Chapter 6 : Antenna Arrays

1

Chapter 6 : Antenna Arrays

• Introduction

• Two-Element Array

• N-element Linear Array: Uniform

Amplitude and Spacing

• N-element Linear Array: Directivity

• N-element Linear Array: Uniform spacing,

Non-uniform Amplitude

• Planar Array

2

Antenna Array: Introduction

• Array is an assembly of antenna elements

arranged in an orderly fashion. The

elements are usually identical.

• Why array? When high gain and/or

narrow beam are required:

– Single element -> Wide beam (low directivity)

– Increasing size -> difficult to build and

expensive

– Useful especially when the element gain is low.

3

Antenna Array: Introduction (2)

• Advantages

– Higher directivity

– Narrower beam

– Lower sidelobes

– Electronic steerable beam

• Types

– Fix direction

– Steerable : Mechanical or Electronic (phased

arrays)

4

Antenna Array: Introduction (3)

• In an array of identical elements, there

are in general five controls that can be

used to shape the overall pattern of the

antenna:

1. Geometrical configuration (linear, circular, etc.)

2. Relative displacement between elements

3. Excitation amplitude of individual elements

4. Excitation phase of individual elements

5. Relative pattern of individual elements

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5

ExamplesVery Large Antenna (VLA)

Airborne Warning and

Control System (AWACS)

6

Two-element Array

• Consider two-element array of horizontal

infinitesimal dipoles (assume no coupling between

elements)

Far-field observationTwo infinitesimal dipoles

7

Two-element Array (2)

cos4

ˆ0

r

elIjk

jkr

E

Recall the far-zone electric field of horizontal infinitesimal dipole in the y-z plane

2

2

21

1

121 cos4

cos4

ˆ21

r

eI

r

eIljk

jkrjkr

EEE

Thus the total electric field becomes:

8

Two-Element Array (3)

cos2

1

drr cos

22

drr rrr 21

Using the far-field approximation

2/cos

2

2/cos

14

cosˆ

jkdjkd

jkr

eIeIr

eljk

E

The total field becomes:

)2

cos2

cos(24

cosˆ0

d

kr

elIjk

jkr

E

difference phase:,; If 2/

02

2/

01 jjeIIeII

)factor(AF)(array factor)(element field total

9

Electric Field Pattern

4/,0 d

10

Electric Field Pattern (2)

4/,2/ d

11

Electric Field Pattern (2)

4/,2/ d

12

Quiz

• Find the far-zone electric field of a two-

element array of infinitesimal circular

loops. Assume that the loops are parallel

to the x-y plane and the two elements are

aligned along the z axis.

• (i) I1=I0, I2=I0, d = λλλλ/4

• (ii) I1=I0, I2=I0, d = λλλλ/2

• (iii) I1=I0, I2=-I0, d = λλλλ/2

• (iv) I1=I0, I2=jI0, d = λλλλ/2

13

N-element Linear Array: Uniform

amplitude & spacing

cos where

1AF

1

)1(

)cos)(1(

)cos(2)cos(

kd

e

e

ee

N

n

nj

kdNj

kdjkdj

L

If the amplitude and spacing are both uniform, the array factor

becomes

14


amplitude & spacing (2)

2sin

2sin

1

1AF

2

1

22

222

1

N

e

ee

eee

e

e

Nj

jj

Nj

Nj

Nj

j

jNthus

If the reference point is the physical center of the array

2

2sin

2sin

2sin

AFsmall:

NN

15



2sinc

2

2sin

2sin

2sin

(AF)small:

n

N

N

N

N

N

KK ,3,2,;,3,2,1

2

2cos

20

2sin

1

NNNnn

N

n

d

nNN

n

n

Normalized AF

Nulls

K,2,1,0

22

cos2

1

m

md

m mm

Maxima

16

Normalized Array Factor

17

Sinc function plot

0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1|sin(x)/x|

18



Nd

Nd

N

N

h

h

782.2

2sin

2

782.2

2cos

391.122

1

2sin

2sin

1

1

HPBW (symmetrical case)

3-dB point

Secondary

Maxima

hmh 2

K,3,2,1;12

2cos

2

12

21

2sin

1

sN

s

d

sNN

s

ss

19



First sidelobe

Nd

N

s

s

3

2cos

2

3

2

1

dB 46.13212.03

2

2

2sin

(AF)

1,

n

ss

N

N

First sidelobe level

20

Example

#2

#1

#3

d

d

z

y

02 II 2/

01

jeII

03 II

A 3-element array of

isotropic sources has the

phase and amplitude

relationships shown. The

spacing between elements

is d=λ/2.

(a)Find the array factor.

(b)Find all the nulls.

21

Broadside Array0cos 2/ kd

4/,10 dN

22

Grating Lobes

nn

kdn

nd

2cos2

cos

,0

,3,2,10,

K

maxd

No grating lobes

Should avoid d=nλ because

dN ,10

23

24

25

Ordinary End-fire Array

kdkdkd cos

4/

10

d

N

kdkdkd 0cos

26

Ordinary End-fire Array (2)

4/,10 dN

27

Grating Lobes

•If d=λ/2, end-fire radiation exists simultaneously in both directions.

•If d = nλ, also broad-side radiation.

•To avoid grating lobes,

2max

d

28

29

30

Phased (Scanning) Array

00 coscoscos0

kdkdkd

4/,10 dN

31

Phased (Scanning) Array (2)

NkdNkd

Nkd

dNkd

dh

782.2coscos

782.2coscos

782.2cos

2cos

782.2cos

2cos

0

1

0

1

0

1

0

1

HPBW

dLdLh

443.0coscos443.0coscos 0

1

0

1

Since N = (L+d)/d,

Not valid for end-fire arrays, i.e., θ0=0,π.

32

Hansen-Woodyard End-fire

Array

0for 92.2

0

N

kdN

kd

0for 92.2

Nkd

Nkd

|cos|||;|cos|||;0For 00 kdN

kd

Additional Conditions:

00 |cos|||;|cos|||;For kdN

kd

which yields

44

1

large:

NN

Nd

33

Hansen-Woodyard End-fire Array (2)

34

Hansen-Woodyard End-fire Array (3)

35

36

37

N-Element Array: Directivity

• Broadside Array

cos;

2

2sin

2sin

2sin

(AF)small:

n kd

N

N

N

N

cos

2;

)sin(

2

cos

cos2

sin

)AF(),(

2

2

2kd

NZ

Z

Z

kdN

kdN

U n

Recall that AF for broadside arrays is given by

The radiation intensity then becomes:

Clearly, the maximum Umax=1 at θ=π/2

38

N-Element Array: Directivity (2)

• Broadside Array (cont’d)

The “average” radiation intensity can be obtained from

Using

0

2

2

0 0

2

0

sin

cos2

)cos2

sin(

2

1

sin)sin(

4

1),(

4

1

4

d

kdN

kdN

ddZ

ZdU

PU rad

dkdN

dZkdN

Z sin2

;cos2

2/

2/

2/

2/

22

0

)sin(1)sin(1 Nkd

Nkd

Nkd

NkddZ

Z

Z

NkddZ

Z

Z

NkdU

39


• Broadside Array (cont’d)

dZZ

Z

NkddZ

Z

Z

NkdU

Nkd

Nkd

22/

2/

2

0

)sin(1)sin(1

For a large array (Nkd/2 -> large),

The directivity is then given by

d

NNkd

U

UD 2

0

max0

dZ

Z

Z2

)sin(

NkdU

0

Ld

d

LdND

dLdNL2122

)1(0

Using L=(N-1)d

Since

40


• Ordinary end-fire Array

)1(cos;

2

2sin

2sin

2sin

(AF)small:

n

kd

N

N

N

N

)1(cos2

;)sin(

2

)1(cos

)1(cos2

sin

)AF(),(

2

2

2

kd

NZ

Z

Z

kdN

kdN

U n

Recall that AF for ordinary end-fire arrays (θ=0) is given by

The radiation intensity then becomes:

Clearly, the maximum Umax=1 at θ=0

41


• Ordinary end-fire Array (cont’d)

The “average” radiation intensity can be obtained from

Using

0

2

2

0 0

2

0

sin

)1(cos2

))1(cos2

sin(

2

1

sin)sin(

4

1),(

4

1

4

d

kdN

kdN

ddZ

ZdU

PU rad

dkdN

dZkdN

Z sin2

);1(cos2

Nkd Nkd

dZZ

Z

NkddZ

Z

Z

NkdU

0 0

22

0

)sin(1)sin(1

42


• Ordinary end-fire Array (cont’d)

0

2

0

2

0

)sin(1)sin(1dZ

Z

Z

NkddZ

Z

Z

NkdU

Nkd

For a large array (Nkd -> large),


d

NNkd

U

UD 4

2

0

max0

NkdU

20

Ld

d

LdND

dLdNL4144

)1(0

Using L=(N-1)d

Thus

43


• Hansen-Woodyard end-fire Array

NkdNkdNkdU

2554.0

871.08515.1

2

22

12

0

For a large array (Nkd -> large),


d

NNkd

U

UD 4805.1

2

554.0

1

0

max0

Ld

d

LD

dLdNL4805.114805.1

)1(0

Using L=(N-1)d

44

45

Example

• Design an 18-element uniform linear array with a spacing of λλλλ/4 between elements. Assume that the array is aligned along the z-axis.

a) Find the array factor for the broadside array case.

b) Find the first null and sidelobe locations of a).

c) Find the phase shift such that the maximum of the array factor is at θ0=45°.

d) Find the first null and sidelobe locations of c).

46

18-element AF (broadside array)

0 20 40 60 80 100 120 140 160 180−50

−45

−40

−35

−30

−25

−20

−15

−10

−5

0

θ [Degree]

|(A

F) n

| [d

B]

18−element array factor (Broadside scan

47

18-element AF (scan array)

0 20 40 60 80 100 120 140 160 180−50

−45

−40

−35

−30

−25

−20

−15

−10

−5

0

θ [Degree]

|(A

F) n

| [d

B]

18−element array factor

48

Quiz

Find the array factor of the 3-element array of isotropic

sources shown below. The spacing between elements

is d=λ/4 and I1 = 1, I2 = -j2, I3= -1.

49

N-element Array: Non-uniform

amplitude, uniform spacing

• Uniform amplitude -> High sidelobe

• Two popular distributions:

– Binomial (maximally flat)

– Tschebysheff (equiripple)

• HPBW: Uniform<Tschebysheff<Binomial

• Sidelobe level:

Binomial<Tschebysheff<Uniform

50

51

52

53

54

55

56

57

58

59

Planar Array

• Linear Array = one-dimensional array,

i.e., can scan the beam only in one plane.

• In order to be able to scan the beam in

any direction, two-dimensional arrays are

needed. Geometries can be planar, circle,

cylindrical, spherical and so on.

60

Planar Array (2)

61

Array Factor

M

m

kdmj

mxxeI

1

)cossin)(1(

1AF

ynxm

kdnjM

m

kdmj

m

N

n

n

SS

eeII yyxx

)sinsin)(1(

1

)cossin)(1(

1

1

1AF

yyy

xxx

y

y

x

x

kd

kd

N

N

M

M

sinsin

;cossin

;

2sin

2sin

2sin

2sin

(AF)n

AF for each linear array along x-axis:

AF for the entire planar array:

For uniform excitation,

i.e., |Im1I1n|=I0,

62

Planar Array Example

5;4/,6/

);22/(;2/

00

NM

dd yxyx

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Chapter 6 : Antenna Arrays