Transmission Lines and E.M. Waves lec 05.docx

8/20/2019 Transmission Lines and E.M. Waves lec 05.docx

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Transmission Lines and E.M. Waves

Prof R.K. Shevgaonkar

Department of Electrical Engineering

Indian Institte of Technolog! "om#a!

Lectre$%

Welcome, in the last lecture we introduce the concept of Loss-Less Transmission Line,

we say if the resistance and the conductance per unit length of the Transmission Line is

zero then there are only reactive elements in the Transmission Line so there is no loss of

power because there is no ohmic element in the Transmission Line. So in an ideal

situation the line is lossless if R !, " !

#Refer Slide Time$ !%$%! min&

Then we introduce the concept of the Low-Loss Transmission Line which is more

practical line where the resistive component R is much more less than 'L and " is much

more less than '(.


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#Refer Slide Time$ !%$)* min&

+f this condition is satisfied then we said that we can create the lines still lossless.

owever since the loss is very small as and when we reuire the calculation of losses

along Transmission Line we can use the value of the attenuation constant , substituting

this condition that R is much more less than 'L and " is much more less than '( we

calculated the propagation constant which can be separated into real and imaginary part

and then we got the value of and / in the appro0imate form.


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#Refer Slide Time$ !%$*1 min&

So, here this uantity was eual to / and this uantity was represented as . owever one

would notice that the condition which you have defined for the low loss is now in terms

of the primary constants of the line. owever in the data sheet for a Transmission Line

the primary constants are rarely mentioned, the parameter which + have mentioned for the

Transmission Line are the effective phase constant on the line or the velocity on the cable

and the attenuation constant of the line either in terms of d2s or in terms of 3epers.

Then one would li4e to convert this condition R 5 'L and " 5 '( in terms of this

secondary parameters or in terms of the relationship between / and . Since these

parameters are available readily in the data sheet if + can establish a condition between

these parameters for low loss nature of the line then + can find out whether a particular

line is low loss at a particular freuency.

So now what we do is starting from this relationship between / and then we can find

out under what condition we can treat the line as a Low-Loss Transmission Line. Ta4ing

this value of + can multiply this uantity here by a suare of L in the numerator and


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suare root of L in the denominator similarly + can multiply this uantity by suare root

of ( in the numerator and suare root of ( in the denominator.

#Refer Slide Time$ !6$7! min&

So now the can be written in terms of this substitution so + get this value of which is

eual to

% R L() L . Similarly multiplying by ( in the numerator and the denominator

in the second term we get

% "L(

) ( . 8ultiplying numerator and denominator by ' this

can be written as

% R % "L( L(

) L ) (ω ω

ω ω +

.


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#Refer Slide Time$ !7$61 min&

9nd as we have seen earlier this uantity L(ω is nothing but / so this + can write as /

in to

% R % "

) L ) (β

ω ω

+

.

3ow from the definition of the low loss

R

Lω is much more smaller than %,

"

(ω is much

more smaller than % so this whole uantity is much more smaller than % so essentially

what we are saying is now is eual to / multiplied very small uantity or in other words

for low loss the condition now is that is much more less than /.

Since we 4now / in terms of wavelengths which is nothing but two

)π

λ . :nce we 4now

the wavelength on the Transmission Line + can find out what is the value of / from the

data sheet + can find out what the attenuation constant is. +f it is given in terms of d2s +

will convert that in the 3epers per meter and if this condition is satisfied that the / is


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much larger compared to then the line can be treated as the Low-Loss Transmission

Line.


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#Refer Slide Time$ !;$!! min&

What does this physically mean we 4now if you travel at a distance of < on Transmission

Line the phase change is eual to )=.

Let us say if + travel the distance for a lossy Transmission Line then the amplitude of the

wave will reduce by e > into the distance traveled which is one wavelength. So if +

consider a wave which travels a distance of one wavelength on Transmission Line then its

amplitude will vary e >0 that is eual to e >


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#Refer Slide Time$ !?$)! min&

Since for low loss condition is much smaller than / this uantity

.)α π

β is much less

than % that means the amplitude of the wave now reduces to

.)

e

α π

β

−

and since this uantity

is very small essentially the amplitude reduction in the wave is negligibly small. So in

other words what we are saying is a line can be treated a low loss transmission line

provided the change in the amplitude of a traveling wave is negligibly small over one

wavelength distance. :f course negligibly small is a very sub@ective number you can

consider one percent as negligible or !.% percent as negligible.

Let us say as a reference value we consider one percent is a negligible uantity so when

the amplitude reduces to one percent of its original value then we say that the line can be

treated as a Transmission Line. Since this uantity is very small essentially when this

number becomes appro0imately one percent that is where the amplitude will reduce by

one percent so if

.)α π

β is appro0imately

%

%!! the wave amplitude will reduce by one


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percent over a distance of one wavelength from here then + can find out what is the

acceptable value of compared to /.

#Refer Slide Time$ !1$*7 min&

So in practice for a given line there is nothing absolute whether the line is lossy line or a

low loss line for a given freuency it may be possible that may be much smaller than /

but when the freuency changes the condition may not be satisfied and the line cannot be

treated as a Low Loss Transmission Line. So for a given loss on Transmission Line

depending upon the freuency the line may be treated li4e a low loss transmission line or

it may not be treated li4e a Low Loss Transmission Line.

Let us ta4e a simple e0ample to find out what are the physical parameters which we will

have on Transmission Line if we @ust ta4e some typical line parameters. So let us say +

have a Transmission Line whose primary constants are given as L !.)*ABm, let us say

the capacitance per unit length is %!! pico Carad per meter, let us say the conductance per

unit length is zero for this Transmission Line. So " ! and we want to 4now what should

be the resistance per unit length of the Transmission Line so that the line can be treated

li4e a low loss transmission line.


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Let us say the freuency of operation is eual to %!! mega hertz. Crom here since + 4now

the value of L and ( and the freuency + can find out what is the value of /. So / is eual

to )= into freuency which is hundred mega hertz which again is %!1 hertz into L

which is !.)* 0 %!-;

or micro henry multiplied by hundred into %!-%)

or the pico Carad.

This will be eual to = radians per meter.

#Refer Slide Time$ %%$%! min&

Dust to ta4e a number if + say is less than one percent of value of / then + can treat the

line as the low loss transmission line. The should be appro0imately %!!

π

. 3ow + can go

bac4 and substitute this value of in the e0pression for that is

% (R

) L


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#Refer Slide Time$ %%$7; min&

and if + substitute the value of

(

L and the value of which is %!!

π

as you have ta4en

one percent of the value of / then + get the value of R which is less than R =

appro0imately 6.%7 ohms per meter.

So if + accept alpha value to be one percent or less of the propagation constant / or the

phase constant / then the resistance per unit length of the Transmission Line should be

6.%7 ohms per meter at the freuency of hundred mega hertz. :f course as + said the

freuency changes then the acceptable value of resistance will change because the line

may not satisfy this condition at that freuency.

now having understood this as we mentioned earlier until and unless somebody

specifically tells you that the line is a lossy line we ta4e a liberty to create the line as a

loss less line because as first order as we have seen that the phase constant for a low loss

line and a lossy line is same, also the characteristic impedance of a low loss line is almost

real and that is same as the characteristic impedance of loss less line.


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So here onwards until and unless somebody specifically say include the losses in the

calculation of Transmission Line we will treat the line to be lossless and carry out all our

analysis for a Loss-Less Transmission Line. So essentially we will assume that

characteristic impedance of Transmission Line is given by this so E!

(

L . This uantity

is a real number and also the propagation constant is eual to the phase constant.

#Refer Slide Time$ %7$!! min&

So we have F @/ that is again eual to @'

(

L . 3ow with these parameters we will again

revisit the voltage and current e0pressions on Transmission Line and then we carry out

the analysis of the standing waves on the Transmission Lines.

"oing bac4 to the original euation of voltage and current as we have seen for a

Transmission Line whose origin has been defined at the load point as L !


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#Refer Slide Time$ %7$6! min&

the voltage and current euation can be given by this. This represent the forward traveling

wave, this represent the bac4ward traveling wave and now we will replace this uantity F

by @/ where 0 will be replaced by -%.

So now we have the voltage and current as a function of distance on the Transmission

Line and F will be replaced by @/ and E! is a real uantity. So we can write e0plicitly the

voltage and current on a Loss-Less Transmission Line. So the voltage is v as a function of

l that is eual to GH e @/l H G- e-@/l, by ta4ing GH e @/l common the same thing can be written

as GH e @/l I% H

-G

G+ e-@)/lJ.

9nd this uantity as we already 4now is nothing but the reflection coefficient at the load

end so this uantity we denote by the reflection coefficient at the load end KL so

-G

G+ as

we have seen earlier is nothing but eual to KL which is eual to

L !

L !

E E

E HE

−

.


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So now the voltage at any location on the Transmission Line can be given as GH e @/l I% H

KL e-@)/lJ.

#Refer Slide Time$ %;$** min&

Similarly + can ta4e the current euation + can substitute F @/l in the current euation +

can get the current that any location on the line +#l&

@ l -@ l

! !

G Ge e

Z Z

β β + −−

9gain ta4ing

@ l

!

Ge

Z

β +

common we can write down here this is

@ l

!

Ge

Z

β +

I% - KL e-@)/lJ where

-G

G+ will be eual to KL.


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#Refer Slide Time$ %1$!1 min&

These two terms as we 4now essentially represent the forward and bac4ward traveling

wave so the whole e0pression here essentially represent super position of the forward and

the bac4ward traveling wave which is nothing but a standing wave on a Transmission

Line.

So here the e0pression for voltage and current represent the standing voltage and standing

current wave on the Transmission Line.

3ow we can investigate certain features for the Transmission Line from here and first

thing what we will note is that there are two terms either in voltage or current so you are

having this term which is having amplitude one and then you arriving at second term

whose amplitude is modulus of this uantity KL plus a phase which is the phase of KL plus

this uantity phase which is minus @)/l. Writing very e0plicitly the comple0 reflection

coefficient in terms of its magnitude and phase let us say + define KL KL @

e Lφ

where Lφ

is the phase of the reflection coefficient at the load end.


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#Refer Slide Time$ %M$7% min&

Then + can write down the current and voltage e0plicitly in terms of this magnitude of

reflection coefficient and the phase. So finally + have two e0pressions here one for

voltage G#l& GH e @/l I% H KL ( ) @ -) l

e φ β

J and the current +#l& will be

@ l

!

Ge

Z

β +

I% - KL ( ) @ -) l

e φ β

J.

#Refer Slide Time$ )!$7! min&


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3ow let us see how the voltage and current varies if + measure the magnitude of the

voltage and current along the Transmission Line, what is the variation of the voltage and

current along the Transmission Line.

So first thing what we will notice is as you move along the Transmission Line this

uantity L is positive because we are moving towards the generator so as + move towards

the generator the phase becomes more and more negative so this uantity # Lφ

- )/l&

phase becomes more and more negative or in terms of a comple0 plane when the phase

become more negative essentially we will move on the cloc4wise direction. So by

moving towards the generator the phase becomes more and more negative the amplitude

of this thing remains constant this term and the total voltage will be the vector sum of this

term and this term is the real term, this term is the comple0 term whose phase is given by

that and whose magnitude is given by KL.

So essentially this is saying that this is the vector of unity one which represent the first

term then + have another vector whose magnitude is KL and the phase of this is # Lφ

- )/l&

and as + move towards the generator the phase becomes more negative but the magnitude

of this remains constant. That means this whole uantity represent the circle where the point moves on this circle as the L changes. So this motion around this is towards

generator and the magnitude of this term in the curly brac4ets is given by the vector

which is the @oining of these two points so this is nothing but I% H K L ( ) @ -) l

e φ β

J magnitude

of this uantity.


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#Refer Slide Time$ )6$!6 min&

The first term is this vector which is unity the second term is this vector which is rotating

as we move towards the generator on the Transmission Line and the magnitude of the

total uantity here essentially varies as we move on the Transmission Line. 3ow the thing

to note is when it moves on a circle at some phase when this uantity is zero or )= or 7=

e @! or e @)= or e @7= that is eual to H%.

So + get a magnitude which is ma0imum which is represented by this point that is nothing

but % H KL. Similarly if this uantity # Lφ

- )/l& is odd multiples of φ e @ odd multiples of =

will be eual to -% so + will get one minus mod K L that is the minimum value which +

will see for this term here which is represented by this point where the two terms cancel

each other.

+ see that the variation of voltage and current is bound by two limits when the two terms

directly add each other that time + will see the ma0imum voltage. +f they cancel each

other i will see the minimum of voltage similarly when these two terms add each other +

will get the ma0imum current and when these two terms cancel each other + will get

minimum of the current.


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So the condition is when this uantity is H% the voltage will become ma0imum but when

this uantity is H% and this uantity is H% there is a minus sign here so when this uantity

goes ma0imum at the same location l this uantity will go minimum thatNs a very

interesting thing now. Oarlier when we tal4ed about lumped circuit wherever we have

voltage higher we also have the current higher. 3ow what we are seeing here is that when

the voltage is ma0imum the current is minimum and vice versa when this uantity

become -% that time the voltage will be minimum but this uantity will become plus so +

will get the current ma0imum. :r in other words if + measure the magnitude of the

current and voltages on the Transmission Line the ma0imum current and ma0imum

voltages do not occur at the same location rather they are staggered in space. Wherever

there is ma0imum voltage there is minimum current and vice versa. So the standing wave

of the voltage and current are shifted with respect to each other in space on the

Transmission Line.

So if + plot the magnitude with the voltage on the Transmission Line so let us say this is

my Transmission Line which is terminated in some load here this is EL and now + plot the

voltage and current on the Transmission Line let us say + plot G the G will have a

variation which will go something li4e that this is the location where magnitude of

voltage is minimum, this is the location where the magnitude of the voltage is ma0imum

and as we saw @ust now that wherever the voltage is ma0imum the current will be

minimum and vice versa so the current will go something li4e that so this is the plot for

voltage G and this is plot for modulus of current. So this is this plot we are having G or

+ as a function of distance l ! and distance is measured towards the generator.


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#Refer Slide Time$ )?$6! min&

So in this location + get voltage ma0imum and current minimum, when + go here + get

current ma0imum and voltage minimum. The important thing is when the standing waves

are present on the Transmission Line then the voltage and current waveforms are shifted

with respect to each other and ma0imum of voltage aligns with minimum of current and

vice versa.

3ow having understood this + can again go bac4 and loo4 at this e0pression of the voltage

and current. +f + go to the location where the voltage is ma0imum that means this uantity

is H% the voltage will be GH multiplied by % H KL because this uantity is H% at the same

location + will have the current which will be minimum as we saw so this will be +#l&

!

G

Z

+

#% - KL& where E! is real since the line is loss less.

So once + 4now the voltage and current + can find out the impedance at this location

where the voltage is ma0imum or the voltage is minimum or when the current is

minimum or the current is ma0imum. The interesting thing to note here is irrespective of


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phase of GH when the voltage is ma0imum or minimum the ratio of G and + is a real

uantity. +f + ta4e a ratio of these two

( )

( )

G l

+ l that is eual to E!

@# ) &

L

@# ) &

L

% e

%- e

L

L

l

l

φ β

φ β

−

−

+ Γ

Γ and

when voltage is ma0imum or minimum this uantity is H% or -% so for ma0imum voltage

# Lφ

- )/l& is even multiple of = that is it is !, )=, 7= and so on, for minimum voltage # Lφ

-

)/l& is eual to odd multiple of = that is =, 6=, *= and so on.

#Refer Slide Time$ 6!$)% min&

So when the voltage is ma0imum this uantity is H% so E ! is real, now this uantity is one

% H KL denominator this uantity is again H% so this uantity is % - K L so the ratio of this

uantity is the real uantity. When the voltage is ma0imum the impedance seen on the

Transmission Line is real irrespective of what the line is terminated in.

Oven if the line is terminated in comple0 impedance if + move on the Transmission Line

and go to the location where the voltage magnitude is ma0imum, at that location the

impedance measured will be always real. Similarly when + go to a location where the

voltage is minimum this uantity will be eual to -%so + will get % - K L you have % H KL


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now. 9nd again this uantity will be real so now we ma4e a very important conclusion

and that is on a Transmission Line wherever there is a voltage ma0imum the impedance

measured is real, wherever there is a voltage minimum the impedance measured is real.

So irrespective of what the impedance with which the line is terminated you will always

find these points on the line where the voltage is ma0imum or minimum and that location

the impedance measured will be purely a real uantity.

What will the value of these ma0imum or minimum impedancesP We can substitute this

so at a location where the voltage is ma0imum and the current is minimum that is the

highest impedance you are going to measure on the Transmission Line. So this uantity

when the voltage is ma0imum at that location we get the ma0imum possible impedance

which we can measure on the Transmission Line. So we can get the ma0imum impedance

which one can see on the Transmission Line and let us call that as E ma0 is nothing but

ma0

min

G

+ .

#Refer Slide Time$ 6)$*) min&


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9nd since we have seen that this uantity when you are having the voltage ma0imum and

current minimum that time the phase difference between them is zero so this uantity is

real uantity so Ema0 is nothing but R ma0 as resistive impedance, from here if + substitute

this eual to plus one + will get R ma0 E!

L

L

%

%-

+ Γ

Γ .

Similarly if + go to a location where the voltage is minimum so this value will be % - K L

but at the same time the current will be ma0imum so that is the lowest value of

impedance you can see on the transmission line. So you get the minimum impedance on

the line which is Emin which will be

min

ma0

G

+ and that will be eual to E!

L

L

%-%H

Γ Γ .

#Refer Slide Time$ 67$%7 min&

So once the load impedance on the line is 4nown the reflection coefficient KL is 4nown its

magnitude is 4nown. + 4now what the ma0imum and minimum value of impedance is +

can see on the Transmission Line. So as we move on the Transmission Line the


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impedance is going to vary as we saw because of impedance transformation but there is a

bound on this impedance variation the lowest value of impedance which one can see on

Transmission Line is Emin or R min is resistive and the ma0imum value which one can see

on the Transmission Line is R ma0 which is given by that.

3ow once we are having the voltage standing wave on Transmission Line at high

freuencies the measurement of phase is rather complicated. Qou can measure the

amplitude of the signal rather reliably but the measurement of phase is little uncertain. So

at high freuency normally we have an effort to estimate the phase not in the direct

manner but in indirect manner. 2y carrying out the measurements of only magnitude 4ind

of uantities we would li4e to estimate the phase of the signal and as we have seen that

the phase of the signal in time get translated into the phase space because the total phase

which we seen on a wave is a combination of space and time or in other words the phase

relationship between the two waves the forward and the bac4ward traveling wave that is

related to the time phase as well as the space phase.

9nd since this total phase governs the location of ma0ima and minima of the standing

wave noting the location of ma0imum and minimum on Transmission Line one can

estimate the phase which is there with the signal. So what we do now is we define a

parameter for the standing wave which is a parameter of only amplitude variation on the

Transmission Line and that uantity is called the voltage standing wave ratio. +t is

essentially a measure of what is the relative contribution of the reflected wave to the

incident wave if the reflected wave is zero then there is no standing wave you will have

only traveling wave if the reflected wave is full then you will have completely developed

standing wave.

So the interference of the two waves the forward and the bac4ward waves are going to

give me this variation of the ma0imum to minimum. So we define this uantity the value

which you get for the ma0imum magnitude on the standing wave and the minimum

amplitude on the standing wave if + call the ma0imum value as G ma0 and the minimum

value as Gmin then the ratio of Gma0 to Gmin magnitude is the voltage standing wave ratio.

9nd this uantity is a very important uantity because without carrying out any phase


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measurement we can measure this uantity on the Transmission Line. Recall reflection

coefficient is a comple0 uantity so if you want to have the complete 4nowledge of the

reflection coefficient then we have to get its amplitude and phase. owever the uantity

which you are defining now is called the voltage standing wave ratio which is measured

by only amplitude measurement.

So by measuring the ma0imum and minimum magnitude of the standing wave we get this

uantity called the voltage standing wave ratio, normally it is denoted by

ma0

min

G

G.

#Refer Slide Time$ 61$7? min&

9nd what is the ma0imum value of voltage which + can see on the line is{ }LG %H

+ Γ

and the minimum value which + can see on the Transmission Line is{ }LG %-

+ Γ .


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The GH cancels so the voltage standing wave ratio is

L

L

%H

%-

Γ

Γ , in short the voltage

standing wave ratio is called as GSWR. So the GSWR for a load whose reflection

coefficient magnitude is LΓ is given by this.


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#Refer Slide Time$ 6M$*! min&

Since the line is lossless the reflection coefficient at the load is

L !

L !

E

E

Z

Z

−

+ and E! is the real

uantity for a Loss-Less Transmission Line so EL can have any comple0 impedance

terminated on the line but E! is real without much effort one can see that the magnitude of

this uantity is all ways less than one. So the mod of LΓ is always less than or eual to %

and that is ma4es a physical sense because what the LΓ

is telling you is the relative

amplitude of the reflected wave compared to the incident wave.


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#Refer Slide Time$ 7!$7) min&

Since we do not have any energy source on the load point the part of the energy only can

get reflected so the amplitude of the reflected wave has to be always less than or eual to

the incident wave. So for any passive loads the LΓ

is always less than or eual to % so in

a condition when EL ! or EL + will get the magnitude of

LΓ

% otherwise thisuantity will be always less than or eual to one so if + want to see very specific load

impedances for which the LΓ

condition will be satisfied. + will see there will be three

cases. (ase one will be when EL ! that means the line is short circuited at the load end

so this is a condition for a short circuited line. + substitute EL ! so + get LΓ

-% in this

case + get LΓ

-% or LΓ

%.


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#Refer Slide Time$ 7)$%! min&

The second case if + ta4e EL that means the line is open circuited then again + can

substitute EL , ta4e EL first common so it will become

!%> E

∞ so this will be eual to

H%. So this will give me LΓ

H% giving me again LΓ

H%.

The third case is that if the line is terminated in an ideal reactance so the third case is if

EL @ times 0 your reactance then again so this is pure reactance then LΓ

will be eual to

!

!

@0 E

@0 H E

−

.

This uantity in general will be comple0 but the LΓ

will be eual to again H%.


30/35

#Refer Slide Time$ 76$7; min&

So we have these three situations when the line is short circuited, when the line is open

circuited at the load end or when the line is terminated in a pure reactance the mod of

reflection coefficient is one. What that means is if the line is either short circuited or open

circuited or terminated in an ideal reactance there will be full reflection from the load end

and that ma4e sense that since + am having short circuit or open circuit or pure reactance

there is no energy absorbing element available at the load end of the line neither short

circuit can absorb power nor open circuit can absorb power nor a ideal reactance can

absorb power.

So whatever power the traveling wave ta4es to the load end there is no option but to ta4e

the entire power bac4 in the reflected waveform. Whatever wave reaches to the load end

that completely get reflected if any of these three conditions are satisfied and that is what

essentially is represented by this that the magnitude of reflection coefficient becomes

eual to one. That means the entire energy with which reaches to the load it gets reflected

so what we see from here is the reflection coefficient is always less than one it becomes

eual to one in this special cases that is short circuit, open circuit or ideal reactance other

wise the magnitude of the reflection coefficient is always less than one.


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3ow if + substitute this into the uantity which we have defined voltage standing wave

ratio now this uantity LΓ

is less than or eual to one. So in the best case when this

uantity is zero we will get a GSWR eual to one otherwise this uantity will be always

greater than one. So as we got the upper bound on reflection coefficient LΓ

. So we can

also define the bounds on the voltage standing wave ratio GSWR and that is when LΓ

! then the GSWR %. owever when LΓ

goes to % that time this uantity is infinity so

we have GSWR its bounds are one eual to or less than eual to infinity.

#Refer Slide Time$ 7;$))&

9nd which is the better caseP The LΓ

! represents no reflected wave on Transmission

Line that means we have full power transfer to the load. So LΓ

going to zero is the best

case as far the reflection is concerned or in other words when GSWR % that is the best

case for the power transfer on the line. 9s the GSWR increases it indicates higher and


32/35

higher value of LΓ

that is higher and higher reflection on the Transmission Line or

lesser and lesser efficiency of power transfer to the load.

So in every circuit design our effort is to ma4e the GSWR as small as possible or as closeas to % as possible. igher the value of GSWR indicates more mismatch on the

Transmission Line or higher value of the reflected wave on the Transmission Line.

So this uantity GSWR is one of the very important uantity in the measurement at high

freuencies. Whenever we design a circuit we essentially try to measure the GSWR on

the Transmission Line and we try to ma4e the GSWR as close to one as possible, ma4ing

sure that the circuit is efficiently transferring power to the load end of the line.

3ow once you have defined this parameter then one can relate this to the ma0imum

and minimum impedance which one can see on Transmission Line. We have already seen

that the ma0imum value of the impedance which we can see on the line is R ma0 E!

L

L

%H

%-

Γ

Γ but now you 4now this uantity is a measurable uantity and that is nothing

but the voltage standing wave ratio .

So the ma0imum value of the resistance or the impedance which we can again see on the

line is nothing but E! into wave standing wave ratio .

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Similarly the minimum resistance which + will see on the line is this is

% ρ so this is eual

to

!E

ρ .

So + get from here R ma0 E! and R min on the line is eual to

!E

ρ . So for any line which is

terminated in arbitrary impedance if + move to a location where the voltage is ma0imum

then + 4now the value of the impedance at that location because measuring the voltage

ma0ima and minima + can find out the value of this voltage standing wave ratio , + 4now

the characteristic impedance align apriori so + 4now the ma0imum value of the

impedance which line will show.

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Similarly at the location where the voltage is minimum the impedance measured will be

R min and that will be the characteristic impedance divide by the voltage standing wave

ratio. 3ow if you recall we had mentioned that if the impedance is 4nown on any point on

Transmission Line then you can always transform that impedance to any other location

on the Transmission Line. Then immediately it stri4es to us that 4nowing the location at

the voltage ma0imum or minimum the impedance value is 4nown there. so now + can

transform either side of Transmission Line towards the load so that + can get the load

impedance.

3ow this essentially opens a measurement techniue for the un4nown impedance at high

freuency if you are having a comple0 impedance its measurement is uite tedious

because we cannot re-measure the phase very accurately. 2ut + have a mechanism of

measuring the phase indirectly as we already mention the phase get reflected into the

standing wave pattern or the location of the voltage ma0ima and minima. So if + measure

the voltage standing wave ratio and the location of the voltage ma0ima and minima + can

always transform this impedance R ma0 or R min to the location of the load which is nothing

but the load impedance.


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Documents

Transmission Lines and E.M. Waves lec 05.docx