In this chapter we will review the principles
of electromagnetic. You might have taken a
course in electromagnetics and vector algebra
before I suppose. So this would be only a
review just to remind you those principles
that you have learnt already.
This is the outline of this chapter. It is
divided into 5 modules. Each of these models
will be approximately 20 minutes to 30 minutes
of lecture time. First in module one, I will
consider Maxwell’s equations. Maxwell’s
equations will be presented in the form of
Faraday’s law, Ampere’s Law, Gauss’s
Law and boundary conditions. We will look
at the derivation of these equations. This
part is very much required. If you would like
to know how electromagnetic fields, that is
electric fields and magnetic fields are interacting
with circuits, the circuits can be in the
form of parallel lines or power conductors
above the ground or it can be traces on a
printer circuit board or it can be component
leads.
We will consider uniform plane waves or transverse
electromagnetic waves in different media.
This is a special case of electromagnetic
fields in which electric and magnetic field
vectors are in a plane perpendicular to the
direction of propagation. We will look into
wave equations, we will consider intrinsic
impedance of the medium, that is the ratio
of electric and magnetic field perpendicular
to the direction of propagation. We will look
into the pure dielectric and the lossy media,
the finite conductivity especially we are
interested in the behaviour of metals when
plane waves are falling on the metals.
Then we will look into the concept of skin
depth, that is how much the electric and magnetic
fields can penetrate a metal or a conducting
media. This section is very much important
when we discuss shielding, shielding of electronic
circuits using a metallic enclosures. Then
we will consider transmission lines in module
3. Here also, the solutions for travelling
waves on transmission lines, we are actually
considering quasi-TEM waves but we are presenting
the equations in terms of circuit parameters
like inductance, capacitance, et cetera. Especially
we are interested in termination in load as
well as termination in another line.
That is when the impedance of the transmission
line are different. We also will consider
transmission line impedance in front of a
boundary. This part is important whenever
we need to consider transient analysis of
transmission lines or grounding conductors.
In module 4, we will look into electric and
magnetic fields from dipoles. We have cases
in which component leads or tracks on a printer
circuit boards are causing electromagnetic
disturbances. So we need to calculate how
much will be emitted from this.
So we will see that any of this can be modelled
as combinations of small electric dipoles
and small loops. We will find the especial
for radiation field, specially the maximum
radiation field. And we will also consider
wave impedance. especially how wave impedance
is different between electric and magnetic
field when you are close to the source. Then
in module 5, we will do several numerical
examples.
First we consider the illustration of line
integral which will come in several of the
Maxwell’s equations. Imagine a vector field
in space all around here. Here the bar above
F indicate that it is a vector, that is it
has not only a magnitude but a direction also.
Consider a path A to B and a small section
of this path which can be indicated as a vector
dl, the direction of this path is tangential
to the small section dl and that is indicated
by a unit vector and a hat symbol is shown
for a unit vector.
Now this vector F and unit vector dl form
an angle theta. Now if you take the dot product
of these 2 vectors, then we get the component
of this vector along line F cos theta. Now
let us see what is the meaning of the line
integral indicated by a path C from A to B
of the vector F. It is defined as line integral
of a vector F along a path C from A to B is
the summation that is the integral of the
product of the tangential components of the
vector, that is F cos theta and the differential
path length dl along the path.
Now you have to remember that both F and theta
can vary along the path. They are not, they
need not be constant. That is why, you need
to have this integration or summation. Now
if the path C is closed, then it is called
a closed line integral which is denoted by
the integral symbol with a circle as well
as a subscript C F dot dl so this is the clothes
line integral.
Now let us look at the meaning of the surface
integral. This is the Cartesian coordinate
system and there consider a small elemental
surface area DS and DS can be represented
as a vector and that is equal to the area
of this small elemental area multiplied by
the unit vector representing that area. And
the unit vector is perpendicular to the small
elemental area, perpendicular to that surface.
Now there is a vector field F in space. Then
dot product of vector F and the unit vector
is F cos theta and the surface integral of
F dot dS that is surface integral of a vector
field F across a surface S is defined as the
summation or integral of the product of the
vertical component of the vector to the surface
F cos theta and the area of the differential
surface dS.
So you are multiplying the component of the
vector field F along with unit vector with
the small elemental area, then summing it
up all along the area. We need to do this
summation or integral because both F and theta
can vary across the surface. If surface S
is closed, it is a closed source integral
denoted by F dot dS and with this symbol,
like this.
Now look at Faraday’s law. Here, concentrate
on this illustration. You can consider it
as a closed plastic bag or a balloon where
only one side is open. Now this is the open
side. Now there is a magnetic flux density
vector field B. Now there is a electric field
vector field E, electric field intensity.
Now what Faraday’s law is stating is that
if you sum up all the component of the E field
along a closed path and multiply it by this
small elemental length, that is if you take
the line integral of E along the closed path,
then that is equal to the surface integral
of the magnetic flux density vector.
That is, if you look at a magnetic flux density
B coming out of this small elemental area
and if you multiply the normal component of
that with the elemental area, you get the
net flux coming out of this limited area.
So you are summing up all those kind of fluxes
coming out of the body of the plastic bag.
Then you take the time derivative of that,
a kind of flux coming out and that is also
called EMF. So line integral of the electric
field along a closed path is equal to the
rate of change of total flux coming out of
the area enclosed by this path.
Now the direction of the induced currents
should be in such a way that the magnetic
flux induced due to that current induced should
be opposing the change in the original flux
enclosed by the closed path. So this is called
Lenz’s law. So we are using Lenz’s law
in finding out in which direction the EMF
should be, the polarity of EMF.
Now for that, consider this diagram, a simplified
diagram where B is assumed to be increasing.
Now this increasing B should be producing
an EMF in such a direction such that the current
produced should be producing an induced magnetic
flux density or magnetic flux that is opposing
this original flux. So using the right-hand
rule, the magnetic flux has to be in this
direction. So this is opposing the original
flux. So this is the correct direction. So
that is how you take care of this negative
sign.
Now in this case, the increasing flux is in
the other direction. So here, to oppose this
flux, you have to have a polarity like this
for the induced EMF. Then only it will produce
a current in this direction and that induced
flux will be opposing this original flux.
So in the modelling of the magnetic field
interaction, you will be using this principle.
Now let us look at the units here. E is in
volts per metre and B is in Weber per square
metre. So Weber per square metre is also called
Tesla. So these are the SI units.
Now this is an illustration connecting the
Faraday’s law or magnetic field interaction
with mutual inductance that you are familiar
in circuit theory. Now imagine some sort of
a current somewhere in space. So this current
is I subscript G. So this current is inducing
some flux in a closed path. Now this is a
imaginary path, this path need not be made
of conductors or anything but of course, if
you want to measure something, then it will
have a conductor here. Otherwise you know
even if you take an imaginary path in here,
Faraday’s law is true.
Now in the case of a circuit in the form of
a closed path, the mutual inductance M between
this and this is defined as M is identical
to rate of change of total enclosed flux here
to rate of change of the current. That is
d phi by DIG. So this is the definition of
mutual inductance. Now you can do some algebraic
relation. You can do like this because this
is the total derivative. So MDIG is equal
to D phi. Now you take the time derivative
on both sides, M DIG by DT equal to D phi
by DT.
So rate of change of flux from Faraday’s
law, we have seen that it is just like a EMF,
a source of voltage. So this is the Faraday’s
law. So D phi by DT can be equated to minus
D by by DT in surface integral of B dot DS.
So you can see that Faraday’s law of magnetic
field interaction can be modelled as a series
voltage source if you are involving a circuit.
So this phenomena of magnetic field interaction
can be replaced by a voltage source. So the
direction of the voltage source need to be
determined by Lenz’s law as we have seen
before and this voltage source VI is given
by M DIG by DT. So we have reduced the magnetic
field interaction into a voltage source, series
voltage source with the circuit. This you
will see in the discussion of crosstalk or
magnetic field interaction.
Then we come to Ampere’s law. So look at
this picture. So here also imagine a plastic
bag with one side open. Now there are several
vector fields here. One is the electric flux
density field D, then we have the free current
field J, current density field J. So these
are you know some of these are entering the
plastic bag and going out of it, it can be
in any direction, you can imagine. If you
look at this equation, here this is a line
integral along this path which is enclosing
the surface as, then this total current density
and the vertical component of the current
density coming out of this.
So if you have current density coming out
here and the small elemental area, if you
multiply it by that, then you get this small
current coming out. So you add up all those
small currents and that is given by this.
So these are the free currents, total free
currents coming out of these surfaces plus
now all the electric flux density is coming
out, the vertical component of that multiplied
by the small area. Actually this is the displacement
current in the Maxwell’s equations. So rate
of change of d dot DS is the displacement
current ID.
So H is given in amperes per metre. So the
line integral of the magnetic field intensity
vector, H around a closed contour C or magneto
motive force or MMF is equal to the sum of
the total conduction current and displacement
current that penetrate the surface S bounded
by the contour C. So this is Ampere’s law.
Now look at Gauss law. Imagine a closed path.
So it can be your plastic bag or balloon where
the end is tied up. So you do not have any
open end. It can be of any shape. So you have
several magnetic flux density fields coming
out on the closed path and going in again
or just going through it. You can have charges
inside, positive charges and you can have
negative charges outside. So there will be
electric flux lines connecting these 2 charges.
Now what Gauss law is stating is that total
electric flux density coming out is equal
to the charge contained within that closed
surface.
That is, you are summing up all the electric
flux density lines multiplied by small elemental
area, summing it up. And so, it is the total
electric flux coming out. That is equal to
the charge contained within this volume. It
does not matter where those charges are. Sum
of all those charges together will be equal
to Q. So this is Gauss law. The net electric
flux D through a closed surface equal to the
net positive charge enclosed by the surface.
Now unit of Q is in coulombs and D in coulombs
per metre squared.
Now instead of electric flux density, if it
is magnetic flux density, then we will see
that closed surface integral of B dot DS equal
to 0, that is because we do not have any magnetic
monopoles. We do not have any positive magnetic
pole or a negative magnetic pole. So that
is why this is always magnetically equal to
0.
Now let us look at the circuit implication
of Gauss’s law of electric field interaction
and how we can tie it up with a mutual capacitance
between 2 metallic bodies. Consider 2 bodies,
one is charged to plus Q positively and the
other is charged to minus Q negatively and
there is a potential difference between them
expressed by V subscript 12. So the capacitance
between them in circuit theory is defined
as C12 equals rate of change of Q with respect
to rate of change of V. So this is the definition
of capacitance, mutual capacitance.
This is the total differential, you can do
this algebraic manipulation. Then you take
the time derivative on both sides, then you
will see that DQ by DT is nothing but the
current I. So it is like a current injection
already you can see that. So tie it up with
the Gauss law. What Gauss law state is that
closed surface integral, so you can take a
closed surface along this. Closed surface
integral of epsilon E which is nothing but
D for a linear material, dot DS equal to Q
and take the time derivative on both sides,
so this is current injection.
So rate of change of field in relation to
conductors can be modelled as a current injection.
So electric field interaction can be modelled
as a current injection. So this electric field
is created by these charges here. So look
at this current source IC. IC is nothing but
C12, DV by DT, rate of change of voltage.
It 
is this one. Now SI units, capacitance is
in coulombs per volt comes from here or we
can call it as Farads, the electric permittivity
epsilon has the unit Farads per metre.
