CC BY
9Journal of Siberian Federal University. Engineering & Technologies 7 (2014 7) 846-852
УДК 621.396.677
Antenna Arrays on Surfaces of Revolution
Alexey А. Erokhina, Vasily S. Pankoa*, Yury P. Salomatova and Mikhail I. Sugakb
aSiberian Federal University 79 Svobodny, Krasnoyarsk, 660041, Russia bSaint Petersburg Electrotechnical University "LETI" 5 Professora Popova Str, St. Petersburg, 197376, Russia
Received 02.08.2014, received in revised form 22.09.2014, accepted 04.10.2014
In this paper explicit analytical expressions are derived which describe radiation pattern of multi ring antenna array and multi ring continuous radiator allocated on an arbitrary surface of revolution. Radiation patterns of semi-spherical, conical and hyperbolical antenna arrays are shown.
Keywords: scanning, ring antenna, antenna array, surface of revolution.
Антенные решетки на поверхностях вращения
А.А. Ерохина, В.С. Панькоа, Ю.П. Саломатова, М.И. Сугакб
аСибирский федеральный университет Россия, 660041, Красноярск, пр. Свободный, 79 бСанкт-Петербургский электротехнический университет "ЛЭТИ" Россия, 197376, Санкт-Петербург, ул. Профессора Попова, 5
В данной статье приведены явные аналитические выражения, описывающие диаграммы направленности многокольцевой антенной решетки и многокольцевого непрерывного излучателя, расположенных на произвольной поверхности вращения. Показаны диаграммы направленности полусферической, конической и гиперболической антенных решеток.
Ключевые слова: сканирование, кольцевая антенна, антенная решетка, поверхность вращения.
1. Introduction
Directional properties of antenna arrays of various kinds are studied in the literature in detail [13]. In [4] and [5] main characteristics for certain class of arrays - ring and multi ring having a planar shape are described. These results can be generalized to the case of antenna arrays which are located not in the plane but on an arbitrary rotational surface (sphere, cone and similar). In this paper we will
© Siberian Federal University. All rights reserved Corresponding author E-mail address: vpanko@sfu-kras.ru
*
derive an expression for radiation pattern of such arrays and will present results for base surfaces of different kind.
2. Radiation pattern of antenna array allocated on a surface of revolution
Let us consider antenna array consisting of N identical elements allocated in z = z0 plane. Radiation pattern of such array is [1]:
N
f (9 9)= F (9 9)e'fe°cos0^I e'k(x"sin0cos9+y"sin0sin9
(1)
where F3(0,9) - radiation pattern of array element; in - amplitude and phase of n-th element excitation; x„, yn - coordinates of n-th element, k - free space wave number.
From this expression it can be seen that location of array plane z0 makes an impact on phase characteristics of array only. Considering a ring array we obtain:
xn sin9cos9 + yn sin9sin9 = Rsin9cos
where R - array radius; a - angular location of "first" element. As a rule, for ring array it is reasonable to assign:
2n (n -1)
9------a
N
i = i eicp» = i e'9", Jn | n|c 0
where i0 - equal amplitude of array elements excitation, - excitation phase.
Phase distribution in ring array with maximum radiation in direction (0„, ^0) can be expressed as
9n = -kR sin 90 cos
2n(n -1)
90--- - a
0 N
To avoid an influence of ring plane location on phase radiation pattern while consideration of multi rings antennas, it is necessary to set a phase distribution as following:
9n = - \ kz0 cos 90 + kR sin 90 cos
9o
2 n(n -1 )
N
Now the expression for radiation pattern of ring array allocated in z = z0 plane can be written as
f (9,9)= F3 (9,p)e'
ikz0 (cos 9-cos 90 )
10 Z e
Radiation pattern of ring array element can be expressed as
F3(9,9) = p (9,9)F (9,9y°(e'<p),
i.e., as the product of three factors. Let is suppose that array elements have identical polarization and phase characteristics p (9,9), O (9,9) but different amplitude patterns - Fn (9,9). Therefore, we can write:
n=1
n=1
. . 2 n(«-1 ) 1 T 2 n(n-l) Ï ikR-l sin0cos| 9---a |-sin0O cos| 9---a I
N
f (0,9)= p(0,(cos0-cos0°>®(0,9)]/o£Fn (6,9)e
"=1
If array element possesses phase center, then we can assign o(0,9) = const in expression for F3 (0,9) and will not pay attention to the influence of element phase characteristic. The same can be said about polarization characteristic. Finally, we obtain
r V! N
f (6,9)= ei[fe°(cose-cos0°)]/0YFn (0,cp)e
2n(n-l) 1 f 2n(n-l) ikR<j sin6cos| 9--^—--a I—sin60 cos| p0--^—-—
Then we will examine antenna array consisting of M rings whose centers is situated on 0Z axis. For radiation pattern of such multi ring array we can write:
M Nm ikRm \ sin6cosl 9-2nt" 1' a.. |-sin0O cos| 9 -
f (6,9)= Y t^m (cOs6-cOs60 )Y Fn (6,9)e L N V N "JJ' ^
m=1
where the following variables describe an every m-th ring array: Im - amplitude of elements excitation (within every ring amplitude is supposed to be uniform); zm - location of array center along 0Z axis; F„m (6,9) - amplitude radiation pattern of n-th element; Rm - radius; Nm - number of elements; am -angular position of "first" element.
This expression allows us to explore the effect of the amplitude distribution for individual ring array Im, rings location along 0Z axis zm, various orientations of elements F^ (6,9) and ring radii (Rm) on radiation pattern of multi ring array allocated on surface of revolution. The simplest case is Im = I0,
zm =0, f (6,9)=Fo (6,9).
3. Continuous circular radiators on a surface of revolution
Expression for normalized radiation pattern of a continuous ring radiator with an arbitrary direction of radiation maximum (00, ^0) can be obtained from (1) in the limit when N ^ œ:
1 r
F (6 9) = — e'kz0cos 6 J eikR0 Lsm eic«^-»)')-sm 60cos(90 -v)]] , 2n 0
where R0 - ring radius; 00 - maximum radiation direction; k - free space wave number. Then after transformation described in [4] we will derive:
F (6,9) = J0 (iR0^sin2 6+sin2 60 - 2sin6 sin60 cos (9-90 ))•
where J0 - Bessel function of first kind of zero order.
Phase distribution in continuous ring radiator with radius R0, situated in z = 0 plane and having radiation maximum in 00, direction:
0(9)= -kR0 sin 60 cos (90 -9). If the ring is located in plane z = z0, then phase distribution becomes:
0(9)= -k [z0 cos 60 +R0 sin60 cos (90 -9)] • - 848 -
n=1
Let us suppose that radiating system consists of a number of continuous ring with different radius, and radiation maximums coincide with each other:
0 = -kz cos 90.
Here we assume that the centers of the rings are located on the axis 0Z. Radiation pattern of such a system can be written as:
M
f (0,p) = 2n£Rmimeib-coseJ0 [kRm8(0,p,0o,<0)] >
m=1
where 8 (0, p, 0o, p0) = ^sin2 0 + sin2 0o - 2sin 0 sin 0o cos (p -p0), Rm - ring antennas radii; M - number of rings.
To get all the rings radiating in the same direction (0o, 90), it is necessary to set
j — J - k [zm cos 00 +Rm sin 00 cos(p0 -<)] m m
Now it is possible to write the expression for normalized radiation pattern in that case:
1 M
Fe(0, <) = -MT^- Z Rmime'hm (cos e-c°se° )J 0 [kRm8(0, <, 00, <0 )] .
Z imRm m=1
m=1
If we assume all rings are located on certain surface of revolution:
f (0, p) = ]i [ z (r )] e'fe(r )(cos0-cos 00 )rJ 0 [kr8 (0, p, 00, <0)] dr,
0
where i(z) - amplitude distribution; R0 - antenna radius; z(r) - function describing the antenna surface.
For direction of radiation maximum (0 = 00, 9 = ^0), this expression can be written in the following form:
f (0o,<0)= j 1 (r)rJ0 [Po,0o,<0 )]dr .
0
This integral does not depend on 0o, 90 and is fully determined by antenna radius and amplitude distribution. Its maximum equals to:
Ro
fmax = j ri (r )dr .
0
This expression determines maximum field of antenna allocated on a surface of revolution. As can be seen in this case, field amplitude is fully determined by antenna aperture R0 (not taking into account shape of amplitude distribution). This fact that antenna can be lengthy does not affect to field amplitude in direction of radiation maximum, because integral does not depend on z(r).
Antenna can have arbitrary shape of surface of revolution. Maximum power is radiated in the direction of surface axis of revolution, and does not depend on surface shape.
4. Examples of antenna arrays on surfaces of revolution
Let us consider various kinds of antenna array's allocated on surfaces of revolution. Fig. 1-3 demonstrates semi-spherical, conical, hyperbolical array and radiation patterns of such arrays for three scan angles and N ^
5. Conclusion
In this paper we derived analytical expression which can be used for calculation of radiation patterns of multi ring antenna arrays and multi ring continuous radiators allocated on an arbitrary surface of revolution.
This work was supported in part by Ministry of education and science of Russian Federation (research in the scope of basic part of government job), Siberian Federal University.
The authors would like to thank; the support by the Ministry of Education of Russia together with the Siberian Federal University and JSC "Academician M.F. Reshetnev Information Satellite Systems", ContractJ№ 02.G25.31.0041.
Fig. 1. Semi-spherical array (top), radiation pattern of semi-spherical ¡array (bottom)
It,
N Or-1-1-r
I.* %i • *
iTim, 0J • • •
A.., .A x .....
[•• *«»1 -i- *» * • •*
jmsk
*.........-2 ^ --A-—1-
......."V -i-io
.........."
a / \ ^ \1 \
! \ 1 4 / \ ' i i \
1 1 \
/ r\l \ N A \ rm / >
\ ? If] u\ Jl 1 A ii A/ \ \ \ \ \ A
1 l\ l!i j V V \j|/U !i
-180-160-140-120-100 -80 -60 -40 -20 0 20 40 60 80 100 120 140 160 180
Fig. 2. Conical array (top), radiation pattern of conical array (bottom)
0
-0.5 - 1
- 1.5
• • • V • • V • • •
• • • • • •
•• • • • • •
•• • • • • ••
■ a a i a • • i a a a
o
x/X
\ ^ \1 * Tp 1 \
f 1 iv / *
' > ' 1 ■ i i
1 Ml SIAm / \
A A i Wl V /» w \> A
f V/ w N. f, J / / m
-180-160-140-120-100 ■
-60 -40 -20 0 20 40 60 80 100 120 140 160 180
B
Fig. 3. Hyperbolical array (top), radiation pattern ofhyperbolical array (bottom)
0
5
-10
-15
-20
-25
-30
References
[1] Antenna Engineering Handbook / John L. Volakis, Editor. // 4th Ed. McGrow Hill, 2007.
[2] C.A. Balanis. Antenna theory: analysis and design. Second ed. John Wiley & Sons, 1997.
[3] С.В. Поленга, Р.О. Рязанцев, Ю.П. Саломатов, В.С. Панько, М.И. Сугак // Журнал Сибирского федерального университета. Серия «Техника и технологии». Т. 4, № 1. С. 40.
[4] Ю. П. Саломатов, М. И. Сугак // Антенны. 2010. № 10. С. 17.
[5] V. S. Panko, Y. P. Salomatov, M. I. Sugak // Journal of Siberian Federal University. Engineering & Technologies 6 (2014 7) pp. 641-647.
