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Theory of the performance of the induction motor under unbalanced conditions Lunn, Edward O. 1933

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•• •  y  j CAT. m. k£*f>±-. i.m Ap.L*X>. THEOHY OF THE PERFORMANCE OF THE INDUCTION TWOT'OT? UNDER UNBALANCED  CONDITIONS  by Edward  A Thesis  0 , Lunn  s u b m i t t e d for  MASTER OF APPLIED i n the  the  Degree  SCIENCE  Department  of ELECTRICAL  The U n i v e r s i t y  of  April,  ENGINEERING  British 1933  e » •  e&luabia  of  TABLE OP CONTENTS I 1. 2.  Introductory. The M a g n e t o m o t i v e Winding.  Force of  a Symmetrical  Polyphase  II 3.  Inductions  4.  Frequency  Motor Reactions of Reactions  and M a g n e t i c  i n Induction  Coupling.  Motors.  Ill 3.  General  Theory  f o r Unbalanced  6.  Balanced Applied Voltage,  All  7. U n b a l a n c e d A p p l i e d V o l t a g e ,  Conditions. Impedances  All  Balanced.  Impedances  Balanced.  8. U n b a l a n c e d A p p l i e d V o l t a g e , P r i m a i - y Impedances U n B a l a n c e d , S e c o n d a r y Impedances B a l a n c e d . 9. U n b a l a n c e d A p p l i e d V o l t a g e , P r i m a r y Impedances B a l a n c e d , S e c o n d a r y Impedances U n b a l a n c e d . x  IV^ ' TO. B l o c k e d  Torque  Tests.  11.  C a l c u l a t i o n s f o r S l i p - T o r q u e Curves Having Unbalanced Applied Voltage.  12.  CalcuationHS Balanced  13.  for  Applied  Slip-Torque Voltage.  Conclusion.  Appendix.  Curves,  f o r Balanced Motor  Unbalanced Primary  1  THEORY OF THE PERFORMANCE 0 7 THE INDUC T I OH" MO TOR UNDER UNBALANCED CONDITIONS  . 1  1.  Introductory.  motors  The g e n e r a l  known.  It  is  also  reduction also circuits  if  o f the motor,  are unbalanced.  motor t e r m i n a l v o l t a g e is  our purpose  to  the  latter is  For t h i s ,  ponents  is very  effective,  application of this  of the  starting  of  This  external  and c o n t r o l  circuits,  case the u n b a l a n c i n g  investigate  manner.  quite  efficiency.  a f u n c t i o n o f the  itative  the  constants  s u c h as  In this  are  induction  i n a r e d u c t i o n o f maximum  and i n a r e d u c t i o n o f  occurs  of  known t h a t a n y u n b a l a n c i n g  applied voltage results  torque and o u t p u t ,  It  characteristics  o p e r a t i n g u n d e r "balanced c o n d i t i o n s  generally the  .  of  current or  t h e s e phenomena i n a  t h e method o f s y m m e t r i c a l  quant-  com-  and a p r o p e r u n d e r s t a n d i n g  theory  is  desirable. of  this  of  is  included  networks  of  A brief  summary o f t h e more i m p o r t a n t p r i n c i p l e s i o l v i n g unbalanced polyphase  slip.  method in  an appendix. • 2. The M a g n e t o m o t i v e Winding. Prefatory  to  Force of a Symmetrical a general  analysis  of  Polyphase unbalanced  conditions,  we s h a l l  c o n s i d e r the magnetomotive  produced by  currents  flowing  distributed winding. phase w i n d i n g i s values  i n the  i n a three-phase,  The d e v e l o p m e n t  shown i n F i g .  conductors  1,  force symmetrical,  of a simple  and b e l o w  are g r a p h i c a l l y  it  three-  the  expressed,  current for  the  instant  considered.  trapezoidal  curves  The r e s u l t a n t  CURRENT  o  W . M . F.  o  Below  this,  show t h e m . m . f .  m.m.f.  is  indicated  three  component  exerted  by each  i n red.  m.m.f. Jhaee.  The l o w e s t  graph  M.M.F.  Fig. indicates in  the m.m.f.  f o r m , and a t  t.  of a single  the  instant  phase.  considered,  It  is  trapeziodal  has a maximum  A amplitude  equal  eziod gives from the the f -  to F .  The F o u r i e r a n a l y s i s  a  the m . m . f . ,  f  a  ,  exerted  by  it  at  of  the  the  angle  a r b i t r a r i l y chosen o r i g i n , which i n t h i s  center  o f phase  trap-  case  is  1 :  ±3- f a i n a s i n 9 + I s i n 3a s i n 3 9 + i s i n ^ a s i n ^ e + • • • • ]  It  w i l l be n o t e d  If  the  currents  harmonic  that  a = 2£ f o r  the  i n the w i n d i n g are  function  of  three-phase sinusoidal, F  winding. a  is  time, A  F so  9  that,  a  expressing  F cos c o t ,  =  a  f as a  a harmonic f u n c t i o n o f  time,  a  O )  fV= 2^ F c o s w t ^  \ s i n n £ s i n n9  Q  Similar  expressions hold  phases,  where  phase  .  .  .  .  . . . (2)  f o r t h e m.m.f. of t h e o t h e r two  the time-phase  of t h e c u r r e n t s and s p a c e -  of the w i n d i n g s a r e d i s p l a c e d — I T and 4 i r r a d i a n s .  3  5  A  O  1 1 5 0 0  f = 24 F cos (cot - -|tr)V Is i n n£ s i n n(9 - l i r ) •ir y Z_ n b 3 b  a  f = 24 F c o s (cot « 4 0 > ^ 3 4ltf 1  c  c  The  (5)  b  1,  s  i  n n  T r s i n n(9 - 1-iT)  b  .  .  .  .  .  . (4)  3  r e s u l t a n t m.m.f. o f the three p h a s e s  a t any a n g l e 9 ,  at time t , i s t h e sum f + f + f . I f t h e c u r r e n t s i n t h e a  t h r e e phases sum  b  c  a r e b a l a n c e d so t h a t  F„ - F = F = F, , the b  c  i s g i v e n by  f, = 2 i F , V cosjcot - ( U - 1 )-§frf\" I s i n n T L s i n n<9 - 0 1 - 1 (5) -ft £7 L 3 'Z. 6 ( 3J) 2  P u t t i n g n=1,3,5, .  .  .  , we h a v e  f, = J l F , f s i n ( e - o ) t ) + —lsin(59+wt) TT  1  L  ' 2 5 _  If  Lsin(79-«t) 49  1- s l n O 19+cot) + • * » • 121 1  t h e c u r r e n t s i n t h e t h r e e phases  ft  ••  are unbalanced,  may b e r e s o l v e d i n t o t h r e e b a l a n c e d , s e t s  «  A  they  o f currents  a c c o r d i n g t o t h e fundamental p r i n c i p l e o f t h e theory o f symmetrical phase  c o m p o n e n t s . We a r e n o t c o n c e r n e d w i t h t h e z e r o  sequence  ordinarily  o f currents, since  this  s e t does n o t f l o w  i n i n d u c t i o n motor w i n d i n g s ,  w o u l d p r o d u c e no r e s u l t a n t m . m . f . study o f F i g .  1.  The p o s i t i v e  and i n any case  a s may b e s e e n f r o m a  phase  sequence  of currents,  I, , g i v e s r i s e  t o f, a s e x p r e s s e d b y e q u a t i o n (5), b u t t h e  n e g a t i v e phase  sequence  o f c u r r e n t s , I , has time-phase z  ( 6 )  displacements the  the r e v e r s e  sign prefixed  to  the is  the m.m.f.  due t o  I  ^|F V  c o s wt  +  f = 2  2  2  o f I , . That i s ,  i n equation  t e r m (N-1 )•§*!", i s given  reversed.  (Jj),  Thus  by  )^irJ\T i s i n n I L s i n n f e - (N-T )-|ir) (7)  or  ^ ^F-sinje  +(cot-<x)) ... __Lsinf59 -  ct b e i n g a c o n s t a n t In equations  angle.  ( 6 ) and (8)  we s e e  higher, harmonic t e r m s ^ f a l l s we may f i x that  our a t t e n t i o n  the m.m.f.  ocity  as f  then,  that  hut  t  I,  r e v o l v i n g at  f *  i n the  off  I j a  the  The r e s u l t a n t  I  2  has  the  for  terms.  It  f, a n d £  2  We s e e vel-  opposite  i s g i v e n by the  terms o n l y )  velocities,  we may o b s e r v e  clear,  m.m.f.  o f co e l e c t r i c a l  In the  ^7 v e c t o r s  the r e s u l t a n t  present,  radians  produces a "negative r o t a t i o n a l "  fundamental  affect  the  becomes  rotational"  m.m.f.  direction.  sum, f, + f .  If  a  A represent  the  same a n g u l a r  oppositp d i r e c t i o n .  same s p e e d  m.m.f.  So,  fundamental  produces a " p o s i t i v e  per second, while  the amplitude o f  rapidly.  on the  due t o  2  that  an angular v e l o c i t y  revolving at  '(cot-*))  +  we  A  o f l e n g t h F, a n d F ( c a n s i d e r i n g 2  revolving with t h e i r corresponding  m.m.f.  how t h e i r r e l a t i v e  magnitudes  • 5 Obviously,  the  ellipse  semi-major  of  locus  F, - F . As l o n g a s  2  axis  F, •+ F ,  o f r o t a t i o n the  angular v e l o c i t y . circular,  ,  2  This  illustrates  phase  winding.  resultant,  a straight  character  of  the p o s i t i v e  wave, to  ahek h e n c e  the  phase  field,  sequence revolving  f i e l d produced by the  of  the  field  i n the  dir-  variable course, velocity.  period being  in a  emphasized, currents  axis  angular  line,  the  The d e d u c t i o n w h i c h s h o u l d be that  is,  wave h a d c o n s t a n t  is  an  has a  R  locus,  z  the  F ,  F, , a l t h o u g h w i t h  When F z : 0 , t h e  locus  2  the  in gereral,  and s e m i - m i n o r  2  same a s  and the m . m . f ,  When F = F, , t h e  is,  R  F,>F  z  ection  o f F = F, + F  single-  however,  produce an  opposite  n e g a t i v e sequence  ~  is a.m.f.  direction  currents.  II 3.  Induction Motor Reactions  s e e n some o f a three-phase  the  effects  d u c t i o n motor a t static  If  machinery, +  E,  the  is  the mutual  a closer  w i l l undergo  in  analysis.  impressed, on a n i n are  the  same  as  ,  mutual)  continuously with respect  inductance  i n magnetic  on t o  flowing  by  = 0  i:R = ohmic r e s i s t a n c e If = i n d u c t a n c e ( s e l f p l u s = 27fx( f r e q u e n c y ) I = current  Coupling.Haying  currents  reactions  and e x p r e s s e d (R + jo)]S)i  one w i n d i n g moves  changes  e.m.f,  standstill,  E where  o f unbalanced  w i n d i n g , we s h a l l p a s s  When a n a l t e r n a t i n g  for  and Magnetic  to  the  p e r i o d i c changes,  c o u p l i n g between  the w i n d i n g s .  other, dtue  Thus  to the  frequency  of  the motor. are  the  the  the  The s e l f -  effective  self-inductance polyphase the  reactions  set  and m u t u a l - i n d u c t a n c e s  polyphase L has  being i  +  set  the v o l t a g e  of  (E + > L ) I  When t h e w i n d i n g s one a n o t h e r ,  does  not  affects erent  shift  phases the  windings,  across  the  open secondary  the  although  that  will  cut  the  the r o t o r a t  currents  machines,  but  diff-  This  mutual e f f e c t s are  merely  i n the  positions.  relations  inductance  is  between  cancelled  *  c o n s t a n t s , 1 and M , a r e  conditions. i n Induction Motors.  frequency  ~  revolving  radians per second,  produced by  respect  phases  the v a l u e s ,  the  total  out-of-phase  rotor is  field  is  of different  time-phase  i n the  "Frequency o f R e a c t i o n s  electrical  primary,  disposed with  position of  fact  i o n m o t o r whose  i n the  ease w i t h a l l p r a c t i c a l  different  unbalanced voltage o f  when a  P  determined under balanced 4.  primary,  joM .  between windings  the  that,  flowing  p  have  and b y d e f i n \ t i o n ,  such a value axe  the phase  the  balanced  I ,  t h e i r magnitudes,  owing to all  is  when a  to.  primary  i m p r e s s e d on the  are symmetrically  as  the mutual e f f e c t s  =  the  of  0,  =  M, has  (balanced) s  is  referred  That i s , that  speed  then  currents,  E  to  E,  open,  The m u t u a l i n d u c t a n c e , balanced  values.  such a value  of voltages,  secondary  depends u p o n t h e  currents the  an  Is  a p p l i e d to  an  induct-  at  a velocity  of  GO -GV  the p o s i t i v e I , i n the p  If  0  rotational  primary winding  a n g u l a r v e l o c i t y 6),, and c o n -  sequently  induce  show l a t e r  that  rotor currents if  the  constants  balanced,  the  rotor currents,  balanced.  But  if  will I  be u n b a l a n c e d ,  and Lj , h a v i n g  5 1  —a>  (J0,+(D  o  with  the  m.m.f.  co, r e l a t i v e  stator  of  s  m.m.f.  revolving at  stator.  at  s  at  velocity thus  rotor,  the  stator;  at  that  I .  a  I  to  5 2  the  to  I  it  is  in  exerts  an  the  -  r  the m.m.f.  ....  '  -co, r e l a t i v e  to  frequency  to  to  the  causing  the  further  number o f  these r e f l e c t i o n s ,  l i m i t e d o n l y by  the  higher  reactance  of  the  the  higher  frequencies. frequencies higher the  We s h a l l s h a l l not  frequencies  higher  We may t a b u l a t e reactions,  exceed  2to -W,,  of  4  the m o t o r .  a  currents  whose  since  currents  of  amplitude,  by the windings  relations  to w r i t e  of  is  to  of negligible  i n o r d e r t o be a b l e  perfocmance  currents  ourselves  offered  the  r e a d i l y when we come the  confine  w i l l be  reactances  to  there  ref-  If  windings  i s unbalanced,  at  rotor,  lections.  the  rotor circuit  thus  exerts  and hence  are produced i n  2 0 J o  the  which i t  stator,  -(2^.-0),) r e l a t i v e  =  step  stator,  r  :  P Z  rotor,  velocity  is,  But  P 1  -<t>, r e l a t i v e  w h i c h may be u n b a l a n c e d , the  produces  and hence  it,  0  e  of  S I  current  components  -©^(cu.-co,) = co ~il 6Pr-. r e l a t i v e  - u> - ((L> -co,)  currents  the I  ~  coming back  0  these  to  velocity  velocity  into  are  4£± a r e  same d i r e c t i o n as  ;V revolves  frequency  i n the  produced, by  velocity  Now,  rotor circuit  are unbalanced,  . and hence  We s h a l l  rotor frequency^  = cOo r e l a t i v e t o  t  the  a n d may be r e s o l v e d  which revolves  a velocity  frequency  of  I ,  constants  the  2  an m . m . f . at  the  of  between  to  the  due  them.  current  to  correlate  them more  the  equations  to  The s u b s c r i p t s  to  determine  1 and 2  refer  8 p o s i t i v e phase sequence and n e g a t i v e phase sequence currents respectively. Current  IP,  )  I p2. Isi Isz  ) )  i *  )  Is.  >  Frequency  V e l o c i t y o f c o r r e s p o n d i n g xu.iu.f. Relative Relative to s t u t o r to r o t o r  <0o  )  + co - COo  -( 2OJ -co,)  + OJo +(to - 2o).)  + CO, - CO,  + to,  0  2TT co,  d  0  6 ) 0 -  2o>,  2co, - 2co,)  -60, -(2co - 3 0 ) , )  +3co - 2co, -00  -2 co - co, _ ( iCOo - CO, )  +  2TT  6)0-  - (0O0  2CO„-C0, 2lT  0  0  0  o  In f i g u r e 3 the d i r e c t i o n s and r e l a t i v e v e l o c i t i e s o f the v a r i o u s m.m.fs. a r e shown.. Those shown b r a e k e t t e d keep s t e p . Fp2  Fp, (<Og)  F£ Qio.-^tQ..)  FA (aJ.-2LO.)  3_jfor  aW, «•  >-  Fs 2-(2a3 -co,') /  0  F,(co.) s  -«  «•  F« (-co,)  Fk1  Fig. 3  Rotor  (2oJ.-«x>,)  t a b u l a t i n g the f r e q u e n c y r e a c t i o n s f o r the c u r r e n t s we a r e considering  ( t h o s e o f f r e q u e n c y not exceeding(2co -co,)), we e>  have the t a b l e i n F i g u r e 4.  REACTION FREQUENCY OF  T h i s i s to be r e a d from the top  Ipz  down to the l e f t , Thus the  IP,  frequency o f r e a c t i o n of I  I*  on I  i s g i v e n i n the t h i r d  OS  I*  column. B l a n k squares  K  indicate  on I  . fote  I's, r  00  CO.  o  %  CO,  CO ^ |Y  CO'  no r e a c t i o n , Tnus t h e r e i s no reaction of I  I*  %  Isi  square down i n the l e f t - hand  I51  CO'  6 /  CO'=2CO -OJ 0  1  Fig. 4  The a c c o m p a n y i n g c l e a r l y how t h e conditions.  I , s  stator  due t o  is  the  In the  set  is  are affected  for balanced  and r o t o r c i r c u i t s .  t o o t h harmonics, but  o n l y one p r e s e n t second s e t ,  the  anced by added e x t e r n a l of  of rotor currents  rotor currents  The f i r s t  and b a l a n c e d ripple  oscillograms  frequency  ,  are balanced,and  In the stator, the  that  .and  Is* » l i t  script  set,  Is  the 2  2,  I ', i s s  d i r e c t i o n of absent *  introduces  T h i s c u r r e n t has  phases,  as has  the  only  1^  1^  the  ; i.e.  component  rotor circuit,ad well  rotor currents  the  of  I'  s  Ii is 2  call  wrongly  the  the  resistance,  are unbalanced,  designated  sequence  a current  a field  as  with I  st  t  present.  i n d i c a t i n g a negative  one w h i c h p r o d u c e s the  the  a l l being  chosen to  current  .  5 (  may seem t h a t  however,  to  This  has b e e n u n b a l a n c e d by e x t e r n a l  result  It  third  the  small  c i r c u i t has b e e n u n b a l -  i n the r o t o r .  I  a  stator  rotor  1 ^ * and o f I»  There i s  voltage  rotor.  in all  is  applied  i n the  same a m p l i t u d e currents  "by u n b a l a n c e d  otherwise,  resistance.  three  show  sub-  c u r r e n t . We h a v e ,  o f n e g a t i v e phase  revolving  rotor.  by the  in a direction  H e n c e , when I i  is  sequence, opposite  balanced,  • 1o III 5.  General Theory f o r Unbalanced C o n d i t i o n s .  considerations analysis  of  will  enable us  the problem o f  Let  E  P 1  foregoing  to b e g i n a m a t h e m a t i c a l  a symmetrically  i n d u c t i o n motor h a v i n g unbalanced c o n s t a n t s and r o t o r c i r c u i t s ,  The  constructed in its  stator  and h a v i n g a n u n b a l a n c e d v o l t a g e  supply.  , E , , be t h e p o s i t i v e and n e g a t i v e s e q u e n c e c o m ponents o f a p p l i e d v o l t a g e , ( E , the zero sequence component, i s d i s r e g a r d e d , s i n c e i t p r o d u c e s no z e r o s e q u e n c e c u r r e n t ) P 2  p 6  E  S |  Ei,  , E  , be t h e p o s i t i v e a n d n e g a t i v e s e q u e n c e c o m p o n e n t s o f r o t o r f r e q u e n c y CO,/2TT , ( s i n c e we s h a l l c o n s i d e r these as the t o t a l r o t o r v o l t a g e s f o r the r o t o r c i r c u i t , they are zero)  S 2  , Esz , be t h e p o s i t i v e and n e g a t i v e s e q u e n c e c o m p o n e n t s o f r o t o r v o l t a g e o f f r e q u e n c y co'/2-rr , (also zero)  Rpo » Rpi » Rpz » be -the s y m m e t r i c a l components o f r e s i s t a n e o f the p r i m a r y c i r c u i t , ( e x t e r n a l r e s i s t ance i n c l u d e d ) Rso » R i » R z t "be t h e c o r r e s p o n d i n g q u a n t i t i e s secondary c i r c u i t * S  L  p o  , L  P I  S  , L  p  z  M  be t h e c o e f f i c i e n t o f m u t u a l i n d u c t a n c e as d e f i n e d i n s e c t i o n 3 ,  p  be  t h e number o f  3W  be  the  power o u t p u t  3T b e  the  torque  + jQ,)be  the  are  of  in lbs.  volt-ampere  t o be  for  for the  the motor  poles, motor,(watts) ft., input.  Equivalent l i n e - t o - n e u t r a l values and c o n s t a n t s ,  the  , be t h e s y m m e t r i c a l components o f s e l f i n d u c t a n c e o f the p r i m a r y c i r c u i t ,  I»so » ^si » ^»S2"»' oe t h e c o r r e s p o n d i n g q u a n t i t i e s • secondary c i r c u i t ,  3@?  for  used.  for a l l voltages,  currents,  11 i  We may now w r i t e  and  secondary  of  the v o l t a g e  Fig.  4 as a g u i d e ,  E  R  =  P O  j co (Lpi Ipj  -t  L polp2 +  +  B  jo), ( L  I ,  *  t  I ,  +  Rso Isz  +  j O ) , ( L a i Is,  +  Lso I  Is.  +  $  +  jco'(L  i;,  +  L«Ua  j ^ ' ^ L s , Isi  +  l s o Isz +  E  £ |  :R  E  S 2  =  E^,  =  Es  = Rsi 1st +  2  s o  I  A I  S  R  S o  +  P 2  J  5  +  P 1  S  I «  S 2  P(  Rso I s z +  S O  s o  There  i s no m u t u a l i n d u c t a n c e  (14),  as may be  equation by  (11)  EP' J e i  =  ,  co,  0  theory  table  2  of  M I , ).. .., .  Ipa-i-  MI. ^.) •  to  MI  S 2  Fig.  P l  © (T1 ) f.  ) = 0  .  (12)  0  ) = 0  . (13)  f  +  0  ) =0  .  !  MI . ) =0  and  (13)  Multiply of  IL, ay  co,  . (1 5)  P 2  4.  by  (14)  I , ) P  (15)  and  are  lEgIeiJ[pi4. Rp^Ipglpiu.  W T T T j . T T T . i l T T  !  1  *  E P Z - I P Z . - RPI Iwlre . Rpalpy.Ip?. j ( I I , I ±I' oIp2lpi+ M I s I ) . "Clio-""' '•'" COo •'• C0 ~' •  .  -. A - •  .  P 1  P  PZ  /: \  .  (17)  .  (18)  0  ).-,.,.  (T9)  0  ).  2  P  P 2  .  t  1  0  - RsoLsiIsi j Rszlszlsi ^ j (Itfplsi I51 + I f e 2 I s i 4 M I | I | ) . 63,  0  CO  =  Adding  Rsilszlsi . j (L Is,Is', + L l i I s , 4 CO' ^  Si  5 2  5a  sl  52  S  50  f f i  2  s  equations  EPI'IPI  to  (16)  .  E  P  2  .  "•'  Rsi Isi l a • Rsnlszls^ . j ( L , I s i Isz+ LSJIS-JIS2+ M I I z ) . CO' ^ Co' ^  :•'  where  I +  L I I .+  Rsalsilsi ,  OS'  0  j (L  Rsolsalsz. cu, • ^  T  0  P  1  Rgilsilszr ' CO,  In  P 2  (21)  =  inclusive,  P + jQ, ,  f  +  conjugate (14)  *  S  I .-f S i  (TO)^  S  term i n equations  (Ig, b e i n g t h e  The r e s u l t s  P  sz  Ish (12) b y L,, (13) b y L*. 63  b y Isz.  s  seen by r e f e r e n c e  by  (10)  P  0  +  i  the  necessary.  Rpo Ipa +•  ^ P I Ipi  the p r i m a r y  the  L  R  I  if  and r e f e r r i n g t o  j w.(I ) I , +  Ip, +  Ep^™ -  for  c i r c u i t s , u s i n g the p r i n c i p l e s o f  s y m m e t r i c a l components,  P I  equations  we  S  •  •  . . .  •  (20)  { 21 )  obtain  . . . . . . .  (22)  12 Q =  (Lpjpjp,)  +  ( L ^ I ^ )  +  (L^.I^)  Is2.Ipz+ Ipz^z )  +v(I.soIs l5 ) 2  •  •  •  •  The sum i n e a c h " b r a c k e t t i s p u r e l y r e a l , is  o f t h e form  •  +  2  •  •  since  (L j;,Is ) 5  •  •  f  •  (23)  e a c h sum  AA o r AB 4 BA , e a c h o f w h i c h i s p u r e l y  real,{(see appendix),  T h e r e f o r e Q, i s p u r e l y r e a l a n d h e n c e  «)Q, p u r e l y i m a g i n a r y -  the r e a c t i v e  the  volt-ampere  input to  motor.  (24) and t h i s  i s also  purely real  - t h e watts  input  to the  m o t o r d i v i d e d b y G J , f r o m e q u a t i o n (22). 0  The c o p p e r l o s s  i n the primary c i r c u i t i s  Hi and s i m i l a r e x p r e s s i o n s secondary  h o l d f o r the copper l o s s e s  c i r c u i t due t o c u r r e n t s  The o u t p u t  I  S 1  i n the  , I . and Isi , I ' i , S 2  s  o f t h e motor i s , p e r phase, W = (input) =  neglecting  a> P 0  -  -  (copper  (copper  the i r o n l o s s e s .  losses  )  losses),  Thus  W  1  j  (Ksoisi Tsi + ^SoXs2^-S2 + Rsilsi Is2+ R52.I52I51 )  d)o-CO, CO,  where to  l  ls + s  t0o-6D  and I s  the currents  ( 25) (26)  a r e the r o t o r copper l o s s e s I , , I s  s  z  corresponding  and Is, , 1 4 . r e s p e c t i v e l y .  13 .. In the equation  2co -£0,  5  o  &> -co, e  is positive  2 C O . - CO, to  fed hack i n t o line  torque  from s y n c h r o n i s m /  standstill.  the  ' f o r a l l motor speeds  Therefore  the quantity  the supply c i r c u i t .  through the p o s i t i v e  in lbs. ft.  ~ i 2co.-oo,  C O o  C O r  T h i s power  sequence  i s t h e power  7  i s drawn f r o m  component. The  i s given by  T « 746 33000 o x 6Q(oJo-co,) W x TT 2TT =  P  ? $ 9  w  (t0 -C0, ) o  = 0.369 P { «^s_ I s ) . . . . . . . . . This  i s a more g e n e r a l  known f o r t h e t o r q u e conditions  :  IT  d i v i d e d by s l i p proportional  statement  o f i n d u c t i o n motors under  ". Equation  general  _  (TO) t o ( T 5 )  i n (28).  This  introduced by s p e c i a l  curve  the torque  is  P 2  P  c u r r e n t s and  procedure  consider  «  A l l Impedances  0, R ,= E ^ , ^ R^ = L  - 0, and the s i x fundamental  PZ  2  equations  i n the  the s i m p l i f i c a t i o n s  conditions.  = 0, R , -  .  i s d e s i r e d , we must  f o r the s e v e r a l  i s a tedious  (\. T r a n c e d A p p l i e d V o l t a g e . E  that  loss  (negative r o t a t i o n a l l o s s e s ) (corresponding s l i p )  e a s e ; we s h a l l , h o w e v e r ,  case,  balanced  to  equations  substitute  this  (28) s t a t e s  the form o f the t o r q u e - s l i p  solve  well  Torque i s p r o p o r t i o n a l to r o t o r copper  ( p o s i t i v e sequence l o s s e s ) (corresponding s l i p ) If  o f the law which i s  . (28)  Balanced. P |  reduce  = L ™ L P 2  to  In s )  =• L  s  z  14 B  P I  E  P Z  = Rpo I i + j ^ ( L p o I P  =  +  P 1  S I  S  Bsj= 0 = . R  s o  E ',= 0 = R  s o  s  E^-'O = R Prom  I  jo>'{L Isz+ j S 2  5 |  So  we f i n d  (30),  (32),  . •  + M Ip) .  .  •  •  .  .  .  .  .  .  .  .  .  . - ( 2 9 )  (30)  » « « . . . .  .  .  .  .  .  . .  . .  .  .  .  .  .  (3D  .  (32)  .  )  s  (33)  s o  i  ra  (33)  0.  z  (331, and ( 3 4 ) ,  .  .  .  .  .  .  . . .  every  Therefore,  in  term i s z e r o , Multiplying  _  equations  leaving  (29)  b y Isi a n d a d d i n g t h e two e q u a t i o n s  (54)  (30) a n d  s  I = 0 and % PSL  .  I '.= 0 . C o m b i n i n g  a n d (31) f o r c o n s i d e r a t i o n . (3D  .  •  .  ( L • I^+-MS -) .  that  .  .  ) V  I '* 0  = 0, and from  (34),  (29)  I  .  MIsi)  - I « + J ' ^ i d s o Ia'+'-'O  S 0  (32)  .  S |  0 ~ Rpo Ip2+ j <^o (Lpo 1^+  E = 0 = R ^ I , + j <4»(Ii^o  and  MI )  by I i P  6)0  so f o r m e d , we  6)0  obtain BELIPI = R p ^ l l p i - i . R?fll?'lsi-i. J [Lfolpilpi + I»soIsiIiM* ^(isr^pi C0  CO  o  whence  t h e r e a l power  total  these  input  Rsolsi I « i » •  losses are  copper  from the i n p u t ,  the torque  and ( 3 D I = S (  ane  I , = E ,(a p  functions  » .  (36)  . .  Subtracting  •  ••  .  .  •  •  •  *  •  *  *  .  .  • •  (37)  w  '  P  RSAL.IS.  .  .  .  .  .  .  .  (38)  f o r I , , we h a v e s  - .jco.MEp. Rpo-c^co.M "* 2  s  • «  Rpalp.Ip,* R ^ L ^ I s , .  Rsolsi I s i 5  0.369 p  = 0.369  Hence  » •  we o b t a i n  OJo-CO,  (29)  is  is  T, =  Solving  to the motor  +  ^ O J ,  and  (35)  COo  a  Rpo Jpi Ipi The  Ipi Isi ^1 •  +  j b ) , and  o f the motor  = jcOoLpo  E (a + jb) P I  Is,T = E p ^ a ^ b ) , sl  constants  2  where  and the s l i p .  a and b  Unbalanced A p p l i e d  li  In this  case,  exception  v o l t a g e . A l l Impedances  equations  that  E  P  i  (29) t o (34) a p p l y ,  has a v a l u e  hut  Ipz a n d I' a r e n o t z e r o .  the  value  for positive  (37) and {38K also, and  (Epii; ) z  The c o p p e r of the  of  given by  case,  equations.  equations  a negative  F o r , from  (30)  Ep^is  R i  P D  (39)  2  a r e , f o r the negative  R Ip2lp2+ R ^ I ^ z I ^ . P0  sequence  Subtracting  components  these losses  t h e n e g a t i v e m e c h a n i c a l power.  f o r m e c h a n i c a l power W-W, + W = (OJ -6J0( 2  and  As  from  i n p u t we have  (40)  = - ^ 7 L 7 ^ ^ l as  term  - R I Tp + | £ Fsolszl^  rftjl  losses  current,  part  an  (29) a n d (31 ) we o b t a i n  i s , i n this  i n t h e power a n d t o r q u e  (34), the r e a l  =  power a n d t o r q u e  But there  w i t h the  i n (30). isz. 0* ^- ^sr" ° i  From  sz  "balanced.  T  pointed  that the  O.369 P  power  ^)IS,_  .  (41)  (Rsok.Is, ( co,  Rso&lk)  (42)  R  out b e f o r e , i s being  negative  a n a means  to'  )  the n e g a t i v e  f e d back  sequence  to t h e s u p p l y ,  into  sign  c u r r e n t does  from  equations  From  (30) a n d (34), I  as a phase  ( 4 2 ) , I may  = E (g + j f ) . P 2  power  Since back  b a l a n c e r and  simultaneously.  S i  (30) a n d (34). Isi was g i v e n S 2  circuit.  work t o f e e d  o f s u p p l y i n g m e c h a n i c a l power f o r torque,  i n (4T) indicates  the supply  t h e motor f u n c t i o n s  In the equation  expression  is  o  =  The c o m p l e t e ,  be d e t e r m i n e d  as  Therefore  E (a + j b ) . P l  since  (a* + b* )•,.•  Is.Isi = Ep,  and i ; i ; = E 2  2  P a  then  T, =. E  and  Iz.-  where  {g"  P l  Ep2  f )  ,  z  4>(s,c)  4>|(2 - s ) , e] .  s and e r e p r e s e n t  ively.  The f u n c t i o n s  so i n g e n e r a l , at  +  slip  and m o t o r c o n s t a n t s  are not  JDi i s n o t  the  equal  same,  respect-  as may "be s e e n  to E P I , a l t h o u g h  this  ,  is  true  standstill. The c u r v e s  in Fig.  5 illustrate  They show T, f o r p o s i t i v e n e g a t i v e sequence  sequence  voltage E  foregoing  voltage E  P I  , and T  . The b r o k e n l i n e  for  z  curve,  T,  is  the  E  O B  = (-0.5  - j O.8660E + ( - 0 , 3  + j o.366)(-E ) = - j  1 .732E  P  E  e  = <-0,5  + i 0.866)Ep,+  -  j 0 . 8 6 6 ) ( - E ) = +j  1.732E  P  That 2\/3  is,  E  O B  =-Eo-,  motor; It  of  the  curves the  form u s u a l l y  were  v o l t a g e has  calculated  motor.  o f the It  is  are  a more s e r i o u s  curves  = (E  P I  -  of F i g . also,  n  PI  from t e s t s  the  The i n c r e a s e d h e a t i n g  The c u r v e i s ,  on a  on t h e  This fact  is  6, c a l c u l a t e d if  applied  f o r the  alternating  by  same  current  m i g h t be  far  unbalanced.  t h e m o t o r on l o a d i s  i m p o r t a n t c o n s i d e r a t i o n t h a n the  torque  illustrated  a p p l i e d v o l t a g e were of  operatio  section.  starting  the r e s u l t s  of  three-phase  any u n b a l a n c i n g o f  were u s e d f o r b r a k i n g p u r p o s e s , if  voltage  shown f o r s i n g l e - p h a s e  that  ~0,  E )  P(  effect  torque.  evident,  from s a t i s f a c t o r y  A  shown i n a l a t e r  be o b s e r v e d t h a t  t h a n o n the p u l l - o u t means  O  t e r m i n a l s B and C .  calculations  will  (-0.5  E  c o r r e s p o n d i n g to a s i n g l e - p h a s e  p  course,  P I  pl  E , a l l i e d between  These  E  P  , so t h a t  clearly.  for  c  c a s e when E £ = -  P 2  the  torque r e d u c t i o n .  a more The  value  17 of  the negative  sequence  current flowing  by m u l t i p l y i n g t h e n e g a t i v e still  admittance  range  to negative  sequence  negative  sequence  factor current torque  equal  to f u l l  degree  characteristics  equations zero  that rotor  are very  Secondary  (10) t o  for this  I' = 0 f r o m SI  case,  we f i n d  i s not sensitive but t h e i r  T = 0.369 p  to a  heating  to such unbalance.  Primary  Impedances  Balanced. Referring to  that  S |  , and L  I .= 0 from sz  ; or i n other words,  i s g i v e n by equation  about  sequence  (2 8) where  will  in.the  the r o t o r  have d i f f e r e n t  5 1  (13) a n d  ( 1 4 ) . That i s , the o n l y c u r r e n t s  but, i n general,  the  ( i . e . an unbalance  S I  s  The t o r q u e  currents  frequencies.  I^-l/.^O;  that  Rs*ISI Isi — Rsol CO,  It  i s worthy o f n o t i c e  E  , s and c o n l y ,  P (  sensitive  (43)  CO'  that  but also  to  Thus, on l o a d , t h e  unbalance,  Impedances  amounted t o  (15), i n w h i c h R , R^, L  a r e I , and  are balanced,  is,  of voltage  Unbalanced A p p l i e d V o l t a g e .  Unbalanced.  be  load current.  have a n  m o t o r s were  a negative  o f o r d i n a r y i n d u c t i o n motors  moderate  8.  component,  o f 0.15) t h e r e w o u l d r e s u l t  6 to 8  them s u c h t h a t  of voltage  sequence  6  I f these  speed  impressed  cause  That i s , they  i m p r e s s e d on  component  •1 5T o f t h e p o s i t i v e  voltage  when l o c k e d , w i l l  o f 6 t o 8 a t 100?& s l i p .  have u n b a l a n c e d v o l t a g e s  by the s t a n d -  over t h e motor  currents). Full  f u l l - l o a d c u r r e n t to flow.  admittance  voltage  (considered constant  on motors o f normal d e s i g n , times  sequence  c a n he o b t a i n e d  I ( i s no l o n g e r a f u n c t i o n o f 3  o f E _ ; and s i m i l a r l y I' . i s P2  S7  18 now a f u n c t i o n o f E , , as w e l l  as  p  may be  seen from e q u a t i o n s  necessary  equations  Epj — Zpolp, +  Zi I _—  E p = Zp.Ip,*  Zpolpit  pt  2  0 =-M,I , +  0  P  0=  0  P2  M  + jto.Lg, Zi -  S  Z  0  P i  0  Zpz  E  Zpo  Ep^  -  •  .  M.'IU  .  .  .  .  Z  PO  R  we  . . .  s and c . for,  + j  P I  writing  the  form, .  •  .  •  *  .  .  .  .  (44)  .  (45)  a>oL  ,  PI  R P2.  Z  + jco'L, , M - - J « * M  .  .  (46) .  (47)  + «j <A>I< . » P2  , M, ~ - j c o . M ,  0  s  This  « . . . . . . . . . .  obtain where A =  -Mo  0  ,  2  . . . . . . . . . . . .  = R  P (  0  P 1  0 , 0  -M,  •  + jco L ,  and M ' = - j c o ' M , 2p  .  0  P  (15);  following  + ZUlz  0  to  0  S[  0  Z =R S  -  s  Z  P O  +  + Z I +  where  ~ R  i(  0  - M' 1^+ p o  i n the  I  0  {10)  of E  ZPO  Zpz - M  Zpi  Zpo  0  0  Z  -M'  0  0  -Jtt,  2s  0  0  0  -M' 0  £  A  I' =  and  Z pi -M,  0  TP  Zpj,  -M  Zpo  0  0  z,  -^pi  D  »  0  0 ,  0  -M'  4\  Evaluating, I =  -EP,M, (MoM' -  Sf  (MoM' I«=  This  (M M'0  is  0  Z  P 0  Zi ) ( M M 0  (  a  Z  P 0  Z  produce current o f  Z  P 0  £  )  .  s  in gernral,  .  .  .  .  .  .  (49)  Zp.Z^Zs Z^  -  (50)  Z  s  )  -  Z  M  Zp,.Z  components,  sequence  a vlotage  s  Z£  statement  an a p p l i e d v o l t a g e that  .  ZreZs )  w i t h the  on s y m m e t r i c a l  balanced network,  network,  EP,M, Z o Z '  EraM' (MOM, -  i n agreement  treatises  ) -  ZpoZi ) ( M M , -  -EP.M'ZP,ZS:  ZfoZs  commonly made  namely,  that  for  o f one sequence  only,  but  component  of  in a will  for an unbalanced either  sequence  19 will 9.  give r i s e  t o c u r r e n t components  Unbala.nr>.flfl A p p l i e d V o l t a g e .  Balanced,  secondary  Primary  Pz  currents is  I  P I  , I , I i, I P2  S  5 2 )  s  It is,  s h o u l d be noted  i n a l l cases,  torque  torque that  value.  the r o t o r constants  are u n -  balanced.  One may t h i n k o f a component  due t o t h e u n b a l a n c e ,  unbalancedi. the  i n this  There i s a g r a d u a l  one p h a s e  60 c y c l e ,  lOh.p.,  Tests.  ondary.  The c o n s t a n t s ,  values,  were p  co Lp 0  =  -Blocked torque  currents are  i n the i n t e n s i t y  balance  of  to the c o n d i t i o n  open-circuited.  The t e s t s were  c a r r i e d out on a  p r i m a r y and s t a r - c o n n e c t e d  reduced to equivalent  0.146  =11.1  OJ.M. =  Ueither  6pole, wound r o t o r i n d u c t i o n m o t o r ,  having a delta-connected  R  single-phase  connection.  increase  o f the r o t o r i s  10. B l o c k e d t o r q u e  of  when t h e s e c o n d a r y  hum, farom t h e c a s e o f p e r f e c t  where  the  are u n -  that  t h e m o t o r be a s q u i e t  In r e a l i t y ,  the r o t o r c u r r e n t s  balanced,  will  (28).  t h e t o r q u e w h i c h we s p e a k o f  i s p u l s a t i n g whenever  torquej  case  values i and the torque  equation,  the e f f e c t i v e  i s , whenever  In this  (10) t o (1.5), we s e e t h a t  I ,, Isz a l l h a v e  g i v e n by the g e n e r a l  sequences.  Impedances  Imparl amies U n b a l a n c e d .  R = R = Lp,^ L p ^ 0. From e q u a t i o n s P|  of both  6.45 t e s t s were  R  5  =  0.275  oj L  s  =  4.67  0  sec-  line-to-neutra  (co =377) 0  taken w i t h balanced  voltage  a p p l i e d and t h e p r i m a r y c i r c u i t u n b a l a n c e d b y added istance. and  The c u r r e n t s were  (50),  voltage,  which,  0  Z Z )  0  I',=  -  r o  s  0  results  ; -  Zp.z^z;  -  Zp.Zp.ZsZi  were  as  .  follows:  A. B  . 2.03 2.03  0  126  0  A B c  1.55 0  The  Z * 1 .12+J11 .1  30.9 5.54 4.48  ft  Zpr 0 2 +J0.45 Zpz-0.2  0  3.72 5.27 V*  r  Z =0.13-j0.26 Pi  A B C  4.78  37.5  0  Z -3.52+jl1.1 z =0.7 t j L 5 4 z ^ o . 7 -j1.54  Z B C  5,w2  Z „ 2 . 34 + 31 T . t  3.37 0  agreement.  calculated values  makes  33.2  Pi  31 . 2 13.1  5.34  F3  3.25  P  PO  13.7  5.5  P1  r  The n e g l e c t  4 3 . 2 13.7. 8.13  -  Zfi-o.51+31. Z^Oi5T-3l.  7.78  I  values  f o r torque are i n f a i r l y  of iron losses  h i g h , but t h i s  t e n d t o make t h e  i s compensated  f o r , to  by the s a t u r a t i o n a t h i g h e r v o l t a g e s ,  the a c t u a l  4.62  -jO.45  Z -4,.5 +J11.1 Z ,-1. +J1.93 Z»*-1. -31.93  some e x t e n t ,  5.25  9  o b s e r v e d and c a l c u l a t e d  close  3.42  10.  6.34 6.71 0  0  114  28.4  , d-i'8j" Z ^ 0 . 7 3 + j 1 1 . 1 0.91 Z -0.13+j0.26  A B  126  ^ 1 .5+31 L I 2W0; 34+jO; 58  Zpf0.34-j0.58  1 .37  A B C  Torque l b s . f t . Calc. Obs.  Current i n fflec. Is.  E x t e r n a l • P r i m a r y Impedance Volts Applied Line Res. Components  61.3  applied  ,  SKMOZPTZ*  (M M, r Z^Zs )  62.3  (49)  reduce to SI  62  from equations  f o r s t a n d s t i l l and b a l a n c e d  I = - E»,K (M M,-  the  calculated  res-  currents greater  than those  which  calculated.  FIGURE  5  T O R Q U E - S U P CURVES Showing t h e e f f e c t o f u n b a l a n c e d v o l + a g e on. balanced machiae. .  TORQUE-SLIP CURVES S h o w i n g e f f e c t of- u n b a l a n c e d V o l t a g e on a m o t o r w i t h unbalanced primary.  • 21 11.  Calculations  f o r s l i p - t o r q u e curves  having unbalanced a p p l i e d v o l t a g e . I n (49)  and  (50)  reduce  this  case  equations  to  - M i BPI •'• • •. M M - Z Z  I= ?  6  (  -  I' =  M  w h i c h becomej  f o r b a l a n c e d motor  M' Q  P4  S  EP*  ,  M ' . z ^ z ;  f o r the p a r t i c u l a r motor under c o n s i d e r a t i o n ,  I « - 0.62/266° 15' s B p , 0.29 5/266° 1 5* + s  i ; = 0.62726g_1_j ( 2 - s ) 0.293/266° 1 5' The d a t a f o r t h e  curves  0.1 18.65 0.2 . 28. 0.5 30. 26.4 0;5, 0.7 \ 2 1 . 8 t.o 16.6  13.3 11 i 6  1.5 1.7 1.8  Calculations  in Fig.  for slip-torque  were c a l c u l a t e d  The r e s u l t s  a r e as  s  Isi  ~0~  0  0.T 0.2 0;5  0.5 0;7 1 .0  giving  0i254E 0;307E  P |  P(  0.409E  the  P1  P 1  prim-  The c u r r e n t and t o r q u e (49),  (50)  and  Is* •  '  0i423E 0 433E  PZ  i  0.429^  6 (  and I'  P t  P 2  +  0i324E  P j  + 0.279E 4 0;2.*3E  n  0.432E  .  P| P|  +O0.217E,,  + 0.t87EP,  0-.422Ep«. + 0.1 7 E 0.409Ep + 0.105E  + 0.102-E« + 0.105 Epj I  P z  P z  0,A31^  + 0. 0 7 T E p 2 + 0.08t E + 0.097 Ew  currents  unbalanced  follows:  P z  P I  0.36lE 0.386E  curves,  from e q u a t i o n s  0.t62E , + 0 i 0 4 5 5 E p  T = K' Ep, j : .  5.  unbalanced a p p l i e d voltage.  values  defined,by  10i3 9-? 9.4-  1.9  They a r e shown p l o t t e d  ary,  K  1 ;3  :  12.  are s  z  s  <2-s)  P I  Z  st  i n terms o f t h e  P|  voltage  (430.  22 components.  The s l i p - t o r q u e  t a i n e d by s u b s t i t u t i n g  13.  Conclusion.  its  application,  its  algebrisation  account  the  being to  effect  the  squirrel-cage  phase  the  utility  network  o f the  slot of  the  quite  does n o t  , such  paper.  ob-  (43).  general  take  It  in  the  into  o r the modconsiderations  c o u l d be  by u s i n g an e q u i v a l e n t  applied  three-  circuit.  the w r i t e r wishes  and o f  Department  It  ratios  this  rotor  assistance  laboratory,  i n equation  of unsymmetrical windings,  machines  for  6 were  b e i n g l i m i t e d o n l y by  involved.  scope  In c o n c l u s i o n , iation  these values  i n t r o d u c e d by  outside  of F i g .  The method o u t l i n e d i s  tedious  ifications  curves  to  express  his  apprec-  g i v e n by P r o f , W . B . C o u l t h a r d ,  the  advice  of  the  of  The U n i v e r s i t y o f B r i t i s h  in  o f D r . H . T i c k e r s , Head  o f M e c h a n i c a l and E l e c t r i c a l Columbia.  Engineering  Appendix. Brief  Statements  o f some o f t h e More I m p o r t a n t  Theorems  :  Used i n the A p p l i c a t i o n o f the Method o f  Symmetrical  Components. 1.  Any u n b a l a n c e d  system  of polyphase  may he r e p r e s e n t e d b y two and n e g a t i v e p h a s e or zero  sequence  three voltage vectors  E  = 1,  E  O A  E  o e  = E + a E,+• a Ea.  E  o c  = E 4 a E, 4 a E  O B  and a r e s i d u a l  a n d Eo  C  are  the system,  = E + E, 4 E ^ Q  2  0  2  Q  0  z  and a i s  a = a,  E  positive  by  2  symmetrical  the v e c t o r  _ 1  +  2 a  O A )  of  voltages  i n an unbalanced three-phase  E , E , , and E a r e the  voltage,  systems  respectively,  Thus, i f  t h e y may be r e p r e s e n t e d  where  symmetrical  sequence,  vector.  currents or  ar = a ,  ar - a ,  j/1  components ,  ( a -  of i  z  2  _ j  ^ )  "* 2  2  etc.  Conversely: Eo = i ( E  o f t +  E ^  E, - i (E +  E j  a V « f V  at)  E i « 1 ( E + a E ^ a JUj M  Exactly  similar  an unbalanced  system.  of l i n e voltage, isolated.  equations ,  may be w r i t t e n  There  or of l i n e  i s no z e r o hhase  c u r r e n t where  T h e r e may be z e r o phase  neutral voltage,  for currents  component  the n e u t r a l  components  or of line-to-ftinfc. current..  in  of  is  line-to-  2.  Any p o l y p h a s e  he r e s o l v e d  s y s t e m o f a s y m m e t r i c a l impedances may  into  two s y m m e t r i a a l s y s t e m s  and a r e s i d u a l  vector; Z - Z +  Z, +• Z  Z = Z +  a*Z, 4 a Z  A  0  B  0  z  Z = Z 4 a Z, 4 a Z c  and  z  0  Z=  1(Z 4  o  Z  A  Z. = 1 ( Z  A  B  4 a  x  a  Z )  +  c  Z  &  +  a  z  Z ) t  3 Z - 1(Z 4 z  a Z 4 a z  A  6  Z ) c  3 3.  In writing voltage  symmetrical  components; the  t e r m must be  constant,  E =Z I 4  Z I , 4 Z, I  E , = Z, I 4  Z I, 4 Z , I  2  Z, I , 4 Z I  2  0  0  0  a  •B - Z I 2  2  0  +  2  e  6  sum o f  or d i f f e r  .  Z,,  an unbalanced network,  F o r an u n b a l a n c e d t h r e e phase  in  terms o f the  I  c  6.  ,  , -T-i  4.  2,2,2,  2.  o  r  that  e.g.:  are wrong.  o f one  sequence  phase  only.  o f any  I  2  ) ,  where  For  sequence  system,  t h e power i s  s y m m e t r i c a l components, by the r e a l Z  each  sequenced.  5.  E , S, + E  1  E = I Z -;f e t c . , e  3, 3.  1 ,  of  of  0, 3,  an\^ a p p l i e d v o l t a g e  may p r o d u c e c u r r e n t s o f a l l  0  subscripts  applied voltages  s e q u e n c e may p r o d u c e c u r r e n t s o f  0  ..  »  as E , - 1 ,  i n terms  by m u l t i p l e s o f  «  For a balanced network,  3(E I +  the  subscripts B.  2  Thus s u c h e q u a t i o n s 4.  or current equations  I o j l , and  I  z  are  the  given  part  of  conjugates  of  I | » I-z r e s p e c t i v e l y . Terms o f  etc.i  the  form  AA , AB 4 BA , ABC 4 ABC , ABC 4 ABC  are p u r e l y r e a l *  terms o f  the  complex  as may.be v e r i f i e d b y e x p a n d i n g  quantities.  in  25 7. R  0  For a three-phase ?  R,, R  loss  Is  s y s t e m h a v i n g component  , a n d component  2  g i v e n by  currents  I,  %  ± the  3 ( R I , I + R ^ . I I + R, I, I + R I 0  t  2  Z  The atoove a r e mere s t a t e m e n t s ; ment and p r o o f s ,  I  resistances,  the r e a d e r  is  2  z  resistance z  I, ).  for a logical  r e f e r r e d to  the  develop-  literature.  BIBLIOGRAPHY FortescuB  Symmetrical Coordinates  Trans. 1913  Wagner and E v a n s  S y m m e t r i c a l Components Westinghouse  p.  A.I.E.S.  1027  publication  E l e c t r i c & Mfg.  Co.  of  

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