Nonholonomic Antenna - WSEAS

2y ago
9 Views
3 Downloads
307.42 KB
6 Pages
Last View : 2m ago
Last Download : 2m ago
Upload by : Aarya Seiber
Transcription

APPLIED COMPUTING CONFERENCE (ACC '08), Istanbul, Turkey, May 27-30, 2008.Nonholonomic AntennaIONEL TEVYCONSTANTIN UDRISTEUniversity Politehnica of Bucharest University Politehnica of BucharestFaculty of Applied SciencesFaculty of Applied SciencesDepartment of MathematicsDepartment of MathematicsSplaiul Independentei 313Splaiul Indpendentei 313060042, BUCHAREST, ROMANIA 060042, BUCHAREST, L ZUGRAVESCUInstitute of GeodynamicsDr. Gerota 19-21020032, BUCHAREST, ROMANIAFLORIN MUNTEANUInstitute of GeodynamicsDr. Gerota 19-21020032, BUCHAREST, ROMANIAAbstract: This paper applies the geometric dynamics tools to describe optimal shape of structures used to couple atransmitter or receiver to a medium in which waves can propagate. Section 1 recalls the laws of regular reflectionand the theory of spherical transmitter. Section 2 analizes the optic geometric dynamics using a reflected vectorfield and an adapted Fermat principle. Section 3 studies the transmitter of revolution and the corresponding holonomic optimal receiver. Section 4 analizes a nonholonomic Ox-symmetric transmitter (special collection of wires)and a holonomic optimal receiver. Section 5 evidentiates a Tzitzeica transmitter with a nonholonomic optimalreceiver (special collection of wires). Section 6 shows that the canonical nonholonomic transmitter admits a holonomic optimal receiver. Section 7 underline possible numerical simulations for practical realization of transmittersor receivers. Section 8 points out that there exist: (i) holonomic or nonholonomic (special collection of wires)mirrors accepting only holonomic optimal receivers; (ii) holonomic or nonholonomic (collection of wires) mirrorsaccepting only nonholonomic (special collection of wires) optimal receivers.Key–Words: Holonomic or nonholonomic mirror, optic geometric dynamics, optimal antenna, optimal transmitterand receiver, optimal solar power station.1 Laws of regular reflectionof the reflected light rays are semi-straight lines starting on the mirror in the direction r. This point of viewseparates the mirrors into two classes: planar mirrorwho produces parallel reflected rays and non-planarmirrors who produce non-parallel reflected rays. Generally, to build an optimal shape receiver associated toa non-planar mirror is strongly dependent on the shapeof the mirror. This problem has simple practical solutions only for particular mirrors (spherical mirror,parabolic mirror, etc).Let us recall the case of spherical transmitter andoptimal receiver [13]. Since a spherical mirror hasan axial symmetry, it is enough to judge in the planexOy. Then the mirror is described byBy reflection we understand a deviation of the direction of radiant flux taking place entirely within or atthe surface (mirror) of a single optical medium. Interfaces, between two media, belonging to the classC 1 produce regular reflections. Otherwise, it appearsreflections of fractal type.Laws of Regular Reflection. The incident andthe reflected rays are: (i) in a normal plane to thesurface of mirror; (ii) on the same side of the surfaceof mirror; (iii) at congruent angles with the normal tothe surface of mirror.Usually, the incident ray (versor) is denoted by s, the unit normal vector to the mirror is denoted by n and the reflected ray (versor) is denoted by r. According the laws of regular reflection, they are relatedby r s 2( s, n) n.x2 y 2 R2 , x R sinand the family of reflected rays isFrom mathematical point of view, r is a versor fieldon the mirror. Traditionally, we accept that the pathsISBN: 978-960-6766-67-1π7x Rcos ty Rsin t .cos 2tsin 2t108ISSN: 1790-2769

APPLIED COMPUTING CONFERENCE (ACC '08), Istanbul, Turkey, May 27-30, 2008.2Reasons regarding the cosinus effect on the intensityof solar radiation show that the optimal shape receivermust be selected using as profiles the orthogonal trajectories of this family of straight lines. Such curvesare described by the algebraic differential systemA Pfaff equation of the formL(x, y, z)dx M (x, y, z)dy N (x, y, z)dz 0x Rcos ty Rsin t11 , 0.cos 2tsin 2tcos 2ty (x)sin 2tdescribes the integral manifolds (curves or surfaces)orthogonal to the field lines of the vector field U (L, M, N ). If (U, rot U ) 0 (completely integrablePfaff equation), then the equation can be written asWe look for parametric solutions of the typex(t) ϕ(t)cos 2t Rcos t F F Fdx dy dz 0, x y zy(t) ϕ(t)sin 2t Rsin tx0 (t)cos 2t y 0 (t)sin 2t 0,with the general solution σc : F (x, y, z) c (holonomic surfaces; by each point we have a unique surface). If (U, rot U ) 6 0, then the Pfaff equation definea nonholonomic surface which is in fact a ”family ofcurves” (by each point, an infinity of curves; spacefilling curves, [11], [16]).Let s be the incident ray. The relation (1) defines r as reflected versor field on a given (holonomic ornonholonomic) mirror σ represented by the normalversor field n. We can extend the reflected versorfield r to a reflected vector field defined on a domainD R3 containing the mirror, at least in two differentways :(i) let S incident vector field, N normal vector field, of class C 1 , satisfying the relations S σ s, N σ n and we define the reflected vector fieldwith the unknown function ϕ(t). We find, ϕ0 (t) Rsint 0 and consequently ϕ(t) R(c cos t).Finally, the family of orthogonal trajectories is characterized byx(t) (c cos t)cos 2t cos tR·Geometric dynamics generated bythe reflected field y(t)π π (c cos t)sin 2t sin t, t ,.R7 7On the other hand, for a spatial spherical mirror, thesurfaces that are orthogonal to reflected rays are obtained by the rotation· around Ox of the previousπ πcurves. The interval ,is imposed to avoid the7 7self-intersections. It follows the vector equationR S 2(S, N )N,r(t, u) (x(t), y(t)cos u, y(t)sin u),· which satisfies R σ r;(ii) the reflected vector field V (X, Y, Z) is ofclass C 1 and satisfies V σ λσ r.Some reasons determine us to work with the vector field V (X, Y, Z) as reflected vector field. Then,according the extended Fermat principle, the reflectedrays are field lines of the vector field V (X, Y, Z), i.e.,solutions of ODE systemπ πt ,, u [0, 2π] .7 7Since these family depends on the parameter c, it appears the idea to select the optimal profile using anoptimization problem relative to c (see [1]-[9], [13]).To improve the previous ideas, we extend the Fermat principle: the reflected light rays are field lines,starting on the mirror, of a vector field V who extend r to a domain in space containing the mirror. Thesefield lines do not intersect. Also, our theory showsthat some practical mirrors produce practical optimalshape receivers.The theory in this paper can be applied to anystructure (antenna) used to couple a transmitter or receiver to a medium in which electromagnetic wavescan propagate. These structures range in from simple straight lines, used either singly or in arrays, toquasioptical devices employing mettalic mirrors or dielectric lenses. Particularly, an optimal multiple wireantenna or a fractal antenna can be part of a nonholonomic surface.ISBN: 978-960-6766-67-1dx(t) X(x(t), y(t), z(t))dtdy(t) Y (x(t), y(t), z(t))dtdz(t) Z(x(t), y(t), z(t)).dtThis ODE system and the Euclidean (Riemannian)structure δij of the space produce the Lagrangian energy density (kinetic potential)µ2L 109¶2dx(t) X(x(t), y(t), z(t))dtISSN: 1790-2769

APPLIED COMPUTING CONFERENCE (ACC '08), Istanbul, Turkey, May 27-30, 2008.µ ¶2dy(t) Y (x(t), y(t), z(t))dtcosinus effect on the intensity of solar radiation imposes that the optimal shape of the receiver in a Solar Power Station is a surface (holonomic or nonholonomic), where the reflected rays (field lines of reflected vector field) fall down perpendicular [1]-[9],[13]. According the theory of Pfaff equations in threevariables [11]-[16], the surface orthogonal to the fieldlines is holonomic if and only if the reflected field isbiscalar, i.e., (V, rot V ) 0 or V λ grad F . If so,the Pfaff equation¶2µdz (t) Z(x(t), y(t), z(t))dt,of least squares type. Let us accept that the pathof a ray moving in an inhomogeneous 3-dimensionalmedium is an extremal of the functional (adapted Fermat principleZ t2t1X(x, y, z)dx Y (x, y, z)dy Z(x, y, z)dz 0L(x(t), y(t), z(t), ẋ(t), ẏ(t), ż(t))dt.has a general solution F (x, y, z) c, i.e., a familywith one parameter of (holonomic) surfaces orthogonal to the field lines of V (X, Y, Z).Let us show that any reflectant surface of revolution produces a biscalar reflected vector field. Withoutloss of generality, any surface of revolution has an implicit Cartesian equation of the formThe extremals are solutions of Euler-Lagrange ODEsd2 x dt2µd2 y dt2µd2 z dt2µ X Y y x Y X x y X Z x z¶¶¶µdy X Z dt z xµdx dt Y Z z yµdx Z Y dt y z¶¶¶dz f dt xdz f dt yy 2 z 2 2h(x) 0,dy f ,dt zwhere h is a function of class C 2 . Consequently, thenormal versor field iswhere2f X 2 Y 2 Z 2( h0 , y, z). n p 2h0 2his the reflected density energy. Every nonconstant trajectory of this dynamical system which has constanttotal energy (Hamiltonian)Also, from theoretical point of view, we can accept s (1, 0, 0). These produce the reflected rayµ2H dxdt¶2µ dxdt¶2µ dxdtö2 r f!2h0 22h0 y2h0 z1 02, 02, 02.h 2h h 2h h 2hExtend r to a domain in R3 , we obtain the reflectedvector fieldis a reparametrized geodesic of a Riemann-JacobiLagrange manifold. In other words we have ageodesic motion in a gyroscopic field of forces [10][12], [14], [15]. The set of all trajectories splits inthree parts: reflected lines or the field lines of V (forH 0); trajectories with positive energy transversalto the reflected lines (for H const 0); trajectorieswith negative energy transversal to the reflected lines(for H const 0).Ã!2h0 22h0 y2h0 zV 1 02, 02, 02.h 2h h 2h h 2hSince two collinear vector fields have the same orbitsand the same family of curves orthogonal to orbits, wereplace V by the potential vector fieldÃW 3Transmitter of revolution,holonomic optimal receivercollinear to V . In case of W , the differential system2h0 (x)dxdydz 20yz2h(x) h (x)We accept that the paths of reflected rays are fieldlines of the reflected vector field V (X, Y, Z) that starton the (holonomic or nonholonomic) reflectant surface (mirror). As field lines starting from differentpoints, these paths do not intersect (comparable withthe situation of a plane mirror when the reflected raysare parallel semi-straight lines). In this context, theISBN: 978-960-6766-67-1!2h(x) h0 2 (x), y, z ,2h0 (x)has the general solution (field lines, reflected rays)ÃZ2y C1 exp1102h0 (x)dx2h(x) h0 2 (x)!ISSN: 1790-2769

APPLIED COMPUTING CONFERENCE (ACC '08), Istanbul, Turkey, May 27-30, 2008.ÃZz 2 C2 exp!2h0 (x)dx.2h(x) h0 2 (x)4Also, the Pfaff equation2h(x) h0 2 (x)dx ydy zdz 02h0 (x)has the general solution (family of surfaces of revolution)Z22y z Let us take a nonholonomic transmitter described bythe Pfaff equationydz zdy f (y, z)dx 0, f ( y, z) f (y, z).It is symmetric with respect to Ox. Thep normal vector N (f, z, y), with N y 2 z 2 f 2Ndetermine the normal versor n . If we take N s (1, 0, 0), then r s 2( s, n) n and2h(x) h0 2 (x)dx c.2h0 (x)Giving a suitable point (x0 , y0 , z0 ), we find the equation of the optimal shape receiver. The details for asphere of radius R with the center at origin are: thefunction h,R2 x2h(x) ;2the reflected versor field,à r 2x2 2xy 2xz1 2 ,,RR2R2the reflected vector field,ÃW W N 2 s 2( s, N )N (y 2 z 2 f 2 , z, y).pTo simplify we take f (y, z) y 2 z 2 . Then wefind: the reflected rays (field lines), x c1 , y 2 z 2 c2 (circles); the family of optimal receivers, zdy ydz 0 or y c3 z (pencil of planes).!;5!R2 2x2, y, z ; 2xTzitzeica transmitter,nonholonomic optimal receiverThe Tzitzeica transmitter has the equation xyz 1.The normal versor is(yz, xz, xy) n p 2 2y z x2 z 2 x2 y 2orbits of reflected rays (field lines of W ),2x2 c1 y 2 R2 , 2x2 c2 z 2 R2 ;family of optimal shape receivers,xR2 ln x2 y 2 z 2 c3 .RUsing the canonical parametrization of the sphere, andimposing the values of the angles to obtain a true mirror, we find c1 0, c2 0. Consequently the orbits of reflected rays are intersections of two ellipticalcylinders.The optimal shape receiver is a surface of revolution obtained by revolving the curvexx2 y 2 R2 ln cRaround Ox. This surface can be seen like a cylindricalsurface with axis Ox and variable radiusrqxRc y 2 z 2 c R2 ln x2 .RThe parameter c is determined by the Creţu conditionR[1]-[9], [13], Rc (x0 ) 8.10 3 R, where x0 is2the maximum point of Rc . It followsIt follows the components of r:2y 2 z 21 p 2 2,y z x2 z 2 x2 y 22z, x2 z 2 x2 y 2 2yp.y 2 z 2 x2 z 2 x2 y 2Now we extend r to a reflected vector field W of componentspy2z2X x2 (y 2 z 2 ) SinceÃrot W 1, Y 2z, Z 2y.x22 2x 2 2x20, 3 , 3yzyy!, (W, rotW ) 6 0,the Pfaff equation of optimal receiversµ¶1x (y z ) 2 dx 2zdy 2ydz 0xs2Rcxx2x ln 2 1.653564, [0.55, 0.80].RR RRThe shape of the optimal receiver is that of a humaneye.ISBN: 978-960-6766-67-1Nonholonomic Ox-symmetrictransmitter, holonomic optimalreceiver22has as solutions only curves (nonholonomic surface).Consequently, the optimal receiver is a nonholonomicsurface (family of wires).111ISSN: 1790-2769

APPLIED COMPUTING CONFERENCE (ACC '08), Istanbul, Turkey, May 27-30, 2008.6Canonical nonholonomictransmitter, holonomic optimalreceiver8Sections 1-6 show that there exist: (i) holonomic ornonholonomic (special collection of wires; space filling curve) transmitters (mirrors) accepting only holonomic optimal receivers; (ii) holonomic or nonholonomic (special collection of wires; space filling curve)transmitters (mirrors) accepting only nonholonomic(special collection of wires) optimal receivers. Section 7 suggests the numerical simulations necessaryfor practical realization of a transmitter or a receiver.Let us show that the canonical noholonomic transmitter zdx dy 0 produces a nonholonomicoptimalreceiver. Indeed,s (1, 0, 0) and n µ¶ z 1 , , 0 give the reflected versor21 zÃ1 z2!1 z22field r ,,0 .1 z2 1 z2Now we extend r to a reflected vector field W (1 z 2 , 2, 0), which verify (W, rotW ) 6 0. Consequently the reflected rays (field lines) are z c1 , 2x (1 z 2 )y c2 and the optimal receiver is(1 z 2 )dx 2dy 0.7ConclusionsAcknowledgements: Partially supported byGrant CNCSIS 86/ 2007 and by 15-th ItalianRomanian Executive Programme of S&T Cooperation for 2006-2008, University Politehnica ofBucharest.Numerical simulationsReferences:The numerical simulations, used to realize an antenna,are based on the discrete version of a Pfaff equationX(x, y, z)dx Y (x, y, z)dy Z(x, y, z)dz 0.[1] C. Călin, T. Creţu, C. Udrişte, Optimization ofheliostatic plants geometry, Tensor, N. S., 54(1993), 146-157.This can be obtained using the midpoint rule whichxn 1 xnconsists in the substitution of x withand2dx by xn 1 xn . It follows the finite difference equation[2] T. Creţu, C. Udrişte, P. Ştiuca, C. Călin,Gh. Macarie, Fixed concentrator with mobile receiver, Analyse of local stainfor solar power station of high temperature, Research Grant, 41-45/05.04.1984, IPB.µX¶xn 1 xn yn 1 yn zn 1 zn,,(xn 1 xn )222µ Yµ Z¶xn 1 xn yn 1 yn zn 1 zn,,(yn 1 yn )222¶xn 1 xn yn 1 yn zn 1 zn,,(zn 1 zn )222 0.[4] T. Creţu, C. Călin, Gh. Macarie, P. Ştiuca,C. Udrişte, The analytical optimization of thetower receiver position in the heliostats field ofsolar power station, Rev. Roum. Physique, 30, 9(1985), 739-798.[5] T. Creţu, C. Călin, Gh. Macarie, P. Ştiuca,C. Udrişte, The focal distances analysis of theheliostat modules for a solar electric power station with tower, Sci. Bull. IPB., 46 (1985), 3242.To find a solution it is necessary to impose an initialpoint (x0 , y0 , z0 ) and two of the sequences xn , yn , zn .Of course, given the convergent sequences yn , zn , weobtain a convergent sequence xn , if the coefficients ofthe Pfaff equation satisfy reasonable conditions. Also,a Von Neumann analysis proves the stability of theprevious finite difference scheme in reasonable conditions.If we need cryptography or packing, we can usespecial sequences; for example, we add the Fibonaccisequences: yn yn 1 yn 2 , zn zn 1 zn 2 , y0 z0 1, y1 z1 2. A fractal antenna (space filling curve) can be obtained only as acurve in a nonholonomic surface.ISBN: 978-960-6766-67-1[3] T. Creţu, V. Fara, C. Călin, Gh. Macarie, Solar radiation flux density distribution reflected bythe flat module heliostat, Rev. Roum. Physique,29 (1984), 765-775.[6] T. Creţu, C. Călin, Gh. Macarie, P. Ştiuca,C. Udrişte, Solar flux density calculation for asolar tower concentrator, Rev. Roum. Physique,33, (1988), 205-210.[7] T. Creţu, C. Călin, Gh. Macarie, P. Ştiuca,C. Udrişte, Optimizarea amplasării ı̂n terena câmpului de captare pentru centrale solaroelectrice cu heliostate şi turn, Sesiunea ştiinţificăjubiliară a Institutului de Mine, Petroşani, 9-10decembrie 1988.112ISSN: 1790-2769

APPLIED COMPUTING CONFERENCE (ACC '08), Istanbul, Turkey, May 27-30, 2008.[8] T. Creţu, C. Călin, C. Lăzărescu, P. Ştiuca,C. Udrişte, Solicitarea mecanică a materialelor cauză a emisiei de electroni de joasa energie, Sesiunea ştiinţifică jubiliară a Institutului de Mine,Petroşani, 9-10 decembrie 1988.[9] T. Creţu, C. Călin, Gh. Macarie, P. Ştiuca,C. Udrişte, A solar flux density on the imageplane calculation using a method of mathematical statistics, Rev. Roum. Physique, 1989.[10] C. Udrişte, Geometric Dynamics, SoutheastAsian Bulletin of Mathematics, Springer-Verlag,24 (2000), 313-322.[11] C. Udrişte, Geometric Dynamics, Kluwer Academic Publishers, 2000.[12] C. Udrişte, Maxwell geometric dynamics, European Computing Conference, VouliagmeniBeach, Athens, Greece, September 24-26, 2007.[13] Udrişte, C. Călin, I. Ţevy, Optimal receiverof solar power station, Proceedings of the 2nd International Colloquium of Mathematics inEngineering and Numerical Physics (MENP2), April 22-27, 2002, University Politehnicaof Bucharest, BSG Proceedings 8, pp. 147-164,Geometry Balkan Press, 2003.[14] C. Udrişte, M. Postolache, Atlas of MagneticGeometric Dynamics, Geometry Balkan Press,Bucharest, 2001.[15] C. Udrişte, M. Ferrara, D. Opriş, EconomicGeometric Dynamics, Geometry Balkan Press,Bucharest, 2004.[16] Gh. Vrănceanu, Opera Matematica, EdituraAcademiei Române, Bucharest, 1969, 1971.ISBN: 978-960-6766-67-1113ISSN: 1790-2769

Pfaff equation), then the equation can be written as @F @x dx @F @y dy @F @z dz 0; with the general solution ¾c: F(x;y;z) c (holo-nomic surfaces; by each point we have a unique sur-face). If (U;rotU) 6 0, then the Pfaff equation define a nonholonomic surface which is in fa

Related Documents:

Random Length Radiator Wire Antenna 6 6. Windom Antenna 6 7. Windom Antenna - Feed with coax cable 7 8. Quarter Wavelength Vertical Antenna 7 9. Folded Marconi Tee Antenna 8 10. Zeppelin Antenna 8 11. EWE Antenna 9 12. Dipole Antenna - Balun 9 13. Multiband Dipole Antenna 10 14. Inverted-Vee Antenna 10 15. Sloping Dipole Antenna 11 16. Vertical Dipole 12 17. Delta Fed Dipole Antenna 13 18. Bow .

Random Length Radiator Wire Antenna 6 6. Windom Antenna 6 7. Windom Antenna - Feed with coax cable 7 8. Quarter Wavelength Vertical Antenna 7 9. Folded Marconi Tee Antenna 8 10. Zeppelin Antenna 8 11. EWE Antenna 9 12. Dipole Antenna - Balun 9 13. Multiband Dipole Antenna 10 14. Inverted-Vee Antenna 10 15. Sloping Dipole Antenna 11 16. Vertical Dipole 12 17. Delta Fed Dipole Antenna 13 18. Bow .

Wire-Beam Antenna for 80m. 63 Dual-Band Sloper Antenna. 64 Inverted-V Beam Antenna for 30m. 65 ZL-Special Beam Antenna for 15m. 66 Half-Sloper Antenna for 160m . 67 Two-Bands Half Sloper for 80m - 40m. 68 Linear Loaded Sloper Antenna for 160m. 69 Super-Sloper Antenna. 70 Tower Pole as a Vertical Antenna for 80m. 71 Clothesline Antenna. 72 Wire Ground-Plane Antenna. 73 Inverted Delta Loop for .

ADVANCES in DATA NETWORKS, COMMUNICATIONS, COMPUTERS 9th WSEAS International Conference on DATA NETWORKS, COMMUNICATIONS, COMPUTERS (DNCOCO '10) University of Algarve, Faro, Portugal November 3-5, 2010 Published by WSEAS Press ISSN: 1792-6157 . www.wseas.org . ISBN: 978-960-474-245-5

Proceedings of the 8th WSEAS International Conference on DATA NETWORKS, COMMUNICATIONS, COMPUTERS (DNCOCO '09) Morgan State University, Baltimore, USA November 7-9, 2009 Recent Advances in Computer Engineering A Series of Reference Books and Textbooks Published by WSEAS Press . ISSN: 1790-5109 . www.wseas.org. ISBN: 978-960-474-134-2

Professional Wireless HA-8089 helical antenna – 470-900MHz Sennheiser A2003 UHF W/B Antenna Sennheiser A5000CP Antenna Sennheiser AD3700 Active Antenna Shure UA830WB UHF Active Antenna Booster Shure UA860/SWB omnidirectional antenna UHF Shure UA870-WB Active Antenna Shure UA874-WB Active Antenna Shure

headliner, down the A-pillar to the instrument panel. For antenna removal, see REMOVAL . For antenna installation, see INSTALLATION . OPERATION ANTENNA BODY AND CABLE The antenna body and cable connects the antenna mast to the radio. The radio antenna is an . 2008 Jeep

A - provider is used by AngularJS internally to create services, factory etc. B - provider is used during config phase. C - provider is a special factory method. D - All of the above. Q 10 - config phase is the phase during which AngularJS bootstraps itself. A - true B - false Q 11 - constants are used to pass values at config phase. A - true B .