radia-
to
tion pattern in a given plane, at a single frequency. relative intensities of phases. at
a
the
If we
the
far-zone field are of interest, and not the
assume that
the
desired radiation pattern is specified
of
directions, determined by angles
(usually large) number plane
Most often, only
considered, instead of
eqn. (9.4)
the
optimization function
can be chosen in the form
where
IEn (cp.) I 1
is
rection determined sity in ample,
that the
tensity of
I I I £0 n <
the normalized electric-field by
direction.
electric
and The
field
-+
IEOn (cp i) I
intensity in
the
di-
the desired normalized inten-
normalizations
can be normalized
both the prescribed and
tain direction.
(9. 8)
can so
be
various.
For ex-
that the maximal in-
attained fields be unity in a cer-
238
Ch.9.
Of
course,
Optimization of antenna radiation pattern
other optimization
functions
can be
constructed,
e.g.,
those which would take care of the polarization of the electric field, phase variations, . . . he max~m~z~ng t
or
Note
that
determination of
simple task as
once
outlined
functions
F
those aimed at . 1 -to-no~se .
in
the
front-to-back ratio,
rat~o.
the
radiation
pattern
is numerically a
the antenna current distribution has been determined, Appendix 3.
usually
p
maximizing . 95
s~gna
Therefore
even
complicated
optimization
do not have too high computer-time requirements,
except if the far-zone field has
to
be computed at a very large number
of directions. It should also be noted that considerable insight into radiation possibilities
of
an antenna structure is needed
optimization function.
before
For example, if a relatively
antenna
is
optimized (shorter
sistive
or
capacitive loading can make
diation
at
an angle of 45° with
deed, at any angle except 0°).
than
constructing the short
cylindrical
about one half-wavelength), no rethe
antenna to have a zero ra-
respect to the antenna axis (and, in-
Therefore, whenever optimizing the pat-
tern of a novel antenna type, it is highly recommendable to analyse the pattern in a number of cases prior to introducing requirements into the optimization function. Finally, should be might
be
in
simultaneous optimization
kept
in mind
that
the
of
admittance
requirements
on
the
and
pattern it
two quantities
to some extent contradictory, without our knowing it.
Appro-
priate choice of the weighting coefficients wa and wp in eqn. (9.1) can be of considerable help in compromizing possible contradictory requirements. In the following sections a number of simple examples of optimization of
the
radiation pattern will be presented.
mization
of
the
driving voltages
of
These cases include opti-
antenna-array
elements,
of
the
loading impedances, or of the antenna shape and dimensions, in order to obtain a
desired radiation pattern, or
and the front-to-back ratio.
to
maximize the directive gain
Sect.9.2. 9.2.
Optimization by varying driving voltages
239
OPTIMIZATION OF RADIATION PATTERN BY VARYING DRIVING VOLTAGES OF ANTENNA-ARRAY ELEHENTS
The
simplest
case
driving voltages
of
of
radiation
these
papers
using
methods
techniques
can
are
optimization
is
to determine
antenna-array 'elements in order to obtain a speci-
fied radiation pattern. many
pattern
This problem has
the be
classical array found
been extensively
design
in References
based on methods
96
techniques. and
97.
treated
in
Review of
Most of these
of
linear algebra, usually involving 98 matrix calculus. In some more recently published papers iterative or . . . hd99,100,101 d H l . f . . opt~m~zat~on met o s are use . owever, on y ~n ew ex~st~ng papers mutual coupling between the array elements is correctly taken 102 into account. In this section we shall present a simple example of determining the driving voltages of array elements for given dimensions and positions
of
these
elements,
taking
rigorously
into
account the
coupling between the antennas. Consider
a
circular array
of
three
half-wavelength monopole antennas,
of
identical radius
circle of radius 0.25 A, as sketched in Fig.9.1. field radiation
pattern
of
the
equidistantly spaced
0.01 A,
located along
a
The desired electric-
array in the horizontal (xOy) plane is
shown in Fig.9.2 by the dashed line. For at
the
angles
purpose
of
optimization, the radiation pattern was
cj>i = (i-l)n/6,
i=1,2, ... ,12,
and
the
sampled
optimization function
FIG.9.1. Sketch of a circular array of three halfwavelength monopole antennas, located above a perfectly conducting ground plane.
240
Ch.9.
Optimization of antenna radiation pattern
y
FIG.9.2. Specified radiation pattern in electric-field strength of the array X sketched in Fig.9.1 (dashed line) and optimized pattern (solid line).
4>
(Fp) , given in eqn. (9. 8), was adopted. Both the desired and attained 4 radiation patterns were normalized so that C3TI/2) =1. In order to
IEn
expedite evaluation
of
the
radiation
pattern,
I
the
far-zone
electric
field at the angles ct>i was expressed in the form
(9. 9)
are
where n=2, V.
J
the antenna driving voltages,
and
T.(<j>.) are trans]
~
These coefficients can be determined knowing the
mission coefficients.
far-zone electric field for three independent driving conditions of the array, in
a
similar manner as were
case of antennas with due
to
symmetry, in
are identical.
determined the Y-parameters in the
concentrated loadings (Section 8.4). the
Note that,
present case many of the coefficients T. (.) J ~ the two-potential equation was used to
In this example
analyse the array. Fixing
the
procedure was
used, to
obtain
v
at v =1 V, the simplex optimization 0 0 the optimal values of the driving volt-
=(0.099- j0.134) V and v =(0.207- j0.874) V. The corresponding 2 1 radiation pattern is shown by solid line in Fig.9.2. Note that the
ages
v
driving voltage
Sect.9.3. largest
Optimization by varying antenna loadings
difference
between
the
from
adopted minimax optimization.
the that
it
is
occurs
specified
of
tes
about
1. 3 dB,
pattern,
at
the
optimal
radiation
several angles
in general very difficult
specified radiation pattern if we
and
241
to
approach an arbitrarily
have only a small number of variable
parameters.
9.3.
OPTIMIZATION OF RADIATION PATTERN BY VARYING ANTENNA LOADINGS
Antenna loadings
can
in
diation pattern,
and
can
some thus
cases have be
of
the antenna radiation properties, as 103 104 Mautz. • For practical applications optimize
also
the feeder.
a large influence on the ra-
efficiently used proposed it
the antenna admittance in order
for
optimization
by Harrington and
is usually desirable to to match the antenna to
In this section we shall present an example of simultaneous
optimization of the radiation pattern and the admittance of an Uda-Yagi array. Consider array,
an
electrically-small,
sketched
inductive loading,
in
loadings, of
Fig.9.3. of
admittance
three-element,
The dipoles
are
symmetrical Uda-Yagi
loaded
by concentrated
admittances Y , Y and Y , and an additional 3 1 2 Y , is provided in parallel with the antenna 0
y
FIG.9.3. Sketch of loaded Uda-Yagi array. All dimensions are in millimeters. Wire diameter is 22 mm.
242
Ch.9.
input
terminals
in order
array
is
designed
to
be
Optimization of antenna radiation pattern
to compensate for
a
the
antenna susceptance.
narrow-band operation at
The
a frequency
f=27 MHz (at which the dipole lengths are only about A/8) with the following
requirements:
(l)
the antenna forward
should be as large as possible, gdb,
should
matched
to
(2)
be as small as possible, and
gain,
gd f'
(3)
the array should be well
a reference admittance Ycref=5 mS (so that it can easily be
matched to a 20 mS coaxial line by means synthesis was loading
directive
the array backward directive gain,
performed
positions),
for
by varying
The optimization function coefficient, R, of
fixed
the
was
of a 4:1 balun).
antenna
dimensions
the magnitudes of
The antenna
(including
the
the
loadings only.
constructed so to combine the reflection
antenna (together with
the compensating admit-
tance Y ) with respect to Ycref' and the optimization functions 0 and (Fp) , given by eqns. (9.2) and (9.3), in a quasi-minimax way, 2
(9. 10) with arbitrarily adopted weighting
coefficients (100, 2
and 5).
The
admittance Y was always chosen so that the antenna admittance be real. 0 The antenna dimensions being fixed, the Y-parameters describing the antenna with respect eqn.(8.43),
can
zone electric
to
its input
and
loading terminals, as given in
be found as outlined in Section 8.4.
field
can
be expressed in
terms
The antenna far-
of the transmission T-
given in eqn. (9.9), where now n=3, v is the antenna 0 driving voltage, and V., j=1,2,3, are the voltages across the loading J terminals. The T-parameters can easily be evaluated in a similar man-
coefficients as
ner as the Y-parameters. The Y- and T-parameters known, the loading voltages and the input admittance
can be evaluated from eqns. (8.45), and
field can be computed from eqn.(9.9). tion can be determined very rapidly.
the
far-zone electric
Therefore the optimization funcThe simplex optimization procedure
yielded the optimal loadings Y =-j1.91 mS, Y =-j2.00 mS, Y =-jl.85 mS 1 2 3 and the compensating admittance Y =-j14.7 mS. For this set of loadings 0 the antenna forward directive gain is gd£=5.2 (i.e., 7.2 dB), the backward directive gain is gdb=0.056 (i.e., -12.5 dB), whence the front-to-
Sect.9.4.
Optimization by varying antenna shape
X
243
Electric-field radiFIG.9.4. ation pattern of the optimal Uda-Yagi array sketched in Fig.9.3 in the xOz, or Hplane ( - - - ) , and in the xOy, orE-plane ( - - -).
back ratio is 19.7 dB, and the modulus of the reflection coefficient is less
than 0. 001.
9. 4.
It
The
array electric-field pattern is plotted in Fig.
should be noted
that
the synthesized array is, actually, al-
most a supergain array, and that it is very sensitive any
dimensions,
loading magnitudes
not unexpected, because
the
and
frequency.
to variations of Of course, this is
array dimensions are relatively small when
compared with the wavelength.
9.4.
OPTIMIZATION OF RADIATION PATTERN BY VARYING ANTENNA SHAPE
90 91 •
Theoretical antenna been
treated
optimization by modifying its shape and size has 90,105-109 in only few papers. In the following subsec-
tions some examples will be presented to illustrate either optimization of
the
radiation pattern only, or simultaneous optimization of radia-
tion pattern and admittance cases optimization function
of wire-antenna of
the
form
given
structures.
In all the
in eqn. (9.1) was used,
with function Fa being the effective reflection coefficient, defined by eqn.(S.S), and the function F =(F ) given by eqns.(9.4)-(9.7). Radiap p 3 tion-pattern optimization was not aimed at synthesizing the whole pattern.
Rather, it was
than some
used either to obtain antenna directivity larger
prescribed value, or
the
best possible front-to-back ratio.
244
Ch.9.
As in other examples care had mittance
to be
of
Optimization of antenna radiation pattern
antenna synthesis presented in this monograph,
be exercised that the requirements on the pattern and adrealistic
and not
contradictory, and
that
the weighting
coefficients w and w be chosen appropriately. a p 9.4.1.
Synthesis
two reflectors.
of
Uda-Yagi arrays with one and two directors and
We shall illustrate synthesis of antennas with respect
to their radiation pattern alone by varying the antenna geometry on two examples
of
small Uda-Yagi arrays.
Such arrays are of interest in the
lower range of very high frequencies (of the order of 100 MHz), because at
these
frequencies
the
whole structure clumsy and
antenna
elements
tend
to
be
large and the
heavy if the antenna has a larger number of
elements. We shall consider an antenna with a driven element, two symmetrically positioned reflectors one more
and
director added.
a single director, and the same antenna with In order to meet the requirements imposed by
the available experimental equipment,
used
to verify
the
theoretical
(Using the theorem of results, both antennas were analysed at 1 GHz. 5 electromagnetic similitude, the antenna can always be scaled to lower or higher frequencies.)
It
was
required that both antennas have maxi-
mal possible directivity and highest possible front-to-back ratio (FBR). In
order
to reduce
the
number of optimization parameters, the size of
the driven element was fixed, at h =70 mm, as well as the distances c = 1 1 c =40 mm of the two reflectors from the antenna plane of symmetry. As 2 the optimization parameters, the heights of all passive elements and their distances from the driven element were considered. The
first
case, of Uda-Yagi
array with
director, is sketched in Fig.9.5.
two
It was adopted that Gd
main beam, in the direction of x-axis), and Gd in
the
opposite
parameters were These values
direction).
The
reflectors and a single
initial
=8 dB* (the 01 =-4 dB (back radiation,
02 values
of
the optimization
h =65 mm, h =h =80 mm and d =d =75 mm (i.e., "A/4). 3 4 1 2 2 based on an educated guess. After 10 simplex itera-
were
*With respect to a semi-isotropic monopole-antenna synthesis.
radiator,
as in
other examples of
Sect.9.4.
Optimization by varying antenna shape
245
z
2
2a
X
I
~-. 4
N
u
u
ty
+----·-··-__ . _ _: I
I
I
I 5
·$-=·FIG.9.5. Sketch of an Uda-Yagi array with two reflectors and a single director; a~3 mm, b/a~2.3.
tions, which amounted tenna was c
1 ~c 2 ~40
mm
antenna (in directive
(prefixed), the
gain
18.7 dB. Y~(42-
32 optimization function computations, an an-
1
h ~70
1
d ~52
direction of in
the
The admittance
j33) mS.
diation
to
obtained with
mm
(prefixed),
mm and
2
d ~77
the x-axis)
2
h ~63
mm.
was
mm,
h
3 ~h 4 ~81
Directivity
of
mm, this
found to be 9 dB, while
opposite direction was -9.7 dB, so that FBR was of
the antenna (which was not optimized) was
The experimental model used for measurement of the ra-
pattern was
practically
the
same
as
the
theoretical, except
that symmetrical antenna was made.
Theoretical and experimental results
for
xOy and xOz planes are shown in Fig.
the
9.6.
radiation pattern in
Theoretical analysis was
tion and n ~4,
2
the
and
performed using the two-potential equa-
polynomial approximation for current distribution, with n ~n ~n ~3.
3 4 5
(Labels
of
the
antenna segments
are
n ~3,
1
given in
246
Ch.9.
Optimization of antenna radiation pattern
X
FIG.9.6. Radiation pattern in electric-field strength of symmetrical equivalent of the optimal Uda-Yagi array sketched in Fig.9.5. The dimensions of the array are given in the text, and frequency f=l GHz. ----- theory; o o o experiment.
Fig.9.5.) Shown in Fig.9. 7 is a sketch of an Uda-Yagi array with two directors and
two
reflectors.
higher directivity Fig. 9. 5.
The
It was than
expected that this structure would have a
the antenna with a single director,
same values of GdO
ceding example.
and Gd0
shown
in
were adopted as in the pre-
1 2 The initial values of the optimization parameters were
The optimal antenna was obtained by the simplex method, which resulted in
h =70 mm (pre1 fixed), h =68 mm, h =60 mm, h =h =85 mm, d =37 mm, d =113 mm, d =80 mm 2 1 4 5 2 3 3 and c =c =40 mm (prefixed). Directivity of the antenna (in the direc1 2 tion of the x-axis) was 9.9 dB, and the front-to-back ratio was about 24 dB,
the
following dimensions of the optimal antenna:
which
is
somewhat
better
than
in
the
preceding example.
The
experimental pattern was obtained on a model practically identical with the
optimal
Fig.9.8,
antenna (except
together
with
the
that
it was symmetrical), and is shown in
theoretical
pattern.
The degrees of the
polynomial approximation of current used were n =3, n =4, n =n =n =n =3. 2 1 3 4 5 6 The theoretical admittance of the optimal antenna sketched in Fig.9. 7 was found to be Y=(33- j43) mS.
Sect.9.4.
247
Optimization by varying antenna shape
u
FIG. 9. 7. Sketch of an Uda-Yagi array with two reflectors and rectors; a=3 mm, b/a=2.3.
y
two
di-
z
X
FIG.9.8. Radiation pattern in electric-field strength of symmetrical equivalent of the optimal Uda-Yagi array sketched in Fig.9. 7. The dimensions of the array are given in the text, and frequency f~l GHz. ----- theory; o o o experiment.
248
Ch.9.
9. 4. 2.
Synthesis
ample, consider
of
the
Optimization of antenna radiation pattern
inclined monopole
antenna.
following design task:
it
As
the
next
ex-
is necessary to synthe-
size a single monopole antenna which is well matched to a 20 mS coaxial feeder at rection
0. 975 GHz
in
fulfil
the
these
and has directive gain of at least 5 dB in one di-
horizontal
plane.
requirements
seemed
The simplest structure which to
be
could
an inclined monopole antenna.
Such an antenna, sketched in Fig.9.9, was therefore considered and optimized. The length of mm,
and
the vertical segment of
the coordinates d
optimization parameters.
and The
the
antenna was fixed at c=20
h of the antenna end t.ere considered as weighting
coefficients in
the
optimiza-
tion function tance was in
this
value
F were adopted to be equal, w =w . The reference admita p adopted to be Ycref=20 mS, and the optimization function Fa
case is simply equal to
of GdOl
was
I Rl, because in eqn. (8.5) nf=l.
adopted to be GdOl =5 dB
in
The
the direction of the x-
axis. It was should
be
fairly obvious "A/4,
3>../4,
that
etc.
the initial total length of the monopole
With monopole
z
length of
A/4, the simplex
d
2
X
2b FIG.9.9.
Sketch of an inclined monopole antenna; a=3 mm, b/a=2.3.
Sect.9.4.
Optimization by varying antenna shape
optimization
resulted
ficient directive
in a
249
practically vertical monopole
with
insuf-
in the x-axis direction of Gd =2.2 dB. For the 1 initial monopole length of 3A./4, after about 50 evaluations of the optimization
gain
function
the
simplex optimization
with h=205 mm and d=98 mm. and sufficient directive
in
an
antenna
Gd =6.27 dB. 1 and experimental results for the antenna admittance versus
Theoretical frequency
resulted
The optimal antenna had VSWR equal to 1.09,
are
gain
in the x-axis direction of
presented in Fig. 9.10.
Theoretical and experimental re-
sults are slightly shifted due to the end effect, which was not included
in
the
was 1.06
theoretical model.
at
Minimal experimental value of
the
VSWR
0.96 GHz, which agrees well with the result of synthesis.
The degrees of
the
polynomial approximation
for
current were n =3 and 1
n =6.
2
9.4.3. element.
Synthesis of Uda-Yagi array with folded monopole as a driven Consider next
reflector and Fig.9.11.
a
It was
folded
the
Uda-Yagi
monopole as
array with
the
driven
two
directors, one
element,
sketched
in
required for the array to have directive gain in the
G,B (mS)
20 18 16 14
FIG.9.10. Conductance (G) and susceptance (B) of the optimal inclined monopole antenna sketched in Fig.9.9, with a=3 mm, b/a=2.3, h=205 mm and d=98 mm. theory; oo, •• experiment.
12 10
8 6
4
2
f ( GHz)
250
Ch.9.
Optimization of antenna radiation pattern
z
dz
d3 -r-
··-
I \.0
..c
7
d1
I
~
..c
I
t-
21 1
2b~
..-
.-
2a ..CN
·~mm
s
I
M
..c
I
.
~r///~
6
I
X
'
FIG.9.11. Sketch of an Uda-Yagi array with two directors, one reflector and folded monopole as the driven element; a=3 mm, b/a=2.3.
forward direction (x-direction) of at least Gdf=9 dB, that in the backward direction of at most Gdb=-4 dB, and to have approximately real admittance at f=1 GHz.
The dimensions
of
the active element were
fixed
at h =70 mm and c=20 mm. The positions and lengths of the passive 1 elements were considered as optimization parameters. The weighting coefficients
in
the
optimization
function
(9.1)
were
adopted
to
be
w =10 and w =1,with Y f=Re(Y), Gd =9 dB (in the direction of x-axis) a p ere 01 and Gd =-4 dB (in the opposite direction). 02 Using
the
simplex method,
the
optimal
structure was
obtained with
lengths h =h =64 mm, reflector length h =84 mm, and director 2 3 4 and reflector positions d =64 mm, d =126 mm and d =81 mm. The forward 1 2 3 directive gain was 9.5 dB, and FBR was 19 dB. The antenna admittance
director
was Y=(7.9+j0.0) mS. The experimental model
dimensions
for
admittance measurements were
the following: h =69.2 mm, h =63.2 mm, h =62.9 mm, h =84.2 mm, c=21.2 1 2 4 3 mm, d =64.2 mm, d =126.4 mm and d =81.2 mm. The theoretical and expe1 2 3
Sect.9.4.
10
Optimization by varying antenna shape
251
G,B (mS) 0
FIG.9.12. Conductance (G) and susceptance (B) of approximately optimal Uda-Yagi array sketched in Fig.9.11; a=3 mm, b/a=2.3, the other dimensions are given in the f text. theory; oo, •• expe( GHz) riment.
-6
rimental
• antenna conductance
and
susceptance,
versus
frequency,
are
shown in Fig.9.12. The experimental model
for
pattern measurements (the symmetric equi-
X
FIG.9.13. Radiation pattern in electric-field strength of symmetrical equivalent of the optimal Uda-Yagi array sketched in Fig. 9.11, with a=3 mm, b/a=2.3 and other dimensions given in the text. - - - t h e ory; o o o experiment.
252
Ch.9.
valent of with
the
the antenna
Optimization of antenna radiation pattern
sketched
optimal antenna.
in Fig.9.11) was practically identical
The theoretical
and experimental radiation
patterns in the xOy and xOz planes are shown in Fig.9.13. 9.4.4.
Synthesis
of
moderately
principle, Uda-Yagi arrays sist
of
respect
considered
width was
to
their radiation
is aimed at
in
110
In
pattern at a single frequency
and
107,
and
increase in band-
in Reference 108, but it seems that no attempt
has been made to optimize the Uda-Yagi array so tent broadband
array.
Theoretical optimization of such
in References 105, 106
demonstrated
Uda-Yagi
narrow-band antennas, because they con-
basically resonant elements.
arrays with was
are
broadband
its radiation pattern and
demonstrating
that
that
it be to some ex-
admittance.
This example
such a synthesis is possible, and that
the results are quite acceptable for practical applications. There was matched to a
a
need
20 mS
for
a
rugged,
simple
symmetrical
antenna,
well-
coaxial feeder (with VSWR less than 1.4) in a fre-
quency range from 440 MHz to 4 70 MHz, dB and FBR not less than 15 dB.
having
directivity larger than 9
Since the frequency range was relative-
ly narrow (about 10%), a solution was sought in the form of an Uda-Yagi
d4
d1
~J
I
11")
..<=
6
I I I
li
'
~
2
1
I
I~
·H. ?Q
rnm
~1'~~~
(Y)
3
I I
li
r-rt
r-~
~
2a ..<=
I
I
.-----
I
~I
•
..<=
t,
I I
li
<:T
..<=
5
I
I
li
X
FIG.9.14. Sketch of Uda-Yagi array with three directors, one reflector and simple monopole as driven element; a=3 mm, b/a=2.3.
Sect.9.4. array.
Optimization by varying antenna shape
For
practical
253
re·asons, asymmetrical equivalent of
the
antenna
was considered, and frequency in theoretical analysis and synthesis was doubled, 0.88
as well
as
and 0.94 GHz
in
was
later experiments, so of interest.
that
An array with
the range between three directors, a
single reflector and a driven element in the form of a simple monopole, shown in Fig.9.14, was considered as the initial structure, with a possibility of increasing the number of elements if needed. The initial synthesis was obtain a
performed using the interactive search, to
reasonable starting point for simplex optimization, with nf=3
(f =0.88 GHz, f =0.91 GHz and f =0.94 GHz). The approximately optimal 1 2 3 antenna thus obtained had the following dimensions: h =74.8 mm, h =h = 2 3 1 68 mm, h =65.8 mm, h =80.5 mm, d =43 mm, d =86 mm, d =129 mm and d = 1 2 4 5 4 3 63 mm. The VSWR of the antenna (with respect to Ycref=43 mS) was less
1. 35
than
9. 7 dB,
in
and
the
whole
FBR larger
properties measured. This antenna was
frequency than
range,
9 dB.
its
directivity larger than
This antenna was realized, and its
The results are shown in Fig.9.15. then used as a starting point for further optimiza-
tion, using the two-potential equation and the optimization function in eqn.(9.1), withY
f as given in eqn.(8.49), w =1, w =0.1, Gd =9.5 dB 01 ere a p (in the x-axis direction) and Gd =-0.5 dB (in the opposite direction). 02 The simplex optimization resulted in somewhat better antenna, having the following dimensions: h =74.9 mm, h =h =68.4 mm, h =65.9 mm, h = 2 3 5 4 1 83.3 mm, d =43 mm, d =84 mm, d =134 mm and d =65 mm. The VSWR of this 4 2 1 3 optimal antenna was less than 1.25 (with respect to Ycref=42 mS) in the
whole frequency range, its directivity (in the x-axis direction) larger than
9. 7 dB,
and
FBR larger
than
ll.5 dB.
These
frequency are shown in Fig. 9. 15 in dashed lines.
parameters versus
The degrees of polyno-
mials used for current approximation were n =3, n =4 and n =n =n =n =3. 1 2 3 4 5 6 The the
simple
antenna shown in Fig.9.14 was thus not able to fulfil all
requirements
sired 15 dB).
(its
FBR was
only
Another director was
11.5 dB, as compared with the detherefore
added
to the structure,
and a folded dipole incorporating a A/4 coaxial transformer (serving to 1 obtain a better match) was adopted as the driven element. ll The resulting antenna, shown in Fig.9.16, did satisfy all
the
requirements,
• (a)
(b)
FIG. 9.15. (a) Conductance (G) and susceptance (B), and (b) directivity (D) and FBR of the antenna sketched in Fig.9.14, versus frequency. ----- theory, approximately optimal antenna of dimensions given in the text; oo, •• experiment, approximately optimal antenna;--the optimal antenna, of dimensions given in the text.
but
its
optimization was
done partly theoretically, partly experimen-
tally, because the available computer was not able to synthesize larger structures than approximately that shown in Fig.9.11.
9.5.
CONCLUSIONS
This, last chapter of
the monograph dealt with some general principles
of synthesis of wire-antenna structures with respect to their radiation pattern,
and with
and admittance.
respect
Relatively
to
simultaneously
large
number
of
their
radiation pattern
examples of
optimization
presented as illustrations were given, most of which were checked experimentally.
They indicated
that
the proposed methods
of wire-antenna
Sect.9.5.
255
Conclusions
FIG. 9.16. Photograph of experimental model for optimization of UdaYagi array with four directors, one reflector and a driven folded monopole, which incorporates a matching network. Lengths and positions of parasitic elements are variable.
synthesis may be considered as reliable engineering methods for modern design
of
such antenna structures.
However, it was
also obvious that
considerable insight, or previous knowledge, was necessary for deciding both on initial values of the optimization parameters, and for adopting the desired antenna parameters. tance when
simultaneous
This is, perhaps, of particular impor-
optimization of radiation pattern and
admit-
tance is desired, because requirements on the radiation pattern and admittance can to some extent be contradictory. Obviously, this
and
the preceding shapters presented only some basic
details of the broad field of computer-aided antenna synthesis.
In the
authors' opinion, this novel approach to design of wire antennas is, in many respects, still in a relatively early stage of development.
APPENDIX 1
Notes on Evaluation of Integrals Encountered in Analysis of Wire Structures Assembled from Straight Wire Segments
Consider
a
straight current-carrying wire
axis, as
shown
in Fig .Al.l.
We wish
segment, located on the s-
to determine
the
p-component of
the electric field due to this segment at the field point P in the case when
the
current distribution along
power-series expansion, as
given
the segment
in eqn. (1.16).
is
approximated by a
Substituting this ex-
pansion into eqn.(l.l4) and multiplying both sides of the equation thus obtained by the unit vector
1p ,
we obtain for the p-component
n
E
p
since
-jw)J
L
(Al.l)
m=O
1p •grad=3/3p,
where
p FIG.Al.l. Straight current-carrying wire segment and the field point P.
258
App.l.
Evaluation of integrals (A1. 2)
a is the wire radius and
~
~ +pi --; -
=
p
s1 s
s
p
(A1. 3)
The auxiliary p-coordinate
was
tion involved in eqn.(A1.1).
introduced to perform
the
differentia-
If we assume that all lengths are in elec-
trical units (i.e., k times the real lengths), we obtain that the integrals in eqn.(A1.1) have the common form
ctp .!s s
p m
The length of of magnitude segment
m
the 1.
+ ms
m-1 a exp(-jr) -) I ds ' ap ra p=O
ra
can be
as
small
as
-4 to typically of the order of magnitude 10 be
(Al. 4)
integration interval (s ,s ) is usually of the order 1 2 When a matching point is in the vicinity or on the
considered,
Pm can
m=O' ... 'n .
pseudosingular,
and
the wire radius, which is
2
10- .
therefore their
Hence the integrals
numerical
integration
should be performed carefully. Using elementary transformations,
the
integrals
Pm
can be expressed
in terms of the integrals
G
u
m
m exp(-jr) r du, a
m=O, ... ,n
(Al. 5)
and
_l_ _d_[exp( -j r a) r dr r a a a
l
(Al. 6)
du'
where -+
-+
t=i •R
s
'
-+ -+ -+ R=r -r p s
'
u=s-t ,
2
2
2
r a= ( u +c +a )
~
,
2 2 ~ c= (R -t ) .
(A1. 7)
App.l.
Evaluation of integrals
259
The integral G
can be evaluated explicitly. The integrands in eqns. 1 (Al.5) and (Al.6) have poles in the complex u-plane at u
The
=
±j (c
2
2
+a )
integrands
~
(Al.S)
of Gm'
softens the influence
m=2, ... ,n, of
the
have
poles
on
am-fold the
zero
at
real u-axis.
u=O,
which
To integrate
these integrals numerically it is very convenient to extract the pseudosingularities.
To
that
purpose
tracted from the integrand
terms
can
term (1/r- 0.5 r)
should
be
sub-
of G , the term um/r from the integrand of 0 - 0.5/r + 0.125 r) from the integrand of s . 0 be integrated explicitly, and the remaining
Gm (m_:::_2), and the term (1/r The subtracted
the
3
parts can be integrated numerically using the Gauss-Legendre quadrature formula. ments not
It appeared that longer
the
than about
five-point formula (for was efficient.
on subseg-
If the poles, given by
eqn. (Al. 8), are distant
from
integration formula
be applied d:i,rectly to the Pm integrals.
can
the integration
n~6)
segment
considered, this All
the integrals, either in eqns.(Al.4), (Al.5) or (Al.6), should be integrated simultaneously, in same trigonometric
or
order
to avoid repeated calculations
square-root functions
of
the
(which require about five to
ten times as much computin8 til'le as algebraic operations). It is possiblo;o/ to reduce in
this manner the average c.p.u. time necessary for evalua- '
tion of one integral Pm
to approximately only ten times that necessary
for evaluation of a single square root.
APPENDIX2
Notes on Hallen's Equation
We shall prove here
that
F(z) given by eqn. (1.25) is the general solu-
tion of eqn.(l.24) for an arbitrary constant z . 0 Let t
2
(z)
J
w(z) t
1
v(z,z') dz'
(A2. l)
.
(z)
Then, according to the known rule of differentiation,
dw(z) = dz
If we
set v(z,z')=kS(z') sin k(z-z'),
t
(z)=z (z =arbitrary constant) 0 0 1 and t (z)=z, w(z) becomes precisely the last term in eqn. (1.25). By 2 2 2 using the above rule twice to obtain dw/dz and d w/dz , it is a simple matter to show that
2
d w(z) = -k 2w(z) +k 2 S(z)
(A2. 3)
di
Substituting (1. 24)
we
adopted w(z)
immediately
represents
c2 sin kz
the
the
both
find
and
the
above
found
that, indeed, the last
term
the
in eqn. (1.25)
Since c cos kz and 1 2 homogeneous equation (k + d /di) F(z)=O and
particular solution of eqn. (1.24). satisfy
2 2 d w/dz into eqn.
2
262
App.2.
are mutually
independent, eqn. (1.25)
is
Notes on Hallen's equation
the
general
solution of eqn.
(1.24). For the
a
given
constants
set
c1
of
two
c2
and
independent conditions can
and
the value of z
be found, thus obtaining
the
solution of eqn.(l.24) corresponding to these conditions. a new constant, z
01
, is adopted instead of z .
0
2
z
J
Suppose that
In that case we have
01
J
S(z')sink(z-z')dz'
0 particular
S(z') sin k(z-z') dz'
+
zo
zo
z
J
+ 2
The
sine
cos kz
integrals stants.
c1
and
01
in the first term on the right-hand side of this equation can
be writ ten in the and
taken do
not
form
(sin kz cos kz' - cos kz sin kz'), and then
in front
of
contain
z
the two integrals any more, so
Therefore, a change in z
c2 ,
(A2.4)
S(z')sink(z-z')dz'.
so that z
0
they
thus
obtained.
sin kz These
are both equal to con-
is actually absorbed in the constants
0 can indeed be chosen arbitrarily.
APPENDIX3
Evaluation of Thin-Wire Antenna Radiation Pattern and Induced Electromotive Force
The most involved part of the antenna analysis is to determine the current distribution along
the antenna.
Once this distribution is known,
other antenna characteristics can be computed with relative ease. luation of mittance
one very important antenna characteristic,
or
impedance, is
known to be the This
same
presented
characteristics,
the
Eva-
antenna ad-
(The admittance is
in the antenna transmitting and receiving modes.)
appendix deals with evaluation
antenna when
in Chapter 2.
the
of
two
other
antenna radiation pattern and
important
antenna
emf induced in the
it is situated in an arbitrary impressed electric field.
Both evaluations
are
based on the known current distribution along the
antenna when it operates in the transmitting mode.
A3.1.
EVALUATION OF RADIATION PATTERN
Consider an arbitrary thin-wire antenna, located in a homogeneous medium and excited by far-field
a suitable generator,
zone,
i.e., for r
overall
antenna
dimensions,
antenna
currents
in terms
of
is
0
as
sketched in Fig .A3 .1.
much greater than the
transverse
the
-+
electric-field vector, E, due -+
to the radius r
In the
wavelength and the to
the
, and can be expressed
0 the magnetic vector-potential as (see, for example, Refer-
ence 96, p.39) (A3. 1) ->-
-+
where ir =r /r is the unit vector directed from the coordinate origin 0 0 0 towards the field point. The radiated magnetic-field vector is given
264
App.J.
Radiation pattern and induced emf
z
FIG.A3.1. Coordinate system for evaluation of far-zone field of a transmitting antenna.
X
by 1-+
-
i
1;
-+-+
ro
xE(r ) , 0
(A3.2)
where ~;=I~!E is the intrinsic impedance of the medium. The dimensions of the wire wavelength, total Thus
when
current the
in
cross-section being much smaller than the
computing the far-field magnetic vector-potential the the wire
can
be
assumed
to be along the wire axis.
far-field magnetic vector-potential can be expressed approxi-
mately as
i sm I m (sm) where,
as
explained
in Section 1.2,
(generally curved) segments, to the wire element
dsm,
the wire -+
r
SID
gin.
is
axis, and
segments
"t•
exp(jk"t'
i SID
is
the
•fro ) dsm
(A3. 3)
antenna is assembled
from N
the unit vector locally tangential
is the distance from the coordinate origin to the
If g(r ) is Green's function, given in eqn.(l.5). 0 are straight instead of curved, "t•="t +s 1. , where SID
ID
SID
the distance between the coordinate origin and the sm-axis ori-
From eqns.(A3.1)
and
(A3.3) we thus obtain for wire antennas as-
sembled from straight segments
Sect.A3.1.
Evaluation of radiation pattern
265
N
E(1 ) =-jksg(r ) 0 0
I
exp(jk1
m=1
I
-r
where le and If Im(sm)
_,_
i~
is
m
sm
•f
ro
) •
(s) exp(jks i ·f ) ds m m sm ro m
(A3.4)
are the unit vectors of the spherical coordinate system. approximated by a polynomial, as given in eqn. (1.16), or
by a combined trigonometric and polynomial expansion, as in eqn.(1.30), the integrals involved in eqn.(A3.4) can be evaluated explicitly. ever,
when
Jism ·fro J<<1,
i.e.,
when
the
field
point
is
How-
close to the
wire-segment equatorial plane, these explicit formulas are inconvenient because
they
involve subtraction of
in significant numerical
errors.
almost-equal terms, which results
Fortunately,
the
integrands in eqn.
(A3.4) are well-behaved functions and the integrals can easily be evaluated numerically. If
the
far-zone
electric
field is known, the antenna directive gain
is given by JE<1o)l2/s (A3.5)
2
Pr/(4-rrro) or, in decibels, (dB)
,
(A3.6)
where p
(A3. 7)
n P input
r
the power fed to the antenna and n the lnput antenna efficiency, which can be computed from eqns.(5.11)-(5.13). The is
the
antenna
radiated
power,
radiation
eqn.(A3.6).
P.
pattern,
if
plotted
in
decibels,
is
determined by
The antenna directivity is, by definition,
D = max{gd(f
ro
)} ,
(A3. 8)
266
App.J.
Radiation pattern and induced emf
and the antenna power gain is given by g(i Both
ro
) ; ngd(ir ) o
the
(A3.9)
directive
and
power gains,. given
are defined with respect to into
the
fectly
whole space.
conducting
an
in eqns. (A3.5) and (A3.9),
isotropic radiator radiating uniformly
In the case of antennas above an infinite, per-
ground
plane,
driven
at
one
or
more points at the
plane, both gains can be defined with respect to a semi-isotropic radiator, radiating uniformly only into the upper half-space. the monopole-antenna
gain
is equal
to
In that case
the gain of the equivalent sym-
metrical dipole antenna.
A3.2.
EVALUATION OF ELECTROMOTIVE FORCE INDUCED IN A RECEIVING WIRE ANTENNA
Consider a
wire
antenna having
nals, as sketched in Fig.A3.2. trary
impressed electric
the antenna terminals equal
to
and Eind in
the are
emf
only
Let the antenna be situated in an arbi-
field,
are
one pair of closely spaced termi-
-+
Ei,
due
to some distant sources.
open, a voltage V
0
If
will appear between them,
Eind induced in the antenna (i.e., V ;Eind' where V 0
0
determined with respect to the reference directions shown
Fig.A3.2).
Assume
that
an
ideal
voltage
generator,
of
voltage
V ;V , is connected between the antenna terminals, in opposition to the g 0 emf induced in the antenna [see Fig.A3.3(a)]. Obviously, no current will flow through the antenna terminals. a
superposition
present,
the
of
two
cases:
(1)
We can represent this case as
the impressed electric
-+
field, Ei,
antenna terminals short circuited [Fig.A3.3(b)],
and
(2)
FIG.A3.2. Receiving antenna in arbitrary impressed electric field.
Sect.A3.2.
267
Evaluation of induced emf
(c)
{b)
(a)
FIG.A3. 3. (a) Ideal voltage generator connected between antenna terminals in opposition to emf induced in antenna. This case can be regarded as superposition of two cases denoted as (b) and (c).
-+
the impressed electric field, Ei, not present, the antenna connected to the voltage generator V [see Fig.A3.3(c)]. Let the current in the aug tenna terminals in the first case be I ,and in the second case I ;I , 02 0 01 with respect to the reference directions shown in Fig.A3.3. Obviously, I
01 .
;I
1 ~ne
;I . Using 02 0 112 currents,
the
reciprocity theorem specialised for the case of
(A3.10) L
where
the
impressed electric
field -+
in
the
first case,
the
corresponding antenna current, and Ei field, namely antenna current in
the
and I the impressed electric 2 2 second case, we can easily obtain
the equation N
I
I
m;l
where
I (s )
m m
is
the
transmitting mode, finally obtain
(s) mm
i
·E.(1')
sm~
current along
driven by
the
ds
(A3.11)
m
the
antenna when it operates in the
generator
of voltage
V . g
Hence we
268
App.3.
-1 I
N
I (s ) i • E. m m sm 1
L
0 m=1
In
the
special
travelling
in
case
the
£.10 exp ( -j k-;:' • ;) ,
when
£.1
direction where
E.10
is
of is
(-;:')
a the
Radiation pattern and induced emf
ds
(A3.12)
m
the
electric field of a plane wave
unit vector
->-
n,
i.e., when
impressed electric
field
E.1 (r ' ) = in
the
coordinate origin, we obtain
E
ind
Note
->= -E
that
io
[ 1 •-
the
N
L
I 0 m= 1 term
i
I
sm m
(s ) exp(-jk-;:' .;) dsm ]. m
(A3.13)
in brackets in eqn. (A3.13) is actually independent
of the current I , and that the sum in this term is equal to the cor0 responding sum in eqn.(A3.3), provided that ;=-i ro
APPENDIX4
Notes on TEM Magnetic-Current Frill Approximation of Coaxial-Line Excitation
This appendix deals with the following topics related netic-current
frill
approximation
of
to
the TEM mag-
coaxial-line excitation: with e-
valuation of near-zone and far-zone fields due to the TEH frill, with a method for determining the antenna admittance based on the complex power of magnetic cu,rrents [which is finally,
with
an
alternative to eqn.(2.27)], and,
the antenna admittance correction which is necessary in
the case when boundary conditions in the excitation region are not satisfied adequately.
A4.1.
NEAR-ZONE FIELD OF TEH HAGNETIC-CURRENT FRILL
The near-zone electric field be
determined
starting
due
to the TEH magnetic-current frill can
from eqn. (2.4).
Let,
for
simplicity, the an-
nular magnetic-current frill be located in the xOy plane, and the field point P
in
the xOz
plane,
as
shown
in Fig.A4.1.
In the case consi-
dered, the surface magnetic currents are circular, of density +
J
-2V
ms
7
p ln(b/a) l~
(A4. 1)
'
according
to eqns.(2.16) and (2.17), 2 2 2 ~ have r=(x +z +p -2xp cos~) , dS=p dp d~
and Fig.2.6. and i
~
From Fig .A4 .1 we
xi =Ci xi )xi =i ci .i )-
r
z
p
r
p
r
z
(i . i ) . Noting that grad g(r) =dgd(r) f , i .! =z/r and i .! =cos l/J, z r p r r r z r p from eqn.(2.4) we obtain for the elemental electric field
i
J
(~ i -cos l/J i ) dgd(r) p dp msrp z r
d~
(A4.2)
270
App.4.
Notes on TEM magnetic-current frill
FIG.A4.l. Coordinate system for evaluating nearzone and far-zone electric field due to annular magnetic-current frill.
y
->-
->-
From Fig .A4 .l we also have i =cos <j> i p
to symmetry,
X
->-
+sin <j> i , y
and cos 1j; dp=-dr.
Due
the y-component of the resultant field, Eiy' is zero, and
for the other two components we obtain 'IT
b
f f cos <j>
2z
P Jms (p)
~ dgd~r)
(A4.3)
dp d<j>
0 a and 'IT
2
b
ff
b
'IT
pJ
ms
(p) dg(r) d<j> = 2
f pJms (p) g(r) 0
0 p=a 'IT
- 2
I
d<j> -
p=a
b
f0 f --!-[pJms (p)jg(r) dpd<j> op
(A4.4)
a
From eqn.(A4.l) pJ (p)=-2V/ln(b/a), so that from eqns.(A4.3) and (A4.4) ms we finally obtain eqns.(2.18) and (2.19).
Sect.A4.1. A4.2.
RADIATION FIELD OF TEM MAGNETIC-CURRENT FRILL
The far-zone field
due
mined in terms
the
only
271
Radiation field
the
of
to the TEM magnetic-current frill can be deter-+
electric vector-potential, Ae.
This vector has
¢-component (see Fig.A4.1) and the magnetic field due
to
the
frill in the far zone has also only this component, -jwA Note
that
(A4.5)
.
e¢
this equation is dual
to
eqn.(A3.1) in Appendix 3.
The
e-·
lectric field has only the 8-component, (A4.6) where r,=l)l/f;
is
the
intrinsic
impedance
of
the medium.
The far-zone
electric vector-potential can be evaluated as
fs Jms exp(jkpfp •fro ) dS
A. e which In
is,
essentially,
eqn. (A4. 7),
-+
r
is
0
an the
(A4. 7)
equation
dual
distance
field point, S is the surface of
between
=sine! +case!, where, for ro x z taken to be located in the xOz plane.
the
J
the frill,
and!
1T
to eqn. (A3.3) in Appendix 3.
IDS
frill
center
simplicity,
the
field
point is
Hence we have
b -2V
f f -p-ln-'(::-'b'-/c-a7") cos¢ exp (jkp cos¢ sin 8) p dp d¢
A e¢
and the
is given by eqn. (A4 .1)
The integration over p in eqn. (A4.8)
can
be
performed
.
(A4. 8)
explicitly, to
obtain
A e¢
Eg(rO)
ln~~~a)
1T
k sin 8 [ 21TT
J exp(jkp
cos¢ sin e)d¢
J Ip:a
(A4.9)
-TT The integral in brackets in eqn.(A4.9) is, essentially, the zeroth-order Bessel function of the first kind.
Hence Ee is finally given by
(A4.10)
272
App.4.
Notes on TEM magnetic-current frill
2 Noting that, for a small argument t, J (t)oel-t /4, an approximate far0 field expression is obtained. It is essentially the same as that presented in Reference 21, and is also valid only if kb<
A4.3.
DETERMINATION OF ANTENNA ADMITTANCE FROM POWER GENERATED BY MAGNETIC-CURRENT FRILL
A rigorous method for determining the antenna admittance
in terms of
the coaxial-line TEM mode must
into account
(see Subsection 2.3.3).
take higher-order modes
All other procedures necessarily involve cer-
tain approximations, and therefore introduce an error in the theoretical results
for
the antenna admittance.
Since eqn.(2.27) is based on ap-
proximation of both the electric and magnetic field at the coaxial-line opening (Fig.2.5) rect.
by the TEM mode, i t is,
also,
theoretically incor-
A somewhat different approach to de terrnining the antenna admit-
tance is presented in
this section.
It is based on the TEM approxima-
tion of the electric field in Fig.2.5, while the magnetic field is, essentially, assumed to contain higher-order modes. The TEM approximation of the electric field results in the equivalent system shown in Fig.2.6(b), where the density of the magnetic currents is given by eqn. (A4.1).
Once the antenna current distribution is ob-
tained, the monopole-antenna admittance is computed as y
(A4.ll)
where Sfrill is the complex power of the magnetic-current frill in Fig. 2.6(b), and the asterisk denotes the complex conjugate.
The frill com-
plex power can be computed as
f frill
sfrill
-+
J
-+
ms
·H*
total
dS
(A4.12)
where -+
H
-+
total
).1
curl A
-+
jwA e
(A4.13)
Sect.A4.3.
A
is
Determination of antenna admittance
the magnetic vector-potential
due
273
to the antenna currents, given
A
in eqn.(l.3), and the electric vector-potential due to the magnetic e currents, given in eqn.(2.3). In has
the only
case
of
the cylindrical antenna
shown
in Fig.2.6(b),
Htot a 1
the q,-component, which is independent of the $-coordinate, due
to symmetry.
Thus from eqns.(A4.1), (A4.11) and (A4.12) we obtain b
y
J Hq, total (p)
2rr V ln(b/a)
(A4.14)
dp ·
a According and
the
to Maxwell's equations, Hq, total (p)
displacement
current
(i.e.,
by
through the annular ring of radii a and p. when a=
the
displacement current
proximately, H
the
is determined by I (0)
flux
of
the
vector jwD)
At lower frequencies and/or
can be neglected, and we have, ap-
(p)=I(0)/(2rrp), so that eqn.(A4.14) is reduced to tota 1 the displacement current is not negligible [for example,
q,
Y=I (0) /V.
If
at higher frequencies
or
for
monopole
lengths comparable with (b-a)
eqn.(A4.14) yields different results from eqn.(2.27). Eqn. (A4 .14)
J,
•
can be obtained also using another approach, as follows.
The antenna admittance
is
defined in terms of the TEM mode at
the
co-
axial-line opening as
(A4.15) (see Subsection 2.3.3), where ITEM(0)=2rrpHTEM(p)#I(O), TEM component
of
If we
the orthogonality property of
recall
-+
and HTEM is the
the total magnetic field at the coaxial-line opening. the
TEM and higher-order
modes in the coaxial line,
J
frill
2
J
HTEM dS '
(A4.16)
frill
substitute HTEM by ITEM(O) / (2rrp)
and evaluate the integral on the right-
hand side of the last equation, we obtain
274
App.4.
Notes on TEM magnetic-current frill b
1 ln(b/a)
J frill
1 + + - i •H dS p ~ total
J H~ total (p) dp
27T ln(b/a)
·
a (A4.17)
Combining eqns. (A4 .15) though
this
and
(A4.17)
extraction of
the
\fe
obtain again eqn. (A4 .14).
TEM magnetic-field
field at the coaxial-line opening resembles
the
mode
Al-
from the total
approach in Subsection
2.3.3, and eqn.(A4.14) is correct, the result is still only an approximation. that
We make
the
here
Htot· a 1 ,
an error in computing
because we assume
electric field at the coaxial-line opening is given solely by
the TEM mode, thus neglecting higher-order electric-field modes. If
the
radial currents at the antenna ends are neglected (which can"'\....-f
be done only when -+
the antenna is very short
potential, A, at
the magnetic-current
which depends on
p and
electric
z
frill
the magnetic vector-
has but
the z-component,
+
_,.
Hence, curl A=-(3Az/3p)i~.
only.
_,.
vector-potential,
),
Ae,
has
only
the
~-component,
Since the from eqns.
(A4.13) and (A4.14) we obtain for the antenna admittance b y
211 {.l[A (p=a) -A (p=b)] - jw V ln(b/a) ~ z z
(A4.18)
J Ae~ dp} Iz=O
a Essentially,
this
in eqn. (A4.14)
equation is simpler than eqn. (A4.14), because
must be
computed
as
the
curl
of
H
tot a 1 the magnetic vector-
potential. As
an
illustration, consider a quarter-wavelength monopole of radius
a=0.01 A, driven by a coaxial line with b/a=2.3.
Eqn.(2.27) yields for
the antenna admittance I(O)/V=(l7.7975-j6.4388) mS.
The first term in
eqn. (A4.18), corresponding to the difference of the two magnetic vectorpotentials, equals (17.7939-j7.5286) mS, and ing
to
eqn.(A4.18)
is
the
(17.7942-j6.5919) mS.
(A4.18) yields practically
As it can be seen, eqn.
same result for the antenna conductance
aseqn.(2.27),
while
for 0. 15 31 mS,
which corresponds to a negative capacitance.
experiments have
the
the
total result accord-
shown
susceptance
that
given
by eqn.(A4.18) is smaller Numerical
this capacitance depends practically only
Sect.A4.4. on
the
Antenna admittance correction
physical
dimensions
independent of the
antenna
of
the
275
coaxial-line cross-section, and is
length and
frequency.
Eqn. (A4 .18)
results which are between those obtained by eqn. (2. 27) and sented
in
Subsection 2.3.3.
yields
those
pre-
It should be noted, however, that evalua-
tion of the right-hand side expression in eqn.(A4.18) is very difficult because the magnetic vector-potential must be computed without approximations (which of which the
involves numerical evaluation of
is singular
electric
Since
the
for
vee tor-paten tial
final
double integrals, one
p=a), and computation of the term containing
correction
involves a
is
triple
practically most
singular often
integral.
insignificant,
eqn.(2.27) is preferred to eqn.(A4.18).
A4.4.
ANTENNA ADMITTANCE CORRECTION WHEN BOUNDARY CONDITIONS ARE SATISFIED INADEQUATELY
If
the
boundary conditions
are
not satisfied adequately, the electric
currents in the equivalent system shown in Fig.2.6(c) are situated in a non-zero
axial
electric field.
Therefore
they generate non-zero com-
plex power, in addition to the power generated by frill.
Taking
veloped in
into
this
the
magnetic-current
account the total genera ted power, a me thad is de-
section for
estimating and, to
some
extent, for cor-
recting the error introduced into the antenna admittance when calculated from eqns. We
(2.27) and (A4.14).
shall assume
that
the coaxial-line dimensions, a and b, are much
smaller than the wavelength, and that the boundary conditions are inadequately satisfied only in the excitation region, in a zone which, measured along
the
z-axis, is much shorter
than
the wavelength.
The com-
plex power generated by the electric currents can be computed as
se
(A4.19)
J
antenna As
can be
field z-axis.
at
seen
from
Fig.2.7,
the antenna surface
the is
z-component of
the
practically identical
Therefore eqn.(A4.19) can be approximated by
total electric to
that at the
276
App.4.
se
Notes on TEM magnetic-current frill
(E +E. ) I(z)* dz ,
- f
z
(A4.20)
~z
L
where
L
to z=d.
is a path along the z-axis in the excitation region, from z=-d This path being relatively short, the antenna current intensi-
ty, I(z), can
be
equal to I(O).
approximately taken to be constant in this region and
Thus we have d
se
-I(O)*
f
(E +E. ) dz
z
2!1V I(O)* ,
~z
(A4.21)
-d where
d
f
!1V
(E +E. ) dz . ~z z
(A4.22)
0
From eqns. (A4. 11)
and
(A4 .12) and the discussion following eqn. (A4. 14)
we can deduce that for kb<<1 the power generated by the magnetic-current frill is very closely equal to 2V I(O)*.
Thus the total power generat-
ed by the electric and magnetic currents is (A4.23)
Sfrill + Se = 2(V+!1V) I(O)* .
Suppose now that we retain the approximate electric-current distribution
along
the antenna, while
which would thus
also
give
correct
we
seek for a new voltage instead of V,
results for
for the antenna admittance.
that (V + !1V) is exactly what we need.
the
total
generated
It is obvious
from
power, and eqn.(A4.23)
Therefore we have for the mono-
pole-antenna admittance y =
_1lQ2_ " I (O) ( 1 _ /1V) V+ 11V
if I 11V/VI <<1. fully
to
V
Although
correct errors
antenna admittance
(A4.24)
V
due
approximate, eqn. (A4.24) can be of
used
success-
the order of few percents in the monopole-
to inadequate satisfaction of the boundary con-
ditions in the excitation region (i.e., due to inaccurate approximation of the current distribution).
However,
the
integral in eqn.(A4.22) is
Sect.A4.4.
very time-consuming cient
to
277
Antenna admittance correction
provide
to
compute, and it is therefore
in advance a sufficiently accurate
always more effiapproximation of
the antenna current distribution. It
should be noted that
measure
of
accuracy of
the
the
ratio f6V /V f can be used as a very good
approximate antenna
current
distribution,
because the smaller is this ratio, the better is the approximation.
APPENDIX 5
Admittance of Monopole Antennas Driven by Coaxial Line
In
this
tance
appendix plots
of
the monopole
infinite,
are presented antenna
showing
sketched
conductance and suscep-
in Fig. 2.S, located above
perfectly conducting ground plane
and fed
in that plane by
means of a coaxial line, versus monopole radius, a, and height, h. medium is assumed plane
and in
the
to
an
The
be a vacuum both above the perfectly conducting
coaxial
line.
These
results were obtained by
the
two-potential equation, formulated to include the end effect, according to Section 3.4.
The coaxial-line excitation was
magnetic-current order mode
as explained
correction was
admittance is
shown
(which includes 0.01,
frill,
In Figs .A5 .l-AS. 3
included.
plotted versus
of
No higher-
the monopole
the normalized monopole height h/1-
the hemispherical end
and a coaxial line
approximated by a TEM
in Subsection 2. 3 .1.
cap), for a/t-=0.0001, 0.001 and
characteristic admittance Yc=20 mS (i.e.,
characteristic impedance Z =SO S"l), which corresponds to b/a=2. 3, where c b is the outer coaxial-line radius. To make these plots applicable for other coaxial-line dimensions, susceptance,
liB
(\vhich
shown
in Fig.A5. 4 is
should be added
to
the
correction in
the values shown in Figs.
A5.1-A5.3), versus the coaxial-line characteristic impedance.
The con-
ductance is practically independent of the coaxial-line dimensions. Accuracy of
the
theoretical results
lieved to be better
than S% when
perimental
except
about 1 mS.
results,
for
presented
in
the plots
is be-
compared with the best available ex-
values
of
susceptance
(B)
less
than
280
App.5.
Admittance of monopole antennas
G,B {mS) 30~--~----~--~----~--~----~--~----~--~
n
I
I
I
I I
10
i\
I
ll I
s/ d . / Ii \ I. \
5
I
4
3
I
/
I j \ 1\
/
I
11
Ii
2
lii ~\\ 1
!i
:1:1
\
/
\'\~I/
I
/
I
I.
i Ii
J
I
\
: ( I
/
1 ·1\.
I
I
\
l\
II I· I
I
'· \
\
I :,,'{-B)l~\ I.
1/
\
.I
\
"KG
!Id I· li
\ .\ lj . 0.5~--~-Hi!--+----+~\r.~d------~~--~-+!l~·~----~~\--841
0.4~~~~,i--+---~+\--!~--~--~-*.!i--r---1-~,-~,: 0. 31---+---+-~,i-+----+----+\--+-J,~-+---·-+--u.lj-+----+-+'· -+--l/
li l!
i
li
I
\!
0.2~~~~.~---~--~-+\~1~1----~---+-41--+----+~,~,~
,,
:j
J 0.1L-L-_L~__L __ _
0.1 FIG.AS.l.
0.2
0.3 0.4
_ L_ _
\\ I1 . Ji 1,. iI h d :j \I A JLL---~----~~~-----~__l-~ 0.5
0.6
0.7
Monopole conductance (G) and susceptance ized monopole height, for a/).=0.0001.
0.8
0.9
1 .0
(B) versus normal-
App.5.
Admittance of monopole antennas
281
G,B (mS)
30
A
20
,~. I I
I I
10
I I
5f-/-
!i
I
~-
\
.
I
1\\
I
·' '\\
B
lj (-B) I.
III·
\
II I·
i!
1
~I
I•
J " I
I
\
I' I
iI
11
i'
!I
--
~
\
I,
d
III· d
B/
\
I
I I
-ll
!
-H
I
\
I \ I
I!
I.
II!
I I
'I\J II
H 0.4
I
\
jl
0.3
~
\
~i
II
li
~\\ \
lj
i!
II 0.2
:
I I _l,
I!
0.1
I
'
I !( -B I(
'f-. _..,
!i
0. 1
I .
I
II
li
0. 2
I
I
I
i\ ;
"I!
0. 3
I
1 :I
I
I'
0. 4
r\
j l1
I
II
0. 5
i
I L'
II
2
I
I
I
,,\
I.
3
~
G
II\
I j I. JI Ii
B I I
4
'
~
il 0.5
0.6
0.7
0.8
0.9
h A
1.0
FIG.A5.2. Honopole conductance (G) and susceptance (B) versus normalized monopole height, for a/;1.=0.001.
282
App.5.
Admittance of monopole antennas
G,B (mS) 301---~r\-..-----.---....--~--.-----r------.--~
/ :\
;1:\
20~--~~--+----+----+----+----+=---+----+----4
Bt
/.I
!~~u~-~j·_\~~'~~---*~--~-~~--~~--~'~~--~
31-----+-~:~'--++----"I
:i I
/
\
"!_
"t
/
~
_../
1/ III~
"~ , / ~ l! \ s/ .r\
! '
2~~~~--~---~-+--~---+----+r-+--+----++--~
I : j(- B) \ lj \
T
J
I !
I
/
11 I
I
\
\
i
\
I(
:
\
I I
-B) \
d
lj
\
I
\
\
.I
I .
I
I
\
\l
I
\ I!
0 5 · If i 0. 41-111---+---ll+---+--~-1+----+----+----+---l+,-+---l.~+----1
7
II
li
II
\i Ii
I'
·I
Ij
·I
J!
h
I
i1
.1,
lj
\!
\I
0.3~--~~--~--rl+,----~---+----~+--+-4,~,-+--~
:!I -l------t+!:-+----1-----+----+.....Jic+-1. \1 li -.j~-r+\I11-----l------1 0 . 21-----1----:l:-1-· 'I
1i 0.1L___ 0.1 0.2
~
i'
li
h . A
il
_L~_ _L __ _~_ _ _ _L __ __ L_ _ _ _~L--L--~L---~-
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
FIG.A5.3. Monopole conductance (G) and susceptance (B) versus normalized monopole height, for a/A=O.Ol.
App.5.
283
Admittance of monopole antennas
t>B ii]I (mS)
150 1-
11-
100
~
1-
11-
50
\
1-
111-
0 1-
11-
-50 1-
20
1\\0
40
zc 80
1\'\
111-
-100
FIG.A5.4. Susceptance correction (t>B) versus istic impedance, Zc.
( n)
100
~
coaxial-line character-
APPENDIX6
Field Intensity Versus Radiated Power, Height and Thickness of a Vertical Monopole Antenna Above Perfectly Conducting Ground Plane.//.;
Of
considerable
interest
field intensity at for a
given
to
communications engineers
a required distance
power radiated by
from
the antenna.
the transmitting antenna, This
field
lated relatively easily for wire-antenna structures tribution
in
pendix 3.
However,
wire
the
antenna,
antenna
or
is very useful
for
its
is known, using
the
frequently
asymmetrical
to have
these
is the electric
the
used
once
can be calcucurrent dis-
formulas derived in Apcylindrical
equivalent
above
symmetrical
ground plane, it
results in the form of a diagram.
Such
a diagram is presented in this appendix. Shown
in Fig.A6.1
is
a
diagram representing
electric-field
intensity of vertical
at a
the
tenna
point on
radiating
trical height, as parameter.
unattenuated
cylindrical
monopole
radiated antennas,
ground at a reference distance of 1 km from the an-
a kh,
reference
power
with
antenna height-to-diameter
the
The diagram was
of
1 kW,
versus the antenna elecratio,
h/2a,
obtained by assuming current distribu-
tion along the antennas of the form (Reference 3, p.254) I(z) = V{Asink(h-z) where V is culated as
the the
centered at
+ B[1- cosk(h-z)]},
driving monopole voltage,
114
the monopole.
there was no need
distribution. that
and power radiated was cal-
flux of the Poynting vector through a large hemisphere Since
with current approximation as rate,
(A6 .1)
Note
in Fig.A6.1
that
the
radiation
pattern of
antennas
in eqn. (A6 .1) is known to be quite accuto use a better approximation for current
the
is based on
commonly used
diagram
corresponding to
the sinusoidal distribution of current,
286
App.6.
and therefore does not 115 na. (This classical in Fig.A6.1.)
take curve
into account
the
Field intensity
thickness of the anten-
coincides with the curve labelled h/2a-+«>
Fig.A6.1 shows that thickness of the antenna becomes
an
important factor for monopole electrical heights larger than about 190°. For a
power
of P kW radiated by a monopole antenna and for a dis-
tance of d km from the antenna, the value of E from the diagram should If the radiated field in the plane of symmetry
be multiplied by
IP/d.
of
antenna
a
symmetrical
is needed
instead, P
should be substituted
by Pd.1po l e /2.
lEI
(mV/m)
440 420 400 380 360 340 320 3001------
I
kh
280 (degrees) 260L.--L-~---L--L-~---L--~~---L--J---L--L--~--~~~ 20 40 60 80 100 120 140 160 180 200 220 240 260 0 FIG.A6 .1. Unattenuated electric-field int~ty on the ground, at 1 km from a ground-based vertic.al cylindrical monopole antenna, per kilowatt ra.diate1l ~r, with height-to-diameter ratio h/2a as parameter. (Ref.113)
APPENDIX 7
Simplex Optimization Procedure 10
Let X=(x , ... ,xn) be a vector in the n-dimensional space, and f(X) a 1 real function of X. We wish to search for a local minimum of f(X) using the simplex optimization procedure.
We shall refer to simplex as to a
polyhedron in
the n-dimensional space, having (n+l) vertices at points
Xi, i=O, ... ,n.
This procedure can be divided into the following steps:
1. form
x0 =(x 01 , ... ,x 0 n)
Given the starting point the
initial
regular
l
simplex, consisting o.f points
the coordinates of which are
l
ln+1 n/2- 1 S + xOk '
k#i
,
rn+f
and the initial stepS,
+ n - 1 S + n/2 xOk '
x0 ,x 1 , ... ,Xn'
i,k=1, ... ,n,
(A7. 1)
k=i
and compute values y.=f(X.), i=O, ... ,n. l
2.
Search
the
l
set {y.} to l
find
the maximal, second maximal and the
minimal value, i.e., yh=f(~) =max{y.},
i
l
y s
=f(X) =max{y.}, s i#h l
y
1
=f(X ) =min{y.}, 1 i l
(A7.2)
and test for termination of the optimization procedure. 3.
Determine the centroid of the points X., i#h, l
n X =.!. c n
I
i=O i#h
X. l
(A7. 3)
288
App.7.
4. X
r
Simplex optimization procedure
Reflect the point Xh with respect to Xc to obtain ;
2X
(A7.4)
- X. --h '
c
and compute yr;f(Xr). If yr
proceed to step 5.
1
If y
Expand the point Xr with respect to Xc to obtain 2X
(A7.5)
r
and compute y ;f(X ). e e I f ye
6.
Contract the point
~ with respect to Xc to obtain
~ ; 0.5(Xh + Xc) ' and compute
(A7.6)
yk;f(~).
If yk
by~
and yh by yk and return to step 2.
If yh
0.5(Xi + x ), 1
i;O, ... ,n,
(A7. 7)
ill,
whereby the simplex is reduced towards the "lowest" point x . 1 y.;f(X.), ill, and return to step 2. l. l.
Compute
In the original Nelder-Mead's algorithm, criterion for termination of the simplex optimization procedure is based on comparison of the values yl.., i;Q, ••• ,n, and the computed value at the centroid, y ;f(X ) • e
c
How-
ever, such a test requires in each iteration one additional evaluation of f(X),
for X;X • c
By avoiding the computation of f(X )
time can be significantly reduced
c
the c.p.u.
(on average for more than 30%) it?
the computation of f(X) is time-consuming, such as
is the case in an-
App.7.
Simplex optimization procedure
tenna optimization.
289
Therefore other tests for termination of the opti-
mization procedure can be adopted, such as:
1.
The largest distance between any two points Xi and Xj is smaller
than a prescribed value. 2.
The difference yh-yl is smaller than a prescribed value.
3.
The maximal prescribed number of iterations is attained.
4. f(X ) is smaller than a prescribed value (i.e., the antenna per1 formance is satisfactory, although the local optimum is not attained).
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Hallen, E. , "Theoretical investigation into the transmitting and rece~v~ng qualities of antennae", Nova Acta Soc. Sci. Upsal., 1938, pp.1-44.
3.
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(Pren-
·Index
Admittance of cylindrical antennas, 32-33 graphs of, 279-283 measured values, 32, 48 Altshuler's antenna, 114 Antenna admittance, 53, 54, 273, 274 corrective terms, 54-55, 275 defined, 25, 31, 40 determination of, 272 measurement of, 63-65 reference, see reference feeder admittance Antenna efficiency, 142 of resistive antennas, 139, 142, 220 Antenna synthesis defined, 168 principles of, 169-170 Antennas as two-terminal networks, 24 Antennas for two frequencies, 228 Belt generator, 44 and TEM frill, 48-49 Boundary conditions in approximate solution, 41-43 Broadband antenna, 97 and antenna size, 212-214 capacitive, 99, 125-128 folded monopole, 223-224 half-octave, 209-213 monopole with parasites, 224228 octave, 209-214 resistive, 114
resistive-capacitive, 214-220 Uda-Yagi, 252-254 Capacitive loading, 118 construction of, 119, 137, 208 measurement of, 129-132 quasidistributed, 98-100 values of, 132, 137 Circumferential distribution of current, 6 Coating over antennas, 100-108 dielectric, 100-108 ferrite, 100-108 Coaxial excitation approximations of, 34 TEM frill, 35 belt generator, 44 higher-order modes, 48 Compensation of susceptance, 207, 214, 230 Concentrated loadings capacitive, 118-128 measured values, 132 defined, 109-111 delta-function, 112 influence of the number of, 123 inductive, 136 measured values, 138 mixed with distributed, 139 other approximations, 113 resistive, 114 used to approximate continuous, 115-117 Current approximation simple polynomial, 11 trigonometric-polynomial, 19
302 various polynomials, 12 Curved segments, 8, 104, 264 Curved wires, 13 Cylindrical antennas equations for, 13-20 long, 20, 88 tubular, 80-81 with flat ends, 80, 85 with hemispherical ends, 80, 84' 2 79 Delta-function generator and Hallen's equation, 29 defined, 25-29 Dielectric-coated antennas, 100108 Directive gain, expression for, 265 Directivity, 265 Discontinuities, defined, 3 Distributed loadings approximation by lumped, 115117 capacitive (quasidistributed), 99 defined, 93 ensuring minimal coupling, 197 in the form of coatings, 100 mixed with concentrated, 139 resistive, 96-98 stepwise approximation of, 201 Effective reflection coefficient, defined, 181 Electric field of general wire structures 265 of magnetic-current frill, 269-271 of magnetic-current ring, 27 of monopoles, 285 Electric scalar-potential, 7, 155-161 Electric vector-potential, 27, 161, 271, 273 End effect, 41-42, 79-88 correction of, 17, 227 Equivalence theorem, 7, 26, 36 Equation of continuity, 8, 38 Excitation of antennas, 23-24 coaxial line, 34 delta-function generator, 25
Index
two-wire line, 56-63 Excitation of wire structures, 3-4 Excitation region, defined, 3, 4' 23 Extended boundary conditions, 9, 13' 2 7' 4 7' 82' 95 Ferrite-coated antennas, 100-108 Half-octave antennas, 209-213 Hallen's equation and delta-function generator, 29 existence of solution, 30 and two-wire line excitation, 57-58 derived, 16 end effect with, 17 for antennas above ground, 163 for antennas with concentrated loadings, 112 for antennas with distributed loadings, 95, 184 for coupled antennas, 195, 225 general solution of, 261 Homogenization of the medium, 7, 26, 37, 93 Impressed field defined, 4 of belt generator, 46 and TEM frill, 49 of magnetic-current frill, 37 and belt generator, 49 of magnetic-current ring, 27 rapidly varying, 23 Inductive loading, 136, 241 construction of, 137 values of, 138 Internal impedance, 93 Junction constraints on currents, 70-73 constraints on field, 73-78 definition of, 3, 69 Kirchhoff's law first, 7, 70 second, 74 Loaded antenna, defined, 91
303
Index
Loadings, definitions of, 91-92 see also capacitive loading, inductive loading, resistive loading Magnetic vector-potential, 7 155-161, 263-264, 273 Matching points, adoption of, 11, 31, 149 Moment method, 10 Monopole folded, 223 inclined, 248 Nonreflecting capacitive termination, 128 resistive loading, 97, 189 Octave antennas, 209-211 Optimal broadband antennas, 206 folded monopole, 223 half-octave, 209-211 limits of VSWR of, 212-214 octave, 209-211 RC, 214-220 Uda-Yagi, 252-254 with distributed loadings, 188-192 with parasitic elements, 224 Optimization of admittance, 179 by varying antenna shape, 220-232 by varying concentrated loadings, 202-214 by varying distributed loadings, 183-192 by varying mixed loadings, 214-220 of mutual admittance, 192-202 of radiation pattern, 235 by varying antenna shape, 243-254 by varying driving voltages, 239-241 by varying loadings, 241-243 simultaneously with admittance, 235-236, 241, 248, 250, 252 Optimization function, examples, 170, 180, 181, 186, 187, 196, 216, 236' 237' 242
Optimization methods, 173 complete search, 175 gradient, 175 simplex, 177, 287 Optimization parameters, 172 Pocklington's equation, 18 for antennas with distributed loadings, 95 Point-matching method, described, 10-11 Polynomial current approximation, 11, 12, 31 discussion of, 12 Power gain, expression for, 266 Proximity effect, correction of, 227 Quasi-distributed capacitive loading, 98 Radiation pattern, 4, 40, 265 evaluation of, 263 optimization of, see optimization of radiation pattern Receiving antenna, 4, 24 emf of, 24, 266 Reference feeder admittance, 183, 192, 221, 226 Resistive loading, 96-97 construction of, 140 nonreflecting, 97, 189 stepwise approximation of, 97, 139, 201 Schelkunoff's equation, 18, 39 for antennas with distributed loadings, 95 Steplike loading, 97 Synthesis of antennas defined, 168 general principles of, 169 TEM frill condition for electric current imposed by, 38 field of, 37, 269-272 near zone, 269-270 radiation, 271 Transmitting antenna, 3, 24 Travelling-wave antenna capacitive, 128, 190
304 resistive, 114, 139, 189 Two-potential equation approximate solution of, 10-13 derived for antennas above ground, 162 for antennas with distributed loadings, 95 for arbitrary structures, 5-9 for cylindrical antennas, 14, 82 Uda-Yagi array broadband, 252 loaded, 241 with folded monopole, 249 with two reflectors, 244 Vector-potential equation, 14 for antennas with distributed loadings, 95 Wire antennas above ground, 154 defined, 3 in homogeneous lossless media, 5 in homogeneous lossy media, 146 on the interface of two media, 150 Wire defined, 3 thin, 3 Wire ends, 3, 69, 79 treatment of, 79-88
Index
ELECTRONIC & ELECTRICAL ENGINEERING RESEARCH STUDIES
ANTEI\INAS SERIES Series Editor: Professor J. R. James, Royal Military College of Sclence,
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Flat Radiating Dipoles and Applications to Arrays G. Dubost
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Analysis and Synthesis of Wire Antennas B. D. Popovi6, M. B. Dragovit, and A. R. Djordjevi6
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