N Element Array

θ

d

d

1 1 max

2 2 max

3 3 max

4 4 max

N N max

22

1 2 max

v Mp t Mp cos k 0 t

v Mp t Mp cos k x t

v Mp t Mp cos k 2 x t

v Mp t Mp cos k 3 x t

v Mp t Mp cos k (N 1) x t

output v v Mp cos t cos t cos 2 t cos 3 t cos (N 1) t

where

k x kdsin

Phasor Addition

A

G

M

Ba

δ/2

δ

nδ/2

nδ

δ=kdsinθ

TOT 0

TOT 0

TOT

cos(k 0 t) cos(k dsin t)v Mp

cos(k 2dsin t) ...cos(k (n 1)dsin t)

cos( t) cos( t ) cos( t 2 ) ...v v

cos( t (n 1) )

v A cos( t )

0

2

V( ) nR sin Where R is distance AP

2 2

sin n2V( ) nV

n sin2

n dsin sin

b( )d

n sin sin

Directivity Index for an n-Element Array

n 1

1

nDI 10log

2 dn sin

21

2 dn

Linear Array

2 2 2n d L L

sin sin sin sin sin sinb( )

d Ld n sin sinn sin sin

Lsinn , n 1,2,3,...

Nulls:

2

0

1 2LDI 10log 10log

b cos d

Piston Array

D

2

1

Dsin2J

bDsin

5 10 15 20

-0.2

0.2

0.4

0.6

J1(Dsin)

Dsin

J1(Dsin) has zero crossings (nulls) at Dsin = 3.83, 7.02, 10.17, 13.32, 16.47, ....J1(Dsin) has extremes (side lobes) at Dsin = 1.84, 5.33, 8.54, 11.71, 14.86, ....

Bessel Function

sincos 2 d

2

sin

sinsin

L

L2

1

sin

sin2

D

DJ

d

d

2

2sin1

2log10

Lfor

2log10

L

Dfor log102

D

... 5, 3, 1, m2

sin

d

m

... 3, 2, 1, m

sin

L

m

0sinJ of roots

... 4.24, 3.24, 2.23, 1.22, z

sin

1

D

Dz

.... 3 2, 1, ,0

sin

md

m

... 4.48, 3.47, 2.46, 1.43, y where

sin

sinsintan

Ly

LL

.... 3.70, 2.68, 1.64, wwhere

sin

D

w

sin4

1,3,5,7,...

hp

n

dn

Lhp

442.0sin Dhp

51.0sin

2-element array continuous line array circular piston

defining parameters element separation distance – d array length – L array diameter - D

beam pattern functionb() =

directivity indexDI

null anglesb() = 0

null

side lobesb()=1max

half power angles b()=0.5

hp

BW=2hp

(only for beam about array axis)

2 4 6 8

0.2

0.4

0.6

0.8

1

Beam width of a piston array

2

1

Dsin2J

b 0.5Dsin

b

Dsin

Dsin1.6

1.6

sin .51D D