43-9
CLOSED FORM EVALUATION OF ELEMENT COUPLING COEFFICIENTS IN
CONFORMAL ARRAYS ON GENERAL QUADRIC CYLINDERS
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
Quadric cylinders (QUACYLs) form a n i m p o r t a n t c l a s s of canonical shapes in
a e r o s p a c e engineering f r o m t h e s u r f a c e modeling point of view. In designing and
analyzing conformal a n t e n n a a r r a y s o v e r a e r o s p a c e bodies, a knowledge of t h e
mutual coupling b e t w e e n individual e l e m e n t s is necessary.
In this paper, a
closed f o r m Geodesic C o n s t a n t Method (GCM) is developed t o model t h e rayg e o m e t r i c a s p e c t s of c o n f o r m a l a r r a y s on QUACYLs. The formulation incorporates
a shaping p a r a m e t e r , p e r m i t t i n g modeling of s u r f a c e s of d i f f e r e n t sharpness.
iviutual coupling results f o r t h e g e n e r a l parabolic cylinder a r e presented t o
i l l u s t r a t e t h e application of t h e formulation.
Introduction
It has been suggested t h a t t h e use of distributed a n t e n n a s on a i r c r a f t wing
for r a d a r applications h a s a significant a d v a n t a g e over t h e conventional single
Distributed a n t e n n a s on wing s u r f a c e s have also been
a n t e n n a c o n c e p t [I].
suggested f o r navigational applications using t h e Global Positioning Systems
(GPS). Such conformal a r r a y s on convex s u r f a c e s directly use t h e theory of
planar arrays. In t h i s paper, w e d e r i v e t h e s u r f a c e r a y g e o m e t r i c p a r a m e t e r s
over g e n e r a l quadric cylinders (QUACYLs). Those p a r a m e t e r s c a n b e used t o
d e t e r m i n e radiation c h a r a c t e r i s t i c s of t h e arrays.
As an example, specific
results on m u t u a l coupling b e t w e e n e l e m e n t p a i r s a r e provided in t h e paper.
QUACYLs, which include circular, elliptic, and g e n e r a l parabolic and hyperbolic
cylinders, form convenient canonical s u r f a c e s t o r modeling many major components
of a e r o s p a c e bodies, such a s fuselage and wings.
Formulation
A g e n e r a l quadric cylinder (QUACYL) is a regular quadric s u r f a c e and can be
expressed in t h e geodesic c o o r d i n a t e s (u,v) in t h e p a r a m e t r i c t o r m
x = f(u)
y = g(u)
z = v
(1)
which in t h e particular c a s e of t h e general parabolic cylinder (GPCYL), c a n b e
expressed a s
x = au
2
y = u
z = v
(2)
w h e r e a is shaping p a r a m e t e r and d e t e r m i n e s t h e "sharpness" of t h e GPCYL. The
effect of a i s i l l u s t r a t e d in F i g . I . All t h e q u a d r i c c y l i n d e r s c a n b e
generalized t o i n c o r p o r a t e t h e shaping p a r a m e t e r in their p a r a m e t r i c equations
1004
in t h e geodesic c o o r d i n a t e system.
The equation f o r t h e right mth-order
geodesic on a QUACYL in t h e geodesic c o o r d i n a t e is expressed a s
T h e subscripts U and v r e f e r t o t h e partial derivatives w.r.t. U and
v respectively. The 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
mutual coupling analysis c a n b e conveniently classified a s [2]
1. Surface-dependent ray geometric parameters which depend on t h e n a t u r e of t h e
s u r f a c e alone and a r e expressed a s a function of t h e coordinates of t h e point a t
which they a r e evaluated. T h e unit s u r f a c e normal v e c t o r and t h e principal
c u r v a t u r e s a r e e x a m p l e s of these.
2. Geodesic-dependent ray geometric parameters depend on t h e individual s p a c e
curves t r a c e d on t h e s e quadric s u r f a c e s in additiorl $0, t h e n a t u r e of t h e
surface, for example t h e F r e n e t - f r a m e field v e c t o r s (t,n,b) and t h e radius of
c u r v a t u r e along t h e geodesic.
3. Interaction-dependent ray geometric parameters which c a n b e described only if
t h e geodesic c o o r d i n a t e s of t h e s o u r c e and observation points a r e given. The
a r c length, t h e generalized torsion f a c t o r and F o c k p a r a m e t e r , and t h e blending
functions a r e e x a m p l e s of interaction-dependent r a y g e o m e t r i c p a r a m e t e r s .
All t h e s e ray g e o m e t r i c p a r a , n e t e r s c a n b e derived in t h e closed f o r m using
One of t h e highlights of our m e t h o d is t h a t all
t h e definition given in [3].
t h e s e expressions a r e now a function of t n e F i r s t Geodesic C o n s t a n t h and t h e i r
a c c u r a c y depends on t h e a c c u r a c y of I alone. For t h i s reason, w e c a l l t h i s
method t h e Geodesic Constant Idethod (GCM).
However, serious difficulties e x i s t in deriving many of t h e i n t e r a c t i o n dependent p a r a m e t e r s ; t h e s e p e r t a i n t o t h e integration of t h e a r c length and
generalized Fock p a r a m e t e r .
W e h a v e been a b l e t o d e r i v e derived t h e s e
p a r a m e t e r s analytically f o r t h e c a s e of t h e QUACYLs. For t h e specific c a s e of
CPCYLs, t h e s e a r e expressed in t h e closed f o r m as
Mutual coupling b e t w e e n finite-dimensional
(0.5
x 0.2 A) slots o v e r
GPCYLs of varying shaping p a r a m e t e r has been c o m p u t e d and t w o specific results
a r e presented h e r e a s a n illustration of t h e m a t h e m a t i c a l model p r e s e n t e d in
this paper. Figures 2 and 3 show t h e mutual a d m i t t a n c e magnitude and phase
plots for increasing separation b e t w e e n t h e two s l o t s along u-axis. T h e z-axis
separation between t h e centroids of t h e slots in t h e Figures 2 and 3, is t a k e n
a s z e r o and 0.5 A, respectively.
1005
Summary
For t h e c a s e s of t h e circular cylinder and t h e GPCYL, t h e expressions
obtained f o r t h e ray p a r a m e t e r s a r e in t h e closed f o r m , since t h e geodesic
c o n s t a n t h is also derivable in t h e closed form. On t h e o t h e r hand, for t h e
c o o r d i n a t e s u r f a c e s of t h e Elliptic-cylinder C o o r d i n a t e System, such a s t h e
e l l i p t i c a n d g e n e r a l h y p e r b o l i c c y l i n d e r s , t h e e x p l i c i t n a t u r e of t h e r a y
p a r a m e t e r s a s a function of h remains s i n c e h c a n b e expressed explicitly in
t e r m s of t h e a s y m p t o t i c s e r i e s expansions of t h e incomplete elliptic integral
functions.
Using d i f f e r e n t shaping p a r a m e t e r s one c a n g e n e r a t e family of QUACYLs of
d i f f e r e n t flatness. The shaping p a r a m e t e r is employed a s a control p a r a m e t e r in
S u r f a c e Modeling for matching t h e quadrics t o t h e a c t u a l surfaces considered.
REFERENCES
[I] B.D. SteinDerg and E. Yadin, "Self-cohering an airborne radio camera", IEEE
Trans. Aerospace & Electronics S y s t e m s (LISA), vol. A S - 1 9 , no. 3, pp. 483490, [way 1983.
121 K.M. J h a , S.A. Bokhari, V . Sudhakar and P.R. Mahapatra: "Ray analysis of
mutual coupling b e t w e e n a n t e n n a s on a general parabolic cylinder", Journees
lnternationales d e Nice sur les Antennas, JINA'88, International Symposium
on Antennas, Universite d e Nice, Nice, France, pp. 70-73, 8-10 Nov. 1988.
[31 P.H. P a t h a k and N. Wang, "Ray analysis of mutual coupling between antennas
on a convex surface", IEEE Trans. Antennas & Propagat. (USA), vol. AP-29,
no. 6, pp. 911-922, Nov. 1981.
0.2.0
Fig. I
G e n e r a l parabolic cylinders (GPCYLs) for d i f f e r e n t value of t h e shaping
p a r a m e t e r a (simulated on a n IBM PC-XT compatible).
1006
360
-60
360
-60
300
300
-BO
-BO
240
240
1 BO
-100
1BO
-100
120
120
-120
-120
60
60
0
-140
0
3
6
0
-140
6
12
9
Arc length s in wovelengths
Arc length s in wovelengths
[ T I
3
0
12
9
I F i g 3 I Mutual
Mutual odmittonce vs. the arc length
admittance vs. the orc length
for separation olong U-oxis between two
far separotion olong U-axis between two
rect. slots (0.5X x 0.2M
rect. slots (0.SX x 0.21)
o
-
on o WtYL of
0.50, z-oxis seporotion -0.0 (constont).
Mogni tude
(-),
Phase
(----)
.
o
- 0.50,
on o GPCYL of
z-oxis seporotion -0.5X (constont).
Mogni tude (-1.
Phose
(----)
.
.-
.
i
- ?
t