In this chapter we will review the principles
of electromagnetic, you might have taken a
course in electromagnetic and vector algebra
before I suppose. So this would be old year
review just to remind you those principles
that you have learned already.
This is the outline of this chapter, it is
divided into 5 modules, each of these modules
will be approximately 20 minutes to 30 minutes
of lecture time. First module 1, I will consider
Maxwell’s equations, Maxwell’s equations
will be presented in the form of Faraday’s
law, Ampere’s law, Gauss’ law and boundary
conditions. We will look at the derivation
of these equations. This part is very much
required if we 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 the printed 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
with 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 does electric and magnetic
fields can penetrate in metals or conducting
medium. This section is very much important
when we discuss shielding, shielding of electronic
circuits using metallic enclosuress. Then
we will consider transmission lines in module
3, here also the solution 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
an 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 field from dipoles, we have cases
in which component leads or tracks on a printed
circuit board are causing electromagnetic
disturbances, so we need to calculate how
much will be emitted from this. So we will
see that any of these can be model as combinations
of small electric dipoles and small loops.
We will find expression for radiation field
especially 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 closed 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 equation. Imagine a vector field
in space all around here F, here the bar above
F indicates 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 vector
dl, the direction of this path is tangential
to this small section dl and that is indicated
by a unit vector and a hat symbol is shown
for 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
the line F Cos theta, now let us see what
is the meaning of the line integral indicated
by 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 differential path length
dl along the path. Now you have to remember
that both F and Theta can vary along the path,
they may not be constant so, that is why you
need to have this integration or submission.
Now, if the path C is closed then it is called
a close 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
closed line integral.
Now let us look at the meaning of the surface
integral, this is the Cartesian coordinate
system and imagine some surface and there
consider a small elemental surfaces 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 unit area and that unit vector is perpendicular
to that 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 the surface integral
of a vector field F across a surface S is
defined as summation or integral of the product
of the vertical component of the vector to
the surface F Cos theta and area of the differential
surface dS.
So you are multiplying the component of vector
field F along the unit vector with this 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 surface 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 an electric field
vector field E electric field intensity. Now
what Faraday’s law is stating is that if
you sum up all the components 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 magnetic flux density B coming
out of this small elemental area and if you
multiply the normal component of that with
elemental area, you get the net flux coming
out of this elemental area.
So you are summing up all those kinds of fluxes
coming out of the body of the plastic bag.
Then you take the time derivative of that
kind of fluxes coming out and that is also
called EMF. So line integral of 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 current should be in such a way
that with magnetic flux induced due to that
current should be opposite to the change in
the original flux enclosed by the close path,
so this is called the 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 you see the right-hand
rule, the magnetic flux has to be in this
direction so this is opposing the original
flux so this is the current direction, so
that is how you take care of this negative
sign. Now in this case 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 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 meter and B is Weber per square
meter, so Weber per square meter 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. This is an imaginary
path, this path need not be made of conductors
or anything but of course if you want to measure
something you need to have a conductor here
always even if you take an imaginary path
air 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 the rate of
change of total enclosed flux here to rate
of change of the current that is d Phi by
d I G, so this is the definition of mutual
inductance.
Now you can do some algebraic relation, you
can do like this because this is a total derivative
so M d I G equal to d Phi, now you take the
time derivative on both sides, M d I G 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 EMF, a source of voltage.
So this is the Faraday’s law, so d Phi by
dt can be equated to minus dy by dt in the
surface integral of B dot dS. So you can see
that how Faraday’s law of magnetic field
interaction can be modelled as a series voltage
source if you are involving a circuit so this
phenomenon of magnetic field interaction can
be replaced by a voltage source.
So the direction of the voltage source needs
to be determined by the Lenz’s law as we
have seen before, and this voltage source
V i is given by M d I G by dt. So we have
reduced the magnetic field interaction into
a voltage source series voltage source with
the circuit, this we will see in the discussion
of crosstalk or magnetic field interaction.
Then we come to Ampere’s law, so look at
this picture here also imagine a plastic bag
with one side open. Now there are several
vectors fields here, one is the electric flux
density field D, then we have the free current
field J current density field J, so some of
these are entering this plastic bag and going
out of it, it can be 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 S then this total
current density and the electrical component
of the current density coming out of this.
So if you have current density coming out
here and this small elemental area if you
multiply 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 through
this surface S plus now all the electric flux
density is coming out, the vertical component
of that multiplied by this small area. Actually,
this is the displacement current in the Maxwell’s
equation, so rate of change of D bar dS is
displacement current I D.
So H is given in amperes per meter, so the
line integral of the magnetic field intensity
vector H allow a closed court contour C or
magneto motive force or MMF is equal to the
sum of the total conduction current and the
displacement current that penetrates the surface
S bounded by the contour C, so this is ampere’s
law.
Now look at Gauss’s 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 of 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 whole area. It does not matter where
those charges are, some 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 meter square. Now, instead of electric
flux density if it is magnetic flux density
then we will see that closed surface integral
of B dot d S = 0 that is because we do not
have any magnetic monopoles, you do not have
any positive magnetic poles or a negative
magnetic pole so that is why this is always
identical equal to 0. Now let us look at the
circuit implementation of the Gauss’ law
of electric field interaction and how we can
tight up with the mutual capacitance between
2 metallic bodies.
Consider 2 bodies, one is charged to plus
Q positively and 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 to rate of
change of Q with respect to rate of change
of V so this is the definition of capacitance,
since it is 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.
Now try it with the Gauss’ law, what Gauss
law states is that closed surface integral
so you can take a close surface along this.
Closed surface integral of Epsilon E which
is nothing but D for a linear material dot
dS equals to Q and take the time derivative
on both sides and 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 this electric
field is created by this charges here.
So look at this current source I c, I c is
nothing but C 12 d V by dt, rate of change
of voltage.
Now SI units, capacitance is in coulombs per
volt from here or we can call it as farad,
the electric permittivity Epsilon has unit
farads per meter.
