Transmission Lines and E.M. Waves lec 06.docx

8/20/2019 Transmission Lines and E.M. Waves lec 06.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 investigated the standing wave patterns on a

Transmission Line. We saw that the voltage standing wave pattern and the current

standing wave patterns are staggered with respect to each other that is wherever there is a

voltage maximum there is current minimum and wherever there is current maximum

there is voltage minimum. We also introduced a very important parameter which is a

measure of reflection on Transmission Line and that is the voltage standing wave ratio

(VSWR is the ratio of the maximum voltage seen on the Transmission Line to the

minimum voltage seen on the Transmission Line. !igher the value of VSWR worse is the

condition on the Transmission Line that is more reflection on Transmission Line.

"lso we esta#lish the #ound on the VSWR that is the VSWR lies #etween $ and % and

smaller the value of VSWR means #etter transmission less reflection on Transmission

Line so more transfer of power to the load. We also esta#lish the #ounds on the

impedance on the Transmission Line that for a given termination load on Transmission

Line there is a maximum and minimum impedance which one can see and that value is

characteristic impedance multiplied #y the VSWR and the characteristic impedance

divide #y VSWR.


2/34

(Refer Slide Time& '&') min

When we do the impedance transformation on Transmission Line the impedances are

#ound #y these two values the maximum value of impedance and the minimum value of

impedance.

Today we will study the impedance transformation on a Loss*Less Transmission Line and

then we will esta#lish a very important characteristic of impedance transformation on

Loss*Less Transmission Line. "nd then we will go to the calculation of power transfer to

the load.

"s we have seen for a general Transmission Line ta+ing the ratio of voltage and current at

any location on Transmission Line we get the impedance at that location.


3/34

(Refer Slide Time& '&- min

"nd we have seen the impedance is given #y this at any location l is e/ual to the

characteristic impedance multiplied #y this impedance transformation term, where this

/uantity is the hyper#olic cosine and hyper#olic sine which are given in terms of the

propagation constant 0 and the length on the Transmission Line.

1ow for a Loss*Less Transmission Line the propagation constant 0 is 23 so if 4 su#stitute

0 5 23 in this expression then 4 get the impedance transformation relationship for a Loss*

Less Transmission Line. "lso we have said that the a#solute impedances do not have any

meaning on a Transmission Line the impedances normali6ed to the characteristic

impedance are the meaningful /uantities so the same expression we have converted into a

normali6ed impedances where we define the impedance #y #ar that is the actual

impedance divide #y the characteristic impedance that is the normali6ed impedance.


4/34

(Refer Slide Time& '7&-$ min

So every impedance which we see on Transmission Line the load impedance, the

impedance at location l all of them are now normali6ed with respect to the characteristic

impedance. Then the normali6ed impedance transformation relationship essentially is

given #y this. "nd as we have said when normali6ed value is e/ual to one that time the

load impedance is e/ual to ', similarly when the normali6ed impedance at location l is

e/ual to one the value at that location is e/ual to '. So either we can use the normali6ed

impedance transformation relationship or we can use the normali6ed transformation

relationship.

!owever, now for a lossless line we su#stitute for 0 5 23 and get the relation for the Loss*

Less Transmission Line. 4f you su#stitute 0 5 23 for the Loss*Less Transmission Line the

cosh 0l is e/ual to the cosh 3l that is nothing #ut e/ual to cos 3l. Where as, the sinh 0l is

e/ual to sinh 23l that is e/ual to 2 sin 3l.


5/34

(Refer Slide Time& '-&7 min

Su#stituting these for hyper#olic cosine and sine in the general expression which we had

got last time we get now the impedance transformation relationship for the lossless line

and that is now the impedance at the location l is e/ual to ' the characteristic

impedance of the line and characteristic impedance let me remind you again is a real

/uantity for a Loss*Less Transmission Line so this /uantity is real. So (l 5

L ''

'

. cos l 2 . sin l

. cos l 2 . sin l L

Z β β

β β

+

+ .


6/34

(Refer Slide Time& '8&' min

4f 4 ta+e ' down here this /uantity

( )

'

. l

Z will #e normali6ed impedance, similarly 4 can

ta+e ' common from the numerator and denominator so the same expression as we had

o#tained earlier in terms of normali6ed impedance will #e e/ual to

L

L

. cos l 2 sin l

cos l 2 . sin l

β β

β β

+

+

.


7/34

(Refer Slide Time& '9&'- min

So either of the impedance transformation relationship can #e used when we transform

the impedance on a Loss*Less Transmission Line from one point to another this is the

a#solute impedance, this is the normali6ed impedance.

:nce we get the impedance transformation relationship then we can esta#lish some very

important characteristic of impedance transformation on Transmission Line and that is we

+now that when you move on a Transmission Line #y a distance of λ

the voltage

characteristic the standing wave characteristic will repeat. So if you ta+e a ratio of

voltage and current at a location we expect that these characteristics would repeat at λ

.

Similarly the special points are if 4 move a distance #y λ we will also see there is a

special distance of ,λ

there is something very special happens and third characteristic is

if 4 terminate the line into the characteristic impedance the impedance will always #e

e/ual to the characteristic impedance irrespective of the length of the line.


8/34

So we have three very important characteristics of the impedance transformation on a

line. The first one that is the impedance value repeats every λ

distance.

(Refer Slide Time& ')&7- min

;oing #ac+ to this expression here let us say 4 have a impedance value of (l at some

location on the line now if 4 move #y a distance of λ

from here that means if 4 go to a

location l < λ

then the impedance at that location would #e where l will #e replaced #y

l < λ


9/34

So, initially let us say at location l, impedance is e/ual to (l, what we want to find out is

the impedance at (l < λ

so 4 replace l #y l < λ

, if 4 use the normali6ed relation

this will #e e/ual to

( ) ( )( ) ( )

.(l cos l< 2 sin l<

cos l< 2 .(lsin l<

λ λ β β

λ λ β β

+

+ .

1ow( )l< λ β 5

( )

l<

π λ λ where

5

π β

λ and this again is e/ual to #eta into 3l plus =

will cancel so 4 will get only >.

(Refer Slide Time& $'&-7 min

So ( )cos l< λ β is nothing #ut ( )cos l


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1ow( )cos


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(Refer Slide Time& $&'- min

Su#stituting now this( )cos l< λ β and ( )

sin l<

λ β into the impedance relation here

we get the impedance at location( )l< λ as

( ) *.(l cos l * 2 sin l

. l< *cos l 2 .(lsin l

β β λ β β

=

+ .

(Refer Slide Time& $7&'- min


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4 can ta+e minus common from numerator and denominator so all this /uantity will #e

plus this is exactly same as the impedance (l at location l on the Transmission Line so

this is nothing #ut e/ual to normali6e impedance at location l. That is very important

characteristic on Transmission Line that is the impedance repeats itself over a distance of

λ

or in other words the impedance transformation there is only memory of λ

distance on Transmission Line how many cycles of λ

have gone on Transmission line

we will never #e a#le to find out from the +nowledge of the impedance.

So no matter what is the length of the Transmission Line essentially A λ

A is the special

information which is availa#le from the impedance transformation on Transmission Line.

So this is one of the very important characteristic for a Loss*Less Transmission Line that

every distance of λ

the impedance characteristic repeats.

The second characteristic is the impedance at a distance of ,λ

if 4 move #y a distance of

,λ . "gain if 4 +now the normali6ed impedance at location l we want to find out what is

the value of this impedance at l < ,λ

so the /uantity 3(l < ,λ

will #e e/ual to 3l plus

#eta is

π

λ so this is

π

λ into ,λ

so that is e/ual to 3l <

π

.


13/34

(Refer Slide Time& $-&$$ min

1ow su#stituting this for 3l < ,λ

in the transformation relation 4 get the normali6ed

impedance at l < ,λ

which will #e e/ual to (l( )cos l< ,λ β which is nothing #ut

( )cos l< π

β which again will #e *sin 3l so this will #e *sine 3l < 2 ( )sin l< ,λ

β so that is

( )sin l< π β so that will #e e/ual to cos 3l divided #y *sine 3l plus 2 .(l cos 3l.


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(Refer Slide Time& $8&- min

4f 4 ta+e the 2 common from here this will #ecome .(l cos 3l < 2 times sin 3l, this will

#ecome cos 3l < 2 .(l sin 3l so that /uantity is nothing #ut

$

.(l of that.

So this is a very important characteristic of Transmission Line that every distance of ,λ

the normali6ed impedance inverts itself note the word normali6e it is not the a#solute

impedance #ecause a#solute impedance if it inverts then dimensionally it will #ecome

admittance the normali6ed impedance does not have any unit it does not have any

dimensions it is a dimensionless /uantity. So for a distance of ,λ

the normali6ed

impedance will invert itself so if 4 have a value of impedance at some location on

Transmission Line if it is greater than ' after a distance of ,λ

it will definitely going to

#e less than ' #ecause the normali6ed impedance will #e the inverted value of the

impedance at the previous location so #y this a distance of ,λ

the normali6ed impedance


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will invert. "gain the impedance will invert after ,λ

so impedance will #ecome same

that is what essentially the previous property that every distance of λ

the impedance is

same.

So when we tal+ a#out the periodicity of the impedance on Transmission Line every λ

the a#solute or normali6ed impedance repeats itself where as every distance of ,λ

the

normali6ed impedance inverts itself and you will see later on when we tal+ a#out the

impedance matching characteristics this property is used extensively for finding out the

impedance transformation which can match impedances on the Transmission Line.

The third characteristic is the matched condition characteristic which you already

discussed #riefly that we tal+ed in general Transmission Line so third characteristic is the

matching condition on the Transmission Line and that is if the line is terminated in the

characteristic impedance then the impedance seen at every point on Transmission Line is

e/ual to the characteristic impedance.

So if 4 ta+e L 5 ' that is L.

5

L

'

.

. 5$, the impedance .(l at any location on

Transmission Line will #e e/ual to .(l which is $ so

cos l 2 sin l

cos sin l 2 sin l

β β

β β

+

+ which is

again e/ual to $ so this is sin3l which is e/ual to $.


16/34

(Refer Slide Time& $B&7 min

So irrespective of the length of the Transmission Line if the line is terminated in the

characteristic impedance then the impedance seen at every point on Transmission Line is

e/ual to the characteristic impedance and if you recall we had discussed this condition

this #ecause this is a very special condition what that means is once the line is terminated

into the characteristic impedance one does not have to worry a#out the impedance

transformation on Transmission Line you can use any piece of Transmission Line and the

impedance at the input of the Transmission Line will #e always same which will #e e/ual

to the characteristic impedance.

We also see that when the L 5 ' that time the reflection coefficient is 6ero so there is no

reflected wave on Transmission Line you have only forward traveling wave on

Transmission Line and as we have argued earlier forward traveling wave always sees an

impedance which is e/ual to characteristic impedance so this result is not very new we

have discussed this earlier when we were tal+ing a#out the general Transmission Lines

and that was if the line is terminated in the characteristic impedance then the impedance

seen at every point on the Transmission Line is e/ual to the characteristic impedance.


17/34

These are three very important characteristics of Loss*Less Transmission Line. let me

summari6e it again, first the impedance transformation repeats of a distance of a λ

all

impedance values repeat every λ

distance the normali6ed impedance inverts itself for

every ,λ

distance and if the line is terminated in the characteristic impedance then the

impedance seen at any point on the line is e/ual to the characteristic impedance.

With this understanding of the impedance transformation now we can go to the power

transfer calculation on the Transmission Line. 4nitially when we start the discussion on

the Transmission Line the purpose was to transfer the power from the generator to theload effectively. 4n the lossless case since there is no lossy element present on the line

ideally the load should #e such that the power ta+en from the generator is completely

transferred to the load.

!owever, as we have seen if the impedance is not e/ual to the characteristic impedance

then there will #e always reflection on Transmission Line that means whatever energy the

generators supplied that energy will reach to the load will not find a condition which is

favora#le for the maximum power transfer the part of the energy will get reflected #ac+

and this energy will come #ac+ and essentially will feed the generator #ac+.

1ow when we tal+ a#out the matching condition or the maximum power transfer

condition there are two issues here, one is when the power is given #y the generator it

should #e maximally transferred to the load it is the efficiency how the power get really

transferred to the load and second issue is when the reflected power come #ac+ and hits

the generator the generator actually is not capa#le of a#sor#ing power it is delivering

power so when this reflected power comes #ac+ with different amplitude and phase it

affects the performance of the generator it changes its phase characteristic amplitude

characteristic so it is desira#le that generator should not see any power coming and

hitting #ac+ so for these two purposes that the power whatever is given #y the generator


18/34

should #e completely transferred to the load and also no power should come #ac+ and hit

#ac+ to the generator we must ma+e sure always that the impedance which the generator

effectively sees e/ual to characteristic impedance so there is no reflection which will lead

to the generator, also at the load end we do something so that the maximum power is

transferred to the load.

These issues we will discuss little later #ut let us ta+e a very general case at the moment

and as+ if 4 have a Transmission Line which is connected to a generator on one end and

the load at the other end how much power will #e delivered to the load. "gain going #ac+

to the original voltage and current e/uations we can write down the power at the location

of the load that means at l 5 '. So what we discuss now is the power delivered to the load.

(Refer Slide Time& &' min

Let us start again with the #asic e/uation. We have voltage e/uation the voltage at l 5 ' is

the load end that is V(' 5 V


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Similarly 4 have current at l 5 ' at the load end that is '

V

.

+

D$ < CLE.

(Refer Slide Time& -& min

So from the general voltage and current relations on the Loss*Less Transmission Line 4

find out the voltage value at the load end, current value at the load end. So the power will

#e nothing #ut half real part of V4 con2ugate so the power delivered to the load p 5

FReDV(' 4G('E.

4f 4 su#stitute for the V(' and 4(' in this that will #e e/ual to F V


20/34

L '

L '

. @ .

. < .is the reflection coefficient so once 4 +now the load impedance 4 can calculate

the mod of reflection coefficient.

(Refer Slide Time& 9&$' min

So the power delivered if 4 +new the value of V


21/34

So one can say that we have now a wave which is going in the forward direction which is

having an amplitude V


22/34

1ow when this voltage wave reaches to the load part of energy is going to travel #ac+ and

there will #e again a traveling wave which is traveling #ac+wards with an amplitude of

V*. So since this wave also sees the impedance which is e/ual to the characteristic

impedance the power carried #y this wave will #e nothing #ut

'

V

.

−

.

So we get the power reflected #y the load in the #ac+ward wave that is Href 5 F

'

V

.

−

.

(Refer Slide Time& 7'& min

So Hforward was the power which was carried towards the load, H reflected is the power which

was ta+en away from the load so difference of these two powers is the one which isdelivered to the load. The net power which is delivered to the load is p is H forward * Hreflected


23/34

that is e/ual to

' '

V V$

. .

+ −

− and

V

V

−

+ as we +now is the reflection coefficient at the

load.

So this /uantity is nothing #ut if 4 ta+e

V+ common this will #e $ @ ACLA


24/34

(Refer Slide Time& 7$& min

This relation is exactly same as what we have derived earlier for the power delivery. So

what we see is when we do the power calculations on the Transmission Line either 4 can

go #y the circuit concept to find out the voltage and current at that location and then

simply ta+e the power delivered will #e half real part of V4G con2ugate or 4 can tal+ in

terms of the traveling waves find out how much power was carried #y the wave and how

much power was ta+en #ac+ in the form of a reflected wave, difference at the two power

will #e the power delivered to the load. So #y using either of the concepts one can find

out what is the power was supplied to the load.

This is the story for the real power that is the power which is actually supplied to the

load. :ne can as+ in general suppose 4 want to calculate the complex power at a

particular location how does that reflect or in general 4 can as+ the /uestion that if 4

calculate the power flow at any particular location on Transmission Line not necessarily

at the load end what does that indicates. So if 4 get the voltage and current at any ar#itrary

location on Transmission Line that is V(l at some location l is V


25/34

Iou have derived the same relation last time for the Loss*Less Transmission Line the

current 4(l will #e '

V

.

+

e 23lD$ < CL e*23lE.

(Refer Slide Time& 77&9 min

1ow again as 4 did in the previous case at the load point 4 have the voltage and current so

the total power the complex power H at that location l is e/ual to F (V 4G 4 can su#stitute

from here for (V 4G and 4 get F

'

V

.

+

D$ < ACLA < 4m (CL e

*23lE. 4t is simple alge#ra you

su#stitute the value of V and 4 in the expression here you get the complex power which

will #e essentially given #y this.

1ow this /uantity is 2ust the imaginary part so the real part of the power which tells you

the actual power delivered at that location is e/ual to this, this is the power which is the

imaginary part of the power what is called the reactive power. So here we have a resistive

power and here we have a reactive power.


26/34

(Refer Slide Time& 7-&7- min

So at a general location on Transmission Line the power is complex and of course that

will #e complex at the load end also. Jut the interesting thing to note here is this resistive

power at location l is exactly same as the power which we have got at the load end which

is this power. Since the line is lossless the resistive power at any location in the line is

same as the resistive power which we will see at the load and that ma+e sense #ecause if

the line is completely lossless then there is no a#sorption of power anywhere on the line

#ut at the load #ecause the load is the one where you have a resistive component and the

power can #e a#sor#ed at that location.

So wherever we are seeing on the Transmission Line it is telling you a power flow #ut

this power flow essentially is the power which is ultimately going to get delivered to the

load. So the resistive part of the power which is telling you the actual power flow should

#e independent of the location on the line #ecause this is the power which is ultimately

given to the load connected to the Transmission Line.

4f you loo+ at the reactive part of the power however this is the function of l and the

reactive part power tells you essentially the energy is stored at different locations on


27/34

Transmission Line so now we are having two things when we calculate the power on

Transmission Line there is a resistive power which tells you the flow of power at any

particular location on Transmission Line and this flow of power is exactly same as what

power would #e delivered to the load. !owever the reactive power tells you the energy

storage at different locations on line and that depends upon the value of voltage and

current at that location and since #ecause of standing wave the voltage and current is

varying along the Transmission Line the energy storage is different at different location

on Transmission Line.

So what we see in general is the reactive power will vary along the length of the line

however the resistive power which is the power delivered to the load will #e independent

of the location of the Transmission Line and that is a very important characteristic. So no

matter where you calculate the real power at the load end or generator end or in the

middle of the Transmission Line for a Loss*Less Transmission Line this power will #e

exactly same as what is ultimately delivered to the load. !aving done this now one can

come to the final /uestion of the analysis of the transmission line that everything we have

done now the impedance transformation relationship we developed we also analy6ed the

power flow on the Transmission Line #ut we have not evaluated final ar#itrary constant

of the voltage or current expression and that is V


28/34

So now from the #oundary condition #y connecting the generator and the load impedance

we calculate the final ar#itrary constant of the voltage expression that is V


29/34

So first thing what we can do is we can transform this impedance at this location and then

treat this circuit as the lump circuit. Let us also ma+e it little general let us say this

voltage source is having a internal impedance which is e/ual to s so let us say 4 connect

an internal impedance here of the voltage source which is given as s. Let us say 4 have

the input terminal which is given #y some ", "K and these are the locations which are

denoted #y J, JK.

:nce 4 +now this L and 4 +now this length then 4 can find out what is the transform

impedance at this location ", "K. Let us say this impedance if 4 see from here that is given

as KL so KL is the transformed impedance seen at the generator end for the length of this

line l as the terminating load L.

(Refer Slide Time& $&-9 min

:nce 4 transform the impedance then the whole circuit is the lumped circuit at this

location 4 do not have to worry a#out the distri#uted elements. so now this circuit is

e/uivalent to having a voltage source which is V S, the internal impedance for the voltage

source that is S and connected to a effective impedance which is KL, once 4 get that then

4 can find out what is the voltage this is the location now ", "K so #y using the lumped


30/34

circuit analysis 4 can find out the voltage at this location and also 4 can find out the

current at this location so the voltage here we call as V" and we call this current as 4" in

this location.

(Refer Slide Time& &-9 min

4 can put the lumped circuit analysis and find out what is V ". So, V" will #e

L

SL S

.K

V. K . ×+

and 4" will #e

"

L

V

.K.


31/34

(Refer Slide Time& 7& min

So from the lump circuit analysis at the generator end 4 +now the voltage and current at

the input end of the Transmission Line. 4 can write down the voltage and current at the

input end of the Transmission Line using the Transmission Line e/uations. So, now

+nowing the load end impedance termination L and a distance l from that essentially 4

have to get the voltage and current at a distance l from the load end. 1ow 4 +now from

the distri#uted elements then that the V at input end of the line V" is nothing #ut V at a

distance l from the load which is e/ual to V


32/34

(Refer Slide Time& -&'- min

So now 4 +now the value of V" from two sides, one is from the lumped element side the

V" is given #y this from the distri#uted elements on transmission line the V" is given #y

this. 4 can e/uate these two values of V" and 4" and from here 4 can solve for the

un+nown /uantity which is V


33/34

(Refer Slide Time& 8&7' min

:nce 4 +now the value of V


34/34

We define very important parameters on Transmission Line what are called the reflection

coefficients and the voltage standing wave ratios. Then we studied the impedance

transformation relationship on a Transmission Line we too+ a special case that was a

Low*Loss or a Loss*Less Transmission Line and then we found some important

characteristics of the Loss*Less Transmission Line, further we investigated the power

flow on Transmission Line and calculated how much power will #e delivered to the load

and ultimately we found out the final un+nown ar#itrary constants which was V

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Transmission Lines and E.M. Waves lec 06.docx