array antennas - theory & design - arraytool · pdf file03-10-2014 · array...

Array Factor Linear Arrays Linear Arrays - Examples Planar Arrays Planar Arrays - Examples Synthesis Multiple Beam Arrays Problems

Array Antennas - Theory & Design

S. R. Zinka

[email protected]

October 16, 2014

Array Antennas - Theory & Design CT531, DA-IICT

mailto:[email protected]


Outline

1 Array Factor

2 Linear Arrays

3 Linear Arrays - Examples

4 Planar Arrays

5 Planar Arrays - Examples

6 Synthesis

7 Multiple Beam Arrays

8 Problems



Cylindrical and Spherical Coordinate Systems

O

ρφ

z

(ρ,φ,z)

X

Y

Z

~r = xx + yy + zz

~r = ρ cos φx + ρ sin φy + zz

~r = r sin θ cos φx + r sin θ sin φy + r cos θz,



Array Factor

ReferencePoint

AF = ∑n

An exp[jk0

(|~r| −

∣∣∣~r− ~r′n∣∣∣)]



Approximation of∣∣∣~r−~r′

∣∣∣ - Cartesian System

In Cartesian coordinate system,

~r′ = x′ x + y′ y + z′ z, and

~r = r sin θ cos φx + r sin θ sin φy + r cos θz.

So, ∣∣∣~r−~r′∣∣∣ = ∣∣r sin θ cos φx + r sin θ sin φy + r cos θz− x′ x− y′ y− z′ z

∣∣=

√(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 θ

r

≈ r(

1− 12

2x′ sin θ cos φ + 2y′ sin θ sin φ + 2z′ cos θ

r

)= [r− (x′ sin θ cos φ + y′ sin θ sin φ + z′ cos θ)]. (1)




∣∣∣ - Cylindrical System ***

In Cylindrical coordinate system,

~r′ = ρ′ cos φ′ x + ρ′ sin φ′ y + z′ z, and


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)




∣∣∣ - Spherical System ***

In Spherical coordinate system,

~r′ = r′ sin θ′ cos φ′ x + r′ sin θ′ sin φ′ y + r′ cos θ′ z, and


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∣∣∣)]

≈∑n

An exp{

jk0 [|~r| − [r− (x′n sin θ cos φ + y′n sin θ sin φ + z′n cos θ)]]}

= ∑n

An 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 you of anything?



Array Factor of a Uniformly Spaced Linear Array

N - Even

N - Odd

AF = ∑n


)= ∑

nAn exp (jkxx′n)

= ∑n

An exp (jkxna) (6)



Array Factor of a Uniformly Spaced Planar Array

AF = ∑n


)= ∑

nAn exp

(jkxx′n + jkyy′n

)= ∑

p∑

qApq exp

(jkxx′pq + jkyy′pq

)= ∑

p∑

qApq exp

[jkx

(pa +

qbtan γ

)+ jkyqb

](7)



Outline

1 Array Factor

2 Linear Arrays


4 Planar Arrays


6 Synthesis


8 Problems



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 ≤(

2π

a− k0

)⇒ k0 sin θ0 ≤

(2π

a− k0

)⇒ θ0 ≤ sin−1

(2π

ak0− 1)

⇒ θ0,max= sin−1(

2π

ak0− 1)



Optimal Spacing

kx0 ≤(

2π

a− k0

)⇒ k0 sin θ0,max ≤

(2π

a− k0

)

⇒ a ≤ 1k0

(2π

1 + sin θ0,max

)

⇒ amax=1

k0

(2π

1 + sin θ0,max

)



Beam BroadeningA

F (in

line

ar sc

ale)



Outline

1 Array Factor

2 Linear Arrays


4 Planar Arrays


6 Synthesis


8 Problems



1 Continuous Linear ArrayDipole Antenna of Arbitrary Length

E

H




Array factor of a continuous linear array antenna oriented along z axis is given as

AF (kz) =

ˆ +l/2

−l/2A (z′) exp (jkzz′) dz′

=

ˆ 0

−l/2

{I0 sin

[k0

(l2+ z′

)]}exp (jkzz′) dz′ +

ˆ +l/2

0

{I0 sin

[k0

(l2− z′

)]}exp (jkzz′) dz′ .

Also, we can prove the following integral identities (one can use Wolfram Online Integrator):

ˆsin[

k0

(l2+ z′

)]exp (jkzz′) dz′ = ejkzz′

jkz sin[

k02 (l + 2z′)

]− k0 cos

[k02 (l + 2z′)

]k2

0 − k2z

and

ˆsin[

k0

(l2− z′

)]exp (jkzz′) dz′ = ejkzz′

jkz sin[

k02 (l− 2z′)

]+ k0 cos

[k02 (l− 2z′)

]k2

0 − k2z


http://integrals.wolfram.com/index.jsp



Using the integral identities provided in the previous slide gives

ˆ 0

−l/2

{sin[

k0

(l2+ z′

)]}exp (jkzz′) dz′ =

jkz sin(

k0 l2

)− k0 cos

(k0 l2

)k2

0 − k2z

+ e−jkz l2

[k0

k20 − k2

z

]

and

ˆ +l/2

0

{sin[

k0

(l2− z′

)]}exp (jkzz′) dz′ = ejkz l

2

[k0

k20 − k2

z

]−

jkz sin(

k0 l2

)+ k0 cos

(k0 l2

)k2

0 − k2z

.

So, array factor of a finite length dipole is given as

AF = I0

(

ejkz l2 + e−jkz l

2

) [k0

k20 − k2

z

]−

2k0 cos(

k0 l2

)k2

0 − k2z

= I0

2k0 ×

cos(

kzl2

)− cos

(k0 l2

)k2

0 − k2z

. (8)




Finally, we need to multiply array factor with element pattern corresponding to a Hertzian dipoleoriented along z direction which is given readily in the below table.

Eθ ηHφ Eφ −ηHθ

Je = x′δ (x′) δ (y′) δ (z′) κ cos θ cos φ −κ sin φ

Je = y′δ (x′) δ (y′) δ (z′) κ cos θ sin φ κ cos φ

Je = z′δ (x′) δ (y′) δ (z′) −κ sin θ 0

Jm = x′δ (x′) δ (y′) δ (z′) −(

κ sin φη

)−(

κ cos θ cos φη

)Jm = y′δ (x′) δ (y′) δ (z′)

(κ cos φ

η

)−(

κ cos θ sin φη

)Jm = z′δ (x′) δ (y′) δ (z′) 0

(κ sin θ

η

)where, κ = − jηk0e−jk0r

4πr and η =√

µ0ε0

So, far-field components of a dipole are given as

Eθ = ηHφ = I0

2k0 ×

cos(

kzl2

)− cos

(k0 l2

)k2

0 − k2z

× jηk0e−jk0r

4πrsin θ (9)



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 (a=λ)Uniform Excitation



5 Discrete Linear End-fire Array (a=λ2 )

Uniform Excitation



Outline

1 Array Factor

2 Linear Arrays


4 Planar Arrays


6 Synthesis


8 Problems



Continuous Planar Array

Array placed in the xy plane:

AF(kx, ky

)=

Ä (x′ , y′) exp

(jkxx′ + jkyy′

)dx′dy′

Array placed in the yz plane:

AF(ky, kz

)=

Ä (y′ , z′) exp

(jkyy′ + jkzz′

)dy′dz′

Array placed in the xz plane:

AF (kx, kz) =

Ä (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

]



Grating Lobe Locations

VISIBLE-SPACE DISK

ARBITRARY SCANSPECIFICATION

kxa = 2µπ &kxb

tan γ+ kyb = 2νπ, where µ, ν = 0,±1,±2, · · · .



Typical Scanning Examples

PLANE PLANE

- CONTOUR

- CONTOUR



Hemispherical Scanning

NORMALTO FACE-2

BEAMDIRECTION

CIRCULARRADIATINGAPERTURE

NORMALTO FACE-1

TRIANGULARPYRAMIDAL

SECTORS

NORMAL TO FACE

1

2

3

NORMAL TO FACE

FOUR TRAPEZOIDALPYRAMIDAL

SECTORS

RECTANGULARPYRAMIDAL

SECTOR

1

2

3

4



Hemispherical Scanning ... Contd

NORMAL TO FACE

1

4

3

2



Hemispherical Scanning ... Contd

1

2

3

PLANE

- CONTOUR

1

2

3

PLANE

4

- CONTOUR



PAWE PAWS



Outline

1 Array Factor

2 Linear Arrays


4 Planar Arrays


6 Synthesis


8 Problems



1 Planar Array (a=λ, b=λ)Uniform Excitation



2 Planar Array (a=λ2 , b=λ

2 )Uniform Excitation



3 Planar Array (a=λ2 , b=λ

2 ) ... ScanUniform Excitation



Outline

1 Array Factor

2 Linear Arrays


4 Planar Arrays


6 Synthesis


8 Problems



Uniform Array - If θnull is given

Array factor of a M element linear array oriented along x axis is given as

AF (kx) =sin(

kxMa2

)sin(

kxa2

) .

So, first null position is given from the below equation:

knullx1 Ma

2= π.

If null-to-null beamwidth, 2θnull , is given, then

⇒ M =2π

knullx1 a

⇒ M=2π

k0 sin θnulla. (10)



Binomial Array

For binomial arrays, array factor is given as

AF (kx) =(

ejkxa − ejkx0a)M−1

. (11)

Reasons for choosing the above array factor are:

• array factor should be a periodic function in kx domain with a period of 2πa

• array factor should have an order of M− 1 in ejkxa domain• all the zeros should present at one single location, i.e., kx = kx0, so that array factor will not

have any side-lobes�

�One can obtain binomial array coefficients by equating (11) to the definition of array factor

AF (kx) = ∑ Amejkxma.



Dolph-Chebyshev Array

Array factor corresponding to Dolph-Chebyshev array is given as

AF (kx) = TM−1

(−c cos

kxa2

)(12)

where Chebyshev polynomial TN (x) is defined as

TN (x) =

{cos

(N cos−1 x

), |x| ≤ 1

cosh(

N cosh−1 x)

, |x| ≥ 1. (13)

The parameter c is decided by the given SLR. If the given SLR is R in linear scale, then c is decidedsuch that

TM−1 (c) = R⇒ cosh[(M− 1) cosh−1 c

]= R⇒ c = cosh

(cosh−1 R

M− 1

). (14)



Dolph-Chebyshev Array Mapping ... Even Order

AF (kx) = TM−1

(−c cos

kxa2

)



Dolph-Chebyshev Array Mapping ... Odd Order

AF (kx) = TM−1

(−c cos

kxa2

)



Dolph-Chebyshev Array ... Contd

Reasons for choosing array factor given by (12) are as given below:

• array factor should be a periodic function in kx domain with a period of 2πa

( 4πa

)for odd

(even) numbered arrays

• array factor should have an order of M− 1 in ejkxa domain

• [−c,+c] region of Chebyshev polynomial should get mapped to[0, 2π

a

]region of kx domain

as shown in the previous mapping figures.



Dolph-Chebyshev Array ... Array Factor Zeros

Array factor zeros are given as shown below:

TM−1

(−c cos

knullx a2

)= 0

⇒ cos[(M− 1) cos−1

(−c cos

knullx a2

)]= 0

⇒ (M− 1) cos−1

(−c cos

knullx,n a2

)= ± (2n− 1)

π

2, where n = ±1,±2,±3, · · ·

⇒ knullx,n a= 2 cos−1

{1c

cos[(

2n− 1M− 1

)π

2

]}. (15)

So, array factor can be written in terms of zeros as

AF (kx) = ∏(

ejkxa − ejknullx,n a

). (16)



Outline

1 Array Factor

2 Linear Arrays


4 Planar Arrays


6 Synthesis


8 Problems



Single Beam Phased Array Antenna System

Radiatingelements

Amplifiers

Phase shifters

Power combiner



Multiple Beam Phased Array Antenna System

Radiatingelements

Power combiner Power combiner



Multiple Beams with Uniform Excitation

Radiatingelements

Power combinerPower combiner



Multiple Beams - Orthogonality

AF

(in li

near

scal

e)



Multiple Beams - Orthogonality

Radiatingelements

Power dividerPower divider Power divider Power divider



FFT Algorithm



FFT Algorithm ... Contd



Example: 8 Point FFT

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

Stage 1Stage 2Stage 3



4 Beam Array

Radiatingelements

-1

-1

-1

-1



4 Beam Array ... Highlighting Butterfly Section

Radiatingelements

-1

-1

-1

-1



Butterfly Implementation in Analog Domain

Hybrid

Hybrid



So, Finally 4 Beam Analog Array is ...

Radiatingelements

Hybrid

Hybrid

Hybrid

Hybrid



Outline

1 Array Factor

2 Linear Arrays


4 Planar Arrays


6 Synthesis


8 Problems



Linear Arrays - I

1 Calculate the array factor of a continuous linear array oriented along z′ axis and having anaperture distribution

A (z′) =

{1, |z′ | ≤ L

20, elsewhere

.

Also calculate (a) the directions in which array factor exhibits zero radiations (i.e., nulldirections), and (b) null-to-null beamwidth.

2 Calculate the array factor of a discrete uniformly spaced linear array oriented along x′ axisand having uniform aperture distribution. Assume that the array is having M number ofelements (M can be either even or odd) and the spacing between any two consecutiveelements is a. Also calculate (a) the directions in which array factor exhibits zero radiations(i.e., null directions), and (b) null-to-null beamwidth.

3 Prove that array factor of any uniformly spaced odd numbered array, located symmetrically

with respect to origin will be periodic in kx domain with a periodicity of 2πa . Also prove that

array factor of any even numbered array located symmetrically with respect to origin will be

periodic in kx domain with a periodicity of 4πa .



Linear Arrays - II

1 For a given linear array oriented along x axis and having an uniform spacing of 15mm,calculate the progressive phase shift required to scan the beam from broad-side direction(i.e., θ = 0◦) to (a) θ0 = 30◦ (b) θ0 = 90◦ . Assume that the operating frequency of the array is10 GHz.

2 For a given linear array oriented along x axis and having an uniform spacing of a = 0.75λ,calculate the maximum scanning that can be done so that grating-lobe doesn’t enter into thevisible-space.

3 If the main beam corresponding to a given linear array has to be scanned to a maximum

angle of θ0 = 40◦ , then what should be the maximum possible uniform spacing between

consecutive elements.



Planar Arrays

1 Calculate the array factor of a discrete uniformly spaced planar array (assume rectangularlattice) placed in the x′y′ plane and having uniform aperture distribution along both x′ and y′axes. Assume that number of elements and spacing along x’ axis are M and a, respectively.Similarly, Assume that number of elements and spacing along y’ axis are N and b,respectively.

2 For a given planar array in xy axis and having a =15mm and b =20mm, calculate the

progressive phase shifts required to scan the beam from broad-side direction (i.e., θ = 0◦) to

(a) θ0 = 45◦ and φ0 = 45◦ . Assume that the operating frequency of the array is 10 GHz.



Array Synthesis

1 Calculate the number of elements required to obtain an array factor with 10◦ null-to-nullbeamwidth. Assume that the array elements are uniformly excited and a = λ/2.

2 Design a 5-element Binomial array having all the zeros at θ0 = 30◦ and having an uniformspacing of a = λ/2.

3 Design a 5-element Dolph-Chebyshev array having SLR of 30 dB and spacing a = λ/2.



Multiple Beam Arrays

1 Develop the 16 point FFT algorithm and construct it’s signal flow graph.

2 For an 8 element array forming 8 beams (assume that the beams are symmetric with respect

to the broadside direction), calculate the progressive phase shifts required for forming the

beams. Also calculate the corresponding excitation values for each array element when all

the beams are simultaneously excited.



References

1 A. K. Bhattacharyya, Phased Array Antennas, Floquet analysis, Synthesis, BFNs, and Active ArraySystems. Hoboken, NJ: John Willey, 2006.

2 Constantine A. Balanis, Antenna Theory: Analysis and Design, , Third Edition, John Wiley &

Sons, 2005.


array antennas - theory & design - arraytool · pdf file03-10-2014 · array...

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