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The application of Lie derivatives in Lagrangian mechanics for the development of a general holonomic… Gustafson, Ture Kenneth 1964

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T H E ;  T  H  E  D  A  P  P  L  I  C  A  T  I  O  E  V  E  L  O  P  M  E  N  T  N  , O P  O  F  L  A  I  G  E  E  N  D  E  R  I  E  R  A  L  V  A  T  I  V  E  S  H  O  L  O  N  O  I  M  I  N  C  L  T  A  G  R  A  N  H  E  O  R  Y  G  I  O  A  N  P  E  M  E  C  H  A  N  I  L  E  C  T  R  I  C  by  TURE  B«AoSeo,  A  THESIS THE  KENNETH  GUSTAFSON  University  of  B r i t i s h  Columbia?  1963  SUBMITTED  IN  PARTIAL  FULFILMENT  OF  REQUIREMENTS MASTER  FOR  THE DEGREE  OF A P P L I E D  in  the  OF  -  SCIENCE  Department  of  E l e c t r i c a l ¥e  accept  required  THE  this  thesis  Engineering a-.s  conforming  to  the  standard  U N I V E R S I T Y OF B R I T I S H  August,  1964  COLUMBIA  C  S  M  A  F  O  R  C  H  I  N  E  S  In the  r e q u i r e m e n t s f o r an  British  mission  for' reference  for extensive  p u r p o s e s may  be  advanced  of  and  the  Library  study,  written  Department  of  the  the  Head o f my  I t i s understood  fulfilment  University  agree that for  The U n i v e r s i t y of B r i t i s h V a n c o u v e r 8, Canada  Department  ffi j f t f c f r  or  t h a t c o p y i n g or r  s h a l l not  /^o^J'/nccJ\JUv\ ^  Columbia,  of •  per-  scholarly  permission*  ^JJL^JAJL^^ &  of  s h a l l make i t f r e e l y  I further  t h i s thesis for f i n a n c i a l gain  w i t h o u t my  Date  degree at  copying of t h i s t h e s i s  g r a n t e d by  representatives.  cation  this thesis i n partial  Columbia, I agree that  available  his  presenting  be  by publi-  allowed  ABSTRACT  A general  a p p r o a c h to the  systems i s d e v e l o p e d . metrical Using the  ideas  are  Tensor concepts  avoided  L i e d e r i v a t i v e s and  equations  resulting  system o f m a c h i n e s . electrical  treatment  i n favour choosing  are  and  are  adopted;  of H a m i l t o n ' s  a holonomic  g e n e r a l , and  These e q u a t i o n s  portion  of e l e c t r i c a l  thus  are  however, Principle.  reference apply  f o r the  system,  to any  F a r a d a y ' s Law  a gradient equation  machine  physical  for  the  mechanical  portion. Transformation  c h a r a c t e r i s t i c s , w h i c h are  independent types,  called  gated.  to t e n s o r  This leads  associated with The general any  equations  equations  and  character  the  and  i-type,are  t o be  of  two  investi-  invariance properties  transformations. of  small o s c i l l a t i o n ,  of motion obtained  i n the  w h i c h a r e b a s e d on thesis,  are  the  derived for  p h y s i c a l system. I n the  the  the  the v - t y p e  found  final  chapter  power s e l s y n system, and  two  examples of a p p l i c a t i o n a r e  the  ii  synchronous machine.  given;  ACKNOWLEDGEMENT  Many discussions expresses  ideas  the  association  with  Thanks of  a  to  Department  author  Mr,  bursary  him at  was  is  are  presented  supervising  thanks  encouragement The  award  the  with  his  Engineering whose  of  due i n  and  The  H,  the  thesis  professor, other  grateful  Dr.  members  University  gratefully  also  i n  of  arose l u .  of  the  B r i t i s h  out  The  of  author  E l e c t r i c a l Columbia,  appreciated. for  his  stimulating  George.  to  the  National  1963.  v i i  Research  Council  for  their  TABLE OP  CONTENTS Page  List  of I l l u s t r a t i o n s  v  Acknowledgement  viii  1.  Introduction  2a  The B a s i c Machine E q u a t i o n s f o r P h y s i c a l C o o r d i n a t e System 2.1  2.2 2.3 2.4 3.  4.  5.  Any 3  The T r a n s f o r m a t i o n of the Machine Equations from a S l i p - R i n g Coordinate System t o Any G e n e r a l P h y s i c a l C o o r d i n a t e System The E x c i t a t i o n and a C o i l Winding The L a g r a n g i a n of M o t i o n  Induction Angles  F u n c t i o n and  the  17  of M o t i o n  i n Machine  and  System  i n True  Coordinate 25  T r a n s f o r m a t i o n Theory  3.3  Tensors  3.4  E x t e n s i o n of the E q u a t i o n s of M o t i o n Machines w i t h R o t a t i n g S a l i e n c y .  27  i n the E q u a t i o n s  of S m a l l  33  of M o t i o n to  36 39  Oscillations  4.1  The V o l t a g e E q u a t i o n s  4.2  The  Mechanical  4„3  The  Combined E q u a t i o n s  Applications Equations  22  25  Loop E q u a t i o n s Systems  Equations  of  Equations  3.2  The  3  10  Pertinent Equations, D e f i n i t i o n s , Notations  The E q u a t i o n s Analysis 3.1  1  39  Equations  of O s c i l l a t i o n  of M o t i o n  t o t h e D e r i v a t i o n of Machine  42 46  System 48  5.1  The  Power S e l s y n System  5.2  Synchronous Machines iii  48 56  6.  Conclusion  Appendix  I  Conditions Guarantee  Appendix  II  The  on an  the  Flux  Tensor  Extremum  Hamiltonian;  Energy  Energy Reference s  iv  and  Co  L I S T OP  ILLUSTRATIONS  Figure 2A  Page The D e t e r m i n a t i o n o f the V o l t a g e i n an m - c o i l by a T r a n s f o r m a t i o n f r o m a F i x e d a-(i System, F i x e d on the R o t o r  4  2B  I n c r e m e n t a l F l u x Changes  10  2C  Virtual  11  2D  Two  General  2E  The  Induction  2F  Virtual  3A  Polarity  4A  Two  5A  Selsyn  5B  Two-Phase S e l s y n U n i t  52  5C  S y n c h r o n o u s Machine  56  5D  (d-q)  57  IIA  S i n g l e C o i l B-H  Rotation Machine C o i l W i n d i n g s Angle  17  Rotations  21  Conventions  Phase I n d u c t i o n  26 Motor  System  -  15  44 48  Reference C o n f i g u r a t i o n Curve  71  v  L I S T OP  TABLES Pages  1  Tensor  Quantities  i n the E q u a t i o n s  2  Comparison of C o n v e n t i o n a l Present Thesis  vi  Ideas  of Motion  w i t h Those of  34 the 68  THE THE  APPLICATION OP L I E DERIVATIVES IN LAGRANGIAN MECHANICS FOR DEVELOPMENT OF A GENERAL HOLONOMIC THEORY OF ELECTRIC MACHINES  1.  There the of  theory  still  INTRODUCTION  r e m a i n . many i m p o r t a n t  of e l e c t r i c  and e s p e c i a l l y  those  the optimal  analyses  An  extension  of s t a t i c (T)  circuit "  a means f o r a g e n e r a l  extension coil the  circuits  o r due t o  terms o f c u r r e n t s o r o f c u r r e n t and s p e e d .  '  m a c h i n e r y by G« K r o n , e m p l o y i n g m a t r i x provides  control  involving non-linear  e f f e c t s due t o s a t u r a t i o n i n t h e m a g n e t i c cross-product  problems i n  m a c h i n e s and power s y s t e m s ; t h e a n a l y s i s  machine and power system s t a b i l i t y ,  criteria,  unsolved  theory  to r o t a t i n g  and t e n s o r a n a l y s i s ,  a n a l y s i s approach.  deals p r i n c i p a l l y with  This  the a d d i t i o n of the r o t a t i n g  as a g e n e r a l i z e d i n d u c t i v e c i r c u i t  element,,  However,  i n c l u s i o n "of m e c h a n i c a l m o t i o n i n an e l e c t r i c a l  introduces  an u n p r o p o r t i o n a l  amount o f d i f f i c u l t y ,  circuit mainly  because o f t h e n o n - l i n e a r i t y r e s u l t i n g , , Kron's a d o p t i o n  of tensors  g e n e r a l i z a t i o n and u n i f i c a t i o n his  helped  achieved  method o f a t t a c k was c o m p l i c a t e d  non-holonomiticity,  i t allowed  him t o d e a l w i t h  p r o b l e m s i n a s e t manner,.  to  power system dynamics would r e a d i l y  This$  of n o n - l i n e a r i t i e s  thereby*  Even  though  by t h e i n t r o d u c t i o n o f  of  inclusion  because of the  a general  class  I t a p p e a r e d as t h o u g h an e x t e n s i o n f o l l o w as w e l l as an  c a u s e d by s a t u r a t i o n and h y s t e r e s i s *  however, d i d n o t o c c u r * The  main b a r r i e r  t o f u r t h e r development l i e s  i n the  2 concept  o f n o n - h o l o n o m i t i c i t y and i t s i m p l i c a t i o n s .  to K r o n , F a r a d a y ' s  Law i s a p p l i c a b l e s o l e l y  to s l i p  According ring  (6)  machines. because  F o r commutator m a c h i n e s an a d d i t i o n a l  applicable  This a l s o i m p l i e s that Lagrange's equations are only f o r these  special  cases  more g e n e r a l Boltzmann-Hamel e q u a t i o n s commutator m a c h i n e s .  This i s a very  means t h a t n e i t h e r L a g r a n g i a n  can be a p p l i e d d i r e c t l y involving  other  than  to s t a b i l i t y  slip  ring  (slip-ring being  frames), the  needed f o r  limiting  restriction for  nor Hamiltonian  mechanics  or o p t i m i z a t i o n problems  machines.  I n the f o l l o w i n g c h a p t e r , employed t o o b t a i n t h e e q u a t i o n s  Kron's equations of motion i n a  are i n i t i a l l y  slip-ring  frame»  By d e f i n i n g a t r a n s f o r m a t i o n f r o m t h i s  general  p h y s i c a l f r a m e , i t i s shown t h a t t h e e q u a t i o n s  motion i n the l a t t e r  remain of the holonomic  f o l l o w e d by a d e r i v a t i o n  of the general  using Hamilton's P r i n c i p l e . s e l e c t e d are r e a d i l y  frame t o any  form.  equations  This i s of motion  seen t o be i n d e p e n d e n t p r o v i n g  In the succeeding a n a l y s i s , L i e deriva/tive  electro-mechanical  chapters, concepts  loop  equations  t h a t any system. i n machine  of t r a n s f o r m a t i o n t h e o r y , the  o f t e n s o r s , and t h e h u n t i n g  Examples o f a p p l i c a t i o n  of  The g e n e r a l i z e d c o o r d i n a t e s  p h y s i c a l machine i s a h o l o n o m i c  concept  enters  o f t h e i n d e p e n d e n c e o f t h e r e f e r e n c e frame and c o i l  velocities.  it  term  equations  are considered.  are i n c l u d e d i n the f i n a l  chapter.  THE  BASIC MACHINE EQUATIONS FOR ANT PHYSICAL COORDINATE SYSTEM  2.1  The T r a n s f o r m a t i o n o f t h e Machine E q u a t i o n s From a S l i p R i n g C o o r d i n a t e System t o Any G e n e r a l P h y s i c a l C o o r d i n a t e System  In motion  t h e g e n e r a l case K r o n o b t a i n e d as t h e e q u a t i o n s o f  f o r an e l e c t r i c a l  machine, a v a r i a t i o n  of the Boltzmann—  (9)  Hamel  equation,  where R^t* a  y  o  c  i s the r e s i s t a n c e i s the inductance  and i n e r t i a  . i s the C h r i s t o f e l and from  C  and i n e r t i a l  Symbol  damping  tensor  tensor  i n a non-Riemannian  space  , the c o o r d i n a t e t r a n s f o r m a t i o n matrix a holonomic  frame t o a g e n e r a l frame o f  reference. The  last  term  the r o t a t i o n The  o f (2-1) t a k e s of conductors  mechanical  Consider by  the i n f l u e n c e  coil  into  and c o o r d i n a t e r e f e r e n c e  equations  t h e case  c o n s i d e r a t i o n s u c h e f f e c t s as  are i n c l u d e d i n (2-1).  of a c o i l i n which a v o l t a g e i s induced  o f an e x c i t a t i o n a p p l i e d  as shown i n F i g . 2A.  frames.  to the " e x c i t a t i o n "  L e t m be t h e i n d u c t i o n  coil  (coil i n  4  which a v o l t a g e let  p© , m  P© » n  i s i n d u c e d ) and  and  p©,  respectively  commutator a x i s ,  the  the  rotor,  speed o f the  a—p  excitation  be,  the  coil.  speed of the  Also m-coil  speed o f t h e n - c o i l commutator a x i s , a l l being  t o d e t e r m i n e the v o l t a g e the  n the  independent.  i n the m — c o i l  system, w h i c h i s f i x e d to the  by  and  I t i s proposed  a transformation  from  rotor.  I 4  Fig.  2A  The D e t e r m i n a t i o n o f the V o l t a g e i n an m - C o i l by T r a n s f o r m a t i o n From a F i x e d oc-g System, F i x e d on Rotor  The  instantaneous  the n — c o i l i s i n d e p e n d e n t n-coil  i s referred.  angular  speed of the  T h i s i m p l i e s t h a t the situated  i n an  e x c i t a t i o n e f f e c t i n the o f the  coordinate  I t depends o n l y on excitation excitation  instantaneous  axis  the  m - c o i l due  frame t o w h i c h angular  a the  to  the  position  and  (n-coil  commutator  axis).  can be  assumed to  be  coil  holonomic  reference  frame.  The  equations  of  motion  i n the  oc-|3 f r a m e ,  which  is  holonomic^ are  e,  From  =  this  R  S  = =  since  t  i *  -"t  S t  R  t  +  a , * ! ! *  the voltage  r> e  t  R  the voltage  s n  ,  1  +  i  L  [ocf3,Tf] i ~ i *  equations •  T  s t  + L  +  dt  S  i n either  r x  L  +  ^  n  axis  i n the  t u  -  . t  »s]  i  (2-2)  a  and p axes  +  - g - ^ pO l  are  . U  i  +  Q-Q— p O l  (s  =  i s  due to  cc,p)  ,  (2-3)  the  excitation  of  (n) the  n - c o i l .  Applying  sinusoidal  flux =  LJL^  Yu's general  variation,  that  I / i i cos  Q c o s  L * ^cos  0  formula  f o r  i s  O " +  X  inductance  i A j ^ s i n  1  0 * s i n 9  (2-4)  4  gives L L  Assuming  K  p  n  n  = =  L i ^  y «  e  n  =  ot =  n  cos  0  n  c o s ( © + 9 0 ° ) c o s  L*js  =  n  L  ^«rv i^ :  + i A ^ s i n  0  n  +  IA  p n  0  s i n ©  n  sin (9+90° )s i n 9  .  a n d Ls.^„ =  Ifl.^ =  +  ( L j cos  cos  +  (-Lj s i n 9  cos  9  n  +  L<^ c o s  +  (-L j cos  s i n 9  n  +  L ^ s i n 9 c o s  d  9  9  L ^ , from  n  9"  +  (2-4)  s i n 9  s i n 9  9 ) n  s i n  9 )p9 n  9  l r ,  )p9  di e  p  =  R ^  n  i  t  t  +  (-L j s i n 9  cos  9 +  +  ( - L j c o s  cos  9  +  ( L j s i n 9  s i n 9  n  9  L ^ cos  n  -  n  +  9  Lq_sin L ^ cos  s i n © 9  9  n  ) ^ ~  n  s i n 9 )p9 n  cos  9  n  )p9  (2-5)  n  i  n  The  voltage  i n the m - c o i l  can be  obtained  from  the  transformation e =  [cos(Q m  w  0 ) , s i n (©*""- ©f| e  Using  (2=5), t h i s  P  gives di ©*)£-{— n  e  m  =  +  + (-L + (-L This i s  e  If 0  m  a  sin ©  d  cos  ( L 4 cos © " c o s  +  cos © " +  m  e sin w  cos  + L  ©*  ©  © +  m  -  m  +  mn  considered  the  mn  L  tt i  as  the  angle  i  © )p© i n  n  n  r i  .  m —  a  rr»n  1  suggests  "  n  i n d u c t i o n coil» coil  the  possibility  of a s s o c i a t i n g  e a c h coiL :. t h e r e b y  g e n e r a l machine e q u a t i o n s  to a holonomic form.  o f t h i s would be previously  ?Sf  a simplification  carried  out  p r i m i t i v e machines.  i n the  i s immersed  of the  reducing The  two  the  consequence  t e d i o u s work t h a t  form of t r a n s f o r m a t i o n s  was  from  Indeed, the b a s i c p r i m i t i v e machines  d i s c a r d e d o r r e p l a c e d by  this  . ^ d t»  mechanical v a r i a b l e s with  by Dro  )p©  n  of motion.  in  J)  a  This r e s u l t  of the  space i n which t h i s  holonomic e q u a t i o n  e  de  P  dt  speed o f the  becomes the  be  m  sin ©  m  sin © c o s  a  L,, s i n © s i n  m  simply  i s  p©™  ) i  m  a more u s e f u l model as p o i n t e d  could out  IU«  A  similar  transformation  of t h e  torque  equation  to  general  7 coordinates torque  will  equation  now be i n the  carried out. ot-|3 f r a m e  (jxp,*]i*i* where  a and p r e f e r  a mechanical negative  of  to  I n most the  cases  rotor.  Expanding  it  is  Thus x  the  two  u  u  - 2  5 x  desired =  to  0 i n any  summations  From  2A.  coordinates this  torque  u  consider  1  1  find  2 ' d6  (2-3)  and u r e f e r s  term by - t , u  along  the  u  to  the  axis;  e  the  torque  reference  in t  the  i ^ i "  Signifying  electromagnetic  t  Fig.  [ap,u]  =  electrical  coordinate. the  of  First,  for  frame  Fig.  associated  with  and  2A  de  oe  a  (2-6) since and  Resolving  i ^ A k * * d e  2  both  i™ and i  (Fig.  n  2A)  into  oc-p  components  gives  _  i  cc ooss ( 0 -  ©)  j  cos(©" -  ©)  i  s i n .(©*" -  ©)  ,  sin(©  ©)  M  n  -  (2-7)  8 The  i n d u c t a n c e s and t h e i r  L  d e r i v a t i v e s are  = L <) cos © +  sin ©  = -L«j cos 9 s i n © + L<^ s i n © cos © L p j j = L,j s m 3^*  =  °^^=-  d  e  0 + L<^cos 0  -L*(cos*© - s i n © ) + L (cos*© 2  <L  2(L  sin*©)  s i n © cos 0 - L ^ c o s © s i n © )  d  = 2 (-La  cos © s i n © + L^_ s i n 0 cos  0)  The t o r q u e t i n the oc-(3 frame i s t  =  (-L  + (-L + (L  a  cos 0 s i n © +  d  s i n 0 cos © )  c o s 2© + L ^ o s  a  20) i  s i n 20 - L«^ s i n 20) i  Using the t r a n s f o r m a t i o n e q u a t i o n s  i^i"*  i*  K  i*  p  (2-7)  i«*i« = c o s ( © - © ) i " i + 2 c o s ( 0 - 0 ) c o s ( © " a  m  + c o s i*i*  a  ( © ©  = cos(© - O ) m  ,  h  w  )  m  i i n  sin(©  - ©)i i m  w  sin (©^£  Hence  m  - 0)) i i * m  m  h  h  n  h  '  0  © ) s i n ( © - ©) h  +  - 0)  i i * n  ©H^i™ + 2 s i n ( © - © ) s i n ( 0  + sin (©" - ©)i i a  + (cos(© -  m  + cos(0* - 0 ) s i n ( 0 i*i*=  w  n  w  + cos(©" - © ) s i n ( 0  ©)i i  n  - ©)i i" w  9  t  =  ( L  s i n 0 cos 0 - L  a  +  (cos*©  s i n 0 cos 0 ) ( i ' i  <l  -' s i n * © ) ( L  B  - i - i * )  - L^) i ^ i *  a  But  i^  i*** I I  -  -2(cos(2©  (sin*{Q -  =  ©) -  m  - ©*"- © ) ) i * i h  +  h  Q))i*i*  cos*(9 w  (sin (© 2  - ©) -  h  cos (9 a  -  n  ©))i i» h  Thus  t  =  (L  d  - L  =  (L  4  - L )  l  ) (  s  i  n  2  s  i  n  2  °  2 2  n  -  G  2  a  cos 2(©  -  Expanding  <  i-i--)  (-cos2(0  - ©)i i ) n  s i n (©•"" + © * -  the double angle  -  n  C  O  S  r n  2  -  r n  n  arguments  cos  ©)i i m  2©(i i*)) a e  -  n  ( s i n2(©  Q  2  2©)i i  -  + s i n 2(©  2cos(2©-© -© )i i'' rn  -  r n  ©)  - ©)  n  r,  and c o l l e c t i n g  hl  + i  n  i  B  )  terms  results  in  t  =  ( L ^ - L^) [ sin© x  =  G  t  n  Two  i  n  n  i  i  w  r  the  i "  impedance  cos© G  results  i  ©""for  of c o i l  matrix.  m  i " +  (sin©  m  cos©  h  + cos©  sin© )  m  n  i i "]  n  n  m  n  f  (2-8)  =,  are observed:  u s e d b y K r o n was  for selecting coefficient  n  where  important  transformation tool  + sin©  n  cos©  no m o r e  than  differentiation,  angular  l)  speed p ©  m  2)  in  The  Christofel  a mathematical & the  m n  i s simply inductive  10 2.2  The E x c i t a t i o n and I n d u c t i o n A n g l e s  A coil  of a C o i l V i n d i n g  i s an a r r a y o f i n t e r c o n n e c t e d w i r e s wound i n  some s p a c e , w h i c h c a n be r o t a t i n g w i t h r e s p e c t t o a c h o s e n inertial  frame.  A coil  commutator b r u s h e s introduced.  winding  consists  from which e f f e c t s  A slip  are observed or  t o t h e a n g u l a r speed  coils  (i.e.,  of the r o t o r .  a p p l i e d through  f l u x p a t t e r n and i n p a r t i c u l a r coil  angular p o s i t i o n of the i — t h  entire and pO  rotor j  (a)  reference,  ( F i g . 2B)„  the r o t o r  2B  rings).  This  s e t s up a  Jc  1  coil  standstill,  y * i s the  w i n d i n g commutator axes w i t h  y * i s allowed to rotate  over the  0* i s t h e r e f e r e n c e a n g l e o f t h e r o t o r  speed.  Rotor a t S t a n d s t i l l Fig.  slip  applied d i r e c t l y to  a f l u x <P ^y } t h r e a d i n g t h e i - t h  winding, the r o t o r b e i n g a t  respect to a f i x e d  case  o f t h e commutator b r u s h e s i s  Consider a rotor with e x c i t a t i o n s its  and t h e  r i n g machine c a n be r e g a r d e d as a s p e c i a l  o c c u r r i n g when t h e a n g u l a r speed equal  of the c o i l  (b)  Virtual Rotation  I n c r e m e n t a l F l u x Changes  11 Let v i r t ' u a l  the  v i r t u a l  angular  excitations pattern  a  are  rotates  s t i l l  rotor  Moreover,  the  coordinate  w i l l  be  v e l o c i t y  and the  now be  given  s  1  s  1  Z  h%(s  =  y  =  at  +  1  8 J y  -  x  flux  Denoting  fl)^  as  the  f l u x  at  seconds.  the  flux  rotor  rotor,  vector  ( y  x  l a t t e r  +  due to  Since the  found  point  a  the  flux at  t\i p © ^ d t ) j  y  f o r  x  q4 =  transformed  p©*dt)  at  1. to  y  i n the  (2-9) (2-10)  s t i l l  machine  since  (2-11)  1  2C  •  r| ^ p © ^ d t  REFERENCE FRAME FO^ X SYSTEM '  R E F E R E N C E FRAME FOR \X oYSTEM  Pig.  the  to  the  by  =  the  •s  found  measured  to  f o r dt  b y p© ' d t  flux  to  acting  given  d i r e c t l y  system  equal  be  connected  w i l l  where  rotation  V i r t u a l  i n the  Rotation  moving  machine  (2-12)  the  12 Prom  (2-9),  the  A  coordinate  =1§J =  t  =  and  transformation,  Slut  S i A f  =  - & < l i  pQ  )*t  6 £ & t =  (2-13)  thus  ^ { y  Taking l e f t  The  a  linear  hand  1  +  C ^ p O ^ d t }  =  Cp^y-}  (2-14)  approximation  to the Taylor  change  observed  expansion  is  i n the  flux  A  The  i s  i  under  system the  i  writing  i s  (2-12)  also  equal  T i s  point  s  this  1  the L i e d i f f e r e n t i a l  coordinate  system  the  due to  rotor  thus  +• (ck*y } - W y } ) = - | ? i ( l t which  of  side  incremental  rotation  By  A  =  as  to  r  of  Cp^y , )  dragged  k  along  p e i ) d t  over by the  ( 2 _ 1 6 )  p9 *.  ' ^ .^ 2  commutator  transformation  y+ x  ( » \ ^ p © ) d t l  (2-17)  13 the  negative  of the L i e d i f f e r e n t i a l  The rotor  flux will  i s stationary  rotate.  o f CP^over  r^pQ-\  s i m i l a r l y be c a u s e d t o r o t a t e  i f the  and t h e axes o f e x c i t a t i o n a r e a l l o w e d t o  Using standard notation  this" .contribution i s  at  Z.$>Ay'} = i™ ( py . dt 0  >  &  —  jftw*  (2-18)  Besides angles the f l u x i s a f u n c t i o n consequence,the  of c u r r e n t .  As a  sum o f t h e time and L i e v a r i a t i o n s must be  b a l a n c e d by c u r r e n t  variations*  3#xdi  That i s  A  (2-19)  Thus  dt: ldP =  ) d T y ,© k  _ a ^ i i l  This  to  ayidt  +  ay^^dt  ( 2  -  2 0 )  k  • + M i d i i  + M  A n f i i i  (2-21)  c a n a l s o be w r i t t e n as  The the  +  last  i-th coil  terms, due t o t h e speed o f t h e space i n w h i c h  i s wound} i n v o l v e s  t h e i - t h commutator a x i s  d i f f e r e n t i a t i o n with  angle only  respect  and i s i n d e p e n d e n t o f a l l  14  commutator axes inductive in  this  i-th  speeds.  I t has t h e c h a r a c t e r i s t i c s  type of volta'ge.  sense w i l l  The commutator a x i s a n g l e y  x  used  be r e f e r r e d t o as t h e i n d u c t i o n a n g l e o f t h e  c o i l w i n d i n g and w i l l  i n which  o f an  i t occurs  be r e l a b e l l e d x  i n any e q u a t i o n s  The c o n t r i b u t i o n -  0  : c a n t h e n be  written | f i p x %  +  The  ^  where px ^  s e c o n d term  on t h e r i g h t  k rtf  (2-23)  hand s i d e  o f (2-22) i n v o l v e s  differentiation  o v e r a l l commutator axes a n g l e s and depends on  commutator axes  a n g u l a r speedso  speeds.  "y-*-" u s e d i n t h i s  I t i s independent  sense w i l l  be r e f e r r e d t o as t h e  e x c i t a t i o n angle of the i - t h c o i l winding with respect  to i t gives r i s e  excitation applied relabelled  under  differentiation  t o t h e a n g u l a r speed o f t h e "y " will A  n o t be  conditions.  ( 2 - 2 3 ) , e q u a t i o n (2-22) can be w r i t t e n  Using rule  since  to the i ' t h c o i l w i n d i n g .  these  of a l l c o i l  i n a chain  form  d&i. aCPjLdi* dt ~ 3 i d t  dflkdy * dy»dt  s  8(Pi.Sx ax d.t  ,  j  J  v  K Z  (7 7  Ve which  s h a l l now a p p l y t h e above r e s u l t s  i s of basic  importance  t o Yu's f o r m u l a ,  i n machine a n a l y s i s .  t h e m - c o i l be' wound i n a space w i t h a n g u l a r speed pQ**.  the  n coil  i s excited  (P = m  L  r  a  r  ,i  p  the f l u x i n t h e m - c o i l  = (Ld cos y c o s m  w n  y  n  + lA  '  I n P i g . 2D  let  If  i s g i v e n by  m n  sin  y s i n m  y")i  18^  n  (2-25)  15  REFERENCE  m - c o i L  Fig. where y ™ and y 1  coils  2D  a r e t h e commutator a x i s  n  respectively.  d  -^= dt  Two G e n e r a l Machine C o i l  Prom  (L^ cos y'cos r  a n g l e s o f t h e m and n-  (2-21)  y" + iA^sin  + . ( - L i c o s y** s i n y * + l A m n  + (-U^sin  Windings  m n  y ^ s i n y") d i dt  sin  y cos w  y^'cos y " + l A ^ c o s y  y")  n  i  n  s i n y ") ( | 2 _ )  ±  (2-26) By  the p r e v i o u s d i s c u s s i o n , y  n-coil  and y ^ t h e  denoted  n  angle of the  i n d u c t i o n angle of the m - c o i l , which i s  by x r a t h e r t h a n y™,  (2-25) and (2-26) t h e n become  m  ou= L ^ o " = ( U ^ c o s 1  i s the e x c i t a t i o n  x cos m  y^ + U  s i n x ™sin y ) i * n  m r ;  (2-27)  dt  ~ a i d-t n  As a s p e c i a l  case  +  Wy 4.t  8  n  the n - c o i l  u  dt  (2-28)  ;  c a n c o i n c i d e w i t h the m - c o i l , y  becoming y w h i c h i s n u m e r i c a l l y e q u a l m  t o x*".  However, the  dy^ a Sx ^ a operators ^^: and J * ^ ^ ^ a r e equal only f o r a s l i p 1  -  -  m  ring  machine, From  (2-23) f o r a number o f c o i l  P Sx  =  x  U  dt  windings (2-29)  dt  m  Here - j ^ - i s t h e speed  of the c o i l  comprising  the m-th  coil  d© ^ winding  and  t h a t of the j - t h space.  connecting these  two  incidence matrix.  speeds w i l l  1  0  px  1  0  0 0  px  b  c  px*  p©*  the i n d u c t i o n a n g l e  1  0  1  0  1  0  1  1  0  1  1  <1>,  be c a l l e d  F o r t h e example o f F i g . 2E i t i s g i v e n by  px 3  The q u a n t i t y  1  <1>  0  (1> = R  0  0  1  0  1  17  ^REFERENCE  \  Fig2.3  The  2E  The  L a g r a n g i a n F u n c t i o n and  The  L a g r a n g i a n f o r any  L = T =  omitting energy  E q u a t i o n s of Motion  electrical  machine  i s g i v e n by  ( K i n e t i c Energy F u n c t i o n )  of the mechanical  (2-30)  the m a c h i n e . energy  and  The the  kinetic  stored  energy.  . In a l l subsequent be  The  capacitive effects within  i s the sum  magnetic  I n d u c t i o n Angle  e q u a t i o n s the f o l l o w i n g n o t a t i o n  will  used; a)  Greek l e t t e r s  r e f e r to mechanical  Jotet? the i n e r t i a Dcteo  quantities  o f the oc'th space  the damping c o e f f i c i e n t  o f the a' t h space  p©**, the a n g u l a r speed o f the a ' t h  space  18 b)  Roman l e t t e r s L  m n  ,  refer  the mutual the  R " k h i c)  currents  is  which a r i s e equations, they  to determine  take  i n both  will  specific  i  1  a  ... i  energy the  the r e s u l t i n g  c a n be c h o s e n  ) o  n  with i ,  indices.  the s t o r e d magnetic  of the c u r r e n t s  enumerated as ( i ,  mechanical  be i n d e x e d  a r e c a l c u l a t e d and t h e i n d i v i d u a l  The o r d e r  the n - c o i l to  i n the m-coil  are r a i s e d i n succession,  contributions  and  the c u r r e n t  or k unless  In order  from  r e s i s t a n c e of the m-coil  e  electrical  j,  summed.  ,  inductance  quantities  m-coil  x and y a n g l e s , and  coil  m  to e l e c t r i c a l  The t o t a l  energy  contributions arbitrarily  stored  energy  then  E  e  = Z(AE J L =  1  f t J L  ) =  .cc' \ Cp.(a') d« J 0  JL  + ^  cc ) a  +  •  da*  JL  ' + Qji* C  ... i " '  , oc")doc"  n  o  =  JCPid'  . . . i " * 1  ,a- )da ,  i  (2-31)  o  using  t h e summation c o n v e n t i o n  and a * as t h e dummy v a r i a b l e  of i n t e g r a t i o n . The  mechanical  E  e n e r g y i n t h e system i s  w  = |  JocccPO^P© " 0  '  (2-32)  and  19  hence the L a g r a n g i a n i s L  =  <Ji(i'  i " ' , a^da 1  1  + |  p  O  *  p©  (2-33)  o The  state  o f any p h y s i c a l l y  c o m p l e t e l y d e s c r i b e d by  (q , © x  realizable  y )*  A  q"*" b e i n g t h e t o t a l  X  y  machine i s  p a s s i n g a r e f e r e n c e p o i n t an t h e i H h c o i l  w i n d i n g l e a d ; ©"*", t h e  a n g u l a r p o s i t i o n o f the i t . h space w i t h r e s p e c t  to a chosen  T  reference; respect (Aq  x  and y *~  the a n g u l a r p o s i t i o n  9  to a chosen referenceo  A©"*";, Ay" *) i s _.a p o s s i b l e  hence t h e d e s c r i p t i o n  of the i ' t h c o i l  M o r e o v e r any  i s h o l o n o m i c and  winding with  "displacement"  " d i s p l a c e m e n t " of, t h e  1  ?  charge  (q , 0 x  y )  X  X  9  state  and  i s a true  (14) "coordinate" system machine  i s a true  Principle  „  v  C o n s e q u e n t l y , any p h y s i c a l l y  electro-mechanical  applies without  S i n c e Ee, = Ee { i  subsidiary « o o i *% y  1  system and H a m i l t o n ' s conditionso  °«<> y  1  realizable  A 9  x  „ „ x }  1  A  0  and  •b  E m .-= E ^ {p©*}$> the v a r i a t i o n the  independent v a r i a t i o n s  AA - ^t.  ( |  +  A E e  E  +  A E  (Aq/% A©  «v)  - A q " |f§v  +  i n the a c t i o n  9  A = ^(L)dt  S )  due t o  , Ay ) i s x  d t  | f > A i - H - 2 E e  +  ^  +  |ff  A©*  A( ©*))dt P  ^E  The two L i e v a r i a t i o n s (Eqn  0  e n t e r through the f l u x  (2-=3l)) and hence f r o m E q n 2  A V  E  e  =  c  (2-16) and A©  0 6  function  (2-18) a r e e q u a l t o (2-35)  20 Substituting last  these  terms by  into  (2—34) and  integrating  parts maintaining fixed  the f i r s t  end p o i n t s t h e s e  ^ = 5<- ft $h*) + <- it l b  and  results:  - sp»  -  (2-36) By  Hamilton's  Principle  are  independent  the  and  dissipation  are:  e  this  i s zero.  equations  n  ft  | i *  +  » «  »-tt|faf  +  D  i  A © ^ and Ay  h  of motion  R  S  Since Aq , including  driving  n  (  MPO*+|f*^  reference space  torque  axis  (3 i f  torque  forces  "  2  3  7  )  (2-38)  ^ =ffi The  x  (2-39)  r- a r i s i n g  from  the dependence o f E  a n g l e y "** a c t s on t h e  i'th coil  does n o t  t ^ the  equal  zero,  and  on  e  the  hence on  externally  the  applied  on the (3-space i s  S= Since referred  i n (2-31)  t o i n the  s  the n - c o i l  M  & {i' n  three equations  torque  equations.  can be  $ U i '  Qi'  n  =  t o a system  ( 2  chosen  as  ~  4 0 )  last  J.  3pq  reduced  ^ l ,  +  summation  Mi  The  »  ...  i"-a"}da'  ... i }  (2-41)  n  (2-34),  (2-38) and  of v o l t a g e equations  and  (2-39) are  a system  thus  of  f ^  =  R  - i  n  •  (2-42)  21. t  The t o r q u e  a)  B  = D„  equation,  Before  can be  b)  Rotation  2F  (2-43)  simplified  Rotor V i r t u a l Rotation  Virtual  from the steady  t h r e a d i n g the m c o i l  as f o l l o w s ,  c)  Excitation Virtual Rotation  Rotations  state value winding  BCD, m the f l u x h a v i n g  A  rv  °  r o t a t e d by ah a d d i t i o n a l  the i ' t h a x i s b e i n g  2 0  the L i e v a r i a t i o n  axis i s  , - * - A ^ ° < -  3 x M *  hand t h e . r o t o r e x c i t a t i o n axes a l o n e rotation,  +  t o s e c t i o n 2o2j, i f t h e r o t o r i s g i v e n a v i r t u a l  r o t a t i o n A9 of f l u x  + J^pfpO')  (2-43),  Fig„ Referring  pO*  A9°  A G  If  (2-44)  on the  other  are g i v e n a v i r t u a l  r o t a t e d by A y , A  the f l u x  change i s (2-45)  22 By making Ay  x  = -r|^A©°% that i s , rotating  c o o r d i n a t e s by -A©°^ t h i s Fig« 4 F ( c ) the sum  change  is +  o f t h e s e two  flux  _ Since the d e r i v a t i v e s for  suitable  The f i n a l  a l l rotor i j c A© *.  excitation  From  1  changes i s z e r o and t h u s  , x. _ d o n  o f (2—43) p a s s under t h e i n t e g r a l s  (2-33)  conditions,  equations of motion are then ^  m  e  w  - "mm  + ^  1  • m . 3$,ndi d i " dt  — R rr»m 1  ^ . t x ^ , d^dy_^ d y dt  n  +  §x* dt  +  A  T  (2-48) t „ = D „  2«4  p(p©») + 2 | f l  p©^ + J „  Pertinent Equations, D e f i n i t i o n s , The f o l l o w i n g p e r t i n e n t  notations w i l l a)  r^j  (2-49)  and N o t a t i o n s  equations, d e f i n i t i o n s ,  and  be u s e d i n s u b s e q u e n t c h a p t e r s ,  Voltage  equations : M  1)J 1  e  e - mn R rr\a  R  For  x  ^^  i d+ t - ^ +  iKV +  the l i n e a r  D  n  n«  d  n  c)CDD ^ d i a i d d+, t nn  a CD* &X_^  h T  c a s e (p = L m  j.  T  D I  ddtt  x  r r i n  i  " ,3  a (Pry, dy * 3 S y »v » d td t  and  r i  L  T  M  h  Sx  n  OL n W  dy  n  23 2) 3)  8 L ,  8y  6)  4~».=  1)  pO '; r  H !T =  V  ^p.  Induction Angle Component)  =  W -  y  Excitation Component)  —  x  =  +  =  Angle  Incident  Gradient  Angle  (i'th  Gradient  Angular  Gradient  p ( p O  +2(^7  ( i ' t h  equations:  D^pp©"  t , =  the  (^7  +  J  linear  Dpppe*  * P =  LnJ.O-^  p  t *  The  P  )  +  J „  p(pQ )  = ^ £ 1 ^ =  Torque Tensor space  =  Electromagnetic Energy Torque on the |3 s p a c e  tr  E )(r^) e  product, +2  -  +2  ft  as  dE*  Qx*'  "8E  e  L  G ^ l j ^  +  U  E ).(r^)  (^  +  P  =  dot  2(sy  p  p  case  =  3)  Induction Matrix  v  =  For  2)  m  Coefficient  Excitation'Coefficient  irinn  1  i^-j.  Torque  v  n  5)  7)  Induction  G  =  dt  0  v  mi,  n  6x"  4)  b)  =  ^ T  m  )  r  fori the  >  ,  ^ i" i» ,  ,  (3  ?  an  Conversion  example,  dE., c)E  a  for  3E»  3 x ' d x ' ax b  9Ee. 5 E  c  7  a  3Ea  dx«-' d x ' d x " ©x b  7  Fig.«  2E  is  24 or  1  1  0  0  0  0  4-2  8E* dx a  dE  3x 9E  where stator on  the  t  s  i s the  and  t  r  rotor.  electromagnetic  i s the  e  1 1 c  A  energy  electromagnetic  conversion  energy  torque  conversion  on  the  torque  3.  3.1  THE  EQUATIONS OP MOTION IN MACHINE SYSTEM ANALYSIS  Loop E q u a t i o n s I t has  valid  f o r any  system. c a n be  i n True C o o r d i n a t e  Systems  been proven t h a t F a r a d a y ' s I n d u c t i o n Lav electrical  machine r e f e r r e d t o a t r u e  C o n s e q u e n t l y methods o f s t a t i c  circuit  e x t e n d e d to i n c l u d e a s y s t e m c o n t a i n i n g  machines.  For  e l e m e n t s , the  an u n s a t u r a t e d general  e  m  is  coordinate  analysis electrical  system i n c l u d i n g c a p a c i t i v e  loop equations  are  Rm  —  =  R ^ i "  = z  m  n  + L  ( p ) i  w n  +1^^0*1  pi"  *  H-I^py"^  +  (3-1)  n  where  (3-2) are  the impedance m a t r i x  be u s e d ; h o w e v e r , s i n c e system, and s i n c e analysis  i s more  elements.  Nodal equations- could  an e l e c t r i c a l  the d r i v i n g  forces  machine  coefficient  inductive  are voltages,  loop  natural.  3L The  i s an  also  torque  tensor  o f p© ^ i n z  r n n  element T (p)«i  have been d e t e r m i n e d , the t o r q u e  m  .= Q ^ x  Therefore equations  n  H.  i- " t s  once t h e l o o p c a n be  n  e  equation  obtained  by  inspection. V =  D^p©^  + J^ptp© *;) + 5  l ^ r ^ i ^ i "  (3-3)  26 Using can be  flow  the  and  of m o t i o n .  towards the  The  Since the  f r o m the  are  currents  increasing  torque  s i g n s o c c u r r i n g i n the references  are  can be  Positive  3A(a)  polarity  Fig.  3A  Polarity  the flows  From Yu's  formula  the  i n the  of  positive since  rj^.  the  These  coils.  Standard P o l a r i t y vention  Convention  x  direction  direction  The  g i v e n by  f o r three  b)  whereas  current  force.  are  is  chosen  f o r c e the  chosen a r b i t r a r i l y ,  equations  shown i n Fig»  as  i n a clockwise  acts  since i t i s a d r i v i n g p  arbitrarily  outward d i r e c t i o n ,  measured p o s i t i v e l y  sense f o r 9  are  machine i f p o s i t i v e ,  is a driving  Positive  convention  o f a l l terms i n t h e  at i t s plus r e f e r e n c e .  d-axis.  reference  polarity  signs  o f the  i n an  the v o l t a g e  coil  y angles  a)  a standard  coil  center  M*M.F i s r e f e r e n c e d  flux<. into  formula  a d o p t e d to d e t e r m i n e the  equations to  Yu's  Con-  27 If  t h e MM/F.direction  chosen r e f e r e n c e d i r e c t i o n all  L  m n  «  Since  equivalent  to  interpreted of turns at the  i n Yu's  setting N  as t h e  formula  end  if N  polarity  convention  L*  This w i l l  = N,„ N  Pj, this  n  Polarity  be  i f N  i s opposite  can  the  noted  by  thus  be number  placing a and  T h i s g i v e s the  of  is  or n e g a t i v e  i s positive  n  to  s i g n appears i n f r o n t  of a p o s i t i v e  i s negative*  n  m n  negative,  w  of the n - c o i l  negative  3*2  a negative  presence  i n a winding.  p l u s end  i n the n - c o i l  at  dot the  standard  o f F i g . 3B(b)«  Transformation  Theory  Many t r a n s f o r m a t i o n s u s e d i n machine a n a l y s i s a r e hypothetical; ease the  equations.  d e f i n e d i n an  transformation, o r any  i  n  Since  = C" i  n  f can  involving  These t r a n s f o r m a t i o n s If  the  s y s t e m t o a new motion i n t h i s  The  an e n t i r e l y  For  voltage  different  transformed  o b t a i n these  transformations  t  true  coordinate  equations  must t h e n  which r e s t r i c t s  old  l e t the  (unbarred)  of  from  equations  restrictions  from the  parameters.  numbers.  system a r e d e r i v a b l e d i r e c t l y  to L a g r a n g e ' s e q u a t i o n s To  current trans-  s e t of  s y s t e m , the  y—angles,  transformation  of the  a r e f r o m one  true coordinate  The  cumbersome  instance, a current  can a l s o i n v o l v e complex  Hamilton's P r i n c i p l e .  formations.  a l g e b r a or  i n v o l v e the x - a n g l e s ,  say.  transformations  new  i n v e n t i o n s which  are h y p o t h e t i c a l , they  c a n be - d e f i n e d i n d e p e n d e n t l y  n  formation,  reducible  they  a r b i t r a r y manner.  other parameters, z  e ^ =-€^"e  voltage  are manipulative  t a s k of d e a l i n g w i t h c o m p l i c a t e d  differential c a n be  t h a t i s , they  the  trans-  current  t o the  be  new  and  28 (barred)  system be r e s p e c t i v e l y  i "  = C £  i  (3-4)  n  e ^ ^ e ^ Dealing with  e  Under the  the v o l t a g e o  m =  1  m  =  *€  +  (  m  we  satisfied  C  R  ±  K  become  f t  +  system t o be  ^51"  a true  energy f u n c t i o n i n the  o n l y i f •€ ^  differentiation  and  or C  then  or both  m  ^  C  (Cilfr)  system  C  2) first  (3-6)  ( p h y s i c a l system)  '  the e = i  &|r£  E =  =  e  The  moved t h r o u g h  a Kronecker  be  the  delta.  satisfied. (3-8)  the  (3-9)  transformations  to  those  (energy) i n v a r i a n t since  m  ^ e  o  can be  T h i s can  = 0 and/or | | ™ = 0  power  m  system.  =  condition restricts  which maintain  new  combined t o g i v e  f o l l o w i n g c o n d i t i o n s must be 1)  The  m n  these  must have  where E i s t h e  The  new  R  \ m  B^O i"+-e£  = («£  f o r the  equations  . n , d_ I d E dt \ 3 i  above t r a n s f o r m a t i o n s  e  In order  (3-5)  «  K  w  i *  C„  l  =  b ^ e _ i  dt = ^ e i d t m  m  =  n  = e ^ i  E  o  Kronecker d e l t a obtained  in  (3-8)  i s not  strictly  a  29 constant  even t h o u g h i t i s a u n i t  derivative,  i s then  iatipn  condition  dt  = 0 allows  combined w i t h - e ^  of (3-8) o c c u r s ,  to give  the v o l t a g e  unchanged and t h e r e f o r e t h e s e interest. to  Ve a r e l e f t  be s a t i s f i e d  true and  system. then  s i n c e t h e time  which i s L i e , i s n o t zero g e n e r a l l y .  The It  matrix  (3-8). equations  = 0; ^  by t r a n s f o r m a t i o n s  possibly  changing  the v o l t a g e  no  remain  differentessentially  = S ^ system t o another  inside ^ I  this  n  i n (3-6)  case,  i t s Lie derivatives  equation  quite  o f |j  are of l i t t l e  f r o m one t r u e  j£ t o be t a k e n  the d i f f e r e n t i a t i o n ,  i n front  £ 0 and €  combined w i t h cjj, t o g i v e S J R .  S^is—under  Since  transformations  with ^  These a l l o w  Crn t o be t a k e n  since enter" t  radically.  rLC" ' Since over  ^ 0, t h e L i e d e r i v a t i v e  a non-zero  S|^3u  and g^:T with  angles  c  a  speed, n  Moreover not a l l the p a r t i a l  vanish.  derivatives  By (3—8) •€ ^ must be t h e i n v e r s e o f  a p o s s i b l e interchange  occurring.  o c c u r r i n g must be  of e x c i t a t i o n  However, s i n c e ~^~r  m  and i n d u c t i o n  = 0 and n o t a l l  d C St and  need be z e r o , t h e L i e d e r i v a t i v e  speed. over and  T h i s i m p l i e s t h a t the L i e d e r i v a t i v e  t h e d i f f e r e n c e between c o i l the o l d system  Consequently and  since these  velocities  t o one  i n t h e new  are the only i n v a r i a n t  referring  to c o i l s rlC  there  reduces  a zero  system speeds.  • € ^ i s a f u n c t i o n o f t h e d i f f e r e n c e between new  o l d i n d u c t i o n angles  Since  must be o v e r  a r e no p r e f e r r e d a n g l e s  and ^ r d  i n t h e same -i.  n  n  space.  R  ' 0, C  m  must be  a f u n c t i o n o f t h e d i f f e r e n c e between new and o l d e x c i t a t i o n angleso Examples o f some s u c h following  page»  transformations  a r e g i v e n on t h e  a)  mn-dq  Transformation,  Id. (y«\ x )  =  4  m  (0,0)  (py*, px ") -  (o, o )  3  (py ,px ) m  All  c o o r d i n a t e s are r e f e r r e d  t r a n s f o r m a t i o n i s on t h e  1 i  r  t o the r o t o r  t  - x*)  ,  + s i n ( x "" - X* )~  -  ,  x ) d  s i n c e the  rotor.  4  w  P  space  - yV r i cos(y* - y )  m  o ) r  P  ,  -sin(x  sin(y*"  4  cos(x" -  x ) d  r  _  _  -  these  dt b)  ( © ,  4  ~ cos(x  In  r  P  -y) ~sin(.y*" - y ) costy*"  n  =  m  .  Synchronous  2 c £ * Machine  o  ;  If  =2«  =  0  Transformations.  pQ )  (py*, p_x ) -  (pO ,  (py ,  (s, p© )  m  d  (y\  px ) a  x )  =  1  (py , a  px ) 1  =  r  s  s  (0,0) =  (0,  p© ) S  31 All  coordinates  formations.  are t i e d  to the s t a t o r  (Note t h a t p x  cos(y* - y ) m  -sin(y  - -  cos(x -sin(x  cos(y  - y )  d  w  - x )  d  p©t)„  =  w  d  - y )'  1  ,  cos(y  d  -  i  sin(x  - x )  ,  m  w  -  d  cos(x  y )_ m  x ^) 1  -  4  em  x ) m  - y ) , c o s ( l 2 0 + y " - y ),  a  w  3  sin(y " - y " ) , sin(l20 a  + y  cos(240 + y  m  - y ) , sih(.240 + y  a  m  1  l  speed.  • m  sin(y  t  trans-  s i s the synchronous  ,  m  t I  for stator  a. a.  y")  1  4?  eg. e  -IT  cos(x  -J?  m  x  cos(240 +  1  -  sin ( y 1  sin(x  x )  cos(l20 +  cos ( y  i  -  a  x  a  a  x ) m  x ) m  -  m  s i n ( l 2 0 •+  x  s i n (240 +  x  1  x )  a  m  a  x ) w  cos(240 + y  a  y )  y a)  sin(240  a  3  s i n (120 + . y  a  - x ) a  a  3  4  J  -  y*)  + x  I n a l l of t h e s e t h e L i e d e r i v a t i v e  a  - x  )  a  - x  )  d  1  1  I  i  y )  1  x )  - x ) , sin(l20 + x  cos (240 + x - - x ) , s i n ( 2 4 0 9  + y  -  1  , sin(x " -  d  »  1  1  a  e  y*),  3  cos(120 + x  formation  1  x )  y*) , cos (120 + y -  1  cos(x  a  e  of the v o l t a g e  0  trans-  i s e q u i v a l e n t t o z e r o whereas t h a t o f t h e c u r r e n t  32 transformation c)  fb-dq  i s not.  That  is ||  w  =  0; sL£"» £  0.  Transformations  -J  1  -j(x*°x )  -j(x -  a  m  x ) 3  1 . j U * " - x*)  , j ( x  w  -  x ) a  Once a g a i n  d-6 ™ dt  The  m  torque  e q u a t i o n i n t r o d u c e s no a d d i t i o n a l c o n d i t i o n s  on t r a n s f o r m a t i o n s from since  one t r u e system  f o r E = E , the- t o r q u e t * = D ^ p Q  +  to another  equation retains p(pO^) +  true,  system,  i t s holonomic  2 ^ E  form. (3-10) (3-11)  where  D  (3-12) (3-13)  pO»= 3  E  = E i n t h e new  angle new  i n t h e new system.  fclp©*  (3-14)  '2  S i t  system, system  (3-15) = ^  , x" " b e i n g t h e i ' t h i n d u c t i o n 1  x  and r^T = r ^ , t h e i n c i d e n c e m a t r i x  i n the  33 3«3  Tensors  In the E q u a t i o n s  A tensor  index  c a n t r a n s f o r m by t h e v o l t a g e  m a t r i x •€, i n w h i c h c a s e transform'by it  will  i twill  be c a l l e d  an i - i n d e x .  (i-tensor),  and  the i n v e r s e of  a l l v—indicies  excitation  manner e x c e p t angles*  except  angles,  f o r a p o s s i b l e interchange  through page.  from the expansion  (3-15). Since  They a r e l i s t e d  the e x c i t a t i o n  how an i n d e x  whereas t h e y The checking for  of motion  o f ( 3 - 6 ) , and f r o m  i n Table  and i n d u c t i o n a n g l e s  transform  detailed  tensor  are not tensor coefficients,  as an i - i  tensor  the torque  i n the v o l t a g e  i n the torque  equation  equation.  t e n s o r n o t a t i o n u s e d above i s u s e f u l f o r  c h a r a c t e r or f o r i n v e s t i g a t i o n g d e t a i l s ,  computational  form the v o l t a g e  as a v — i t e n s o r  (3-10)  1 on t h e f o l l o w i n g  c a n be s e e n f r o m t h e t a b l e t h a t i n g e n e r a l  t e n s o r elements t r a n s f o r m  trans-  above o r below i t .  q u a n t i t i e s n e i t h e r a r e the i n d u c t i o n o r e x c i t a t i o n  It  of i n d u c t i o n  o f i n d u c t i o n and e x c i t a t i o n  t o show e x p l i c i t l y  be p l a c e d e i t h e r  so -€y£  i n t h e same  Tensor q u a n t i t i e s o c c u r r i n g i n the equations can be d e t e r m i n e d  case  combination  I f (3-8) i s s a t i s f i e d ,  a l l tensor i n d i c e s transform  I f i t i s necessary l (  C, i n w h i c h  ( v - t e n s o r ) or a  f o r a p o s s i b l e interchange  forms a ' V ' o r a n i " w i l l  or i t can  Any t e n s o r q u a n t i t y c a n have; a l l  of v and i i n d i c e s ( v - i t e n s o r ) , , is  transformation  a v-index,  the c u r r e n t t r a n s f o r m a t i o n matrix  be c a l l e d  i-indices  of Motion  purposes matrix n o t a t i o n i s b e t t e r . equation  however,  In t h i s  i n a t r u e r e f e r e n c e frame a r e  •(e) = ( (R) + p ( L ) ) ( i ) , p = | T  (3-16)  34 Tensor a)  Voltage  Transformation  Quantities  Properties  Equa;tion  V  1  3i  m  —  'w  m  T  dj&m = o c  m  I—i tn p V  ^  b)  ( 3<Dwi U"»  Torque  ^rp  <T>  m  C C T B  n  Equation  P * »  t?  (  a < P , r n  Table  I  Tensor  H  T  w  Quantities  A m a t r i x m u l t i p l i c a t i o n .occurs the this  electromagnetic index  for  (t ) p  torque  clarity  =  the  (D„)(p©*)  i n the for  since torque  T  Equations  each m  n  |3 i n t h e  p i s  a triado  equations  + (J^pMp©*)  +  of  Motion  calculation  of  Retaining  i n a true  frame  are  2((^E) (r^) ) 0  p  (3-17) For  the  unsaturated  case  this  is  35 (t»)  =  (D„)(p© ) +  (J p)(.p©*) + 2 ( ( i f ( T ) ( i ) )  3  The  w  transformation equations  For these  are  (e)  (-ef  (3-20)  =  (e) (3-16) and  (3-18) become  *  +  (3-31)  (€)*p(L)(C)(T)  )(p© ) +'(J p)(p0 ) + P  & p  (3-5)  (3-19)  (€)_*'(R) CC) (I) (D  and  (C)(i)  •  (t,)=  (3-4)  (i) =  transformations  (e) =  (3-18)  p  1 1  M  ((D (C) (T) (C)(i)) t  t  ?  (3-22) I f p© ^ does n o t  equal  z e r o and  r e p l a c e a b l e by("€!) e x c e p t excitation angles is  the m a t r i x di.(  (•e)  (L)  p  the  (3-8)  holds  so t h a t  f o r an i n t e r c h a n g e transformed  of c o e f f i c i e n t s  torque  (C) i s  o f i n d u c t i o n and tensor  (C)*' ( T ) ( C ) B  o f p© ^ in' '(3.-21) b e f o r e p o s s i b l e  C)  Q^f- ( i ) c o n t r i b u t i o n s a-.re a d d e d  once the v o l t a g e  equations  torque  can be  equations  interchange  0  In t h i s  have been t r a n s f o r m e d ,  o b t a i n e d by  of i n d u c t i o n and  case the  inspection after  excitation  then. transformed  a possible  coordinates„  I n v a r i a n c e . , a p r o p e r t y u s u a l l y a s s o c i a t e d w i t h the v e c t i o n o f two result.  As  tensors  of o p p o s i t e  an example u n d e r  (3—4)  types'  9  and  i s no  longer  can  to  i f i t is realized  another  (3-8)  For  .  m  and  a t r a n s f o r m a t i o n f r o m one  is satisfied  and  the  '  (3-23)  t h a t power and  be d e f i n e d o n l y i n a 'true system f o r w h i c h e  p h y s i c a l l y measurable.  a general  (3-5)  e ^ i ^ e . - e ^ C ^ i ' ^ - e J c S e ^ (3-23) i s u n d e r s t a n d a b l e  con-  i  energy m  are  true  system  invariance property follows.  36 F o r an u n t r u e quantities; in  the  (m)  system. to the  system e  invariants  system  torque  In t h i s  t  p  form  abstract Power  r e f e r e n c e d t o the r e a l  j(C™C™e^i^)dt referenced  sense  b o t h power and  i s a covariant  t e n s o r ; however  the K r o n e c k e r  Mechanical  energy  Saliency  and  i t i s thus  similarly  equation i s e s s e n t i a l l y  reference  conversion  t o a Machine  dependencies  With  i s e x p r e s s e d by  a  o f the  9  (2-48) and  assumed f i x e d w i t h r e s p e c t t o the  and hence ©  This  i n the mechanical c o o r d i n a t e s .  In the e q u a t i o n s of motion  s a l i e n c y was  a,nd  invariant.  t o the e l e c t r o m a g n e t i c e n e r g y  i n the e q u a t i o n s of motion  spaces.  equation.  actually  dependence o f t h e f l u x upon a n g u l a r d i s p l a c e m e n t s , 0'  any  are  i t s transformation  }  angles are t r a n s f o r m e d  torque  in particular  delta  E x t e n s i o n of t h e . E q u a t i o n s of M o t i o n Rotating Saliency  machine  (m)  o f i n v a r i a n e e o c c u r s i n the t o r q u e  torque which i s c o v a r i a n t  3.4  are  o f any t r a n s f o r m a t i o n .  hence the e n t i r e applies  and ^ ( e ^ i ^ d t  i s likewise  system.  ma,irix. i s always invariant.  m  is •€™C™e^i*  energy  Another The  i  t h e y a r e n o t t h e t r u e power and. e n e r g y .  The (m)  R  (2—49)  chosen  were i r r e l e v e n t  and  could  be n e g l e c t e d . I n g e n e r a l t h e n the f l u x magnetic  energy  t h u s the  are i n f l u e n c e d by m e c h a n i c a l  as the m e c h a n i c a l generalizations  and  energy./  of  stored  rotations  These added v a r i a t i o n s  (2-22) and  electroas  are L i e  the  9  (2-=34) b e i n g  ^^m&^Sfr+ZVx _  CLS-L d i * ~ di*. d t  well  ,S +  3 p  (3-24)  CD.JL L  dy dt  a  1 +  CD JL d © ? B©^ a t  . +  3  d© ? ^dt  CD^J  (3-25)  37 •t  AA=  5<^»  •a^  »R.  ^ i -+  o  P r o c e e d i n g as i n 2 « 3 , t h e e q u a t i o n s o f m o t i o n a r e e a s i l y f o u n d t o be M  „ e  _ u tr> -  -j n , 3.(Pun  "mn  1  +  at  (3-27)  t , = 'D-^P©*+ J„p(pO*)  + 2 ^ E  e  ^ + | | | (3-28)  As a s p e c i a l  case t o be u s e d l a t e r ,  formula f o r a r o t a t i n g L  =  d-axis. L*  Yu's  I t can be w r i t t e n i n t h e form  cos X c o s  Y  m  m n  consider  n  + iAfn^sin X s i n m  Y  n  (3-29) where X =  x*" - S " >  (3-30)  Y- =  y* -  S*  (3-31)  m  N  x a n d  y  n coils  a r e t h e i n d u c t i o n and e x c i t a t i o n abg3.es o f t h e m and  n  respectively.  6 >  which w i l l  X  angle of the i - t h c o i l ,  i s defined  as ds dQ» dt - 8 0 ? dt ^  e  sa  -*--'- y S-'enc  an  e  _  " »  saliency  dt  ft^-ao*-  i n c i d e n c e m a t r i x , has i n t h i s  e l e m e n t s w h i c h a r e one's i f (3 r e f e r s d-axis i s located.  the  by  x  1  "Ip"'  be c a l l e d  (T\O\ U-^J case,  t o t h e space i n w h i c h t h e  Otherwise the elements are zeroes.  (3-27)  and  (3-28) a r e t h e n  (3-33)  (3-34) J.  £  ^ ^ O ^  T  p©^ + J  p ( p © * ) •+  ??  2\y,E  e  <r  Here  pX = m  p The f a r m  y  h  ^  = ft  of these  (x* -s*)  =  1  (  y  *  ~  s  h  )  =  equations  ( r ^ p © * - fjp© *) = < T ^ p © 5  (  p  y  n  ~  ^  p  q  i s identical  P  (3-35)  ?  )  (  t o (2-48) and  3  "  3  6  (2-49)  w i t h (Tp&s  the g e n e r a l i z e d i n d u c t i o n  The t o r q u e  t e n s o r f o r t h e (3 spa.ce, as b e f o r e , i s t h e a r r a y o f  coefficients  o f p©  e x c l u d i n g those  angle  from  incidence matrix.  pT -.<,  )  39 4.  THE  EQUATIONS OP  SMALL OSCILLATIONS  To d e r i v e the h u n t i n g e q u a t i o n s , the g e n e r a l n e t w o r k loop equations w i l l , be  e  and  the torque  equations  i n the f o l l o w i n g  form  usedo .  m ~ m r,  w  R  1  +  , aCPm d i d i " dt  n +  cXPn.Sx dxdt  a(p dy ^ dt  1  +  ay-  w  x  l C "  C J  m  -n,,  (4-1) tp 4ol  -  p©* + J  The V o l t a g e  p j l  p ( p © * ) + 2( ^ E ) . ( r | ) ^  Equations  If.small variations system  of machines s e v e r a l  w h i c h c o n t r i b u t e t o the a)  The  c o n d i t i o n s change s l i g h t l y ,  " h u n t i n g " o f the  can be  slightly  spaces from  are analogous  e- = !fS  m  o f the c o i l "GI^^  steady  given  and  winding =  and  by  *{P^*  applied voltage e c a n  angle  the  and/or o f  to f l u x v a r i a t i o n s w i t h © ^  (4-3).  2) SCe^^l^C A y - V The  a l l of  system,  o f the c o i l  changed  and hence a r e L i e v a r i a t i o n s  1) tC  to a  ( i , e , , the i n s t a n t a n e o u s n o n - o s c i l l a t i n g v a l u e s ) .  These v a r i a t i o n s y  i n v o l t a g e are a p p l i e d  angular p o s i t i o n s  the r e f e r e n c e angles state values  (4-2)  (4-4)  o n l y be  a f u n c t i o n of the r e f e r e n c e  to which i t i s a p p l i e d . -  The  above v a r i a t i o n  Therefore i s as a  result  |f£ b)  The  a n d / o r o f the The  (I?  steady  coil  ^  The  winding  c)e»v> * ; i Si* A  l  /-a =  (  R  w  commutator  are  ,  A  a  coil  axes can  spaces  change  slightly.  (4-1)  oscillating  about  their  steady  variation  . C>i*di  +  o f the  o b t a i n a b l e from  c u r r e n t s c a n be  state values giving  (4-5)  state v e l o c i t i e s  resulting variations  c)  + Ay-)  ,, d C D E x d Cpm d y K d i ^ d x ^ dt SJL'dy-*. d t ft  1  -A  a  m  w P  +  +  }  A  l  (4-7) d) about  The  their  rates  steady  o f change o f c u r r e n t s can be  oscillating  s t a t e v a l u e s to give  3 di^)  d t  d  l  (  dt e) the  '  From t h e p r e s e n c e  c u r r e n t can be  of c a p a c i t a n c e , the  integral  of  oscillating  A( C> ( $ i d t )  $  i"dt) = ^  A(i i ) = c % h  Ai  n  (4-9)  n  In voltage  analogy  oscillation  0  with  (2—22) t h e f i r s t  equations i s  approximation  of  the  41 Ae  m  =  -  e | n o  ) = §p  ApO>  Apy - + | ^e_j»A i "• 4 A  +  o + Ae is m  steady  *the v o l t a g e v a r i a t i o n state value  w  = (Rm„ + ^  c^i)  A1  (4-10)  m  i n t h e m' t h l o o p ,  and e ^ t h e a c t u a l  Substituting  Ae  e*, + £ £ e  i  8  " +H^A  .  >  J  L  -^f  J l —  m o  b e i n g the  value.  (4-2) through. (4-9) i n t o  p +^ h  e  (4-10) ,  5i^< S T * ' ~ d t c5i ' cVC -1dT  +  +  - -h^TApy^  PX  +  | ^ (  n  - A ©  ?  F  l  ,  + Ay-) (4-11)  There a r e t h r e e •\ H  (T,  distinct • Si"  contributions _ . ^.(Dvn • di^d'i*  p  n  i±  A  1  oscillation  d  t  _ i +  c  .mr,  impedance  Here s „ ( p ) = z m  \ •. J A l  p  arising  from the e f f e c t o f  (-g J a c t i n g m  W n  oh  oscillating  ( p ) -ip g>j!^A  0  last  term v a n i s h e s f o r u n s a t u r a t e d systems.  (gpT  P +  oscillating  )  I  A  w*n ( p ) A i * 3 a v a r i a t i o n  currents,,  c  \  m  a  the  b)  n  cyp  equation.  d i * , 3 [.cXPmxfix* , dt d i ^ d x " V dt  a a< op\ d y a (i a  d±  i n t h e above  Ay g n  reference  (|y^ + ^ " P ) 1 ? A p © i p  oscillating  voltages  arising  from  axes.  voltages arising  eo.il spa,ees.  due t o  T  H  E  42 For  the unsaturated  Ae^=  case  z  m  n  Cp^= L ^ i " * ,  (p) A i  n  (4-10) becomes  + V ^ ^ i '  1  1? A p © * + f f ^  +  Apy" +  1?  A  y  *»  AO >  (4-12)  where (P)  with  =  ( *rm + L ^ R  h  + V ^ p y * *  p + G^^JfJpO*  7 ^  As  an example  c o n s i d e r t h e i n d u c t i o n motor o f F i g . 4A  fixed  commutator  axes*  py  = 0 and gj^X =  applied voltages  a r e a f u n c t i o n o f time  equation  to  Ae*=  reduces  (Rv««+ L m « P  + G ^ ^ J p G ^ A i "  only,,  + 0  since the  0  The g e n e r a l  „i l$(ApQ*) n  m w  (4-13)  4.2  The M e c h a n i c a l For  the  the voltage  loop equations  between d i f f e r e n t must d e a l w i t h the  Equations  equations  oscillation  was t a k e n j machines.  individual  results9  the torque  inverse,  e t c . ) on t h e t o r q u e  a v a r i a t i o n of  containing  spaces  relationships  i n contrast  and i n t h i s  This separation arises  the torque  f p r t h e p »-.th s p a c e ) .  t  Torque v a r i a t i o n s ,  machine  (| ^°  equations  equations  are separateda  when one c a l c u l a t e s L  of O s c i l l a t i o n  sense computationally  t e n s o r f o r t h e component Since  f o r e a c h 0,  tensor i s a t r i a d i c .  f  spaces  a matrix  Operations  (transpose,  t e n s o r a p p l y ' : f o r each p  individually. Proceeding equations  as i n t h e d e r i v a t i o n  c o n t r i b u t i o n s ti» t a r g u e  of the v o l t a g e  oscillations  arise  oscillation  from  43 a)  Variations  due  to o s c i l l a t i n g  ft ©  mechanical  angles  Q  , mechanical  accelerations,  angular p  speeds,  (p©*) <>  pO  From  and m e c h a n i c a l  }  (4—2)  the  sum  angular  of these  contri-  butions i s  ft*  A(P(P^))  + |§fr A(pO>) = § g * A © * +  b)  A variation  |||  A i A =  due  ||^  A variation  (4-14)  w  to o s c i l l a t i n g  A i A  due  D^ApO'  J Ap(p©*)  =  2 (  ^||  currents  x ) a ( > l V A i  given  (4-15)  A  to o s c i l l a t i n g  r e f e r e n c e axes  +  i^  3x> \  dx<>  x-  = 2 ^r(( ^E)„ (^) )Ay  (4-16)  1  p  To equal  a first  t o the  mechanical  approximation  applied  Atf* = ! © *  the  sum  torque v a r i a t i o n  oscillation  equations  +  +  are  J p)Ap©  of t h e s e v a r i a t i o n s  At^ = t ^ - t  p  o  <,  ^  |  A©*  +  (D  w  + V (V^ S  The  E). (^)Ay' -  '+ 2( ^ - C P ^ P . . ^ ) ^ A i V  $©*  is  thus  + 2y{(  } i  w  3t,  by  :  C0 ).(r|)^A  2( c)  +  + J^p E  A  (4-17)  p)Ap© 2  Ay^ + CFUAi^r^  (4-18)  44 For unsaturated This  systems (p^ss L £ ji. i •** and E = ^ L  ^  gives  Ai  (4-20) In m a t r i x n o t a t i o n r e t a i n i n g contributions where t F i g . 4A?  a r e ( V | j i ) ( A y ) and  i s the t r a n s p o s e  the m e c h a n i c a l  ((T)^(i  operator.  ) +  consider  ( t ) the e l e c t r o m a g n e t i c  S  F i g . 4A  Two  and t  K  a  1  b  1  c  0  d  0  r 0 0 r  l 1  Phase I n d u c t i o n Motor  energy c o n v e r s i o n torque v e c t o r i s on t h e s t a t o r  these  (T)J(i))*(Ai)  As an example  t h e two phase i n d u c t i o n motor,,  index  , that acting  where t on t le  8  rotor.  i s that  acting  45  J*  V  Ay  Ay Ay  a  c d  a ^ _a_ ax* ay' o  The  o  zeroes are p r e s e n t = 0 f o r 3 = S and  sec ond  (T) , R  since t  i = c,d)  contribution,  stator;  o  o  +  denote  the t o r q u e  Bx  c  +  D y V ' Sxi+ax*^ c  i s independent  s  and  (T)  g  similarly  for  as the t o r q u e  t e n s o r the  rotor;  o f c or d For  angles the  t e n s o r f o r the  and  ( i ) as  the  column v e c t o r  i  ((T)jj(i)  c  +- ( T ) * ( i j )* ( A i ) i s t h e n  "(T)  s  (T)  r  equal  to  (i)  (i)  (Ai)  +  The  torque  (T)*  oscillation  equation  i n matrix  fornv-for t n e  u n s a t u r a t e d case i s  (At ) 3  =(§||A9*)+  +  ((T + T * ) ( i ) * ) ( M ) +  (vJjO(Ay)  (D + Jp) (ApO) (4-21)  46 4.3  The  Combined E q u a t i o n s The  system  Ae  of  Motion  equations  *^(p)Ai"  +  ($fep +^^?A0^ (^r | *«5^ ||f VAM^y +  Z^(p)ApO* +  A©*  + ViftrtfAi*  +  y  +  ^  (4-22) where Z  =  (D  & o t  + J^p)  and  3 3 c) Vu=C§^C + g p ^ ^ T  a s e t of c o u p l e d m a t r i x d i f f e r e n t i a l the  steady  the  initial  state  The to  and  hunting  conditions oscillation  state  c) + QJ^)  equations which  of a machine  system  form  determine once  a r e known. equations  can be  give a s i n g l e m a t r i x e q u a t i o n of  c o n v e n i e n t l y combined  oscillation  "(Ae)"  (Ai ) H  (At)_  (Ap© ) p  3< LEQ 3y*  3y*  (Ay")  (4-23)  ¥hen s a t u r a t i o n  i s absent  this  simplifies  to  47  (Ae)  ^Sx  (At)_  ((T+T*)(i))*,  3y i m  +  U y »  I n the g e n e r a l to  commutator  m  V  +  (D + J p ) +  1  hunting  '  P  (Ay")  ^2>y  p  w  So*  (Ai")  (ApO*)  p  ^ ' (4-24)  equations  (Ay) and  axes and hence a r e c o n s t r a i n e d  (Apy) r e f e r  externally-  5.  5.1  APPLICATIONS TO THE DERIVATION  The Power S e l s y n Two  System  i n d u c t i o n motors i n t e r c o n n e c t e d  form a s e l s y n system. The  as shown i n F i g . 5A  The t r a n s m i t t e r i s d r i v e n e x t e r n a l l y .  r e c i e v e r r u n s a t t h e same  Fig. The  OF MACHINE SYSTEM EQUATIONS  loop equations  5A  speed w i t h  Selsyn  an a n g l e  System  f o r a single  phase c o n n e c t i o n a r e  o  -P ^ t t i i  0  en 0 —  of l a g 6 .  p t_ _j  r n  R 35  + pL  35  (5.1)  49 The i n d u c t a n c e s  are o b t a i n e d  f r o m Yu's f o r m u l a  (2-27)  - s i n y*py )  (5-2)  pt, = L ^ ( c o s  x*-p - s i n x p x - )  (5^3)  pt, = L nr  n r  (cos  y p - siny py )  (5-4)  p-L = L  r n  (cos  x p - sin x px )  (5-5)  p k m i = mJi(cos L  Am  A  rn  p^xi=  y  A  r  "px-= px  x  px  r  0  r  px = 0 h  Substituting  L  + L  rr,r»+  r  ( i = m,l n,r)  L  (5-6)"  f  rr)  i s g i v e n by " 1  0  0  0 ~  0  1  0  0  0.  0  1  0  0  0  0  1  " p © ^ p©  _p°  (5-7) =  n  L  0  (5-1),  ,  P rY,rn  0  1  p'e'  (5-6) and' (5-7) i n t o  R  r  r  P JL1= i X P  —  .  J  r  pt,33= P ( U i a  The i n c i d e n c e m a t r i x  4  p  a  0  ,L  m J l  -  0 0  -L  r n A  ( cos x  f ,^hn -  p  , L sm  h r  s i n . y  (cos  x pe^) A  y p A  l  p y < )  ,-L (cos y p - sin y py ) ,B + pL x p - s i n x pe ') r  P^r.n  +  (cos  n r  r  r  3 3  r  3 3  r  (5-8) The t o r q u e  t e n s o r f o r the two r o t o r  spaces i s  ,  0  6  o,  o L^&sin  0 x%  0j_  0  0;  0  ,0  0,  0  ,o  o,  sin  x ,0 r  (5-9)  r  50 The  corresponding torque t  A  =  (J  a a  P + D )  pO  t  r  =  (J  r r  p + D )  pO  i a  equations 2  f o r the  + L^sin  x*i™i*  - L  x i°i  system  are  (5-10) r r  Assuming t h a t angles, set  and  that  For  a l l angles  sin  r  f o r a system  3  a r e measured i n t e r m s o f  the machine o p e r a t i o n  of equations  p h a s e s c a n be  n r  i s balanced,  electrical  an  analogous  o p e r a t i o n on a g r e a t e r number  of  obtained., as f o l l o w s s  a two  phase l a g t h o s e  phase a r r a n g e m e n t , the o f the  first  by  c u r r e n t s i n the  an a n g l e  of  (TC/2).  In  second matrix  form  i*(D~ i (2)  i (l)  i (l)  i (l)  w  n  (5-11)  3  n  i (2) n  i (D 3  i (2) 3  •-]  where t h e b r a c k e t e d numbers r e f e r for  a polyphase  machine, adding  t o the phase number.  the  factor  —  one  Similarly  obtains  TP  i (o)  i (o)  i (o)  i'-(o)  X  l  £  «  1  i (o) h  L (o-) n  e  p  a 6  e  i (D h  i (p-D x  e  3  e  -jang-i  o  a  -qanp—a.  (5-12)  51  i"(*>) = 1  e  P i  _ i  ,  (  5  -  1  3  )  where <J) i s the phase number, the l o w e s t b e i n g p i s the number o f p h a s e s p r e s e n t , n r e f e r s t o the n ' t h c o i l . Note t h a t a two system  system i s a c t u a l l y  a semi-quarter  phase  (p=4). Taking  of  phase  zero  a tensor approach, t h i s matrix  as a t r a n s f o r m a t i o n of t h e i  =  n M  C ™ £ i  R  c a n be  thought  form (5-14)  W  i n w h i c h the u n b r a c k e t e d the b r a c k e t e d  equation  indicie^Vj  indic£v(| r e f e r (dead) t o the  t o the  coil  phase number.  number Here3.=  and Os  and <S  Since  the  =  e _ i  6^co)  p  (5-15)  t r a n s f o r m a t i o n i s assumed t o be  phase r e p r e s e n t a t i o n t o a t r u e s i n g l e e  The  *k)=  C  mn(5e>tp—  Z  Z  f*F\  Dropping z  fits.)  = b  _ _  multi-  (5-16)  impedance t e n s o r i s t h e n Z  <£)(?> ("Ho> O /P~ the  Pnm  Or  e  true  phase r e p r e s e n t a t i o n  '  £s> ruoo  transformed  f r o m one  mrn*xp m  e  ^  r i ^  Z  P  ~  rrn«)h ) t?  ^S_e /P"  ±>  0(B)(0)  .  (5-18)  (o) indic.e*-J z  FfUoo^y,  i n terms of 1^-=  p  gj^ ^-^)  (5-18)  1  inductances Lnwm ja e ^ ^ (  P  ^  (5-19)  52  In m a t r i x form t h i s i s 1 2 (z>=  "  1  .n  if ( p - i ) «e  S r  . « n  -3£n e P  e -r  e  p  -jaTTP-i 6  These  results  a two phase impedance lu's  system  tensor  formulae  braeketed  will  9  t o an e q u i v a l e n t  of a s i n g l e  phase  phase  two—phase machine  stator  number  single  coils,  (fig.  v  rotor  9  5c)  e d); s  i 0U r  Fig,,  J  (5-20)  now be u s e d t o r e d u c e t h e e q u a t i o n s o f  "s" d e n o t e s  index  p  5B Two-P&ase  Selsyn  Unit  system,.  The  i s o b t a i n e d from coils  and t h e  53  0  (Zmn)=s(l)  s(2)  R  0  r(l) L  T(2)  S r  p(cos  x )  L  r  -L<. p(sin  t  ss  6 r  p(cos  L  s r  s r  r r  p(cos  0  x )» r  p(sin y )  r  r  p ( s i n x ) , R +pL  r  y ), -L  p(sin y ), L  r  s r  x ),L  r  +pL  s s  L  £ r  r  s r  p(cos  y ) r  0  rr  » ^rr P^rr +  (5-21  The b a l a n c e d 1  1  j  single  j  jHavi_ng o b t a i n e d machine j) l o o p  tained  0  1  0  j  e q u i v a l e n t i s thus - s(l)  1 0  0 0  0 0 1  These  phase  s(l)  R  r(l)  L  -L  Using  , R,  m A  pe~  i ^  = §  ~ ™P  0  1  0  0  i=°_  L  + pL pe"  n r  5  L  ,  hrP  R  matrix  (5-22)  of a single  a g a i n be o b t a i n e d , elements ob-  .1  T l  (5-23)  e  + pL  3 3  3 3  the r e f e r e n c e a x i s of  t h e n - a x i s by  0  (5-24)  1  0  0  0  1  0  0  0  e  • six -**) 1  e =o 3  \,R  h n  + L P  •L (p-jp©M. Lnr e  i  mA  L  impedance  by r o t a t i n g  ?  U  \  of  ( 5 - 7 ) ,  »-Lrn P  0 0  S X  e  This g i v e s , using>the i n c i d e n c e matrix  0  p  e i y  V  0  e  L  the n o t a t i o n of P i g . (5A)  are best handled  into  ~1  -  , L  i X  equations  1—coil  i  e  a r e g i v e n by ('5-1) w i t h t h e m u t u a l  0  the  L  f o r t h e s y s t e m c a n once  0  These  +  t h e e q u i v a l e n t impedance  equations  i n (5-22).  r(l)  ss P ss t s r P r srP " » Brr+P rr  A  ^  r  X  ?  (p-jpQ ) r  L  *R  n  33  r  ( +j(py^yM (p-jpy )  e ^ y - y  +L  a  )  P  A  3 3  (5-25J  54 where x - x i  d€ 2~  since  p©  = p©  r  the  possibility  equations As  These e q u a t i o n s  = 0 f o r both  The these  y x-y  =  r  an  and  r  px  rotation  L e t t i n g w-p©  state  •dx  ^  i  J  ~0 0  A  n r  »  e  0 0 e  the torque  r  =  D  A  =  D  p©  r  r r  l  X A  pO  The  machine  =  + J + J  py =0; A  wL^=  XJLJ,  m m4  (5-26)  this  S  +  J 6  (5-27)  0 0 0_  are  r r  p  (p© )  +  A A  p  (p© )  - jL  r  1  o f the  jL  i i m  m  J  n r  selsyn  L  e  5  i"i~  i £ >  0(p-r l)'^^ f  e  o  s  c  i H  + (T •+ T**)  »  Sx^  (i))\  set of equations  these  ' t i  a  z  f n n  o  n  can be  studied using  which apply d i r e c t l y  s  1  (5-28) 3  system  i s i n a t r u e r e f e r e n c e system.  Umnlp)  (At)  S  of small o s c i l l a t i o n ,  (Ae)  from  equations  stability  equations  In  and  0 0 0_  0 0  •0 0  at  = w(s)  -is  CT)  dQP^  r  0  0  t  condition: py =  .= w ( l - v )  r  t  i s included i n  obtains  Rmm+jXrom  and  system  .  r  assume a s t e a d y  p=jw.  r e a c t a n c e , one  py  i n a true  (5-16).  o f commutator  through  example  and  A  (5-24) and  are  Since  equations  M.*  V  ^  J  ( p ) i s g i v e n by  3 e, dy'  since  =  the  the  0,  are  +  BLirTir,  8y"  -n P  (Ay)  (-Vi^ )  (5-25).  Directly  (5^29)  55  3x  >  m  1  r  m  ~0  0  n  0  0  (5-30)  3  i'Apy" =  J  m  rr  e  0  n jL e A(py -py ) 3 |_ -jL.rr-Apy 3 6  r  (5-31)  A  n r  1  ( (T + T ) ( i ) ) t  = (i,) ^ (T + -T*)  t  (i-,  i  % i  B  )  0  ,  0  0  ,  J'L^JL  , 0 , 0 _0_,_ 0 _  (i<"  +jL  i  f  i , 3  m i  0  V4  P  A  y  i =  % i  1  ) 0  ,  0  0  ,  0  0  , -jL  ,  0  ,-jL e°  i  ,  3  0  , -jL h r  e  *JL 6  h r  Vi  '  -jL  m  n  ,  J  68 e  0  (5-32)  A  r  r r  e  j  6  i  5  Ay - j L  n  e  r  j  i ^  &  0 -jL  n r  i  i e  f t  3  j £  (jAy*-jAy )  (5-33)  r  The h u n t i n g e q u a t i o n s f o r t h e s i n g l e  phase e q u i v a l e n t o f t h e /  two—phase  selsyn^system'are 0  Ae^ A.e„  -LmxCp-JP^U^e  0 At  ; 3  At.  L  f- miP  ,  L  e"  r  (p^jpoO^Baa + L  56  •  0  j6 •3  T  -jL  n r  e  l  •T  ,-jL  n r  0  (p+jp(y -yO o  3 3  •  e  , 0  ,  o  h r  e"  i6 • i l  ,DJU  +JJUL- p,  »  0  >D  R R  A(py -pyM  j 6  r  -3't Apy  A  rr  -;jL i iV (jAy^;jAy*) h r  n  6  Aifv' 3 6  Ai  n r  0  jL  Ai^  (p-jpyO.» j x L i % - r j L e  0  A  •  thus  (5-34)  ApQ  0 +J  5  R R  p  1  ApO  3  56  The  brush  axis angles  and t h e i r  are  constrained externally  and a r e t h u s  T h i s example i l l u s t r a t e s the  equations  deriving study  a general procedure  equations.  g e n e r a l beha.vior  variations  known i n a l l e q u a t i o n s . f o r s e t t i n g up  o f m o t i o n f o r any s y s t e m , t r a n s f o r m i n g  the hunting  their  speeds as w e l l as t h e i r  To s o l v e t h e s e  i s complicated  them, and  e q u a t i o n s and  by t h e n o n - l i n e a r i t y  involved.  (1-6,10,13,15,16) 5o2  Synchronous  Machines  In. t h e c o n v e n t i o n a l axes a r e c o n n e c t e d as  stationary.  a n a l y s i s o f s y n c h r o n o u s machines the d-q  t o the r o t a t i n g  This i s convenient  s a l i e n t p o l e s , which are viewed f o r handling  single  machine  p r o b l e m s ; however, when m u l t i - m a c h i n e p r o b l e m s a r e a t t e m p t e d , complicated This  interconnection matrices c a n be a v o i d e d  arise.  by c o n s i d e r i n g .a common d - a x i s  complete  system and a p p l y i n g t h e e q u a t i o n s  saliency  to i n d i v i d u a l  viewed with  machines.  i t s ; stator  of motion w i t h  M o r e o v e r , e a c h machine  s t a t i o n a r y , - r a t h e r than  ment i n w h i c h t h e s t a t o r r o t a t e s .  Fig.  f o r the  5C S y n c h r o n o u s Machine  f r o m an  rotating c a n be  arrange-  I.  57  I-  The  hew r e f e r e n c e  O L — (3 s y s t e m  i s tied  scheme i s i l l u s t r a t e d  to the r o t a r .  the  synchronous  the  d-q s y s t e m s a r e p a r a l l e l y  chronous along  speed.  i n F i g . 5C.  The d-q system  F o r a system  i s rotating at  of interconnected  d i s p l a c e d , each r o t a t i n g  machines-, a t t h e syn-  s p e e d . C o n s e q u e n t l y i n t e r c o n n e c t i o n c a n be made  t h e d-q a x i s and l o o p  a n a l y s i s used  The  directly  t o o b t a i n t h e system  equations. The be  general  impedance m a t r i x  d e r i v e d u s i n g Yu's g e n e r a l  conventional  theory  will  f o r a single  formula  machine w i l l  (3-29) and ( 3 - 3 3 ) .  be c o n s i d e r e d  as a s p e c i a l  when t h e ( d - q ) axes a r e c o n s t r a i n e d t o move w i t h standard  polarity  Fig. S the  5D  i n this  impedance pL  m n  =  of  (L»  i s x , using  cfor  elements are t h e r e f o r e : cX ct m  mri  n  The obtained  the r o t o r .  The  F i g . 5 D . w i l l be u s e d .  ( d - q ) - (aQ-3) R e f e r e n c e case  case  now  +lA  sX sr)p w  n w  Configuration cosine  and s f o r s i n e ,  58 RY  + (-L^sX^Y" + L w ^ s Y * ) + (-L^ eX sY m  + L  n  n  From F i g .  ff-  ,sX cY ) f f ^ m  i m  nn  n  (  5  _  3  5  )  5D  pt^*  (Ld c X c Y m  + U  n  + (-L3 sX cY  n  + (-L* cX *sY  n  rT,  + U  mn  r  sX sY ) m  r n h  cX sY ) m  m n  + U^sX  nh  p  n  px"  n  , n  cY )  ( p y " - py~)  n  (5-36)  (m,n = d,q) pi  = (Li  cX cY m  m n  +.U  n  -(-Li^sXW  sX sT ) p w  m |  n  + L ^ p i ^ s T " ) px-  (5-37)  (m=ds,q; n=f,g,h)  P  L  m  n  ^  (L^ cX cT m  + L^sX-sY* ) p  m  n  + (-I/ cX sY m  n  m n  + LV sX cY ) m  n  n  ( p y * - py*)  (5-38)  (m=fjg,h; n=q,d) V^mrT pt  m n  L  <  W P  = Ifl- p m n  (m,n,=f,g) .  (m,n=h)  (5-39)  ptc,h = pLh-P = pLh^= p L f h = 0 In  a l l these These  chine more  could  py^px^p©  r  be o b t a i n e d b y u s i n g  the synchronous  o f 2%2 b)«  the procedure i s  However,  ma-  tedious. sequence e q u a t i o n  i t i s uncoupled From  of  also  transformations  The z e r o since  elements  (5-40)  section  p@^ e x c l u d i n g  i s omitted  from the remaining  3;5 t h e t o r q u e contributions  tensor  i n the present  s e t of equations. elements are the c o e f f i c i e n t s  from pY* .  F o r the r o t o r  are -(-I/W^cY*  + IA^CX^SI"  analysis  )  (m=d,q,i,n=d,q,f,g,h)  space  they  59 The above using  p r o c e d u r e c a n be e x t e n d e d t o m u l t - m a c h i n e  systems  loop-analysis. The  conventional  by r e s t r i c t i n g Physically space.  (x  4  s y n c h r o n o u s machine  , y  the d-q a x i s  Assuming N  = N  d  i stied  d  d  t o t h e oc -(3« a x i s  and u s i n g L ^ = TJ ;  <1  are obtained  1* = 0, p X = -px~= -pe-*" ,pY =0.  ) such t h a t ;  d  equations  1/^= I A ^ ,  nm  n  i n the s t a t o r "the i m -  pedance m a t r i x i n s t a n d a r d n o t a t i o n i s R +pL£ , M p f  ,  f3  Mf p , R g + L p 3  M  £ a  ,  9  G  0 p  , M  3 d  0  M  ,R  f  M  p  *  h q  _ 2 _ i HE _ Ri+LiPjM^p© ^  p©  1  yR^+L^p  r  d  In the l i t e r a t u r e ,  f o r a generator  o f x^, p © , and t r  Adopting t h i s  (5-41)  M  ~M p©  references  0  0  .» >±P*!h.  p  £ d  reference  r  (alternator),  are opposite  t o those  scheme and p a r t i t i o n i n g ,  the p o l a r i t y  shown i n Pig« 5D  the equations f o r  an a l t e r n a t o r a r e i' = i i ii l i = \  e 5 0 S>_  f  f  3  I  a  ed  ,  0  , -R -L p,  0  (z .) = - R - p L , - M p £  (  £  -M p ^9 0 rx  <*3>  =  -M  f d  f3  a  9  0  P  -M p© f i  5  (z )  -M p  a  £d  -M a p s  o  ,-TR, —pL •  y -^Mgip  , M  , -Mg p©  , -M^V  d  9  K < l  p©  , o , o -M  -R -L p 4  a  ; M p©  -M p©  r  , -Rq-L^p  d  r  c l  (5-41)  (e_) i s t h e n t h e d and q a x i s g e n e r a t e d v o l t a g e The f i r s t iminate =  matrix equation i n the p a r t i t i o n e d  l ! : from t h e s e c o n d s e t g i v i n g  (z ) ( z j ' M e J 8  s e t c a n be u s e d  - (z ) (z ) a  x  _ : L  (z ) a  ( i ) + (z«)' i a  a  (5-42)  60 Written i n f u l l  this i s L^(p)p©  G(p)pE  +  G(p)p© E r  r  (5.-43)  -r -L (p)p d  it  4  a s s u m i n g r<j.=r L (p) =  p (L  (M  a  d  ft :  a  f J  f -2M <iM M a  p (L L -  (M  a  f  L^(p) =  a  £9  ) )  + L (M a) £  3  f  f  f  9  g  hq  9  £ <  p (L L -  (5-44)  f  a  3  £  g  G(p)pO E r  i s seen t h a t  t-r-Li(p)p  , L^(p)p ©  -L (p)pQ  , -r-L^(p) p  r  d  that'all  open c i r c u i t  quantities.  r  (5-45)  f o r t h e a r m a t u r e a x e s , t h e impedance  e x a c t l y t h e same f o r m as p r e v i o u s l y ,  (i° = ^  £  these e q u a t i o n s i n t h e form - G(p)pE  a  ) ) + p ( R L + R^Lg) + R B £  £g  f  it  Ma + .BaMja  £q  (M  2  e  £  (M .f p  G(p) = P ( L M L - M  ¥riting  a  +R R  Rh + LhP  E = e  +Rg (Msa f)  + p(R (M *f  £  9  + p(R L + R L )  2  £ g  £d  quantities  (z.,), t h e o n l y d i f f e r e n c e  a r e r e p l a c e d by s h o r t  I n p e r u n i t n o t a t i o n , adding the zero  (i - + i + i 3  b  c  ) , where i " , 3  i  t e n s o r has  b  i are  t h e phase  c  9  being  circuit  sequence  equation  quantities)  v3  w h i c h r e m a i n s unchanged  throughout^ Parks equations are obtained.  e = G ( p ) p E - z ( p ) i + x ^ p ) 1°L d  d  = G(p)p© E r  e < L  e =  r  d  d  (5-46)  z (p)i^ q  -z i°  0  0  The  torque equation  t = D r  The  <- x ( p ) p © i -  r  r  P  ©  r  + J  electromagnetic  (3-04) i * e .  (p© ) r  r r P  +  ^"?i i m  n  energy c o n v e r s i o n torque t  e be  overcome  by t h e p r i m e  mover.  r  i s the torque t o  (6)  61  o , -Mfi,  o -M  M»  gi  M  0  H  0  -M  d  i  = Using  (i^-)(-K  i -M f  £i  once a g a i n t h e f i r s t i i^(x,(p)  - x (p))  d  in  ii -M  Park's  d  c L  ) + i (M i  partition  h c i  i  V l  + M^)  (5-47)  of (5-4l) to e l i m i n a t e ( i ) 1  + i ^ p )  e  (5-48)  £  form of the equations  o b t a i n e d as f o l l o w s .  dependence-(R)(i)  (5-46) and  F o r the u n s a t u r a t e d  m a t r i x ' Has b e e n o b t a i n e d  (e) =  ( 1  p e * - u n i t system.  A different  angular  i  9  3 <  4  i t s coefficients  case,  (5-48) can be  once t h e  impedance  no l o n g e r d i s p l a y  ¥e c a n w r i t e f o r t h e  alternator  £^  r  (L)(i) + (T) (i)p©  any  (5-49)  r  where (L) • = +L  0  £  0 0 +M  (TL =  -M (L)(i), coils  Mfa M g d  0~ 0  0  0  0  0  £d  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  o  0  %  £d  -M  M  gi  g i v e s the f l u x  i n t h e same a x i s .  due t o e x c i t a t i o n  K  0  -M  d  (5-50)  0  i n each c o i l  due  to e x c i t a t i o n  ( T ) ( i ) , g i v e s the f l u x r  of a l l c o i l s  at r i g h t  angles  i n each  of a l l coil  t o i t and hence  62  is  a cross-flux. After  written e  elimination  currents,  (5-49) c a n be  -^p©  d  d  4^ p©  e« -= - r i ^ + p ^ l  = -ri  ) , the r o t o r  1  (6) sequence added —  w i t h the zero  = -ri +py  d  of ( i  (5-51)  d  +pvy  6  0  Here  .y In  = G(p)E - x ( p ) i  d  this  t  form the torque  = i^-  r  For unit  (5-48) i s  (5-53)  Also  d  f  state  the a l t e r n a t o r  angle  lags  p=0 and p © = l r  i n the per  and x ^ ( p ) = x^, G(p) =  d  i s the " i n t e r n a l  f  = E f . Assuming  bus whose f i e l d  a t steady  x (p) = x  I f E = E p j - j where E G(p)E  equations  -  an a l t e r n a t o r  system  (5-52)  \  d  d  that  generated voltage",  i s connected  t o an  then  infinite  o f t h e g e n e r a t o r by 6 so t h a t  E  =  e sin 6 (5-54)  e cos 6 the  equations  o f m o t i o n become  =  e sin £ e c o s &-  t*=  i  d  E^  x<^  _-x  i ^ (xo-Xd)  The h u n t i n g from  —r  + E  (5-55)  -r _  d  £  i ^  (5-56)  i ^  e q u a t i o n s f o r a s y n c h r o n o u s machine  (4-23) b y t h e f o l l o w i n g  changes  c a n be  ( c . f . 3 -33; 3 - 3 4 ) .  obtained  63  r  p In the r e p r e s e n t a t i o n used here the r e f e r e n c e to  the f i e l d  axes and t h u s o s c i l l a t e  (AY  n  =0)«  axes r e m a i n  tied  The h u n t i n g  equ-  a t i o n s become  "(Ae)  (z*m(p))  ?  p  )  ^ d Y ^ ^ ' i "  (At) For  these From  (Ai) (ApO)  equations  the r e f e r e n c e s  o f Fig„  5D w i l l be. used,  (5-54)  0 0 0 as e co s 8 3Y  d ein c>Y  n  n  |-e s i n For  (5-58)  a machine c o n n e c t e d t o an i n f i n i t e -feus S = y'°S y* = y * - y fc  4  (y^-y ) 4  where y ^ i s t h e r o t o r a n g l e chosen r e f e r e n c e  3Y Hence  Using  ?  o f t h e bus w i t h  respect  to the  this  n  0 0 3 e to _ 0 BY -e cos 6 e sin6 n  The  voltage variation  Ae  (5-57)  A -'e  f  - "  0  o e s i n £> § c o s £>_  (5-59)  Ae i s  —  0  ^Ae  0  0  0  0  e cos £ -e s i n &_  f  Ae s i n 6 _Ae cos6_  (5-60)  64 where A & i s t h e v a r i a t i o n  o f & o t h e r than L i e , the l a t t e r  a l r e a d y been i n c l u d e d i n t h e e q u a t i o n s  A& = L{y~-  y*) - M y ^ - y ) 0 6  = Ay"= A 9  The g e n e r a l i z e d i n c i d e n c e m a t r i x f  0  9  0  pX PX  pX  h  je^  -i  ^\  ~i  ^  i s g i v e n by  r  (5-62)  the r e m a i n i n g  terms o f (5-57) a r e  0  n  c)I  O i r y  (5-61)  r  (p© )  ~  r  n  0 0 0 0 -1 —e cos B e sin6•—1  0  -  8  Here  0  px  Therefore  of motion.  having  n r  0 0 0 e cost -e s i n 6  0  0  0  0  0  0  0  G  0  0  19  G  0  0  0  0  i*  0  0  0  M  d  o  -M  i  a  (5-64)  4  ii  0  d  (5r63)  0 0 0 M i + h  K  -M ^P,^=  £ < i  i  it)\(T%  - M dLi 9 _ M i  £  !  a  +  (T)  (5-65)  c  d  )  r  —t  t  -M d i ^ 3  Mhqi 4  M  i + h  h c i  (M  (M^Md) i  i + M £  £ c L  j i ) + (M^-Ml) 2  3 <  (5-66)  ^ i  d  65 Substituting  these  i n (5-54) the  t h e p o l a r i t y o f F i g . 5D  are on t h e f o l l o w i n g  Although t h i s d e r i v a t i o n proach  can be  extended  o s c i l l a t i o n s equations  with  page^  i s f o r a s i n g l e m a c h i n e , the  to a multi-machine  system  0  ap-  Ae  £  0  >  0  > £dP  7  0  9 Rg+pLg 9  0  9 M  ?  0  R +pli£ J £  M  £ g  P  Ae s i n £  M  A e cos S  ~M  f d  p £ d  £ P 3  o  0  0  M  1  p©  ,R  ,  MgdP  r  ~M p©*\  ?  n  +pL  M  9 d  Mh^P  r  p  0  h  9 Mhq.P©  g d  S  d P  hqP  ©  f  0  Ai  9  , "0  Ai  h  Ai  d  ,M  1  R^+L^p  0  A i  9  r M^p© ^  o R^+L^p 9 -M  M  ,  h < L  ,-M  i  r  h  i -M di -M i - ( • M i +Mqdi ), Dp + J p + (Mq_^Md) i £  £ d  g  a  Ai^  a  d  At  ™M£ i^ d  9  -=Mc,di^- ,  »•  ML, i  h  a  £ d  £  A©  d  ( 5 - 6 7 )  T  67  6. CONCLUSIONS By s e l e c t i n g holonomie and  derived*  Lie  coordinates,  s y s t e m , and a p p l y i n g H a m i l t o n ' s P r i n c i p l e ,  torque  chanical  a s e t of independent  equations  i.e., a the voltage  a p p l i c a b l e t o any p h y s i c a l machine were  Contributions arising  f r o m i n t e r a c t i o n s between me-  r o t a t i o n s and m a g n e t i c f l u x e s were shown t o be due t o  variations. Once t h e b a s i c e q u a t i o n s  were d e r i v e d , t h e h u n t i n g  and  transformation properties followed readily.  the  independance  the  current transformation matrix,  of the v o l t a g e  i o n o f the system e q u a t i o n s . formation properties The those  i s f r o m one t r u e  transformation matrix, (C), f o r a general  (€!), and transformat-  R e s t r i c t i o n s a r i s i n g when t h e t r a n s -  system t o another  leads to the i n v a r i a n c e  t a b l e f o l l o w i n g compares t h e c o n v e n t i o n a l those  presented  i n the present  t h e s i s i n d i c a t e s s e v e r a l p o s s i b l e areas  gation: s t a b i l i t y approach^  Of i m p o r t a n c e i s  o f power..  o f Kron, w i t h The  equations  and o p t i m i z a t i o n s t u d i e s u s i n g  the r e a l i z a b i l i t y  of a true  ideas,  thesis. of i n v e s t i -  the Hamiltonian  s y s t e m , w h i c h may n o t  n e c e s s a r i l y be a p h y s i c a l system a l t h o u g h  the converse  true;  problems.  and t h e i n v e s t i g a t i o n  principally  of n o n - l i n e a r  i s always  68. Conventional  Present  Slip  ring only  machines  Any p h y s i c a l  L a g r a n g e -,s equations  Slip  ring only  machines  Any p h y s i c a l machine t a k i n g i n t o account Lie variations  Quasi-^holonomic  F i x e d commutator a x i s machines  Non—holonomic  Anything which i s not Any hypotheteifjLl machine i n c l u d e d i n t h e above, f o r w h i c h t h e t r a n s hypothetical or formation matrices physical (C). and (•€) f r o m a p h y s i c a l machine satisfy  Holonomic  system  r  No  Thesis  such system  dt r ' u  Direct a p p l i c a t i o n Slip, r i n g of l o o p o r N o d a l analysis Moving  saliency  —  —  dt ^  Any t r u e s y s t e m ; i n c l u d e s a l l p h y s i c a l systems  machines  t  Non-holonomic  I n v a r i a n e e of tensor products  machine  Holonomic Follows from properties  —  standard references  (C) and  (£)  Polarity convention  No  F o l l o w s from the b a s i c equations  General comment  System e q u a t i o n s o b t a i n e d General equations d e r i v e d by K r o n are c o m p l e t e l y g e n e r a l , apply only to a including non-linearities, l i n e a r system, i e , , and a r e l e s s c o m p l i cated i n form. b  Table  2„  Comparison  of C o n v e n t i o n a l Ideas w i t h Thesis  t h o s e of t h e  Present  69 APPENDIX I  The by  CONDITIONS ON THE FLUX TENSOR TO GUARANTEE AN EXTREMUM  e x i s t a n c e o f an extremum o f t h e a c t i o n  t h r e e t e s t s w h i c h c a n be a p p l i e d  extend  the r e s u l t s  static  circuit  of r e a l i t y  i s guaranteed  to the L a g r a n g i a n .  tests  f o r mutual  These  i n d u c t a n c e din \ \  theory.  -\ \  The  three t e s t s w i l l  be d e a l t w i t h i n t u r n .  \ '»  a) b)  The E u l e r E q u a t i o n s jThe L e g e n d r e  j  the  ! '  are s a t i s f i e d .  *  |  I f S = $ L(t,q -,q )dt  Teste  ;  o  q u a d r a t i c forni whose m a t r i x  is(  1  i s an extremum  ) = (E^i,^i)  I  . ; ! ; '  must be/  //  positive  I ,/  definite.  Thus t h e m a t r i x  ML .  is  positive  this  Si'  ' d i  di'  '  a  definite.  to zero.  c)(P.  '  c5i ' * "  A necessary  t o be t r u e i s t h a t  or equal  a  and s u f f i c i e n t  the minor d e t e r m i n a n t s  That i s  dO,  di' ' di  £  >  0  LSi' S i . £  F o r the. u n s a t u r a t e d case  this i s  condition f o r  be g r e a t e r  than  70 L  The  >0  J a c o b ! Te/st:  (  d~E  3i?,di  r n  > °  »  II  c)  6 i  A l l terms  u d*E di d,£ 8  \ '  s  of the  / 6 E ^di di A  £  \  form;  A  V  r  must b e . g r e a t e r t h a n o r e q u a l t o z e r o .  l  For  no  "dT  ai^  F H  i s included  siderations  0  saturation this i s (L  This  "  )  That i s  )(L  r r  s s  -  (L ) r s  i n the above  o r as can be  di  G )  )  £  ^ 0/  s i n c e L-£  s  = L  proved d i r e c t l y  r  E  e)i di s  £  _  r  -  aor _ di  s  _  _a E £  d± di T  s  _  d^s  ~ Dir  a r  since  by  energy  con-  71 APPENDIX I I  THE HAMILTONIAN; ENERGY AND CO-ENERGY  The'Lagrangian implicit (closed  o f an e l e c t r o m e c h a n i c a l system i s an  f u n c t i o n o f time  and hence i f d i s s i p a t i o n  system) t h e H a m i l t o n i a n  Hamiltonian  f o r the e l e c t r i c a l  La,grangian  i s conserved.  i s neglected  I f H i s the  p a r t o f t h e system and L  e  the  c)L H  = cpiiH i s this is  case  i s not equal  E  t o E g e n e r a l l y and hence t h e e n e r g y @  g e n e r a l l y n o t a c o n s t a n t under f r e e  oscillations  of the  (14)  machine.  It will  An  fluctuate  interpretation  generalizing  t h e case  Fig.  IIA  so as t o m a i n t a i n  o f the H a m i l t o n i a n  of a s i n g l e  coil.  Single Coil  B H Curve  H a constant.  can be made by  (Pig. IIA).  i In t h i s The  case  co-energy  the energy  i s ^ ( p d i which i s equal  i s d e f i n e d as C  =Cpi  - E  =tpi-  i \(p(i)di  t o t h e shaded  area,  w h i c h i s the unshaded a r e a o f the r e c t a n g l e , and Hamiltonian  0  Thus as a g e n e r a l i z a t i o n , is  the  co-energy  of the  system  d e f i n e d as C  and  i s the  the  Hamiltonian  E  =  (PJL I  i s equal  1  - E  to the  e  = H generalized  co-energy.  73 REFERENCES  L  Adkins,, B., The G e n e r a l T h e o r y o f E l e c t r i c a l M a c h i n e s ( B o o k ) J o h n ¥ i l e y & Sons I n c . , New Y o r k , 1957. s  2.  Bewley,  L.V. „. T e n s o r A n a l y s i s o f E l e c t r i c a l C i r c u i t s and M a c h i n e s (Book),, R o n a l d P r e s s Co., New Y o r k , .1961.  3.  C o n c o r d i a , C , Synchronous New Y o r k , 1951.  4.  G i b b s , W.J., T e n s o r s i n E l e c t r i c a l Machine Chapman & H a l l , Tendon, 1952.  5.  ,Keller, Ernest Dover  6.  Kron,G„,  7.  K r o n , G.., Non-Rtemannian Dyna.mics o f R o t a t i n g E l e c t r i c a l M a c h i n e r y , J o u r n a l c f M a t h e m a t i c s and Physics', X I I I 2, May 1934, pp. 103-1.94.  8.  K r o n , G., S e r i e s Review,  9.  K r o n , G., The A p p l i c a t i o n o f T e n s o r s To t h e A n a l y s i s o f R o t a t i n g E l e c t r i c a l M a c h i n e r y (Book) T  Machines  ( B o o k ) , J o h n Wiley-&  Theory  (Book),  G. , M a t h e m a t i c s o f Modern E n g i n e e r i n g P u b l i c a t i o n s I n c . New Y o r k .  Tens o r s F o r C i r c u i t s New Y o r k , 1959.  ( B o o k ) , Dover  Sons,  ', V o l I I  Publications Inc.,  of P u b l i c a t i o n s i n the General E l e c t r i c S c h e n e c t a d y , New Y o r k , P a r t s I - X V I I I , 1935-38.  10.  P a r k , R o H . , Two R e a c t i o n T h e o r y o f S y n c h r o n o u s M a c h i n e s , A I E E T r a n s a c t i o n s , P a r t I , V o l . 48, 1929, pp. 716-73Q, P a r t I I , V o l . 52, 1.938, pp. 3 52-354.  11.  S c h o u t e n , J.A., R i c c i C a l c u l u s ( B o o k ) , 2nd e d . , S p r i n g e r - V e r l a g , B e r l i n s , G o t t i n g e n , 1.954.  12.  Schouten, J . A . T e n s o r s O x f o r d , 1951o  For Physicists  (Book), Clarendon P r e s s , ;  13.  T a k e u c h i , T . J . , M a t r i x Theory of E l e c t r i c a l Ohm-Sha, T.okyo, 1962.  14.  W h i t t a k e r , E.T., A n a l y t i c a l Dynamics ( B o o k ) , U n i v e r s i t y P r e s s , 4 t h e d . , 1961.  15.  W h i t e , D.C., and Woodsen, H H , E l e c t r o m e c h a n i c a l E n e r g y C o n v e r s i o n (Bo\ok), J o h n Wil^y^fe Sons, New Y o r k , 1959.  16.  Wood,  17.  Yu, Yao-nan, The Impedance T e n s o r o f t h e G e n e r a l M a c h i n e , T r a n s . A I E E , p t . I (C o m m u n i c a t i o n and E l e c t r o n i c s ) , V o l . 75, May, 1956, pp. 181-1.87.  18.  Yu, Yao-nan, The Torque T e n s o r o f t h e G e n e r a l M a c h i n e , Power A p p a r a t u s and S y s t e m s , F e b . , 1963, pp. 623-629.  0  (Book),  Cambridge  0  Theory of E l e c t r i c a l B u t t e r w o r t h , I9\58.  W o S o ,  Machinery  Machines  (Book),  London,  

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