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