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NASA Technical Mern:oraiidnm 871 ,5

ti _ 

A Numerical Method for Approximating

Antenna Surfaces Defined -b

Discrete-Surface Points

(NASA-TM-871.25) A NUMERICAL METHOD FOR 986-10381

APPROXIMATING ANTENNA SURFACES DEFINED SX 

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

^--

EctF D 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 surfaces have been reported in open literatures, 1-3 all these

techniques are rather complicated and involve lengthy computations. 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 aperture 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 approximation 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 (x o ► y o` z o ) + ( x - x

o ) ax + ( y - y o ) 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
I€

ii^f

ii

where the partial derivatives of r are evaluated at ro(xo,

y o ,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 - x o ) 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,

y o ,z o ). 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

n x = 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
4

partial 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 z 4 z5

ax x 4 - x5

( z 2 - z 0 ) ( z 0 - z 7 )

8 2 z (y 2 - y0 ) (y0 - y7) ^- N

ay (Y2 - Y7)

az z 2 z7

ay 

N y

2 - y7

( z^- z 3 ) ( Z 6 z8)

a 2 z (x, - xg) ( x g - x8)

N ^-+

axay y2 - y7

where (xl ► Y,, <,0, (x2 ,y 2 ,z 2 ) ... (x8 ,:y 8 za ) are the coordinates 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 ,p oint of incidence are

located by searching an array of discrete surface points defining 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 algorithm has been programmed and applied to the antenna configuration 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 surface 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

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 .croowaves. 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 Pattern 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.

1m X

(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
-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.

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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

12, Sponsoring Agency Name and Address Technical Memorandum

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 University 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 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.

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. 221G1 1

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