-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