CALIFORNIA STATE UNIVERSITY, NORTHRIDGE
ANALYSIS OF SMALL PARABOLIC REFLECTOR
A thesis submitted in partial satisfaction of the
requirements for the degree of Master of Science in
Engineering
by
Louis Parker Anderson, Jr.
January, 1983
ERRATA
The following typographical errors were found after the manuscript
had gone to press:
1.
Page 24; line #16 should read;
.
2.
; J
<P
=
A
J .
s
e<P
Page 38; line #16 should read;
a
,.,
..
E = . . .. + jsine
8
4f 3k
0
J·
3.
Page 83; line #3 should read;
W.V.T. Rusch . . .
4.
Page 84; line #3 should read;
The same as above.
5.
Page 89; line #3 should read;
+1
f(x)dx .
J
-1
6.
Page 89; line #13 should read;
2
a n w;f (a2(Zi
. . . 2
+ 1))
i=O
7.
Page 90; line #4 should read;
• • • a=
8.
2.10
11
,
•••
Page 90; line #7 should read;
F = 0.380
9.
Page 91; line #15 should read;
. • . integration routine are for . . .
10. Page 70; 1ast 1 ine should read; "3 not
\f
The Thesis of Louis P. Anderson is approved:
William MacDonald
EdmondS. Gillespie
California State University, Northridge
ii
Acknowledgment
I would like to express my deepest gratitude to Prof. E. S.
Gillespie and Prof.
Sembiam R. Rengarajan for their helpful guidance
throughout the course of my thesis research.
In addition, I am
indebted to Mr. Pete Johnson, who suggested the topic of study and
Mr. Foster Beasley who V·Jas invaluable in helping to construct the
computer program.
Lastly, I would like to thank Miss Cynthia Anderson
for her cooperative attitude in the typing of the manuscript.
iii
Table of Contents
Page No.
ABSTRACT
1.0
INTRODUCTION
1
2.0
APERTURE FIELD METHOD
5
2.1
Formulation for Uniform Illumination
5
2.2
Formulation for Space Tapered Illumination
14
2.3
Computation of Radiation Patterns
18
3. 0
4.0
INDUCED SURFACE CURRENT
24
Formulation for Huygen Source Illumination
24
3.2
Computation of Radiation Patterns . . . . .
37
RESULTS
. . . . . . . . 42
Discussion of Computed vs. Measured
Radiation Patterns
....... .
42
4.1.1
Aperture Fields (Uniform)
43
4.1. 2
Aperture Fields (Space Tapered)
46
4.1. 3
Induced Surface Currents
49
ENHANCEMENT OF ANALYTICAL MODEL
....... .
53
53
5.1
Discussion
5.2
Feed Characteristics (Qualitative)
54
5.3
Monopulse Aberrations .
56
5.4
Strut Blockage Analysis Using Induced
. 66
Field Ratio Hypothesis
6.0
. . . . . . . . .
3.1
4.1
5.0
~lETHOD
CONCLUSIONS
. 82
iv
Table of Contents (continued)
Page No.
REFERENCES .
APPENDIX A.
.
83
DEVELOPMENT OF SURFACE DEFINITIONS
USED FOR INDUCED SURFACE CURRENT
METHOD
APPENDIX B.
. . . . . . . . . . .
. . . . . 85
APPLICATION OF GUASS-LEGENDRE
QUADRATURE TO NUMERICAL EVALUATION
OF RADIATION INTEGRALS
APPENDIX C.
. . . . . . . . . . 88
SMALL APERTURE PARABOLIC REFLECTOR
COMPUTER PROGRAM
91
C.1
Description . .
91
C.2
Program Listing
92
v
List of Illustrations
Figure No.
-2.1
2.2
2.3
Title
Page No.
Aperture Field Method Coordinate System
6
Small Parabolic Reflector: Aperture
Blockage (Boresight View) .
. . . . . . 11
Small Parabolic Reflector:
Blockage (Top View) . .
. . . . . . 12
Aperture
Induced Surface Current Method
Coordinate System
25
3.2
Huygen Feed . . . . .
32
4.1
Aperture Field Method Uniform Aperture
(E Plane) . . . . .
.
3.1
4.2
.
Aperture Field Method Uniform Aperture
(H Plane) .
.. .
. . . .
44
. .
45
4.3
Aperture Field Method Space Tapered
Aperture (E Plane). .
. . . . . . . . . . . 47
4.4
Aperture Field Method Space Tapered
Aperture (H Plane). .
. . . . . . . 48
4.5
Induced Surface Current Method
Huygen Illumination (E Plane) .
. 50
Induced Surface Current Method
Huygen Illumination (H Plane) .
. . . . . . . 51
4.6
Small Parabolic Reflector Monopulse
Feed Coordinate System . . . . . .
57
5.2
Small Parabolic Reflector Monopulse Feed
58
5.3
Geometry to Define Induced Field Ratio
68
5.4
Scattering Geometry of Right Circular Cylinder.
69
5.5
Generalized Geometry of Conductive Strut
73
5.6
Generalized Geometry of Conductive Strut
(Oblique View). . . . .
. . . . . . . . . . . 74
5.7
Equivalent Aperture Defined by Projecting
Strut Onto X Y Plane . . . . . . . . . .
78
Geometry of Strut of Arbitrary Cross Section
80
5.1
1
5. 8
1
vi
List of Illustrations (continued)
Figure No.
A.l
Title
Parabolic Reflector Surface Definition . . . .
Page No.
86
List of Tables
4.1.1
Aperture Fields (Uniform)
43
4.1. 2
Aperture Fields (Space Tapered)
46
4.1. 3
Induced Surface Currents . . . .
49
vii
ABSTRACT
ANALYSIS OF SMALL
PARABOLIC REFLECTOR
by
Louis Parker Anderson, Jr.
Master of Science in Engineering
January, 1983
The purpose of this study is to investigate the radiation pattern
distribution produced by scattering from an electrically intermediate
sized parabolic reflector antenna.
This investigation has been
prompted by the fact that very little has been written on pattern
prediction for this size class of antennas.
Three methods are employed
for the actual numerical calculation of the patterns.
These methods
are compared by considering the deviations from ex{sting radiation
patterns obtained experimentally.
Two methods are based on a prior
knowledge of the field distribution over the aperture; the scattered
field in the far zone being expressible in terms of the integrals of
the field vectors over the aperture surface.
literature as the aperture-field method.
This is known in the
The problem is first attacked
by assuming a u,ni formly illuminated circular aperture.
viii
Secondly, the
aperture fields are modified in accordance with the natural or "space"
tapering of these fields due to the parabolic reflector.
The third
method, termed as the induced surface current method, computes
scattered fields by integration of surface currents induced on the
reflector by known incident fields from sources located at the focus
of tbe parabola.
Due to the complexity of the integrands generated
by the latter two techniques, it was necessary to numerically integrate
the expressions to arrive at the far zone field values.
Included in
all three models is the effects of aperture blockage due to the feed
assembly.
All the techniques employ the assumption of a "point source"
feed; the simplest case being the superposition of electric and
magnetic dipoles forming a "Huygen's source" illumination.
These
three methods and their implementation comprise Sections 2.0 and 3.0.
In Section 4.0 the accuracy achieved by these methods is compared
to measured data.
Lastly, the techniques for achieving greater
accuracy in the results for the scattered field values is discussed
in Section 5.0.
ix
1
1.0
INTRODUCTION
This thesis is concerned with the calculation of far-zone
radiation patterns of an intermediate-sized parabolic reflector antenna.
To limit the complexity of the problem, certain judicious assumptions
are made concerning the sources and the boundary conditions at the
scattering surface.
Perturbation effects on the scattered field due to
aperture blockage caused by the supporting feed waveguides is considered in all calculations.
The degradation of the observed pattern
in comparison with the unblocked aperture is studied.
The formulation
of the pattern integrals for both the aperture field and induced
surface current techniques are derived from the more general problem
of determination of fields arising from a prescribed set of sources in
a homogenious medium.
The integration of the vector Helmholtz
equations,
s 2 I= -jwJl J e 2
H - s H = -jws Jm +
IJ X IJ X E
\1 X IJ X
-
IJ
x Jm
IJ X
Je
is achieved by the use of the vector Green's theorem.
magnetic and electric currents, respectively.
Here Jm' Je are
The general solution for
the field equations in terms of the sources for a time periodic field
is obtained:
+ (~'I)
VI);) dS
2
Here the Green•s function is chosen to be the eigensolution for a
spherical wave
e-jsr
lji=--
r
where:
s = w I v0 £ 0
If we consider our antenna to be in free space, this general
solution can be reduced by virtue of the fact that the sources are 'in
an unbounded region V; this results in the second integral vanishing.
Use of the equations of continuity to express the field in terms of
the current distributions alone yield
E =
p
j
jjj{(Je · v)v
+ s2 J
-
jw£
Jm x V}e
-jsr
dT
e
r
v
The corresponding magnetic field is derived directly from Maxwell •s
4n w£
equations as:
Hp = n(R x Ip) ; here n = free space wave impedance
1
Here R1 is the radial distance to the observation point as shown
in Figures 1.1 and 2.1.
The definition of the induced current
distribution J in explicit terms for integration over the reflector
surface is required.
The exact method for the determination of
J
is
to use the boundary condition for a perfect conductor at the reflector
surface; namely:
n
X
(Ei +
r)
=0
Here Ei and Es are incident and scattered field respectively.
This
will lead directly to an integral equation for J as follows:
-jsr
~ X Ei = ~ X 4 j
J . · V) + S2 J } e
dS
n w£
s
s
s
r
This equation has been shown {1} to be valid for obtaining the
ff {(
exact distribution of currents over the reflector surface for
reflectors up to twenty wavelengths in extent.
It is beyond the scope
3
of this thesis to solve this equation.
The procedure for the solution
of the above equation is outlined in {2}.
In all cases the physical
optics approximation is employed, i.e.,
Js = 2( ~ x Hi)
Here -i
H is the incident magnetic field and n is the normal to the
differential surface dS.
shadow regions.
The currents are assumed to be zero in the
For smooth reflectors this approximation is valid in
the limit of zero wavelength {3}.
The major advantage of this approach
is the reduction in complexity of software required for the solution
of the problem.
This allows direct numerical integration of the
radiation integrals (i.e., vector potential of current distribution)
to obtain the desired field values.
been reduced to:
2)
definition of the surface description in terms
1)
of the radius vector
The remainder of the problem has
p and
the differential surface element ndS, and
specification of the incident Hi field in the same coordinate
variable.
As the name implies in the induced surface current method, the
far field of the antenna is computed from the induced currents over
the reflector surface.
Whereas in the aperture field method, the
problem is reduced to that of scalar diffraction through a planar
aperture.
The generalized diffraction field can be obtained by the use
of the Helmholtz-Kirchhoff {3, 4} integral
where
-jBr
ljJ
e
= -=--r
4
Here U denotes the particular component of the aperture field under
consideration.
This equation was obtained from the general solution for the field
equations in terms of the sources {5}.
If the fields Er and Hr are
known over any surface that surrounds the reflector completely, the
scattered field Ep at some external point P is given as:
The reduction to the planar geometry is facilitated by the
assumption that the aperture field has negligible cross polarization
components and for the most part completely linearly polarized.
Additionally, across the surface of the aperture deviations from
constant phase are assumed to be small.
Thus what is termed as
Fraunhofer diffraction is considered only in the aperture field method.
For aperture field method the computation of two cases will be
considered:
1)
plane wave illumination of the circular aperture for both blocked
and unblocked conditions; (the blockage attributable to the feed).
2)
the extension of the above to a tapered distribution over the
aperture; the taper being the result of the natural taper due to
the parabolic reflector.
The complete description and implementation of both techniques of
solution will be applied to an existing parabolic reflector with known
far field patterns, for verification of the theory.
5
2.0 APERTURE FIELD METHOD
2.1
Formulation for Uniform Illumination
A paraboloid of rev.olution converts a spherical wave from an
isotropic point source at the focus into a constant phase wave-front
at the aperture plane.
Th1s surface of revolution has the property
that the entire family of rays reflected from the illuminated area
comprising the reflecting surface all lie in one hemisphere of space
The aperture field method formulates an approximation to the
{5}.
secondary pattern of the reflector via diffraction of an incident wave
radiating through a circular hole in an infinite ground plane; this
aperture being of same diameter D as the reflector.
The far zone
radiated fields are thus described by consideration of the scattered
field due to diffraction through a general aperture in free space:
IR1
E : : 2j
P
\
X
A
here,
n
Ifsa {~ ra X
=
nR1
X
(~
X
Ha)} ejS(r•. Rl)dS
Ie-~(3R
normal to aperture plane
Ia, ~=field distributions over aperture plane
The radial vectors
r•
to the source point and R1 to the observation point are defined in Figure 2.1. Assuming a Huygens source
radiator, the fields at the aperture plane can be represented in terms
of currents by way of the physical optics approximation:
-
J5
= 2(n
A
;-;-a
x H )
The radiation fields for Z>O follows from the expression for magnetic
6
FIGURE 2.1.
Aperture Field Method Coordinate System
NOTES:
1.
2.
lies in xy plane.
E vector denotes incoming wave polarization traveling in +Z
~erture
dil~ection.
'
"
3.
Feed assemblv not shown.
4.
Rl
= R Rl; 1 = r r;
r' = r r'.
I
7
Substituting the physical optics approximation, this becomes expressible
as:
l
A = -1 n x
A
ff
W eJ.8 (-.
r
·
AR )
1
l-
dS · e jRsR
sa .
We allow the radiation integral to be represented by the standard
2n
notation
N
=
flsa Wejs(r•.
Rl)ds
for further computations.
The aperture surface Sa is confined to the
two dimensional xy plane.
The radius vector
r•
to any current point
in this plane can be represented as:
A
A
r• = X •e X + y •ey
Since we are interested in free space fields at P(r, 8, $),we express
in spherical coordinates the unit radius vector from the origin:
A
A
A
R1 = ex sin e cos $ + ey sin
e sin ¢ + e2 cos e.
The radiation integrals become for the two dimensional aperture distriA
bution (i.e., Z e 2 = 0):
Nx =
JJ
H~(x•, y•)ejS(x• sine cos¢+ y• sine sin ¢)dx• dy• ex
sa
NY
ffsa Hya(x•, y•)ejS(x•
=
sine cos¢+ y• sine sin ¢)ctx• dy• ey
Recall the expression for total electrical field is defined in terms of
magnetic vector potential, i.e.:
A
IUe,
cp)
= -jwJ.l Ae ee - jw].l A¢ e¢
this will be used for final construction of the field solutions.
Subsituting the radiation integrals, the total magnetic vector potential
becomes allowing the surface normal to be in the Z direction:
8
To express this potential in spherical coordinates the relationships
are used:
A
cos
~
+ e6 cos
e cos
~
+
e~ sin~
ey = er sin e sin
~
+ ee cos
e sin
~
+
e~ cos~
ex = er sin
Q
The spherical far field description contains only TEM components (i.e.,
no Nz component):
{~ 6 cos e(!Nxl sin~
A= lTI
-j NY I cos
~)
+
~¢(1 Nx) cos ~-
NY lsin;p)}
e-j sR
R
X
The separated field components are:
I6 =
--1
E~ = -~
~ -
n case{ !Nx/ sin
n{
INx j
cos
j NY
cp -
These are the general field distributions for diffraction through
an aperture in the xy plane.
To evaluate the special case of aperture
bounds being circular, the primed coordinates will represent polar
measurements in the xy plane such that B1 = TI/2 and the source point is
described asP= P(r
dx dy
1
1
->
r
1
1
1
dr
,
1
~ ).
d~
Thus the surface element is transformed as:
1
The transformation to rectangular coordinates being:
x =r
1
cos cp
Y1 = r
1
sin<jl
1
The radiation integrals are expressed using these transformations, thus:
{J H~ ejSr sine
= - sa
1
cos(~ - ~ )rldrldcp
I
r dr d~
I
INx I
INY j = JJsa HaY ej Sr sine cos (
1
1
1
cp - cp
)
1
I
9
Assume the aperture field to be a linearly polarized plane wave,
thus:
Ha
y
( v X E~)
=0
From this, we see that the only significant contribution from the
radiation integrals comes from the Hy ·component,; i :e;: •
N
y
JJ
= __fu__
Til•
ejsr• sine- cos(cp- cp•) r• dr• dcp•
sa
the specific integral under consideration is derived using the boundary
conditions of the physical aperture of the problem, thus:
2n a
J J ejsr•
0
sine cos(cp- cp•) r• ctr• dcp•
0
This is reducible by the following relations:
21r
J ejXcos
c; de;
0
a
J uv
Jv
-1 (u)du
0
For the case of v = 1 the double pattern integral is thus reducible to:
2na 2 JJ(sasine)
sasine .
therefore:
2
N = 2na Eo J1(sa~ine)
Y
fi
sas 1 n e·
-.
The corresponding field components are thus expressed as:
f
I
. 2 Eo cos e cos,!, JL (sa~in e)
e = JS a
'~'
s as 1 n e
I
e JS
R
R"
ee
10
E<P = j sa 2 E s i n<P
o
Ill(
sa~ i n e)
sas1n e
I
e- j sR
R
~
<P
for the elevation and azimuth principal planes respectively.
These are the expressions for the unobstructed circular aperture
far zone field distributions.
The particular problem of interest
contains aperture blockage caused by the necessity of physical placement of the feed at the focus of the reflector.
This blockage in turn
modifies the diffracted field produced by uniform illumination.
The
case under study can be reduced to that of two main shadow
contributions 1.
The rectangular strip waveguide assembly
2.
The square feed assembly
This approach assumes zero field strength in the shadow region, (i.e.,
"null field hypothesis").
Here each contribution is subtracted from the unobstructed field for
the total resultant field, i.e.:
Etotal = Eunblocked - Ewaveguide - Efeed
aperture
strip
The projected shadow of each blockage component is formulated by first
expressing their respective radiation integrals:
1.
Waveguide Strip:
+WS
- +L e
2
N( s) = f:D_
ej s ( x sin e cos
y
2.
Tl
J J
-W~
2
-L
I
n
-W.f
e
<P
+ y sin
sin
<P
+ y s1n e s1n
<P)
dx dy
I
1
1
dyl
e
Feed Assembly:
. ( x s1n e cos
E +WfJQ. eJS
N(f) = ~j't
2
Y
1
1
•
I
•
•
<P
)
dx
-!l,
y 2
here: Le = a - £/2
Direct integration of the radiation integrals in all cases can be
11
A.op
= 0.675"
!
I :
I
;-.-11
1
FIGURE 2.2.
= 0.350"
Small Parabolic Reflector:
(Boresight View)
Aperture Blockage
12
.\op = 0.675"
Waveguide Support
Structure
r -- -
-r-"'-~....::...-----+-----~~~-,
I
f = i. 380"
I
A!)erture Plane
-~--¢::::==~--=::::::::-4-.::::::::::::..__-l::::::==::L-_j:_~,ll
j
~-t-------o=
FIGURE 2.3.
4.20"
Small Parabolic Reflector:
(Top View)
Aperture Blockage
13
accomplished entirely analytically, i.e.:
+L
sin e, cos
1.
t1>
dx •
J
e J• oy I S i n 8 S i n
e
IJ
,f.,
'~'
dy'
-Le
w
= 2L~ Ws Eb [sin(sLe sin e sin rp) sin(sy sin e cos t/>)J
Ws .
S- s 1n 8 cos t1>
n
Sle sin e sin t1>
2
Most similarly, the feed assembly expression is transformed as:
N(f) = Wf i Eo
2.
y
·[sin(s~ sin
t
8
.
sin t/>) sin(s+ sin e cos
.
s2 s 1n e s 1n
n
--wf
t1>
r~>)]
.
s2 s 1n e cos
t1>
The complete radiation field for the two dimensional blockage shadow
due to the waveguide strip in the elevation plane becomes:
r(s) =
2jle Ws E0 [sin(sLe sin
8
s L sin e sin rp
-jsR e
A
X
cos e cos
e sin
t1>
e R
w
rp)
sin(~
sin o co:
•l]
s Ws sin e cos
2
likewise the azimuth plane field:
w
-(s) = 2jle Ws E0 [sin(sLe sin a sin •i sin( s-f sin
E t/>
w
A
sLe sin e sin tl>
s-f sin 8 cos rp
The shadow contribution due to the feed blockage becomes for the
elevation plane:
w
I(:) = j~~f
9..
A
-js R
Eo sin ( sf sin e sin t1>) _s_in--;-(-;-s---'2=-f-s_1_·n_e_co_s_cfl_) ]
wf
.
cos
e
cosp~
[
sf sin e sin rp
s2 s1n e cos
tl>
14
Likewise for the azimuth plane:
E0
w
[sin(S~
--f
sin e cos ¢
sin 8 sin¢) sin (13
W
Q,
•
•
f .
132 Sln 8 Sln¢
2 s 1n s cos <P
A.
)J s1n¢e
.
A
<P
The components of the aperture shadow region being defined
0
explicitly, we can now arrive at the expression for the total field
by superposition of the individual terms:
E
total
_
E
.
e
=JSE
= E(ap) - ~s)
{-_J 1(sasin
o
Wf.Q.
- -2
E¢ = jS E
- E(f)
LeWs [-s,_·n_(_sL_e_u_) sin(si- v)J
7T
Ws
Basin a
SL u
s2 v
W
e
[sin(s¥u) sin(s 2f v)
-jsR
W
cos e cos </> e R
e8
QQ,
f
f.J2 u
·- 2 v
8)
---
]l
ft
l~
sin(~-j- v)]
-{J 1(sasin8)
0
s~ v
2
sasin e_
w
wf~~,
[sin(st u) sin(s-f v)
W
f
2n
B U
By v
t
-
]J·s1n¢
A
e¢
Here use of the substitute variables are employed:
u = sin e sin ¢ ; v = sin 8 cos <P
2.2
Formulation for Space Tapered Illumination
The nature of the parabolic reflector manifests itself as a
natural amplitude taper in aperture plane {6}. This effect is termed
as "space loss" or "space taper" due to the parabolic surface.
To
15
study this effect on the secondary far field distributions, the
aperture field caused by a uniform incident plane wave becomes space
tapered and outlined by Jones {7} to be of the form:
~
j E
= - -0
4A.f2
(1 +cos w)
2
. (f +
e-Js
z0 ) e
A
X
The aperture phase constant B(f + Z0 ) is
d~fined
in figure 3.1.
Projected onto the reflector aperture, the radial coordinate is
introduced:
This aperture distribution now includes space taper and the Stokes
inclination factor {4}, a characteristic of Huygen source fields.
Note
that phase and not amplitude is held constant at the aperture due to
Fermat 1 S principal.
The aperture magnetic field becomes:
rA
..a
H =- e
n
y
The corresponding radiation integral is thus expressed:
a
r eJ·sr sin@ cos(~-~~)
'~'
'~'
J
-----;~=----
0
2 2
( 1 + ({f)
dr dcp
)
The angular variable of the radiation integral can be reducible by
usage of the Bessel relation as before:
)!
a
_ 21Tj E0
N -
y
nA. f
e
-js(f + z
o
0
r J 0 (srsin e)
2
(1
+ (!f) )
2
dr
16
The secondary far field radiation patterns for the aperture are
related for the ex polarized incident field:
L8
E<P
-- j i3n e-ji3R
ZnR
- ji3n e-ji3R
Zn R
-
IN
I
t y cos e cos
INY.I
si n
<P
<P
The field expressions are found by direct substitution of the
space tapered form of the radiation integral:
a
2n E
0
<P
J
0
r J 0 ( srs i n e)
2 2 dr
(l+({f))
l
R
and
Here the phase constant of the aperture is represented by the
dummy variable:
c = 13( f + z0 )
The blockage terms due to the feed assembly remain unchanged
since the field taper here is attributable to the parabolic reflector
surface only.
Thus all that is required for pattern computation is the
normalization of the primary aperture terms for the principal planes.
This procedure is covered for both plane wave illumination and space
tapered aperture cases in the following section.
the total fields including blockage become:
The expressions for
17
E8
=
!
2jEo
a
j,
Af
A
e
~~
r Jo{13rsin8) dr - LeWs ( sin(BLeul sin{13 2 v)
13!is__ v
(1 +(!f)2)2
13Leu
2
f
-jc
0
Q,
_ Wf<
2
(sin(~2
u)
sin(eY
13.&. u
2
vi) j cos8
cos¢
13y v
e-ji3R
R e8
A
X
E
<P
=
2jEo
A
~
a
j,
Af
e
-jc
J
0
r J o ( 13 rs i n8)
dr - LeWs ( sin(Bleu)
(1 +(_I_)2)2
13Leu
2f
£
_ Wf< (sin(B2 u)
2
sin(~Wf
13.&. u
2
v))
13 2 v
e-ji3R
X
R
A
e<P
as before the substitute variables are represented by:
u
=
s in 8 s in
<P ;
v
=
sin 8 cos
<P
l
!is_
sin(B 2
sin¢
v))
13~
2 v
18
2.3 Computation of Radiation Patterns
Typical engineering measurements of antenna radiation patterns are
made in the principal planes (i.e., yz and xz planes) of the reflector.
To compare our theoretical prediction with those of actual measured
data, we must convert our field expressions above into that of principal
plane normalized power patterns.
Initially we observe that the plane in which the incident E field
vector lies (i.e., E plane or elevcition plane) is described by the
azimuthal variation being constant at the value
of~
= 0.
total fields are comprised of the e8 directed component.
of the unblocked aperture this becomes:
I
8
Thus the
For the case
- IJl (sa~ln8
. ) 1e -jSR e~
= jsa2Eocos8
sas1n8
R
8
The normalized pattern factor is defined as the pattern divided by
its maximum value:
n=O,l,2
The limit is evaluated using L1 Hospital •s rule
lim JI(Sasin8) = lim d~ {Jl(sasin8)}
sas i n8
8+2n1T
8+2nrr
recognize also that
such that the unobstructed normalized E plane pattern becomes
expressible as:
E = cos 8 2Jl(Sa~in8)
8
sas1n8
19
The corresponding power pattern follows directly, i.e.,
= 20 Log
j2J 1 (sa~ in e)
sasw e
10
cos
el
The plane containing the magnetic field vector (H plane or azimuth
plane) is described by the azimuthal variation being constant at the
value
of~
A
= n/2.
Therefore, the field is comprised of only the
e~
directed component and the H plane field becomes:
I
~
= _j 8a2E
I J (
.
e)
e
1 Ba~ 1 n
'
o}
sas 1n
l
e
-jSR e
A
R
~
The unobstructed normalized pattern factor is found similar·ly:
E
~
= ~=
Emax
2Jl(Basin e)
Basin e
the corresponding power pattern:
P<P
2JJ(Basin e)
Basin e
= 20 Log 10
Obtaining the power pattern for the blocked aperture will also be
based on the linear superpositioning of the unblocked aperture pattern
with those of the field and its associated waveguide support structure
shadow patterns.
It must be noted that during the normalization
process, vital information with regards to the physical area of the
blockage is lost.
This will be compensated for by using area ratio
multiplier terms to compensate the derived blockage terms .
....
The elevation plane ee directed components for the blockage
become:
1.
Waveguide Strip
E(s) = 2jleWsE 0
e
A
{
t
Ws sin e) }cose e -jBR
sin(B2
Ws .
R
By Sl n e
(~
= 0)
20
Since
Ws
sin e)
L1·m sin(s~
Ws .
e4nTI s ~ s1n e
=1
= 0,
n
1, 2 ...
the pattern factor becomes:
Ws
= sin(s~ sin e) cos e
sW ssine
2
The correction factor is found for the physical dimensions of the
problem by taking the ratio of the total area of the reflector with
that of the blockage region:
Here
= 0.850
t
Arefl = 'IT ( 2. 1)
1
t
11
A.Op
= o.600
11
= 0.675"
2 = 13.85 in 2
Astrip = Wsl~ = 0.99 in
Le = 2a-
wf
Ws = 0.295"
a= 2.1 11
2
= 3.35 in
Le = a - t/2 = 1.68 in
• A = Astrip = 0.071
1
Arefl
'·
The modified pattern factor is therefore:
{
Ws
E(~) = A J sin ( s2 sin e ) cos e
1t
SW2s s i n e ~
l
2.
Feed Assembly
¥
J
-jSR
E( f) = j W-ft Eo { s i n( s s in e ) c0 s e e
Wf s1n
. e
e
R
A. "
s-
2
The pattern factor follows by first calculating the correction term
Afeed
'
0
=
W-ft
=
0.51 inl
21
. A - Ateed = 0.037
2 - Arefl
thus
~{f)
Ee
Wf sin e) lcos 8_
= A sin ( BT
2{
BWf
.
T s1n
8
The total elevation plane pattern factor for the obstructed aperture
becomes:
-{T)
Ee
=
f~_tl-
l
Basin e
the power pattern follows
Pe = 20 Log 10
Ws
A sin(s2 sin e) - A2
1
BWs sin e
2
Wf ~in e)
sin(~(
By s1n e .
l
case
as~
IEe \
In similar fashion, the azimuth plane power pattern becomes (¢ = n/2)
p = 20 L0 g
¢
10
f2Jl(Ba~in
e)_ A
Bas1n e ·
1
t
sin(Ble~in
e)_ A sin(BI sin e)~
BLesln e
2
Bt sine j
The normalization of the tapered aperture is accomplished:
a
_( rJo(Brsin e l d
- _ Ee _ cos~o (1 + (r/2f)Z)2 r
Ee - -E- r.1
r dr
max
(1 + ( r/2f)2)2
J
0
the normalization integral is reducible by direct integration, thus:
a
/
r dr
(1 + (r/2f)2)2
L
(a/2f) 2
2
_ 21
- 2f 1 + (a/2f)2J- 2f k
0
The normalized E plane field can now be evaluated
•
!)
22
likewise for the H plane fields the normalization is accomplished:
dr
The final for field normalized patterns for the blocked tapered
aperture becomes:
(1)
(~
Elevation Plane
= 0)
r.
Ws
A sin(s2 sin e) _ A sin ( s2Wf sine~ os e
2
Wf .
1
sW s sin e
Sy
Sln e
2
the power patterns are only defined for Pe:
IE(~) j .
Pe = 20 Log 10
(2)
~(T)
E
~
Azimuth Plane
a
I
_ 1
- 2f2k
(~
= n/2)
~
2
d
A sin(sLesin e)
r - l
Slesin e
- Az -=-s- .:. . :in-'-'('-£"s..£::::.2-=-s. n.:.___:_e-'--) ]"
,.:...c.·
s2
s1 n e
the corresponding power pattern is defined only for
P~
= 20 Log 10
\
P~:
E(;) 1
These comprise the power pattern formula for the space loss tapered
~
blocked aperture.
Since the incident field is ex directed, the block-
age terms exhibits the most perturbational effects on the elevation
plane fields.
It is expected that the beam degradation effects due to
the blockage producing an inverse taper (amplitude depression) are {14}
1)
Sidelobe level increasing (over that of uAiform unblocked illumination)
2)
Gain reduction (loss in main lobe directivity)
3)
Beamwidth narrowing
23
The radiation integrals produced for the case of the space loss
tapered aperture are evaluated numerically.
For maximum accuracy of
the approximations used for these integrals, it has been found {13}
that the method of Guassian quadratures yields low errors in comparison
with other more standard methods of numerical integration (see Appendix
B for details).
Thus the integrals will be approximated by forms as:
a
N
f
2 dr
0
~
~
i=l
w~
This method, though, is limited to prediction of close-in sidelobe
structure.
Outside these limits, errors become larger rapidly.
This
is possible due to reflector edge scattering effects not accounted for
in the formulation.
The computed power patterns, therefore, will be
constrained to ±30° off boresight field of view.
! .
j
24
3.0
INDUCED SURFACE CURRENT METHOD
3.1
Formulation for Huygen Source Illumination
The far field of a general scattering surface can be expressed by
the integral form derived by Silver {5} as:
E = -jw]l e -jsRJJ {J - (J
p
4TIR
S
S
R )R } e-j!3(p ··R1) dS
1
1
s
The approximate current distribution to be assumed from physical optics;
-i
A
Js = 2(n x H )
here
n = reflector surface normal
Hi = incident field
Maxwell's equations allow the physical
opti~s
current to be expressible
in terms of the incident electric field:
J
2
= -
s
n
A
A
•
n x (e x
P
E1 )
Observation of the integrand yields the· fact that {Js - (Js • R1)
R1} is just the component of the surface current Js perpendicular to R1
in the observation direction, thus:
A
here J 8
= Js • e8
;
J<l> = Js • e8
A
The far field can now be reduced tof_its e8 and e<l>
_
A
A
A
A
EP ={(I ··e 8 )e 8 + (I· e<l>}e<l>}
components
e-j8R
R
the radiation integral being represented by the dummy variable I;
I
=
-jw]lj 1~
2Tin
s
X (
~
X
p
ri )ej 8 (p. R.)
dS
25
/
Aperture .Plane
I
I
FIGURE 3.1.
Induced Surface Current Method Coordinate System
26
From Figure 3.1, it is seen that the reflector surface is described by
the radius vector
The symmetric case under study, (i.e., parabolic reflector) can be
described as a surface of revolution about the axis of radiation; thus
the radius vector will vary with respect to subtended angle only
P = p(l/J)ep
The incident field Ii produced by the feed being placed at the origin
of the coordinate system can also be expressed in terms of its
spherical coordinates (p,
.
f
Il = I (1/J,
~. ~)
-jSp
~ )-=-e_ _
p
The feed coordinates are expressible in terms of their separated
components
If = El/J etjJ
+
S:
e~
The far field coordinates are just the mirror image of its associated
feed coordinate system.
The relations of the spherical/rectangular
transformation in both systems become:
x =sin
8
cos¢ = x•
y =sin
8
sin¢ = y• = sin
z = cos
8
=-z• =-cos
=sin~ cos~
l)J sin~
1/J
The primed rectangular coordinates are those of the feed location. Note
that in the case of offset feed positions (as in offset reflectors or
monopulse systems) the primed coordinates are translated and/or rotated
with accordance to its respective element position {8}.
The case
under study considers only sum (I) patterns for computation.
'
0
27
For computational purposes, explicit definition of the radiation
integral amplitude and phase terms is required.
The phase term is con-
sidered initially by expressing the individual components of the dot
product in both spheri ca 1 and rectangular reference systems:
p
= p sin
ljJ
cos t; ex + p sin
ljJ
sin t; ey - pcos
ljJ
ez
and
Rl = sin e cos
ex + sin e sin
cp
ey + cos e ez
cp
The total phase term can be expressed thus:
p
.
~
Rl = p(ijJ) {ep • R1}
= p(lji) {sin
cos t; sin e cos
ljJ
cp
+ sin
ljJ
sin
t;
sin e sin
cp -
cos
ljJ
cos e}
This now explicity expresses the phase term in the coordinate reference
frame.
Attention. is now focused onto the amplitude term.
A generalized
approach using differential geometry techniques is used for the
definition of the reflector surface vector properties.
Initially the
amplitude term is simplified by the vector triple cross product
re 1at ion, i . e. ,
~
~
-f
-f
-f
n x ( ep x E ) dS = {( n • E ) ep - ( n • ep ) E } dS
A
A
A
The normal to the reflector surface can be expressed in feed coordinates
(see Appendix A)
~ dS
·-
= op · X
'dlji
~ dlji
31;
dt;
here.
The reflector surface under study is a symmetric paraboloid of
revolution abo.ut the Z axis; described in terms of the feed vector p
28
is expressible as:
- 1 +2cos
f 1/J
p (''')
't'
-
= f sec 2 ~2
Thus the associated surface derivatives become
]£. =
31/J
ptan ~2 ·, .£2_
=0
31;
The normal to the reflector surface can now be explicity expressed
2
A
dA
A
3A
ndS = {-p sin 1/J e p + p sin 1/J 3 ~ e¢ + p 3 ~ e~} dl/J
dt;
For the parabolic reflector this becomes:
A
n dS
=
2
1/J
A
A
p sin 1/J {tan 2 eljJ - ep}
The expansion of the cross product amplitude term follows by consideration of the now defined individual dot product terms:
-f
a
3
( n • E ) ep dS = psi n 1/J El/J 3 ~ + PEt; a~
A
A
Further defined for the symmetric parabola
-f
A
A
(n • E )ep dS = (p
2
Y!__
A
sinljJ El/J tan 2 )e p
likewise
(~ •
~P )If dS = -p 2 s i nljJ ( El/J eljJ
Thus the amplitude term becomes:
a
a
-f
A
n x ( ep x E ) dS = p{ ( Elj! s i nlj! a~ + Et,: a~ ) ep + ( p El/l s i nl/J ) el)J
A
A
A
A
+ ( p El; s i nl/J ) et;
}
dljJ dl;
or more specifically for the reflector type under consideration:
~ x (~p x Ef)dS = {(El/J tan !)~p + El/l ~ + ~ ~~}p 2 sinl/J dl/J dl;
If we allow the
Ep
=
El/J tan
~ubstitute
variable
!
we can formulate the total radiation integral as:
. &
29
1jimax
I
= -j
A.
t;;max
f. f.
1)!ml n
.;m1 n
or more compactly as:
I -T
- -j
·~~
s
The actual computation of the radiation integral will be
facilitated by the expression of the amplitude vector
r
in terms of
the rectangular coordinate reference frame since the spherical unit
vectors vary over the surface of integration. I is expressible in
terms of the far field unit vectors by first considering the quantity
in primed feed coordinates:
A
A
A
ep = sin1)! cos.; e'X + s in1J! sin.; e'y + COS1ji e'z
el)J = COS1ji cos.; e'X - COS\jJ sin.; e'y s i n1ji e'z
e.; = -(sin.; e~ + cos.; e;)
A
A
A
A
A
The appropriate translation expressing the mirror image relation
betvJeen the feed and far field coordinate systems:
A
A
A
A
e z = -e•z
ey = e'y
The amplitude term can now be expressed in far field rectangular unit
ex = e'X
vectors, with the feed coordinates as a scalar parameter:
A
Ex = ( Ep sinljJ cos.;
+ EI/J COSlj; cos.;
E.; sin.; )ex
A
rY
= ( Ep s inlJ; sin.; + EljJ COSlj; sin.; + E.; cos.; )ey
Ez
= ( E1Ji s inljJ - Ep COSlj; )ez
30
These quantities are now employed to resolve the corresponding
rectangular components of the radiation integral.
To define in its
entirity the complete radiation integral, it would be judicious to
consider the reduction of the phase term to its simplest form allowing
=
<P
<P
(lji,
1;,
e, <P) = S{(p· R1)- p}
Factoring the scalar part of surface vector, we have:
Once terms are grouped and simplified; this becomes:
<P
= Sp(lji) { 1 + coslji case - sin1)i sine cos(!; - q,)}
Thus, the final field is calculated by recognizing that the field
is expressible in terms of the superpositioning of its coordinate
components:
EP(e, <P) = Ee ee
+ E<P e<P
This can be expressible in terms of the rectangular components of the
defined radiation integrals:
. e-jSR
- Iz sine) R ee
h
Ee
=
(IX
COS8
E<P
=
(Iy
COS<jl
+ IY case s i nq,
COS<jl
IX sin¢)
e -jSR
R e<P
h
The radiation integrals being defined in the resolved component forms
I =~
E eJ.<P psin1)i dlji di;
X
A
S
X
. Il
Iy = ~
EY eJ psinlji dlji di;
Iz
E eJ psinljl dljl di;
2
ffs
= -~ Jfs
"<P
"<IJ
The field distribution of the illuminating source set at the focal
point of the reflector must now be defined to completely specify the
entire radiation integral.
The actual feed to be modeled is a four
port rectangular waveguide configuration for monopulse pattern
31
excitation in both principal planes.
This would require modification
of the radiation integral phase and amplitude terms to accurately model
the incident field.
This is addressed in Section 5.
The computed
results will employ the simplifying assumption of a generalized Huygen's
source radiator as the feed.
This may be represented as the orthogonal
superpositioning of electric and magnetic dipoles located at the focal
point (and phase center)_of the parabola.
The polar components of such
a feed with the electric dipole polarized along the X' axis and
magnetic dipole (see Figure 3.2) situated in the X' z• plane are:
-
. •. e-j8p
EljJ = E0 cost;. ( P1 cos•~1• + P.>
eJa)
-~
p
A
elji
and
The quantities are defined here as:
P1 = excitation amplitude of electric dipole
P2 excitation amplitude of magnetic dipole
=
a
= phase difference between orthogonal dipoles
Assuming the reflector to be in the far field of the feed. the
condition of plane wave polarization is such that the excitation
amplitudes are equal and no phase difference exists.
Pl = P2 ;
a
=0
such that
Eljl = E0 cost;.(l + cosljl)
e -j8p
e-j(3p
Et;. = -E 0 sint;.(l + ~osljl)
A
eljl
P
P
A
et;.
Thus:
32
/
to\agnet'
,.,.
.c
//
;;;pole (Loop)
;..,'
FIGURE 3.2.
Huygen Feed
33
The formulation of the radiation integral for the specific case under
study is now complete.
The individual amplitude term components are:
Ex = E0 (1 + cos*)
Ey = 0
Ez
The
= E0 (sin*
corr~sponding
I X ~ -Zjf
E
A
0
c?s~)
radiation integral components become
JJs
·
sin* ej<P
d* d~
Iy = 0
Iz = -Z~f E0
J~
sin* tan ~cos~ ej<P
d* d~
These results support the premise that if a symmetric parabola is
illuminated with plane wave polarization only axial and polarized
components of surface current exist {3, 8}.
The axial component Iz is usually ignored in the majority of field
computations due to its non-contribution to the on axis peak field
intensity, albeit it does modify the sidelobe structure {3}.
It can
also be seen from the established field formulation that this component
also has no contribution to the azimuthal component (E<P) of the field
~
·~
because e<jl is rotating in a plane normal to ez.
This is consistant with
results derived from aperture field techniques to expect the contribution due to the axial component to figure promenently in the increase
in sidelobe intensity in the elevation plane due to the blockage taper
effect.
The aperture of the reflector as described in Section 2.1,
contains blockage attributable to the waveguide feed assembly necessary
for illumination.
Initially, the field distribution for the case of the
unblocked aperture is calculated.
The projection of the reflector
surface onto the aperture will be useful for the simplification of the
34
radiation integrals.
Relations for the transformation of trigonometric
integrals are used {9}.
if tan
! =z
then
2z
2dz
sinl/J = r+z2 and dl/1 = 1 + :z2
Recognize also that
tan t = __!:__
2 2f
and
=r
psinlj!
Consideration of the radiation pattern sidelobe structure close to
boresight (i.e., +30° scan off of boresight) allows the approximation
of the radiation integral to simplify numerical evaluation as:
Bp(w)U + cosl/J case}"' 2!3f if case "' 1
The components of the radiation integral now become:
2n
I
X
= -2j
A.f
E e-2jsfj
0
0
1
a
0
and
Here u
re~resents
u
the reduced variable
= psinw sine = rsine
The transformation of the circumfiremtial integration over
by use of the Bessel relation {7}.
2n
ejsu cos(~ - <P) cos n~ d~
J
0
= 2njn cos
n<P
Jn(su)
~
is aided
35
The cases of the polarized and axial surface current component
contributions of the radiation integral become reducible respectively
as:
2rr
J ej su cos ( ~ - <P)
d~
= 2rr J o ( su)
0
and
2rr
Jf ejsu cos(~-¢)
cos~ d~ = 2rrj cos¢ J (su)
1
0
The reduced radiation integrals become respectively:
Ja
= -4rrjE0 e-ZjSf
IX
r Jo(srsine) dr
o (1 + ({f)2)2
A.f
"'fc
= 2rrE 0 e-Zjsf
cos'~'
Iz
f2
r 2Jl(Srsine) dr
r 2 2
A.
.. o 0.+ (2T))
The axial component of the radiation integral is dependent on the
observation angle¢.
From this fact, we can deduce that the contribu-
tion to the sidelobe level degradation will be observed only in the E
plane(¢= 0) radiation patterns.
The component parts of the resolved radiation integrals being
established, the far field pattern distribution can now be written;
-jsR
E = (Ix case cosq, - I 2 sine)e R e8
8
. e-jsR
E<P = (-Ix sin¢) R e<P
A
A
Substituting our formulated radiation integrals, the field distributions
become:
Ee =
-jSEoe-2jSf
f
cos•l2 cose
e -jsR
X
A
ee
R
·1
l -
.
- 2j sf sin¢ ~
a r J a ( srs1ne
. ) dr e j S R e
E = -2JSEo
¢
f
( 1 + ( {f) 2 ) 2
R
<P
0
A
36
These are the induced surface current derived far zone fields of
a uniformly illuminated parabolic reflector.
As in Sections 2.1 and
2.2, our prime concern is that of the fields created by the parabolic
reflector with the aperture blocked by the necessary feed assembly.
We shall, as before, employ the "null field" hypothesis for the effect
of the blockage.
Due to the fact of the radiation integrals in the
above expressions being in terms of the reflector surface as projected
onto its respective aperture, we can justifiably use the model of the
blockage being represented as rectangular apertures in an infinite
ground plane as the null fields.
These have been developed in Section
2.1 and can be substituted directly:
Ee
::;:
-jsEoe -jsR
fR
COS<jl
[ e-2jSf
a
+ js~ne
J
0
!
a
2cosej r J 0 { s rs i ne}
(1 + (_r__)2)2
2f
0
r 2J J ( s rs i ne }
(1 + (_r__)2)2 dr
2f
J
!is_
LeW sf
'IT
Wftf
- --z;;:-
J sin(SLeu) sin(s 2 v}
l
I
Sleu
t
sin(~2
u)
su
2
dr
sws v
2
Wf
sin{s2 v)
sWf v
2
l
COS8
J case J
ee
37
likewise
a
0
-l
sin(Sleu)
Sleu
Ws
sin(s2u)
Ws
LeWs
r J 0 ( f3 rs i ne ) dr
f
(1 + (_!'_)2)2
2f
~
l--\~fe; ~sin(s2u)
s;ru
2n
(
98~
as before:
u = s i ne s i n<P
V
= S i n8
COS<jl
It can be noted that with the exception of the constant
coefficients of the radiation integrals, the two techniques (i.e.,
aperture field method vs. induced current method) are essentially
equivalent; their only difference being that of the axial component of
the radiation integral.
It is the effect of this additional component
on the sidelobe structure in the elevation plane blocked aperture field
distributions for which this study has been inspired.
These fields can
now be normalized for computation of the power patterns.
3.2
Computation of Radiation Patterns
As before, in the previous chapter on aperture field methods,
interest is focused on the computation of
fields of the reflector.
th~
principal plane radiation
In particular, the normalized power patterns
will be generated to _determine these distributions.
Initially, we shall seek to normalize the field distributions of
the unblocked reflector aperture.
The elevation plane fields are first
38
= 0).
(~
calculated
~2
Ee "-jSEa:-2jBf
-3.
cos ef_
0
e- j sR
x
A
ee
R
The normalized form of which is derived from:
-E
E
= _s_a_
8
Emax
Examination of the maximized field is accomplished by observing
that maximum intensity will be at e = 0.
Thus for the
~ =
0 principal
plane field:
now
I (o)
=
.
-4rrJE 0 e
-2j sf
A.f
X
a
J
0
r dr.
(1 + ({f)2)2
From results derived in Section 2.3, the integral is reducible to
a constant quantity:
• E
.fE -2jsf
· · max
= - 8rrJ
a
A.
-jsR
k _e--=-_
R
The normalized field pattern for the elevation plane thus becomes:
2
r J1(srsine)dr
(1 + (;f)2)2
as before
(:f)2
k
= --=-=----;;:;1 + (~)2
2f
The corresponding power pattern can be found by taking note of the fact
39
\
here
-[
8
= Ee/Emax
now
-E •
8
- * =
Ee
~cos2a
2f k
J r(1 J 0+(~rsina)dr}
({f)2)2 '
(\
2
+ fsina
4f3k
0
Thus for the unblocked elevation plane power pattern we have:
2
Jar
p = 10 Log
lease
J 0 (srsine;dr}
8
10
2f2k
(1 + (J::_) 2) 2 '
1
o
2f.
J
The power pattern for
the~=
+ lsine
{4f3k
2
2
r J1(srsine)dr·l
22
o (1 + (_I_)
2f )
.
Ja
n/2 principal (azimuth) plane follows
similarly as:
-'jsR
.fE -2jsf e-jsR
k --:::-Emax = E~ (o) = -Ix(o)e R e = - 8n J Aoe
R e<P
A
thus
I -
P~ = 20 Log 10 j E~
I
1
= 20 Log 10
Note that in the elevation plane, the contribution of the axial component becomes predominant only in the far out sidelobes.
This is due
to the sine coefficient controlling this contribution.
The case of the apertvre being blocked by the feed assembly and
its supporting waveguide is attacked in the same manner as in Section
2.1, using the "null field" hypothesis.
The elevation plane fields
must first be grouped into real and imaginary parts before the computation of the power pattern is accomplished.
Thus:
40
-(T)
Ee
=
The blockage is still modelled as two dimensional radiating apertures.
Thus once terms are grouped we have for the normalized elevation plane
field:
a
Ws
Wf
r Jo(Brsine) dr _A sin(S2sine) _ A sin(B2sine)
2 sWf ine
1
BWs sine
( 1 + (___.!:._)2)2
2f
2
2 s
f
0
l
x cose
The corresponding power pattern is handled as the previous case of the
unblocked reflector:
Pe
X
I *I
Ws
= 10 Log I(f_1_ Ja r Jo(srsine)dr _A sin(s 2 s_i_D_tl_ A
1
2
10 l2f2k · o (1 + (___.!:._) 2) 2
sWs sine
Wf
2f .
2
2
2
2
sin(s2sine) l
e) +(sine Ja r Jl(srsine) dr )
Wf
cos
r 2· 2
= 10 Log 10 Ee • Ee
4f 3k
By sine
o (1 + ( 2f) )
The power pattern for the azimuth plane is found to be that of the
results derived by the aperture field method as in Section 2.2, i.e.:
a
1
2f 2k
£
x s i n ( s2 s i ne )
£ .
s2 s 1ne
~
r J 0 (ersine) dr _A sin(BLesine) _ A
2
1
2 2
(1 + ({f) )
sLesine
41
The area corre1ation factors A1 and A2 are as defined in the Section
2.3 pattern computation. The integrals are also approximated by the
Guassian quadrature method as well.
42
.4. 0 RESULTS
4.1 Discussion of Computed vs. Measured Radiation Patterns
The primary effort of the theoretical analysis presented was twofold:
1.
To find the most simplistic model possible to adequately
predict secondary radiation pattern characteristics of the
antenna system under study.
2.
To observe the effect of the longitudinal (i.e., axial)
component of surface current on the sidelobe structure of
the secondary radiation patterns.
Measured radiation patterns have been superimposed on computed
results for direct comparison.
The computed results comprise both
unblocked and blocked aperture conditions to aid in the evaluation of
pattern degradation effects of the feed structure blockage.
Four
major pattern parameters are used to evaluate the accuracy of the
computed results as compared to measured data.
power (3 dB) beanMidth, (2)
sidelobe level and (4)
the main beam.
They are:
( 1)
beamwidth between first nulls, (3)
half
first
location of first sidelobe peak relative to
Accuracy is judged to be sufficient if the computed
value is within 10% of the measured value.
These parameters are
presented in tabular form to supplement the graphicaJ presentation
of the data.
Acronyms are used in the tables for sake of brevity.
These are HPBW (half power beamwidth), BWFN (beamwidth between first
nulls), SLL (sidelobe level) and SLA (sidelobe angle).
43
4.1.1 Aperture Fields (Uniform)
The computation of the radiation patterns of a 6\ diameter
uniform aperture was done as a first iteration for a heuristic approach
to the analysis of the reflector.
TABLE 4.1.1
E PLANE (EL)
H PLANE (AZ)
Parameter
Thea.
Meas.
%Dev.
Thea.
Meas.
%Dev.
HPBW (deg)
9
11
18
9
14
36
12.5
16
22
15.5
20
23
BWFN (deg)
20
25
20
22
39
44
SLA
15.1
17.5
14
15
22
32
SLL
(dB)
(deg)
Examination of both tabular and graphical formats (Table 4.1.1
and Figure 4.1) indicate a less severe variation in between theoretical
and measured results in the elevation plane.
This seems to indicate
an aperture taper in the H plane of the feed which has not been allowed
for in this analysis.
This aspect can be accounted for by a more
detailed formulation of the primary (feed) pattern.
The feed presents
a slight defocusing effect in both planes since it is designed for
monopulse excitation.
This characteristic and the filling of the first
null in both E and H planes will be discussed in Section 5.2.
It can
be noted that a 1 dB drop in gain is observed as referenced to the
theoretical unblocked aperture.
This effect is due to the inverse
taper caused by shadowing from the feed structure.
Due to expected
minimal correlation between computed and measured patterns the measured
patterns are not superimposed in Figures 4.1 and 4.2.
44
A~H~i,H'.JA
!
L.....:-----
-5
.--.
~
,_,
·-·
CL
~
"""
[I
- 15
-20
w
:::- -25
H
1-
ti
_j
t: . :• >·:, -.
I
\:~._-.,...--.\--+-----+-.-
r::
I
~
- t:.J
-30
PAT TE:PN
tc
//' unb lock~d
!
·. \
I ·. · .·. .·\.\
t=
_.-...:b:.;l.._oc.:..:!.<:...:e:.:d_ _ _ _ ____..;
- //
. l. /
I
1
·--y,
•
-
L'
.
I-----;
'
I
'-
f-
~
r
r-
i::
r
r-
1.....0
.:L
-::'S
-40
--45
rj-
r
r-
,_r
·I
1
-·
-';13
I
0
FIGURE 4.1.
:
:
I
~
·,I
r-----~~------~--~------~~~~~·----~-----~
c::
1
i :!1
l ·1
:
1
~I I I
' I i :1; I
;
i I
r-
••
I
:
38
Aperture Field Method- Uniform Aperture (E Plane)
45
RNTEl',.Jf'.IR PRTTERI\1
-5
.;;-13
(i -1 s
w
=
w
:> -25
H
I-
I
t
-'
w
0.:
-::o
-35
i
-50
s
FIGURE 4.2.
10
.'"".,
20
II
I I
I;
1
30
Aperture Field Method - Uniform Aperture (H Plane)
46
4.1.2 Aperture Fields (Space Tapered)
The initial studies of a uniformly illuminated circular aperture
indicate that a tapered aperture distribution would more closely
approximate the measured data.
This is due to the characteristic of
beam broadening and lower sidelobe levels associated with the taper.
The primary illumination pattern is still held to be an isotripic
radiator, to study the natural amplitude (i.e., "space 11 ) taper caused
by the reflector itself.
The computed results are in tabular form in
Table 4.1.2.
TABLE 4.1. 2
E PLANE (EL)
H PLANE (AZ)
Parameter
Thea.
Meas.
%Dev.
Theo.
Meas.
%Dev.
HPBW (deg)
10
11
9
11
14
21
SLL
14.5
16
9
19
20
5
BWFN (deg)
22.6
25
10
25
39
36
SLA
16.5
17.5
16.5
22
25
(dB)
(deg)
6
Comparison of the theoretical half power beamwidths in the
principal planes indicate a narrower beamwidth in the E plane.
This
can be attributable to the blockage taper being more prominent in the
plane containing the polarization vector.
As observed in the results
of the uniform aperture, results of the H plane computed field still
leave something to be desired inasmuch as defocusing and aperture taper
of the primary feed is not allowed for in the analysis.
The assumption
of uniform feed excitation in the E plane seems to be valid though,
since computed results seem to be in good agreement with measured data.
This is most likely because actual feed openings are of narrower width
in theE plane, (i.e.,
11
a 11 dimension), thus causing less lateral
47
RNTEt'iNA PATTERN
~
:>'-..
r ..:::::...___
.,......,·-.
-s
; -10
u
~ -15
F
· ·. ·. .
~
l
~';-,,\\
t=---
<..
F=
~-\
-~ '\
.'
..-
i'
i-
\-:r;
-30
-'
w
u.:
-35
'-
!....
f"\ \ / .. /
i-
I \ /~~
,.-
E
1-
·, ; I
r-
1 : :'
I ..
I
r-
-40
.
I
F
..
r=
:;_----i----~-·
,.-
I I
I
1
II
1.
:-\
I
I
I
:: !
s
l\
i
I. '
I
t---1
l
II
/
L_
I' \l Jl _____ _ j
iI
i. f
;
i.
II
I
·~I
i
,
I
1...----j
I '
I
I
I
!
!;
~4ethod-
J
I
!
\ j!
..
"\
·,\
\,
---1I
/
15
Aperture Field
(E Plane)
I
_:_j
~~~==---~~.
f\
I
!
I
i
I
....J._j
38
SCAN ANGLE
FIGURE 4. 3.
I
I
I
•
-"'1-:;.X'""'
'\.J
l,...l
I
i,,j'
i-
0
j
~
~-·;x; ~ ~I_
1
r~
·. ·' I I
t=
~
.
J'
.
1.1-. I . __! __ .. I
:-
~ -25
I-I
·1·
.
·
-. ·;:ebal.:cr>::d •.
·, \ '.
t:
l
blocked
a( .:1 ~~;3)
Space Tapered Aperture
48
RNTD·JNA
PRTTERf\J
-5
~ -10
"
a::-15
w
..,.
......
cr -20
l.LJ
~·
1-1
.
-25
l-
ei --=-C"
....J
·..J'-1
w
a::
-35
-40
0
5
!S
SCAi··.l ANGLE 8
---
FIGURE 4.4.
. II
-J
C dec~
Aperture Field Method - Space Tapered Aperture
(H Plane)
49
defocusing as compared to the defocused effect observed in the H plane.
Filling of the first null in either plane is not predicted by this
approach and is in need of being properly addressed.
4.1.3 Induced Surface Currents
The rigorously correct surface current approach is used to
determine the effect of the longitudinal current component on computed
radiation patterns.
fields method.
This term has no counterpart in the aperture
The fact that the longitudinal component of current
only contributes to the E plane (elevation) field distribution can be
observed by examination of the computed radiation patterns.
The
primary effect this method predicts is the occurrence if null filling
of the first null in this plane, albeit the predicted value is off
from the measured results by 6 dB
(~23%)
in null depth.
Consideration
of the comparative values of the major pattern parameters of Tables
4.1.2 and 4.1.3 respectively indicate that this method holds a slight
edge in accuracy over that of the aperture fields approach.
The severe
disagreement with H plane measured results can be attributable to the
defocused nature of the monopulse feed as well as the primary pattern
taper in the H plane as before.
Inadequacies in the assumption that
the currents and fields on the shadowed portions of the reflector
surface are non-radiative are most likely to blame for lack of null
filling in the computed H plane results.
This issue shall be addressed
rigorously through consideration of strut blockage analysis using the
Induced Field Ratio hypothesis.
50
ANTENNA PRTTERI'·l
,t.---.::::--....
.
'·)(::.:-5
;; - !0
F
[
..
·-~~ '~.
~
·
..~.\
-45
C'
·J
!8
jC'
• .J
25
SCAN At·JGLE8(deg)
FIGURE 4.5.
Induced Surface Current Method - Huygen Illumination
(E Plane)
51
ANTEN~·.JA
PRTTERI'l
-s
;;;-!0
·~
o::-15
w
..,..
0
0...
-C:·p
~w
w
> -25
1-1
i-
§
f=
r
[
t:f::
r
r
F
c
I
I _·:·p
.....J ..,<.)
r
w
F=
r
t
0::
-35
-45
-5\3
;-
§
t
t
I
f-
!=
r! I
Q
!
I
5
18
2S
SC=lN ANGLE 9(d ec~)
FIGURE 4.6.
Induced Surface Current Method - Huygen Illumination
(H Plane)
52
TABLE 4.1.3
E PLANE (EL)
H PLANE (AZ)
Parameter
Thea.
r~eas.
%Dev.
Thea.
r~eas.
%Dev.
HPBW (deg)
11
11
0
10
14
29
SLL
14.5
16
9
19
3
BWFN (deg)
23
25
8
20.5
25 .·.
39
36
SLA
16.5
17.5
6
16.5
22
25
(dB)
(deg)
53
5.0
ENHANCEMENT OF ANALYTICAL MODEL
5.1 Discussion
The discrepancy between computed and measured results can best be
attributable to two main reasons:
1.
Measurement error
2.
Insufficiently rigorous approach to formulation of antenna
analysis model.
The latter reason will be examined by discussion of the physical
characteristics of the actual antenna system under analysis.
The
analysis presented in the previous tvm chapters were based on two
major simplifying assumptions:
1.
The reflector feed origin of radiation (i.e., phase center) was
located at the focus of the parabola; the feed itself being an
isotropic point source.
2.
The struts producing distinct shadows and being non-radiative.
The reflector under consideration is an amplitude comparison
monopulse tracking antenna.
The four horn feed system (Figure 5.2)
consists of an open ended waveguide cluster (WR 51) centered about the
focus of the reflector.
An effort has been made to adjust for the
optimum placement of their relative phase centers by usage of an iris,
foreshortening each opening in the a dimension (i.e., H plane).
11
11
This
creates an inductive loading at the guide opening and must be compensated, for by capacitive tuning.
This is physically implemented by
tuning screws at each port.
A more rigorous approach is presented for the monopulse feed array
by treating the problem as the superpositioning of laterally defocused
5!4
radiators.
The slight directivity of the open ended waveguide is also
addressed through the formulation of the open ended waveguide far zone
radiated field expressions in order to adequately define the source
illumination characteristic.
Secondly, the Induced Field Ratio hypothesis is investigated as a
rigorous solution to the waveguide support structure blocking problem.
The struts under consideration have a width on the order of a half
wavelength (Ao/2).
There is no expectation in this case, for deep,
clearly defined optical shadows "cast by the various waves impinging
11
on the structure.
The IFR hypothesis does not employ the concept of
shadows and takes into account cross section, tilt, polarization and
frequency.
This approach will be investigated for the specific case
at hand.
5.2
Feed Characteristics (Qualitative)
The fact of the monopulse feed being inherently defocused prompts
the need for both qualitative and quantitative investigation of its
effect on the secondary radiation patterns produced by the feed/
reflector system.
Initially, a qualitative view of these effects
could be useful as a tool in the development of a formulation to model
these characteristics.
The expected perturbations on the secondary
fields are:
1.
Main lobe broadening
2.
First null filling due to merger of first sidelobe with main
beam.
These statements can be substantiated by observation of the effect
of displacement of a single feed point described in the results of
55
Ruse {10} i.e.:
1.
Main lobe scan off axis, less than the squint (angular
displacement) of the feed.
2.
Sidelobe level increase on the side nearest to the boresight
axis, more commonly called "coma" lobes; a corresponding
decrease in the sidelobe level on the other side of the main
lobe.
3.
Decrease in gain level relative to a focused condition.
The resultant sum pattern would reflect main lobe broadening,
null filling of first null due to the merger of first sidelobe with
the main beam and an overall increase in the magnitude of the envelope
enclosing the sidelobes.
This is a consequence of the increase in
the sidelobe structure of the individual beams produced by each off
axis feed horn.
It shall be shown in Section 5.3 that this lateral
shift in feed placement off the focal point produces a phase error
cubic in nature which manifests itself as beam degradation and a beam
shift in the opposite direction to that caused by the feed squint {23}.
There is an additional perturbational effect on the secondary
·pattern due to the finite aperture size of the feed ports themselves.
These ports can be viewed as rectangular apertures with no variation
of field intensity in the E plane dimension and a half sinusoidal
variation in field intensity across the H plane dimension.
As a
result of this taper, beam broadening and a decrease in sidelobe
levels in the H plane is expected relative to the E plane pattern
intensity levels.
These aspects will also be formulated to adequately
describe the amplitude characteristic of the primary illumination.
56
With both phase and amplitude characteristics of the monopulse
feed system formulated, the principle of pattern multiplication can be
implemented to detenn:ine the resultant illumination function for the
generation of sum and difference radiation pattern integrals.
These
integrals are developed on the basis of the induced surface current
approach (i.e., physical optics).
5.3 Monopulse Aberrations
The radiation pattern integrals· developed in the
~~evioas
s:ections implied a single feed antenna located at the focal point of
the parabolic reflector.
The actual antenna system undei study
incorporates monopulse tracking which can be viewed as an array of
four displaced feeds clustered about the focal point of the reflector.
The development of the modified radiation pattern integrals for this
is presented for usage with the induced surface current approach for
far fielrl radiation pattern prediction.
Consider, initially, a single feed point displaced from the
origin (Ftgure 5.1).
feed is given by
~.
The radius vector to the phase center of the
The new radius vector from the displaced feed to
an element of surface area
p' = p
-
ds on
the reflector can be defined as:
£
where:
£
= £x
ex + £y e y
and as before:
p
= psinw
cos~
ex + psinw
sin~
The vector ;• makes new
ey -
pcos~
ez
angles~· and~'
with respect to the axis
of the displaced feed so that the incident electric field on the
57
~\
(
>
/
'j /
NOTES:
1.
2~
\I
.:-K·-----!
~ -;
1./
~
I
FIGURE 5.1.
'
i
Small Parabo·lic Reflector:
System
Si = displaced nth source
S = focal point source
~lonopulse
Feed Coordinate
58
~0.320"-..l
l.o:-
a----11
'
I
I
~ d
FIGURE 5.2.
I
----1
Small Parabolic Reflector Monopulse Feed
NOTES:
=0.128 11
=0.160 11
1.
c:
2.
d =distance between feed phase centers
X
E:
y
59
reflector surface becomes:
-i
A
e -jsl P - ~I
h
E = ( ElJ! 1 eljJ, + Et; , et;
1 )
IP
_
E:
I
The case under study has the characteristic that the feed
displacements from the focal point are small compared to the distance
between the feed and the reflector surface.
Thus from the geometry
of the feed as in Figure 5.1 and the usage of surface projection onto
the reflector aperture the relations are developed
p
1
s i nljJ 1
=r
for n feed ports.
- r::
t;'=t;-l;
on
Small feed displacements dictate that
1--;111-;;-1« 1
and the above relations are reducible to
lJ! ' "' lJ!
l;
I
= l; - l; on
Here ~on is the circumferential angle of the nth port feed.
The
denominator of the incident field expression relates the path loss of
an isotropic source.
From the assumptions made above
r::
is small and
the path loss term can be approximated as
jp--
sl
jp-j
The phase term, though, must remain intact since even small feed
displacements can cause significant phase changes at the reflector
surface.
The monopulse feed can now be modelled as the superposition
of an array of four identical feeds, each with equal amplitude
excitation.
The incident field can be written as:
4
n
I= 1
4
=
~
(ElJ!
~ljJ
+ Et: et;')
L
n
=1
e
-jsjp -
r::
n
I
60
The summation can be viewed as the array factor for the four port
feed.
The radius vector to each port is represented by en.
The total
derivation will be based on four identical isotropic point sources,
and the amplitude terms will remain the same as in the previous
The phase term can be expanded as such:
chapters.
p
+
p2I
z
I~
where:
= psinw coss', Py = psinw
Px
sin~·,
pz = -pcosw
The substitution of these results into the general form of the
radiation pattern integral yields the result:
£r1 ±
~j
I
e-jB{(P'n • R1) - p'nl
n
\~hat
J dS
=1
remains now is the reduction of the phase term into nonprimed
quantities.
Initially the evaluation of the phase term is started by
expansion of the square root:
Pl
n
=
=
I
p2I
x
IP 2
+ p 2 I + p 2 ' - 2r.
y
z
- 2p£nx'sinw
~x·
cos~;'
E:
nx
I
-
2p
IE:
y ny
1
+
E: 2
nx'
- 2p£ny'sin.P sinf;' +
+
£~
E:
2
ny
I
!z
I
\i
Expansion of this factor by the binominal theorem yields the
approximate form:
P~ ~
P -
Enx1sin~ coss 1 - Eny•sin~ sin~· + O{E: 2)
The small value of jt::l has allowed the consolidation of all terms
jEj 2 and higher and are considered to be negligible.
expression for p' is reduced:
A
P~ = P- €n = P- E:nx'ex - E:ny'ey
The vector
61
Reca 11 i ng that
A
R1 =sine cos<f> ex+ sine sin<P ey + cose ez
the evaluation of the total phase term is straight forward.
The
leading term in the phase portion requires the evaluation of the dot
product:
p'n
. R1
= psin1J! sine
cos(~ - <P)-
pcos1ji cose
-E nx' COS<fl Sine - Eny' s i n<P sine
(p'n • R1) -pIn =p{l + COS1/J cose - s i n1jJ sin~
- {Enx' sin1J! cos~ • + Eny' sin1J! s.in~ •
cos(~
- <P)
+ Enx, cos<P sine + Eny , s i n<P sine }
Observe that the leading term is just that of a feed at the focal
point of the reflector.
In the previous chapters it was stated that
for pattern evaluation close in to boresight (approximately ±30°) cose
~
1.
{(p~
Thus the phase term can be approximated as:
· R1)-
p~} "' 2f- psin1Ji sin~ cos(~ - <P)
- {Enx' sin1Ji
cos~'
.
+ Eny , sin1Ji
sin~
•
+ Enx' cos<f> sine+ Eny' sin<P sine}
Substitution of the explicit phase term into the radiation
pattern integral can now be accomplished.
Thus:
Kn(e,
n
=1
+ Enx' sin1jJ
<t>)/ j
E e-jS{psin1jJ sin~ cos(~ - <P)
s
cos~·
+ Eny' sin1jJ
sin~'}
dS
62
here:
K (e ,
n
</>)
= e-j S{£ nx coS</> sine + ~:: ny s i n<t> sine}
1
1
The observation variables e and
are factorable since the radia-
<f>
tion integral is evaluated over the source coordinates only (i.e., *
and
~ ).
1
The elemental surface area dS is described as:
d~
ps i n·* d*
since
d~
' = ps i n* d*
= d~~. The radiation integral can be reduced to a one
d~
dimensional integration as in the previous chapter by first using the
aperture projection transformations:
r = psin* = tan ¢/2
*min
=0
*max
=a
The evaluation of the
integration is aided by extensive
~
algebraic manipulation of the phase portion of the radiation integrals.
The trigonometric identities:
cos(a
±
s) = cosa coss
±
sina sins
sin(a ± s) = sina coss ± sins cosa
and also:
A case + B sine = JA 2 + BL cos(e - o)
where:
o = tan- 1(!)
are employed.
The exponential can first be rearranged to fit the
standard form for the transformation.
exponential becomes:
The bracketed term of the
63
The transformation now becomes:
!}= en cos
(I; -
n: n)
here:
c~
=
and:
2 2
r sin e + 2rsin8 sinlji{enxl cos(<P ~ t; 0 n) + e ny
2
+ e~ sin 1J!
2
2
2
n = £ nx' + £ ny
or more compactly as:
C~ = r 2sin 2e + 2enrsine sin1J! cos(<P£
tan on
sin(<P - t; on )}
1
1
= rsin8 sin<P +sinlj!(enx' sin
rsin8 cos<!> + s inlj! (enx • cos
t;
0
2
en)+ e~ sin 1J!
n
l;on
t; 0 n
+ £ny cos
- eny sin
1
1
i;a 0 )
t; 0 n)
tan 1;; n = .:.DL
enx
Expansion of the variable C using the binominal theorem allows
1
the approximation:
C"' rsine +en sinlj! cos(<!> -
since terms O(e2) are negligible.
t; 0 n-
Cn)
Reduction of the
1J!
angular
dependence is now accomplished by approximating:
(r/f)
. _
Slnlj! - 1 + (r/2f)2
"' r/f{ 1 - (r/2f)2 + (r/2f)4 . . . }
The first two leading terms of the series are retained as a first
order approximation since they are the primary contributors to pattern
degradation {10}.
·
C"' rs1ne +
e 0 r(r
2
- 2f) cos (<P2f3
t;
0
n- en )
Thus the entire radiation integral is put into a form reducible
by the Bessel relation of the previous chapter.
Allowing for the
proper phasing of each feed due to the microwave arithmetic of the
64
monopulse function desired (i.e.,
2:
and
l'l
patterns in the principal
planes) the pattern integrals become:
Copolarized component:
2
4
a
SE:nr( r - 2f)
2f3
cos(p-~on-r;;o)} dr
Ix~
ejvnKn(e,rp)J rJo{Srsine+ (1 +· (r/2f)2)2
Z:
n =1
o
Axial component:
4
I 2 ~ 2rrj
L
n =1
·
eJv nl<ri(e, rp)
a
J
0
2
SE: 0 r{ r - 2f)
2
r JI{Srsine +
2f3
cos(q,- ~on- sn)
( 1 + ( r 1 2f) 2) 2
x cos on(r) dr
here v n = phase of nth feed port.
The effect of a single displaced feed is such that the beam
"squints" off axis a distance slightly less than the angular displacement of the feed from boresight of the reflector.
There is a
corresponding increase in the sidelobe levels on the side of the
pattern away from the beam squint.
These are termed "coma" lobes.
The superposition of these effects results in serious degradation of
the far field sum pattern of the four horn array.
It has been shown
{11} that as the separation between the individual feeds in reduced,
the amount of squint that each feed produces is reduced, and a
corresponding decrease in beamwidth of the secondary pattern is
observed.
Intuitively, this fs.-logical:since the amount of narrowing
observed of the secondary pattern would be in the limit the point at
which all four feed horns are placed at the focus; which is, of course,
physically unrealizable.
The narrowest separation found to be
achievable is approximately d = 0.5A (Figure 5.1) which could be
aided by dielectric loading of the feed horns to allow the principal
mode (TE 10 ) to propagate.
The antenna f/D ratio can be adjusted in
65
addition to achieve a desired illumination taper.
The radiation integrals produced thus far have assumed point
source excitation of the reflector.
The physical characteristics of
the actual feed port openings are such that each can be modelled as
rectangular apertures in a ground plane.
The aperture field of each
individual port is assumed to have no variation in field intensity in
~
the E plane (ex) and a half sinusoidal variation in field intensity in
the H plane (ey).
The fact that the reflector surface is in the far
zone of these apertures allows the usage of their secondary (i.e.,
diffraction) patterns as the element factor of the four horn array.
The array factor being developed, the entire feed field definition can
thus be approximated by pattern multiplication.
gular guide openings are defined in Silver
{5}
The field of rectanand are correspondingly
modified for the case of dielectrically loaded feed ports for TE 10 mode
propagation:
I
E~
= M sin
~~{1 +
~
K10 cosw} Vn(w~~n)·ew
and
E~,
= Mcos
~~{cos~+
K10 1
Vn(~~~~) e~
here
K
10
= ~ /sr - {>- 0 /2a)2
Vn(~ ~~) = [cos(na/~
I
M
sin cos~·~
na/>- Slnw cos ~n) - n/2
n
= -nna
] [
s i n(1t b/A s i nlJ; s i n ~ n) ]
nb/>- sin~ sin ~n
2
2v'ErA.
b
The angular transformations are used as be.fore in the development of
the element factor:
~
1
= ~ - ~on
66
These expressions have assumed zero reflection at the waveguide/
free space interface.
The radiation integrals produced to approximate
the complete feed fields would require numerical evaluation of a
double integration inasmuch:i1S analytical reduction to a single
integration as in the previous analysis would be difficult.
For
propagation above cutoff in a "perfect" dielectric
Er
>(~~y
This yields a value of € r
ports.
>
1.2 for the dielectric loading of the feed
5.4 Strut Blockage Analysis Using Induced Field Ratio Hypothesis
The preceding analysis
~n
Sections 2.0 and 3.0 has
ap~roximated
the secondary field perturbational effects due to the feed support
structure aperture blockage asatwo dimensional rectangular aperture,
180° out of phase with respect to the reflector aperture fields.
is commonly known as the "null field hypothesiS
11
•
This
This approach,
though, is not sufficiently rigorous enough to predict effects such
as null filling and boresight cross polarization degradation (this due
to the strut not being parallel or perpendicular to the aperture
electric field).
This is because the "null fields" formulation fails
to take into account depth, cross section or tilt of the feed
structure.
Rusch {15} has developed the concept of the "Induced Field
Ratio" in order to attack the problem of strut blockage rigorously.
The "Induced Field Ratio" or "IFR" has been defined as referenced
to an infinitely long cylindrical scatterer illuminated by an incident
plane wave.
Rusch defines the IFR explicitly as "the ratio of the
67
forward scattered field to the hypothetical field radiated in the
forward direction by the plane wave in the reference aperture of width
equal to the cylinder on the incident wave front
11
•
A major assumption
of the theory is that the strut currents generated due to the plane
wave component of the paraboloid focal region 'fields are the same
currents that would flow on an infinite cylindrical structure of the
same cross section in free space immersed in an infinite plane wave
with the same polarization and direction of incidence as the local
geometrical ray incident upon that part of the strut as it emerges from
the aperture {15, 16}.
Consideration of the geometry of the reflector/
feed system reveals that theE vector is perpendicular to the feed
support structure axis.
the case of the
H vector
The mathematical definition of the IFR for
parallel to the longitudinal axis of the strut
as proportional to the total currents generated over the strut surface,
i.e.:
IFRH =
2(-r
1
1
- -r 2 )H 0
f--1: H (~ • ~)ej(3p'cos(G?' - TI/2) dS
J--J'
Z X
s
The definition of the surface normal, source and observation
vectors are shown in Figure 5.3.
At this point, it would be judicious
to review the basics of two dimensional scattering by cylinders.
Harrington {17} has treated the problem using cylindrical wave transformations for the analytical solution of the simple case of the Z axis
aligned cylinder.
Understanding of this treatment will provide
valuable insight into the development of the IFR formulation.
Consider a perfectly conducting, infinitely long cylinder, aligned
coaxial with the Z axis (see Figure 5.4).
If an incident plane wave is
polarized transverse to the Z axis (i.e., perpendicular to the cylinder
'
d
68
_str~.-L t
·
FIGURE 5.3.
d v."?~
~et:tl.:-•1
Geometry to Define Induced Field Ratio
69
t-
1
I
I
"-I
.,.:.::::1)
1-----r-- i
FIGURE 5.4.
Scattering Geometry of Right Circular Cylinder
70
axis) it can be expressed as:
-too
H e-jSx = H
0
0
'
n
L.
= -oo
The total field at an observation point P(r,
~,
Z) is:
H =Hi + Hs
z
z
z
here Hi, Hs are the incident and scattered fields respectively.
The
scattered field from the cylinder is represented in the form of an
outward traveling wave, i.e.:
H~"" H ~ j-n bn H~ 2 )(Sr)ejn~
0
n
=
-oo
To determine the coefficients bn' the boundary condition that the
tangential electric field I must be zero at the cylinder surface must
be met, i . e. :
E~ =o@ r=a
here a is the radius of the cylinder.
TheE~
component can be deter-
mined by Maxwell's equations:
E=
j~£
l
~z
v x Hz
l
when expanded the bracketed term becomes:
=~ (l
r
aH_z_) _ ~ ( ~)
r
3~
~
ar
The circumferential component is thus:
I~= -~~(a:rz)
= js H
W£
0 n
I:fnl J • (sr)
= -oo
n
which must satisfy the boundary condition.
2
H0 n
+ bn
(Ba) J
~!
~_J"~J~(Ba)
u
\1
=
bn
H~ )'
-Jr)(sa)
H(2)'(sa)
n
Thus:
ejr.~ =
·
o
71
The total field now becomes :
.-n
J
n
= -oo
Determination of the currents generated on the cylinder are found using
the induction theorem at the point r
JS
~)
= (Ffi
X
= !Hz
~z x •r
l
r
= a:
=a
+:a
J (/) =J S(/)
e =H 0
'L..
n = -co
This can be found to be reducible using the Wronskian of Bessel •s
equation such that:
+m
z:
J = 2j H0
1ft3a
<p
n .:;:
~-oo
Thus we can see that normal incidence of a plane wave produces
directed currents.
e<p
At large distances from the strut (i.e., sr ._co)
the large arguement approximation is used for the scattered field
expression, i.e.:
Hs -
z Sr-+
oc
Jiirj
..,. -H 0
·-
';T(3
r
e -jsr
The magnitude of the ratio of the scattered and incident field is
found to be:
jn<p
Jr)(sa)
e
2
n = -co H( ) • (sa)
n
72
Rusch's form of the IFRH is similar, with the exception that allowance
is made for non-normal incidence.
Here the IFR is determined for an
equivalent cylinder with linear dimensions of the cross section reduced
by the factor cosa., i.e., for the case of a right circular cylinder:
-j-oo
IFR
H
= sacosa.
-1
n =
-co
The formulation of the strut blocking equations is initially aided
by consideration of a generalized geometry of the perfectly conducting
strut in Figures 5.5 and 5.6.
The strut longitudinal axis lies in the plane
~~
= ~0 .
The strut
also lies entirely on one side of the Z axis with at least one end
touching said axis.
The (X', Y
1
,
Z') coordinate system coincides with
that at the induced surface current method (Figure 3.1) and is centered
at the paraboloid prime focus with the Z' axis directed toward the
reflector along the reflector axis of symmetry as before.
lies at angle a. with respect to the r
Z in the plane
1
= ~0 •
~~
1
The strut
axis which is perpendicular to
The end of the strut lying closer to the Z'
axis has coordinates {r •, Z
and the other end has coordinates {r 2 ',
1
1
z2 where r 1 ~ 0, r 2 • > 0 and r 1 • < r 2 From this we find that the
angle a. is thus expressible as:
1
1
)
1
1
)
-
tan -1
a -
•
~z2,' - zl'j
T
-.-
r2
r1
the strut length is found from the geometrical distance formula
L
=
{
I
I
l(Z2 - z1)
2
+ ( rz
-
I
I
rl)
2l
~
The oblique double prime coordinate system (X", Y", Z") is shown to
have its origin located at the end of the strut farthest from the
reflector aperture {q
1
,
Z1').
The X" axis lies in the
plane~~= ~
0
73
/
/
I
t--'
I \
I
I
I \:, ,_
1
--&.,., I
--~--~~~.--------~--~~------------------~--~I
· .,
, ' l
/
nf
l
,
1:'
/
/
t·
J::_____
_
I
_ _ _v
cl
/
-"'
~! ~o·
I
-~~
__.....
tz
,... i
l:z.
,..., L 01 (_
"""
- .
I
I
I
I
·7 I
c:,
FIGURE 5.5.
+
t..:
t..
Generalized Geometry of Conductive Strut
qo
~
74
I
/.4'
't
I I/
1/
/I
I
I
!I
L:.
-------f----¥'?-~X~r--L.r~.:-'f'
'I
,
FIGURE 5.6.
• !J
1'
. .'-,._/_ _ _ __,__
••
~-
Generalized Geometry of Conductive Strut
(Oblique View)
75
with an incident plane wave emerging from the reflector with the wave
traveling the positive Z (i.e., negative Z') direction.
The oblique
view is depicted in Figure 5.6 with the angle cp" being measured in the
X'' yu plane as referenced to the X axis.
Note that as referenced to
11
the ISC coordinate frame,
s = C{J
1
•
Initially, for a heuristic approach, the strut will be assumed to
have a circular cross section, a restriction which will be removed once
the complete formulation has been developed.
The
H vector
incident plane wave is also assumed to lie in the plane cp'
along the strut axis).
of the
= cp 0 (i.e.,
For a. f: 0 both axial and circumferential
components of surface current are induced.
The scattered field due
to the strut currents is:
E(P)
=
.
-Jwll
41T
-. SR
e
J
R
j
21T
•
L
f
adcp"
dZ"(J
e
scp''
l
0
~
0
for a strut of circular cross section.
+J
cptrans
sZ" eZ"
lejSp·Rl
(
trans)
Here P is a field point with
coordinates (R 1 , e, cp), R is the distance from the origin at the focus
to the field point, Pis the vector from the focus to the integration
point on the strut and R1 is the unit vector from the focus to the
field point P. Only the transverse components (as referenced to R1)
contribute to the scattered field.
From the earlier development of two
dimensional circular cylinder theory, the two components of current can
be extrapolated for the oblique cylinder.
j ncp"
r n -,--=-r-,....:e=-----
= 2jHo e -jss i na.Z"
(31T a
These components are:
n =
.,.co
H(2) '(sacosa.)
n
76
and
_ -2jHo -jssinaZ"
JsZ" - 1rSa e
sina
8acos2a
n
=
-oo
Note that the Z" component of current exists when a 'f 0.
Insertion of these quantities into the scattered field radiation
the~~~
integral and evaluating over
_ j 8 e- j 8R j Po
E(P) - 21T
R
e
integration yields the result:
~-
aciFRH(D, o,a) +
saD
[~
-
(
A + A .) s i na
ee e
e~ ~ sacos2a
A
A
.,
l
r2
X 2a
Jf
nHa(r )ejsr'Bo
r-1
1
Here
-·
A
ac = ee {sino co sa sine + sino s ina cose cos ( ~ - ~ 0 )
A
- coso cose sin ( ~- ~o)}
- e¢ {sino sina sin ( ~- ~o)}
A
as= ee{coso cosa sine+ coso sina cose
cos(~ -~ 0 )
+sino cose sin ( ~- ~o)} + e<P{sino cos ( ~- ~6)
-coso sina sin
Ae
= cosa
Acp
=-
(~- ~o)}
cose cos ( ~- ~o)- sina sine
co Sa s in ( ~ - ~ o)
also
P0 = (SZi -Sri tana )(cose - 1)
B0 =sine cos ( ~- ~o) + tana (cose - 1)
E
= sina
sine cos ( ~- ~o) - cosa eose
F
= s i ne
si n (
D
= (E 2 +
2
F h
o
= tan -l
!~ ~
~
-
~ o)
77
the generalized IFR functions are
+oo
-1
IFRH(D, o, a) = sacosa
-·n=...;oo
and
+oo
-1
njejna
JFRH(D, o, a) = sacosa
n=-oo
L
L
H~
2
Jn(saD)
) •(sacosa)
The remaining integral is an integral of Ha(r•), the focal plane H
field in the r• direction.
Note that when e = 0, P lies on the
boresight axis and the scattered field expression E(P) is simplified
to the result:
E(R, 0, 0)
·s
=-in
-jsR
e R
e~'(IFRH)2a
A
J2r• nHa(r•)dr•
r•1
where as before IFRH is the H polarization IFR of a circular cylinder.
Except for the IFR factor, it can be noted that this is the expression
for the field of a rectangular aperture (see Figure 5.7) as defined
by the projection fo the strut on the x•y• plane.
In the optical
limit IFRH-1 and the radiation integral of the scattered field reduces
to the null field result.
However, for typical strut widths on the
order of a wavelength the IFRH is far from -1 and a considerably
different result is obtained.
The radiation integral of the scattered
field also provides more precise angular information about the strut
tilt than does the conventional
11
flat shadow representation.
11
Consideration of the actual geometry of the waveguide support
structure reveals that the struts are rectangular in cross section.
Thus the development of the IFR in terms of a generalized cross section
must be considered.
Some of the analytically reduced results must
now be carried out numerically for this case.
For the case of H
78
-~
I
•
U'
J
FIGURE 5.7.
Equivalent Aperture Defined by Projecting Strut
onto X'Y' Plane
79
polarization,, it is convenient to formulate the three necessary
numerical integrals in terms of HZ"' total axial component of H field
at the surface of the conducting strut.
Thus:
Hz;, (x_") = e-jSsinaZ" Hzu(£")
where£" is the path length around the strut periphery in the X"Y"
plane as shown in Figure 5.8.
The function J' s Z" (£ ") must
. be obtained
numerically in tabular form using method of moments techniques. The
functions IFRH (0, o, a) and JFR(n) are used in formulating the
scattered field expression:
A
nwD J FRH ( 1)
J FRH ( 2)
X w
l
lrz
dr'
r'1
Here the generalized IFR functions are expressible as:
A
IFRH(n) = 2wcosa
JFRH(l)
=
f
cos(c;o,1
-
o) Hz"(£")]
H
e jsx"Dcos e jsy"Dsino
[
0
nO
2cosa
and
~j'---- tf.
a
[HZ"(£")] jsx"Dcos jsy"Dsino
£".
JFRH(2) = 4cosa :Y ;(£")
H0
e
e
d(T)
A
Q,"
d(·-)
A
80
,,
' cf
l j
I
I
r'
FIGURE 5.8.
Geometry of Strut Arbitrary Cross Section
81
where
<Pn
X" axis.
<Pn
= 0.
is the angle between the strut surface normal at
£"
and the
For the special case of struts of rectangular cross section
For a circular cylinder:
+ro
L
r.=-oo
.-n
J
H~ 2 ) '(sacosa.)
Generally, H' Z" must be obtained numerically in tabular form
H;-
using method of moments.
Rusch indicates through comparisons of
computed radiation patterns that struts of rectangular cross section
generally exhibit more extreme pattern degradation such as null filling
and higher sidelobe levels than a circular cross section strut of
equivalent cross sectional area.
For example, the circular strut of
equal cross sectional area exhibits 20% less H polarized incident wave
blocking than its rectangular counterpart {21}.
Further information on the formulation of the rectangular strut
scattering problem can be found in {22}.
82
6.0 CONCLUSIONS
The effect of axially directed surface currents on the far field
secondary radiation patterns of a small parabolic reflector has been
investigated.
This work was inspired by the fact that these currents
have a negligible effect on the close in sidelobe structure of large
reflectors (i.e., D > 10\) but do effect sidelobe structure of small
reflector secondary patterns.
Previous works have not concerned
themselves with the analysis of this size class of parabolic reflectors.
In addition, the effect of aperture blockage due to the reflector feed
has been included in all computations.
Primarily, the axial component seems to contribute exclusively
to null filling in the plane of incident wave polarization for a
linearly polarized feed.
Analytical procedures for the rigorous modelling of the feed
and its associated blockage has also been presented for the case under
consideration.
The formulation for monopulse radiation integrals
based on vector diffraction theory (i.e., Straton Chu formulation) has
been presented for determination of the effect of
currents on both E
and~
th~
far field radiation patterns.
axially induced
A rigorous
formulation for determination of the effects of aperture blockage has
been investigated in the literature, the results of which have been
presented for the case of H polarized incidence.
These results are
to be used for further enhancement of the existing computer model.
83
REFERENCES
1.
A. C. Ludwig,
Reflectors
2.
11
Calculation of Scattered Patterns from Asymmetrical
USC PhD Dissertation 1969.
11
,
R. L. Tanner et. al.,
Problems
11
Numerical Solution of Electromagnetic
IEEE Spectrum, September 1967.
11
•
3.
W. U. T. Rusch, Analysis of Reflector Antennas, Academic Press 1973.
4.
J. B. Marian, Classical Electromagnetic Radiation, Academic Press
1965.
5.
S. Silver, Microwave Antenna Theory and Design, MIT Radiation
Laboratory Series, Volume 10, 1949.
6.
G. A. Thiele et. al., Antenna Theory and Design, John Wiley & Sons,
1982.
7.
E.
t~.
T. Jones,
Antennas
11
Paraboloid Reflector and Hyperboloid Lens
IRE Transactions on Antennas and Propagation, Volume
11
,
AP-2, July 1954.
8.
D. F. Difonzo,
11
0ffset and Symmetrical Reflector Antennas;
Polarization and Pattern Effects
9.
11
,
M. S. Thesis, CSUN, 1972.
M. Abramowritz et. al., Handbook of Mathematical Functions, National
Bureau of Standards Applied Mathematics Series No. 55, 1964.
10. J. Ruse,
11
Lateral Feed Displacement in a Paraboloid
11
,
IEEE
Transactions on Antennas and Propagation, Volume AP-13, Sept. 1965.
11. D. Gonzalez,
11
Analysis of Dual Frequency Antenna
11
,
General Dynamics
Engineering Research Report, ERR-P0-292-8, December 1966.
12. E. Kreysizig, Advanced Engineering Mathematics, John Wiley and Sons,
Inc., 1962.
13. C. C. Allen,
Calculations
11
11
,
Numerical Integration
~1ethods
for Antenna Pattern
IRE Transactions on Antennas and Propagation,
84
REFERENCES (continued)
Volume AP- 7 , December 1959.
14.
J. D. Draus Antennas, McGraw-Hill, 1950.
15.
W. U. T. Rusch et. al., "Reflector Antenna Theory and Synthesis"
course notes USC technical seminar, May 1982.
16.
J. Ruse, "Feed Support Blockage Loss in Parabolic Antennas",
Microwave Journal, No. 11, No. 12, 1968.
17.
R. F. Harrington, Time Harmonic Electromagnetic Fields, McGraw-Hill,
1961.
18.
R. K. Chugh et. al., "Comparison of Romberg and Guass Methods for
Numerical Evaluation of Two Dimensional Phase Integrals", IEEE
Transactions on Antennas and Propagation, Vol. AP-24, July 1976.
19.
B. Carnahan et. al., Applied Numerical
~1ethods,
John Wiley and
Sons, 1969.
20.
A. H. Stroud et. al., Guassian Quadrature Formulas, Prentice Hall,
1966.
21.
W. V. T. Rusch et. al., "Forward Scattering from Square Cylinders
in the Resonance Region with Application to Aperture Blockage",
IEEE Transactions on Antennas and Propagation, AP-24, March 1976.
22.
K. K. Mei et. al., "Scattering by Perfectly Conducting Rectangular
CylinderS
11
,
IEEE Transactions on Antennas and Propagation, AP-13,
March 1963.
23.
R. C. Hansen, Microwave Scanning Antennas, Vol. I, Academic Press,
1964.
85
APPENDIX A
Development of Surface Definitions Used for Induced Surface Current
Method
The differential geometry description of an arbitrary surface is
defined by the curvelinear coordinates u and v such that we can
describe a radius vector r which transverses the curvel inear surface:
r = r(u, v)~r
can be represented differentially as the vector path element along the
surface:
dr = ~ du + ~ dv
au
av
An element of surface area is expressible as {12}.
dS = .£I:_ x R du dv
au
av
By the definition of the cross product the vector dS is normal to
the surface r( u1 , u2 ).
the origin:
ar X~
av.
n = au
u x*l
av
Thus the unit norma 1 is directed inward toward
~
The relationship ndS is now expressible in terms of the reflector
surface as described in Figure A.l:
~dS
= -ap
3\jJ
X
~ di/J d~
a~
The cross product derivatives are:
ap - ~ ~
a;j; - aljJ ep + peljJ
and
.:can be derived by considering the radius vector in rectangular coordinates using the spherical to rectangular transformation.
86
(b)
(a)
- T
-
I
i
if
I
FIGURE A.l.
Parabolic Reflector Surface Definition
(a) Cross Section
(b) Differential Surface
87
p = psin~
cos~
ex
+
The derivative with respect to
~
pcos~
+ sin~ _ap
a~
sin~ rcos~
e2
becomes:
J ex
+ sin1)J -~~ J ~y + [ ;~
aa~ = cos~ [ pcos~
+
ey +
psin~ sin~
A
psin~ J ~Z
COS1)J -
To solve explicitly for~~ will require conversion of the rectangular
unit vectors into its representative spherical components using
ex =
sin~ cos~
ep +
cos~
cos~
ey =
sin~ sin~
ep +
cos~
sin~ e~
ez =
cos~
ep -
sin~
e~
+
e~
sin~ e~
sin~ e~
Substituting into the derivative and grouping terms
:~
+
"!
sin.p cos
cos;;f:~
2
1:;
~COSo/
sin;; - psin.p
+ sino/
Jl~P
:~]
+ !cos;
+ sin.p sin
cos 2~:;[;~
2
1:;
rCOSo/ + sin.p
sin.p + pcos.p
l
J
+cos~ sin 2 ~~cos~ + ~ sin1)JJ- sin1)J[~ cos~- psin~J ~.c);
Reducing further
ap - ap
~- ~
A
eP +
pe~
~is
derived in a similar manner.
a~
:~ J
88
APPENDIX B
Application of Guass-Legendre Quadrature to Numerical Evaluation of
Radiation Integrals
The justification for the usage of a non-standard method of
numerical evaluation for the solution of the radiation integrals
described in the text can be found in the literature {13}.
The
integrals considered for evaluation are of the form:
I
JC2rr~a.
0
f(r) ejurcos(~ - k)drd~
0
The variables u and k being e and
~
dependent, respectively.
To
evaluate the relative accuracies of the numerical approximations used,
f(r) was set to unity.
I
This will yield the closed form solution:
= 2ira J 1( ua)
ua
which makes it relatively simple to compare errors in the numerical
integration.
To determine the order of the Guassian quadrature to be
used, plats of number of integration points vs. value of ua {18} were
employed.
The maximum value of this quantity for the case under study
is found as:
ua
= sasin emax = 9.77
This value corresponds to usage of N > 300 formulas over the
entire range for the accuracy of the approximation of
be~
1.0%.
For
evaluation of the radiation integrals, normalization to the (-1, +1)
interval used by the Guass-Legendre quadrature
textbooks {19}.
i~
described in standard
The development of the algorithms to be computed is
outlined as follows:
89
The standard Guass-Legendre quadrature used for the numerical
integration is given by:
+1
n
f
f(y)dx"'
L
wi F(Zi)
-1
i=O
where the Zi are the abscissa points generated by Legendre polynomials.
The radiation pattern integrals must be normalized to the above form.
A general finite limit integral is transformed as:
b
n
f(x)dx"' b2a
wi F Z;(b-a)2+ (b+a)
J
L
i=O
The radiation integrals to be transformed are:
a
a
r 2J 1( Crs ins
r J 0 (Crsins~ d
and
B
(1 + (Dr)2)
r
( 1 + (Dr)
a
A/
f
0
0
the constants being defined as:
1
c = 21T/..
A = 2kf2
B
dr
=
1
4kf3
0 =
Since b - a = a
a
1
2f
k
=
a 2
( 2f)
1 + (~) 2
2f
n
J f(r)dr"' I L
w;t I(Zi + 1)
o
i=O
Thus the basic integrals are approximated as:
a
n
a
ac
2(Zj
+
l)J
(2'(Z;
+ !)sins)
r Jo(Crsins) d _ Aa
A
0
w·1
(1 + (Dr)2)7 r- 2
(1 + (a~(Zi + 1))2)2
0
i=O
or more compactly as:
j
n
"' G
L
wi
UlJ 0 (EUisins)
1 + (FU;)2)2
i=O
likewise the second radiation integral reduced for numerical evaluation:
a 2
.
r
J!(Crsins~
8
(1 + (Dr)2 2 "' H
i=O
0
J
90
the lumped constants are defined:
a3
H = 32kf3
E
= Tia
F
=
A.
a
4f
for the case under study a= 210 11 , f = 1.38 11 ,
= 0.791
k = 0.366
G
H = 0.301
E = 3.1h
F = -.380
U.=(Z.+1)
1
1
.\
= 0.675 11 , thus:
91
APPENDIX C
Small Aperture Parabolic Reflector Computer Program
C.1
Description
This section contains a listing of the computer program which has
been developed to yield the far field power patterns for a small
aperture parabolic reflector.
The program comprises all three methods
of pattern evaluation as described in the text; the method to be used
is selected by the user before the program is run.
The program was written for computation on a HP 9845B desktop
computer.
The programming language which the code is written is an
enhanced version of BASIC and outputs the computed results in a
graphical format (i.e., antenna power patterns).
The input data is a set of tabular values for:
1.
Coefficients used in the polynominal approximation algorithms used
for computation of J 0 (X) and J1(X).
This is included in the
subroutine BESL.
2.
Weights and abscissas used in the Guass Legendre numerical
integration routine for N = 256(GQUAD).
This is a separate
subroutine labeled as XDATA.
These tabular values for both subroutines are found in Reference 9.
Note that all constant coefficients have been set for the specific
dimensions of the antenna under study, albeit the user can change
these parameters upon referral to the text and Appendix B.
The main program consists of the following alpha-numeric variables
used in the computation of radiation patterns in each of the methods:
z.1
ith abscissa of the Guass Legendre Quadrature.
w.1
ith weight of the Guass Legendre Quadrature.
92
PA
Azimuth power pattern.
PE
Elevation power pattern.
PAB
Blocked azimuth power pattern.
PEB
Blocked elevation power pattern.
ET1
Elevation term #1 (ISC method).
ET2
Elevation term #2 (ISC method).
BTE1
Elevation blockage term #1.
BTE2
Elevation blockage term #2.
SUM0
. qua dra t ure sum con t a1n1ng
..
1.th t erm of Guass1an
Jo(X).
SUM1
C.2
ith term of Guassian quadrature sum containing
Program Listing
The following is a listing of the BASIC program used to compute
the principal plane secondary far field patterns of the small parabolic
reflector:
93
~EM
-~-~*~~·~--~~-~~-T~~~~~~~~~~~---~~~-~~~--·~~~~~-~~~~-~~-*~~*---~~~~~~·--~
~Er·1
~
I::E:·'!
-.io
·Hl
'5•J
r~~::: c~·,JG?t=t;·1 ,:·on~·.-;-;::::: irtE :.=:::..:r~J.;f;:V :;·~Dif•TII.Ht ?AT-:-~;:;·~{ :)F r1 ;:-~::;·1E
!='IJ•:·u:; ~·t1:=;:t=t:::(1 L.! :. ?S?':_~:ro~: ;=I~{TE~·tN1 ~liTH Ft~·:J :~~TH,)UT Ff~·~?.TURE
~<C::M .., BLOCKAGE .DUE ~,J TnE "'EED ::TrWCTIJ:~E.
TnREE ME-1-'0D::: )F •:OMPUTUIG
PEi-' -' "!"HE i=<j;;• "IELD F'ATTE!"tlS IH THE ?RltKI?AL 0 '!_:=tt1E'5 APE AVAILABLE
~
10
20
2:0
IN THIS PROGRAM.
RRS DESCRtBED
~0
~E~
;-,j
;:EM "
1. t·ler. h 1;
•JN! i"C:RML'(
?Q
100
11•)
RE!·1 •
~;. i·l~t:.h·3;
It!DUCED '3URF;=tc·::
REM
120
139
1~~
T~ESE
~S
FOLL0~t3:
l LL'J!~ PlATED C l ?•:'ULAR A 'ERTI.;RE
C:.:~J;·E~~,··::
+~~~~~~**~*~•~++~~~·~*~~**~***~~~~-~~~~~-~~~-~~-~~~~4~*~~~~~~~~~~~~~~~~~~
OPTI•}tl Bt=t·;e: 1
D!~
T~a~40~~·,P(~00>,Pt{~J0)
DIM W<i99>,Z(609>
CJM x<30e),Ja~~eo~·.;t:3o~~
tS0
CCi:
P'l.l.4110:: ~ F'~(4(l(t ·,?Eli:~'·-..:..);.)_.~ F.;b( J.(tO-'
16~
PE~
•~~~~~~·++~~~~~--•~·~~~~~*~+~~~~~----~~·~~~-~~--~~+·--~~~·~~~~~~~~~~~-~~
t7:a
180
~·Et-1
~
~EM
*•+~~+~·~~+~~~~~~~·~~~~~~~+~~~~~~~~~~+-~•~-~*~~~~~~+~~~~~++~~~~~~~*~~~
1'~13
:2~t(1
At=.971
A2= • ..):3{
210
P•:l='\,37;:
22~)
•:-!=·~. ;:~:;;
~-:::~
!)=
24(•
t 5. s·;..z
'·/=:3. ?50
2'50
'~=.
~e•)
•>•t·~.
2.7(1
280
::ECRIB!i-~G
GE~)r'l-~iRV
:;c::
~~~~TEttt~A
:-;·:·;iE:i1
Ut1DE~
;"';;[
'5
h.= 1
F''=J. 141S69
l·:.o
300
319
~:!JN:;THr~T·;
!NPu:·
~=-;:o
~EM
~~~~*+~~~-*~~~~~~+~~~~~~~~~+~~~~~~-~~-++•~*i·+~-~~~+~~~+~~~~+~+~~
~EM
•
"330
A~ALYSIS
DE7E?MJNE METHOD OF
[~~PUT ·•rt~F·UT METHOD 1.2 OR 3
!F M€th=1 TYEN ~OTO •39
I:" ALL. . ·= ;,.t:. :~·.l·J ..- .. ·,I: . . ~:.'
FG? r~t 7(t ~5~
349
3:.0
3~0
=.:r.:
:E~~
~
·-.~~· •
~;~(1
;:'Eti
~
l;·lit>~t:;~tT!'HI
~00
?E~
~~~~-~•~~··-~~+~~~~~~~~~~~*~+•~~·+~-~~~~~·~~~-~~~~~·+~·•~•+~~~~-+~
.-·
PPitlT
4.;;9
I~'
:riP~;:
tUr 1 S~·:(~!...
!>,:o.r>
tiE:;T :
~:.~J
3•.. rn2=e
440
FOP T•.1 TQ M STEP
Th•T+i"i.· 1:~0
4';9
460
~~0
·~ut=~;·:.-:..s:::;E:iD-;.E
~~~1 •• rlr:r:·,.:1TCF::::
.::::·~:,
410
~-:=tE;:_::_.-;:;_
''~~~th
PRINT
PEM
.1
~
~*~~~+~*··~*~+~*+~**~+~~~*++~~·+~~+~~~+*~~*··~~~-+-~~~~~~~-+~+~+~
3tci=A1+SiNCP~~SINCThl•/(P~-'SlNCTh:
3~~-2=Al~SIN(~~SIN\Th)).~~a~SI~~(7h'~
Bt.;,1=A1+SHI•.U"'''>!il· Tl1;.
J .. -riJ,;,.;J,H
Br..a~:=H2-i+:3INc.'·/•3!~~lTi'1):·
!F ;1er.. ~,·-.>l
•:-~LL
:_t(ji
G~·Tl)
;:':3()
THE!'~
~~oro
ff_(Th, K, Pt
('-/4i!~3!t"i
Th>.
7h))
'57~3
~1.
Bt...~~,
E•t.o?~ ~
3t.
:~
-,
IF r·Ter.. h~- ··l THEN 071:3
CAL~
580
.:,qr.•.::s.·::l~ Th,
t~,
·::•.• m~J. 3•Airt1
~ ~.J(~--',
:•.:-"'!-) .:•
~~~
~~M
~+~~~+~~·+~~~~*~~~*~~+~~~~~~~+~~*-~~-~~~~~~-+*~~~~~~*~~~~*~~~~~~
tSdO
P:~~·~
~
51~
?EM
~~+~+~~~-~+*~~~+~*~~~~~~~~~~~~~~~+-~~~+--~4+~~~-"'!-+~~~~~-~~~~~~·+
,:;c;·1PI) l"E F 0~4E;: ~..; ~;~;_~4S
F"o:-•.L ·.
=2~1~Lt.:;7
...:.as
Ph(~. ;=;;~j~L(~T·:
F':a.o•."K)=20*LG
P~-b~~:J=20~LGT
GOT(I 78•3
=JR
1
·;F'~C ~
:~~.~I.Hu•:1*(:1JS·~ih.
~~PEF'EI'
-1
HFE.~TUP.E
:'
BS·"G%:3um0))
<.F'EIS{I::;~·:.um0-B· -?l-!'t-3.2~
~ES•G~£um0-3~·~1-3~~2)iCQ:;(Th)
Ht-tHL ·:.:·3
94
!F
~70
e·3~J
700
710
THEN GOTO 300
M~•~•-3
CALL.
~89
Gqtjad~Th~K,Sum~,Suml,W(+l.:~~~)
Et 1 =G ... _:;~.ur~e
E:t~=(G+Sit-~·~Th)*-':;tJrt~1:
··2
~EM
~~¥~~~~-~~*~+~~~•~~••~~~+~*~-•*~~~-~~~~~~~-~*~~*~~~~~~~~*~~~~~~~
720
~~~
•
730
REM
~~~~~~i·~•~•*~*~•~~~·~~4++~~·~~••+•~*•+*+•-+*~~~~**•·~~·~~~~-~·~~
~Q~PUTE
POWER PATTERNS FOR INDuCED SURfACE CURRE iTS
74e
Pi.,_.K~~=ZO*LGTtAES~
750
P~lK\=10*LGT(ABS((Et1-COS~7h))·~2+E.~2)~
P~b~K)•2B•LGTCAES<Etl-Bta1-Sta2:)
( '-.Et-1-Bt~!l--Bt..ot:!!; *-CC·S.:. -~h:-. J ..~;:+~t..2})
7eO
F'-tt•~..t::'=10~t.G.T·-.r=tB·::(
770
780
Th~(K'~Th+180.•Pj
?·;.~)
1'-- =K + 1
:;.ue
·tE:-;r r
:31~1
r l=t'."-l
330
::C5~
8~0
• SET UP FOP PLOT SUBROUTINE
RE~
:<rul n=::J.
~m~x=3~
8~0
~$t~p=5
880
:3'?0
·~·mtn=-5'1
Ymax=.a
·3ee
~·=-t-:,:.=5
·?t,a:
CALL To!rttf)l
9~0
GJSUB 980
MAT P=ZER•400l
MAT ~b=ZE~<400>
930
940
·•::o
C~L~
·~•),3US
~(!.)
EetD
-;.~e
1000
(.t<l~P~:_~),Pbl.*},?-~<~:.~P-:a.b( '!".:0:•
Temp~~~l,P•.*),PO\~).P~(~~~P~b(•)l
'?60
.·?80
·~80
GRAF·H I •:S RtJU TINE
THIS SUBROUTINE IS FOR
:;CLEAR
P~
1 ~)..;. . .)
·.1s2•·scAH AHI:LE
·,'li='REL~l-IVE
10.;;)
1070
PLOTTER IS ·GRAPHICS"
GRAPHIC:;
DEG
1130
114t)
1150
1160
1170
11:39
tt·:.o
1100
1210
1220
12:30
!240
~ARP~
PLGTS
ittTER r·;; 1~
.-U="."ttliErHtH PAT;E~:t·l"
1(15~)
! 11 I)
1120
LIHE~~
·::fr1i'tDAPD
1 >) 1 J
lt)2(t
1 (':3(t
1<)80
1090
1!00
Er.l;. :·
15,100,15~~0
LCCATE
SCALE
A::~;ES
FOR
._.j~g)·•
PG~ER~~B)''
·~mir.,~~max,Ymin,·(m~x
:-~;:.t. .:-r~.-~·5,··,"':=-t.:-p/5,:~;ro!r:,·,.·rnln.5,S,~
X=Xm~r,
TO
~m~x
STEP :<Atep
i10VE :-<,'lrntfl
DRAW ·~, '•'m;,.;,
rlE:<r ;;
CSI!E ::
LORG 5
C'CIR :·<=>-~mi
MOVE
''
TO :::ma~< ST~P :~;:;1: o:p
X,Ymjn-~Ym~x-Ymln)/15
LABEL USING "DDD";X
:lE:,-:r :~
i"'o)R 'i=';'rn i ,, TO 'lrn.~x ''>TEP
NO'·lE i<rn1
"l:>t
~p
!'1, ·.~
DRAU \rus.:<, 'l
•iE:;T i
12.;(!
;_tjRI;
!;::70
F0P ·t:Ymir, TO
1
12:30
i'liJ\·:E :-\nt; n- (
!290
LABEL
~SING
Ym~x
STEP '(step
i·::ma;..:-:.~m
in)/ ~3, ....
"DDD";"l
1~00
• d
1310
CSI<:E
l-320
MOVE
:3.~
Xmin1·(Xma~-!min)/3,Ymin-~'tm~x-f~t~)~6
95
1330
l340
LABEL :o<:ll
7",
Ym-i.x -·,·m; n) .· ·3
1:360
;-H)'..'E :.:':n in·- (
.... r. l F.: '?1)
Li'tBEL Yt:S
1"37~)
C:~IZE
13:3~
r·1C'·.·'E :•=:m in"· ( >;n·,.a,.., -~;min)...- 3, '-;'r.-•.a.::.:.. + ( 'lm.a~"~- .-·m 1 n) . - 1 ~
1390
L!JIR 0
1400
1410
t·::se
~(:r.-:
-ilK -~-:m1 n '
4
14:;.0
LABEL rl.f
LHtE T'i'PE
t·HJ VE :-~m 1 r., '(m! r.
"'OR J•'l Tl:• K1
144•)
1450
~IEXT
J
tHO
f'li)VE
:.-;ri• in, \ min
1420
14;"1)
1481)
14'~0
15 ('I)
1511)
1520
1530
154;)
15510
t:s.:t
I'm~,..,.(.
PLOT Tha(J),PCJ'.1
1
UNE TYPE 3
-FOR J=1 Tl) Kl
PLOT Tha~J),PbCJ),l
nE:-(T J
!..INE TYPE 1
FRAME
E:<IT •~RAPHIC:3
:JUMP GRAPH!C3
C'ETURtl
1:;7')
1~:;~
l ':'?~
1600
1610
1620
1630
1640
1650
SUB Udiff<Th,K,Bt~1.BI~2,Bt~1,Bt~2>
OPTIOH 3A~-E 1
COM XC300:, J•)0::3•30:··, J1 <:300>
COM PaC400>,P~'490>,P~bC400!,P•~•400)
(:1'3. 5
:•<(K)=CooSl ~l(Th.>
•:ALL B-t~ l C<JK>, Jo<:K;•, Jl <Kl •
1660
Ca.t:::Z*·-·lf.K,• ··:.-~-:.K·..
PeiK ::le~~·~T(A£S(Cat~COS~Th)))
~ ;...~. k) =~:~j~L~~T •.. H B:3 ··. C ~t :- :.
1670
16813
l ~·~9
1 ;"c)<)
l; 1'.)
17 21)
?~t~~:·=;.0~LGT(~ES~~~-~t-!!e!-Pt~~)tCOSlTh.··
i=-
:s.C· ·. :.;__:.
=2ti~L·:T·: ~3;
·:Cat -Bt sl-:a
.=a,;;~· ~
:;:;.;BEND
17'3;3
1741)
1 7'5•)
. 3UB
Gquad'Th,K,Sum~,Su~l,~,~>,Z<+))
17€-r)
1(70
OPTION
1780
t-~:2'36
1 7·~·)
S•.,m0=0
3um1=0
18€Bj
1:31'.)
1:321:1
CCM
EA~;E
1
:<~·300:,Jo(J0e~.-Jt(30e>
Wfn.:e.~e
t.fno:1=0
1830
F 2 .380
184•)
t:350
1:360
187'0
1S80
18'?0
1'300
t '? 1•)
C:zJ,l1•<3. !41596
FOR I= 1 TO H
X'~'"E•ZC[)~SIK<ThJ
CALL
B~sllXIK),Jo(K',J1•K.>l
Wfnc~=W(!)*Z(!)*]o(K)/(l+CF~Z(!))~2'A~
Sum,:t=S•.a•0+~H.. nc 1:3
IF Heth•2 THEH
G~TO
1~38
Wfncl=W(I)*Z(!)A~~!t~K·
0::-•..uul
t·~zv
\l~(F*~~!jA~)~~.)
=S·•.,m 1 +Wfn·.: l
1'?"30
HE> T I
1'?40
t·no
;uBEND
REM +*~~·*~·•****•*+~+•**~*•*~~~*~~~•-~++*~++*~**~~~•*~~•***~~~~**~~~~~~~~
REM • THIS SUBROUTINE COMPUTES THE :EORTH AHD FIRST O~DER ~ESSEL FHCd.
REM • QF THE FIRST KIHD
1'~80
~EM
!9'5J
1%13
~~4~~~~·4~~*~~~+++~•~•*~**~~**•**~~~*~~*~~·*•~~~~*~**•*~~~w+~*~~*~~~+~~
96
t·:o :.-o
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