induction motor

1
Induction Motor

• By
• Ashvani Shukla
• Manager(CI)
• BGR ENERGY

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• 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.

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• 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.

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

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• 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
  load through the shaft.
• 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.

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• 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
  torque to the load. This shaft is made up of
  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

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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.
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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
made up of stamping which is about 0.4 to 0.5 mm
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.
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• 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.

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• 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
  motors have the advantage of adapting any number
  of pole pairs. The below diagram shows squirrel
  cage induction rotor having aluminum bars short
  circuit by aluminum end rings.

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• 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.

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• 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.

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

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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
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• 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.

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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.

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• 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.

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• 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.

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• 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.

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• 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.

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• 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.

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• 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
  loads (loaded pumps, compressors, and conveyors).

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• 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.

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• 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.

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• 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.
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• 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.

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

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• The no load current and the no load angle
  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.

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• Maximum Quantities from Circle Diagram

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• 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.

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• 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-
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• 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
  receives mechanical energy and delivers
  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.

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• 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.

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• 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.

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• 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,