This is module 2.3 of the continuation of
electromagnetic principles. In this chapter
we will see about transmission lines.
Especially we will see travelling or transmission
lines, the solution for that. What will happen
when transmission lines are terminated in
the load and what will happen when transmission
lines are terminated in another line. And
also we will look into expressions for transmission
line impedance in front of a boundary that
is in the presence of multiple reflections.
Transmission lines support a transverse electromagnetic
field structure between these conductors so
this is the basic assumption.
For example, structure cross-section that
can support TEM fields, they are wire above
are conducting plane, 2 parallel conductors,
a co-axial cable or it could be any generic
coaxial structure, its shape may not be very
regular as long as the cross-section is the
same and the line is fairly large compared
to the wavelength and the cross-section is
very small compared to the wavelength, it
is possible to have a TEM field structure.
So by the TEM field structure we also assume
that we can uniquely define a voltage between
conductor 1 and 2, which is an integral of
electric field along any path so it will be
path independent, whatever path you will take
between 1 and 2 it should be in the same value
or voltage. So there is a decoupling between
electric field and magnetic field when there
is a TEM field structure that is the current
in the conductor is uniquely defined by a
close integral of magnetic field intensity
around this conductor so that will uniquely
give the current, so these are some of the
basic conditions.
Now, first let us concentrate on this part
trying to find the correspondence between
the electric and magnetic fields with voltages
and currents. So this is a transmission line
structure; two parallel conductors so this
is the length, the TEM wave is propagating
along this Z direction, so E and H fields
are perpendicular to the direction of propagation
so let us assume the E vector to be like this
and H vector is coming out of the paper let
us say then this is the direction of the power
flow. The medium of transmission line is characterized
by these properties; conductivity, magnetic
permeability and electric permittivity.
We can have a circuit representation of this
transmission line, before that as we said
before for pure TEM, the voltage between 1
and 2 is defined as integral of E dot dl,
line integral of electric field between the
conductors at C are path independent. Similarly,
the current across one conductor will be close
integral of H dot dl and the power in the
Z direction is given by E cross H dS, which
is nothing but voltage times the current,
so instantaneous power flow or pointing vector
in the Z direction is given by the product
of voltage and current at z at time T, so
these are the connections between voltage
and current.
Now we can define, we can take a very small
section of the transmission line let us say
it is Delta Z in this direction and represented
in terms of circuit parameters that we are
more familiar with say for example, we can
imagine an inductance which is related to
the magnetic field interaction and we can
define capacitance more related to the electric
field interaction, so C and L are values per
meter. So it can be absolute values for Delta
Z to be multiplied by Delta Z. Then because
of this conductivity there is some leakage
of current when there is a voltage difference
between these two, some current will be flowing
through the due to the conductivity of the
medium so that is represented by conductance
G per unit length multiplied by Delta C, so
this structure can support the TEM, L, C and
G.
Now in practice we have always some small
resistance however good this conductor is,
but there is some series resistance. This
series resistance is not part of TEM, this
says that there will be voltage drop across
this so you can have instruction of TEM wave
structure, but if R is small enough still
we can assume that TEM solution is approximately
valid, G does not cause any problem for TEM
structure, so the resistance times Delta C
that is the series resistance component. Now
we can derive Telegrapher’s equation for
the transmission line which are nothing but,
first we equate the voltage drop across a
close loop like this that is obtained by this,
voltage drop across this, voltage drop across
this.That should be equal to the difference
in voltage between these 2 ports so that is
written over here.
Then we can write the 2nd equation by the
current entering and leaving this node over
here. So the current entering and leaving
and difference is this much and that should
be equal to the current entering over here,
so you have this negative sign and G V Z t,
voltage across this plus C dV by dt, so we
got the current also, so this is the Telegrapher’s
equation. If you look at these parameters
R, L, G and C of that R is property of conductor
material and conductor cross-section, whereas
L, G and C that depends also on the type of
medium that you are using, whether it is air
or some other medium.
Now consider a lossless transmission line,
it does not have any loss that is no attenuation
that is R = 0 that is the amplitude are same,
all around the transmission lines and load
distortion G is equal to 0, no dissipation
of the current along the way of the transmission
line. So under this condition the previous
equations simplify into these 2 equations
rate of change of voltage with respect to
space minus L d I by dt. So it is only spatial
derivative of current equal to - C time derivative
voltage. So you can manipulate these 2 equations
by typing the 2nd derivative and substituting
one into other which you can do at home and
see that it is true, then you will get the
wave equation, the second degree differential
equation.
And here it is 1by L C special derivative
of voltage 2nd time derivative of voltage.
Now we can find the solution for this wave
equation using the normal roots in differential
equations and one can see that one of the
solutions is of the form, voltage at any point
in time equal to the voltage that has happened
at a time earlier at C equal to 0. For example,
voltage at space Z or along the line and that
time t equal to the voltage that would have
been there at z = 0 at a time z by V earlier
so z by V is the time required for the wave
to travel up to this point and V is the speed
of the wave.
So this has the form of forward travelling
wave, so V is the speed and Delta t is equal
to Z by V. So this is one of the solutions,
a forward travelling wave without any attenuation
distortion is one of the solutions for this
transmission line. Now comparing with the
original wave equation we get this V square,
V square is the velocity and 2nd derivative
of voltage with respect to space and 2nd derivative
of voltage with respect to time. Now comparing
we get this speed v is nothing but square
root of 1 by square root of LC, so this is
the expression for this speed, so this speed
is related to the parameter L and C.
Another possible solution for wave equation
is backward travelling wave so you can say
that the wave that is present at C would be
travelling backwards. So if it is a backward
travelling wave so this kind of reflection
in transmission lines that we will see later,
so it is T + Z by V, so the total solution
will be the sum of the forward travelling
wave and the backward travelling wave. So
we denote that by the symbols + and - in the
superscript.
It is also possible to find wave equations
in terms of the current waves, so we can find
the wave equations in terms of the current,
then do the same procedure then you will see
that that also is some of forward and backward
current waves and we can define a impedance
at 0 which is defined as square root of L
by C and related to the forward and backward
travelling waves. So here this symbol is negative
when we are representing in terms of voltage
and impedance. This also you can derive it
and you can see for yourself that this is
true, there is a kind of a homework.
Now let us look at transmission lines boundary
condition that is when transmission line is
abruptly entered either by a lump impedance
or it can be another transmission line that
we will see in the next page. Suppose, it
is terminated in a lumped impedance and this
point this reflection point here, so we call
it as R so we call I R, V R and Z R, Z R is
the impedance so V R is the net voltage over
here, so this is the sum of forward and backward
reflected waves over here and the impedance
square root of L by C is obtained either as
a ratio of voltage the forward travelling
voltage and the forward travelling current
Z 0 or it is the ratio of backward travelling
voltage and backward travelling current with
a “-” sign.
Now, if we say V R by I R, so boundary condition
says that V R by I R the net current here
and net voltage here should be equal to Z
R, so this is given by summation of forward
and backward waves divided by summation of
forward and backward currents. You can manipulate
this and easily find that this is nothing
but from this we can say that if we try to
find V R - divided by V R + so this is reflection
coefficient, reflection coefficient is the
wave that is reflected back divided by the
wave that is incident. So that is the reflection
coefficient for voltage, so that we defined
with this symbol Rho. So voltage reflection
coefficient from this group of expressions
you can derive as the termination impedance
- the characteristic impedance Z 0 divided
by the sum of those impedances, so difference
divided by sum, so this is voltage reflection
coefficient.
Now we can derive another reflection coefficient
also, so intuitively a force current reflection
coefficient will have a negative sign in front.
So we can verify that over here that is because
I R is – V R reflection divided by Z 0 and
I R positive is V R positive divided by Z
0. So you get current reflection coefficient
which is negative of voltage reflection coefficient.
Now the incident power is product of the forward
moving voltage and current waves and reflected
power is product of backward moving voltage
and current waves. So if you take the ratio
of reflected power divided by incident power
and simplify it, we will get an expression
for power reflection coefficient or we can
obtain power reflection coefficient as a product
of voltage reflection coefficient and current
reflection coefficient, so this is the expression,
so here this is the square.
Now consider a case of transmission line boundary
conditions when it is terminated in another
transmission line. This situation can happen
quite often in nature, you are connecting
2 cables of different characteristics, so
let us say that cable 1 has impedance Z 0
so any current at just left out this junction
we call it as I 0, and any voltage just left
of this junction we call it as V 0 so the
medium 0, then the current that is going into
the second conductor, we call it as I 1 and
voltage just on the right we call it as V
subscript 1 like that, and this transmission
line has an impedance Z 1. So note that these
2 currents are the same because some of the
energies reflected back from this point and
these 2 voltages are different from these
2 voltages.
Now reflection coefficient, for defining reflection
coefficient we can if these lines are quite
long, we can consider it as termination with
other impedance that is equal to the characteristic
impedance of this line. Now here it is assumed
that this line is quite long and not influenced
by what is happening over here, then the reflection
coefficient is something like we derived earlier
in the previous figure difference in the impedances
divided by the sum of the impedances and current
reflection coefficient with a negative sign
in front. And the power reflection coefficient
that is power that is reflecting here, that
coefficient can be obtained by this expression
so this is very similar to the previous derivation.
Now what is new is that we need to find what
is going into this. Of course, whatever is
remaining after reflecting is going into this
one so that is called transmission coefficient
with a subscript T. So voltage transmission
coefficient is defined as the forward wave
that is going into this medium, so we call
it as V 1 +, it is not the same as V 1 you
have to understand that, V 1 is a combination
of forward and backward waves, so V 1 + divided
by V 0, forward wave that is coming in this
direction along the medium 0. So that is equal
to so the voltage at this point, the voltage
is V 1 + has to be V 0 + of the forward wave
and reflective wave that will be the net voltage
over here divided by forward wave that is
going into this medium.
So which will be equal to 1 + you can manipulate
this and find out that this is the reflection
coefficient into this medium, so that will
be nothing but if you write out these equations
and you will see that this is 1 and this is
this reflection coefficient, V0 – by V0
+. So you will get as 2 Z 1 divided by Z 1
+ Z 0, so this is what you will be getting.
Similarly, for the current reflection coefficient
we can write in the similar manner ratio of
the forward wave going into this divided by
the wave falling into this, so that will be
equal to the sum of the forward waves + reflected
waves divided by the wave that is going into
the 2nd medium, and again that is written
as 1 + Rho 0 I, so you will see that 2 Z 0
divided by Z 1 + Z 0, you will get it like
that.
So the expressions are very similar except
that for the voltage transmission coefficient,
so this is the transmission coefficient for
the voltage transmission coefficient you have
Z 1 here, and for the current transmission
coefficient we have Z 0 over here. Then total
power transmission coefficient, this is multiplication
of voltage transmission coefficient and multiplication
of current transmission coefficient. While
talking I might have said that reflection
coefficient but these are not reflection coefficient,
transmission coefficient, the way to identify
is that you look at subscript T, T means transmission
coefficients. So, this is the power transmission
coefficient into this line.
Now transmission line input impedance, often
we have situations in which we are connecting
a short line or it can be a long line also
to do an impedance, but they are not infinite
transmission lines so you have reflects from
here, reflects back so you may have multiple
reflections like that. So in that case often
it is of interest to see, what is the impedance
at a given point X as you are looking into
the load side say for example, if you are
connecting 2 instruments similarly, we always
talk of impedance matching this has got relevance
to that. And also when you are connecting
equipment to the ground there also this is
applicable. So a generic transmission line
has characteristic impedance which is given
by square root of R + J mega L divided by
G + J mega C.
So writing the impedance as Z equal to X as
instantaneous voltage at X divided by instantaneous
voltages, instantaneous current at X, V x
by I x so this V x and I x includes all forward
and backward waves. If we consider it like
that, so you can write it out and it will
be a long algebraic expression and simplify
and finally you may end up in this type of
relationships. Now, of that special case is
one that is shown out here when X = 0, so
this is the input impedance as seen from the
source side.
So you have a load and you have a line and
you are connecting this load and the line
together to the source, so the source will
see an impedance that is given by this expression
simplified expression. So that is the input
impedance from the source side that is equal
to the characteristic impedance times and
this is hyperbolic function and this is the
propagation constant Gamma, which is frequency
dependent. So from this expression you can
immediately see they are similar the numerator
and denominator except that this L and C are
appearing at different places.
So when the characteristic impedance is same
as the load impedance, suppose we use a cable
that has the same characteristic impedance
as the load, then you see that the numerator
and denominator are the same and it becomes
1, the input impedance is nothing but the
characteristic impedance. So you get the matched
impedance, so this is one of the reasons why
we always can connect instruments together.
Often it may have 50 ohms impedance with many
of the standard machine equipment that we
use in the lab, so if we use that 50 ohms
cable and 50 ohms impedances then we can match.
If the load impedance is not equal to characteristic
impedance then input impedance will be varying
from a very long value to a high-value because
this is propagation constant so it is periodic,
and that depends upon the electrical length
of the line. So we will look into some special
cases when Z L is not equal to Z C these kinds
of effects.
Input impedance of transmission lines; let
us look into some special cases. For simplicity
we assume a lossless transmission line that
is R = 0 and G = 0, it means that attenuation
Alpha = 0 therefore, propagation constant
Gamma equal to J Beta, where Beta is 2 Pi
by L. Then from the trigonometric identity
we know that hyperbolic transformation in
J Beta L equals J times Beta L. When the length
of the line is extremely small compared to
the wavelength that is Beta L far less than
1, and also beta L far less than the characteristic
impedance and the load, then we find that
input impedance is load impedance always,
this we can verify that by looking at the
previous expressions because beta L when it
is too small, this will be equal to 0.
So it means that when the line is extremely
short compared to the wavelength, input impedance
will be always equal to terminating impedance.
Now let us look at some of the cases, suppose
transmission line is connected to the ground
and short, Z L equal to 0, then in this particular
case from the previous expression.
From here you can get that when Z L is equal
to 0, you get an expression that is equal
to Z input impedance is equal to Z C Tan beta
L, so this is what you will be getting. Now
the length L can be different electrically
depending upon the frequency, so assume a
case in which this is a quarter wavelength,
Lambda by 4, then Beta L equal to Pi by 2.
So substituting this we see that Tan Pi by
2 is infinity that is Sin Pi by 2 by Cos Pi
by 2, so we get the input impedance as equal
to infinity even though it is a short-circuit,
it should be zero normally we assume but it
is not, it is infinity so the earth will look
like an open circuit. So even though you think
that you have earth it is not really a short-circuit,
it is an open circuit.
Now consider another case in which you are
deliberately leaving it open, Z L = infinity.
Again from the previous expression for input
impedance for this particular case, we can
signify the expression to be in this form
and assume that your length is one fourth
wavelength then you will find that input impedance
is equal to 0. So even though it is open circuit
would, from the equipment side it looks like
short-circuit, so this is a kind of contradiction.
So this kind of funny behaviour can happen
at various frequencies and this type of behaviour
can happen with lines length of only 78cm
at 100 megahertz, at higher frequencies it
will be even at lesser length.
