ENERGY CONVERSION ONE (Course 25741)

1
ENERGY CONVERSION ONE (Course 25741)

• CHAPTER SEVEN
• ..INDUCTION MOTORS

2
ROTOR CIRCUIT MODEL

• The rotor current
• IR ER/ RRjXR
  (1)
• IRER/ RRjs XR0 or IRER0 / RR /s j XR0
  (2)
• Note from last equation, can treat rotor effects
  due to varying rotor speed as caused by a varying
  impedance supplied from a constant voltage
  source ER0
• Equivalent rotor impedance from this point of
  view
• ZR, eq RR / s jXR0
  (3)
• rotor equivalent circuit using this
  convention shown next
  ?

3
ROTOR CIRCUIT MODEL

• Rotor voltage is ER0 constant rotor impedance
  ZR,eq contains effects of varying slip

4
ROTOR CIRCUIT MODEL

• Plot of current flow in rotor from equations
  (1) (2) shown below

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FINAL EQUIVALENT CIRCUIT

• Note at very low slips resistive term RR/sgtgtXR0
• ? rotor resistance predominates
  rotor current
• varies linearly with slip
• at high slips, XR0 much larger than RR/s rotor
  current approaches steady-state value as slip
  becomes very large
• To develop a single, per-phase, equivalent
  circuit for induction motor ? should refer to
  rotor part of model over stator side
• Rotor circuit model, referred to stator side
  (shown in last equivalent circuit of rotor) in
  which effect of speed variation is concentrated
  in impedance term

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FINAL EQUIVALENT CIRCUIT

• Reminding that in an ordinary transformer,
  voltages, currents and impedances on secondary
  side of device can be referred to primary side by
  means of turns ratio of transformer
• VpVsa Vs
• IpIsIs / a
• Zs a2 Zs
• Same sort of transformation employed for
  induction motors rotor circuit
• If the effective turns ratio of induction motor
  aeff
• The transformed voltages are
• 
  E1ERaeff ER0
• and rotor current is I2IR
  / aeff
• And rotor impedance become Z2 aeff2 (RR/ s
  j XR0)

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FINAL EQUIVALENT CIRCUIT

• If the following definitions employed
• R2 aeff2 RR X2 aeff2 XR0
• the equivalent circuit of induction motor is
  shown as below (per-phase equivalent circuit)

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FINAL EQUIVALENT CIRCUIT

• Rotor resistance RR locked-rotor reactance XR0
  are difficult or impossible to determine directly
  on cage rotors effective turns ratio aeff also
  difficult to determine for cage rotors
• fortunately, can make measurements that directly
  provide referred resistance and reactance R2 X2
  , (though RR, XR0 and aeff not known separately)
  
• measurement of these parameters covered later

9
POWER TORQUE IN INDUCTION MOTORS-LOSSES

• since induction motor is a singly excited
  machine,? its power torque relationships is
  different from sync. machines
• Losses power-flow Diagram
• the input is electric power and the output
  mechanical power (while rotor windings are short
  circuited)
• As shown in power flow Figure (next), Pin is in
  form of 3 phase electric voltages currents
• 1st losses is stator winding losses I2 RPSCL
• 2nd Hysteresis Eddy currents loss in stator
  Pcore
• Power remained at this point transferred to rotor
  through air gap is called air-gap power PAG

10
POWER TORQUE IN INDUCTION MOTORS-LOSSES

• Part of power transferred to rotor lost as
• I2 RPRCL rest converted from electrical
  to mechanical form Pconv, friction windage
  losses PFW stray losses Pmisc subtracted ?
  Pout

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POWER TORQUE IN INDUCTION MOTORS-LOSSES

• Note in practice core loss is partially related
  to stator and partially to rotor, however since
  induction motor operates at a speed near
  synchronous speed, relative motion of magnetic
  field over rotor surface is quite low (frequency
  of induced voltage s fe) rotor core losses
  are very tiny
• These losses in induction motor equivalent
  circuit represented by a resistor RC (or GC) ,
• If core losses are given as a number (X Watts)
  often lumped with mechanical losses subtracted
  at point on diagram where mechanical losses are
  located

12
POWER TORQUE IN INDUCTION MOTORS-LOSSES

• The higher the speed of an induction motor, the
  higher its friction, windage, and stray losses,
  while the lower the core losses ? sometimes
  these 3 categories of losses are lumped together
  and called rotational losses
• Since component losses of rotational losses
  change in opposite directions with a change in
  speed, total rotational losses of a motor often
  considered constant
• EXAMPLE A 480, 60 Hz, 50 hp, 3 phase induction
  motor is drawing 60 A at 0.85 PF lagging
• The stator copper losses are 2 kW, and rotor
  copper losses are 700 W, friction windage
  losses are 600 W, core losses 1800 W, and stray
  losses negligible. Find
• (a) PAG (b) Pconv (c) Pout (d)
  efficieny of motor

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POWER TORQUE IN INDUCTION MOTORS-LOSSES EXAMPLE

• (a) PAGPin-PSCL Pcore
• Pinv3 VT ILv3 (480) (60) (0.85)42.4 kW
• PAG42.4-2-1.8 38.6 kW
• (b) PconvPAG-PRCL38.6-700/100037.9 kW
• (c) PoutPconv-PFW-Pmisc37.9-600/1000-0
• 37.3 kW 37.3/ 0.746
  50 hp
• (d) efficiency of motor
• ? Pout/Pin x 100 88

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POWER TORQUE IN INDUCTION MOTORS

• Employing the equivalent circuit, power torque
  equations can be derived
• Input current I1 Vf/ Zeq
• R1 jX1 1/GC-jBM 1/R2/s jX2

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POWER TORQUE IN INDUCTION MOTORS

• ? stator copper losses, core losses and rotor
  copper losses can be found
• stator copper losses (3 phase)PSCL 3 I12 R1
• core losses Pcore 3 E12 GC
• ? PAGPin-PSCL-Pcore
• only element in equ. cct. where air gap power can
  be consumed is resistor R2/s , air gap power
  can also be given PAG3 I22
  R2/s (1)
• Actual resistive losses in rotor circuit
• PRCL3 I22
  R2 (2)
• PconvPAG- PRCL3I22 R2/s-3I22 R2 3I22 R2
  (1/s-1)
• Pconv 3I22 R2
  (1-s)/s

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POWER TORQUE IN INDUCTION MOTORS

• Note
• from equations (1) (2) ?
• rotor copper losses air gap
  power x slip
• The lower the slip the lower the lower rotor
  losses
• And if rotor is stationary s1 air gap power is
  entirely consumed in rotor, this is consistent
  with the fact that output power in this case
  would be zero since ?m0, ?
  PoutTload ?m0
• PconvPAG-PRCLPAG-sPAG(
  1-s)PAG (3)
• If friction windage losses and stray losses are
  known, output power
• PoutPconv-PFW- Pmisc
  

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POWER TORQUE IN INDUCTION MOTORS

• Induced torque Tind as torque generated by
  internal electric to mechanical power conversion
• It differs from available torque by amount equal
  to friction windage torques in machine
• TindPconv/?m also called developed torque of
  machine
• Substituting for Pconv from (3) for ?m, (1-s)
  ?sync
• ? Tind (1-s)PAG/ (1-s)?sync PAG/?sync
  (4)
• So (4) express induced torque in terms of
  air-gap power sync. Speed which is constant ?
• 
  PAG yields Tind

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POWER TORQUE IN INDUCTION MOTORS

• SEPARATION of PRCL Pconv in induction motor Eq.
  cct.
• Part of power coming across air gap consumed in
  rotor copper losses, the other part converted
  to mechanical power to drive motor shaft
• it is possible to separate these two different
  uses of air-gap power present them separately
  in the equivalent circuit
• Equation (1) is an expression for total air-gap
  power, while (2) gives actual rotor losses, the
  difference between these two is Pconv must be
  consumed in an equivalent resistor
• RconvR2/s-R2 R2(1/s-1) R2 (1-s)/s

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POWER TORQUE IN INDUCTION MOTORS

• The per-phase equivalent circuit with rotor
  copper losses power converted to mech. form
  separated into distinct elements shown below

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POWER TORQUE IN INDUCTION MOTORS TORQUE
EXAMPLE

• a 460 V, 60 Hz, 25 hp, 4 pole, Y connected
  induction motor has following impedances in O
  /phase referred to stator circuit
• R10.641 O R20.332 O
• X1 1.106 O X2 0.464 O XM26.3 O
• total rotational losses are 1100 W, assumed
  to be
• constant
• core loss is lumped in with rotational losses.
  For rotor
• slip of 2.2 at rated voltage rated
  frequency, find
• Speed (b) stator current (c) P.F. (d)
  Pconv Pout
• (e) Tind Tload (f) Efficiency

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POWER TORQUE IN INDUCTION MOTORS TORQUE
EXAMPLE

• (a) nsync120 fe/p120 x60/41800 r/min
• ?sync1800 x 2p x 1/60 188.5 rad/s
• rotors mechanical shaft speed
• nm(1-s) nsync(1-0.022) x 18001760 r/min
• ?m (1-s) ?sync (1-0.022) x 188.5 184.4
  rad/s
• (b) to find stator current, consider eq.
  impedance of cct. Then combine referred rotor
  impedance in parallel with magnetization branch,
  and add stator impedance to the combination in
  series
• The referred rotor impedance is
• Z2 R2/s j X2 0.332 / 0.022 j0.464
• 15.09j0.46415.1/_ 1.76? O
• combined magnetization plus rotor impedance
  is
• Zf 1/1/(jXM) 1/Z2 1/ -j0.038
  0.0662/_ -1.76?
• 
  1/0.0773/_ -31.1?12.94/_31.1 ?

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POWER TORQUE IN INDUCTION MOTORS TORQUE
EXAMPLE

• Total impedance
• Ztot ZstatZf
• 0.641j1.10612.94/_31.1? O
• 11.72 j7.79 14.07 /_33.6? O
• Resulting stator current
• I1Vf/Ztot 206/_0? / 14.07 /_33.6
• 
  18.88/_-3.6 A

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POWER TORQUE IN INDUCTION MOTORS TORQUE
EXAMPLE

• (c) motor power factor
• PFcos 33.6 0.833 lagging
• (d) Input power of motor
• Pinv3 VT VL cos ?v3 x 460 x 18.88 x 0.833
• 
  12530 W
• PSCL3 (18.88)2 (0.641)685 W
• air-gap power PAGPin-PSCL12530-68511845 W
  and the power converted is
• Pconv(1-s)PAG (1-0.022)(11845) 11585 W
• PoutPconv-Prot11585-110010485 W14.1 hp

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POWER TORQUE IN INDUCTION MOTORS TORQUE
EXAMPLE

• (e) induced torque is
• TindPAG/ ?sync 11845 / 188.5 62.8
  N.m
• output torque TloadPout/?m
• 
  10485/184.456.9 N.m
• (f) motors efficiency
• ? Pout/ Pin x 100 10485/12530 x 100
• 83.7

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INDUCTION MOTOR TORQUE CHARACTERISTIC

• How does its torque change as load changes?
• How much torque can supply at starting
  conditions? how much does the speed of induction
  motor drop as its shaft load increases?
• it is necessary to understand the relationship
  among motors torque, speed, and power
• the torque-speed characteristic examined first
  from physical viewpoint of motors magnetic field
  then a general equation for torque as function
  of slip derived

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INDUCTION MOTOR TORQUE CHARACTERISTIC

• Induced Torque from a Physical Viewpoint
• Figure shows a cage rotor of an induction motor
• Initially operating at no load ? nearly sync.
  speed

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INDUCTION MOTOR TORQUE CHARACTERISTIC

• Net magnetic field Bnet produced by magnetization
  current IM flowing in motors equivalent circuit
• Magnitude of IM and Bnet directly proportional
  to E1
• If E1 constant, then Bnet constant
• In practice E1 varies as load changes, because
  stator impedance R1 and X1 cause varying voltage
  drops with varying load
• However, these drops in stator winding is
  relatively small so E1 ( hence IM Bnet)
  approximately constant with changes in load
• In Fig (a) motor is at no load, its slip is very
  small therefore relative motion between rotor
  and magnetic field is very small rotor
  frequency also very small

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INDUCTION MOTOR TORQUE CHARACTERISTIC

• Consequently ER induced in rotor is very small,
  and IR would be small
• So frequency is very small, reactance of rotor is
  nearly zero, and maximum rotor current IR is
  almost in phase with rotor voltage ER
• Rotor current produces a small BR at an angle
  just slightly greater than 90? behind Bnet
• Note stator current must be quite large even at
  no load, since it supply most of Bnet
• That is why induction motors have large no load
  currents compared to other types of machines

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INDUCTION MOTOR TORQUE CHARACTERISTIC

• The induced torque, which keeps rotor running is
  given by
• Tind k BR x Bnet or Tindk BR Bnet sind
• Since BR is very small, Tind also quite small,
  enough just to overcome motors rotational losses
  
• suppose motor is loaded (in Fig (b)) as load
  increase, motor slip increase, and rotor speed
  falls. Since rotor speed decreased, more relative
  motion exist between rotor stator magnetic
  fields in machine
• Greater relative motion produces a stronger rotor
  voltage ER which in turn produces a larger rotor
  current IR

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INDUCTION MOTOR TORQUE CHARACTERISTIC

• Consequently BR also increases, however angle of
  rotor current BR changes as well
• Since rotor slip get larger, rotor frequency
  increases frsfe and rotor reactance increases
• (? LR)
• Rotor current now lags further behind rotor
  voltage (as shown) BR shift with current
• Fig b, shows motor operating at a fairly high
  load
• Note at this situation, rotor current increased
  and d increased
• Increase in BR tends to increase torque, while
  increase in d tends to decrease the torque (dgt90)
• However since the effect of first is higher than
  the second in overall induced torque increased
  with load

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INDUCTION MOTOR TORQUE CHARACTERISTIC

• Using Tindk BR Bnet sind derive output
  torque-versus-speed characteristic of induction
  motor
• Each term in above equation considered separately
  to derive overall machine behavior
• Individual terms are
• 1. BR directly proportional to current flowing
  in rotor, as long as rotor is unsaturated
• Current flow in rotor increases with
  increasing slip (decreasing speed) it is plotted
• 2- Bnet magnetic field in motor is
  proportional to E1 therefore approximately
  constant (E1 actually decreases with increasing
  current flow) this effect small compared to the
  other two ignored in drawing
• 3- sind d is just equal to P.F. angle of
  rotor plus 90? d?R90
• And sindsin(?R90)cos ?R which is P.F. of
  rotor

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INDUCTION MOTOR TORQUE CHARACTERISTIC

• Rotor P.F. angle can be determined as follows
• ?R atan XR/RR atan sXR0 / RR
• PFR cos ?R PFRcos(atan sXR0/RR)
• plot of rotor P.F. versus speed shown in fig (c)
• Since induced torque is proportional to product
  of these 3 terms, torque-speed characteristic can
  be constructed from graphical multiplication of 3
  previous plots Figs (a,b,c) and shown in fig (d)

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INDUCTION MOTOR TORQUE CHARACTERISTIC

• Development of induction motor torque speed

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Development of induction motor torque speed
35
Development of induction motor torque speed
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Development of induction motor torque speed
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INDUCTION MOTOR TORQUE CHARACTERISTIC

• This characteristic curve can be divided into
  three regions
• 1st region is low-slip region in which motor
  slip increases approximately linearly with
  increase load rotor mechanical speed decreases
  approximately linearly with load
• In this region rotor reactance is negligible, so
  rotor PF is approximately unity, while rotor
  current increases linearly with slip
• The entire normal steady-state operating range of
  an induction motor is included in this linear
  low-slip region

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INDUCTION MOTOR TORQUE CHARACTERISTIC

• 2nd region on curve called moderate-slip region
• In moderate-slip region rotor frequency is
  higher than before, rotor reactance is on the
  same order of magnitude as rotor resistance
• - In this region rotor current, no longer
  increases as rapidly as before and the P.F.
  starts to drop
• - peak torque (pullout torque) of motor occurs
  at point where, for an incremental increase in
  load, increase in rotor current is exactly
  balanced by decrease in rotor P.F.

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INDUCTION MOTOR TORQUE CHARACTERISTIC

• 3rd region on curve is called high-slip region
• In high-slip region, induced torque actually
  decreases with increased load, since the increase
  in rotor current is completely overshadowed by
  decrease in rotor P.F.
• For a typical induction motor, pullout torque is
  200 to 250 of rated full-load torque
• And starting torque (at zero speed) is about 150
  of full-load torque
• Unlike synchronous motor, induction motor can
  start with a full-load attached to its shaft

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INDUCTION MOTOR INDUCED-TORQUE EQUATION

• Equiv. circuit of induction motor its power
  flow diagram used to derive a relation for
  induced torque versus speed
• TindPconv/?m or TindPAG/?sync
• Latter useful, since ?sync is constant (for fe
  and a number of poles) so from PAG ? Tind
• The PAG is equal to power absorbed by resistor
  R2/s , how can this power be determined?

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INDUCTION MOTOR INDUCED-TORQUE EQUATION

• In this figure the air-gap power supplied to one
  phase is PAG,1fI22 R2/s ?
• for 3 phase PAG3I22 R2/s

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INDUCTION MOTOR INDUCED-TORQUE EQUATION

• If I2 can be determined, air-gap power induced
  torque are known
• easiest way to determine Thevenin equivalent of
  the portion of circuit to left of arrow E1 in
  eq. cct. figure
• VTH Vf ZM/ ZMZ1 Vf j XM / R1jX1jXM
• Magnitude of thevenin voltage
• VTH Vf XM / vR12(X1XM)2
• VTH Vf XM / X1XM , ZTH Z1ZM /Z1ZM
• ZTHRTHjXTH jXM(R1jX1)/R1j(X1XM)

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INDUCTION MOTOR INDUCED-TORQUE EQUATION

• Thevenin equivalent voltage of induction motor