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Dear students myself Devaprasad Kastha welcome
you to this introductory course on electrical
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machine one. You all know that electrical
machines are basically electromechanical energy
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conversion devices which convert electrical
energy to mechanical energy and vice versa.
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More than 90 percent of all electricity generated
is by means of conversion from mechanical
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energy to electrical energy, and again more
than 70 percent of all electricity generated
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is finally converted to mechanical form through
again this electrical machines; that goes
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on to show you the importance of electrical
machines in the field of electrical engineering.
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Now there are different types of electrical
machines that are used, although their basic
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operating principles are somewhat similar,
they are very different from their point of
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view of construction and usage. In this course
we will talk about two types of electrical
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machines, namely the transformer and the dc
machine. This is as I said is an introductory
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course, hence does not assume any previous
knowledge on the part of the student regarding
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the material; however, we do assume some working
knowledge on the basic principles of the working
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of electrical circuits and basic methodologies
for electrical circuit analysis.
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The text book for this course are Alexander
S. Langsdorf. The name of the book is “Theory
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of Alternating Current Machinery.” This
is second edition; the publisher is Tata Mcgraw-Hill,
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and the year of publication is 1993, this
is one text book. The other one is by professor
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D. P. Kothari and sir I. J. Nagrath. The name
of the book is “Electric Machines.” This
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is third edition; the publisher again is Tata
Mcgraw-Hill, year of publication is 2004.
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Other than these two books we will also have
occasion to refer to other text books, but
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that I will let you know as and when we use
them.
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The first electrical machine that we will
pick up are transformers. Now as you will
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very soon see it is not really a electromechanical
energy conversion device. It in fact, converts
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the voltage level of the AC supply. So, before
we go into the constructional features or
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the operating principle of a transformer,
I would like to show why a voltage transformation
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is required at all. Now the present electrical
power generation plants work on the principle
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of economy of scale; that means the larger
the power plant the cheaper will be the electricity
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generated by it, at least that is the belief;
however, you can easily understand that such
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a large power plant cannot really be located
near a dense populated city.
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There are many constraints; for example, space,
environmental pollution, availability of raw
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material, water, and then coal. So, all these
constraints usually determine where a power
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plant will be located, and it can be very
long distance away from the point where the
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electricity will be ultimately used, and also
all the electricity generated by a large power
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plant cannot be used at a single place. So,
that brings in the question of transmitting
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large amount of electrical power that is in
thousands of megawatts over very long distance
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more than hundreds of kilometers, and this
has to be done efficiently in order to make
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the economy work.
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Now we will see how the voltage level of transmission
affects the efficiency of power transmission.
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Let us say there is a load
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which needs P amount of power at a voltage
of V, and this power is being transmitted
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from a generating station which is some L
meters away from the load. You can easily
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find out the current drawn by the load I;
I is given by P by V. Now, the load is connected
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to the source using a transmission line; transmission
lines are usually designed to operate at a
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constant current density J. Therefore, the
cross-sectional area of the conductor a will
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be equal to I by J. The resistance of the
transmission line R, then is given by rho
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L by a which is equal to rho L J by I where
rho is the resistivity of the transmission
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line material, L is the length, and a is of
course, the cross sectional area.
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So, what is the total power loss in transmission?
Power loss P L equal to I square R, but I
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is given by P by V; therefore, this is also
equal to, but I again itself is given by P
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by V. So, we see in order to transmit a given
quantity of power P the power loss P L by
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P, the percentage power loss P L by P is proportional
to L by V. So, larger the voltage V smaller
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will be the power loss. So, that says that
the power should be transmitted at the maximum
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possible voltage. So, how do we determine
this voltage?
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Now, in order to transmit this power from
the generating station to the load we will
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need to erect transmission towers, and larger
the voltage higher will be the height of the
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transmission tower. So, there will be some
cost associated with constructing these transmission
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towers. So, when we reach a breakeven point
that is the cost of erecting those transmission
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towers exceeds the power loss that will be
saved, that is the voltage which will be chosen
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for transmitting a given amount of power,
or whatever voltage is permitted by the present
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state of technology. Whatever may be the criteria
this voltage usually comes to a substantially
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large value.
Nowadays the maximum transmission voltage
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is in excess of 400 kilovolt; however, generating
alternating voltage at this voltage level
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of 400 kilovolt or even 100 kilovolt is not
very convenient, because after all, this power
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has to be generated in the confined space
of a generator and providing for insulation
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in that confined state at that high voltage
level is problematic. Therefore, generation
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of electrical power will be usually at a much
lower voltage in the range of 11 kilovolt
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possibly. Similarly when you want to consume
this power at the load end; it is not convenient
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to use the same 400 kilovolt or 200 kilovolt
or whatever is the transmission voltage, because
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that is neither convenient from the construction
point of view of the load nor is it safe.
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Therefore, the load will also like to see
the voltage at a much lower level; that brings
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into the question that both at the generating
end and at the load end you need to change
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the voltage level, and this is done using
transformers. So, the general construction
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will be you have this generating station which
will feed a transformer; the transformer will
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be connected to the transmission line on the
load end. There will be another transformer
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which will connect to the load. So, this is
the idealistic construction of a transmission
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system for electrical power from the generating
station to the load; of course, you can understand
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that there will be some power loss associated
with the transformers themselves.
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And since the purpose of this transformer
is to raise the transmission voltage level
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in order to reduce loss, the transformers
themselves should be very efficient; otherwise,
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the purpose will be defeated. Fortunately
transformers can be constructed to give really
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very high efficiency in the large power range;
it is not unusual to make transformers with
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efficiencies larger than 98 percent, and this
scheme works. In fact, it is the possibility
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of easy voltage transformation at very high
efficiency that has made alternating current
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system almost universally acceptable over
the direct current transmission system; that
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is it.
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Now let us see how this voltage transformation
can be achieved? The simplest way to do it
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is to put a potential divider; that is you
put a potential divider, and connect the load
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here. Well, it can work only for stepping
down the voltage; of course, you can understand
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it is not possible to increase the load voltage
above the supply voltage, not only that, there
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will be continuous power loss in the potential
divider resistance. So, this is not very efficient.
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If it is an alternating circuit then this
resistance can be replaced by an inductive
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potential divider. That will solve the problem
of power loss, because an inductor in steady
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state will not consume any power, but then
this circuit is not just an inductive potential
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divider; it is something more than that.
If you exchange the position of the load and
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the source; that is if you supply the power
at this point and connect the load here, the
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load will see a voltage which is larger than
the supply voltage. So, this construction
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as we will see is actually called an auto
transformer, and we will have occasion to
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discuss it in more detail at later point of
time. Here unlike a resistive potential divider
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the power transfer takes place both conductively
through this path and also by magnetic induction
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because of flux coupling between the two sections
of the windings. This is the principle of
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transformer, and since magnetic coupling does
not require physical connection between the
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windings it is entirely possible to separate
this auto transformer arrangement into two
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different windings.
For example, it will then look like this.
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These two windings are magnetically coupled,
and power transfer takes place through this
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magnetic field. Now this arrangement has some
advantage over this; just consider that there
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is a short circuit somehow between these two
terminals. What will happen? If this side
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voltage is much larger than this; for example,
this side if it is 11 kV and if this side
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is just 400 volts, as soon as there is a short
circuit a single short circuit the load will
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see the full 11 kilovolt applied across it
and will consequently be destroyed probably.
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The same fault here; the same fault here will
not cause any such catastrophic failure.
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In order to have this kind of a catastrophic
failure here both end, at both end there should
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be a short circuit the possibility of which
is far less. So, this construction is preferred
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over this construction of a transformer particularly
in cases where the voltage levels on two sides
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are widely different. This is called the two
winding transformer or simply the transformer.
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Now that we have mentioned the transformers
work on the principle of magnetic coupling
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it will be worthwhile to recapitulate whatever
we have studied regarding magnetic circuits,
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magnetic coupling, induced voltage in the
basic electrical technology course once more.
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So, what is a magnetic circuit? By magnetic
circuit we mean this kind of a construct.
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You have a core of some regular shape on which
some winding is put. Now if there is no current
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flowing into this coil we do not expect any
flux lines to be generated in this core; however,
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when you switch this source on and a current
does flow then magnetic lines of force will
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circulate inside this core. The flux density
or the magnet field intensity inside the core
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will depend on the magnitude of the current
i into the number of turns in the coil N.
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This quantity multiplication of N and i is
called the magneto motive force or MMF, and
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this magneto motive force causes a flux to
circulate.
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So, we can see an analogy with an electrical
circuit where the magneto motive force takes
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the place of EMF or voltage source in an electrical
circuit and the flux which is the effect takes
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the place of current which is the effect in
a electrical circuit. From this analogy we
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call this a magnetic circuit; we will see
further analogies later on. But before we
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get into how to draw this magnetic circuit
let us recapitulate some fundamental laws
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regarding magnetic fields.
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We all know that if a conductor carries some
current i then at a point near the conductor
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there will be a magnetic field. A magnetic
field can be characterized by a magnetic field
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intensity vector H or a magnetic flux density
vector B. The relationship between these two
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is very simple. B is equal to mu naught mu
R H, where mu naught is the permeability of
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free space and is given by 4 pi into 10 to
the power minus 7; mu R is a dimensionless
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quantity. It is called the relative permeability
of the material with which the core is made;
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for free space mu R is one; for ferromagnetic
material it can be in thousands.
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Now the magnetic flux density, the incremental
magnetic flux density at any point near a
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conducting wire due to a small length of the
wire d L is given by the Biot-Savart’s law,
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which says you can find out the incremental
flux density d B; from the formula d B equal
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to mu naught mu r by 4 pi into i dL cross
r by r cube, where r is the position vector
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of the point from the incremental length dL.
Please note that the d B is a vector, and
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its direction will be determined by the cross
product of i dL and r. In this case r is in
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this direction, and i dL is in this direction;
therefore, the direction of B will be out
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of this paper, on the top.
The total flux density B at that point due
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to length of the wire can be obtained by integral
equal to integral over length mu naught mu
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R by 4 pi i dL cross r by r cube. While this
gives a general formula for determining the
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flux density near a conducting wire in practice
this integration is difficult to perform except
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for very simple structure of the current carrying
conductor.
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Therefore, for solving magnetic circuits we
use a different formula which is the ampere’s
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circuital law. This says that the line integral
of magnetic flux intensity H over a closed
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path is equal to the current total current
enclosed. There may be several of them the
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total current enclosed; that is integral over
a closed path H dot dL equal to the total
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current enclosed. This law is very convenient
for finding out the flux density and the magnetic
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field intensity in magnetic circuits; let
us see why.
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The structure of a magnetic circuit as we
have said consists of a regular core and a
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winding. Now when this winding is excited
there will be flux line inside the core. Since
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the relative permeability of the core is normally
much higher than air or free space almost
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all the flux lines will be confined inside
the core. Also since the total amount of flux
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crossing the core at any position of the core
is fixed the flux density along this flux
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line is constant, and hence the H over this
path is constant. Now if we choose the closed
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path of the circuital law to be something
inside this core we know that H over this
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length is constant; therefore, we can find
out H from the simple formula that if there
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are total N number of conductors carrying
current I then the total current enclosed
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is N I, and if the length of the mean length
of the path to the core is L then H is given
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by NI by L.
So, in a magnetic circuit finding out the
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value of H and hence the value of B which
is nothing but mu naught mu R H is relatively
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simple. This of course makes the assumption
that the leakage flux outside the core is
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almost negligible. Now this gives you a method
of relating the flux with the MMF in I. That
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is the total flux a phi is equal to the flux
density B multiplied by the cross-sectional
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area of the core A which is then is given
by mu naught mu R H is given by N I by L into
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A, or this is given by N I divided by mu naught
Phi is equal to B into A which is mu naught
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mu R; H is N I by L mu naught mu R A.
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So, if N I is the MMF then flux phi can be
written as phi equal to MMF N I by reluctance
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R where the reluctance R of the core is given
by L divided by mu naught mu R A. So, if we
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extend the analogy that the MMF N I is equivalent
to a source of potential, and flux is equivalent
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to a current in a magnetic circuit then the
quantity reluctance is equivalent to resistance
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in an electrical circuit; therefore, a magnetic
circuit can be represented by a MMF source
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N I which sends flux through a reluctance
R.
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The corresponding equivalent electrical circuit
is a voltage source E sending a current through
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a resistance R; that is why this kind of a
structure with a core with a winding on it
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is referred to as a magnetic circuit because
of its similarity with an electrical circuit.
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In practice though this magnetic circuit is
somewhat more complicated than a electrical
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circuit. In a electrical circuit the resistance
is normally constant; however, in almost all
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practical magnetic circuits the core material
is made up of ferromagnetic material.
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One property of this ferromagnetic material
is that the relationship between the flux
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density and H magnetic field intensity is
not linear; that is if we plot the flux density
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B versus the magnetic field density H. If
it was linear then it would have been a straight
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line; however, for most material most practical
ferromagnetic material it is not so. Initially
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it follows a straight line, but as B increases
after some point this starts deviating from
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a straight line. In fact, B does not increase
as fast as H, and beyond some point even if
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you keep on increasing H, B practically do
not increase.
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This is due to the special constructional
feature of this magnetic material which are
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made up of very tiny molecular level magnetic
dipoles; however, almost all ferromagnetic
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material exhibits these characteristics which
is called the B-H characteristics, and that
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B H characteristics has a prominent saturation
phenomenon where the relationship between
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B and H is nonlinear, and after a critical
value of B further increase in H does not
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really increase B to a very great extent.
Therefore, for ferromagnetic material the
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practical operating point that realizes the
true potential of the material is somewhere
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in this junction in this region where the
B H curves starts bending. This is called
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the knee point of operation.
Because beyond this point even if you put
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a large MMF that is H there will not be much
increase in the magnetic flux density or the
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total magnetic flux circulated; obviously,
in this region the relative permeability of
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the material which is mu R equal to B divided
by mu naught H does not remain constant; therefore,
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reluctance of the magnetic core which is given
by R equal to L divided by mu naught mu R
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A does not remain constant for all values
of B. This is what makes analysis of magnetic
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circuit somewhat more difficult compared to
analysis of a electrical circuit, let us see
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why.
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A magnetic core need not always be made of
a single material. In many cases a magnetic
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core in addition to the ferromagnetic core
material we will also possibly include an
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air gap. Now the relative permeability of
the air gap and the magnetic core are very
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different. So, how we are going to find given
a number of turns and the current flowing
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00:39:50,769 --> 00:39:59,339
through it; how are we going to find out what
is the total flux circulated? In this case
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where the magnetic circuit has two different
types of material with two different relative
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permeability. In an electrical case it would
have been simple. This is basically a series
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circuit; we could have drawn it for electrical
case; we could have drawn it as a MMF source
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N I, reluctance of the magnetic path magnetic
core R core in series with the reluctance
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of the air gap or air.
The problem with ferromagnetic material is
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that the value of R c is not known at priority,
because it depends on the value of the flux
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density as we have already seen. The value
of R a more or less remains constant, it is
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not a function of the flux density it is linear;
the B H characteristics of air gap is linear.
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Therefore solving this involves is a little
more involved, and let us see how we will
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solve this kind of a circuit. Although the
field intensity may be different the total
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flux crossing the core material and the air
gap of course same. If we neglect the fringing
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effect at the air gap then the flux density
in the core and flux density in the air should
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00:41:48,259 --> 00:42:03,519
also be same. The total MMF N I
can be written as flux magnetic field intensity
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00:42:03,519 --> 00:42:11,999
H c in the core multiplied by the length of
the core plus field intensity in the air gap
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00:42:11,999 --> 00:42:22,690
multiplied by the length of the air gap.
But the relationship between intensity in
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00:42:22,690 --> 00:42:35,269
the air gap and the flux density is constant;
therefore, N I equal to H c into L c plus
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00:42:35,269 --> 00:42:47,140
H is equal to B air by mu naught since mu
R for air gap is one into L a, but B a is
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equal to B c. Therefore, we can write N I
equal to H c L c plus B c by mu naught L,
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a or B c equal to mu naught into N I minus
H c L c by L a. This is one relationship relating
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B c to H c which is linear. The other relationship
between B c and H c is given by the B H curve
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of the material.
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Now we plot them together H c on this axis
and B c on y axis. This is the intrinsic B
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H characteristics of the material, and the
other one given by the circuit which is can
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00:44:11,059 --> 00:44:19,819
be considered as a magnetic load line. The
intersection of these two curves will give
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00:44:19,819 --> 00:44:37,710
you the operating point H c zero and B c zero.
So you see, although solution of a magnetic
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circuit like this is somewhat more involved
than solving an equivalent electrical circuit,
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but it can still be handled in a graphical
manner, and this approach is used. At this
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point we think that we have discussed sufficiently
about magnetic circuit. So, we will move to
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00:45:02,369 --> 00:45:11,569
a different topic which is the induced voltage
in a coil.
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Let us say we have a magnetic core, and we
are using a coil to set up magnetic flux in
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this direction. If the current flowing is
into the coil is direct current then this
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flux will be a direct flux; however, if the
core is excited with alternating current then
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the flux will also be alternating. We have
seen the relationship between the flux and
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the current which is linear if the reluctance
of the material can be assumed to be constant.
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The question now comes if I put another coil
around this leg, what will happen? This has
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been studied by Michael Faraday, and the famous
law says that there will be a voltage induced
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00:46:46,769 --> 00:46:55,109
in this coil one, two. The magnitude of the
induced voltage e will be proportional to
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the rate of change of flux density d phi d
t multiplied by the number of turns in the
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00:47:02,660 --> 00:47:15,240
coil N, and the polarity of the induced voltage
will be such that it will try to oppose the
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very cause which it is due. Let us look at
it a little more carefully what that means.
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Let us say this is our coil, and there is
a flux phi in the upward direction. This is
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a coil with let us say N turns, and there
is a flux phi in this direction, and let the
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phi be varying in a sinusoidal fashion phi
equal to phi max sin omega t. So, by Faraday’s
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00:48:17,549 --> 00:48:30,109
law the induced voltage e will be such that
magnitude e will be proportional to N d phi
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00:48:30,109 --> 00:48:54,660
d t equal to N omega phi max cos omega t,
and the direction of the polarity of the induced
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voltage will be such that it will tend to
oppose the very cause which it is due; what
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is the cause that the changing of the flux.
Now let us see at t equal to 0 the flux was
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0, and as time progresses t becomes positive;
this flux tries to increase.
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So, in order to oppose that cause, what the
induced voltage will try to do? If you close
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this switch this induced voltage E will send
a current through this loop which will tend
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00:49:37,559 --> 00:49:44,730
to cancel this current which will tend to
cancel this flux; therefore, the circulating
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00:49:44,730 --> 00:49:50,539
current will be generating a flux which will
be in the opposite direction. So, what will
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00:49:50,539 --> 00:49:56,730
be the direction of that current which we
can cancel this flux; obviously, the current
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00:49:56,730 --> 00:50:07,740
will be in this direction. Therefore, this
coil with this kind of a flux can be replaced
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00:50:07,740 --> 00:50:17,579
by an emf source E where the terminal 2 at
time 0 will be positive, and the terminal
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00:50:17,579 --> 00:50:28,349
1 at time 0 will be negative, and the potential
E 21 in that case will be given by N omega
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00:50:28,349 --> 00:51:00,630
phi max cos omega t. So, if we want to find
out the rms value, then E 21 equal to omega
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00:51:00,630 --> 00:51:22,990
is given by 2 pi f into N phi max divided
by root 2.
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00:51:22,990 --> 00:51:49,150
Or the induced voltage E equal to 2 pi by
root 2 is 4.44 f phi max N. So, this is the
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00:51:49,150 --> 00:52:02,980
formula of induced rms voltage in a coil with
number of turns N which links a alternating
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00:52:02,980 --> 00:52:11,750
flux phi. Generally alternating quantities
in ac circuit analysis are represented by
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00:52:11,750 --> 00:52:20,250
phasors; therefore, if we draw the phasors
and take the flux phasor to be the reference
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00:52:20,250 --> 00:52:30,589
phasor then the induced voltage E will be
leading it by an angle 90 degree. Here I would
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00:52:30,589 --> 00:52:41,030
like to bring to your attention some of the
conventions that are used by the authors of
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00:52:41,030 --> 00:52:47,430
different books. The quantity that we have
been calling induced voltage in fact by many
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00:52:47,430 --> 00:52:55,579
authors is called the counter induced voltage,
because here we have got this e by simply
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saying e equal to N d phi d t.
And we have incorporated the last part of
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00:53:06,259 --> 00:53:12,539
the Faraday’s law that is it opposes the
very cause which it is due by taking its proper
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00:53:12,539 --> 00:53:21,319
polarity. This many authors call the counter
induced voltage. The actual Faraday’s law
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says e equal to minus N d phi d t. So, in
our case this e will be opposing the supply
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00:53:34,230 --> 00:53:39,970
voltage in the case where e is taken with
a negative sign. This is called the induced
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voltage; therefore, in our case the KVL of
the coil will be written as V equal to e;
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00:53:46,849 --> 00:53:54,079
in this case the KVL will be written as V
plus e equal to 0, that is the only difference.
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00:53:54,079 --> 00:54:01,609
In fact, if there is no scope of confusion
then we will keep calling this as the induced
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00:54:01,609 --> 00:54:08,940
voltage although many authors in some text
books will be calling it the counter induced
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00:54:08,940 --> 00:54:11,030
voltage.