Both deterministic and statistical analysis methods for investigating large aperture antenna radiation patterns are examined. The effects of surface errors on radiation patterns are discussed

Agrawal, P. K.

Cockrell, C. R.

Staton, L. D.

Breton 

NASA Technical Reports Server

DEVELOP?IENT OF ELECTROMAGNETIC ANALYSIS HETHODS 
FOR LARGE APERTURE ANTENNAS 
C.  R. Cockrell and L. 3. Staton 
NASA Lanelcy Research Center 
P. K.  ~ ~ r a w a l *  
RCA - Moorestown, N. J. 
LSST 1ST ANNUAL TECHNICAL REVIEW 
Nove~ber 7-8, 1979 
* Formerly with George Washington University a t  NASA Langley Research Center 
https://ntrs.nasa.gov/search.jsp?R=19800010876 2020-03-21T19:48:58+00:00Z
OBJECTIVES 
TO ESTABLISH AND VERIFY TECHNIQUES FOR PREDICTION OF EM 
PERFORMANCE OF LARGE SPACE ANTENNAS 
TO DEVELOP STAT1 STlCAL AND DETERMINI STIC MODELING TECHNIQUES- 
INCLUDING EFFECTS OF SURFACE ROUGHNESS, DISTORTION, AND 
SECMENTAT I ON 
DESCRIPTION OF WORK 
EXTENS ION OF PRESENT TECHNIQUES OF APERTURE INTEGRATION FOR LARGE, 
SEGMENTED REFLECTORS 
* Planar Segmentation 
* Curved Segmerrtation 
ANALYSES OF DEXRMINI  STICALLY D l  STORTED LARGE REFLECTORS WHOSE SHAPE 
I S  DETERMINED B Y  SAMPLED POINTS 
UN lFlED APPL iCATlON OF STAT1 STICiiL CORRELATION THEORY FCR BOTH SURFACE 
ROUGHNESS AND LARGE SCALE SURFACE D N  lATIONS 
CONSTRUCTION AND TkSTING OF EXPER IMENTAL MODEL TO VER IFY THEORETICAL 
MODEL 
GEG#t;Al OF THE REFLECTOR A N T E N N A  
The g e o m e t r y  o f  f i g u r e  1 is  u s e d  t o  compute  t h e  r a d i a t i o n  
p a t t e r n s  o f  r e f l e c t o r s .  The  a p e r t u r e  i n t e g r a t i o n  t e c h n i q u e  i s  
employed  i n  o b t a i n i n g  t h e s e  p a t t e r n s :  F i r s t ,  r a y s  a r e  t r a c e d  
f r o m  t h e  f e e d  t o  t h e  r e f l e c t o r  s u r f a c e  a n d  t h e n  t o  a n  a p e r t u r e  
p l a n e  l o c a t e d  i n  f r o n t  of t h e  r e f l e c t o r .  Nex t ,  u s i n g  g e o m e t r i c  
o p t i c s  f o r  e a c h  r a y ,  t h e  t a n g e n t i a l  e l e c t r i c  f i e l d  is found  a t  
many p o i n t s  i n  t h e  a p e r t u r e  p l a n e  and,by n u m e r i c a l l y  p e r f o r m i n g  
a d o u b l e  i n t e g r a t i o n  o v e r  t h e  a p e r t u r e  p l a n q t h e  s e c o n d a r y  f a r -  
f i e l d  r a d i a t i o n  p a t t e r n  i s  computed ,  The r e l a t i v e l y  s l o w  l a t e r a l  
v a r i a t i o n  o f  t h e  f i e l d s  i n  t h e  a p e r t u r e  p l a n e  compared  v i t h  t h a t  
of t h e  s u r f a c e  c u r r e n t s  o n  t h e  r e f l e c t o r  a l l o v s  a  more e c o n o m i c a l  
c o m p u t a t i o n  t h a n  d o e s  t h e  d i r e c t  i n t e g r a t i o n  o f  s u r f a e e  c u r r e n t s .  
Aperture Plane 
-- - ----- X 
---+-u 
Figure 1 
COXPUTED RADIATIOX PATTERNS FOR SEGHEXTED 
REFLECTORS US IilC APERTURE INTECRATIOi? >lETUOD 
Shown i n  f i g u r e  2 a r e  c a l c u l a t e d  H-p lane  r a d i a t i o n  p a t t e r n s  
f o r  a s p h e r i c a l  r e f l e c t o r  wh ich  i s  a p p r o x i m a t e d  b y  a number o f  
p l a n a r  h e x a g o n a l ,  t r i a n g u l a r ,  a n d  s q u a r e  f a c e t s  p l a c e d  on  t h e  
i d e a l  s p h e r e  s u c n  t h a t  t h e  R>lS s u r f a c e  d e v i a t i o n  i n  e a c h  case is  
t h e  same (.012m). On t h e  a v e r a g e  t h e  s u r f a c e  d e v i a t i o n  i s  more 
s l o w l y  v a r y i n g  o v e r  t h e  r e f l e c t o r  a p p r o x i m a t e d  by p l a n a r  n e x a g o n s  
t h a n  o v e r  t h a t  a p p r o x i m a t e d  by p l a n a r  t r i a n g l e s .  The p a t t e r n  f o r  
t h e  h e x a g o n a l  a p p r o x i m a t i o n  a g r e e s  b e t t e r  w i t h  t h e  p a t t e r n  f o r  
t n e  e x a c t  s p h e r i c a l  r e f l e c t o r  ( n o t  shown).  The p a t t e r n  f o r  t h e  
s q u a r e  a p p r o x i m a t i o n ,  howeve r ,  h a s  h i g h e r  s i d e  l o b e s  w h i c h  a r e  
a p p a r e n t l y  c a u s e d  by t h e  r e g u l a r  s u r f a c e  d e v i a t i o n  a s s o c i a t e d  
v i t h  t h i s  s h a p e .  
0 0.1 0.2 0.3 i . 4  0.5 
Angle off the Reflector Axis (Degrees) 
F i g u r e  2 
EFFECTS OF LARGE SCALE SURFACE ERRORS 
ON EN PERFOKXAHCE 
An e x a m p l e  o f  l a r g e  s ca le  d e t e r m i n i s t i c  t y p e  p h a s e  e r r o r  
( q u a d r a t i c )  e f f e c t s  on t h e  r a d i a t i o n  p a t t e r n s  o f  a n  a p e r t u r e  
a n t e n n a  is shown i n  f i g u r e  3. The s o l i d  c u r v e  c o r r e s p o n d s  t o  
z e r o  e r r o r  (A-0); i .e . ,  t h e  a p e r t u r e  d i s t r i b u t i o n  v a r i e s  o n l y  i n  
a m p l i t u d e .  The c a s e s  A=X/& and  A - X f G  c o r r e s p o n d  t o  i n c r e a s i n g  
q u a d r a t i c  p h a s e  e r r o r  w i t h  tihe same a m p l i t u d e  d i s t r i b u t i o n .  In -  
c r e a s e  i n  p h a s e  e r r o r  r e s u l t s  i n  a n  i n c r e a s e  i n  power r a d i a t e d  
i n  t h e  s i d e  l o b e s .  
Phase 
Aperture 
F i g u r e  3 
ENVELOPES OF R A D I A T I O N  PATTERNS F O R  
QUASI-RANUOM P t l A S E  E R R O R S  
F i g u r e  4 i l l u s t r a t e s  f o r  a c i r c u l a r  a p e r t u r e  a n t e n n a  
( d i a m e t e r  0 )  how t h e  r a d i a t e d  p o w e r  i n  t h e  s i d e  l o b e s  i n c r e a s e s  
f o r  a c lass  o f  i r r e g u l a r  p h a s e  e r r o r s  ( s e e  i n s e t )  w h i c h  may b e  
c h a r a c t e r i z e d  as  q u a s i - r a n d o m  ( r e f .  1). T h e  c u r v e  l a b l e d  0 i s  
e r r o r - f r e e  a n d  t h e  r e m a i n i n g  c u r v e s  r e p r e s e n t  i n d i c a t e d  a m o u n t s  
o f  maximum p e a k  t o  p e a k  p h a s e  e r r o r s .  A p e r t u r e  t o l e r a n c e s  
c o r r e s p o n d i n g  t o  t h e s e  p h a s e  e r r o r s  a r e  shown i n  t h e  p a r e n t h e s e s .  
T h e  g e n e r a l  e f f e c t  o f  s u c h  i r r e g u l a r , b u t  d e t e r m i n i s t i c ,  p h a s e  
e r r o r s  i s  a l m o s t  a l w a y s  t o  r a i s e  t h e  side l o b e  l e v e l s  a n d  h e n c e  
t o  l o w e r  t h e  a n t e n n a  beam e f f i c i e n c y .  
Phase Error 
peak to Peak (talerance) 
1 I I I I I L I i 
0 10 20 30 40 50 60 70 80 
(nDl~) Sin 9 
F i g u r e  4 
RADIATION PATTERNS OF 20-SEGMENT APERTURE 
ANTENNA WITH GAUSSIAN DISTRIdUTED PUASE ERROR 
F i g u r e  5 s h o w s  t h e  c a l c u l a t e d  r a d i a t i o n  p a t t e r n s  o f  2 0 - s e g m e n t  
a p e r t u r e  a n t e n n a s  o f  w i d t h  D w i t h  a s s u m p t i o n  o f  u n i f o r m  a m p l i t u d e  
o n  a l l  s e g m e n t s  a n d  G a u s s i a n  d i s t r i b u t e d  p h a s e  e r r o r  on  e a c h  
s e g m e n t .  T h e  s o l i d  c u r v e  r e p r e s e n t s  t h e  r a d i a t i o n  p a t t e r n  when 
n o  phase e r r o r  e x i s t s  a c r o s s  t h e  a p e r t u r e .  The  d o t t e d  c u r v e  
r e p r e s e n t s  t h e  e x p e c t e d  ( e n s e m b l e  a v e r a g e )  p a t t e r n  w a i c h  i s  
o b t a i n e d  t b r o u g h  random p r o c e s s  a n a l y s e s .  The  r e m a i n i n g  
r a d i a t i o n  p a t t e r n s  come f r o m  two members  o f  t n e  e n s e m b l e  o f  
2 0 - s e g m e n t  a p e r t u r e  a n t e n n a s .  
Phase 
(~DIA) Sin 9 
F i g u r e  5 
SECONUARY PATTERS BEAM EFFICIENCY v s .  RMS 
SURFACE ERROR I N  WAVELENGTHS FOR DIFFERENT CORRELATION LENGTHS 
The e f f e c t s  o f  s u r f a c e  e r r o r s  o n  t h e  beam e f f i c i e n c y  o f  a  
l a r g e  a p e r t u r e  a n t e n n a  c a n  b e  i l l u s t r a t e d  by a p p l y i n g  t h e  t h e o r y  
d e v e l o p e d  by  R u z e  ( r e f .  2 ) .  I n  R u z e ' s  w o r k  t h e  p h a s e  e r r o r  
( s u r f a c e  e r r o r )  i s  c h o s e n  f r o m  a G a u s s i a n  p o p u l a t i o n  w h i c h  i s  
s t a t i s t i c a l l y  u n i f o r m  o v e r  t h e  e n t i r e  r e f l e c t o r  s u r f a c e . T h e  auto- 
c o r r e l a t i o n  f u n c t i o n  o f  t h e  p h a s e  i s  a l s o  t a k e n  t o  h a v e  a  
G a u s s i a n  f o r m  w i t h  a  c o n s t a n t  v a r i a n c e  ( s q u a r e  o f  t h e  c o r r e l a t i o n  
l e n g t h ) .  B a s e d  o n  h i s  w o r k  t h e  beam e f f i c i e n c y  ( a s s u m e d  t o  b e  
1 0 0  p e r c e n t  f o r  t h e  u n p e r t u r b e d  a p e r t u r e )  i s  p l o t t e d  i n  f i g u r e  6 
as a  f u n c t i o n  o f  t m s  s u r f a c e  e r r o r  w i t h  n o r m a l i z e d  c o r r e l a t i o n  
l e n g t h  ( C / A ) a s  a p a r a m e t e r .  I t  i s  s e e n  t h a t  t h e  s m a l l e r  C I A  
v a l u e s  ( i . e . ,  m o r e  r a p i d l y  v a r y i n g  s u r f a c e s )  g i v e  r i se  t o  m o r e  
s t r i n g e n t  r e q u i r e m e n t s  o n  s u r f a c e  rms e r r o r  f o r  n i g h  beam 
e f f i c i e n c y .  F o r  m o r e  s l o w l y  v a r y i n g  s u r f a c e  ( l a r g e  CIA) t h e  
s u r f a c e  r m s  e r r o r  c a n  b e  r e l a x e d  a n d  s t i l l  a l l o w  r e l a t i v e l y  h i g h  
beam e f f i c i e n c y .  
RMS Surface Error in Wavelengths 
F i g u r e  6 
R E A L  P A R T  O F  T H E  A U T O - C O R R E L A T I O N  F U N C T I O A  O F  
2 0 - S E G M E N T  A P E R T U R E  A N T E N N A  W I T H  
G A U S S I A i t  D I S T R I B U T E D  P H A S E  E R R O R  
F i g u r e  7 s h o v s  t h e  r e a l  p a r t  o f  t h e  a u t o - c o r r e l a t i o n  f u n c t i o n  
o f  a  20-segment  a p e r t u r e  a n t e n n a  w i t h  t h e  a s s u m p t i o n  o f  u n i f o r m  
a m p l i t u d e  a n d  G a u s s i a n  d i s t r i b u t e d  p h a s e  e r r o r  on e a c h  s e g m e n t .  
The s o l i d  c u r v e  i s  t h q  z e r o  e r r o r  c a s e  a n d  t h e  d o t t e d  c u r v e  i s  
t h e  e n s e m b l e  a v e r a g e  o b t a i n e d  t h r o u g h  random p r o c e s s  a n a l y s e s .  
The two r e m a i n i n g  c u r v e s  a r e  t h e  r e a l  p a r t s  o f  t h e  a u t o - c o r r e l a -  
t i o n  f u n c t i o n s  o f  two members o f  t h e  e n s e m b l e  o f  20 - segmen t  
a p e r t u r e  a n t e n n a .  The F o u r i e r  t r a n s f o r m s  o f  t h e  t o t a l  a u t o -  
c o r r e l a t i o n  f u n c t i o n s  ( g i v e n  i n  p a r t  i n  f i g u r e  7 )  a r e  t h e  power 
p a t t e r n  c u r v e s  g i v e n  i n  f i g u r e  5. 
F i g u r e  7 
RATIOS OF STANDARD DEVIATION TO THE MEAN 
AS A FUNCTION OF AUTO-CORRELATION SEPARATIOA UISTANCE 
By o b s e r v i n g  f i g u r e  7 ,  i t  c a n  b e  s e e n  t h a t  t h e  a u t o -  
c o r r e l a t i o n  f u n c t i o n s  f r o m  members o f  a n  e n s e m b l e  o f  2 0 - s e g m e n t  
a p e r t u r e  a n t e n n a s  c a n  d e v i a t e  f r o m  t h e i r  e n s e m b l e  a v e r a g e  
a u t o - c o r r e l a t i o n  f u n c t i o n  ( d o t t e d  c u r v e  i n  f i g ~ r t  7 ) .  T h e  
r a t i o  o f  t h e  t a n d a r d  d e v i a t i o n  t o  mean f o r  t h e  2 0 - s e g m e n t  
a p e r t u r e  a n t ~ n n a  a s  a  f u n c t i o n  o f  a u t o - c o r r e l a t i o n  s e p a r a t i o n  
d i s t a n c e  i s  g i v e n  b y  t h e  c i r c l e s  i n  f i g u r e  8. By i n c r e a s i n g  
t h e  h u m b e r  o f  s e g m e n t s  w i t h i n  e a c h  a n t e n n a ,  t h e  a u t o - c o r r e l a t i o n  
f u n c t i o n s  o f  t h e  s e p a r a t e  a n t e n n a s  a p p r o a c h  a r b i t r a r i l y  c l o s e  t o  
t h e e n s e m o l e  a v e r a g e  ( e x p e c t e d )  a u t o - c o r r e l a t i o n  f u n c t i o n  p r e -  
d i c t e d  b y  random p r o c e s s  a n a l y s e s .  The  i n c r e a s e  i n  t h e  number  
o f  s e g m e n t s  c l e a r l y  r e d u c e s  t h e  r a t i o  o f  s t a n d a r d  d e v i a t i o n  t o  
t h e  means  shown b y  t h e  s q u a r e  a n d  t r i a n g l e  s y m b o l s  i n  f i g u r e  8. 
- 
0 " A  
0 0 O A  
0 0 
A& 
a a n ~ ~ ~ A '  
&AnnA 
Symbols 
I I I I I I I I I - I I 1 I I I 
0 4d 8d 12d I6d 2W 24d 28d Z d  606 64d 68d R d  76d 80d A 
Separation Distance 
Figure 8 
CONCLUDING REMARKS 
Deterministic Analyses 
* assumes exact knowledge of surface 
* t rue  radiation patterns 
* both scale distort ions 
a range of possible distortions can lead to wide variance about 
the designed pattern 
Raildom Process Analyses 
* assumes statistical knowledge of burface 
+ average radiation patterns 
* both scale distortions, provided correlation region is small 
compared to the aperture size 
large c~r re la t ion  region can lead to large variance about the 
average pattern 
General Effects of both Analyses 
0 broaden main beam 
4 side lobe radiation increases with phase amplitude and variation 
- requirement of h igh beam efficiency in a very large antenna 
demands utmost care in EM analysis 
MPER IMENTALLY VER IFY SEGMENTED REFLECTOR PROCRAM 
EX PER IMEMTAUY VER t FY SAMPLED PO I NT PROGRAM 
CONTINUE ANALYSES OF ANTENNAS THROUGH AUTOCORRELATI ON TECHNIQUES 
CONTINUE SEGMENTED REFLECTOR ANALYSES 
PUBLICATIONS 
"A PRELIMINARY STUDY OF A VERY LARGE SPACE RADIOMETR IC ANTENNA," 
NASA TM MK)4? , JAN. 1979, P. K. AGRAWAL 
"A MEMOO FOR PATTERN CALCULATION FOR RERECTCR ANTENNAS WHOSE 
GEOMETRY I S DESCR IBED BY  A FINITE NUMBER OF DISCRETE 
SURFACE POINTS," IEEE APS SYMPOSIUM, JUNE 1979, P. K. AGRAWAL, 
J. F. KAUFFMAN, AND W. F. CROSWELL 
a * PRELIMINARY DESIGN OF LARGE REFLECTORS w ITH FLAT FACETS~NASA TM 
80164, P. K. AGRAWAL, M S. ANDERSON, M.F. CARD 
1 .  d r a g o n e ,  C .  and Hogg, U. C . :  Vide-Angle  R a d i a t i o n  Due t o  
Rough Phase F r o n t s ,  B e l l  Sys t em T e c h n i c a l  J o u r n a l ,  
Vol. XLII, No. 5 ,  September  1963 .  
2 .  Ruze,  John: Antenna T o l e r a n c e  Theory - A Reviev,  P r o c .  I L E E ,  
Vol. 5 4 ,  pp. 6 3 3 - 0 4 3 ,  A p r i l  1 9 6 6 .  


Development of electromagnetic analysis methods for large aperture antennas

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