Lecture 21 EE743

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Lecture 21 - EE743
3-Phase Induction Machines Steady-state model
Professor Ali Keyhani
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Three-phase induction machines

• 3-phase winding is located on the stator (stator
  winding)
• 3-phase winding is located on the rotor (rotor
  winding)
• Torque is produced from interactions of two
  rotating flux waves

cr - axis
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Cutaway Views of Squirrel-cage Induction Motor
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Rotating stator flux wave

• The flux variation of coil a-a when it is
  energized can be written as

• B(t) time changing flux produced by va(t)
• For sinusoidal va(t) max B
• fs supply frequency
• x variable distance traveled from the ref
• P Nbre of poles
• D Diameter, (Px/D is the angle in radians)
• winding distribution around the stator
  peripher

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Rotating stator flux wave

• For coils bs-bs and cs-cs

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Rotating stator flux wave
total flux wave in the air gap of the machine
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Frequency of induced current in the rotor windings

• Rotor at stand still (?m0)Btot,s(x,t) is
  rottating in the air gap at the electrical speed
  of ?s2?fs fs stator frequencyThe equivalent
  mech. speed of the rotating B(x,t) isThe eq.
  mechanical speed is called synch. speed?sync
  is mech. speed of rotating Bs(x,t)Bs(x,t)
  rotates at ?sync with respect to statorThis
  magnetic field will induce voltage in short
  circuited rotor windings (rotor stand still)

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Frequency of induced current in the rotor windings

• Br(x,t) rotates at ?sync with respect to stator.
• Suppose rotor is rotating at ?mThe speed of
  Br(x,t) (flux wave) produced by rotor with
  respect to rotor is?r(mech) rotor traveling
  (rotating) flux wave speed.

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Frequency of induced current in the rotor windings

• The elctrical eq. speed of rotor rotating flux
  wave isfr is the frequency of induced
  voltage (current) in the rotor winding

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Torque Production

• We have two rotating flux waves in the air gap of
  the machine
• Bs(x,t) of the stator rotates at ?sync mech
  speed with respect to stator
• Br(x,t) of the rotor rotates at ?r(mech) speed
  with respect to rotor
• Br(x,t) of the rotor rotates at ?sync mech speed
  with respect to stator
• Interaction of two magnetic fields will generate
  torque
• Motor action
• ?sync and ?m are rotating in the same direction
• ?sync gt?m

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Calculation of slip

• Let us multiply and divide by ?s (stator electric
  speedbut

N
?sync
S
?sync
N
Tem
?m
Text
S
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Calculation of slip

• Lets define the slip as or frsfNote
  ?sync in rad/sec ?m in rad/secbut in rpm