8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
1/35
Transmission Lines and E.M. Waves
Prof R.K. Shevgaonkar
Department of Electrical Engineering
Indian Institte of Technolog! "om#a!
Lectre$%
Welcome, in the last lecture we introduce the concept of Loss-Less Transmission Line,
we say if the resistance and the conductance per unit length of the Transmission Line is
zero then there are only reactive elements in the Transmission Line so there is no loss of
power because there is no ohmic element in the Transmission Line. So in an ideal
situation the line is lossless if R !, " !
#Refer Slide Time$ !%$%! min&
Then we introduce the concept of the Low-Loss Transmission Line which is more
practical line where the resistive component R is much more less than 'L and " is much
more less than '(.
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
2/35
#Refer Slide Time$ !%$)* min&
+f this condition is satisfied then we said that we can create the lines still lossless.
owever since the loss is very small as and when we reuire the calculation of losses
along Transmission Line we can use the value of the attenuation constant , substituting
this condition that R is much more less than 'L and " is much more less than '( we
calculated the propagation constant which can be separated into real and imaginary part
and then we got the value of and / in the appro0imate form.
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
3/35
#Refer Slide Time$ !%$*1 min&
So, here this uantity was eual to / and this uantity was represented as . owever one
would notice that the condition which you have defined for the low loss is now in terms
of the primary constants of the line. owever in the data sheet for a Transmission Line
the primary constants are rarely mentioned, the parameter which + have mentioned for the
Transmission Line are the effective phase constant on the line or the velocity on the cable
and the attenuation constant of the line either in terms of d2s or in terms of 3epers.
Then one would li4e to convert this condition R 5 'L and " 5 '( in terms of this
secondary parameters or in terms of the relationship between / and . Since these
parameters are available readily in the data sheet if + can establish a condition between
these parameters for low loss nature of the line then + can find out whether a particular
line is low loss at a particular freuency.
So now what we do is starting from this relationship between / and then we can find
out under what condition we can treat the line as a Low-Loss Transmission Line. Ta4ing
this value of + can multiply this uantity here by a suare of L in the numerator and
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
4/35
suare root of L in the denominator similarly + can multiply this uantity by suare root
of ( in the numerator and suare root of ( in the denominator.
#Refer Slide Time$ !6$7! min&
So now the can be written in terms of this substitution so + get this value of which is
eual to
% R L() L . Similarly multiplying by ( in the numerator and the denominator
in the second term we get
% "L(
) ( . 8ultiplying numerator and denominator by ' this
can be written as
% R % "L( L(
) L ) (ω ω
ω ω +
.
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
5/35
#Refer Slide Time$ !7$61 min&
9nd as we have seen earlier this uantity L(ω is nothing but / so this + can write as /
in to
% R % "
) L ) (β
ω ω
+
.
3ow from the definition of the low loss
R
Lω is much more smaller than %,
"
(ω is much
more smaller than % so this whole uantity is much more smaller than % so essentially
what we are saying is now is eual to / multiplied very small uantity or in other words
for low loss the condition now is that is much more less than /.
Since we 4now / in terms of wavelengths which is nothing but two
)π
λ . :nce we 4now
the wavelength on the Transmission Line + can find out what is the value of / from the
data sheet + can find out what the attenuation constant is. +f it is given in terms of d2s +
will convert that in the 3epers per meter and if this condition is satisfied that the / is
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
6/35
much larger compared to then the line can be treated as the Low-Loss Transmission
Line.
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
7/35
#Refer Slide Time$ !;$!! min&
What does this physically mean we 4now if you travel at a distance of < on Transmission
Line the phase change is eual to )=.
Let us say if + travel the distance for a lossy Transmission Line then the amplitude of the
wave will reduce by e > into the distance traveled which is one wavelength. So if +
consider a wave which travels a distance of one wavelength on Transmission Line then its
amplitude will vary e >0 that is eual to e >
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
8/35
#Refer Slide Time$ !?$)! min&
Since for low loss condition is much smaller than / this uantity
.)α π
β is much less
than % that means the amplitude of the wave now reduces to
.)
e
α π
β
−
and since this uantity
is very small essentially the amplitude reduction in the wave is negligibly small. So in
other words what we are saying is a line can be treated a low loss transmission line
provided the change in the amplitude of a traveling wave is negligibly small over one
wavelength distance. :f course negligibly small is a very sub@ective number you can
consider one percent as negligible or !.% percent as negligible.
Let us say as a reference value we consider one percent is a negligible uantity so when
the amplitude reduces to one percent of its original value then we say that the line can be
treated as a Transmission Line. Since this uantity is very small essentially when this
number becomes appro0imately one percent that is where the amplitude will reduce by
one percent so if
.)α π
β is appro0imately
%
%!! the wave amplitude will reduce by one
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
9/35
percent over a distance of one wavelength from here then + can find out what is the
acceptable value of compared to /.
#Refer Slide Time$ !1$*7 min&
So in practice for a given line there is nothing absolute whether the line is lossy line or a
low loss line for a given freuency it may be possible that may be much smaller than /
but when the freuency changes the condition may not be satisfied and the line cannot be
treated as a Low Loss Transmission Line. So for a given loss on Transmission Line
depending upon the freuency the line may be treated li4e a low loss transmission line or
it may not be treated li4e a Low Loss Transmission Line.
Let us ta4e a simple e0ample to find out what are the physical parameters which we will
have on Transmission Line if we @ust ta4e some typical line parameters. So let us say +
have a Transmission Line whose primary constants are given as L !.)*ABm, let us say
the capacitance per unit length is %!! pico Carad per meter, let us say the conductance per
unit length is zero for this Transmission Line. So " ! and we want to 4now what should
be the resistance per unit length of the Transmission Line so that the line can be treated
li4e a low loss transmission line.
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
10/35
Let us say the freuency of operation is eual to %!! mega hertz. Crom here since + 4now
the value of L and ( and the freuency + can find out what is the value of /. So / is eual
to )= into freuency which is hundred mega hertz which again is %!1 hertz into L
which is !.)* 0 %!-;
or micro henry multiplied by hundred into %!-%)
or the pico Carad.
This will be eual to = radians per meter.
#Refer Slide Time$ %%$%! min&
Dust to ta4e a number if + say is less than one percent of value of / then + can treat the
line as the low loss transmission line. The should be appro0imately %!!
π
. 3ow + can go
bac4 and substitute this value of in the e0pression for that is
% (R
) L
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
11/35
#Refer Slide Time$ %%$7; min&
and if + substitute the value of
(
L and the value of which is %!!
π
as you have ta4en
one percent of the value of / then + get the value of R which is less than R =
appro0imately 6.%7 ohms per meter.
So if + accept alpha value to be one percent or less of the propagation constant / or the
phase constant / then the resistance per unit length of the Transmission Line should be
6.%7 ohms per meter at the freuency of hundred mega hertz. :f course as + said the
freuency changes then the acceptable value of resistance will change because the line
may not satisfy this condition at that freuency.
now having understood this as we mentioned earlier until and unless somebody
specifically tells you that the line is a lossy line we ta4e a liberty to create the line as a
loss less line because as first order as we have seen that the phase constant for a low loss
line and a lossy line is same, also the characteristic impedance of a low loss line is almost
real and that is same as the characteristic impedance of loss less line.
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
12/35
So here onwards until and unless somebody specifically say include the losses in the
calculation of Transmission Line we will treat the line to be lossless and carry out all our
analysis for a Loss-Less Transmission Line. So essentially we will assume that
characteristic impedance of Transmission Line is given by this so E!
(
L . This uantity
is a real number and also the propagation constant is eual to the phase constant.
#Refer Slide Time$ %7$!! min&
So we have F @/ that is again eual to @'
(
L . 3ow with these parameters we will again
revisit the voltage and current e0pressions on Transmission Line and then we carry out
the analysis of the standing waves on the Transmission Lines.
"oing bac4 to the original euation of voltage and current as we have seen for a
Transmission Line whose origin has been defined at the load point as L !
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
13/35
#Refer Slide Time$ %7$6! min&
the voltage and current euation can be given by this. This represent the forward traveling
wave, this represent the bac4ward traveling wave and now we will replace this uantity F
by @/ where 0 will be replaced by -%.
So now we have the voltage and current as a function of distance on the Transmission
Line and F will be replaced by @/ and E! is a real uantity. So we can write e0plicitly the
voltage and current on a Loss-Less Transmission Line. So the voltage is v as a function of
l that is eual to GH e @/l H G- e-@/l, by ta4ing GH e @/l common the same thing can be written
as GH e @/l I% H
-G
G+ e-@)/lJ.
9nd this uantity as we already 4now is nothing but the reflection coefficient at the load
end so this uantity we denote by the reflection coefficient at the load end KL so
-G
G+ as
we have seen earlier is nothing but eual to KL which is eual to
L !
L !
E E
E HE
−
.
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
14/35
So now the voltage at any location on the Transmission Line can be given as GH e @/l I% H
KL e-@)/lJ.
#Refer Slide Time$ %;$** min&
Similarly + can ta4e the current euation + can substitute F @/l in the current euation +
can get the current that any location on the line +#l&
@ l -@ l
! !
G Ge e
Z Z
β β + −−
9gain ta4ing
@ l
!
Ge
Z
β +
common we can write down here this is
@ l
!
Ge
Z
β +
I% - KL e-@)/lJ where
-G
G+ will be eual to KL.
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
15/35
#Refer Slide Time$ %1$!1 min&
These two terms as we 4now essentially represent the forward and bac4ward traveling
wave so the whole e0pression here essentially represent super position of the forward and
the bac4ward traveling wave which is nothing but a standing wave on a Transmission
Line.
So here the e0pression for voltage and current represent the standing voltage and standing
current wave on the Transmission Line.
3ow we can investigate certain features for the Transmission Line from here and first
thing what we will note is that there are two terms either in voltage or current so you are
having this term which is having amplitude one and then you arriving at second term
whose amplitude is modulus of this uantity KL plus a phase which is the phase of KL plus
this uantity phase which is minus @)/l. Writing very e0plicitly the comple0 reflection
coefficient in terms of its magnitude and phase let us say + define KL KL @
e Lφ
where Lφ
is the phase of the reflection coefficient at the load end.
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
16/35
#Refer Slide Time$ %M$7% min&
Then + can write down the current and voltage e0plicitly in terms of this magnitude of
reflection coefficient and the phase. So finally + have two e0pressions here one for
voltage G#l& GH e @/l I% H KL ( ) @ -) l
e φ β
J and the current +#l& will be
@ l
!
Ge
Z
β +
I% - KL ( ) @ -) l
e φ β
J.
#Refer Slide Time$ )!$7! min&
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
17/35
3ow let us see how the voltage and current varies if + measure the magnitude of the
voltage and current along the Transmission Line, what is the variation of the voltage and
current along the Transmission Line.
So first thing what we will notice is as you move along the Transmission Line this
uantity L is positive because we are moving towards the generator so as + move towards
the generator the phase becomes more and more negative so this uantity # Lφ
- )/l&
phase becomes more and more negative or in terms of a comple0 plane when the phase
become more negative essentially we will move on the cloc4wise direction. So by
moving towards the generator the phase becomes more and more negative the amplitude
of this thing remains constant this term and the total voltage will be the vector sum of this
term and this term is the real term, this term is the comple0 term whose phase is given by
that and whose magnitude is given by KL.
So essentially this is saying that this is the vector of unity one which represent the first
term then + have another vector whose magnitude is KL and the phase of this is # Lφ
- )/l&
and as + move towards the generator the phase becomes more negative but the magnitude
of this remains constant. That means this whole uantity represent the circle where the point moves on this circle as the L changes. So this motion around this is towards
generator and the magnitude of this term in the curly brac4ets is given by the vector
which is the @oining of these two points so this is nothing but I% H K L ( ) @ -) l
e φ β
J magnitude
of this uantity.
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
18/35
#Refer Slide Time$ )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.
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
19/35
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.
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
20/35
#Refer Slide Time$ )?$6! min&
So in this location + get voltage ma0imum and current minimum, when + go here + get
current ma0imum and voltage minimum. The important thing is when the standing waves
are present on the Transmission Line then the voltage and current waveforms are shifted
with respect to each other and ma0imum of voltage aligns with minimum of current and
vice versa.
3ow having understood this + can again go bac4 and loo4 at this e0pression of the voltage
and current. +f + go to the location where the voltage is ma0imum that means this uantity
is H% the voltage will be GH multiplied by % H KL because this uantity is H% at the same
location + will have the current which will be minimum as we saw so this will be +#l&
!
G
Z
+
#% - KL& where E! is real since the line is loss less.
So once + 4now the voltage and current + can find out the impedance at this location
where the voltage is ma0imum or the voltage is minimum or when the current is
minimum or the current is ma0imum. The interesting thing to note here is irrespective of
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
21/35
phase of GH when the voltage is ma0imum or minimum the ratio of G and + is a real
uantity. +f + ta4e a ratio of these two
( )
( )
G l
+ l that is eual to E!
@# ) &
L
@# ) &
L
% e
%- e
L
L
l
l
φ β
φ β
−
−
+ Γ
Γ and
when voltage is ma0imum or minimum this uantity is H% or -% so for ma0imum voltage
# Lφ
- )/l& is even multiple of = that is it is !, )=, 7= and so on, for minimum voltage # Lφ
-
)/l& is eual to odd multiple of = that is =, 6=, *= and so on.
#Refer Slide Time$ 6!$)% min&
So when the voltage is ma0imum this uantity is H% so E ! is real, now this uantity is one
% H KL denominator this uantity is again H% so this uantity is % - K L so the ratio of this
uantity is the real uantity. When the voltage is ma0imum the impedance seen on the
Transmission Line is real irrespective of what the line is terminated in.
Oven if the line is terminated in comple0 impedance if + move on the Transmission Line
and go to the location where the voltage magnitude is ma0imum, at that location the
impedance measured will be always real. Similarly when + go to a location where the
voltage is minimum this uantity will be eual to -%so + will get % - K L you have % H KL
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
22/35
now. 9nd again this uantity will be real so now we ma4e a very important conclusion
and that is on a Transmission Line wherever there is a voltage ma0imum the impedance
measured is real, wherever there is a voltage minimum the impedance measured is real.
So irrespective of what the impedance with which the line is terminated you will always
find these points on the line where the voltage is ma0imum or minimum and that location
the impedance measured will be purely a real uantity.
What will the value of these ma0imum or minimum impedancesP We can substitute this
so at a location where the voltage is ma0imum and the current is minimum that is the
highest impedance you are going to measure on the Transmission Line. So this uantity
when the voltage is ma0imum at that location we get the ma0imum possible impedance
which we can measure on the Transmission Line. So we can get the ma0imum impedance
which one can see on the Transmission Line and let us call that as E ma0 is nothing but
ma0
min
G
+ .
#Refer Slide Time$ 6)$*) min&
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
23/35
9nd since we have seen that this uantity when you are having the voltage ma0imum and
current minimum that time the phase difference between them is zero so this uantity is
real uantity so Ema0 is nothing but R ma0 as resistive impedance, from here if + substitute
this eual to plus one + will get R ma0 E!
L
L
%
%-
+ Γ
Γ .
Similarly if + go to a location where the voltage is minimum so this value will be % - K L
but at the same time the current will be ma0imum so that is the lowest value of
impedance you can see on the transmission line. So you get the minimum impedance on
the line which is Emin which will be
min
ma0
G
+ and that will be eual to E!
L
L
%-%H
Γ Γ .
#Refer Slide Time$ 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
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
24/35
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
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
25/35
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$7? min&
9nd what is the ma0imum value of voltage which + can see on the line is{ }LG %H
+ Γ
and the minimum value which + can see on the Transmission Line is{ }LG %-
+ Γ .
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
26/35
The GH cancels so the voltage standing wave ratio is
L
L
%H
%-
Γ
Γ , in short the voltage
standing wave ratio is called as GSWR. So the GSWR for a load whose reflection
coefficient magnitude is LΓ is given by this.
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
27/35
#Refer Slide Time$ 6M$*! min&
Since the line is lossless the reflection coefficient at the load is
L !
L !
E
E
Z
Z
−
+ and E! is the real
uantity for a Loss-Less Transmission Line so EL can have any comple0 impedance
terminated on the line but E! is real without much effort one can see that the magnitude of
this uantity is all ways less than one. So the mod of LΓ is always less than or eual to %
and that is ma4es a physical sense because what the LΓ
is telling you is the relative
amplitude of the reflected wave compared to the incident wave.
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
28/35
#Refer Slide Time$ 7!$7) min&
Since we do not have any energy source on the load point the part of the energy only can
get reflected so the amplitude of the reflected wave has to be always less than or eual to
the incident wave. So for any passive loads the LΓ
is always less than or eual to % so in
a condition when EL ! or EL + will get the magnitude of
LΓ
% otherwise thisuantity will be always less than or eual to one so if + want to see very specific load
impedances for which the LΓ
condition will be satisfied. + will see there will be three
cases. (ase one will be when EL ! that means the line is short circuited at the load end
so this is a condition for a short circuited line. + substitute EL ! so + get LΓ
-% in this
case + get LΓ
-% or LΓ
%.
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
29/35
#Refer Slide Time$ 7)$%! min&
The second case if + ta4e EL that means the line is open circuited then again + can
substitute EL , ta4e EL first common so it will become
!%> E
∞ so this will be eual to
H%. So this will give me LΓ
H% giving me again LΓ
H%.
The third case is that if the line is terminated in an ideal reactance so the third case is if
EL @ times 0 your reactance then again so this is pure reactance then LΓ
will be eual to
!
!
@0 E
@0 H E
−
.
This uantity in general will be comple0 but the LΓ
will be eual to again H%.
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
30/35
#Refer Slide Time$ 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.
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
31/35
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;$))&
9nd which is the better caseP The LΓ
! represents no reflected wave on Transmission
Line that means we have full power transfer to the load. So LΓ
going to zero is the best
case as far the reflection is concerned or in other words when GSWR % that is the best
case for the power transfer on the line. 9s the GSWR increases it indicates higher and
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
32/35
higher value of LΓ
that is higher and higher reflection on the Transmission Line or
lesser and lesser efficiency of power transfer to the load.
So in every circuit design our effort is to ma4e the GSWR as small as possible or as closeas to % as possible. igher the value of GSWR indicates more mismatch on the
Transmission Line or higher value of the reflected wave on the Transmission Line.
So this uantity GSWR is one of the very important uantity in the measurement at high
freuencies. Whenever we design a circuit we essentially try to measure the GSWR on
the Transmission Line and we try to ma4e the GSWR as close to one as possible, ma4ing
sure that the circuit is efficiently transferring power to the load end of the line.
3ow once you have defined this parameter then one can relate this to the ma0imum
and minimum impedance which one can see on Transmission Line. We have already seen
that the ma0imum value of the impedance which we can see on the line is R ma0 E!
L
L
%H
%-
Γ
Γ but now you 4now this uantity is a measurable uantity and that is nothing
but the voltage standing wave ratio .
So the ma0imum value of the resistance or the impedance which we can again see on the
line is nothing but E! into wave standing wave ratio .
#Refer Slide Time$ 71$67 min&
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
33/35
Similarly the minimum resistance which + will see on the line is this is
% ρ so this is eual
to
!E
ρ .
So + get from here R ma0 E! and R min on the line is eual to
!E
ρ . So for any line which is
terminated in arbitrary impedance if + move to a location where the voltage is ma0imum
then + 4now the value of the impedance at that location because measuring the voltage
ma0ima and minima + can find out the value of this voltage standing wave ratio , + 4now
the characteristic impedance align apriori so + 4now the ma0imum value of the
impedance which line will show.
#Refer Slide Time$ 7M$66 min&
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
34/35
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.
8/20/2019 Transmission Lines and E.M. Waves lec 05.docx
35/35
Top Related