Download - Butler Matrix
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-Har
ish
Raj
ago
pal
an
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O
bje
ctiv
e
E
lem
ents
of
Butl
er m
atri
x
S
tudy o
f 2x2, 4x4 a
nd 8
x8 b
utl
er m
atri
ces
T
esti
ng a
nd r
esult
s fo
r 4x4
C
oncl
usi
on a
nd f
utu
re s
cope
![Page 3: Butler Matrix](https://reader034.vdocuments.mx/reader034/viewer/2022050704/54f895f44a79597b198b517a/html5/thumbnails/3.jpg)
Antenna Beam Scanning Methods
Mechanical Scanning Electronic Scanning
Phase Scanning Frequency Scanning
Series Feed Parallel Feed
(Blass m
atrix) (Butler matrix)
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Principle of Butler matrix
Inci
dent
wav
efro
nt
Ant
enna
s
0
-90
-90
0
1
2
1
2
1’
2
’
1
’
2’
BE
AM
B
EA
M
LE
FT
RIG
HT
I 1= A
1e j0
------(1)
I 2 =A2e jπ
/2 ------(2)
and A1 = A
2 -------(3)
I 1’= I1+ I2ejπ/2
or I1’= A
1ej0+ A
2ejπ-------(4)
I 2’= I1ejπ/2+ I2,
I 2’= A
1ejπ/2+A2ejπ/2,
Or I 2’= (A1 + A
2 )ejπ/2 -------(5)
A1and A
2
amplitudes of antenna currents
I 1and I2
Input antenna currents I1’and I2’
output antenna currents
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Characteristics of Butler matrix
-Number of beams = Number of antenna elements =N
-Number of Hybrid rings = N/2log2N
-Number of Phase shifters =
N/2 (log2N-1)
-Low insertion loss
-Uniform
antenna array illumination
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4 x 4 Butler matrix
1L = A1 ∠
450 + A2 ∠
900 + A3 ∠
1350+ A4 ∠
1800
2R = A1 ∠
1350+ A2 ∠
00 + A3 ∠
2250+ A4 ∠
900
2L = A1 ∠
900+ A2 ∠
2250+ A3 ∠
00+ A4 ∠
1350
1R = A1 ∠
1800+ A2 ∠
1350+A3 ∠
900+ A4 ∠
450
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2R
1R
1L
2L
=
e-j3/4π
e-j0π
e-j5/4π
e-jπ/2
e-jπ
e-j3/4π
e-jπ/2
e-jπ/4
e-jπ/4
e-jπ/2
e-j3/4π
e-jπ
e-jπ/2
e-j5/4π
e-j0π
e-j3/4π
A1
A2
A3
A4
Phase matrix of 4 x 4 Butler matrix
2R
0 -
135 9
0-4
5
A1
1R
=0 -
45 -9
0-1
35
A2
1L
0
45 9
0135
A3
2L
0
135 -9
045
A4
Phase progression
![Page 8: Butler Matrix](https://reader034.vdocuments.mx/reader034/viewer/2022050704/54f895f44a79597b198b517a/html5/thumbnails/8.jpg)
8 x 8 Butler matrix
1 2
3 4 5 6
7 8
9
Equation for 1L port.
At point 1: A1 + A5∠90
At point 2: A2 + A6∠90
At point 3: A1∠45 + A5∠135
At point 4: A3 + A7∠90
At point 5: A2∠45 + A6∠135
At point 6: A4 + A8∠90
At point 7: A1∠45 + A5∠90 + A3∠90
+A7∠180
At point 8: A2∠45 + A6∠135 + A4∠90
+A8∠180
At point 9: A1∠112.5 + A5∠157.5
+A3∠157.5 + A7∠247.5
At port 1L: A1∠112.5 + A5∠157.5 + A3∠157.5 + A7∠247.5 + A2∠135 + A6∠225 + A4∠180 + A8∠270
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Phase m
atrix of 8 x 8 Butler matrix
Phase progression
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Beam pattern for 8x8 Butler matrix
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Components of Butler matrix
•Qudrature hybrid
•Fixed phase shifter
•Helical antenna
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Qudrature Hybrid
Branch line coupler
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MicrostripLines
Basic Structure
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Desig
n O
f H
ybrid (B
ranchline C
ouple
r)Frequency = 1GHz Wavelength(in air) = 30 cm.
Substrate used for PCB manufacturing is FR-4 Glass Epoxy.
For FR-4 board, Relative dielectric constant (εr) = 4.4
The height of the dielectric (d) = 1.6 mm
Characteristic impedance Zo = 50 Ω
Zo1 = Zo/√2 = 35.35 Ω
Formulae:
1. E
ffec
tive
Die
lect
ric
const
ant(εe) = εr+1+ εr-1 (1 + 12 d / W
)-1/2
2
2
2 . W
/d = 2/π[B –1 –ln (2B –1) + (εr –1)/2εr ln (B –1) + 0.39 -0.61/εr]. …
W/d > 2
where
B = 377π
2Zo(εr)
1/2
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Calculations
For 50 Ω
B = 5.646312
W/d = 1.91335
W = 3.06136 m
m
ε eff = 3.33024
λmicrostrip = λo/√εeff = 16.43886 cm
Length of track = λmicrostrip/4 = 4.1097 cm
For 35.35 Ω
B = 7.98629
W/d = 3.26475
W = 5.2236
ε eff =3.48619
λmicrostrip = λo/√εeff = 16.0674 cm
Length of track = λmicrostrip/4 = 4.0168 cm
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Fixed Phase shifter
Semi rigid cable
•O
ute
r C
onducto
r (C
opper)
-minimizes the power loss
-maximizes the mechanical integrity
-provides the desired interface with connections
•C
ente
r conducto
r(Silver pla
ted c
opper)
-acts as primary signal carrier
-provides excellent high frequency conductivity
•D
iele
ctr
ic m
ate
rial (P
oly
tetr
afluoro
eth
yle
ne )
-maintains the spacing and geometry of the cable
-assures mechanical integrity during form
ing and
bending or under pressure
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Design of phase shifter
L1
a1
L2
a2
Differential measurement
L1 provides phase shift of a1o
L2 provides phase shift of a2 o
(L1-L2) provides phase shift of (a1-a2) o
L1 = 10 cm,
a1 = 97.2
o
L2 = 15 cm,
a2 = 18.5
o
(15-10) cm
(18.5-97.2) o
5 cm
-78.7
o
1 cm
-15.7
o
22.87 cm
360o
λc-wavelength in cable = 22.8 cm
λa-wavelength in air = 30.0 cm
Velocity factor = λc/λa = 76%.
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Helical antenna
Helical geom
etr
y
D = 110 mm
d = 2 mm
L = 353.5 mm
S = 75 mm
A = 450 mm
C = πD = 345.5mm
α= arc tan S/πD = 12.24o
N =
num
ber
of
turn
s =
6
1.2 ≥Cλ≥0.8,
14o ≥α≥12o and n ≥4
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Transmission And Radiation Modes Of Helix
1.
Norm
al mode-
The field radiated by the antenna is m
aximum in a plane norm
al to the helix
and minimum along its axis.
2. Axial mode
The field radiated by the antenna is maximum along its axis.
Power Beam W
idth (HPBW) = 52
= 36o
cλ√nsλ
![Page 20: Butler Matrix](https://reader034.vdocuments.mx/reader034/viewer/2022050704/54f895f44a79597b198b517a/html5/thumbnails/20.jpg)
Impedance m
atching
With axial feed the term
inal impedance (resistive)is given by
R = 140Cλ
(Ω)
Gradually tapered transition from helix to coaxial line
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Testing and Results
Testing of Phase shifter
Testing of 4 x 4 Butler matrix
Beam form
ation of 4 x 4 Butler matrix
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Tes
ting
of
Hy
bri
d
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PC
B L
ayou
t
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Applications
-tra
ckin
g o
f ra
dio
sourc
es
-direction fin
din
g
Futu
re s
cope
-Adaptive a
rray a
nd s
mart
ante
nna -D
igitiz
ation
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Ref
eren
ces
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