Chapter 7 : Antenna Synthesis

1

Chapter 7 : Antenna Synthesis

• Continuous sources vs. Discrete sources

• Schelkunoff polynomial method

• Fourier transform method

• Woodward-Lawson method

• Triangular, cosine and cosine-squared

amplitude distributions

2

Continuous sourcesRecall the array factor

If the number of elements increases in a fixed-length array,

the source approaches a continuous distribution.

In the limit, the array factor becomes the space factor, i.e.,

N

n

nj

n kdea1

)1( cos; AF

)'()1( )'(zj

n

nj

nnezIea

The radiation characteristics of continuous sources can

be approximated by discrete-element arrays, i.e.,

2/

2/

)]'(cos'[')'(

l

l

zkzj

n dzezI nSF

3

Schelkunoff polynomial

method

)cos( kdjjeejyxz

The array factor for an N-element, equally spaced, non-

uniform amplitude, and progressive phase excitation is given

by

Let

N

n

nj

n kdea1

)1( cos; AF

N

n

N

N

n

n zazaaza1

1

21

1LAF

which is a polynomial of degree (N-1).

4


method (2)

1|||| zezzj

Thus

Note that

)())(( 121 NN zzzzzza LAF

cos2

cos dkd

z is on a unit circle.

where z1,z2,…,zN-1 are the roots. The magnitude then

becomes

|||||||||| 121 NN zzzzzza LAF


method (3)

5

VR=Visible Region

IR=Invisible Region

β = 0


method (4)

6

VR=Visible Region

IR=Invisible Region

β = π/4


method (5)

7


method : Example

8

Design a linear array with a spacing between the elements

of d=λ/4 such that it has zeros at θ=0,π/2,π. Determine the

number of elements, their excitation, and plot the derived

pattern.


method : Example pattern

9

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]

4−element array factor

Fourier Transform Method

10

The normalized space factor for a continuous line-source

distribution of length l can be given by

k

kkk

dzezIdzezI

zz

l

l

zjl

l

zkkj z

1

2/

2/

'2/

2/

)')cos(

coscos

')'(')'()(SF

where kz is the excitation phase constant of the source. If

I(z’)=I0/l,

k

kkl

k

kkl

Iz

z

cos2

cos2

sin

)( 0SF

Fourier Transform Method (2)

11

Since the current distribution extends only over -l/2≤z’≤ l/2,

')'()()( '

dzezIzj SFSF

The approximate source distribution Ia (z’) is given by

dedezI

zjzj '' )(2

1)(

2

1)'( SFSF

The current distribution can then be given by

elsewhere0

2/'2/)(2

1)'(

)'(

'lzldezI

zI

zj

a

SF

2/

2/

' ')'()()(l

l

zj

aaa dzezI SFSFThus

Fourier Transform Method : Example 7.2

12

Determine the current distribution and the approximate radiation

pattern of a line source placed along the z-axis whose desired

radiation pattern is symmetrical about θ=π/2, and it is given by

elsewhere0

4/34/1)(

SF

Fourier Transform Method :

Example


Linear Array

14

For an odd number of elements, the array factor is given by

M

Mm

jm

mea )()( AFAF

where

For an even number of elements,

1

2

12

12

12

'

mMdm

Mmdm

zm

Mmmdzm ,,2,1,0,'K

Elements’ locations

M

m

mj

m

Mm

mj

m eaea1

]2/)12[(1

]2/)12[()()( AFAF

coskd

Even-number

Odd-number


Linear Array (2)

15

For an odd number of elements, the excitation coefficients can

be obtained by

MmM

dedeT

ajmjm

T

Tm

)(2

1)(

1 2/

2/AFAF

where

For an even number of elements,

coskd

Mmde

mMde

amj

mj

m

1)(2

1

1)(2

1

]2/)12[(

]2/)12[(

AF

AF


Example

16

Same as Example 7.2 with d = λ/2; non-zero only

2/cos2/ kd

4/34/

thus

therefore

2

2sin

2

1

2

1 2/

2/

m

m

deajm

m

0101.00588.0

010.00518.02170.0

0455.00895.03582.0

0496.00578.00.1

73

1062

951

840

aa

aaa

aaa

aaa

Result


Example

Quiz

A 5-element uniform linear array with a

spacing of λλλλ between elements is designed

to scan at θθθθ=ππππ/3. Assume that the array is

aligned along the z-axis.

a) Find the array factor

b) Find the angle of the grating lobe.

c) Find the condition such that there exists no

grating lobe.

Quiz solution

0 20 40 60 80 100 120 140 160 1800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

θ [Degree]

|(A

F) n

| [d

B]

5−element array factor

d=λ

d=5λ/8

d=λ/2

Woodward-Lawson Method• Sampling the desired pattern at various discrete

locations.

• Use composing function of the forms:

as the field of each pattern sample

• The synthesized pattern is represented by a finite

sum of composing functions.

• The total excitation is a sum of space harmonics.

mmmmmm NNbb sin/)sin(or /)sin(

Woodward-Lawson Method:

Line-source

2/'2/)'(cos'

lzlel

bzi mjkzm

m

M

Mm

jkz

mmeb

lzI

cos'1)'(

number) odd 12(for ,,2,1,0

number)even 2(for ,,2,1 where

MMm

MMm

K

K

m

m

mm kl

kl

bs

coscos2

coscos2

sin

)(

Let the source be represented by a sum of the following constant current source of length l.

Then the current source can be given by

The field pattern of each current source is given by

Composing function


Line-source (2)

dmmb )(SF

lkz lz

2' '||

For an odd number samples, the total pattern becomes

bm can be obtained from the value at the sample points θm, i.e.,

In order to satisfy the periodicity of 2π and faithfully reconstruct the desired pattern,

M

Mmm

m

m kl

kl

b

coscos2

coscos2

sin

)(SF


Line-source (3)

samples oddfor ,2,1,0,cos K

ml

mmm

Thus the location of each sample is given by

Therefore, M should be the closest integer to M=l/λ.

sampleseven for

,2,1,2

12

2

12

,2,1,2

12

2

12

cos

K

K

ml

mm

ml

mm

m


Example

5,,2,1,0),2.0(cos)(cos 11 Kmmmm

Same as Example 7.2; for l = 5λ.Since l = 5λ, M = 5 and ∆ = 0.2.

m θθθθm bmm θθθθm bm

0 90 1

1 78.46 1 -1 101.54 1

2 66.42 1 -2 113.58 1

3 53.13 1 -3 126.87 1

4 36.87 0 -4 143.13 0

5 0 0 -5 180 0

Woodward-Lawson Method: Example (2)

Composing functions for line-source (l = 5λ)


Linear array

m

m

mm kdN

kdN

bf

coscos2

sin

coscos2

sin

)(

The pattern of each sample (uniform array) can be written as (assuming l = Nd)

Composing function

M

Mmm

m

m kdN

kdN

b

coscos2

sin

coscos2

sin

)(AF

dmmb )(AF

For an odd number elements, the array factor becomes

bm can be obtained from the value at the sample points θm, i.e.,


Linear array (2)

sampleseven for

,2,1,2

12

2

12

,2,1,2

12

2

12

cos

K

K

mNd

mm

mNd

mm

m

samples oddfor ,2,1,0,cos K

mNd

mmm

The location of each sample is given by

The normalized excitation coefficient of each element is given by

M

Mm

zjk

mnmneb

Nza

cos1)'(


Example

Element number

n

Position

z’n

Coefficient

an

±1 ±0.25λλλλ 0.5696

±2 ± 0.75λλλλ -0.0345

±3 ± 1.25λλλλ -0.1001

±4 ± 1.75λλλλ 0.1108

±5 ± 2.25λλλλ -0.0460

Same as Example 7.2; for N=10 and d = λ/2.

The coefficients can be found to be

To obtain the normalized amplitude pattern of unity at θ=π/2,

the array factor has been divided by 4998.0na


Example

0 20 40 60 80 100 120 140 160 1800

0.2

0.4

0.6

0.8

1

1.2

1.4

θ [Degree]

No

rma

lize

d m

ag

nitu

de

Line−source

Linear Array (N=10,d=λ/2)

Triangular, cosine, cosine

squared distributions

32

Mutual Coupling

• Consider two antennas

2221212

2121111

IZIZV

IZIZV

2

1

2221

1211

2

1

I

I

ZZ

ZZ

V

V

2112 ZZ If reciprocal,

Equivalent Circuit

Two-port NetworkT-Network Equivalent Circuit

Mutual Coupling: 2 antennas

01

111

2

I

I

VZ

02

112

1

I

I

VZ

01

221

2

I

I

VZ

02

222

1

I

I

VZ

impedancepoint driving

active

21

ddZ

1

21211

1

11

I

IZZ

I

VZ d

2112 ZZ

for reciprocal networks

:, 2211 ZZ

Input impedance

2

12122

2

22

I

IZZ

I

VZ d

2

12

1

21 on depends ;on depends :Note

I

IZ

I

IZ dd

Mutual Coupling: 2 antennas (2)

• As I1 and I2 change, the driving point impedance changes.

• In a uniform array, the phase of I1 and I2

is changed to scan the beam.

• As the beam is scanned, the driving port impedance in each antenna changes.

• In general, Z11,Z12=Z21,Z22 can be calculated using numerical techniques.

• For some special cases, they can be calculated analytically.

Mutual Coupling: N antennas

NNNNN

N

N

N I

I

I

ZZZ

ZZZ

ZZZ

V

V

V

M

L

MOMM

L

L

M

2

1

21

22221

11211

2

1 IZV

jkIj

iij

k

I

VZ

,0

In an N-element uniform array )1(

0

nj

n eII

N

n

nj

n

Nj

N

j

d

eZ

eZeZZI

VZ

1

)1(

1

)1(

11211

1

11

L

changes. as changes :Note 1 dZ

Mutual Coupling: 2 dipoles

Mutual Coupling: 2 dipoles (2)

Chapter 7 : Antenna Synthesis

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