Antennas and Propagation Array. Alberto Toccafondi

Transcription

1 Antennas an Propagation Array Alberto Toccafoni

2 Two-element array z Ø Array of two ientical horizontal wire antennas positione along the z-axis Ø Consier a reference antenna on the origin of the coorinate system anapply the translational phase shift to erive the far fiel of each translate antennas E "#$ r = jk e+,-" 4πr f "#$ (θ,φ) f "#$ θ, φ = 2ζI ; k cos π 2 cos θ sinθ θb I 7 x θ y r E 7 r = jk e+,-" 4πr 2ζI cos π 7 2 cos θ k sinθ θb e,- C DE" 7 = z I 8 2 E 8 r = jk e+,-" 4πr 2ζI cos π 8 2 cos θ k sinθ θb e,-c IE" 8 = z k = 2π λ

3 Ø Total electric far fiel the two antennas Two-element array z E KLK r = E 7 r + E 8 r E KLK r = jk e+,-" 4πr 2ζI cos π ; 2 cos θ k sinθ θb I 7 I ; e,- C DE" + I 8 I ; e,- C IE" I 7 θ r f "#$ θ,φ E KLK r = jk e+,-" 4πr f "#$(θ, φ) c 7 e,-c DE" + c 8 e,-c IE" c 7 = I 7 I ; c 8 = I 8 I ; x I 8 2 y E KLK r = jk e+,-" 4πr f "#$ θ, φ AF θ, φ element factor array factor Ø array factor AF θ, φ = c 7 e,-c D E" + c 8 e,-c I E" complex coefficients

4 Two-element array E KLK r = jk e+,-" 4πr f "#$ θ, φ AF θ, φ z Ø Particular cases AF θ, φ = c 7 e,-c QRS T + c 8 e +,-C QRS T I 7 θ r c 7 = c 8 = c c 7 = c 8 = c AF θ, φ = 2c cos(k cos θ) AF θ, φ = 2jc sin(k cos θ) x I 8 2 y

5 N-element array z Ø Array of N ientical, mutually ecouple antennas positione in a 3D-space E KLK r = jk e+,-" 4πr f "#$ θ,φ c ; + c 7 e,-c D E" Y+7 + c 8 e,-c I E" + c Y+7 e,-c \]D E" I θ r AF θ, φ = V c W e,- C XE" WZ; x y Y+7 AF θ, φ = c ; c 7 c 8 c Y+7 E e,-c _E" e,- C \]DE" = c E s c ; c = vector of complex weights c W = c W e,c X c Y+7 s = e,-c _E" e,- C \]DE" steering vector (array mainfol vector) for irection of propagation for transmitting arrays or irections of arrivals for receiving arrays

6 Pattern Multiplication Principle z Ø Array of N ientical, mutually ecouple antennas positione in a 3D-space E KLK r = E "#$ r AF θ, φ = jk e+,-" 4πr f "#$ θ, φ AF θ, φ f e" θ, φ I θ r Ø Raiation intensity y U e" θ, φ = 2ζ k 8 4π 8 f e" θ, φ 8 = 2ζ k 8 4π 8 f "#$ θ, φ 8 AF θ, φ 8 x Y+7 U e" θ, φ = U "#$ θ, φ AF θ, φ 8 Ø Array factor can ramatically alter the irectivity properties of the singleelement antenna

7 Equally space, linear array z N- Ø Equally space array along the z-axis at locations z W = n n =,,2 N W = nz Y+7 AF θ, φ = V c W e,-c XE" WZ; Y+7 = V c W e,-wcjlkt WZ; Ø Equally space array along the x-axis at locations x W = n n =,,2 N Y+7 W = nxm AF θ, φ = V c W e,-c XE" Y+7 = V c W e,-wcknwtjlko x 3 2 θ y r WZ; WZ; Ø Equally space array along the y-axis at locations y W = n n =,,2 N Y+7 Y+7 W = nym AF θ, φ = V c W e,-c XE" = V c W e,-wcknwtknwo WZ; WZ; Ø Equally space array along a generic irection um at locations u W = n n =,,2 N W = num W E r = n cos γ γ: angle between r an um

8 Uniform linear array Ø Uniformly excite, linearly phase, equally space, linear array along the z- axis at locations z W = n n =,,2 N W = nz Y+7 AF θ, φ = V c W e,-c XE" WZ; Y+7 = V c W e,-wcjlkt WZ; c ; = c c 7 = ce,c c 8 = ce,8c c Y+7 = ce, (Y+7)c Y+7 AF θ, φ = c V e,w(-cjlkttc) WZ; efine Y+7 ψ = kcosθ + α z x N- 3 θ 2 y r AF ψ = c V e,wx Ø Geometric series WZ; Y+7 AF ψ = c V e,wx WZ; = c x e,yx e,y8 = c e,x e,x 8 e +,Yx 8 e,yx 8 e +,x 8 e,x 8 = ce,(y+7)x 8 sin N ψ 2 sin ψ 2

9 Uniform linear array Ø Uniformly excite, linearly phase, equally space, linear array along the z- axis z N- AF ψ = ce,(y+7)x 8 sin N ψ 2 ψ = kcosθ + α sin ψ 2 Ø Reference point for the phase in the physical center of the array AF ψ = c sin N ψ 2 sin ψ 2 x 3 2 θ y r Ø Maximum value of AF occurs when ψ = AF = cn Ø Normalize arrayfactor (universal arrayfunction) AFy ψ = e,(y+7)x sin N ψ 2 8 N sin ψ 2

10 Uniform linear array Ø Perioicity AFy ψ = AFy ψ + 2π z N- Ø The array factor is a pattern that has a rotationally symmetry about the line of the array. In this case is inepenent of φ. The complete structure is etermine by < θ < π. This efine the visibility region. k + α < ψ < k + α Ø Length of visibility region: 2k. Exactly one perio of the normalize AFy ψ appears in the visibility region if = 8. x 3 2 θ y r Ø Length of visibility region: 2k. Exactly one perio of the normalize AFy ψ appears in the visibility region if = 8. Ø Nulls of AFy ψ sin N ψ 2 = ψ = ± 2lπ N l =,2, l N, 2N, θ W = cos +7 λ 2π α ± 2lπ N

11 Uniform linear array Ø Plot of AFy ψ AF y ψ 2π z N x 3 2 θ y r α ψ visible region k + α k+ α Ø AFy ψ has a main lobe at ψ = ± 2mπ Ø Main beam scanning ψ = = k cos θ # + α θ # = cos +7 α k

12 Uniform linear array 2k Ø Broasie array Ø α = Ø N=8 Ø = o 35 o 45 o φ 8 o 5 B 5 o 35 o 45 o 9 o

13 Uniform linear array 2k Ø Orinary enifire array Ø α = ± 8 Ø N=8 Ø = o 35 o 35 o 9 o 9 o 5 45 o B 5 45 o φ o

14 o 35 o 35 o 9 o 9 o 5 45 o B 5 45 o φ o 2β

15 Linear array N=3

16 Linear array N=4

17 Linear array N=5

18 Linear array N=6

19 Linear array N=7

20 Linear array N=8

21 Linear array

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24 g(θ) Δθ 3 B Fig Mainlobe with an sielobe level of uniform array. φ φ = ψ /(k)=. ( π/n)/( π/λ) Δθ 3 B.886 λ N

25 g(θ) Δθ 3 B Fig Mainlobe with an sielobe level of uniform array. φ φ = ψ /(k)=. ( π/n)/( π/λ) R 2log AF(ψ) AF() 3.26 B

26 Grating lobes N= o 35o 45 o φ o B o