Fundamentals of Electrical Engineering

Prof. Debapriya Das

Department of Electrical Engineering
Indian Institute of Technology, Kharagpur

Lecture – 51
Magnetic Circuits

So, we are back again. So, far we have studied DC circuit, then single phase AC circuits
along with you your what you call that regulance as well as your maximum power
transfer theory you have seen, and also the three phase circuits.

(Refer Slide Time: 00:34)

Now, we will start Magnetic Circuits right. First thing is that magnetic circuit we will
find things are very simple, only one or two things we have to understand. And next we
will find things are easy, particularly in magnetic circuit with coupling right, so mutual
coupling. So, and we have to understand that, and we will find things are very easy.

So, just a magnetic circuit actually, the area around a magnet is called the magnetic field
right that you know and it is in this area that we are what you call that the effect of the
magnetic force produced by the magnet can be detected right.
(Refer Slide Time: 01:13)

So that is your now and another thing is the electromagnetic system is an essential
element of all rotating electrical electric machines in our electrical engineering we see,
and electro mechanical devices as well as static devices like transformer right. So, and
our magnetic field actually is a coupling medium allowing interchange of energy in
either direction between electrical and mechanical system right, so that is your what you
call that at your it is what a actually magnetic field.

(Refer Slide Time: 01:44)

So, next is your the magnetic effects of electric current. Now, this is suppose here now H
is actually called magnetic field intensity or magnetic force, and first let me tell you, this
is a conductor right, this is a conductor circular conductor. And flux means just try to
understand that your flux line and this thing right. This flux line means you take a paper,
and current is entering into the plane or into the paper right.

So, if the if this is the direction of the current, if my thumb is the direction of the current,
then flux line will be the curling this curling finger right. So, this is this is the direction
of the current, this flux means that your current is entering into the into on the on the
paper right that mean in the plane, and this direction this is the direction of the flux.

So, if I say this current is entering into the plane right or in the paper your notebook say
it is entering, then this is the direction of the flux right that is clockwise direction. And
that is why this one that is why this is the flux line, and this is that is why this is the flux
line, it is clockwise direction, this is actually right hand rule right. So, I just we have to
understand one or two few things right. And this is any from here to here, it is taken
some distance are right, this is your what to call the magnetic flux line.

So, H is actually magnetic field intensity or sometimes we call magnetizing force right.
And your what you call and this is a r is this a we later will come to that r, it is radius
right. So, in this case, you are just let me clear it. So, then H actually it is uniform this
magnetic field is uniform around the conductor, so, it is basically tangential to this point,
and it remains constant right.
(Refer Slide Time: 03:37)

So, this is that your flux surrounding current right. So, a long straight conductor carrying
current it whatever I said it is written here right into the plane, the current causes a
magnetic field to be established in the space you are surrounding it right.

(Refer Slide Time: 03:48)

A line of flux is a closed path around the current such that the magnetic force is
tangential to it is all point around the line. This is H right, this is a H right, H is the
magnetic intensity or magnetizing force right.
(Refer Slide Time: 04:04)

So, the direction of flux I told you the right hand rule, I will just explain you, you go
through it whatever I said, same thing it everything it is here.

(Refer Slide Time: 04:13)

So, now H defined as the causative current, we will come to come later what is causative
current, per unit length of the flux line you are enclosing the current right. Now, in this
case because of symmetry H is uniform along each flux line. So, this is actually if you
look into that, this flux line is you are called your concentric right and uniform. So, H
actually at every flux is tangential and uniform right. So, this is your what you call so, in
this case because of symmetry, H is uniform along each flux lines.

(Refer Slide Time: 04:51)

Since, H is tangential all along a flux line, for any flux line in radius r right, so H is equal
to I upon 2 pi r ampere per meter that means, this is the radius of a flux line of r say here
in the diagram.

(Refer Slide Time: 05:11)

This is the radius of the flux line your r right, this is your radius of the flux line is r right
and your this is uniform, it is a circular. So, it is circumference is 2 pi r that is actually
length l is equal to 2 pi r. Therefore, H is equal to it will be I upon 2 pi r, it is ampere per
meter is Ampere’s law right so, let me clear it. So, this is actually your what you call H is
equal to I upon 2 pi r ampere per meter that is Ampere’s law.

(Refer Slide Time: 05:40)

Now, the flux density B is established by this field intensity right is a property of the
medium. So, for air or any non-magnetic medium, the ratio of magnetic flux density to
magnetizing force is a constant, that is if your B is the flux density, it is Tesla or Weber
per meter square it is unit right. And H right the magnetic field ampere per meter or we
will later we will see it is ampere tons per meter that B by H is constant, then its flux
density by magnetizing force ratio is always constant right.
(Refer Slide Time: 06:14)

And this constant is defined as mu 0, so the permeability of free space or the magnetic
space constant right, it is called permeability of free space or the magnetic space
constant.

(Refer Slide Time: 06:27)

And mu 0 is equal to either 4 pi into 10 to the power minus 7 Weber per ampere meter or
it is Henry per meter either this unit or this unit Weber per ampere meter or Henry per
meter. So, B is equal to mu 0 into H, B is the flux density, it is mu 0 into H and it is
Weber per meter square or Tesla it is unit is right. So, for all media other than free space,
actually B is equal to mu 0 mu r into H right.

(Refer Slide Time: 06:58)

So, mu where mu r is the relative permeability and is defined as mu r is equal to flux

density in the material divided by flux density in a vacuum right. So, it is a
dimensionless quantity, because numerator also flux density, denominator also flux
density, so it is basically dimensionless constant. So, flux density in material, so flux
density in a vacuum.

(Refer Slide Time: 07:20)

So, mu r varies with the type of magnetic material, and since it is a ratio of flux densities
I told you, it has no unit, it is a dimensionless. Also, we can write that B is equal to mu
into H; instead of mu 0, we can write mu into H in any material right your what you call
in any in a any particular you are what you call in material.

(Refer Slide Time: 07:44)

So, where mu is equal to mu 0 into mu r, it is called absolute permeability right. So, mu

is equal to mu 0 into your mu r. And the flux passing through the area for uniform flux
density phi is equal to B into A; B is the flux density. Suppose, Weber per meter square
an A is the area, it is meter square so, phi is equal to B into A, it is Weber right. So, this is
simple these are all these simple formula that B is equal to mu H, mu is equal to mu 0 mu
r, and phi is equal to B into A.
(Refer Slide Time: 08:23)

Now, magnetic circuits, now, you consider let us consider a toroidal ring of
ferromagnetic material, where mu greater than 1, it maybe iron, it maybe cobalt, and
nickel etc right. So, it is consider a toroidal ring.

(Refer Slide Time: 08:37)

Now, this is a circular ring so, what we will do is we will take your what you call that
you are mean path, this is actually mean flux path, we will take the mean distance right.
So, from here to your mean radius is capital R right, and this is the core circular core, so
it is diameter is d right. If it is diameter is d, so area will be pi by 4 d square. If it is a
meter, then it will be your meter square, if d is in meter. This is that cross sectional area
right, this is circular one. This is the mean flux your what you call mean your flux path.
Now, let me clear it let me move little bit up right.

(Refer Slide Time: 09:26)

Now, if you look into this if you look into this diagram, this is the coil wound on this you
are what you call on this toroidal ring. Now, current actually entering it, this current is
entering this ring has N number of turns right. So, whenever, now here we have to
understand your I cannot revolve this 1 to 90 degree, I will tell you that first thing is that
this is the leak[age] leakage flux part is also solve, and then this is the mean flux path.

Now, question is that it is a coil, if you have a coil, you wrap the coil like this, you grabs
the coil like this, and this is the current is entering right, this is the current entering, this
is the curren, and finally the current is leaving.

So, if you grabs the conductor like this, then this will be your the direction of the your
mean flux path, this is the direction of the flux path right. If I for example, this is your
what you call if you grab it this conductor, this coil this right if you grab it like this, then
this will be the direction of the flux path. So, if I make it like this, so this that is why this
is my the dire[ction] this is the I mean in the direction of the current, you grabs it right,
and then this thumb will be your direction of the flux path right.
So, in this case if you make it like this by right hand, if you grab this way your what you
call this winding by right hand, then thumb will be the direction of the flux path. So, this
is the mean flux path, and there will be some leakage so, this line also so some leakage
flux path right. So, it is actually coming it is actually direction will be like your in the
direction of the thumb, if you grabs it right hand. So, this is actually I, and this is N-
turns, and this is the only thing need to understand the direction of the flux path. So, and
this is you call mean flux path whatever core will take, we will take the your what you
call the mean radius or mean length.

Suppose, for example, suppose from here, suppose the thickness of this one, suppose if
you take your what you call thickness of this one whatever it is, you take the mean path,
you would calculate from here to here the mean path right. Because, we will go for mean
your what you call assuming that is uniform cross section, so your mean follow the mean
flux path. So, this is my mean flux path. So, direction of the you are what you call flux
line later we will show you that is analogy to DC circuit voltage current resistance to
your what you call magnetic circuit what are those quantity for analogous to that right.

(Refer Slide Time: 11:53)

So, that means, your this one, your the radius mean radius R, circular cross section the of
the diameter is d. So, in this case, the ring termed as core is excited by a coil wound, it
with N-turns carrying the current I told you right.
(Refer Slide Time: 12:06)

So, that means, the flux lines in the core enclose a current of F m is equal to N I right.
So, this is actually here it is number of turns is N, it is I right. So, your magnetizing force
or sometimes we call m m f that is your F m is equal to say N I ampere turns; if I is
ampere and N is turn, so its unity call ampere turn.

(Refer Slide Time: 12:28)

Now, which is the causative current I told you, this causative current means cause of the
existence of a magnetic flux in a magnetic circuit right that is why we call it is a
causative current, so establishing this flux. So, this is known as magnetomotive force,
sometimes we call it is m m f right, so magnetomotive force. So, F m is actually
magnetomotive force, sometimes we call it is at m m f.

(Refer Slide Time: 12:59)

So, by symmetry, H in this core is constant, because it is a right round each flux line and
for the mean flux your mean flux line of radius capital radius capital R, the magnetizing
force or magnetic intensity right, it can be written as H is equal to N I upon 2 pi R.
Earlier you are writing I upon 2 pi r, but here you have N number of turns, so H is equal
to N I upon 2 pi R. So, N I is ampere turn, and this the length, so ampere turns per meter
and then N I is equal to F m, here we have define F m is equal to N I right.
(Refer Slide Time: 13:34)

So, here you substitute, it will be F m upon 2 pi R ampere turns per meter or we can
write H is equal to F m upon l ampere turns per meter that l is equal to actually length of
the mean flux path that is that circumference that is you 2 pi R. So, H is equal to F m
upon l right.

(Refer Slide Time: 13:56)

So, next is the mean flux density, it is we know, B is equal to mu into H, so but H is
equal to F m upon l. So, B is equal to mu into F m upon l Weber per meter square or
Tesla either or mu this is the unit.
(Refer Slide Time: 14:08)

Now, flux is equal to A into B; A is the cross sectional area, and B is the flux density. So,
A into B that is B is equal to mu F m upon l, so mu F m upon l right. So, that means, my
phi is equal to we can write F m divided by l upon A mu right or we can write F m upon
R or we can write is equal to P into F m right. So, R actually we call l upon A mu M it is
ampere turns per Weber its unity, it is called reluctance of the magnetic circuit right, this
is called the reluctance of the magnetic circuit; or P is equal to 1 upon R, it is called
Weber per ampere turns right.

So, here it is R is actually ampere turns per Weber, and P is the reciprocal of R, the
reluctance it will be just Weber, this unit will be Weber or ampere turns of this physical
permeance of the magnetic circuit right. So, actually in the DC circuit, if you compare
that F m actually like your what you call the voltage in DC circuit emf right, here it is m
m f.

Now, phi actually in DC circuits say if it is a current, here analogous to magnetic circuit
is a flux. And R is the resistance in DC circuit, whereas in your magnetic circuit is the
reluctance right, it is just a analogy to that analogous to that right. And permeance in the
DC circuit g is equal to 1 upon R, the conductance here the permeance that is 1 upon R
that is Weber per ampere turns, so permeance of the magnetic circuit right.
(Refer Slide Time: 15:42)

So, that means, if I put it in a tabular form that analogue the analogous thing, electrical
circuits say emf, E is a volt in magnetic circuit, it analogies F m right. Current here is I
ampere in magnetic circuit in analogous is phi flux Weber. Now, resistance R ohm in
electric circuit in per magnetic circuit, reluctance R ampere turns per Weber or Henry or
your reciprocal right (Refer Time: 16:03) Henry to the power H 2 the power minus that
means, 1 upon Henry right.

And current I is equal to E by R that is you know emf upon R. And here flux is equal to
also it is analogous to that F m upon R right, because F m is analogous to E and R
reluctance R is analogous to resistance. So, phi is equal to F m upon R so, this is actually
called mmf upon reluctance right. Here it is emf upon resistance that is mmf upon
reluctance. And R is equal to you know rho l upon A right in the case of reluctance, it is l
upon A mu right, so l upon A mu 0 mu r; mu is equal to mu this is your what you call this
is the analogy from the electrical circuit and magnetic circuit.

When we will solve the magnetic circuit, we will follow the same way the way you do
super position or now we will phi current. Here also we can draw the magnetic circuit,
and we can solve like this right. So, things are very simple, only thing is that the right
hand rule little bit you we probably apply for various purposes the right hand rule. So,
the because the polarity why do you make this magnetic equivalent circuit like electrical,
you have to know which will be the plus, I mean analogous to your electrical circuit will
be plus and which will be minus that will actually from the right hand rule will be
determinant we will see that.

(Refer Slide Time: 17:18)

So, now if we make the DC circuit magnetic circuit like this, so this is F m, and this is
phi, this is R.

(Refer Slide Time: 17:44)

Actually, you how you have taken it that if you come to this your what you call this
diagram, so if you look into that I could grabs the conductor and thumb will be direction
of the flux right, so that means, flux line is suppose, this is my flux line, this is my flux.
So, as it is direction current actually leads the positive terminal for your (Refer Time:
17:50) your what you call for electric circuit say DC circuit right. For positive terminal
that your what you call any take any voltage source that current actually living the
positive terminal right that means, this is my plus, and then this is my minus.

So, this will be my F m, F m is equal to this N I right your what you call by your what
you call l that is the ampere turns per meter, this is your that means, you have to see the
whose direction you have to take the flux. So, if you grab the conductor, the flux is
coming out, and then you put the reluctance and close the circuit, and this is your what
you call the direction of the phi. So, phi actually is equal to your this is actually your F
m. So, phi actually F m upon your reluctance right, and F m is equal to N I upon l.

So, whenever flux I mean whenever you take in a direction, you see that flux is leaving
like this, then this will be your plus, this is minus for analogous. But, if you see in other
way, if you can see direction of the flux is like this. Then this will be plus, and this will
be minus, this is the direction of the flux got it, I think you have got it right so, this I will
make it.

So, once you understand this, you will find magnetic circuit is very simple right. So that
is why this circuit is drawn analogous to your say DC circuit say this is analogous to DC
circuit so, your phi is equal to F m upon reluctance. So, this way you can draw the
circuit, even you will see one or two problem that how to solve using this kind of circuit
right. So, it is a DC circuit analogous of magnetic system right.
(Refer Slide Time: 19:29)

Next is it is very simple thing next is fringing, it is fringing flux, which is air gap. And
this one core, this is another core in between some air gap is there. So, it is actually if
you look into that, the flux density is not uniform right, the particular the corner it will
taking some different path, not directly going from this to that right, it is taking some
different path.

(Refer Slide Time: 19:55)

So that means, at an air gap in a magnetic core right, the flux fringes is out into
neighbouring area your sorry neighbouring air paths as shown in figure 4. So, it is like
this, it is taking like this right so, this is basically your non-linear path right. So, there
being of reluctance you are comparable to that of the gap right. The result is non uniform
flux density in the air-gap that is decreasing outward right, so I mean decreasing your it
is outwards. So, in that case, your what you call your that is your enlargement of the
effective air gap area, and they decrease in the average gap your what you call average
gap flux density. The fringing effect also disturbs the core flux patterns to some depth
near the gap right.

(Refer Slide Time: 21:15)

And the effect of fringing increases with the air gap length I mean it is I mean it depends
on the length right. So, I mean if it is the suppose if this air gap length increases, the
fringing effect also will increase. So, this type of although we will not study in detail for
this course, we will not study this, just to give an idea that flux density is not uniform
right, an effective air because of this fringe effective your area is getting decrease right,
so, this is called fringing.
(Refer Slide Time: 21:25)

Now, next is the hysteresis loop, here we have to understand something what is
hysteresis loop, because magnetic circuit you will see this one, so this hysteresis loop.
So, magnetic you will see this a your what you call hysteresis loop suppose, this is in
figure 5, it has been drawn.

(Refer Slide Time: 21:31)

Now, suppose let a ferromagnetic material, which is completely demagnetized that is one
in which B is equal to H is equal to 0. Now, generally if you want to make that your what
will if you want to completely demagnetize the ferromagnetic material, what you have to
do is before going to the figure, you have to make B is equal to H is equal to 0 that
means, I has to be 0. If H is equal to 0, I has to be I mean your 0 all right.

So, what you can do is so this by making this B is equal to H is equal to 0. By reversing
the magnetizing current say I, a large number of times while at the same time gradually
reducing the current to zero that means, several times you have to see you have to
reverse the direction of the your magnetizing current, I mean large number of times, and
while at the same time gradually you have to reduce in the current to zero, then only you
will get B is equal to H is equal to 0. And be subjected to your what you call me
subjected to increasing values of magnetic field strength H right, and the corresponding
flux density B measured right.

So, in this case, what will happen that you have completely you are what we called
demagnetize that ferromagnetic material, so that means, it is starting from 0. Now, what
you can do is now what else I am do is that you are you are trying after making this, we
are trying to increase the H right. So, H will what you call if H means you are increase
the current right then this one we will follow the path o g b right.

So, in that case, what will happen that your H, now increasing values of magnetic field
strength H and the corresponding flux density B measured right, so that means, you are
increasing the H value now. Then we will starting from the origin that 0, because we
have totally you are what you call demagnetize the ferromagnetic material, now we are
increasing the H that means, we are increasing the current I say right.
(Refer Slide Time: 23:30)

So, in that case, what will happen after going on increasing the current right, the domains
actually begins to align and the resulting relationship between B and H is shown by the
curve o g b right. So, this is the curve o g b right, this is the middle line right, this middle
your curve right o g b right, suppose it has reach up to H.

(Refer Slide Time: 23:52)

Now, if you come to this, now at a particular value of H as shown in o y, most of the
domains will be you are aligned, and it becomes difficult to increase the flux density any
further. The material is said to be saturated, thus by the saturations flux density that
means, this you are increasing. Now, current is increasing, you will reach a value that
value that is o y, this is o, and this is your y, o y right.

So, after that even if you increase your you try to increase H max that is I, you will find
that flux density is not increasing, so this is the maximum value after saturation right.
And this is your maximum flux density, and you can say this has been you are what you
call it is saturated. So, that is why it is written here that you are what you call that after
that flux density will not increase, the material is said to be saturated; now it is saturated.
Thus is by is the saturation flux density B r.

So, this is my H max, and this is my B ma[x] this is your what you call this is your what
you call the B max, it is saturated right. Now, and one thing is there, this B r do not read
at it as saturated right, it is actually later we will see, it is actually B residual right, so
anyway we will come to that. So, this one, so just 1 minute, let me this thing right.

(Refer Slide Time: 25:23)

So, after that what you will do that your now if the values of H is now reduce it right you
try to reduce, it is found that flux density follows the curve b c right.
(Refer Slide Time: 25:39)

Now, when you try to when you try to reduce b c or what you call try to reduce the
current now, suppose why I am moving in this the curve reducing the current, so it is
moving in this direction. So, you will find when H is equal to 0, some flux will be there
in the material, this is actually call residual flux right. So, this o c actually o to c that is
your B r, it is actually residual flux right. So, although H is equal to 0, but some residual
magnetism will be there, this is called your what you call that residual flux right, and this
is residual flux density.

Now, further if you reverse the direction of the current right, now this side you are
moving, you will come to some point o d right that this point o d. So, some value of will
be there in the reverse direction that H value or you are reversing the direction of the
current, at that time you will find that this value right your what you call this flux density
right, this value you are slowly and slowly coming to 0. At that time, you will find B is
equal to 0 at this point right.

So, what is happening, first it is we are from here, we have increasing the it will reach to
the saturation, we call saturation flux density. Then this your B max, then you are
reversing the direction of the current. When you are reversing the that means, current is
decreasing now current is we are decreasing, not reversing, we are decreasing the
current. So, here at this point, it will come some residual flux will be there. Further we
are reversing the direction of the current, so it will come to this portion, where you will
get H value will be negative, and you will find your flux density B at this point is 0 right.

Same philosophy will happen, if you further go on your what you call that your increase
the current in the negative direction I mean this thing, the way it has reached here same
way it will the right, and it will get as get a point there like your saturated point you will
get there. And again further you are moving again in this direction, that again you are
you are what you call go with your this negative current this side, you are slowly and
slowly you are decreasing and coming to this point, so some residual magnetism again
will be here.

And finally, again after this point same as before, see it will come like this, and finally it
will come like this. So, this way your B H loop will be completed right. So, this is
actually your what you call that your B H curve of a with a ferromagnetic material right.

So, this is what we have this thing right, what you have understood. This did not much
we will just as is a magnetic circuit, so you have to explain, then you have to your what
you call little bit understand on that, so that means, everything is given here, that from
here it will start, but finally, it will move like this right. So, just step by step you go
through whatever I have written here and whatever I will said right, and that is the
philosophy behind this right.

(Refer Slide Time: 28:25)

So, here also everything is whatever I said everything is written here right. And one thing
is there magnetic field strength that is o d required to remove the residual magnetism is
called coercive force right. So, this time in this value, this is this value actually this is the
value, this is o d, this is that your this is your o d this, portion this portion right.

This is your whatever free step is required right to make the flux density is 0, this o d is
required that is your H is equal to o d, this will be negative value, this is actually called
coercive force right let me clear it. So, this is this is actually this one that magnetic field
strength o d required to remove the residual magnetism is called the coercive force right.

(Refer Slide Time: 29:27)

So, hysteresis and eddy-current losses, when magnetic materials undergoes cyclic
variation of flux density right, hysteresis and eddy-current power losses occur in them,
which are together known as core loss. Core loss means hysteresis loss plus eddy-current
loss right, we will see later in the single phase transformer after this topic. So, and appear
in the form of heat right.
(Refer Slide Time: 29:50)

So, core loss is important in determining temperature rise, rating and efficiency of
transformer, we will see that right, machines and other your AC-operated electro your
what you call AC-operated in electromagnetic devices.

(Refer Slide Time: 30:07)

So, now power loss on the account of hysteresis is actually it is an empirical formula
right key or K h into f into B max to the power n into your what you call V watts right.
So, your V is not the voltage, V is the value of the material right. So, K h is characteristic
constant of your core material, n is equal to Steinmetz exponent, range 1.5-2.0, typical
value is 1.6, so it is called Steinmetz constant right. And V is equal to volume of the
material in meter cube. This V is actually meter cube volume right. And f is equal to
supply frequency is hertz, this f in hertz right. This is the formula for the your what you
call for the your hysteresis loss right.

(Refer Slide Time: 30:53)

Similarly, for eddy current power loss is given by it is K e f square B max square into V
watt. For some material, it can be derived, but here this is your what to call we are not
giving that derivation and other thing, just you keep it in your mind that K e is the f
square B max square into V watt. So, K e is the characteristic constant of the core right.
So, up to this, your what you call that you are regarding little bit on magnetic circuit.
(Refer Slide Time: 31:25)

Now, next is magnetically coupled circuit. So, when two loops with or without contact
between them affect each other through the magnetic field generated by one of them,
they are said to be magnetically coupled right. Suppose, you have two loops with or
without contact between them, but affect each other through the magnetic field generated
by one of them, they are said to be magnetically coupled. Then two loops are there, if
one loop is carrying that your what you call that time varying current, and another loop
that some flux will be it will links some flux to the other coil right, and they are said to
be actually magnetically coupled.

(Refer Slide Time: 32:04)

Now, mutual inductance, now, when two inductors or coils right are in a close proximity
to each other, the magnetic flux will see later in the circuit diagram, the magnetic flux
caused by current in one coil links with other coil right, thereby inducing voltage in the
latter right. And this phenomenon is known as mutual inductance, that means, when two
inductors or two coils are there in a close proximity to each other, the magnetic flux
caused by current in one coil links with the other coil, thereby inducing voltage in the
latter; this phenomenon is known as mutual inductance right.

(Refer Slide Time: 32:42)

So, for example, suppose this is simple circuit, this current is current sources shown say
it is i t for easy understanding, and voltage across v’s beyond this. This is the phi right
that flux your what you call this is a time varying your what to call time varying current,
and this is the flux phi right, and it has number of your N turns.

So, in your what you call the magnetic flux produced by a single coil with N turns. So,
in this case also, if you your what you call that this is the your what you call this is the
direction of the your what you call direction of the flux right, and this is your N turn.
(Refer Slide Time: 33:20)

So, in that case what according to Faraday’s law right, according to your Faraday’s law,
so your the voltage v induced in the coil is proportional to the number of turns N and the
time rate of change of the magnetic of the flux that is we know, the v is equal to N into d
phi by d t right. So that means, v is equal to N into d phi by d t right but, question is that
this phi the flux phi it depends on the current I. If I decreases, phi will decrease; if I
increases, phi will increase right. So, this phi actually that phi actually is a for your what
you call it say is a function of I.

(Refer Slide Time: 34:09)

So, in that case what will happen that your that means, this equation this N into d phi by
d t, we can write as a chain rule right. So, d phi by d t, we can write d phi by d i into d i
by d t right. So, it is d phi by d i and d i by d t. So, this term N d phi by d i, this is
actually L, the inductance that means, v is equal to L into d i by d t, which you have seen
earlier also right, so L into d i by d t. And your L is equal to N into L is equal to N into d
phi by d i right; or in general, sometimes we can write L let me clear it.

(Refer Slide Time: 34:49)

That means, sometimes you write that your L is equal to right N phi by your i that means,
L i is equal to N phi right. This is also we require for numericals that L i is equal to N
phi.
(Refer Slide Time: 35:10)

So, that means, v is equal to L and L is equal to N into d phi, so inductance of the coil
commonly called as self-inductance. So, this is actually self-inductance for this kind of
coil. And in the here there is no other coil in the your what you call in the vicinity of this
right, no other coil.

(Refer Slide Time: 35:29)

So, next is your suppose this is the your what you call this is your two coils in the your
what you call mutual inductance M 2 1 of coil 2 with respect to coil 1. Now, here a
current source is i 1 t is there, this side it is not there, so it is volt[age] earlier voltage
here is measured v 2. And two coils are there so, if my total flux linkage, if total flux
linkages say my phi say this phi 1 for this coil 1, this is coil 1, this is my coil 1, and this
is my coil 2, this inductance is L 1, inductance is L 2.

So, total flux is phi 1 so, phi 1 1 is the flux linkages your what you call your this coil 1
right. In addition to that, phi 1 to another path right, it will all it will phi 1 to also link phi
coil 1 as well as your coil 2, this is actually mutual flux. So, this phi 1 is equal to your
phi 1 1 plus phi 1 2 what actually you are doing, phi 1 is the total flux right that links this
your what you call links the coil 1, but phi 1 2 actually links the coil 2.

This is your this is the other part linking the coil. So, phi 1 1 plus phi 12 it links the coil
1, and phi 1 2 only links the coil 2. So, phi 1 is equal to phi 1 1 plus phi 1 2 right and
whenever you say mutual and number of turns of the coil 1 is N 1, and number of the
turns in coil is N 2 right.

And mutual inductance M 2 1 means 2 1 means coil 2 with respect to coil 1. When you
say M 2 1 right that means, coil 1 is excited by some current i 1 right, and that is the
thing. And we are trying to find out the mutual inductance M 2 1 that is that is this one M
2 1. So mutual inductance M 2 1 of coil 2 with respect to coil 1 whenever, we write M 2
1 means, it is actually what you call mutual inductance of the coil 2 with respect to coil
1, this way you will read rights. This meaning of the suffix 2 1 is like that, the mutual
inductance M 2 1 whatever writing here with respect to your what you call of coil 2 with
respect to coil 1 right.

So, thank you very much, we will be back again.