Transmission Lines Video Lecture - Electrical Engineering (EE) | Best Video for Electrical Engineering (EE)

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so the no to sahaja yoga why they just
need all of us kilala
in this lecture I want to take a look
back at what we have done and then make
the final step from what we have been
doing has waved theory to something more
practical which is the subject of
transmission lines so let us look at
where we started we started with statics
and we concluded that there were two
equations which governed electric field
one was curl of E was equal to zero the
other was divergence of D was equal to
Rho because curl of E was equal to zero
this allowed us to say the electric
field is derived from a potential
so e was equal to minus grad Phi and we
invented this concept of the
displacement vector because then any
material has charges in it which are
bound to where they are to the atoms in
which they belong but in the presence of
an electric field there is an induced
polarization and we worked it out in
this course that you can show that D is
equal to epsilon e that is D is
proportional to e so the correct
equation here is divergence of E is
equal to Rho total over epsilon not
but by introducing the concept of
induced electric field and saying that
the induced electric field is
proportional to the electric field
itself we could write this as divergence
of E is equal to Rho bound plus Rho free
divided by epsilon knot and we
identified this Rho bound by epsilon
knot this piece with a divergence of T
induced and so we came up with a new
equation which was divergence of epsilon
naught times e minus e induced was equal
to Rho free and that is this equation so
this is nothing but D so this was
statics and the final achievement of
statics was a Poisson equation which
gave us that del squared Phi was equal
to minus Rho over epsilon naught now
there are some things that I didn't do
here because they were purely
mathematical in nature one of them was
that you can prove that if you have any
volume V and if on the surface of this
volume on every point you specify either
Phi or del Phi del n then the solution
inside exists and is unique so that
means that this equation has is
well-defined and has unique solutions
so that was statics that is
electrostatics when we went to magneto
statics and when we went to magneto
statics the picture was exactly reversed
we found that we could write curl of B
was equal to MU naught J and we could
write divergence of B was equal to zero
now I discussed with you that the
magnetic field is not a new field in
fact if you go back and look through
some of my older lectures the magnetic
field is nothing more than the electric
field when you apply a relativity to it
so it is it is actually a side effect of
the electric field now you take this
equation and once again apply to
materials this equation becomes mu
naught J bound plus mu naught J free and
once again you can attach an implied
field to this bound checker --nt and you
can say that this is equal to curl of B
induced which allowed us to say curl of
B minus B induced divided by nu naught
was equal to J free and we identified
this as a new vector which we called H
so we had four equations one was curl of
E was zero divergence of D was Rho curl
of H is J divergence of B was zero and
these four equations
took care of time independent phenomena
however once you went to time dependent
phenomena two of these equations were
found inadequate so we added extra terms
first this curl equation turned out to
have a correction to it which was -
there will be del T and the effect of
this - del V del T could be seen because
if this equation was still correct
which meant B was the curl of some other
function a substituting it back in here
you found that II was minus grad Phi but
substituting this curl back into this he
also got a minus del a del T so the
electric field was not only due to a
gradient of a potential it was also due
to a time dependence of the vector
potential
similarly this equation was found to be
unsatisfactory you found you needed a
new term which was del D del T so that
gave us the final system of equations
and let me write them down once so that
we can see them curl of e is equal to
minus del V del T Faraday's law
divergence D is equal to the free charge
Gauss's law
divergence of B is zero there is no such
thing as magnetic charge curl of H is
equal to the free charge plus del D del
T that is what we call Maxwell's
equation D is equal to epsilon iva
epsilon is the permittivity B is equal
to MU H mu is the permeability and you
have one more equation which connects
Rho and J which is divergence of J is
equal to minus del Rho del T so these
are the equations that define the field
and there is one equation that is coming
outside the system which happens to be
at least as important as these which is
the force equation the force on any
charged particle is Q each plus V cross
so these are the system of equations
that we have been looking at through
this whole course initially we ignored
the time derivatives and we did statics
then we took the time derivatives into
account we got Faraday's law induced
e-m-f motional EMF and then finally we
got waves we will spend quite some time
on waves now and what I want to do is to
make those wave solutions interesting to
someone who is not going to be in
communication theory because really if
you are not going to do AM and FM
transmission why should you learn about
wave theory well there is a good reason
for that like why you should learn
and it has to do with the fact that even
if you do not do free space wave
propagation every electrical engineer
like it or not
is going to have to deal with
transmission lines so what is the
transmission line up to now we have
looked at ways which travel in one
dimension basically we know that there
are more complicated solutions but we
have looked only at one-dimensional
solutions you have the electric field in
some direction magnetic field 90 degrees
to it and he cross H is the direction in
which the wave is traveling now the
first thing of interest to us is that if
you had a torch and you shone light from
this torch you know that the light
spreads what that means is if you if you
are standing somewhere and you shine a
torch at the wall the wall the spot will
appear on the wall if you shine the same
torch at a faraway building the spot
will be much bigger and if you shine it
at a mountain you won't see anything at
all
so what it means is that this beam is
spreading out and we work this out in
terms of pointing theorem we said that
there is an inverse square law for
intensity namely volume integral of e
cross H dot es is constant assuming that
your light bulb is in one place you are
asking how much how spread out is my
electric and magnetic field of my wave
what you have is e cross H dot d
s is independent of how far away I
measure it if I measure using a surface
like this I will get one answer if I
measure using a surface like this I will
get the same answer that is because the
same wave is going through there are no
sources here so I should get whatever
came in must leave a cross H is power
power density if you like we saw last
time that it has the units of watts per
meter square so this power density is
inversely proportional to the area of
the surface and the area of a sphere is
proportional to the square of the radius
so power density is proportional to one
over R square that explains a lot of
things it explains the sun's why the Sun
is not brighter than it is how how light
spreads out it explains a great number
of things but it is an inconvenience
supposing I want information to travel
from point A to point B without
spreading out well of course I can I can
design very good optics and make sure
that the light does not diverge and we
do do that when we have point-to-point
microwave links our transmitters have
only one watt of power and the links may
be tens of kilometers apart so it is
possible to have very low power sources
going through air without spreading much
however that is not always the best
approach so what we would prefer to do
is to have cables we prefer to take this
wave package it into a cable and send it
from point A to point B
like we have a computer and we have a
cable an Ethernet cable and it connects
to a hub this cable is also taking
time-dependent information one hundred
million bits per second from this point
to this point and the wave that is
traveling through this cable is forced
to stay inside the cable and so it does
not obey the inverse square law that is
the strength of the wave at this point
is not inversely proportional to the
distance it attenuates there are losses
in this cable but otherwise if it were
lossless the amount of energy here would
be equal to the amount of energy that
left here so what we really want is the
same wave that travels through space
except we want to bottle it into a pipe
we would like to look at what solutions
one can have if I take a wave and I make
it travel inside a metallic pipe
unfortunately there is bad news let us
look at this along the axis so it is got
some shape this is X this is y so this
pipe has some cross-section now my
electric field and magnetic field are
what they are so let us say I have my
electric field this way and my magnetic
field is this way now you can see that
there are places where the electric
field will be tangential to the wall now
we already discussed that tangential
electric field is always continuous
since it is always continuous the
electric field inside the metal is also
what it is at the wall which means you
have a very strong electric field inside
a conducting material similarly if you
look at this point your magnetic field
has a normal component and just as we
show that the tangential component of
the electric field is always continuous
the tangential the normal component of a
magnetic field is also always continuous
so you find that whatever magnetic field
is here is also inside the material but
you know that amperes law tells us that
if you have an electric field you have
current if you have a magnetic field you
have an induced electric field and
therefore you have again lots of current
so if you have current conductivity your
Sigma is extremely high you cannot
really afford to have a large electric
field which means that this kind of
solution is not possible if you try to
package what is called a plane wave the
kind of wave we've been looking at and
put it into a pipe the plane wave will
be destroyed in centimeters it will just
not survive if a plane wave is to
survive it must change it must be a
different kind of wave it must actually
vary not only in Zed but it must also
vary in X and in Y and how much did vary
it must vary by becoming zero wherever
the electric field tries to be
tangential and the magnetic field must
become zero wherever it is trying to be
normal there are such solutions and the
study of waves
flowing through pipes it's a very
important subject it is called waveguide
theory for instance if you are doing
radar and you you have a building where
you have generated your strong radio
frequency signal and you have your your
radar antenna well there will be some
100 meters or few hundred meters between
the antenna and the place where you're
generating the signal so this high
frequency signal has to be taken there
and it may be 10 kilowatts of power that
must be given to this antenna it has to
go through a pipe and such waves that go
through pipes and drive an radar antenna
these are wave guides but as I said the
problem with wave guides problem with
waves flowing in pipes is that your
electric field in your magnetic field
are not the simple fields that we have
looked at so far they have more
complicated fields for instance if you
had a cross section that was rectangular
so you had X and you have Y then you
could imagine that you would have an
electric field that was zero at the two
ends and slowly became bigger biggest
and became smaller such a solution is
quite possible and these kinds of
solutions exist and they are called
transverse electric and transverse
magnetic mode x' you see the names te
and TM modes that is all it means it
means that there is
variation in the X and y direction a
transverse electric mode is one which
does not have any electric field in the
Z direction a transverse magnetic mode
is a mode that does not have any
magnetic field in the Z direction unlike
all these modes which are quite
complicated and not really of importance
to low frequency applications there is
one special mode and this special mode
can travel at low frequencies it can
travel through pipes and it looks like a
plane wave and it is that wave that I
want to talk about today
let us go back to the wave equation so
we had del Square D minus 1 over C
square del square d del t square is
equal to 0 now I am going to write this
del squared has two parts it is actually
equal to the second derivative with
respect to X plus the second derivative
with respect to y plus the second
derivative with respect to Z that is
what del squared is so I am going to
separate out these two pieces and keep
this separate so del squared e del X
square plus del square e del Y square
plus del square e del Z squared minus 1
over C square del square e del T squared
is equal to 0
I just put brackets these two equations
are identical whether I just written out
the del squared collected to terms and
put the third term with the time
derivative now I am going to make a
guess I am saying I have a long tube so
whatever shape I have in X that shape
remains what it is as I go down in Z so
if I cut here I will still see the same
shape and if I go further I will still
see the same shape so I guess that the
electric field has a dependence on X and
Y which is independent of how it depends
on Z and T so I write the electric field
as some vector which depends on x and y
alone times e to the J Omega t minus KZ
this is perfectly reasonable I do this
because I can see that in time and in Z
my structure is symmetric if I shift in
z problem looks the same so I can look
for sines and cosines in Z by shift in
time the problem looks the same so I can
look for sines and cosines in T but I
cannot shift in x and y because I have a
pipe so that problem is more complicated
if I substitute this in here and I will
call this e perp I can write this as del
square XY acting on a perp multiplied by
e to the J Omega t minus KZ plus the del
del Z squared will act on K K Z so I
will get a minus K squared plus Omega
squared over C squared e / e to the J
Omega t minus KZ is equal to zero so all
I have done is have substituted this
form into this equation this part I do
not know what is going to do is just
going to act on this unknown function
but this part I know what is going to do
because I know the form of the
dependence on time and the dependence on
Z so I pulled it out let me get rid of
this exponential because it is common so
I remove it
now if I want to compare this problem
with a plain wave problem I would need
to get rid of this piece without this
piece this is nothing but my
one-dimensional wave equation this piece
is what is going to complicate my life
so I look for solutions where this piece
goes away so I try to find solutions
where the del squared X Y acting on this
amplitude which depends only on X Y is
equal to 0 if I could find it then the
rest of it is done this is a
one-dimensional wave equation so it is
my plane wave bottled into a pipe and
packaged and sent now the question is
how do I solve this problem I have gone
from one problem to another but the
second problem does not look any easier
than the original problem so I want del
square del X square of my perp plus del
square del Y squared of a perp is equal
to 0 and E perp cross
the normal to the surface is equal to 0
on boundary because the tangential
component must be 0 now what I am going
to do is I am going to look at this and
say this does not depend on time at all
so I will guess T perp is a vector is
equal to minus gradient of some scalar
after all it looks like electrostatics
to me so why not try it even this looks
like electrostatics it says there is no
tangential electric field so the wall is
a equipotential so let me put this back
into this what do I get I get del square
del X square plus del square del Y
squared acting on minus del Phi del X X
hat minus del Phi del Y Y hat is equal
to 0
and in addition pi equals constant on
surfaces because I need tangential
component to go to zero so Phi is
constant tangential component is also
zero now because these are all
derivatives without any coefficients
I can always commute them I can pull
this del Del X outside and I can put
these inside so I can write this as X
hat del Del X plus y hat del Del Y
acting on del square del X square plus
del square del Y squared of 5 because
after all second partial derivatives
commute
so it does not matter in which order I
do del Y del X del a del del Y del Del X
or del Del X del Del Y the order does
not matter so I will do this operation
after I do this operation but when you
look at this equation what does it say
it says minus gradient of del square Phi
equals 0 which can be easily satisfied
it is okay if del squared Phi is equal
to 0 so I am really looking for solving
this problem I am really looking for
something very simple I am saying I will
guess that the electric field looks like
this some function of X Y times e to the
J Omega t minus KZ and this function of
X Y is derived from a potential and this
potential satisfies Laplace's equation
let us look for an example and see
whether all of this makes sense or not I
will start with a central conductor and
a metallic pipe the outside I am going
to hold a ground inside set some voltage
V and there is a dielectric now really I
shouldn't say I am holding it at a
voltage V it is going to carry a wave
which means that the voltage is not
constant voltage is a function of Z but
when I look for the solution of this
problem I am looking for del square Phi
is equal to 0 Phi equals 0 on outer wall
Phi equals V on inner one this is a
well-known problem we have already done
it this is nothing but what we did for
calculating capacitance of two cylinders
so you can easily find a solution and
you know that there is a radially
symmetric er this electric field er will
satisfy del squared Phi equals 0 there
is no tangential electric field on the
outside surface there is no tangential
electric field on the inside surface so
it satisfies all the conditions we want
and therefore del squared XY of a perp
is zero and the boundary conditions are
satisfied what does that mean it means
that for this particular form we are
solving only this equation now that
means my solution is going to look like
this the electric field
which is a function of X Y Z and T is
really and T are along our direction
that er you can work out is going to go
like 1 over R so you can say e naught
over R e to the J Omega t minus KZ what
will the corresponding H look like this
will be the curl of e so when you look
for the curl of e is going to take the Z
derivative of e R that will be in the
theta direction H of X Y Z T will again
look like some H naught over R in the
theta direction e to the J Omega t minus
KZ and when you put these into the wave
equation you get this solution which
means Omega is equal to plus or minus H
now these are very special waves these
are not the waves that are used for
taking high-energy radar signals they
have quite a few nice properties for one
thing the line that connects Omega and K
the curve that connects Omega and K if
you plot it you find that it is a
straight line now this is very important
because if you look at what happens if I
have a circuit and I have a switch and I
close the switch when I close the switch
current will start to flow now the speed
with which this signal that the switch
is closed goes through this circuit
depends on how well the signal
propagates through these wires if the
signal propagates at different
velocities at different frequencies then
what will happen is out here in time the
signal the switch just closed and
instantly the signal came on and
continued but out here it may very well
be that the signal will look like this
that is to say the fact that the switch
closed was not a very sharp start in the
current it starts gradually there is a
rise time to the signal
and this rise time due to the signal
comes because the wire is not carrying
all frequencies with the same velocity
but this particular solution we have is
a special solution it does in fact care
carry all frequencies at the same
velocity because the velocity is nothing
but Omega over K it is equal to C so all
frequencies travel together so if I take
the Laplace or Fourier transform of this
signal it is a spectrum every part of
that spectrum travels together I can
take the inverse transform here and I
will get back the same shape it won't be
one one frequency is delayed with
respect to the other which is the main
cause of these kinds of effects so that
is one very important advantage of such
waves and these waves are called
TEM waves transverse electromagnetic
waves and they are the waves that live
in what we call transmission lines
now in a certain sense transmission
lines have come to rule Electrical
Engineering no matter which branch of
Electrical Engineering you are in that
is because the state of the art has kept
progressing till however fast or however
slow your circuits are they are pushing
the limits and the limits are always the
limits of signal propagation through
circuits so you can see this happening
through the different technologies that
are happening in different branches of
Electrical Engineering
for instance naturally in communications
transmission lines are required wave
guides and transmission lines are
automatically required but if you look
at for example a PC main motherboard now
that does not look that is not
communications after all what about it
well the size of the lines on a PC
motherboard a few centimeters and the
speed of the clock on the motherboard
which are hundreds of megahertz combined
together so that in fact signal delays
become very crucial so once again you
find transmission lines are true sure if
you do not take proper care about delays
you do not properly understand your
transmission lines you cannot design a
good motherboard whatever an IC chip
well these distances of lines are now in
tens of microns to hundreds of microns
but the speeds are much higher because
the speeds are much higher once again
signal propagation delays are important
and you have transmission lines once
again so you comes lower
in frequency and so you come to power
lines now you are talking about 50 60
Hertz but your power lines are very long
your power lines can be crossing a
country and once it starts getting
approaching a large fraction of a
thousand kilometers once again here a
transmission line so the theory of
transmission lines and more generally
the theory of wave guides has come to
dominate Electrical Engineering no
matter which branch you're in and it is
the same theory the theory cannot change
because you are talking about power
lines as opposed to communications in
all these cases what is important is
that you are able to separate the del
squared of e ex and ey and this the
solution along Z which is del square del
Z squared minus 1 over C square del
square del T square of the behavior in
tnz so I gave one example of this which
was the cylindrical two wire guide that
is what you call a BNC cable it is also
found in the TV coaxial cable and all of
those are machine to great precision so
that they can carry these signals up to
very high frequencies but since we are
talking about low frequency FX you can
also talk about power lines because you
can consider two wires one of which is
that one voltage and the other is it
another so you have current flowing into
this and coming out of this while these
are the return paths of this current
then we are solving for x Y electric
fields
by solving this capacitance problem it
is a solvable problem in fact there are
exact solutions available and if you
solve this problem and substitute it
back into this equation you have got the
transmission line that is used by power
lines and this is nothing but the the
electromagnetic wave that travels
through the hundreds of kilometers in
which power is transmitted so the power
is actually going into the board but
electro electrostatic and magnetic
static fields are set up due to the
charges and currents that get
established on this on these wires so it
is an important thing to realize that
transmission lines are a very unifying
idea and they are in fact like you might
say the the final step in a basic
electro magnetics course we have learnt
about static fields we have learnt about
magnetic magnetostatic fields we learnt
about motional EMF we learned about
waves and when we went to waves it was
like we went to a new subject but
actually now we have come all the way
back we are talking about using those
waves inside circuits we are talking
about power lines carrying power we are
talking about Ethernet cables carrying
signals and those circuit elements are
in fact waves again and perhaps the most
surprising part of it is to solve for
this wave you need electrostatics you
are not using wave theory wave theory is
the simple part you have to do
electrostatics now if you are doing PC
motherboard you would have
a high permittivity epsilon of maybe 10
12 high permittivity material dielectric
on which you have thin copper strips the
bottom part of this plate is actually
fully copper and it is connected to
ground this is what is called a two
layer board and of course standard PC
motherboard will not be to layer before
layer six layer or higher but this is
the simplest way of looking at it so
what happens is if you look at this
structure and look at it from sideways
you have your dielectric at the bottom
you have copper and this copper is
grounded at the top you have no copper
current signals flow in into this copper
strip that is they are flowing this way
that is into the board in this picture
when this when charges get built up on
this strip you get capacitance
you have to solve this problem this
problem would be the problem of having
an air dielectric and an epsilon much
much greater than epsilon not with
copper strip in the middle this problem
corresponds to solving this having
solved it and actually there are some
problems I won't go into them so it is
only an imperfect solution you can make
having solved them you are then left
with a transmission line the
transmission line carries signals along
this wire at essentially the speed of
light actually half the speed of light
approximately similarly in an IC chip I
mean the same ideas go whether it's in a
PCB or an IC chip or even in a micro
strip antenna now if you have done your
transmission line course or even if you
haven't when you get to it
the language will be very different you
will not see wave theory at all in fact
you will see a distributed RLC Network
you will be told that transmission line
can be viewed as a two-port Network
and you can put can put an equivalent
circuit this distance would be DZ this
would be C over 2 C over 2 DZ and this
is LD Z and you will be told that this
represents the transmission line an
ideal transmission line for a short
distance DZ and then you combine many
many many such sections and you get the
behavior of a transmission line which
does not look very much like this
equation at all this equation came from
studying Maxwell's equations whereas
this is more of a equivalent circuit so
where does this picture fit in with this
picture well the way to look at it is as
follows imagine that we have a parallel
plate capacitor which is very long this
direction is Z in this direction the
direction between the plates is X and we
do not care too much what is happening
in Y so
we have again the possibility of a TEM
wave because we can solve for
capacitance of this arrangement so we
will have electric fields and then
electric fields will change directions
and that is what is our wave the wave is
going in the Z direction now I want to
convert from electric and magnetic
fields to voltages and currents because
that is what circuits know and that is
what is that equation the equation has
to do with inductors and capacitors well
the idea is we have solved for a
potential we looked for e equals minus
grad Phi so therefore we can talk about
the voltage across these plates we can
calculate integral e dot DL from point
one to point two we can also talk about
a dot DL from 0.3 to 0.4 and let us say
that this distance is DZ
so in that case I can talk about the
difference between a dot DL from three
to four and a dot DL from one to two so
I can do integral three to four e dot DL
minus integral 1 to 2 e dot DL which is
nothing but loop integral e dot DL
because electric field is up so these
two component sections do not do
anything electric field is orthogonal to
DL what is that that is nothing but
surface integral curl of e dot d s which
is apply Faraday's law minus del V del T
dot es I will pull the del Del T out it
is minus del Del T of surface integral B
dot DL but we know that surface integral
B dot d s is nothing but flux and flux
is Li so we can write this as minus del
Del T of Li let us see what we have
written
this l4 this flux is the flux between of
a region which is DZ y so this
inductance is L DZ and this difference
is voltage at Z plus DZ minus voltage at
Z and I am going to take that DZ out and
bring it this side divided by DZ is
equal to minus del Del T of Li
this is nothing but the Z derivative of
voltage which gives us a circuit
equation circuit equation is Del V del Z
is equal to minus L times del I del T
similarly you can get an other equation
the other equations will use amperes law
and it will give you del Y del Z is
equal to minus C del V del T and these
two equations together will give you the
transmission line equations the the
lesson to be learned here is all the
work that we have done in
electromagnetic Theory statics motional
EMF waves has finally reached here it
allows us to define a wave that is
bottled into a pipe and that wave that
is bottled into a pipe a useful variant
of that wave is what we call a TEM wave
and that TEM wave can be modeled by
circuit concepts you can talk about
inductance per unit length and
correspondingly you can talk about
capacitance per unit length and you can
instead of talking about electric and
magnetic fields talk about voltages and
currents and when you do that this is
nothing but the theory of transmission
lines and the theory of transmission
lines is the practical as
lect of electromagnetic theory
you
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