A simple numerical method for the quadratic approximation of a discretely defined reflector surface is described. The numerical method was applied to interpolate the surface normal of a parabolic reflector surface from a grid of nine closest surface points to the point of incidence. After computing the surface normals, the geometrical optics and the aperture integration method using the discrete Fast Fourier Transform (FFT) were applied to compute the radiaton patterns for a symmetric and an offset antenna configurations. The computed patterns are compared to that of the analytic case and to the patterns generated from another numerical technique using the spline function approximation. In the paper, examples of computations are given. The accuracy of the numerical method is discussed

Acosta, R.

Lee, R. Q.

English

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NASA Technical Mern:oraiidnm 871 ,5
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A Numerical Method for Approximating
Antenna Surfaces Defined -b
Discrete-Surface Points
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(NASA-TM-871.25) A NUMERICAL METHOD FOR 	 986-10381
APPROXIMATING ANTENNA SURFACES DEFINED SX 	 j
DISCRETE SURFACE POINTS (NASA) 10 p
HC A02/MF A01	 CSC'L 20H	 Unclas	 _€ _
G3/32 27537
°^ Richard Q. Lee and Roberto Acosta
	
'	 J,ewes Research Center
Cleveland, Ohio
	
.	 OCT r
^--
	
EctFD	 o
95
Prepared for '.the
1985 Antenna Applications Symposium
c6sponsored by the University of Illinois
and the Rome Air Development Center
	
`	 Monticello, Illinois, Septem?,er 18-20, 1985
k
WASA
A NUMERICAL METHOD FOR APPROXIMATING ANTENNA SURFACES
DEFINED BY DISCRETE SURFACE POINTS
Richard Q. Lee and Roberto Acosta
National Aeronautics and Space Administration
Lewis Research Center
Cleveland, Ohio 44135
1. INTfRODUCTION
Reflector antennas on earth orbiting spacecrafts generally
cannot be described analytically. The reflector surface is
subjected to large temperature fluctuation and gradients, and
is thus warped from its true geometrical shape. Aside from
distortion by thermal stresses, reflector surfaces are often
co
	 purposely shaped to minimize phase aberations and scanning
LL,	 losses. To analyze distorted reflector antennas defined by
discrete surface points, a numerical technique must be applied
to compute an interpolatory surface passing through a grid of
discrete paints. Although numerical techniques for analyzing
reflector surface distortions and doubly curved reflector sur-
faces have been reported in open literatures, 1-3 all these
techniques are rather complicated and involve lengthy computa-
tions. In this paper, a simple numerical technique based on
Taylor's expansion and finite differences approximation is
presented. By applying the numerical technique to approximate
the surface normals of a distorted reflector surface, the aper-
ture field over.a near-field plane is obtained using geometric
optics technigde. The aperture integration method is then
applied to find the radiation patterns.4
2. FORMULATION
The proposed numerical Technique is a quadratic approxima-
tion of a distorted surface defined by a grid of nine points.,
Let (x,y,z) be the coordinate of any point, P, on the surface
which is described by
z = z(x,y)
The position vector of P is given by
.a..	 ..
r= xx +YY +zz
To obtain an approximation for the surface z(x,y), the position
vector is replaced by its Taylor's expansion around ro(xo,
y o ,z () ), the closest point to the point of incidence. That is
r0 ,y
► z) = ro (xo ► y o` z o ) + ( x - x
o ) ax + ( y - yo) ay
2-^•	 2a
+I (x,.xo)2a2 +(y-yo)2 a2
ax	 ay
2.a
+ 2(x - x o "r(l - yo ) axay + O(x3,y,3)
2
h
r^
I€
ii
^f
ii
a
where the partial derivatives of r are evaluated at ro(xo,
yo ,z o ). All terms higher thar the quadratics of x and y
have been neglected as indicated by 6(x 3 , y 3 ). The surface
may then be approximated by
z(x rY ) =z r a zo + (x - xo ) ax + ( Y-YO) ay
2	 7
+ 2 
(x - xo) 2 as z 
+ 2(x - xo) (Y ^- Yo) axay
ax
+ (y - yo)2 a 
2
yz + 0(x3,Y 3)
All the partial derivatives of z are evaluated at ro(xo,
yo ,zo ). The normals to the interpolatory surface can be
found by taking the gradient of z(x,y) as follows:
n _	
x+azy +z
ax	 ay
2
nx = x + (x - xo) 
a^c2 + (y
	 y o ) axay + 0(x2,y2)
22
n y 	 ay + (x - xo ) axay + (y _ yo) 
a 
2 + 0(x2 ► Y2)
ay
n z = -1
where ( n x ,n y ,n z ) are the x, y, and z components of n with
the normal pointing toward the source. Finite differences can
be applied to approximate the partial derivatives. Since the
3
4partial derivatives are of second order, a grid of nine points
is required.
The partial derivatives may be expressed as
( z4 - z0)
	
(z0 - z5)
a 
2 
z
	 (x4 - x0 )	 (x0 - x5)
ax  N
	
(x4 - x5)
2
az	 z4 	 z5
ax	 x4 - x5
( z2 - z0 )	 ( z0 - z7 )
82 z	 (y2 -
 y0 )	 (y0 - y7)
^- N
ay 	 (Y2 - Y7)
az z 2	 z7
ay 
N y
2 - y7
( z^- z 3 ) ( Z6 z8)
a2 z	 (x, - xg)	 ( x g - x8)
N ^-+
axay	 y2 - y7
where (xl ► Y,, <,0, (x2 ,y2 ,z 2 ) ... (x8 ,:y 8 za ) are the coordi-
nates of the points around the closest point (xo'yo'zo).
3. DISCUSSION AND RESULT
In applying the numerical technique to compute the surface
normals, the nine clesist points to the , point of incidence are
located by searching an array of discrete surface points defin-
ing the distorted antenna surface. Ray-tracing is then used to
i
.
compute the reflected field from the approximated normals. The
secondary far field patterns are obtained from the Fourier
transformation of the aperture field distribution.
To illustrate the numerical I .echnique, the .numerical algo-
rithm has been programmed and applied to the antenna configura-
tion shown in figure 1, The surface grid points for an
artificia'ily "enlarged" parabolic surface are generated. Thus,
the boL^-A ary points along the rim of the actual reflector sur-
face can be included in the computation. The computed secondary
far field patterns for the symmetric and the offset cases are
shown in figures 2 and 3 respectively. In the figures, the
E-plane radiation patterns computed from an analytic expression
of a parabolic surface are drawn with solid lines, while that
computed numerically are drawn with dots and broken lines. The
E-plane radiation patterns computed numerically from a grid of
50x50 surface points of -0.7 ), spacing was found to compare
exactly to that computed from an analytic expression. For the
offset case, the aperture surface is described by fewer points;
therefore, a higher density of target points is required to
produce the same accuracy. With a scan step of X/3 for the
f	 symmetric case and 1/6 for the offset case, the computer time
required to compute the secondary radiation patterns numerically
is 25 sec for the symmetric case and 17 sec for the offset case,
respectively. For comparison purpose, the radiation pattern
+ A
5
for the offset case has been computed using spline function
approximation. 5 The computed E-plane pattern is shown in
figure 3.
REFERENCES
1. Agrawal, P.K., Kauffman, •J.F., and Cr,osriell, '4.7., (1979) A
Method for .Pattern Calculation for Reflector Antennas Whose
Geometry is Described by a Finite Number of Discrete Surface
Points, in IEEE International Sym oslum on Antennas and
Propagation, Vol. 1, IEEE, Piscataway, NJ, pp. 67-70..
2. Chan, K.K., and Raab, A.R., (1981) Surface--Current Analysis
of Distorted Reflector Antenna, IEE Proc., Part H: p .croo-
waves. Ogt Antennas, 128:206-212.
3. Sletten, C.J., (1981) Ray Tracing Method fo p Doubly Curved
Reflector Surfaces, Proc. IEEE, 69:743-744.
4. Lam, T.C., Lee, S.W., and Acosta, R., (1984) Secondary Pat-
tern Computation of an Arbitrarily Shaped Main Reflector,
UILU-ENG-842541. NASA Grant NAG3-419, Electromagnetics
Laboratory Scientific Report No. 84-7, University of
Illinois.
5. Steinbach, R.E., and Winegar, S.R., (1985) Interdisciplinary
Design Analysis of a Precision Spacecraft Antenna, in
Structures, Structural Dynamics and Materials Conference,
26th, AIAA, New York, pp. 704-712.
6
1	 ^	 3
A	 X 5
0
G	 1
X
Enlarg ed section
of reflector surfaces.
X1m(33.3A )
' i .
(a) Antenna configuration.	 rb) Grid of surface points describing
the reflector surface.
Figure 1 -Antenna configuration used in the study.
0	 — -- EXACT
	• • • • •	 50X50 GRID
-- 30X30 GRID
'K
r `	
-12
`I
;r
9	 .18
,I
^	 W
f ° -24
a
	
7 
-30
	 r^
^	 S
:	 a
j	 -42	 t
f„	 -48	 l ;	 f
-54
-10.0	 -7,5	 -5.0	 -2.5	 0	 2.5	 5.0	 7.5	 10.0
ELEVATION ANGLE, deg
Figure 2. - E - plane pattern for the symmetric case,
t
*.I
U^,
-12
:o
ui
a
-2a
a.
a
P
x
-36
•
^----- EXACT
••••••• 50x50 GRID
-- — — 30230 GRID
r. •w•rra SPI IMF FIIMO.TInM
-a2
I
-aa •
-5	 •
-10.0	 -7.5	 -5.0	 -2.5	 0	 2.5	 5.0	 7.5	 10.0
ELEVATION ANGLE, dog
Flgure 3. - E - plane pattern for the offset case.
al
,I
D
	 J
i
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'i
1. Report No,	 2, Government Accession No, 3. Recipient's Catalog No,
NASA TM-87125
4. Tale anu B.ubiftie t5, Report Palo
A Numerical Method for Approximating Antenna
Surfaces Defined by Discrete Surface Faints c. Performing Organization Code
506-58-22
7. Author(s) 9. Performing Organization Report No,
Richard Q.	 tree and Roberto Acosta E-2738
10. WorK Olt No.
9. Performing Organization Name and Address
National Aeronautics and Space Administration
11. Contract or Grant No.
Lewis Research Center
Cleveland,  Ohio	 44135 13, Type of Report and Period Cuvsred
Technical Memorandum12, Sponsoring Agency Name and Address
National Aeronautics and Space Administration
Washington,	 D.C.	 20546
w
14,Sponsoring Agency Code
15. Supplementary Notes
Prepared for the 1985 Antenna Applications Symposium, cosponsored by the Univer-
sity of Illinois and the Rome Air Development Center, Monticello, 	 Illinois,
September 18-20, 1985.
16. Abstract
A simple numerical method for the quadratic approximation of a discretely defined
reflector surface is described.	 The numerical method was applied to interpolate
the surface normal of a parabolic reflector surface from a grid of nine closest
surface points to the point of incidence. 	 After computing the surface normals,
the geometrical optics and the aperture integration method using the discrete
Fast Fourier Transform (FFT) were applied to compute the radiaton patterns for a
syrometric and an offset antenna configuratons. 	 The computed patterns are com-
pared to that of the analytic case and to the patterns generated from another
numerical technique using the spline function approximation. 	 In the paper,
examples of computations are given. 	 The accuracy of the numerical method is
discussed.
17, Key Words (Suggested by Author(a)) 18, Distribution Statement
Numerical method Unclassified - unlimited
Reflector antenna STAR Category 32
19, Security Classif, (ot this report)
Unclassified
20, Security Classif, (of this page)
Unclassified
21. Nn, of pages 22, Price"
t
For Sale by the National Technical Information Service, Springfield, virgha. 221G11
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