• TABLE OF CONTENTS
HIDE
 Title Page
 Acknowledgement
 Table of Contents
 Abstract
 Introduction
 Possible scanning methods and development...
 Feeds and calculation of radia...
 Calculation of radiation from paraboloidal...
 Noise performance
 Secondary pattern asymmetry and...
 Theoretical analysis of secondary...
 Optimum feed
 Summary and future work
 Appendix
 Reference
 Biographical sketch
 Copyright














Title: New scanning techniques in conopulse angle tracking radar
CITATION THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00082362/00001
 Material Information
Title: New scanning techniques in conopulse angle tracking radar
Physical Description: 162 leaves : ill. ; 29 cm.
Language: English
Creator: O'Brien, Gregory G., 1966-
Publication Date: 1995
 Subjects
Subject: Tracking radar   ( lcsh )
Automatic tracking   ( lcsh )
Electrical Engineering thesis, Ph. D
Dissertations, Academic -- Electrical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis (Ph. D.)--University of Florida, 1995.
Bibliography: Includes bibliographical references (leaves 157-161).
Statement of Responsibility: by Gregory G. O'Brien.
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00082362
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 002046784
oclc - 33430150
notis - AKN4720

Table of Contents
    Title Page
        Page i
    Acknowledgement
        Page ii
    Table of Contents
        Page iii
        Page iv
        Page v
    Abstract
        Page vi
        Page vii
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
    Possible scanning methods and development of antenna design
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
    Feeds and calculation of radiation
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
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        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
    Calculation of radiation from paraboloidal reflector
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
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        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
    Noise performance
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
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        Page 101
        Page 102
    Secondary pattern asymmetry and its effects
        Page 103
        Page 104
        Page 105
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    Theoretical analysis of secondary pattern asymmetry effects
        Page 128
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        Page 132
        Page 133
        Page 134
        Page 135
        Page 136
        Page 137
        Page 138
    Optimum feed
        Page 139
        Page 140
        Page 141
        Page 142
        Page 143
        Page 144
        Page 145
        Page 146
        Page 147
        Page 148
    Summary and future work
        Page 149
        Page 150
        Page 151
    Appendix
        Page 152
        Page 153
        Page 154
        Page 155
        Page 156
    Reference
        Page 157
        Page 158
        Page 159
        Page 160
        Page 161
    Biographical sketch
        Page 162
        Page 163
    Copyright
        Copyright
Full Text











NEW SCANNING TECHNIQUES IN CONOPULSE ANGLE TRACKING RADAR


By

GREGORY G. O'BRIEN















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA


1995















ACKNOWLEDGMENTS


I thank my advisor and committee chairman, Dr. Peyton Z.

Peebles, Jr., for his patient supervision, informative

counseling, and methodical guidance throughout this study. I

also thank Dr. Thomas E. Bullock, Dr. Leon W. Couch, II, Dr.

Ewen M. Thomson, and Dr. Kermit N. Sigmon for their time and

interest in serving on my supervisory committee. Finally,

thanks go to my parents for their continuous love,

understanding, and support throughout my school years.















TABLE OF CONTENTS



ACKNOWLEDGEMENTS ........................................... ii

ABSTRACT ................................................... vi

1. INTRODUCTION ............................................. 1

1.1 Conical Scan and Monopulse............................ 1
1.2 Conopulse .............................................3
1.2.1 The Conopulse Concept ........................ 3
1.2.2 Conopulse Implementations ....................4
1.3 Research Objectives................................... 8

2. POSSIBLE SCANNING METHODS AND DEVELOPMENT OF ANTENNA
DESIGN.................................................10

2.1 Possible Scanning Methods............................ 10
2.1.1 Rotation of Two Squinted Beams ...............10
2.1.2 Nutation of Two Squinted Beams ...............12
2.2 Selection of Circular Polarization................... 17
2.3 Proposed Antenna Design...............................18

3. FEEDS AND CALCULATION OF RADIATION ......................21

3.1 Dominant Mode Horn....................................22
3.2 Dual Mode Horn........................................25
3.3 Hybrid Mode Horn......................................28
3.4 Dielectric Rod.......................................30
3.5 Diffraction-Limited Feeds and Dielectric Rod.........33

4. CALCULATION OF RADIATION FROM PARABOLOIDAL REFLECTOR ....39

4.1 Surface Geometry .....................................40
4.2 Aperture Distribution Method......................... 45
4.2.1 Effect of Lateral Feed Displacement ..........52
4.2.2 Numerical Integration ........................ 53
4.2.3 Accuracy of Simpson's Rule ...................55
4.3 NEC Computer Program...............................57
4.4 Effect of Limitations on Aperture Distribution
Method ............................................ 58


iii












5. NOISE PERFORMANCE ........................................60

5.1 Quantification of Noise Performance.................. 60
5.2 Minimum Lateral Displacement.........................68
5.2.1 Minimum Size of Central Waveguide ............68
5.2.2 Minimum Size of Feed ......................... 71
5.3 Effect of Lateral Displacement....................... 72
5.4 Effect of Feed Size................. .............. .82
5.5 Effect of Paraboloid Size............................ 89
5.6 Performance of Dielectric Rod Antenna versus Other
Feeds...............................................99

6. SECONDARY PATTERN ASYMMETRY AND ITS EFFECTS ............103

6.1 Sources of Secondary Pattern Asymmetry..............104
6.2 Secondary Pattern Asymmetry......................... 110
6.3 Computation of RL(t) for an Asymmetric Pattern.......113
6.4 Computation of DC Error Signals..................... 117
6.5 Study of Effects of Secondary Pattern Asymmetry.....119
6.5.1 R (t) ..................... ......................119
6.5.2 DC Errors ...................................121
6.5.3 Sensitivity versus Target Rotation Angle ....126

7. THEORETICAL ANALYSIS OF SECONDARY PATTERN ASYMMETRY
EFFECTS...............................................128

7.1 Gaussian Pattern with Elliptic Secondary Pattern
Asymmetry..................... ................. 129
7.1.1 Equations of Model .......................... 129
7.1.2 Effect of Secondary Pattern Asymmetry on
DC Error Signals .........................130
7.1.3 Comparison of Gaussian Pattern with
Elliptic Secondary Pattern Asymmetry
with Symmetric Gaussian Pattern........130
7.2 Pattern with Arbitrary Asymmetry.................... 134
7.3 Validity of Models..................................135

8. OPTIMUM FEED .......................................... 139

8.1 Criteria for Optimum Feed........................... 139
8.2 Optimization of Parameters.......................... 141
8.2.1 Lateral Displacement .......................141
8.2.2 Feed Size................................... 141
8.2.3 Paraboloid Size............................ 142
8.2.4 f/D Ratio ..................................142
8.3 Recommended Feed.....................................143










9. SUMMARY AND FUTURE WORK ........

9.1 Summary ...................
9.2 Future Work.................

APPENDIX A CALCULATION OF RADIATED

APPENDIX B CALCULATION OF RADIATED
PARABOLOID REFLECTOR ............

APPENDIX C CALCULATION OF RADIATED
PARABOLOID REFLECTOR ............

LIST OF REFERENCES ................

BIOGRAPHICAL SKETCH ...............


FIELDS FROM

FIELDS FROM



FIELDS FROM
............

FIELDS FROM
............


FEEDS


......149

......149
......151

......152


....153


...155

...157

... 162














Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


NEW SCANNING TECHNIQUES IN CONOPULSE ANGLE TRACKING RADAR

By

Gregory G. O'Brien

May, 1995



Chairman: Dr. Peyton Z. Peebles, Jr.
Major Department: Electrical Engineering


Conopulse angle tracking radar is a hybrid of monopulse

and conical scan. A possible antenna design was developed

with the antenna design consisting of a paraboloid reflector

and two offset waveguide feeds.

System performance with noise was investigated and

quantified by the rms value of angle tracking error. With

diffraction-limited feeds, it was found that rms error of

approximately 0.052 beamwidths could be obtained. The noise

performance of these feeds was deteriorated by low antenna

efficiencies caused by excessive spillover radiation.

With a dielectric rod antenna, better performance was

achieved with a rms error about 0.037 beamwidths. The

dielectric rod obtained its. superior performance largely due









to better antenna efficiency, which resulted from its more

directive radiation properties.

Previous conopulse literature has assumed rotationally

symmetric secondary patterns. However, in a practical

system, there is some asymmetry in the patterns with this

asymmetry possibly having a harmful effect on system

performance.

Secondary pattern asymmetry was investigated and

quantified by the normalized maximum deviation in 3-dB

beamwidth (versus azimuth angle). Secondary pattern

asymmetry was relatively small. At the best noise f/D ratio,

the normalized maximum deviation of the dominant mode horn

and dielectric rod were about 4 percent and 0.3 percent,

respectively.

The effects of secondary pattern asymmetry were

examined. It was found that secondary pattern asymmetry did

not cause dc errors and had negligible effect on sensitivity

versus target rotation angle.

To provide a theoretical basis for secondary asymmetry

effects, a model was developed involving gaussian patterns

with the theoretical results closely matching actual results.

On the basis of different criterion, it was concluded

that the dielectric rod is the best feed for this system and

the dominant mode horn is the best feed of the diffraction-

limited feeds.


vii
















CHAPTER 1
INTRODUCTION

A tracking radar system measures coordinates of a

target: range, azimuth angle, elevation angle, and possibly

doppler frequency. The data generated by the tracking radar

can be used to determine target trajectory and predict future

target position. There exist a variety of tracking radar

systems, which can be grouped in the categories of sequential

lobing and simultaneous lobing. The most established

sequential lobing technique is conical scan, while the most

commonly used simultaneous lobing method is monopulse.


1.1 Conical Scan and Monopulse


In a conical scan system, an offset antenna beam is

rotated continuously about the boresight axis, which is the

axis of rotation. When the target is not on the boresight

axis, the echo signal will be amplitude modulated at a

frequency equal to the rotation frequency of the antenna beam

(Skolnik, 1980). The conical scan modulation is extracted

from the echo signal with the angle tracking information

contained in the modulation envelope. The percentage of

amplitude modulation is proportional to the angle between the

boresight axis and target axis (Sakamoto, 1975). The phase









shift of the envelope signal relative to the antenna rotation

signal depends on the direction of the target relative to a

reference axis (Sakamoto, 1975). The conical scan modulation

is passed to two servo-control systems, which continuously

position the antenna boresight axis at the target. Two

servos are necessary since tracking requires two dimensions.

The advantage of conical scan is its simplicity.

Conical scan requires only one antenna beam and one receiver

channel to generate complete two-dimensional target data.

The main disadvantage of conical scan is its susceptibility

to target amplitude fluctuations (subsequently referred to as

simply amplitude fluctuations) from pulse to pulse. If the

amplitude fluctuations are random, there is a decrease in

signal to noise ratio and possibly false tracking

measurements (Sakamoto, 1975). These pulse to pulse

amplitude fluctuations can severely degrade the accuracy of

conical scan, especially if the fluctuations are close to the

rotation frequency of the antenna beam (Sakamoto, 1975).

In contrast, monopulse makes the angle measurement on

the basis of one pulse and will not be degraded by amplitude

fluctuations. Monopulse radar generates the equivalent of

two overlapping antenna beams for each angular coordinate

with the overlapping point being the boresight axis.

In monopulse (Skolnik, 1980), the received signals from

the equivalent beams are combined to form the sum and

difference signals with amplitude fluctuations causing the

same amount of amplitude variation on both these signals









(Sakamoto, 1975). The sum and difference signals are

combined in a phase-sensitive detector to generate an error

signal with the amplitude fluctuations being removed by

forming the ratio of the difference to sum signals. The

magnitude and direction of the error signal determine target

position.

Monopulse tracking systems have the advantage of freedom

from amplitude fluctuations that are common to conical scan,

while the main disadvantage is its complexity. Monopulse

requires at least three, and usually four, antenna beams and

three channels of receiver equipment (Sakamoto, 1975).

1.2 ConoDulse

Conopulse is an angle tracking technique which is a

hybrid of conical scan and monopulse (Sakamoto, 1975;

Sakamoto and Peebles, 1978; Bakut, 1966). Conopulse has the

desirable property of ideally being free from angle errors

due to amplitude fluctuations, while having a simpler

receiver (two receiver channels) and simpler antenna

(conceptually) than monopulse (Sakamoto and Peebles, 1978).

The main disadvantage of conopulse is there is no practical

antenna design at the present time that will allow the

realization of a conopulse system (Sakamoto and Peebles,

1978).

1.2.1 The Conopulse Concent

In conopulse, two independent beams (patterns) are

simultaneously scanned about the boresight axis, instead of

one beam as in conical scan (Sakamoto and Peebles, 1978).









The general conopulse receiving system is shown in figure

1.1. Both received voltages v, and v2 contain full angle

tracking information and could be processed individually as

in conical scan with amplitude fluctuations causing the

amplitudes of the received voltages v, and v2 from each

antenna beam to vary by the same factor.

Bean 1







Processor
A n tenna


Figure 1.1 A general conopulse receiving system. (Source:
Sakamoto and Peebles, 1978.)


The key idea is to form a difference and sum of received

voltages v and v, and then take the ratio of the difference


amplitude fluctuations modulate the received voltages v and

v2 by the same factor, the difference and sum signals will

also be modulated by this same factor. By forming the ratio

of the difference to sum signals, angle errors due to

amplitude fluctuations are theoretically removed.


1.2.2 Conooulse Implementations


In conopulse, a combination of monopulse and conical
scan signal processing methods are used in processing the

received beam voltages to form two error signals. These
received beam voltages to form two error signals. These






5

error signals drive two servomechanisms, which position the

antenna in elevation and azimuth coordinates (Sakamoto and

Peebles, 1978).

Conopulse has two possible implementations with these

implementations differing in which form of signal processing

is done at the intermediate frequency (IF) first. In the

first implementation, monopulse methods are done first with

this implementation being known as a monopulse-conical scan

(MOCO). The second implementation performs conical scan

methods first and is consequently known as conical scan-

monopulse (COMO).

From the literature (Sakamoto and Peebles, 1978), MOCO

is capable of removing wideband target fluctuations with a

maximum spectral extent of half the radar pulse rate, while

the COMO system can only remove narrowband target

fluctuations with a maximum spectral extent equal to the

frequency of beam scanning. For this reason, the MOCO

implementation may be preferred for many angle tracking

applications and this study will only consider MOCO

implementations from this point on.

The functions performed by a MOCO system are shown in

figure 1.2, while the block diagram of a possible MOCO

practical implementation is shown in figure 1.3. In figure

1.2, DUP represents a duplexer and LPF a low pass filter.

A hybrid network forms the sum I and difference A
signals from the received beam voltages v, and v2. Next, the

ratio R=Al/ is generated in the same way as monopulse with













Feed 1 Feed 2


V1 V2



Hybrid




|A











cos -0,1 -t sin OC



LPF LPF



e x e y



Figure 1.2 Functional block diagram for a monopulse-conical
scan implementation of a conopulse system. (Source:
adapted from Sakamoto and Peebles, 1978.)





















Feed 1 I-. l L 1- J


Hybrid Local IAGC En.
Osc. Amp. Det. Phase l
Det.
Feed 2
DUP RF Mixer
Amp. Amp. Conical scan
2 A detectors


I Tnsmittr Monopulse normalization




Figure 1.3 Block diagram of a possible practical conopulse
system of the MOCO type. (Source: adapted from Sakamoto
and Peebles, 1978.)


the effects of target fluctuations being removed by this

ratio. The ratio R is a video pulse train at the radar pulse

rate, which is amplitude modulated by the scanning of the

antenna beams. Since the radar pulse rate is much higher

than the scan frequency, a low pass filter will smooth the

pulse train and produce a voltage RL that is proportional to

the amplitude modulation from beam rotation.

Two conical scan detectors (product device and low pass

filter) process the nearly periodic voltage RL to generate dc

error signals e, and e,. These error signals should measure

the projections of the target on the x and y axes.











1.3 Research Objectives


The first objective of this study is to develop a

practical antenna design, which will provide two independent

scanning beams that are necessary to implement a conopulse

radar. The term scanning is the method for mechanically

moving the feeds and the antenna beams, which results in

angle measurements.

The method of scanning should also reduce undesirable

target amplitude fluctuations. For example, with a single

linear polarization, rotation of the beams will vary

polarization and cause changes in target cross section, which

is polarization sensitive. The fluctuations in target cross

section may cause amplitude fluctuations at the scan rate in

the received beam voltages. These amplitude fluctuations may

result in inaccurate angle measurements.

Previous conopulse literature (Peebles and Sakamoto,

1980a) has derived a lower bound on the variance of angle

tracking error due to noise. However, this literature has

not quantified these angle errors with an actual antenna

implementation. Angle errors due to noise will be quantified

for an antenna design with actual feeds.

Previous conopulse literature has assumed rotationally

symmetric secondary patterns. However, in a practical

system, there is some asymmetry with this asymmetry possibly

having a harmful effect on system performance. The amount of






9

asymmetry will be quantified and the effects of this

asymmetry will be investigated.















CHAPTER 2
POSSIBLE SCANNING METHODS AND DEVELOPMENT OF ANTENNA DESIGN


The lack of a practical antenna implementation is one of

the main reasons that conopulse has not been built into a

hardware system.


2.1 Possible Scanning Methods


Various scanning techniques were examined to achieve the

two independent scanning beams that are necessary for

implementation of conopulse, including nutation of two

squinted beams. The term scanning describes the mechanical

method of moving the feeds. A possible scanning method must

be physically realizable and generate dc outputs that are

proportional to the azimuth and elevation angles. Previous

conopulse literature (Sakamoto, 1975; Sakamoto and Peebles,

1978) has shown that rotation of two beams could be used to

implement a conopulse system. The possibility of nutating

two beams was also examined.


2.1.1 Rotation of Two Sauinted Beams


A possible scanning method is the rotation of two

squinted antenna beams with each beam having the same axis of

rotation as shown in figure 2.1. Figure 2.1 illustrates the

angular positions of the antenna beams and the target in









y Beam I
Nose


K; 1


Target


Beam 2
Nose
Figure 2.1 Geometry of angles that locate beam and target
positions relative to the radar boresight axis.
(Source: Sakamoto and Peebles 1978.)


space with the boresight axis at the origin. The azimuth

error axis is the x axis and the elevation error axis is the

y axis. The position of the target depends on its target

offset angle OT and its target rotation angle O. The beam

nose of each antenna is squinted off the boresight axis by a
constant angle O with the scan position of beam two being

1800 from that of beam one. The angular positions of the

antenna beams are denoted 0, and 02, while the instantaneous

target angles relative to the beam noses are denoted 01 and

02 (Sakamoto and Peebles, 1978). After low pass filtering, a

conopulse system should have dc outputs e. and e, that are

proportional to the projections of the target on the x and y

axes:
e, = kr cos ,
e =T (2-1)
e, = kyOT sin T

where k. and k, will subsequently be known as the dc error

signal constants.









The rotation of two squinted beams has been examined in

the literature (Sakamoto and Peebles, 1978) and it has been
demonstrated that the dc error signals e, and e, satisfy (2-

1).

2.1.2 Nutation of Two Sauinted Beams


Another possible scanning technique is the nutation of

two antenna beams together so that each beam has the same y

coordinate and each beam has a different axis of nutation

(figure 2.2) with the axis of nutation for each beam being

offset from the boresight axis. Nutation of antenna beams

means that polarization will not change as the beams are

scanned and a single linear polarization could be used

without target fluctuations due to polarization rotation

during beam scanning. However, mechanically, it is much more

complicated for nutation (requires a flexible joint) of

antenna beams as compared to rotation (rotary joint simpler

than flexible joint). This conopulse implementation is

physically realizable, but the implementation had to be

analyzed to see if proper dc outputs are generated. The

following analysis shows that this nutation technique does

not produce the proper dc outputs.

Figure 2.2 shows the target location in space relative

to the boresight axis, located at the origin. The x axis is

the azimuth axis, while the y axis is the elevation axis and
the horns are separated by 8,. The target location depends

on its offset angle T8 and its rotation angle OT. The noses

of beam 1 and beam 2 are squinted off the nutation axes by








angles 8eq and ,,q with the angular frequency of scanning being

o),= 2xf,. The rotation angles of beams 1 and 2 are 0#(t) and

2i(t), while the instantaneous target angles relative to beam
positions are denoted by 0,(t) and 02(t).




y (angle)
Benam I Beam 2






C- O, t x (angle)














Figure 2-2. Geometry of angles that locate beam and target
positions relative to the radar boresight axis.

First, the signal RL(t) is calculated. Then, RL(t) is
passed through conical scan detectors to see if dc outputs e,
and e, are generated that satisfy (2-1). This analysis

assumes gaussian antenna beams. The one-way voltage gaussian








pattern is defined as


G(0) = G exp -- 0 (2-2)

where G, is the one-way voltage maximum, a2 =2.776/ ,2 with ,B

being the 3-dB beamwidth of the pattern, and 0 is the angle
from the beam maximum.

The instantaneous target angles were obtained using
right triangles and Pythagorean theorem:


12 (t) = {Oi sin(w,t) 0, sin( )}2 + + 0, cos(ot) 06 cos(r)}2
S(2-3)
622(t) = {Oq1 sin(wos) 0 ( + {+2 l + 6q1 cos(ot) 0T cos(^)}


The received signals for gaussian beams v,(t) and v,(t) have the
following form:
v,(t) = c p(- e,(t
(2-4)
v2(t) = c,exp -0(t)2

where c, depends on the parameters of the radar equation.

The difference signal A(t) and sum signal X(t) will have the
following forms respectively:


A(t) = v(t)-v2(t)= c, exp 2 (t)- exp 022 (t)
( / 2-5)
(t) = vi(t + 2(t= ci exp( 2(t) + exp 22(t)








After expansion of terms, the final signal RL(t) will
form:


have the


A(t)_
RL(t)=- )
=(t)


Kexp (-2{-,, cos(ct) + 0T, cos( )x

exp {+ .O, cos(ot) T0r cos(Or)J
L- ( 2


exp(- 2I-O"q cos(),t) + O, cos(or)c J

+ exp2- {+6,0 cos(cot) 0e0e cos(OP)J


where the factor K equals


K(=c expF 2 e1 2 + 2- +e 2 2q,q r sin(m,t)sin( OT)
K= c, exp -a2 lcos(o0tc
-20,i,ocos( ,t)cos(Or)


(2-6)








(2-7)


The signal RL(t) can then be simplified to the following form:


RL(t)= tanh -{-,~0 cos(,t) + 0, cos(4r)}.


(2-8)


Now, the signal RL(t) is passed through conical scan
detectors to see if proper dc outputs are generated. For
small angles, the signal RL(t) can be approximated as


RL () ---0,e0 cos(ot) + e,,T cos(T))}


(2-9)


After multiplication of RL(t) by cos(o,t), the resulting signal

ecos(t) equals

ecos(t) = -2{-0,Oq, cos (ct) + 0,0, cos(0r)cos(Ct)} (2-10)


I









Using a trigonometric identity, ecos(t) can be expressed as


ecos(t)= -- { 2 - Q1cos(2ot) + OT cos(o,)cos(O),t) (2-11)
2 I _2 2 1


After passing through a low pass filter with cutoff frequency
less than o,, the azimuth dc error signal e, equals



ex=- 2 2(2-12)


The dc error signal e, is not proportional to the azimuth

angle and (2-1) is not valid.
Similarly, after multiplication of R,(t) by sin(m,t),

the resulting signal esin(t) equals

esin(t) = OX{-0, q, cos(ot))sin(ot) + 0qOT cos(,)sin(o,t)} (2-13)



Using a trigonometric identity, esin(t) can be expressed as


esin(t)=- a---~sin(2 o,t) + O cos(,)sin(cot) (2-14)



At the output of a low pass filter with cutoff frequency less
than C),, the elevation dc error signal e, equals

e,=0 (2-15)


The dc error signal e, is not proportional to the elevation


angle and (2-1) is not valid.









Since nutation will not generate dc outputs that are

proportional to the azimuth and elevation angles, nutation of

two squinted beams can not be used to implement a conopulse

system.


2.2 Selection of Circular Polarization


Polarization is a property of an electromagnetic wave,

which describes the time varying amplitude and direction of

the electric field vector. More specifically, polarization

is the figure traced as a function of time by the extremity

of the electric field vector at a fixed location in space

when viewed in the direction of propagation. In general, the

figure that is traced out as is an ellipse and it is traced

out in a clockwise or counterclockwise direction. When the

ellipse becomes a line or circle, the polarization is known

as linear and circular polarization respectively.

In many radar systems, linear polarization is used,

because it is the easiest to implement. However, with linear

polarization, rotation of the beams would rotate the plane of

polarization and cause undesirable target amplitude

fluctuations at the scan rate. Target fluctuations at the

scan rate are the most undesirable fluctuations and can cause

a slowly varying dc error (Sakamoto, 1975).

Theoretically, the monopulse normalization (part of

conopulse signal processing) removes these target

fluctuations. However, in a practical system, target

fluctuations may not be completely removed and the result can









be inaccurate measurements in either e, and/or e, (Sakamoto

and Peebles, 1978). In addition, these fluctuations could

limit the dynamic range of the monopulse normalization

network (Sakamoto and Peebles, 1978).

Circular polarization can be decomposed into two

orthogonal linearly polarized waves, which are 90 degrees out

of phase. Beam rotation will vary the orthogonal

polarizations and the radar cross section will fluctuate for

each polarization. However, circular polarization is

expected to have less amplitude fluctuations than linear

polarization, because the received signal depends on both

polarization components, which is the reason that circular

polarization is preferred for transmission.


2.3 Proposed Antenna Design


An antenna design (figure 2.3), found during this study,

suggests one possible way to simultaneously rotate two

squinted antenna beams. The path from the radio frequency

(RF) unit to feed 1 will be described. The path involving

feed 2 behaves in manner similar to path 1. The energy flows

from rectangular TE10 waveguide to the rotary joint with the

rotary joint consisting of transitions in waveguides from

rectangular TE10 to circular TM01 and back to rectangular

TE10 (Fink, 1947; Rizzi, 1988). In a rotary joint, it is

necessary to have a circular waveguide with a rotationally

symmetric mode, because the polarization of the rotating part

must be the same as the polarization of the fixed part.












rotary joint

'4--- feed support


Figure 2.3 Proposed Antenna Design

After passing the rotary joint, the energy passes up the

rectangular waveguide and changes direction by passing

through two rectangular waveguide bends. Next, the

rectangular TE10 waveguide is transformed to a circular TEll

waveguide via a mode transducer. Finally, the energy passes

from the circular TE11 waveguide to circular waveguide horn,

where the energy is radiated toward the paraboloid reflector.









As previously noted, the path from the RF unit to feed 2 is

similar, but the rotary joint is located above the

paraboloidal reflector as opposed to below it.

The principal parts of the antenna are a paraboloidal

reflector and two "offset" waveguide feeds. The term

"offset" means that the feeds are laterally displaced from

reflector axis.

The proposed antenna design has the following

advantages:



Making bends in rectangular TE10 waveguide is a

standard technique.

Mode transducers exist which can convert

rectangular TE10 to circular TE11 (Rizzi, 1988).

Since TE10 and TEll are the dominant modes in

rectangular and circular waveguides, respectively,

there will be less change of mode conversion of a

desired mode to undesired mode(s).


There is the following disadvantage of the proposed

antenna design:



Since the rotary joint requires two waveguide

transitions, it will be bulky and will cause more

aperture blockage than a rotary joint made only of

circular waveguide.














CHAPTER 3
FEEDS AND CALCULATION OF RADIATION


There are various types of horn antennas including

pyramidal horns, sectoral horns, and conical horns. The

conical horn (figure 3.1) is capable of handling any

polarization excited by the dominant TE11 mode since it has

axial symmetry. In particular, the conical horn radiates

circular polarization very effectively (Johnson and Jasik,

1984). In addition, the conical horn achieves radiation

patterns with more symmetry than pyramdidal horns or sectoral

horns. For these reasons, conical horns are chosen as the

feeds.

The dual mode and hybrid mode horns, shown in figures

3.1b and 3.1c, can simultaneously achieve radiation patterns

with a high degree of rotational symmetry and also low levels

of cross polarization in the intercardinal planes (Johnson

and Jasik, 1984).

Different conical horns are considered in this study,

because system performance will differ depending on the type

of horn.

This study will use aperture field methods to calculate

the radiated fields of the feeds. With this method, aperture

field techniques are used to compute the radiated fields from

the fields across the aperture.












TEnI TE,,

(o)




TE TEI/TM11

(b)




TE -- -- HE,


(c)
Figure 3.1. Conical horns. (a) Dominant. (b) Dual-mode. (c)
Hybrid mode. (Source: Johnson and Jasik, 1984.)


In this study, the investigator is primarily

concerned with the part of the feed pattern that illuminates

the reflector and the calculated radiation will only need to

be accurate for the main beam and nearby sidelobes.

In this chapter, all expressions for the radiated fields

of the feeds will pertain to the far field of the feed. The

coordinate system for the feeds is shown in figure 3.2.


3.1 Dominant Mode Horn


The dominant mode horn is excited by the dominant mode

TEll and radiates the dominant mode. Since there are

different boundary conditions on the E and H planes, this

horn has a large amount of asymmetry in the feed radiation.































Figure 3.2. Radiation coordinate system. (Source: adapted
from Potter, 1963)


This horn should be large enough to support the dominant

mode, but small enough to prevent the existence of higher

order modes (Johnson and Jasik, 1984):



1.84

where k=21r/A, A is the wavelength, and a is the horn

aperture radius.


When the horn semi-flare angle Of is less 30, it has

been shown that the radiation characteristics can be








calculated from Lommel functions as (Narasimhan and Rao,
1971b)


Ee = C exp(-jkR) *[U,(2s,d)+jU2(2s,d)]sinO

E# = exp(-jkR) [U,(2s,d)+ jU2(2s,d)]cos0cos 0 (3-2)
R

where Ci is a normalization constant, R is the distance from
the feed in spherical coordinates, U.(2s,d) is the Lommel

function of two variables, s=nra2//IL, d=kasine, and L is the
axial length of the horn from the cone apex.


The Lommel functions can be in integral form or series form.
The integral form of the Lommel functions is (Hu, 1960)

1
U,(2s,d) = 2sf Jo(d)cos[s(l '2)]d
0
0 (3-3)
U2(2s,d) = 2sf Jo(d)sin[s(1- 521)]4
0
where Jo is a zero-order Bessel function of the first kind.


The series form of Lommel functions is (Hu, 1960)


U (2s, d) = Jd (d) J (d) + ) 5(d)-...
(3-4)
U, (2s, d) = d 2 J(d) J4(d) Y+ (d)-...

When calculating the radiation characteristics of the
dominant mode horn, this study used the integral form of the
Lommel functions, because it is exact. The series form will







25

have some error when the series does not have an infinite

number of terms. However, highly convergent series can be

obtained for the series form of the Lommel functions.


3.2 Dual Mode Horn

By adding a higher order mode (TM11) to the dominant

mode, it is possible to obtain a radiation pattern with a

high amount of rotational symmetry and also a low amount of

cross polarization (Johnson and Jasik, 1984).

The modes must have proper relative phases and

amplitudes to cancel the electric field at the boundary of

the aperture (Turrin, 1967). At the aperture boundary, when

the modes are properly phased, the TM11 mode will cancel the

0 component of the magnetic field due to TEll (Johnson and

Jasik, 1984). This cancellation causes the # component of

the magnetic field to vanish as well as the # component of

the electric field. Under this condition, there is nearly

perfect symmetry of the primary pattern and low levels of

cross polarization.

In the original dual mode horn (Potter, 1963), the TM11

mode was generated by a step change in the radius of the

waveguide as shown in figure 3.3. This horn will be

subsequently referred to as the Potter horn. For this horn,

let the input waveguide before the step change be denoted b

and the waveguide after the step change be denoted a. The

input waveguide b should be sized to allow the TE11 mode to














0 0

DB DB
20 20


40 nA
900 0 900
TEl1 TMI I MULTIOua









Figure 3.3. The dual-mode conical horn. (Source: Love,
1976c.)



propagate but not the TM11 mode (Johnson and Jasik, 1984):



1.84

The waveguide after the step change should be large enough to

allow TM11 to propagate, but small enough to prevent the

propagation of TE12 (Johnson and Jasik, 1984):


3.83

(3-6)






27

The radiated electric field components EE, Efrom the

feed are (Potter, 1963; Johnson and Jasik, 1984)


ELCos(= (+11cose -a J,(kasinO) e-R.
k KE kasin R
ksinfJ ka

(3-7)

E#=(' +cosO Ji' (kasinO) e-j u
E k ksine 9 R
K,,. )

where K,,a is 1st root of J1' and equals 1.841,

Pfl, =k2-K2n KlEa is the 1st root of J1 and equals 3.832,
i, = =k2-K21E J, is the Ist-order Bessel function of the
first kind, J,' is the 1st derivative of J, with respect to

its argument, a equals an arbitrary constant defining the
relative power in TE11 and TM11 modes.

Potter (1963) states that a value of a equal to 0.653
will make the E and H plane half power beamwidths equal and
cause the phase centers to be coincident. The Potter horn is
not broadband, because the TE11 and TM11 modes have different

phase velocities due to the different boundary conditions on
the E and H planes (Love, 1976a).
The use of a step discontinuity near the horn throat is
not the only method to convert TEll to TM11 energy in a horn.

Turrin (1967) has designed multimode horns without a
discontinuous step with these horns having larger bandwidths
than the Potter horn. Turrin obtained the radiation
characteristics of these horns based on Potter's results.











3.3 Hybrid Mode Horn


The desirable properties of rotational beam symmetry,

low levels of cross polarization, and low sidelobe levels can

be achieved in a conical horn that transmits the hybrid HEll

mode. The hybrid mode horn has the polarization

characteristics of a Huygen's source as shown in figure 3.4.

E-DIPOLE M-DIPOLE RESULTANT






APERTURE FIELDS

Figure 3.4 The Huygen's source with ideal feed polarization
characteristics. (Source: Love, 1976c.)



The hybrid mode is a mixture of TE11 and TM11 and can be

expressed as .TE11 + y TM11, where y denotes the mode-

content factor (Johnson and Jasik, 1984). To support the

hybrid mode, the inner surface can be an anisotropic surface,

which may be obtained by cutting circumferential grooves into

the surface.

The hybrid mode horn has equal boundary conditions on

the E and H planes, which causes the TEll and TM11 to

propagate with the same phase velocity (Love, 1976a). The

equal boundary conditions also cause equal E and H plane






29

beamwidths and a rotationally symmetric radiation pattern

(Simmons and Kay, 1966).

For equal E-plane and H-plane patterns and zero cross

polarization, the size of the feed should be (Knop and

Wiesenfarth, 1972)

ka>2n (3-8)


With the horn flare angle less than 30 degrees, the

radiation characteristics of the hybrid mode horn are

(Narasimhan and Rao, 1970b)


E- C1 exp(-jkR) (1+ cos 9) (b3W(2sd)+ b2Wo2(2s,d) in
E= |sine
R 2 +b4W'(2s,d) + boWo(2s,d))
(3-9)
E q, exp(-jkR) (1+ cos 9) (b3W03(2s,d)+ bW,2 (2s,d)
R 2 + +b1W,'(2s, d)+ bWo (2s, d))



where bo =-0.007259, b = 0.6525, b2 =0.2708, b3 = 0.08399, and the

function Wo"(2s,d) is defined as



Wo"(2s,d)= (1- 2) J(d)ej)('-) (3-10)
0


The function Wo"(2s,d) can be expressed in terms of the Lommel

function and its derivatives (Hu, 1960)


Wo(2sd(2s, d) (2/ j)" (2s, (3-11)
Sd(2s)" 2s (2s)" 2s









Since the field components E,,E, are identical, the

radiation will be rotationally symmetric.


3.4 Dielectric Rod


Dielectric aerials produce a single lobe radiation

pattern directed along the feed axis and the directivity of

the feed is proportional to the length of the dielectric

aerial (Kiely, 1953). Two of the more important types of

dielectric aerials are (Kiely, 1953)



solid dielectric rod

dielectric horn, which is similar to conventional

metal electromagnetic horn, except metal walls are

replaced by dielectric walls


The basic principle of dielectric aerials is to

transform the launcher aperture into a larger aperture

(James, 1972) by transmission of a surface wave. Dielectric

aerials transport the surface wave power from the launcher to

the radiating aperture. The bounds of the surface wave will

be much larger than the physical aperture (James, 1967).

For conventional feeds, the size of the aperture is

determined by the physical aperture of the metal horn. With

the dielectric aerials, the effective size of the aperture

will be determined by the surface wave, which exists outside

the dielectric rod.







31

The proposed dielectric aerial (figure 3.5) consists of

a metal wave guide operating in TEll mode which will excite a

HE11 mode surface wave in the dielectric rod. In general,

there are three sources of radiation: end aperture radiation,

taper radiation, and launcher radiation. The proposed design

consists of a nearly optimum taper so that taper radiation is

negligible. For the design shown in figure 3.5, the end

aperture radiation is 87.2% and the launcher radiation is

12.8%.



0-7?o O 448go 0-395Ao
01- 1
__ ___ 0-33X0

48L o 2Xo 2Xo I 2Xo
TaDer radiation negigiole
Launcher radiation 12'8%
End aperture radiation 87-2%

Figure 3.5 Dielectric rod antenna with optimized profile.
(Source: adapted from James, 1972.)



The approach used in this research to characterize the

radiation characteristics of the end aperture of the

dielectric rod is based on Huyghens Principle and elementary

ray theory (Kiely, 1953). The mathematical expression

contains the parameters rod length, rod diameter, and

wavelength. This approach is applicable to dielectric rods

excited in the HE11 mode. Using this approach, the radiation








pattern from the aperture at the end of the rod can be
expressed as (Kiely, 1953)


(K 1)sin (K-cos )
E9 = I cos( -sin 0) sin 0
(K-cos0)sin -R(K- ) "0

nL '(3-12)
(K-1)sin (K- cos)
o _____d______
=-- cos(--sin 0) cos
(K-cosO)sin IL-(K-1) o



where K is a constant equal to 1.0, L/Ao is rod length in

terms of free space wavelengths, and di/A is the diameter of

the dielectric rod in terms of free space wavelengths.


The constant K is chosen as 1.0 to match the rod to

free space (Kiely, 1953). When the rod is matched to free

space, the level of the sidelobes and back radiation is

decreased.
The launcher radiation can be considered the same as

from the dominant mode horn (James, 1972) and was given in

(3-2)


Ee = C exp(-jkR) [U,(2s,d)+ jU,(2s,d)]sin 0
R

E = xp(-kR) [U, (2s, d) + jU2(2s, d)] cos cos 0


The total radiated field components can be obtained by

summing the end aperture radiation and launcher radiation







33

with the appropriate weighting (Brown, 1957). When summing

with the appropriate weighting (87.2% end aperture radiation

and 12.8% launcher radiation), the maximum values of the end

aperture radiation and launcher radiation must have the same

values. Since the expression for the end aperture radiation

given in (3-12) has a maximum of one, the normalization
constant C, in (3-2) should be chosen so that launcher

radiation has a maximum of one.


3.5 Diffraction-Limited Feeds and Dielectric Rod


A feed, which is diffraction-limited, has radiation

properties that are governed by the size of the aperture with

radius a. With a larger aperture, diffraction-limited feeds

have more directive radiation properties. In this study,

diffraction-limited feeds include: dominant mode horn, dual

mode horn, and hybrid mode horn. The feed patterns of these

types of feeds are shown in figures 3.6-3.8.

Since the radiation properties of the dielectric rod are

largely independent of the rod diameter d (figure 3.9), the

dielectric rod is not diffraction limited. Rather, the

radiation properties of the dielectric rod depend on its

length L as shown in figure 3.10.









































10 20 30 40 50

9 (degrees)


60 70 80


V .-- ... _


10 20 30 40 50

0 (degrees)


60 70 80


Figure 3.6 Effect of feed size on feed patterns of dominant
mode horn: (upper) E-plane pattern, (lower) H-plane
pattern.


.......... ............ ........... ................. ..... ...................




........... ......... ............ .............
--


5-t


I--
-10

--10


" -20
ve


U


-10
5-


1=7
-30


i_. -40

50


. . . .. S

... ... ... ..
I I I ~I I I
......... ............................ .......`s. ........


. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . ....

. . . . . .. . . . . . . . . . .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..


I


n





,nn


ZILJ


- J
























-10 ==-- -- -
_ 10 ........... ............ ........... ............. ........ ............. ...........
-20 ........... -- ................. .....-- .... -...... ... .... .. ........... ........

-30 ............. --


-40 v---- - ---- -:--- tT ------- --: t.................... ...

-50


20 30 40 50
9 (degrees)


60 70 80


10 20 30


40 50
0 (degrees)


60 70 80


Figure 3.7 Effect of feed size on feed patterns of dual
mode horn: (upper) E-plane pattern, (lower) H-plane
pattern.


I--
Cq


0


A


-10

-20

-30

-40

-50
0


................. ........ ..........
.. .. .. .. ...... . .... ... ........ .. .... ................ .... ...... ..
.. ..... ... ............- %"*. ... o .. %. .

S. . .. . . . . . .. ............ ... ..........: ...... ...... ...... ... ... ..... .. ... .


I i i I


. = i I



































5- -15 .... -............ ............ ........... ............ ...........
-25----^-* ***-----**--- -----,-----,------,---- --...-----,-----.......


S-20
-10 .... ..... ............ ... ..... ... ............ ........... .......................





-30 .- :-:..-*** -. **** **- ***\ *-- .. ..........


-35 ........... - *** **............ **- ..........-- .... . .... .......... ....... ... ...........
-35 ........... ........... ........... ............ .... ...... ............ ...........


0 ........... ... ......... ........... .... .. ........ ..... .. ...... ..:.. .


-35 .. .. .......... .......... . ............ ...... .





-45
0 10 20 30 40 50 60 70 80
6 (degrees)




a =1.0 1

--- a=1.4






Figure 3.8 Effect of feed size on E-plane and H-plane feed
patterns of hybrid mode horn.
















0

-5

-10

S-15

1_ '-20

-25
0




0


-10


Ir -20

-30
-30
1 -1


60 70 80


40 50
0 (degrees)


Figure 3.9 Effect of dielectric rod diameter on feed
patterns of dielectric rod: (upper) E-plane pattern,
(lower) H-plane pattern.


10 20 30 40 50
8 (degrees)


.. . . .. . ............ .. .. ... .... ... .. ... . . . ..... . . ... . . .



. . . . . . . . . . . . . . . . . . . . . . . . . . .... . ..:. .

. . . . . . . . . . . . . . .


d = O.33A0,

L=1O.8A0

d=O.66A0,

L=1O.8A.0


c


-


-























0 (degrees)


-20 . . . ........ I...." .... ... ....... ............ ........


2... ...:.............................. ........
20 ........ ...................... .. .............. .............................


.


0 (degrees)



d = 0.331o,
L=10.81A

d =0.33Ao1
L=21.6A0





Figure 3.10 Effect of dielectric rod length on feed patterns
of dielectric rod: (upper) E-plane pattern, (lower) H-
plane pattern.















CHAPTER 4
CALCULATION OF RADIATION FROM PARABOLOIDAL REFLECTOR


For some of the analyses in this study, it will be

necessary to calculate a reflected (from paraboloid)

secondary field based on knowledge of the primary feed

radiation characteristics and the reflector configuration.

In this study, there are two methods for secondary field

calculation.

The first and major method is the aperture distribution

method (Balanis, 1982; Collin and Zucker 1969b) combined with

analysis by Lo (1960). The aperture distribution method

presented in Balanis, Collin and Zucker assumes the feed is

located at the focus of the paraboloidal reflector. The

effect of laterally displacing the feeds (from focus) is

based on analysis by Lo. The aperture distribution method

(presented in Balanis, Collin and Zucker), combined with

analysis by Lo, will subsequently be referred to as simply

the aperture distribution method.

A second technique for secondary field calculation is to

use a computer program, which is a modification of the

Numerical Electromagnetic Code (NEC) Reflector Antenna Code

(Rudduck and Chang, 1982; Chang and Rudduck, 1982) and will

subsequently be referred to as the NEC computer program.









For much of the analyses in this study (e.g. secondary

pattern asymmetry), the aperture distribution method will be

used. The NEC computer program is used mainly to verify the

accuracy of the aperture distribution method.


4.1 Surface Geometry


One of the most popular radar antennas is the

paraboloidal reflector, which will be subsequently referred

to as a paraboloid. Some properties of the paraboloid and

its surface geometry will be stated. Next, some important

relationships for the analysis of paraboloid reflector

systems will be given.

The surface of a paraboloid is generated by the rotation

of a parabola about its axis. A source of energy called a

feed directs energy toward the paraboloid (figure 4.1) with

the feed being located near the focus of the paraboloid.




Poroabolic
Surface


Vertex
or Focus Beam axis







Figure 4.1. Parabolic-reflector antenna (Source: Skolnik,
1980.)











The paraboloid has the following desirable properties:



all rays generated from the focus are reflected by

the paraboloid surface parallel to the reflector

axis

the distances traveled by all rays from the focus

to the reflector surface and reflected back to a

plane perpendicular to the reflector axis are equal



These properties of the paraboloid cause a point source

of energy directed from the focus to generate a reflected

plane wavefront of equal phase.

The geometry of a paraboloid of diameter D and focal
length f (Balanis, 1982) is shown in figure 4.2.


So
I
I------.-.----
i---~-- !I


Figure 4.2. Two-dimensional configuration of a paraboloidal
reflector. (Source: adapted from Balanis, 1982.)









Using the second property



OP+PQ= constant= 2f (4-1)

From figure 4.2


OP = r'
(4-2)
PQ= r'cos' (4-2)

Upon substitution of (4-2) into (4-1), (4-1) can be expressed

as



r'(1+cos0')= 2f (4-3)



or


r'= 2f 0' o (4-4)
1+cos


Using a trigonometric identity, r' can also be expressed as


r'=fsec2 P 5) O0 (4-5)



Equations (4-4) and (4-5) are important equations, because

they relate the distance r' traveled by a ray to the

reflection point to the angle 0'.

The relationship between the subtended angle of the
reflector 80 and the f/D ratio is important for the analysis








of paraboloidal systems. From figure 4.2


o= tan(_D/2
Oo=,. L--t -


(4-6)


By geometry (Balanis, 1982), zo can be expressed in terms of

f and D as


S=f(D2) f (4-7)
4f 16f
Upon substitution of (4-7) into (4-6), the subtended angle of
the reflector equals


e f tan(f
S= tan-' 2 D

(fDY 16,


(4-8)


Figure 4.3 illustrates the relationship between the subtended
angle of the reflector and the f/D ratio.


80







01)
o 20
40 ,-- - ^.. ...i '............. ....... ............................... .



0 0.5 1 1.5 2 2.5 3
f/D

Figure 4.3 Subtended angle of paraboloid reflector as
function of f/D ratio.









The relationship between 0'and the radial distance p can

be derived in the following manner. From figure 4.2


p = r'sin O' (4-9)



Upon substitution of (4-5) into (4-9)


sin e'
p=f (4-10)
cos 24


Using the trigonometric identity


0'
sin 0'= 2sin -cos-- (4-11)
2 2


Upon substitution (4-11) into (4-10), the following

expression is obtained


0'
2sinm-cos-
f 2 2 = 2f tan (4-12)
2


Equation (4-12) can also be expressed as


'= 2 tan-'21 (4-13)
12f/









Another important relationship is the relationship
between r' and p. Using a trigonometric identity, (4-5) can

be expressed as

= f 1+tanI 2] (4-14)



From (4-12), the following expression is obtained


tan( --) = (4-15)
.2) 2f


Combining (4-14) and (4-15), r' can be expressed as



r'= f 1+[-) (4-16)



4.2 Aperture Distribution Method


In the aperture distribution method (Balanis, 1982,

Collin and Zucker, 1969b), the reflected field is calculated

over a plane orthogonal to the reflector axis. In most

cases, the plane is taken as passing through the focus and is

known as the aperture plane as shown in figure 4.4. The

reflected field is usually obtained by ray tracing methods

(Balanis, 1982). Next, equivalent sources are generated over

the aperture plane, where, the equivalent sources are usually

assumed to be zero outside the projected cross-sectional area

of the reflector upon the aperture plane. Finally, the

secondary radiated fields are calculated from equivalent










sources by aperture methods. The effect of laterally

displacing the feed will be based on analysis by Lo (1960).


/'"* J I Aperture plane

/ I I


(r'. o')


S/ /
1..: ," ,, ',,


P(x. y. :)


J




(projected cross-
sectional area of
reflector-
aperture plane)


Figure 4.4. Three dimensional geometry of a paraboloidal
reflector system. (Source: Balanis, 1982.)


It should be noted that the notation for the coordinate

system in chapter 3 is different than the notation for the

coordinate system in chapter 4. The coordinates R,9,# in

chapter 3 become r',e',t' in chapter 4.

Balanis (1982) states the aperture distribution makes

the following approximations:

* The current density is set equal to zero on the shadow

size of the reflector.


The discontinuity of the current density over the

reflector rim is not taken into account.








Aperture blockage by the feed and direct radiation from

the feed are not taken into account.

Balanis (1982) states that these approximations give
accurate results over the main beam and nearby minor lobes.

For more accurate results in all regions, especially far

minor lobes, Balanis states geometrical diffraction methods
can be used.

For purposes in this study, the magnitude of the
electric field will be frequently computed. The square of
the magnitude of the electric field radiated from the feed
equals (Balanis, 1982)


(,r'., ',, ) E. eI, -', o,, ) +JE,.(r', o%. ,)'
'I E,.o(',' + E,.( ,0 ')1] (4-17)


The magnitude of the electric field normalized by its maximum

value is the feed voltage pattern, while the square of the
magnitude of the electric field normalized by its maximum
value is the feed power pattern. It should be noted that the
voltage pattern has a phase and the sidelobes can have
different phase than the main lobe.

Consider a y-polarized feed located at the focus of a
paraboloid with feed voltage pattern JG/(f,0'). The feed

power pattern G,(0',0') is defined as


G,( ,') = I O(' O,')j / (0Io,)

{IEO. (0', O')21 + E, (', ( ') 2} / I(o,o0)I (4-18)








The feed patterns of the different horns are obtained from
the radiated electric field components E,,E,. given in chapter

3. The incident electric field with a direction normal to

the radial distance, can be expressed as (Balanis, 1982)

-e'
E.(r',0 ',')= ,C (g', ') (4-19)

C 7=[, J- (4-19a)
27r
where e is a unit vector orthogonal to ^ and parallel to the
plane generated by and ai as shown in figure 4.5, 7 is the

intrinsic impedance of free space, and P, is the total

radiated power.


y



Sa









X

Figure 4.5. Unit vector alignment for a paraboloidal
reflector system. (Source: adapted from Balanis, 1982.)









Before finding the electric field over the aperture E ,

the electric field E, at the reflection point r'is first

obtained. The reflected field is


e
Ef (e 0')r' (4-20)

where ^ is a unit vector representing the polarization of the

reflected electric field.


It can be shown (Collin and Zucker, 1969b) that


Sa, sin cos 0' (1 cos ') a,(sin2 cos + cos2 ('s)
er= (4-21)
1 si sin 2 '


where a, and a, are unit vectors in the x and y directions as

shown in figure 4.5.

On the aperture plane, the electric field equals (Collin

and Zucker, 1969b)


S-jkr'(1+cos9')
E, C = 2rC G,(',-')e (4-22)



By comparison of (4-22) and (4-20), it is seen that the

phase of the electric field over the aperture plane is

different than the phase of the electric field at the

reflection point due to the extra distance traveled by the

ray. The amplitude of the electric field over the aperture

plane is the same as at the point of reflection. This fact

occurs, because the spreading of energy only takes place from









the feed to the reflector (Collin and Zucker, 1969b). Since

the rays are collimated after reflection, there is no more

spreading of energy back to the aperture plane.

The electric field over the aperture plane can also be

expressed as


^E =E. + a,^E, (4-23)


where E, and E, denote the x- and y- components of the

reflected electric field over the aperture plane.


Using (4-21),(4-22) and (4-23), the x- and y- components of

the reflected electric field equal


E. -jkr'(I+cos 9') sin cos -cos0') (4
E,=cJ e sin cos '(1-cos9')
E = C1Gf(0',9)e *-* r ( (4-24a)
r' 1-sin 0' sin2 '
e- kr'(+c ') sin2 O' cos 8' +cos2 0'
E,=-C GF,(,f')f? ) - 2 (4-24b)
Ir'I -sin 8'sin2

The x- component for a y polarized source arises due to

depolarization from the reflector. Depolarization from the

reflector becomes very small with large size reflectors and

large f/D ratios (Collin and Zucker, 1969b; Balanis, 1982).

From the reflected electric field components on the
aperture plane (E, and Eoy) as given in (4-24), the

equivalent sources are obtained over the aperture plane

(Balanis, 1982, chapter 13). The radiated electric field

components can be calculated from the equivalent sources

(Balanis, 1982, chapter 13). After conversion of Balanis's








equations to cylindrical coordinates (paraboloid reflector
has circular aperture), the radiated electric field
components are


Eo =- (1- cos 0)J (+E, cos + E, sin
4Sr (4-25a)
x eik[psin cos()' -) p9pd9'


E- =ke (1-cosO)J (-E sin 0+ EY cos)
sO (4-25b)
Xejk[psincos(#)'-)lpapap'


where So is projected cross-sectional area of reflector

aperture plane as shown in figure 4.4.


It should be noted that E. and E, are in terms of the

coordinates r',6',%', while the integration is over the
coordinates p,#'. The conversion of r'in terms of p was given
in (4-16), while the conversion of O'in terms of p was given

in (4-13).
The square of the magnitude of the secondary electric
field radiated from the parabaloid equals (Balanis, 1982)


IE(r, 0, )f2 = IE(r, 0,0)12 +E, (r, 0, )2
[1EO(, O2 + E (o, 02 (4-26)



The magnitude of the electric field normalized by its maximum
value is the secondary voltage pattern, while the square of









the magnitude of the electric field normalized by its maximum

value is the secondary power pattern. In a similar manner to

the feed voltage pattern, the secondary voltage pattern also

has a phase.

4.2.1 Effect of Lateral-Feed Displacement


The material presented for the aperture distribution

method in Balanis (1982), Collin and Zucker (1969b) has

assumed that the feed is located at the focus. For the

proposed antenna, the two feeds will be laterally displaced

from the focus. The effect of laterally displacing the feed

is based on analysis by Lo (1960).

The assumption will be made, as by Lo, the magnitude of
the aperture electric field E, for a laterally displaced feed

is the same as E for a feed located at the focus. If the

lateral displacement is small compared to the focal length,

this assumption is reasonable.

Lo shows that laterally displacing the feed will change

the phase term of (4-25). Lo derives the phase term for a

feed displacement from the focus being small compared to
focal length f. The radiated electric field components with

the laterally displaced feeds are still calculated in terms

of the coordinate system in figure 4.4.

Based on Lo's analysis, the phase term for a feed of
lateral displacement d, in the x direction is


k[psin cos(' -)) d, sin cos f']


(4-27)










The effect of the "new" phase term is beam shift and beam

degradation. Ruze (1965) has suggested that this phase term

ignores field curvature and astigmatism.

Upon substituting the phase term (4-27) into (4-25), the
radiated electric field components for a laterally displaced

feed change from (4-25) to


Ee =k (1- cos O)f (+E cos0 +E sin
s, (4-28a)
x e [psin cos(- #)-d, sin E'cos#' pd


E =jke (1- cos )-f (-E sin +E, cos )
s, (4-28b)
x ejk[psin Ocos(#'-)-d, sinO'cos 'pap


Since the feeds are laterally displaced from the focus,
the secondary fields radiated from the reflector were

calculated using (4-28).

4.2.2 Numerical Integration


The aperture distribution method requires computation of

double integrals. The majority of the work in this study was

done with the programming language MATLAB. MATLAB can do one

dimensional integrals, but not two dimensional integrals.

Therefore, the double integral had to be calculated by

numerical integration techniques.








There exist a variety of numerical integration
techniques including trapezoidal rule and Simpson's rule

(Atkinson, 1989). This study utilized Simpson's rule,

because it is robust and has better accuracy than the

trapezoidal rule.

Simpson's rule is a well known method for evaluating the

integral

b
I(f)= f(x)dx (4-29)
a


The assumption will be made that the interval [a,b] is always

finite. Simpson's rule breaks the integral in equation (4-

29) into a sum of integrals over small subintervals.

After some derivation (Atkinson, 1989), the composite
Simpson's rule approximation I1(f) is obtained:


I.(f)=[fo +4f, + 22 +4f3 +2f4+...+2f,2 +4f,~_ + f] (4-30)


where h=(b-a)ln with n being the number of points, f0
denotes f(x0),f, denotes f(x,), etc..


The error E,(f) with Simpson's rule (Atkinson, 1989)

equals


E(f) h (b a)f(4)(rl) r [a,b] (4-31)
180
where f(4) denotes the fourth derivative of a function.










4.2.3 Accuracy of Simpson's Rule


The Simpson's rule in this study used 235 subintervals.

It is known that there will be some error with Simpson's rule

due to the finite number of subintervals.

For the purpose of verifying the accuracy of Simpsons's

rule, it will be useful to express the phase term of (4-27)

in an alternative form. Based on analysis by Ruze (1965),

the phase term (4-27) can be expressed as


Akpcos(4'-a) (4-32)



In equation (4-32), the parameters A and a may be determined

as


A2 =u2_ 2uu COS+ u (4-33a)
M(p) M'(p)


usin
tana= (4-33b)
ucos0 -u,/ M(p)


where u,, u and M(r) are defined as


u, =- = tan , u = sin M(p) = 1+ (p / 2f)2

Upon substitution of the phase term (4-32) into (4-28),

the radiated electric field components for a paraboloid of








radius a are


Jke-yk7 2xa
E =ke-i(1 -cosO)J (Eacos + E, sin4)
00 (4-34a)
x ejkos(#'-a) pap'

E, = (1-cos0) f (-Esin +Ecos )
00 (4-34b)
Xe jkpA''-) pcpa


For a rotationally symmetric E, and E, (4-34) reduces

to

E = jke-jr (1-cos6)j(Eacos0+E0 sin@) Jo(krA)pap (4-35a)
4rr 0



E = -r (1-cos0) (-E,,sinE0+E cos0) Jo(krA)p9p (4-35b)
0



MATLAB can do one dimensional integrals with a relative

error of one part in a thousand as stated in the MATLAB

numerical integration function. With a rotationally
symmetric E, and Eay, the computed value of the double

integral with Simpson's rule was compared with the MATLAB

value of the one dimensional integral. This verification was

done for a variety of aperture electric fields and for a
variety of far field coordinates 9 and #. For all cases of

verification, the accuracy of Simpson's rule was better than

one percent.









4.3 NEC Computer Program


The computer program used in this study is a modification

of the Numerical Electromagnetics Code (NEC) Reflector

Antenna Code originally developed by R.C. Rudduck at Ohio

State (Rudduck and Chang, 1982; Chang and Rudduck, 1982).

The basic approach for the NEC Reflector Antenna Code is

a combination of aperture distribution techniques and

geometrical theory of diffraction (GTD). Aperture

distribution techniques are used to compute the main lobe and

near sidelobes, while GTD techniques are used to compute far

sidelobes and backlobes. The basic aperture distribution

method was described in the previous section. For a more

complete description of the NEC Reflector Antenna Code and a

listing of the computer program, one should refer to

references (Rudduck and Chang, 1982; Chang and Rudduck,

1982).

To obtain the secondary pattern using the computer

program, it is necessary to generate an input data file. The

most important quantities in the input data file are: the

size of the paraboloid, the focal length, the location of the

feed, and the primary pattern.

A sample input data file is included in appendix C. For

this example the symmetric primary pattern was entered in

&increments of 1 degree. For the primary pattern, the first

column corresponds to values of 8 The second column

corresponds to the normalized amplitude of the electric field









from the feed, while the third column represents the phase of

the electric field as it leaves the feed (0 degrees).

The most important limitation is the feed must be

located near the focus (Rudduck and Chang, 1982; Chang and

Rudduck, 1982).


4.4 Effect of Limitations on Aperture Distribution Method


Since the NEC computer program uses geometrical

diffraction methods in addition to the aperture distribution

method, this method was compared with the aperture

distribution method. With a 2 wavelength diameter small

flare angle hybrid mode feed located at the focus of a 40

wavelength paraboloid with an f/D of 1.0, the NEC computer

program was compared with the aperture distribution method

(upper part of figure 4.6) for a phi cut 0 equal to zero

degrees.

As shown in the figure, the aperture distribution method

gives accurate results over the main beam and nearby

sidelobes. However, this method is not accurate for the far

sidelobes and backlobes due to edge diffraction.

With the same parameters as in the upper part of figure

4.6, the feed is now displaced to 1.28 wavelengths (lower

part of figure 4.6). The same comments apply to the lower

part of the figure as were made about the upper part of the

figure.







59











0

oo 20 ----**-**** *--_ -**--------i--------------- ------------- ----,----
00 -20 .... .... ........ .... ... ...... .................. ................ ...............



Z -40 .............. ............... .. ........... .. ......... ..... .
Ic

0 0
1 -60
0 1 2 3 4 5 6

180- (degrees)


0 0



0 -30



-40 I
00
-50 ---i
IL 0 1 2 3 4 5 6

180- (degrees)




NEC

----- Aperture
Distribution






Figure 4.6 Comparison of NEC computer program and aperture
distribution method: (upper) hybrid mode feed at focus,
(lower) hybrid mode feed displaced 1.28 wavelengths.















CHAPTER 5
NOISE PERFORMANCE


The purpose of this chapter is to investigate the noise

performance of the system with different feeds. In this

chapter, the following parameters are considered: type of

feed, f/D ratio, lateral displacement of feed, feed size,

and paraboloid size.

It will be shown that the noise performance of the

diffraction-limited feeds (defined in chapter 3) is similar.

The diffraction-limited feeds are dominant mode horn, dual

mode horn, and hybrid mode horn. The dielectric rod has

different characteristics than diffraction-limited feeds.

For this chapter, it should be noted that the term

pattern refers to the secondary pattern unless explicitly

stated.


5.1 Ouantification of Noise Performance


With the assumption that the target is detectable, this
study will focus on the normalized angular rms error due
to noise. The lower bound on the angular rms error caused by
to noise. The lower bound on the angular rms error caused by









noise (in one tracking coordinate) for a conopulse system is

(Peebles and Sakamoto, 1980a)


oTs 1
e,- k4 (5-1)



where 0Q is the 3 dB beamwidth of the one-way voltage sum

pattern, k, is the one-way voltage normalized error pattern

slope, and S/N is the ratio of peak signal power to noise

power at the output of a filter matched to the location of

the target.


Equation (5-1) was obtained by Cramer-Rao analysis

method and is independent of the signal processor connected

to the antenna. With large single-pulse signal-to-noise

ratio and any form of RF pulse, Peebles and Sakamoto (1980b)

showed that this lower bound will be approached with the use

of a sufficiently short-duration range gate. The error slope
k can be defined as (Peebles and Sakamoto, 1980a)



k = (3L JdG (0) j=0 (5-2)



where G. is the one-way voltage sum pattern evaluated on the

boresight axis and GA(O) is the one-way voltage difference


pattern.









The signal to noise ratio S/N depends on the antenna
efficiency qa and sum pattern G, as (Peebles and Sakamoto,

1980a)


SIN=Kq,2G,4 (5-3)


where K is a constant that depends on the parameters of the

radar equation.



Later, in this chapter, the normalized angular rms error
a,/63 will be quantified versus different parameters. For

the results in chapter 5, the constant K equals 100. This

constant results in a signal-to-noise ratio SIN of 20 dB
when both the antenna efficiency n, and the sum pattern on

boresight G, equal one.

It should be noted that all calculated results for noise

performance will use the exact patterns as opposed to an

approximation. The calculation of the sum and difference

patterns assumes zero mutual coupling between the patterns of

the feeds.

All results will have the restriction that the value of
the sum pattern G, must be greater than one. If the value of

the sum pattern is less than one, a double hump occurs in the
sum pattern with each hump being larger than Gz as shown in

figure 5.1. A double hump is undesirable in applications,

which require narrow beamwidths such as multipath.



























1

0.8

S0.6

t 0.4

0.2

n


-3 -2 -1 0 1 2

0 (degrees)


0
8 (degrees)


Figure 5.1 Sum pattern with different values on boresight
due to different squint angles: (upper) sum pattern on
boresight less than one, (lower) sum pattern on
boresight greater than one.


L.................. ...... ....... ...






.... ......... .
........ ........ .. .... .. ...






64

For the purpose of analyzing noise performance, it will
be useful to define some quantities. The crossover value 1/4

is the ratio of the value of one offset pattern on the

boresight axis to the maximum of the pattern. The crossover

value equals half the value of the sum pattern on boresight
Gr. The squint angle O8 is the angle between the maximum of

one offset pattern and the antenna boresight, while 8B is the

3-dB beamwidth of one pattern. Barton (1988) gives the

crossover loss as a function of the normalized squint angle
6,18, as shown in figure 5.2 for two types of patterns.


Normalized offset angle, 6k183


Figure 5.2 Conical-scan crossover loss. (Source: Barton,
1988).









The beam deviation factor BDF is defined as


BDF= -- (5-4)
-1 (J
tan

where d, is the lateral displacement of the feed from the

focus.


From the radar equation (Skolnik, 1980), it is clear

that the received signal-to-noise ratio is proportional to
the square of the antenna efficiency a,. Since the rms error

a. due to noise is inversely proportional to the square root

of the signal-to-noise ratio, it is necessary to calculate

antenna efficiencies when calculating noise performance.

Ludwig (1963) has analyzed the antenna efficiency of a

paraboloidal reflector system. Ludwig assumes that there is

not any blockage from the feed or any errors from surface

irregularities. In addition, the assumption will be made

that there is a uniform phase distribution across the

aperture and that the feed voltage pattern equals zero for

S> 90.

The expressions developed by Ludwig are for a feed

located at the focus. In a similar manner to chapter 4, the

assumption will be made that the magnitude of the fields over

the aperture are the same as for a feed located at the focus.

This assumption is reasonable for cases where the lateral

displacement is small compared to the focal length.









The antenna efficiency will depend on the amplitude

distribution of the fields across the reflector aperture

plane. In addition, energy losses due to spillover and

cross-polarization will affect the antenna efficiency.

Ludwig quantified these factors by the following
efficiencies: 7, (spillover), 77 (illumination), and 77

(cross-polarization). Based on analysis by Ludwig, these

factors are defined as






o
0


o
I {IA( 6')l+l B(') 2asin dO
7= 2cot (5-6)
f2 A(0') + B(') sin OdW
0


fjA(')j+ B(')jf sin 6 dO'
7Ox= o (5-7)
2 IIA(' )12 + B(1 )12 } sin O' dO'
0
o

where 80 is the subtended angle of the reflector, and

IA('),IB(e')I are the E and H plane feed voltage patterns,

respectively.


The overall efficiency q, can be defined as the product

of the individual efficiencies:


70 = 71,1 7i(57


(5-8)








The spillover efficiency 7, is the ratio of power intercepted

by reflecting surface to the total power radiated by the

feed. The uniformity of the amplitude distribution of the

fields over the aperture plane is measured by the
illumination efficiency 77 and is maximized with a uniform

distribution. The cross-polarization efficiency 7,

quantifies the uniformity of the polarization over the

aperture plane.

To verify the computed results, the computed results
were compared with known results for a few special cases.

With the assumption of a rotationally symmetric primary

pattern and zero cross polarization, Balanis (1982) showed
that the antenna efficiency can be calculated as



77 = cot2 ( G(0') tan"- dO' (5-9)


Consider a class of rotationally symmetric feeds with
feed power patterns defined as (Balanis, 1982)


(Go(") cos"(6) 0 5 0' ;/ 2(
Gf(0')=2 (5-10)
/ 0 /2:5 / 0'5 ;r

where G(")= 2(n+1) .


The quantity of G() (voltage pattern) for the cases of

n=2 and n=4 of (5-10) was then substituted for the E and H

plane feed voltage patterns given in equations 5-5 5-8.
The value of 77 in Ludwig's method was then compared with









equation (5-9) for the same two cases. There was almost

exact agreement between the computed results and the known

results.


5.2 Minimum Lateral Displacement


An important point is that there is a minimum lateral

displacement in conopulse systems. The smallest lateral

displacement of the waveguide horns occurs when each horn is

as close as possible to the central transmission line without

touching the transmission line (see figure 2.3). Thus, the

minimum lateral displacement is equal to


d, d
d, + (5-11)
2 2
where d. is the lateral displacement, d, is the maximum

dimension of the transmission line, and d, is the diameter of

the waveguide horn.



The minimum lateral displacement occurs with the

smallest transmission line and the smallest possible

waveguide horn. However, there are practical limitations on

the minimum size of the transmission line and waveguide

horns.


5.2.1 Minimum Size of Central Waveauide


There are a variety of transmission lines including

twisted pair wire, coaxial cable, waveguide, and fiber optic









line. For radar applications, the two most important types

of transmission lines are coaxial cable and waveguide.

Coaxial cable can be smaller than waveguide. In the

past, coaxial cable has been used in some low power radar

systems. However, the coaxial cable may break down with high

power. Another problem with coaxial cable is that there can

be large attenuation at high frequencies.

For the proposed conopulse system, the system will

operate in the 1-10 GHz region with peak power levels on the

order of MW and average power levels of several kW. The

coaxial cable could shatter at these power levels and have

relatively large attenuation losses at these frequencies.

Therefore, coaxial cable will not be used for this system.

There are two major types of waveguide: rectangular

waveguide and circular waveguide. Waveguide will be used as

the transmission line for this system, because it has the

following properties:

simple mechanical structure

high power handling capability

low attenuation losses at high frequencies



In order for a waveguide mode to exist, the waveguide

must have certain dimensions. The minimum possible size of

the waveguide is when the dominant mode only can exist. For

reasons given in section 2 on the development of the antenna

design, the rectangular waveguide will be used as the

"central waveguide".









Consider a rectangular waveguide (figure 5.3) with a
representing the longest dimension and b the shortest
dimension. For TE10 mode, the cutoff wavelength Ac equals 2a

(Ramo, 1984). Although the dimension b does not affect the

cut off frequency, it does matter for some other reasons.




Tr
'' -


b/ /




Figure 5.3 Coordinate system for rectangular waveguide.
(Source: Ramo, 1984)


Ramo (1984) has suggested it is desirable to have a
large range between the cutoff frequency of TE10 and the next
higher mode. If b>0.5a, the TE01 mode has the next lowest

frequency (Ramo, 1984).

The shorter dimension b will be chosen as half the
longer dimension a. A waveguide of these dimensions is the

size used in most practical rectangular waveguides (Ramo,

1984). Since the "width" is twice the "height" for the TE10

mode, the smallest possible rectangular waveguide has the

dimensions


b = 0.25A
a= 0.50 (5-12)
a = 0.50X









The feeds must rotate around the maximum dimension of
the central waveguide d, which equals


d,= (0.25)2 +(0.50A)2 =0.56A (5-13)


5.2.2 Minimum Size of Feed


The minimum size of the feed will depend on the type of

feed. In order to illustrate the effect of feed size on

minimum lateral displacement, the dominant mode horn will be

considered.

The smallest dominant mode horn will occur when the feed

has no flare and the feed reduces to an open ended waveguide.

As given by (3-1), the smallest dominant mode horn has

diameter equal to 0.58 wavelengths. The minimum lateral

displacement from equations 5-11 and 5-13 is

d, = 0.28A + 029A = 0.57A (5-14)


The minimum lateral displacement of the evaluated horns

is shown in table 5.1.


Table 5.1 Minimum lateral displacement of evaluated feeds.

Feed Type Minimum Lateral Displacement
(wavelengths)
dominant mode horn 0.57
dual mode horn 0.89
hybrid mode horn 1.28
dielectric rod 10.63









5.3 Effect of Lateral Displacement


For the study of the effect of lateral displacement, the

minimum size feeds were considered with a paraboloid diameter

of 40 wavelengths. To determine the effect of lateral

displacement, the noise performance of each type of feed at

minimum size was examined at various lateral displacements

from the focus: minimum lateral displacement, 1.2 times the

value of the minimum lateral displacement, and 1.4 times the

value of the minimum lateral displacement.
Since the BDF and the 3-dB beamwidth of one pattern 68

are nearly constant, increasing the lateral displacement by

factors 1.2 and 1.4 approximately increases the ratio of
0,18B by this same amount. These values of lateral

displacement were selected, because they were large enough to

see changes caused by variations in the lateral displacement.

For diffraction-limited feeds (figures 5.4-5.9) and tables

5.2-5.4, it was found that the noise performance became worse

with increasing lateral displacement. With these feeds, the

smallest rms error occurred when the sum pattern equals one

(crossover value equals 0.5). For increased lateral

displacement, a larger f/D ratio is required to obtain a

crossover value of 0.5, because increased lateral
displacement results in larger values of 0,/O8 (crossover

value depends largely on the ratio of 0,/Oa). With a larger

f/D ratio, there is decreased antenna efficiency due to more

spillover radiation.














0.9

0.8

0.7

0.6

0.5


.... i'\1. ^ ........ ................

..... ...... .....................

\ \ :
...... .... .... ....................



I- \
.5 1 1.5 2 2.5

f/D



5
4 -*---1--** ----^---- ----'----I






1 .. ..
. . . .. . .. . .


0.5 1 1.5

f D


2 2.5


D.5 1 1.5 2 2.5

fID



i.6

: /
1.4 .......... .......... ........ .. ...........


1 ......... ...... '. ..................
)2' : :


.",


0.5 1 1.5

fID


2 2.5


d- = 0.58A
d- = (0.58 1.2)A
-.-. ,=(0.58*1.4)


Figure 5.4 Effect of lateral displacement on dominant mode
horn: (upper left) normalized error pattern slope,
(upper right) crossover value, (lower left) aperture
efficiency, (lower right) normalized rms error due to
noise.


.......... :............ .... .........

/ /.



.. ... .. ............. ..........
:1i"


B


0.4















1.2


a)
(U 1

a 0.8


0.6


0.4
0.


5


1 1.5

flD


1


0.98


Q 0.96


0.94


0.92
0.=


0.7


0.6 ..... .... ......." ...

&\\ \ \
0 .5 .....***** **.. .... ... ......... .........
-- 0.5 ... ...,


0.4 ......... .... \......


0.3
0.5 1 1.5 2 2.5

fID


0.5


5


1 1.5

fID


2 2.5


1 1.5

fID


d, = 0.58A.

d, =(0.58*1.2)A

d.=(O.58e1.4)p.


Figure 5.5 Effect of lateral displacement on dominant mode
horn: (upper left) squint angle, (upper right) beam
deviation factor, (lower left) squint angle normalized
by 3-dB beamwidth of one pattern, (lower right) 3-dB
beamwidth of sum pattern.


.... . .


I '

'



......... ...


: ....


. . . . . . . . . . .


1.


I I


:1 .^r-* i


c


'


-













6 0.9
5 ..:...... ............ 0.8 .............. ..... : ..........
5 .... .................. ......... 0.8 ............
t5 0 .8 ..... .
i :
..... 0 .... ....


..\ . / /.
-. .............: 0.7 ... ...... - *. .... .... ......
2 ... ..... .............0.5

1 0.4
1 2 3 4 1 2 3 4
f/D f/D



0.4 -- 0.6

0.3 \ 0.4 ....... ........... /
\ 0.4 ... .... .............. . ..........
S0.2 ....---- ......--....----..

..0.2 ..... ............
0.0.2 .......... ..................
0.1 ----- --- -- ...........

0 0
1 2 3 4 1 2 3 4
f/D f/D



d, = 0.902
d-- = (0.901.2)A
d, = (0.90 1.4)A




Figure 5.6 Effect of lateral displacement on dual mode
horn: (upper left) normalized error pattern slope,
(upper right) crossover value, (lower left) aperture
efficiency, (lower right) normalized rms error due to
noise.





























1 2 3 4
fID


,... :.- ........... ...........

\ \\


............. .. .... % .............



2 3 4

fID


1 2 3

fID


d- = 0.9 0

d- =(0.90 1.2)A

.-. =(0.90*1.4)A


Figure 5.7 Effect of lateral displacement on dual mode
horn: (upper left) squint angle, (upper right) beam
deviation factor, (lower left) squint angle normalized
by 3-dB beamwidth of one pattern, (lower right) 3-dB
beamwidth of sum pattern.


1.2


ai 1
0)

a, 0.8


0.6


0.


'. "-\
. .. .. .... ....- ........



............. \ ............


f/D


.. ..\ ............. .............


\ : .

\ ". ........ .............
v ^ '. .. .-


U.


I


"





















% "





......... ....... .. .........



2 3 4 5

fID


U'


1 2 3

f/D


0.9 8 .
U.3 ~ ~ --- --- --- ---
0.8 ............. ... ............ .. ....

S0.7 "

/ /:
0.6 ....... ... .. ../ ......... .........

0.5 . ..."..... ......... .........

0.4
1 2 3 4 5
fID


4 5


1 2 3
fID


S d, = 1.28A

d- = (1.28 *1.2)A

-.-. d, = (1.28 *1.4)A


Figure 5.8 Effect of lateral displacement on hybrid mode
horn: (upper left) normalized error pattern slope,
(upper right) crossover value, (lower left) aperture
efficiency, (lower right) normalized rms error due to
noise.


0.4


0.3


2 0.2


0.1


.4 ....... I. ...... ... ... ......... ....



n'2

02 ----**** ---- - -- *.- ---***


4 5


*


. . o..o.. ..... .........





................ ... ....... ... ....
*.. . . . ......
---- -- -: ---------:---------- -----



- -- - '-- - --* *-- ''-- - - -

































2 3 4 5
f/D


2 3 4 5
flD


0.995 -



0.99 ...


0.985
1


2 3 4 5
fID


3 ......u . ........... ..........


. ............ .........
3 ****** ,-\ \ -



2 ......... .. .. ...... ........



1 2 3 4 5
flD


d, = 1.28A
--- d,=(1.28*1.2)A

-.-. d= (1.28*1.4)A


Figure 5.9 Effect of lateral displacement on hybrid mode
horn: (upper left) squint angle, (upper right) beam
deviation factor, (lower left) squint angle normalized
by 3-dB beamwidth of one pattern, (lower right) 3-dB
beamwidth of sum pattern.


.......... -\ >.. q. \...................


.......... .........
.. .. .. .. .. .. .


n,


1


0.7


0.6


0.5


0.4


0.3
1


........ .... ... ....... .........
\ ...
\ : .
\ \*


S.


...... ... .. .. .. .


m


I










Table 5.2 Effect of lateral displacement on dominant mode
horn with parameters at minimum rms error.


d, (A)
0.58
0.58-1.2
0.58.1.4


k,
4.5716
5.0002
4.4993


1/4
0.5208
0.5018
0.5208


7.
0.3872
0.2939
0.2122


CFOe/03
0.0521
0.0675
0.0965


flD
0.7487
0.9100
1.1178


Table 5.3 Effect of lateral displacement on dual mode horn
with parameters at minimum rms error.


d, (A)
0.90
0.90-1.2
0.90.1.4


k
4.4742
4.8678
4.4697


1/4 10,
0.5218 0.3741
0.5050 0.2858
0.5206 0.2099


aG / 03
0.0549
0.0705
0.0983


fID
1.2515
1.4900
1.7918


Table 5.4 Effect of lateral displacement on hybrid mode
horn with parameters at minimum rms error.

d, (A) k, 1/4 77, a9 e/e flD
1.28 4.5603 0.5180 0.3761 0.0543 1.7984
1.28-1.2 4.4708 0.5210 0.2761 0.0746 2.1890
1.281o.4 4.4522 0.5213 0.2103 0.0983 2.5700


For the dielectric rod (figures 5.10-5.11 and table

5.5), the noise performance became better with more lateral
displacement due to larger values of k .


Table 5.5 Effect of lateral displacement on dielectric rod
with parameters at minimum rms error.

d, (A) k 1/ 77, ao/ 3 flD
0.63 1.2492 0.8171 0.8155 0.0368 1.4205
0.63.1.2 1.6946 0.7436 0.8144 0.0328 1.4163
0.63.1.4 2.5581 0.6455 0.7868 0.0298 1.3457


It should be noted that the appearance of the beam

deviation factor plots is due to the finite "resolution" when


calculating this quantity.


L




















6

5

4

3

2


0.8


S0.6


0.4


..... .... .....s......................

.. . .. . .......... ........
.........


0.5 1 1.5 2 2.5
fID




I


0.5


1 1.5
f/D


U.Y

0.8

0.7

0.6

0.5

0.4
0.





0.07

0.06

0.05

0.04

0.03

0.02
0


2 2.5


,5


1 1.5
flD


.5


1 1.5

flD


d, = 0.63A
= (0.63 1.2)A

-.-. =(0.63*1.4);





Figure 5.10 Effect of lateral displacement on dielectric
rod: (upper left) normalized error pattern slope,
(upper right) crossover value, (lower left) aperture
efficiency, (lower right) normalized rms error due to
noise.


- -
........ .. % ....... .

/ -
e .~..:.~ ...... ....................
I ..
I.r.......


.. ....... N....


2 2.5


2 2.5


. .. ..
*i .

/ ...........
S*
.... .... . .


A^


n,


I,


1
























... ... ..........
\ \
.... .. ... ...... .......... .........




S.-. *
......... .. ......
: : :
:


.5 1 1.5 2 2.5
fID




8


1 1.5
fID


2 2.5


d, = 0.63A

d, =(0.63*1.2)A

d,=(0.63e1.4);L


Figure 5.11 Effect of lateral displacement on dielectric
rod: (upper left) squint angle, (upper right) beam
deviation factor, (lower left) squint angle normalized
by 3-dB beamwidth of one pattern, (lower right) 3-dB
beamwidth of sum pattern.


1.4

1.2

1

0.8

0.6


.4
0


0


0.7

q 0.6

S0.5

0.4

0 3


1 1.5
fID


1

0.98

S0.96

0.94

0.92


0.5






6

m 5
0)
0)
4

S3

a 2

1
0.


.. .. ... . .
.......\.......... ....... ........i


2 2.5


2 2.5


0.5


5


1 1.5
fID


.,..,


*


t3_


# I


r......... .................... ..


.. . .. . . .. . .

.. .. .. ... .. .. ... .. .. .
V~~~'~'~~' '~''~~~
... ... ... ...I ... ... ... ..









5.4 Effect of Feed Size

For investigation of the effects of horn size, the

paraboloid diameter was chosen as 40 wavelengths and the

feeds were at minimum lateral displacement. The dominant mode

horn was examined for horns with radii equal to (in

wavelengths): 0.30, 0.45, and 60. The dual mode horn was

studied with the radii (wavelengths) of 0.62, 0.72, and 0.82,

while the small flare angle hybrid mode horn was investigated

with the radii (wavelengths) of 1.0, 1.2, and 1.4. The

dielectric rod has only one size given in figure 3.5 and was

not studied for the effect of horn size.

Again, the minimum rms error occurred with a crossover

value equal to 0.5. It was found that the rms error generally

became better with larger horn size as shown in figures 5.12-

5.17 and tables 5.6-5.8 at the expense of larger f/D ratios.

This was due in large part to increased antenna efficiencies

with larger horn sizes.


Table 5.6 Effect of feed size on dominant mode horn with
parameters at minimum rms error.

a (l) 1/ 4 ao / e fID
0.30 4.5716 0.5208 0.3872 0.0521 0.7487
0.45 4.9876 0.5034 0.4309 0.0459 0.9500
0.60 5.0342 0.5016 0.5203 0.0377 1.1600


Table 5.7 Effect of feed size on dual mode horn with
parameters at minimum rms error.

a (,) k 1/4 ao/e, f/D
0.62 4.4742 0.5218 0.3741 0.0549 1.2515
0.73 4.9894 0.5012 0.3960 0.0504 1.4162
0.84 4.4538 0.5231 0.3945 0.0520 1.5712




















5 .... 1 1.. ... ..........5 2 2.....







f"
3 3
4.................

S. .\ .
3 ...... . :. ......................

2 ............ ..... ................


0.5 1 1.5 2 2.5
fID




A - ------------


0.9

0.8

0.7

- 0.6

0.5

0.4
0.


5


1 1.5
flD


n


0.5

0.4

0.3

0.2

0.1
0.


,5


1 1.5
f/D


2 2.5


1.5

fID


a = 0. 30A
--- a = 0.45A
a = 0.60A


Figure 5.12 Effect of feed size on dominant mode horn:
(upper left) normalized error pattern slope, (upper
right) crossover value, (lower left) aperture
efficiency, (lower right) normalized rms error due to
noise.


.......... ...... ...
. '-. .-

.......... ..... .. . ........: ...........
/ /
: / /

;.... ............ ..........
/ /


2 2.5


............. ...... .......... .......

.... . .. ...........

....... .. .... .. .... .... .....

..". ......,


MI. I *


* I *


I I


















1


0.98


0.96


0.94


I


0.5 1 1.5 2 2.5 0.5

flD


1 1.5
fID


I


2 2.5


0.5


1 1.5 2 2.5

fID


1 1.5 2 2.5

f/D


a = 0.30A
-----a = 0.451
-.-.-.a a=0.60


Figure 5.13 Effect of feed size on dominant mode horn:
(upper left) squint angle, (upper right) beam deviation
factor, (lower left) squint angle normalized by 3-dB
beamwidth of one pattern, (lower right) 3-dB beamwidth
of sum pattern.


0.8


0.6


... ... ... ...... ... i..........


\,

......... ....-." ....-.... .'i', ^... .
~~~~~.
'-


\..... "
"\ \
\ \ \


.... . .. . ..\. ... ...... . .. ...... ..

\. \ .

\- ,._.:.


0.5
0.5


c ____ 1


r,


c


--


.. . . . . . . . . . . . . . . . . . ..
. . . . . . . . . .. . . .
.. .. . .. .. .... .. .. .. .. .


0.4


f q%='


-I

















5

4
- 3


2

1


0.5

0.4 ... ........ .. .....................
\\









f I
0 .2 .. ... ... ........... ... ..........




0.1


flD


I :
*. --








1 2 3
fID


fID


a = 0.62A
a = 0.73A
a=0.84;


Figure 5.14 Effect of feed size on dual mode horn:
(upper left) normalized error pattern slope, (upper
right) crossover value, (lower left) aperture
efficiency, (lower right) normalized rms error due to
noise.


Ki -- -


2 3 4
flD


v 'I ..... .. ............ .............

... ................ ....... .....

... .... .... .. ..... .... ..

... ..\ ............ ..............






















" ". '". ...........................


....... \ . .............
\ \.


........ ......... ,, .............



2 3 4

flD


1


0.8


0.6


0.4
1


\



........ .. .............
: ,N s
%" "'


0.7


0.6


0.5


0.4

A 1%


\*
...^ V............... .............


\ \ :

.. . . . . * **, .. . . .. . . . . . . ..




2 3

flD


a = 0.62A
a = 0.73A
a= 0.84A


f/D


f/D


Figure 5.15 Effect of feed size on dual mode horn: (upper
left) squint angle,(upper right) beam deviation factor,
(lower left) squint angle normalized by 3-dB beamwidth
of one pattern, (lower right) 3-dB beamwidth of sum
pattern.


























1 2 3 4 5

f/D






5.


2 3

flD


0.9

0.8

0.7

0.6

0.5


1 2 3 4 5

flD






n03


0.2

b"


4 5


2 3

f/D


4 5


Figure 5.16 Effect of feed size on hybrid mode horn:
(upper left) normalized error pattern slope, (upper
right) crossover value, (lower left) aperture
efficiency, (lower right) normalized rms error due to
noise.


n i


5

4

3

2

1


..... .. N ..... ......... .
'


-
..............


/ ,..
./7.......I.....

*1i


n


0.4


S0.3


0.2


0.1
1


JI
.......... ......... / .....

.. .. .. . .. .. . ... ..


' ............... .........


.-


!


I


U


I






















a,
a)

S0.8
'0

0.6


0.4
1


0.995 f-


0.99F


I -.r


2 3 4 5 1
fID


u. -I

0.6 ......... .. ......... ..........


0 .5 ......... .... ... .......* .. ........


0.4 ..... .... ... .. .. .....


0.3
1 2 3 4 5

fID


2 3 4 5
fID


3 ..

3 ............ \ '... .....................
2 .5 ....... ......... ......





1.5
1 2 3 4

f/D


Figure 5.17 Effect of feed size on hybrid mode horn: (upper
left) squint angle,(upper right) beam deviation factor,
(lower left) squint angle normalized by 3-dB beamwidth
of one pattern, (lower right) 3-dB beamwidth of sum
pattern.


- --i- -- *--- --' s --***- --

\\
.... ... .. .. : .....................



\ \ \
"'........


... ... . ........


... ....... ...


....... .................


A A I"l









Table 5.8 Effect of feed size on hybrid mode horn with
parameters at minimum rms error.

a (A) k 1L /4 0e3 flD
1.0 4.5603 0.5180 0.3761 0.0543 1.7984
1.2 4.4653 0.5229 0.4046 0.0506 2.0873
1.4 5.0227 0.5004 0.4497 0.0442 2.3000



5.5 Effect of Paraboloid Size


To investigate the effect of paraboloid size, each type

of feed was investigated for its smallest size and at minimum

lateral displacement. The size of the paraboloid reflector

was chosen as 20, 40, and 60 wavelengths.

For a given f/D ratio, it was found that the noise

performance was largely independent of paraboloid size as

shown in figures 5.18-5.25 and tables 5.9-5.12. For a given

feed with its primary pattern, the antenna efficiency is a
function of the subtended angle of the reflector 8o. As

given by equation (4-8), the subtended angle is a function of

the f/D ratio only. At a given f/D ratio, the antenna

efficiency does not depend on paraboloid size.



Table 5.9 Effect of paraboloid size on dominant mode horn
with parameters at minimum rms error.

D (A) k 1/, 47, o/ e, flD
20 4.5703 0.5209 0.3872 0.0521 0.7487
40 4.5716 0.5208 0.3872 0.0521 0.7487
60 4.5714 0.5208 0.3872 0.0521 0.7487

















































1.5
fID


0.9

0.8

0.7

0.6

0.5
4


U.'


0.5 1 1.5
flD


2 ... *... ***.. ..... .. .............


1 ......... . .. ......... .............
1


0
0.5 1 1.5
flD


D = 20A
--- D = 40A
D = 60A I


Figure 5.18 Effect of paraboloid size on dominant mode
horn: (upper left) normalized error pattern slope,
(upper right) crossover value, (lower left) aperture
efficiency, (lower right) normalized rms error due to
noise.


fID


0.5

0.4


0.3


0.2

0.1
0.


5


..... ...................................

....... ..... *.............4..............

.. ...... ................. ..............

.........................................
.......i.................


d


.............. 7 ........ .................


......... ... ............ i..............


............~ ....


.. .. * * ... ......... ........ .......






91










2.5 1

c2.i------:--- --2------1-----

S2 ..... .. ........ ............. .. ..
0.98 4 -----:*----
0 1.5 ....... ..... ". ..... ............. 0 9
Q 0.96 ............. ........... .............



0 0.92
0.5 1 1.5 2 0.5 1 1.5 2
flDo flDo



0.7 8
fID flD






D0.7- -0



0.5 .1 1 1.5 2. 0.5 1 1.5 2
















(upper left) squint angle, (upper right) beam deviation
factor, (lower left) squint angle normalized by 3-dB
0.4 .. . 2 .. . .









beamwidth of one pattern, (lower right) 3-dB beamwidth
of sum pattern.0.5 15 2
fID fID



D = 20A
D = 40A





Figure 5.19 Effect of paraboloid size on dominant mode horn:
(upper left) squint angle, (upper right) beam deviation
factor, (lower left) squint angle normalized by 3-dB
beamwidth of one pattern, (lower right) 3-dB beamwidth
of sum pattern.

























3
5 ---- *---- -----i-----i----4---------






1
1 1.5 2 2.5
flD


............. -- 7 --........- ..
...................................






1.5 2 2.5
fID


flD flD


D = 20A
---- = 40A
D = 60A


Figure 5.20 Effect of paraboloid size on dual mode horn:
(upper left) normalized error pattern slope, (upper
right) crossover value, (lower left) aperture
efficiency, (lower right) normalized rms error due to
noise.






















* --- -- --- -_--


..5 .............. ..... ..



0
0~ ------------------- "
1 1.5 2 2.5
flD


....i...iii.. . . ... ... ..............







1.5 2 2.5
fID


1 1.5 2 2.5
f/D


0
1 1.5 2 2.5
flD


Figure 5.21 Effect of paraboloid size on dual mode horn:
(upper left) squint angle, (upper right) beam deviation
factor, (lower left) squint angle normalized by 3-dB
beamwidth of one pattern, (lower right) 3-dB beamwidth
of sum pattern.




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