CC BY
9Journal of Siberian Federal University. Engineering & Technologies 6 (2014 7) 641-647
УДК 621.396.677
Directional Characteristics of Circular Scanning Aperture
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 17.07.2013, received in revised form 23.08.2014, accepted 10.09.2014
In this paper explicit analytical expressions are derived which describe basic directional characteristics of circular scanning aperture: radiation pattern for different kinds of amplitude distribution, beamwidth, part of radiated power contained in the main lobe, directivity. Consideration is based on the theory of continuous ring radiator within the scalar approximation.
Keywords: scanning, circular aperture, ring antenna
Характеристики направленности круглой сканирующей апертуры
В.С. Панькоа, Ю.П. Саломатова, М.И. Сугакб
аСибирский федеральный университет Россия, 660041, Красноярск, пр., Свободный, 79 бСанкт-Петербургский государственный электротехнический университет «ЛЭТИ» Россия, 197376, Санкт-Петербург, ул. Попова, 5
В данной статье приведены явные аналитические выражения, описывающие основные характеристики направленности круглой сканирующей апертуры: диаграмма направленности для различных видов амплитудного распределения, ширина главного лепестка, часть излучаемой мощности, содержащейся в главном лепестке, коэффициент направленного действия. Рассмотрение основано на теории непрерывного кольцевого излучателя в скалярном приближении.
Ключевые слова: сканирование, круговая апертура, кольцевая антенна.
© Siberian Federal University. All rights reserved Corresponding author E-mail address: [email protected]
1. Introduction
Directional properties of the radiating circular aperture are studied in the literature in detail [1, 2]. However, all presented results usually describe a uniform-phase aperture, while characteristics of the scanning aperture are determined numerically. In this paper we present explicit analytical expressions describing the main directional characteristics of circularscanning aperture: radiation pattern for different kinds of amplitude distribution, beamwidth, part of radiated power contained in the main lobe, directivity. Also we analyze the directivity clianging while scanning. All consideration is based on the theory of continuous ring radiator within the scalar approximation.
2. 10 irectional characteristics
Radiation pattern
Radiation pattern of circular aperture can be written in the form of Huygens-Green integral [3]:
n 2n
/(0,n) = J J /(r,n')eftrsinecos(n'-n)rJsp'fl'r r (1)
0 0
where h(r, cp') - field amplitude ond phase distribution over the aperture, k = 2n / X, (0, 9) - angles of spherical coordinate system, r, pi' - integration point coordinaJes (Fig. 0).
Various kinds of amplitude vnd phasa distribution are considered in [4]. For circular scanning apertuee examination we suppose axially Jymmetpic amplitude distribution.
|/(a,n')| = /( r),
and phase dnstributio n which establishes maximum radiation in direction (0O, p0), i.e.
*F(r,e0,n0,n') =&nsine0cos(n0 -sp')-
Then
f (0,9) = J/(r = rpK-eeos(tt,'-,)-(1ne0eo^-,')-\d(^,dr . 0 0
Internal integral here represents radiation pattern of a continuous ring radiator, and it can be written as [50]:
F(r,9,cp,90,cp0) 0r./sin2 9 + sin2 9 j - 2 srn 9 sm 90 cos (cp -p0)),
where J0(z) is the Bessel function of the first kind of order zero. Then radiation pattern of circular aperture takes the form:
R
f (6,9) = J 2 (r) rJ0 [kr -8(6,9,0o, 9c )] dr 0
where
5(9,cp,(^0,cf>0)) == (sin2 8 +sin2 0O -2sin0sin90 co s(cp -cp0).
Now we can obtain the expression of tadiation pattern for scanning circular aperture with uniform amplitude distribution 7(r) = n0:
wi e ) SH.71|[jfefgt>(e,cp, 90, (Po) ] (2)
¿-(Wr,^ ^ 9,p,9r, pr) ' (2)
where Jy(z) is tlie Bessel function of the first kind of first order.
For oxially symmetric amplitude disdributions which taper Cowards aperture edge:
I (r ) = (1 -A) + A
2
1 - r ' R
(3)
where 1 - A - excitation level at the aperture edge, n = 1,2..we can obtain analogically
/(e,p) = ( 1 - A) A- (u) + -+- A„+1 (u), (4)
n +1
where A,B (a) = ,/„ (m)«!/^jj , u =kri)c5((3,cp), J+
. - the Bessel function of the fi rst; kind of n-th or-
v / » v / J y 2 j '
der.
Expressions (22), (4) represent radiation patterns of circular aperture for arbitrary direction of radiation maximum (90, (p0). F"ig. 22 demonstrate s cut section of radiation pattern in cp = 0 plane for aperture withR = 4X and amplitude distribution of (3), whete A= r and n = 1.
Using of presented expressions, we can determine other directional properties of circular aperture: beamwidth for arpitiary direction of main lobe; directivity; fraction of radioted power toncentrated withrn the main lobe.
Beamwidth of scanning circular aperture
Beamwidth of circular aperture radiation pattern in different cut sections can be derived as [5]:
0.365X „ „ n
2A0=^-' 0¿00 <-, 9 = 9o = const. (5)
R-J 0.5 cos 0o 2
Here R - aperture radius. Fig. 3 represents change of beam width depending on the direction of the radiation maximum 90 (pp0 = 0).
Fraction of power, concentrated in main lobe
Fraction of radiated power, concentrated within main lobe of circular aperture radiation pattern can be written as
F(0), dB 0 -i
-5 --10 --15 -
-20 -
-25 --30
\/
I \ - \
ft ' ÛI
\
u
e0 = oo e0 = 20o
9°° = 40o
0o = <soo
-30
0
60
90
Fig. 2. R^adiation pattern of circular aperture R = 4X with taper amplitude distribution (3) for different direction ofradiation maximum 00
o
0
10
20
30
T
40
50
60
70
- R = 2L
--R = 3L
---R = 4L
90
Qn,C
Fig;. 3. Beamwidth of circular aperture radiation pattern for different R and radiation maximum direction
C
2n A9
lo = j ( F2 (6,cp)sin6ii6iicp . 0 0
In [3] it is stated that factor (1 + cos 0) / 2 is excluded from (1). This factor for small 0 can be approximated as o/cosP). St all2ws us to make a suggeetion that circular aperture with uniform amplitude distribution consists of "elemeatary" radeators with radiation narterns sn the form of V/os9 . In this way we can obtain:
° k R2 J kR sin P v ! By changing of variable t = pRsin 0:
871
°Ak=F i
kR sin AO T2
Po =
o2 t)
t
dt.
(6)
For power allocated in the whole radration pattern, supposing radiation in upper semi-sphere only A0 = Tt / 2,
kR r2 1
Pl=kiR 2| t dt.
These expressions can be transformed in the following manne r. Sinci
* 2
jxJC^dx^— [ JO^xJ + jfCxJ+und
x S 1
i Jn (X) dx =
„ x in
1 + Jo2(x) + Jx (x) — 211 (x )
k=1
([4], p. 38), we can get
i i|2(x)dx = 2) + J2(x)—ai2(x)].
By integrating xf (6) and using the results from [5], we can write: 471
Po =
rt
nZ =
(PAO
Jl + R (PR sin AA0) - J]2: PR sin AO)
■[l + J—(a^-J^R)
Fraction of power, concentrated within main lobe, will be equal to: 4=1 +f-R sin A0) - J\ (kbRf sin A9)
fz l + J (nR)~J?(kR) '
Halfbeamwidth can be found from (5): 0.365W2
A0 =-
2R
(7)
(8)
(9)
For lnrge apertures R >>X half ba3mwidth hiias small values A « 1. Therefore in (9) we can substitute sinus by its argument. Considering that limn J01lkR) —t0:
R—x
R^rl + J2 (0.36571^/2) -R (0.3(55tt-V/2) ,
or
A
Pv
= 0.869.
Directivity
Talking into account definition of PE in (8) we can derive directivity of circular aperture with uniform phase distribution:
(kR)2
D =
4n
, while — >> 1.
Pe 1+./r2(£R )-Jf (WR) X
For apertures with large electrical radii lim Jr 1 (kR) ^ 0, then obtain well known formula [1] D = 4n
X 2
where SA - aperture ghyficar area.
Usee's examine changing of circulae aprrture directivity wMle scamning. Without loss of generality in (2) we always can assign cp0 = 0, then radiation pattern takes a form:
F (9,9r ) =
2 J1 [kR8(9,9,9r)]
kR8(9,9,9r)
Directivity can be found numerically for every certain 00. Results are presented in Fig. 4.
3. Conclusion
In this paper we derived explicit analytical expressions, which can be used for defining of directional properties of circular scanning aperture without numerical computing.
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 EducatioR of Russia together with the Siberian Federal University and JSC "Academician M.F. Reshetnev Information Satellite Systems", Contract № 02.G25.31.0041.
D, dBi 28 26 -24 -22 20 -18 -
0 10 20 Fig. 4. Directivity of scannirg circular aperture
30
50
T
60
~r
70
— R = 2L
— R = 3L
— R = 4L
80
90
References
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[2] Balanis C.A. Antenna theory: analysis and design. Second ed. John Wiley & Sons, 1997.
[3] Корнблит С. СВЧ-антенны. Оптические принципы в приложении к конструированию СВЧ антенн. М.: Связь, 1980.
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