Array Antennas - Analysis - Arraytool5 Planar Arrays - Examples Array Antennas - Analysis EE533, School of Electronics Engineering, VIT. IntroductionLinear ArraysPlanar ArrayLinear

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]


Outline

1 Introduction

2 Linear Arrays

3 Planar Array

4 Linear Arrays - Examples

5 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 Factor

ReferencePoint

AF = ∑n

An exp[jk0(|~r| −

∣∣∣~r− ~r′n∣∣∣)]



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)



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)



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)



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 Factor of a Uniformly Spaced Linear Array

N - Even

N - Odd

AF = ∑n


′n + jkzz

′n)

= ∑n

An exp (jkxx′n)

= ∑n

An exp (jkxna) (6)



Array Factor of a Uniformly Spaced Planar Array

AF = ∑n


′n + jkzz

′n)

= ∑n


′n)

= ∑p

∑q

Apq exp(

jkxx′pq + jkyy′pq

)= ∑

p∑

qApq exp

[jkx

(pa +

qbtan γ

)+ jkyqb

](7)



Outline

1 Introduction

2 Linear Arrays

3 Planar Array





Continuous Linear Array

AF (kx) =ˆ ∞−∞

A (x′) exp (jkxx′) dx′



Discrete Uniformly Spaced Linear Array

AF (kx) = ∑n

An exp (jkxx′n) = ∑n

An exp (jkxna)



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)



Maximum Scan Limit

kx0 ≤(

2πa− k0

)⇒ k0 sin θ0 ≤

(2πa− k0

)⇒ θ0 ≤ sin

−1(

2πak0− 1)

⇒ θ0,max = sin−1

(2πak0− 1)



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

)



Outline

1 Introduction

2 Linear Arrays

3 Planar Array





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′



Visible Space in kxky domain

VISIBLE-SPACE DISK



Discrete Uniformly Spaced Planar Array

VISIBLE-SPACE DISK

ARBITRARY SCANSPECIFICATION

AF = ∑p

∑q

Apq exp[

jkx

(pa +

qbtan γ

)+ jkyqb

]



Typical Scanning Examples

PLANE PLANE

- CONTOUR

- CONTOUR




1

2

3

PLANE

- CONTOUR

1

2

3

PLANE

4

- CONTOUR




TRIANGULARPYRAMIDAL

SECTORS

NORMAL TO FACE

1

2

3

NORMAL TO FACE

FOUR TRAPEZOIDALPYRAMIDAL

SECTORS

RECTANGULARPYRAMIDAL

SECTOR

1

2

3

4



A Practical Phased Array Antenna Radar System



Outline

1 Introduction

2 Linear Arrays

3 Planar Array





1 Continuous Linear Arrayλ2 Dipole Antenna



2 Uniformly Spaced Discrete Linear Array (a=λ2 )Uniform Excitation



2 Uniformly Spaced Discrete Linear Array (a=λ)Uniform Excitation



4 Uniformly Spaced Discrete Linear Array - Scan



5 Uniformly Spaced Discrete Linear End-fire Array



Outline

1 Introduction

2 Linear Arrays

3 Planar Array





We will comeback to this section when we discuss aperture antennas


Array Antennas - Analysis - Arraytool5 Planar Arrays - Examples Array Antennas - Analysis EE533, School of Electronics Engineering, VIT. IntroductionLinear ArraysPlanar ArrayLinear

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