Transmission Lines and E.M.

Waves
Prof R.K. Shevgaonkar
Department of Electrical Engineering
Indian Institute of Technology Bombay

Lecture-6

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. Higher the value of VSWR worse is the
condition on the Transmission Line that is more reflection on Transmission Line.

Also we establish the bound on the VSWR that is the VSWR lies between 1 and ∞ and
smaller the value of VSWR means better transmission - less reflection on Transmission
Line, so more transfer of power to the load. We also establish the bounds 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 by the VSWR and the characteristic impedance
divide by VSWR.
(Refer Slide Time: 02:08 min)

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

bound by 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 establish a very important characteristic of impedance transformation on
Loss-Less Transmission Line. And then we will go to the calculation of power transfer to
the load.

As we have seen for a general Transmission Line, taking the ratio of voltage and current
at any location on Transmission Line we get the impedance at that location.
(Refer Slide Time: 02:45 min)

And we have seen the impedance is given by this Z at any location l is equal to the
characteristic impedance multiplied by this impedance transformation term, where this
quantity is the hyperbolic cosine and hyperbolic sine which are given in terms of the
propagation constant γ and the length on the Transmission Line.

Now for a Loss-Less Transmission Line the propagation constant γ is jβ; so if I substitute
γ = jβ in this expression then I get the impedance transformation relationship for a Loss-
Less Transmission Line. Also we have said that the absolute impedances do not have any
meaning on a Transmission Line the impedances normalized to the characteristic
impedance are the meaningful quantities so the same expression we have converted into a
normalized impedances where we define the impedance by bar; that is the actual
impedance divide by the characteristic impedance that is the normalized impedance.
(Refer Slide Time: 03:51 min)

So every impedance which we see on Transmission Line the load impedance, the
impedance at location l all of them are now normalized with respect to the characteristic
impedance. Then the normalized impedance transformation relationship essentially is
given by this. And as we have said, when normalized value is equal to one, that time the
load impedance is equal to Z0, similarly when the normalized impedance at location l is
equal to one the value at that location is equal to Z0. So either we can use the normalized
impedance transformation relationship or we can use the normalized transformation
relationship.

However, now for a lossless line we substitute for γ = jβ and get the relation for the Loss-
Less Transmission Line. If you substitute γ = jβ for the Loss-Less Transmission Line the
cosh γl is equal to the cosh βl that is nothing but equal to cos βl. Whereas, the sinh γl is
equal to sinh jβl that is equal to j sin βl.
(Refer Slide Time: 05:23 min)

Substituting these for hyperbolic 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 Z at the location l is equal to Z0 the characteristic
impedance of the line and characteristic impedance; let me remind you again is a real
quantity for a Loss-Less Transmission Line so this quantity is real. So Z(l) =
 Z cos l  j Z0sin l 
Z0  L .
 Z0 cos l  j ZLsin l 
(Refer Slide Time: 06:20 min)

Z l
If I take Z0 down here this quantity will be normalized impedance, similarly I can
Z0
take Z0 common from the numerator and denominator so the same expression as we had
 Z cos l  j sin l 
obtained earlier in terms of normalized impedance will be equal to  L .
 cos l  j Z L sin l 
(Refer Slide Time: 07:05 min)

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

the impedance on a Loss-Less Transmission Line from one point to another this is the
absolute impedance, this is the normalized impedance.

Once we get the impedance transformation relationship then we can establish some very
important characteristic of impedance transformation on Transmission Line and that is

we know that when you move on a Transmission Line by a distance of  the voltage
2
characteristic the standing wave characteristic will repeat. So if you take a ratio of

voltage and current at a location we expect that these characteristics would repeat at  .
2

Similarly the special points are if I move a distance by  we will also see there is a
2

special distance of  there is something very special happens and third characteristic
4
is, if I terminate the line into the characteristic impedance the impedance will always be
equal to the characteristic impedance irrespective of the length of the line.
So we have three very important characteristics of the impedance transformation on a

line. The first one that is the impedance value repeats every  distance.
2

(Refer Slide Time: 08:35 min)

Going back to this expression here let us say I have a impedance value of Z(l) at some

location on the line now if I move by a distance of  from here that means if I go to a
2

location l +  then the impedance at that location would be where l will be replaced by
2

l+ 
2

So, initially let us say at location l, impedance is equal to Z(l), what we want to find out is

the impedance at Z(l) +  so I replace l by l +  , if I use the normalized relation

2 2


this will be equal to 

 Z(l) cos l+ 
2  
 j sin l+   
2 . 


 
cos l+ 
2  
 j Z(l)sin l+   
2  
 
Now  l+  2 = 2  l+  2  where  =
2

and this again is equal to beta into βl plus λ

will cancel so I will get only π.

(Refer Slide Time: 10:53 min)

So cos l+   2  is nothing but cos   l+  , similarly sin l+   2  is nothing but

sin   l+  .

Now cos  +  = -cos θ and sin  +  = -sin θ. So this quantity cos l+   2  is nothing
but –cos βl.

So if I substitute into this I get, cos l+   2 = cos   l+  = –cos βl

Similarly sin l+  2  = sin   l+  = –sin βl.

(Refer Slide Time: 12:05 min)


Substituting now this cos l+ 
2  and sin  l+  2   into the impedance relation here


we get the impedance at location l+  2 as Z l+  2    -cos
-Z(l) cos l - j sin l 
 l  j Z(l)sin l 
.

(Refer Slide Time: 13:05 min)

I can take minus common from numerator and denominator; so all this quantity will be
plus this is exactly same as the impedance Z(l) at location l on the Transmission Line so
this is nothing but equal to normalize impedance at location l. That is very important
characteristic on Transmission Line that is the impedance repeats itself over a distance of
 or in other words the impedance transformation there is only memory of 
2 2

distance on Transmission Line how many cycles of  have gone on Transmission line
2
we will never be able to find out from the knowledge of the impedance.

So no matter what is the length of the Transmission Line essentially |  | is the special
2
information which is available 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  the impedance characteristic repeats.

2

The second characteristic is the impedance at a distance of  if I move by a distance of

4
 . Again if I know the normalized impedance at location l we want to find out what is
4

the value of this impedance at l +  so the quantity β(l +  ) will be equal to βl plus
4 4
2 2 
beta is so this is into  so that is equal to βl + .
  4 2
(Refer Slide Time: 15:11 min)

Now substituting this for βl +  in the transformation relation I get the normalized
4

impedance at l + 
4
which will be equal to Z(l) cos l+  4  which is nothing but
cos  l+   which again will be -sin βl so this will be -sine βl + j sin  l+   so that is
2 4

sin  l+   so that will be equal to cos βl divided by -sine βl plus j Z(l) cos βl.
2
(Refer Slide Time: 16:25 min)

If I take the j common from here this will become Z(l) cos βl + j times sin βl, this will
1
become cos βl + j Z(l) sin βl so that quantity is nothing but of that.
Z(l)

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

4
the normalized impedance inverts itself note the word normalize it is not the absolute
impedance because absolute impedance if it inverts then dimensionally it will become
admittance the normalized impedance does not have any unit it does not have any

dimensions it is a dimensionless quantity. So for a distance of  the normalized

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

Transmission Line if it is greater than Z0 after a distance of  it will definitely going to

4
be less than Z0 because the normalized impedance will be the inverted value of the

impedance at the previous location so by this a distance of  the normalized

4

impedance will invert. Again the impedance will invert after  so impedance will
4
become same that is what essentially the previous property that every distance of  the
2
impedance is same.

So when we talk about the periodicity of the impedance on Transmission Line every 
2

the absolute or normalized impedance repeats itself whereas every distance of  the
4
normalized impedance inverts itself and you will see later on when we talk about 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 briefly that we talked 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
equal to the characteristic impedance.

ZL
So if I take ZL = Z0 that is ZL = =1, the impedance Z(l) at any location on
Z0

 cos l  j sin l 
Transmission Line will be equal to Z(l) which is 1 so   which is
 cos sin l  j sin l 
again equal to 1 so this is sinβl which is equal to 1.
(Refer Slide Time: 19:43 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
equal to the characteristic impedance and if you recall we had discussed this condition
this because 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 about 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 be always same which will be equal
to the characteristic impedance.

We also see that when the ZL = Z0 that time the reflection coefficient is zero; 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 equal to characteristic impedance so this result is not very new we
have discussed this earlier when we were talking about 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 equal to the characteristic impedance.
These are three very important characteristics of Loss-Less Transmission Line. Let me

summarize it again, first the impedance transformation repeats of a distance of a  all

2

impedance values repeat every  distance the normalized impedance inverts itself for
2

every  distance and if the line is terminated in the characteristic impedance then the
4
impedance seen at any point on the line is equal to the characteristic impedance.

With this understanding of the impedance transformation now we can go to the power
transfer calculation on the Transmission Line. Initially when we start the discussion on
the Transmission Line the purpose was to transfer the power from the generator to the
load effectively. In the lossless case since there is no lossy element present on the line
ideally the load should be such that the power taken from the generator is completely
transferred to the load.

However, as we have seen if the impedance is not equal to the characteristic impedance,
then there will be 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
favorable for the maximum power transfer the part of the energy will get reflected back
and this energy will come back and essentially will feed the generator back.

Now when we talk about the matching condition or the maximum power transfer
condition there are two issues here, one is when the power is given by the generator; it
should be maximally transferred to the load it is the efficiency how the power gets really
transferred to the load and second issue is when the reflected power come back and hits
the generator the generator actually is not capable of absorbing power it is delivering
power so when this reflected power comes back with different amplitude and phase it
affects the performance of the generator it changes its phase characteristic amplitude
characteristic so it is desirable that generator should not see any power coming and
hitting back so for these two purposes that the power whatever is given by the generator
should be completely transferred to the load and also no power should come back and hit
back to the generator we must make sure always that the impedance which the generator
effectively sees equal 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 but let us take a very general case at the moment
and ask if I have a Transmission Line which is connected to a generator on one end and
the load at the other end how much power will be delivered to the load. Again going back
to the original voltage and current equations we can write down the power at the location
of the load that means at l = 0. So what we discuss now is the power delivered to the
load.

(Refer Slide Time: 24:20 min)

Let us start again with the basic equation. We have voltage equation the voltage at l = 0 is
the load end that is V(0) = V+ into 1 + ΓL e -2βl
but l = 0 at the load end so this quantity
will be {1 + ΓL} so this is the voltage at the load end. This is we are talking about the
voltage and current at the load end of the line.
 V
Similarly I have current at l = 0 at the load end that is {1 + ΓL}.
Z0

(Refer Slide Time: 25:24 min)

So from the general voltage and current relations on the Loss-Less Transmission Line I
find out the voltage value at the load end, current value at the load end. So the power will
be nothing but half real part of VI conjugate so the power delivered to the load p =
½Re{V(0) I*(0)}.

If I substitute for the V(0) and I(0) in this that will be equal to ½ V+ multiplied by I
conjugate which is V+ conjugate divide by Z0 conjugate where Z0 is the real quantity so
2
V
this will be equal to multiplied by the conjugate of this so this will be equal to if I
Z0
take the real part of this then that will be equal to {1 - |ΓL|2}. So once I know the
impedance at the load I know the reflection coefficient and let me write there ΓL =
Z L – Z0
is the reflection coefficient so once I know the load impedance I can calculate
Z L + Z0
the mod of reflection coefficient.
(Refer Slide Time: 27:10 min)

So the power delivered if I knew the value of V+ then I can find out what is the power
delivered to the load, how do we find out V+ which we will discuss in a minute.
However at this location if I know the amplitude of the forward traveling wave and if I
know the load impedance I can calculate the value of the power delivered to the load.

Here we have calculated the power which is from the circuit point of view; that if I know
the voltage and current at a particular location I can apply the relation that the power
delivered at that location is VI conjugate the real part of that and from there we get this
power delivered to the load. We can use a little different argument to come to the same
answer and that is on the Transmission Line the power was supplied by the generator in
the form of a traveling wave which was going towards the load and as we already said
that the traveling wave always sees an impedance which is equal to characteristic
impedance. So if the traveling wave had an amplitude V+ it is as if this wave is supplying
a power to Z0 which is the real quantity in a lossless case.

So one can say that we have now a wave which is going in the forward direction which is
having an amplitude V+ this wave always sees an impedance which is equal to Z0 so the
 2
V
power carried by this wave will be half of ½ . So the power carried by forward
Z0
2
V
wave Pfor will be equal to ½ because this is the voltage V+ which is traveling on
Z0
this; it is seeing an impedance equal to the characteristic impedance Z0 which is the real
quantity.

(Refer Slide Time: 29:13 min)

2
V
So the power delivered to the forward wave is ½ where ½ is the factor for the rms
Z0
value of the power.

Now when this voltage wave reaches to the load part of energy is going to travel back
and there will be again a traveling wave which is traveling backwards with an amplitude
of V-. So since this wave also sees the impedance which is equal to the characteristic
2
V
impedance the power carried by this wave will be nothing but .
Z0
 2
V
So we get the power reflected by the load in the backward wave that is Pref = ½ .
Z0

(Refer Slide Time: 30:22 min)

So Pforward was the power which was carried towards the load, Preflected is the power which
was taken away from the load so difference of these two powers is the one which is
delivered to the load. The net power which is delivered to the load is p is Pforward - Preflected
  2 V 2 
1V  V
that is equal to    and as we know is the reflection coefficient at the
2  Z0 Z0  V
 
load.

2
So this quantity is nothing but if I take V  common this will be 1 – |ΓL|2
(Refer Slide Time: 31:22 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 I can
go by the circuit concept to find out the voltage and current at that location and then
simply take the power delivered will be half real part of VI* conjugate or I can talk in
terms of the traveling waves find out how much power was carried by the wave and how
much power was taken back in the form of a reflected wave, difference at the two power
will be the power delivered to the load. So by 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. One can ask in general suppose I want to calculate the complex power at a
particular location how does that reflect or in general I can ask the question that if I
calculate the power flow at any particular location on Transmission Line not necessarily
at the load end what does that indicate. So if I get the voltage and current at any arbitrary
location on Transmission Line that is V(l) at some location l is V+ ejβl{1 + ΓL e-j2βl}.
You have derived the same relation last time for the Loss-Less Transmission Line the
V  jβl
current I(l) will be e {1 + ΓL e-j2βl}.
Z0

(Refer Slide Time: 33:27 min)

Now again as I did in the previous case at the load point, I have the voltage and current
so the total power the complex power P at that location l is equal to ½ (V I*) I can
2
V
substitute from here for (V I*) and I get ½ {1 + |ΓL|2 + 2Im (ΓL e-j2βl)}. It is simple
Z0
algebra: you substitute the value of V and I in the expression here you get the complex
power which will be essentially given by this.

Now this quantity is just the imaginary part. So the real part of the power which tells you
the actual power delivered at that location is equal 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.
(Refer Slide Time: 35:35 min)

So at a general location on Transmission Line, the power is complex and of course that
will be complex at the load end also. But 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 makes sense
because if the line is completely lossless then there is no absorption of power anywhere
on the line but at the load because the load is the one where you have a resistive
component and the power can be absorbed at that location.

So wherever we are seeing on the Transmission Line it is telling you a power flow but
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
be independent of the location on the line because this is the power which is ultimately
given to the load connected to the Transmission Line.

If you look 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
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 be delivered to the load. However 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 because 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 be 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 be
exactly same as what is ultimately delivered to the load. Having done this now, one can
come to the final question of the analysis of the transmission line that everything we have
done now; the impedance transformation relationship we developed we also analyzed the
power flow on the Transmission Line but we have not evaluated final arbitrary constant
of the voltage or current expression and that is V+ and that we did not do so far that is
because we were always taking the relationship which were for the impedances and for
that we had a ratio of voltage and current and the absolute value of V+ did not play any
role.

However as we see now when we do the power flow calculation on Transmission Line
we require the knowledge of V+ because now we cannot talk about the relative quantities
we have to absolutely find out the voltages and currents at the load end or at any other
location on the line and therefore we now have to absolutely evaluate the quantity V+
because without knowledge of V+ we will not be able to tell the absolute power delivered
to the load for a given generator and the load conditions.
So now from the boundary condition by connecting the generator and the load impedance
we calculate the final arbitrary constant of the voltage expression that is V+ and then
completes the analysis of the voltage or current on Transmission Line. So now what we
discuss is we discuss evaluation of the arbitrary constant V+.

Let us take the general case: I have a generator here which is having a voltage Vs then I
have a Transmission Line which is of length l and then I connect a load impedance to this
Transmission Line which is equal to ZL.

(Refer Slide Time: 40:12 min)

The characteristic impedance of this line is Z0 which is given a priori, some voltage wave
is going to travel on this Transmission Line, which will get reflected from here it will
come back so on the Transmission Line we will have the standing waves so there will be
impedance transformation from ZL to some value when see from the generator. And now
by using that impedance transformation relationship and matching the boundary
condition at the input we want to find out what is this quantity V+ for the given circuit.
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 make it little general - let us say this
voltage source is having an internal impedance which is equal to Zs so let us say I connect
an internal impedance here of the voltage source which is given as Zs. Let us say I have
the input terminal which is given by some A, A' and these are the locations which are
denoted by B, B'.

Once I know this ZL and I know this length then I can find out what is the transform
impedance at this location A, A'. Let us say this impedance if I see from here that is given
as Z'L so Z'L is the transformed impedance seen at the generator end for the length of this
line l as the terminating load ZL.

(Refer Slide Time: 41:57 min)

Once I transform the impedance then the whole circuit is the lumped circuit at this
location I do not have to worry about the distributed elements. So now this circuit is
equivalent to having a voltage source which is VS, the internal impedance for the voltage
source that is ZS and connected to an effective impedance which is Z'L, once I get that
then I can find out what is the voltage this is the location now A, A' so by using the
lumped circuit analysis I can find out the voltage at this location and also I can find out
the current at this location so the voltage here we call as VA and we call this current as IA
in this location.

(Refer Slide Time: 42:57 min)

Z 'L
I can put the lumped circuit analysis and find out what is VA. So, VA will be  VS
Z'L  ZS
VA
and IA will be .
Z 'L
(Refer Slide Time: 43:44 min)

So from the lump circuit analysis at the generator end I know the voltage and current at
the input end of the Transmission Line. I can write down the voltage and current at the
input end of the Transmission Line using the Transmission Line equations. So, now
knowing the load end impedance termination ZL and a distance l from that; essentially I
have to get the voltage and current at a distance l from the load end. Now I know from
the distributed elements then that the V at input end of the line VA is nothing but V at a
distance l from the load which is equal to V+ ejβl{1 + ΓL e-j2βl} and the current IA which is
V  jβl
nothing but I at the location l from the load that is e {1 + ΓL e-j2βl}.
Z0
(Refer Slide Time: 45:05 min)

So now I know the value of VA from two sides, one is from the lumped element side the
VA is given by this from the distributed elements on transmission line the VA is given by
this. I can equate these two values of VA and IA and from here I can solve for the
unknown quantity which is V+ where every other quantity is known here. The
propagation constant β is known, the length of the line is known, the Γ which is related to
the load impedance that reflection coefficient is known, the characteristic impedance is
known so substituting the VA and IA in this two equations I can finally solve for the
quantity which is the V+ quantity so I get the final expression for the V+ and that is V+ =
Z'L VSe-j l
.
 ZS +Z'L  1  Le j2 l 
(Refer Slide Time: 46:30 min)

Once I know the value of V+ then my problem is completely solved. I can substitute that
in to the power calculation and I can find out what is the power delivered to the load or
what is the power at any particular location on the line.

Now this is a complete solution to the voltage and current on the Transmission Line so let
me summarize what we have done so far in the Transmission Line.

To start with we made a case that; when we increase the frequency the concept of lump
element is not adequate because the space has to be brought into picture and then we
introduce the concept of the distributed elements, then in the framework of distributed
elements we wrote down the voltage and current relations taking the limit the either sides
of the circuits tends to zero so that the model is valid at any arbitrary high frequency. We
got the differential equation for voltage and current we solved the voltage and current
equations and then we got a general solution for voltage and current on Transmission
Line. Then we impose the boundary conditions that were the impedance boundary
conditions on Transmission Line and then from there we evaluated certain arbitrary
constants on Transmission Line.
We defined 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 took 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 be delivered to the load
and ultimately we found out the final unknown arbitrary constants which was V+ on
Transmission Line so that we can now calculate the absolute power delivery to the load
for given conditions.

So this now essentially completes the first part of analysis of Transmission Line. Here
onwards we will go to more applications of Transmission Line, we will go and discuss a
graphical representation of Transmission Line or graphical tool for analyzing the
problems on Transmission Line and later on we will go to applications of Transmission
Line at high frequencies.