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 DiffractionLimited 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
diffractionlimited 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 3dB
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 servocontrol 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 twodimensional 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 phasesensitive 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 monopulseconical 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 monopulseconical
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 (21)
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 22. 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 (21). This analysis
assumes gaussian antenna beams. The oneway voltage gaussian
pattern is defined as
G(0) = G exp  0 (22)
where G, is the oneway voltage maximum, a2 =2.776/ ,2 with ,B
being the 3dB 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(23)
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
(24)
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)
( / 25)
(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( 2IO"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)
(26)
(27)
The signal RL(t) can then be simplified to the following form:
RL(t)= tanh {,~0 cos(,t) + 0, cos(4r)}.
(28)
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))}
(29)
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)} (210)
I
Using a trigonometric identity, ecos(t) can be expressed as
ecos(t)=  { 2  Q1cos(2ot) + OT cos(o,)cos(O),t) (211)
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(212)
The dc error signal e, is not proportional to the azimuth
angle and (21) 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)} (213)
Using a trigonometric identity, esin(t) can be expressed as
esin(t)= a~sin(2 o,t) + O cos(,)sin(cot) (214)
At the output of a low pass filter with cutoff frequency less
than C),, the elevation dc error signal e, equals
e,=0 (215)
The dc error signal e, is not proportional to the elevation
angle and (21) 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) Dualmode. (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 semiflare 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 (32)
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 (33)
U2(2s,d) = 2sf Jo(d)sin[s(1 521)]4
0
where Jo is a zeroorder Bessel function of the first kind.
The series form of Lommel functions is (Hu, 1960)
U (2s, d) = Jd (d) J (d) + ) 5(d)...
(34)
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 dualmode 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
(36)
27
The radiated electric field components EE, Efrom the
feed are (Potter, 1963; Johnson and Jasik, 1984)
ELCos(= (+11cose a J,(kasinO) eR.
k KE kasin R
ksinfJ ka
(37)
E#=(' +cosO Ji' (kasinO) ej u
E k ksine 9 R
K,,. )
where K,,a is 1st root of J1' and equals 1.841,
Pfl, =k2K2n KlEa is the 1st root of J1 and equals 3.832,
i, = =k2K21E J, is the Istorder 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.
EDIPOLE MDIPOLE 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 Eplane and Hplane patterns and zero cross
polarization, the size of the feed should be (Knop and
Wiesenfarth, 1972)
ka>2n (38)
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))
(39)
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)(') (310)
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, (311)
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%.
07?o O 448go 0395Ao
01 1
__ ___ 033X0
48L o 2Xo 2Xo I 2Xo
TaDer radiation negigiole
Launcher radiation 12'8%
End aperture radiation 872%
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 (Kcos )
E9 = I cos( sin 0) sin 0
(Kcos0)sin R(K ) "0
nL '(312)
(K1)sin (K cos)
o _____d______
= cos(sin 0) cos
(KcosO)sin IL(K1) 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
(32)
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 (312) has a maximum of one, the normalization
constant C, in (32) should be chosen so that launcher
radiation has a maximum of one.
3.5 DiffractionLimited Feeds and Dielectric Rod
A feed, which is diffractionlimited, has radiation
properties that are governed by the size of the aperture with
radius a. With a larger aperture, diffractionlimited feeds
have more directive radiation properties. In this study,
diffractionlimited feeds include: dominant mode horn, dual
mode horn, and hybrid mode horn. The feed patterns of these
types of feeds are shown in figures 3.63.8.
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) Eplane pattern, (lower) Hplane
pattern.
.......... ............ ........... ................. ..... ...................
........... ......... ............ .............

5t
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) Eplane pattern, (lower) Hplane
pattern.
I
Cq
0
A
10
20
30
40
50
0
................. ........ ..........
.. .. .. .. ...... . .... ... ........ .. .... ................ .... ...... ..
.. ..... ... ............ %"*. ... o .. %. .
S. . .. . . . . . .. ............ ... ..........: ...... ...... ...... ... ... ..... .. ... .
I i i I
. = i I
5 15 .... ............ ............ ........... ............ ...........
25^* ***** ,,, ...,.......
S20
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 Eplane and Hplane feed
patterns of hybrid mode horn.
0
5
10
S15
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) Eplane pattern,
(lower) Hplane 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) Eplane 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. Parabolicreflector 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. Twodimensional configuration of a paraboloidal
reflector. (Source: adapted from Balanis, 1982.)
Using the second property
OP+PQ= constant= 2f (41)
From figure 4.2
OP = r'
(42)
PQ= r'cos' (42)
Upon substitution of (42) into (41), (41) can be expressed
as
r'(1+cos0')= 2f (43)
or
r'= 2f 0' o (44)
1+cos
Using a trigonometric identity, r' can also be expressed as
r'=fsec2 P 5) O0 (45)
Equations (44) and (45) 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=,. Lt 
(46)
By geometry (Balanis, 1982), zo can be expressed in terms of
f and D as
S=f(D2) f (47)
4f 16f
Upon substitution of (47) into (46), the subtended angle of
the reflector equals
e f tan(f
S= tan' 2 D
(fDY 16,
(48)
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' (49)
Upon substitution of (45) into (49)
sin e'
p=f (410)
cos 24
Using the trigonometric identity
0'
sin 0'= 2sin cos (411)
2 2
Upon substitution (411) into (410), the following
expression is obtained
0'
2sinmcos
f 2 2 = 2f tan (412)
2
Equation (412) can also be expressed as
'= 2 tan'21 (413)
12f/
Another important relationship is the relationship
between r' and p. Using a trigonometric identity, (45) can
be expressed as
= f 1+tanI 2] (414)
From (412), the following expression is obtained
tan( ) = (415)
.2) 2f
Combining (414) and (415), r' can be expressed as
r'= f 1+[) (416)
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 crosssectional 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] (417)
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 ypolarized 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 (418)
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', ') (419)
C 7=[, J (419a)
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' (420)
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= (421)
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)
Sjkr'(1+cos9')
E, C = 2rC G,(',')e (422)
By comparison of (422) and (420), 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, (423)
where E, and E, denote the x and y components of the
reflected electric field over the aperture plane.
Using (421),(422) and (423), 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 '(1cos9')
E = C1Gf(0',9)e ** r ( (424a)
r' 1sin 0' sin2 '
e kr'(+c ') sin2 O' cos 8' +cos2 0'
E,=C GF,(,f')f? )  2 (424b)
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 (424), 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 (425a)
x eik[psin cos()' ) p9pd9'
E =ke (1cosO)J (E sin 0+ EY cos)
sO (425b)
Xejk[psincos(#)')lpapap'
where So is projected crosssectional 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 (416), while the conversion of O'in terms of p was given
in (413).
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 (426)
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 LateralFeed 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 (425). 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']
(427)
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 (427) into (425), the
radiated electric field components for a laterally displaced
feed change from (425) to
Ee =k (1 cos O)f (+E cos0 +E sin
s, (428a)
x e [psin cos( #)d, sin E'cos#' pd
E =jke (1 cos )f (E sin +E, cos )
s, (428b)
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 (428).
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 (429)
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] (430)
where h=(ba)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] (431)
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 (427)
in an alternative form. Based on analysis by Ruze (1965),
the phase term (427) can be expressed as
Akpcos(4'a) (432)
In equation (432), the parameters A and a may be determined
as
A2 =u2_ 2uu COS+ u (433a)
M(p) M'(p)
usin
tana= (433b)
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 (432) into (428),
the radiated electric field components for a paraboloid of
radius a are
Jkeyk7 2xa
E =kei(1 cosO)J (Eacos + E, sin4)
00 (434a)
x ejkos(#'a) pap'
E, = (1cos0) f (Esin +Ecos )
00 (434b)
Xe jkpA'') pcpa
For a rotationally symmetric E, and E, (434) reduces
to
E = jkejr (1cos6)j(Eacos0+E0 sin@) Jo(krA)pap (435a)
4rr 0
E = r (1cos0) (E,,sinE0+E cos0) Jo(krA)p9p (435b)
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
diffractionlimited feeds (defined in chapter 3) is similar.
The diffractionlimited feeds are dominant mode horn, dual
mode horn, and hybrid mode horn. The dielectric rod has
different characteristics than diffractionlimited 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 (51)
where 0Q is the 3 dB beamwidth of the oneway voltage sum
pattern, k, is the oneway 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 (51) was obtained by CramerRao analysis
method and is independent of the signal processor connected
to the antenna. With large singlepulse signaltonoise
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 shortduration range gate. The error slope
k can be defined as (Peebles and Sakamoto, 1980a)
k = (3L JdG (0) j=0 (52)
where G. is the oneway voltage sum pattern evaluated on the
boresight axis and GA(O) is the oneway 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 (53)
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 signaltonoise 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
3dB 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 Conicalscan crossover loss. (Source: Barton,
1988).
The beam deviation factor BDF is defined as
BDF=  (54)
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 signaltonoise 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 signaltonoise 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
crosspolarization will affect the antenna efficiency.
Ludwig quantified these factors by the following
efficiencies: 7, (spillover), 77 (illumination), and 77
(crosspolarization). Based on analysis by Ludwig, these
factors are defined as
o
0
o
I {IA( 6')l+l B(') 2asin dO
7= 2cot (56)
f2 A(0') + B(') sin OdW
0
fjA(')j+ B(')jf sin 6 dO'
7Ox= o (57)
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
(58)
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 crosspolarization 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' (59)
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 (510)
/ 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 (510) was then substituted for the E and H
plane feed voltage patterns given in equations 55 58.
The value of 77 in Ludwig's method was then compared with
equation (59) 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, + (511)
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 110 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 (512)
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 (513)
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 (31), the smallest dominant mode horn has
diameter equal to 0.58 wavelengths. The minimum lateral
displacement from equations 511 and 513 is
d, = 0.28A + 029A = 0.57A (514)
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 3dB 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 diffractionlimited feeds (figures 5.45.9) and tables
5.25.4, 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 3dB beamwidth of one pattern, (lower right) 3dB
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 3dB beamwidth of one pattern, (lower right) 3dB
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 3dB beamwidth of one pattern, (lower right) 3dB
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
0.581.2
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
0.901.2
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
1.281.2 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 5.105.11 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 3dB beamwidth of one pattern, (lower right) 3dB
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 5.65.8 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 3dB
beamwidth of one pattern, (lower right) 3dB 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 3dB beamwidth
of one pattern, (lower right) 3dB 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 3dB beamwidth
of one pattern, (lower right) 3dB 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 5.185.25 and tables 5.95.12. 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 (48), 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: 21
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 3dB
0.4 .. . 2 .. . .
beamwidth of one pattern, (lower right) 3dB 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 3dB
beamwidth of one pattern, (lower right) 3dB beamwidth
of sum pattern.
3
5  * ii4
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 3dB
beamwidth of one pattern, (lower right) 3dB beamwidth
of sum pattern.
