-Har
ish
Raj
ago
pal
an
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
Antenna Beam Scanning Methods
Mechanical Scanning Electronic Scanning
Phase Scanning Frequency Scanning
Series Feed Parallel Feed
(Blass m
atrix) (Butler matrix)
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
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
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
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
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
Phase m
atrix of 8 x 8 Butler matrix
Phase progression
Beam pattern for 8x8 Butler matrix
Components of Butler matrix
•Qudrature hybrid
•Fixed phase shifter
•Helical antenna
Qudrature Hybrid
Branch line coupler
MicrostripLines
Basic Structure
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
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
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
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%.
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
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
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
Testing and Results
Testing of Phase shifter
Testing of 4 x 4 Butler matrix
Beam form
ation of 4 x 4 Butler matrix
Tes
ting
of
Hy
bri
d
PC
B L
ayou
t
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
Ref
eren
ces