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Steady-state characteristics at subsynchronous speeds of an SCR-controlled synchronous motor Kano, Takashi 1971

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STEADY-STATE CHARACTERISTICS AT SUBSYNCHRONOUS SPEEDS OF AN SCR-CONTROLLED SYNCHRONOUS MOTOR  by  TAKASHI KANO B.Sc,  Doshisha  A THESIS SUBMITTED  U n i v e r s i t y , 1969  IN PARTIAL FULFILMENT OF  THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE  i n t h e Department o f Electrical  We accept  this  Engineering  t h e s i s as conforming  required  to the  standard  Research S u p e r v i s o r  Members o f t h e Committee  Head o f t h e Department  Members  o f the Department  of E l e c t r i c a l  Engineering  THE UNIVERSITY OF BRITISH COLUMBIA  A p r i l , 1971  In  presenting  this  an a d v a n c e d d e g r e e the L i b r a r y I  scholarly  by h i s of  at  written  thesis  that permission  It  financial gain  permission.  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a  J2  Columbia  / l / w / s / f f /  the  requirements  B r i t i s h Columbia, for  I agree  r e f e r e n c e and copying of  this  shall  that  not  copying  or  for  that  study. thesis  by t h e Head o f my D e p a r t m e n t  is understood  Department  Date  of  for extensive  p u r p o s e s may be g r a n t e d  for  fulfilment of  it freely available  representatives.  this  in p a r t i a l  the U n i v e r s i t y  s h a l l make  f u r t h e r agree  for  thesis  or  publication  be a l l o w e d w i t h o u t my  ABSTRACT Okada's three-phase s t a r - c o n n e c t e d  circuit  SCRs i n s e r t e d i n the n e u t r a l p o i n t i s a n a l y z e d method.  The  three-phase synchronous motor w i t h  with  three  u s i n g Take-uchi's  delta-connected ^-function  three delta-connected  SCRs  i n s e r t e d i n the n e u t r a l p o i n t of the armature w i n d i n g s i s then i n v e s t i g a t e d . By  c o n t r o l of the f i r i n g  possible.  The  of the SCRs, o p e r a t i o n at subsynchronous speeds i s  a n a l y s i s of the s t e a d y - s t a t e o p e r a t i o n o f the  synchronous motor i s e x p e r i m e n t a l l y  checked.  ii  SCR-controlled  TABLE OF CONTENTS Page ABSTRACT  i i  TABLE OF CONTENTS  i i i  LIST OF ILLUSTRATIONS  iv  ACKNOWLEDGEMENT  V  NOMENCLATURE  vi  1.  INTRODUCTION  1  2.  CHARACTERISTICS OF THREE-PHASE DELTA-CONNECTED SCRs  4  2.1. 2.2. 2.3. 2.4. 2.5. 2.6.  3.  ANALYSIS OF THE SCR-CONTROLLED MOTOR 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7.  4.  Introduction A n a l y s i s f o r Case 1 A n a l y s i s f o r Case 2 A n a l y s i s f o r Case 3 Comparison o f A n a l y t i c a l and E x p e r i m e n t a l R e s u l t s f o r ThreePhase D e l t a - C o n n e c t e d SCRs R e l a t i o n s h i p Between t h e F i r i n g Angle and t h e E x t i n c t i o n Angle  Introduction Mode 1 O p e r a t i o n o f the Motor S a l i e n t P o l e Motor E q u a t i o n s , Mode 1 O p e r a t i o n Round-Rotor Motor, Mode 1 O p e r a t i o n Mode 2 O p e r a t i o n o f the Motor T r a n s i t i o n between Mode 1 and Mode 2 S t a r t i n g Torque  EXPERIMENTAL VERIFICATION MOTOR 4.1. 4.2. 4.3. 4.4.  .5.  v  4 5 9 13'.." 13 14 18 18 21 23 27 28 28 35  OF THE OPERATION OF THE SCR CONTROLLED 36  System Components '. 36 D e t e r m i n a t i o n o f y = 0 P o s i t i o n o f the D i s t r i b u t o r 38 Comparison o f A n a l y t i c a l and E x p e r i m e n t a l Current Waveforms . 42 Speed-Torque C h a r a c t e r i s t i c s 48  CONCLUSIONS  51  APPENDIX .  52  REFERENCES  59  iii  LIST OF ILLUSTRATIONS Figure  .  Page  1.1  Motor c o n f i g u r a t i o n w i t h c o n t r o l l o g i c  2  2.1  Three-phase d e l t a - c o n n e c t e d SCRs  4  2.2  Three-phase d e l t a - c o n n e c t e d SCRs under Case 1 o p e r a t i o n w i t h SCR conducting  1  5  2.3  Waveforms f o r Case 1  15  2.4  Waveforms f o r Case 2  16  2.5  E x t i n c t i o n angle —  3.1  Circuit  3.2  Waveforms f o r c i r c u i t  3.3  Motor w i t h r o t a t i n g D.C. f i e l d  firing  a n g l e curves  17  t o g e n e r a t e SCR gate p u l s e s  19  o f F i g . 3.1  20  under Mode 1 o p e r a t i o n , SCR^  conducting 3.4  21  E q u i v a l e n t motor w i t h r o t a t i n g armature under Mode 1 o p e r a t i o n , SCR^ c o n d u c t i n g  22  3.5  E q u i v a l e n t motor w i t h m e c h a n i c a l commutator  3.6  Motor under Mode 2 o p e r a t i o n , SCR^ and 5CR^ c o n d u c t i n g  29  3.7  D e f i n i t i o n of angle y  30  4.1  E x p e r i m e n t a l set-up  37  4.2  D i s t r i b u t o r and synchronous motor  39  4.3  D i g i t a l phase s h i f t e r  40  4.4  Armature c u r r e n t waveforms  4.5  Field  4.6  Armature c u r r e n t waveforms  4.7  Field  4.8  Armature c u r r e n t waveform  4.9  Speed-torque c h a r a c t e r i s t i c  (y = 40 deg.)  49  4.10 Speed-torque c h a r a c t e r i s t i c  (y = 80 deg.)  49  c u r r e n t waveforms  c u r r e n t waveforms  and b r u s h e s  a t speed o f 720 r.p.m  at speed o f 720 r.p.m a t speed o f 900 r.p.m  a t speed o f 900 r.p.m. a t speed o f 1080 r.p.m  iv  23  43 44 45 46 47  ACKNOWLEDGEMENT I w i s h t o e x p r e s s my s i n c e r e thanks t o t h e p e o p l e who have g i v e n me a s s i s t a n c e throughout t h e c o u r s e o f t h i s p r o j e c t .  Especially, I appreciate  the guidance and encouragement g i v e n by Dr. H. R. Chinn and Dr. Y. N. Yu, my supervisors. Thanks a r e due t o Mr. D. G. Mumford and Mr. E. S t r u y k f o r t h e i r c a r e f u l p r o o f r e a d i n g , and t o M i s s L . M o r r i s f o r h e r t y p i n g . I  am g r a t e f u l f o r the a s s i s t a n c e o f Mr. W. R. B l a c k h a l l , Mr. C.  Chubb and Mr. D. G. Daines f o r p r e p a r i n g the e x p e r i m e n t a l s e t - u p , and o f Mr. H. H. B l a c k f o r p r e p a r i n g the photographs  i n the t h e s i s .  A l s o , I w i s h t o thank Dr. B. J . K a b r i e l f o r h i s v a l u a b l e comments. The  f i n a n c i a l s u p p o r t o f the N a t i o n a l Research C o u n c i l o f Canada  and t h e U n i v e r s i t y o f B r i t i s h  Columbia  i s gratefully  v  acknowledged.  NOMENCLATURE current  through phase windings  a, b and c  d- and q - a x i s c u r r e n t f i e l d current load  inductance  d- and q - a x i s components o f armature i n d u c t a n c e field  inductance  magnitude o f mutual i n d u c t a n c e between f i e l d •j^,  differential  and armature w i n d i n  operator  number of machine p o l e - p a i r s load resistance field output  (Chapter 2 ) ; armature r e s i s t a n c e  (Chapter 3)  resistance torque i n Newton-meters  phase v o l t a g e  amplitude  v o l t a g e s o f phase a, b , c l i n e - l i n e v o l t a g e between t e r m i n a l s A and B d- and q - a x i s v o l t a g e f i e l d voltage firing  angle  extinction  angle  angle between q - a x i s and b r u s h a x i s i n e q u i v a l e n t motor w i t h mechanical power f a c t o r  commutator and brushes angle  a n g l e between q - a x i s and a x i s o f r o t a t i n g magnetic a n g l e between q - a x i s and armature w i n d i n g electrical  field  o f phase a  a n g u l a r v e l o c i t y o f the motor i n r a d i a n s p e r second  vi  power s u p p l y f r e q u e n c y i n r a d i a n s p e r second to.  f r e q u e n c y o f the c u r r e n t waveform i n r a d i a n s p e r second  0)  m e c h a n i c a l motor speed i n r e v o l u t i o n s p e r second  1  mech  vii  1  1.  INTRODUCTION  E l e c t r i c a l motors w i t h speed c o n t r o l a r e r e q u i r e d a p p l i c a t i o n s , such as paper and steel" m i l l d r i v e s . motors  a r e o f t e n used f o r such p u r p o s e s .  D.C.  f o r many  and A.C.  commutator  However!',' m e c h a n i c a l commutators  and t h e i r s p e c i a l w i n d i n g c o n n e c t i o n s a r e major d i s a d v a n t a g e s because o f maintenance  and s e r v i c i n g . Synchronous  hand,do not r e q u i r e commutators.  and i n d u c t i o n motors, on the o t h e r  But a synchronous motor o p e r a t e s a t a  f i x e d speed, the synchronous speed, and an i n d u c t i o n motor o p e r a t e s a t speeds s l i g h t l y  lower than the synchronous  speed.  T h e i r speed can be  c o n t r o l l e d o v e r a wide range o n l y by changing the s o u r c e s u p p l i e s . T h y r a t r o n s and mercury  a r c r e c t i f i e r s were used to c o n t r o l the  speed o f e l e c t r i c a l motors, known as T h y r a t r o n motors Stromrichtermotoren rectifiers  (2).  With the development  of s i l i c o n  Methods f o r c o n t r o l l i n g  the speed o f A.C. motors  v a r i a t i o n o f the s u p p l y f r e q u e n c y by means of SCRs Krause, e t . a l .  using  and L i p o , e t . a l .  involve  to obtain  and M i y a - i r i ,  et.al.  ( 7 ) , (8) o p e r a t e d a synchronous motor i n a s i m i l a r  manner.  Miya-iri, et.al.  and D.C.  power s u p p l i e s to d r i v e synchronous motors.  ( 9 ) , (10) and Sato (11) a l s o r e p o r t e d u s i n g  one f r e q u e n c y to A.C.  inverters  Tsuchiya, e t . a l .  o p e r a t e d a synchronous motor w i t h a c y c l o - c o n v e r t e r , which  (12)  c o n v e r t s A.C. at  a t some o t h e r f r e q u e n c y .  T h i s t h e s i s p r o j e c t examines F i g . 1.1, which was  i s being  converter-inverter  ( 3 ) , ( 4 ) , (5) used t h i s arrangement  a v a r i a b l e - f r e q u e n c y s u p p l y to d r i v e an i n d u c t i o n motor, (6)  controlled  (SCRs), or t h y r i s t o r s , the speed c o n t r o l of A.C. motors  re-examined.  schemes.  (1) o r K o l l e c t o r l o s e r  originally  the motor c o n f i g u r a t i o n shown i n  t e s t e d by Okada at D o s h i s h a U n i v e r s i t y .  Three  Three Phase Power Supply Q O O  6 o  6 CD  JO lQ Co N j  o  CO  o Co  'Co  Q  icj Co 33  CD  6  3  SCRs, d e l t a - c o n n e c t e d , a r e i n s e r t e d i n the n e u t r a l p o i n t of t h e three-phase armature w i n d i n g o f the synchronous motor. as f o l l o w s :  Firing  SCR^, the magnetic  The p r i n c i p l e o f t h i s motor i s  field  caused by the armature  i s a p p r o x i m a t e l y i n the d i r e c t i o n from A towards SCR to  2  i s fired  so t h a t the d i r e c t i o n of the magnetic  align i t s e l f  B and C. magnetic  B.  i n a d i r e c t i o n from A towards  SCR^ then e x t i n g u i s h e s and o n l y SCR f i e l d produced by the armature  from B towards  C.  With SCR^ c o n d u c t i n g , field  rotates clockwise  a p o s i t i o n somewhere between 2  remains  conducting.  resulting  from the armature  has r o t a t e d about 120 degrees from the o r i g i n a l d i r e c t i o n .  r o t a t i n g magnetic  2  field.  c o n t r o l l e d by f i r i n g  The  c u r r e n t i s then i n the d i r e c t i o n  Now t h e magnetic f i e l d  same p r o c e d u r e f o r SCR  current  current  R e p e a t i n g the  and SCR^, and so on, the s e t o f SCRs p r o v i d e s a The speed o f the r o t a t i n g magnetic  f i e l d may be  these SCRs i n the above sequence w i t h p u l s e s d e r i v e d  from the p o s i t i o n and speed o f the r o t o r . In circuit  this  t h e s i s , the d e l t a - c o n n e c t e d SCRs i n a three-phase A.C.  (13) i s a n a l y z e d u s i n g T a k e - u c h i ' s <j>-function method ( 1 4 ) .  b a l a n c e d three-phase c i r c u i t  w i t h o u t mutual  v e r i f i c a t i o n of the a n a l y s i s i s g i v e n . to  the motor c o n f i g u r a t i o n o f F i g . 1.1.  e x p e r i m e n t a l s e t up of the system, analytical  c o u p l i n g i s assumed.  A  Experimental  The a n a l y s i s i s extended i n Chapter 3 The d e s c r i p t i o n o f the d e t a i l e d  the t e s t r e s u l t s and comparison w i t h  r e s u l t s a r e g i v e n i n Chapter 4.  2. 2.1  CHARACTERISTICS OF THREE-PHASE DELTA-CONNECTED  SCRs  Introduction B e f o r e the motor c o n f i g u r a t i o n of F i g . 1.1 i s a n a l y z e d , we  investigate in  t h e c h a r a c t e r i s t i c s o f three-phase  the n e u t r a l p o i n t o f a b a l a n c e d  three-phase  shall  d e l t a - c o n n e c t e d SCRs i n s e r t e d s t a r - c o n n e c t e d l o a d as shown  i n F i g . 2.1.  C  o  B  o  1  F i g . 2.1 Three-phase d e l t a - c o n n e c t e d SCRs The gate p u l s e s to the SCRs a r e s y n c h r o n i z e d w i t h firing  a n g l e , a, and t h e e x t i n c t i o n a n g l e , g ,  the.supply frequency.  The  f o r each SCR a r e measured  from  the n e g a t i v e - p o s i t i v e z e r o - c r o s s i n g p o i n t o f the l i n e - l i n e v o l t a g e . Three modes corresponding  o f SCR o p e r a t i o n a r e d e f i n e d as Modes 1, 2 and 3,  to the' number o f SCRs c o n d u c t i n g a t the same  time.  5  Three cases may be c o n s i d e r e d depending upon t h e f i r i n g r e l a t i o n t o the power f a c t o r o f the l o a d . In Case 2 Modes 1 and 2 o c c u r Note t h a t SCR f i r i n g  2.2.  alternately.  c o n t r o l i s completely  angle i n  I n Case 1 o n l y Mode 1 o c c u r s . I n Case 3 Modes 2 and 3 o c c u r . lost  i n Case 3.  A n a l y s i s f o r Case 1 As mentioned above f o r Case 1, o n l y one SCR i s c o n d u c t i n g a t any  time.  Assume t h a t o n l y SCR^ i s c o n d u c t i n g as shown i n F i g . 2.2 and t h a t  the l o a d i n each phase c o n s i s t s o f a r e s i s t o r R and an i n d u c t o r L i n s e r i e s .  B  O  ;  Fig.  2.2  Three-phase d e l t a - c o n n e c t e d SCRs under Case 1 o p e r a t i o n w i t h SCR^ Let  conducting  the phase sequence be a, b and c, v,„ = v - v, , and v,_ be ^ AB a b AB r  the r e f e r e n c e phasor.  The v o l t a g e a p p l i e d t o the l o a d between A and B i s  6  SiV  v. (t) = B  A where co = s u p p l y f r e q u e n c y o A  s i n u t (U. - U.)  i n r a d i a n s per  (2.1)  second,  V = magnitude o f the phase v o l t a g e , A a = firing  angle,  A 3 = e x t i n c t i o n angle, = u n i t step f u n c t i o n U(t - ~ ) > o A g. = u n i t step f u n c t i o n U(t - — ) o  .  T a k i n g the L a p l a c e t r a n s f o r m of e q u a t i o n  s s i n a + co V  A f i  (s)  = /3 V  [  (2.1),  cos a  1 + co o  5  s  ex (- — P  s s i n B + co cos o 2 2 s + co o The  s)  o  exp(-^s):] r  (2.2)  w  o  network e q u a t i o n i s v^Ct)  = 2[R + L p ] i ( t )  Ad where p = -J-JT > the d i f f e r e n t i a l  operator.  T a k i n g the L a p l a c e t r a n s f o r m o f e q u a t i o n •  V  A B  (s)  (2.3),  = 2[RI(s) +L{sI(s)  AiJ  (2.3)  - i ( t )}]  (2.4)  O  where i ( t ) i s the l o a d c u r r e n t at the i n i t i a l time o f s w i t c h i n g , namely t o _ o or  cx/co . o  From e q u a t i o n s  (2.2)  T /  v  I(s)  and  (2.4)  pr „ v3 V  r  s s i n a + co cos a o  •i r , [  +  2  (s + ft) (s  -  • s sm  +  2, + co ) o  e  . x  p  a o g + co cos g O / r r exp((s + ft) (s + CO ) o  (  -  a -  . s  )  o „ —  p  o  (2-5) vi S)J  7 i(t ) o  +  7T^  e x p (  - o t  s )  where ft = R/L. We define the  f(a,t) =  <f>-function (10) as  sin(co t + a - x) ~ sin(a .  - x)exp(-ftt) •  2  Z 2+ ~ /ft  (2.6)  2  10  o  i.e. , s s i n a + to cos 3 <£{<Ka,t)}=  (s + ft)(s  2  + to ) o  (2.7)  2  where X = tan "''(a) L/R) o Using <j)-functions, equation (2.5) becomes I(s) =-^7-  [^{<Ka,t)>  exp(- — o  s)  -XWB.t)}  exp(-^-s)] o  i(t) + ~ r exp(-t s) s + ft o  (2.8)  From equation (2.6), the inverse Laplace transform of equation (2.8) i s /3~V 1 ( t )  =  [sin^t _ ) x  (  U  i  - u ) 2  2/R" + w L" 2  - sin(ct - x)exp(t - — ) U CO  n  1  + sin(3 - x)exp(t - —)'U ]  o + i ( t )exp(t - - f ) U O  CO  CO  /  o  n  X  o By d e f i n i t i o n , i ( t ) i s zero and the e x t i n c t i o n angle 3 must be less than . 2TT a + -y . 0  oi For — $ co 0  3 t < — , U.. i s unity and U„ i s zero, hence, co 1 2 0  (2.9)  8  yr i(t) = —  —  [  v  a) t - a i  S  (  n  t - x) - s i n ( a - ) e x p ( - f t —  u  )]  x  (2.10)  o where i ( t ) % 0. The equation  extinction  (2.10).  angle g i s determined by s e t t i n g i ( t ) t o zero i n  Therefore,  sin(g - x )  e x  8  P(^—)  =  sin(a - x )  e x  ex P(^—)  o Note t h a t  (2.11)  o  6 cannot be e q u a l t o a. E q u a t i o n (2.10)  o f the n e g a t i v e p o r t i o n  gives the p o s i t i v e p o r t i o n  of i , .  o f i , and the magnitude  The p o s i t i v e p o r t i o n s o f i , and i a r e  b  b  e  o b t a i n e d i n the same way. In  general, , iCt) -  ,[sin(co t - y) {U(t  2^ — ) to  o  2TT  9  - s i n ( a + n4f  J  - )exp{-ft(t X  a + n-r— ) }U(t to o g.  +  + s i n ( 3 + n- ^ - ) e x p { - f t ( t  2TT  T1  2  J  X  to o  0  — ) } U(t  - U(t  , 2TT tu  — ) )  , 2TT  a + n-r— ) to o g  +  to o  ^ — )]  (2.12)  where i ( t ) > 0 and n = 0, 1 and 2 f o r the p o s i t i v e p o r t i o n s o f i . i . and i a b c respectively. The  negative portion  c u r r e n t flows from C t o A. i.e., i = - i ' a c where i < 0. a  o f i i s o b t a i n e d when SCR conducts s i n c e t h e a j  9  Therefore, the t o t a l current  V  t  ;  "  i  i s given by  5-5- [sin(co t - H U ( t 2/R + co L o  - U(t - -2-)} "o  X  n  2  2  2  0  U  - s i n ( a - )exp{-ft(t - — ) }U(t - — ) + sin(g - )exp{-fi(t - — ) > U ( t - — ) ] * CO CO CO CO o o o o X  X  {U(t - — ) - U(t - -£-)} CO CO o o  •  a + 4r B + 4? — - ) - U(t - — — - ) ) o "o  y^v  , /o o o [sin(co t - ^ - ){U(t zA + co L ° o . X  2  2  2  3  U  , 4TT a + — —))U(t CO o  - s i n ( a - x)exp{-fi(t  B + 4^-  + sin(B - x)exp{-Q(t  —)}U(t  CO  .  a +  g + 4f — ) ] * CO  o  , 4TT a + -5-) - U(t CO o  {U(t  2.3.  4TT  — —) 'CO o  o  4TT  B + -~CO  -)}  (2.13)  o  Analysis f o r Case 2 The e s s e n t i a l difference between Case 1 and Case 2 i s i n the value  of the extinction angle 3.  In Case 1, the extinction angle g f o r one phase i s  2TT  always less than a + —^r, which i s the f i r i n g angle of the following phase.  How-  ever, i n Case 2, two SCRs may be conducting under Mode 2 operation 2 4 greater than or equal to a + y but less than a +  g is  To obtain the steady-state following. as follows:  current  so that  through phase a, consider the  Let the f i r i n g sequence of the SCRs for alternate Modes 2 and 1 be SCR  2  and SCR , SCR only, SCR and SCR^ SCR only, SCR 3  3  3  ±  ;L  and SCR  SCR only, and so on. Then the voltage applied to the load of phase a i s 2  v (t) = V s i n ( c O t o  - |)(U  X  - U) 2  + 4j  V sin(co t Q  - |) ( U  2  - Ug)  2>  10  + V sin(co t -  +  V  sin(co t o  (U,- U.) +  T-)(U_ 6 5  V s i n (to t) ( U . - U _ )  U,)  (2.14)  o  2TT  77 —)  .Ct  .  where U , = U ( t 1  too 8 -  U  £ U(t  0  2  — 3  co  U, * U(t o  „  4  3_)  u ( t  4  co  o , 2TT  u  and  5  i « « - - ^ - > o A  8  U, = U(t - — ) D CO o  are unit step  functions.  Note that, f o r the above, phase voltage  and one-half of the l i n e - l i n e  voltage  are applied a l t e r n a t e l y to the load of phase a for Modes 2 and 1 , respectively. The c i r c u i t equation i s v  [R  (t) =  + Lp]i 3.  a.  From equations  (2.14)  and  (2.15)  (t)  (2.15),  a piecewise solution o f . i (t) (see Appendix) Si  gives i (t) =  —  a  /R /R  [sin(to t - -5- - X) o  ++ co L  c  co t - a + — _ )exp(-ft ° • =2-)] o to . o  - s i n (a -  x  2TT  * for  a  C0_ —  4TT  0  " T  B  $ t <  '  T (0  (2.16)  11  i  (t) =  [- -J-  , •R  sm(u t - —  - ) X  + co L o 5TT - s i n ( a - - 7 - - x) exp (-ft — 0  +{sin(B - |1 - ) + ^ | s i n ( g -  ~  3~  A  )  co o  3—) ]  _ )}exp(-ft - 2 — co o  x  x  (2.17)  4TT for  1  ( ) t  a  3— <  CO.  o  t <  5-  CO  o  — [sin(to t - -r - x)  =  •R  + to L 5TT  s i n (a -  -  %  x) exp (-ft  O  - a + —  )  CO .  4TT +{sin(B - f- - ) + ^  s i n  X  •3  -{sin(a - f ' - ' x )  <  8  " T• '  . ,  }  )  e x  P(-^  0 t  g  +  3  M  )  4TT co t - a - x)>exp(-ft-5 )] -J CO o  + "7T s i n ( a -  O  X  Z  _ < : — i i . 6  J L ,  F O R  CO  1 (t) = a  [2  /R  ll + co L  [-5- s i n (to  ~  2  D  IT  X  ) +  - {sin(a r J - X) +  co t - a + —  x) exp (-ft-  5  sin(a -  :  )  CO  o  nr o sin(g - | L _  / J  o  t - x)  5TT  -  CO  o  0  - s i n (a - -7 ~ +{sin(6 - f-  t  4TT -  -j-  x  X  )  }  e  x  p  (  cot-B+4^" _ p ^ _ 1_) o  CO t — 0 1 ) }exp (-ft-5-^ )  (2.18)  12  2jL +{sin(  B  -  - ) - - | sin(3 - ^  ]  - X)}exp(-ft-2  x  ^ 2ir  2TT  for  —  CO  < t<  CO  o 1  () t  3.  =  /2  (2.19)  o  ~ [sm(co^t - - - x) 6 2 2 co L o  °  1  - s i n (a-.  ,  - x) exp (-ft  2-TT  o  )  co t - 3 + ^ +{sin(3 - j  1  ~ X) + 4  s  i  n  (  B  ' "^3 " X)}exp(-ft^  -{sin(a - T - X) + —T s i n ( a O  Z  x)} P(-fie x  j  —  Tr  ^  )  )  2  CO  o r +{sin(3 - —  6  - x) " i 1  /T sin(3 2  9  C 0 t - 3 + - )}exp(-ft^ o  3  W  u  -{sin(a + -| - ) X  — ^-)  X  j sin(a + ~  - x)}exp(-ft-  _  t  _ 21  a  —) ]  2  o (2.20)  for  ^ .< t < CO  CO  o  o  a + (t) = 0  i  for — co  a  °  $ t <  — 3 CO .  0 The e x t i n c t i o n angle 3 i s obtained by s u b s t i t u t i n g 3 f o r co t and o equating i = 0 i n equation ( 2 . 2 0 ) . 3.  (2.21)  13  2.4.  A n a l y s i s f o r Case 3  By d e f i n i t i o n , a t l e a s t  two SCRs a r e c o n d u c t i n g  a t any time.  T h i s i m p l i e s t h a t phase v o l t a g e s a r e always a p p l i e d to a l l phase l o a d s , and  that f i r i n g  c o n t r o l o f the SGRs i s l o s t .  Thus, the c i r c u i t equations  e q u a t i o n f o r phase a i n Case 3 i s g i v e n by  (2.22) and (2.23).  a  ( t ) = V sin(oo t - 7) o 6  (2.22)  a  ( t ) = (R + Lp) i ( t ) a  (2.23)  v  v  Hence the s t e a d y - s t a t e c u r r e n t through the l o a d o f phase a i s  i («> a  -  /3~> 2~2 /R + OJ L  Since f i r i n g firing  2.5.  o  x  c o n t r o l i s no l o n g e r e f f e c t i v e , terms c o n t a i n i n g the  a n g l e ct do not appear i n e q u a t i o n  (2.24).  Comparison o f A n a l y t i c a l and E x p e r i m e n t a l R e s u l t s f o r Three-Phase Delta-Connected To v e r i f y  Fig.  (2.24)  sin(u t - j - )  SCRs  the a n a l y s i s i n the p r e v i o u s s e c t i o n s t h e c i r c u i t of  2.1 was s e t up i n the l a b o r a t o r y w i t h Resistor R  20.45 ohms  Inductor L  11.3  mh  Phase v o l t a g e  120  volts  Load power f a c t o r  0.98 l a g .  The SCRs used were type GE20C.  14  Fig.  2.3 shows t y p i c a l c u r r e n t waveforms f o r Case 1.  Fig.  2.3(a) i s the waveform c a l c u l a t e d  Fig.  2.3(b) shows  The f i r i n g  from e q u a t i o n  (2.13).  the e x p e r i m e n t a l c u r r e n t and v o l t a g e waveforms.  a n g l e i s 114 degrees.  The a n a l y t i c a l and the e x p e r i m e n t a l  results  agree q u i t e w e l l . T y p i c a l c u r r e n t waveforms f o r Case 2 a r e shown i n F i g . 2.4. waveform c a l c u l a t e d  from e q u a t i o n s  The  (2.16) to (2.21) i s shown i n F i g . 2 . 4 ( a ) .  The c o r r e s p o n d i n g e x p e r i m e n t a l c u r r e n t and v o l t a g e waveforms a r e shown i n Fig.  2.4(b).  The f i r i n g  a n g l e f o r t h i s case i s 14 degrees.  was o b t a i n e d between the a n a l y t i c a l and the e x p e r i m e n t a l  2.6. R e l a t i o n s h i p  Good agreement  results.  Between the F i r i n g Angle and the E x t i n c t i o n  The e x t i n c t i o n a n g l e 3 i s a f f e c t e d by b o t h  the f i r i n g  Angle a n g l e a and  the power f a c t o r o f the l o a d , and the e x t i n c t i o n a n g l e i n t u r n determines a c t u a l case o b t a i n e d , i . e . , Case 1, 2 o r 3. firing  The r e l a t i o n s h i p between the  a n g l e , the power f a c t o r o f the l o a d , the e x t i n c t i o n a n g l e and the case  o b t a i n e d i s shown i n the e x t i n c t i o n a n g l e - f i r i n g for  v a r i o u s power  angle curves o f F i g . 2.5  factors.  As l a g g i n g power f a c t o r approaches u n i t y , the e x t i n c t i o n decreases  f o r any f i x e d f i r i n g  angle.  angle i n c r e a s e s .  c o n s i d e r a l o a d o f power f a c t o r 0.5 l a g g i n g . values of f i r i n g 180 degrees,  a n g l e s o f 0 to 30 degrees,  respectively.  factor,  F o r example,  Cases 3, 2 and 1 o c c u r f o r 30 to 104 degrees  and 104 to  A l s o the v a l u e s o f e x t i n c t i o n a n g l e s f o r Cases 2  1, i n the example, a r e 262 and 223 degrees,  values of f i r i n g  angle  A l s o f o r a g i v e n l a g g i n g power  the e x t i n c t i o n a n g l e d e c r e a s e s as the f i r i n g  and  the  a n g l e s o f 60 and 120  degrees.  r e s p e c t i v e l y , f o r corresponding  15  6.0-j oJ  3.0  UJ CK  0-  O  i-3.0-  -6.0  —I  1-  j —  120 240 THETfl (DEG.) (a)  THETA  (46 Deg./Div.) (b)  (a) (b)  F i g . 2.3 Waveforms f o r Case 1 C a l c u l a t e d ' l o a d c u r r e n t waveform E x p e r i m e n t a l c u r r e n t and v o l t a g e waveform  360  8.0  ...  .  "  ^  in  ° I  V  //  \  A  .  '  /  / THETA  (46  Deg./Div.)  (b) F->'g„ 2.4  (a) (b)  Waveforms f o r Case 2  Calculated load current waveform Experimental current and voltage waveform  360  330 H  FIRING ANGLE  (deg.)  F i g . 2.5 E x t i n c t i o n angle - f i r i n g  angle curves  M  18  3. 3.1.  SCR-CONTROLLED MOTOR  Introduction The  now  ANALYSIS OF THE  d e l t a - c o n n e c t e d SCRs d i s c u s s e d i n the p r e v i o u s c h a p t e r  be c o n s i d e r e d i n r e l a t i o n  to a synchronous  motor.  The  will  delta-connected  SCRs are i n s e r t e d i n the n e u t r a l p o i n t of the armature w i n d i n g o f the synchronous  motor.  The  f i e l d winding  i s D.C.  excited.  The motor w i t h c o n t r o l l o g i c i s shown i n F i g . 1.1. c o n s i s t s o f a synchronous an AND as  gate and a 21.6  The  motor w i t h SCRs, a d i s t r i b u t o r , a phase  KHz  crystal  clock.  The  SCR  system shifter,  gate p u l s e s a r e o b t a i n e d  follows:  The phase s h i f t e r output which then remains  i s used t o s e t an R-S  i n the ON  state u n t i l  flip-flop,  F i g . 3.1,  r e s e t by a p u l s e which o c c u r s a t  the p o s i t i v e - n e g a t i v e z e r o - c r o s s i n g p o i n t o f the l i n e - l i n e power s u p p l y voltage.  The r e s u l t i n g R-S  power s u p p l y f r e q u e n c y . a d i s c , each  d i s t r i b u t o r c o n s i s t s o f t h r e e lamps mounted  to each o t h e r and  motor armature.  when l i g h t  The  output i s a p e r i o d i c p u l s e o f the  lamp b e i n g a l i g n e d w i t h a photo-diode  d i s c s are f i x e d  motor s h a f t .  flip-flop  on a n o t h e r d i s c .  A t h i r d d i s c w i t h two  through a s l o t  slots  c u t i n i t i s mounted on  i n the r o t a t i n g d i s c .  p u l s e s i g n a l s a r e o b t a i n e d from a s s o c i a t e d c i r c u i t r y . of an output p u l s e i s r e l a t e d frequency i s r e l a t e d 21.6  KHz  crystal  At these The  the  photo-diode instants,  time of o c c u r r e n c e  to- the r o t o r p o s i t i o n w h i l e the p u l s e r e p e t i t i o n  to the motor speed.  clock output.  These  a d j u s t a b l e i n p o s i t i o n w i t h r e s p e c t to the  L i g h t from a lamp i s r e c e i v e d by the c o r r e s p o n d i n g  passes  on  T h i s p u l s e i s then used  to gate  These p u l s e s a r e a p p l i e d to the AND  y i e l d b u r s t s o f f i r i n g p u l s e s , F i g . 3.2. t h a t the w i d t h o f the output p u l s e s from  the  gate to  The d i s t r i b u t o r s l o t s a r e c u t so the d i s t r i b u t o r i s o n e - t h i r d the  Phase  Voltage  (a) Phase Voltage (b)  _  SCR ogate  0  OSCR cathode  i  Phase  Shifter  Output  Mono Stable  Distributor Output  Clock Output Fig.  3.1  C i r c u i t t o generate SCR g a t e p u l s e s  Line-line Supply voltage  Phase shifter output  R-S flip-flop output  Distributor output  SCR gate pulses  Fig.  3.2  Waveforms f o r c i r c u i t o f F i g . 3.1  21  p e r i o d f o r a g i v e n speed, i . e . , the b u r s t o f f i r i n g p u l s e s to each SCR gate may occupy up t o o n e - t h i r d o f a p e r i o d . Mode 1 o p e r a t i o n o c c u r s when one SCR i s c o n d u c t i n g . is  fired,  i t s e x t i n c t i o n a n g l e i s dependent upon i t s f i r i n g  power f a c t o r o f the l o a d . next SCR i s f i r e d .  angle and the  Thus, the SCR may s t i l l be c o n d u c t i n g when the  The r e s u l t i s Mode 2 o p e r a t i o n w i t h two SCRs c o n d u c t i n g ,  as d e f i n e d i n S e c t i o n 2.1. discussed i n d e t a i l  Once an SCR  The a n a l y s i s o f  each o f these Modes w i l l be  i n the f o l l o w i n g s e c t i o n s .  The motor w i l l be assumed  to have two p o l e s and t o o p e r a t e a t a c o n s t a n t speed.  3.2.  Mode 1 O p e r a t i o n o f the Motor One SCR i s c o n d u c t i n g under Mode 1 o p e r a t i o n , as shown i n F i g . 3.3.  A  o  B  o Fig.  3.3  Motor w i t h r o t a t i n g D.C. f i e l d under Mode 1 o p e r a t i o n , SCR  1  conducting  The d i s t r i b u t o r has been made so that f i r i n g pulses cannot be applied to more than one SCR at any. time.  Thus, bursts of f i r i n g pulses are applied  to the gates of SCR^, SCR^, and SCR^, i n turn, after each 120 degrees rotation of the rotor.  Since the r e p e t i t i o n frequency of the output pulses  from the d i s t r i b u t o r i s related to the motor speed, the magnetic f i e l d set up by the armature windings rotates at that speed. Consider the period during which SCR^ i s conducting.  The motor  configuration of F i g . 3.3 with rotating f i e l d , f o r Mode 1, may be replaced by a motor with a rotating armature, as shown i n F i g . 3.4.  This second motor configuration may be further replaced by a motor with a mechanical commutator and brushes as shown i n F i g . 3.5.  Since the width  of the d i s t r i b u t o r output pulses i s one-third the period f o r a given motor speed, the commutator consists of two segments attached  to the armature  windings a and b, and each commutator segment spans 120 degrees.  o  o  B  A Fig.  3.5  Equivalent motor with mechanical commutator and brushes 3.3  S a l i e n t Pole Motor Equations, Mode 1 Operation The s a l i e n t pole motor w i l l be discussed i n t h i s s e c t i o n .  Using the transformation matrix a 0 [A] =  y|  1//6  d sin 9 q cos 6  1//6 sii<e cos  (6  1/^6 g ) — Y )  sin cos  (6  (6  .  +  +  2TT.  — )  ~ )  24  whose inverse i s  o  q  /372  tA]  -1  sm  cos 6  sin(6 - ^ )  cos (6 - y )  s i n (6 +  cos(0 + -^)  (3.2)  where 0 i s the angle, measured counter clockwise, between the armature winding of phase a and the q-axis , the motor e q u a t i o n i n d-q coordinates are v.  R +  v  toL.  /  3  d  f ^  — M p 2 f  -OJL  L P  R + L  i  p q  vr  R  — M p 2 f  f  +  L  o.  vanishes s i n c e ( i  a  (3.3)  i ' f  f P  F  i  q  + i , + i ) i s equal to zero, b c ^  From equation (3.1) •  *  sin 6  *d  s i n ( 0 - ^|)  sin(6 + y )  fi  (3.4)  / 3 i  cos 6  q  Since i  c  i s zero and i ,  b  ,„  2TT.  cos (6 - — )  i s euqal to - i  a  , 2TT.  cos (6 +  —)  when SCR, i s conducting, equation 1  (3.4) becomes 2n  sin6 - sin(8 - — )  (3.5)  2IT  cosG - cos (6 - —~) Define an angle  as the angle between the q-axis and the M-axls,  the axis of the r o t a t i n g magnetic f i e l d . 0 and  TJJ  Is  Then the r e l a t i o n s h i p between  25  (3.6)  9 = \\i Substituting  equation  (3.6) i n t o e q u a t i o n  ( 3 . 5 ) , we g e t  s i n TJJ [ 1  (3.7)  a'  \p  cos From e q u a t i o n (3.2)  f  •  V  cos 6  sm  a  v  •„  V  b  V  / 3  c  2TK  /.  2TT  d  N  sin(6 - - j )  cos(6  g)  sxn(9 + — j ) .  cos (9 + y )  (3.8) V  q  where v . v, , v , v . and v a r e unknown, a' b ' c ' d q Equation *  (3.8) may be r e w r i t t e n  with equation  (3.6) as  >  V  ^ cosijj + — s i n 4J /3  — sinijj - y cos i|> /3 . , 1 ,  a  (3.9)  / 3 1_ sinij; + — cos ij; 2 2  V, b  Although v  and  &  <3 c o s ^ - TJ- s i n i|; 2  a r e unknown, the l i n e v o l t a g e v^g i s known.  V  AB  =  V  Hence,  a " b V  = 7l sinib v , + 7l cosib v d q S u b s t i t u t i n g v ^ and v^ from e q u a t i o n  (3.10)  (3.3), equation  (3.10) becomes  /3 V  AB  =  ^  S  ±  n  ^  +  L  d  p ) i  d  "  u L  q V f P2 f ^f a a q/~2 i  +  M  i  K  f'  + /2 cosi|) [ t o L j i , + ( R + L p ) i + / - M,a>i>] aa q q v z r r  (3.11)  S i n c e i , and i a r e known i n terms o f i , d q a  2 2 a v , = 2[R + to(L, - L )sin2ij)]i + 2(L,sin.iJ> + L cos i | i ) — r ~ AB d q a . d q dt d  AT  i  r  di — — dt f  + /i Also  m^cosi> f  i ^ + / J sirnjj M f f  (3  from e q u a t i o n s (3.3) and (3.7) V  = /3 M  f  f P  i  d  + (R  = /3 toM.cosiJj i f a  f  + L  f  )i  P  f  di + /3 M.sinil' — : f d t  d i r R,-i,- + L,- ~ ; — f f f d t f  T  (3  From e q u a t i o n s (3.12) and (3.13), the machine e q u a t i o n s f o r Mode 1 are  di dt d  i  dt  a  ll  G  '12  H  ll  H  12  =  f  (3 G  £  22  21  H  21  H  22  +  sin  D = 2[L L f  v 4 -  2  M  2  f  sin i|>] 2  G. = - [ 2 L { R + <o(L - L ) s i n 2 ^ } - -| M w sin2;l;]/D 11 r d q 2 r 2  n  G  H  H  C  = -/3M [coL cos^ - R simM/D f  f  f  v "[2RfLv "I ^f ^ 3 / V° 12 " 21 " Mf Sln* /D  G  G  1 2  21 0 1  = -2/3  M [OJL  11 =  H  9 0  22  COS  ^ - sini); {R +  S i n 2  22 ~  H  r  £  v  = 2L /D  AB  D  u)(L,  d  -  L  q  )sin2ijj} ]/D  27  The torque e q u a t i o n i n d-q c o o r d i n a t e s i s , i n g e n e r a l , T - p [ / f  M  i  f  f  i  q  a  +  d  where P i s the number o f p o l e - p a i r s . equation  q  )  i  d  i  q  ]  (3.15)  S u b s t i t u t i n g equation  M C O S O yJ J £  f  1  i - i f a  +  v  However, the i n s t a n t a n e o u s  (3.16)  ( L , - L )sin2ip i ] d q a r  the above e q u a t i o n , P, M ^ , ( L , - L ) and i  J  are nonnegative values.  torque may be n e g a t i v e depending upon the  v a l u e o f i\i, the r o t o r p o s i t i o n , and the v a l u e o f the f i e l d  3.4.  (3.7) i n t o  (3.15), we get  T = P[/3 In  - L  current i ^ .  Round-Rotor Motor, Mode 1 O p e r a t i o n The machine equations f o r a r o u n d - r o t o r  by s e t t i n g  type motor a r e o b t a i n e d  ( L ^ - L^) to zero i n the above a n a l y s i s , so t h a t di dt  A  l l  A  B  12  11  B  12  (3.17)  _ii  d  A  dt  21  A  22  f  where L = L , = L ' c d q F = 2[LJL  .  I  - I M^sin ^] 2  i t  c  = -[2RL  - |  A  n  A  12  = -/3 M  A  2 1  = -2/3 M [ u L  A  22  = -L2R L - I  f  f  B a u  f  L /F f  OJM  2  sin2iJ;]/F  [coL cos \\> - R siniJ;]/F f  f  c  AB  c  f  COSIJJ  - R sin^]/F  OJM s i n 2 ^ ] / F 2  21  B  22.  28  B  12 " 21  B  B  2 2  =  - v / J  M  f  S  ±  n  ^  / P  4 2 L /F c  The t o r q u e e q u a t i o n f o r a r o u n d - r o t o r motor i s t h e n  T - P [ /4 M , i , i ] = /3 P M costo i . i V 2 r f q f f i  (3.18)  Y  •3.5.  Mode 2 O p e r a t i o n o f t h e Motor Two SCRs a r e c o n d u c t i n g under Mode 2 o p e r a t i o n .  diagram  f o r Mode 2 i s shown i n F i g .3.6.  same as e q u a t i o n (3.3)  V  R  +  =  f  -coL  coL  R + L p q  _/ 2 f M  /f fP  L p d  q  v  The machine e q u a t i o n s  •  d  V  The machine  M  q  d  A  d (3.19)  i q R  P  + L  f  f P  The t o r q u e e q u a t i o n f o r t h i s mode o f o p e r a t i o n i s a l s o t h e same as e q u a t i o n  (3.15) T-P[/§ M  3.6.  f  i  f  i  q  +  ( L - L ) i i d  q  d  q  ]  .  (3.20)  T r a n s i t i o n between Mode 1 and Mode 2  We w i l l now d i s c u s s t h e o c c u r r e n c e o f t h e above two modes d u r i n g n o r m a l motor o p e r a t i o n . The a n g l e ^ i s a f u n c t i o n o f t i m e . <J> = wt +  L e t i t be e x p r e s s e d by  O '  where time i s measured from t h e i n s t a n t t h a t t h e commutator segments  (3.21)  30  make c o n t a c t w i t h the b r u s h e s , F i g . 3.5. measured f o r each  c o n d u c t i n g SCR.  Note t h a t time i s i n d e p e n d e n t l y  L e t y be the a n g l e o f t h e b r u s h e s  c l o c k w i s e from the q - a x i s , as shown i n F i g . 3.7.  advanced  Then, ty^ i s e q u a l to  - C | + y) and * = cot - | - y  (3-22)  S i n c e the c o n d u c t i o n p e r i o d i s 120 degrees m e c h a n i c a l , ij> has a v a l u e between - y -  Y  and - - y. Next c o n s i d e r the v o l t a g e a p p l i e d  operation. 120  under Mode 1  Because the motor speed i s n o t n e c e s s a r i l y the synchronous  degrees m e c h a n i c a l  r o t a t i o n does n o t c o r r e s p o n d t o 120 degrees  phase s h i f t o f the s u p p l y v o l t a g e . SCR  to the armature  electrical  Commutation o c c u r s when the c o n d u c t i n g  e x t i n g u i s h e s and the next SCR i s f i r e d .  i n r o t a t i o n which i s —^co i n time.  speed,  T h i s may o c c u r a f t e r 120 degrees  S i n c e any machine speed co may be e x p r e s s e d  by  (3.23)  6  O Fig.  3.7  D e f i n i t i o n o f angle y  31  where m and n a r e i n t e g e r s , and co^ i s the synchronous speed, the d i f f e r e n c e between the m e c h a n i c a l and e l e c t r i c a l phase a n g l e s a f t e r one commutation i s g i v e n by 2TT _ 2TT_ _ "o  "l  U  o ii  3  OJ  ^ 2TT 3  n-m 2TT m 3  (3.24)  In g e n e r a l , a f t e r K commutations n-m 2IT  Note t h a t a f t e r  (3.25)  3m commutations, i . e . , m m e c h a n i c a l r e v o l u t i o n s , n i s  n which means t h a t  3 m  = ( n - m ) 2TT  the o r i g i n a l phase r e l a t i o n s h i p s occur a f t e r every m  revolutions. I n g e n e r a l , the o r i g i n a l phase r e l a t i o n s h i p s o c c u r a f t e r every m/P r e v o l u t i o n s f o r a machine w i t h P p o l e - p a i r s . Thus, the v o l t a g e a c r o s s the b r u s h e s i s  v  s  = S3 V sin(oj t + o  K ^ ^ + m 3  ?  )  (3.26)  where £ i s the phase angle o f the l i n e - l i n e v o l t a g e a t the b e g i n n i n g o f the 3m  commutation.  completed  K i s an i n t e g e r  (K = 0, 1,2,...) i n d i c a t i n g  t h e number o f  commutations. The c o n d i t i o n s f o r Mode 1 o p e r a t i o n to change to Mode 2 w i l l be  considered next.  Under Mode 2 o p e r a t i o n , v o l t a g e i s a p p l i e d t o a l l windings: v.  sin 6  sm(6  cos 6  cos (9  ,„  -) 2TT  X  -)  sm(9  + -y)  , „ ,  cos (9 +  2n\  —)  32 cos (co t. - ll) + O o  (3.27)  sin(co t - i\> + O o  As mentioned before, the e x t i n c t i o n angle of an SCR i s dependent upon i t s f i r i n g angle and' the load power factor.  This means that an SCR may be con-  ducting beyond the normal f i r i n g period of the SCR, where the f i r i n g period corresponds  to one-third of a rotation because of the mechanical  distributor.  Thus, SCR^ i n F i g . 1.1 may be conducting when SCR enters i t s f i r i n g period. 2  If SCR i s f i r e d while SCR^ i s s t i l l conducting, then the mode changes from 2  Mode 1 to Mode 2. One of the necessary conditions to change from Mode 1 to Mode 2 i s that armature current continues to flow through SCR^ beyond i t s f i r i n g period. The other necessary condition i s that gate pulses must be applied to SCR while SCR^ i s conducting.  2  At the instant that the f i r i n g period ends f o r  SCR^, the f i r i n g period f o r SCR  2  commences.  Hence, time t i s again zero and  K i s incremented by one. Then, the phase angle of the supply voltage at this instant i s , from equation (3.26),  a  «  2=H ± + m J  (3.28)  2  K  K  A burst of f i r i n g pulses i s applied to the gate of SCR when 6 i s 2  a $ 6 < 7T  (3.29-)  These two conditions are not s u f f i c i e n t to change the mode of operation because of voltages generated i n the armature windings. T h e a d d i t i o n a l condition for changing the mode i s that the current i n the t h i r d winding, i.e.,  the phase winding c i n this example, must become negative.  Since  the i n i t i a l current i n the third phase winding i s zero, the t h i r d condition i s di dt  °  < 0  (3.30)  33  From e q u a t i o n (3.1)  (3.31)  . [sin(6 + I ) ! , + cos(e+ I ) i ]  i (t)  1  c  3  1  3  d  3  q  Hence,  fT  d i  ^  = /j  9  9  [to cos(e+ y - ) i  d  d l  - co s i n (  + sin(e+ 3 )  4 co[cos(6 + ^jf) i , - s i n ( e + % ) 3 i d 3  2  2  3  L  <  v  + ^f) i  c  2  2^  6  - LoL^i^  + ~ )  - Ri  j — ^  9  + cos(  e  d  + +f  i  n  -£]  i ] q  L„v, - /•£ M.V. - L ^ R i , + < C _ L i f d / f f. £ d q f g — 3 2 d f " 2 f  • /« . ^  — sm(9 + — )  cos(6  9  H  L  1  M.R.i f f f  M  •/I  toM^i^ (3.32)  0  When these  conditions are s a t i s f i e d , Since  the i n i t i a l  operation  changes from Mode 1 t o Mode 2.  c o n d i t i o n s o f i , and i f o r Mode 2 a r e d q s i n ip  = ft i  (3.5) COS  lj>  q At  t h e i n s t a n t .that o p e r a t i o n  changes from Mode 1 to Mode 2, t h e c u r r e n t  through t h e w i n d i n g s a r e  i  = / — ( s i n 0 i , + cos 0 i ) / 3 d q  a  2 /*3  i  b  {sin(ij/ - -^OsiniJ; + c o s ( ^ - -r)cos ip}l 0 b a  = / f  [sin(8 - ^ j ) i + cos(0 d  -  ]  34  —  i  [sin(ij) - - ^ r ) s i n \p + cos(iJj - -^~) cos <J>]i  = / - [sin(6 + % i , V 3 3 d  c  2 =  ~ i  =  a  + cos(e + %  + -r)sin  [sin(i|;  q  ]  TT  TT  —  i  3  a  + cos  (iji  +  T;)COS  ijj]i  = 0  Next, the c o n d i t i o n to change o p e r a t i o n from Mode 2 to Mode 1 i s t h a t the t h y r i s t o r SCR^ e x t i n g u i s h e s w h i l e SCR^ c o n t i n u e s the c u r r e n t through phase a o f the armature i s z e r o . and s u f f i c i e n t  conducting, i . e . ,  Hence, the n e c e s s a r y  c o n d i t i o n to change from Mode 2 t o Mode 1 i s  i  a  (t) = / — / 3  [ s i n 6 i , + cos 9 i ] d q  S i n c e time i s a g a i n zero a t the b e g i n n i n g  (3.3 3)  o f t h e f i r i n g p e r i o d f o r SCR2,  2TT  \p i s r e p l a c e d by \j> + — j . Substituting i  i  &  = 6, ±^=x^ and i ^ - i ^  d  i n t o equation  sm(9  p  sm(9 +  —p  cos (9  p  cos (9 +  —p  (3.4),  / 3 i  where 9  =  q  - — H o  (3.34)  r  J  Then,at the i n s t a n t t h a t o p e r a t i o n changes from Mode 2 t o Mode 1 the c u r r e n t s are g i v e n by •  >  s i n ij; =  1  q  SI COS ]\).  (3.35)  3.7.  Starting  Torque  Equation instant  (3.22)  shows t h a t time  The  r o t o r p o s i t i o n at the i n s t a n t o f s t a r t i n g may  p o s i t i o n such  t h a t (- — - y)  from e q u a t i o n  (3.10),  T  s  = P[/3M  Note t h a t s t a r t i n g  of  the  t h a t the commutator segments b e g i n t o make c o n t a c t w i t h the  f o r each SCR.  starting  i s always measured from  £  f  S i|» $ ( ^ - -  starting  cos  torque T  4». i , i +  l  y) .  f a  g  torque i s zero when  q  )sin2ijj.  field  ty^.  Then,  (3.  i ] 2  a  or i r / 2 .  T h i s means t h a t  rotor position.  i s always s y n c h r o n i z e d w i t h  means o f the d i s t r i b u t o r so damper w i n d i n g s ,  l  i s -TT/2  t o r q u e i s dependent upon the i n i t i a l  the r o t a t i n g  any  i s g i v e n by  (L, - L  d  D e f i n e t h i s a n g l e as  be  brushes  A l s o the  t h a t o f the r o t o r  t h a t the motor i s s e l f - s t a r t i n g  u n l i k e the normal synchronous motor.  by  even w i t h o u t  speed  36  4.  EXPERIMENTAL VERIFICATION OF  To v e r i f y Fig.  1.1  was  OPERATION OF  THE  SCR  CONTROLLED MOTOR  the a n a l y s i s of Chapter 3 the motor c o n f i g u r a t i o n of  s e t up  i n the l a b o r a t o r y .  i s - shown i n F i g . 4.1.  The  a c t u a l experimental  Comparison of the e x p e r i m e n t a l  a n a l y t i c a l r e s u l t s w i l l be  4.1.  THE  discussed  i n this  set  r e s u l t s with  up the  chapter.  System components  Gate P u l s e s . an SCR  gate.  One  firing  a n g l e a and  Various  types o f p u l s e s i g n a l s may  type c o n s i s t s o f gate p u l s e s end  o f the power s u p p l y .  SCRs, e s p e c i a l l y where i n d u c t i v e l o a d s  pulses  to 180  are i n v o l v e d .  to g e n e r a t e these f i r i n g  a short-pulse,  commencing a t the d e s i r e d f i r i n g power s u p p l y ,  may  be  short-duration pulse  i s t h a t i t may  short-duration pulses, p u l s e s has  firing  been d e s c r i b e d Motor and  experiment.  The  fail  o f an SCR  i n Section  SCRs.  gate.  nine i s o l a t e d  pulses.  to f i r e  the SCR.  F u l l - l o a d current  trans-  connection.  degrees of  pulses the  The  With a b u r s t l o g i c to o b t a i n  of these  3.1. synchronous machine i s used f o r t h i s  follows:  Output power Nominal i n p u t v o l t a g e  Pulse  D.C.  d i s a d v a n t a g e of the s i n g l e  i s assured.  A UNITEC U-132  s p e c i f i c a t i o n s are as  ending a t 180 The  firing  Because the d e l t a -  or a b u r s t of s h o r t - d u r a t i o n  a n g l e a and  a p p l i e d to an SCR  desired  degrees i n phase  formers a r e used to overcome the problem of the common ground With a"pulse transformer,  to  are most d e s i r a b l e f o r  c o n n e c t e d SCRs cannot share a common ground c o n n e c t i o n , power s u p p l i e s are r e q u i r e d  applied  t h a t commence a t the  at the i n s t a n t c o r r e s p o n d i n g Such l o n g - d u r a t i o n  be  : ( t h r e e - p h a s e 1-1):  1/4  H.P.  220 v o l t s :  1.3  amps.  F i g . 4.1  Experimental  set-up  38  Number of poles  : 4  Field excitation  : 1.5 amps, a t 24 v o l t s  Synchronous speed  : 1800 r.p.m.  Three GE20C SCRs are delta-connected and i n s e r t e d i n the n e u t r a l p o i n t of the armature windings. Distributor.  The f u n c t i o n of the d i s t r i b u t o r i s to detect the  motor speed and the r o t o r p o s i t i o n with respect to the armature windings. The c o n s t r u c t i o n of the d i s t r i b u t o r has been described i n Section 3.1. and i s shown i n F i g . 4.2. Phase S h i f t e r .  D i g i t a l l o g i c i s used to s h i f t phase i n s t e a d of  an i n d u c t i o n r e g u l a t o r . The l o g i c used, F i g . 4.3, i s a s . f o l l o w s : The negativep o s i t i v e zero-crossing point of a l i n e - l i n e supply voltage i s detected by a comparator.  At t h i s i n s t a n t a pulse t r i g g e r s a decade counter to count the  21.6 KHz c r y s t a l clock pulses u n t i l i t reaches the number s e t by a thumb-wheel switch.  This number corresponds  to the desired angle a.  The output pulse of  the phase s h i f t e r occurs at the i n s t a n t corresponding to t h i s angle ex.  Pulses  f o r the Other two phases are derived from t h i s phase s h i f t e r output pulse by delaying i t s u c c e s s i v e l y f o r one-third of a period Of the power supply. Torque Meter.  A UNITEC U-235 Eddy Current Prony Brake i s used to  measure the motor torque.  I t works by the i n t e r a c t i o n of a permanent magnet  d i s c and eddy currents i n a r o t a t i n g copper d i s c . 4.2.  Determination of y = 0 P o s i t i o n of the D i s t r i b u t o r The angle Y has been defined as the angle between the brush-axis  and the q-axis of the equivalent motor f o r the f i r i n g period of the SCR under c o n s i d e r a t i o n , Section 3.6.  Since the value of the angle Y a f f e c t s the  operation of the motor, i t i s necessary to determine i t s value. (3.14),  From equation  Phase  Voltages (b)  (a)  2 s i n if) +  2[L,  d  2^3  When b o t h  [toM  [2R (L  +  [2L (L  f  f  if) and  sin  d  if; + L  2  (4.2)  current.  IJJJV,,  f  r  sin 'if; + L c o s  to are z e r o , e q u a t i o n  =  R  f ^  +  L  if) - M  r  s i n ^ {R + to(L, - L  )sin2ip}]i  (4.1)  2  f  sin if>]  (4.1)  2  becomes  f  shows t h a t the f i e l d  A l s o from e q u a t i o n  AB  2  i|>) - 3M  2  2  sinV>' vA1J  R  i>) - -|coM s i n 2 i j ; ] i  2  q  f  - /J Mf  iji + L c o s  2  cos  q  ( 4  -  2 )  c u r r e n t i s independent of the armature  (3.22) Y =  Therefore,  2  2  d  V  Equation  q  cos i|> (L s i n .  £  +  cos .  L  r  -f  (4.3)  the procedure to determine y = 0 p o s i t i o n i s as f o l l o w s : 1)  Supply o n l y enough three-phase v o l t a g e to t u r n on any the f i e l d  2)  circuit  with  open.  While manually t u r n i n g the r o t o r , observe the v o l t a g e waveforms induced  i n the r o t o r c i r c u i t .  t h a t t h e r e i s no  induced  o c c u r s when the angle 3)  SCR  Adjust  v o l t a g e i n the. f i e l d  circuit.  such This  if; i s z e r o .  the p o s i t i o n of the d i s c s c o n t a i n i n g the lamps and  photo-diodes w i t h t h a t one of  F i n d the r o t o r p o s i t i o n  a slot  lamp and  the  r e s p e c t to the armature f o r each phase such i t s corresponding  p h o t o - d i o d e i s i n the m i d d l e  of the r o t a t i n g d i s c when if> i s z e r o .  the angle y i s z e r o .  T h i s o c c u r s when  42  4.3.  Comparison of A n a l y t i c a l and  Experimental  From the a n a l y s i s of Chapter'3, o b t a i n the waveforms of the f i e l d compared w i t h e x p e r i m e n t a l Section  C u r r e n t Waveforms  a computer program was  and armature c u r r e n t s .  The  w r i t t e n to  results  r e s u l t s o b t a i n e d from the s e t up d i s c u s s e d i n  4.1. The  armature c u r r e n t waveforms at the motor speed  of 720  which i s t w o - f i f t h s the synchronous speed, i s shown i n F i g . 4.4. shows the armature c u r r e n t c a l c u l a t e d have a p e r i o d of 83 msec.  m/P  (3.14) and  F i g . 4.4(a) (3.21), t o  a p e r i o d of 81 msec.  As d i s c u s s e d i n S e c t i o n  the o r i g i n a l phase r e l a t i o n s h i p s of the s u p p l y v o l t a g e r e p e a t s r e v o l u t i o n s f o r a motor w i t h P p o l e - p a i r s .  p o l e - p a i r s and P a r e two.  Hence, one  r e v o l u t i o n corresponds or 83 msec.  armature c u r r e n t v a l u e s a r e 3.0 e x p e r i m e n t a l v a l u e s are 2.9 720  The  and -2.9  e x p e r i m e n t a l r e s u l t which has may  be a t t r i b u t e d  and  30 degrees,  amps, r e s p e c t i v e l y .  amps, r e s p e c t i v e l y .  The  to e x p e r i m e n t a l e r r o r .  w i t h the e x p e r i m e n t a l  of 900  corresponding  field  The  The  current result  corresponding  difference i n values  angles y and  a were s e t to 40  a n a l y t i c a l r e s u l t s a r e i n good agreement  results.  a n a l y t i c a l and  the motor speed  The  The  Fig. 4.5(a) shows the c a l c u l a t e d  a p e r i o d of 81 msec.  respectively.  and  to one p e r i o d of the waveform  a p e r i o d of 83 msec, w h i l e F i g . 4.5(b) shows the  The  two  c a l c u l a t e d maximum.and minimum  and -2.8  r.p.m. i s shown i n F i g . 4.5%  which has  S i n c e the motor has  every  i s o p e r a t i n g a t t w o - f i f t h s the synchronous speed, b o t h m  which i s 60/720 s e c ,  at  from e q u a t i o n s  r.p.m.,  F i g . 4.4(b) shows t h a t the c o r r e s p o n d i n g e x p e r i -  mental armature c u r r e n t has 3.6,  are  the e x p e r i m e n t a l  c u r r e n t s are a l s o compared a t  r.p.m., i . e . , a t o n e - h a l f the synchronous speed.  The  3D 2D  1  Q: Cfc o  Ui  ct ^>  I  •7.0 -2.0 -3.0 40  TIME(ms) (a)  Q \  U) CL  o  f\i  •  UJ Q; Q:  ft 0 Oft  o  Uj  S  j  Q;  • »  : TIME (10 ms/ . F i g . 4.A  .  (  b  Div.)  )  Armature c u r r e n t waveforms at speed o f 720 (a) (b)  A n a l y t i c a l c u r r e n t waveform E x p e r i m e n t a l c u r r e n t waveform  r.p.m  44  3.02.0  fe  7.0  ct ^> o 0.0 Q —i Uj  •7.0 40  TIME(ms) (a)  \  s o Q  •—i  TIME  (10ms/Div.)  00 F i g . 4.5  F i e l d c u r r e n t waveforms a t speed of 720 r.p.m. (a) A n a l y t i c a l c u r r e n t waveform (b)  E x p e r i m e n t a l c u r r e n t waveform  80  45  1.0\UJ  ct  0.0  ID  °£  -1.0  1  •2.0  75  32 T  IME(ms)  (a)  \ to  I  •  • -••  g  ct ct  :D  o  Uj  § ?  • •  ct TIME  (5  ms/Div.)  (b) rig.  4.6  Armature c u r r e n t waveforms a t speed o f 900 r.p.m. (a) A n a l y t i c a l current_waveform (b) E x p e r i m e n t a l c u r r e n t waveform  48  46  F i g . 4.7  F i e l d current waveforms at speed of 900 r.p.m. (a) A n a l y t i c a l current waveform (b) Experimental current waveform  47  TIME (10 Fig.  4.8  ms/div.)  Armature c u r r e n t waveform  a t speed o f 1080  r.p.m.  48  c a l c u l a t e d armature c u r r e n t i s shown i n F i g . 4.6(a) which has a p e r i o d o f 33 msec.  The c o r r e s p o n d i n g e x p e r i m e n t a l r e s u l t , shown i n F i g . 4 . 6 ( b ) , has  a p e r i o d o f 31 msec.  F o r o n e - h a l f the synchronous speed, the same phase  r e l a t i o n s h i p s o f the v o l t a g e r e p e a t every o n e - h a l f r e v o l u t i o n o f the motor. Note t h a t the e x a c t v a l u e o f the p e r i o d at h a l f - s y n c h r o n o u s speed i s 33.3 msec.  F i g . 4.7(a) shows the c a l c u l a t e d f i e l d  corresponding experimental r e s u l t . r e s u l t s a r e i n good  4.4.  Speed-Torque  c u r r e n t and F i g . 4.7(b) the  The a n a l y t i c a l and the e x p e r i m e n t a l  agreement.  Characteristics  The speed-torque c h a r a c t e r i s t i c s o f the motor has been i n v e s t i g a t e d f o r two y v a l u e s . speed o f 1080  The r e s u l t  f o r y of 40 degrees i s shown i n F i g . 4.9.  A  r.p.m., which i s t h r e e - f i f t h s the synchronous speed, i s  o b t a i n e d f o r a l o a d of 0.225 Newton-meters. speed o f 900 r.p.m., which i s h a l f  The motor o p e r a t e s s t a b l y a t a  the synchronous speed, w i t h a l o a d  between 0.31  and 0.46  Newton-meters.  The motor i s u n s t a b l e f o r a l o a d  between 0.46  and 0.52  Newton-meters, except t h a t a s t e a d y speed of 771 r.p.m.,  which i s t h r e e - s e v e n t h s o f the synchronous speed, i s o b t a i n e d at 0.475 Newton-meters.  I n c r e a s i n g the l o a d torque p a s t 0.52  motor becomes s t a b l e a g a i n but changes t w o - f i f t h s the synchronous speed, u n t i l meters.  F i g . 4.10  Newton-meters, the  i t s speed to 720 r.p.m., which i s the l o a d torque reaches 0.85  shows the speed-torque c h a r a c t e r i s t i c f o r y of 80 degrees.  O p e r a t i o n a t t h r e e - q u a r t e r s the synchronous speed, 1350 0.21  Newton-meters.  Newton-  r.p.m., o c c u r s at  The motor o p e r a t e s s t a b l y a t a speed of 1200  t w o - t h i r d s the synchronous speed, f o r a l o a d between 0.21 The motor becomes u n s t a b l e beyond  0.36  I t i s not s t a b l e a t 900 o r 720 r.p.m.  and 0.36  Newton-meters and the speed Note t h a t the motor was  r.p.m., i . e . , Newton-meters. changes.  s t a b l e a t these  49  1100k  1000  500  8-  800  Ui  k»  700  to 6*00 0^  04  06  0.8  TORQUE (N. m) Fig. 4.9  02  Speed-torque characteristic (T=t0deg.)  0.4  0.6 TORQUE  0.8  (N.m)  Fig. t.10 Speed-torque characteristic (t~ = B0deg.)  50  speeds f o r a y o f 40 degrees. synchronous  Thus the motor seems to run at m/n  times the  speed, where m and n a r e i n t e g e r s of v a l u e 1, 2, 3, ...  The  v a l u e s of m and n r e l a t e the motor speed and the s u p p l y f r e q u e n c y , r e s p e c t i v e l y , to the f r e q u e n c y of the c u r r e n t waveforms as f o l l o w s : ... 2nPw mech  = mto. ' I  (4.4)  and to = nco. o 1  (4.5)  where to , = m e c h a n i c a l speed i n r e v o l u t i o n s p e r second, mech -  - •  io = f r e q u e n c y p f the c u r r e n t waveform i n r a d i a n s p e r second (0  q  = s u p p l y f r e q u e n c y i n r a d i a n s per second  P = number o f p o l e - p a i r s Compare F i g . 4.4 w i t h F i g . 4.8 which show the armature  current  f o r motor o p e r a t i o n a t two- and t h r e e - f i f t h s the.synchronous and 1080 r.p.m., r e s p e c t i v e l y .  waveforms  speed,  720  The s t a b l e o p e r a t i o n of the motor a t any  speed depends upon the v a l u e of y.  In g e n e r a l , y must be c l o s e to 90  degrees f o r h i g h speed o p e r a t i o n , and s m a l l f o r low speed  operation.  51  5. The  three-phase  CONCLUSIONS  star-connected c i r c u i t with three delta-connected  SCRs i n s e r t e d i n the n e u t r a l p o i n t was method.  The  a n a l y s i s was  A three-phase  analyzed using Take-uchi's  c o n f i r m e d by  experiment.  synchronous motor w i t h t h r e e d e l t a - c o n n e c t e d SCRs  i n s e r t e d i n the n e t u r a l p o i n t of the armature windings f o r speed  control.  instances related  The  was  f i r i n g p u l s e s a p p l i e d to the SCR  The  then  investigated  gates o c c u r a t  to the motor speed,the r o t o r p o s i t i o n , and  of the v o l t a g e s u p p l y .  rj>-function  the z e r o - c r o s s i n g s  armature c u r r e n t waveforms t h a t r e s u l t a r e  l o n g e r p e r i o d i c a t the s u p p l y frequency but at some o t h e r frequency to the motor speed. m/n m  th  Stable o p e r a t i o n at  where m and n a r e : . i n t e g e r s . dependent upon i n i t i a l mentally  t h a t the  ' of the c u r r e n t waveforms i s l a r g e ,  The  starting  torque  being  a n a l y t i c a l r e s u l t s were e x p e r i -  verified. The  yield  i n v a l u e such  The motor i s s e l f - s t a r t i n g ,  rotor position.  related  a subsynchronous speed, which i s  o f the synchronous s p e e d , i s p o s s i b l e i f m i s low harmonic component o f the frequency  no  above scheme o f speed  continuous  speed  variations  c o n t r o l o f a synchronous motor does not  f o r a voltage supply with  i n c o n t r a s t w i t h c o n v e r t e r - i n v e r t e r schemes, but  fixed  frequency,  the l a t t e r r e q u i r e more SCRs.  F u r t h e r r e s e a r c h on the S C R - c o n t r o l l e d synchronous motor c o u l d i n v e s t i g a t e the t r a n s i e n t  and  stability  characteristics.  52  Appendix Equations (2.16) to (2.21) are obtained as follows: Taking the Laplace transform of equation (2.14) we get  V» "  Ssin(av [  4^)+co  cos ( a - "5-?)  XT  6  S + c o  2 o  a-  ^?  «*- ^ >  6  s  Ssin(3-  cos (3-  -i-7r)+co  4—5  o  T^)  *  + co  S  0  exp(- — — - S)] CO o  4^)+^  =• Ssin(3cos (8»J „ 3 o + — V[ T — r  1  S + c o  .„ +V[  ^r)  3  8~ - ^ r  cos ( a - •?) 6 o 6 / cn - 2 — ^ exp(- - S ) c  b  +  to  Ssin(8- • -|)+co cos(8| "2 — 2 o  S  N  Ssin(8~ —f)+w cos (86 o -j—  o  o  ^r)  6  1  e  x  p  8exp(- - j — B )  + co  N  o  Ssin(a+ -^Ho^cos (a+ -j—  *  2  S + c o  o  o  8- —? 3 „ -, S>]  , ( -—  S + c o  o  2  S + c u  o  o  S s i n ( a - -?)+co  r  cos ( a - - £ ) 3 o 3 , a - 2 — 2 «cp<- —  S s i n ( a - -^)+to  , 3 exp(- - — S )  o  a +  exp(  —  CO  —S)]  o  Ssin(ot+ + V  '  y)+  u  cos(orf  ^ 2 S+co  o  f)  cc+ ^ |  - » < " - To*  Ssin(8S )  T)+IO  cos (8-  - S +Jc oT i a  7)  W  fo  S  >1  (1.1)  „, , S) ]  53  S i m i l a r l y , f o r e q u a t i o n (2.15), we g e t : V ( ) = RI (S) + S [ L I (S) - i ( t ) ] S  3.  cl  3.  SL  S i n c e zero time was chosen such t h a t the i n i t i a l (1.2)  (1.2)  O  current i s zero, equation  becomes  v (s)  v (s)  a  V  S )  =  a  R + LS  L(S +  =  ft)  (  I  ,  3  )  D e f i n e the 4>-function, <j)(a, t ) , by sin(co t + a - x)  sin(a - x) P( fit)  -  e x  -  <{>(a,t)^  (1.4) co + ft 2  2  0  i.e.  . S s i n a + co cos a ZO(a,t)}=  where  ft  = R/L  = t a n "^(co L/R) . o ( 1 . 1 ) and ( 1 . 3 ) then become  Equations  and  (1.5)  ^ j (S + ft)(S + u ) o  X  a- —  I (S) = I a L  UU(or  +  4r,t)}exp(-  3-  — — 3 - S)-&<K  co o  o  tfUCe-^.tnexpC-  -rr-  1  R -  2jL, )]exp(t  D  —  u  3[£{<Kcr-£,t:)}exp(-  S)-Z{<}»(B- ^ , t ) }exp( o  /+ ' T ^ [X(<j>(3- ^ , t ) } e x p (  co o  o  3  a+ 2  J  CO  CO  o a+ +  7  L  |-,t)}exp( 2,  — 3- S) ]  3 - S)-^{<t.(a+• ^,t)}exp(  J  S) ]  — r " - S) ] o  g- —  i  L  ^  S)-ZU(a- | , t ) } e x p ( - ^-S)]  ° + £  —  —  o  — 3-  co  0  J,t)>exp(-  S)-Z{K36  S)] %  (1.6)  54 T a k i n g the i n v e r s e L a p l a c e t r a n s f o r m o f e q u a t i o n  3- —  a- —  i  a  ( t ) 1 [ < a - ^f,t-  V~ r )u  T  ( B  "  -  "r^V  An  g_ o  o  Bo  &  +  - ll,  m  — o  B- — -^ ) U  t  ( 1 . 6 ) , we o b t a i n  - • (a  4  +  + — t- V>U ^ 5  (1.7) I  +  [  ^  (  a  f  |  i  _ _ 3  t  )  o 2u  where  A  a  1^ = U ( t  " ~T  —-) "o  U. £ U ( t 2  U, £ U(t - - i ) to  J  „  4  u  (  t  A  '5  -1  3 (0  <t  U  o  4 u(t  o  a + to o  u A u(t - JL) 6  o  —r  i)  u  +  (  B  _ | , t - f ) U o  6  ]  55  From e q u a t i o n s  (1.4)  and ( 1 . 7 ) , t h e p i e c e w i s e s o l u t i o n o f i ( t ) i s as f o l l o w s : a  When S C R . and S C R . a r e c o n d u c t i n g , U = 1 and U „ = U . = U . = U = U , = 0, z J 1 z j 4 5 o C  s i n ( t o t - — - x)  sin(a  _  o  jr - )exp(-Q—-  )  x  =  ^  V  / R  +A  2  (1.8)  2  2TT  a - — for  When S C R  u  5  e x t i n g u i s h e s and S C R  2  $  co o  = U " = 1 and  remains c o n d u c t i n g ,  3  t <  „ 4TT B - — co o =  2  - u - 0, 6  a -—  g  —  4 TT +  2  L *  (  3  B  s  r [  j-7TR + coY~2 L  t  w  }  To *3 . , . ~ sxn(co t L  0  2-rr  / 4ir \ . , - - x)-sm(a  c co t - a + — 5ir •> „ o 3 7- ~ x)exp(-fi -) n  3  N  6  u  o  ..An  ~ /r o + {sin(B - ^ f - ) + - ^ 8 1 ^ 8 - - 3 X  X  -  c o t - B + -5— )}exp(-ft-^ 1-) ]  B for  — CO  o  3- $ t < i CO  o  56  When  SCR  3  and  SCR.j^ a r e c o n d u c t i n g ,  a  u  =  u  ±  = 2  U  3  1  a  n  d  8 -  —  '  =  o  U  4  =  U  5  =  U  6  °'  =  — o  Air 5u  +  3  ,t  ) - <j> (a - ^ t  ")]  3'  *(a - - , t - — ) o  I  a. ~3~ 2  5TT  [sin(to t - — - x) " s i n ( a O  /R  2  +  TT  2  U  y - x)exp(-fi  D  ~  O  O  A  OJ  t  t  )  L  2  8 + {sin(8 - y  -  X  )  + y  sin(8 - y  "T  - )}exp(-ft.-2  2_)  X  o  - {sin(a " f  " X) + y  sin(a - y  - x)}exp(-ft - y — — ) o a for — £  oe  5 c  = U, = 0, o  -  2rr  ~  to  o  and U  (1.10)  $ t <  to  When S C R ^ e x t i n g u i s h e s and S C R ^ remains c o n d u c t i n g ,  ]  o  =  =  =  = 1  57  3 -  —  - O  + £ [••(<» -  t - -) o  - •(B - — ,  S +  " ^  (  3 '  e  T  T  A  fc  -  V /3 ZZZZ IT "L* o  -  t -  6 ^ - y — ) ] o  ^  —  )  . , „ . . , 5TT H<J> t ~ X) - s i n ( a — 7  , 2TT / ^ Oto t - a + 3—  c  r  .  O  s1I  AT  0  +' {sin(g - f- - ) + ±£ s i n ( g -  o  X  N e x  - {sin(a - - -x)  + y  - f i  ATT  t - 3 - )}exp(-fi-2 _ o  +  x  t -  to  sin(a - y  - x)}exp(-ft-  )  CO  co  4TT  iv  n  x) P(  -7-  -L)  a  )  5  o + {sin(3 - ^  - x) - -T- sin(8 - 4f - )}ex (-ft-5 X  :L)  P  co  2TT  Q  ,  " T  8  for  CO  ••  —  < t <  o  When SCR. and SCR. are conducting. 1 I  -  .  V . 5TT . 1 (t) = - [<fr (a - -7-, t a L 0 t  U. = U„ = U„ = U, = U 1 2 3 4 5  —  R  3,  f  . , 9ir ^ ) - 4,(3 - — 0  -  3 co  g _ 4^ TT 3 \ , , ' TT a , — r •(« - 3, t - — ) ] o o 11  +  • y3V -2L  r  1  /Q  5TT  ~ ~T'  }  r  —  *  t  D  to  ]  0  N  )]  °  +  to  2TT  3~ —  o = 1 and U, = 0, 6  58  3 o  ^ o  o  V  /  ,  T  o  , 2TT 3,  ^  5ir  TT  [sin(to t - 7O  -  D  x)  W  x) *p(-  " s i n ( a - -7 0  e  sin(3 - ^  - X) + y  -  ~  a  ~3~ )  fi  to  t + {sin(3 - ^  o  3 + ^  x)}exp(-f2^  _  7r  L) o  ir  /3~  - { s i n ( a - g- -  x)  + y  4ir sin(a  ^o*" ~  a  j - xH *p(-^  )  e  o + {sin(3 - ^  - ) - - y sin(3 - ^ X  - )>exp(-^  i-)  X  o 2TT  /T - {sin(a + f  t - a  o  to  " X) " " y s i n ( a + y  - )}exp(-ft-2  =i-) ]  X  (1.12)  o .2 a + -r-ir for 3- * t< to o When SCR^ e x t i n g u i s h e s and SCR^  i s not c o n d u c t i n g ,  ^o  to  t h e r e i s no c u r r e n t  through  the l o a d o f phase a. Hence, when o n l y SCR^ i s c o n d u c t i n g , i  (t) = 0  4TT f o r f o  $ t <  — - ^ 0  (1.13)  . REFERENCES 1.  W. Alexanderson, "The Thyratron Motor", AIEE Trans., Vol. 53, 1934, pp. 1517^1531.  2.  M. Stohr, "Die Typenleistung K o l l e c t o r l o s e r Stromrichtermotoren bei der Einfachen Sechsphasenschaltung", Archiv fur Electrotechinik,. XXXII. Band., 11.  3.  Heft., 1938, pp. 691-720.  P.C. Krause, T.A. Lipo, "Analysis and Simplified Representations of a R e c t i f i e r - I n v e r t e r Induction Motor Drive", IEEE Trans, on Power Apparatus and Systems, V o l . PAS-88, No. 5, May 1969, pp. 588-596.  4.  T.A. Lipo, P.C. Krause, H.E. Jordan, "Harmonic Torque and Speed Pulsations i n a R e c t i f i e r - I n v e r t e r Induction Motor Drive", IEEE Trans, on Power Apparatus and Systems, V o l . PAS-88, No. 5, May 1969, pp. 579-587.  5.  P.C. Krause, J.R. Hake, "Method of Multiple Reference Frames Applied to the Analysis of a R e c t i f i e r - I n v e r t e r Induction Motor Drive", IEEE Trans, on Power Apparatus and Systems, V o l . PAS-88, No. 11, November 1969, pp. 1635-1641.  6.  S. M i y a - i r i , Y. Tsunehiro, "The Operation of the Damper Winding i n a D.C. Commutatorless Motor", IEE of Japan, Vol. 87-8, No. 947, August 1967, pp. 1601-1609.  7.  P.C. Krause, T.A. Lipo, "Analysis and Simplified Representation of R e c t i f i e r Inverter Reluctance-Synchronous Motor Drives", IEEE Trans, on Power Apparatus and Systems, V o l . PAS-88, No. 6, June 1969, pp. 962-970.  8.  T.A. Lipo, P.C. Krause, " S t a b i l i t y Analysis f o r Variable Frequency Operation of Synchronous Machines", IEEE Trans, on Power Apparatus and Systems, V o l . PAS-87, No. 1, January 1968, pp. 227-234.  9.  S. M i y a - i r i , Y. Tsunehiro, "Research on the Commutatorless Motor with IEE of Japan, V o l . 82, No. 890, November 1962, pp. 1741-1750.  SCR",  60  10.  S. M i y a - i r i , Y. T s u n e h i r o , "The A n a l y s i s of a Commutatorless Motor as a D.C.  Motor and i t s C h a r a c t e r i s t i c s " , IEE of Japan, V o l . 85-9, No.  September 11.  1965j pp. 1585-1594.  924,  '  N. S a t o , "A Study of the Commutatorless Motor", IEE of Japan, V o l . 84-8  4  No. 911, August 1964, pp. 1249-1257. 12.  T. T s u c h i y a , H. S a s a j i m a , K. T a t s u g u c h i , " B a s i c C h a r a c t e r i s t i c s of S e r i e s Commutatorless Motor", IEE of Japan, V o l . 89-9, No. 972, September  1969,  pp. 1773-1778. 13.  T. Okada, "Three Phase A.C. Rectifiers",  14.  C o n t r o l Systems w i t h D e l t a - C o n n e c t e d C o n t r o l l e d  IEE o f Japan, V o l . 86-10, No.'973, October 1966, pp. 1702-1711.  T . J . T a k e - u c h i , "Theory of SCR C i r c u i t and A p p l i c a t i o n t o Motor Tokyo E l e c t r i c a l E n g i n e e r i n g C o l l e g e P r e s s ,  1968.  Control",  

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