Reflector Antennas. Summary. 1.Basic Concepts of Antennas. 2.Radiation by Aperture Antennas. 3.Reflector Antennas. 4.Design and Analysis Methods

Transcription

1 Reflector Antennas DR. RAFAEL ABRANTES PENCHEL Summary 1.Basic Concepts of Antennas 2.Radiation by Aperture Antennas 3.Reflector Antennas 4.Design and Analysis Methods 1

2 Basic Concepts of Antennas Coordinate System Elevation Plane z θ da = r 2 sin θ dθdφ θ : elevation 0 θ 2π φ : azimuth 0 θ 2π φ y a θ a φ rdθ x Azimuth Plane a θ r sin θ dφ 2

3 Radian One radian (1 rad) is defined as the plane angle with its vertex at the center of a circle of radius r that is subtended by an arc whose length is r. r 1 rad r C = 2πr 1 rad 57, Steradian One steradian (1 sr) is defined as the solid angle with its vertex at the center of a sphere of radius r that is subtended by a spherical surface area. r 1 sr A = r 2 r r equal to that of a square with each side r. A = 4πr 2 3

4 Radiation Pattern A mathematical function or a graphical representation of the radiation properties of the antenna, such as : Field or Power Phase Polarization, etc. as a function of angular space coordinates θ, φ. Radiation Pattern Rectangular Polar three-dimensional 4

5 Radiation Pattern Field Pattern Power Pattern E = E θ, φ E max P θ, φ E θ, φ E max 2 Radiation Pattern Power Pattern Power Pattern (db) U θ, φ E θ, φ E max 2 U db θ, φ = 10log 10 U θ, φ 5

6 D (db) D (dimensionless) 06/06/2015 Radiation Pattern P db θ, φ = 10log 10 P θ, φ P θ, φ E θ, φ E max 2 Radiation Pattern 20 D 0 = (dimensionless) (degrees) D 0 = (db) (degrees) 6

7 Radiation Pattern + + Major lobe Minor lobes Side lobe + + Nulls + Back lobe + Radiation Pattern Isotropic Omnidirectional Directional 7

8 Radiation Pattern Isotropic Vertical Horizontal Radiation Pattern Omnidirectional Vertical Horizontal 8

9 Radiation Pattern Vertical Horizontal Directional Field Regions Far-field (Fraunhofer) region Radiating near-field (Fresnel) region Reactive near-field region R 1 = 0.62 D 3 λ D R 1 R 2 R 2 = 2D 2 λ 9

10 Field Regions Reactive Near-Field Phases of Electric and Magnetic fields are often near quadrature Wave impedance highly reactive (imaginary); High content of non-propagating stored energy; Radiating Near-Field Fields are predominantly in phase; Fields do not yet display a spherical wavefront; Region where near-field measurements are made; Wave impedance is active (real) and reactive (imaginary); Field Regions Far-Field e jkr Fields exhibit spherical wavefront : r E r = H r 0 E θ ηh φ η = μ ε E φ ηh θ TEM So, the phase is constant for any θ, φ. Ideally, the pattern does not change with distance r; Ideally, the wave impedance is strictly active (real); Power predominantly real; Propagating energy. 10

11 Radiation Power Density The quantity used to describe the power associated with an electromagnetic wave is the average Poynting vector: W rad = 1 2 Re E H W/m 2 P rad = 1 2 Re E H ds W Isotropic Radiator An isotropic radiator is an ideal source that radiates equally in all directions. Although it does not exist in practice, it provides a convenient isotropic reference. P rad = π π W 0 r 2 sin θ dθdφ 0 = W 0 4πr 2 W 0 = P rad 4πr 2 W/m 2 11

12 Radiation Intensity It is the power radiated from an antenna per unit solid angle. This is a far-field parameter mathematical expressed as: U = r 2 W rad W/sr U = r2 2η E r, θ, φ 2 W/sr U 0 = P rad 4π Beamwidth 0.75 HPBW HPBW (Half-Power Beamwidth) the angle between the two directions in which the radiation intensity is onehalf value of the beam in the plane containing the direction of the maximum of a beam. 12

13 Beamwidth FPBW 0.25 FNBW (First-Null Beamwidth) is the angular separation between the first nulls of the pattern. Other beamwidths are those where the pattern is 10 db from the maximum. Directivity The ratio of the radiation intensity in a given direction to the radiation intensity averaged over all directions. The average radiation intensity is equal to the total power radiated by the antenna divided by 4π. D θ, φ = U U 0 = 4π P rad U θ, φ U θ, φ 1 2η E θ 2 + E φ 2 13

14 Directivity D = 1.67 D = 1 Isotropic D = Dipole /2 D = Input Impedance The impedance presented by an antenna at its terminals or the ratio of the voltage to current at a pair of terminals or the ratio of components of the electric to magnetic fields at a point. ~ Radiated wave ~ Z A Z A = R A + jx A 14

15 Antenna Efficiency The ratio of the total power radiated (P rad ) by an antenna to the net power accepted by the antenna (P in ) from the connected transmitter. It takes into account losses at the input terminals and within the structure of the antenna. Reflection Efficiency e 0 = e r e c e d Conduction Efficiency Dielectric Efficiency Antenna Efficiency Generator (P in ) ~ Radiated wave (P rad ) Generator ~ Z 0 Z A e 0 = e cd 1 Γ Γ = Z A Z 0 Z A + Z 0 15

16 Gain The ratio of the intensity, in a given direction, to the radiation intensity that would be obtained if the power accepted ( P in ) isotropically. G θ, φ by the antenna were radiated = 4π P in U θ, φ P rad = e cd P in Gain The gain of the antenna is closely related to the directivity, it is a measure that takes into account the efficiency of the antenna as well as its directional capabilities. G θ, φ = e cd 4π P rad U θ, φ G θ, φ = e cd D θ, φ 16

17 Return Loss [db] 06/06/2015 Bandwidth The range of frequencies where the antenna characteristics are within an acceptable value of those at the center frequency. 1. Input impedance 2. Return Loss 3. Beamwidth 4. Gain, etc. Bandwidth f c = 4.5 GHz RL < 10dB BW 2.25GHz f 1 4 f c 5 f f (GHz) BW ±25% 17

18 Polarization The property of an electromagnetic wave describing the time-varying direction and relative magnitude of the electric-field vector. E y E E z; t = E x z; t a x + E y z; t a y ψ E x E x z; t = E xo e j ωt+kz+φ x E x z; t = E xo cos ωt + kz + φ x E y z; t = E yo e j ωt+kz+φ y E y z; t = E yo cos ωt + kz + φ y Linear Polarization The field vector (electric or magnetic) possesses: 1. Only one component (E x or E y ); 2. Two orthogonal linear components that are in time phase or π out-of-phase. E y E ψ ψ = tan 1 ε y ε x E x 18

19 Linear Polarization E xo 0, E yo = 0 φ = φ y φ x = ±nπ ψ = π 2 n = 0, 1, 2, 3, Linear Polarization E xo = 0, E yo 0 φ = φ y φ x = ±nπ ψ = π 2 n = 0, 1, 2, 3, 19

20 Linear Polarization E xo = E yo E xo 0, E yo 0 φ = φ y φ x = ±nπ n = 0, 1, 2, 3, ψ = tan 1 ψ = π 4 ε y ε x Circular Polarization The field vector electric possesses all of the following: 1. Two orthogonal linear components the same magnitude (E x = E y ) ; 2. The two components must have a time-phase difference of odd multiples of 90 ; a. If positive multiples of 90 clockwise (CW); b. If negative multiples of 90 counterclockwise (CCW); 20

21 Circular Polarization E xo = E yo 0 φ = φ y φ x = n π 2 Clockwise (CW) n = 0, 1, 2, 3, Circular Polarization E xo = E yo 0 φ = φ y φ x = 1 + 2n π 2 Counterclockwise (CCW) n = 0, 1, 2, 3, 21

22 Elliptical Polarization Two orthogonal linear components: 1. If the two components are of different magnitude, the timephase difference between the two components must not be 0 or multiples of 180 (or it will then be linear); 2. If the two components are of the same magnitude, the time-phase difference between the two components must not be odd multiples of 90 (or it will then be circular); Elliptical Polarization φ = π 2 E yo = 2E xo E xo E yo 0 φ = φ y φ x nπ, > 0 Clockwise (CW) n = 0, 1, 2, 3, 22

23 Elliptical Polarization φ = π 2 E yo = 2E xo E xo E yo 0 φ = φ y φ x nπ, < 0 n = 0, 1, 2, 3, Counterclockwise (CCW) DR. RAFAEL ABRANTES PENCHEL IWT IWT Elliptical Polarization φ = π 6 E yo = E xo E xo = E yo 0 φ = φ y φ x n 2 π, > 0 n = 0, 1, 2, 3, Clockwise (CW) 23

24 Polarization Loss Factor (PLF) The amount of power extracted by the antenna from the incoming signal can be measured by PLF define as: PLF = ρ w ρ a 2 = cos ψ p 2 ψ p ψ p = π 2 PLF = 1 PLF = cos 2 ψ p PLF = 0 Effective Area (or Aperture) The ratio of the available power (P T ) at the terminals of a receiving antenna to the power flux density (W i ) of a plane wave incident on the antenna from that direction, the wave being polarization-matched to the antenna. A e = P T = λ 2 W i 4π D θ, φ Capture A e = A c A s A L Scattering Loss 24

25 Maximum Effective Aperture The maximum effective aperture antenna (A em ) is related to its maximum directivity (D o ) by: A em = λ2 4π D o Aperture efficiency (e ap ) is defined as the ratio of the maximum effective area (A em ) to its physical area (A p ): e ap = A em A p 0 e ap 1 Radiation Characteristics of Aperture Antennas 25

26 Introduction source : source : Introduction The mathematical tools developed to study the radiation characteristics of apertures can be use in analyze o many kind of aperture antennas: Waveguide or Horns antennas (square, rectangular, circular, elliptical, or any other configuration) Reflector antennas 26

27 Field Equivalence Principle The principle by which sources, such as an antenna, are replaced by equivalent sources. The fictitious sources are said to be equivalent within a region because they produce the same fields within that region. E 1, H 1 ε 1, μ 1 S E 2, H 2 ε 2, μ 2 Meio 2 Meio 1 n V E eq, H eq ε 1, μ 1 J S = n H 1 M S = n E 1 S E = 0, H = 0 ε 1, μ 1 Free Space n V Far-Field Approximations z S R r r y A = μ 4π x J s e jkr R ds F = ε 4π M s e jkr R ds 27

28 Far-Field Approximations z r S r R z S r ψ R r y y x Green s Function G R = e jkr R x Phase Amplitude R r r cos ψ R r Far-Field Approximations A ε e jkr 4π r F ε e jkr 4π r M s e jkr cos ψ ds M s e jkr cos ψ ds z S R r ψ r y E θ jωηf φ x 28

29 σ = σ = 06/06/2015 Aperture Equivalence A waveguide aperture is mounted on an infinite ground plane, the tangential components of the electric field over the aperture are known (E a ). E a F ε e jkr 4π r M s = 2n E a M s e jkr cos ψ ds E θ jωηf φ E φ +jωηf θ Rectangular Aperture b y φ θ a x z E θ abe o E A = E o y jk 2π e jkr r sin φ M S = +2E o x sin X X sin Y Y E φ abe o jk 2π e jkr r cos θ cos φ sin X X sin Y Y X = ka 2 sin θ sin φ Y = kb 2 sin θ cos φ 29

30 Sinc sinc X sin X = X X = ka 2 sin θ sin φ Rectangular Aperture b y φ θ a z x a = 5λ b = 5λ X = ka 2 sin θ sin φ Y = kb sin θ cos φ 2 E φ = 0; E θ sin Y Y E θ = 0; E φ cos θ sin X X 30

31 Uniform Aperture If the phase and polarization of the aperture electric field are uniform, then, the maximum radiation occurs in the normal direction of the aperture plane, regardless of its geometric shape. D o θ = 0 = 4π P rad U θ, φ U θ, φ 1 2η E θ 2 + E φ 2 Rectangular Aperture b y φ θ a z x a = 5λ b = 10λ X = ka 2 sin θ sin φ Y = kb sin θ cos φ 2 E φ = 0; E θ sin Y Y E θ = 0; E φ cos θ sin X X 31

32 Rectangular Aperture E φ cos θ sin Y Y Rectangular Aperture E φ cos θ sin Y Y 32

33 Rectangular Aperture E θ sin X X Rectangular Aperture E θ sin X X 33

34 Circular Aperture y a φ θ x E A = E o y M S = 2E o y z E θ 2πa 2 E o jk 2π e jkr r sin φ J 1 ka sin θ ka sin θ E φ 2πa 2 E o jk 2π e jkr r cos θ cos φ J 1 ka sin θ ka sin θ J 1 Bessel function of first order Bessel Function J 1 ka sin θ ka sin θ 34

35 Circular Aperture a = 5λ y a φ θ z x E φ = 0; E θ J 1 ka sin θ ka sin θ E θ = 0; E φ cos θ J 1 ka sin θ ka sin θ Circular Aperture E φ cos θ J 1 ka sin θ ka sin θ 35

36 Circular Aperture E θ J 1 ka sin θ ka sin θ Uniform Aperture For any planar aperture the Directivity D θ, φ is a function of: 1. Phase and Amplitude of aperture fields (E A ); 2. Polarization of aperture fields; 3. Area of aperture (a, b); What are the conditions for maximum directivity?? 36

37 Uniform Aperture The first and second conditions for maximum directivity are uniform polarization and phase of aperture electric field E A (and, consequently, M s ), so that, the sum of all contributions generates a vector with maximum amplitude. y x dx dm dy s z E dm s dx dy Uniform Aperture If the phase and polarization of the aperture electric field are uniform, then, the maximum radiation occur in the normal direction to aperture plane, regardless of its geometric cross shape. The last condition for maximum directivity is uniform amplitude of the aperture electric field. 37

38 Uniform Aperture If the phase and polarization of the aperture electric field are uniform, then, the maximum radiation occurs in the normal direction of the aperture plane, regardless of its geometric shape. The last condition for maximum directivity is uniform amplitude of the aperture electric field. A planar aperture with amplitude, phase and polarization uniform is said Uniform Aperture. Reflector Antennas 38

39 Introduction Plane wavefront Spherical wavefront Introduction The operating principle of most reflector antennas is the same as lighthouse, lanterns, etc.: collimation of energy. The parabolic reflector antenna transforms an incoming plane wave into a spherical wave converging toward the focus. From another point of view, a spherical wave generated by a point source placed in the focus is reflected into a plane wave as a collimated beam. 39

40 Confocal Conics Confocal conics are conic sharing a common focus. The classical reflective surfaces (paraboloid, ellipsoid and hyperboloid) are generated by the rotation of conics around their. Parable Ellipse Hyperbola Introduction Directive Systems Single Reflector Front-Fed Off-set Reflector (Classical or Shaped) Dual Reflector Circularly Symmetric Classical Reflector Shaped Reflector Dual Reflector Offset Classical Reflector Shaped Reflector Omnidirectional Systems Single or Dual Classical Reflectors Shaped Reflector(s) 40

41 Circularly Symmetric Paraboloid Paraboloid Feed Front-Fed Ellipsoid Dual-Reflector Offset Reflector Antennas Paraboloid Dual-Reflector Offset Paraboloid Single-Reflector Offset Feed Ellipsoid

42 Omnidirecional Dual-Reflector Dual-Reflector Omnidirectional Feeders Single Reflector Feeder Dual Reflector Feeder Ku Band LNB C Band Ku Band LNB Ku Band 42

43 Radome The Radome is a structural weatherproof made to protect the antenna surfaces from weather. They are constructed of dielectric material to be transparent to in antenna s frequency operation. Front-Fed Paraboloid Simple Geometry; Aperture blockage by feeder and its supports; Low manufacturing cost; Parable Focus Satellite reception, microwave point-to-point links, etc.; 43

44 Front-Fed y y z x x z The reflector antena s aperture can be defined by the projection of the main reflector on a constant phase plane. The approximate analysis can be made using Aperture Method (ApM) with aperture field given para by Geometrical Optics; Front-Fed P a d r F θ F θ E f y E A x z r F = 2f cos θ F + 1 r F = fsec 2 θ F 2 Maximum directivity: Uniform Phase: Constant optical path length (r F + PA = const.); Uniform amplitude: Feeder with D θ F sec 4 θ F 2 Uniform Polarization: tan θ E 2 = d 4f r F + PA = 2f Feeder with circularly symmetric radiation pattern; 44

45 Front-Fed y Uniform Aperture in Free Space x E A = E o y M S = +E o x z H A = E o η x J S = +E o y E θ 2πa 2 E o jk 2π e jkr r cos φ 1 + cos θ J 1 ka sin θ ka sin θ E φ 2πa 2 E o jk 2π e jkr r sin φ 1 + cos θ J 1 ka sin θ ka sin θ Front-Fed y x z Maximum directivity: D o θ = 0 = π λ d 2 Uniform Aperture in Free Space D θ, φ = π λ d cos θ J 1 Ω Ω D o = 43,92dBi D o = 49,94dBi Ω = k d sin θ 2 θ 3dB θ 3dB dB Beamwidth: θ 3dB 58.4 λ d deg. 45

46 Front-Fed: Aperture Efficiency Aperture efficiency (e ap ) is defined as the ratio of the maximum directivity D o (or effective area A em ) to an uniform aperture with the same geometry and radiated power (or its physical area A p ): ε ap = A em = D o A p π λ d 2 = D o λ πd 2 Front-Fed: Aperture Efficiency ε ap = cot 2 θ E 2 The aperture efficiency is a function of the subtended angle θ E and the feed pattern G f θ of the reflector. Thus for a given feed pattern, all paraboloids with the same ratio f d have identical aperture efficiency. 0 θ E f d = 1 4 cot θ E 2 G f θ tan θ 2 dθ 2 46

47 Front-Fed: Aperture Efficiency To illustrate the variation of the aperture efficiency as a function of the feed pattern and the angular extent of the reflector function G f θ is defined as: G f θ = G o cos n θ 0 θ π 2 π 0 θ π 2 Front-Fed: Aperture Efficiency ε ap = ε s ε t ε p ε x ε b ε r Aperture Efficiency: Spillover Efficiency (ε s ) Taper Efficiency (ε t ) Phase Efficiency (ε p ) Polarization Efficiency (ε x ) Blockage Efficiency (ε b ) Random error Efficiency (ε r ) 47

48 Front-Fed: Aperture Efficiency Spillover Efficiency (ε s ): fraction of the total power that is radiated by the feed, intercepted, and collimated by the reflecting surface; Taper efficiency (ε t ): uniformity of the amplitude distribution of the feed pattern over the surface of the reflector. Spillover ε s = θ E G f θ sin θ dθ 0 π 0 G f θ sin θ dθ C. A. Balanis, Antenna Theory: Analysis and Design Front-Fed: Aperture Efficiency 82% C. A. Balanis, Antenna Theory: Analysis and Design As the feed pattern becomes more directive (n increases), the angular aperture of the reflector that leads to the maximum efficiency is smaller. For a given feed pattern (n = constant): There is only one reflector with a given angular θ E aperture or f d ratio which leads to a maximum aperture efficiency. Each maximum aperture efficiency is in the neighborhood of 82 83%. n θ E f d A B db

49 Front-Fed: Design Example 1. Design a Front-Fed reflector antenna with maximum gain G o = 50dBi; G o = ε s ε t π λ d 2 2. Determine θ E for a feeder with n = 9; = 0.8 π λ d 2 d 113λ A B db 10 log cos 4 θ E 2 cosn θ E ε ap = 0.8 A B 11dB 10 log cos 4 θ E 2 cos9 θ E 11 θ E Determine f d for a feeder with n = 9; f d = 1 4 cot 39 2 = f 80λ Front-Fed: Aperture Method The Aperture Method is useful to provide physical understanding and initial reflector antenna design. However, the method has several inaccuracies, for example: It does not take into account feed and supports blockage. Diffractions at border reflector Coupling effects between reflector surface, supports and feeder; Direct fields from feeder; 49

50 Axis-symmetric Dual-Reflector y x Paraboloid Paraboloid z Another way to obtain uniform Ellipsoid Hyperboloid phase in the aperture is using two reflectors. The main reflector is a paraboloid and the subrefletor is a ellipsoid (Gregorian) or hyperboloid (Cassegrain). Gregorian Cassegrain Dual-Reflector: Gregorian Paraboloid 6.5m Gregorian Parable Focus Ellipse Focus Ellipsoid Paraboloid 50

51 Dual Reflector: Gregorian Andrew Antenna 7.6m, Ku-band Optics Configuration: Gregorian Frequency Transmit: 14 to 14.8 GHz Receive: 10.7 to GHz Antenna Gain Transmit: 59.4dBi (14.5 GHz) Receive: 58.3 dbi (12.75 GHz) 3 db Beamwidth Transmit: 0.18 Receive: Dual-Reflector: Cassegrain Paraboloid Parable Focus Hyperbola Focus Hyperbooid Source: Vertex Antenna 6.1m C-band Frequency: Rx 3.6 to 4.2 GHz / Tx. 5.8 to 6.4GHz Gain: Rx 47.1 dbi (4.2GHz) / Tx.: 50.0 dbi (6.4 GHz) 3dB Beamwidth Rx.: 0.82 / Tx.:

52 Dual Reflector: Cassegrain GDSatcom 13.2M Ka-Band Antenna Optics Configuration: Cassegrain Frequency Transmit: 28 to 30 GHz Receive: 18 to 20 GHz Antenna Gain Transmit: 69dBi (30 GHz) Receive: 66dBi (20 GHz) 3 db Beamwidth Transmit: 0.06 Receive: 0.09 Source: Single Reflector x Dual Reflector Advantages Easy access to the feeder; Less thermal noise. The feeder is pointed at the sky. Greater control over the aperture fields (2 reflectors); More efficient Disadvantages Problems to access the feeder Higher manufacturing cost; More complex, bigger and directive feeder; Support structures for subreflector surface; Aperture blockage; 52

53 Dual-Reflector Symmetric Paraboloid Paraboloid Parable Focus Parable Focus Ellipse Focus Ellipsoid Hyperbola Focus Hyperbooid Offset Reflector Paraboloid Geometry more complex then Front-Fed. No aperture blockage by feeder and its supports; Satellite communications, radar, etc. ; This geometries generates cross-polarization; Feed 53

54 Offset Reflector Source: Offset reflector antennas can use over a feeder in order to produce a multibeam oppering in many satellites at the same time. Source: Frequency range: 10,7-12,75 GHz Antenna width / height: 91 cm / 70 cm Multi beam capability: +/- 20 orbital Gain (db): 36 dbi 3 db beamwidth: 2,3 Offset Reflector Offset reflector antennas can use with many feeder in order to produce a shaped coverage area. 54

55 Offset Dual-Reflector Paraboloid Geometry very complex. No aperture blockage; Satellite communications This geometries generates cross-polarization; Ellipsoid Antennas with very high efficiency Paraboloid Hyperbooid Offset Dual-Reflector: Gregorian Paraboloid Ellipsoid Frequency : TD-120 Ku 1.20 m Transmit: to 14.5 GHz Receive: 10.7 to GHz Antenna Gain Transmit: GHz Receive: GHz

56 Offset Dual-Reflector: Cassegrain Paraboloid Hyperbooid Hyperbola Focus Parable Focus Omnidirectional Dual-Reflector Subreflector Generatrix TEM coaxial horn TEM Shaped Reflector Generatrix 56

57 Omnidirectional Dual-Reflector Application WiMax LMDS (Local Multipoint Distribution Service) 5G (28GHz / 38GHz) Main benefits Broadband behavior High gain antennas Compact structures Shaping procedure Evaluation of an ordinary differential equation Concatenation of short local conic sections Omnidirectional Dual-Reflector OADG (Gregorian) OADH (Hyperbola) OADE (Ellipse) OADC (Cassegrain) 57

58 Omnidirectional Dual-Reflector OADG (Omnidirectional Axis-Displaced Gregorian) Parable Focus Real Ring Caustic Ellipse Focus The ray mapping is made between the negative side of the ellipse and the positive side of the parable. Omnidirectional Dual-Reflector OADE (Omnidirectional Axis-Displaced Ellipse) Ellipse Focus Parable Focus Real Ring Caustic The ray mapping is made between the positive side of the ellipse and the positive side of the parable. 58

59 Omnidirectional Dual-Reflector OADC (Omnidirectional Axis-Displaced Cassegrain) Hyperbola Focus Parable Focus Virtual Ring Caustic The ray mapping is made between the positive side of the hyperbola and the positive side of the parable. Omnidirectional Dual-Reflector Parable Focus OADH (Omnidirectional Axis-Displaced Hyperbola) Virtual Ring Caustic Hyperbola Focus The ray mapping is made between the negative side of the hyperbola and the positive side of the parable. 59

60 Directive Radiation Pattern Directive Radiation Pattern 60

61 Cosecant Square Radiation Pattern r = h = h csc β sin β P r 1 = G P r G t t r 2 λ 4π P r 1 r 2 ; P r 1 csc 2 β 2 G t csc 2 β P r const. Cosecant Square Radiation Pattern 10m θ N θ o 10m 200m Picocell 61

62 z( ) z( ) z( ) 06/06/2015 Cosecant Square Radiation Pattern G csc 2 θ x( ) x( ) x( ) Shaping Procedure Subreflector z D S V S O E P D B M 1 0 V M Main Reflector M n 1 M n n 1 n D M M N N 62

63 z( ) z( ) z( ) 06/06/2015 Cosecant Square (93 < θ < 115 ) 8 6 E.I x( ) E.II x( ) Parameters OADC Real Caustic OADC Virtual Caustic D M ( ) 17,515 17,521 V M ( ) 8,566 8,482 VOL.( ) 4,295 4,275 Cosecant Square (93 < θ < 115 ) H.I x( ) 5 H.II Parameters OADC Real Caustic OADC Virtual Caustic D M ( ) 16, V M ( ) 7,834 17,308 VOL.( ) 3,057 8,299 63

64 Shaped Offset Shaped Reflector Ω G, β x y O Feeder z Shaped Offset Reflector Aplication Satellite Communications Restrict cellular coverage area Main benefits Optimized coverage Low co-channel interference Numerical Technique Monge-Ampère Differential Equation Finite Differences Local Confocal Quadric Surfaces 64

65 Shaped Offset: Super-Elliptical Coverage η 1 Região com Restrição Região de Cobertura Shaped Offset: Super-Elliptical Coverage Região com Restrição Região de Cobertura 65

66 Shaped Offset: Generic Coverage Shaped Offset: Generic Coverage 66

67 Shaped Offset: Generic Coverage Shaping Procedure Confocal Quadrics z P A is of the Quadric Surface V j,k r Polar Grid j,k θ O φ j,k y x j,k 67

68 Shaped Offset: Contour Feeder A entad r x x x z θ c Ω θ o O o z c Ω y, y, y Ca D stante Far-field z Shaped Offset: Contour m φ = cot α 2 Re u = cot α u 2 Re u 2σ + m 2σ = 1 68

69 Diretividade [dbi] Im[1/ ''] 06/06/2015 Shaped Offset: Feeder I θ = I 0 cos 2n θ, n = 9.6 Shaped Offset: Circular Contour α c = 8, α o = Re[1/ ''] GO GO ±3dB PO (0.9GHz) PO (1.8GHz) PO (7.2GHz) ψ = 0,69, L G = 3dB [graus] 69

70 X [cm] Diretividade [dbi] 06/06/2015 Shaped Offset: Circular Contour α c = 8, α o = GO GO ±3dB PO (0.9GHz) PO (1.8GHz) PO (7.2GHz) Z [cm] ψ = 0,69, L G = 3dB [graus] Shaped Offset: Circular Contour 0.9GHz 1.8GHz 3.2GHz 70

71 Im[ ''] X [cm] Ganho [dbi] Ganho [dbi] Im[ ''] Ganho [dbi] Ganho [dbi] X [cm] Ganho [dbi] Ganho [dbi] Im[ ] X [cm] 06/06/2015 Shaped Offset: Circular Contour L G = 0dB L G = 12dB L G = 20dB Re[ ''] Re[ ''] Re[ ] GO GO ±3dB PO (0.9GHz) PO (1.8GHz) PO (7.2GHz) GO GO ±3dB PO (0.9GHz) PO (1.8GHz) PO (7.2GHz) GO GO ±3dB PO (0.9GHz) PO (1.8GHz) PO (7.2GHz) [graus] [graus] [graus] Shaped Offset: Circular Contour L G = 0dB L G = 12dB L G = 20dB Z [cm] 20 GO GO ±3dB PO (0.9GHz) 15 PO (1.8GHz) PO (7.2GHz) Z [cm] 25 GO GO ±3dB 20 PO (0.9GHz) PO (1.8GHz) PO (7.2GHz) Z [cm] 25 GO GO ±3dB 20 PO (0.9GHz) PO (1.8GHz) PO (7.2GHz) [graus] [graus] [graus] 71

72 Im[1/ ''] X [cm] Diretividade [dbi] Diretividade [dbi] 06/06/2015 Elliptic Coverage Contour α o = 18 α u = 8, α = GO GO ±3dB PO (0.9GHz) PO (1.8GHz) PO (7.2GHz) Re[ ''] ψ = 0,69, L G = 3dB [graus] Shaped Offset: Elliptical Contour Z [cm] α o = 18 α u = 8, α = GO GO ±3dB PO (0.9GHz) PO (1.8GHz) PO (7.2GHz) ψ = 0,69, L G = 3dB [graus] 72

73 Im[1/ ''] Diretividade [dbi] 06/06/2015 Shaped Offset: Elliptical Contour 0.9GHz 1.8GHz 3.2GHz Shaped Offset: Super-Elliptical Contour α o = 18, σ = 2 α u = 8, α = GO GO ±3dB PO (0.9GHz) PO (1.8GHz) PO (7.2GHz) Re[1/ ''] ψ = 0,69, L G = 3dB [graus] 73

74 X [cm] Diretividade [dbi] 06/06/2015 Shaped Offset: Super-Elliptical Contour X [cm] α o = 18, σ = 2 α u = 8, α = GO GO ±3dB PO (0.9GHz) PO (1.8GHz) PO (7.2GHz) -5 ψ = 0,69, L G = 3dB [graus] Shaped Offset: Super-Elliptical Contour 0.9GHz 1.8GHz 3.2GHz 74

75 Numerical and Asymptotic Methods GO (Geometrical Optics): Asymptotic Method; Extremely low computational cost; It does not take into account diffractions effects; It does not take into account electromagnetic coupling effects; Phase and polarization of feeder; Direct fields from feeder; Initial design of reflector and lenses antenna; Numerical and Asymptotic Methods ApM (Aperture Method) : Asymptotic Method; Low computational cost; It takes into account some diffractions effects; It does not take into account border diffractions; There are some techniques to describe more accurately the edge currents and, consequently, border diffraction effects; Analysis of electromagnetic scattering of aperture antennas with big electrical dimensions. 75

76 Numerical and Asymptotic Methods PO (Physical Optics) : Asymptotic Method; Medium computational cost; It takes into account some diffractions effects; It does not take into account correctly border diffractions; There are some techniques to describe more accurately the edge currents and, consequently, border diffraction effects; Analysis of electromagnetic scattering of bodies with big electrical dimensions. Numerical and Asymptotic Methods MoM (Method of Moments) : Numerical Method; High computational cost; It takes into account accurately all diffractions effects; It takes into account electromagnetic coupling effects; Analysis of electromagnetic scattering; 76

77 Numerical and Asymptotic Methods FEM (Finite Element Method) : Numerical Method; Extremely high computational cost; It takes into account accurately all diffractions effects; It takes into account electromagnetic coupling effects; Many electromagnetic analysis; References [1] C. A. Balanis, Antenna Theory: Analysis and Design, Third Edition, John Wiley and Sons, Inc., New York, [2] S. Silver (ed.), Microwave Antenna Theory and Design, IEE Electromagnetic Wave Series Vol. 19, Peter Peregrinus, London, [3] A. W. Love (ed.), Reflector Antennas, IEEE Press, New York, [4] A. W. Rudge and N. A. Adatia, Offset-Parabolic-Reflector Antennas: A Review, Proceedings of the IEEE, vol. 66, n. 12, pp , December [5] P. Clarricoats and G. Poulton, High-efficiency microwave reflector antennas a review, Proceedings of the IEEE, vol. 65, pp , Oct

78 Thank you!! 78