{A i ' ABSTRACT

MODIFICATION OF SCATTERING FROM THICK
CYLINDERS AND RADIATION FROM LOOPS
BY IMPEDANCE LOADING

BY

John R. Short

The modification of the scattering from a thick, slotted cylinder
and the radiation and circuit pr0perties of a circular 100p antenna by
impedance loading are investigated in this thesis.

The modification and control of the scattering of a plane elec-
tromagnetic wave by a thick, conducting, infinitely long cylinder loaded
with several impedance -backed longitudinal slots is investigated in
Part I of this thesis. The incident plane wave is polarized with its
electric field vector perpendicular to the cylinder axis. The slots are
electrically narrow and the electric fields across them are assumed to
be constant. Within this assumption an exact theory is deve10ped.
Synthesis procedures are deve10ped to find load impedances and purely
reactive load impedances that cause the scattered field to vanish in one
or more desired directions. Synthesis procedures are also deve10ped
for finding a single purely reactive load impedance that produces mini-
mum scattering in one direction and load impedances which result in
zero scattering in one direction at several frequencies. The frequency
dependence and bandwidths of the different loadings are also considered.
Extensive numerical results are presented. The theoretical predictions
are confirmed with an experiment.

The modification of the radiation fields and circuit pr0perties
(impedance) of a loaded, circular -100p antenna is investigated in Part
II of this thesis. An improved theory for the 100p antenna is deve10ped
which includes a finite gap excitation. "Effective" gap widths are de-
fined for the cases of a 100p antenna driven by a two-wire line or a
coaxial line. Excellent agreement between theoretical antenna adrnit-

tances is found. The maximum and minimum gain attainable from a

100p loaded by a single impedance is presented for 100ps up to five
wavelengths in circumference. A procedure is deve10ped to facilitate

the design of loaded 100p antennas that have specified radiation patterns.
Several examples of loaded 100p antennas which have relatively direc-
tive patterns with reapect to unloaded 100p antennas are given. A scheme
for matching a 100p antenna to a transmission system is presented and

its efficiency is compared to that of a conventional base tuning network.

MODIFICATION OF SCATTERING FROM THICK
CYLINDERS AND RADIATION FROM LOOPS
BY IMPEDANCE LOADING

by

John R.6\ghort

A THE SIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Electrical Engineering and Systems Science

1971

To My Parents

ii

ACKNOWLEDGMENTS

The author wishes to express his deep appreciation to his major
professor, Dr. K. M. Chen, for his guidance and never-ending support
through the course of this work. He also wishes to thank the other
members of his guidance committee, Drs. D. P. Nyquist, J. S. Frame,
J. Asmussen, and G. Kemeny for their time and interest in this work.

. A Special note of thanks is extended to Dr. D. P. Nyquist who
introduced the author to electromagnetics and stimulated him with
many enlightening discussions throughout his graduate education.

The research reported in this thesis was supported in part by
the Air Force Cambridge Research Laboratory under Contract No.
F19(628)-70-C-0072.

Finally, the author thanks his fiance, Mary, for correcting

the Spelling in the manuscript.

iii

Chapter:

I.

II.

III.

TAB LE OF CONTENTS

LIST OF FIGURES

LIST OF TABLE .

PART I. MODIFICATION OF EM SCATTERING

FROM A THICK CYLINDER BY

MULTI-SLOT IMPEDANCE LOADING

IN TR ODUC TION

THEORETICAL FORMULATION OF PLANE WAVE
SCATTERING BY A MULTI- LOADED, SLOTTED
CYLINDER .

2.

NNNNNN

1

KIO‘U'IIfiUON

Formulation of the Problem and Boundary
Conditions

Superposition - - .

Scattering from a Solid Cylinder . .
Radiation from a Cylinder with N Driven Slots
Scattering from a Cylinder with N Loaded Slots
Bistatic Scattering Cross Section . . .
Generalizing the Theory -

THE SYNTHESIS OF LOAD IMPEDANCES THAT
RESULT IN ZERO OR MINIMUM SCATTERING IN

ONE

3.

3.

3.1
3.2

3

4

OR MORE DIRECTIONS OR FREQUENCIES

Zero Scattering in N Directions

Zero Scattering in N/Z Directions Using
Purely Reactive Load Impedances .
Minimum Scattering Using Purely Reactive
Load Impedances. . ,
Zero Scattering at N Different Frequencies

iv

Page
vii

xiv

r—It—Ir—a
UJNOCDKJU'IUJ

15
15
16

18
19

IV.

VI.

NUMERICAL RESULTS AND DISCUSSION

Numerical Method

Effect of Slot Width - -
Zero Scattering in Several Directions
Zero Backscattering by a Cylinder Loaded
with Two Purely Reactive Irnpedances .

with Equal Purely Reactive Irnpedances
. 6 Frequency Dependence of the Modified
Scattered Field . .

e 9 999+
QWNv—I

EXPERIMENT AND COMPARISON TO THEORY .

5. 1 Experimental Model and EXperiment
5. 2 Comparison of Theory with Experiment

CONCLUSIONS

APPENDIX A

APPENDD§ B

APPENDIX C

REFERENCES

III.

PART II. MODIFICATION OF RADIATION FIELDS

AND CIRCUIT PROPERTIES OF A LOOP
ANTENNA BY MULTI-IMPEDANCE LOADING

INTRODUCTION
THEORY OF THE LOADED LOOP ANTENNA

2. 1 An Integral Equation for the Current 0n the
Loaded L00p Antenna

2. 2 Fourier Series Solution for the Current on
a Loaded L00p Antenna

2. 3 Radiation Fields of a Loaded L00p Antenna

IMPEDANCES, CURRENTS, AND RADIATION
FIELDS OF A LOOP ANTENNA EXCITED BY
A FINITE GAP GENERATOR . . .

3. 1 Numerical Method . .

3. 2 Effect of Finite Gap Generator .

3. 3 Comparison of "Finite Gap” Theory with
Experimental Results .

3. 4 Input Impedances. Currents and Radiation

Fields of L00p Antennas

Scattering by a Cylinder Symmetrically Loaded

23
23
24
25
47
57
64
76

76
78

83
85
88
91
95

99

102

102

106
111

115

115
118

122

129

IV. MODIFICATION OF RADIATION FIELDS AND
INPUT IMPEDANCES OF LOOP ANTENNAS
BY MULTI-IMPEDANCE LOADING . -

4. 1 A LOOp Loaded with a Single Impedance

4. 2 Maximum and Minimum Gain of a L00p
Loaded with a Single Impedance . .

4. 3 Modification and Design of Radiation Patterns
of L00ps by Multi-Impedance Loading

4. 4 A Double Loaded Matched L00p

V. C ONC LUSIONS

REFERENCES

vi

139
139
143

144
157

164

165

Figure

2.1

LIST OF FIGURES

PART I
Page
An infinite cylinder with N longitudinal slpts
illuminated by a plane EM wave with its E -field
vector perpendicular to the cylinder axis, (a) Front
view. (b) Cross Section view. 4

Superposition for an illuminated cylinder with
N loaded slots, (a) Illuminated slotted cylinder,
(b) Illuminated solid cylinder, (c) Driven slotted cylinder

(a) Normalized backscattering cross section and
(b) Normalized forward scattering cross section, for a
solid cylinder as a function of cylinder size, ka

Normalized bistatic scattering cross section patterns,
a' (0 )/1ra, for solid thick cylinder with (a) ka: .0
(b)ka=.,50 and(c)ka= 10.0 . . . . . . . .

Slot impedance for zero backscattering as a function
of cylinder size ka, (a) Normalized resistive part of
load impedance, (b) Normalized reactive part of
load impedance . . ..

Normalized bistatic scattering cross section patterns,
0(9 )/1ra, for a thick cylinder loaded with one load at
0: -180° with (a) ka: 2. 0, (b) ka: 5. 0, and (c) ka: lO. 0

Slot impedance for zero scattering in directions 0 =l70°
and 190° as a function of cylinder size ka, (a) Normalized
resistive part of load impedance, (b) Normalized reactive
part of load impedance.

Normalized bistatic scattering cross section patterns,

0(9 )/1ra, for a thick cylinder loaded with two loads at

0: -I80° and 6: 190° with (a). ka: 2. 0, (b) ka: 5.0, and
(c)ka=10..0....
Slot impedance for zero scattering in directions 9 290° ,
and 180° as a function of cylinder size ka, (a) Normalized

resistive part of load impedance, (b) Normalized reactive
part of load impedance.

vii

26

27

29

30

31

32

33

Figure

4.8

4. 10

4.11

4. 12

4. l3

4. 14

4.15

4. 16

Page
Normalized bistatic scattering cross section patterns,
a" (6 )/1ra, for a thick cylinder loaded with two loads at
6 =170° and 190° for zero scattering in the directions
0 =90° and 180° with (a) ka=2. 0, (b) ka=5. 0, and
(c)ka=10.0................. 34

Slot impedance for zero scattering in the directions

0 =135°, 180°, and 225° as a function of cylinder size ka,

(a) Normalized resistive part of load impedance,
(b) Normalized reactive part of load impedance .

Normalized bistatic scattering cross section patterns,
6(0 )/1ra, for a thick cylinder loaded with three loads at
0 =170°, 180°, and 190° for zero scattering in the
directions 0 =135°, 180°, and 225° with (a) ka=lO. 0,
(b) ka=5. 0, and (c) ka:2. 0 . . . . . . . . .

Slot impedance for zero scattering in directions

0 =90°, 135° . 180° , and 225° as a function of cylinder
size ka, (a) Normalized resistive part of load impedance,
(b) Normalized reactive part of load impedance .

Normalized bistatic scattering cross. section patterns,
0(6 )/1ra, for a thick cylinder loaded with four loads at
9 =165°, 175°, 185°, and 195° for zero scattering in
the directions 0 =90°, 135°, 180°, and 225° with
(a) ka=lO. 0, (b) ka=5. O, and (c) ka =2. 0 . .

Slot impedance for zero scattering in directions

9 =90°, 135°, 180°, and 225° as a function of cylinder
size ka, (a) Normalized resistive part of load impedance,
(b) Normalized reactive part of load impedance .

Normalized bistatic scattering cross section patterns,
(7(9 )/1ra, for a thick cylinder loaded with four loads at

0 =0°, 90°, 180°, and 225° for zero scattering in the
directions 0 =90°, 135°, 180°, and 225° with (a) ka=lO,
and(b)ka=2.0..................
Normalized bistatic scattering cross section pattern,
0(9 )/1ra, for a thick cylinder ka: 5. O loaded with four

loads at 0 =0°, 90°, 180°, and 270° for zero scattering
in the directions 9 =90°, 135°, 180°, and 225°

Relative backscattering cross section as a function
of slot position with ka=5. 0 .

viii

35

36

37

38

39

40

41

44

Figure

Page
Relative backscattering cross section as a function of
the first slot position with ka=5. O . 45
Relative backscattering cross section as a function of
slot position with ka=5. O . . .. . . . . . . . . . 46
Existence of a solution for zero backscattering from a
cylinder loaded with two purely reactive loads
with 9 :180° and 6 =6 =0. 10 radians . 48

1 1 2

Purely reactive impedances for zero backscattering
from a two slot cylinder as a function of cylinder size ka.
(x1, x2) is first solution and (x'l, x'z) is the second solution

Normalized bistatic scattering cross section patterns,
0(9 )/1ra, for a thick cylinder, ka: 2. O, with two purely
reactive loads located at 6 =l60° and 180° . (a) First
solution, (b) Second solution . . . . . . . .

Normalized bistatic scattering cross section patterns,
0(0 )/1ra for a thick cylinder, ka: 5. O, with two purely
reactive loads located at 6 =160° and 180° . (a) First
solution, (b) Second solution . . . . . . . .

Normalized bistatic scattering cross section patterns,
0(0 )/-rra for a thick cylinder, ka=lO. 0, with two purely
reactive loads located at 9 =160° and 180° . (a) First
solution, (b) Second solution . . . . . . . .

Purely reactive impedance for zero backscattering from
a two slot cylinder as a function of cylinder size ka .

Purely reactive impedances for zero backscattering from
a two slot cylinder, as a function of slot position.
(x1, x2) is one solution and (x' , x'z) is the second solution

Relative backscattering cross section as a function of
slot position with purely reactive loading and ka: 5. 0

Relative backscattering cross section as a function of
purely reactive load impedances for cases N=l, 2, 3,
4 w1th zl=22=z3=z4=3x . .
Normalized bistatic scattering cross section patterns
for thick cylinders, ka: 2. 20, loaded with purely
reactive loads. (a) Minimum backscattering when N=1,
(b) Minimum backscattering when N=2 .

ix

50

51

52

53

54

55

56

58

59

Figure

4. 29

4. 30

4.31

4. 32

4. 33

4. 34

4. 35

4. 36

4. 37

4. 38

4. 39

Page

Normalized bistatic scattering cross section patterns
for thick cylinders, ka=2. 20, loaded with purely

reactive loads. 6:0. 10 radians in all cases.
backs cattering when N=3,
N=4, (c) Maximum backscattering when N=4 .

Optimum minimum and maximum backs cattering cross
section as a function of slot position for doubly and
singly loaded cylinders

Optimum slot reactance for minimum backscattering
cross section as a function of slot position . . .

Relative backscattering cross section as a function

of the first slot position, .' 0 .for one through four

slots with fixed equal purely reactive loading impedances,
zl=zz=z3=z4= JX .

Short circuited TEM parallel plane line.
(b) Schematic,
resistance

(a) Geometry,
(c) Short circuited line with series

Relative backscattering cross section as a function of
cylinder size ka for cylinders with (a) One slot, (b)
Three slots

Relative scattering cross section as a function of cylinder
size ka. (a) Backscattering cross section, (b) Bistatic
scattering cross section of 0 :l70° . . . . . .

Normalized bistatic scattering cross section patterns,
(7(9 )/1ra, for a thick cylinder loaded with impedances
Zl/=a6 z /a5= 2332+ja61252 at 61:17O° and e Z=19oo
wilthfil =Z—62=.005radians...

Normalized bistatic scattering cross section patterns,
«(0 )/1ra, for a thick cylinder loaded with impedances
/a6= z /a5= 2332+j2612$2 at 91:170° and e z=190°

w11th61=2-§=.005radians. . . . . . . . .

Relative backscattering cross section and load reactance
as a function of cylinder size.

Relative backscattering cross section as a function of
cylinder size ka for cylinders. (a) Broad- band loading,
(b) Large constant purely reactive loading -

Scattering model and experimental arrangement .

(a) Minimum
(b) Minimum backs cattering when

60

61

62

63

66

68

69

71

72

73

75

77

 

 

 

Page
Optimum minimum and maximum backs cattering
cross section as a function of slot position . . . . . 79
Cylinder with curved parallel plane line short circuited
at 4). (a) Cross-section view, (b) Equivalent Circuit
forcavityload................ 80
Relative backscattering cross section as a function of
the length of TEM cavity backing slot . . . . . . . 82

PART II

L00p antenna loaded with N impedances: (a) Geometry,
(b)Schematic................103
Coordinate system for the fields radiated by a 100p
antenna...................112
Real and Imaginary parts of 1/ao, 1/ a1, l/az, 1/a3
for Q = 12. (a) Real parts. (b) Imaginary parts . . . 116
Input susceptance of 100p antenna as a function of gap
widthforS2=10( )and52=12(----). . . . . 120
Input susceptance of 100p antenna as a function of kb
for 6b/a = 1. o, 10. o, and 20.0 withfl = 12 (i. e.,
a/b='0.0155) . ......... . 121
Comparison of finite gap theory and experimental con-
ductances of a 100p driven by a two wire line . . . . 124
Comparison of finite gap theory and eXperimental
susceptances of a loop driven by a two wire line . . . 125
Comparison of finite gap theory and experimental
admittances of a 100p driven by a coaxial line . . . . 126
Comparison of finite gap theory with 60b/a = 13. 6 and
experimental admittances of a 100p loaded at <1) 2 180°
wich1=00(i.e.,agap)- . . . . . . . 128
Admittance of circular 100p antenna driven by a finite
gap generator Gob/a = 12.1 and (S2 = 12 —), ($2 =
18 - II c- -) o o o o o o o a o o o o o o o o a 130
Impedance of circular 100p antenna driven by a finite
gap generator bob/a = 12.1 andQ =12 . . . . . . . 131
Magnitude and phase of the current distribution on a
loop antenna with kb = 5.0, $2 = 12, 5b/a = 12, (
magnitude), ( ----- phase) . . . . . . . . . . . 133

xi

Figure
3. ll

Magnitude and phase of the current distribution 2on
a 100p antenna with kb= 10. 0, :2- - 12, 5b/a :

 

( magnitude) (— - - - phase)
Gain patterns of a 100p antenna with kb = 1. 0, S2 = 12,
and 5b/a = 12. (a) 5: 0° plane, (b) 5 = 90° plane,

and (c) 0- " 900 plane

Gain patterns of a loop antenna with kb = 1.5, {2 = 12,
and 5b/ a : 12. (a) 5: 0° plane, (b). 5 : 90° plane,
and (c) 0— - 90° plane .

Gain patterns of 100p antenna with kb 5. 09 = 12,

and 5 5/2: 12. (a) 5 - 0° plane, (b) 5 90° plane,
and (d) 0 = 90° plane . . .
Gain patterns of 100p antenna with kb 0.0 $2 = 12,

and 6 b/2= 12. (a) 0— - 0° plane, (b) 4) = 90 plane,
and (8) 0: 90° plane . .

(a) Maximum and Minimum ogain of a loop antenna

kb - 1.0 loaded at 0 - -°180 with a purely reactive
load. (b) Optimum 0ad reactances for maximum

and minimum gain

(a) Maximum and Minimum gain in direction 0: 90°,
4): 90° of a 100p antenna loaded at 4) 180° with a
purely reactive load as a function of°kb. (b) Optimum
load rgactances for max. and min. gain in direction

=90 4>=90

(a) Maximum and Minimum gain in direction 9: 90°,

5: 180° of a 100p antenna loaded at 5 -180° with a

purely reactive load as a function of 13b? (b) Optimum

load reactances (for max. and min. gain in direction
=°,90 0:180 . .

Gain pattern of 100p antenna kb= l. 0 loaded at 0: 180°
with a purely reacgive load for minimum and maximum
gain at: (a) <1) = 90 and (b) (11:180o

Page

134

135

136

137

138

145

146

147

148

Gain pattern of a 100p antenna kb 2 5. 0 loaded at (b = 180°

with a purely reactive load for minimum and maximum
gain at 4) = 90

Load impedances as a function of load position for

a 100p agtelana of size kb= 1.0 811th the poattern Specified
as f (90,0 )- 1.0, f2(90°, 160) = f3(90°, 200° ) = 0.0,
and :12, 5b/a:1o . . . . .

Gain pattern of a 100p antenna, kb= 1.0, oloaded

with three impedances located at (bl = 85 , 02 1800
and 4) 275° ()The pattern is specified by f12(90°, 0° ..)
0.29:90",150°):£3,(9o°2oo°:) .

xii

149

153

154

Figure

4.8

409

4.10

4.11

4.12

Gain pattern of a 100p antenna, kb = 0.3, loaded

with three impedances at 01 = 85° .02 = 1800, and

03 = 275° oThedpatterno is specified by ...fl(90 0° )=
.,0 f2(90°, 160 ) = f 3(900, 2000 )=

Gain pattern of a 100p antenna, kb= 1.0, loaded
with five impedDances of 01 = 60° , 02 120°, 03 =
180° , 4) 240 , and ($35 = 300° and the p)attern0
Specifie e11 bar 11(90°, 0°) : 1. 0,1 (90° 900):
f3(90°,150 )= f4(90°, 210° )= f 5190°, 270 )=

Base tuning network .

Efficiency of 100p loaded with two impedances at

4’1 = 0° and (1)2 = 180° to match antenna with 300 ohm
input impedances. Compared to efficiency of base
matching network.

(a) Load reactances necessary for 300 ohm input im-
pedance to 100p. (b) Matching network reactances

necessary to match a 100p antenna 9- — 12, 6b/a— - 12.1

to 300 ohms .

xiii

Page

155

156

160

162

163

LIST OF TABLE

PART II
Page
Table 3. 1 Input susceptance of circular 100p antenna
as a function of terms retained in series
solution with kb = 2.5 and S2 = 12.0 . . . . . . 119

xiv

PART I

MODIFICATION OF EM SCATTERING FROM A THICK
CYLINDER BY MULTI-SLOT IMPEDANCE LOADING

CHAPTER I

INTRODUCTION

The modification of the electromagnetic wave scattered by a
thick infinite cylinder using the technique of impedance loading is in-
vestigated in this study.

When a conducting body is illuminated by an electromagnetic
wave a surface current is induced on the conducting body. This Sure.
face current, in turn, reradiates a scattered field. Distributed or
lumped impedances can be installed on the surface of the conducting
body which will alter the amplitude and phase of the induced surface
current, thus modifying the scattered field. This method of modifying
the field scattered by a conducting body is known as impedance loading.
The modification or control of electromagnetic scattering has appli-
cations in the areas of antenna design, electromagnetic compatibility,
scattering cross section modification and others.

The history and development of the impedance loading technique
can be found in a review paper by Schindler, Mack, and Blacksmith. 1
Since the writing of this review paper, scattering modification by the
impedance loading technique has received considerable attention?"17
A review of the literature shows that nearly all the studies have been
concerned with impedance loaded thin wire rods or 100ps. To present,
there have been only two published studies on impedance loading of
electrically thick objects. Liepa and Senior18 considered a conducting
sphere loaded with a single impedance backed circumferential Slot, *
and the authors15 considered a thick cylinder loaded with a single im-
pedance backed longitudinal Slot. There is a noticeable lack of any
information on the modification and control of the field scattered by a

thick object loaded with more than one impedance.

 

*Chen and Vincent 6’ 10also loaded a Sphere with two loaded wires
which is another technique of loading an object. This differs from the

discreet surface loading considered inlthis study.
1

This study is concerned with an electrically thick conducting
cylinder loaded with N impedance-backed longitudinal slots. The
cylinder is illuminated with a normally incident plane electromag-
netic wave polarized with its electric field vector perpendicular
to the cylinder axis. The incident field induces a circumferential sur-
face current on the surface of the cylinder. The control of the sur-
face current, Which in turn controls the scattering, is accomplished
by the longitudinal loaded slots which intersect the induced current.

The purposes of this study are l) to determine the extent of the
control over the scattering that can be accomplished with different
types of loading, 2) to develop and analyze procedures for determining
optimal loadings which result in zero or minimum scattering in desired
directions, 3) to determine what procedures lead to broad band loadings
and 4) to develop broad banding techniques.

In Chapter II the basic theory used in analyzing the scattering
from the loaded cylinder is developed. The theory is of a general form
into which all multi-loaded scatters fall. 19’ 2°

Chapter III deals with general synthesis procedures for finding
optimal impedances that result in zero or minimum scattering in one
or more desired directions. First, a procedure is deve10ped for
finding N load impedances which cause zero field to be scattered
in N directions. This synthesis procedure is similar to the one
used by Strait 21’ 22’ 23 in synthesizing radiation patterns of loaded
antennas and arrays. Secondly, a procedure is developed for finding
N purely reactive load impedances that result in zero field scat-
tered in N/2 desired directions. After this, procedures are deve1-
0ped for finding a single purely reactive load impedance that produces
minimum scattering in one direction and a set of N load impedances
which result in zero scattering in one direction at Ndi fferent fre-
quencies.

Numerical results of these procedures and other loading schemes
are presented and discussed in Chapter IV. The frequency dependence
and bandwidths of these loading techniques are also considered.

The theoretical predictions are confirmed with an experiment.
This is described in Chapter V.

Chapter VI summarizes the work presented in this study.

CHAPTER II

THEORETICAL FORMULATION OF PLANE WAVE SCATTERING
BY A MULTI-LOADED, SLOTTED CYLINDER

2. l. Formulation of the Problem and Boundary Conditions

A perfectly conducting cylinder of infinite length and radius a has
N impedance-backed longitudinal slots cut on its surface as indicated
in Figure 2. l. The center of the nth Slot is located at 0 = 0 In The
nth Slot has angular width °n and is loaded with an impedance Zn that
is lumped in this slot region on the cylinder surface. The cylinder is
illuminated by a plane electromagnetic wave which is linearly polarized
with its E -field vector perpendicular to the cylinder axis. This inci-
dent wave induces a circumferential surface current K0 (0 ) on the
cylinder, which in turn, radiates a scattered electromagnetic field.

The tangential E -field must vanish at the cylinder surface except
in the Slot regions since the cylinder is assumed to be perfectly con—
ducting. The slots are taken to be electrically narrow and thus the
tangential E -field is assumed to have a constant, uniform distribu-
tion within each slot region. The potential difference across the nth

slot is

0n.-5 /2 _
V = -S n Ee(r=a )ad9 = a6 E9(r=a',0=0 )
en+5n/2 n n

and is a slot voltage Since the slots are electrically narrow so that the
quasi-static approximation is valid. The boundary condition on the
tangential E -field at the surface of the illuminated slotted cylinder is
V
11

a6

. 15140-0 |<5/2
Ele (r=a+) + E: (r=a+) =( n n n

n=1,2,...,N

 

LO elsewhere (1)

 

(a)

 

N
<
co)

 

 

 

 

1 +
E A
o 1'
e 2 Ph‘. 9 . Z)
1 ._ 9 0:0
Hi x
a
(b)
Figure 2.1.

An infinite cylinder with N’longitudinal slots illuminated by

a plane EM wave with its E -field vector perpendicular to
the cylinder axis.

(a) Front view. (b) Cross Section view.

where E19 and E; are the 0 components of the incident and scattered

E -fie1ds, respectively. The nth load impedance is defined by

V
11

Zn K9 (0 =0n)

- Zn Hz(r=a, 0 :0 n) (2)

It is noted that the physical dimension of the load impedance Zn is

ohm-mete r.

2. 2, Superposition

The field scattered by a cylinder with N impedance—backed, longi-
tudinal Slots can be obtained by the superposition of the field scattered
by an unloaded solid cylinder illuminated by a plane wave and the field
radiated by a cylinder with N longitudinal slots having slot voltages
Vn’ n=1, 2,. . .,N impressed across the Slots. A mathematical state-

ment of this superposition is

Es : EC+Er (3)
"111° : fi°+fir (4)

Where E S and H 8 represent the field scattered by a slotted cylinder
illuminated by a normally incident plane wave, E C and 11° represent
the field scattered by a solid cylinder illuminated by the same incident
plane wave, and E r and Hr represent the fields radiated by a Slotted
cylinder driven by slot voltages Vn' The excitation of the nth slot, Vn'
must be determined in accordance with the total surface current on the
illuminated slotted cylinder at the location of the slot and the impedance
backing the slot. This superposition is indicated schematically in
Figure 2. 2.

The boundary condition (1) for the illuminated slotted cylinder
can be separated into the boundary condition for the illuminated solid

cylinder

 

3v

 

#00330 003on :00,qu on
£00510 0:00 003E533 3v £00510 003on 003E535 A3
.3on 00003 2 new? H00G2>0 000.235.32 :0 now sofiumomuoaom

 

 

 

 

l 1 3V 1 An
0.
+ 7m nm N
5 am e
L r L r

.N .N 3:30

v I

 

E; (r=a+) + E

C

+
9 (r=a) = O (5)

and the boundary condition for the driven slotted cylinder

 

 

Fvn 5n
- <
aén for '0 9n I 2
r + n : 1, Z, O 0., N
Ee (r=a ) :(
‘L 0 elsewhere (6)

 

Boundary conditions (5) and (6) define the scattering and radiation prob-

lems to be discussed in the following two sections.

2. 3. Scattering from a Solid Cylinder

Consider a perfectly conducting cylinder of radius a which is il-
luminated by a normally incident plane electromagnetic wave with an
E-field vector perpendicular to the cylinder axis. The geometry of the
problem is defined in Figure 2. 2(b).

The incident plane wave can be represented by the following field

 

expansions
H 1 : e-ka _ e-Jkr cos 0
z
co
-2 H" ( e ) J (kr)
— EOn J cos n n
n=0 (7)
i i 1
He _ Hr : Ez : O (8)
El : _J 8 H1
9 060 3r 2

: jgo E °0n (-j)n cos (n0) ng(kr)
n=0 (9)

E 1 —_. _:.1__ L .3)... Hi
1‘ 1.060 r 80 z
00
_ j _1_ _.n .
— (060 r °0n( J) n Sln(n0)Jn(kr)
n=0 (10)

where Jn (kr) is the nth—order Bessel function of the first kind and €On

is the Neumann factor and equals unity for n=0 and is equal to 2 other-
wise. The impedance of free-space to iS 120“ ohms, and k is the free-

space wavenumber. The ert ti me-dependence factor is implied.

The solution for the fields scattered by a perfectly conducting

 

 

 

 

infinite cylinder illuminated by a plane wave are well known25’ 2° and
are given by
C i n Hnm (111-)
- _ _' 1
Hz — °0n( J) cos (n0) Jn(ka) H (2), (ka)
n=0 n (11)
c c c
Hr'He‘Ez’O (12)
00 (2)
EC : 1 —1— 2 e (-')n+ln Sin (n0 ) J' (ka) H“ (kr)
r we r 0n J n (”T
O _ H (ka)
n—O n
(13)
° H (2)' (111-)
EC - g 6 (-j)n+1 cos (n0) J' (ka) n
0 7 0 On 11 H (2)' (ka)
n=0 n (14)

(2)

where Hn (kr) is the nth-order Hankel function of the second kind.

2. 4. Radiation from a Cylinder with N Driven Slots

The field radiated by a cylinder with N longitudinal Slots which
have voltages V1, V2, . . . , VN impressed across them [see Figure
2. 2 (c)] can be found by solving the boundary value problem subject to

boundary condition (6).

9

The magnetic field has only a 2 component which is governed by
the wave equation,

2 2 r
(V +k)I-Iz=0 (15)

The cylinder and Slots are infinitely long and the excitation is assumed

uniform axially, thus the radiated field has no z-dependence, that is
8 _

 

 

—a-; : 0 . The wave equation for H: becomes
2 2
8 l 8 l 8 2 r
[ + —— + + k ] H = 0
M2 r 8r r2 39 2 z

This partial differential equation can be solved by the method of

separation of variables. The appropriate solution is

00
H: : Z [An cos (n0 ) + Bn sin (110)] Hum (kr)
n=0 (16)

where An and En are unknown coefficients to be determined by boundary
condition (6) and HS) (kr) is the nth order Hankel function of the second

kind which represents an outward traveling cylindrical wave. The other
components of the radiated field can be determined from Equation (16)
and Maxwell's equations. Coefficients An and BH are found by applying

boundary condition (6) to the E; component of the field [See Appendix

A] and are

 

 

 

 

 

 

 

 

n6
6 N sin( m )
A : .2 0n (2)1, E V n62 cos (n0 m)
n J «ago Hn (ka) m=1 m ( m) (17)
2
e- N Sin (nfim )
B : .2 0“ ml. 2 v n62 sin (n0 )
n J «ago H (ka) _ m m m
n 111-1 ( 2 1 (18)

The field radiated from the cylinder with N driven slots is now

completely determined and is given by

10

 

 

 

 

 

 

 

 

 

 

1 n6
° ‘2’ (kr) N sin( m)
1131‘ - 1 Z n 2 V ——-——Z 0 0 ))
0 - 21ra €0n H (2)' (ka) m nfim cos(n( ' m
n:0 n m=1 ( 2 ) (19)
n6
r 1 on (2) (kr) N sin( 2m ) '
Er - wakr Z n (2), Z Vm n6 Sln(n(0-0m))
n=1 I{n (ka) m=1 ( m)
2 (20)
r _ r _ r _
Ez - Hr _ H0 - ° (21)
n6
r °° Hn ‘2) (kr) N sin; 2‘“)
H = ——-L— 25 Z Vm cos (n(0-9 ))
z 21151130 60n Hn n(2)' (ka) °m m
“=0 m 1("'2—") (22)

2. 5. Scattering from a Cylinder with N Loaded Slots

The Slot voltages Vn which excite the slots of the driven cylinder
must now be determined in view of our intent to combine the results of
the preceding two sections. The voltages Vn can be expressed in terms
of the impedances backing the Slots, Zn’ and the total surface current

on the illuminated slotted cylinder. From Equation (2).

V
n

-2 H (r=a, 0:0 )
11 Z 11

=3

.2 [H1+H°+Hr] r

 

N

-zn [—KnO + Z Vm Ynm]
m=1 (23)

11

where

7:
1|

-[Hiz +H:=]ra
9:6

11

n0

 

0° . p HZ( ) (ka)
.2 60p (-J) cos(p 0n) [Jp(ka)- -J p(ka)J Ha), (ka)
p=0 Hp

 

 

(D
_ :5 p+1 cos(pG n)
_ - 1Tka '
p:

(Z)
0 HI) (ka) (24)

which is the value of the surface current on the unloaded solid cylinder

evaluated at the position of the nth slot, and the terms

 

 

 

p6
sin( am) Hi?) (ka)
z _-.i__ 0 -0
ynm Zflaéo 0 (P5m ) H (2)' (ka) COS (p( n m”
P 2 P (25)

are the self and mutual short circuit radiation admittances of the slots.
The dimensions of these admittances are mho/meter.

Equation (23) can be written in matrix form as

 

r ‘r " " ' 1
V114r Y1 V12 ' ' ‘ y1N V1 K10
y21 "22Jr Y2 ' ° ' y2N V2 K20
L YNl y1512 ' ' ' yNN+YI\L bVN~ LKNO_ (25)

 

 

 

 

 

where Yn = l/Zn and is the load admittance of the nth slot. The v01-
tages Vn are found by solving the above matrix equation. From the
superposition picture it is seen that the voltages, Vn’ depend on the

Short circuited radiation admittances of the driven slotted cylinder

12

and the surface current on the illuminated solid cylinder.

The fields scattered by an infinitely long, perfectly conducting
cylinder with N impedance loaded longitudinal slots is now completely
determined and can be obtained by the superposition of the results of
the preceding two sections in accordance with Equations (3) and (4),
and the solution to Equation (26).

In the radiation zone the scattered field behaves as an outward
traveling TEM cylindrical wave, which can be observed by replacing
Hn(z) (kr) and its derivative with their principal asymptotic forms for

large arguments. 2'7 This procedure yields

 

 

 

 

 

7‘ (k /4) m I JP (ka)
sr __ -j r-w E g e cos(pO) '
E9 - - wkr e ’ 0 Op H (2) (ka)
p—0 p
N
60p )p+1 sin(—755531.)
+ 21ra H(2)' XV 136 C03 (P(9 -9m))]
(ka) m- 0 m )
Hp 2 (27)
sr sr sr sr
Er _ Ez _ Hr : H9 : O (28)
SI.‘ 51'
Hz : E9 /§0 (29)

where the second superscript denotes radiation zone fields.

2- 6- Bistatic Scattering Cross Section
The bistatic scattering cross section per unit length of the illumi-
hated slotted cylinder is given by

2
8

(7(9) = Iim 21rr {911—9—1—
r-roo E (30)

he re 6 defines the direction in Wthh the scattered field is received,

a.
nd the illuminating plane wave is incident from the direction 9 : 180° .

Using Equation (27), the bistatic scattering cross section can be
I.i‘-‘»‘l:en

l3

 

 

 

N Z
a(9):-E— 50+: vms]m
m=l (31)
where
°° J' (ka)
S = Z 6 C08(p0)—R-——-—
0 OP H (2) (ka)
and
a. Pém
1 +1 sin(—r—) cos (p(9 -6m))
srn : Zwag Z 60p”)p p6 (2)'
O :0 ( m) H (ka)
P 2 P (33)

Looking at Equation (31) from the superposition picture, So cor-
responds to the contribution of the solid cylinder to the bistatic
scattering cross section. The Sm coefficients multipled by the appro-
priate slot voltages correspond to the contribution of the driven

slotted cylinder to the bistatic cross section.

2- 7. Generalizing the Theory
The physical interpretation of the quantities KnO’ Ynm’ So’ and

S allows the theory just developed to be interpretated in a more
general manner. Consider, from the point of view of superposition,
the scattering of an EM wave by a multi-loaded conducting body of any
arbitrary shape. If the geometry of the loading is such that unique
load voltages can be defined, then a matrix equation, having a form
identical to Equation (26), relating the load voltages to the short cir-
Quit radiation admittances of the structure and the surface current on
a. 8 imilar unloaded structure can be formulated. The values of the
radiation admittances and the surface current may be found exactly,
as Was done in this chapter, or by some approximate method.
S:i‘:b-'=1'.i.lar1y a result for the bistatic scattering cross section of the

1“-1].ti-loaded body can be formulated in terms of the load voltages as

as done in Equation (31). Harringtonlg’ 20 has pursued this idea

14

in his multiport network parameter representation.

The loading techniques and synthesis procedures deve10ped in

the next chapter are completely general and can be applied to conducting
bodies of arbitrary shapes.

Mme; 4...... .. O- u
§dn5 _,

CHAPTER III

THE SYNTHESIS OF LOAD IMPEDANCES THAT RESULT
IN ZERO OR MINIMUM SCATTERING

FOR ONE OR MORE DIRECTIONS OR FREQUENCIES

In this chapter procedures are developed for finding load imped-
ances that result in zero scattering in one or more directions and load
impedances that result in zero scattering in one direction at several
frequencies. Procedures are also deve10ped to find purely reactive .

load impedances that result in zero or minimum scattering in one or

more directions.

3. 1. Zero Scattering in N Directions
A set of N load impedances thatcauses the field scattered by the

N—slotted, loaded cylinder to vanish in N directions may be determined by
the following synthesis procedure.

If the load admittances Y1, Y2, . . . , YN are assumed to be un-
knowns, matrix Equation (26) contains 2N complex unknowns, the N
load admittances and the N slot voltages. Since matrix Equation (26)
contains only N complex equations, N complex constraint equations may
be chosen to completely determine the problem.

It can be seen from Equation (31) that

SO(G=901)+§ VmSm(9=901) = 0
mzl

(34)
Implies the radiation zone field scattered by the loaded cylinder will
var-:11 311 in the direction 6 =6 01. Similarly a system of N complex con-
Str aint equations, involving the N unknown slot voltages, which force

12 .
he radiation zone scattered field to vanish in directions 601, 9 02, . . . ,

B
0N can be expressed as

15

 

l6

- I. (-
C11 ch ClN c10
C21 C22. C2N C20
0 c 1"”
N1 N2 NN J N0 (35) E
.J. b b ..

 

 

 

 

 

where Cij = Sj (9 :901).

The load admittances that result in zero scattering in the N direc-
tions are found from Equation (23) to be

rm_ ‘T-"“-"_‘
1

N
Yn : [KnO- Z ynmvm1/Vn
m” n=1,2,...,N (36)
where V1, V2, . . . , VN are the slot voltages found by solving Equation

(35).

The admittances found by the above procedure may have negative

re a1 parts which are difficult to physically realize. If the positions of

the slots are free to be changed, it is possible in many cases to find

to slot positions such that the load admittances will have positive real

parts, This t0pic will be considered in Section 4. 3.

In some cases when the load admittances have negative real parts
it may be more practical to consider purely reactive loading.
3" 2- Zero Scattering in N/Z Directions
Using Purely Reactive Load Irnpedances
In the previous section, the introduction of N complex load ad-
mittances led to the elimination of the scattered field in N directions
be Qa-use N complex constraint equations were allowed to be introduced.
If N purely reactive load impedances are considered, it is possible to
eliminate the scattered field in N/Z directions only. This is due to the

:E
act that N purely reactive load impedances gives the same number of

£8“

 

17

degrees of freedom as N/2 complex load impedances and thus only
N/Z complex constraint equations for zero scattering are permitted.
A set of N purely reactive load impedances that result in zero
field scattered in N/Z directions may be determined by the following
synthesis procedure.
The condition that the N load admittances be purely reactive is

equivalent to N real constraint equations, and may be written

Y +Y*=O n=1,2,...,N

, — ”“4“‘31
' I

where Yn* is the complex conjugate of Yn. Using Equation (36) this

condition can be written
N

.c * _ * 3k ’3 :
(Kn0an< + Kn0 Vn) Z (ynm van + ynm Vrn Vn) O
m: l

[MM-upw- rJHJ a .. r

n=1,2,...,N (37)

which is a set of N real nonlinear constraint equations involving the
slot voltages. (Note: N real equations are equivalent to N/2 complex
equations. ) To completely determine the problem N/2 complex con—
straint equations must still be chosen.
The scattered field will vanish in the N/Z directions 6 01, 6 02’
- - - , GON/Z if the slot voltages satisfy

 

 

 

 

 

 

r c “ r ‘
C:11 12 ' ' ' CIN 1 C10
. VZ
1L CN/Zl cN/Zz . . . cN/lei _CN/ZO.L
v
_ NJ (38)

Equations (37) and (38) comprise a system of nonlinear equations
W ~
hlch can be solved for the slot voltages, provided a solution exists.
Lee again, the load admittances may be found by substituting the

18

solution of Equations (37) and (38) into Equation (36). These load ad-
mittances will be purely susceptive and will cause the scattered field
to vanish in directions 9 01, 6 02, . . . 6 0N/2'

The special case of two slots (i. e. , N=2) is worked out in detail
in Appendix B. The admittances are found to be the solutions to a
quadratic equation. The position of the slots on the cylinder surface
is a crucial factor in determining whether or not solutions to this prob-
lem exist. This topic is discussed in detail in Section 4. 4.

The non-existence of a purely reactive loading that results in zero
scattering in one or more directions does not imply that the reduction
of the scattering to levels other than zero in these directions by a
purely reactive loading is impossible. In many cases the scattering
can be significantly reduced below the unloaded level by purely reac-

tive loading.

3. 3. Minimum Scattering Using Purely Reactive Load Irnpedances
In general, the field scattered by a loaded cylinder cannot be
reduced to zero in a given direction when the load impedances are
purely reactive and all equal (i. e. , jX = Z1 = Z2 = . . . = ZN). This does
not, however, rule out the possibility of reducing the scattered field
to a minimum in a given direction by a suitable choice of the loading
reactance. An optimum reactance, XOp’ for minimum bistatic scat-
te ring cross section can be determined by differentiating the bistatic
S Cattering cross section with respect to X and setting this derivative
e qual to zero. The optimum reactance is found by solving the resulting
e quation.
Applying this procedure to the case N21 yields the following
1' e 8 u1t.

= 1/Z[G¢~(GZ+4I] (39)

XOP
Whe re

sz + F2) - (c2 + D2) (42+ BZL

G :
(c2 + D2) (BE - FA) + D(E2 + F2)

19

I : AAZ+ Bz)+ (13E - FA)
(c2 + D2) (BE - FA) + 130::2 + F2)

and the constants on the right hand side of the above two equations
are real and defined by

A+jB s (e

O 0) In»
C+jD : y11(90)

_ 0 1
Equation (39) yields two solutions one of which results in a mini- !
mum and the other a maximum bistatic scattering cross section in the "
direction 9 :6 O .
Other procedures can also be developed for determining the op-
timal loading reactances for minimum scattering by cylinders with

more general configurations of purely reactive loadings.

3- 4. Zero Scattering at N Different Frequencies..

Many times it is of interest to modify the scattering properties
of an object at several different frequencies or over a large band of
frequencies. Consider the problem of the synthesis of N load imped-
ances that reduce the scattered field to zero in one direction at N

different frequencies to 1, (oz, . . . , wN.
At the first frequency, w l’ the constraint equation is
N
Clo+ Z Vm(w1) C1m : 0
m=l
Whe re CI'
The sec J

ond constraint equation is
N

c20+ Z Vm(“’2)CZm = 0
m=1

=Sj(9:90, wzwl).

20

where Czj = Sj (9 :9 0’ to 21.32). The slot voltages in the first equation,
{Vm (w l), mzl, 2, . . . , N} , and the slot voltages in the second equa-
tion {Vm (wz), mzl, . . . , N} are in general not equal and must be con-
sidered as independent variables. Likewise all other slot voltages at
the other frequencies must be treated as independent variables.

Using matrix Equation (26) all the voltages can be eliminated
from the problem and the constraint equations written directly in terms

of the admittances. Consider the nth constraint equation

 

 

 

 

 

 

N 1 .‘
Cn0 + Z Cnm Vrri (wn) : 0 i
m=l 1
Evaluating matrix Equation (26) at wzwn and augmenting it with the E‘
above constraint equation yields 4 '
r " " 1 r “
y11 + Y1 y12 ' ‘ ‘ y1N V1(“’n) K10
y21 y22+Yz ' ' ' y2N V2(“’n) K20
VNI yN2 ' ' ' YNN+YN KNo
Cn1 Cn2 ° ° ° CnN VN(wn) -Cn0
b ’ b ’ ‘ (40)

Where all the coefficients y nm and K n0 are evaluated at wzwn and the
load admittances Yn are assumed to be frequency independent.
The first matrix in nEquation (40) becomes N by N+l at this step.

E (in ation (40) however, can be rewritten as

21

 

V11+Y1 3'12 ' ' ‘ y1N “K10 Vlwn)
3’21 y22+Y2 ' ° ' y2N ‘Kzo V2(wn)

= 0
YN1 yN2 ' ' ' yNN+YN'KNO VN(“’n)
JLc:r11 an . . . an cno L1

 

 

 

which will have a solution only if the determinant of the coefficient
matrix vanishes. Similar arguments hold for all N frequencies and

this gives a system of N nonlinear equations in terms of the N load

admittance s .
y11+Y1 Viz ° ° ’ YIN ’Klo
+. . . . _
V21 3’22 Y2 y2N K20
= o
yN1 yN2 ' ' ' YNN+YN 'KNo
C:nl Cn2 ' ° ° CnN Cn0 (41)

 

 

for n = l, 2, . . . , N with all the Ynm and K n0 coefficients

evaluated at (0:an .

Provided a solution exists, this set of equations can be solved for the

load admittances Y1, Y2, . . . , Yn which will give zero scattering in the
direction 9 :9 O at the freq uenc1es ml 2, . . N'

The special case of zero scattering in one direction at two fre-

, w . , w
quencies is examined in detail in Appendix C. The impedances are
found to be solutions to a quartic equation.

Throughout this section it has been assumed that the scattering
was forced to zero in the same direction , 9 :9 0' at all N frequencies.

Examining the theory shows that this restriction is not necessary.

22

The scattering can be reduced to zero in one direction at one frequency,
a different direction at the second frequency, and so forth. However,
at any one frequency the scattering is still reduced to zero in only one
direction.

An assumption has been made through the develOpment of this
last procedure that the load impedances are constants with respect to
frequency. The practical application of this procedure as a broad
band technique is thus limited by the frequency dependence of the load

impedances actually available for implementation.

l “ . 'zm‘im'wstrJ‘agr, ‘.

CHAPTER IV

NUMERICAL RESULTS AND DISCUSSION

In order to gain a firm understanding of the theory and proce-
dures deve10ped in the previous two chapters a considerable amount
of numerical results were calculated. This chapter deals with the

presentation, interpretation, and discussion of these results.

4. 1. Numerical Method
The series 80’ Sm' ynm
the bistatic scattering cross section and the slot impedances were evalu-

ated on the Michigan State University CDC 6500 computer. Series S

, and Kno involved in the expressions for

O,
Sm, and Kno converged rapidly and the computations were straight

forward. Thirty terms were retained in these series. This gave
eight-digit accuracy over the range of cylinder size considered (1. e. ,
15 ka5 13). The theory is not limited to this range but for larger
cylinders it may be necessary to retain more terms to attain this ac-
curacy.

The evaluation of series Ynm' the self and mutual radiation ad-
mittance of the slots, is complicated by the slow convergence of its
imaginary part. Mathematically the imaginary part of ynm approaches
infinity as the slot width 6m approaches zero. Physically this implies
an infinite stray capacitance existing at the slot with an infinitesimal
gap width. The real part of ynm remains finite for any slot width cor—
responding to the existence of a finite radiation resistance for a slot
radiator. Thus the numerical calculation of ynm requires special
attention.

The actual computation of ynm is accomplished by summing 300
terms of the series. The first M terms of the series are treated
exactly, where M depends on ka and varies from 95 to 149. In the next

(ISO-M) terms the approximation

23

24

H1112) (ka) Yn (ka) Yn_l (ka)

~

 

 

Hn‘z)' (ka) — Y1; (ka) Yn (ka) ka

is made since I Yn (ka)] >> I Jn (ka)| for n>>ka. The Bessel func-
tions are replaced by their asymptotic expressions for large order

in the last 150 terms. This leads to the approximation

(2) -1
Hn (ka) 2 ( n-1)n‘1/2(eka) - L
Hn(2)' (ka) n 2n ka

 

 

 

where e = 2. 71828. . . . The real part of Ynm has eight-digit accuracy
while the imaginary part may be in error by as much as one per cent,

but in most cases the error is much less than this.

4. 2. Effect of the Slot Width

As discussed in the previous section, the slot width has a signi-
ficant effect on the imaginary part of ynm‘ It is thus reasonable to
expect the slot width to have some effect on the load admittances re—
sulting from the synthesis procedures. Numerically, extensive cal-
culations were performed for different load configurations at several
values slot width 6. It was found the real parts of the load admittances
obtained from the impedance synthesis procedure are only slightly af-
fected by the slot width, however, its effect on their imaginary parts
is more significant.

It was also found that the bistatic scatterirg patterns for zero
scattering in several directions are nearly identical for different slot
widths.

One more remark on the slot width is also important. The value
of 6 limits the size of a cylinder that can be considered with this theory
since the slot width 6a is assumed to be electrically narrow. If
6a< X/lO the slot can be considered to be electrically narrow and it

follows that the electrical cylinder size is limited by

ka<

 

Tr
56

25

4. 3. Zero Scattering in Several Directions

Numerical results of the impedance synthesis procedure of
Section 3. l are presented and discussed in this section. The slot con-
figuration and the directions in which zero scattering is desired are
specified and the impedances necessary to realize these scattering
modifications are calculated using Equations (35) and (36). It is of
interest to examine numerically the effect the number of slots and
their location (relative to the incident wave and directions of zero
scattering) have on the synthesized load impedances.

The superposition picture is useful in interpreting the numeri-
cal results. It should be remembered, the modified scattered field
of a loaded cylinder is the superposition of the field scattered from a
solid cylinder and the fields radiated by a series of driven slotted
cylinders whose driven slots are located at positions corresponding
to the loaded slots on the loaded cylinder [See Equation (31)] . The
slot voltages driving the slotted cylinders are determined by the sur-
face current on the unloaded cylinder, the short circuit radiation ad-
mittances of the slots, and the load impedances [See Equation (26)] .
The impedance synthesis procedure yields impedances such that the
radiation zone field ”radiated" by the driven slotted cylinders has ex-
actly the same amplitude and is 180° out of phase with the field scat-
tered from the unloaded cylinder in directions where the total scat-
tered field has been constrained to be zero.

Consider a brief description of the surface current and the bi-
static scattering pattern of a thick solid cylinder illuminated by a
plane wave whose E -field vector is polarized perpendicular to the

cylinder's axis. 25’ 26

The amplitude of the surface current is nearly
constant in the region about the center of the illuminated side of the
cylinder. Progressing toward the shadow region, it decreases nearly
linearly until it becomes slightly irregular in the center of the shadow
region. The backscattering and forward scattering cross section as a
function of electrical cylinder size ka are displayed in Figure 4. l and the
bistatic scattering cross section patterns for ka equal to 2, 5, and 10

are shown in Figure 4. 2. The backscattering cross section, forward
scattering cross section, and bistatic scattering cross section are

normalized to the geometric-optics value of the backscattering cross

tn

1T3

2122

TT3

26

 

 

 

 

 

 

I ' I T r I I T . T ‘7 I V
E1
1 8 - 9 4
l 6 _ q
iii ii

1.4 .. q
1 Z )- 4
1.0 - ..
0.8 " ..(
0.6 b .4
0.4 I- fl
0.2 _ ..

1 1 i 1 1 1 1 1 1 1 _1 1

0 Z 4 6 8 10 12

ka
(a)

I T I I I I I T T I I I
18 " '1
16 . ..
14 b q
12 ‘
10 1
8 +
6 ..
4 d
2 .

 

 

 

 

Figure 4. 1. (a) Normalized backscattering cross section and

(b) Normalized forward scattering cross section, for a
solid cylinder as a function of cylinder size, ka.

 

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Figure 4. 2. Normalized bistatic scattering cross section patterns,
0(6 )/-rra, for solid thick cylinder with (3.) ka: 2. o,
(b) ka=5. 0, and (c) ka=lO. 0 .

28

section, Ira. It is seen that the bistatic scattering pattern of a

thick cylinder is fairly uniform in the region around the backscat-

tered direction while the remainder of the pattern consists of many

sublobes with one very large lobe in the forward scattered direction.
The final component in the superposition picture is the radiation

pattern of a thick driven slotted cylinder. 29’ 30

The pattern has a
fairly uniform amplitude over a region of about 60° on either side of
the slot, then falls off to a much smaller value and becomes non-
uniform on the side of the cylinder opposite the slot.

Slot loading impedances, calculated by the procedure described
in Section 3. 1, that result in zero scattering in one, two, three and
four directions are displayed as a function of electrical cylinder
size ka in Figures 4. 3, 4. 5, 4. 7, 4. 9, 4. 11, and 4. l3. Correspon-
ding bistatic scattering cross section patterns for ka equal to two, five,
and ten are shown in Figures 4. 4, 4. 6, 4. 8, 4. 10, 4. 12, 4. l4, and
4. 15. The load impedances are normalized to the slot width 6a. These
normalized impedances are the wave impedances of the slot fields evalu-
ated at the center of the slots. It should be noted, the scales of the
impedances vary from figure to figure. The bistatic scattering cross
section patterns are normalized to the geometric-optics value of the
backscattering cross section, 17a.

The resistive parts of the load impedances are negative over a
large range of cylinder size for many slot configurations. This means
that a device with negativeresistance characteristics such as a. tunnel-
diode must be used in implementing these impedances. The reactive
parts are in general inductive and decrease in amplitude with in-
creasing frequency (i. e. , negative 310pe). In most cases both the
resistive and reactive parts become increasingly smooth and uniform
as the cylinder size increases.

Figure 4. 3 displays the slot impedance necessary for zero back-
scattering from a cylinder with one loaded slot located at 9 :180° as
a function of electrical cylinder size ka. Figure 4. 5 displays the slot
impedances necessary for zero scattering in directions 9 =170° and
190° from a cylinder with two slots which are located at 9 =170° and
190° . The resistive and reactive parts of the load impedance in

Figure 4. 5 are more nonuniform than those in Figure 4. 3. The load

R/a6 , ohms

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Figure 4. 3. Slot impedance for zero backscattering as a function of

cylinder size ka.
(a) Normalized resistive part of load impedance.
(b) Normalized reactive part of load impedance.

 

 

 

 

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attering cross section
cylinder loaded with one

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for
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Figure 4. 4. Normalized bistatic sc

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Figure 4. 5. Slot impedance for zero scattering in directions 6 =170° and
190‘ as a function of cylinder size ka.
(a) Normalized resistive part of load impedance.
(b) Normalized reactive part of load impedance.

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static scattering cross section patterns,
for a thick cylinder loaded with two loads at
and

' and 6 =l90° with (a)ka=2. 0, (b) ka=5. 0,

0(9 )/rra.

Figure 4. 6. Normalized bi

R/a6 , ohms

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Slot impedance for zero scattering in directions 6 =90" , and
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part of load impedance. (b) Normalized reactive part of load
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Figure 4. 9. Slot impedance for zero scattering in the directions
9 =135' , 180°, and 225° as a function of cylinder size ka.
(a) Normalized resistive part of load impedance.
(b) Normalized Reactive part of load impedance.

 

 

 

 

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Slot impedance for zero scattering in directions 8 =90° , 135°,
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(a) Normalized resistive part of load impedance.

(b) Normalized reactive part of load impedance.

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Figure 4. 12. Normalized bistatic scattering cross section patterns,
«(9 )/1ra, for a thick cylinder loaded with four loads at
6 =165°, 175°, 185°, and 195° for zero scattering in
the directions 9 =90°, 135°, 180°, and 2.25" with
(a) ka=lO. 0, (b) ka=5. 0, and (c) ka=2. 0 .

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Figure 4. l3. Slot impedance for zero scattering in directions 9 =90° , 135°.
180° , and 225‘ as a function of cylinder size ka.

’3; Normalized resistive part of load impedance.
b Normalized reactive part of load impedance.

 

 

 

 

 

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42

resistance in Figure 4. 5 is always positive which makes the implemen-
tation of this impedance easier than the impedances shown in Figure

4. 3. The bistatic scattering patterns corresponding to Figures 4. 3 and
4. 5 are nearly identical, but with a reduction of the scattered field
over a slightly larger angular excursion (i. e. , null width) in the

second case.

The cylinder considered in Figure 4. 7 has the same slot configu-
ration as the cylinder considered in Figure 4. 5, however, the direc-
tions of zero scattering are now taken to be 9 =90° and 180° . Com-
parison of these two figures shows that the load resistance in Figure
4. 7 is no longer always positive, but takes on large negative values.
Examining the corresponding bistatic scattering patterns indicates
that forcing the scattered field to be zero at 9 :90° has produced large
enhanced scattering in directions other than those constrained to have
zero scattering. The null widths have also markedly decreased. The
enhancement of the scattering is to be expected whenever the load
impedances have large negative resistive parts since this will, in
general, result in an increase in scattered power above that scattered
by the unloaded solid cylinder.

I This enhancement of the scattering, as shown in Figure 4. 8, can
also be explained by the superposition picture. The field scattered from
the thick solid cylinder and the fields ”radiated'' from two slotted cylin-
ders with slots located at 0 =170° and 190° , respectively, must sum
to zero at 6 =90° and 180° . The scattering enhancement arises from
the condition that the fields sum to zero at 9 =90° . The "driven" slots
are located at 170° and 190° so that 90° is in the region where their
radiation patterns have fallen off to a small value. Large driving vol-
tages are thus necessary to produce enough radiation in the direction
0 =90" to cancel the field scattered from the solid cylinder. Hence
in directions other than those constrained to have zero total scatter-
ing, the total scattered field from the loaded cylinder may be very
large. This type of argument indicates that large negative load re-
sistances and enhanced scattering in some directions other than those
constrained to have zero scattering might be expected whenever the
directions of zero scattering do not lay in the same region as the slots

are located. Figures 4. 12, 4. l4, and 4. 15 can also be explained by

43

this type of argument.

Figure 4. 9 diSplays the slot impedances necessary for zero scat-
tering in directions 6 =135° , 180° , and 225° from a cylinder with
three slots located at 6 =170° , 180° , and 190° as a function of ka.
The load resistances are in general negative but are very flat and have
fairly small amplitudes. The absolute value of the normalized load
resistances are less than 400 ohms for Z< ka< l3 and in particular
I Rzl /a6< 10 ohms for 5< ka< l3. Irnpedances that have a very simi-
lar form to these have been realized using Negative Impedance Con-
verters. 1 The corresponding bistatic scattering patterns are shown
in Figure 4. 10 and exhibit the widest null widths considered. It should
be noted that in this case the directions of zero scattering and the slot
positions lay in the same region and furthermore the slot positions
are in the center of the illuminated region of the cylinder.

Figures 4. 11 and 4. 13 display impedances that result in zero
scattering in directions 0 =90° , 135° , 180°, and 225° from a cylin—
der with four slots. The cylinder considered in Figure 4. 11 has all
its slots located in the center of the illuminated region of the cylin-
der while the cylinder considered in Figure 4. 13 has its four slots
equally Spaced around the cylinder so that one slot is in the center of
the illuminated region, one in the center of the shadow region, and the
remaining two slots are located on the borders between the two re—
gions. The load resistances in Figure 4. 13 are predominantly more
negative than those in Figure 4. 11. Also, the load reactances are
more irregular in Figure 4. 13 where all the slots are not located
in the center of the illuminated region. Considering the correspon-
ding bistatic scattering patterns, again it is seen that the larger the
amplitude of the negative load resistances is, the greater the enhanced
scattering is in directions other than those constrained to have zero
scattering.

Figures 4. 16, 4. l7, and 4. 18 examine the change in backscat-
tering cross section when a loaded cylinder is rotated while the load
impedances and relative slot positions are held constant. The cylin-
der considered in Figure 4. 16 has a load configuration identical to
the one considered in Figure 4. 3 and cylinder size of ka=5. 0. The

load impedance is taken to have the value of the impedance shown in

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Figure 4. 3 when ka=5. 0, hence, the backscattering is zero when
6 :180‘ . Likewise Figures 4. 17 and 4. l8 correspond to the load
configurations and impedances described in Figures 4. 7 and 4. 9 when
ka=5. 0. It is seen that changing the number of slots does not signifi—
cantly change the result. These results are typical of the results ob-

tained for other cylinder sizes.

4. 4. Zero Backscattering by a Cylinder Loaded
With Two Purely Reactive Impedances

Numerical results of the impedance synthesis procedure of Sec-
tion 3. 2 for the case of two slots (N=2) are presented in this section.
This synthesis procedure yields purely reactive load impedances. The
difficulty with this procedure is that the constraint equations are non-
linear.

The load reactances for the case of two slots may be found in
terms of a quadratic equation which is derived in Appendix B. Since
the load reactances are solutions to a quadratic equation, real solutions
do not always exist. The existence of a solution depends on the cylin-
der size, slot configuration, and direction of zero scattering. The
existence of solutions to Equation (B-7) for zero backscattering from
several different size cylinders each having one slot located at 6 :180°
and the second slot‘s position varied from 9 =O° to 0 =170° is indicated
in Figure 4. 19. An "x” indicates a solution exists for the particular
position of the second slot and cylinder size described by the position
of the ”x". Likewise, the absence of an "x" indicates no solution exists
for that particular geometry. Examining this figure it appears that
there is an area on the shadow side of the cylinder where no solution
exists when the second slot is located in this area. As the cylinder
size increases, the size of this area also increases. This trend was
examined and found to continue for larger values of ka than shown in
this figure.

Solutions for purely reactive loading impedances that result in
zero backscattering from a cylinder with slots located at 0 :160°
and 180° exist over the entire range of cylinder size 15 ka5 12.

These reactances are displayed in Figure 4. 20. The two solutions to

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49

the quadratic equation which determines these load reactances are
labeled (X1, X2) and (X' ,X'Z). The unprimed reactances are consid-
erably smoother than the primed set. The reactances have negative
slope and thus can not be realized by passive elements32, but work has
been done on realizing such reactances using active elements.

Comparison of the bistatic scattering patterns resulting from
cylinders loaded with the two different sets of reactances is made in
Figures 4. 21, 4. 22, and 4. 23. The shape of the scattering patterns
and the null widths differ considerably between the two cases.

Figure 4. 24 displays the reactances that produce zero backscat-
tering from a cylinder with slots located at 6 :175° and 185° . A
solution exists only in the regions 15 ka5 2. 11 and 2. 535 ka5 3. O4
and it is very irregular. Due to the symmetry of the two slot loca-
tions with respect to the incident wave and the directions of zero scat-
tering the two solutions are degenerate and thus only one distinct set
of load reactances exists. Comparison of Figures 4. 2.0 and 4. 24 shows
that a relatively small change in a slot configuration may drastically
change the region over which solutions for zero backscattering exist.

Purely reactive loading impedances that result in zero backscat-
tering from a cylinder whose two slots are located 20° apart are dis-
played as a function of the first slot position in Figure 4. 25. Solutions
exist over only a small region mainly in the center of the illuminated
side of the cylinder. Although not shown by the figure, no solutions
exist in the region 0501 S 40° .

Figure 4. 26 examines the change in backscattering cross sec-
tion when a cylinder loaded with two reactive slots located 20° apart
is rotated with respect to the incident wave. The load reactances are
held constant and are chosen to give zero backscattering when 61 =160°
[See Figure 4. 20]. The two curves represent the two different solu-
tions that are produced by the synthesis procedure. It can be seen
that the position of the slots is considerably more critical in the case
of the second solution.

An examination of the numerical results of this section seems to
indicate that the best positions the slots can be located in, such that
a solution to the synthesis procedure exists, are in the center of

the illuminated region of the cylinder. Of the slot configurations

 

 

 

          

 

 

 

 

 

 

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two-slot cylinder as a function of cylinder size ka. (x x2)

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and 180 ° .

(b) Second solution.

Normalized bistatic scattering cross section
6(9 )/-na for a thick cylinder, ka

reactive loads located at 9 =160°

(a) First solution.

Figure 4. 22.

 

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(b)

Figure 4. 23. Normalized bistatic scattering cross section patterns,
«(6 )/Ira for a thick cylinder, ka=lO. 0, with two purely
reactive loads located at 9 =160° and 180° .
(a) First solution. (b) Second solution.

10000

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two-slot cylinder as a function of cylinder size ka.

 

12

Figure 4. 24. Purely reactive impedance for zero backscattering from a

55

 

 

 

 

 

 

 

 

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Figure 4. 25. Purely reactive impedances for zero backscattering from a two-
slot cylinder, as a function of slot position. (x1, x2) is one
solution and (iii, x'z) is the second solution.

56

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considered, one was found that yielded solutions over the entire re-
gion 15 ka5 13.

It should be pointed out, that even though no solution exists for
a purely reactive loading that results in zero backscattering, a suit-
able set of load reactances can usually be found which significantly

reduce s the backs catte ring.

4. 5. Scattering By a Cylinder Symmetrically Loaded
With Equal Purely Reactive Irnpedances

It has been shown that the backscattering cross section of a
discretely loaded cylinder is strongly dependent on the orientation
of the cylinder with respect to the incident wave. Thus if a cylinder
is slowly rotating about its axis, or is randomly orientated with re-
spect to the incident wave, the impedance loading schemes previously
discussed must be modified to remain effective. One method to
overcome this problem is to adjust the loading impedances as the
position of the cylinder changes. A second, simpler method might be
to symmetrically load the cylinder with several loadings in an at-
tempt to reduce the sensitivity of the backs cattering to the cylinder
orientation. This second method is now considered.

This section examines the scattering modifications that can be
obtained by a cylinder loaded with one, two, three, and four purely
reactive slots located symmetrically around the cylinder. Further-
more, the restriction is added that all the reactances loading a cylin-
der are to be equal (i. e. , X:X1=X2:X3:X4).

Figure 4. 27 displays the relative backscattering cross section
of cylinders of size ka:2. 2 having one, two, three, and four sym-
metrically located, purely reactive loaded slots. Since all the slots
of a cylinder are loaded by reactances having equal value, the net
effect is very similar to that of a single load. The result for N=3
is superimposed and indistinguishable from the result for N21 over
the portion of the curve where X< O. The curves approach asymp-
totic values ranging from approximately -7db for the case N22 to
-5db for N=4 for large values of inductive and capacitive loading.

The greatest minimization of the backscattering is attained by the cyl—

inder with two slots. The ability of the loading to minimize the

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64

backscattering degenerates as the number of slots is increased to

three and then four. Figures 4. 28 and 4. 29 display the corresponding
bistatic scattering cross section patterns for the cases of minimum
backscattering when N: l, 2, 3, and 4 and maximum backscattering when
N=4. Only the upper half of the scattering patterns are displayed

since the lower halves are identical to the upper halves due to symme-
try.

Figure 4. 30 displays the optimum minimum and maximum back-
scattering cross section as a function of slot position for cylinders of
size ka=2. 20 loaded with one and two purely reactive slots. In the
case of a single slot, it was found in general, the control over the
scattering was markedly decreased for ka>5 and also when the slot
is located in the shadow region of the cylinder. The introduction of
the second slot considerably increases the slot positions where signi-
ficant reduction in backscattering can be accomplished. Figure 4. 31
displays the load reactance required to obtain the Optimum minimum
backscattering from a single slotted cylinder as described in the
previous figure. The technique described in Section 3. 3 was em-
ployed in calculating the results in the last two figures.

The backscattering cross section as a function of slot orienta-
tion for fixed, purely reactive loading is displayed in Figure 4. 32.
Cylinders with one, two, three, and four slots are considered and in
all cases the load reactances are chosen to minimize the backscattering
at 0 1:180“ . Significant reduction of the backscattering is obtained
over the largest region of slot orientations when the cylinder with
three slots is considered. In all cases some slot orientations exist
where enhancement rather than reduction of the backscattering is

experienced.

4. 6. The Frequency Dependence of the Modified Scattered Field

The techniques that have been discussed in this chapter are con-
cerned with modifying the field scattered by a cylinder at one fre-
quency only. It is usually desired, however, that the scattering be
modified over a band of frequencies. In this section the frequency
dependence of the fields scattered by loaded cylinders will be dis-

cussed.

65

The frequency dependence of the field scattered by a loaded
cylinder depends directly on the frequency dependence of the load im-
pedances. For example, if a set of load impedances can be found that
have exactly the same frequency dependence as the desired loading
that results in zero backscattering, zero backscattering will be at-
tained at all frequencies.

Three types of load impedances will be considered: (1) the
short circuited TEM parallel plane line; (2) the short circuited TEM
parallel plane line in series with a resistance; and (3) the constant im-
pedance (i. e. , an impedance that is constant with respect to fre-
quency).

The geometry of the short circuited TEM parallel plane line is
shown in Figure 4. 33. The input impedance of the line is purely re-

active and given by33

Z. = j Z tank! ohm-meter
1n 0

where

ZO = g0 d ohm-meter

is the characteristic impedance of the line, I is the length of the
line, d is the separation between the parallel planes and {,0 is the im-
pedance of the medium between the parallel planes. This type of
impedance is easily implemented behind the slots in a cylinder sur-
face as a load impedance. This topic is discussed in detail in Chap-
ter 5.

The short circuited TEM parallel plane line with a resistance
in series yields an impedance with a constant resistance and a reac-
tance which behaves as a short circuited line. This type of impedance
could be easily implemented by installing a narrow resistive strip
along each input terminal of the parallel plane line dis cussed above.

No attempt is made to explain the implementation of a constant
impedance. It is presented to give a comparison of the frequency de-
pendences of differing cylinders and loadings. All end effects at the

junction of the load impedance and the cylinder surface are neglected.

66

 

(a)

o

 

 

 

 

 

 

 

LL

(C)
Short circuited TEM parallel plane line.

(a) Geometry (b) Schematic (c) Short circuited line
with series resistance.

Figure .4. 3 3.

67

The bandwidth of a loaded scatterer is defined as the frequency
band over which the backscattering cross section is reduced by
10db or more below the level of the backscattering cross section of
an unloaded cylinder of tte same size. The backscattering vs. fre-
quency curves are not symmetric about the point of zero or minmum
backscatte ring. Hence, it is convenient to define an upper half band-
width, UHBW, which is the portion of the bandwidth. above the point of
zero or minimum backs cattering and similarly a lower half bandwidth,
LHBW. The bandwidths are given in percent of the frequency of zero
or minimum backs cattering.

The frequency dependence of the backscattering from a one-
slot cylinder loaded with a constant impedance is shown in Figure
4. 34(a). The backscattering cross section of the loaded cylinder is
normalized to the backscattering cross section of an unloaded cylin—
der of the same size and plotted as a function of ka, which is linearly
proportional to frequency. The load impedance is obtained from
Figure 4. 3 with ka=6. 5 which results in zero backscattering at this
frequency. The UHBW is 43% while the LHBW is 18% for an overall
bandwidth of 61%. Figure 4. 34(b) displays the frequency dependence of
a three—slot cylinder loaded with a constant impedance obtained from
Figure 4. 9 with ka=6. 5. The bandwidth is extremely narrow. Com-
paring Figures 4. 3 and 4. 9, the impedances necessary for zero back-
scattering, shows them to be very similar for ka> 4, yet the corre-
ponding curves in Figure 4. 34 differ greatly.

Figure 4. 35(a) compares the frequency dependence of the back-
scattering from a two-slot cylinder loaded first with a TEM line in
series with a constant resistance and secondly a constant impedance.
In both cases the value of 1:12 load impedances at ka=6. 4 is set equal
to the value obtained from Figure 4. 5. This results in zero scat-
tering in directions 9 =170° and 190° at ka=6. 4. The characteristic
impedance of the TEM line is ZO/a6 = 240v ohms which corresponds
to an air-filled parallel plane line with the planes separated by twice
the slot width, that is d=2a6.

In the case of the TEM line in series with a constant resistance
the UHBW is 14% and the LHBW is 25%, while in the case of the con-
stant impedance, the UHBW is 49% and the LHBW is 43% which is a

68

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Figure 4. 34. Relative backscattering cross section as a function of
cylinder size ka for cylinders with (a) one slot,
(b) three slots.

 

69

 

 

 

 
 

 

 

    
  
  

 

 

 

 

 

  
 

 

 

 

 

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Figure 4. 35. Relative scattering cross section as a function of cylinder

size ka. (a) Backscattering cross section. (b) Bistatic
scattering cross section of 9 :170° .

70

significant improvement. Comparing the frequency dependence of
the short circuited line (i. e. , tank! which has a positive 810pe) and
the constant reactance to the reactance in Figure 4. 5(b) (which has
a negative 810pe), shows that the constant reactance matches the de-
sired reactance in Figure 4. 5(b) better than the short circuited line
does. This explains the wider bandwidth in the case of the constant
impedance. Figure 4. 35(b) displays the bistatic scattering cross
section at 9 =170° as a function of frequency for the same constant
impedance loading used in Figure 4. 35(a). The UHBW is 38% and
the LHBW is 58% with general shape of the curve similar to the back-
scattering cross section curve shown above.

Figures 4. 36 and 4. 37 describe the frequency dependence of
the bistatic scattering pattern of a cylinder loaded with the same con-
stant impedance load configuration that was a consideration in the
previous figure. The bistatic scattering patterns are normalized to
the geometrical-optics value of the backscattering cross section.

The frequency dependence of the backscattering cross section
of a cylinder loaded with two purely reactive loads that yield zero back-
scattering when ka26. 5 is displayed in Figure 4. 38(a). Three differ-
ent types of load reactances are compared: (1) A short circuited TEM
parallel plate line with Zo/a6 : 120w ohms, 1 l/a = 0. 197, and
1 2/a = 0.402., (2) A short circuited TEM parallel plate line with
ZO/a6 = Z401r ohms, 1 1/a : O. 159, and I 2/a = O. 440. , (3) A constant
reactance with Xl/aé = 12.67. and XZ/aé = -220. 5. The bandwidths
for the three types of loading are: (l) UHBW 3%, LHBW 3%, (2) UHBW
5%, LHBW 4%, and (3) UHBW 18%, LHBW 12%. The values of the
load reactance for the two cases of short circuited TEM lines are
shown in Figure 4. 38(b). Comparing these reactances with the de-
sired reactance function displayed in Figure 4. 20 explains the dif-
ferences in bandwidth for the three different types of loading. Com-
paring the curves for constant reactance loading and those for con-
stant impedance loading (i. e. , with non-zero resistances) shows that
in most cases the purely reactive loading has a significantly narrower
bandwidth than general impedance loading.

It has been seen that the reactances necessary for zero back-

scattering generally have a negative slope as a function of frequency

71

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6? {...—...... .

«66:— =. a

’6 :9»:

9’3}:

isles/5%.

./ I” ‘G l

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O’O’fla a

$44.3... ,

0": 6.. o .

Mummflwg, 4......

ummnn/u/fl “Nun 1mm“

mmmumss/m... . .an . _ .
\\\\ § II t ......t... 1 .. ,_ 4 =5? .. , t u 52
‘\\\\\ ... II x _ 4 .4. L, .= =2: 45’ 4;. f4: ; n == =— a,
es s 444 . a/ l 44. i .3. 4 =5: 4?; t E. a _ =2}.
“‘ \N = u. I ’ 4 t4 ,r :— —= 59 4.. 4 c. = ——= pr

\s§s~ss44 a _ .O 4 m4 4%. i. & _“_._==.fi-5:~ 4.4,.:44H.4mm_u= E... E:

 

05 radians.

igz/aé
Z“ .

atterns, «(9 )/1Ta,
5

1/a6
1:

with 6

Z=l90°

for a thick cylinder loaded with impedances Z

Normalized bistatic scattering cross section p
2332+j26129 at 61:170° and 6

Figure 4. 36.

72

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A; "I. _ H W. ....\+.
.41 . _- , &.;N . _
-tmr .541: ......1.
a n . r . ...; : . . n
.w _. 1. .14 . a”
L .1 + ... .. 5‘...
1 .1? Jt. .
. ., ...:.....,.f~.+..._ ... f
. Mluunl: 1.1.x. ..
: . ,_ _ ..
fiwi 5:: ,
..IJ i

 

 

 

 

 

 

 

 

 

ka

 

 

 

 

 

 

   

I

I
I:
II
I
l

67
4; ’71
.72.
1'1.

.2”

Q

\

-.mgpw yj,._,_w MM§®

’77
I
c

’ O

4
avail/ll"!

:4
’0
’1

/
z

“3.4%

41a,» . . .10 .u . . .

. 4 4.04.. a. w: 1... .../x 5%,. Wa/v b. . ,4. . ...), ..,/../.H,..,,.M/.a,./ ”NEW. ._ _ “a . rag/”Harri!” .
_. a .. ._ s , ... . .4,\ ... .344 .4 . . q. .... , .... . . .I
...;4:; 3;: 42% a 5 see: . . ..

 

Z /a6
65 radians.

O.

1/a6
1:52:

Normalized bistatic scattering cross section patterns, o'(6 )/Tra,
2332+j26129 at 91:170° and 92:190° with 6

for a thick cylinder loaded with impedances Z

Figure 4. 37.

Load Reactances (ohms)

73

 

 

 

 

 

 

 

 

      
 

 

 

 

 

 

 

 

a 10 I I I I I I I I I r 1* I
'3
" solid 4
m C I“ .
/ cylinder
.b 0 M" I A
15. '
‘8 ‘ "
0
U)
m -10 _ ..
U!
o
H
U I- 4
DD
.5
3 -20 .. _,
1“; shorted TEM line - I'
E :- w1th Zo/a6:1201rfl N: ...
9 =l60°
8 -30 ....-- shorted TEM line 1 _,
f. with Zo/a6:2401rfl 92:180°
.2. .. 6 =6 :0. 05 rad .,
... 1 2
2 _._.. constant reactance
; 1 1 1 1 1 1 1 1 1 1 1 1
0 2 4 6 8 10 12
ka
(a)
I I I: I I I I I
4000 - .I .
O
.. _: l
l
2000 F' I
0 Zo/aoslzossz
/
- XI — . /
I
x2 -"- /
~2000 .. .
' "z _ '— .
Z 6:240 52
-4000 _ 0/“l " i
1 1 1 :I 1 1 1
0 Z 4 6
ka

(1))

Relative backscattering cross section and load reactance

Figure 4. 38.
as a function of cylinder size.

74

while a short circuited TEM line has a tank! dependence, which has
a positive SIOpe. Furthermore, in many cases large reactances are
required which force the short circuited line to be Operated near
antiresonance (i. e. , k! ~ 11/2). In this region tank! is a rapidly
changing function of frequency which is undesirable from the view-
point of bandwidth. Increasing the characteristic impedance of the
line moves the operating region away from the antiresonance point
thus in most cases increasing the bandwidth. 13

Examing Figure 4. 27 suggests a broad band reduction of ap-
proximately -5dB in the backs cattering might be attained with very
large purely reactive loading. This is confirmed for the case of a
constant reactance of Xl/aé : 20, 000 ohms in Figure 4. 39(b). The
practical problem with this scheme is in realizing a very large reac-
tance which is constant with respect to frequency.

The loading technique developed in Section 3. 4 for zero scat-
tering in one direction at several different frequencies suggests a

method which might lead to broadband scattering modifications.

75

 

 

 

    

 

 

 

 

 

 

 

 

 

I I I I I I I I I I I l
23 10 _ ‘
£0 0 _
b o ‘
d 0
.2
8
O "' u
U)
3 -10 .
0 d
L.
U
DO "' —:
.5
:3 -20 , .
3 h constant impedance NtZ -
3 Zl/aéz-969+jl70052 01:158°
:2 - z /a6:680+j423§2 0 :180° -
U z 2
cu - _
.o -30- (SI—62-0.05 q
0
.2
4‘; l 1 l l I 1 L l l l l L
'53 0 2 4 6 8 10 12
0: ka
(a)
... I I I I I I I I I I I I
to
3 10 — .:
an o
b o .. ..
§ 0
‘5
d)
m h A - d
U) NV __.
U)
2 -10 .. ..
U
DO
.5 b st ..
E, o e l=0°
u -2 _ _ a d
8 02-180
In
,5 1- 61—62‘0 05 rad .1
g constant reactances
g ’30,. Zl/a5=ZZ/a6
g :jZO, 00052 ‘1
3 1 1 1 1 1 1 1 1 1 1 1 1
m 0 Z 4 6 8 10 12
ka

0?)

Relative backscattering cross section as a function of
cylinder size ka. for cylinders.

(a) Broad-band loading. (b) Large constant purely reactive
loading.

Figure 4. 39.

76

CHAPTER V

EXPERIMENT AND COMPARISON TO THEORY

To confirm the preceding theoretical predictions a series of
backscattering measurements were performed on a cylinder with one

purely reactive loaded slot.

5. 1. Experimental Model and Experiment

The experimental model [See Figure 5. 1(a)] consisted of a
cylindrical brass tube, 7/8-inch OD, 3/4-inch ID, and 36 inches
long, with a l/8-inch wide longitudinal slot cut in its surface. The
slot impedance is implemented by installing a curved parallel plane
TEM line interior to the cylinder. The inner wall of the slotted cylin-
der forms one of the conductors of the line, with the outer surface of
a brass cylinder of CD l/Z-inch, installed coaxial with the slotted
cylinder forming the other conductor. One end of the line Opens at
the slot in the cylinder's surface while the other end is short circuited.
The short location is adjustable so that the length of the line can be
varied, which in turn varies the slot impedance.

The experiment is conducted inside an anechoic chamber at
frequencies ranging from 8. 4 to 9. 4 GHz. The experimental arrange—
ment and block diagram of the test instrumentation are shown in
' Figure 5. 1(a) and (b), respectively. The source separation method34
is used to measure the backscattering cross section of the cylinder.

The horn antenna does not illuminate the cylinder with a plane
wave. The amplitude and phase of the incident wave vary along the
axis of the cylinder. It was found that by placing the cylinder about
ten wave lengths in front of the horn the consequences of the nonuni-
form illumination and the end effects (arising from the finite length
of the scattering model) were small, while the detection system

provided the desired sensitivity.

77

 

 

 

 

 

us" I. I
1!— 1 l ..
(K i :3 7‘8 >
f...

 

..m-I-H "_ 36H

 

 

 

adjustable short

a) Experimental model of slotted cylinder

 

 

R. F. absorber
covers 6 walls

 

b) Anechoic Chamber

horn
antenna

V

load fre q. direct. hybrid amp.
isolator mete r coupler T det.

matched

 

 

term.

 

c) Block diagram of instrumentation

Figure 5. l. Scattering Model and Experimental Arrangement.

78

5. 2. Comparison of Theory with Experiment

The first comparison is made with data which depends on the
slot orientation but only indirectly on the value of the load impedance.
Figure 5. 2 compares experimental data and theoretical calculations for
the maximum and minimum backscatte ring cross section, which can
be achieved by one purely reactive load, as a function of slot location.
The experimental points were determined by setting the position of
the slot, then determining the maximum and minimum possible back-
scattering by varying the short position. The theoretical results were
calculated from Equation (31) with the reactances for maximum and
minimum backscattering calculated from Equation (39). The agree-
ment between experimental and theoretical results is excellent.

In order to compare results directly involving values of the load
reactance, a mathematical model must be deve10ped for the impedance
backing the slot in the experimental scattering model. The load im-
pedance of the slot is modeled as a short circuited TEM parallel
plane line [See Section 4. 6] in series with a lumped reactive imped-
ance which accounts for end effects and the right angle bend at the
input end of the line [See Figure 5. 3(b)] . The length of the parallel
plane line is taken to be the mean length of the curved line

I
1 = iii-13¢ = 0.7937so cm.

where a' is the inner radius of the outer cylinder, b is the outer ra-'
dius of the inner cylinder, and <1> is the angular displacement of the
adjustable short [See Figure 5. 3(a)] . The separation of the plates

of the line is
d: a' - b: 0.13175 cm.

The approximate model of the impedance backing the slot in the ex-

perimental scattering model is

Z1 : jX + jl. 197 tan (0. 007937k¢) ohm-meters.

E

79

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4 4 4 4 L 4 4 4 4 4 4 4 4 4 4 4 4

 

 

80

(a)

 

 

 

 

 

 

 

 

EN]

 

 

21 ngod tan k!

l

 

(b)

Figure 5. 3. Cylinder with curved parallel plane line short circuited at ((3.
(a) Cross-section view. (b) Equivalent circuit for cavity load.

81

Figure 5. 4 compares theoretical and experimental results of
backscattering as a function of the angular length of the cavity. The
experimental points are obtained by setting 9 1:180° and observing the
backscattering while varying the position of the short, (I). The theo-
retical calculations involve first calculating the slot impedance for a
given 4) using the above expression, then calculating the backscattering
cross section from Equation (31). The lumped end effect reactance
XE was obtained by matching the position of the first minimum point
of the theoretical and experimental results which required a 12 degree
shift. This corresponds to an inductive reactance of O. 427 ohm meters.
The agreement between the theoretical and experimental results is
again excellent. This [indicates that not only is the theory valid, but

the approximate model for the curved parallel plane line is reasonable.

82

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om o a
m
m

CHAPTER VI

CONCLUSIONS

In the preceding chapters the scattering behavior of a con-
ducting cylinder loaded with N impedance backed longitudinal slots
and illuminated with a normally incident plane electromagnetic wave
polarized with its E -field vector perpendicular to the cylinder axis
has been considered. The slots were assumed to be electrically nar-
row but finite and with constant electric fields across them. Under
this assumption the analysis was exact.

It has been shown that the field scattered by a cylinder loaded
with N slots can be:

1. reduced to zero in N directions when the load

impedances are complex and can take on all
positive and negative values,

2. reduced to zero in N/2 directions when the

loading impedances are purely reactive,

3. reduced to zero in one direction at N different

frequencies.

Synthesis procedures have been deve10ped for finding load
impedances that produce the above results. For Case (1) the proce-
dure is straightforward, involving only the solution of a system of
linear algebraic equations. On the other hand, the constraint equa-
tions involved in the procedures for Cases (2) and (3) are nonlinear.
This complicates the procedures, and in fact, solutions to these equa-
tions do not always exist.

It was found the position of the slots on the cylinder surface
is a critical factor in whether or not solutions exist to the last two pro-
cedures. The case of a cylinder loaded with two purely reactive slots
has been numerically examined in detail. A set of slot positions has

been found such that solutions exist for zero backscattering over the

83

84

entire range of cylinder size considered (i. e. , 15 ka5 13). These
numerical results seem to indicate that the best positions for the slots
are in the center of the illuminated region of the cylinder. For these
slot positions, purely reactive loadings which significantly reduce the
scattered field in the desired directions can usually be found even when
no solution exists for zero scattering in these directions.

The positions of the slots were also found to be important
factors in determining the form of the bistatic scattering cross section
patterns and null widths for all types of loading impedances. Two dif-
ferent slot configurations having the same number of slots and both
being constrained to have zero scattering in the same directions, may
have grossly different bistatic scattering patterns and null widths.

The frequency dependence of the fields scattered by the loaded
cylinders was considered for three types of load impedances.

l. a short-circuited TEM line.

2. a short-circuited TEM line in series with a

constant resistance.
3. a constant impedance (i. e. , constant with
respect to frequency).
Considerably wider bandwidths are, in general, obtained with load im-
pedances which have resistive parts rather than purely reactive load
impedances. Bandwidths of nearly 1:1 are demonstrated.

An experiment has been performed which confirms the theory.

The basic advantage of multiple impedance loading, over load-
ing by a single impedance, is that it gives additional degrees of free-
dom which can be used to control the scattering of an object over both

space and frequency domains.

85

APPENDIX A

DERIVATION OF FOURIER COEFFICIENTS

In this appendix the Fourier Coefficients (l7) and (18) for the
problem of radiation from a cylinder with N driven slots are derived.
The 2 component of the fir-field is given in terms of the un-

known coefficients An and En.

H: : Z [Ancos (n6 ) + Bn sin (n6 )] HLZV (kr)
n:0 (16)

The coefficients are determined from the boundary condition at the

cylinder surface ..

 

 

V 6
m for [6 -6 | < m
a6 m 2
Eg(r:a+) :4 m m: 1,2,...,N
_ O elsewhere (6)
L

 

The E; component of the field can be determined from Equation (16)

by using Maxwell's equation for a source free region

 

VX-I-Ir : jwe Er
This gives
1‘ _ .1 8 1‘
E0 _ weo 8r Hz
(1)
I
: jgo Z [An cos (n6)+ Bn sin (n0)] 1411(2) (kt)
n:0 (A-l)

where the prime denotes a derivative with respect to kr.

86

Using (A-l), boundary condition (6) can be expressed in terms

of the unknown Fourier coefficients

 

a)
I
E; (r:a+) : j go 2 [An cos (n9 ) + Bn sin (n9 )] Hn(2) (ka)

n:0

V 6
m for '9 -6 I <_Ll’l_

a6 m 2

:4 m m=l,2,...,N

D

b

0 elsewhere (A— Z)

 

The coefficient An can be found by multiplying (A-Z) by cos
(p9 ), integrating with respect to 6 over the domain [-1r, 1r] , and using

the orthogonality property of the sine and cosine functions. For the

2

case 11:]: 0 this gives

6+5 /2

m m V
S 711‘— cos (n9 ) d6
9 -6 /2 a m
l m m

n6
N Sin( 2m)
2 Vm n6 cos (nem)
m=l ( 2m)

j g0 An Hnm' (ka)1r

 

 

I
”It.

 

 

 

 

so
n6
2 1 EN: Sin(2m)
A = , , V -——-——-—cos(n9 ).
n JZwaLO Hn(2) (ka) m-l m (nfim) m
— 2 (A-3)

For the case n:0

 

jgo A0 Hom' (ka)21r

3M2

87

SO

 

N
A0 : ijlaz; Z Vm
0 Hn (ka) m=l (A-4)
with the Ne umann factor
1 for n=0
6On

2 otherwise,

Equations (A-3) and (A-4) can be combined into the final expression

for the coefficient

 

 

 

 

n5
EOn § sin( am)
A : , Vm cos (n9 )
n ijaLO Hn (2)1 (ka) 1 m( n6m ) m
m 2 (17)

Similarly coefficient Bn is also found from .Equation (A-Z)

by multiplying it by sin(pe ) and integrating over the same domain.

The result of this derivation is Equation (18).

APPENDIX B

PURELY REACTIVE IMPEDANCE LOADS
FOR ZERO SCATTERING IN ONE DIRECTION
FROM A TWO-SLOT CYLINDER

In this appendix an equation is derived whose solutions are the
purely reactive load impedances that cause the field scattered from
a cylinder loaded with these loads to be zero in a direction 9 0' The
constraint equation that forces the scattered field to be zero in the

direction 9 0 is[See Equation (38)]

(311(9 0)V1+ (312(9 0)V2+ (510‘9 0) Z 0 (13-1)

Instead of using the nonlinear constraint Equation (37) to force the load
impedances to be purely reactive, a different procedure will be used
which directly determines an equation for the load reactances. It

can be shown that both procedures yield identical results.

Matrix Equation (26) for the case N=2 is

 

 

 

 

 

 

1 * '1 - 1

y11 + Y1 y12 V1 : K10

Yzl y22 + Y2 V2 Kzoq (13-2)
__ d u- d L. —.

Equation (B-Z) may be used to eliminate the slot voltages from

Equation (B-l). This results in a constraint equation which directly

involves the load admittances.

AYl + BYIYZ + CY2 + D = 0 (13-3)

whe re

88

89

A : c10y22+ CIZKZO
‘ C10

C = C10Y11+ C11K10

C11 C12. C10

D: Y11 y12 'Klo

3'21 V22. 'Kzo

 

 

The real part of the load admittances are now set equal to

zero so that

Y

1 jfi1

Y

2 jgz

where [51 and [3 2 are the load susceptances. This step is equivalent
to enforcing constraint Equation (37). The complex coefficients of

Equation (B-3) are written

A=A +jA.
r 1

B=B+jB.
r 1
C=C+jC.
r 1
D=D+jD.
r 1

where the subscripts r and i refer to the real and imaginary parts
of the coefficients, respectively.

With these definitions, complex constraint Equation (B-3)
can be separated into real and imaginary parts. This results in two

real equations

'AiBl ‘ Brfl 1F3 2 ' C152 + Dr Z 0 (13-4)

Ar'fll ' 3181‘3 2 + CrBZ + Di = 0 (13-5)

90

Solving these equations for [3 1 and [3 2 gives

 

 

 

F3 : Dr - C8‘32
1 Ai+BrpZ (B-6)
and
-G1JGZ-4F(A D +A.D.)
[3 = r r 1 1
2 2F (B-7)
where
F = B C + B.C.
r r 1 1
G :

A.C -A C.+B D.-B.D
1 r r 1 r i 1 r
The solutions to Equation (B-6) and (B-7) are the susceptances
that give zero scattering in directions 9 :9 0. There are two possible
sets of susceptances to achieve the same purpose. The purely reac-
tive load impedances that result in zero scattering in direction 9 =9 0
are Zl = -j/[3l and Z2 = -j/[3 2.
Equations (B-6) and (B-7) have suitable form for programming

on a digital computer.

91

APPENDIX C

ZERO SCATTERING IN ONE DIRECTION
AT TWO DIFFERENT FREQUENCIES

In this appendix an equation is derived whose solutions are

load impedances which result in zero scattering in one direction

 

 

 

 

6 :9 0 at two different frequencies, (.31 and (.12. Equation (41) for
the case N=Z gives
y11 + Y1 y12 ‘Klo
y.21 y22 + Y2 'Kzo z 0
C11 C12 C1o “’ z “’1
6 = 6 0
y11 + Y1 y12 “K10
3’21 Yzz + Y2 ‘Kzo : 0
C21 C22 C20 “’ z “’2
9 : 9 0
Evaluating these determinants leads to

AlYl + BlYlYZ + CIYZ + D1 : 0 (C-1)
A2Y1+ BZYIYZ + CZYZ + DZ : O

(C-Z)
where

A1 = [C10V22+ C1.7.K201w = 101

9:90

form

where

92

 

 

 

 

B1 : C10(“=“’1' 9 :90)
C1 = [C10Y11+ C11K1011.) 21.11
9 = e 0
C11 C12. C10
D1: y11 y12 'Klo
y21 Yzz ‘Kzo w = “1
e = 90
A2 = [Czoyzz + C22K20]w = 1.12
e = e 0
B2 : C20(w=w2, 9 :90)
C2 2 [C20y11+ C21K1011.) 21.12
e = 90
C321 C22 C20
D2: y11 Y12 'Klo
Val y22 ’Kzo w = wz
e = e 0

Equations (C-1) and (C-2) are easily manipulated into the

Y1+LYZ+ M : 0 ((3-3)
2
Yz+ NYZ+ P _ o ((3-4)

Define

and

where the

93

Cle - CZBI

AIBZ - AZBI

 

DIBZ - DZBI

M :
A1132 ' A231

AL+BM-C

 

 

Z Z 2
N :
BZL
AM-D
P _ Z
BZL
Y1: G1+inl
Y2 : G2+JBZ

L=L+jL.
1‘ 1
M: M +jM.
r 1
N: N +jN.
1‘ 1

P

P +jP.
1‘ 1

subscripts r and i refer to the real and imaginary parts of

the coefficients, respectively. Separating Equations (C-3) and (C-4)

into real and imaginary parts and performing some alegebra these

e quations be come

 

G1 = -Ler+ LipZ-Mr
51: -LrBZ-L1G2'Mi
G PiDNrfiZ

: +
2 N1 292

(C-5)

 

 

 

 

 

4 3 5N12 er 2 (N1 Nr N1)
‘32 + ZNiflz +( 4 ' 2 'Pr) 82 + 4 ‘ NiPr + 4 F32
(N N.P. P.2 . P )
+ r 1 1 _ 1 _ ___1_'__ z 0
4 4 4 (C-8)

Equation (C-8) is a quartic equation which may have real or complex

roots. The only roots that are solutions to this problem are the real

roots. Substituting the real roots of Equation (08) into Equations

(C-S), (C-6), and (C-7) gives admittances

Y1: GI+JB1

Y2 : G2+Jp2

which result in zero scattering in direction 9

wzwl, andw 21112.

: 9 0 at frequencies

REFERENCES

l. J. K. Schindler, R. B. Mack, and P. Blacksmith, Jr. , "The con-
trol of electromagnetic scattering by impedance loading, " Proc. IEEE,
_52, 993-1004 (August 1965).

2. K. M. Chen, "Minimization of backscattering of a cylinder by
double loading, " IEEE Trans. Ant. Prop., AP-13, 262-270
(March 1965).

3. K. M. Chen, ”Reactive loading of arbitarily illuminated cylinders
to minimize microwave backscattering, " Radio Sci. , 690, 1481 (1965).

4. . R. F. Harrington and J. L. Ryerson, "Electromagnetic scat-
ter1ng by loaded wire 100ps,” Radio Sci. , 1, 347-352 (March, 1966).

5. K. M. Chen, "Minimization of end-fire radar echo of a long thin
body by impedance loading, " IEEE Trans. Ant. Prop. , AP-l4,
~ 318-323 (May 1966).

6. K. M. Chen and M. Vincent, ”A new method of minimizing the
radar cross section of a sphere, " Proc. IEEE, 5_4, 1629-1630 (Novem-
ber 1966).

7. K. M. Chen, J. L. Lin, and M. Vincent, ”Minimization of back-
scattering of a metallic loop by impedance loading, " IEEE Trans.
Ant. PrOp. , AP-15, 492-494 (May 1967).

8. M. Vincent and K. M. Chen, "A new method of minimizing the back-
scattering of a conducting plate, " Proc. IEEE, fl, 1109-1111 (June
1967).

9. J. L. Lin and K. M. Chen, "Minimization of backscattering of a
loop by impedance loading - theory and experiment, " IEEE Trans.
Ant. Prop. , AP-16, 299-304 (May 1968).

10. M. C. Vincent and K. M. Chen, "Modification of backscattering of
a sphere by attaching loaded wires, " IEEE Trans. Ant. Prop. , AP- 16,
462-468 (July 1968).

11. C. L. Chen, ”Modification of the backscatte ring cross-section
of a long metal wire by impedance loading, " IEEE 1968 G-AP Inter-
national Symposium

12. J. E. Clark and J. L. Tauritz, "Cross section control of a thin
wire loop by impedance loading techniques, " IEEE Trans. Ant. PrOp. ,
AP-l7, 106-107 (January 1969).

95

96

13. D. P. Nyquist, J. R. Short, and K. M. Chen, "Broad-band reduc-
tion of backscattering from a thin cylinder, " IEEE Trans. Ant. Prop. ,
AP-l7, 248-250 (March 1969).

14. O. D. Sledge, "Scattering of a plane electromagnetic wave by a
linear array of cylindrical elements, " IEEE Trans. Ant. Prop. ,
AP-17, 169-175 (March 1969).

15. J. R. Short and K. M. Chen, "Backscattering from an impedance
loaded cylinder, " IEEE Trans. Ant. PrOp. , AP-17, 315-323 (May 1969).

16. R.J. Coe and A. Ishimaru, "Optimum scattering from an array
of half-wave dipoles, " IEEE Trans. Ant. PrOp. , AP-18, (March 1970).

17. J. L. Tauritz, "A useful graphical representation in the theory
of loaded scatterers, " IEEE Trans. Ant. Prop. AP-18, 826-829
(November 1970).

18. V. V. Liepa and T. B. A. Senior, "Modification of the scattering
behavior of a sphere by reactive loading, ” Proc. IEEE, §_3_, 1004-
1011 (August 1965).

19. R. F. Harrington, "Theory of loaded scatters, " Proc. IEE (London),
IE, 617-623 (April 1964).

20. R. F. Harrington, Field Computation By Moment Methods (Macmillan,
New York, 1968), Ch. 6.

21. B. J. Strait and K. Hirasawa, "Array design for specified pat-
tern by matrix methods, ” IEEE Trans. Ant. Prop. , AP-17, 237-239
(March 1969).

22. B. J. Strait and A. T. Adams, "Analysis and design of wire anten-
nas with applications to EMC, " IEEE Trans. Electromag. Compati-
bility, EMC-12 , 45-54 (May 1970).

23. B. J. Strait and K. Hirasawa, ”On long wire antennas with multi-
ple excitations and loading, ” IEEE Trans. Ant. PrOp. , AP-18,
699-700 (September 1970).

24. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York,
1941), 371—372.

25. R. W. P. King and T. T. Wu, The Scattering and Diffraction of
Waves (Harvard University Press, Cambridge, Mass. , 1959),
Chapter 2.

26. J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, Electro-
magnetic and Acoustic Scattering by Simple Shapes (North-Holland
Publishing Co. , Amsterdam, 1969), 103-111.

27. M. Abramowitz and I. A. Stegun, Handbook of Mathematical
Functions (Dover, New York, 1965) 364, eq. (9. 2. 4).

97

28. M. Abramowitz and I. A. Stegun, Handbook of Mathematical
Functions (Dover, New York, 1965), 365, eq. (9.3.1).

29. C. H. Papas and R. W. P. King, ”Currents on the surface of an
infinite cylinder excited by an axial slot, " Quart. Appl. Math. , 1,
175-182 (July 1949).

30. G. Sinclair, "The patterns of slotted-cylinder antennas, " Proc.
IRE, 36, 1487-1492 (December 1948).

31. E. L. McMahon, "Circuit realization of impedance loading for
cross section reduction, " Scientific Report No. 8, AFCRL-70-0514,
Air Force Cambridge Research Laboratories, Laurence G. Hanscom
Field, Bedofrd, Mass. , (September 1970).

32. C. G. Montgomery, R,H. Dicke, and E. M. Purcell, Principles
of Microwave Circuits (Dover, New York, 1965), 97.

33. S. Ramo, J.R. Whinnery, and T. Van Duzer, Fields and Waves
in Communication Electronics (Wiley, New York, 1965), 33, 375-377.

34. P. Blacksmith, Jr., R.E. Hiatt, and R.E. Mack, "Introduction
to radar cross-section measurement, " Proc. IEEE, 5_3, 901-920
(August 1965).

PART II

MODIFICATION OF RADIATION FIELDS AND
CIRCUIT PROPERTIES OF A LOOP
ANTENNA BY MULTI-IMPEDANCE LOADING

98

CHAPTER I

INTRODUCTION

The modification of the radiation and circuit prOperties of a cir-
cular wire 100p antenna by lumped impedance loading is investigated
in this study.

When a wire antenna is excited, a current is induced on the
antenna. This current radiates an electromagnetic field and deter-
mines the input impedance of the antenna. Lumped impedances can
be installed along the antenna to modify the magnitude and phase of the
antenna current which, in turn, modifies the antenna radiation and
input impedance. The modified current can assume quite an irregular
distribution and hence a fairly accurate theory must be deve10ped.

The problem of determining the current distribution on a cir-
cular 100p antenna can be formulated in terms of two coupled integral
equations. 1 These equations reduce to a single one dimensional inte-
gral equation after the "thin wire” approximation is introduced. Fourier
series solutions to the thin wire integral equation have been studied by
Hallenz, Storer3, and Wul. An iterative solution has been considered
by Adachi and Mushiake4' 5.

based on the work of Hallen, Storer, and Wu.

The theory formulated in this study is

A point exists in Hallen's series where the terms become very
large and for some antenna dimensions infinite. Hallen concluded that
the series was divergent and could only be used as an asymptotic series.
He suggested that the problem arose from the one dimensional approxi-
mation. Storer extended Hallen's result by summing the first five terms
of Hallen's series exactly and using an approm'mate integral technique
to sum the remaining terms. The troublesome point in Hallen's series
now occurred under an integral which Storer evaluated as a Cauchy
principle value. Wu questioned Storer's technique and re-examined
Hallen's solution. He pointed out that Hallen's difficulty did not arise
from the one dimensional integral equation, but arose from other ap-
proximations made. Wu used a less approximate Kernel and modified

Hallen's solution eliminating the troublesome point.

99

100

All of the above authors assumed a 6-function (or slice) voltage
generator in their model of the 100p antenna. This leads to an infinite
input susceptance and thus a divergent series for the input admittance
of the antenna. Wul suggested a possible procedure for calculating
the apparent input admittance of a half 100p antenna above a conducting
ground plane driven by a coaxial line, but unsolved problems still
exist in applying this procedure to the 100p antenna.

King, Harrison and Tingley 7 have calculated values for the
input admittance of, and current distribution on, moderate size 100p
antennas using Wu's theory. They retain twenty terms in Wu's series.
The number of terms retained in the divergent series for the 100p sus-
ceptance appears to be somewhat arbitrary. For example, the sus-
ceptance of a thin 100p one half wave length in circumference increases
by more than 20% when thirty terms rather than twenty are retained.

In this study further reference to this theory will be made as the twenty
term theory.

An alternate approach to modeling the voltage driver is taken
in this study. The generator is assumed to be of finite size and to
exist over a finite gap along the 100p. This leads to a convergent series
for the admittance of the 100p and the number of terms retained in the
series is determined by the desired accuracy of the solution. A dis-
cussion and justification of the "finite gap" theory is given in Chapter
III of this study. It is shown that by introducing the finite gap into the
theory, the agreement between the theory and experimental results is
improved. Very recently, Ito, Inagaki, and Sekiguchi8 published a
paper on arrays of 100p antennas where they also introduced a finite gap
generator.

Multiloaded 100p antennas have been investigated by Iizukag’ 10
and Harringtonl 1. Iizuka deve10ped his theory by the superposition of
Storer's results and found a significant discrepancy between his theory
and his measured admittances. Harrington based his results essen-
tially on Hallen's series and did not include Wu's correction. He did
not compare his theoretical results with any experimental results. Both
authors deve10ped their theories for more than one loading, but restricted
their results and discussions to singly loaded 100ps. Furthermore,

most of the existing results are confined to resonant loOps of one wave

length in circumference.

101

When a 100p is loaded by more than one load the number of
variables (i. e. , load resistances, load reactances, position of the
loads, etc.) becomes overwhelming. To overcome this problem,
synthesis procedures are deve10ped in this study to facilitate the de-
sign of a multiple -loaded 100p antenna.

The major purposes of this study are (l) to deve10p an improved
theory for the loaded 100p antenna, and (2) to deve10p and analyze pro-
cedures for the design of a multiple loaded 100p antenna that results in
desired radiation or circuit characteristics.

In Chapter 11 an integral equation for the multi-loaded 100p
excited by a finite gap generator is developed and a Fourier series
solution is obtained.

Numerical methods used in evaluating the theory results are
discussed in Chapter III. In addition, a comparison of the "finite gap"
theory to other existing theories and experimental results is made. A
brief discussion of the characteristics of unloaded 100p antennas is also
presented. Since the "finite gap" theory is more applicable to larger
100ps than Storer's theory or the twenty term theory, some examples
of impedances, currents, and radiation fields of large loop antennas
are also presented.

Chapter IV deals with procedures for determining (l) the load-
ings necessary for a Specific modification of the radiation pattern of a
100p antenna, (2) the Optimum reactive loading to produce maximum
gain in a Specified direction from a 100p loaded with a single impedance,
and (3) the set of reactive loadings that leads to a Specified input
impedance.

Chapter V summarizes the results obtained in this study.

CHAPTER II

THEORY OF THE LOADED LOOP ANTENNA

2. 1. An Integral Equation for the Current on the Loaded L00p Antenna

A transmitting circular loop antenna of radius b and constructed
of perfectly conducting wire of radius a is loaded with N impedances as
Shown in Figure 2. l. The 100p is excited by a finite-gap, voltage gen-
erator which produces a uniform, impressed electric field in the gap
region H] < 60/2. The nth load impedance Zn is lumped into a gap
region of angular width 6n whose center is located at d: = ¢n' There
are two components of surface current density induced on the 100p,
K¢(¢, LIJ) flowing around the leap in the <1) direction and K¢(¢>, 1p) flowing
about the wire in the 4: direction. Integral equations for the currents on
the 100p can be obtained from the boundary condition on the tangential
component of the electric field at the 100p surface.

The problem is greatly simplified by assuming a thin wire 100p
whose gap generator and load impedances are restricted to regions
small with reSpect to a wave length, that is

3.2 << b2 and ka << 1 (1)

and

6nb << 1 for all n. (2)

where k = w Viz-5: = 211/). is. the prOpagation constant. Harmonic
time dependence of the form ert is assumed. .

Under the thin wire assumption, the Ll} component of surface
current will be small in comparison to the 4) component of surface cur-
rent and can be neglected. It is also reasonable to assume the total

current flowing in the loop is

1,144 = Zwa K¢(¢).

The integral equation for I¢(¢) can be derived from the boundary
condition on the <1) component of the electric field at the 100p surface
which is

102

103

 

 

 

 

2
o b _ x
. 3 Finite gap
0 0 generator

 

(a)

 

(b)

Figure 2. l. LOOp antenna loaded with N impedances: (a) Geometry,
(b) Schematic.

104

:1" - E2] : 0 --- at the surface of the 100p (3)

where E1 is the impressed electric field at the surface of the 100p and

[E

E: is the induced electric field at the surface of the loop maintained by

the current and charge on the antenna.

The impressed field is

- Vo Po(¢) --- for (43' < 60/2
i _
12¢ — I¢(¢m) zm me - ¢m1 :rJf-gml <13...” (4)
0 --- elsewhere on the 100p
where
1
qb— --- for |¢| < {Sn/2
an = (5)
0 --- elsewhere

is a unit area pulse function. The distance from the center of the 100p

to the observation point of the field on the loop surface has been taken

to be b everywhere which is consistent with the thin wire assumption.
The induced field is

a 13<I>

 

 

E4> ‘ 'EW'5“A¢ ‘6’
where
m?) = 4,2 5 naive-13m dV'
0 V
and

_. 1* .... -ij
A¢(r) = fiSV$-J(r')eR dV'

 

are the scalar potential and the <1) component of the vector potential,
respectively. p(}") and JG") are volume charge and current densities
and R = I; - :‘l is the distance between the source point r" and field
observation point 1". For the 100p

_._, I (¢')
J(r’) dV' "‘ I?” %;ra— ad1):' bdd)l

105

p(?')dv' -> 32%)- adqn bd¢'

where q(¢') is the total charge per unit length on the 100p and is related

to the total current on the 100p by the continuity equation

31(¢')

%‘—%r' = 49901“)-

The potentials maintained by the current and charge on the 100p are

 

now
w 81 (cw)
M) = 'j... 41,160,, S —§TW(¢-¢')d¢' (7)
A¢<¢1=M5 www- ¢')cos(¢- ¢')d¢' (8)
where
1r -ij
W(¢-¢') = 72—5 E—f— dw'. (9)
-1r

Substituting equations (7) and (8) into equation (6) yields

saw) - ————zl ‘9 Tr 74—81(4)“ W<<1> MW
4) jw41rtob 53 -1r 4)

jwuo
"ZEB'S‘WI ¢(¢')W(¢- ¢')COS(¢- ¢')d¢'
-1r

Integrating the first integral in the above expression by parts and

noting that %- W(¢ - (b') = - 3—3; W(<(> - cb') gives

a _. I l _ 1 1
E¢(¢) — 115:1 4,01 )[l‘lbafz + kb cos (4- ¢ 1] W(¢ <1 N4»
(10)
where to = Vpo7 50 is the intrinsic impedance of the medium. Sub-
stituting equations (4) and (10) into equation (3) results in the following

integral equation for the current on the loaded 100p antenna.

vopom = g I¢(¢ )2 P (<1 -¢ m1+4igylx¢ (¢')M(¢ -¢')d¢'

m m m
m=1 (ll)

106

The kernel is now

 

 

 

2
MM» -¢’) = [kb cos<¢ -¢')+i<’%3:_,7] WW -¢*) (12)
where
11' -jkle
W(¢ -¢') = Z—L-S e—R—dw' (13)
--TI’ 1
with
R1 = R/b
2' J4b2 sinZ [(¢-¢')/2] + 4a2 sin2 (LIN/2)
b
= J4 sin2[(¢-¢')/2] + Az/bZ (14)
where

A = 2asin(41'/2).

The approximate expression for R given above is consistent with the
thin wire assumption and retains the essential characteristic of the

singularity in the integrand of W(¢ - ¢').

2. 2. Fourier Series Solution for the Current on a Loaded Loop Antenna
A solution of integral equation (1 1) can be obtained in the form of
a Fourier Series. The current I¢(¢'), the kernel W(¢ - tb'), and the

pulse function Pm(¢ - dam) are expanded in Fourier series.

(I)

I¢(¢') : Z Ine-jmb' (15)
n=-oo
(I)
W(¢_¢I) : Z Kne‘Jn(¢‘¢') (16)
n=-oo
i -jn(¢>-<1>m) 1
me - 4>m) -- Bnme < 7)

113-0)

107

where In’ Kn' and Bnm are unknown Fourier coefficients which are

defined by the following integrals.

I = lynx (¢')ejn¢'d¢' (18)
11 Zn -174)
_ 1 " . ' (¢- ')
Kn Z? SWWW - ¢ )eJn 4’ dc)
= K-.. (19)
1 1' j(¢‘¢m)
Bnm : 2.1—r- -1r Pm“) - (pm) e d¢ (20)

Bnm is easily evaluated by substituting equation (5) into equation (20)

and performing the integration. The result is

 

 

 

n6
1 sin( 2m)
nm : 21rb n6 (21)
m
(T)

A Fourier series expansion of the kernel M(¢ - (b') is obtained by sub-

stituting equation (16) into equation (12) and is

MN) _¢1) : f ane‘Jn(¢"¢') (22)
n=-oo
where
2
an = % (Kn+1 + Kn_1) - E—b Kn (23)

Substituting equations (15), (17), and (22) into integral equation
(11) gives

00 N 00 .
-jn¢ 'Jn(¢‘¢m)
V0 2 Bnoe - Z‘ I¢(¢m) Zm Z: Bnme
nz-(I) m=1 nz-oo
jLO 1T 00 -jn¢' 00 “11033499
= as) 2 Inc 2 a: 8 d4"

n=-oo l 00

108

.1 C. -
_ o - Jn¢
‘ 2b 3 In ane

nz-cx)

The following expression for In is derived from the previous equation

by using the orthogonality prOperty of the ejn n¢ functions.

_ 2b J'mbm
In - jg an[anor:11<|>(¢m)zm Bnme ] (24)

The series determining the current on the loaded 100p antenna is given

by equations (15), (21), and (24) and is

 

 

I¢(¢) = .:i_ v O[—-—+2 n: Sin T “:91“

112-1")

N I (4. mm —1—+z m Si“(::m) C°S(“‘: tn.)
21.4.0 2 (“2m ) ]}

The values of the current at the positions of the loads must

 

 

 

(25)

now be determined. Load voltages are defined as

V = I¢(<1>m)Zm (26)

m

where Vm is the voltage drOp across the mth load impedance Zm' It

is also convenient to define

 

n6
, 0° sin( m) cosLnOb-(b fl
y(¢-¢m) = 53%;ng n61: an m ](27)
1. )

which has the same dimension as admittance but is actually the current

 

distribution on an unloaded 100p antenna driven by a generator of unit
voltage located at o : ¢n' For the case of the voltage generator located
at <1) = 4’0 = 00 the notation y(¢) will be used and it is implied that

6 = 60. Evaluating equation (25) at the N positions of the load im-
petctllances and noting that 143(4)m )- - Vm Ym whereY = l/Zm and is the

load admittance gives N simultaneous equations:

n: 1,2,...,N (28)

With the notation

Ynm = Y(¢n - ¢m). Yn

Mn) (29)

the previous set of simultaneous equations can be written in matrix

form as
r- 1- - r- “-1
y11+Y1 y12 . . . y1N V1 Y1
Y21 y22+Y2 Y2N V2 y2
= v0 . (30)
_ yN1 yN2 YNN+YN_ _VN_ _YN_

 

 

 

 

 

 

which can easily be solved for the load voltages.
The current on the loaded 100p antenna is now completely de-

termined and given by

N
I¢<¢) = voym - Z vmyw -¢m) <31)

m=1

The correSponding input admittance is

I (0)
_ 9 _ _ Z —
Yin — V _ yo mem (32)
o :1
where
Vrn : Vm/Vo

is the normalized load voltage.

The above results can be interpreted in terms of a superposi-
tion picture which will be useful later. Equation (31) shows that the
current on the loaded 100p antenna is the superposition of N + l cur-
rents. The first current (corresponding to the first term on the right

hand Side of equation (31)) is equivalent to the current on an unloaded

110

100p driven at 41 = 00 by V0, the second current is equivalent to the

current on an unloaded 100p driven at <1) = 411. by - V , etc. The load

1
voltages V1, V2, . . . , V are determined by matrix equation (31).

N
The input impedance and current distribution of the loaded 100p
antenna depend on the coefficient
kb 2

- __ .. L
8‘n ‘ 2 (Kn+l + Kn-l) kb Kn (23)

where Kn has been defined as [see equations (13) and (19)]
l 11 1r -jkle . ,
K S (S e R1 d4”) eJn(<1>"<1> )d¢> (33)

n (211') -1r -1T

and

 

R

1 J4 Sin2((¢-¢')/2) + 4(a2/b2) sinZ W/Z)

Wul’ 11’ 12 has evaluated this integral under the assumption a2 << b2

and obtained

kb
K0 2 ;lr— 1n(8?b) - %5: [520(x) + j Jo(x)] dx (34)
n — 77— K0( b ) (_b—) In] - 2 . o 2n(x) J 2n(x) x
3
where ( 5)
-1
Cn = 1n(4n) + V - 2.: Tull—+1- (36)

m=0

and y = 0. 5772. . . iS Euler's constant, :(x) and Ko(x) are the modi-
fied Bessel functions of the first and second kinds of order 0, Jn(x) is

the Bessel function of the first kind and order n, and 52n(x) is the
* 13, 14

Lommel-Weber function defined by
1 1r
fln(x) : :7- 5, sin(x sine - n9) d0 . (37)
o

 

*
Some authors define the negative of this function as the Weber function
denoted En(x). that is En(x) = -Qn(x).

111

2. 3. Radiation Fields of a Loaded LOOp Antenna

The electromagnetic fields radiated by a loaded 100p antenna
can be obtained by integrating term by term the Fourier series expan-
sion of the current on the 100p. The electromagnetic fields in terms

of the vector potential are given by

'f:(?) -ij + m _ (38)

 

j(“L060
in?) = i v x15. (39)
where
KG) = 2'; Svfffi") 6-11m dV' (40)
and R = I? -;'I

Consider the loaded 100p antenna which lies in the 9 = 90° plane
with its center at the origin of a Spherical coordinate system as Shown
in Figure 2. 2. DrOpping terms of higher order than l/r and making
the following standard radiation zone approximation

r - r '15 --- for the phase term

R :.'-
r --- for the amplitude term

equations (33), (34), and (35) become

E(?) = -jw(8A9(¥) + $A¢(?)) (41)

EEG-3 = -jk (-3A¢(}') + 3.149(3) (42)

—> -> P- ~jk1‘ 211. h ’ ‘ .-+l

A(r) = 3397—5 I,,(<1>')<1>'eJ"‘r 1‘ bd¢' (43)
0 .

which is valid in the radiation zone where r >> b.

The integral that results from substituting equation (31), the
current on the loaded 100p, into equation (43) can be integrated exactly
in terms of Bessel functions of the first kind. A similar integral for
the unloaded 100p has been evaluatedls’ 16 and the result for the loaded
100p can be obtained by superposition in accordance with equation (31).

The resulting radiation fields of the loaded 100p antenna are

112

11
P(r. 9. <1) b 4)

”I

 

 

I
I
I
I
l
l
l
|
h‘ 9 I
l
I
l
l
|
|

Figure 2. 2. Coordinate system for the fields radiated by a 100p
antenna. -

113

 

 

 

 

 

 

 

 

 

 

 

E(r) = 8E9+$E¢ (44)
-> -v 0 0
H<r1 = - 9134/40 + «we/40 (451
where
. -'kr
_ ka 6 J S -
E9 7 11 r Vo [F00 - VmFGm (46)
m=l
. -jkr __
E4) = 1:1” e r vo [17.10 -S va¢m] (47)
m:l
and
n6
( n sin(6 2m)l
Fem = cosG n: #211 2[Jn-l(kb Sine)
( 2 m)
+ J +1(kb sin 0] sin (Md) - 41111))
00 n sin(nam) J (kb sine)
= cos 9 Z gl— n62 n k?) sine sin (M43 - ¢m)) (48)
n=1 n m
1 2 )
n6
J l(kzb Sine) (.) :sin(— m) l
F¢m = - 1:1ng [Jn_1(kb sine)

 

(2m) 2

- J +1(kb Sin 9)] coS (nu) - ¢n))

 

 

n6
J'(kb sine) . n Ski—m)
0 2a +§ g—l— n62 Jr'l(kb sin 0) cos (n(<1) -¢n))
o n=l n ( 2m (49)

 

where J n(Z) is the Bessel function of the first kind and J' n(z): 553— J n(z).
Examination of equations (44) through (49) indicates that the radiation
fields are outward traveling Spherical waves. It also indicates E9

and E4) are in phase implying that the radiation is linearly polarized.

114

A parameter which is useful in describing antenna radiation is

the gain of the antenna. The gain of an antenna is defined by

0(9) <1>) =

411 power density per unit solid angle in direction 0, 4)

 

total input power to antenna

(50)

The gain of an antenna differs from its directivity by a factor which

takes into account the efficiency of the antenna.

For the loaded 100p antenna the gain is given by

 

 

 

N 2
2
4(kb) Z —
9‘9”” t, «G. F00 ' VmFGm +
0 1n m=I

where Gin is the input conductance of the 100p.

F¢O - SVchbm
n=1

 

(51)

CHAPTER III

IMPEDANCES, CURRENTS, AND RADIATION
FIELDS OF A LOOP ANTENNA EXCITED
BY A FINITE GAP GENERATOR

This chapter deals with the numerical method used in calculat-
ing the results of the loaded 100p antenna with the finite gap excitation
which was developed in the previous chapter. Theoretical results
based on the finite gap excitation and theoretical results based on the
6-function generator are compared with existing experimental results.
Finally, some examples of impedances, currents, and gain patterns

of large 100p antennas excited by finite gap generators are presented.

3. 1. Numerical Method

Numerical results based on series expansions for the radiation
fields, current distributions, and input admittances of 100p antennas
have been evaluated on a CDC 6500 computer system. These series
depend on coefficients an which are functions of the kb and a/b. These
coefficients also depend on the Kn integrals [see equation (23)] which
were evaluated using Wu's expression as given in equations (34), (35),
and (36). A standard M. S. U. computer library subroutinel7 was used
to generate the Bessel functions of the first kind. The modified Bessel
functions were calculated by polynomial approximationsl8 while a

13’ 14 was used to evaluate the Lommel-Weber

series expansion
functions.

Examples of the first four coefficients are Shown in Figure 3. 1
as functions of kb with S2 : 21n(21rb/a) = 12. Increasing the value of 52
(i. e. , constructing the 100p with thinner wire) tends to Sharpen the
peaks of the an while decreasing 52 tends to flatten the peaks.

The series determining the radiation fields [equations (48) and
(49)] are rapidly convergent if kb is not too large. In this study twenty

terms were retained to insure accurate results for kb 5 10.

115

116

 

 

 

 

 

 

 

 

 

 

 

1 1 1 1 1 1
3 -+
2 ..
«1C 2
\
"‘ 3
“a 1 -‘
4.)
1..
(6
CL 0
".3
0)
m -l- _
-Zt- —(
1 1 1 1 1 1 1 1
0 2 3 4 5
kb
(81)
1 1 1 1 T 1 1 1 1

n
F

 

 

Imaginary part of l/a

 

 

 

 

(b)

Figure 3.1. Real and imaginary parts of l/a , 1/a , l/az, l/a
for $2 = 12. (a) Real parts. (b) Imaginary parts.

117

The series determining the input admittance is given by expres-
sions (27), (29), and (32). The real part of this Series which corre-
Sponds to the input conductance converges very rapidly independent of
the gap width. For example, with kb = 2. 5 three digit accuracy can be
obtained for the conductance rataining only four terms in the series.

On the other hand, the imaginary part of this series which correSponds
to the input susceptance is somewhat more troublesome. Consider
briefly the convergence pr0perties of this series. The input admittance

of the unloaded loop is found from eq. (32) to be

(I) 1'160
_ _ _. l Sing—T) l 7
Y " 41.4%”; n6. 5;] ‘5?"
2

in

 

For large n with I) >> kb

and when n >> b/a the dominant term in Kn is the term involving the
product of the modified Bessel functions. Retaining the first term in

an asymptotic expansion for this product19 results in the following ex-

pression:
1 — na na 1 b
Kn N 318.113" 1011.— "’ 2.71:
Hence
hi ~ 2nkb(§) —1-
n n
and the nth term in the admittance series has the following asymptotic

form for large n

n6
sin(——i—)
a 2 l . (53)

 

When 60 > 0 the series converges,20 but when 60 = 0 (which is the case
of the 6-function generator), the series diverges. Physically this im-
plies an infinite stray capacitance existing at a gap with an infinitesimal

gap width. It is interesting to note that (53) depends only on the wire

118

sin (n 60/ 2)
size ka and the term , which arose from expandirg the 100p
(n 150/2)

 

excitation in aFourier series, and does not depend on the 100p size kb.
For small 11 the terms are also a function of kb. The above observa-
tions imply that the detailed behavior of the susceptance is determined
strongly by the manner of excitation of the 100p at the driving point. A
comparison of the input susceptances obtained from equation (52) with
50 = 0. 0 and 60 = 0. l rad. as a function of the number of terms re-
tained in the series is made in Table 3. 1.

The actual computation of the input admittance Yin’ the current
distribution I¢(¢), and the self and mutual short circuit admittances of
the loading points ynm was accomplished by summing 1001 terms of
their respective series. The first 61 coefficients were calculated
exactly. The following approximate expression was used in calculating
the remaining coefficients.

_. 1— na — na 1 kb
Kn * F K011?) 101T1+27119 154’

The last term in the above expression arises from integrating the first
term in a series representation of the Lommel-Weber function. 13 This
method of calculation assures at least two digit accuracy. Only slightly
over one second of computer time was needed to generate the 1001

coefficients and sum them.

3. 2. Effect of Finite Gap Generator

The gap Size has no effect on the conductance of Small and
moderate size 100ps and only a Slight effect on large 100ps in the range
of kb = 10. 0. Physically this is to be expected since the conductance
is prOportional to the total power radiated by the 100p and Should not be
Significantly affected by the gap size.

The effect of the gap Size on the susceptance is shown in
Figures 3. 2 and 3. 3. Decreasing the gap size tends to make the 100p
susceptance more capacitive. Figure 3.2 shows that the susceptance
of thick 100ps is more sensitive to gap size than thinner 100ps. It can
be seen that the effect of the gap increases with increasing 100p size.
For very small 100ps the susceptance is nearly independent of gap width.

This can be explained by the fact that the current is essentially uniform

119

 

 

 

 

 

 

 

Number of terms Susceptance, millimhos
in partial sum 60 = 0‘ 0 50: 0.1 rad. ’
2 -0.868 -0.868
4 -l.36 -l.36
6 -0.666 -0.673
8 -0.366 -0.378
10 -0.182 -0.199
12 -0.0527 -0 0762
14 0.0451 0.0153
16 0.123 0.0865
18 0.187 0.144
20 0.242 0.191
22 0.289 0.230
24 0.331 0.264
26 0.368 0.292
28 0.401 0.317
30 0.432 0.338
40 0.553 0.408
50 0.643 0.441
60 0.715 0.452
70 0.775 0.450
80 0.827 0.443
90 0.874 0.433
100 0.915 0.425
200 1.19 0.428
300 1.36 0.425
400 1.48 0.423
500 1.57 0.423
1000 1.86 0.424
Table 3. 1. Input susceptance of circular 100p antenna as a function of

terms retained in series solution with kb = 2. 5 and S2 = 12. 0.

Susceptance B, millimhos

 

 

 

 

 

 

I T T T Y T 1 r r I T 1
600 d
1
500 d
.0
q
4.0 .1
d
3.0 d
d
2.0 .1
an)
1.0 4
.1
° 1
.1
'L ‘
kb 2 0.1
_6.0 1- -------------------o q
- .1
.7.0 )- 4
)- .

-8.0 1-

 

 

 

0. 0 0. 04 0. 08 0.12 0.16 0. 20 0. 24
Gap width, 6, radians

Figure 3. 2. Input susceptance of 100p antenna as a function of gap
width for $2 = 10 ( ) and f2 = 12 ( ----- ).

 

 

 

 

T I T I I T F T T I I I
]-
41- ' _l
"' 1
3' q
"’ ‘ all
21- .1
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0 . ?-16
IE 1’ III
a
53’ 4 , .
03 ’ 10.0
0 I I
«‘1‘ O . V
E. . I
U h )1 I d
g \ I 20.0
U) ’11- / \p’ q
1
b l q
-21. q
)- 4
-3b I
'4‘ q
J l 1 l L I J j l l l l

 

 

 

.2 .4 .6 .8 1.01.21.41.6 1.8 2.0 2.2 2.4
kb

Figure 3. 3. Input susceptance of 100p antenna as a function of kb for
6b/a = 1. o, 10. o, and 20. 0 with)? = 12 (i.e., a/b =‘ 0.0155).

122

on small 100ps and hence there is no charge build up on the 100p.
Thus, changing the gap width does not change the distribution of cur-
rent and charge and hence does not change the input susceptance of

the 100p.

3. 3. Comparison of "Finite Gap" Theory with Experimental Results
Theoretical analysis of wire antennas usually makes use of an
idealized generator, whether it be a 6-function generator or a finite
gap generator, that eliminates the transmission line which is usually
present in practice. The theoretical admittance of the antenna is thus
an intrinsic quantity of the antenna and idealized generator. On the
other hand, the measured admittance of an antenna is the apparent
admittance the antenna presents to a transmission line when connected
as a terminating load. The theoretical admittance neglects:
a. Electromagnetic coupling between the antenna and
transmission line near the junction between them.
b. Changes in the characteristic impedance of the
transmission line near the junction due to the fact
that the line is not infinitely long.
and, in addition, for the case of a two wire transmission line driving
the antenna:
c. The absence of the antenna in the gap between the two
conductors of the transmission line.
For the case of a dipole antenna driven by a two wire line.
King21 (this paper references other related works) has derived a
terminal-zone correction to relate the ideal, theoretical admittance
to the apparent admittance of the antenna. Likewise, correction terms
have been found for a mon0pole over a ground plane driven by a coaxial

22’ 23’ 24 No such terminal correction terms have been determined

line.
for the 100p antenna.

It is noted that the terminal zone corrections for the dipole
depend mainly on the transmission line geometry and not on the length
of the dipole and that the dominant term in the correction is a lumped
Shunt susceptance. It has been seen in the last two sections that the

high order terms in the series determining the admittance of the 100p

123

depend on the gap size and the 100p wire size but not on the size of the
100p,‘ and that changing the gap width affected the 100p admittance in
the same was as a lumped susceptance shunting the 100p would. In
view of the Similar characteristics of the terminal zone corrections
and the finite gap generator, it is pr0posed that "effective" gap widths
be defined which result in theoretical 100p admittances which corre-
spond to apparent measured admittances. For the case of the 100p
driven by a two wire line the effective gap width is taken to be the dis-
tance between the centers of the two wires of the line. For the case of
a half 100p over a ground plane driven by a coaxial line the effective
gap width is taken to be the inside diameter of the outer conductor of
the coaxial line. The justification for these effective gap widths is
that they yield theoretical admittances that compare very well to
measured admittances.

Experimental measurements of the admittance of 100p antennas
driven by a two wire line has been reported by Kennedy. 25 A compari-
son of her measured conductances with theoretical results calculated
from equation (52) is Shown in Figure 3. 4. The 6-function theories and
the finite gap theory give essentially identical conductances. Figure 3. 5
compares measured susceptances, twenty term, 6-function theory sus-
ceptances, and susceptances resulting from the finite gap theory with
two different gap widths. In the region 4. 0 < wkb < 6. 0 the susceptances
of the twenty term, 6-function theory fall between those of the two
finite gap theories and are not Shown. A detailed description of the
experiment is found in the above reference, but it is worth noting that
the eXperiment was performed at a constant frequency of 750 MHz
with the 100p size being changed from wkb = 1. 48 with fl = 21m?) =
8. 16 to ‘1ka = 9. 38 with $2 = 11. 84. A comparison of Kennedy? z(isata

’ It

is found that the finite gap theory gives susceptances that are in better

and Storer's 6-function theory can be found in the literature.

agreement to the experimental data than Storer's theory.
Iizukag

of a circular 100p antenna imaged into a conducting ground plane and

has reported measurements of the admittances of half

driven by a coaxial line. A comparison of the finite gap theory, the
twenty term, 6-function theory, and Iizuka's measured admittances is

shown in Figure 3. 6. The experiment was performed at 600 MHz and

124

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l l l l I l l I J J L
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
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Figure 3. 6. Comparison of finite gap theory and experimental
admittances of a 100p driven by a coaxial line.

127

once again the physical size of the 100p was changed to effect changes
in kb. (2 : 21n(21rb/a) takes on values which range from approximately
8 through 14 while kb changes from 0. 2 to 2. 2. The gap width is taken
to be the inside diameter of the outer conductor of the coaxial line which
results in 5Ob/a = 13. 6. Excellent agreement between theory and
measured admittances is obtained. Increasing the gap width to 6ob/a :
27. 2 reduces the susceptance by approximately 0. 2 millimhos over the
range of kb considered. Reducing the gap width to 5013/3. = 6. 8 increases
the theoretical susceptance by roughly 0. 15 millimhos. Thus it
appears a gap width equal to the inside diameter of the outer conductor
of the coaxial line is optimum.

Values for the susceptance calculated by the twenty term theory
are denoted by an "a " in Figure 3. 6. At kb = 1.0 and 2. 0 the twenty
term susceptance and the finite gap susceptance are essentially identi-
cal and no 13 is indicated. It is seen the twenty term theory does not
compare to the measured susceptances as well as the finite gap theory.
At kb = 0. 2 the twenty term susceptance is approximately 0. 6 millimhos
more capacitive than the measured value while at kb = 2. 2 the twenty
term value is approximately 0. 4 millimhos more inductive than the
measured value. This can be explained as follows. A numerical
study of the twenty term, 6-function theory and the finite gap theory
over the range of parameters considered in this experiment has shown
that truncating the 6-function series at twenty terms yields approxi-
mately the same susceptance as using the finite gap theory with 60 =
0. 15 radians. However, in Iizuka's experiment 60b/a was held con-
stant while b was varied, hence 60 varied as kb was changed. In the
experiment with kb 2 0. 2, 6o :' 1. 07 radians while at kb 2 2. 2, 60 ='
0. 1 radians. It is concluded that the inaccuracy in the twenty term,
6-function theory is introduced by not taking into account change in the
excitation of the 100p.

Iizuka9

with a lumped impedance Zl z 00. The load impedance is implemented

has also reported admittances of a loop loaded at <1) = 1800

by simply removing a short segment of the 100p at 4) 2 180°. The
theoretical gap width for the load, 61, was taken to be identical to the
physical gap width used in the experiment which was . 012). at 600 MHz.

(:1 Theory _

Twenty term 0 D D

 

 

 

 

 

 

 

 

 

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I
I
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I
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I Iizuka's
10' 1: Experiment . . . "I
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1' I
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2 .:
m“ 6.. ,' :
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<3 I I
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h ' ' «I
'b
1 4111 1 l 1 ’1 1 1 1 1
0.2 0.4 0.6 0.8 1.01.21.4 1.6 1.8 2.0 2.2
kb

Figure 3. 7. Comparison of finite gap theory with Gob/a = 13. 6 3‘96]
experimental admittances of a 100p loaded at (I) = 180
with Z1: 00 (i.e., a gap).

129

Calculations made using equations (30) and (32) are compared with
Iizuka's measured admittances in Figure 3. 7. Once again, excellent
agreement between the finite gap theory and the measured admittances
is obtained. Again, it is found that the twenty term admittances do

not agree with the measured admittances as well as the finite gap theory
does. Iizuka found a fairly large discrepancy between his theory and
his measured values (which are used in Figure 3. 7) and it may be con-
cluded that the finite gap theory is a significant improvement in the

theory of the loaded 100p antenna.

3. 4. Input Irnpedances, Currents and Radiation Fields of LOOp
Antennas

The electrical prOperties of small 100p antennas with kb S 0. l

26

are well known. The radiation resistance of a small 100p is

"Q

R =' 73 (kb)4 (55)
and the input reactance is inductive with an inductance of
. 8b
L _ “Chem—3) _ 2) (56)

The current on a small 100p is uniformly distributed and the radiation

fields have the form,

-jkr

e

 

—> A
E N 4) r sine . (57)

The input impedances of and the current distributions on 100ps
of moderate size up to kb = 2. 5 have been tabulated by Storer3 and
Kingb’ 7. Radiation patterns in the plane of the 100p for 100ps of size
kb = 1, Z, and 3 have been calculated by Sherman27 under the assump-
tion of a cos (kbcb) current distribution. Rao28 has reported the radia-
tion patterns of 100ps of size kb : l. 5, Z. 0, and 2. 5 with both measured
results and theoretical results calculated by using the first 5 terms of
Storer's series solution.

The input admittances of 100p antennas of size 0 5 kb 5 10 and
$2 = Zln(21rb/a) : 12 and 18 are di6played in Figure 3. 8. The corre-

sponding input impedance for the case S2 = 12 is shown in Figure 3. 9.

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132

These impedances and admittances were Gob/a : 12. 1 which is the
effective gap width for a 300 ohm, two wire line.

The current distributions on large 100ps of size kb = 5.0 and
10. 0 are diSplayed in Figures 3. 10 and 3. 11. It can be seen that the
current distribution differs greatly from a cosinusoidal distribution.
The current distributions for gap widths of dob/a = 6 and 12 are shown
in Figure 3. 10. The current distribution for oob/a : 6 is plotted over
the range 00 < 4) < 80 which is the only region where it significantly
differs from the other distribution. A numerical study over a range of
100p sizes of O < kb S 5 showed that the gap width only affected the cur-
rent distribution in and very near the gap region itself.

LOOp antenna gain patterns for 100p sizes of kb : l, 1.5, 5, and
10 are shown in Figures 3.12, 3.13, 3.14, and 3.15. The figures dis-
play the patterns in: (a) the d) = 00 plane; (b) the <1) 2 90° plane, and
(c) the 9 = 900 plane which is the plane of the 100p sometimes called the
E-plane of the loop. The 153-field is polarized in the 4) direction in the
(I) = O0 and 9 = 900 planes and has both a 9 and (b component in the d) = 900
plane. The ratio of IE9] to the magnitude of the total field is also

shown in part (b) of each figure.

133

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CHAPTER IV

MODIFICATION OF RADIATION FIELDS AND
INPUT IMPEDANCES OF LOOP ANTENNAS
BY MULTI-IMPEDANCE LOADING

Several procedures are deve10ped in this chapter that determine
the loadings necessary to realize Specific modifications of the radiation

and circuit prOperties of loaded 100p antennas.

4. l. A L00p Loaded with a Single Impedance

A simple, approximate formula is deve10ped in this section that
shows the essential characteristics of radiation patterns of 100ps of
size kb S l. 0 loaded with a single impedance.

Examining Figure 3. l and equations (48) and (49) indicates that
the radiation fields of 100ps of size kb 5 l. 0 can be approximated fairly
well by the first three terms of their series solutions. With this series

truncation and approximating the Bessel functions by

. x 2
J00!) = 1-(2')

n

)

N|><

l
3,4

Jn(x)

and using the relations
J3 (x) - Jl(x)
1

equations (48) and (49) become

I!

I

 

Fem £- 2%: cost) sin(¢-¢m)
.. 41:32 sine cosB sin (2(4) -c|>m)) (58)

139

140

 

 

. ' Z . Z
F¢m : _ 41:30 sme + 7337(1 - g—(kb) sm 9) cos (4) -¢m)
- 41:32 sine cos (2(4) -¢m)) (59)

When a 100p is loaded with a single impedance the load is
usually located at ct] = 1800. This retains the symmetry of the 100p,
and for kb 5 1 it is the point at which maximum current occurs with the
exception of points near the driving point. With the restriction N = l
and (bl = 180°

Fe : FOO-VIFBI

: _15_[1+Vl] cosG sincb

 

 

 

Z l
- 41;: [l - V1] C089 sinB sin(2¢) (60)
and
F4) = F¢O -V1F¢1
:’ 41:30 [1 - V1] sine + Z—ja—l-[l + V1](1-%(kb)2 sin29)cos¢
.. 1%[1 - V1] sine cos(th) (61)
where
Y Y Z
V1 z _y11"+lYl : 1+;lllzl (62)

is found from equation (30).
Consider two cases, a small 100p kb = 0. l and a resonant 100p

kb =1.0. First, when kb 2 0.1and52 :12

ll.

( ) 7.4+j9.1x10'4 2' 7.4

_1__
a
O

-0.075+_jl.8x10-5 =‘ —o.o75

l
(at—1‘)

141

(‘5‘) g .o.02.2+j2.4x10‘9 =‘ -0.022

3.1;: 7.4x1o'7-j6.3x10‘3 mhos
. -7 . -3
3'11 : 8.0xl0 -36.0x10 mhos

and equations (60) and (61) become

Fe = -j0. 037[1+‘\71] cose sin¢ + o. 0005[1-'\71] cose sine sin(Zcp)
=' j0.037[1+ VI] cose sincb (63)
F4) 2' -o.18[1-V1] sine-jo.o37[1+Vl] cos¢ (64)

For the case of small 100ps the terms arising from (l/az) are always
small in comparison with the terms arising from (l/ao) and quantity
[1 + V1] appears in both terms. Thus, the (l/az) terms are drOpped
in equations (63) and (64). This shows that the approximation of drOp-
ping higher order terms is very accurate for very small 100ps.

The radiation fields of the small loaded 100p (<1)l = 1800) take on

two limiting cases. First, when VI =' O or 1 + VII =' 0

Fe=0

F ~ sine

<1>

which is the pattern of the unloaded 100p. This gives an omidirectional

pattern in the x-y plane and a sine pattern in the x-z plane. VI = 0

implies Z1 = O and 1+ VI = 0 implies 1:. -0. 01 -j81 ohms. The
second limiting case arises when '1 - VII :' 0. The fields then become
Fe ~ cos 9 sincb
F ~ cos
4, <l>

This gives a cos¢ pattern in the x-y plane and an omidirectional pat-
tern in the x-z plane which is just a rotation of the first limiting case.

ll - V115 0 arises when '21] -* 00 or when 21 = -0.67 + j3300 ohms.

Now consider the second case with kb = 1. 0 and $2 : 12.

142

(—) =‘ 0.67 +j0.061 =' 0.67

O
(—1—-) - 21+°3 o
a — - . J .
1
1 . .
(—-) = ‘0-29+JO.003Z = -o.29
a
2
yl ~_= .5.0x10'3’.j3.7x10'3
yll =' 5.1x10"°’+j4.1x10'3
With these coefficients
F6 =' -(l.5+jl.l)[l +71] cose sincb +0.075[1 -71] cose sine sin(Zcb)
(65)
17¢ =’ -o.17[1-‘\71]sine -(1.5+j1.1)[1+‘i71](1 -%—sin20)cos¢
+ o. 075 [1 - V1] sine cos (up)
= -o.17[1-Vl] 1-0.44 cos(2¢) sine
- (1.2 -j0.89)[1 +71] (1 +0.23 cos (29)) cos¢ (66)

In the x-y plane (9 : 900)

179:0

Fcb
O

and in the x-z plane ((1: = O , 1800)

-0. l7[ 1 -V1] (1 - 0.44 cos (24») - (0. 93 +j69)[ l +Vl] cos¢

Fe = 0
F4) = -o.095[1-V1] sine¥(1.z+jo.89)[1+‘\71](1+o.23 cos(29))
oO
fore :
180°

Once again it is seen that there are two limiting cases of the loaded
radiation pattern.
None of the equations deve10ped in this section are used in

actual calculations.

 

143

4. 2. Maximum and Minimum Gain of a LOOp Loaded with a Single
Impedance

Many times it is of interest to determine the maximum and
minimum gain or directivity of an antenna, in a given direction. For
the case Of the loaded antenna it is also Of interest to determine the
Optimum loadings that result in the maximum or minimum gain. If
no restrictions are placed on the load impedances, it is found that many
times the optimum load impedances have large negative real parts.

TO eliminate this in this discussion, the load resistance is assumed to F1
have a fixed value and only the reactive part of the load is optimized.
For simplicity, the additional restriction is made that the direction in
which the gain is to be Optimized is in the plane 9 = 90°.

The gain of a loOp loaded with one impedance is found from

 

equation (51) to be

F v W T‘F'“ no“... .-L'be;:.\.£m

! . _,— '.

"I" ... _ _
Mn, «I

2 l
G(9=9OO,¢) : 8(kb)_ 3

C6" Y. +Yfk
In In

(67)

where the * sign denotes the complex conjugate of the quantity, and

F4) = F¢o --\71F¢1

and from equation (30)

Y1

V =
+Gl

l

 

Y11 +JB1

where G1 is the fixed load conductance and B1 is the load susceptance

to be Optimized. After some algebra it is found

 

 

Z

Z A+CB +DB

8 b l l
G(e=9o°.¢) = “£2, 2 (68)

o H+IB1+JB1

where
A"lF (2| +GIZ-2Rea1[F*F (*‘LGH‘FIF 12! l2
" ¢o y11 1 ¢0 ¢1Y1y11 1 4’1 yl

C = 2Imag[Y11] IF¢olZ-Zhnag[F:oF¢1Y1]

144

2
IF¢OI

2 Z :1:
- 2 Real [y1(y11+Gl)]

m
ll

ZRea1[yo] ly11+Gll

2
I = 4Rea1 [yo] Imag [3'11] - 2 Imag [Y1]
J = ZRea1[Y0]

are real constants and "Real" and ”Imag" are Operators which retain

only the real and imaginary part of a complex quantity, rexpectively.

 

Differentiating equation (68) with reSpect to B1 and setting this result

equal to zero gives

(DI - Jcmi‘ + 2 (DH - JA) 131+ (CH - IA) = o (69)

 

which can easily be solved for B1.

Several examples Of the Special case Of a purely reactive
loading (i. e. , R1 = 0) are now considered. Figure (4. l) diSplays the
maximum and minimum gain and correSponding optimum load reac-
tances of a loop with kb = 1 loaded at 61 = 180° as a function of the
position where the gain is maximized. The unloaded IOOp gain is also
diSplayed. Figures (4. 2) and (4. 3) display the maximum and minimum
gains in the directions 4) = 900 and 4> = 1800 and corresponding optimum
reactances as a function of lOOp size kb. CorreSponding gain patterns
for kb = l. 0 and kb : 5. 0 are given in Figures (4.4) and (4.5).

It was found that by introducing a resistance into the load im-

pedance, the ability to modify the radiation is increased in many cases.

4. 3. Modification and Design of Radiation Patterns of LOOpS by Multi-
Impedance Loading

A method of determining the load impedances necessary to pro-
duce Specific modifications in the radiation pattern of a loaded lOOp
antenna is deve10ped in this section. The idea is simply to Specify
the radiation pattern in N directions which determines a set of N Simul-
taneous linear, algebraic equations in terms of the N unknown, nor-
malized load voltages. This set of equations may be solved, and the

load impedances are determined from the normalized load voltages.

 

 

LL.

145

 

 

 

 

  

— — —unloaded
lOOp gain

 

llllll

 

Max. and min. gain G(6 = 900, (1)), dB

 

120 150 180

 

 

 

 

 

 

 

 

 

I I l
3-
E -
..r: 2"
o
M
m“ 1"
o
U -
L1
1': o
o
m -
a) -
a
e -1' -
:3 _ .-
8
a. -2- ..
O _ ..(
-3- XBmaX q
IJIIIJJJIIIIIII
O 30 60 90 120 150 180
¢, degrees
(b)
Figure 4. 1. (a) Maximum and Minimum gain of a loOp antenna kb :

1.0 loaded at (bl = 180° with a purely reactive load.

(b) Optimum load reactances for maximum and minimum

gain.

 

 

146

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

In
'0
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o
o _
O~
n 0
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.8. -
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C!
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«’é o
E
I j I I T I I
3 I. x xGmin xGminT
Gmin .-
E
)- ....
.r: 2
o _ ..
M
. 1 " -
o
U _ -
s:
3‘3. 0
o
«I
Q) - u-I
H
E '1 ‘ "‘
D _
E .
g- '2 ”- xGmax xGmax
x
_ Gmax
-3 _
'- I
l I l l l l l l l .
O l 4 5
kb
(b)
Figure 4. 2. (a) Maximum and Minimum gain in direction 6 = 900,

¢ = 90° of a IOOp antenna loaded at ¢ = 180° with a
purely reactive load as a function Ofokb. (b) Optimum

lnarl roam-ances for max. and min. gain in direction

147

 

180°), dB

 

 

 

— - —unloaded 100p

 

 

max. and min. gain G(9 = 900. (b

 

 

1 1 1 1 1 1 ,
2 3 4 5 -"
kb :

(a) j

[‘95

 

Optimum reactance, K ohms

 

 

 

 

 

 

 

 

 

 

_3 _ x(3min _,
- q
I I l l I l l l
O l 2 3 4 5
kb
(b)
Figure 4. 3. (a) Maximum and Minimum gain in direction0 9: 900,

(b = 1800 of a 100p antenna loaded at (to 20180 with a
purely reactive load as a function of kb. (b) Optimum

load reactances for max. and min. gain in direction
e—- .. 90° , ¢=180°.

148

 

 

 

 

 

 

 

 

 

 

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-15” —
lllllll] llllllJl
0 3O 60 90 120 150 180
¢. degrees
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91
L3 rnax
.5 ‘10" "
m
no
-15“. —
11111111111111111
0 30 60 90 120 150 180

¢. degrees
(b)

Figure 4. 4. Gain pattern of 100p antenna kb = 1. o loaded at 6 = 180°
with a purely reactive load for minimum and maximum
gain at: (a) 6 = 90° and (b) 6 = 180°.

149

 

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150

Consider a lOOp antenna loaded with N arbitrary load impedances.
The radiation pattern function Of either the 9 or 4) component of the

radiation zone electric field may be written as

F(9.¢) = Fo(9.¢) - ianwwn (70)
n=l

[See equations (44) through (49) where Fn represents either Fen or
F¢n] . It is convenient to work with pattern functions that have been P
normalized to the unloaded pattern function. With this in mind, the (
following normalization is defined: _
F(9, 9)
Pow. ¢)

an. 6)
f = —-—-
n Fo(9, 4))

 

3""

and equation (70) becomes

f = 1- ivnfn (71)

n=1

The normalized unloaded loOp radiation pattern is a unit Sphere SO that
changes in the unloaded lOOp pattern are easily Specified. This elimi-
nates the need to know the exact values of the unloaded lOOp radiation
pattern and for most applications the general shape Of the unloaded
pattern is all that is needed.

The value of the normalized pattern is Specified in N directions.

With equation (71) this yields the following System of N equations

i(em,6m) = 1- ivnfn(em,6m) (72)
n21 m: 1,2,...,N.
Defining
fm = f(6m.¢m) and fmn = fn(9m.¢m) (73)

equations (72) can be written in matrix form as

151

r— “ r " L1
1 -f1 fll £12 . . . le v1
1 " f2 f21 f22 V2
-.- I ' (74)
1 - fl‘f. 5N1 . . . fNNJ _de

 

 

 

 

 

 

The term (l-fn) represents the difference between the unloaded and
loaded radiation patterns in the direction (On, 6n). Equation (74) is
easily solved and the load admittances are found from equation (30) to
be

Yn : (yn - i Ynmvm)/Vn' (75)

Care must be taken in choosing the directions in which the radi-
ation pattern is Specified because the matrix in equation (74) can become
singular when tOO much symmetry is introduced into the problem.

This can be seen by examining equations (48) and (49).

Consider the design of a pattern which is relatively directive
with respect to the pattern of the unloaded IOOp. An attempt will be
made to

a. maximize the front to back ratio Of the loaded pattern

b. minimize the beam width of the loaded pattern

eliminate or minimize the need for negative load
impedances.

For Simplicity the following discussion will be restricted to the pat-
tern in the plane Of the lOOp (i. e. , 9 = 90°). The problem is, given
N loads, to determine the best set of N load positions and the best set
of N pattern Specifications to accomplish the above criteria.

It was found that one maximum point Should be Specified and
remaining points Specified as zeroes in the pattern. In many cases,
an attempt to Specify the pattern more exactly than this results in the
develOpment of large lobes in directions where none are desired.
Little success was Obtained using only two loadings while with three or
more loadings favorable results were Obtained. For numerous exam-

ples considered, it was determined that the modified pattern Shape was

152

predominantly determined by the directions in which the pattern was
Specified and the positions of the loadings had only a Slight effect.
However, the values of the load impedances necessary to produce a
given pattern modification are strongly affected by the positions of
the loads.

Consider now, a lOOp of size kb = 1.0 loaded with three imped-
ances. It was determined that a Slightly better directivity could be
Obtained in the 4) = 00 direction than in the direction 4) = 1800. The
best pattern Specifications for several different loading positions were F3
found to be {1(90", 0°) = 1. o, f2(90°, 160°) = o. o and {3(90", 200°) = 0.0. ’
Moving the points the pattern was Specified at farther apart, it only
slightly decreased the beam width, B. W. , (the angle between half

power points in the plane of the 100p) with the back lobe increasing

 

more rapidly. The load reactances resulting from the above pattern
Specification are shown as a function of load position in Figure 4. 6.
It can be seen that all the load resistances are positive in the region
80° < 6 < 950. Figure 4. 7 diSplays the resulting gain pattern of the
lOOp when the loads are located at ((11 = 85°, (pa = 1800, and 413 = 2750.
This pattern has a B. W. = 1100 which is approximately the same as
that for the unloaded lOOp. The loaded IOOp has a front to back ratio,
F. B. R. , (ratio of the magnitudes Of the electric fields evaluated at
6 = 0° and 6 = 180°) Of 50 while the unloaded lOOp has a F. B. R =‘ 1.
The lOOp load impedances are Z1 = Z3 =' 387 - j 1180 Ohms and Z2 =
108 - j 140 Ohms and the input impedance is Zin :' 224 - j 23 ohms.

Next, the effect of changing kb was investigated. It was found
that for the above pattern Specifications and load configuration the
resulting load impedances had positive real parts and were very smooth
functions Of kb for kb greater than 0. 3 and less than 1. 1. Outside this
region the loop impedances become quite irregular functions Of kb.
The pattern retained its basic Shape up to kb < 1.3 for these pattern
Specifications and load positions.

An example of a small lOOp (kb = 0.3) with these same pattern
Specifications and load positions as discussed above is shown in Fig-
ure 4. 8. The load impedances are Z1 = Z3 =' 39 + j 291 Ohms and
Z2 =' 678 - j459 and the input impedance is Zin = 885 + j 40 ohms. The

very small values Of gain arise from the fact that the gain Of an

153

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

m 2.0 — —(
E
.1:
o
M 1.5 ‘- _.
a;
8 1.0 _. _
«3
+3
3
m
.U is”
m
O a. -.
a 0 a I
it .
0 3O 6O 90 120 150 180
¢1, degrees
I I 4T I *r 1 I I I I I I I I I I I
1 5_ (1)2 : 1800 -
_ O
E (113 — 360 -4)1
g 10" '-
M
a} 0.5— _.
‘5 <2
«I
‘5’
w 0
o
1..
'3 -0.5)— ..
o
A
x and:x
-1.5" 1 3 ""
lllllllllllilllll
0 3O 6O 90 120 150 180
(1)1 degrees
Figure 4. 6. Load impedances as a function Of load position for a

lOOp antenna. of size kb = 1. O with the pattern Specified
as f (90°, 0 ) = 1. 0, {2(900, 160°) = £3(90°, 200°) = 0. 0,
and : 12, 6b/a = 10.

 

 

154

 

 

 

 

 

 

 

 

 

 

 

 

 

5
lrIIIII IIIIIIII5
CG
'0
a; 07(—' 0
O
II
o -5"' -'5
95
O
.5 ‘10 “-10
:0
OD
14111111L11111411
0 30 60 900 30 60 9O
0, degrees
5
I I I I I I I I I IT I I I I I
m
"o
3 ...
o“
O
o" ‘
II
5?.
O -
.5
«3
60
11111411111111/
0 30 60 90 120 150 180
ct), degrees
Figure 4. 7. Gain pattern of a lOOp antenna, kb : 1. 0, loaded with

three imcpedances located at (1)1 = 85°, 4) = 1800, and
63 = 275 . The pattern is specified by 31(900, 0°) 2
1. 0, {2(900, 160°) = 13(900, 200°) = 0. 0.

gain (3(6). (1) = 900), dB

155

 

 

 

 

 

 

 

I I l I I 1 F I I I I I I I I T
(1) CG
'0 "U
A -20 I— '1 '20 I:
00 00
II --—' 25 CI:
-2 - -
g s“ - I
g -30 - - -30 0
-g .5
b0 Cd
1 1 J 1 L 1 I 1 L 1 I l 1 I l l °°
0 30 60 90 0 30 60 90

 

 

 

 

 

 

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-(
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'0
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.9.
O.
O
0‘ :-
ll
SE
0 _.
.E
«I
on
llLlllIIilllLlllI
0 30 60 90 120 150180

(1), degrees

Figure 4. 8. Gain pattern of a 100p antenna, kb— - 0.3, loaded with
three impedances at (b1: 850 , 6:3—00 - 180° and (1)3 -.?.750

The pattern is Specified by f1(90 = 1 0 {2(3900 1600 1:
f 3,(900 200°):

 

156

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

IIITIIII IITIIIWI
m
'o
. 0/ ‘ 0
0"
o
I!
.9. '5- "5
Si
O
.5 '10- -(-10
m
on
I IJ I I I I I
0 30 90 900 30 6O 90
9, degrees
lOIIIIIIrIIIIIIIIII
CO
PO
3 -
o“
o
O\ ‘
ll
Si
U -
.5
c0
60
B.W. 280°
_30_ F.B.R=9.5 _-
IIIIIIIJIIIIILI
0 30 60 90 120 150 180
4), degrees
Figure 4.9. Gain pattern of a100p antenna, kb =1.0, loaded with

five impedances of 61— 60°, 6221200, (I) — -0180
¢4= 2400 o, and ¢5f= 31000 and the pattern Specified by
“(90°, 0° )- -1. 0, f2’6900 90° )- - f (90°, 150)

14(90°, 210°)- - f 5(90, 270°):

90°), dB

gain G(0, <1)

 

‘1...

157

antenna is equal to its directivity times its efficiency. The extremely
small values of gain imply that the radiation resistance of the loaded
lOOp is very small compared with the input resistance of the loaded
100p.

Figure 4. 9 diSplays the gain pattern Of a lOOp (kb = l. 0) loaded
with five impedances as indicated. The load impedances are Zl =
25 = 244 +j226 ohms, Z2 = Z4 = 274 + j779 ohms, and Z3 = -121+
j397 ohms and the input impedance is Zin = 230 - j 50. Some effort
was made to eliminate the one negative load resistance, but not all
possible load configurations were explored. It can be seen that the

beam width is much narrower than the case of three loads but at the

expense Of increased side lobes.

4. 4. A Double Loaded Matched LOOp.

It is well known that the maximum power will be transferred
to an antenna if the impedance the antenna presents to the transmis-
sion System or generator driving it is the complex conjugate of the
impedance Of the transmission system or the impedance the generator
presents to the antenna. When this condition exists the antenna is said
to be "matched". In many cases there is a practical problem of match-
ing the antenna. In particular, with electrical small antennas which
characteristically have very small input resistances and large reac-
tances the impedance transformers necessary to match the antenna are
in many cases very lossy. This leads to a very low efficiency for the
radiating system (which in this case is taken to include the antenna and
its matching network).

It has been Shown by Harrison29 and Nyquist, Chen and

30’ 31' 32’ 33 that the efficiency of electrically small dipoles,

others
and slot antennas can be improved by impedance loading.

In this section the possibility of increasing the efficiency of a
small lOOp by impedance loading is considered. The efficiency of the
loaded loop is compared to that of an unloaded lOOp of the same Size
but matched at its input terminals with an impedance matching network.
The lOOpS themselves are considered lossless but the loading imped-

ances and matching impedances are assumed to be lossy inductors

and/or capacitors.

2 1" ‘l fig?

 

1;__

‘1‘

158

A different analytic technique than previously used is used to
determine the values of the impedances necessary to match the lOOp.
Consider a lOOp loaded with two lumped inductors or capacitors

which have finite Q. The Q of the loads is defined as

 

 

 

Ix I I13 I
1 1
= __ : (76)
1 R1 G1
Ix I Is I
2 2
Q = = (77) '9“:

where R1 and R2 are the resistances that arise from the nonideal ele-

_. 1 r1; 'Lr-‘f‘J‘a’Nd-‘I‘.’ ~.

ments. The admittance of the loadings can be expressed in terms of

their Q's and susceptance as (;

 

(78) ...-r

l

1 .

Y (79)

2

1 .
6; Ile +JBZ

A constraint equation on the load voltages is Obtained from

equation (32) as

Yin = y0 - Vlyl (80)

where Yin is the Specified, desired input impedance Of the 100p, and

the normalized load voltages are related to the load admittances by

Y Y Y

11 1 12 1 = 1 (81)
2 V2
which is found from equation (30). The normalized load voltages can

be eliminated from equation (80) by using equation (81). The resulting

equation is

AY1+CY1YZ+DY2+E = 0 (82)

where A, C, D, and E are constants defined in this section as

2
A ‘ y22(3’o ‘ Yin) ' y2

C

II

(Y0 " Yin)

159

2
D _ y11(YO -Yin) -yl
y11 Y12 ”Y1

Y21 y22 "Yz

 

 

'Yl 'Vz (Yo -Yin)
Defining the notation
A = Ar +jA1

c = cr+jci

D = Dr+jD1

E = Er+jE1

and writing
IBII = s 1 where Slzdzl

IBZI 2 where 82: 11

I
U)
N

complex equation (82) can be written as two real equations. These

equations may be manipulated into the form

 

 

131 =PBZ+R (83)
2
BZ+SBZ+T _ o (84)
1 Dr 1 11:)1 r)
E(EESZ - D) -fi(b—Z_SZ+D
P:-1_ 162:. A1 1__‘°_*1_3 +Ar
F lel' 'H Q11
sf. 12:1
R: _ F ‘F
1Ar 1 1A1 r
flora A ) 'fi(oTSI+A)
s — Lil—s +Ar)+5+—1—— P-l—s +Dr
‘ H Q1 1 P HP (2.2 2

 

 

 

 

__ R Ai r E1
T ‘ 1113621514rA )+HP
and r 1 i
C r C C
F = s S -C -——S -——-s
°1°2 12 Q1 1 Q2 2
1 r r
' C C
H = s s -C1+--S +—S
QIQZ12 Q1 1 OZ 2

0"
a:
ll
H
-
0)
II
I
g—a

d. stz-l.

The efficiency Of this matching technique is given by

1[ 2 2 2 1
IO R. - I R - I R
eff . z l ( )l m l (¢1)I 1 l (¢2)| z x 100%

l 2
3 IRON R.

 

n
_ 2 lBll _ 2 IBZI
e [1-lvll 57531-1sz m1XIOOO/o (85)

The impedance transform network shown in Figure 4. 10 can be

used to base tune the unloaded lOOp.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

...—— th oe
Z _) Z1 Zunloaded
in t 10013

c 4‘16

 

Figure 4. 10. Base tuning network.

 

 

161

The tuning impedances are assumed to be nonideal capacitors or in-

ductors which have finite Q's. These impedances can be represented as

Z lth' + .X (81 X 86
= —— J = —+1) ( 1
t1 ot1 t1 ot1 t1
Z .'__'_..X .(2.,). ...,
t2 QtZ t2 QtZ t2
where 81: :tl and S2 = :l:2..

Specifying a desired input impedance and given an unloaded loop im-

pedance, the values of th

cuit theory. The exact solution is found to be the roots Of a quadratic

and X 2 may be found by conventional cir-

equation. It can be shown that the efficiency of this network is given by

 

 

Z Runloaded
t1 100p
eff = . (88)
Zt1 + Zunloaded Rin
lOOp

This assumes lzll 7! 0.

A numerical study was conducted, and it was determined that
in the case Of small lOOpS the loadings necessary for a purely resistive
input impedance are always inductive. However, when one load is
placed at (bl = 00 and the second at <|>2 = 1800 the loading necessary at
the driving point becomes capacitive. This configuration is desirable
Since capacitors are usually less lossy than inductors.

A comparison is made in Figure 4. ll of the efficiencies of a
loop loaded at 61 = 0° and °2 = 180° so that the input impedance is
300 ohms and the base matching network required to match the unloaded
lOOp input impedance to 300 Ohms. A Q of 300 Oth was assumed for
all the elements. The correSponding reactances are Shown in Figure
4. 12. It is seen that the efficiencies are nearly identical for the loaded
loop and the matching network. In reality the matching network is
probably more efficient at least at lower frequencies where the Q of

capacitors is greater than 300.

 

 

 

tl ‘ ' ‘ n.- .3 : ".i. {wavy—n

162

 

 

 

 

 

 

100 j
80 -
p
a‘J
1:
8 ..
H 60
Q)
0.
>‘; b
o
G
.2
.8 40 -
m
o
20 '-
_ Loaded lOOpe _
- —— Base matching
network
I I I I I I I I
0.0 0.2 0.4 0.6 0.8 1.0

kb

Figure 4. 11. Efficiency Of lOOp loaded with two impedances at (b) = 00
and (b = 1800 to match antenna with 300 Ohm input im-
pedances. Compared to efficiency Of base matching
network.

 

 

 

 

 

 

 

 

 

 

 

 

 

E
I: -2
O
M
C -4
<13
+J
U
«I
Q)
m
-6
-81
“a”
.c: 0
o
M
a?
8
Id -2
4.)
o
«s
Q)
m
-4
Figure 4. 12.

 

 

(b)

(a) Load reactances necessary for 300 ohm input im-
pedance to 100p. (b) Matching network reactances
necessary to match a lOOp antenna 82 = 12, 6b/a = 12. 1
to 300 Ohms.

CHAPTER V

CONC LUSIONS

In the preceding chapters, the circuit and radiation properties
Of a circular lOOp antenna were considered. A refined theory Of the
loaded lOOp antenna was deve10ped which included a finite gap excitation. .
The effects of the finite gap excitation were considered and effective m
gap widths corresponding to the situation when the loop is driven by a
two wire line and a coaxial line are prOposed. A comparison of the

finite gap theory including the effective gap widths is made with exist-

 

ing experimental input admittances. Excellent agreement between E j
theory and experiment is obtained. "l

Synthesis procedures have been deve10ped to facilitate the de-
sign Of multi-loaded lOOp antennas. First, an expression for the maxi-
mum and minimum gain attainable from a lOOp loaded with a single
impedance was deve10ped. This expression gives upper and lower
bounds on the amount of antenna gain pattern modification that can be
accomplished by a Single load impedance. It was found for the case of
a purely reactive loading located at <1) = 1800, that in the plane of the
lOOp the antenna gain cannot, in general, be greatly modified in the
directions near <1) = 00 and 1800 but can, in most cases, be significantly
modified in directions near 6 = 90° and 270°.

A procedure was then deve10ped for the design of Specified
radiation patterns. Examples were given Showing that relatively di-
rective patterns could be easily designed.

Finally, the possibility of loading a small loop antenna to pro-
duce a desired input impedance was considered. This technique does
not appear promising as a means of improving the efficiency Of the

radiating System.

164

 

 

10.

11.

12.

13.

REFERENCES

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Math. Phys., 2, 1301-1304, (Nov.-Dec. 1962).

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J. E. Storer, "Impedance of Thin-Wire Loop Antennas, ” Trans.
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S. Adachi and T. Mushiake, ”Theoretical Formulation for Cir-
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S. Adachi and Y. Mushiake, "Study Of Large Circular LOOp An-
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R. W. P. King, C. W. Harrison, Jr., and D. G. Tingley, "The
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R. W. P. King, C. W. Harrison, Jr., and D. G. Tingley, "The
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K. Iiauka, "The Circular LOOp Antenna Multiloaded with Positive
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K. Iizuka and F. LaRussa, "Table of the Field Patterns Of a
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R. E. Collin and F. J. Zucker, Antenna Theory, Pt. I. (McGraw-
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E. Jahnke and F. Emde, Tables Of Functions, (Dover, New York,
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165

 

 

14.

15.

l6.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

166

G. N. Watson, A Treatise on the Theory of Bessel Functions
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R. E. Collin and F. J. Zucker, Antenna Theory, Pt. I. (McGraw-
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R. W. P. King and C. W. Harrison, Jr., Antennas and Waves
(M. I. T. Press, Cambridge, Massachusetts, 1969), 569-576.

 

G. Gilbert and G. Baker, Bessel Function (Fortran), M. S. U.
Computer Lab. No. 00000223.

 

M. Abramowitz and I. A. Stegun, HandbOOk_ Of Mathematical E
Functions (Dover, New York, 1965), 378-379. '

 

M. Abramowitz and I. A. Stegun, Handbook of Mathematical
Functions (Dover, New York, 1965), 378, Eq. (9.75).

 

H. K. Crowder and S. W. McCuskey, TOpics in Higher Analysis
(Macmillan, New York, 1964), 190 and 199.

 

K. Iizuka and R. W. P. King, "Terminal-Zone Corrections for
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(U. S. ), 66D, 775-782 (Nov. -Dec. 1962).

R. W. P. King, Theory of Linear Antennas (Harvard University
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T. T. Wu, "Input Admittance of Infinitely Long Dipole Antennas
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P. A. Kennedy, "LOOp Antenna Measurements, " IRE Trans.
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J. B. Sherman, "Circular LOOp Antennas at Ultra-High Frequen-
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B. R. Rao, "For Field Patterns of Large Circular LOOp An-
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30.

31.

32.

33.

167

C. J. Lin, D. P. Nyquist, and K. M. Chen, ”Short Cylindrical
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C. J. Lin, D. P. Nyquist, and K. M. Chen, "Parasitic Array of
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T. Z. Hsieh, D. P. Nyquist, and K. M. Chen, ”The Short-Slot
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T. Z. Hsieh, D. P. Nyquist, and K. M. Chen, ”The Loaded
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Commission V, (1971).

 

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III)

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111(1)