chapter 4. antenna synthesis 4.1 basic principle for antenna synthesis 4.2 line source synthesis...
TRANSCRIPT
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