MW TX LINES

Tx line can be analyzed by: a. Maxwell’s field equations-involves three space variables in addition to time variable

b. Distributed circuit method-involves only one space variable in addition to time variable

Tx Line Equations & Solutions

Transmission Line

Types of TL

Analysis of Tx Line

The dielectric between the conductors has a not zero conductivity, which is responsible of a current flowing from one conductor to the other through the insulator.This phenomenon is accounted for by the conductance G∆z

Analysis of Tx Line

Analysis of Tx Line

If we ignore loss the equation become:

These two equations are called Telegrapher equations or the Transmission line equations

Analysis of Tx Line

WAVE EQUATIONS PHASE VELOCITY and WAVE EQUATION

TRANSMISSION-LINE EQUATIONS

Summation of voltage drop around the central loop

Transmission-Line Equations

(z, t) (z, t)(z, t) i(z, t) R (z, t) (1)

Rearranging, dividing by z and omitting (z,t) we get

- (2)

i vv z L z v z

t z

v iRi L

z t

• The summation of current at point B:

(z , t)i(z, t) (z , t)G (z , t)

(z, t) (z, t)[ (z, t) ]G [ (z, t) ]

i(z, t) (3)

Rearranging, dividing by z, omit

v zv z z C z i z

tv v

v z z C z v zz t z

c z

ting (z,t) and summing

z equal to zero (4)i v

Gv Cz z


Differentiating eqn (2) wrt ‘z’ and eqn (4) wrt ‘t’ and combining, the final transmission line equation in voltage form:

Differentiating eqn (2) wrt ‘t’ and eqn (4) wrt ‘z’ and combining, the final transmission line equation in current form:

2 2

2 2(RC LG) (5)

v v vRGv LC

z t t

2 2

2 2(RC LG) (6)

i i iRGi LC

z t t


Transmission line equations are applicable to the general transient solution. Voltage and Current on the line are the functions of both position z and time t. The instantaneous line voltage and current are:(z, t) ReV(z)e (7)

i(z, t) ReI(z)e (8)

The phasor give the magnitudes and phases of the

sinusoidal function at each position of z and expressed as:

V(z) V e V e (9)

I(z) e

j t

j t

t t

t

v

I

e (10)

Where, = +j is the propagation constant

tI


2 22

2

Substitute j for in equaton (2), (4), (5) and (6)

and divide each equation by e , the Tx equation

in frequency domain

(12) and (13)

(14) and

j t

t

dV dIZI YV

dz dz

d V d IV

dz dz

22

I (15)

Where,

(16) ; Y=G+j C (17)

and = = +j (18)

Z R j L

ZY


2 22 2

2 2

For Lossless line, R=G=0 and transmission line equation:

(19) and (20)

(20) and LCI (22)

dV dIj LI j CV

dz dz

d V d ILCV

dz dz


Solutions of Transmission-Line Equations

22

2

One possible solution of equation (14) i. e:

is:

V(z) V e V e V e e V e e (23)z z z j z z j z

d VV

dz

V and V- are complex quantities

The term e-jβz , wave travelling in positive z direction and ejβz , wave travelling in negative z direction

The quantity βz is called the electrical length of the line in radians


22

2

0 0

00

One possible solution of equation (15) i. e:

I is:

(V e V e ) (V e e V e e ) (24)

Characteristic Impedance of the line is defined as:

1

z z z j z z j z

d I

dz

I Y Y

Z R j LZ

Y Y G j C

0 0 (25)R jX

Magnitude of Voltage and Current wave shown in Fig


2

At microwave frequencies R L and G

y using Binomial Expansion, the Propagation

Constant can be expressed as:

= (R )(G ) ( ) 1 1

1 11 1

2 2

C

B

R Gj L j C j LC

j L j C

R Gj LC

j L j C

(27)

2

1

1

12

R Gj LC

j L j

jC L

R GL C

LC

C


The Attenuation and Phase Constant are:

(28)

(29)

1

2

LC

C LR G

L C


1/2 1/2

0

0Characteristic Impedance Z :

1The Phase V

(R )Z = 1 1

(G )

1 11 1 (30)

2 2

11

2

(elocity= 31) p

j L L R G

j C C j L j C

L R G

C j L j C

L R G

C j L j

L

C

vLC

C

The product LC is independent of the size and separation of the conductors and depends on only on the Permeability μ and Permittivity ε of the insulating medium

If a lossless transmission line used for microwave frequencies has an air dielectric and contains no ferromagnetic materials, free-space parameters can be assumed

The numerical value of for air insulated conductor is approximately equal to the Velocity of Light in vacuum


1/ LC


8

0 0

1 13 10 / sec

When the dielectric of lossy MW Tx line is not air

the Phase Velocity is smaller than the Velocity of

1

phase velocityhe Phase Velocity Factor=

ight in vacum

p

r r

actual

v c x mLC

cL

T

v

of l

ight

1r

r r

veloci

vv

c

ty

REFLECTION COEFFICIENT AND TRANSMISSION COEFFICIENT

The travelling wave along the line contains two components:a. One travelling in the positive z directionb. Other travelling in the negative z direction

If the load impedance is equal to the line characteristic impedance; the reflected travelling wave does not exist

Reflection Coefficient


A wave experiences partial transmittance and partial reflectance when the medium through which it travels suddenly changes. The reflection coefficient determines the ratio of the reflected wave amplitude to the incident wave amplitude.

Standing Wave FormationStanding Wave Formation

The animation depicts two waves moving through a medium in opposite directions. The blue wave is moving to the right and the green wave is moving to the left.

As is the case in any situation in which two waves meet while moving along the same medium, interference occurs. The blue wave and the green wave interfere to form a new wave pattern known as the resultant. The resultant in the animation below is shown in black.

The resultant is merely the result of the two individual waves - the blue wave and the green wave - added together in accordance with the principle of superposition.


The result of the interference of the two waves above is a new wave pattern known as a standing wave pattern. Standing waves are produced whenever two waves of identical frequency interfere with one another while traveling opposite directions along the same medium.

Standing wave patterns are characterized by certain fixed points along the medium which undergo no displacement. These points of no displacement are called nodes (nodes can be remembered as points of no displacement). The nodal positions are labeled by an N in the animation above.

The nodes are always located at the same location along the medium, giving the entire pattern an appearance of standing still (thus the name "standing waves").


A careful inspection of the above animation will reveal that the nodes are the result of the destructive interference of the two interfering waves. At all times and at all nodal points, the blue wave and the green wave interfere to completely destroy each other, thus producing a node.

Midway between every consecutive nodal point are points which undergo maximum displacement. These points are called antinodes; the anti-nodal nodal positions are labeled by an AN. Antinodes are points along the medium which oscillate back and forth between a large positive displacement and a large negative displacement. A careful inspection of the above animation will reveal that the antinodes are the result of the constructive interference of the two interfering waves.


• Solve the transmission line problem from the receiving end rather than the sending end, since the voltage to current relationship at the load point is fixed by the load impedance


voltage and Current travelling along tx line

V V e V e (1)

I e e (2)

z z

z z

Incident

I I


CIRCUIT DIAGRAM

Transmission line terminated in a load impedance


0 0

0

Current wave interms of Voltage

V V I e e (3)

and Current at receiving end, line length l

V V e V e (4)

V I e

z z

l ll

ll

Z Z

Voltage

Z

0

0

Ve (5)

V V e V e load Impedance Z (6)

I V e V e

l

l ll

l l ll

Z

The Z


(7)

The Reflection Coefficient at the receiving end:

Reflected Voltage or CurrentReflection Coefficient =

Voltage or Current

ref ref

inc inc

Incident

V I

V I

V

0

0

(8)l

ll

l

Z Ze

V e Z Z

If the load impedance and characteristics impedances are complex then the Reflection Coefficient is also complex


=Phase angle between inciden

t and refl

(9)

whe

ected

voltages at the receiv

re, 1 ; ne

ing e

ver greater than unity

generalized reflection coefficien

n

t

i

d

l

l

j

l

l l

The

e

s defined as:

(10) z

z

V e

V e

• Let z=l-d, then the Reflection Coefficient at some point located at a distance d from the receiving end is:

• Very important equation fro determining the reflection coefficient at any point along the line


(l d)2 2

(l d) = = (11)

Reflection Coefficient at that point can be expressed in

terms of Reflection Coefficeint at the receiving end as:

d

ld d

d l

l

l

V e V ee e

V e V e

e

2 ( 2 )2 2 2 (12)lj dd j d dle e e

For lossy line, both the magnitude and phase of the reflection coefficient are changing in an inward-spiral way:


Reflection Coefficient for lossy line

For a lossless line α=0, the magnitude of Reflection Coefficient remains constant, only Phase of Г is changing circularly toward the generator with an angle of -2βd.


Reflection Coefficient for lossless line

It is evident that Гl will zero and there will be no reflection from the receiving end when the terminating impedance is equal to the Characteristic Impedance of the line.

Terminating impedance that differs from the characteristic impedance will create a reflected wave traveling towards source.

The reflection, upon reaching the sending end, will itself be reflected if the source impedance differs from the line characteristic at the sending end.


TRANSMISSION COEFFICIENT

Transmission line terminated in its Characteristic Impedance is called Properly Terminated line otherwise Improperly Terminated line.

There is a Reflection Coefficient at any point along an improperly terminated line.

Incident power minus Reflected power is equal to the power transmitted to the load

Transmission Coefficient

2 20 transmitted to Load=1- (13)

T=Transmission

Voltage or Current

Voltage or C

Coef

urr

ficient

(1e

4)nt

tr tr

inc in

l

c

l

V ITransmittedT

Incident

ZPower T

Z

V I


Power Transmission on a line

0 0

0

the travelling wave at the receiving end

V e V e = V e (15)

VV V and e e = e (16)

(16) x Z and substituting in (15)

V e

V e

l l ltr

z z ltr

l

l

ll

l l

Let

Z Z Z

Z Z

Z

0

(17)l Z


0

0

0

, V e V e = V e

V eV e 1+ =

V e V e

V 1+ = =T

V

(182

T= )

l l ltr

lltr

l l

l tr

l

l

l

Now

Z Z

Z

Z

Z

Z

Z


2 2

0 0

Power carried by two waves in the side

of the incident and reflected waves

( ) ( )P P -P (19)

2 2

l l

inr inc ref

The

V e V e

Z Z


2

0

2

2

The power carried to the load by the transmitted waves:

( ) P (20)

2

and using equation (17) and (18)

T (1 )

P =

P

lt

i

rtr

n

l

r

l

l

r t

V e

Z

Putt n

Z

i g

Z

(21)

relation varifies the previous statementThis


STANDING WAVE AND STANDING WAVE RATIO

Transmission-line equation consist of two waves traveling in opposite directions with unequal amplitude. FIG

Standing Wave

The equation we have derived:

V(z) V e V e V e e V e e

V e cos sin V e cos sin (1)

(V e V e )cos (V e V e )sin

z z z j z z j z

z z

z z z z

z j z z j z

z j z

Standing WaveStanding Wave Pattern/Amplitude of Voltage Wave

0

2 20

2 2 1/2

no loss line we can assume V e and V e real

voltage equation V V e (2)

Where,

V {(V e V e ) cos

(V e V e ) sin }

z z

js

z z

z z

For

The

z

z

(3)Phase Pattern of Standing Wave

V e V earctan tan

V e V (4)

e

z z

z zz

Standing WaveThe Maximum Amplitude

maxV V e V e V e (1 ) (5)

occurs at z=n where, n=0, 1, 2...

z z z

This

The Minimum Amplitude

minV V e V e V e (1 ) (6)

occurs at z=(2n-1) / 2 where, n=0, 1, 2...

z z z

This

Standing Wave Pattern in a Lossy Line

Voltage Standing Wave Pattern in a Lossless Line

Standing Wave

1

distance between and two successive

maxima and minina is one-half wave length

occurs at z=n where, n=0, 1, 2...

n nz (7)

2

there is n

/ 2 2

Note o zeros i that, n the minimum

The

This

n z

Distance Between Maxima and Minima

Standing Wave

max

min

,

e e e (1 ) (8)

e e e (1 ) (9)

z z z

z z z

Similarly

I I I I

I I I I

Standing Wave Positive and Negative wave have equal

amplitude: V e V e i.e. Magnetude

of Reflection Coefficient is unity, the Standing

Wave with zero Phase is:

V 2V e cos

z z

zs

When

z

0

Pure Standing Wave for Cu

(12)

I 2 e sin (13)

Called Pure Stand

rren

ing W v

t

a e

zs j Y V z

• The Voltage and Current Standing Waves are 900 out of phase along the line. The point of zero current are called the current nodes

• Voltage and Current nodes are interlaced and a quarter wave apart

Standing Wave

• The voltage and current may be expressed as real functions of time and space:

Standing Wave

0

(14)

(15)

the amplitudes of these two e

The voltage is maximu

(z, t) Re[V (z)e ] 2V e cos

quations

m at the

vary sinu

(z, t) Re[I (z)e ] 2

soidally

with ti

instan

me;

cos

t w

V e sin s

h

n

en

i

j t z

z

s

j ts s

sv

i Y z t

z t

the current

is zero and vice versa. Fig.

Pure Standing Wave of Voltage and Current

STANDING WAVE RATIO

Standing Wave Ratio

Standing results from the simultaneous presence of waves traveling in opposite directions on a transmission line. The ratio of the maximum of the standing wave pattern to the minimum is defined as the standing wave ratio ρ

Standing Wave Ratio

max max

min min

voltage or currenttanding Wave Ratio=

voltage or current

(16)

MaximumS

Mnimum

V I

V I

• Standing wave results from the fact that two traveling wave component add in phase at some points and subtract at other points

• The Standing wave ratio of a pure traveling wave is unity and that of a pure standing wave is infinite

• Standing wave ration of Voltage and current are identical

• When the standing wave ratio is unity, there is no reflected wave and the line called a FLAT LINE

Standing Wave Ratio

The standing wave ratio can not ne defined on a lossy line because the standing wave pattern changes markedly from one position to another

Low loss line the ration remains fairly constant and it may be defined for some region

For a lossless line, the ration remains same throughout the line

Standing Wave Ratio

o Since the reflected wave is defined as the product of and incident wave and its reflection coefficient, the standing wave ratio is related to the reflection coefficient by :

o FIGo The standing wave ratio is a positive real number

and never less than unity. o The magnitude of the reflection coefficient is

never greater than unity

Standing Wave Ratio

1 (17)

1

1 and vice versa (18)

1

1 1

LINE IMPEDANCE AND ADMITTANCE

Line ImpedanceFIG

0

(z) Impedance of Tx Line Z= (1)

I(z)

V=V +V V e +V e (2)

I=I +I (V e -V e ) (3)

At the sending end z=o (2) and (3)

V

z zinc ref

z zinc ref

s s

VThe

Y

I Z

0

0

0

+V (4)

V -V (5)

V ( ) (6)2

V ( ) (7)2

s

ss

ss

I Z

IZ Z

IZ Z

Line Impedance

0 0

0 00

0 00

0

V and V in (2) and (3)

V= [( )e ( )e ] (8)2

I= [( )e ( )e ] (9)2

The line Impedance at point z from sending end

( )e ( )e

( )e (

z zss s

z zss s

z zs s

zs

Substituting

IZ Z Z Z

IZ Z Z Z

Z

Z Z Z ZZ Z

Z Z Z

0 )e zs Z

Line Impedance

0

0 00

0 0

0

The line Impedance at point z=l from Receiving

end interms of and

( )e ( )e (11)

( )e ( )e

The line Impedance can be expressed

interms of and ; At z=l,

s

l ls s

r l ls s

l

Z Z

Z Z Z ZZ Z

Z Z Z Z

Z Z

0

V

V e +V e (12)

V e -V e (13)

r l l

l ll l

l ll

I Z

I Z

I Z

Line Impedance

0

0

V e +V e (12)

V e -V e (13)

Solving these two equation for V and V

V ( )e (14)2

V

l ll l

l ll

lll

l

I Z

I Z

IZ Z

I

0( )e (15)2

llZ Z

Line Impedance

0 0

0 00

these results in (2) and (3)

and letting z=l-d

[( )e ( )e ] (16)2

[( )e ( )e ] (17)2

d dll l

d dll l

Substituting

IV Z Z Z Z

II Z Z Z Z

Z

Line Impedance

0

0 00

0 0

00

The line Impedance any from Receiving

end interms of and

( )e ( )e (18)

( )e ( )e

line impedance at the sending end can

be found from (18). Let, l=d

(

l

d dl l

r d dl l

ls

Z Z

Z Z Z ZZ Z

Z Z Z Z

The

Z ZZ Z

0

0 0

)e ( )e (19)

( )e ( )e

l ll

i ll l

Z Z

Z Z Z Z

These equations can be simplified by replacing the exponential by Hyperbolic or Circular functions

Line Impedance

00

0

00

0

Hyperbolic function obtained from

e cosh( ) sinh( )

At any point from the sending end in terms of

Hyperbolic function

Z cosh( ) Z sinh( )Z= Z

Z cosh( ) Z sinh( )

Z Z tanh( )= Z

Z Z tanh( )

z

s

s

s

s

The

z z

z z

z z

z

z

(21)

Line Impedance

0 00

0 0

00

0

00

0

( )e ( )e (18)

( )e ( )e

, from (18), from the Receiving End

in terms of Hyperbolic function

Z cosh( ) Z sinh( )Z= Z

Z cosh( ) Z sinh( )

Z +Z tanh( )= Z

Z +Z

d dl l

r d dl l

l

l

l

Z Z Z ZZ Z

Z Z Z Z

Similarly

d d

d d

d

(22)tanh( )l d

Line Impedance0 0

00

0

00

0

0

The impedance of a lossless line Z

be expressed in terms of circular function

Z cos( z) sin( z)Z= R

cos( z) jZ sin( z)

Z -jR tan( z)= R (25)

-jZ tan( z)

ZZ= R

s

s

s

s

R

can

jR

R

R

0

0

00

0

cos( ) sin( )

cos( ) jZ sin( )

Z +jR tan( )= R (26)

+jZ tan( )

l

l

l

l

d jR d

R d d

d

R d

Impedance in Terms of Reflection Coefficient

0

0

2

0 2

-ZRe (18) and substituting =

Z

line impedance looking from Receiving end

1 Z=Z (27)

1

The Reflection Coefficient at a distance d

from receiving en

ll

l

dl

dl

Zarranging

Z

e

e

( 2 )2 2

0

d,

(29)

Then the simple equation at a distance d from

1load, Z=Z (30)

1

lj dd dl le e e


2

0

is a complex quantity (31)

, 2

Impedance variation along the lossless line

1 1 (cos sin )Z(d)=

1 1 (cos sin )

(d) jX(d) Z(

jl

dl l

j

j

e

e d

The

e jR

e j

R

d) (32)dje


2

0 2

2

0 2

0 2

2

1 2 coswhere, Z(d) (33)

1 2 cos

1 R(d)= (34)

1 2 cos

2 sin X(d)= (35)

1 2 cos

2 sin(d)=arctan arctan (

1

R

R

R

X

R

36)


, = 2 , then = 2 if,

However,cos( 2 ) cos and

sin( 2 ) sin

, Z(d)=Z(d+ )=Z(d+ ) (37)2

, impedance along the lossless line

will be repe

l l

l l

l l

Since d d

then

Hence

ated for every interval at

a half-wavelength


Since, the magnitude of a reflection coefficient is

related to the standing wave ratio as

1 1 and (38)

1 1

The line impedance at any location from

the receiving end can be written

0

( 1) ( 1)eZ=R (39)

( 1) ( 1)e

j

j

Determination of Characteristic Impedance

Determination of Characteristic Impedance and Propagation Constant of a given transmission line - take two measurementa. Measure the sending end impedance with the receiving end short-circuited and record the result:b. Measure the sending end impedance with the receiving end open circuited and record the result:

0 tanh( l) (40)scZ Z

0 coth( l) (41)ocZ Z

• Then the Characteristic Impedance of the measured transmission line is:

• And the propagation constant of the line can be computed from:

Determination of Characteristic Impedance

0 (42)oc scZ Z Z

1arctan (43)sc

oc

Zj h

l Z

NORMALIZED IMPEDANCE The Normalized Impedance is defined as:

Lower case letters are commonly designated for normalized quantities in describing distributed transmission line circuits

{Equations (39), (40) and (44)}Normalized impedance fro a lossless line has the following significant features:

0

1 (44)

1

Zz r jx

Z

1. Maximum Normalized Impedance

Normalized Impedance

maxmaxmax

0 0 min

max

1 (45)

1

Where is the positive real value and it is

equal to Standing Wave Ratio the location

of any max voltage on the line

VZz

R R I

z

at

2. Minimum Normalized Impedance


minminmin

0 0 max

min

1 1 (46)

1

Here, is a positive real number but

reciprocal of Standing Wave Ratio at the

location of any min voltage on the line

VZz

R R I

z


max min

max max

min min

3. For every interval of half-wavelength

distance along the line z or z is repeated

z (z) z (z / 2) (47)

z (z) z (z / 2) (48)


max min

max

min

maxmin

4. Since, V and V are separated by a

quarter-wavelength, z is equal to reciprocal

of z for any /4 separation:

1z (z /4) (49)

z (z)

LINE ADMITTANCE• When a transmission line is branched, it is better

to solve the line equations for the line voltage, current and transmitted power in terms of admittance rather than impedance.

• The characteristic admittance and the generalized admittance are defined as:

0 0 00

0 0

1 (50)

1 (51)

Admittance can be written:

1y= (52)

Y G jBZ

Y G jBZ

Normalized

Y Zg jb

Y Z z

MW TX LINES

Documents

Transcript of MW TX LINES