that was the transformer; transformer as you understand we discussed it quite early was

one of the two port devices that we were supposed to discuss and it takes the electrical energy

from one side called the primary and delivers the electrical energy on another port called

the secondary and that is also the electrical energy and we modelled the transformers, studied

its equivalent circuit and got to know how it operates the principles of the transformers

and the various non-idealities which go up to make a practical transformer.

Today we shall begin our discussion of another two port device which is the DC machine. The

DC machine is also a two port device like a transformer. However, it has one fundamental

difference and that is the voltage or the effort on one side is related to the flow

or the through variable of the other port. Likewise, the flow on the electrical side

is related to the across variable or the potential variable or the effort variable on the port

two side.

And another point to be noted is that in a DC machine one port is electrical in nature

meaning the quantities that are going to be applied to one port are electrical quantities

and the quantities that are applied in the other port are the mechanical quantities because

it is in the mechanical domain mechanical rotational domain so torque and omega will

be the effort and flow variables that is the potential and the through variables, potential

of the kinetic variables and in the case of the electrical domain it will be voltage and

the current as usual.

Hence, we are going to discuss another two-port device called the DC machine. There is going

to be energy which gets energy which flows into the machine through one port and we will call that was port 1

and energy flows out through another port and that is called port 2.

One port is electrical in nature meaning port 1 let us say is electrical domain, port 2

is in the mechanical domain mechanical rotational domain. This means the potential variable

and the kinetic variable on the electrical side is the voltage e and the current i in amps,

volts and amps, the product is going to be the watts and the potential variable on this

side is torque in Newton meter and omega in radians per second. So this is volts, this

is amps, this is torque in Newton meter and this is radian per second.

So e on one side, torque on the other side are the effort variables or potential variables,

i on the electrical side, omega on the mechanical side are the flow or the kinetic variables;

the product of effort to flow on each side is always going to be watts that is the power

flow. Now, if the energy is flowing in this direction as shown which means from the electrical

to the mechanical side then the same device is called a motor. If the energy is flowing

from port 1 to port 2 that is the electrical energy is converted into mechanical energy

and used for driving something then the DC machine is called the motor. The same instrument

or the equipment can be used in a way wherein the energy flows from the mechanical to the

electrical meaning something rotates the mechanical shaft of the machine and that is going to

induce the electrical electrical side parameters and therefore the energy from the mechanical

side can flow that is energy from the mechanical side can flow to the electrical side and such

under such conditions the device is called a generator.

as the as the effort On the effort on the flows on the electrical side are DC values

the motor and the generator both in the case of motor and the generator the machine is

called a DC machine.

This is the machine that we are going to focus now and try to understand its principle of

operation, the basics, the concepts what makes it happen, what makes it rotate, what makes

it generate. First we consider the DC machine as a generator so we discuss the DC generator

and the same machine will be later discussed as a motor that is a transverse energy from

the electrical to the mechanical domain; most the principles that we study about the DC

machine as a generator will also be applicable to the same machine which will be in the motoring

mode. So these two functions of the DC machine we will try to focus in this class and the

coming classes. but before that we need to know one or two small principles like we had

the Faraday's law of electromagneticmagnetism in the case of the transformers which form

the bases, here also of course we will be using Faraday's laws of electromagnetism because

the energy is passing now through three domains. You see that the energy.......... let us say

if it is in the case of the generator, ultimately you will need to have the energy in the electrical

domain, the energy emanates from the mechanical domain as far as this equipment is concerned

and it passes through the electrical domain through an intermediary medium and that is

the magnetic domain. So energy passes from the mechanical domain into the magnetic domain

and from the magnetic domain to the electrical domain. Therefore, three domains are involved.

And in the case of motor the energy emanates from the electrical domain and goes into the

magnetic domain and then from the magnetic domain it goes into the mechanical domain

and causes the mechanical movement that would be the green arrows indicate the motoring

direction for the energy flow.

You see that electrical domain to the magnetic domain, magnetic domain to electrical domain

the principles here are very similar to that of a transformer. All the transformer principles

the Faraday's laws or the electromagnetics all those things are valid. There are few

rules that we need to know form mechanical to magnetic domain and that is one is of course

the Faraday's law and another is the lorentz force lorentz force. These two laws are applicable

in this transformation of this domain.

Now let us take let us take a piece of magnet and let us call it as north pole and somewhere

there on the left side you will see the south pole for that piece of magnet. Let us take

another piece of magnet and let me give it some space let us take another piece of magnet

and place it at some distance in line with the north pole but the south pole facing the

north pole and this has a north pole also of course because no magnetic element can

be with an isolated pole so it has its own north and south poles. But we are interested

in this that is until here

Now here if you see the magnetic lines of force this is at quite some distance it is

not quite near it because this is much much further than the distances that we are indicating

here so that this north pole does not have much interaction with the field lines which

are occurring here.

Now if we take the field lines there is a flux from the north to the south, it starts

going from north to the south, it is quite strong north to south. But in this direction

in this direction the field in an orthogonal direction is zero. is zero Now in an arbitrary

direction at an angle the field is going to exist but it is going to keep decreasing;

let us say we take the we we draw a circle let me draw a circle as shown like that so we have a circle here and if one travels

along the periphery of the circle at this point the field is strong on positive at a

point here the field this is at a point it is orthogonal to this north south axis and

the field there is zero in that direction, at any other direction let us say at when

when we are at a point here then the field field will field will be a bit reduced compared

would be a bit reduced compared to the position on this because it is not directly in line

and then if we take a point somewhere here the field will still further be reduced and

still further if we take a point somewhere here the field is positive still but still

further reduced and so on till it becomes zero at this point.

And then further if I take a consider a point on this circle circumference here in that

direction the field starts becoming negative because if we consider a point here with a

respect to that point the field is going to the point not away from the point and therefore

corresponding to this point the field direction is negative and it is shown by this negative

arrow. And as it as this point started in the circumference as this point starts travelling

like this this would be going in this fashion with increasing field gradually till it becomes

maximum and negative at this point.

So you see that there is a the field goes in this fashion in a sinusoidal fashion here.

Now with a maximum when it is at the north pole negative maximum when the position is

at when a point is at the south pole, zero at both the orthogonal points. So when the

point starts moving at this point we see again the mirror image of this and then when this

starts moving again towards the orthogonal axis you see the field decreasing field starts

decreasing until it is zero at the orthogonal axis and as the point further continues it

becomes positive that is this is the positive direction, this is zero, this is the positive

direction so field again changes direction and you see that radially there is different

amplitudes till it reaches the positive maximum when the point has again reached this point.

So you see that if we if there is a point which is situated on the circumference of

the circle starting from the north pole the point located near the north pole if the field

is moving away from the point it is we are putting putting it on the right side of this

axis and the field is on the uh on the south pole the field is entering the point and then

that is considered as negative. So a point which moves around the circumference like

that starts with the field which is positive and then keeps decreasing as the point is

moving along the axis there decreasing in this sense till it becomes zero and then it

goes negative in this portion of the segment keeps going so on till the field is negative

maximum till it is negative maximum as shown here nearest to the south pole point and as

the point traverses along the circumference towards again the orthogonal axis this traverses

the amplitude starts reducing further again till it becomes zero when it is at the when

the point is at the orthogonal point and then this goes again in this direction changes

direction and becomes positive max when the field again goes to this point. This is how

the field would look like when a point traverses on the circumference as shown here when we

place a north and south pole. So this would be the zero of the field phasor the field

vectors. So this would be the picture of the field vector field vector spatial picture.

Now let us have a north pole and a south pole and let me have a conductor with some as such there is no such current

in the conductor let us let us say that there is a field which is existing from north to

south in this direction. Now this is a conductor. Now this conductor if it is stationary the

field that is linking the conductor is also constant, there is no d phi by dt so as there

is no d phi by dt there is no induced voltage in the conductor and therefore there is no

current flow in the conductor.

If the field is changing, if the amplitude of the flux here, amplitude of the flux phi

here is changing then there is a d phi by dt and this results in an induced emf in the

conductor which will result in a current flow through the conductor in a specific direction.

Therefore, what is essential is a changing field for a current for a voltage to be induced

in the conductor and thereby a current to flow in the conductor.

There are two ways in which to produce a changing field as far as the conductor is consider

considered concerned. Let us say the conductor is fixed at a point. Now the flux phi here

can change with respect to time; flux is the function of time, then amplitude here changes

and therefore d phi by dt is finite not zero which will induce a voltage across the conductor

and thereby a current to flow. The other way is to have flux fixed, flux is fixed, the

flux amplitude is fixed; you have a permanent manner north and south pole and therefore

the distant the distance between them is also fixed and therefore the flux phi is fixed.

Under such condition the only way that you can have a change in flux is of the conductor

moves. If the conductor moves let us say to this position then the flux at that position

which is at a radial angle is going to be lesser and if it is moves still further the

flux is going to be lesser, if it is at the orthogonal axis the flux there is going to

be zero and let us say it continues to move. So the flux amplitude that is going to cut

the conductor is going to keep changing as the conductor moves along the circle. so that

is Therefore, the moving to the conductor is going to give you the d phi by dt which

is going to induce a voltage and thereby a current to flow. This is the other method

in which induced emf can also be produced in the conductor.

In the case of DC machines the flux here is kept constant by means of the fixed permanent

magnets let us say or something that produce constant flux and thereby the only way that

you can produce an induced emf in this conductor is by moving the conductor along the periphery

which means that this conductor is going to see varying amplitudes of the flux at different

amplitudes different points on the circumference of the circle and therefore a d phi by dt

and therefore an induced emf and therefore a current to flow through the conductor. This

comes by from the Faraday's laws of electromagnetism where you know that the induced emf which

is equal to Nd phi by dt.

An alternative expression in terms of the conductor length can also be derived which

is.......... let us just modify the................ and then we have this, these are the magnets

and then of course there is the conductor, so this conductor is positioned like this

and then you have all the flux lines linking the north and the south poles.

Now if this conductor is having a length L and then there is a cross section area A c