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Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
Array Antennas - Analysis
S. R. [email protected]
School of Electronics EngineeringVellore Institute of Technology
July 24, 2013
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
mailto:[email protected]
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
Outline
1 Introduction
2 Linear Arrays
3 Planar Array
4 Linear Arrays - Examples
5 Planar Arrays - Examples
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
Outline
1 Introduction
2 Linear Arrays
3 Planar Array
4 Linear Arrays - Examples
5 Planar Arrays - Examples
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
Cylindrical and Spherical Coordinate Systems
O
ρφ
z
(ρ,φ,z)
X
Y
Z
~r = xx̂ + yŷ + zẑ
~r = ρ cos φx̂ + ρ sin φŷ + zẑ
~r = r sin θ cos φx̂ + r sin θ sin φŷ + r cos θẑ,
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
Array Factor
ReferencePoint
AF = ∑n
An exp[jk0(|~r| −
∣∣∣~r− ~r′n∣∣∣)]
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
Approximation of∣∣∣~r−~r′∣∣∣ - Cartesian System
In Cartesian coordinate system,
~r′ = x′ x̂ + y′ ŷ + z′ ẑ, and~r = r sin θ cos φx̂ + r sin θ sin φŷ + r cos θẑ.
So, ∣∣∣~r−~r′∣∣∣ = ∣∣r sin θ cos φx̂ + r sin θ sin φŷ + r cos θẑ− x′ x̂− y′ ŷ− z′ ẑ∣∣=
√(r sin θ cos φ− x′)2 + (r sin θ sin φ− y′)2 + (r cos θ − z′)2
=√
r2 − 2rx′ sin θ cos φ− 2ry′ sin θ sin φ− 2rz′ cos θ + x′2 + y′2 + z′2
≈√
r2 − 2rx′ sin θ cos φ− 2ry′ sin θ sin φ− 2rz′ cos θ
= r
√1− 2x
′ sin θ cos φ + 2y′ sin θ sin φ + 2z′ cos θr
≈ r(
1− 12
2x′ sin θ cos φ + 2y′ sin θ sin φ + 2z′ cos θr
)≈ [r− (x′ sin θ cos φ + y′ sin θ sin φ + z′ cos θ)] . (1)
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
Approximation of∣∣∣~r−~r′∣∣∣ - Cylindrical System ***
In Cylindrical coordinate system,
~r′ = ρ′ cos φ′ x̂ + ρ′ sin φ′ ŷ + z′ ẑ, and~r = r sin θ cos φx̂ + r sin θ sin φŷ + r cos θẑ.
So, following the same procedure given in the previous slide∣∣∣~r−~r′∣∣∣ ≈ r− (x′ sin θ cos φ + y′ sin θ sin φ + z′ cos θ)≈ r− (ρ′ cos φ′ sin θ cos φ + ρ′ sin φ′ sin θ sin φ + z′ cos θ)≈ r− (ρ′ sin θ (cos φ′ cos φ + sin φ′ sin φ) + z′ cos θ)≈ r− (ρ′ sin θ cos (φ− φ′) + z′ cos θ) . (2)
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
Approximation of∣∣∣~r−~r′∣∣∣ - Spherical System ***
In Spherical coordinate system,
~r′ = r′ sin θ′ cos φ′ x̂ + r′ sin θ′ sin φ′ ŷ + r′ cos θ′ ẑ, and~r = r sin θ cos φx̂ + r sin θ sin φŷ + r cos θẑ.
So, following the same procedure given in the previous slide∣∣∣~r−~r′∣∣∣ ≈ r− (x′ sin θ cos φ + y′ sin θ sin φ + z′ cos θ)≈ r− (r′ sin θ′ cos φ′ sin θ cos φ + r′ sin θ′ sin φ′ sin θ sin φ + r′ cos θ′ cos θ)≈ r− (r′ sin θ sin θ′ (cos φ′ cos φ + sin φ′ sin φ) + r′ cos θ′ cos θ)≈ r− (r′ sin θ sin θ′ cos (φ− φ′) + r′ cos θ′ cos θ). (3)
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
So, Array Factor in Cartesian Co-ordinate System is ...
AF = ∑n
An exp[jk0(|~r| −
∣∣∣~r− ~r′n∣∣∣)]≈∑
nAn exp
{jk0 [|~r| − [r− (x′n sin θ cos φ + y′n sin θ sin φ + z′n cos θ)]]
}= ∑
nAn exp [jk0 (x′n sin θ cos φ + y
′n sin θ sin φ + z
′n cos θ)]
= ∑n
An exp (jk0 sin θ cos φx′n + jk0 sin θ sin φy′n + jk0 cos θz
′n)
= ∑n
An exp(jkxx′n + jkyy
′n + jkzz
′n)
(4)
For continuous arrays, the above equation reduces to
AF =˚
A (x′ , y′ , z′) exp(jkxx′ + jkyy′ + jkzz′
)dx′dy′dz′ . (5)
Does the above equation remind of something?
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
Array Factor of a Uniformly Spaced Linear Array
N - Even
N - Odd
AF = ∑n
An exp(jkxx′n + jkyy
′n + jkzz
′n)
= ∑n
An exp (jkxx′n)
= ∑n
An exp (jkxna) (6)
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
Array Factor of a Uniformly Spaced Planar Array
AF = ∑n
An exp(jkxx′n + jkyy
′n + jkzz
′n)
= ∑n
An exp(jkxx′n + jkyy
′n)
= ∑p
∑q
Apq exp(
jkxx′pq + jkyy′pq
)= ∑
p∑
qApq exp
[jkx
(pa +
qbtan γ
)+ jkyqb
](7)
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
Outline
1 Introduction
2 Linear Arrays
3 Planar Array
4 Linear Arrays - Examples
5 Planar Arrays - Examples
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
Continuous Linear Array
AF (kx) =ˆ ∞−∞
A (x′) exp (jkxx′) dx′
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
Discrete Uniformly Spaced Linear Array
AF (kx) = ∑n
An exp (jkxx′n) = ∑n
An exp (jkxna)
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
Discrete Uniformly Spaced Linear Array
AF (kx) = ∑n
An exp (jkxx′n) = ∑n
An exp (jkxna)
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
Discrete Uniformly Spaced Linear Array
AF (kx) = ∑n
An exp (jkxx′n) = ∑n
An exp (jkxna)
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
Discrete Linear Array - Progressive Phasing
AF (kx − kx0) = ∑n
An exp [j (kx − kx0) x′n] = ∑n
An exp [j (kx − kx0) na] = ∑n
An exp (−jkx0na) exp (jkxna)
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
Maximum Scan Limit
kx0 ≤(
2πa− k0
)⇒ k0 sin θ0 ≤
(2πa− k0
)⇒ θ0 ≤ sin
−1(
2πak0− 1)
⇒ θ0,max = sin−1
(2πak0− 1)
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
Optimal Spacing
kx0 ≤(
2πa− k0
)⇒ k0 sin θ0,max ≤
(2πa− k0
)
⇒ a ≤ 1k0
(2π
1 + sin θ0,max
)
⇒ amax =1
k0
(2π
1 + sin θ0,max
)
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
Outline
1 Introduction
2 Linear Arrays
3 Planar Array
4 Linear Arrays - Examples
5 Planar Arrays - Examples
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
Continuous Planar Array
Array placed in the xy plane:
AF(kx, ky
)=
˚A (x′ , y′) exp
(jkxx′ + jkyy′
)dx′dy′
Array placed in the yz plane:
AF(ky, kz
)=
˚A (y′ , z′) exp
(jkyy′ + jkzz′
)dy′dz′
Array placed in the xz plane:
AF (kx, kz) =˚
A (x′ , z′) exp (jkxx′ + jkzz′) dx′dz′
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
Visible Space in kxky domain
VISIBLE-SPACE DISK
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
Discrete Uniformly Spaced Planar Array
VISIBLE-SPACE DISK
ARBITRARY SCANSPECIFICATION
AF = ∑p
∑q
Apq exp[
jkx
(pa +
qbtan γ
)+ jkyqb
]
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
Typical Scanning Examples
PLANE PLANE
- CONTOUR
- CONTOUR
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
Typical Scanning Examples
1
2
3
PLANE
- CONTOUR
1
2
3
PLANE
4
- CONTOUR
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
Typical Scanning Examples
TRIANGULARPYRAMIDAL
SECTORS
NORMAL TO FACE
1
2
3
NORMAL TO FACE
FOUR TRAPEZOIDALPYRAMIDAL
SECTORS
RECTANGULARPYRAMIDAL
SECTOR
1
2
3
4
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
A Practical Phased Array Antenna Radar System
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
Outline
1 Introduction
2 Linear Arrays
3 Planar Array
4 Linear Arrays - Examples
5 Planar Arrays - Examples
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
1 Continuous Linear Arrayλ2 Dipole Antenna
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
2 Uniformly Spaced Discrete Linear Array (a=λ2 )Uniform Excitation
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
2 Uniformly Spaced Discrete Linear Array (a=λ2 )Uniform Excitation
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
2 Uniformly Spaced Discrete Linear Array (a=λ)Uniform Excitation
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
2 Uniformly Spaced Discrete Linear Array (a=λ)Uniform Excitation
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
4 Uniformly Spaced Discrete Linear Array - Scan
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
5 Uniformly Spaced Discrete Linear End-fire Array
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
Outline
1 Introduction
2 Linear Arrays
3 Planar Array
4 Linear Arrays - Examples
5 Planar Arrays - Examples
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT
Introduction Linear Arrays Planar Array Linear Arrays - Examples Planar Arrays - Examples
We will comeback to this section when we discuss aperture antennas
Array Antennas - Analysis EE533, School of Electronics Engineering, VIT