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 shownВ 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 решето

Erokhin, Alexey А.

Panko, Vasily S.

Salomatov, Yury P.

Sugak, Mikhail I.

Ерохин, А.А.

Панько, В.С.

Саломатов, Ю.П.

Сугак, М.И.

Siberian Federal University Digital Repository

– 846 –Journal of Siberian Federal University. Engineering & Technologies 7 (2014 7) 846-852 ~ ~ ~УДК 621.396.677Antenna Arrays on Surfaces of RevolutionAlexey А. Erokhina, Vasily S. Pankoa*, Yury P. Salomatova and Mikhail I. SugakbaSiberian Federal University 79 Svobodny, Krasnoyarsk, 660041, Russia bSaint Petersburg Electrotechnical University “LETI” 5 Professora Popova Str, St. Petersburg, 197376, RussiaReceived 02.08.2014, received in revised form 22.09.2014, accepted 04.10.2014In 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 вращенияА.А. Ерохина, В.С. Панькоа, Ю.П. Саломатова, М.И. СугакбаСибирский федеральный университет Россия, 660041, Красноярск, пр. Свободный, 79 бСанкт-Петербургский электротехнический университет “ЛЭТИ” Россия, 197376, Санкт-Петербург, ул. Профессора Попова, 5В 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. IntroductionDirectional properties of antenna arrays of various kinds are studied in the literature in detail [1-3]. 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– 847 –Alexey А. Erokhin, Vasily S. Panko. Antenna Arrays on Surfaces of Revolutionderive 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 revolutionLet us consider antenna array consisting of N identical elements allocated in z = z0 plane. Radiation pattern of such array is [1]:( ) ( ) ( )0 sin cos sin sincos1, , n nNik x yikznnf F e I e θ ϕ+ θ ϕθ∋=θ ϕ = θ ϕ ∑   , (1)where ( ),F∋ θ ϕ – radiation pattern of array element; nI  – amplitude and phase of n-th element excitation; хп, уп – coordinates of п-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:( )2 1sin cos sin sin sin cosn nnx y RNπ − θ ϕ+ θ ϕ = θ ϕ− −α  , where R – array radius; α – angular location of “first” element.As a rule, for ring array it is reasonable to assign:0n ni in nI I e I eϕ ϕ= =  , where I0 – equal amplitude of array elements excitation, φп – excitation phase.Phase distribution in ring array with maximum radiation in direction (θ0, φ0) can be expressed as( )0 02 1sin cosnnkRNπ − ϕ = − θ ϕ − −α  . 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:( )0 0 0 02 1cos sin cosnnkz kRN π −  ϕ = − θ + θ ϕ − −α     . Now the expression for radiation pattern of ring array allocated in z = z0 plane can be written as( ) ( ) ( )( ) ( )0 00 02 1 2 1sin cos sin coscos cos01, ,n nN ikRN Nikznf F e I e π − π −    θ ϕ− −α − θ ϕ − −α    θ− θ      ∋=θ ϕ = θ ϕ ∑  . Radiation pattern of ring array element can be expressed as( ) ( ) ( ) ( ),, , , iF p F e Φ θ ϕ∋ θ ϕ = θ ϕ θ ϕ  , i.e., as the product of three factors. Let is suppose that array elements have identical polarization and phase characteristics ( ),p θ ϕ , ( ),Φ θ ϕ  but different amplitude patterns – ( ),nF θ ϕ . Therefore, we can write:– 848 –Alexey А. Erokhin, Vasily S. Panko. Antenna Arrays on Surfaces of Revolution( ) ( ) ( ) ( ) ( )( ) ( )0 00 02 1 2 1sin cos sin coscos cos ,01, , ,n nN ikRN Ni kznnf p e I F e π − π −    θ ϕ− −α − θ ϕ − −α     θ− θ +Φ θ ϕ        =θ ϕ = θ ϕ θ ϕ∑  . If array element possesses phase center, then we can assign ( ), constΦ θ ϕ =  in expression for ( ),F∋ θ ϕ and will not pay attention to the influence of element phase characteristic. The same can be said about polarization characteristic. Finally, we obtain( ) ( ) ( )( ) ( )0 00 02 1 2 1sin cos sin coscos cos01, ,n nN ikRN Ni kznnf e I F e π − π −    θ ϕ− −α − θ ϕ − −α     θ− θ        =θ ϕ = θ ϕ∑ . 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:( ) ( ) ( )( ) ( )0 002 1 2 1sin cos sin coscos cos1 1, ,m m m mm mmn nNM ikRN Nikz mm nm nf I e F e π − π −    θ ϕ− −α − θ ϕ − −α    θ− θ      = =θ ϕ = θ ϕ∑ ∑ , (2)where the following variables describe an every m-th ring array: Iт – amplitude of elements excitation (within every ring amplitude is supposed to be uniform); zm – location of array center along 0Z axis; ( ),mnF θ ϕ  – amplitude radiation pattern of п-th element; Rm – radius; Nm – number of elements; αm – angular position of “first” element.This expression allows us to explore the effect of the amplitude distribution for individual ring array Iт, rings location along 0Z axis zm, various orientations of elements ( ),mnF θ ϕ  and ring radii (Rm) on radiation pattern of multi ring array allocated on surface of revolution. The simplest case is Iт = I0, zm =0, ( ) ( )0, ,mnF Fθ ϕ = θ ϕ .3. Continuous circular radiators on a surface of revolutionExpression for normalized radiation pattern of a continuous ring radiator with an arbitrary direction of radiation maximum (θ0, φ0) can be obtained from (1) in the limit when N → ∞:( ) ( ) ( )0 0 002sin cos sin coscos01,2ikRikzF e e dπ θ ϕ−ψ − θ ϕ −ψθ  θ ϕ = ψπ ∫, where R0 – ring radius; φ0, θ0 – maximum radiation direction; k – free space wave number.Then after transformation described in [4] we will derive:( ) ( )( )2 20 0 0 0 0, sin sin 2sin sin cosF J kRθ ϕ = θ+ θ − θ θ ϕ−ϕ . where J0 – Bessel function of first kind of zero order.Phase distribution Ф(φ) in continuous ring radiator with radius R0, situated in z = 0 plane and having radiation maximum in θ0, φ0 direction:( ) ( )0 0 0sin coskRΦ ϕ = − θ ϕ −ϕ . If the ring is located in plane z = z0, then phase distribution becomes:( ) ( )0 0 0 0 0cos sin cosk z RΦ ϕ = − θ + θ ϕ −ϕ   . – 849 –Alexey А. Erokhin, Vasily S. Panko. Antenna Arrays on Surfaces of RevolutionLet us suppose that radiating system consists of a number of continuous ring with different radius, and radiation maximums coincide with each other:0coskzΦ = − θ . 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:( ) ( )cos 0 0 01, 2 , , ,mMikzm m mmf R I e J kRθ∑=θ ϕ = π δ θ ϕ θ ϕ  ∑  , where ( ) ( )2 20 0 0 0 0, , , sin sin 2sin sin cosδ θ ϕ θ ϕ = θ+ θ − θ θ ϕ−ϕ , Rт – ring antennas radii; М – number of rings.To get all the rings radiating in the same direction (θ0, φ0), it is necessary to set( )0 0 0cos sin cosm mik z Rm mI I e − θ + θ ϕ −ϕ = . Now it is possible to write the expression for normalized radiation pattern in that case:( ) ( ) ( )0cos cos 0 0 0111, , , ,mMikzm m mMmm mmF R I e J kRI Rθ− θΣ==θ ϕ = δ θ ϕ θ ϕ  ∑∑. If we assume all rings are located on certain surface of revolution:( ) ( ) ( )( ) ( )00cos cos0 0 00, , , ,Rikz rf I z r e rJ kr drθ− θθ ϕ = δ θ ϕ θ ϕ      ∫ , where I(z) – amplitude distribution; R0 – antenna radius; z(r) – function describing the antenna surface.For direction of radiation maximum (θ = θ0, φ = φ0), this expression can be written in the following form:( ) ( ) ( )00 0 0 0 0 0 00, , , ,Rf I r rJ kr drθ ϕ = δ θ ϕ θ ϕ  ∫ . This integral does not depend on θ0, φ0 and is fully determined by antenna radius and amplitude distribution. Its maximum equals to:( )0max0Rf rI r dr= ∫ . 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.Alexey А. Erokhin, Vasily S. Panko. Antenna Arrays on Surfaces of Revolution4. Examples of antenna arrays on surfaces of revolutionLet us consider various kinds of antenna arrays 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. ConclusionIn 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”, Contract № 02.G25.31.0041.  ThiAs can be into accounto field amAntthe directio Letdemonstratthree scan   s expressioseen in thit shape of plitude in denna can hn of surfac4 us considees semi-spangles and Fig. 1. S-180 -160-30-25-20-15-10-50n determins case, fieldamplitude irection of ave arbitrae axis of re. Examplesr various kherical, conN → ∞. emi-spherica-140 -120 -100maxf&es maximum amplitudedistributionradiation mry shape ofvolution, an of antenninds of anteical, hyperl array (top), r-80 -60 -40( )00RrI r d= ∫ field of a is fully de). This factaximum, b surface ofd does not a arrays onnna arraysbolical arr adiation patte-20 0 20r . ntenna allotermined b that antenecause integ revolutiondepend on  surfaces o allocated oay and radirn of semi-sp40 60 80cated on ay antenna ana can be lral does no. Maximumsurface shaf revolution surfaces ation patteherical array 100 120 140 surface of perture R0 engthy doet depend o power is pe. n of revolutirns of such (bottom) θ ,°160 180revolution.(not takings not affectn z(r). radiated inon. Fig.1-3 arrays for       Fig. 1. Semi-spherical array (top), radiation pattern of semi-spherical array (bottom) F-180 --30-25-20-15-10-50ig. 2. Conica160 -140 -120 -10l array (top), r0 -80 -60 -40 adiation patte -20 0 20rn of conical 40 60 80 1array (bottom00 120 140 16 ) θ ,°0 180  Fig. 2. Conical array (top), radiation pattern of conical array (bottom)Fig. 3. Hyperbolical array (top), radiation pattern of hyperbolical array (bottom) F-180 --30-25-20-15-10-50ig. 2. Conica160 -140 -120 -10l array (top), r0 -80 -60 -40 adiation patte -20 0 20rn of conical 40 60 80 1array (bottom00 120 140 16 ) θ ,°0 180   Fig. 3. Hyperbolical array (top), radiation pattern of hyperbolical array (bottom)  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", Contract № 02.G25.31.0041.  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 СФУ. Техника и технологии. Т. 4. № 1. С. 40.Где год? [4] Ю. П. Саломатов, М. И. Сугак // Антенны. 2010. № 10. С. 17. θ ,°-180 -160 -140 -120 -100 -80 -60 -40 -20 0 20 40 60 80 100 120 140 160 180-30-25-20-15-10-50Alexey А. Erokhin, Vasily S. Panko. Antenna Arrays on Surfaces of RevolutionReferences[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.

Антенные решетки на поверхностях вращения

Антенные решетки на поверхностях вращения

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