Chapter 4. Antenna Synthesis 4.1 Basic principle for antenna synthesis 4.2 Line source synthesis (Fourier transform, woodward-lanson sampling) 4.3 Linear.

Chapter 4. Antenna Synthesis

4.1 Basic principle for antenna synthesis

4.2 Line source synthesis (Fourier transform, woodward-lanson sampling)

4.3 Linear array synthesis (Fourier series, woodward-lanson sampling)

4.4 Low sidelobe synthesis (Dolph-Chebyshev, Taylor)

Synthesis Problems

Given affordable SLL, No. of elements, how to synthesize?

Ideal case: narrow beam, constant side-lobe envelope

Approaches: Dolph-Chebyshev, Taylor Line Source….

Secret behind: to synthesize a polynomial like pattern….

Dolph-Chebyshev Linear Array The Chebyshev polynomials:

)coshcosh(

)coscos(

)coshcosh()1(

)(1

1

1

xn

xn

xn

xT

n

n

1x11 x

1x

188)(

34)(

12)(

)(

1)(

244

33

22

1

0

xxxT

xxxT

xxT

xxT

xT )()(2)( 11 xTxxTxT nnn

Property used:

)][cos()cos( nTn

Chebyshev Polyminals

Symmetrically Excited Array

2/

1

2/)1(

10

]2

)12cos[(2

)cos(2

)(P

mm

P

mm

mi

mii

f

,mm ii

, P odd

, P even

)][cos()cos( nTn Property:

)(f is P-1 th polynomial of )2

cos(

)2

cos(0

xx Let

Choose appropriate to match the coefficients to those in Chebyshev polynomial, we obtain,

mi

)]2

cos([)( 01

xTf P

Chebyshev Polynomial Example

Synthesis StandardsSLL=-20log R (dB)

so ]cosh)1cosh[()( 01

01 xPxTR P

]cosh1

1cosh( 1

0 RP

x

1cos

1

1

optd

1

cos1

2

1

optd

Optimum spacing,

Broadside: Endfire:

)]1ln(1

1cosh[ 2

RR

P, where

Beamwidth and Directivity

In general, dHP h

1cos2 (broadside)

)1(cos 1

dHP h

(endfire)

, where

]cosh1

1cosh[

]2

cosh1

1cosh[

cos21

1

1

RP

RP

n

An approximation for the broadside:L

RHP

)2ln(

1

Beambroadening factor: )2ln(637.0866.0

)2ln(1

RR

bHP

HPR

RD

2

2

1

2

Directivity:

Example No.1Five element array (P=5), -20dB Side-Lobe, Half-Wavelength Spaced Dolph-Chebyshev Array

Synthesized Array Factor

Example No.2Optimum Spaced 10-Element, -30dB Side Lobe,Dolph-Chebyshev Endfire Array

Taylor Line Source Method)()( 22

02 xaxTxP NN

where a is a constant and, wLL

x

cos

So the pattern maximum is, RxTwP NN )()0( 02

Transform:

For the side-lobe region:

)],(coscos[)( 220

12 xaxNxP N

Main beam region:

)],(coshcosh[)( 220

12 xaxNxP N

1220 xax

1220 xax

Transformed Chebyshev Polynomial

Basic Properties of ZerosZeros of the transformed function in the side-lobes are given by:

,2

)12(cos

10 N

nx

axn

1n

For the main beam, )],cosh1

cosh[ 10 R

Nx

Introduce A so that, RA 1cosh1

),cosh(0 N

Ax

For large N,

20 )/(

2

11)/cosh( NANAx

2]2

)12([

2

11

2

)12(cos

N

n

N

n

NnANa

xn ,)2

1(

2

1 22

and

Selection of Constant aa is selected so that the first zero location remains fixed as N increases,

Na

2

22 )2

1( nAxn

Then

The pattern should have the following format,

1

222

1

22 ])2

1([)(

nnn nAxxx

Ideal Pattern FactorNormalizing the pattern to unity at x=0,

A

Ax

n

A

n

Ax

xf

n

n

cosh

)cos(

)21

(1

)21

(1

)(22

1 2

2

1 2

22

R

AwLwf

22)/(cos)(

A

wLAwf

cosh

)/(cosh)(

22

(side lobes)

(main beam)

Realistic Pattern Factor

n

nAxn

22 )2

1( nn 1

nn 22 )21

(

nA

nwhere

1

12

2

)/(1

)/(1sin),,(

n

n

n

nx

xx

x

xnAxf

1

12

2

)/(1

)/(1

/

)/sin(),,(

n

n

n

nLw

ww

Lw

LwnAwf

In format of w,

nL

nALwn

22 )2

1( nn 1

nn

Woodward-Lawson Equivalent

,)( )/(2

n

nsLjnea

Lsi

2L

s

,)](Sa[)(

n

n nL

wL

awf

nL

wsn

),,()( nAnfwwfa snn 0na nn for

,)](Sa[),,()(1

1

n

nn

snww

LnAnfwf

1

1

)2cos(),,(21)(n

n

nsL

nAnfL

si

The source current is

0

)1()1)(1(

])!1[(

),,(

1

12

22 n

m mx

n

nnnn

n

nAnfnn

nn

Beamwidth and Directivity

2/1

2121 )2

(cosh)(cosh2

R

RL

HPiw

2/1

21211 )2

(cosh)(coshsin2R

RL

HPw

(ideal Taylor)

(realistic Taylor)

iww HPHP

Example

A 10-Wavelength Taylor Line Source with –25 dB side lobesand 5n

Pattern and Current Source