17-3
CLOSED FORM SURFACE RAY ANALYSIS FOR ANTENNAS LOCATED ON A
CLASS OF AIRCRAFT WINGS
R M J h a , SA Bokhari, V Sudhakar and Pi? Mahapatra
D e p a r t m e n t of Aerospace Engineering
Indian I n s t i t u t e of Science
Bangalore, 560 012 India
ABSTRACT
An a i r c r a f t wing has been modeled a s a hybrid of a right circular cylinder and a
g e n e r a l p a r a b o l i c cylinder. This hybrid s t r u c t u r e model is successfully
employed along with t h e Geodesic C o n s t a n t Method (GCM) developed by t h e a u t h o r s
for t h e r a y - t h e o r e t i c EM analysis. S u r f a c e ray p a r a m e t e r s have been derived in
closed f o r m and mutual coupling results h a v e b e e n obtained f o r wings designed
for d i f f e r e n t flight regimes.
Introduction
A i r c r a f t wings h a v e a l a r g e and a l m o s t f l a t s u r f a c e a r e a , and a r e o f t e n
used by a i r c r a f t designers for locating a n t e n n a s over them. With t h e growing
t r e n d towards utilization of higher frequencies of t h e EM s p e c t r u m for aerospace
communication, wings h a v e c o m e t o b e t r e a t e d as electrically large s c a t t e r e r s
and their c u r v a t u r e s and thicknesses c a n no longer b e ignored in mutual coupling
calculations b e t w e e n a n t e n n a s l o c a t e d over them.
In this paper, a new method has been developed t o d e r i v e t h e s u r f a c e ray
g e o m e t r i c p a r a m e t e r s required in t h e high-frequency c o m p u t a t i o n s of a n t e n n a
c h a r a c t e r i s t i c s for radiators located on a n a i r c r a f t wing which is modeled by
f i n i t e sections of g e n e r a l parabolic cylinder (GPCYL) and t h e right circular
cylinder. This hybrid quadric cylinder (h-QUACYL) is t r e a t e d by t h e Geodesic
C o n s t a n t Method (GCM) which expresses a l l t h e s u r f a c e ray g e o m e t r i c p a r a m e t e r s
in t e r m s of t h e First Geodesic Constant, which itself c a n b e d e t e r m i n e d in
closed form.
Actual results have been p r e s e n t e d f o r wings designed f o r
d i f f e r e n t flight regimes.
Formulation for Hybrid QUACYL Wings
As a n e x a m p l e of a h-QUACYL, an a i r c r a f t wing, modeled by t h e sections of
g e n e r a l parabolic a n d circular cylinders (Fig. I), c a n b e described by t h e t w o
s e t s of hybrid p a r a m e t r i c equations [I]. The GPCYL s e c t i o n is described as
x = au
y = U2
z = z
with t h e truncation limit for t h e GPCYL given by
t h e circular cylindrical section is expressed as
x =
9 cos m
with t h e required matching
+ ps i n b
conditions p = a u t
/uI 5 ut.
y = L
CH2654-2/89/0000-0356
$1.00 01989 IEEE
356
z = z
and
0
5#
(1)
On t h e other hand,
(2)
< n w h i c h implies
t h a t a semi-circular cylinder is employed a s t h e matching section.
Equations ( I ) and (2) d e s c r i b e a hybrid cylindrical a i r c r a f t wing where t h e
t w o d i f f e r e n t cylinders i n t e r s e c t along t h e z-axis. Although t w o d i f f e r e n t s e t s
of c o o r d i n a t e s y s t e m s have been used f o r parametrizing t h e a i r c r a f t wing, t h e
use of common rectangular c o o r d i n a t e bases (x,y,z) t o d e s c r i b e t h e hybrid
ouadric s u r f a c e leads t o uniformitv in t h e s u r f a c e ray analysis. The junction
transition point (Fig. I ) is d e t e r m i n e d a s
Zt
= Zf
-
bfJVZf
-
Pbf
SSl
2,)
(3)
+
Since ut (and h e n c e $ ) is known a priori, e q (3) leads t o t h e c o m p l e t e
determination of t h e transiiion point in both t h e c o o r d i n a t e systems. T h e ray
analysis over t h e c o m p l e t e ray p a t h from t h e s o u r c e point S t o observation
point P is t r e a t e d by separately analyzing t h e a r c s e g m e n t s ST on t h e GPCYL
and T P on t h e circular cylinder. The ray g e o m e t r i c p a r a m e t e r s required for t h e
high frequency a n t e n n a c h a r a c t e r i s t i c s calculations [2,31 c a n now be easily
c a l c u l a t e d on t h e individual QUACYLs while t h e integral p a r a m e t e r s like t h e a r c
length a r e derived a s sum of t h e individual components over t h e t w o surfaces.
Another consequence of t r e a t i n g t h e ray g e o m e t r i c p a r a m e t e r s on t w o
different s u r f a c e s is t h a t t h e moving F r e n e t - f r a m e field vectors, i.e., t h e
tangent, normal and binormal vectors, obtained at t h e s o u r c e (on t h e G P C Y L ) and
observation (on t h e circular cylinders) points via t w o d i f f e r e n t c o o r d i n a t e
systems, a p p e a r a s dyadic pairs in t h e e l e c t r i c field expression required in t h e
mutual coupling calculations.
Numerical Results and Discussion
In a e r o s p a c e engineering, a i r c r a f t wings a r e usually designed f o r t h r e e
d i f f e r e n t flight regimes. The thickness of a wing is quantitatively expressed
in t e r m s of t h e thickness-to-chord r a t i o (T-C ratio) which is a n i m p o r t a n t
p a r a m e t e r in defining t h e wing shape. The t y p i c a l values of t h e T-C r a t i o in
t h e supersonic, high subsonic and low subsonic r e g i m e s a r e 3, 8, and 12%
respectively.
High frequency mutual coupling results have been presented in Fig. 2 and 3
for t h e variation of t h e location of t h e observation s l o t s along t h e u - p a r a m e t e r
line since in this principal direction t h e c u r v a t u r e of t h e wing s u r f a c e is nonzero.
Our study shows t h a t under identical s e t of conditions t h e m u t u a l
coupling is not sensitive t o t h e variation in t h e thickness-to-chord r a t i o of
t h e wing due t o t h e relatively weak c u r v a t u r e effects of t h e GPCYL portion of
t h e wing.
This, in t u r n , m i n i m i z e s c o n s t r a i n t s o n t h e s t r u c t u r a l a n d
aerodynamic engineers in choosing s u i t a b l e wing shapes.
357
REFERENCES
[ I ] R.M. J h a , S.A. Bokhari, V. Sudhakar and P.R. Mahapatra, "Kay analysis on a
class of hybrid c y l i n d r ~ c a la i r c r a f t wings", Electronics L e t t e r s (GB), vol.
24, pp. 21-22, Jan. 1988.
N. Wang, W.D. a u r n s i d e and R.G. Kouyoumjian, "A uniform GTD
solution for t h e radiation from sources on a convex surface", IEEE Trans.
Antennas & Propagat. (USA), Vol. AP-29, no. 4, pp. 609-22, July 1981.
[21 P.H. P a t h a k ,
[3] P.H. Pathak and N. Wang, "Ray analysis of mutual coupling b e t w e e n a n t e n n a s
on a convex surface", IEEE Trans. Antennas & Propagat. (USA), Vol. AP-29,
no. 6, pp. 911-22, Nov. 1981.
Trailing edge-,Junction
line
parabolic
cylinder
Fig. I
Mutual coupling b e t w e e n s l o t s on a hybrid cylindrical a i r c r a f t
wing.
358
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