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TRANSMISSION LINES

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WHAT IS TRANSMISSION LINE THEORY ANALYSIS OF CIRCUITS AT MICROWAVE FREQUENCIES SINCE THESE LINES ARE

USED TO CONNECT ANTENNA TO RECEIVER / TRANSMITTED, PAs, etc

CONVENTIONAL CIRCUIT THEORY USES LUMPED CONSTANTS TO REPRESENTCAPACITANCES AND INDUCTANCES

THIS IS OK FOR LOW FREQUENCIES. AS FREQ INCREASES TWOEFFECTS BECOMEPROMINENT

INDUCTANCES OVER SHORT LENGTHS BECOME SIGNIFICANT

CAPACITANCE BETWEEN CONDUCTORS BECOME SIGNIFICANT

VOLTAGE AND CURRENT TRAVEL AS WAVES ALONG THE TRANSMISSION LINEAND INDUCTANCE & CAPACITANCE EFFECTS COMBINE AT EACH POINT ALONGTHE CONDUCTOR

CONSEQUENTLY, IMPENDANCE OFFERED BY A SHORT LENGTH OF CONDUCTOR ISSIGNIFICANT

HENCE DISTRIBUTED CIRCUIT THEORY IS EMPLOYED TO ANALYZE TRANSMISSIONLINES @ MICROWAVE FREQUENCIES

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NECESSITY FOR CHANGED APPROACH

~ RzLz

Gz

CzRs

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ELEMENTARY SECTION OF Txn LINE

LINE LENGTH IS LARGE COMPARED TO IN THE DIRECTION OF +ve Z AXIS

EACH SECTION HAS INFINITESIMALLY INCREMENTAL IMPEDANCE Z

AT THE START IS THE GENERATOR OR SOURCE

AT THE END IS THE LOAD ZL

~

~

~

~

~

~

~

~

~

ZL

A B

v(z,t) v(z+z,t)

i(z,t) i(z+z,t)

Rz RzLz Lz

Gz Cz Cz

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DERIVATION OF Txn LINE EQUATIONS

APPLY KIRCHOFFS VOLTAGE LAW IN THE INNER LOOP

~

~

~

~

~

~

~

~

~

RL

A B

v(z,t) v(z+z,t)

i(z,t) i(z+z,t)

Rz RzLz Lz

Gz Cz Cz

),(),(),(

),(),(),(

)(),(

)(0

)],([),(),()(),(

tzi

t

LtzRi

z

tzv

z

tzvtzzvtzi

tz

zLtzi

z

zR

tzzvtzit

LtzizRtzv

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DERIVATION OF Txn LINE EQUATIONS

APPLY KIRCHOFFS CURRENT LAW AT POINT B, WE GET,

~

~

~

~

~

~

~

~

~

RL

A B

v(z,t) v(z+z,t)

i(z,t) i(z+z,t)

Rz RzLz Lz

Gz Cz Cz

),(]),[()})(,(),({[)(

)})(,(),(){[(0

]),[(]),[()(]),[()(),(

tzitzziztzv

z

tzv

t

zC

ztzvz

tzvzG

tzzitzzvt

zCtzzvzGtzi

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DERIVATION OF Txn LINE EQUATIONS

DIVIDING BOTH SIDES BY z

t

vCGv

z

tzi

ztzvzt

tzvt

Cztzvz

GtzGvztzi

z

tzitzzi

z

ztzvz

tzvt

zC

z

ztzvz

tzvzG

),(

)})(,({),())(,(),(),(

),(),(

)(

)})(,(),({[)(

)(

)})(,(),(){[(

0

may be neglectedmay be neglected

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DERIVATION OF Txn LINE EQUATIONS THUS WE HAVE

AND

DIFFERENTIATE wrt z DIFFERENTIATE wrt t

2

2)(

t

vC

t

vG

t

zi

2

2

2

2

2

2

2

2

2

2

)(

}{}{

)(

t

vLC

t

vLGRCRGv

z

v

t

vLC

t

vLG

t

vRCRGv

z

v

t

vCGv

tL

t

vCGvR

ztiL

ziR

zv

t

vCGv

z

tzi

),(

t

iLRi

z

v

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TRANSMISSION LINE EQUATIONS

2

2

2

2

)(

t

vLC

t

vLGRCRGv

z

v

2

2

2

2

)(

t

iLC

t

iLGRCRGi

z

i

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SIMPLIFICATION OF Txn LINE EQUATIONS We have

AND

SUBSTITUTING FOR i & v IN BOTH EQUATIONS, WE GET

where, Impedance Z = R+jLand Admittance Y = G+jC

t

vCGv

z

tzi

),(

t

iLRi

z

v

ZIILjRdz

dV

LIejRIedz

dVe

t

IeLRIe

z

Ve

tjtjtj

tjtj

tj

)(

)(

YVICjGdz

dI

CVejGVedz

dIe

t

VeCGVe

z

Ie

tjtjtj

tjtj

tj

)(

)(

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SIMPLIFICATION OF Txn LINE EQUATIONSWe have, Impedance Z = R+jL and Admittance Y = G+jC and Txn Line eqn given by

SUBSTITUTING v = Vejt IN THE ABOVE EQUATION, WE GET

where,

2

2

2

2

)(t

vLC

t

vLGRCRGv

z

v

Iz

I

SIMILARLYVz

V

OR

CjGLjRnoteVLGRCjLCRG

LCVeVeLGRCjRGVedt

Vde

tVeLC

tVeLGRCRGVe

zVe

tjtjtjtj

tjtj

tj

tj

2

2

22

2

2

2

2

2

2

2

2

2

2

,

))(()]()[(

)(

)(

ZY

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INTERPRETATION OF Txn LINE EQN INSTANTANEOUS v & i VARY BOTH IN TIME AND IN SPACE

THEY CAN BE EXPRESSED AS

WHERE V(z) & I(z) ARE PHASORS WITH COMPLEX MAGNITUDE AND PHASE COMPONENTS

THEY CAN BE EXPRESSED AS

is attenuation constant and is measured in Nepers/unit length

phase constant and is measured in Radians/unit length

})(Re{),(

})(Re{),(

)

)

tj

tj

ezItzi

ezVtzv

j

eIeIzIeVeVzV

zz

zz

)()(

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CHARACTERISTIC IMPEDANCE One solutions for the transmission line equations is as given below

Z0 is the characteristic impedance of the medium given by

zzeVeVzV

)(Wave

travelling

inve z

direction

Wave

travelling

in +ve z

direction

zz

zz

zz

zz

zz

eVeVZ

I

eVeVZ

I

eVeVZ

I

ZIeVeVzzV

eVeVzV

0

1

)(

)(

Y

ZZ 0

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We have,

Since R

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We have,

Since R

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Phase Velocity is the velocity at which the wave travels. In free space and air, the phasevelocity denoted by vp is 3x108 m/s

It is given by

In free space

In a lossy medium, relative velocity is given by

CONCEPT OF PHASE VELOCITY

rr

r

p

p

cv

smxLC

v

LCv

1

/10311

1

8

00

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General solution of the transmission line equations is as given below

V+ and V- are the forward and reflected waves Reflected wave arises due to mismatch in the characteristic and load impedances

Reflected wave = 0 if Z0 = Z

if Z0 Z then there will be reflections and the reflection will be complex

REFLECTION COEFFICIENT

zzzz

eVeVZ

I

eVeVV

0

1

~ Z

Zg Ps

Z0

Prs

Pinc

PrefPtr

ld

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Reflection coefficient is defined as

At the receiving end, let the travelling waves be

EXPRESSIONS FOR REFLECTION COEFFICIENT

eV

eV

eVeVZ

I

eVeVeV

0

1

~ Z

Z0 Ps

Z0

Prs

Pinc

PrefPtr

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At the load end, Z is given by

is the reflection coefficient at the receiving end or the load end

The reflection coefficient depends on the load impedance

EXPRESSIONS FOR REFLECTION COEFFICIENT

0

0

0

0

ZZ

ZZ

eV

eV

eVeVeVeV

eVeVeVeV

eVeVeVeV

ZZ

eVeV

eVeVZ

I

VZ

lll

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The reflection coefficient is a complex quantity and can be expressed as

At a distance of d from the load, the reflection coefficient will be given by,

This may also be expressed as

EXPRESSIONS FOR REFLECTION COEFFICIENT

j

ll e

d

ld

d

d

d

d eeeV

eeV

eV

eV

2

)(

)(

)(2222 djdl

dj

l

d

ldleeee

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Transmission coefficient is defined as

EXPRESSION FOR TRANSMISSION COEFFICIENT

20

2

0

0

00

0

0

1

12

lL

L

Ltr

trl

ll

L

LL

l

ll

L

L

l

l

Z

ZT

ZZ

Z

V

VT

Ve

eVeVBut

ZZ

ZZZZ

eV

eVeV

ZZ

ZZ

eV

eV

I

I

V

V

currentorvoltageIncident

currentorvoltagedTransmitteT trtr

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Voltage Standing Wave results from the fact that two travelling wave components add in

phase at some points along the line and subtract at other points

Since and are real, the voltage standing wave may be expressed as

VOLTAGE STANDING WAVE

)tan(tan

)(sin)(cos

)sin()cos(

)sin()cos()sin()cos(

1

2/12

22

2

0

0

zeVeV

eVeV

zeVeVzeVeVV

eVV

zeVeVjzeVeVV

zjzeVzjzeVV

eeVeeVeVeVV

zz

zz

zzzz

j

s

zzzz

zz

zjzzjzzz

zeV

zeV

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VSWR is defined as

Vmax occurs when both forward and reverse waves add and Vmin occurs when both forward

and reverse waves subtract.

Substituting,

Therefore, Alternatively,

VOLTAGE STANDING WAVE RATIO

min

max

min

max

I

I

V

V

currentorvoltageMinimum

currentorvoltageMaximumVSWR

zzeVeVV

maxzz

eVeVV

min

1

1

1

1

1

1

1

1zzz

zzz

zz

zz

eVeVeV

eVeVeV

eVeV

eVeV

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Vmax occurs when z = n Vmin occurs when z = (2n-1)/2

Inductive load Maxima near load Capacitive load Minima near load

LOCATION OF MAXIMA & MINIMA OF VSWR

zzzzzzz

zzzzzz

eVeVeVzjeVeVzV

zjzeVzjzeVV

eVeVeVeVeVeVV

)sin()cos(

)sin()cos()sin()cos(

/2

Vmax

Vmin

Vmax

Vmin

/2

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POINTS TO REMEMBER

VSWR (s11) Reflected Power (%) Reflected Power (dB)

1.0 0.000 0.00 -Infinity

1.5 0.200 4.0 -14.0

2.0 0.333 11.1 -9.55

2.5 0.429 18.4 -7.36

3.0 0.500 25.0 -6.003.5 0.556 30.9 -5.10

4.0 0.600 36.0 -4.44

5.0 0.667 44.0 -3.52

6.0 0.714 51.0 -2.92

7.0 0.750 56.3 -2.50

8.0 0.778 60.5 -2.18

9.0 0.800 64.0 -1.94

10.0 0.818 66.9 -1.74

15.0 0.875 76.6 -1.16

20.0 0.905 81.9 -0.87

50.0 0.961 92.3 -0.35

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POINTS TO REMEMBER

Generally, reflection coefficient = < 1 and is complex

IF LOAD IS PERFECTLY MATCHED TO THE LINE. = 0 ; VSWR = 1;

IF THEN THE LINE IS A SHORT CIRCUIT. = -1 ; VSWR = ;

IF, THEN THE LINE IS AN OPEN CIRCUIT. = +1; VSWR = ;

VSWR >1. It is a dimensionless ratio.

Typical VSWR values of a Troposcatter communication system

1:1 is perfect and is rarely achieved.

1.2 : 1 is very good and practical systems target this figure

1.5 : 1 usually gives a minor alarm in systems

2 : 1 is excessive and results in major alarm.

Return Loss is given by RL = -20log10 || dB (Loss due to reflection of signal)

Insertion Loss is given by -20log10 |T| dB (Loss due to insertion of load)

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POINTS TO REMEMBER

Maximum power is delivered o the load if= 0 & no power if = 1

As || increases, VSWR increases as does degree of mismatch between line &

load

Distance between successive minima of voltage maxima is /2 and distance

between maxima and minima is /4

At voltage maxima, Z = Z0 * VSWR

At voltage minima, Z = Z0 / VSWR

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General solution of the transmission line equations is as given below

LINE IMPEDANCE

zzzz

zz

eZ

Ve

Z

VeIeII

eVeVV

00

~ Zl

Zg

Ig

Z

Prs PrefPtr

l

Zs

Vs V Vr

Zr

Ir Il

z d

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We have

From the sending end, z = 0 and

We get and

Hence,

Impedance at any point on the line calculated in terms of Zs(from the sending end) is

LINE IMPEDANCE

zzzz

zz

eZ

Ve

Z

VeIeII

eVeVV

00

VVZI ss

0

2ZZ

IV s

s 02

ZZI

V ss

z

s

z

s

s

z

s

z

s

s

eZZeZZZ

I

I

eZZeZZI

V

00

0

00

2

2

z

s

z

s

z

s

z

s

s

s

eZZeZZ

eZZeZZZ

I

VZ

00

000

VVZIs 0

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At z = , line impedance can be expressed as in terma of Z

and Z0

Solving these two equations we get,

Putting ( - z) = d, impedance at any point on the line calculated in terms of Z(from the

load end) is

LINE IMPEDANCE

lss eZZIV 02

lss eZZIV 02

ll

ll

eVeVZI

eVeVZI

0

dldl

d

l

d

l

eZZeZZ

eZZeZZZZ

00

000

)(

0

)(

0

0

)(

0

)(

0

2

2

z

s

z

s

s

z

s

z

eZZeZZZ

I

I

eZZeZZI

V

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e = cosh() sin()

Impedance from sending end is given by,

Impedance from the receiving end is

LINE IMPEDANCE IN TERMS OF HYPERBOLIC

FUNCTIONS

dZZ

dZZZ

dZdZ

dZdZZZ

tanh

tanh

sinhcosh

sinhcosh

0

00

0

00

zZZ

zZZZ

zSinZzZ

zSinZzZZZ

s

s

s

s

tanh

tanh

cosh

cosh

0

00

0

00

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If the load end is open circuited,

Since all the power is reflected, the incident voltage and

the reflected voltage are equal at the load.

Due to the open circuit, current is zero at the load

OPEN CIRCUITED LINE

coth

tanh1

tanh10

0

00 jZ

ZZ

ZZZZ

/2

jV

VeeVV

jj

2

jV

V

2

jV

IZ

2

0

jV

Vee

Z

VI

jj

20

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If the load end is short circuited,

Since all the power is reflected, the incident voltage and

the reflected voltage are equal at the load.

Due to the open circuit, current is zero at the load

Impedance of a short circuited line is purely reactive

At = \4, z is infinity(since current is 0) repeating at \2

SHORT CIRCUITED LINE

tanh

tanh

tanh0

0

00 jZ

ZZ

ZZZZ

/2

V

VeeVV

jj

2

V

V

2

V

IZ

2

0

V

VZee

Z

VI

jj

2

0

0

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If the load end is short circuited,

If the load end is open circuited

Solving these two equations we get,

Normalized Impedance is calculated as

SHORT & OPEN CIRCUITED LINE

ocsc

ocsc

ZZZ

ZZZ

0

20

tanh

tanh

tanh0

0

00 Z

ZZ

ZZZZ

coth

tanhtan/1

tanhtanh

0

0

00

0

00 Z

ZZZZZ

ZZZZZZ

1

1

0

Z

Zz

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For a loss line, at = /4

For = 0, Zin= Z0. Hence, if Z1 is chosen carefully, the line will look as if it is terminated in Z0.whereas, the load impedance remains unchanged.

Hence, adding a quarter wavelength section of appropriate characteristic impedance to the

line will match the load impedance

QUARTER WAVE TRANSFORMER

Z

Z

jZlZ

jZZZZin

2

1

1

01

sincos

sincos

ZZ0 Z1Zin

/4