Array Antennas - Analysis

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1 S. R. Zika School of Electroics Egieerig Vellore Istitute of Techology July 24, 2013

2 Outlie 1 Itroductio 2 Liear Arrays 3 Plaar Array 4 Liear Arrays - Examples 5 Plaar Arrays - Examples

3 Outlie 1 Itroductio 2 Liear Arrays 3 Plaar Array 4 Liear Arrays - Examples 5 Plaar Arrays - Examples

4 Cylidrical ad Spherical Coordiate Systems Z O (ρ,φ,z) z X ρ φ Y r = xˆx + yŷ + zẑ r = ρ cos φˆx + ρ si φŷ + zẑ r = r si θ cos φˆx + r si θ si φŷ + r cos θẑ,

5 Array Factor Referece Poit ( )] AF = A exp [jk 0 r r r

6 Approximatio of r r - Cartesia System I Cartesia coordiate system, r = x ˆx + y ŷ + z ẑ, ad r = r si θ cos φˆx + r si θ si φŷ + r cos θẑ. So, r r = r si θ cos φˆx + r si θ si φŷ + r cos θẑ x ˆx y ŷ z ẑ = (r si θ cos φ x ) 2 + (r si θ si φ y ) 2 + (r cos θ z ) 2 = r 2 2rx si θ cos φ 2ry si θ si φ 2rz cos θ + x 2 + y 2 + z 2 r 2 2rx si θ cos φ 2ry si θ si φ 2rz cos θ = r 1 2x si θ cos φ + 2y si θ si φ + 2z cos θ r r ( x si θ cos φ + 2y si θ si φ + 2z cos θ r [r (x si θ cos φ + y si θ si φ + z cos θ)]. (1) )

7 Approximatio of r r - Cylidrical System *** I Cylidrical coordiate system, r = ρ cos φ ˆx + ρ si φ ŷ + z ẑ, ad r = r si θ cos φˆx + r si θ si φŷ + r cos θẑ. So, followig the same procedure give i the previous slide r r r (x si θ cos φ + y si θ si φ + z cos θ) r (ρ cos φ si θ cos φ + ρ si φ si θ si φ + z cos θ) r (ρ si θ (cos φ cos φ + si φ si φ) + z cos θ) r (ρ si θ cos (φ φ ) + z cos θ). (2)

8 Approximatio of r r - Spherical System *** I Spherical coordiate system, r = r si θ cos φ ˆx + r si θ si φ ŷ + r cos θ ẑ, ad r = r si θ cos φˆx + r si θ si φŷ + r cos θẑ. So, followig the same procedure give i the previous slide r r r (x si θ cos φ + y si θ si φ + z cos θ) r (r si θ cos φ si θ cos φ + r si θ si φ si θ si φ + r cos θ cos θ) r (r si θ si θ (cos φ cos φ + si φ si φ) + r cos θ cos θ) r (r si θ si θ cos (φ φ ) + r cos θ cos θ). (3)

9 So, Array Factor i Cartesia Co-ordiate System is... ( )] AF = A exp [jk 0 r r r A exp { jk 0 [ r [r (x si θ cos φ + y si θ si φ + z cos θ)]] } = A exp [jk 0 (x si θ cos φ + y si θ si φ + z cos θ)] = A exp (jk 0 si θ cos φx + jk 0 si θ si φy + jk 0 cos θz ) = A exp ( jk x x + jk y y + jk z z ) (4) For cotiuous arrays, the above equatio reduces to AF = A (x, y, z ) exp ( jk x x + jk y y + jk z z ) dx dy dz. (5) Does the above equatio remid of somethig?

10 Array Factor of a Uiformly Spaced Liear Array N - Odd AF = A exp ( jk x x + jk y y + jk z z ) = A exp (jk x x ) = A exp (jk x a) (6) N - Eve

11 Array Factor of a Uiformly Spaced Plaar Array AF = A exp ( jk x x + jk y y + jk z z ) ) = A exp ( jk x x + jk y y = p = p ( ) A pq exp jk x x pq + jk y y pq q A pq exp q [jk x ( pa + qb ) ] + jk y qb ta γ (7)

12 Outlie 1 Itroductio 2 Liear Arrays 3 Plaar Array 4 Liear Arrays - Examples 5 Plaar Arrays - Examples

13 Cotiuous Liear Array ˆ AF (k x) = A (x ) exp (jk x x ) dx

14 Discrete Uiformly Spaced Liear Array AF (k x) = A exp (jk x x ) = A exp (jk x a)

15 Discrete Uiformly Spaced Liear Array AF (k x) = A exp (jk x x ) = A exp (jk x a)

16 Discrete Uiformly Spaced Liear Array AF (k x) = A exp (jk x x ) = A exp (jk x a)

17 Discrete Liear Array - Progressive Phasig AF (k x k x0) = A exp [j (k x k x0) x ] = A exp [j (k x k x0) a] = A exp ( jk x0 a) exp (jk x a)

18 Maximum Sca Limit ( ) 2π k x0 a k 0 ( ) 2π k 0 si θ 0 a k 0 θ 0 si 1 ( ) 2π 1 ak 0 θ 0,max = si 1 ( ) 2π 1 ak 0

19 Optimal Spacig k x0 k 0 si θ 0,max a amax = ( ) 2π a k 0 ( ) 2π a k 0 ( ) 1 2π k si θ 0,max ( ) 1 2π k si θ 0,max

20 Outlie 1 Itroductio 2 Liear Arrays 3 Plaar Array 4 Liear Arrays - Examples 5 Plaar Arrays - Examples

21 Cotiuous Plaar Array Array placed i the xy plae: AF ( k x, k y ) = A (x, y ) exp ( jk x x + jk y y ) dx dy Array placed i the yz plae: AF ( k y, k z ) = A (y, z ) exp ( jk y y + jk z z ) dy dz Array placed i the xz plae: AF (k x, k z) = A (x, z ) exp (jk x x + jk z z ) dx dz

22 Visible Space i k x k y domai VISIBLE-SPACE DISK

23 Discrete Uiformly Spaced Plaar Array VISIBLE-SPACE DISK ARBITRARY SCAN SPECIFICATION AF = p ( A pq exp [jk x pa + q qb ) ] + jk y qb ta γ

24 Typical Scaig Examples PLANE PLANE - CONTOUR - CONTOUR

25 Typical Scaig Examples PLANE 4 PLANE - CONTOUR CONTOUR

26 Typical Scaig Examples TRIANGULAR PYRAMIDAL SECTORS RECTANGULAR PYRAMIDAL SECTOR FOUR TRAPEZOIDAL PYRAMIDAL SECTORS 1 1 NORMAL TO FACE NORMAL TO FACE 3 2

27 A Practical Phased Array Atea Radar System

28 Outlie 1 Itroductio 2 Liear Arrays 3 Plaar Array 4 Liear Arrays - Examples 5 Plaar Arrays - Examples

29 1 Cotiuous Liear Array λ 2 Dipole Atea

30 2 Uiformly Spaced Discrete Liear Array (a= λ 2 ) Uiform Excitatio

31 2 Uiformly Spaced Discrete Liear Array (a= λ 2 ) Uiform Excitatio

32 2 Uiformly Spaced Discrete Liear Array (a=λ) Uiform Excitatio

33 2 Uiformly Spaced Discrete Liear Array (a=λ) Uiform Excitatio

34 4 Uiformly Spaced Discrete Liear Array - Sca

35 5 Uiformly Spaced Discrete Liear Ed-fire Array

36 Outlie 1 Itroductio 2 Liear Arrays 3 Plaar Array 4 Liear Arrays - Examples 5 Plaar Arrays - Examples

37 We will comeback to this sectio whe we discuss aperture ateas

Antenna Engineering Lecture 8: Antenna Arrays

Antenna Engineering Lecture 8: Antenna Arrays Atea Egieerig Lecture 8: Atea Arrays ELCN45 Sprig 211 Commuicatios ad Computer Egieerig Program Faculty of Egieerig Cairo Uiversity 2 Outlie 1 Array of Isotropic Radiators Array Cofiguratios The Space