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SCALAR  CONTROL  OF  INDUCTION  MOTOR  DRIVES    We   have   seen   that   applying   balanced,   sinusoidal   3-­‐phase   supply   to   a   3-­‐phase   sinusoidal   distributed  winding  produces  a   rotating  MMF  wave  and  hence  rotating  magnetic   flux   in   the  airgap  at   the  synchronous   frequency.  The  rotating  magnetic  flux  will  induce    

(i)   EMF  on  the  stator,  𝐸!"  (ii)   EMF  on  the  rotor,  𝐸!  

 The  induced  EMF  on  stator   is  known  as  the  back  EMF  (or  also  known  as  the  airgap  voltage),   is  proportional  to  the  airgap  flux  and  the  frequency  of  the  applied  voltage,  thus    

𝐸!" = 𝑘!𝑓𝜙!"               (1)    The  stator  voltage  equation  can  be  written  as    

𝑉! = 𝑅!𝐼! + 𝑗2𝜋𝑓𝐿!"𝐼! + 𝐸!",           (2)    where  𝑅!𝐼!  and   𝑗2𝜋𝑓𝐿!"𝐼!  represent   the   voltage   drop   due   to   stator   resistance   and   stator   flux   leakage  respectively.  The  induced  EMF  on  the  rotor  circuit,  on  the  other  hand,  will  be  at  slip  frequency  since  the  rotor  rotates  at  slip  frequency  with  respect  to  the  rotating  magnetic  flux.  The  induced  EMF  on  rotor  can  be  written  as    

𝐸! = 𝑘!𝑓!"𝜙!"             (3)    Since  the  rotor  bars  are  shorted  by  end  rings,  rotor  current  will  flow,  hence  the  rotor  voltage  equation  can  be  written  as    

𝐸! = 𝑅!𝐼! + 𝑗2𝜋𝑓!"𝐿!"𝐼!           (4)    

where  Llr  is  the  rotor  leakage  inductance  to  represent  for  the  leakage    rotor  flux,  and  Rr  is  the  rotor  resistance.    The  rotor  current,  in  turn,  induced  the  rotor  flux,  which  rotates  at  a  slip  frequency  with  respect  to  the  rotor  and  therefore  at  synchronous  frequency  with  respect  to  the  stationary  stator.  The  torque  production  can  be  considered  as  a  result  of  the  interaction  between  the  rotor  flux  and  the  airgap  flux.    The  per-­‐phase  equivalent  circuit,  which  we  have  derived  before,  is  shown  in  Figure  1(a).  Figure  1(b)  shows  the  corresponding  phasor  diagram.                      

Figure  1:    (a)    Per-­‐phase  equivalent  circuit,  (b)  Phasor  diagram    It  can  be  shown  that  the  steady  state  torque  of  the  induction  motor  is  given  by    

𝑇! = 𝐾𝐼!𝜙!"𝑠𝑖𝑛𝛿,           (5)    

𝛿  𝐸!"  𝐼!  

𝐼!  𝐼!  

𝑉!  

𝑉!  

𝐼!  𝐼!  

𝐼!  𝐸!"  

+  

_  

+  

_  

𝐿!"  𝐿!"  

𝐿!   Rr/s   𝜃  𝑗𝜔𝐿!"𝐼!  

(a)  

(b)  

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where  K  is  a  constant,  and  𝛿  is  the  angle  between  𝐼!  and  𝐼!.  For  small  slip  operation,  𝛿  is  close  to  90o  (i.e.  𝜃 ≈ 0)  and  hence  (5)  can  be  written  as    

𝑇! = 𝐾𝐼!𝜙!"             (6)    When  the  rotor  is  running  at  small  slip,  𝑅! ≫ 2𝜋𝑓!"𝐿!",  hence  combining  (3),  (4)  and  (6),  we  can  write    

𝑇! = 𝐾′𝜙!"! 𝑓!"             (7)    To  ensure  maximum  torque  capability  at  any  frequency,  it  is  therefore  necessary  to  maintain  the  airgap  flux,  𝜙!"  at  its  rated  value.  With  constant  𝜙!",  the  torque  is  proportional  to  the  slip  frequency,  fsl.  We  have  seen  that  in  the  per-­‐phase  equivalent  circuit,  the  current  that  flows  through  Lm  (i.e.  magnetizing  current  Im)  is  responsible  for  the   airgap   flux  production.   Thus   from   the  per-­‐phase   steady-­‐state   equivalent   circuit   point   of   view,   in  order   to  maintain  the  rated  airgap  flux  at  any  frequency,  it  is  therefore  necessary  to  ensure  that  Im  is  at  its  rated  value  at  any  frequency.  From  the  equivalent  circuit,  the  magnetizing  current  Im  can  be  written  as  (8).    

𝐼! = !!!!"!!

  !  𝐼!,!"#$% =!!,!"#$%

!!!!"#$%!!         (8)  

 According   to   (8),   the   magnitude   of   the   magnetizing   current   can   be   maintained   constant   at   its   rated   by  maintaining  the  ratio  Eg/f  equals  to  Eg,rated/frated.  If  the  frequency  is  reduced,  Eg  has  to  be  reduced  proportionally  to  maintain  constant  Im.  If  operation  at  small  slip  is  considered,  and  the  ratio  of  Eg/f  is  maintained  constant,  the  motor  characteristics  at  different  synchronous  frequencies  are  as  shown  in  Figure  2.                                    

Figure  2:    Constant  airgap  characteristics  at  different  frequencies    At   high   speed,  where   the   induced   back   EMF,   Eg,   is   large   and   the   voltage   drop   across   the   stator   leakage   and  resistance  are  relatively  small;  under  this  condition,  Eg/f  is  maintained  constant  by  maintaining  Vs/f  constant.  In  other  words,  we  can  assume      

!!!≃ !!

!               (9)  

 However  at  low  speed,  Eg  is  small  and  thus  the  voltage  drop  across  the  stator  impedances  is  significant  and  approximation  (9)  cannot  be  used.  If  (9)  is  assumed,  then  the  rated  flux  cannot  be  maintained  hence  torque  capability  will  be  reduced.        In  order  to  improve  the  torque  capability  at  low  speed,  the  following  method  can  be  used:  

𝜔!"#$,!"#$%  𝜔!"#!  

𝜔!"#!  

𝑇!"#$  

T  

𝜔!  (rad/s)  𝜔!"#$,!"#$%  𝜔!"#$!   𝜔! ,!"#$%  𝜔! ,!  

𝑇!"#$%  

𝑇!  

𝑇!"#$%𝜔!"#$,!"#$%

=𝑇!

𝜔!"#$!=

𝑇!𝜔!"#$!

 

𝑇!  

𝜔!"#$!  𝜔! ,!  

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   (i)   Boosting  the  voltage  at  low  frequency:      

To   accurately   boost   the   voltage,   stator   current   needs   to   be   measured.   The   voltage   drop   across   the  stator  impedance  is  then  calculated  and  added  to  the  stator  voltage.  Alternatively,  one  can  approximate  the  amount  of  voltage  boost  needed  at  low  speed,  which  depends  on  the  stator  current  and  hence  on  the  load.  Low  frequency  voltage  boost  can  be  either  a  linear  boost  of  a  non-­‐liner  boost  (Figure  2)  

 

 Figure  3:    Voltage  boost  at  low  frequency  

 ii)     Stator  current  control  

It  also  possible  to  control  the  magnetizing  current,  Im,  in  order  to  ensure  rated  magnetizing  current  at  all  times.  The  relationship  between  the  stator  current  and  the  magnetizing  current  can  be  obtained  from  the  per-­‐phase  equivalent  circuit.  Thus  the  magnetizing  current  can  be  indirectly  controlled  via  the  stator  current.  This  can  be  accomplished,  for  example,  using  a  current-­‐controlled  voltage  source  inverter.    

 From  the  per-­‐phase  equivalent  circuit,  

𝐼!!!"!!"!

!!!

!" !!!!! !!!!

𝐼!           (10)  

which  gives      

𝐼!!!"!!!

!!!

!"!!"!!!!

𝐼!             (11)  

 Let  𝐿!" = 𝜎!𝐿!  ,  where  𝜎!  is  the  rotor  leakage  factor,  then  we  can  write  (11)  as  

   

𝐼!!!"!!!

!!!

!" !!!!!!

!!!!!!

𝐼!           (12)  

 Recognizing  that  𝜔!"#$ = 𝑠𝜔    and  𝜏! =

!!!!  ,  (12)  can  be  written  as  

 

𝐼!!!!!"#$!!!!

!!!"#$!!

!!!!!!!!

𝐼!           (13)  

 Constant  magnetizing   current   Im   can  be  obtained  by   controlling   Is   according   to   (13).  With   Im   set   to   its  rated   value   and   motor   parameters   assumed   constant,   (13)   indicated   that   Is   is   a   function   of   slip  frequency.  One  possible  scheme  is  shown  in  Figure  4.  The  speed  controller  generates  the  slip  frequency,  

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which   is  fed  to  the  function  generator  to  produce  the  stator  current  magnitude  according  to  (13).  The  stator  current  reference  generator  generates  the  3-­‐phase  current  references  based  on  this  magnitude  and   the   synchronous   speed,  which   is   obtained  by   adding   the   slip   speed  with   the   rotor   speed.   Three-­‐phase  stator  currents  are  synthesized  using  current  controlled  scheme  as  discussed  in  earlier  module.  As  can  be  seen  from  (13),  the  generation  of  the  stator  current  reference  is  highly  dependent  on  the  motor  parameters   (Rr,   Lr  and   Lm),  which  will   change  with  operating   temperature.   If  motor  parameters   varies  from  their  nominal  values,  Im  will  not  be  at  its  rate  value.  

 

 Figure  4:    Constant  magnetizing  current  with  stator  current  control  

 Open-­‐loop  V/f  control    For   low   cost,   low   performance   drive,   open-­‐loop   constant   V/f   control   is   normally   employed.  With   open-­‐loop  speed  control,  the  rotor  speed  will  be  less  than  the  synchronous  speed  by  slip  speed.  In  other  words,  the  desired  speed,  𝜔!!∗ ,  will  differ  from  the  actual  speed,  𝜔!,!,  by  slip  speed  𝜔!"#$!,  as  shown  in  Figure  5.  To  improve  on  the  speed  regulation,  slip  speed  has  to  be  estimated  and  added  to  the  reference  speed  –  this   is  known  as  the  slip  compensation  technique.  According  to  Figure  5,  the  new  reference  speed,  𝜔!!∗ ,   is  obtained  by  adding  𝜔!!∗  with  the  estimated  𝜔!"#$!.  With   the  new   reference   speed,   the  new   rotor   speed  𝜔!!  will   be   approximately   equal   to  𝜔!!∗ .   In  actual,  𝜔!"#$!  will  be  slightly  higher   than  𝜔!"#$!;   if   the   load  torque   is  constant,   then,    𝜔!"#$!  =  𝜔!"#$!.  A  typical  open-­‐loop  constant  V/f  control  scheme  is  as  shown  in  Figure  6.                                  

Figure  5:      Slip  compensation    

𝜔!!∗  

𝑇!"#$  

T  

𝜔!  (rad/s)  𝜔!"#$!  𝜔!!  

𝑇!  

𝜔!!∗ = 𝜔!!

∗ + 𝜔!"#$!  

𝜔!! ≈ 𝜔!!∗  

𝑇!  

Motor  characteristic  AFTER  slip  compensation  

Motor  characteristic  BEFORE  slip  compensation  

𝜔!"#$!  

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Figure  6:    Constant  V/f  drive  with  slip  compensation    How  is  the  slip  speed  estimated?      The  slip  frequency  is  proportional  to  the  torque,  hence  it  can  be  estimated  by  estimating  the  torque.  The  torque  is  estimated  from  the  air-­‐gap  power,  which  is  obtained  by  subtracting  the  input  power.  Thus,    

𝑇! =𝑃!"#!!"#𝜔!

 

 Input   power,   on   the   other   hand,   is   calculated  by   subtracting   the   input  DC  power  with   the   inverter   losses,   as  shown  in  Figure  7.      

   

Figure  7:    Airgap  power  estimation    Closed-­‐loop  speed  control  by  slip  compensation  Speed  regulation  can  be  improved  by  employing  closed-­‐loop  speed  control  system  with  tachometer  feedback,  as  shown  in  Figure  8  [2].  The  reference  and  actual  speed  are  compared  and  the  error  is  fed  to  the  speed  controller,  

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which  generates  the  slip  frequency.  The  slip  frequency  is   limited  to  its  maximum  value  and  added  to  the  rotor  frequency  that  gives  the  synchronous  frequency;  the  slip  frequency  is  limited  in  order  to  avoid  the  synchronous  frequency   from   reaching   the   breakdown   frequency.   Using   the   synchronous   frequency,   constant   V/f   is  implemented.    

   

Figure  8:      Closed-­‐loop  speed  control  by  slip  compensation      Further  readings:    [1]      Power  Electronic  Control  of  AC  Motors  –  J.M.D.  Murphy  and  F.G.  Turnbull,  Pergamon  Press,  1988    [2]      Modern  Power  Electronics  and  AC  Drives  –  BK  Bose,  Prentice  Hall,  2001  [3]      Power  Electronics:  Converters,  applications  and  design  –  Ned  Mohan,  TM  Undeland,  WP  Robbins,  John  Wiley,  2003