# induction motor - PowerPoint PPT Presentation

View by Category
Title:

## induction motor

Description:

### three phase induction motor,single phase motor, emf equation , torque eqn , theory of operation ,circuit diagram – PowerPoint PPT presentation

Number of Views:4863
Slides: 44
Provided by: srkark6676
Tags:
Transcript and Presenter's Notes

Title: induction motor

1
Induction Motor
• By
• Ashvani Shukla
• Manager(CI)
• BGR ENERGY

2
• INTRODUCTION
• One of the most common electrical motor used in
most applications which is known as induction
motor. This motor is also called as asynchronous
motor because it runs at a speed less than
synchronous speed. In this, we need to define
what is synchronous speed. Synchronous speed is
the speed of rotation of the magnetic field in a
rotary machine and it depends upon the frequency
and number poles of the machine. An induction
motor always runs at a speed less than
synchronous speed because the rotating magnetic
field which is produced in the stator will
generate flux in the rotor which will make the
rotor to rotate, but due to the lagging of flux
current in the rotor with flux current in the
stator, the rotor will never reach to its
rotating magnetic field speed i.e. the
synchronous speed. There are basically two types
of induction motor that depend upon the input
supply - single phase induction motor and three
phase induction motor. Single phase induction
motor is not a self starting motor which we will
discuss later and three phase induction motor is
a self-starting motor. Now in general we need to
give two supply i.e. double excitation to make a
machine to rotate. For example if we consider a
DC motor, we will give one supply to the stator
and another to the rotor through brush
arrangement.

3
• Working Principle of Induction Motor
• But in induction motor we give only one supply,
so it is really interesting to know that how it
works. It is very simple, from the name itself we
can understand that there is induction process
occurred. Actually when we are giving the supply
to the stator winding, flux will generate in the
coil due to flow of current in the coil. Now the
rotor winding is arranged in such a way that it
becomes short circuited in the rotor itself. The
flux from the stator will cut the coil in the
rotor and since the rotor coils are short
circuited, according to Faraday's law of
electromagnetic induction, current will start
flowing in the coil of the rotor. When the
current will flow, another flux will get
generated in the rotor. Now there will be two
flux, one is stator flux and another is rotor
flux and the rotor flux will be lagging to the
stator flux. Due to this, the rotor will feel a
torque which will make the rotor to rotate in the
direction of rotating magnetic flux. So the speed
of the rotor will be depending upon the ac supply
and the speed can be controlled by varying the
input supply. This is the working principle of an
induction motor of either type.

4
• Types Induction Motor
• SINGLE PHASE INDUCTION MOTOR
• Split phase induction motor
• Capacitor start induction motor
• Capacitor start capacitor run induction motor
• THREE PHASE INDUCTION MOTOR
• Squirrel cage induction motor
• Slip ring induction motor

5
• Construction of Three Phase Induction Motor
• The three phase induction motor is the most
widely used electrical motor. Almost 80 of the
mechanical power used by industries is provided
by three phase induction motors because of its
simple and rugged construction, low cost, good
operating characteristics, absence of commutator
and good speed regulation. In three phase
induction motor the power is transferred from
stator to rotor winding through induction. The
Induction motor is also called asynchronous motor
as it runs at a speed other than the synchronous
speed.
• Like any other electrical motor induction motor
also have two main parts namely rotor and stator
• Stator As its name indicates stator is a
stationary part of induction motor. A stator
winding is placed in the stator of induction
motor and the three phase supply is given to it.
• Rotor The rotor is a rotating part of induction
motor. The rotor is connected to the mechanical
• The rotor of the three phase induction motor are
further classified as Squirrel cage rotor,
• Slip ring rotor or wound rotor or phase wound
rotor.
• Depending upon the type of rotor construction
used the three phase induction motor are
classified as Squirrel cage induction motor,
• Slip ring induction motor or wound induction
motor or phase wound induction motor.

6
• The construction of stator for both the kinds of
three phase induction motor remains the same and
is discussed in brief in next paragraph. The
other parts, which are required to complete the
induction motor, are Shaft for transmitting the
steel.
• Bearings for supporting the rotating shaft.
• One of the problems with electrical motor is the
production of heat during its rotation. In order
to overcome this problem we need fan for cooling.
• For receiving external electrical connection
Terminal box is needed.
• There is a small distance between rotor and
stator which usually varies from 0.4 mm to 4 mm.
Such a distance is called air gap.
• Stator of Three Phase Induction Motor
• The stator of the three phase induction motor
consists of three main parts Stator frame,
• Stator core,
• Stator winding or field winding.
• Stator Frame

7
It is the outer most part of the three phase
induction motor. Its main function is to support
the stator core and the field winding. It acts as
a covering and it provide protection and
mechanical strength to all the inner parts of the
induction motor. The frame is either made up of
die cast or fabricated steel. The frame of three
phase induction motor should be very strong and
rigid as the air gap length of three phase
induction motor is very small, otherwise rotor
will not remain concentric with stator, which
will give rise to unbalanced magnetic pull.
8
Stator Core The main function of the stator core
is to carry the alternating flux. In order to
reduce the eddy current loss, the stator core is
laminated. These laminated types of structure are
thick. All the stamping are stamped together to
form stator core, which is then housed in stator
frame. The stamping is generally made up of
silicon steel, which helps to reduce the
hysteresis loss occurring in motor.
9
• Stator Winding or Field Winding
• The slots on the periphery of stator core of the
three phase induction motor carries three phase
windings. This three phase winding is supplied by
three phase ac supply. The three phases of the
winding are connected either in star or delta
depending upon which type of starting method is
used. The squirrel cage motor is mostly started
by star delta stater and hence the stator of
squirrel cage motor is delta connected. The slip
ring three phase induction motor are started by
inserting resistances so, the stator winding of
slip ring induction motor can be connected either
in star or delta. The winding wound on the stator
of three phase induction motor is also called
field winding and when this winding is excited by
three phase ac supply it produces a rotating
magnetic field.

10
• Types of Three Phase Induction Motor
• Squirrel cage three phase induction motor The
rotor of the squirrel cage three phase induction
motor is cylindrical in shape and have slots on
its periphery. The slots are not made parallel to
each other but are bit skewed (skewing is not
shown in the figure of squirrel cadge rotor
beside) as the skewing prevents magnetic locking
of stator and rotor teeth and makes the working
of motor more smooth and quieter. The squirrel
cage rotor consists of aluminum, brass or copper
bars (copper bras rotor is shown in the figure
beside). These aluminum, brass or copper bars are
called rotor conductors and are placed in the
slots on the periphery of the rotor. The rotor
conductors are permanently shorted by the copper
or aluminum rings called the end rings. In order
to provide mechanical strength these rotor
conductor are braced to the end ring and hence
form a complete closed circuit resembling like a
cage and hence got its name as "squirrel cage
induction motor". The squirrel cage rotor winding
is made symmetrical. As the bars are permanently
shorted by end rings, the rotor resistance is
very small and it is not possible to add external
resistance as the bars are permanently shorted.
The absence of slip ring and brushes make the
construction of Squirrel cage three phase
induction motor very simple and robust and hence
widely used three phase induction motor. These
of pole pairs. The below diagram shows squirrel
cage induction rotor having aluminum bars short
circuit by aluminum end rings.

11
• Advantages of squirrel cage induction rotor-
• Its construction is very simple and rugged.
• As there are no brushes and slip ring, these
motors requires less maintenance.
• Applications Squirrel cage induction motor is
used in lathes, drilling machine, fan, blower
printing machines etc.

12
• Slip ring or wound three phase induction motor
In this type of three phase induction motor the
rotor is wound for the same number of poles as
that of stator but it has less number of slots
and has less turns per phase of a heavier
conductor. The rotor also carries star or delta
winding similar to that of stator winding. The
rotor consists of numbers of slots and rotor
winding are placed inside these slots. The three
end terminals are connected together to form star
connection. As its name indicates three phase
slip ring induction motor consists of slip rings
connected on same shaft as that of rotor. The
three ends of three phase windings are
permanently connected to these slip rings. The
external resistance can be easily connected
through the brushes and slip rings and hence used
for speed control and improving the starting
torque of three phase induction motor. The
brushes are used to carry current to and from the
rotor winding. These brushes are further
connected to three phase star connected
resistances. At starting, the resistance are
connected in rotor circuit and is gradually cut
out as the rotor pick up its speed. When the
motor is running the slip ring are shorted by
connecting a metal collar, which connect all slip
ring together and the brushes are also removed.
This reduces wear and tear of the brushes. Due to
presence of slip rings and brushes the rotor
construction becomes somewhat complicated
therefore it is less used as compare to squirrel
cage induction motor.

13
• Advantages of slip ring induction motor - It has
high starting torque and low starting current.
control speed.
• Application
• Slip ring induction motor are used where high
starting torque is required i.e in hoists,
cranes, elevator etc.

14
Difference between Slip Ring and Squirrel Cage
Induction Motor
Slip ring or phase wound Induction motor Squirrel cage induction motor
Construction is complicated due to presence of slip ring and brushes Construction is very simple
The rotor winding is similar to the stator winding The rotor consists of rotor bars which are permanently shorted with the help of end rings
We can easily add rotor resistance by using slip ring and brushes Since the rotor bars are permanently shorted, its not possible to add external resistance
Due to presence of external resistance high starting torque can be obtained Staring torque is low and cannot be improved
Slip ring and brushes are present Slip ring and brushes are absent
Frequent maintenance is required due to presence of brushes Less maintenance is required
The construction is complicated and the presence of brushes and slip ring makes the motor more costly The construction is simple and robust and it is cheap as compared to slip ring induction motor
This motor is rarely used only 10 industry uses slip ring induction motor Due to its simple construction and low cost. The squirrel cage induction motor is widely used
Rotor copper losses are high and hence less efficiency Less rotor copper losses and hence high efficiency
Speed control by rotor resistance method is possible Speed control by rotor resistance method is not possible
Slip ring induction motor are used where high starting torque is required i.e in hoists, cranes, elevator etc Squirrel cage induction motor is used in lathes, drilling machine, fan, blower printing machines etc
15
• We had mentioned above that single phase
induction motor is not a self starting and three
phase induction motor is self starting. So what
is self starting? When the machine starts running
automatically without any external force to the
machine, then it is called as self starting. For
example we see that when we press the key the fan
starts to rotate automatically, so it is self
starting. Point to be note that fan used in home
appliances is single phase induction motor but it
is self starting. How? We will discuss it how.
Why is Three Phase Induction Motor Self Starting?
• In three phase system, there are three single
phase line with 120 phase difference. So the
rotating magnetic field is having the same phase
difference which will make the rotor to move. If
we consider three phases a, b and c, when phase a
is magnetized, the rotor will move towards the
phase a winding, in the next moment phase b will
get magnetized and it will attract the rotor and
than phase c. So the rotor will continue to
rotate.

16
Working Principle of Three Phase Induction Motor
• Why Single Phase Induction Motor is not Self
Starting?
• But what about single phase. It will be having
only one phase still it makes the rotor to
rotate, so it is quite interesting. Before that
we need to know why single phase induction motor
is not a self starting motor and how the problem
is overcome. We know that the ac supply is a
sinusoidal wave and it produces pulsating
magnetic field in uniformly distributed stator
winding. Since pulsating magnetic field can be
assumed as two oppositely rotating magnetic
fields, there will be no resultant torque
produced at the starting and due to this the
motor does not run. After giving the supply, if
the rotor is made to rotate in either direction
by external force, then the motor will start to
run. This problem has been solved by making the
stator winding into two winding, one is main
winding and another is auxiliary winding and a
capacitor is fixed in series with the auxiliary
winding. This will make a phase difference when
current will flow through the both coils. When
there will be phase difference, the rotor will
generate a starting torque and it will start to
rotate. Practically we can see that the fan does
not rotate when the capacitor is disconnected
from the motor but if we rotate with hand it will
start to rotate. So this is the reason of using
capacitor in the single phase induction motor.
There are several advantages of induction motor
which makes this motor to have wider application.
It is having good efficiency up to 97. But the
speed of the motor varies with the load given to
the motor which is an disadvantage of this motor.
The direction of rotation of induction motor can
easily be changed by changing the sequence of
three phase supply, i.e. if RYB is in forward
direction, the RBY will make the motor to rotate
in reverse direction. This is in the case of
three phase motor but in single phase motor, the
direction can be reversed by reversing the
capacitor terminals in the winding.

17
• Working Principle of Three Phase Induction Motor
• An electrical motor is such an electromechanical
device which converts electrical energy into a
mechanical energy. In case of three phase AC
operation, most widely used motor is Three phase
induction motor as this type of motor does not
require any starting device or we can say they
are self starting induction motor.
• For better understanding the principle of three
phase induction motor, the basic constructional
feature of this motor must be known to us. This
Motor consists of two major parts Stator Stator
of three phase induction motor is made up of
numbers of slots to construct a 3 phase winding
circuit which is connected to 3 phase AC source.
The three phase winding are arranged in such a
manner in the slots that they produce a rotating
magnetic field after AC is given to them. Rotor
Rotor of three phase induction motor consists of
cylindrical laminated core with parallel slots
that can carry conductors. Conductors are heavy
copper or aluminum bars which fits in each slots
they are short circuited by the end rings. The
slots are not exactly made parallel to the axis
of the shaft but are slotted a little skewed
because this arrangement reduces magnetic humming
noise can avoid stalling of motor.

18
• Working of Three Phase Induction Motor
• Production of Rotating Magnetic Field
• The stator of the motor consists of overlapping
winding offset by an electrical angle of 120.
When the primary winding or the stator is
connected to a 3 phase AC source, it establishes
a rotating magnetic field which rotates at the
synchronous speed.
• Secrets behind the rotation
• According to Faradays law an emf induced in any
circuit is due to the rate of change of magnetic
flux linkage through the circuit. As the rotor
winding in an induction motor are either closed
through an external resistance or directly
shorted by end ring, and cut the stator rotating
magnetic field, an emf is induced in the rotor
copper bar and due to this emf a current flows
through the rotor conductor.
• Here the relative velocity between the rotating
flux and static rotor conductor is the cause of
current generation hence as per Lenz's law the
rotor will rotate in the same direction to reduce
the cause i.e. the relative velocity.

19
• Thus from the working principle of three phase
induction motor it may observed that the rotor
speed should not reach the synchronous speed
produced by the stator. If the speeds equals,
there would be no such relative velocity, so no
emf induction in the rotor, no current would be
flowing, and therefore no torque would be
generated. Consequently the rotor can not reach
at the synchronous speed. The difference between
the stator (synchronous speed) and rotor speeds
is called the slip. The rotation of the magnetic
field in an induction motor has the advantage
that no electrical connections need to be made to
the rotor. Thus the three phase induction motor
is Self-starting. Less armature reaction and
brush sparking because of the absence of
commutators and brushes that may cause sparks.
Robust in construction. Economical. Easier to
maintain.

20
• Classification of Squirrel Cage Induction Motor.
• NEMA in United States and IEC in Europe have
classified the design of the squirrel cage
induction motors based on their speed-torque
characteristics into some classes. These classes
are Class A, Class B, Class C, Class D, Class E
and Class F.
• In Class A Design
• A normal starting torque.
• A normal starting current.
• Low slip.
• In this Class, pullout torque is always of 200 to
300 percent of the full-load torque and it occurs
at a low slip (it is less than 20 percent).
• For this Class, the starting torque is equal to
rated torque for larger motors and is about 200
percent or more of the rated torque for the
smaller motors.

21
• In Class B Design
• Normal starting torque,
• Lower starting current,
• Low slip.
• Induction Motor of this Class produces about the
same starting torque as the class A induction
motor and this starting torque is with about 25
percent less current.
• Pullout torque is always greater than or equal to
200 percent of the rated load torque. But it is
less than that of the class A design because it
has increased rotor reactance.
• Again Rotor slip is still relatively low (less
than 5 percent) at full load.
• Applications of Class B design are similar to
those for design A. But design B is preferred
more because of its lower starting-current
requirements.

22
• In Class C Design
• High starting torque.
• Low starting currents.
• Low slip at the full load (less than 5 ).
• Up to 250 percent of the full-load torque, the
starting torque is in this class of design.
• The pullout torque is lower than that for class A
induction motors.
• In this design the motors are built from
double-cage rotors. They are more expensive than
motors of Class A and B classes.
• Class C Designs are used for high-starting-torque

23
• In Class D Design
• In this Design of Class motors has very high
starting torque (275 percent or more of the rated
torque).
• A low starting current.
• A high slip at full load.
• Again in this class of design the high rotor
resistance shifts the peak torque to a very low
speed.
• It is even possible at zero speed (100 percent
slip) for the highest torque to occur in this
class of design.
• Full-load slip (It is typically 7 to 11 percent,
but may go as high as 17 percent or more) in this
class of design is quite high because of the high
rotor resistance always.

24
• In class E Design
• Very Low Starting Torque.
• Normal Starting Current.
• Low Slip.
• Compensator or resistance starter are used to
control starting current.
• In Class F Design
• Low Starting Torque, 1.25 times of full load
torque when full voltage is applied.
• Low Starting Current.
• Normal Slip.

25
• Circle Diagram of Induction Motor

The CIRCLE DIAGRAM means that it is figure or
curve which is drawn has a circular shape. As we
know, the diagrammatic representation is easier
compared to theoretical and mathematical
descriptions. Actually, we do not have that much
time or patience to go through the writings so we
prefer diagrammatic representation. Also, it is
very easy to remember the things which are shown
in picture. As we know, A PICTURE IS WORTH 1000
WORDS. This also holds good here and we are to
draw circle diagram in order to compute various
parameters rather than doing it mathematically.
Importance of Circle Diagram The diagram
provides information which is not provided by an
ordinary phasor diagram. A phasor diagram gives
relation between current and voltage only at a
single circuit condition. If the condition
changes, we need to draw the phasor diagram
again. But a circle diagram may be referred to as
a phasor diagram drawn in one plane for more than
one circuit conditions. On the context of
induction motor, which is our main interest, we
can get information about its power output, power
factor, torque, slip, speed, copper loss,
efficiency etc. in a graphical or in a
diagrammatic representation.
26
• Test Performed to Compute Data Required for
Drawing Circle Diagram
• We have to perform no load and blocked rotor test
in an induction motor. In no load test, the
induction motor is run at no load and by two watt
meter method, its total power consumed is
calculated which is composed of no load losses
only. Slip is assumed to be zero. From here no
load current and the angle between voltage and
current is required for drawing circle diagram
and calculated. The angle will be large as in the
no load condition induction motor has high
inductive reactance. In block rotor test, rotor
is blocked which is analogous to short circuit
secondary of a transformer. From this test, we
need to calculate short circuit current and the
lag angle between voltage and current for drawing
circle diagram. Also, we need rotor and stator
copper loss.

27
• Procedure to Draw the Circle Diagram
• We have to assume a suitable before drawing it.
This assumption is done according to our
convenience.

28
calculated from no load test is plotted. This is
shown by the line OA, where O0 is the no load
power factor angle.
• The short circuit current and the angle obtained
from block rotor test is plotted. This is shown
by the line OC and the angle is shown by OB.
• The right bisector of the line AC is drawn which
bisects the line and it is extended to cut in the
line AE which gives us the Centre.
• The stator current is calculated from the
equivalent circuit of the induction motor which
we get from the two tests. That current is
plotted in the circle diagram according to the
scale with touching origin and a point in the
circle diagram which is shown by B.
• The line AC is called the power line. By using
the scale for power conversion that we have taken
in the circle diagram, we can get the output
power if we move vertically above the line AC to
the periphery of the circle. The output power is
given by the line MB.
• The total copper loss is given by the line GM.
• For drawing the torque line, the total copper
loss should be separated to both the rotor copper
loss and stator copper loss. The line DE gives
the stator copper loss and the line CD gives the
rotor copper loss. In this way, the point E is
selected.
• The line AD is known as torque line which gives
the torque developed by induction motor.

29
• Maximum Quantities from Circle Diagram

30
• Maximum Output Power
• When the tangent to the circle is parallel to the
line then output power will be maximum. That
point M is obtained by drawing a perpendicular
line from the center to the output line and
extending it to cut at M. Maximum Torque
• When the tangent to the circle is parallel to the
torque line, it gives maximum torque. This is
obtained by drawing a line from the center in
perpendicular to the torque line and extending it
to cut at the circle. That point is marked as N.
Maximum Input Power
• It occurs when tangent to the circle is
perpendicular to the horizontal line. The point
is the highest point in the circle diagram and
drawn to the center and extends into S. That
point is marked as R. Conclusion of Circle
Diagram
• This method is based on some approximations that
we have used in order to draw the circle diagram
and also, there is some rounding off of the
values as well. So there is some error in this
method but it can give good approximate results.
Also, this method is very much time consuming so
it is drawn at times where the drawing of circle
diagram is absolutely necessary. Otherwise, we go
for mathematical formulas or equivalent circuit
model in order to find out various parameters.

31
• Torque Slip Characteristics of Induction Motor

The torque slip curve for an induction motor
gives us the information about the variation of
torque with the slip. The slip is defined as the
ratio of difference of synchronous speed to the
speed at any mechanical load to the synchronous
speed of the machine. The variation of slip can
be obtained with the variation on speed that is
when speed varies the slip will also vary and the
torque corresponding to that speed will also
vary. The curve can be described in three modes
of operation-
32
• Motoring Mode In this mode of operation, supply
is given to the stator sides and the motor always
rotates below the synchronous speed. The
induction motor torque varies from zero to full
load torque as the slip varies. The slip varies
from zero to one. It is zero at no load and one
at standstill. From the curve it is seen that the
torque is directly proportional to the slip. That
is, more is the slip, more will be the torque
produced and vice-versa. The linear relationship
simplifies the calculation of motor parameter to
great extent. Generating Mode In this mode of
operation induction motor runs above the
synchronous speed and it should be driven by a
prime mover. The stator winding is connected to a
three phase supply in which it supplies
electrical energy. Actually, in this case, the
torque and slip both are negative so the motor
electrical energy. Induction motor is not much
used as generator because it requires reactive
power for its operation. That is, reactive power
should be supplied from outside and if it runs
below the synchronous speed by any means, it
consumes electrical energy rather than giving it
at the output. So, as far as possible, induction
generators are generally avoided.

33
• Braking Mode In the breaking mode, the two leads
or the polarity of the supply voltage is changed
so that the motor starts to rotate in the reverse
direction and as a result the motor stops. This
method of breaking is known as plugging. This
method is used when it is required to stop the
motor within a very short period of time. The
kinetic energy stored in the revolving load is
dissipated as heat. Also, motor is still
receiving power from the stator which is also
dissipated as heat. So as a result of which motor
develops enormous heat energy. For this stator is
disconnected from the supply before motor enters
the breaking mode.
• If load which the motor drives accelerates the
motor in the same direction as the motor is
rotating, the speed of the motor may increase
more than synchronous speed. In this case, it
acts as an induction generator which supplies
electrical energy to the mains which tends to
slow down the motor to its synchronous speed, in
this case the motor stops. This type of breaking
principle is called dynamic or regenerative
breaking.

34
• Torque Slip Characteristics of Single Phase
Induction Motor

35
• From the figure, we see that at a slip of unity,
both forward and backward field develops equal
torque but the direction of which are opposite to
each other so the net torque produced is zero
hence the motor fails to start. From here we can
say that these motors are not self starting
unlike the case of three phase induction motor.
There must be some means to provide the starting
torque. If by some means, we can increase the
forward speed of the machine due to which the
forward slip decreases the forward torque will
increase and the reverse torque will decrease as
a result of which motor will start.
• From here we can conclude that for starting of
single phase induction motor, there should be a
production of difference of torque between the
forward and backward field. If the forward field
torque is larger than the backward field than the
motor rotates in forward or anti clockwise
direction. If the torque due to backward field is
larger compared to other, then the motor rotates
in backward or clockwise direction.

36
• Torque Equation of Three Phase Induction Motor

The torque produced by three phase induction
motor depends upon the following three
factors Firstly the magnitude of rotor current,
secondly the flux which interact with the rotor
of three phase induction motor and is responsible
for producing emf in the rotor part of induction
motor, lastly the power factor of rotor of the
three phase induction motor. Combining all these
factors together we get the equation of torque as-
Where, T is the torque produced by induction
motor, f is flux responsible of producing induced
emf, I2 is rotor current, cos?2 is the power
factor of rotor circuit. The flux f produced by
the stator is proportional to stator emf E1. i.e
f ? E1 We know that transformation ratio K is
defined as the ratio of secondary voltage (rotor
voltage) to that of primary voltage (stator
voltage).
37
Rotor current I2 is defined as the ratio of rotor
induced emf under running condition , sE2 to
total impedance, Z2 of rotor side,
and total impedance Z2 on rotor side is given by
38
• Putting this value in above equation we get,

We know that power factor is defined as ratio of
resistance to that of impedance. The power factor
of the rotor circuit is
Putting the value of flux f, rotor current I2,
power factor cos?2 in the equation of torque we
get,
39
• Combining similar term we get,

Removing proportionality constant we get,
Where ns is synchronous speed in r. p. s, ns Ns
/ 60. So, finally the equation of torque becomes,
40
• Derivation of K in torque equation. In case of
three phase induction motor, there occur copper
losses in rotor. These rotor copper losses are
expressed as Pc 3I22R2 We know that rotor
current,

Substitute this value of I2 in the equation of
rotor copper losses, Pc. So, we get
The ratio of P2 Pc Pm 1 s (1 - s) Where
P2 is the rotor input, Pc is the rotor copper
losses, Pm is the mechanical power developed.
41
Substitute the value of Pc in above equation we
get,
On simplifying we get,
42
• The mechanical power developed Pm T?,

Substituting the value of Pm
We know that the rotor speed N Ns(1 - s)
Substituting this value of rotor speed in above
equation we get, By calculating and substituting
the all values we get the equation.
43
• Equation of Starting Torque of Three Phase
Induction Motor
• Starting torque is the torque produced by
induction motor when it is started. We know that
at start the rotor speed, N is zero.

So, the equation of starting torque is easily
obtained by simply putting the value of s 1 in
the equation of torque of the three phase
induction motor,