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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. S c , Doshisha U n i v e r s i t y , 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of E l e c t r i c a l Engineering We accept t h i s t hesis as conforming to the required standard Research Supervisor Members of the Committee Head of the Department Members of 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 p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced deg ree a t t he U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r ee t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t he Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada Date J2 / l / w / s / f f / ABSTRACT Okada's three-phase star-connected c i r c u i t with three delta-connected SCRs inse r t e d i n the neutral point i s analyzed using Take-uchi's ^-function method. The three-phase synchronous motor with three delta-connected SCRs inser t e d i n the neutral point of the armature windings i s then investigated. By c o n t r o l of the f i r i n g of the SCRs, operation at subsynchronous speeds i s p o s s i b l e . The analysis of the steady-state operation of the SCR-controlled synchronous motor i s experimentally checked. i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF ILLUSTRATIONS i v ACKNOWLEDGEMENT V NOMENCLATURE v i 1. INTRODUCTION 1 2. CHARACTERISTICS OF THREE-PHASE DELTA-CONNECTED SCRs 4 2.1. Introduction 4 2.2. Analysis f o r Case 1 5 2.3. Analysis f o r Case 2 9 2.4. Analysis f o r Case 3 13'.." 2.5. Comparison of A n a l y t i c a l and Experimental Results f o r Three-Phase Delta-Connected SCRs 13 2.6. Relationship Between the F i r i n g Angle and the E x t i n c t i o n Angle 14 3. ANALYSIS OF THE SCR-CONTROLLED MOTOR v 18 3.1. Introduction 18 3.2. Mode 1 Operation of the Motor 21 3.3. Sali e n t Pole Motor Equations, Mode 1 Operation 23 3.4. Round-Rotor Motor, Mode 1 Operation 27 3.5. Mode 2 Operation of the Motor 28 3.6. T r a n s i t i o n between Mode 1 and Mode 2 28 3.7. S t a r t i n g Torque 35 4. EXPERIMENTAL VERIFICATION OF THE OPERATION OF THE SCR CONTROLLED MOTOR 36 4.1. System Components '. 36 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 38 4.3. Comparison of A n a l y t i c a l and Experimental Current Waveforms . 42 4.4. Speed-Torque C h a r a c t e r i s t i c s 48 .5. CONCLUSIONS 51 APPENDIX . 52 REFERENCES 59 i i i LIST OF ILLUSTRATIONS Figure . Page 1.1 Motor configuration with co n t r o l l o g i c 2 2.1 Three-phase delta-connected SCRs 4 2.2 Three-phase delta-connected SCRs under Case 1 operation with SCR1 conducting 5 2.3 Waveforms for Case 1 15 2.4 Waveforms f or Case 2 16 2.5 E x t i n c t i o n angle — f i r i n g angle curves 17 3.1 C i r c u i t to generate SCR gate pulses 19 3.2 Waveforms for c i r c u i t of F i g . 3.1 20 3.3 Motor with r o t a t i n g D.C. f i e l d under Mode 1 operation, SCR^ conducting 21 3.4 Equivalent motor with r o t a t i n g armature under Mode 1 operation, SCR^ conducting 22 3.5 Equivalent motor with mechanical commutator and brushes 23 3.6 Motor under Mode 2 operation, SCR^ and 5CR^ conducting 29 3.7 D e f i n i t i o n of angle y 30 4.1 Experimental 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 current waveforms at speed of 720 r.p.m 43 4.5 F i e l d current waveforms at speed of 720 r.p.m 44 4.6 Armature current waveforms at speed of 900 r.p.m 45 4.7 F i e l d current waveforms at speed of 900 r.p.m. 46 4.8 Armature current waveform at speed of 1080 r.p.m 47 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 i v ACKNOWLEDGEMENT I wish to express my sincere thanks to the people who have given me assistance throughout the course of t h i s p r o j e c t . E s p e c i a l l y , I appreciate the guidance and encouragement given by Dr. H. R. Chinn and Dr. Y. N. Yu, my supervisors. Thanks are due to Mr. D. G. Mumford and Mr. E. Struyk for t h e i r c a r e f u l proof reading, and to Miss L. Morris for her typing. I am g r a t e f u l for the assistance of Mr. W. R. B l a c k h a l l , Mr. C. Chubb and Mr. D. G. Daines f o r preparing the experimental set-up, and of Mr. H. H. Black for preparing the photographs i n the t h e s i s . Also, I wish to thank Dr. B. J . K a b r i e l f o r h i s valuable comments. The f i n a n c i a l support of the National Research Council of Canada and the U n i v e r s i t y of B r i t i s h Columbia i s g r a t e f u l l y acknowledged. v NOMENCLATURE current through phase windings a, b and c d- and q-axis current f i e l d current load inductance d- and q-axis components of armature inductance f i e l d inductance magnitude of mutual inductance between f i e l d and armature windin • j ^ , d i f f e r e n t i a l operator number of machine pole-pairs load r e s i s t a n c e (Chapter 2); armature resi s t a n c e (Chapter 3) f i e l d r e s i s t a n c e output torque i n Newton-meters phase voltage amplitude voltages of phase a, b, c l i n e - l i n e voltage between terminals A and B d- and q-axis voltage f i e l d voltage f i r i n g angle e x t i n c t i o n angle angle between q-axis and brush axis i n equivalent motor with mechanical commutator and brushes power f a c t o r angle angle between q-axis and axis of r o t a t i n g magnetic f i e l d angle between q-axis and armature winding of phase a e l e c t r i c a l angular v e l o c i t y of the motor i n radians per second v i to. 1 0) m e c h power supply frequency i n radians per second frequency of the current waveform i n radians per second mechanical motor speed i n revolutions per second v i i 1 1. INTRODUCTION E l e c t r i c a l motors with speed co n t r o l are required f or many 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 . D.C. and A.C. commutator motors are often used f o r such purposes. However!',' mechanical commutators and t h e i r s p e c i a l winding connections are major disadvantages because of maintenance and s e r v i c i n g . Synchronous and induction motors, on the other hand,do not require commutators. But a synchronous motor operates at a fi x e d speed, the synchronous speed, and an induction motor operates at speeds s l i g h t l y lower than the synchronous speed. Their speed can be c o n t r o l l e d over a wide range only by changing the source s u p p l i e s . Thyratrons and mercury arc r e c t i f i e r s were used to control the speed of e l e c t r i c a l motors, known as Thyratron motors (1) or K o l l e c t o r l o s e r Stromrichtermotoren (2). With the development of s i l i c o n c o n t r o l l e d r e c t i f i e r s (SCRs), or t h y r i s t o r s , the speed co n t r o l of A.C. motors i s being re-examined. Methods f o r c o n t r o l l i n g the speed of A.C. motors involve v a r i a t i o n of the supply frequency by means of SCRs using converter-inverter schemes. Krause, e t . a l . (3), (4), (5) used t h i s arrangement to obtain a variable-frequency supply to drive an induction motor, and M i y a - i r i , e t . a l . (6) and Lipo, e t . a l . (7), (8) operated a synchronous motor i n a s i m i l a r manner. M i y a - i r i , e t . a l . (9), (10) and Sato (11) also reported using i n v e r t e r s and D.C. power supplies to drive synchronous motors. Tsuchiya, e t . a l . (12) operated a synchronous motor with a cyclo-converter, which converts A.C. at one frequency to A.C. at some other frequency. This thesis p r o j e c t examines the motor configuration shown i n F i g . 1.1, which was o r i g i n a l l y tested by Okada at Doshisha U n i v e r s i t y . Three Three Phase Power Supply Q O O 6 o 6 CD 6 JO lQ Co Nj o CO o Co 'Co Q icj Co 33 CD 3 SCRs, delta-connected, are inse r t e d i n the neutral point of the three-phase armature winding of the synchronous motor. The p r i n c i p l e of t h i s motor i s as follows: F i r i n g SCR^, the magnetic f i e l d caused by the armature current i s approximately i n the d i r e c t i o n from A towards B. With SCR^ conducting, SCR 2 i s f i r e d so that the d i r e c t i o n of the magnetic f i e l d rotates clockwise to a l i g n i t s e l f i n a d i r e c t i o n from A towards a p o s i t i o n somewhere between B and C. SCR^ then extinguishes and only SCR2 remains conducting. The magnetic f i e l d produced by the armature current i s then i n the d i r e c t i o n from B towards C. Now the magnetic f i e l d r e s u l t i n g from the armature current has rotated about 120 degrees from the o r i g i n a l d i r e c t i o n . Repeating the same procedure f o r SCR 2 and SCR^, and so on, the set of SCRs provides a r o t a t i n g magnetic f i e l d . The speed of the r o t a t i n g magnetic f i e l d may be co n t r o l l e d by f i r i n g these SCRs i n the above sequence with pulses derived from the p o s i t i o n and speed of the r o t o r . In t h i s t h e s i s , the delta-connected SCRs i n a three-phase A.C. c i r c u i t (13) i s analyzed using Take-uchi's <j>-function method (14). A balanced three-phase c i r c u i t without mutual coupling i s assumed. Experimental v e r i f i c a t i o n of the analysis i s given. The analysis i s extended i n Chapter 3 to the motor configuration of F i g . 1.1. The d e s c r i p t i o n of the d e t a i l e d experimental set up of the system, the test r e s u l t s and comparison with a n a l y t i c a l r e s u l t s are given i n Chapter 4. 2. CHARACTERISTICS OF THREE-PHASE DELTA-CONNECTED SCRs 2.1 Introduction Before the motor configuration of F i g . 1.1 i s analyzed, we s h a l l i n v e s t i g a t e the c h a r a c t e r i s t i c s of three-phase delta-connected SCRs in s e r t e d i n the n e u t r a l point of a balanced three-phase star-connected load as shown i n F i g . 2.1. C o B o 1 F i g . 2.1 Three-phase delta-connected SCRs The gate pulses to the SCRs are synchronized with the.supply frequency. The f i r i n g angle, a, and the e x t i n c t i o n angle, g , f o r each SCR are measured from the n e g a t i v e - p o s i t i v e zero-crossing point of the l i n e - l i n e voltage. Three modes of SCR operation are defined as Modes 1, 2 and 3, corresponding to the' number of SCRs conducting at the same time. 5 Three cases may be considered depending upon the f i r i n g angle i n r e l a t i o n to the power f a c t o r of the load. In Case 1 only Mode 1 occurs. In Case 2 Modes 1 and 2 occur a l t e r n a t e l y . In Case 3 Modes 2 and 3 occur. Note that SCR f i r i n g c o n t r o l i s completely l o s t i n Case 3. 2.2. Analysis f o r Case 1 As mentioned above f or Case 1, only one SCR i s conducting at any time. Assume that only SCR^ i s conducting as shown i n F i g . 2.2 and that the load i n each phase consists of a r e s i s t o r R and an inductor L i n s e r i e s . B O ; F i g . 2.2 Three-phase delta-connected SCRs under Case 1 operation with SCR^ conducting Let the phase sequence be a, b and c, v,„ = v - v, , and v,_ be r ^ AB a b AB the reference phasor. The voltage applied to the load between A and B i s 6 v . B ( t ) = SiV s i n u t (U. - U.) (2.1) A where co = supply frequency i n radians per second, o A V = magnitude of the phase voltage, A a = f i r i n g angle, A 3 = e x t i n c t i o n angle, = unit step function U(t - ~ ) > o A g. . = unit step function U(t - — ) o Taking the Laplace transform of equation (2.1), s s i n a + co cos a V A f i ( s ) = / 3 V [ 5 1 e x P ( - — s) s + co o o s s i n B + co cos o 2 2 r w s + co o o e x p ( - ^ s ) : ] (2.2) The network equation i s v ^ C t ) = 2[R + L p ] i ( t ) (2.3) A d where p = -J-JT > the d i f f e r e n t i a l operator. Taking the Laplace transform of equation (2.3), • V A B ( s ) = 2 [ R I ( s ) + L { s I ( s ) - i ( t )}] (2.4) AiJ O where i ( t ) i s the load current at the i n i t i a l time of switching, namely t o - _ o or cx/co . o From equations (2.2) and (2.4) pr „ s s i n a + co cos a T / v v3 V r o . a . I ( s ) • i r [ , + 2 + 2, e x p ( - - s ) (s + ft) (s + co ) o o • a o (2-5) s s m g + co cos g „ - O / p vi r r exp(- — S)J (s + ft) (s + CO ) o o 7 i ( t o ) + 7 T ^ e x p ( - t o s ) where ft = R/L. We define the <f>-function (10) as sin(co t + a - x) ~ sin(a - x)exp(-ftt) f(a,t) = 2 . • (2.6) i.e. , Z 2 ~ 2 /ft + 10 o s sin a + to cos 3 <£{<Ka,t)}= (s + ft)(s2 + to2) (2.7) o where X = tan "''(a) L/R) o Using <j)-functions, equation (2.5) becomes I(s) =-^7- [^{<Ka,t)> exp(- — s) - X W B . t ) } e x p ( - ^ - s ) ] o o i ( t ) + ~ r exp(-t s) (2.8) s + ft o From equation (2.6), the inverse Laplace transform of equation (2.8) is /3~V 2/R" + w2L" 1 ( t ) = [ s i n ^ t _ x ) ( U i - u2) - sin(ct - x)exp(t - — ) U n + sin(3 - x)exp(t - —)'U ] CO 1 CO / o o + i ( t )exp(t - -f)U n (2.9) O CO X o By definition, i ( t 0 ) i s zero and the extinction angle 3 must be less than . 2TT a + -y . oi 3 For — $ t < — , U.. is unity and U„ is zero, hence, co co 1 2 0 0 8 yr v a) t - a i ( t ) = — — [ S i n ( u t - x) - s i n ( a - x ) e x p ( - f t — )] (2.10) o where i ( t ) % 0. The e x t i n c t i o n angle g i s determined by s e t t i n g i ( t ) to zero i n equation (2 .10) . Therefore, 8 ex sin ( g - x ) e x P ( ^— ) = s i n ( a - x ) e x P ( ^ — ) (2.11) o o Note that 6 cannot be equal to a. Equation (2.10) gives the p o s i t i v e p o r t i o n of i , and the magnitude of the negative p o r t i o n of i , . The p o s i t i v e portions of i , and i are b b e obtained i n the same way. In general, , 2^ 0 , 2TT i C t ) - ,[sin(co t - y) {U(t — ) - U(t — ) ) o to tu 2TT , 2TT 9 a + n-r- a + n-r-- s i n ( a + n4f - X)exp{-ft(t — ) }U(t — ) J to to o o g. + T12TT g + ^ + sin ( 3 + n-2^ - X)exp{-ft(t — ) } U(t — ) ] (2.12) J to to o o where i ( t ) > 0 and n = 0, 1 and 2 for the p o s i t i v e portions of i . i . and i a b c re s p e c t i v e l y . The negative portion of i i s obtained when SCR conducts since the a j current flows from C to A. i . e . , i = - i ' a c where i < 0. a 9 Therefore, the total current i i s given by V t ; " n 5-5- [sin(co t - X H U ( t - - U(t - -2-)} 2/R2 + co2L2 0 Uo "o - sin(a - X)exp{-ft(t - — ) }U(t - — ) + sin(g - X)exp{-fi(t - —)>U(t - — ) ] * CO CO CO CO o o o o {U(t - — ) - U(t - -£-)} CO CO o o • y^v , a + 4r B + 4? /o o o [sin(co t - ^ - X){U(t — - ) - U(t - — — - ) ) zA2 + co2L2 ° 3 Uo " o o . , 4TT . 4TT a + — a + — - sin(a - x)exp{-fi(t — ) ) U ( t —) CO 'CO o o B + 4^- g + 4f + sin(B - x)exp{-Q(t —)}U(t — ) ] * CO CO o o , 4TT 4TT a + - 5 - B + -~-{U(t -) - U(t -)} (2.13) CO CO o o 2.3. Analysis for Case 2 The essential difference between Case 1 and Case 2 i s in the value of the extinction angle 3. In Case 1, the extinction angle g for one phase is 2TT always less than a + —^r, which i s the f i r i n g angle of the following phase. How-ever, in Case 2, two SCRs may be conducting under Mode 2 operation so that g i s 2 4 greater than or equal to a + y but less than a + To obtain the steady-state current through phase a, consider the following. Let the f i r i n g sequence of the SCRs for alternate Modes 2 and 1 be as follows: SCR2 and SCR3, SCR3 only, SCR3 and SCR^ SCR± only, SCR;L and SCR2> SCR2 only, and so on. Then the voltage applied to the load of phase a i s v (t) = V s i n ( c O o t - | ) ( U X - U 2) + 4j V s i n ( c o Q t - |) (U 2 - Ug) 1 0 + V sin(co t - ( U , - U . ) + V sin (to t) ( U . - U _ ) + V sin(co t - T - ) ( U _ - U , ) ( 2 . 1 4 ) o 6 5 o 2TT . .Ct 77 where U , = U(t — ) 1 to o 8 - — U 0 £ U(t 3 2 co U, * U(t -o „ 4 u ( t 3_) 4 co o , 2TT u 5 i « « - - ^ - > o A 8 and U, = U(t - — ) are unit step functions. D CO o Note that, for the above, phase voltage and one-half of the lin e - l i n e voltage are applied alternately to the load of phase a for Modes 2 and 1 , respectively. The c i r c u i t equation i s v (t) = [R + Lp]i (t) ( 2 . 1 5 ) a. 3. From equations ( 2 . 1 4 ) and ( 2 . 1 5 ) , a piecewise solution o f . i (t) (see Appendix) Si gives i a ( t ) = — [sin(to ot - -5- - X) /R + R  co L c co t - a + — - sin (a - _ x)exp(-ft ° • =2-)] (2.16) o to . o 2TT 0 4TT * a " T B ' T for — $ t < C0_ (0 11 i (t) = , [- -J- sm(u t - — - X) •R + co L o 5TT ~ A 3~ - sin(a - - 7 - - x) exp (-ft — ) 0 co o +{sin(B - |1 - x ) + ^ | sin (g - _ x)}exp(-ft -2—- 3—) ] (2.17) co o 4TT f o r 3— < t < 5-CO. CO o o 1a ( t ) = — [sin(to t - -r - x) •R + to L 5TT % - a + — - s i n (a - x) exp (-ft ) O CO . 4TT +{sin(B - f- - X) + ^  s i n < 8 " T• ' X ) } e x P ( - ^ 0 t M g + 3 ) •3 . , 4TT co t - a -{sin(a - f ' - ' x ) + "7T sin(a - - x)>exp(-ft-5 )] (2.18) O Z -J CO o 6 _ F O R J L , t < : — i i . CO CO o o 1 (t) = ~ [-5- s i n (to t - x) a [2 ll  2 0 /R + co L 5TT co t - a + — - s i n (a - -7 x) exp (-ft-5 : ) D CO o ~ nr o c o t - B + 4 ^ " +{sin(6 - f- - X ) + s in (g - | L _ x ) } e x p ( _ p ^ _ 1_) o IT / J 4TT CO t — 01 - {sin(a r J - X) + s i n ( a - -j- - X ) }exp (-ft-5-^ ) 12 2jL +{sin( B - - x) - - | sin(3 - ^ - X)}exp(-ft-2 ] (2.19) 2TT ^ 2ir for — < t< CO CO o o 1 ( t) = ~ [sm(co^t - - - x) 3. / 2 2 2 co L o ° 6 , 2-TT - s i n (a-. 1 - x) exp (-ft - ) o co t - 3 + ^ T r +{sin(3 - j 1 ~ X) + 4 s i n ( B ' "^ 3 " X)}exp(-ft^ — ^ ) -{sin(a - T - X) + —T sin(a - x)} e xP(-fi- 2 ) O Z j CO o r /T 9 C 0 t - 3 + — +{sin(3 - — - x) " 1i sin(3 - - X)}exp(-ft^ ^-) 6 2 3 Wo u t _ a _ 21 -{sin(a + -| - X) j sin(a + ~ - x)}exp(-ft-2 — ) ] o (2.20) for ^ .< t < CO CO o o a + — i (t) = 0 for — $ t < 3 (2.21) a co ° CO . 0 The extinction angle 3 i s obtained by substituting 3 for co t and o equating i = 0 in equation (2.20). 3. 13 2.4. Analysis f o r Case 3 By d e f i n i t i o n , at l e a s t two SCRs are conducting at any time. This implies that phase voltages are always applied to a l l phase loads, and that f i r i n g c o n t r o l of the SGRs i s l o s t . Thus, the c i r c u i t equation f o r phase a i n Case 3 i s given by equations (2.22) and (2.23). v (t) = V sin(oo t - 7) a o 6 (2.22) v (t) = (R + Lp) i (t) a a Hence the steady-state current through the load of phase a i s (2.23) i a ( « > -/3~> 2~2 /R + OJ L s i n ( u o t - j - x ) (2.24) Since f i r i n g c o n t r o l i s no longer e f f e c t i v e , terms containing the f i r i n g angle ct do not appear i n equation (2.24). 2.5. Comparison of A n a l y t i c a l and Experimental Results f o r Three-Phase Delta-Connected SCRs To v e r i f y the analysis i n the previous sections the c i r c u i t of F i g . 2.1 was set up i n the laboratory with Resistor R Inductor L Phase voltage Load power fa c t o r The SCRs used were type GE20C. 20.45 ohms 11.3 mh 120 v o l t s 0.98 l a g . 14 F i g . 2.3 shows t y p i c a l current waveforms for Case 1. F i g . 2.3(a) i s the waveform calculated from equation (2.13). F i g . 2.3(b) shows the experimental current and voltage waveforms. The f i r i n g angle i s 114 degrees. The a n a l y t i c a l and the experimental r e s u l t s agree quite w e l l . T y p i c a l current waveforms for Case 2 are shown i n F i g . 2.4. The waveform c a l c u l a t e d from equations (2.16) to (2.21) i s shown i n F i g . 2.4(a). The corresponding experimental current and voltage waveforms are shown i n F i g . 2.4(b). The f i r i n g angle f o r t h i s case i s 14 degrees. Good agreement was obtained between the a n a l y t i c a l and the experimental r e s u l t s . 2.6. Relationship Between the F i r i n g Angle and the E x t i n c t i o n Angle The e x t i n c t i o n angle 3 i s a f f e c t e d by both the f i r i n g angle a and the power f a c t o r of the load, and the e x t i n c t i o n angle i n turn determines the actu a l case obtained, i . e . , Case 1, 2 or 3. The r e l a t i o n s h i p between the f i r i n g angle, the power factor of the load, the e x t i n c t i o n angle and the case obtained i s shown i n the e x t i n c t i o n angle - f i r i n g angle curves of F i g . 2.5 f o r various power f a c t o r s . As lagging power f a c t o r approaches unity, the e x t i n c t i o n angle decreases f o r any f i x e d f i r i n g angle. Also for a given lagging power f a c t o r , the e x t i n c t i o n angle decreases as the f i r i n g angle increases. For example, consider a load of power fa c t o r 0.5 lagging. Cases 3, 2 and 1 occur f or values of f i r i n g angles of 0 to 30 degrees, 30 to 104 degrees and 104 to 180 degrees, r e s p e c t i v e l y . Also the values of e x t i n c t i o n angles f or Cases 2 and 1, i n the example, are 262 and 223 degrees, r e s p e c t i v e l y , for corresponding values of f i r i n g angles of 60 and 120 degrees. 15 6 . 0 - j o J 3 . 0 U J CK O 0 -i - 3 . 0 -- 6 . 0 — I 1 - j — 120 240 THETfl (DEG.) (a) 3 6 0 THETA (46 Deg./Div.) (b) Fig. 2.3 Waveforms for Case 1 (a) Calculated'load current waveform (b) Experimental current and voltage waveform 8.0 ^ in ° I " ... . V / \ . ' A / / / THETA (46 Deg./Div.) (b) F->'g„ 2.4 Waveforms for Case 2 (a) Calculated load current waveform (b) 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. ANALYSIS OF THE SCR-CONTROLLED MOTOR  3.1. Introduction The delta-connected SCRs discussed i n the previous chapter w i l l now be considered i n r e l a t i o n to a synchronous motor. The delta-connected SCRs are i n s e r t e d i n the n e u t r a l point of the armature winding of the synchronous motor. The f i e l d winding i s D.C. excited. The motor with co n t r o l l o g i c i s shown i n F i g . 1.1. The system consists of a synchronous motor with SCRs, a d i s t r i b u t o r , a phase s h i f t e r , an AND gate and a 21.6 KHz c r y s t a l clock. The SCR gate pulses are obtained as follows: The phase s h i f t e r output i s used to set an R-S f l i p - f l o p , F i g . 3.1, which then remains i n the ON state u n t i l reset by a pulse which occurs at the positive-negative zero-crossing point of the l i n e - l i n e power supply voltage. The r e s u l t i n g R-S f l i p - f l o p output i s a p e r i o d i c pulse of the power supply frequency. The d i s t r i b u t o r consists of three lamps mounted on a d i s c , each lamp being aligned with a photo-diode on another d i s c . These discs are f i x e d to each other and adjustable i n p o s i t i o n with respect to the motor armature. A t h i r d d i s c with two s l o t s cut i n i t i s mounted on the motor shaft. Light from a lamp i s received by the corresponding photo-diode when l i g h t passes through a s l o t i n the r o t a t i n g d i s c . At these i n s t a n t s , pulse s i g n a l s are obtained from associated c i r c u i t r y . The time of occurrence of an output pulse i s r e l a t e d to- the rotor p o s i t i o n while the pulse r e p e t i t i o n frequency i s r e l a t e d to the motor speed. This pulse i s then used to gate the 21.6 KHz c r y s t a l clock output. These pulses are applied to the AND gate to y i e l d bursts of f i r i n g pulses, F i g . 3.2. The d i s t r i b u t o r s l o t s are cut so that the width of the output pulses from the d i s t r i b u t o r i s one-third the Phase Voltage _ (a) Phase Voltage 0 (b) Phase Shifter Output Distributor Output Clock Output i Mono Stable SCR ogate OSCR cathode F i g . 3.1 C i r c u i t to generate SCR gate pulses Line-line Supply voltage Phase shifter output R-S flip-flop output Distributor output SCR gate pulses F i g . 3.2 Waveforms for c i r c u i t of F i g . 3.1 21 period f o r a given speed, i . e . , the burst of f i r i n g pulses to each SCR gate may occupy up to one-third of a period. Mode 1 operation occurs when one SCR i s conducting. Once an SCR i s f i r e d , i t s e x t i n c t i o n angle i s dependent upon i t s f i r i n g angle and the power f a c t o r of the load. Thus, the SCR may s t i l l be conducting when the next SCR i s f i r e d . The r e s u l t i s Mode 2 operation with two SCRs conducting, as defined i n Section 2.1. The analysis of each of these Modes w i l l be discussed i n d e t a i l i n the following sections. The motor w i l l be assumed to have two poles and to operate at a constant speed. 3.2. Mode 1 Operation of the Motor One SCR i s conducting under Mode 1 operation, as shown i n F i g . 3.3. A o B o F i g . 3.3 Motor with r o t a t i n g D.C. f i e l d under Mode 1 operation, SCR1 conducting The distributor 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^, in turn, after each 120 degrees rotation of the rotor. Since the repetition frequency of the output pulses from the distributor is 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^ is conducting. The motor configuration of Fig. 3.3 with rotating f i e l d , for Mode 1, may be replaced by a motor with a rotating armature, as shown in Fig. 3.4. This second motor configuration may be further replaced by a motor with a mechanical commutator and brushes as shown in Fig. 3.5. Since the width of the distributor output pulses is one-third the period for 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 B o A F i g . 3.5 Equivalent motor with mechanical commutator and brushes 3.3 Salient Pole Motor Equations, Mode 1 Operation The s a l i e n t pole motor w i l l be discussed i n this section. Using the transformation matrix a 0 1//6 [A] = y| d s i n 9 q cos 6 1//6 s i i < e g ) cos ( 6 — Y ) 1/^6 . 2TT. s i n ( 6 + — ) c o s ( 6 + ~ ) 24 whose inverse i s tA] -1 o /372 sm q 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 equation i n d-q coordinates are v. v R + L dP toL. -OJL R + L p q / 3 vr — M p 2 f F — M p 2 f R f + L f P f ^ i q i ' f i vanishes s i n c e ( i + i , + i ) i s equal to zero, o. a b c ^ From equation (3.1) (3.3) • * *d fi / 3 i q s i n 6 sin(0 - ^ |) sin(6 + y ) ,„ 2TT. , 2TT. cos 6 cos (6 - — ) cos (6 + — ) Since i i s zero and i , i s euqal to -i when SCR, i s conducting, equation c b a 1 (3.4) (3.4) becomes 2n sin6 - sin(8 - — ) 2IT cosG - cos (6 - —~) (3.5) Define an angle as the angle between the q-axis and the M-axls, the axis of the rotating magnetic f i e l d . Then the relationship between 0 and TJJ Is 25 9 = \\i - (3.6) Su b s t i t u t i n g equation (3.6) into equation (3.5), we get s i n TJJ cos \p [ 1 a ' From equation (3.2) f - • V a Vb / 3 V c sm cos 6 • „ 2TK /. 2TT N sin(6 - - j ) cos(6 g) sxn(9 + — j ) . cos (9 + y ) v d V q where v . v, , v , v. and v are unknown, a' b' c' d q Equation (3.8) may be rewritten with equation (3.6) as /3 . , 1 , /3 * > V a / 3 V, b — sinijj - y cos i|> ^ cosijj + — s i n 4J 2 1_ 2 <3 2 sinij; + — cos ij; cos^ - TJ- s i n i|; Although v & and are unknown, the l i n e voltage v^g i s known. Hence, VAB = V a " Vb = 7l sinib v, + 7l cosib v d q Sub s t i t u t i n g v^ and v^ from equation (3.3), equation (3.10) becomes (3.7) (3.8) (3.9) (3.10) / 3 VAB = ^ S ± n ^ + L d p ) i d " u L a i a +/~2 M f P i f ^ q q V 2 f K f' + /2 cosi|) [toLji, + ( R + L p ) i +/- M,a>i>] a a q q v z r r (3.11) Since i , and i are known i n terms of i , d q a 2 2 d i a v A T, = 2[R + to(L, - L )sin2ij)]i + 2(L,sin.iJ> + L cos i|i)—r~ AB d q r a . d q dt d i f + /i m^cosi> i ^ + / J sirnjj M — — f f f dt Also from equations (3.3) and (3.7) V f = /3 M f P i d + (R f + L f P ) i f d i d i f = /3 toM.cosiJj i + /3 M.sinil' — : r R,-i,- + L,- ~;— f a f T d t f f f d t (3 (3 From equations (3.12) and (3.13), the machine equations f o r Mode 1 are di a dt G l l d i f = dt G21 sin + '12 22 H l l H12 H21 H22 AB (3 D = 2[L £L - 4 M 2 s i n 2 i | > ] f v 2 f G.n = -[2L C{R + <o(L - L )sin2^}- -| M 2 w sin2;l;]/D 11 r d q 2 r G 1 2 = -/3Mf [coLf cos^ - RfsimM/D G 0 1 = -2/3 M £ [ O J L COS ^ - sini); {R + u ) ( L , - L )sin2ijj} ]/D 21 r v d q G22 ~ "[2RfLv "I f^ S i n 2 ^ 3 / D -H11 = V° H12 " H21 " Mf Sln* /D H 9 0 = 2L /D 22 v 27 The torque equation i n d-q coordinates i s , i n general, T - p [ / f M f i f i q + a d - L q ) i d i q ] where P i s the number of p o l e - p a i r s . S u b s t i t u t i n g equation (3.7) in t o equation (3.15), we get (3.15) T = P[/3 M £ C O S O J J i - i + ( L, - L )sin2ip i ] 1 f y f a v d q r a J In the above equation, P, M ^ , (L, - L ) and i are nonnegative values. However, the instantaneous torque may be negative depending upon the value of i\i, the rotor p o s i t i o n , and the value of the f i e l d current i ^ . (3.16) 3.4. Round-Rotor Motor, Mode 1 Operation The machine equations for a round-rotor type motor are obtained by s e t t i n g (L^ - L^) to zero i n the above a n a l y s i s , so that d i dt d_ii dt A l l A12 A21 A22 f where L = L, = L ' c d q . F = 2[LJL - I M^sin 2^] I c i t A n = -[2RL f - | OJM 2 sin2iJ;]/F A 1 2 = -/3 M f [coLfcos \> - R fsiniJ;]/F A 2 1 = -2/3 M f[uL c C O S I J J - R s i n ^ ] / F A 2 2 = - L 2 R fL c- I OJM2 sin2^]/F B 11 B 12 21 B 22. AB (3.17) B u a L f / F 28 B12 " B 2 1 = - v / J M f S ± n ^ / P B 2 2 4 2 L c /F The torque equation f o r a round-rotor motor i s then T - P[ /4 M , i , i ] = /3 P M costo i . i V 2 r f q f Y f i (3.18) •3.5. Mode 2 Operation of the Motor Two SCRs are conducting under Mode 2 o p e r a t i o n . The machine diagram f o r Mode 2 i s shown i n F i g . 3.6. The machine equations same as equation (3.3) • V d R + L dp -coL q / fMfP A d V q = coLd R + L p q i q v f _ / 2 M f P R f + L f P (3.19) The torque equation f o r t h i s mode of o p e r a t i o n i s a l s o the same as equation (3.15) T - P [ / § M f i f i q + ( L d - L q ) i d i q ] . (3.20) 3.6. T r a n s i t i o n between Mode 1 and Mode 2 We w i l l now discus s the occurrence of the above two modes during normal motor o p e r a t i o n . The angle ^ i s a f u n c t i o n of time. Let i t be expressed by <J> = wt + O ' (3.21) where time i s measured from the i n s t a n t that the commutator segments 30 make contact with the brushes, F i g . 3.5. Note that time i s independently measured for each conducting SCR. Let y be the angle of the brushes advanced clockwise from the q-axis, as shown i n F i g . 3.7. Then, ty^ i s equal to -C| + y) and * = cot - | - y (3-22) Since the conduction period i s 120 degrees mechanical, ij> has a value between - y - Y and - - y. Next consider the voltage applied to the armature under Mode 1 operation. Because the motor speed i s not n e c e s s a r i l y the synchronous speed, 120 degrees mechanical r o t a t i o n does not correspond to 120 degrees e l e c t r i c a l phase s h i f t of the supply voltage. Commutation occurs when the conducting SCR extinguishes and the next SCR i s f i r e d . This may occur a f t e r 120 degrees i n r o t a t i o n which i s —^ co i n time. Since any machine speed co may be expressed by (3.23) O 6 F i g . 3.7 D e f i n i t i o n of angle y 31 where m and n are integers, and co^ i s the synchronous speed, the di f f e r e n c e between the mechanical and e l e c t r i c a l phase angles a f t e r one commutation i s given by 2TT _ 2TT_ _ "o ^ 2TT " l U o i i 3 O J 3 n-m 2TT m 3 (3.24) In general, a f t e r K commutations n-m 2IT Note that a f t e r 3m commutations, i . e . , m mechanical re v o l u t i o n s , n i s n 3 m = ( n - m ) 2TT which means that 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. In general, 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/P revolutions f o r a machine with P p o l e - p a i r s . Thus, the voltage across the brushes i s (3.25) v = S3 V sin(oj t + K ^ ^ + ? ) s o m 3 (3.26) where £ i s the phase angle of the l i n e - l i n e voltage at the beginning of the 3m commutation. K i s an integer (K = 0, 1,2,...) i n d i c a t i n g the number of completed commutations. The conditions for Mode 1 operation to change to Mode 2 w i l l be considered next. Under Mode 2 operation, voltage i s applied to a l l windings: v. s i n 6 s m ( 6 -) s m ( 9 + -y) ,„ 2T T X , „ , 2n\ cos 6 cos (9 -) cos (9 + — ) cos (co t. - ll) + O o sin(co ot - i\> + O 32 (3.27) As mentioned before, the extinction 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 fi r i n g period corresponds to one-third of a rotation because of the mechanical distributor. Thus, SCR^ i n Fig. 1.1 may be conducting when SCR2 enters i t s f i r i n g period. If SCR2 is fired while SCR^ is s t i l l conducting, then the mode changes from Mode 1 to Mode 2. One of the necessary conditions to change from Mode 1 to Mode 2 is that armature current continues to flow through SCR^ beyond i t s f i r i n g period. The other necessary condition is that gate pulses must be applied to SCR2 while SCR^ is conducting. At the instant that the f i r i n g period ends for SCR^, the f i r i n g period for SCR2 commences. Hence, time t is again zero and K is incremented by one. Then, the phase angle of the supply voltage at this instant i s , from equation (3.26), a « K 2=H 2 ± + K (3.28) m J A burst of fi r i n g pulses i s applied to the gate of SCR2 when 6 is a $ 6 < 7T (3.29-) These two conditions are not sufficient to change the mode of operation because of voltages generated i n the armature windings. The additional condition for changing the mode is that the current in the third winding, i.e., the phase winding c i n this example, must become negative. Since the i n i t i a l current in the third phase winding is zero, the third condition is di ° < 0 (3.30) dt 33 From equation (3.1) i c ( t ) . [sin(6 + I 1 ) ! , + cos(e+ I 1 ) i ] 3 3 d 3 q (3.31) Hence, d i fT 9 9 d l H 9 9 d i n ^ = /j [to cos(e+ y-) i d + sin(e+ 3 ) - co s i n ( 6 + ^ f ) i + cos( e + +f -£] 4 co[cos(6 + ^jf) i , - sin ( e + % ) i ] 3 i d 3 q L„v, - /•£ M.V. - L^ R i , + <C_L ci M.R.i 2 • /« . 2 ^ f d / 2 f f. £ d q f g — sm(9 + — ) — 3 1 3 2 L d L f " 2 M f f f f 2^ v - L o L ^ i ^ - Ri cos(6 + ~ ) j — ^ •/I t oM^i^ (3.32) < 0 When these conditions are s a t i s f i e d , operation changes from Mode 1 to Mode 2. Since the i n i t i a l conditions of i , and i f o r Mode 2 are d q s i n ip = ft i q C O S lj> (3.5) At the i n s t a n t .that operation changes from Mode 1 to Mode 2, the current through the windings are i = /— (s i n 0 i , + cos 0 i ) a / 3 d q 2 /*3 {sin(ij/ - -^OsiniJ; + cos(^ - -r)cos ip}l 0 b a i b = / f [sin(8 - ^ j ) i d + cos(0 - ] 34 — [sin(ij) - -^r)sin \p + cos(iJj - -^~) cos <J>]ia = ~ i a i = / - [sin(6 + % i , + cos(e + % i ] c V 3 3 d 3 q 2 TT T  = — [sin( i|; + -r)sin + cos (iji + T;)COS i j j]i = 0 Next, the condition to change operation from Mode 2 to Mode 1 i s that the t h y r i s t o r SCR^ extinguishes while SCR^ continues conducting, i . e . , the current through phase a of the armature i s zero. Hence, the necessary and s u f f i c i e n t condition to change from Mode 2 to Mode 1 i s i (t) = / — [sin 6 i , + cos 9 i ] a / 3 d q (3.3 3) Since time i s again zero at the beginning of the f i r i n g period for SCR2, 2TT \p i s replaced by \j> + — j . Subs t i t u t i n g i & = 6, ±^=x^ and i ^ - i ^ i n t o equation (3.4), i d / 3 i q sm(9 p sm(9 + —p cos (9 p cos (9 + —p (3.34) where 9 = - — H r o J Then,at the i n s t a n t that operation changes from Mode 2 to Mode 1 the currents are given by • > = SI s i n ij; 1 q COS ]\). (3.35) 3 . 7 . S t a r t i n g Torque Equation ( 3 . 2 2 ) shows that time i s always measured from the i n s t a n t that the commutator segments begin to make contact with the brushes for each SCR. The rotor p o s i t i o n at the ins t a n t of s t a r t i n g may be any p o s i t i o n such that (- — - y) S i|» $ ( ^ - - y) . Define t h i s angle as ty^. Then, from equation ( 3 . 1 0 ) , s t a r t i n g torque T g i s given by T = P [ / 3 M £ cos 4». i , i + (L, - L ) s i n 2 i j j . i 2 ] ( 3 . s f l f a d q l a Note that s t a r t i n g torque i s zero when i s -TT/2 or i r / 2 . This means that s t a r t i n g torque i s dependent upon the i n i t i a l rotor p o s i t i o n . Also the speed of the r o t a t i n g f i e l d i s always synchronized with that of the rotor by means of the d i s t r i b u t o r so that the motor i s s e l f - s t a r t i n g even without damper windings, unlike the normal synchronous motor. 36 4. EXPERIMENTAL VERIFICATION OF THE OPERATION OF THE SCR CONTROLLED MOTOR To v e r i f y the analysis of Chapter 3 the motor configuration of F i g . 1.1 was set up i n the laboratory. The actual experimental set up is- shown i n F i g . 4.1. Comparison of the experimental r e s u l t s with the a n a l y t i c a l r e s u l t s w i l l be discussed i n t h i s chapter. 4.1. System components Gate Pulses. Various types of pulse s i g n a l s may be applied to an SCR gate. One type consists of gate pulses that commence at the desired f i r i n g angle a and end at the i n s t a n t corresponding to 180 degrees i n phase of the power supply. Such long-duration pulses are most des i r a b l e for f i r i n g SCRs, e s p e c i a l l y where inductive loads are involved. Because the d e l t a -connected SCRs cannot share a common ground connection, nine i s o l a t e d D.C. power supplies are required to generate these f i r i n g pulses. Pulse trans-formers are used to overcome the problem of the common ground connection. With a"pulse transformer, a short-pulse, or a burst of short-duration pulses commencing at the desired f i r i n g angle a and ending at 180 degrees of the power supply, may be applied to an SCR gate. The disadvantage of the s i n g l e short-duration pulse i s that i t may f a i l to f i r e the SCR. With a burst of short-duration pulses, f i r i n g of an SCR i s assured. The l o g i c to obtain these pulses has been described i n Section 3.1. Motor and SCRs. A UNITEC U-132 synchronous machine i s used f o r t h i s experiment. The s p e c i f i c a t i o n s are as follows: Output power : 1/4 H.P. Nominal input voltage (three-phase 1-1): 220 v o l t s F u l l - l o a d current : 1.3 amps. F i g . 4.1 Experimental set-up 38 Number of poles : 4 F i e l d e x c i t a t i o n : 1.5 amps, at 24 v o l t s Synchronous speed : 1800 r.p.m. Three GE20C SCRs are delta-connected and inserted i n the neutral point of the armature windings. D i s t r i b u t o r . The function of the d i s t r i b u t o r i s to detect the motor speed and the rotor position with respect to the armature windings. The construction 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 instead of an induction regulator. The l o g i c used, F i g . 4.3, i s as.follows: The negative-positive zero-crossing point of a l i n e - l i n e supply voltage i s detected by a comparator. At this instant a pulse triggers 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 set 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 instant corresponding to th i s angle ex. Pulses for the Other two phases are derived from this phase s h i f t e r output pulse by delaying i t successively for 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 in t e r a c t i o n of a permanent magnet disc and eddy currents i n a rotating copper disc. 4.2. Determination of y = 0 Position of the Distributor The angle Y has been defined as the angle between the brush-axis and the q-axis of the equivalent motor for the f i r i n g period of the SCR under consideration, Section 3.6. Since the value of the angle Y affects the operation of the motor, i t i s necessary to determine i t s value. From equation (3.14), Phase Voltages (b) (a) 2 2 2 [ L , s i n if) + L cos I J J J V , , - /J M R sinV>' vA1J d r q . r f f AB 2^3 [toM£ cos i|> (L sin. 2 iji + L c o s 2 if) - M r s i n ^ {R + to(L, - L )sin2ip}]i + [ 2 R f ( L d s i n 2 if; + L q c o s 2 i>) - -|coM2 sin2ij;]i f + [ 2 L f ( L d sin 2'if; + L q c o s 2 i|>) - 3M2 sin2if>] (4.1) When both if) and to are zero, equation (4.1) becomes V f = R f ^ + L f ( 4 - 2 ) Equation (4.2) shows that the f i e l d current i s independent of the armature current. Also from equation (3.22) Y = - f (4.3) Therefore, the procedure to determine y = 0 p o s i t i o n i s as follows: 1) Supply only enough three-phase voltage to turn on any SCR with the f i e l d c i r c u i t open. 2) While manually turning the r o t o r , observe the voltage waveforms induced i n the rotor c i r c u i t . Find the rotor p o s i t i o n such that there i s no induced voltage i n the. f i e l d c i r c u i t . This occurs when the angle if; i s zero. 3) Adjust the p o s i t i o n of the discs containing the lamps and the photo-diodes with respect to the armature for each phase such that one lamp and i t s corresponding photo-diode i s i n the middle of a s l o t of the r o t a t i n g d i s c when if> i s zero. This occurs when the angle y i s zero. 42 4.3. Comparison of A n a l y t i c a l and Experimental Current Waveforms From the analysis of Chapter'3, a computer program was w r i t t e n to obtain the waveforms of the f i e l d and armature currents. The r e s u l t s are compared with experimental r e s u l t s obtained from the set up discussed i n Section 4.1. The armature current waveforms at the motor speed of 720 r.p.m., which i s two-fifths the synchronous speed, i s shown i n F i g . 4.4. F i g . 4.4(a) shows the armature current c a l c u l a t e d from equations (3.14) and (3.21), to have a period of 83 msec. F i g . 4.4(b) shows that the corresponding experi-mental armature current has a period of 81 msec. As discussed i n Section 3.6, 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 supply voltage repeats every m/P revolutions for a motor with P p o l e - p a i r s . Since the motor has two pole-pairs and i s operating at two-fifths the synchronous speed, both m and P are two. Hence, one r e v o l u t i o n corresponds to one period of the waveform which i s 60/720 s e c , or 83 msec. The c a l c u l a t e d maximum.and minimum armature current values are 3.0 and -2.8 amps, r e s p e c t i v e l y . The corresponding experimental values are 2.9 and -2.9 amps, r e s p e c t i v e l y . The f i e l d current at 720 r.p.m. i s shown i n F i g . 4.5% Fig. 4.5(a) shows the c a l c u l a t e d r e s u l t which has a period of 83 msec, while F i g . 4.5(b) shows the corresponding experimental r e s u l t which has a period of 81 msec. The d i f f e r e n c e i n values may be a t t r i b u t e d to experimental e r r o r . The angles y and a were set to 40 and 30 degrees, r e s p e c t i v e l y . The a n a l y t i c a l r e s u l t s are i n good agreement with the experimental r e s u l t s . The a n a l y t i c a l and the experimental currents are also compared at the motor speed of 900 r.p.m., i . e . , at one-half the synchronous speed. The Q: Cfc o Ui ct >^ I 3D 2D 1 •7.0 -2.0 -3.0 40 TIME(ms) (a) Q \ U) CL o f\i UJ Q; Q: o Uj S Q; • ft 0 Oft j • : » TIME (10 ms/ Div.) . . ( b ) F i g . 4.A Armature current waveforms at speed of 720 r.p.m (a) A n a l y t i c a l current waveform (b) Experimental current waveform 3.0-44 2.0 fe 7.0 ct >^ o Q — i U j 0.0 •7.0 40 TIME(ms) (a) 80 \ s o Q •—i TIME (10ms/Div.) 00 F i g . 4.5 F i e l d current waveforms at speed of 720 r.p.m. (a) A n a l y t i c a l current waveform (b) Experimental current waveform 45 UJ ct ID °£ 1 1.0\-0.0 -1.0 •2.0 75 32 48 T IME(ms) (a) • • -• • • • \ to I g ct ct :D o Uj § ? ct TIME (5 ms/Div.) (b) r i g . 4.6 Armature 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 46 Fig. 4.7 Field current waveforms at speed of 900 r.p.m. (a) Analytical current waveform (b) Experimental current waveform 47 TIME (10 ms/div.) F i g . 4.8 Armature current waveform at speed of 1080 r.p.m. 4 8 c a l c u l a t e d armature current i s shown i n F i g . 4.6(a) which has a period of 33 msec. The corresponding experimental r e s u l t , shown i n F i g . 4.6(b), has a period of 31 msec. For one-half the synchronous speed, the same phase r e l a t i o n s h i p s of the voltage repeat every one-half r e v o l u t i o n of the motor. Note that the exact value of the period at half-synchronous speed i s 33.3 msec. F i g . 4.7(a) shows the ca l c u l a t e d f i e l d current and F i g . 4.7(b) the corresponding experimental r e s u l t . The a n a l y t i c a l and the experimental r e s u l t s are i n good agreement. 4.4. Speed-Torque C h a r a c t e r i s t i c s The speed-torque c h a r a c t e r i s t i c s of the motor has been investigated f o r two y values. 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 speed of 1080 r.p.m., which i s t h r e e - f i f t h s the synchronous speed, i s obtained f o r a load of 0.225 Newton-meters. The motor operates stably at a speed of 900 r.p.m., which i s h a l f the synchronous speed, with a load between 0.31 and 0.46 Newton-meters. The motor i s unstable f o r a load between 0.46 and 0.52 Newton-meters, except that a steady speed of 771 r.p.m., which i s three-sevenths of the synchronous speed, i s obtained at 0.475 Newton-meters. Increasing the load torque past 0.52 Newton-meters, the motor becomes stable again but changes i t s speed to 720 r.p.m., which i s two-fifths the synchronous speed, u n t i l the load torque reaches 0.85 Newton-meters. F i g . 4.10 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. Operation at three-quarters the synchronous speed, 1350 r.p.m., occurs at 0.21 Newton-meters. The motor operates stably at a speed of 1200 r.p.m., i . e . , two-thirds the synchronous speed, f o r a load between 0.21 and 0.36 Newton-meters. The motor becomes unstable beyond 0.36 Newton-meters and the speed changes. I t i s not stable at 900 or 720 r.p.m. Note that the motor was stable at these 1100k 1000 49 500 8- 800 Ui k» 700 to 6*00 0^ 04 06 TORQUE (N. m) Fig. 4.9 Speed-torque characteristic (T=t0deg.) 0.8 02 0.4 0.6 0.8 TORQUE (N.m) Fig. t.10 Speed-torque characteristic (t~ = B0deg.) 50 speeds f o r a y of 40 degrees. Thus the motor seems to run at m/n times the synchronous speed, where m and n are integers of value 1, 2, 3, ... The values of m and n r e l a t e the motor speed and the supply frequency, r e s p e c t i v e l y , to the frequency of the current waveforms as follows: ... 2nPw = mto. ' (4.4) mech I and to = nco. (4.5) o 1 where to , = mechanical speed i n revolutions per second, mech - - • io = frequency pf the current waveform i n radians per second ( 0 q = supply frequency i n radians per second P = number of pole-pairs Compare F i g . 4.4 with F i g . 4.8 which show the armature current waveforms for motor operation at two- and t h r e e - f i f t h s the.synchronous speed, 720 and 1080 r.p.m., r e s p e c t i v e l y . The stable operation of the motor at any speed depends upon the value of y. In general, y must be close to 90 degrees f o r high speed operation, and small f o r low speed operation. 51 5. CONCLUSIONS The three-phase 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 neutral point was analyzed using Take-uchi's rj>-function method. The analysis was confirmed by experiment. A three-phase synchronous motor with three delta-connected SCRs ins e r t e d i n the n e t u r a l point of the armature windings was then in v e s t i g a t e d f o r speed c o n t r o l . The f i r i n g pulses applied to the SCR gates occur at instances r e l a t e d to the motor speed,the rotor position, and the zero-crossings of the voltage supply. The armature current waveforms that r e s u l t are no longer p e r i o d i c at the supply frequency but at some other frequency r e l a t e d to the motor speed. Stable operation at a subsynchronous speed, which i s m/n of the synchronous speed,is p o s s i b l e i f m i s low i n value such that the th ' m harmonic component of the frequency of the current waveforms i s large, where m and n are:.integers. The motor i s s e l f - s t a r t i n g , s t a r t i n g torque being dependent upon i n i t i a l rotor p o s i t i o n . The a n a l y t i c a l r e s u l t s were experi-mentally v e r i f i e d . The above scheme of speed c o n t r o l of a synchronous motor does not y i e l d continuous speed v a r i a t i o n s for a voltage supply with f i x e d frequency, i n contrast with converter-inverter schemes, but the l a t t e r require more SCRs. Further research on the SCR-controlled synchronous motor could i n v e s t i g a t e the transient and s t a b i l i t y c h a r a c t e r i s t i c s . 52 Appendix Equations (2.16) to (2.21) are obtained as follows: Taking the Laplace transform of equation (2.14) we get Ssin(a- 4^)+co cos ( a - "5-?) a - ^? Ssin(3- -i-7r)+co cos (3- T ^ ) V » " v [ 6 XT 2 6 «*- ^ s> 4—5 * S + c o o S + co o 0 exp(- — — - S)] CO o =• Ssin(3- 4^)+^  cos (8- ^ r ) 8~ - ^ r S s i n ( a - -^)+to cos ( a - -£ ) »J „ r 3 o 3 , 3 3 o 3 , a „, , + — V[ T — 1 exp(- - — S ) - 2 — 2 «cp<- — S) ] S + c o o S + c u o o o Ssin ( a - -?)+co cos ( a - •?) Ssin(8~ —f)+w cos (8- ^ r ) 8- —? .„ r 6 o 6 / cn c N 6 o 6 , 3 „ N -, +V[ - 2 — ^ exp(- - S ) -j—1 e x p ( - — S>] b + to o S + c o o Ssin(8- •2-|)+coocos(8- 8- Ssin(a+ -^Ho^cos (a+ | "2 — 2 e x p ( - - j — B ) -j—2 * S + co o S + c o o o a + — exp( — S ) ] CO o Ssin(ot+ y)+u cos(orf f) cc+ ^ | Ssin(8- T)+IO cos (8- 7 ) + V ' ^ 2 - »< " - T * S ) - J T i W f S>1 S + c o o S+co o o a (1.1) 5 3 S i m i l a r l y , f o r equation (2.15), we get: V ( S) = RI (S) + S[LI (S) - i (t )] (1.2) cl 3. 3. SL O Since zero time was chosen such that the i n i t i a l current i s zero, equation (1.2) becomes v a(s) v a(s) V S ) = R + LS = L(S + ft) ( I , 3 ) Define the 4>-function, <j)(a, t ) , by sin(co t + a - x) - s i n ( a - x ) e x P ( - f i t ) <{>(a,t)^ (1.4) co2 + ft2 0 i . e . . Ssin a + co cos a ZO(a,t)}= ^ j ( 1 . 5 ) (S + ft)(S + u ) o where ft = R/L and X = tan "^ (co L/R) . o Equations ( 1 . 1 ) and ( 1 . 3 ) then become a- — 3- — I (S) = I UU(or 4 r,t)}exp(- — — 3 - S ) - & < K R - 2jL, t)]exp(- — — ^ S) ] a L o co D co o o + tfUCe-^.tnexpC- - r r - 1 S)-ZU(a- |,t)}exp(- ^ -S)] ° u o 3- — + £ [£{<Kcr-£,t:)}exp(- S)-Z{<}»(B- ^ , t ) }exp( r " 3 - S) ] o o /- g- — a+ — + ' iT^ [X(<j>(3- ^ , t ) } e x p ( 3 - S)-^{<t.(a+•2^,t)}exp( 3- S) ] L J CO J CO o o a+ — + 7 |-,t)}exp( 3- S ) - Z { K 3 - J , t ) > e x p ( - S)] ( 1 . 6 ) L 2, co0 6 % 54 Taking the inverse Laplace transform of equation (1.6), we obtain a- — 3- — i a ( t ) 1 [ T < a - ^ f,t- V~)ur ( B" - "r^ V g_ An o o B- — o o B- — + — + & m - ll,t - ^ ) U 4 - • (a + t - V > U 5 ^ + I [ ^ ( a f | i t _ _ 3 ) u + ( B _ | , t - f ) U 6 ] o o 2u A a " ~T where 1^ = U(t — - ) '5 "o U. £ U(t 2 U, £ U(t - - i ) J to o „ 4 u ( t - 1 3 <t (0 o A a + —r U 4 u(t i ) to o u6 A u(t - JL) (1.7) o 55 From equations (1.4) and (1.7), the piecewise s o l u t i o n of i (t) i s as follows: a When S C R . and S C R . are conducting, U = 1 and U „ = U . = U . = U C = U , = 0, z J 1 z j 4 5 o sin(to ot - — - x) _ s i n ( a jr - x )exp(-Q—- ) = V ^ / R 2 + A 2 (1.8) 2TT „ 4TT a - — B - — for $ t < -co co o o When S C R 2 extinguishes and S C R 3 remains conducting, = U" 2 = 1 and = u 5 - u 6 - 0, a - — g — 4 TT + 2 L * ( B 3 s t w } 2-rr To / c co t - a + — -r *3 . , . 4ir \ . , 5ir n •> „ o 3N [ ~ sxn(co t - - x)-sm(a 7- ~ x)exp(-fi -) j-7T- Y~2 L 0 3 6 u R + co L o . . A n -~ / r o c o t - B + -5— + {sin(B - ^ f - X) + - ^ 8 1 ^ 8 - - 3 - X)}exp(-ft-^ 1-) ] B — fo r 3- $ t < i -CO CO o o 56 When S C R 3 and SCR.j^ are conducting, u ± = u 2 = U 3 = 1 a n d U 4 = U 5 = U 6 = °' a — 8 - — ' o o 5u ,t Air 3 ) - <j> (a - ^ t 3' ")] + I * ( a - - , t - — ) o / R 2 + TT2L2 a. 2 t t 5TT U O ~ A ~3~ [sin(to t - — - x) " s i n ( a y - x)exp(-fi ) O D O OJ 8 " T + {sin (8 - y - X ) + y sin ( 8 - y - X)}exp(-ft . - 2 2_) o - {sin (a " f " X) + y s i n ( a - y - x)}exp(-ft - y — — ) ] (1.10) o o 2rr £ a e - ~ f o r — $ t < to to o o When S C R ^ extinguishes and S C R ^ remains conducting, = = = = 1 and U c = U, = 0, 5 o 57 3 - — - O O 6 - ^ + £ [••(<» - t - - ) - •(B - — , t - - y — ) ] o o S - ^ + " ^ ( e 3 ' fc — ) , 2TT T T A - c to t - a + — V r /3 . , „ . . , 5TT N / ^ O 3 n ZZZZ IT s1IH<J> t ~ X) - s i n ( a — 7 x ) e x P ( - f i ) CO "L* o ATT . 0 AT o co t - 3 + - 7 -+' {sin(g - f- - X) + ±£ sin(g - - x)}exp(-fi-2 _ -L) o iv 4TT to t - a - {sin(a - - -x) + y sin(a - y - x)}exp(-ft-5 ) o + {sin(3 - ^ - x) - -T- sin(8 - 4f - X)}exP(-ft-5 :L) ] co 0 Q 2TT , 2TT 8 " T • • ° + 3~ for — < t < — CO to o o When SCR. and SCR. are conducting. U. = U„ = U„ = U, = U r = 1 and U, = 0, 1 I 1 2 3 4 5 6 - — R - — . V t . f 5TT . 3, . , 0 9ir ^ * 3 1 (t) = - [<fr (a - -7-, t ) - 4,(3 - — t )] a L 0 to D co 4 TT g _ ^ 11 • y3V r 1 / Q 5TT 3 \ , , ' TT a N, + -2L ~ ~T' — } r •(« - 3, t - — ) ] o o 58 3 - ^ o o o o , 2TT V / , T ^ 3, TT 5ir W o ~ a ~3~ [sin(to t - 7- - x) " s i n ( a - -7 x) e*p(- f i ) O D 0 to t 3 + ^ 7 r + {sin(3 - ^ - X) + y sin(3 - ^ - x)}exp(-f2^ _ L) o ir /3~ 4ir o^*" ~ a - {sin(a - g- - x) + y s i n ( a j - xHe*p(-^ ) o + {sin(3 - ^ - X) - - y sin(3 - ^ - X ) > e x p ( - ^ i-) o 2TT /T o to t - a =-- {sin(a + f " X) " " y s i n ( a + y - X)}exp(-ft-2 i-) ] (1.12) o .2 a + -r-ir f o r 3- * t< ^ -to to o o When SCR^ extinguishes and SCR^ i s not conducting, there i s no current through the load of phase a. Hence, when only SCR^ i s conducting, 4TT i (t) = 0 f o r f $ t < — - ^ (1.13) o 0 . REFERENCES 1. W. Alexanderson, "The Thyratron Motor", AIEE Trans., Vol. 53, 1934, pp. 1517^1531. 2. M. Stohr, "Die Typenleistung Kollectorloser Stromrichtermotoren bei der Einfachen Sechsphasenschaltung", Archiv fur Electrotechinik,. XXXII. Band., 11. Heft., 1938, pp. 691-720. 3. P.C. Krause, T.A. Lipo, "Analysis and Simplified Representations of a Rectifier-Inverter Induction Motor Drive", IEEE Trans, on Power Apparatus and Systems, Vol. PAS-88, No. 5, May 1969, pp. 588-596. 4. T.A. Lipo, P.C. Krause, H.E. Jordan, "Harmonic Torque and Speed Pulsations in a Rectifier-Inverter Induction Motor Drive", IEEE Trans, on Power Apparatus and Systems, Vol. 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 Rectifier-Inverter Induction Motor Drive", IEEE Trans, on Power Apparatus and Systems, Vol. PAS-88, No. 11, November 1969, pp. 1635-1641. 6. S. Miya-iri, Y. Tsunehiro, "The Operation of the Damper Winding in 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 Rectifier-Inverter Reluctance-Synchronous Motor Drives", IEEE Trans, on Power Apparatus and Systems, Vol. PAS-88, No. 6, June 1969, pp. 962-970. 8. T.A. Lipo, P.C. Krause, "Stability Analysis for Variable Frequency Operation of Synchronous Machines", IEEE Trans, on Power Apparatus and Systems, Vol. PAS-87, No. 1, January 1968, pp. 227-234. 9. S. Miya-iri, Y. Tsunehiro, "Research on the Commutatorless Motor with SCR", IEE of Japan, Vol. 82, No. 890, November 1962, pp. 1741-1750. 60 10. S. M i y a - i r i , Y. Tsunehiro, "The Analysis 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, Vol. 85-9, No. 924, September 1965j pp. 1585-1594. ' 11. N. Sato, "A Study of the Commutatorless Motor", IEE of Japan, Vol. 84-8 4 No. 911, August 1964, pp. 1249-1257. 12. T. Tsuchiya, H. Sasajima, K. Tatsuguchi, "Basic C h a r a c t e r i s t i c s of Series Commutatorless Motor", IEE of Japan, V o l . 89-9, No. 972, September 1969, pp. 1773-1778. 13. T. Okada, "Three Phase A.C. Control Systems with Delta-Connected Controlled R e c t i f i e r s " , IEE of Japan, Vol. 86-10, No.'973, October 1966, pp. 1702-1711. 14. T.J. Take-uchi, "Theory of SCR C i r c u i t and A p p l i c a t i o n to Motor Control", Tokyo E l e c t r i c a l Engineering College Press, 1968. 

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