IMPEDANCE - LOADED RECEIVING ANTENNAS A

WITH MWMUM BACKSCATTERING
AND MAXIMUM RECEIVED POWER

Thesis for the Degree of» Ph. D.
MICHIGAN STATE UNWERSETY
HOWARD JOSEPH DECK
1968

THESIS

This is to certify that the

thesis entitled

IMPEDANCE-LOADED RECEIVING ANTENNAS
WITH MINIMUM BACKSCATTERING
AND MAXIMUM RECEIVED POWER

presented by
Howard Joseph Deck

has been accepted towards fulfillment
of the requirements for

1311- D. . degree in___E- E_-_

5% ca

/ Major professor

Date AURUSt 2. 1968

 

0-169

© Howard Joseph Deck 1969

 

ALL RIGHTS RESERVED

ABSTRACT

IMPEDANCE-LOADED RECEIVING ANTENNAS
WITH MINIMUM BACKSCATTERING
AND MAXIMUM RECEIVED POWER

by Howard Joseph Deck

In addition to serving its recognized intended purpose, a
conventional receiving antenna of almost any variety acts inher-
ently as a scatterer, re-radiating a portion of the electromagnetic
energy impinging on it. In many practical applications this re-
radiation is inconsequential in its effect on the environmental
surroundings of the antenna. In other situations, however,
such secondary-source effects are undesirable and, in fact, may
be highly detrimental to overall system performance. This is
true, for example, in situations where several receiving antennas
(presumably operating independently) exist in close proximity to
each other.

The purpose of this thesis is to achieve a reduction in the
amount of electromagnetic power backscattered from a dipole
receiving antenna, utilizing techniques of impedance loading.

This approach involves inserting lumped impedances directly

HOWARD JOSEPH DECK

into the antenna itself in an effort to modify the distribution of
induced current on the antenna in such a way as to effect the
reduced scattering desired.

With two such identical “auxiliary impedances " placed
symmetrically about the center-load impedance, an approximate
closed-form solution is found for the antenna current distribution.
The condition of zero broadside backscattering is established in
the form of a constraint equation between the auxiliary impedances
Z and the center-load impedance ZL. Using this constraint to
eliminate either Z or ZL, the received power delivered to ZL is
expressed in terms of a single impedance parameter and is
subsequently maximized with respect to this single complex

variable. This procedure yields values of Z and ZL for an

optimum receiving antenna whose backscattered field in the

 

broadside direction vanishes and (by virtue of the negative real
parts of the Optimum auxiliary impedances obtained) which pro-
duces an infinite amount of received power.

Several cases are investigated in which one of the four

real variables contained in Z = R + jX and Z = R + jX

L L L13

arbitrarily specified. In each case the remaining unspecified

impedance parameter values are determined for a near-optimum

 

receiving antenna for which the condition of zero backscattering

 

HOWARD JOSEPH DECK

is satisfied but for which the received power is finite and
maximum.

In general, using a purely reactive auxiliary impedance
(R Specified as zero), the backscattering constraint between Z
and ZL leads to a trivial receiving antenna for which RL= 0. How-
ever, a very interesting phenomenon occurs at a specific position
of the auxiliary load. At one particular frequency, with the value
and placement of the auxiliary reactance properly selected, the
resulting antenna is not only invisible to electromagnetic illumi-
nation at normal incidence, but is incapable of delivering any
power to the center impedance ZL. Furthermore, this invisible
frequency-rejection characteristic is independent of the choice
of ZL.

Relaxing the backscattering constraint between Z and ZL'
it becomes possible to utilize reactive loading to achieve minimum
backscattering. Two such cases are considered: one in which
ZL is selected as the complex conjugate of the input impedance of

the antenna, and another in which R is arbitrarily Specified

L

while XL is chosen as the negative of the input reactance. Corres-
ponding to the latter case, a fairly extensive correlation between

theory and experiment is given for a fixed-length antenna for

several different positions of loading.

HOWARD JOSEPH DECK

Experimental results are also presented which sub-
stantiate the existence of the invisible frequency-rejection

characteristic predicted theor etically.

IMPEDANCE-LOADED RECEIVING ANTENNAS
WITH MINIMUM BACKSCATTERING
AND MAXIMUM RECEIVED POWER

BY

Howard Joseph Deck

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering

1968

Copyright by
HOWARD JOSEPH DECK

1968

ACKNOWLEDGMENTS

The author wishes to express his sincere appreciation
to his major professor, Dr. K. M. Chen, for his guidance in
the research effort and for his careful review of the written
thesis. He also wishes to thank the other members of his guidance
committee: Dr. J. S. Frame, Dr. J. B. Kreer, Dr. R. J. Reid,
and Dean L. W. Von Tersch, for reading the thesis, andto
acknowledge the helpful suggestions of Dr. D. P. Nyquist in
regard to the experimental part of the research.

Finally, the author wishes to express a Special thanks to
his wife, Ann, for her help in the preparation of the figures, and
for the encouragement and understanding that only a wife can give.

The research reported in this thesis was supported in part
by the Air Force Cambridge Research Laboratories under contract

AF19 - (628) - 5732.

ii

TABLE OF CONTENTS

ACKNOWLEDGMENTS..................

LISTOFTABLES......... ...........

LIST OF FIGURES . ........ . . . ..........

INTRODUCTION.... ...........
I THE DOUBLE- LOADED ANTENNA ..........

1.1 Formulation of the Problem. . . ........
l. 2 Induced Current on the Receiving Antenna . . .

1. 2.1 Integral Equation for the Induced
Current
1.2.2 Solution for the Induced Current .....

1. 3 Current Distribution on the Transmitting
Antenna. 0 O O O C O O O O C O O O O O O O O

1.4 Input Impedance of the Transmitting Antenna . .
l. 5 Reduction to the Unloaded Dipole Antenna . . .
1. 5.1 Current Distribution on the Unloaded

Receiving Antenna . . . . . . . . .
1.5.2 Current Distribution on the Unloaded
Transmitting Antenna . . . . . . . . .
1. 5. 3 Input Impedance of the Unloaded
Antenna.................
II THE OPTIMUM RECEIVING ANTENNA. . ..... .

2.1 The Condition for Zero Backscattering--the
constraint Equation O O O C O C O O O O C C C O

2. Z The Condition for Maximum Received Power .

iii

Page

19
2.2

23

24
2.5

26

27

32
35

TABLE OF CONTENTS

(continued)
Page

2.3 Selection of the Optimum Parameters . . . . . . 39

2.4 Theoretical Results. . . . ....... . . . . . 43
HI NEAR-OPTIMUM RECEIVING ANTENNAS ....... 50
3.1 Pure Reactive Loading . . . . ..... . . . . . 53
3.2 Pure Resistive Loading. . . . . ........ . 58

3.3 Center-Load Resistance Specified . . . . . . . . 67

IV REACTANCE-LOADED RECEIVING ANTENNAS
WITH MINIMUM BACKSCATTERING . . . . . . . . 75
4.1 The Conjugate-Match Condition. ......... 77
4.2 Load Resistance Specified . . ......... . 83

V EXPERIMENT AND RESULTS . . . . . . . . . . . . . 91
5.1 The Experimental Reactance-Loaded

Receiving Antenna. . . . . . . . . ....... 91

5.2 The Experimental Setup. . . . . . . . . . . . . . 97

5. 3 Experimental Results and Comparison
WithTheory...................102

5. 3.1 Performance of the Reactance-Loaded
Receiving Antenna as a Function of
the Auxiliary Reactance. . . . . . . . . 102
5. 3. 2 Experimental Verification of the Invisible
Frequency-R ejection Receiving Antenna 109

REFERENCES. ............. . ....... . . . 111

APPENDIX: Received Power of the Double-Loaded Antenna 112

iv

Table

LIST OF TABLES

Parameter values for the Optimum Receiving

Antenna in ohms (£303.: 0.001). . . . . . . . .

Reactance-loaded receiving antennas with
minimum backscattering--the conjugate-

match condition (fioa = 0.001) . . . . . . . . .

Reactance-loaded receiving antennas with
minimum backscattering- -1oad resistance

SPeCi-fied ($03 : O. 001) o o a o o o o o o o o o 0

Experimental results of the search for the
invisible frequency-rejection antenna . .

Page

48

82

88

110

3.5

3.6

3.7

LIST OF FIGURES

Page
The double-loaded receiving antenna. . ...... 4
Center—load resistance for the Optimum
Receiving Antenna. . . . . ..... . . . . . . 46

Optimum Receiving Antenna parameter
values for Sch: 2.5 (poa : 0.001). . . . . . . . 47

Auxiliary reactance and loading position
for a receiving antenna with a frequency-
rejection characteristic at (o = [3 v

(poa=o.001)..........°.°....... 57

Received power for the resistance-loaded
Near-Optimum Receiving Antenna
(fi0a=0.001) .. .. .. ............ 63

Auxiliary resistance for the resistance-
loaded Near-Optimum Receiving Antenna
(floa : o. 001) o o o o o o o o o o o o o o o o o o 64

Center-load resistance for the resistance-
loaded Near-Optimum Receiving Antenna
(floa=0.001)............ ..... .65

Center-load reactance for the resistance-
loaded Near-Optimum Receiving Antenna
(poa : o. 001 )0 C O O O ..... O O ....... 66

Received power for the Near-Optimum
Receiving Antenna for which the load
resistance is specifiedu-RL: 50 ohms
(fi0a=0.001).............. ..... 71

Center-load reactance for the Near-Optimum
Receiving Antenna for which the load
resistance is specified--R = 50 ohms

(poa=o.001)........L..... ...... 72

vi

LIST OF FIGURES

(continued)

Figure Page

3. 8 Auxiliary resistance for the Near-Optimum
Receiving Antenna for which the load
resistance is Specified--R = 50 ohms

(00a = 0.001) . ...... L. .......... 73

3. 9 Auxiliary reactance for the Near-Optimum
Receiving Antenna for which the load
resistance is specified--R = 50 ohms

(Boa: 0.001) ..... ..L.. ..... .... 74

5.1 The experimental reactance-loaded
receiving antenna . . . . . . . . ....... 93

5. 2 Assembly drawing of the experimental

reactance-loaded receiving antenna ,,,,, 95
5. 3 Block diagram of the experimental

equipment setup ...... . ......... 98
5.4 The anechoic chamber ............... 99
5. 5 Experimental and theoretical results for a

reactance-loaded receiving antenna with

XL .-. .xin (X) and d/h = 0.3 ,,,,,,,,,, 104
5. 6 Experimental and theoretical results for

a reactance-loaded receiving antenna

w1th XL = .xin (X) and d/h = 0.4 ,,,,,,, 105
5. 7 Experimental and theoretical results for

a reactance-loaded receiving antenna

with XL = -Xin(X) and d/h = 0.5. ....... 106

LIST OF FIGURES

(continued)

Figure Page

5. 8 Experimental and theoretical results for
a reactance-loaded receiving antenna

w1thXL= -Xin(X) and d/h: 0.6 . . . . . .. 107
5. 9 Experimental and theoretical results for

a reactance-loaded receiving antenna
with XL = ..xin (X) and d/h = 0. 7 ....... 108

viii

INTRODUCTION

In addition to serving its rec0gnized intended purpose, a
conventional receiving antenna of almost any variety acts inher-
ently as a scatterer, re-radiating a portion of the electromagnetic
energy impinging on it. In many practical applications this re-
radiation is inconsequential in its effect on the environmental
surroundings of the antenna. In other situations, however,
such secondary-source effects are undesirable and, in fact, may
be highly detrimental to overall system performance. This is
true, for example, in situations where several receiving antennas
(presumably Operating independently) exist in close proximity to
each other.

The purpose of this thesis is to achieve a reduction in the
amount of electromagnetic power backscattered from a dipole
receiving antenna, utilizing techniques of impedance loading.

This approach involves inserting lumped impedances directly
into the antenna itself in an effort to modify the distribution of
induced current on the antenna in such a way as to effect the

reduced scattering desired.

On the presumption that this objective can be accomplished
without seriously degrading the primary function of the antenna,
the results might well provide a new impetus to the area of
antenna design. Not only would such a capability be a potential
aid in relieving the interference problem between independently-
acting receiving dipoles, but would provide a means for making
such structures invisible to radar. Moreover, an impedance
loading technique which reduces the broadside re-radiation from
individual dipole structures might very well provide a practical
means for reducing the mutual coupling between the elements

of transmitting antenna arrays, as well as receiving arrays.

I THE DOUBLE-LOADED ANTENNA

1.1 Formulation of the Problem

The configuration of the antenna fundamental to this
study is shown in Figure 1.1. The antenna assumed is a per-
fectly conducting cylinder of radius a and length 2h. A coordinate
system is chosen such that the cylinder lies along the z-axis with
its center at the origin. At locations z = id two identical im-
pedances Z are inserted ”in-line" into infinitesimal gaps in the
cylinder. These symmetrically placed loads will be referred to
as "auxiliary impedances. " When operating as a receiving
antenna, it is assumed that the illuminating electromagnetic
field is a plane wave, normally incident with its E-field parallel
to the z-axis. In this mode, a receiving load impedance ZL is
inserted into the antenna at z = 0.

The dimensions of interest are

h<x and Ba<<l (1.1)
O 0

where X0 is the free space wavelength and [30 = 21r/Xo is the wave
number. For antennas longer than two wavelengths the approxi-

mating form which is used for the antenna current distribution

om:

cm

Figure l. l.

I(z)

 

 

 

 

2a

The double -1oaded receiving antenna

becomes too inaccurate. The second restriction given in (1.1)
implies that the cylinder is thin so that it can be assumed that

only an axial current is induced.

l. 2 Induced Current on the Receiving Antenna

1. 2. 1 Integral Equation for the Induced Current

The electric field E0 illuminating the receiving antenna
is assumed to be normally incident and parallel to the antenna.
It is further assumed that E0 varies sinusoidally with time. Since
the tangential component of the total electric field must be con-
tinuous at the surface of the cylinder, the following equation is

valid on the surface.
E + E0 = ZLI(0)6(z) +ZI(d)6(z-d) +ZI(-d)6(z+d) (1.2)

E: is the tangential electric field maintained by the current and
charge on the cylinder, I(p) is the induced current at z = p, and
6(z-p) is the Dirac delta function which is identically zero for
2 f- p. Equation (1. 2) indicates that the total tangential electric
field vanishes on the surface of the cylinder while maintaining
voltages ZI(-d), ZLI(O), and ZI(d) across the infinitesimal gaps
located respectively at z = -d, z = 0, and z = d.

The electric field produced on the surface of the antenna

by the antenna itself is given by

E = -—-i)- -j0.)A (1.3)
z

where (1) is the scalar potential at the surface maintained by the
antenna charge and AZ is the axial component of the vector potential
at the surface supported by the antenna current. The scalar
potential ¢ can be eliminated from (1. 3) by using the Lorentz

condition

6A
2

82 = -pro€o¢ (L4)

 

where the symbols (0, p0, and 60 represent, respectively, the

angular frequency of the incident electric field, the permeability
of free space, and the permittivity of free space. Differentiating
both sides of (1.4) with respect to z and substituting the resulting

expression for 2% into (1. 3) gives the following equation for E:.

. 32A 2
Ea : -J—w— [ z + [3 A] (1-5)
z ‘32 az2 o z
o

 

Combining equations (1. 2) and (1. 5) and using the fact that by
symmetry I(-d) = I(d), the following differential equation is ob-

tained for AZ.

GZAZ 2 jpoz
Z + ‘30 AZ 2 (.0 [ZLI(O)6(Z) + ZI(d){5(z-d)

 

 

dz
+ 6(z+d)} - E0] (1'6)

The general solution of (1.6) is given by

_ ;i -
Az(z) - V0 [Clcosfloz +Czsmfloz

+ E0 NBC- 2%- ZLI(O)sin fiol z| - :- ZI(d) {sinflol z-dl +sinl30|z+d| }]
(l . 7)
where V0 is the velocity of light in free space and is equal to Hm .
The first two terms on the right side of equation (1. 7) correspond to
the complementary solution; C1 and C2 are complex constants. The
last three terms are particular integrals stemming from the in-
homogeneity of the differential equation.
In addition to (1. 7) another equation may be written for the

tangential component of the vector potential in which Az(z) is ex-

presSed in terms of the induced current which produces it; viz. ,

p, h
o
A (z) = — S I(z')K (z,z')dz' (1.8)
2 411' -h a

where the kernel Ka(z,z') is given by

 

-jpo\/(z-z')z+ a2
Ka(z,z') = e . (1.9)

\/(z-z')Z + a2

 

Based upon the evenness of the function I (z), it can readily be
demonstrated from (1. 8) that Az(z) is also an even function. This
implies that C2 = 0 in equation (1. 7).

Equating (l. 7) with (1. 8) yields the following integral

equation for the induced current. Notice that P'ovo = 12017.

h

 

S I(z')Ka(z,z')dz' =

-h

, E 2 1(0)

g—gmlcospow—fi—E- é meow-5:91 {sinfiOIz-dl+sinsolz+dl}]

(1 . 10)
It is expedient at this point to express C1 in terms of the
vector potential at some fixed point on the antenna. This potential

can later be conveniently determined from equation (1. 8). Selecting

z = h as the fixed point and substituting this value of 2 into (1. 7),

. _ 1 . 1 .
JVOAZUI) - C1008 poh +Eo/po- 2 ZLI(0)sm[30h- 2 ZI(d){2 SinBoh cos flod}

(1.11)

or

c = secpoh{jvoAz(h)-Eo/po+lz I(O)sin[30h+ZI(d)sinBohcos[30d}.

1 2 L
(1.12)
The result of putting this expression for C1 back into equation (1.10)
is
h j 1
1 l I: L. ° _ _ ‘
Sh I(z )Ka(z, 2 )dz 30 [sec [30h{jvoAz(h) ISO/[50+ 2 ZLI( O)81nf30h

+ ZI(d)sinfioh cos 50d}cosfioz +Eo/[30-é- ZLI(0)sin Bo|z|

l

-EZI(d){sinfio|z-d|+sin130|z+d|}]. (1.13)

A more convenient form of the integral equation is obtained by

substituting z = h into (1.13) and subtracting the result from (1. 13).

This procedure yields
h

I(z')K (z,z')dz' =
1.. d

72% secfioh[ (ijAz(h) -EO/poxcospoz - cos Boh) +% ZLI(0) sin [30(h- IZI)

+% ZI(d) g(z)] (1.14)

where the new kernel is

K (z,z') = K (z,z') - K (h,z') (1.15)
d a a

and g(z) = 2 sinfioh cos (30d cos (302 - cos Boh{sinfio| z-d| + sinfiol z+d| }.
(1.16)

Equation (1. 14) is an integral equation for the induced current on

the double-loaded receiving antenna, valid for -h 5 z 5 h. Unlike

equation (1.13) the integral in (l. 14) is proportional to vector

potential difference; as such, the latter equation is automatically

satisfied at the ends of the antenna. Notice that Az(h), 1(0), and

I(d) on the right side of (l. 14) are functions of the induced current

being sought and are still unknown.

1. 2. 2 Solution for the Induced Current

Although an exact, closed form solution to integral equa-
tions of the type represented in (1.14) has never been found, a
number of methods for obtaining approximate solutions have been

advanced in the last several years--notably schemes devised by

lO

[1,2,3] [4,5,6]

1
, King and Middleton[ ], Chen , Duncan and

.[8]

7
Hinchey[ 1, and Mei . Most of these methods are inherently

King

numerical in nature, based upon techniques such as iteration,
numerical integration, or solution by series. Such methods are
unsuitable here since the solutions would be too complicated to
make further theoretical development feasible.

The technique chosen for obtaining an approximate solution
to equation (1. 14) is based on a recent method of King[ 3]. This
procedure yields a solution in closed form as desired, and pro-
vides a suitable degree of accuracy. The rationale for this method
will be indicated briefly as it is used.

The key to the solution technique afforded by King's modi-
fied method lies in closely examining the difference kernel
Kd(z, z') appearing in integral equation (1. 14). The kernel is

written in terms of a real and an imaginary part and expanded as

 

 

 

 

 

follows:
Kd(z,z') = KdR(z,z') +j KdI(z,z') (1.17)

where

cos $0J(z-z')2+ a2 cosf3o\l(h-z')2+a2
KdR(Z.Z') = 13° —-—-—-———-—— - Z—T— (1. 18)

fioJ(z-z') + a BOJ(h-z') + a
and

s1n00J(h-zI)2+ a2 sinpoJ(z-z')Z+ a2

 

.. .(l.l9)

K (z,z') =f5 ————-———— _ fi—
dI o $0J(h-z') +a fioJ(z-z') +a

11

It can be readily shown from (1.19) that |KdI(z, z')|/[3o < 2. On

the other hand examination of (1. 18) indicates that as z' approaches

z, |KdR(z, z')| becomes very large. Indeed, at z' = z, the latter
quantity increases without bound as the radius of the antenna approaches
zero. This sharp peak in KdR(z, 2') indicates that the principal con-

tributions to the part of the integral that has K (2, 2') as its kernel

dR

come from elements of current very near z'= z. Likening K z')

dR(z’
to a Dirac delta function, moreover, suggests, in view of the right

side of equation (1.14), that the form of the current distribution be

assumed as
I(z) = Cc(cosfioz - cos 50h) + CSSLnBO(h- |z|) + Cig(z) (1.20)

where Cc' CS, and Ci are complex constants to be determined.
Notice that this assumed form for I(z) satisfies the boundary con-
ditions at the ends of the antenna. The substitution of (l. 20) and

(l. 17) into integral equation (1.14) yields
h
t_ ‘ _ i I 1
5h {Ccicosfioz cos fiOhHCSsmflom Iz l1+Cig(z )}{KdR(z.z)

+ jKdI(z, z')}dz' = 3% sec [30h[ (jvoAz(h) - EO/fioHcos [302

- cos Bah) +-:— ZLI(0)sin {30(h- I2 I) +% ZI(d)g(z)]. (1 . 21)

Finally equation (1. 21) is broken up into the following three equa-

tions:

12

h

Sh CssinBo(h-|z'|)KdR(z, z')dz' = 8% sec floh[% ZLI(o)sinBO(h-|z|)]

(1.22)
h -. 1
Sh Cig(Z')KdR(Zs z')dzt : 3%; Sec Boh[EZI(d)g(z)] (l o 23)
h h
Sh Cc(cosfiozi_cosi30h)KdR(z, z')dz' + Sh {Cc(cospozu_ COsfioh)

+ Cssinfio(h- | z'l) +Cig(z')}{jKdI(z, z')}dz' = S—‘Z, sec fioh[ (jvoAz(h)

- E H3 )(cosB z-cosB h)]. (1.24)
o o o o

The justification for the first two equations in this group is based on
the characteristics of the function KdR(z, z') mentioned above. The

sharp peak in K (2, 2') at z' = z also accounts for the presence of

dR
the first integral on the left side of equation (1. 24). Notice that the
remaining terms from the integrand of (1. 21) not already accounted
for are included in the second integral in (l. 24). The justification

for the presence of this latter integral is based on numerical con-
siderations. It can be shown numerically that the results obtained

by integrating the products (cos [Soz'- cos fioh)KdI(z, 2'), sin 00(h- I z'l ) °
KdI(z, z'), and g(z')KdI(z, 2') over the length of the antenna closely

approximate, in each case, a function which is proportional to the

shifted cosine function (cos fioz - cos fioh).

13

Three complex definite integrals which arise in the pro-

cess of solving for the constants in equation (1. 20) are defined

here.
h
Tcd = Slh (cos fiozhcos (30h)Kd(0, z')dz' (1.25)
h
TSd = Sh (sin [30(h- |z'|)Kd(0,z')dz' (1.26)
h
Tid : Sh g(z')Kd(O,z')dz' (1.27)

The letter "R" or "I" appended to any of these "T-integrals" will
be understood to denote "the real part of" or "the imaginary part
of, " respectively.

Proceeding with the determination of CC, Cs, and Ci’
equations (1. 22), (l. 23), and (l. 24) are first rewritten with z

set to zero.

h

Sh Cssin 5001- lz'|)KdR(0, z')dz' : 5% sec (30111::- ZLI(0)sin Bob]
(1. 28)

h -.
Sh Cig(zt)KdR(0, z')dzI = 3%)- sec [30h[ ZI(d)sinfio(h-d)] (1, 29)

h h
jib Cc(cos [3025 cos fioh)Kd(O, z')dz' +511 {CsSinpo(h- I z' I)

+ cig(z')}{1KdI<O.z'>}dz' = s,- sec 00111 (jvoAzm

- Eo/ [30)(1 - cos Boh)] (l. 30)

14

CS and C1 are found immediately from equations (1. 28) and (l. 29)

as

, Z I(0)Sinf3 h]
"J L O
C = — secfih[ (1.31)
s 30 0 2Tst

C = isecfioh‘: (1.32)

 

ZI(d)sin BO(h-d):l

T

° 0
l 3 idR

Using (1. 31) and (l. 32) equation (1. 30), after rearrange-

ment, yields

 

.. Li.
CC — sec Boh

30 [(jvoAz(h) - ISO/(30)“ - cos Bah)

 

 

cd
j ZLI(0)TSdIsm Boh + ZI(d)TidIsm (30(h-d)
2 Tst TidR

(1.33)

Substitution of these constants into equation (1. 20) pro-
duces the following expression for the induced antenna current.
-J'

I(z) = 80 sec (30h

 

Tcd

_j zL1(0)TsdIsinsoh + ZI(d)TidIsin[30(h-d) ].

2 Tst TidR

[(jvoAz(h)- E0/ {501(1 - cos 50h)

 

 

ZLI(0)sini3 h
2 T
s

 

[cos Boz-cosfloh] + sin(50(h-|z|)

dR

ZI(d)sinfi (h-d)
+ ° g(z) (1.34)

T idR

 

15

Notice that the quantities Az(h), 1(0), and I(d) on the

right side of equation (1. 34) are still unknown. The currents

1(0) and I(d) can be determined from (1. 34) itself by successively

letting z = 0 and z = d and solving the resulting two equations

simultaneously. Making the indicated substitutions in equation

(1. 34) gives the system of equations

 

 

 

 

K11(0) + KZI(d) = F1 (1.35)
K3I(0) + K4I(d) = F2 (1.36)
where
ZLsinfl h
Kl = j30 cos [30h + 2T T [jTSdIU -cos[30h)- Tcdsm (30h)
cd st
(1. 37)
Zsinfio(h-d)
K.2 = T T, [JTidIH - cos Boh)- ZTCds1nBo(h-d)] (l. 38)
Cd ldR
ZLsinfioh
K3 = —_2T T [JTSdI(cos (30d _cos 50h)- Tcdsmfio(h-d)] (l. 39)
cd st
Zsin[30(h-d)
K4: J30C“‘30h + T T. [JTidI(COSpod- “56011)
cd ldR
- 2T cosfldsinfi (h-d)] (1.40)
cd 0 0
(iv A (h) - E /0 )(l-cos (3 102
o z o o 0
Fl = T (1.41)

cd

16

(jvoAz(h) - Eo/ [30)(1 - cos floh)(cos Bod- cos 60h)

F2 = T . (1.42)
cd

 

Solving (1. 35) and (l. 36) yields

KF -KF

 

 

2 2
1(0) = (1.43)
K1K4 ' K2K3
K F .. K F
1 2 3 1
I(d) = (1.44)
K1K4 " K2K3

Equations (1.43) and (1.44) are used to eliminate 1(0) and
I(d) from equation (1. 34). After a considerable amount of alge-
braic manipulation the expression for the antenna current

becomes

I(z) = K5[ jvoAz(h) - Eo/ [30H Kc(cosfioz - cos (30h) +KssinBo(h- |z|)

 

+ Kig(z)] (1.45)
where
-j(l -cos 00h)
K = (1.46)
5 30Tch1dRTst(K1K4'K2K3)
. . . 2 . . 2
Kc - Z[ ZL{Sinfloh smfiodsm 130(h-d)}-j60Tstcos(30d s1n 130(h-d)]
. . 2
+ zL[-J 15 TidRsm 00h] - 900 TidRTstcos 00h (1.47)

7"
ll

{-sinfloh}{ZZLU-cosfiod)sinzflo(h-d) + ZL[ -j15 TidRU-cosfiohfl}

(1.48)

l7

Sinflo (h- d)
K. ={—————} {ZZ Lsinfi oh[sin[3 ooh(l-cosf5 d)-sin[3 o---d(l cosB 0h)']

1

+ Z[ j60Tst(cos[50d - cosfiohfl} . (l . 49)

The vector potential at the end of the antenna is the only remain-

ing unknown in equation (1. 45). Another expression relating

A (h) to the induced antenna current is obtained by setting z = h
z

in equation (1. 8).

s h
A (h) = —3 S I(z')K (h,z')dz' (1.50)
z 411 -h a

Substituting equation (1.45) into (1. 50) yields

in K
o 5 ,
4 [JvoAz(h)- ETD/[30H KcTca+ KSTsa+ KiTia] (1.51)

 

A (h):
z

where the new complex constants introduced here are defined as

h

Tca = Sh (cos fiozh cos 50h) Ka(h' z')dz' (l. 52)
h

Tsa = 5h sinfio(h-|z'|)Ka(h,z')dz' (1.53)
h

Tia = Sh g(z')Ka(h,z')dz' . (1.54)

Multiplying equation (1. 51) by jvo, subtracting Eo/ (30 from each

side, and rearranging,

18

-E0/ pO

l-j30K [KT +KT +K,T,]'
5 cca ssa 11a

 

[ijAz(h) - 130/130] =
(l . 55)
Putting (l. 55) back into equation (1.45) completely determines
the antenna current. After rearrangement and simplification,
the induced current on the receiving antenna may finally be ex-

pressed as

I(z) = K[ Kc(cosfioz - cosfioh) + Kssinflo(h-|z|) + Kig(z)].
(1 . 56)

The complex coefficients in equation (1. 56) are functions of the
antenna dimensions, the value of the center-load impedance, and
the value and position of the symmetrically placed auxiliary
impedances. Kc' KS, and Ki are given explicitly in terms of
these parameters in equations (1.47), (1.48), and (1.49). The
function g(z) is given by equation (1.16). Factor K can be ex-
pressed as

- (115-) (l - cosfloh) E

O
K : O — (1.57)
+ + +
Z(AZ_ B) cz D (30

 

where
A = - sinflohsinflom-d){2sin(30(h-d)[ F4sinf30d +Tsa(1 - cosfloh)(l - cosfiod)]

- TiaF3u - cosfioh) + jcosfioh[ TidIF3- ZTSdIU - COSROdlsmfiom-dlll

(1.58)

19

B = j60TS Rs1nflo(h-d){2F4cosflods1nfio(h-d)

d

+ [cosflod - cosfioh][ Tia“ - cosfioh) - jT cosfioh]} (l . 59)

idI

C = j30TidRsinBoh{F4sin(30h .+ [ l - cosfioh][ Tsa(l - cosfioh)
- stdICOSpohn (l . 60)
D = lBOOTidRTstF‘iCOSfioh (1.61)
F3 = sinfioh - sinflod - sinfio(h-d) (l. 62)
F4 = (Tca+ Ted) cosfioh - Tea (1. 63)

l. 3 Current Distribution on the Transmitting Antenna

In the design techniques developed in the next three
chapters for optimizing receiving antenna performance it will
be quite desirable to have at hand an expression for the input
impedance of the double-loaded antenna. In order to determine
the ratio of applied voltage to current at the driving point of the
transmitting antenna, it is necessary to first solve for the
current distribution on the driven antenna. The solution for
this current is included here since the mathematical develOp-
ment closely parallels the technique of the preceding section

used in finding the induced current on the receiving antenna.

20

When operating as a transmitting antenna the center-
load 21. shown inFigure 1.1 is replaced by a sinusoidal voltage
driver having a specified terminal voltage V. Proceeding
exactly as before the continuity of the tangential component of

the total electric field at the surface of the antenna is expressed

as

E: = .. V6(z) + th(d)6(z-d) + ZIt(-d)6(z+d). (1.64)

As before E: represents the tangential electric field maintained
on the antenna surface by the current and charge on the antenna,
It(p) is the current at z = p, and 5(z-p) is the Dirac delta func-
tion.

This equation is quite similar to equation (1. 2) which
gives E: for the double-loaded receiving antenna. In fact the
latter equation is obtainable from the former simply by setting
the incident field to zero and replacing the factor ZLI(0) with -V.
Since these factors are not functions of the z coordinate and since
the symmetry condition It(-d) = It(d) applies in the case of the
transmitting antenna as well as for the receiving antenna, it
follows immediately that the integral equation for the current
on the transmitting antenna is identical to that found for the

receiving antenna with ZLI(0) replaced with -V and E0 = 0.

21

Thus, from equation (1.14),

h t
S. I (z')Kd(z, z')dz' =
-h

333' secpohl (jvoAz(h>><cospoz - cospoh) % Vsinfio(h- Iz I)

+% ZIt(d)g(z)]. (l. 65)

Again the current is assumed to be of the form

It(z) = Cc(cosf30z - cossoh) +css1n00(h- | z I) +Cig(z), (1.66)

given in the previous section as equation (1. 20).

It follows, therefore, that the results of the previous
section prior to the elimination of I(0) are, subject to the sim-
ple modification indicated above, immediately applicable here.
Thus, substituting -V for ZLI(0) and setting E0: 0 in equation

(1. 34), the current on the transmitting antenna is

 

 

 

 

t _. 1 -VTSdIsin(30h
1 (z) = "3‘16 secsoh (ijAz(h))(1 - (269.0011)-) 21
Cd st
t
21 (d)T. sins (h-d)
+ Tldl o - [c0813 z-cosfi h]
idR ° °
[-Vsinsoh] l l [th(d)sin)30(h-d) ]
+ -———— sinfl (h- z ) + g(z)
2T.st ° TidR

(1.67)

22

The terms It(d) and Az(h) are eliminated from the right side

of equation (1. 67) by the same process as was used in the
previous section. It(d) is determined by substituting z = d

into equation (1. 67) after which equation (1. 50) is used to deter-
mine Az(h). When the details of this procedure are carred out,

equation (1. 67) becomes

It(z) = Kt[K:(cosi30z- cossoh) +Kstsinf30(h-Izl) +K: g(z)] (1.68)

where
Kt _ Vtanfioh (1 69)
' BZ +D ’

t . 2 .
KC - Zsm fio(h-d)[ (1 - cosfioh)(Tia- ZTSaCOSpod)-Jcosfloh(TidI

- 2T Icosfiodfl + 30T.l cosfioh[ TSdIcosfioh +sta(l - cosfloh)]

sd dR
(1. 70)
t . .
Ks — Zsmfio(h-d)[ -2F4cosfiodsinfio(h-d) +(cosf30d
- cosfiothTidIcosfioh - Tia“ - cosfloh})] + 330F4TidRcosfioh
(l. 71)

K: = Zsinfio(h-d)[ F4sinfio(h-d)+(cos[30d - cosfioh)(TSa{l - cosfloh}

- jT Icosfloh)]. (1.72)

sd
1. 4 Input Impedance of the Transmitting Antenna
An analytical form for the input impedance of the double-

loaded transmitting antenna is readily obtained from the result

23

of Section 1. 3. The current at the center of the transmitting

antenna is found by substituting z = 0 into equation (1. 68).

1t(0) = Kt[K:(l - cospoh) +K:s1nsoh +K: {zsinpo(h-d)}] (1.73)

. . . t t t t
After subst1tut1ng the express1ons for K , Kc’ Ks’ and K.l
defined in the previous section and simplifying, equation

(1. 73) becomes

t _ AZ +C
1(0) _ vl:m:| . (1.74)

Thus, the input impedance of the double-loaded transmitting

antenna is given by

V BZ + D
Z. = -—— : ——-— (1.75)
1n It(0) AZ + C

where the factors A, B, C, and D, defined by equations (1. 58),

(l. 59), (l. 60), and (1. 61), are functions of the antenna dimen-

sions and the position of the auxiliary impedances.

1. 5 Reduction to the Unloaded Dipole Antenna

Closed form solutions for the current distributions on
the double-loaded transmitting and receiving dipole antennas
were developed in the preceding sections. These results can
be readily reduced for application to the ordinary unloaded

dipole antenna simply by setting the values of the auxiliary

24

impedances to zero or by stipulating that these impedances be
located at the very ends of the antenna. Reduction of the results
to the unloaded dipole is included here for the sake of complete-
ness.
1. 5. 1 Current Distribution on the Unloaded
Receiving Antenna
The current on the unloaded receiving dipole antenna

is found by specifying Z = 0 in equation (1. 56). Thus,

I(z) = K(Z=0)[ KC(Z=0)(cosfioz - cosfioh) +KS(Z=0)sin[30(h- ) 2')

+ Ki(Z=0)g(z)] (1.76)

where the complex coefficients are given by (1.47), (1. 48),
(1.49), and (l. 57) with Z set to zero. After inserting the
expressions for these constants and simplifying, equation

(1 . 76) becomes

I(z) = Ko[ (-Z sinzfioh +j6OTs cosfloh)(cosfioz - cosfloh)

L dR
+ stinpohu - cossoh)sinpo(h-|z|)] (1.77)

where K =
o

 

Eo
(1 - cosfioh) (F)
o —
j3OZLsinfloh[ F4sinfioh+(l-cosfloh)(Tsa{l-cosi30h}-jTSdIcos[30h)]

+1800 TSdRF4cos [30h

(1.78)

25

Equation (1. 77) gives the current induced on an ordinary receiving
dipole antenna with center-load ZL situated parallel to the field
E0 of a normally incident electromagnetic plane wave. This
expression is very similar to the result determined by Chen and

[4]

Liepa , differing only in the use of TS in place of Ts and

dR d

the inclusion of the term in the denominator of Ko involving the

factor TS It is expected that equation (1. 77) is a better repre-

dI'
sentation of the true current distribution than the result of Chen
and Liepa. This statement is based upon the results of a direct
comparison of the two methods (the technique of Chen and Liepa
as compared with King's modified method) of separating the
integral equation for the current on the transmitting antenna.
This test consists simply of evaluating both sides of the integral
equation numerically at several points on the antenna--first
using one solution form and then the other. For antennas of

lengths one wavelength or less this comparison indicates a

substantial margin of accuracy in favor of King's method.

1. 5. 2 Current Distribution on the Unloaded
Transmitting Antenna

The current on the unloaded transmitting antenna is

found by specifying Z = 0 in equation (1. 68). Thus,

It(z) = Kt(Z=0)[K:(Z=0)(cosfloz- cosfloh)+KSt(Z=0)sin[30(h-lzl)

+ K: (Z=0)g(z)] (1.79)

26

where the complex coefficients are given by (1. 69), (l. 70), (1. 71),
and (l. 72) with Z set to zero. After inserting the expressions

for these constants and simplifying, equation (1. 79) becomes

It(z) = K:[ (T ICOSpoh +sta{l - cosfioh})(cosfioz - cosfloh)

 

sd
+ jF4s1nf50(h-'zl )] (1.80)
t V tanfioh
where K = . (1. 81)
o éoTstF4

Equation (1. 80) gives the current distribution on an
ordinary transmitting dipole antenna center-driven by a slice
generator of potential difference V. This expression is in exact

[31.

agreement with King's result

1 . 5. 3 Input Impedance of the Unloaded Antenna

As the values of the auxiliary impedances are reduced
to zero the expression for the input impedance of the double-
loaded transmitting antenna found in Section 1. 4 becomes that
of an ordinary unloaded dipole. Thus, substituting 2:0 in

equation (1. 75),

D 60TstF4

in C Sinfioh[ (TSdIcosfioh-l-JTsafl-cosfloh})(1-cos[30h)

cosfloh

+jF481ni30h]
(1. 82)
Alternatively, (l. 82) may be obtained as the ratio V/It(0)

using equation (1 . 80).

II THE OPTINIUM RECEIVING ANTENNA

Traditionally the load impedance connected to the termi-
nals of an ordinary dipole receiving antenna is chosen purely
from the standpoint of absorbing as much power as possible
from the antenna. The fact that the receiving antenna acts as
a scattering element re-radiating in virtually every direction
some portion of the energy intercepted from the incident field
is noted as an inherent secondary effect in the system and is
generally given no further consideration.

It is well known that the load impedance which will re-
sult in the extraction of maximum power from a dipole receiving
antenna of specified dimensions is equal to the complex con-
jugate of the input impedance of the antenna. In recognition of
this fact and in the interest of simplicity and expediency, one of
the most popularly used dipole receiving antennas is the nomi-
nal half-wave dipole, corresponding in length to the so-called
first resonance condition in which the reactive part of the input
impedance vanishes. With an antenna length corresponding to

one of the resonant lengths the conjugate match condition is

27

28

achieved with a purely resistive load, obviating the need for
reactive tuning at the antenna terminals. In addition to the ob-
vious physical advantage of being the shortest antenna in the
resonant length class the half-wave dipole exhibits a reasonably
convenient value of input resistance--approximately 72 ohms.

In the half-wave dipole example it is noteworthy that no
real effort has been given to truly maximize the received power
with respect to each of the available design parameters. Rather,
the length of the antenna is selected for reasons not directly re-
lated to power expectations; not until after the physical dimensions
of the antenna are specified is an attempt made to maximize the
power receiving capability.

Chen[ 4’ 6], in his efforts to reduce the radar cross section
of thin metallic cylinders, has demonstrated that the backscattered
power from objects of this shape can be reduced very substantially
using a central impedance loading technique. Although Chen's
final objectives in this study are distinctly different from the con-
siderations associated with the design of a receiving antenna, the
physical system in his work may, nevertheless, be properly re-
garded as an ordinary dipole receiving antenna. By controlling
the distribution of the induced current on such cylinders through
his choice of values for the center-load impedance, Chen is able
to theoretically achieve the condition of zero broadside backscatter-

ing. Although the load impedances corresponding to the zero

29

backscatter condition predicted by his work are complex and, in
fact, passive for cylinders less than one wavelength long, cylinders
loaded in this manner do not make effective receiving antennas.
Compared with a cylinder having the same dimensions but with a
conjugate-matched center load, the receiving power capabilities
of the former are on the order of 20 dB less efficient. Moreover,
this is probably a grossly conservative estimate of the actual
inefficiency; when the work of Chen is recast using the more
accurate approximation to the induced cylinder current afforded
by King's modified method, it turns out that the resistive part of
the center load required for the condition of zero backscatter
vanishes altogether. This result is shown in Section 2. 3 where
it follows as a reduced case in the study of the double-loaded
receiving antenna.

It is clear from the above discussion that the specification
of zero backscattered power and the simultaneous requirement of
maximum received power, or at least a power receiving capability
comparable to that obtained in the conjugate-match case, are
mutually incompatible objectives insofar as an ordinary dipole
antenna is concerned. What is needed in order to accomplish
these goals simultaneously is the inclusion of some additional
degrees of freedom in the antenna system. This added flexibility

is provided by the double-loaded receiving antenna introduced in

30

Chapter 1. The three new real parameters corresponding to the
resistive and reactive components of the auxiliary impedance Z
and the distance d at which these impedances are symmetrically
placed about the center of the antenna allow for considerable con-
trol over the antenna current distribution. Consequently, by adapt-
ing this antenna configuration the designer gains rather wide
latitudes in his ability to demand a particular scattering character-
istic while simultaneously insisting that his antenna deliver a res-
pectable amount of power to the receiving load.

It is appropriate at this point to lay down some specific
performance requirements for the double-loaded receiving antenna
and to proceed with the determination of the parameters in the
system which will produce these desired characteristics. The
first and most stringent set of performance specifications to be
considered is expressed explicitly in the following definition of
the optimum receiving antenna.

Definition: The Optimum Receiving Antenna is defined
as a double-loaded dipole receiving antenna
of given dimensions and having the placement
d of the auxiliary impedances Z specified for
which the broadside backscattered power
vanishes and for which, subject to the first

constraint, the power delivered to the center-

load ZL is maximized with respect to each

of the impedance parameters Z and ZL'

31

Although the received power is clearly a function of not

only Z and Z . but also the antenna dimensions a and h and the

L
placement d of the auxiliary impedances, notice that the latter
group of parameters is held fixed in the definition of the optimum
receiving antenna. This rather artificial limitation of variables
reduces to a manageable task the analytical work involved in

determining the optimum values of Z and Z without actually

L
narrowing in any way the scope of the investigation. Were it
attempted to retain any combination of the parameters a, h, and
d as variables in this definition, the procedure for maximizing
the received power would become extremely difficult due to the
highly implicit manner in which these parameters appear in the
equation for the induced antenna current. On the other hand the
influence which these quantities have on the power delivered to
ZL can eventually be determined rather easily by displaying
graphically the manner in which the received power of the optimum
receiving antenna changes as one or more of these parameters is
varied over some appropriate range of values.

With both Z and ZL completely variable it turns out theoreti-
cally that it is possible to obtain infinite received power for any
choice of the parameters a, h, and (1. Thus, the problem of

choosing from the set of all optimum receiving antennas the one

yielding the most received power does not arise. In Chapter 3 ,

32

however, some one of the impedance parameter values is assigned,
thus reducing by one the number of degrees of freedom in the maxi-
mization of received power. For the resulting near-optimum
receiving antennas (to be defined almost exactly the same as the
optimum antenna but with some one of the impedance parameters
fixed), the maximum obtainable value of received power varies

as a, h, and d are varied and so the distinction of a precise defi-

nition for a near-optimum receivingantenna becomes much more

 

important.

2. l The Condition for Zero Backscattering--The Constraint
Equation

The electric field produced in the broadside direction
and in the radiation zone of a thin cylindrical radiating element

in free space is given by

s -J50Ro h
E : -j30p _e___ SI I(z')dz' . (2-1)
0 R
o -h

This is a well known result from elementary antenna theory. In
equation (2.1) E8 is the electric field at an observation point
located a distance Ro from the center of the radiating cylinder;
I(z') represents the complex current distribution on the cylinder
of length 2h. As before 60 = 211/ RC is the wave number associated

with the free space wavelength X0.

33

The power density at this same far-zone observation point

 

is
l s 2
P = —- .
s 2;, IE I (2 2)
0
or, substituting (2. 1) for ES,
1 300 2 h 2
P = — 5 I(z')dz' (2.3)
s Zéo R0 -h

 

 

Here, 130 = m is the impedance of free space and is equal to
12011 ohms.

If the radiating element in question happens to be a dipole
receiving antenna which is being illuminated by a parallel electric
field at normal incidence, as illustrated in figure 1. 1, then
equation (2. 3) re presents the power density of the backscattered
field maintained by this receiving antenna. Specifically, when
(l. 56) is substituted for I(z') in equation (2. 3) the backscattered

power density from the double-loaded receiving antenna is found

 

 

to be:
1 300 2 h
— — - . - '
PS — 2L R 5 K[ Kc(cosfioz' cosfioh)+KSSlnf30(h I2 I)
o o -h
2
+Kig(z')]dz' . (2-4)

Performing the indicated integration, this becomes

34

 

Kc(sinfiOh-flohcosfioh)+Ks( l-cosfloh)+2Ki(cosflod
2
- cosfioh) . (2.5)

After substituting the defining relations for the K-coe’fficients from
Chapter 1 into (2. 5) and carrying out a moderate amount of re-
arrangement and simplification, the backscattered power density
in the far-zone of the double-loaded receiving antenna can be

written:

2
L

s 1511R02 Z(AZL+B)+CZL+D

2 2
= (l-cosfioh) (ED/F50) Z(FZL+G)+HZ +L (2 6)

 

P

 

where A, B, C, and D are defined in (l. 58) through (1. 61) and

where
F = sinflohsinflo(h-d)[ sinflo(h-d){sinfiod(sin(30h-i30hcosi30h)

- (l-cosflohHl-cosflodn + F3(cosfiod-cosf30h)] (2. 7)
G = j60Tstsinflo(h-d)[ (cosBOd-cosfioh)z-cosf30d(sinfloh

-(30hcos[30h)sinfio(h-d)] (2. 8)
H = leTidRsinfloh[ (l-cosfioh)2-sinfioh(sinfioh-f30hcosfioh)] (2. 9)

L =~900TidRTstcosfioh(s1nfioh-fiohcosfioh) . (Z- 10)

To make the backscattered field vanish the numerator of

(2. 6) is set to zero, imposing the following condition on the

35

ultimate selection of the impedance parameters Z and ZL'

z[FZ +G]+HZ +L = 0 (2.11)

L L
Once the antenna dimensions and the placement of the auxiliary
impedances have been specified, Z and ZL must be chosen sub-
ject to this constraint equation if the condition of zero backscat-
tering is to be achieved.

The fact that in equation (2. 11) the coefficients F and L
are real while G and H are imaginary places one rather severe
restriction on the implementation of a zero-backscattering re-
ceiving antenna: specifying Z as being purely reactive in the
constraint equation leads ultimately to an antenna which is
incapable of delivering any power whatsoever to the receiving

load ZL' This will be demonstrated in Chapter 3 in connection

with the investigation of some near-optimum cases.

2. 2 The Condition for Maximum Received Power

The received power delivered to the center-load ZL of
a dipole receiving antenna is given by
l 2
PR— 2 |1(0)| RL (2.12)

where 1(0) is the complex current at the center of the antenna

and R is the real part of the center load Z

L L '

36

The current existing at the center of the double-loaded

receiving antenna is obtained from equation (1. 56) with z = 0.

= - + ‘ + ' - -
1(0) K[KC(1 cosfioh) Kssmfloh 2Kis1nf30(h d)] (2 13)
After substituting the defining expressions for the K-coefficients
and simplifying, equation (2.13) becomes

J(MZ +N)

Z(AZL+ B) + CZL+ D

1(0) (2.14)

 

where the new coefficients appearing in the numerator of I(O)

are defined as

J = - 4Tstcosfioh(l - cosfiohHEo/flo) (2.15)
M = (l - cosfiod)sin2(30(h-d) (2. 16)
N = -j15TidR(l- cosfioh) . (2.17)

The received power yielded by the double-loaded antenna is ob-

tained by substituting (2. 14) into equation (2. 12).

p.51;
R-2

J(MZ +N)
ZL(AZ +C) +BZ +D

 

(2.18)

 

 

For an ordinary unloaded dipole antenna equation (2. 18)

reduces to

JN
CZL+D

 

, or (2.19)

 

 

37

 

 

 

R 2
L V'
PR ‘ 2 2 +21 (2‘20)
L g
where V' = JN/C and Z'g = D/C . (2.21)

In Equation (2. 20) V' and Z'g can be interpreted as the voltage
driver and its series impedance in an equivalent circuit repre-
sentation of the receiving antenna system as viewed from the
antenna terminals. The maximum power transfer theorem from
classical circuit theory, applicable in this situation since V' and

, requires that Z be chosen as the

Z'g are independent of Z L

L
complex conjugate of Z'g in order to effect maximum power trans-
fer to the load ZL. As evidenced by comparing (2. 21) with (l. 82)
in Chapter 1, Z'g in Equation (2. 20) corresponds to the input
impedance of the unloaded antenna; thus, this is just a reiteration
of the well known conjugate-match condition for an ordinary dipole
receiving antenna.

It should be emphasized that in the application of the
maximum power transfer theorem, it is presumed that the speci-
fied generator impedance Z'g is passive; the theorem accordingly

predicts the value of a second passive impedance Z which, when

L

connected across the terminals of the equivalent network composed

of the series combination of V' and Z'g, results in the absorption

38

of maximum power by Z Should the real part of Z'g be nega-

L'
tive, on the other hand, this theorem per se has no applicability
to this simple generator-load network; for the delivery of maxi-

mum power to Z in the latter case one should obviously choose

L
ZL = - Z'g.
Acknowledgment of the implications of the above state-
ments is helpful in visualizing the unique power delivering
capabilities of a multiply—loaded dipole receiving antenna. Notice
that the expression given in (2. 18) for the received power of a

double-loaded antenna can also be cast in the form of (2. 20)

wherein

V' = J(MZ + N)/(AZ + C) and Z'g = (BZ + D)/(AZ + C). (2. 22)

As before, Z'g corresponds to the antenna's input impedance--
this time, of the double-loaded antenna, as evidenced by com-
paring (2. 22) with equation (1. 75) in Chapter 1. With the value
of Z fixed in (2. 22) exactly the same arguments hold regarding
the selection of ZL as were made above in connection with the
unloaded antenna. However, whereas the input impedance of an
ordinary unloaded antenna is inherently passive, so that the
value of ZL can always be selected meaningfully on the basis

of the maximum power transfer theorem, the use of auxiliary

impedances having negative real parts in the implementation

39

of a multiply-loaded antenna immediately admits the possibility
of producing a receiving antenna having an input impedance with
a negative real part. In this situation one can theoretically ob-
tain an infinite amount of received power by choosing 2L = -Zin°
That the received power of the double-loaded antenna can
be expressed, innately, in the form shown in (2. 20) has not been
established; this follows from the fact that equation (2. 18) for
PR was derived using only an approximate solution for the in-
duced antenna current. It is possible, however, to demonstrate
the validity of (2. 20) for the double-loaded receiving antenna on
the basis of fundamental considerations, in a manner which re-

quires no foreknowledge of the current distribution. This is

shown in the Appendix.

2. 3 Selection of the Optimum Parameters
In accordance with the definition for the optimum re-
ceiving antenna, values for the impedance parameters Z and

ZL are sought, subject to the constraint that the broadside

backscattered field maintained by the antenna vanishes, such

that the received power delivered to ZL is maximized with

respect to each of the parameters Z and Z Elimination of

L'
the backscattered field is achieved by imposing the condition

of constraint equation (2.11) on the relationship between Z

and ZL' Equation (2.11) can be rearranged to give either

40

HZL+L
Z = - W or (2.23)

G2 + L
ZL - {rs-:11] - (“4’

The expression formulated in Section 2. 2 for the re-
ceived power produced by the double-loaded antenna is repeated

below for convenience.

P _ RL J(MZ +N) (2 18)
R’ 2 ZL(AZ+C)+BZ+D '

 

 

Upon substituting constraint (2. 24) into equation (2. 18), PR be-

comes

2

R 2
P :__L J{}(z+az+6} (2.25)

R 2 622+ .192 + 6

 

E

where
= FN + HM
HN

> (2. 26)

= AG-BF

63(3ng
II

= AL+CG-BH-DF

 

6 =CL-DH J

As is evident from equation (2. 25) the Optimum values of Z
occur at the poles of the current 1(0) and are determined as

the solutions of the complex quadratic equation

41

c[z]:p+ .B[Z]Op+ 6 : 0. (2.27)

The corresponding optimum values of the center-load impedance,

[Z , are obtained from constraint equation (2. 24). Notice

Llop

that forcing the denominator of P to vanish in (2. 18) or (2. 25)

R

is equivalent to requiring that Z - zin’ as discussed in the

L:

previous section, where from equation (1. 75)

BZ +D
ZL‘ ' Zin ' {XE-FE] ' (2‘28)

Thus, the quadratic expression for [Z]opin (2. 27) can be formu-
lated, alternatively, simply by equating the right-hand sides of
(2. 24) and (2. 28).

There are several considerations in regard to the opti-
mum receiving antenna which deserve some comment at this
point. First of all, there is no guarantee that the numerical
values of the optimum parameters found from equations (2. 27)
and (2. 24) will make sense physically; only if the real part of

[Z is positive will these results be meaningful insofar as a

L]op
receiving antenna is concerned. Whether the Optimum values of
RL are positive or negative depends entirely upon the placement

d of the auxiliary impedances and the specified dimensions of the

antenna. Furthermore it would be naive indeed to expect the

42

real parts of [ 210p and [ZL]op to be positive simultaneously in
view of the unique power delivering capability required Of this
antenna.

Secondly, notice that although the denominator in equation
(2. 6) for the backscattered power density vanishes at the Optimum

values of Z and Z Ps remains zero because the condition of

L’
constraint equation (2.11) is satisfied; that the denominator of
(2. 6) approaches zero as the impedance parameters approach
their optimum values is merely indicative of the fact that the
current distribution on the antenna is accordingly increasing
without bound.

Finally, in the interest of completeness and in order to
provide some degree of corroboration of the results obtained
here with the work of other investigators in the area of scatter
reduction from thin metallic cylinders, the constraint equation
for zero backscattering derived in Section 2.1 is reduced to yield
the following two special cases. First, specifying Z = 0 in

(2. 24) gives the zero-backscatter value of impedance for a

center-loaded cylinder .

~j60Tst[ l-fiohcotfloh]

L
Z = - - = . (2. 29)
L H 2cosfioh - 2 +flohs1nfloh

 

43

, (2. 29) is in exact

[4.6]

Except for the factar'I; in place of Ts

(1 dB

agreement with the impedance value predicted by Chen
for eliminating the broadside backscattered field from a center-
loaded cylinder. That the form given in (2. 29) differs in its
real part from Chen's result is a consequence of the difference
between Chen's method and King's modified method for approxi-
mating the induced antenna current.

In the second special case center-load Z is set to zero,

L
corresponding to a cylinder loaded with two impedances Z

placed symmetrically at distances d about the center. For this

situation constraint equation (2. 23) is used and

N

ll

1
Olt“

-j15TidRcosfioh(s1nfioh-fiohcosfioh)

sin130(h-d)[ (cosflod-cosfioh)2-cosflod(sinfloh-Bohcosfloh)sin,BO(h-d)] .

(2. 30)

Again, by virtue of the T-integral term (Tid instead 0f Ti 1,

dB
Chen's [ 5] value for Z, in addition to having a reactive com-

ponent identical to (2. 30), also possesses a resistive component.

2. 4 Theoretical Results
Numerical values of the optimum impedance parameters

are given in Figures 2. l and 2. 2 and Table 2.1 for several

44

representative antenna lengths, according to the formulas developed
in Section 2. 3. Whereas the quadratic equation (2. 27) yields two
values for [Z]Op’ it will be noted that only one set of optimum
parameter values is indicated in the graphical and tabular data
which follow. The reason for this is that, invariably, over the
range of antenna lengths investigated, the real part Of one of the
two values of [ZL]op Obtained from (2. 24) and (2. 27) has an ex-
tremely small magnitude--typically, in the range between 10-
and 10..9 ohms. The magnitude of the second value of [RL]op
obtained from these formulas, on the other hand, is much larger--
in most cases, greater than one ohm. Therefore, for practical
reasons, the value of [ZL]OP with the smaller real part is dis-
carded, and all numerical results shown here correspond to the
larger value of [RL]op'
The curves in Figure 2.1 show the Optimum center-load
resistance [RL]op for all possible positions of the auxiliary im-
pedances [Z]op. Notice, in each case, that [RL]op is positive
over only a portion of the complete range of loading positions.
Since the results corresponding to negative values of [RLjop
have no meaning in the case of a receiving antenna, a tabular ,
rather than graphical, format has been chosen for representing

only those sets of optimum parameter values which have appli-

cability to a receiving antenna.

45

In addition, one complete set of optimum parameter speci-
fications is furnished by the curves in Figure 2. 2, corresponding
to the case of Sch 2 2.5. This is included merely to provide some

indication of the manner in which the complement set of parameters--

[R10p» [X10p. and[

over all possible values.

XL]op--vary as the loading position d ranges

46

 

15
10 q)-
5 dr-
fioh=4.0
/ o 5

 

 

 

lRLlo - ohms
//

 

 

 

 

 

 

-5 cut)—
-10 .1-
-15 u)-
1 1 1 1 1 1 1 1
-20 r 1 r 4'1 r l I Fr 1
0 0.2 0.4 0.6 0.8 l.
d/h

Figure 2. l. Center-load resistance for the Optimum Receiving Antenna
(Boa = 0. 001)

47

 

 

   

     

 

 

104 .1...
[X lop
103 -(-
[XL]op
Q
E
.1:
O
102 --
IR 1., O
10 d-
2 1 m 1 1 4 1 1 L 1 L
I r I I 1 1 T‘ W
0 0. Z 0. 4 0. 6 0. 8

d/h

Figure 2. 2. Optimum Receiving Antenna parameter values for (3 h = 2. 5
(60a = 0. 001) °

1.0

48

 

 

 

 

 

 

 

 

 

 

Table 2. 1. Parameter values for the Optimum Receiving Antenna
in ohnns (floa.= 0.001)
h
(30 d/h lRLlop [XL]op [R]op [X]op
0.5 0.2 0.00380 7560 -0.0508 20800
0.3 0.00714 5450 -0.0423 10600
0.6 0.0130 3340 -0.0187 4530
0.9 0.0931 2640 -0.0548 7380
1.0 0.1 0.0967 8020 -5.66 44300
0.2 0.0777 4340 -l.10 12200
0.3 0.0159 3110 -0.0964 6080
1.5 0.1 0.405 5850 -28.1 35100
0.2 0.295 3130 -4.62 9200
0.3 0.0866 2210 -0.551 4410
2.0 0.1 1.56 4690 ~151 32700
0.2 0.955 2460 -18.0 7820
0.3 0.311 1720 -2.15 3520
2.5 0.1 4.18 3990 -768 37000
0.2 2.59 2040 -67.6 7380
0.3 0.894 1380 -7.11 2980
3.0 0.1 11.2 4130 -10700 74800
0.2 6.85 2030 -300 8320
0.3 2.52 1330 -22.3 2700
3.5 0.1 11.0 2230 -6l700 -350000
0.2 6.58 1030 -4810 19400
0.3 2.60 632 -66.9 2800
4.0 0.4 0.238 245 -1.90 1020
0.5 1.39 126 -3.32 515
0.6 2.28 45.2 -2.77 375
0.7 2.86 -11.3 -2.48 362
0.8 3.10 -48.6 -2.55 469
0.9 2.88 -6l.4 -3.23 979

 

 

 

 

 

 

 

Table 2. 1 (continued)

49

 

 

 

 

 

 

50h an. [RLlop [XLJOP [Rlop 1x10p
4.5 0.4 0.601 -6.13 -8.53 1070
0.5 39.4 -120 -92.8 385
0.6 73.2 -l97 -72.3 248
0.7 102 -249 -67.9 230
0.8 125 -283 -81.1 304
0.9 134 -291 -134 678
5.0 0.5 220 -466 -403 471
0.6 443 -451 -309 203
0.7 577 -374 -276 175
0.8 636 -294 -320 269
0.9 634 -252 -458 714
5.5 0.5 1400 -1070 -266 1020
0.6 1220 502 -726 714
0.7 734 600 -643 582
0.8 534 539 -403 702
0.9 450 462 -224 996
6.0 0.5 20900 4240 -22.6 1100
0.6 296 1450 -219 1290
0.7 141 882 ~186 1050
0.8 91.9 644 -57.5 716
0.9 70.0 496 -23.3 822

 

 

 

 

 

 

 

 

(1‘ I

w

"M'.' TIBETJ"

 

 

 

III NEAR-OPTIMUM RECEIVING ANTENNAS

In this chapter and the next, several special cases of the
double-loaded receiving antenna are considered in which one or
more of the impedance parameters--R L' XL, R, X--is arbitrarily
specified. Although, by definition, the optimum receiving antenna
of Chapter 2 is unattainable when a pre-condition exists on Z or

Z by proper utilization of the remaining unspecified parameters,

L’
it is possible to achieve performance characteristics which, although
less spectacular than those of the Optimum receiving antenna, may
be very desirable for certain applications.

As shown below, when only one of the four impedance para-
meters is specified beforehand, it is always possible to achieve
the condition of zero backscattering; furthermore, unless the

specified pre-condition is that R = 0 or R = 0, one degree of free-

L
dom will remain with respect tO which the received power can be
maximized. Antennas such as these will be called near-Optimum

receiving antennas, since they produce no backscattering, but have

only a finite power delivering capability.

50

51

The condition for zero backscattering found in Section 2. l

is repeated below for convenience.

Z[FZ +G] +HZ p+1_ = 0 (2.11)

L L

As pointed out in Section 2. 3, constraint equation (2. 11) can be

rearranged to give either

F ‘1
HZL-l- L

z = - iii—+73 (2.23)

L. L ..J

I- a

G2 + L
Z = - — . .

°r L FZ + H (2 24)

b .3

 

 

Given that either Z or ZL is partially or totally specified before-
hand, the condition for zero backscattering can be enforced by
choosing the remaining free impedance value according to (2. 23)

or (2. 24). However, constraint equation (2. 11) contains one rather

important implication about the interrelation of Z and Z . which is

L
not Obvious from any of the forms above; not until the right-hand
sides of (2. 23) and (2. 24) are written in such a way that the real
and imaginary parts can be distinguished, does this feature become
apparent.

Noting from the defining relations (2. 7) through (2. 10) that
the coefficients in (2. 11) are not strictly complex (F and L are real,

while G and H are imaginary), equations (2. ll), (2. 23), and (2.24)

can be expanded and simplified to give:

52

 

 

 

 

- + 1 _ 1 =
fRRL X(fXL g) hXL+£ 0 (3.1a)
1 1 :
fXRL+R(fXL+g ) +hRL 0 (3.1b)
2 2
(f1 +g'h')R -fh'(R +X )+(ff—g'h')X +g'l
L . L L L
Z = ' 2 2 +3 2 2
I l
(fRL) +(fXL+g) (fRL) +(fXL+g)
(3.2)
(11 + 'h')R -1 '(R2+Xz)+(ff- 'h')X +1111
ZL:' 2L 2 H J 2 j2 (3‘3)
(fR) +, (fX+h') (fR) + (fX +h')
where Z and ZL have been represented as Z = R + jX and ZL =

RL+ jXL, and where the convention of designating the imaginary
part Of a complex number by a prime and the real part without a

prime has been adopted in writing
F=f, G=jg', H=jh', sz. (3.4)

The important consideration revealed by (3. 3) is that the arbi-

trary specification Of R = 0 implies, in general, that RL = 0.

The single exception to this implication is considered in Section 3. 1.
It is clear from these preliminary comments that the con-

dition of zero backscattering can always be satisfied while arbi-

trarily specifying any 922 of the impedance parameters-~RL, XL’

R, or X--(although the particular pre-condition that R = 0 or, of

course, RL= 0 leads to a trivial receiving antenna). On this basis,

the following definition is given for a near-Optimum receiving

antenna .

53

Definition: A Near-Optimum Receiving Antenna is defined as

 

a double-loaded dipole receiving antenna of given
dimensions, having the placement d of the auxiliary
impedances Z specified, and having no more than one

of the impedance parameters--RL, XL' R, X--

arbitrarily pre-selected, for which the broadside
backscattered power vanishes and for which, subject
to the above constraints, the power delivered to the

center-load ZL is maximized with respect to each of

the impedance parameters Z and ZL.

3.1 Pure Reactive Loading
With the pre-condition that R = 0, the auxiliary impedances

are purely reactive and the backscattering constraint as represented

by the two real equations in (3.1) reduces to

(1xL+ g')X + h'XL- 1 = 0 (3.5a)

(fX+h')RL = 0 - (3.5b)

Equation (3. 5b) is satisfied for either RL2 0 or X = -h'/f. Dis-
carding the first as trivial, and substituting the second into (3.5a)

yields the following result.
(ff +g'h') = 0 (3.6)

Since the terms in (3.6) are independent of the impedance para-
meters, but are functions only of the antenna dimensions and the

placement of the auxiliary loads, (3. 6) is not satisfied, in general.

54

for an arbitrary loading position; it must be concluded, therefore, accord-
ing to the definition, that a near-optimum antenna does not exist in the
case of pure reactive loading.

The implications of (3.6), nevertheless, are rather interest-
ing. If a, h, and d can be chosen in such a way as to satisfy this
equation, a receiving antenna can be produced for which the back-
scattered field automatically vanishes independent of the choice of
ZL’ simply by choosing X = -h'/f. Not at all surprising, however,
is the fact that these very same conditions simultaneously force the
current at the center of the antenna to vanish; this is readily demon-
strated by using (3. 6) in the numerator of equation (2.14) for 1(0)
with Z = -jh'/f.

In spite Of the fact that (3. 6) leads to a trivial receiving
antenna at the design frequency (corresponding to 00 = 211/ 1.0).
such a loading scheme suggests a practical means for providing
signal suppression at some other frequency. A double-loaded re-
ceiving antenna requiring only reactive auxiliary loading which has
the ability to reject a given (unwanted) signal frequency from the
receiving load, while at the same time achieving the condition of
radar invisibility at the same frequency, might well be desirable
in certain applications. The condition which (3. 6) places on the

loading position d for such an antenna is investigated below.

55

Using the defining relations (2. 7) through (2.10), the left-

hand side of (3. 6), after simplifying, can be written

1 1 = ' ' _ .
ff +g h 900 TidRTstsmpohsmpo(h d)

‘ . ' . 2
{(s1nfiOh-fiohcosf30h)sinflo(h-d)-(l-cosfioh)(cosfiod-cosfioh)} .
(3. 7)

The result of setting the’right side of (3. 7) to zero indicates that,
given the antenna dimensions a and h, loading position (1 must be

chosen such that

QcosBod +Ssinfiod + T = 0 (3.8)

where Q = (sinfioh-fiohcosfioh)sinfioh - (l-cosfioh)

S

-(sinfioh-Bohcosfiohkosfloh . (3. 9)

T

(1 - cosfioh)cos[30h

 

The substitution of cosBod = All - sinzfiod in (3. 8) leads to the

following quadratic equation in sinflod.

2 2
sinzfi d +(_2‘.?.’_S_I_2.) sinfi d +(12—L—gf) = 0 . (3.10)
° 3 +0 ° 5 +Q

In general, equation (3. 10) produces only one meaningful result
for d; a second value, obvious from inspection of (3. 7) is the tri-
vial solution d = h.

The value and the position of the auxiliary reactance re-

quired in the realization of a double-loaded receiving antenna

56

having a rejection characteristic at the frequency corresponding

to 00 are shown in Figure 3.1 as a function of antenna length. The
fact that these results are based on only an approximate solution

for the antenna current should be appreciated. It might well be, for
example, that in practice a non-zero value of R is required to imple-
ment such an antenna. Moreover, it is possible that the conditions
determined here for making 1(0) vanish are merely indicative of a

minimum of 1(0) and not a true null.

X - kilohms

 

 

10

9-1- .-
8-ir- 4
7d)- -1
61'- -

 

 

 

 

 

 

d/h
4 cu— —l
3 d- -1
) -- -
x—/
1 ch- -
0 I I l L l l
I T T T I T
0 1 3 5 4 5 6

50h

Figure 3. 1. Auxiliary reactance and loading position for a receiving
antenna with a frequency-rejection characteristic at
00 = Bovo (Boa = 0. 001)

1.0

0.6

58

3. 2 Pure Resistive Loading

When the auxiliary impedance is specified to be of the form
Z = R + j0, the broadside backscattered field will automatically
vanish if, setting X = 0 in (3. 3), the center-load impedance is
chosen as

2

 

1 11 _ 1 11
ZL “ +2“); ”’1' Jflz—flz— - (“1’
(fR) +h' (fR) +h'

From (2. 25) in Section 2. 3, the power received by an antenna

loaded in this manner is

 

" 27
_ RL(Z) J{}<zz+ dz + 6}
PR _ T 2 . (3.12)
Oz + .82 + 6 Z=R+j0

 

 

Using RL(Z = R) from (3. ll). equation (3. 12) becomes

2
z 2
PR : Hz"— XRZ+ as +8 . -(11 gfliz'm . (3.13)
GR + .DR +6 (fR) +h'

In (3.13) the received power is a function of the single variable
R. At this point, according to the definition of a near-Optimum
receiving antenna, R is sought such that the function PR is maxi-
mum. Before this maximization procedure can be carried out,
however, the right side of (3.13) must be rearranged somewhat.

After breaking up the complex terms into real and ima-

ginary parts:

59

 

 

 

 

 

 

X=k+j0 G=c+jc' \
= 0+ja' .8 = d+jd' B, (3.14)
B : b +j0 6 = e + je'
J
equation (3. 13) can be expanded and simplified to the form
r- -1
2 u R4+ R2+ R
LIL 4 u2 u0 “’1
PR: 2 4 3 2 2 (3'15)
v4R + v3R + sz + le +v0 yZR + y()
where
2 2 W
: = '
u4 k v4 c + c
u2 = a'z+ Zbk v3 = 2(cd + c'd')
u0 = b2 v2 = d2+ d'z+ 2(ce + c'e') g (3. 16)
W1 = -(fl + g'h') v1 = 2(de + d'e')
y = f2 V = 82+ 6'
2 2 0
Y0 = h' .2
After carrying out the multiplication indicated in (3. 15), this
:1:
equation can be put into the form
f' 6 n}
Wl ”'2 < 5:303,nR
PR - -—Z—-— 6 n > (3.17)
E b R
:0 n
L“ .J

 

 

 

*Rational functions similar to the one in (3. 17) are encountered
in the next section and in Chapter 4. As in the case of (3.17),
these functions are to be maximized or minimized with res-
pect to a single variable. Thus, although the numerator in
(3.17) is actually a fifth degree polynomial, it is purposely
indicated as sixth degree, so that the results obtained here
in connection with maximizing PR (R) can be applied again
later by simply ran-defining the coefficients ai and bi in (3. 18).

where
a6 = 0
a5 = u4
= 0
a4
a3 = uz
a2 = 0
a1 = 110
a0 = 0
To find the

60

b6 : v4"2 0

b5 = v3Y2

b4 : v4"0 +V2Y2

b3 = v3yo +yly2 > (3.18)
I"2 : szo +V0V2

bl = V1"0

ID0 : V0”0

 

J

maxima of the function in (3. 17), it is sufficient to

differentiate the quantity within the braces; the critical points

of PR

occur at the zeros of the resulting rational function of R.

The differentiation is carried out most readily if the summing

notation indicated in (3. 17) is retained in the process--the

 

 

 

result is
F6 W 6 6 .
2‘, a Rn 2‘, Z n[a b.-a.b ]R1+n-1
n . n 1 1 11
£411.30 _ 1:0 11:1 (3. 19)
dR 6 n) 6 n 2
E b R E b R
n n
:0 11:0
L J
Setting the right side of (3.19) to zero implies that
u 6 m 1
E { E n[anbm-n-tl- a'm-n+lbn]} R = 0
m=0 n=l
P (3. 20)
(ai, bi: 0 for i<0 and i>6)

 

61

where the summation indices have been altered in order to
show each term AiRi of the polynomial explicitly. The leading
coefficient A11 vanishes identically so that the degree of the
polynomial in (3. 20) is actually only ten.

Although clearly got an Odd function of R, PR(R) exhibits
characteristics not unlike an odd function: given that the
quantity (ff + g'h') is positive, it is evident from inspection of
(3.13) that R and PR have unlike signs (that, in general,

(ff + g'h') is positive follows from (3. 3) in view of the fact that
in Chapter 2 [R]Op and [RL]Op have unlike signs); moreover, PR
approaches zero as R becomes infinitely large through either posi-
tive or negative values; finally, in numerically extracting the roots
of the polynomial in (3. 20), only two real critical points Of the
function PR are obtained, and in every case considered, these
occur as i IRCI. It follows, therefore, that +IRCI corresponds

to the only minimum of PR (R), and - chl corresponds to the

only maximum. Of the two roots, only R = - IRCI leads to a

receiving antenna which is meaningful; R +|Rc| yields negative

values of PR by virtue of requiring negative values of center-load
resistance.

Accordingly, there exists but one set of impedance para-
meter values for the resistive-loaded near-optimum receiving

antenna. These results are presented graphically for antennas

of several different lengths in the figures which follow. The

62

curves for ZL represent plots of equation (3. 11) with R = -IRCI.

The received power is shown by curves of power gain, in which

the performance Of this near-Optimum antenna is compared with
that of an ordinary dipole antenna of the same dimensions having

a conjugate-matched center load. The pronounced dip in each

of the curves of RL and PR occurs at the particular loading posi-
tion d for which (3. 6) is satisfied. In order to sustain the condition
of zero backscattering, the equations in (3. l) demand that RL

vanish for this placement Of the auxiliary loads.

PR - d‘B gai-n

63

 

 

 

 

 

 

 

 

 

 

 

 

 

«I(-

-50

0.4 0.6 0.8

Figure 3. 2. Received power for the resistance-loaded Near-Optimum
Receiving Antenna (Boa = 0. 001)

64

 

 

 

-106
-105 d1-
-,, \
'2 -104 «Ill--
1
m

 

3 j ‘ 2'5
-10 -1- ‘

 

 

 

 

4.0
6.0 6.0
102 I M 1 L L 1 L. 1 1
" 1 j I l 1 i l I I
0 0.2 0.4 0.6 0.8 1.0
d/h

Figure 3. 3. Auxiliary resistance for the resistance-loaded Near-

Optimum Receiving Antenna (Boa = 0. 001)

65

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

000 0.5
1 ur- 6.0
1.5
4.0
2,5
100 ...).
tn
.5.
o 4.0
_. )
as 1
10 -- \
l 1 1 1 1 1 1 1 4
l l I I I I I T
0 0,2 0.4 0 6 0.8
d/h

Figure 3. 4. Center-load resistance for the resistance-loaded Near-
Optimum Receiving Antenna (00a = 0. 001)

1.0

66

10

 

10 ‘-

 

 

 

XL - ohms

 

 

 

 

 

 

 

 

 

K; Z
3 \ 25 /fi

 

\
10Z -)-
6.069 6.09 6.06) 6.06
J I _l l _l 1 I l
10 1 I I I i I T
0 0,2 0,4 0.6 0.8 l.

d/h

Figure 3. 5. Center-load reactance for the resistance-loaded Near-
Optirnum Receiving Antenna (00a = 0. 001)

 

67

3. 3 Center-Load Resistance Specified
With RL arbitrarily specified, the condition of zero back-
scattering is achieved by choosing Z according to constraint

equation (2. 23), repeated below in the expanded form given in (3. 2).

(a +g'h')RL -fh'(Ri +xz L.)+(11 g'h')XL +g'f
z = -
(1R L21+<£x L'Z+g) +<rXL+ g')

(3.2)

 

 

Using either of these forms for Z, the received power of the double-

loaded antenna can be expressed as a function of Z alone; substi-

 

 

L
tuting (2. 23) into equation (2. 18) for PR gives
N N 2,
R J{ dz + B}
L L
13R = 2 ~ 2 ~ N (3.21)
ch+ fizL+6
where
ngN-HM =0+ja' ‘
'5 = GN - LM = b +j0
3’ = AH- CF = c +jc' . (3.22)
'3 =AL+BH~CG-DF =d+jd'
<3 =BL-DG =e+je' J

 

The tilde distinguishes the script symbols which appear in (3. 21)
from the otherwise identical notation used in (2. 25). Note, how-
ever, that (for convenience) the very same symbols which were
introduced previously in (3.14) are used again in (3. 22) for desig-

nating the real and imaginary parts of these script coefficients.

68

After expanding and simplifying, equation (3. 21) can be put
into the same form as (3. 17), which was formulated in connection

with the resistance-loaded antenna; viz. ,

 

 

 

 

r 6 n W
RLIJI2 n20 a'nxL
pR =——é-—( 6 Xn > (3.23)
“fob“ L J
where
a6 = 0 b6 = 0 3
a5 = 0 b5 2 0
a4 = 0 b4 = c2+ 0'
a3 = 0 b3 = 2(Cpl+ c'ql) (3. 24)
a2 = a'2 b2 2 pi + q: + 2(cp0+ c'qo)
a1 = ~2a'b b1 = 2(plpo+ qlqo)
a0 = a'ZR:+bZ b0 = p3 + g(z) J
and p1 = 2c'RL+ d' q1 = -(2cRL+ d)

2 2 (3. 25)
Z - : - ' 7 0
p0 (cRL+ dRL+e) q0 (c RL+ d RL+ e )

With RL arbitrarily specified in (3. 23), PR is a function

Of the single variable X Determination Of the particular value

L.

of XL for which PR(XL) is maximum establishes the near-optimum

receiving antenna for this case. Since (3. 23) has exactly the same

69

form as (3. 17), it follows that the results obtained in Section 3. 2
for maximizing the function in (3.17) are also applicable here.
Thus, replacing R with XL in (3. 20), the equation whose solutions
correspond to the critical points of the function in (3. 23) is

11 6
Z {Z n[a;n

Bxfizo
m=0 n21

bm--n +1 - a'm-- n+1 bn

(3.26)

(ai, biEO for i<0 and i>6)

In view of the coefficients which are zero in (3. 24), (3. 26) actually

reduces to the fifth degree equation
5
E A X _ = 0 (3. 27)

where Am is still given by the summation within the braces in
(3. 26).
Several comments can be made at this point about the be-

havior of the power function PR(X On the basis of equations

L).

(3. 21) and (3. 23) and the presumption that R is positive. pR(XL)

L

is a non-negative function which tends to zero as IXL' becomes

infinitely large. Furthermore, unless R is specified as having

L
the value [RL]op' the function remains finite. It follows that at

least one of the critical points contained in (3. 27) corresponds to

a max1mum of PR(XL).

70

In the numerical work undertaken in connection with this

particular antenna, parameter R has been assigned, successively,

L
the values Of l, 5, 10, 50, and 72 ohms; for each case the real
solutions of (3. 27) and the corresponding values Of PR(XL) have
been computed for each Of several antennas with lengths corre-
sponding to the range 0. 5 _<_ (30h _<_ 7. 0, over the entire range of
possible loading positions. Although it is difficult to draw any
general conclusions about the behavior of PR(XL) on the basis of
the numerical results, a remark about the number of real solutions
Obtained from equation (3.27) can be made: in none of the cases
considered do more than three real solutions to (3. 27) occur, and
with RL equal to 50 ohms and 72 Ohms, only the one real solution
guaranteed by the odd degree of the equation is Obtained.

Shown in the following figures is the complete set of im-
pedance parameter values corresponding to the near-optimum
receiving antenna for which RL= 50 ohms. Using the real solution
of (3. 27), curves for the auxiliary impedance are obtained from
equation (3. 2). As in Section 3. 2, the received power is expressed
as power gain relative to an ordinary dipole antenna of the same
dimensions having a conjugate-matched center load. Similar to
the situation described in connection with the resistive-loaded
antenna Of Section 3. 2, the received power vanishes at loading
positions which satisfy equation (3.6). This time RL’6 0, but ‘

1(0) = 0; this case corresponds exactly to the frequency rejection

antenna of Section 3. l.

71

 

 

 

 

 

 

 

 

 

 

 

 

1.0

30
4.0 fioh = 6.0 6.0
20 J-
2,5
10 "- 4.0
1.5
2.5
d)-
0 /5
a /
~14
«11
DD
93 -10 --
I
m
0.1
_20 «II-
_30 up ‘
-40 up
_50 1 1 1 4 1 m 1
r I T I l T l l
0 0.2 0.4 0.6 0.8
d/h

= 50 ohms

Figure 3. 6. Received power for the Near-Optimum Receiving Antenna
for which the load resistance is specified--R

(130a = 0. 001)

XL - ohms

72

 

 

 

 

 

 

 

 

104 -
6.0

103 d1-
102 1h

6.0 6.09 4.0 4.09

10 I J_ J h L [A l l l
I 1 1 u 1 1 W 1 1
0 0.2 0.4 0.6 0.8
d/h

Figure 3. 7.

Center-load reactance for the Near-Optimum Receiving
Antenna for which the load resistance is Specified" -

RL = 50 ohms " (003. = 0. 001)

1.0

73

 

 

 

 

 

 

 

-105
_104 .41. 4.0
E
“g _103 ‘1-
CL”.
2
-10 ‘1' 4.0
2.5
1.5
l
10 J l Ah I L L _l I l
' r I I I I I I 5
0 0.2 0.4 0.6 0.3

Figure 3. 8.

d/h

Auxiliary resistance for the Near-Optimum Receiving
Antenna for which the load resistance is specified--

11L = 50 ohms (Boa = 0. 001)

 

1.0

74

 

 

 

 

 

 

 

106 ‘ L
E
,g 4
o 10 ‘1" \
. \
><
floh=1.5
2.5
1034-“- \‘
x
4.09 4.0 6.0 6.0
7- n
w : : . 1 : : : : :
0 0.2 0.4 0.6 0.8

d/h

Figure 3. 9. Auxiliary reactance for the Near-Optimum Receiving
Antenna for which the load resistance is specified--
RL = 50 chins (floa = 0. 001)

 

1.0

IV REACTANCE-LOADED RECEIVING ANTENNAS
WITH MINIMUM BACKSCATTERING

As pointed out in Section 3.1, with Z restricted to the form
Z = 0 + jX, insistence on the condition of zero broadside back-

scattering leads to the trivial receiving antenna for which R p: 0.

L
However, if the backscattering requirement is relaxed somewhat,
so that Z and ZL are no longer constrained by equation (2.11),
the case of pure reactive loading, clearly of practical interest,
can be reconsidered.

With R = 0 and no pre-conditions on the remaining three
impedance parameters, an entirely new problem arises in which

the fundamental quantities describing the antenna performance are

of the form

PR = PR(R x ‘ X) (4.1a)

P

n
"U
’55
>4
5

.1
s s L’ L' (4 b)

In spite of the many degrees of freedom existing at the outset,
it is not obvious how one should go about using them in order to

yield the receiving antenna with the best performance. More to

75

76

the point, it is not clear just what the 258.2 performance character-
istics might be.

Certainly the ultimate selection of RL' XI.’ and X must
provide an antenna which has a power delivering capability which
is at least comparable to that of its ordinary conjugate-loaded
counterpart,and at the same time must achieve a reduction in the
backscattered power which can be regarded as significant. Beyond
this, the particular approach to be used for "maximizing" the
function of three variables in (4. la) while simultaneously ”mini-
mizing” the function of the same three variables in (4. lb) is some-
what arbitrary. Clearly, unless the maximum value of (4.1a) and
the minimum value of (4. lb) occur for the same parameter values,
there is no "solution" for this problem which could be termed
unique.

A "best" antenna could probably be found from inspection
and comparison of the two functions in (4. l) for all possible com-

binations of values for R , X , and X. In dealing with functions

I. I.-
of three variables, however, such an approach would be, at best,
highly impractical. The alternative is to choose a method which
offers a more limited scape for studying the variations of PR and
PB. Such a method is provided, for example, if (on the basis of

some reasonable criteria), the number of variables involved can

be reduced at the outset. In the case of the functions in (4. 1),

77

this can be accomplished by the immediate elimination of Z by

L

stipulating that for each and every value of X, Z p be chosen to

L
yield maximum received power. The auxiliary reactance can be

subsequently selected in such a way that with R = RL(X) and

L
XL: XL(X)’ the function in (4.1b) is minimized. Although this
procedure may very well fail to yield the "best" reactance-

loaded receiving antenna, it does lead to an antenna which achieves
the stated objectives with at least moderate effectiveness. More-
over, this process for selecting the impedance parameters is

well suited for purposes of experimentation, particularly since

RL can be arbitrarily specified, if desired.

4. l The Conjugate-Match Condition
Given the value of auxiliary reactance X, the center-load

— 2* x
L- in( )

impedance which yields maximum received power is Z
(the asterisk denotes the complex conjugate). This follows from
the maximum power transfer theorem when the real part of Zin
is positive, as discussed in Section 2.. 2. With this condition

on ZL' the backscattered power density becomes a function of

X alone, and from (2. 6) is given by

 

_- 2—
(l-cosp h)2(E H3 )2 Z(FZ +G) +HZ +I.~
P = o o o L L (4.2)
s lsz 2 Z(AZL+B) +CZL+D *
0 Z = . (Z)
_ _ L in

 

 

Z=jX

78

where, by equation (1. 75) for zin’

_ —B(jX) +D
ZL ‘ [A(jX) +c] ' (4‘3)

After substituting (4. 3) into (4. 2) and rearranging, PS can be

written as

(1 -cos[30h)2(E0/ [30)2

 

 

P :
S 151TR 2
o
2 Z 2
(uZX + ulX +uo) + J(sz + le +vo)
2 (4.4)
Z(wZX + w1X + wo)
With A, B, C, and D written as
A=a+ja' C=c+jc'
. (4-5)
B=b+jb' D=d+jd'
the coefficients in (4.4) are
= I I
u2 a g + bf 1
u1 = -a'l +bh'-cg' +d'f
u0 = of + d'h'
v2 = ag' - b'f
v1 = -a1 - b'h' + c'g' + df > (4. 6)
= _ :1 3
v0 0 + dh
WZ = ab + a'b'
w1= ad'-a'd+bc'-b'c
w = cd + c'd'

 

79

where f, g', h' and I are as defined in (3. 4). Finally, after

expanding and collecting terms, (4. 4) can be put into the form

 

 

 

 

r 6 n w
2
(l-cosp h) (E [[3 )2 Z anX
o o 0 n=0
p z < __ > (4.7)
S 6011' R 2 6 n
o 2) b X
K n=0 n J
where
= = W
a6 0 b6 0
a5 = 0 b5 2 0
2. Z 2
a4 — 0.2 + v2 b4 - wz
a3 = 2(u1u2+ VIVZ) b3 = 2w1w2 P . (4. 8)
— 2 + 2 + 2( + ) b - z + 2
a.2 ' u1 V1 u0‘12 Vo"z 2 ‘ “’1 W0W2
a1 = 2(u0u1+ vovl) b1 2 Elwow1
2 2 2
a0 — u0 + v0 b0 — wo J

Using the results of Section 3. 2, the critical points of
the function PS(X), according to (3.20), correspond to the real

solutions of the following equation.

 

11 6 m I

Z { Z n[a'nbm-ni-lm am-n+lbn]}x = 0
m=O n=1 5

(4.9)
(a.,b. ‘=' 0 fori<0 and i> 6)
l l
J
Because a , a , b , and b are zero and since the coefficient

6 5 6 5

of X7 vanishes identically, (4. 9) reduces to the sixth degree

equation

8O

2 A x = o (4.10)

where Am is still given in (4. 9) by the summation within the
braces.

When the backscattered power density function is computed
at each of the real solutions of (4.10), the minimum value sought
is readily determined by inspection of the numerical results. These
computations have been made for several different antennas with
lengths in the range of O. 5 E poh _<_ 6. O and over the entire range
of possible loading positions. In general, equation (4.10) pro-
duces either two or four real values of X. However, there are
intervals of load-placement, for the shorter antennas, where all
six solutions of (4.10) are complex.

Table 4. 1 shows the results of the numerical investigation
for a few selected antenna lengths. The fact that the ”best" case
often shifts from one minimum of (4. 7) to another as the position
of the auxiliary load is varied results in a definite lack of con-
tinuity in the "best antenna" values of X, ZL, PR’ and PS. Prin-
cipally for this reason, a tabular, rather than graphical, format
has been chosen for representing the findings in this study. En-
tries for P8 in this table are given in terms of power gain, and
represent a normalization of this minimum value of backscattered

power density with respect to that produced by an ordinary dipole

81

antenna having a conjugate-matched center-load. The correspond-
ing values of received power, computed from (2.18), are repre-
sented similarly. Values of ZL are obtained from (4.3).

Of the nine loading positions considered (for each antenna
length, d/h = 0. l, O. 2, . . . , 0. 9), only those cases which resulted
in a decrease in P3 of 5 dB or more and which simultaneously
produced a loss in received power no greater than 5 dB are shown
in Table 4.1.

Although not apparent from the data in Table 4. 1, as d
approaches the particular loading position predicted in (3.10)

and shown in Figure 3.1, this antenna approaches the invisible

frequency-rejection antenna described in Section 3.1.

82

 

 

 

 

 

 

 

 

 

 

 

 

Table 4. 1. Reactance-loaded receiving antennas with minimum
backscattering--the conjugate-match condition
(fioa = 0.001)
poh d/h X RL XL PR PS
mhms) JOhHlS) (ohms) (dB gain) (dB gain)
1.5 0.2 9290 0.00214 3120 -4.70 -13.8
0.4 2840 0.0196 1740 -4.42 -7. 85
0.5 2130 0.0417 1450 -4.38 -6.55
2.5 0.1 37100 0.0190 3990 -0.597 -17.0
0. 7380 0.0376 2040 -3.77 -23.4
4.0. 0.1 -12700 0.0774 1980 +7.83 -9.12
0.2 -16200 0. 500 323 ‘ +9.63 -4. 54
0.4 1020 0.0491 245 I -2.74 -30.0
o. 5 515 0.251 126 ( +3.54 -20. 0
0.6 375 0.423 . 45.5} +5.29 -17.4
' 0.7 361 0.581 40.8) +6.59 -l6.2
0.8 469 0.725 -48.1) +7.92 -15.4
i 0.9 978 0.353 -60. 9! +9.59 -14. 8
g r _fi
6.0' 0.1 6340 32.0 2300 +24.4 -l.57
‘ 0.2 1240 52.2 1020 +23.6 -7.54
g 0.3 611 90.8 , 422 +20.7 -13.9
g 0.4 495 183 g -342 +16. 3 -22. 5

 

 

 

 

 

83

4. 2 Load Resistance Specified

When the value of RL is arbitrarily specified, conjugate-
matched loading cannot be achieved. In this case, on the basis of
a corollary of the maximum power transfer theorem, maximum

received power occurs when X is chosen as the negative of the

I.-
(series) input reactance Xin(x)°
The backscattered power density from a reactance-loaded

receiving antenna having a center-load impedance selected in this

manner is, from (2.6).

 

 

 

 

I Z
2 2
P _ (l-cosBoh) (EC/so) Z(FZL+ G) + HZL+ L
s 15'n'R02 Z(AZL+ B) + CZL+ D z _R -X (Z)
_. _) L- L‘J in
Z = jX
(4.11)
where, by equation (1. 75) for zin’
. B(jX) + D
= - ———-- . 4.12
21.. RL 3‘9”“ [A(jX) + c] ( )

When A, B, C, and D are written as shown in (4. 5) and the re-

sult is simplified, (4. 12) becomes

2 . 2
[{pzX + 91X + po}RL] +J[qZX +q1X + qo]
z = (4.13)
L x2+ x +
p2 91 p0

 

84

where
2 2
= I = - I I W
p2 a + a q2 ab + a b
pl : 2(ac' - a'c) q1 = ad +a'd' - bc - b'c' ) (4-14)
2 2
— l : .. ' '
p0 c + c q0 Cd + c d J

 

After substituting (4.13) into (4. 11) and rearranging, PS can

be written as

(1-cosfloh)2'(Eo/[30)Z

 

 

 

P :
s 1511'R 2
o
3 2 . 3 2
[u3X + uZX + ulX + no] - J[ v3X + vZX + le + v0]
[w X3+w X2+w X+w ]-j[y X3+y X2+y X +y]
3 2 1 0 3 2 1 O
(4.15)
where
= 1
u3 fq2+ g pZ I
u = fq+g'p+h'q-£p
2 1 1 2 2 ? (4.16)
= 1 i -1
ul fqo+ g po+ h ql pl
11 = h'q - 1p
0 0 0 J
= W
V3 (”’sz
v = (fp +h'p )R
2 1 2 L P (4.17)
= I
v1 (fpo+ h pl)RL
= I
V0 0“ p0)RL

 

85

aqz+ (a'RL+ b')p2

aq1+ c'qz+ (a'R p+ b')p1- (cR

+
L d)pz

L
aq0+ c'ql+ (a'RL+ b')p0— (cRL+ d)pl

I ..
c q0 (CRL+ d)p0

_ i
a q2+ (aRL+ b)p2

-' + + + + ' +'
441 “12 (aR b)pl (cR d)pZ

L L

+ + ' +I
L b)p0 (cRL d)?1

-I +
aq0 cq1+(aR

Cq0+ (C'RL+ d')pO

 

 

J

V

(4.18)

(4.19)

The terms f, g', h’, and f appearing in (4. 16) through (4.19)

are defined in (3.4).

Ps can finally be put into the form:

P

where

S

 

 

 

r 6 n \
15‘trRo2 g b Xn

K n20 n J

= u2 + v2

3 3

= 2(u2u3+ vzv3)

= u: + v: + 2(u1u3+ v1v3)

= 2(u0u3+ vov3+ u1u2+ vlvz)

= u: + vi + 2(u0uz+ VOVZ)

= 2(uou1+ vovl)

= u: + V:

When (4.15) is expanded and simplified,

(4.20)

(4.21)

 

86

2 2
= +
b W3 y3
b5 = Z(WZW 3+ y2y3)
b - Z + 2 +2( + )
’ W2 y2 W1W 3 y1Y3
b3: 2(wow w3+ yoy3+w1w 2+ ylyz) > . (4.22)

2
b =w +YI+Z(WW )

2 1 o ""24r y0"2
b1: 2(""o""1+"o"1)

2 2
b — w0+y0

 

J

By (3. 20) in Section 3. 2, the critical points of the func-

tion PS(X) correspond to the real solutions of the following

 

equation.
\
11 6 m
?0 1:31 n[anbm-nfl- a"m-n+lbn] X = 0
m‘ ‘ > (4.23)
(ai,bi50for 1<0 and i>6) J

11
Inasmuch as the coefficient of X vanishes identically, (4. 23)

is actually the tenth degree equation:

10

z A xmz o. (4.24)
m

m=0

Similar to the conjugate-matched case of Section 4.1,
P3 is computed at each of the real solutions of (4. 24), and
the minima of this function are found from inspection of these

results. As in the previous section, the particular local

87

minimum of Ps which provides the best balance between PR and
P8 is selected as the "best" case. For each of four values of
assumed load resistance (1, 10, 100, and 1000 ohms), these
"best” cases are shown in Figure 4. 2 for antennas of several
different lengths. Only those loading positions are included

at which Ps < -5 dB for at least one of these values of R _ and,

L

at the same time, where PR > -5 dB for at least one of these

values of R As before, P and P8 are indicated as gain

L' R
quantities, normalized to the performance of the ordinary
conjugate-loaded receiving antenna. Values of XL are obtained
from (4. 12).

Again, with X a free variable in a situation involving
backscatter reduction, this antenna approaches the invisible

frequency-rejection antenna of Section 3. l as d/h nears the

value shown in Figure 3.1.

88

 

 

 

 

 

Table 4. 2. Reactance-loaded receiving antennas with minimum
backscattering--load resistance specified (poazo. 001)
(30h d/ h RL X XL PR Ps
(ohms) (ohms) (ohms) (dB gain) (dB gain)
1.5 0.8 1 2090 941 -2.30 -2.76
10 1890 1230 -1.12 -8.57
100 2120 920 -14.9 -26.0
1000 692 -252 -4.51 -13.3
0.9 1 3560 646 -5.00 +4.32
10 3470 698 -1.29 -3.39
100
1000 2880 262 -2.03 -7.26
2.5 0.4 1 284 1190 -33.3 -72.6
10 284 1190 -23.3 -52.6
100 284 1190 -13.4 -32.7
1000 284 1180 -4.22 -13.5
0.5 1 264 1150 -32.5 -71.1
10 264 1150 -22.5 -51.1
100 264 1150 -12.6 -31.2
1000 264 1140 -3.60 -12.2
0.6 1 284 1100 -31.9 -70.0
10 284 1100 -21.9 -50.0
100 284 1100 -12.0 -30.1
1000 283 1080 -3.13 -11.2
0.7 1 358 1060 -31.4 -69.1
10 358 1060 -21.4 -49.1
100 358 1060 -11.6 -29.2
1000 358 1040 -2.76 -10.4
0.8 1 564 1020 -31.0 -68.4
10 564 1020 -21.0 -48.4
100 564 1020 -11.1 -28.5
1000 564 990 -2.44 -9.82
0.9 1 1350 988 -30.6 -67.7
10 1350 988 -20.6 -47.9
100 1350 985 -10.8 -28.0
1000 1350 892 -2.16 -9.35

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 4. 2 (continued)

89

 

 

 

50h d/h RL x xL PR P
(ohms) (ohms) (ohms) (ngain) (dB ain ‘

4.0 0.2 1 6930 1200 -21.6 -46.8
10 6930 1200 -11.8 -27.0
100 6730 1190 -3.55 -8.85
1000 1640 534 -2.81 -8.36

0.3 1 -1570 361 -18.3 -43.9
10 -1570 361 -8.58 -24.2
100 -1520 373 -0.729 -6.34

1000 884 543 -10.8 ~25.9

0.5 1 514 126 +1.64 -15.9
10 -510 348 -18.5 -42.3

100 -510 347 -8.78 -22.6
1000 -513 303 -1.21 -5.04

0.6 1 374 46.2 +4.48 914.5
10 373 47.8 -2.58 -ll.8

100 -609 462 -10.4 -25.5
1000 -612 415 -2.19 -7.28

0.7 l 361 -10.2 +6.21 -l4.2
10 357 -5.44 0.00 -10.7

100 -1250 587 -1l.5 -27.4

1000 -1260 542 -3.02 -8.91

0.8 l 468 -47.5 +7.74 -l4.2
10 465 -4l.7 +2.06 -10.1

100 19100 711 -12.5 -28.9

1000 18400 669 -3.78 -10.2

0.9 l 978 -60.1 +9.43 -14.2
10 973 -54.7 +4.19 -9.61

100 2570 816 -13.4 -29.9

1000 2560 775 -4.53 -11.1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 4. 2 (continued)

90

 

 

 

56h d/h RL x XL P PS
(ohms)— (ohms) (0mg (dB gain) (dB gain) ‘
6.0 0.1 1 6350 2300 +15.2 -25.8
10 6170 2300 +22.0 -8.49
100 3640 2460 +16. 9 -1.44
1000 1210 3120 +10.7 -0.117
0.2 1 1250 1020 +12. 3 -36.0
10 1240 1020 +20.9 -l7.4
100 1180 1020 +22. 3 -5. 58
1000 639 1090 +14.6 -3.41
0.3 1 612 421 +7.04 -47.1
10 612 421 +16.2 -27.9
100 606 422 +20. 5 -13. 5
1000 476 433 +15.0 -9.25
0.4 1 496 -342 -0. 308 -61. 8
10 496 -342 +9. 28 —42. 2
100 495 -342 +15. 9 -25. 5
1000 464 ~341 +13.4 -18. 3

 

 

 

 

 

 

 

 

 

 

 

V EXPERIMENT AND RESULTS

In this chapter experimental results for the reactance-
loaded receiving antenna are presented and correlated with the
theoretical work of the previous chapters. A fairly extensive
comparison of theory and experiment is made for an antenna of
one particular length, in which, for several positions of auxiliary
loading and with a specified load resistance of 100 ohms, the
theoretical and experimental values of received and backscattered
power are compared, as the value of the auxiliary reactance is
varied over a rather wide range. In addition, an experimental
search for the invisible frequency-rejection receiving antenna
described in Section 3.1 is conducted for antennas of several

different lengths.

5. l The Experimental Reactance-Loaded Receiving Antenna

All of the experimental work has been performed using
a specially designed reactance-loaded monopole antenna, mounted
on a rectangular ground plane (6'): 8') which forms one wall of
a completely enclosed anechoic chamber. The choice of a test
frequency of 1. 5 GHz allows the receiving monopole to be opera-

ted in the far zone of the transmitting monopole on the given

91

92

aluminum ground plane, and, at the same time, permits con-
struction of an electrically thin experimental receiving antenna
from metallic tubing large enough to make the fabrication reason-
ably easy. At 1. 5 GHz, corresponding to a wavelength of 20
centimeters in free space, brass tubing with an outside diameter
of 1/4 inch gives an electrical thickness of poa = 0.1.
Photographs of the experimental receiving monOpole
are shown in Figure 5.1. The antenna sections in Figure 5. lb
appearing in multiple lengths are necessary so that loading
position d can be varied continuously over a wide range of values.
A variable auxiliary reactance is provided across the gap con-
taining the white teflon spacer by the longitudinally-slotted
cylindrical section of tubing immediately adjacent and to the
left of the gap. This slotted section contains a sliding brass
disc which is threaded to a 1/ 16 inch brass rod running the entire
length of the section. Viewed from the gap end, this section

represents a shorted length of coaxial transmission line, and

effects a reactance across the gap of

X = t 1 5.1
R0 anfio s ( )

where RO is the characteristic resistance of the lossless coaxial
transmission line and 1 s is the distance between the shorting

disc and the left edge of the gap. The characteristic resistance

 

(a)

 

 

 

........' ( llill‘llullllllumg'l ()1) 1'
«noon *4 mu 1

(b)

Figure 5.1. The experimental reactance-loaded receiving antenna

94

of an air-filled transmission line composed of concentric cylin-

ders is well known and is given by

g d
o 2
z _ l _ ,
R0 2" :1 d1 K (5 2)

where go: 120w ohms is the characteristic impedance of free

is the inside diameter of the outer cylinder, andd is

space,d l

2
the outside diameter of the inner cylinder. Using the nominal
values of diameter indicated in the assembly drawing of Figure

5. 2, equation (5.2) gives R0: 66 ohms. However, when the

actual measured diameter of the threaded inner conductor is used,
(5. 2) gives R0= 71 ohms. Moreover, if the effective value of d1

is approximated as the average diameter (on the basis of assuming
a perfect V-cut thread), equation (5.2) yields a characteristic
resistance of almost 80 ohms.

Lack of knowledge of the amount of stray capacitance
existing directly across the teflon spacer adds a further compli-
cation to the problem of predicting the exact value of auxiliary
reactance effected across the gap, as a function of the stub
length 1 3' Means for indirectly determining approximate values
of R0 and this parallel gap capcitance from the experimental
results are discussed in Section 5. 3. 1.

Some mention of the criterion used for selecting the

(fixed) length of the coaxial shorting- stub section should be made.

95

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Disregarding the effect which the gap capacitance has on the total
value of the auxiliary reactance, an adjustable stub with a maxi-
mum length of one-half wavelength (10 cm at l. 5 CH2) would be
required in order to provide the capability of achieving all values
of X (-oo < X < oo ). A section of this length would immediately
restrict the applicability of the experimental receiving mono-
pole to antennas for which poh > 1r, even before provisions for
mounting and implementing a means for varying d were considered.
The particular length chosen for this section (6 cm) allows the
line to be tuned over all positive values of X, according to
equation (5. 1), and a large portion of the negative values. The
presence of the stray capacitance across the gap at location d
extends the range of achievable negative values of true reactance
even more.

With the length of the shorting-stub section finalized, the
remaining sections of the experimental antenna assembly have
been carefully designed (with a minimum number of parts in
each set of like parts) to provide a range of applicability of 3. 0 5
(Sch 5 5. 0, and such that the position of the auxiliary loading can
be varied continuously over the range 0.2 E d/h E 0. 9 (with the
exception that the lower limit on d/h is about 0. 35 for fioh = 3. 0).
The larger values of d/h are obtained by reversing the stub

tuner section end-for-end.

97

5. 2 The Experimental Setup

A block diagram of the experimental setup used in the
measurement of the received and backscattered power of the
reactance-loaded test antenna is shown in Figure 5. 3. Figure
5. 4 contains photographs of the anechoic chamber.

Operating at a frequency of l. 5 GHz, the experimental
receiving antenna described in Section 5.1 is located 36 inches
(nearly five wavelengths) from the transmitting antenna. A third
element positioned on a line with the transmitting and receiving
monopoles and lying 18 inches beyond the receiving antenna
will be referred to as a scattering detector, and is used in the
measurement of the backscattered power density produced by
the test receiving antenna.

Measurement of the backscattering is accomplished by
a simple cancellation method. With only the scattering detector
and the transmitting antenna in the chamber, the signal from the
scattering detector is added to a second signal of equal amplitude
but Opposite phase derived from the same oscillator which drives
the transmitting antenna. The relative amplitude of this sum is
monitored by a heterodyne measurement scheme. When the
test receiving antenna is inserted in place, the scattering detector
responds to what can be considered the sum of two electric fields:

one equal to the field which existed before the test antenna was

98

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(b)

Figure 5. 4. The anechoic chamber .

100

added to the chamber, and one caused directly by the broadside
re-radiation of the test antenna. By first establishing a reference
level of scattering with an ordinary unloaded monOpole of the
same dimensions as the test antenna, this scheme provides a
very efficient means for determining the test antenna's back-
scattered power gain.

The measurement of received power is complicated
somewhat by the practical impossibility (using commercially
available cylindrical transmission line test equipment) of locating
the center-load impedance right at the ground plane. In the
experimental search for the invisible frequency rejection antenna,
the load impedance which appears at the driving point of the re-
ceiving monopole is not critical, since this phenomenon is pur-

portedly independent of Z On the other hand, in an experimental

L'
study of the receiving antenna of Section 4. 2, a means for insuring

a constant specified value of R at the driving point is essential.

L

The problem of transforming what is inherently a parallel
combination of resistance and adjustable reactance (using com-
me rcially available resistive terminations, stub reactance tuners,
air lines, and tees), occurring at some distance away from the
ground plane, into a desired series combination right at the
driving-point is resolved as follows.

The input impedance to a lossless air transmission line

of length It terminated in an impedance Zp is given by

101

Z - R chos fiolt +jROSlnfioft (5 3)
- + . 0 O
f o Rocos fiolt ijsmfiolt

 

where R0 is the characteristic resistance of the line. If It is

selected such that for n a positive integer

+ n( ) , (5.4)

7
NIO

then
pi = g (1 +Zn) (5.5)

o t

and, using (5.5), equation (5. 3) becomes

R0
2 : ? . (5'6)
P

Now if Zp consists of resistance R in parallel with reactance

Xp, (5. 6) can be expanded and written as

R3; R:
Z. = i— ‘J '3;- - ‘5'”
P P
Furthermore, notice that
z = R -j(R2/X) forR =R . (5.3)
1 o o p p o

In the experimental study of the reactance-loaded
antenna of Section 4. 2, using a tee and an adjustable coaxial
air line of 50 ohms characteristic resistance, a 50 ohm termi-

nation (corresponding to Rp in (5. 8) ) can be connected in

102

parallel with a reactance tuning stub to achieve a center-load

impedance of

ZL = 50+jXL

(XL = -2500/xp)

(5.9)

at the driving point of the receiving antenna, simply by choosing

the length of the adjustable air line according to (5.4).

5. 3 Experimental Results and Comparison with Theory

5. 3. 1 Performance of the Reactance-Loaded Receiving
Antenna as a Function of the Auxiliary Reactance

For a fixed length antenna corresponding to [Bob = 4. 0,
gain measurements of backscattered power density and received
power have been made for each of the loading positions: d/h =
0.2, 0. 3, 0. 4, . . ., 0. 9. In each case, with RL= 50 ohms and
the center-load reactance adjusted for maximum received power,
the auxiliary reactance has been varied over the complete range
of attainable values. The measured values of Ps and PR are
shown in the following figures for several different positions of
auxiliary loading. Curves based upon the theoretical work in
Section 4. Z are shown for each case.

As pointed out in Section 5.1, the actual value of auxiliary

reactance is the combination of the reactance given by equation

(5. l) in parallel with some unknown stray reactance, owing to

103

the stray capacitance across the gap at the position of the load.

In order to correlate the length of the coaxial shorting stub £8

with the total value of reactance existing across this gap, this
stray reactance must somehow be determined first. Knowing the
length [I 8] min for which the experimental value of P3 is minimum,
this problem is resolved simply by assuming the experimental
value of minimizing auxiliary reactance to be the same as the
theoretical value, and, by using [£8] min in (5. l), choosing the
value of parallel stray reactance accordingly.

Since the configuration of the conducting surfaces of the
test antenna changes as the loading position is varied, the amount
of stray reactance also varies. For this reason, the assumed
value of stray reactance, as computed by the above scheme, is
different in each of the cases shown.

As it turns out, the above correlation procedure is rela-
tively insensitive to small changes in the assumed value of the
characteristic resistance of the coaxial shorting-stub section.

In all cases, Ro has been taken as 75 ohms.

104

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109
5. 3. 2 Experimental Verification of the Invisible Fre-
quency-Rejection Receiving Antenna

According to the theoretical work in Section 3. 1, there
exists a particular value of loading position d such that, with
the proper value of auxiliary reactance, the backscattered field
and the current at the center of the antenna will vanish simul-
taneously, independent of the value of Z L' Using the same setup

as in the previous experiment, and with Z arbitrary but

L
adjustable, a search for this invisible frequency-rejection antenna
has been made. With the initial position of the load selected
according to the information in Figure 3.1, both the position and
the reactance of the auxiliary load have been carefully adjusted

to achieve minimum values of PR and P8 simultaneously. The
results of this investigation are shown in Table 5.1. With the
center-load reactance tuned for maximum received power in

each case (i. e., with XL(X, d) = -xin (X, d) ), the entries for

PR and P8 shown in this table represent the simultaneous
minimum values of these power gain quantities as a function

of the two variables X and d. Except for the case of Bob 2 5. 0,
where no minimum of PR could be found, these results are

clearly indicative of a phenomenon almost exactly like that

predicted in Section 3. l.

110

 

 

 

Table 5.1. EXperimental results of the search for the invisible
fr equency- r ejection antenna.
theor etical exp er imental
p h d/h d/h PR Ps
° (dB gain) (dB gain)
3.5 0.369 0.371 -20.0 -24.0
. 0.382 0.390 -12.7 -25.5
4.5 0.399 0.405 -Z.5 -21.0
5.0 0.420 0.427 -15.5

 

 

 

 

 

 

 

 

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

R EF ER ENCES

R. W. P. King, The Theory of Linear Antennas. Cam-
bridge, Massachusetts: Harvard University Press,

1956.

 

R. W. P. King, "Linear Arrays: Currents, Impedances,
and Fields, " IRE Trans. on Antennas and PrOpagation
(Special supplement), vol. AP-l7, pp. 5440-5457,
December 1959.

R. W. P. King, ”Dipoles in Dissipative Media, ” Gordon
McKay Laboratory, Harvard University, Technical
Report 336, pp. l-12, February 1961.

K. M. Chen and V. Liepa, "The Minimization of the Back
Scattering of a Cylinder by Central Loading, ” IEEE
Trans. on Antennas and PrOpagation, vol. AP-lZ,
pp. 576-582, September 1964.

K. M. Chen, "Minimization of Backscattering of a Cylin-
der by Double Loading, ” IEEE Trans. on Antennas and
Prepagation, vol. AP-13, pp. 262-270, March 1965.

K. M. Chen, "Reactive Loading of Arbitrarily Illiminated
Cylinders to Minimize Microwave Backscatter, " Journal
of Research NBS/ USNC-URSI, vol. 69D, no. 11,
pp. 1481-1502, November 1965.

R. H. Duncan and F. A. Hinchey, ”Cylindrical Antenna
Theory, ” Journal of Research NBS, vol. 64D, no. 5,
pp. 569-584, September-October 1960.

K. K. Mei, "On the Integral Equations of Thin Wire Antennas, "
IEEE Trans. on Antennas and Propagation, vol. AP-l3,
pp. 374-378, May 1965.

R. W. P. King, ”The Linear Antenna--Eighty Years of

Progress, "Proc. IEEE, vol. 55, pp. 2-16, January
1967.

111

APPENDIX

RECEIVED POWER OF THE DOUBLE-LOADED ANTENNA

Based on the continuity of the tangential component of
the total electric field at the antenna surface, the vector po-
tential Az(z) maintained on the surface of the double-loaded

receiving antenna by the induced antenna current I(z) must

 

 

satisfy:
BZAZ 2 “302
322 + (50 A2 = w [ZLI(O)6(z)+ ZI(d) {6(2-d)+5(z+d)}_Eo].

(A. 1)
The symbols in this equation are carefully defined in Chapter 1
where (A. 1) is derived as equation (1. 6).
Since (A. 1) is a linear equation it can be written as

the sum of the following two equations.

 

 

 

 

Z R . 2

3 AZ 2 R Jpo IR

822 + 5., A, = w [2 (d){5(Z-d)+6(z+d)}-Eo] (A.2)

82A: Z T jfloz T

322 +130 AZ = w [ZLI(0)6(z)+ZI (d){6(z-d)+5(z+d)}]
(A.3)

Where Az(2) = Aim) + A:(z) (A.4)

112

 

113

and I(d) = IR(d) + IT(d) (A.5)

Noting that equation (A. 1) applies to the receiving antenna
system depicted in Figure l. 1, comparison of the right side of this
equation, in turn, with the right-hand sides of (A. 2) and (A. 3) readily
indicates the physical significance of the latter two equations: in
equation (A. 2), A:(z) is the surface vector potential supported by
the induced current IR(z) on a double-loaded receiving antenna for
which the center impedance is specified as zero; equation (A. 3), on
the other hand, relates the surface vector potential A:(z) to the cur-
rent distribution IT(z) existing on a double-loaded transmitting antenna

which is being center-driven by a voltage source having the value
V = -I(0)ZL . (A.6)

Using equation (1. 8), the vector potential functions A:(z)
and A:(z) are individually expressed in terms of the current distri-

butions supporting them.

h

p.
Aim = fi ShIR(z')Ka(z,z')dz' (A.7)
p. h
AzT(z) = 2% Sh IT(z')Ka(z,z')dz' (A.8)

Adding these equations, according to (A.4), gives

p. .
Az(z)=-4—: 5:{IR(z') + IT(z')} Ka(z, z')dz' . (A. 9)

114

In view of equation (1. 8), it follows from (A. 9) that
I(z) = IR(z) + IT(z) . (A. 10)
Thus, in particular,
1(0) = 1R(0) + 1T(0) . (A. 11)
Now, in terms of the quantities defined in connection with

the transmitting antenna described by equation (A. 3), the input

impedance of the double-loaded antenna can be written

Z = V . (A. 12.)

in IT(O)

 

By combining (A. 12) with (A. 6), V is eliminated to give

-I(0)ZL

Z.
in

T

I (0) = (A.13)

After substituting (A. 13) into (A. 11) and rearranging, I(0) becomes
Zi IR(0)
1(0) = —9—— . (A.14)

+
2'1. Zin

Equation (A. 14) can be rewritten in the form:

VI

1(0) = m (A. 15)
L s
where Z' = Z. and V' = Z, IR(0) , (A.16)
g in 1n

so that the received power of the double-loaded antenna can be

expressed as

115

 

 

R 2
L V'
PR - ‘2‘" 2727 (“7)
I- g

The terms V' and Zlg appearing in (A. 17) are properly interpreted
as the voltage driver and its series impedance in an equivalent cir-
cuit representation of the double-loaded receiving antenna system
as viewed from the antenna terminals. ' As is evident from (A. 16),
these two quantities, although clearly functions of the auxiliary
impedances Z, are totally independent of the center-load impedance
ZL.
That equation (A. 17) applies, innately, in the case of a double-
loaded receiving antenna constructed of perfectly conducting material,

is therefore established. Significant in its own right, this result is

contributory to the work in Section 2. 2.

 

   

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