In this next lecture on the modeling and analysis
of machine scores let us go ahead and look
at the elementary ideas on how we can model
electrical machines.
We know that all electrical machines has an
electrical port through which you are going
to give electrical supply and then there is
a mechanical port 
at which point you are going to connect the
load.
This is what we had already seen in the last
lecture and we start off from there.
The aim of our exercise as I mentioned last
time is to derive relationships between the
electrical side, the inputs you give to the
electrical side and the response of the system
because of that.
If you look at the electrical side you know
that electrical machine has machine windings.
If it is a DC machine it has an armature,
if it is an induction it has three phase easy
windings, an alternator also has three phase
AC, and this is in its elementary form.
It is nothing but a wire loop having an input
electrical port where you can then connect
an AC supply or a DC supply.
In the simplest form let us say we have a
wire loop and then you have a DC supply and
you are going to connect this to this wire
loop.
So how do you now write equations for this?
It is very simple being a DC circuit it is
very elementary what you would say is the
applied voltage V is equal to the resistance
of this loop multiplied by the flow of current,
let us call it IA and this as VA.
So you have VA here and this wire loop has
a resistance R, there is a current that is
flowing that is IA and these three are then
related by this equation.
Now this is fairly straightforward there is
nothing else to do we have already arrived
at the model for this wire loop as far as
it is DC.
Now supposing you are now going to vary this
DC voltage in some manner and you want to
find out how this current is going to change.
You may say that well I have an equation VA
is equal to R into IA and if I change VA,
IA is going to change in proportion.
That is not really true, because if you are
going to change this applied voltage no doubt
the current will change if you change slow
enough yes, IA you will see is going to change
in proportion to V, but if you are going to
change this VA rapidly then you need to consider
some more effects and to be more relevant
to the case of electrical machines.
Suppose you have an AC supply where this voltage
is now going to change fairly rapidly in the
case of AC supplies that we have India you
can, the AC supply is going to change at the
rate of 50 Hz.
So if that is going to change then, the relationship
between VA and IA is not just this.
Why it is not just that?
That is because any flow of this kind of IA
in a wire loop is primarily going to generate
in addition to the other effects.
It is going to generate a magnetic field.
Now it is this magnetic field that it is going
to generate, that is what is going to cause
the difference between a DC excited wire loop
and an AC excited wire loop.
Now let us look at an experiment which you
might have done earlier to see what this magnetic
field is.
Now this is an experimental arrangement of
an experiment that you may have done in your
high school days, an experiment intended to
demonstrate the existence of a magnetic field.
Now field is an idea which has been proposed
in order to explain the observation of action
that happens at a distance.
Due to something that happens here some other
action happens somewhere else even though
apparently there is no physical connection.
Now you are going to see an experiment where
due to the existence of a magnetic field ion
gets aligned along certain ways.
These then demonstrate that there is an effect
of the flow of certain amperes in a loop on
the ion which has no physical contact with
the wire loop.
Now can I, so in this experimental arrangement
what we have is a wire loop.
Having several turns and in the horizontal
plane of the wire loop I mean horizontal level
we have then added some small particles of
ion and upon excitation of the wire loop with
DC flow of current and if you now gently tap
this these small particles of ion they then
turn to align themselves along certain lines
which you can see, you can see that there
is several lines here and then similarly several
lines here and in the centre there are broader
flow lines here, right.
You can see that they have aligned themselves
and these lines are then known as the magnetic
flux lines which then represent the existence
of a magnetic field.
Now it is easy to look at these arrangements
in a simple loop like this, but in an electrical
machine this would be far more difficult because
the arrangement of the loop is not so simple
and therefore if you are going to look at
electrical machines and you want to find out
how the field lines like this arrange themselves
one has to do it by some sort of simulation
which we shall see a few examples of that
now so that every time we do not have to look
at these kind of experiments.
We can then do completely by simulation alone.
So here we have some simulation results, if
you want to simulate how, simulate and analyze
how magnetic field distributions are there
in a particular region of space you generally
use simulation software which go under the
generic name called finite element analysis
based simulation software, and here we have
the results from a simulation of the system
that we have just now seen which is a coil,
this area designates the coil effectively,
this shows that you have a plane and a coil
that goes in and out of that the distribution
of the lines is what you have seen on this
plane, okay.
So this area which I have marked in red shows
the section that you get if you slice this
coil as you have seen here by this plane,
horizontally if you slice it then you see
this rectangular area as the one having several
turns and some flow of loop current through
that, in this case the simulation has been
done for a fifty turn loop having i equal
to 10 ampere flowing through each turn.
Now with this you can see how the field lines
are, this is a, these simulation results of
how the magnetic field lines are going to
look like and here you can see the magnetic
field lines as you have seen in the experiment
the field lines go circular near this and
then they take bigger loops here and here
still bigger loop.
If you allow it this will go and complete
somewhere far away.
Similarly on the other side this goes and
completes far away.
So you have these outputs can be simulate.
Now what we see here are magnetic flux lines
which denote the existence of a magnetic field.
Now in addition to this you also describe
something called magnetic flux density which
is an indication of how many flux lines are
there in a given area.
If you now take an area which is going to
be at this place, for example if you take
here the magnetic flux lines are going to
be loops like this 
as you have seen here and if you now take
an area which is at, which is equal to, which
is inclined at an angle 90 degrees to the
flow of these lines.
Then how many such lines are there in a given
unit area that is indicated by the magnetic
flux density?
What I want to say here is the magnetic flux
density 
is nothing but flux lines per unit area.
So this is generally given the symbol B, so
B denotes the magnetic flux density, if there
are more flux lines in a given area that means
flux density higher if there is less flux
lines then flux density is small.
So obviously in the regions here you find
that the flux density is more because there
are more flux lines and as you go outside
far away from this loop you find that the
flux lines are fewer.
So in all these areas flux density is small.
Now an indication of this is also there from
the simulation software and that you see in
the next plot.
Which shows how the magnetic flux density
looks like.
Now this indicates how the software has given
an output regarding the flux density in different
areas, you see that if in areas where you
have deep red which is here, deep red areas
the flux density as you can see from here
is about 6.179 into 10 to the power of (-4),
flux density is measured in the units of vapor
per meter square is also called as in the
units of Tesla, whereas flux itself, flux
lines, flux is measured in vapors which is
abbreviated as WB and flux density is abbreviated
as E.
Now you see that the flux density in regions
here are higher than the flux in regions outside.
So if you go far away the flux density decreases
you can see on the scale here flux density
which is in the regions of red is about 6.179
into 10 power (-4) and as you go down to the
green regions it falls down to 1.5 and so
on and finally in the regions which are really
far away you have flux densities which are
very, very small.
Very little flux is there.
Now these levels of flux are really small
and not of much use in electrical machine.
You want higher flux and if you want higher
flux you need to change the medium in which
this is being the magnetic field is being
established and for that what one does is
to put ion.
Now the next plot here shows that the region
in between this wire loop is being filled
with ion and now you can see that flux density
in the ion is very high whereas in the regions
around it is very low.
If you look at the levels the deep red areas
here now have a flux density of about 1.2
web air about per meter square as is indicated
by the legend here and whereas regions that
are close to the edges here that goes to about
0.4 web air per meter square and in the regions
outside it is really small as compared to
what is there inside.
Now if you look back at the earlier figure
nowhere in the area around the wire loop you
have v which is in the range of 1.2 vapor
per meter square.
The maximum is only 0.7 into 10 power (-3)
which means the flux levels the flux densities
that are obtained in a pure air when it is
around this wire loop it is very small whereas
when you put ion the flux density levels dramatically
go up.
Also you notice another thing that most of
the flux is now confined to the ion and it
appears as if there is not anything outside
of the ion.
That is not really so if we now focus on the
areas which have small flux density and zoom
into the areas that have smaller flux density.
Now you see the next plot you see that there
is a flux density variation even in those
areas which appeared to be uniformly low in
this region.
This appears uniformly low flux density is
not really so, there is a gradation there
also you can see that around the wire loop
areas, this is the wire loop one section and
this is the other section which is going in,
so here there is a slightly higher flux density
whereas if you go around flux density is still
lower and you find some flux lines that are
encircling this section of the wire loop.
Not all the flux lines as you see here.
Here if you see it appears that all the flux
lines go through ion only, but whereas on
deeper look you can see that there are some
flux lines that are circling this area of
wire alone.
So if you now put ion then you have much higher
level of flux and then most of that higher
level of flux is confined to ion areas which
are not ion spaces that are not ion have much
lower density of flux.
So with this understanding let us now get
back to our equations, so in effect from all
these things what we have seen is that even
a DC current flowing through a wire loop generates
magnetic field, higher the flow of amperes
more the magnetic field that is going to be
generated 
and the magnetic field is enhanced by the
presence of iron right, so if you are now
going to take, let me draw this ion structure
symbolically, you have ion and then you had
a wire loop.
This was what we had stimulated in the stimulation
software.
You have certain number of turns here and
you had certain number of turns here, right.
We are now looking at exciting this and what
we have seen is that the flux levels in the
ion here are dramatically higher than the
flux levels had there been no ion and those
dramatically higher flux levels are confined
to ion whereas rest of the space is not having
very high flux density levels.
So which means that if I now shape this ion
material in different ways you would still
have high flux densities in those regions
of ion and which therefore means by providing
an appropriate shape to this ion I can make
this high flux density area occupy whichever
space I want; however I want to shape it.
For example, if I am now going to have an
ion core that looks like this 
and then put certain number of turns here
I can still be sure that in all these areas
you will have high flux density.
So shaping of high flux density regions is
another important aspect in electrical machines.
So we have now seen that current reduces magnetic
field, use of ion 
increases dramatically 
the magnetic field for the same excitation,
for same current that flows.
There is no use in saying that by increasing
this amount of flow of current you are increasing
the field for a given current with and without
the iron there is a dramatic difference and
because iron high field density exists in
the iron and not anywhere else you can now
shape iron in order to get high flux density
regions where desired.
So these are all important aspects of the
generation of magnetic field inside the machine.
So now that there is a magnetic field invariably
iron is used in electrical machines because
you want high magnetic field for low flow,
right you do not want to generate large excitation
and have a large magnetic field.
You want to generate as much magnetic field
as you can for small current.
Now if there is a field then and if you are
going to change the excitation the field is
also going to change and if the field is going
to change then there are some basic laws of
electromagnetism which you will know, you
would have heard as Faraday's laws, they now
come into existence, which says that whenever
there is a change in the field associated
with the loop there is an induced EMF.
To be more exact this law says that an EMF
is induced in a wire loop which is equal to
the rate of change of flux linkage in that
loop associated with that loop, and how do
you define this term flux linkage?
Flux linkage is defined as the number of let
us call that as ?, the flux linkage as ?, then
? is defined as the number of turns in the
loop, which is N multiplied by the flux lines
that are going through that loop, which is
five, so if this is going to undergo a change
then an EMF is induced in the loop, which
is proportional or which is equal to the rate
of change of flux linkages.
And therefore, the simple equation that we
had Va=Ria can now be written as V=R x ia
+ the rate of change of flux linkages.
So a new term has to be added if one is going
to consider that the excitation is going to
change with respect to time.
So how do we now find out how much is the
value of ?, for that we take recourse to again
laws of electromagnetics.
There are laws of electromagnetics which you
can refer to books on electromagnetics.
Let us start with a simple equation that is
necessary for us.
We will look only at what is required for
us.
There is one law, which says that the ? over
a closed loop of H.dl both being vectors is
equal to the surface ? of J.da where J is
the current density in the region da, in the
region A over which means we are going to
integrate.
H is then called as magnetic field intensity.
Now, how does this apply to the situation
we have here.
You have an iron look now if you see this
arrangement let me draw three dimension extension
of this.
So that one can understand what is going on
three dimensional pictures and this wire loop
now goes around like this.
So if you now take a section of this along
the horizontal and look at that what would
you see is an iron core with a coil here,
this coil here and then this core.
This is the simulation that we have done in
the FE analysis.
We have taken a coil here and a coil here
allowing this current to flow and all that
we have seen.
Now in this you are going to have a magnetic
filed generator and we have seen that magnetic
flux lines go around this iron core in this
manner we have seen this in simulation in
the graphs that we have in the pictures.
Now h.dl around the close loop, so this is
now your closed loop around which flux close.
So if you integrate h along this closed loop
this equations says that must be equal to,
if you look at surface enclosed by this loop
which is this, in the surface enclose by this
loop you find out what is J, that is how flow
of amperes is distributed in that, and then
integrator that.
If you integrate it around this you would
basically get number of turns times multiplied
by whatever current is flowing, if I call
this as ia, this ia what is flowing around
the loop.
And if you integrate over that surface it
is nothing but J which is ampere density in
each of this turns multiplied by the area
of each of those turns of nothing but ia itself.
In the rest of areas there is no flow it is
open, so all these integration would gives
zero.
There is flow of current only in the region
where there is the wire loop.
So this is nothing but nia, and this h? that
is h.dl ? around this closes loop, h is the
same inside this iron material assuming that
the area of the sectional area of the iron
remains the same.
And therefore, this left side boils down to
h multiplied by the length of this loop which
is same as the length of the root through
which the flux flows inside the iron core.
So you have basically h.hlc is equal to nia
which therefore means that h is nothing but
nia/L. This term nia is then called as the
magneto motive force or mmf for short, numerator
is called as magneto motive force.
And the h is then measured in ampere per meter
in si units, which is also sometimes written
as ampere turns per meter, because of the
numbers of turns n that appears here.
Having got this how do we know go head and
find out what ? is.
We know that ? is nothing but numbers of turns
times the flux that is going through the iron
core, you have lot of flux lines going through
the iron core which we have seen in this simulated
result also.
And those flux lines number of such flux lines
multiplied by the number of turns in this
is what is going to give you ?. This in turn
can be written as numbers of turns times flux
density in the core we assume for now that
the flux density in this core is uniform,
that means all through this area of the iron
core remember this picture is drawn by the
taking a cross section of this.
So all through this area flux density remains
the same.
If that is the case then flux is given by
bxarea of the core, you must remember that
flux density b is given by flux lines per
unit area, it is measured in Weber per meter
square which is also called as tesla.
So nbac, b on the other hand we know from
the physics of the materials that b and h
are interrelated depending on the material.
Now let us assume therefore, that b = ÁH,
where Á is the permeability of the material.
You would definitely remember that if is the
material is air that is then called as Á0
or which is same as the free space, Á0 has
value a equal to 4?x10-7.
Now b = Áh, and therefore, if you substitute
that then you have ? = NAcÁH.
And now we have already derived an expression
for H here, so we substitute this there and
therefore that is NAcÁNia/L which is nothing
but ÁN2AcIa/L.
So if you have a flow of current ia as you
have in this expression then you get a flux
linkage which is ÁN2Acia/L. And therefore,
if you substitute this in the expression for
Va then you have Va = ria+d/dt(ÁN2Acia/L).
Obviously in this expression N2Ac and L do
not change with respect to time, and therefore
they can be taken out of the differentiation.
Now the question is what do you do with Á,
Á may be dependent on ia in a nonlinear material.
And therefore, it may not be feasible to take
it out of the differentiation ia in term depends
upon T, and therefore Á you can say it depends
upon T. But in situations where it does not
depend upon ia, we say that those material
are linear materials in that case you can
write this as ria +ÁN2Ac/Lxdia/dt.
And it is this term that you now call as inductance.
So you can now write this simplified form
as ria +Lxdia/dt which is the expression that
you would have been very familiar with from
the first course is on circuit analysis.
So this is a sense is the birth of the inductor,
the inductor is an interpretation of the phenomena
involving magnetic filed on or it describes
the effect of the magnetic phenomena on the
electrical circuits.
And this inductance is going to be very important
in determining the behavior of electrical
machine as we will see in the several lectures
that are going to come.
We then need to get an expression for ? so
that we can substitute in this equation. ? as
we have seen earlier can be describes as N?
and which is nothing but N multiplied by flux
density into the area of core, we must remember
that flux density is measured in units of
Weber per meters square, it is nothing but
the number flux lines the unit area.
So if you multiply that by the area of the
core, then what you get is flux and that multiplied
by number of turns is then you flux linkage.
B can now be written as ÁH where Á is then
the permeability of the material 
which is given by Á0xÁr where Á0 is the
permittivity of free spaces; Ár is the relative
permeability of the material.
So all this you would have seen definitely
in earlier course.
So here you end up with ?=NÁhAc, and we have
already derived in the expression for h which
is nothing but Nia/L. So you substitute that
expression here, so it is ÁN2ia/LxAc, this
is then expression of the flux linkage.
And one can then substitute this expression
in your expression for voltage.
So you have V=ria+ d/dt(ÁN2iaAc/L).
If you 
look at terms here N2Ac and L are not going
to change with respect to time.
And therefore, they can be taken out of the
derivative operations and this expressions
can be simplified as ria+N2Ac/Lxd/dt(Áia).
Á is something that may change with respect
to operation which we will see later on.
But enlarge in elementary electrical circuits
electromagnetic circuits also Á is a considered
to be a Fickĺs number, and therefore, that
can also be taken out of the differentiation
process.
So you have ÁN2Ac/L, and it is in fact this
expression that is now known as the inductance.
And therefore, you write this as ria + Lxdia/dt.
This is expression that would have seen in
first courses on electrical networks where
applied voltage if it is Ac can be written
as ria+Ldi/dt.
So this in effect is the birth of the induction
the inductions then represents is an interface
between electrical circuits and the magnetic
field.
It is representation of the phenomena that
involve electromagnetic fields on the electrical
network.
And this induction as we shall see in the
lectures to come is an important aspect in
determining the behavior of the electrical
machines.
It is therefore necessary for us to understand
what is inductance is, and how to write equations
for circuits involving inductances in next
few lectures, with that we will end now and
continue in the next lecture.
