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Induction motor simulation for the computer aided design of induction motor drives Peabody, Frank Gerald 1981

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INDUCTION MOTOR SIMULATION FOR THE COMPUTER AIDED DESIGN OF INDUCTION MOTOR DRIVES by Frank G e r a l d Peabody B.A.Sc. U n i v e r s i t y o f B r i t i s h Columbia, 1978 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department of E l e c t r i c a l E n g i n e e r i n g ) We a c c e p t t h i s , t h e s i s as conforming t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA August 1981 © Frank G e r a l d Peabody, 1981 In presenting th is thes is in pa r t i a l fu l f i lment o f the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by h is representa t ives . It is understood that copying or pub l ica t ion of th is thes is fo r f i n a n c i a l gain sha l l not be allowed without my wri t ten permission. Department of h)e^jyieuz) te^f-t^e^r-,y> 4. The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date Ju)y_ 2.7} iqg/ ABSTRACT An i n v e r t e r i n d u c t i o n motor d r i v e s i m u l a t i o n program f o r use i n power e l e c t r o n i c d e s i g n has been w r i t t e n . The v a r i o u s components which must be r e p r e s e n t e d i n an i n v e r t e r i n d u c t i o n motor s i m u l a t i o n program a r e d e s c r i b e d . The i n d u c t i o n motor e q u a t i o n s a r e c o n s i d e r e d and a s i m p l i f y i n g t r a n s f o r m a t i o n i s used i n the s i m u l a t i o n program. The second o r d e r e f f e c t s o f s a t u r a t i o n , l e a k a g e i n d u c t a n c e v a r i a t i o n and r e s i s t a n c e v a r i a t i o n under d i f f e r e n t o p e r a t i n g c o n d i t i o n s have been a n a l y z e d . The e f f e c t o f leakage i n d u c t a n c e and r e s i s t a n c e v a r i a t i o n have been implemented i n the s i m u l a t i o n program. The v a r i o u s v o l t a g e s o u r c e s t h a t a r e used w i t h i n d u c t i o n motor d r i v e s a r e c o n s i d e r e d and implemented i n t o t h e computer program. The program i s demonstrated t o be e x t r e m e l y u s e f u l as an a i d i n the d e s i g n o f i n v e r t e r i n d u c t i o n motor d r i v e systems. i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF ILLUSTRATIONS v LIST OF TABLES v i i i ACKNOWLEDGEMENT i x 1. INTRODUCTION 1 1.1 A p p l i c a t i o n s of T h e s i s R e s u l t s 1 1.2 H i s t o r y 2 1.3 O u t l i n e of T h e s i s 3 1.4 Summary 4 2. MOTOR REPRESENTATION 5 2.1 B a s i c E q u a t i o n s 5 2.2 Measurement o f Parameters 7 2.3 T r a n s f o r m a t i o n s of Motor E q u a t i o n s 11 2.4 D i s c u s s i o n o f T r a n s f o r m a t i o n s 13 2.5 T r a n s f o r m a t i o n Used 14 3. SECONDARY EFFECTS 18 3.1 I n t r o d u c t i o n t o Secondary E f f e c t s 18 3.2 S k i n E f f e c t on R e s i s t a n c e and Leakage Reactance 18 3.3 C a l c u l a t i o n s f o r T y p i c a l Motors 24 3.4 Leakage Inductance V a r i a t i o n Due t o C u r r e n t Displacement 27 3.5 S a t u r a t i o n of Leakage Reactance 30 3.6 S a t u r a t i o n of Main Reactance 31 3.7 Other E f f e c t s 33 4. VOLTAGE SOURCE REPRESENTATION 35 4.1 I n t r o d u c t i o n 35 4.2 F u l l V o l t a g e L i n e S t a r t i n g . 35 4.3 C y c l o c o n v e r t e r s 36 4.4 I n v e r t e r s 39 i i i 5. NUMERICAL METHODS 44 5.1 I n t r o d u c t i o n 44 5.2 Nu m e r i c a l Methods 44 5.3 The T r a p e z o i d a l Rule 45 5.4 T r a p e z o i d a l Method w i t h V a r i a b l e Time Stop 46 6. RESULTS AND DISCUSSION 50 6.1 I n t r o d u c t i o n 50 6.2 F u l l V o l t a g e L i n e S t a r t i n g 51 6.3 Other V o l t a g e Sources 51 6.4 Time Step M o d i f i c a t i o n 51a 6.5 V a r i a t i o n i n R e s i s t a n c e and Inductance 52 6.6 P r a c t i c a l A p p l i c a t i o n of Program 52 7. CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER RESEARCH 76 7.1 Summary 76 7.2 Recommendations f o r F u r t h e r Research 77 REFERENCES 79 APPENDIX 1 82 i v LIST OF ILLUSTRATIONS F i g u r e Page . 1 Diagram of the i n d u c t i o n motor 5 2 C l a s s i c a l p e r phase e q u i v a l e n t c i r c u i t 8 3 R e l a t i o n s h i p o f r o t o r c u r r e n t s 15 4 S t a t o r f l u x diagram 19 5 Diagram of r o t o r bar 20 6 T y p i c a l s h a l l o w bar r o t o r 25 7 T y p i c a l deep bar r o t o r 26 8 E f f e c t o f v a r i a b l e f r e q u e n c y on leakage f l u x 28 9 R o t o r - s t a t o r f l u x p aths 31 10 S a t u r a t i o n c u r v e 31 11 Diagram t o i l l u s t r a t e s a t u r a t i o n 32 12 C y c l o c o n v e r t e r c i r c u i t 36 13 C y c l o c o n v e r t e r v o l t a g e waveform , 37 14 Diagram t o i l l u s t r a t e c o s i n e c r o s s i n g c o n t r o l s t r a t e g y 38 15 B a s i c i n v e r t e r c i r c u i t 39 16 B a s i c i n v e r t e r v o l t a g e waveform 40 17 Development of b a s i c i n v e r t e r waveform 41 18 T r i a n g l e i n t e r c e p t method 43 19 I l l u s t r a t i o n of p r e d i c t i o n s u s i n g d i f f e r e n t s t e p times 47 20 Step waveform 48 21 Phase v o l t a g e a p p l i e d d u r i n g s i m u l a t i o n 54 22 S i m u l a t i o n r e s u l t o f f u l l v o l t a g e l i n e s t a r t i n g s 55 23 Measured v a l u e s o f c u r r e n t f o r f u l l v o l t a g e 55 l i n e s t a r t i n g , f i r s t maximum. v Page 24 Measured v a l u e s o f c u r r e n t f o r f u l l v o l t a g e 56 l i n e s t a r t i n g , second maximum 25 Measured v a l u e s o f c u r r e n t f o r f u l l v o l t a g e 56 l i n e s t a r t i n g , t h i r d maximum 26 C y c l o c o n v e r t e r s i m u l a t i o n r e s u l t s 57 27 F i x e d p u l s e w idth v a r i a b l e f r e q u e n c y s i m u l a t i o n 58 p u l s e width=.05333 seconds 28 F i x e d p u l s e w idth v a r i a b l e f r e q u e n c y s i m u l a t i o n 59 p u l s e width=.0012 seconds 29 Torque and speed p u l s e width=.00533 seconds 60 30 Torque and speed p u l s e width=.0012 seconds 61 31 T r i a n g u l a r m o d u l a t i o n 60 Hz v o l t a g e waveform 62 32 T r i a n g u l a r m o d u l a t i o n 60 Hz c u r r e n t waveform 63 33 F i x e d p u l s e w idth f i x e d f r e q u e n c y v o l t a g e waveform 64 34 S i m u l a t i o n o f c u r r e n t waveform u s i n g v a r i a b l e t i m e s t e p 65 35 V o l t a g e source used f o r s i m u l a t i o n o f v a r i a b l e 66 i n d u c t a n c e s and r e s i s t a n c e s 36 C u r r e n t s i m u l a t i o n f o r no l o a d f u l l v o l t a g e 67 w i t h parameter v a r i a t i o n s 37 C u r r e n t s i m u l a t i o n f o r f u l l l o a d v o l t a g e and i n i t i a l 68 speed o f 1800 rpm. 38 V o l t a g e waveform f o r 4 Hz harmonic e l i m i n a t i o n method 69 39 C u r r e n t s i m u l a t i o n o f 4 Hz harmonic e l i m i n a t i o n method 70 40 Measured c u r r e n t of 4 Hz harmonic e l i m i n a t i o n method 70 41 Torque p u l s a t i o n s f o r 4 Hz harmonic e l i m i n a t i o n 71 mo d u l a t i o n method 42 Torque p u l s a t i o n s f o r 4 Hz t r i a n g u l a r m o dulation method 72 v i Page 43 V o l t a g e waveform o f 30 Hz harmonic e l i m i n a t i o n method 73 44 C u r r e n t s i m u l a t i o n of 30 Hz harmonic e l i m i n a t i o n waveform 74 45 Measured c u r r e n t o f 30 Hz harmonic e l i m i n a t i o n waveform 74 46 V o l t a g e waveform of 60 Hz harmonic e l i m i n a t i o n method 75 47 C u r r e n t s i m u l a t i o n o f 60 Hz harmonic e l i m i n a t i o n method 76 48 Measured c u r r e n t of 60 Hz harmonic e l i m i n a t i o n method 76 v i i LIST OF TABLES T a b l e Page I I n d u c t i o n Motor Parameters 50 I I Comparison Between S i m u l a t i o n and A c t u a l Peak 51 C u r r e n t s Measured For F u l l V o l t a g e L i n e S t a r t i n g v i i i ACKNOWLEDGEMENTS I would l i k e t o ex p r e s s my thanks t o Dr. Dommel f o r the c o n f i d e n c e he showed i n my a b i l i t y and t h e i n s p i r a t i o n he p r o v i d e d . I would a l s o l i k e t o thank P r o f e s s o r Berg a t t h e U n i v e r s i t y o f C a l g a r y , P r o f e s s o r Lauw a t t h e Oregon S t a t e U n i v e r s i t y and P r o f e s s o r Nasar a t t h e U n i v e r s i t y o f Kentucky who have h e l p e d w i t h my u n d e r s t a n d i n g o f t h e t o p i c and a l s o p r o v i d e d encouragement. I would a l s o l i k e t o thank Dr. S c o t t Meyer of t h e B o n n e v i l l e Power A d m i n i s t r a t i o n who h e l p e d w i t h the r u n n i n g o f computer s i m u l a t i o n s , the American Department o f Energy f o r p r o v i d i n g computer f a c i l i t i e s and Westinghouse Apparatus S e r v i c e f o r d o i n g motor t e s t s . I would a l s o l i k e t o thank the NSERC and t h e B.C. S c i e n c e C o u n c i l f o r p r o v i d i n g the f i n a n c i a l s u p p o r t which a l l o w e d me t o r e s e a r c h t h i s a r e a . i x 1 1. INTRODUCTION 1.1 A p p l i c a t i o n o f T h e s i s R e s u l t s The i n c r e a s i n g use of power e l e c t r o n i c d e v i c e s i n a l l areas of e l e c t r i c a l e n g i n e e r i n g has n e c e s s i t a t e d t h e development o f power c i r c u i t d e s i g n a i d s . One of the most s u c c e s s f u l r e c e n t developments i n power system a n a l y s i s , which would prove u s e f u l f o r power e l e c t r o n i c s , i s the use of d i g i t a l computers t o s i m u l a t e the e l e c t r i c a l c h a r a c t e r i s t i c s of c i r c u i t s a t d i s c r e t e i n t e r v a l s of t i m e . The computer s i m u l a t i o n a l l o w s the d e s i g n e r t o s i z e components p r o p e r l y and t o p r e d i c t the performance of the c i r c u i t b e f o r e i t i s b u i l t . Once the c i r c u i t i s b u i l t the c h a r a c t e r i s t i c s can be compared t o t h e computer s i m u l a t i o n . D i f f e r e n c e s between p r e d i c t e d and a c t u a l r e s u l t s w i l l show e r r o r s i n c i r c u i t i m p l e m e n t a t i o n such as s t r a y c a p a c i t a n c e o r i n d u c t a n c e . Such a check on c i r c u i t i m plementation i s a f u r t h e r a i d t o t h e c i r c u i t d e s i g n e r . Thus h a v i n g a power e l e c t r o n i c c i r c u i t model would p r o v i d e a v a l u a b l e t o o l . One power e l e c t r o n i c a r e a which has become p a r t i c u l a r l y important i s i n d u c t i o n motor d r i v e s . The a v a i l a b i l i t y o f h i g h power 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 , and more r e c e n t l y , t r a n s i s t o r s has made p o s s i b l e v a r i a b l e speed motor d r i v e s u s i n g e c o n o m i c a l i n d u c t i o n motors i n f a r g r e a t e r numbers of a p p l i c a t i o n s . The i n d u c t i o n motor i s one of the most i n t e r e s t i n g and most c r u c i a l elements t o model i n o r d e r t o p r e d i c t t h e performance o f the c i r c u i t . I t i s thus r e a s o n a b l e t o model the i n d u c t i o n motor f i r s t and t o s i m p l i f y the r e s t of the c i r c u i t as a simple v o l t a g e s o u r c e i f the computer model i s t o be of immediate use. At a l a t e r time the o t h e r c i r c u i t elements can be m o d e l l e d and i n c l u d e d i n the program t o form a complete power e l e c t r o n i c a n a l y s i s program. 2 1.2 H i s t o r y One o f the f i r s t attempts a t m o d e l l i n g t r a n s i e n t c o n d i t i o n s i n the i n d u c t i o n motor was by Lyon [1] who i n t r o d u c e d the concept of symmetrical components f o r t r a n s i e n t a n a l y s i s . U s i n g some o f the a n a l y s i s o f Lyon, L e v i n e [2] c a l c u l a t e d the s h o r t c i r c u i t c u r r e n t s i n an i n d u c t i o n motor u s i n g the a c t u a l v o l t a g e s and c u r r e n t s i n t h e s t a t o r and r o t o r which, however, made the s o l u t i o n o f the e q u a t i o n s v e r y d i f f i c u l t . The e q u a t i o n s used by L e v i n e can be c o n s i d e r e d as t h e b a s i c s e t o f e q u a t i o n s as t h e y c o n t a i n elements t o r e p r e s e n t a l l the windings and t h e i r r e l a t i o n s h i p i n t h e i n d u c t i o n motor d i r e c t l y . N e i t h e r L e v i n e nor Lyon c o u l d e a s i l y model r o t o r speed v a r i a t i o n s . S t a n l e y [3] used the i d e a s of Park [4] t o s i m p l i f y the e q u a t i o n s t o a more workable form. Even S t a n l e y ' s e q u a t i o n s r e q u i r e d the use o f analogue computers t o o b t a i n good r e p r e s e n t a t i o n of i n d u c t i o n motor b e h a v i o u r . Krause [5] was a b l e t o s i m u l a t e most o p e r a t i n g c h a r a c t e r i s t i c s o f i n d u c t i o n motors i n c l u d i n g f u l l v o l t a g e l i n e s t a r t i n g and u nbalanced phase v o l t a g e c o n d i t i o n s . R e c e n t l y d i g i t a l computers have become more p o p u l a r i n t r a d i t i o n a l analogue a r e a s , such as power system a n a l y s i s [ 6 ] . D i g i t a l computers are more f l e x i b l e and may s o l v e d i f f e r e n t problems s i m u l t a n e o u s l y . In a d d i t i o n , analogue computers are e x p e n s i v e and cannot handle o t h e r b u s i n e s s r e q u i r e m e n t s such as a c c o u n t i n g . De S a r k a r [7] m o d e l l e d the b a s i c e q u a t i o n s of L e v i n e u s i n g a d i g i t a l computer program. However, the b a s i c e q u a t i o n s can be s i m p l i f i e d t o reduce computing time w i t h no l o s s o f important i n f o r m a t i o n . Robertson and Hebbar [8] u s e d a v a r i a t i o n o f S t a n l e y ' s t r a n s f o r m a t i o n t o reduce the computation t i m e . Secondary e f f e c t s such as v a r i a t i o n i n r e s i s t a n c e and s a t u r a t i o n a r e known t o be important but o n l y r e c e n t l y have t h e s e e f f e c t s been s i m u l a t e d i n d i g i t a l programs. 3 Gautam Nath [9] used f a c t o r s based on s l i p f r e q u e n c y t o a l t e r t h e r o t o r r e s i s t a n c e and leakage i n d u c t a n c e . A l t h o u g h r o t o r r e s i s t a n c e i s d i r e c t l y r e l a t e d t o r o t o r c u r r e n t f r e q u e n c y the leakage i n d u c t a n c e depends on t o t a l f l u x l i n k a g e s r a t h e r than r o t o r s l i p . None of the p r e v i o u s l y mentioned d i g i t a l models have attempted a v a r i a b l e f r e q u e n c y model which t a k e s i n t o account t h e important secondary e f f e c t s which are r e q u i r e d t o o b t a i n a c c u r a t e r e s u l t s . T h i s i s the o b j e c t i v e o f t h i s t h e s i s . 1.3 O u t l i n e o f T h e s i s The b a s i c e q u a t i o n s which r e p r e s e n t t h e i n d u c t i o n motor are f i r s t p r e s e n t e d and e x p l a i n e d . The p r o c e d u r e t o o b t a i n the motor parameters i s des-c r i b e d a l o n g w i t h a d i s c u s s i o n o f the assumptions on which i t i s based. The v a r i o u s t r a n s f o r m a t i o n s t o s i m p l i f y t h e d i f f e r e n t i a l e q u a t i o n s a r e t h e n p r e s e n t e d w i t h a d i s c u s s i o n as t o the advantages of each. Important secondary e f f e c t s which cause v a r i a t i o n s i n t h e motor p a r a -meters, and thus performance, a r e e x p l a i n e d . Measurements and c a l c u l a t i o n s o f th e parameters o f t y p i c a l motors a r e p r o v i d e d t o show t o what degree t h e secondary e f f e c t s w i l l i n f l u e n c e motor performance. A survey of the v a r i o u s v o l t a g e s o u r c e i n v e r t e r s i s g i v e n w i t h t h e v a r i o u s v o l t a g e waveforms t h a t t h e y produce. N u m e r i c a l s o l u t i o n t e c h n i q u e s a r e then s u r v e y e d as t o t h e i r advantages. P a r t i c u l a r emphasis i s p l a c e d on t h e t r a p e z o i d a l r u l e as t h i s has p r o v e d t o be one o f the most s u c c e s s f u l s o l u t i o n t e c h n i q u e s . The d e s i r a b i l i t y o f v a r y i n g the s t e p s i z e d u r i n g computation i s d i s c u s s e d and recommendations made. R e s u l t s o f v a r i o u s computer s i m u l a t i o n s a r e p r o v i d e d and comparisons w i t h a c t u a l t e s t c a ses a r e made. Comparisons a r e p r o v i d e d f o r cases w i t h and w i t h o u t secondary e f f e c t s i n o r d e r t o show t h e i r importance. P r e d i c t e d 4 r e s u l t s were e x t r e m e l y c l o s e t o these o b t a i n e d f o r the a c t u a l t e s t case s i m u l a t e d which i n d i c a t e s the a c c u r a c y o f t h e model. F i n a l l y , t h e m o d e l l i n g o f o t h e r c i r c u i t elements i s e x p l o r e d and the f e a s i b i l i t y o f i n c l u d i n g them i n t h e i n d u c t i o n motor model a t a f u t u r e date i s c o n s i d e r e d . 1.4 Summary The m a t e r i a l i n t h i s t h e s i s p r o v i d e s a v a l u a b l e a n a l y s i s and d e s i g n t o o l f o r use i n power e l e c t r o n i c s . I t r e l a t e s t h e s t a n d a r d measurement p r o -cedures t o the b a s i c e q u a t i o n s o f the i n d u c t i o n motor, e x p l o r e s v a r i o u s s i m p l i -f y i n g t r a n s f o r m a t i o n s , models a l l important a s p e c t s o f i n d u c t i o n motor b e h a v i o u r and p r o v i d e s f o r the r e p r e s e n t a t i o n o f a l l v o l t a g e s o u r c e s . There i s a l s o p r o v i s i o n i n t h e programs t o t e s t maximum speed changes a c h i e v a b l e under c o n d i t i o n s o f v a r i o u s v o l t a g e s o u r c e s and d i f f e r e n t l o a d s . The computer model d e v e l o p e d i s v e r y u s e f u l as i s o r can be i n c o r p o r a t e d i n t o a more g e n e r a l power e l e c t r o n i c s program. 5 2. MOTOR REPRESENTATION 2.1 B a s i c E q u a t i o n s The three-phase i n d u c t i o n motor i s s i m p l y a s e t of c o i l s on a s t a t o r and a s e t o f c o i l s on a f r e e l y r o t a t i n g s h a f t . The machine i s d e s i g n e d so t h a t a l l c o i l s have a h i g h degree of mutual c o u p l i n g . M e c h a n i c a l power i s deve l o p e d by the i n t e r a c t i o n o f t h e s t a t o r and r o t o r magnetic f i e l d s . The r o t o r magnetic f i e l d i s produced by c u r r e n t i n d u c e d i n the r o t o r by the s t a t o r ' s magnetic f i e l d . The c o i l s of the i n d u c t i o n motor can be r e p r e s e n t e d d i a g r a m m a t i c a l l y as shown i n F i g . 1. The e l e c t r o m a g n e t i c e q u a t i o n s a r e g i v e n i n E q u a t i o n 1. The symbols used t h e r e and throughout the r e s t o f t h i s paper are as f o l l o w s : FIGURE 1 v s" a RS+pLSS pLSM pLSM pMSRcos0 pMSRcos ( 0+y^) 2ir pMSRcos( 0-J-) . s 1 a V pLSM RS+pLSS pLSM pMSRcos ( ©""I11) pMSRcos 0 pMSRcos(0+|^) . s *b v 8 C = pLSM pLSM RS+pLSS 2n pMSRcos ( 0+y11) 2ir PMSRCOS ( 0-y-) pMSRcos© . s 1 c v r a pMSRcosG 2n pMSRcos( O-y-) pMSRcos ( e+y) RR+pLRR pLRM pLRM . r 1 a pMSRcos(0+|^) pMSRcosG pMSRcos (0-|^) pLRM RR+pLRR pLRM . r v r C pMSRcos(0 - | ^ ) pMSRcos ( G+y) pMSRcos 0 pLRM pLRM RR+pLRR . r 1 c _ — LSS - per phase s t a t o r s e l f i n d u c t a n c e LSM - per phase mutual i n d u c t a n c e between s t a t o r windings LRR - e q u i v a l e n t p e r phase r o t o r s e l f i n d u c t a n c e LRM - e q u i v a l e n t per phase r o t o r mutual i n d u c t a n c e MSR - maximum v a l u e of ind u c t a n c e between r o t o r and s t a t o r winding RS - per phase s t a t o r r e s i s t a n c e RR - e q u i v a l e n t p e r phase r o t o r r e s i s t a n c e 7 2.2 Measurement o f Parameters The c l a s s i c a l e q u i v a l e n t p e r phase model of the i n d u c t i o n motor ( f i g . 2) i s important because t h e v a l u e s o f t h e e q u i v a l e n t c i r c u i t a r e u s u a l l y the o n l y parameters t h a t a r e a v a i l a b l e or can be measured f o r the i n d u c t i o n motor. The s t a n d a r d t e s t s t o o b t a i n the parameters, DC r e s i s t a n c e t e s t , b l o c k e d r o t o r t e s t and no l o a d t e s t , g i v e the f o l l o w i n g v a l u e s f o r the e q u i v a l e n t c i r c u i t : D.C. R e s i s t a n c e t e s t phase p e r phase ^ h a s e (2) B l o c k e d Rotor T e s t Z. b r phase Pwr phase - R. > phase R, '2 (3) assume X, =X, (4) 2 No Load T e s t V / 2 2 phase = /R, + (X +X,) X T = —*• 1 m l nL I , phase - A 2 The above v a l u e s a r e then s u b s t i t u t e d i n t o the c l a s s i c a l p e r phase e q u i v a l e n t c i r c u i t o f F i g . 2. R, L, W W V \ A - / 1 5 W 0 W V R, 1 FIGURE 2 The v a l u e s a t L^, and 1^ a r e u s u a l l y g i v e n as impedances a t 60 Hz even by authors d o i n g t r a n s i e n t s t u d i e s . However, i t makes more sense t o use i n d u c t a n c e s as the 60 Hz impedance may not even be used d u r i n g a t r a n s i e n t s t u d y . The v a l u e s f o r R^ and R 2 a r e j u s t RS and RR i n the p r e v i o u s m a t r i x e q u a t i o n s . L j and 1^ when added t o g e t h e r i s t h e apparent s t a t o r s e l f i n d u c t a n c e L g . The v a l u e o f 1^ i s 3/2 MSR and s i n c e h-^h^ LRR i s e q u a l t o LSS. These v a l u e s can be deduced by l o o k i n g back a t t h e o r i g i n a l m a t r i x e q u a t i o n s (eqn. 1) and s u b s t i t u t i n g t h e e q u a t i o n s f o r c u r r e n t i n a b a l a n c e d three-phase machine where r e q u i r e d . . s . s . s i + i , + l = 0 a b c (6) r r r i + i , + i = 0 a b c For example, under no l o a d (s=0) t h e r e a r e assumed t o be no r o t o r c u r r e n t s . Then: V = p ( L S S i + LSMi + LSMi ) - RSi a b c a = p ( L S S i - LSMi ) - R S i (7) a a a = (p(LSS - LSM) - R S ) i a and t h e r e f o r e LSS - LSM = L^ + L^ and RS = R^. There i s some c o n f u s i o n i n the l i t e r a t u r e as t o s i g n c o n v e n t i o n . S i n c e LSM i s a n e g a t i v e v a l u e t h e v a l u e o f LSS - LSM i s l a r g e r than LSS or LSM but some au t h o r s w r i t e LSM a l r e a d y w i t h a n e g a t i v e s i g n and thus have L. + L = LSS + LSM (8) l m The v a l u e o f 3/2 MSR = Lj, can be seen from l o o k i n g a t the f i r s t l i n e o f e q u a t i o n 1 [ 2 ] . V a S = (RS + pLSS)i_^+ p L S M i b S + p L S M i c S + pMSRcosOi.^ + pMSRcos(0 + ^i-^ + pMSRcos(0 - | ^ ) i S 3 C 10 S u b s t i t u t e : c o s ( x ± y) = c o s x c o s y + s i n x s i n y s i n ( x ± y) = s i n x c o s y ± c o s x s i n y = p(LSS - LSM) i S + pMSR[cos0i T + c o s ( 0 + - | ^ ) ( - i r - i r ) + cL SL J 3, C c o s f e + i l x - i a r - i b r ) ] = p(LSS - LSM)i S + pMSR[(cos0 - c o s ( 0 + | ^ ) ) i * - c o s ( 0 + 4 ^ ) ^ 8i o Si o JD 2ir . r - i - c o s ( 0 + — 1 I 7 r* J 3 ' c s r 1 1 p(LSS - LSM) i + pMSR[(cos0 + - c o s 0 + - c o s 0 - - s i n 0 /3 r 1 /3 r 1 /3 r n + — s i n 0 ) i + (- c o s 0 - - s i n O ) i , + (- c o s 0 + - s i n 0 ) i 2 a 2 2 D 2 2 C J = p(LSS - LSM)i S+ pMSR[(2cos0)i + - i c o s 0 ( i , _ r + i ) + - (- l f a + xc ) sxn0] 3 r /3 r p(LSS - LSM)i S + pMSR[(- c o s 0 ) i r + - s i n 0 (- i r + i r ) ] Si a. Si Ct D C = p(LSS - LSM)i S + p | MSR[cos0i r - s i n 0 i r < -90°1 a 2 L a r a J = p(LSS - LSM)u S + p \ MSR[cos0I r s i n ( s t ) - s i n 0 i rsin(wt-90°)1 a 2 L m r J .= p(LSS - LSM)i S + p -5 M S R [ c o s 0 i r s i n ( w t ) + s i n 0 i r c o s ( w t ) ] = p(LSS - LSM)i S + p ~ MSRI r s i n ( 0 + wt) (9) a 2 iti p and s i n c e s i n t O j . + wt) i s j u s t s i n ( w t ) s h i f t e d i n time the r o t o r p o s i t i o n a n g l e has no e f f e c t . 11 I t i s i m p o r t a n t t o u n d e r s t a n d t h a t a l t h o u g h the t e s t s are conducted f o r b a l a n c e d c o n d i t i o n s , t h e i n f o r m a t i o n o b t a i n e d i s s t i l l v a l i d f o r unbalanced v o l t a g e s o u r c e s as l o n g as the motor i s e l e c t r i c a l l y b a l a n c e d . Other t e s t s may be conducted t o o b t a i n v a l u e s f o r s a t u r a t i o n o f t h e m a g n e t i z i n g branch by v a r y i n g the v o l t a g e a t no l o a d . S a t u r a t i o n of the leakage i n d u c t a n c e may be t e s t e d by v a r y i n g t h e v o l t a g e a t b l o c k e d r o t o r a l t h o u g h t h i s t e s t may damage the motor i f not done c a r e f u l l y . I f a v a r i a b l e f r e q u e n c y s i n u s o i d a l s u p p l y i s a v a i l a b l e t h e n the s k i n e f f e c t o f the s t a t o r and r o t o r may be measured. 2.3 T r a n s f o r m a t i o n s o f Motor E q u a t i o n s I t i s p o s s i b l e t o s i m p l i f y the d i f f e r e n t i a l e q u a t i o n s through the use o f t e n s o r a n a l y s i s and t h e c o n n e c t i o n m a t r i x as devel o p e d by Kron [11]. The important f e a t u r e s o f the t r a n s f o r m a t i o n s used i n motor t h e o r y a r e now e x p l a i n e d w i t h a g e n e r a l i z e d example. C o n s i d e r a s e t of e q u a t i o n s r e p r e s e n t i n g the v o l t a g e and c u r r e n t i n a motors w i n d i n g s . (10) The Z's may be f u n c t i o n s o f time o r can be d e r i v a t i v e and i n t e g r a l o p e r a t o r s . A change i n v a r i a b l e s can be made as f o l l o w s : v l C l l C12 V ± 1 °11 °12 V V 2 C21 C 2 2 V L2 C21 C 2 2 (11) The t r a n s f o r m a t i o n m a t r i x [C] i s assumed t o be n o n s i n g u l a r , r e a l and independent of a l l v o l t a g e s and c u r r e n t s . U s i n g t h e d e f i n i t i o n s o f e q u a t i o n (10) can be w r i t t e n a s : 12 c c 11 12 V z z 11 12 V C C _21 22_ M z z _21 22 V c c 11 12 -1 z z 11 12 C C 11 12 V V C C 21 22 z z 21 22 C C 21 22 V (12) I f a new m a t r i x i s d e f i n e d as: z • 11 z ' 12 " C11 C12 _ -1 " Z11 Z12 _ " C11 C12 _ z * _21 z • 22 _ _°21 C22_ _Z21 Z22_ C22_ (13) Then the e q u a t i o n o f 12 can be w r i t t e n as: V z • z * 11 12 Z ' Z ' 21 22 i 4 ' 1 i * _ 2 _ (14) E q u a t i o n 14 then r e p r e s e n t s the r e l a t i o n between v o l t a g e and c u r r e n t i n t h e new f i c t i o n a l machine. The power g o i n g i n t o t h e motor i s : P = [ i , i 2 ] (15) The power g o i n g i n t o the motor and i n t o t h e f i c t i o n a l machine can be made the same and i t i s m a t h e m a t i c a l l y c o n v e n i e n t i f t h e power can be e x p r e s s e d i n the form of e q u a t i o n (15) f o r both c a s e s . A t r a n s f o r m a t i o n h a v i n g t h i s p r o p e r t y i s power i n v a r i a n t . S u b s t i t u t i n g e q u a t i o n (12) i n t o e q u a t i o n (15) g i v e s : 13 P = C 1 1 ° 1 2 C C 21 22 X 2 t | -C C 11 12 G C 21 22 _ C 1 1 C 1 2 _ t " C 11 C 1 2 _ " V l " C C _ 2 1 22 C C _ 2 1 22 _V2_ ( 1 6 ) and thus [ C ] t [ C ] = u n i t m a t r i x f o r power i n v a r i a n c e . I t i s now j u s t a matter o f c h o o s i n g a frame o f r e f e r e n c e o r t r a n s f o r m a t i o n which w i l l s i m p l i f y computation w i t h o u t making i t d i f f i c u l t t o o b t a i n the a c t u a l c u r r e n t s and v o l t a g e s d e s i r e d . 2 . 4 D i s c u s s i o n o f T r a n s f o r m a t i o n s S i n c e t h e b a s i c d i f f e r e n t i a l e q u a t i o n s which d e s c r i b e t h e i n d u c t i o n motor are d i f f i c u l t t o s o l v e , many t r a n s f o r m a t i o n s have been d e v e l o p e d t o s i m p l i f y them. Most o f t h e s e t r a n s f o r m a t i o n s o r i g i n a t e from Park's [ 4 , 1 2 ] i d e a of h a v i n g a p a i r o f axes a f f i x e d t o the r o t o r and u s i n g t r a n s f o r m a t i o n s t o p u t a l l o t h e r terms i n t h i s r e f e r e n c e frame. There a r e t h e e x c e p t i o n s of c o u r s e , such as De Sa r k a r [ 7 ], who suggests t h a t w i t h modern computers i t i s j u s t n ot worth the e f f o r t t o do t h e t r a n s f o r m a t i o n s . The most a v i d d e v e l o p e r o f d i f f e r e n t r e f e r e n c e frames i s Krause [ 5 , 1 3 ] who c o r r e c t l y saw t h a t Park's o r i g i n a l t r a n s f o r m a t i o n was j u s t one of a f a m i l y w i t h t h e p o s i t i o n o f the r e f e r e n c e frame b e i n g the d i s t i n g u i s h i n g element. The one o t h e r t r a n s f o r m a t i o n not r e l a t e d t o Park's work i s t h a t o f F o r t e s c u e ' s s y m m e t r i c a l c o o r d i n a t e s [ 1 4 ] . A l t h o u g h the p r o c e d u r e o f s y m m e t r i c a l components s i m p l i f i e s s t e a d y - s t a t e problems v e r y w e l l , i t becomes i n c r e a s i n g l y more c o m p l i c a t e d when t r a n s i e n t a n a l y s i s and then v a r y i n g r o t o r f r e q u e n c y a r e i n c l u d e d . I f no s i m p l i f i c a t i o n o f the problem, such as i g n o r i n g s t a t o r r e s i s t a n c e [15], a r e made then the method o f sym m e t r i c a l components p r o v i d e s almost no advantage when s o l v i n g t h e problem w i t h d i g i t a l computers. 2.5 T r a n s f o r m a t i o n Used The a, 3, y, o t r a n s f o r m a t i o n s u g g e s t e d by [8] l e a v e s the s t a t o r v a r i a b l e s e s s e n t i a l l y unchanged but m o d i f i e s the r o t o r v a r i a b l e s so t h a t they a r e t r a n s f o r m e d t o a n o n - r o t a t i n g r o t o r i n o r d e r t o remove t h e r o t o r p o s i t i o n dependence o f t h e i n d u c t a n c e m a t r i x . The s t a t o r c u r r e n t s a r e d e f i n e d i n t h e new r e f e r e n c e frame by e q u a t i o n ( 1 7 ) . . s . s . s 1 =1 - 1 a a o . s . s . s x = x - i (17) 3 b o . s . s . s I = i - l Y c o s s s s S i n c e i + i . + i = 3 i and t h e z e r o sequence c u r r e n t i s t h e same i n e i t h e r a b e o r e f e r e n c e frame then e q u a t i o n 18 i s a l s o t r u e ( i S + i S ) + ( i Q S + i S ) + ( i S + i S ) = 3 i S (18) o p o Y o o and t h e r e f o r e e q u a t i o n 19 i s t r u e . . s . s . s 1 + i + i = o Y i / = - i S - i S (19) 3 a Y R e f e r r i n g t o F i g u r e 3 t o show the r e l a t i o n between t h e r o t o r c u r r e n t s r r r r r r i , i . , i and t h e t r a n s f o r m e d c u r r e n t s i , i„ , i one can deduce t h e a b c a 3 Y r e l a t i o n s h i p s between t h e v a r i a b l e s . 15 FIGURE 3 These a r e found t o be; COS0 4TT c o s ( 0 + — ) c o s ( 0 + | 2 ) 2TL cos( 0 + — ) COS0 2T\ cos( 0 + — ) 4 IT cos( 0 + — ) 2ir. cost 0 + — ) COS0 . r X 3 2 . r l a 3 . r X b 2 3 2 . r l c (20) As the r o t o r i s s h o r t c i r c u i t e d t h e r e w i l l be no zero sequence c u r r e n t f l o w i n g and s i m i l a r l y t h e r e w i l l be no ze r o sequence c u r r e n t f l o w i n g i n the s h o r t c i r c u i t e d f i c t i t i o u s s t a t i o n a r y r o t o r w i n d i n g . T h e r e f o r e e q u a t i o n (21) i s t r u e . . r . r . r I = - I - I 3 a y (21) E q u a t i o n (21) can then be s u b s t i t u t e d i n t o e q u a t i o n (20) t o e l i m i n a t e the second row. F o r the f i r s t row: 16 r 2ir % . r , _ 4ir. . r 3 . r cos i + c o s ( 0 + — ) i g + c o s ( 0 + — ) x = 2 Xa 2 IT 4TT 2IT r 3 r [cos - cos( +•=-)] i + [ c o s ( 0 + 3-) - c o s ( 0 + zr-)] i = ^ i o (X J y Z a 2 1 3 r 1 /3" 1 r t r - [cos + - cos + - s i n ] i a + [- - cos0 + - s i n 0 + - s i n 0 ] i J = ± & 2 . IT. . r 2 . r — cost + —) i + — s i n Q = l 3 6 a /3 and s i m i l a r l y f o r the t h i r d row t o g i v e : (22) 2 r 2 n r r ( — s i n ) i + — c o s ( 0 - —) i = i 3 /3 ° (23) r r r S i n c e i , = - i - i t h e r e has been no l o s s o f i n f o r m a t i o n and t h e m a t r i x b a c e q u a t i o n s have been reduced. The c o n n e c t i o n m a t r i x f o r t h e c u r r e n t t r a n s f o r m a t i o n has now been e s t a b l i s h e d . The reduced form o f the c o n n e c t i o n m a t r i x which i s used t o c a l c u l a t e [R]' and [L ] * i s e q u a t i o n ( 2 4 ) . 0 1 . . . s i 0 I I OC • 0 I | i S | (24) . s 1 a 1 1 0 . s x b 1 -1 -1 . s X c = 1 0 1 . r X a 0 0 0 . r X c 0 0 0 — — — 2 Tf — COS(0 + ^ -) /3 6 2 — s i n 0 /3 — s i n 0 /3 — c o s ( 0 -/3 6 When the c o n n e c t i o n m a t r i x i s a p p l i e d t o t h e b a s i c e q u a t i o n o f (1) the n t h e e q u a t i o n s of (25) a r e o b t a i n e d . [ V ] = [R'l [ i ' ] + [ L ' ] p [ i ' ] where: 17 [V]' -s s s V + V ° + V a b c s s V - V, a b s s V - v. c b [i ] 1 = . s 1 o . r 1 a (25) |3RS 0 0 [ R] ' = 0 2RS RS 3 /3 -MSRW 2 r 0 RS 2RS -3/3 -MSRW 2 r 0 0 0 2RR /3 RR+3 —(LRR+LRM)W 2 r RR-3 -(LRR+LRM)W 2 r 2RR [L] ' 3(LRR-2LRM) 0 0 0 0 2(LSS+LSM) LSS+LSM 3MSR |MSR LSS+LSM 2(LSS+LSM) |MSR 3MSR 0 3MSR |MSR 2(LRR+LRM) (LRR+LRM) |MSR 3MSR (LRR+LRM) 2(LRR+LRM) T h i s c o n c l u d e s the p r o c e d u r e f o r o b t a i n i n g t h e motor parameters and t r a n s f o r m i n g them i n t o a s i m p l i f i e d form f o r use i n a computer s i m u l a t i o n . 18 3. SECONDARY EFFECTS 3.1 I n t r o d u c t i o n t o Secondary E f f e c t s I t has been the p r a c t i c e i n p r e v i o u s i n d u c t i o n motor models t o have the motor parameters o f r e s i s t a n c e , leakage i n d u c t a n c e and mutual i n d u c t a n c e c o n s t a n t under a l l o p e r a t i n g c o n d i t i o n s . I t has a l s o been t h e p r a c t i c e t o i g n o r e the e f f e c t of space harmonics and non-symmetrical motor w i n d i n g s . Secondary e f f e c t s a r e those which cause t h e above s i m p l i f i c a t i o n s t o be i n c o r r e c t . 3.2 S k i n E f f e c t on R e s i s t a n c e and Leakage Inductance S k i n e f f e c t i s due t o magnetic f l u x i n t h e c o n d u c t o r s which causes g r e a t e r c u r r e n t d e n s i t y towards the s u r f a c e o f the c o n d u c t o r . There a r e e s s e n t i a l l y two d i f f e r e n t magnetic p a t h s i n t h e motor: a i r o r o t h e r non-magnetic m a t e r i a l and s t e e l , and magnetic m a t e r i a l . These two d i f f e r e n t magnetic p a t h s d i r e c t the f l u x through some c o n d u c t o r s and around o t h e r s . T h i s i s i l l u s t r a t e d i n F i g . 4. As can be seen i n the f i g u r e , the c u r r e n t i n t h e lower p a r t o f t h e co n d u c t o r i s l i n k e d by a l a r g e r number of f l u x l i n k a g e s . The a l t e r n a t i n g c u r r e n t w i l l f i n d g r e a t e r impedance i n t h i s p a t h and w i l l t e n d t o f l o w near the t o p of the b a r . T h i s i n c r e a s e s the r e s i s t a n c e o f the bar t o a l t e r n a t i n g c u r r e n t . (The r o t o r i s d e s i g n e d i n t h i s way so as t o i n c r e a s e s t a r t i n g t o r q u e . ) I f t h e f l u x was u n i f o r m then the c u r r e n t would t e n d t o fl o w on the s u r f a c e o f the c o n d u c t o r and a l s o i n c r e a s e the e f f e c t i v e r e s i s t a n c e , but t o a l e s s e r amount than i n t h e p r e v i o u s c a s e . The amount of s k i n effect,and variation of r e s i s t a n c e w i t h f r e q u e n c y , w i l l depend on the magnetic p a t h around the c o n d u c t o r . The amount o f s k i n e f f e c t f o r the most common type of conductor/magnetic p a t h found i n an e l e c t r i c motor FIGURE 4 w i l l now be d e r i v e d . T y p i c a l dimensions o f s m a l l i n d u c t i o n motors w i l l then be s u b s t i t u t e d i n t o the e q u a t i o n s t o o b t a i n approximate v a l u e s o f v a r i a t i o n i n r e s i s t a n c e w i t h f r e q u e n c y [ 1 6 ] . To c a l c u l a t e the s k i n e f f e c t the v a l u e f o r magnetic f l u x , c u r r e n t d e n s i t y and r e s i s t i v i t y must be s u b s t i t u t e d i n t o the e q u a t i o n f o r v o l t a g e drop a l o n g the c o n d u c t o r . The magnetic f l u x and c u r r e n t d e n s i t y , which are m u t u a l l y dependent, are f i r s t c a l c u l a t e d . F l u x d e n s i t y a t any h e i g h t y f o r a conductor i n an open s l o t i s g i v e n by 20 10 o (26) T h i s s i t u a t i o n i s i l l u s t r a t e d i n F i g u r e 5 which a l s o g i v e s the d e f i n i t i o n s f o r the symbols used. The r.m.s. f l u x l i n k i n g the conduc t o r f i l a m e n t a t h e i g h t y i s g i v e n by: ip = / Bdy o The r.m.s. v o l t a g e i n d u c e d by t h i s f l u x i s : (27) E = - j2irfTp = - j2irf / 13dy y (28) The c u r r e n t d e n s i t y a t h e i g h t y i s the net v o l t a g e (impressed minus induced) d i v i d e d by the r e s i s t i v i t y : J = V - E/p ^conductor | ^-magnetic material (29) • current filament d = depth of bar w = width o f s l o t and c o n d u c t o r f = fr e q u e n c y p = r e s i s t i v i t y ' y = d i s t a n c e from bottom of bar J = rms c u r r e n t d e n s i t y a t h e i g h t y, i n amperes/cm 2 = dc r e s i s t a n c e (P/wd ohms per cm of a x i a l l e n g t h ) FIGURE 5 21 The change o f c u r r e n t d e n s i t y w i t h depth i n the conductor i s g i v e n a s : d J dE = 2jrf B (30) dy pdy J p and a l s o : d 2 J .2irf dB , , 4.2irf r 4 i r J p v 9-P 10 From e q u a t i o n (26) and — = i J L . j dy 9 y 10 2 dB _ 4jr_ d J ^ 2 9 dy dy 10 J = ^ ( ^ ) (32) 10 P The g e n e r a l s o l u t i o n o f t h i s e q u a t i o n i s : B = P c o s h ( l + j ) 2ir / — y + Q s i n h ( l + j ) 2ir / — ^ i|> (33) p10' ' P 1 0 9 S u b s t i t u t i n g l i m i t s i n t o e q u a t i o n (33) g i v e s P = 0 when B = 0 B = when y = d w10 4-u o 4 i r 1 10~ 9 thus Q = w s i n h ( l + j ) 2 IT / 9 p10 22 and s i n h ( l + j ) 2TT J—^ y 4 IT I r P10 n B = — — [ * — J webers p e r s q . cm. (34) 10~w / f s i n h ( l + j ) 2TT / — ^ d p10 and s i m i l a r l y f o r e q u a t i o n (31). J = M c o s h ( l + j ) 2TT / — y + N s i n h d + j ) 2TT / - f p10 p10 S u b s t i t u t i n g l i m i t s i n t o e q u a t i o n (35) d J — = 0 where y = 0 dy 9 y (35) and from e q u a t i o n (30) and (35). M ( l + j ) 2, / X = 4 * 1 1 0 " 9 ( J _ 2 J L £ ) / P i o 9 / — r p ^ w s i n h d + j ) 2ir /-p 1 0 9 (1 + j ) 2ir / — ^ - I c o s h d + j ) 2TT / — ^ y p10' r ' p 1 0 9 J = ^ ^ (36) s i n h d + j ) 2TT / d p10 The t o t a l v o l t a g e drop a l o n g t h e bar i s t h e sum of the IR drop, where the c u r r e n t i s g i v e n by e q u a t i o n (36) above, and the r e a c t i v e component produced by the f l u x c r o s s i n g the s l o t . The r e a l component of t h e v o l t a g e w i l l g i v e t h e e f f e c t i v e r e s i s t a n c e of the bar f o r a l t e r n a t i n g c u r r e n t , and the j component w i l l g i v e the e f f e c t i v e r e a c t a n c e . The IR drop a l o n g t h e bar a t any h e i g h t y i s : = Jp v o l t s p e r cm (37) The average drop o v e r the e n t i r e b ar and thus the e f f e c t i v e r e s i s t a n c e i s : 23 d ( 1 + j ) 2TT / — ^ p i d c o s h d + j ) 2TT / — ^ y dy V = I / J dy = , P 1 ° / P 1 ° r d J wd J i ° ° s i n h ( l + j ) 2 i r / — ^ d p 1 0 p i s i n h d + j ) 2TT J'—^S, d £ i f i L - - E j - i R (38) wd dc wd s i n h d + j ) 2TI / — — d p 1 0 The t o t a l rms f l u x c r o s s i n g t h e s l o t above the h e i g h t y and thus l i n k i n g the c u r r e n t below y i s c a l c u l a t e d from e q u a t i o n s (27) and ( 3 5 ) . c o s h d + j ) 2TT / — ^ d - c o s h d + j ) 2ir / — ^ y ] f „_ 4TT 1 - 1 0 r p 1 0 P 1 0 * = J Bdy = I- ~ ~  Y (1 + j ) 2TT J—^ w s i n h d + j ) 2TT F ' p 1 0 ' ' p 1 0 9 d (39) The t o t a l v o l t a g e a p p l i e d t o t h e bar a t t h e h e i g h t y i s ; V = p j + j 2Trfi|j (40) S u b s t i t u t i n g e q u a t i o n s (36) and (39) into(40),,- g i v e s : -9 r~r j 2TT f (4ir I 1 0 ) c o s h d + j ) 2 i r / — ^ - d p 1 0 -V = p — (41) (1 + j ) 2TT / — ^ w s i n h d + j ) 2 i r / — — d 9 J ' / 9 p 1 0 p 1 0 which can be w r i t t e n as V = V , + V r e a l r e a c t i v e ( s i n h 4ir P 1 0 v >iT / f a / f cosh 4 [ / - cos 4 IT / p 1 0 / p 1 0 24 2TT / — ( s i n h 4ir / — ^ - r d - s i n 4TT / — — d V . r p 1 0 p 1 0 p10 , r e a c t a n c e = j I R d c [ ^ ^ Z I ; Z Z Z ~ J (43) cosh 4ir J——— - cos 4TT , / -p i e r : p i o 9 3.3 C a l c u l a t i o n s f o r T y p i c a l Motors The e f f e c t i v e r e s i s t a n c e and i n d u c t a n c e f o r a t y p i c a l s m a l l motor w i l l now be c a l c u l a t e d . From F i g u r e 6 the depth o f the bar i s shown t o be a p p r o x i m a t e l y .85 cm. The f r e q u e n c y o f t h e r o t o r c u r r e n t s a t stopped r o t o r w i l l be t h e f r e q u e n c y o f the s t a t o r c u r r e n t s d i v i d e d by the number of p o l e p a i r s . The f r e q u e n c y o f the c u r r e n t s f o r t h e f o u r p o l e motor o f F i g u r e 6 can v a r y between 0 t o 60 Hz when connected t o f u l l l i n e v o l t a g e (from stopped r o t o r t o synchronous s p e e d ) . The r o t o r c o n d u c t o r s can be c o n s i d e r e d as c o n d u c t o r s i n open s l o t s due t o s a t u r a t i o n o f t h e r o t o r s t e e l on i t s s u r f a c e . v r e a i and v r e a c t i v e c a n b e approximated as: V i = I R , [1 + 4 - - p 1 ° r ] (44) • r o p i l rin AS AUS -I ( 2 » / - ^ d ) 2 ( 2 , / ~ ^ d ) 4 3 2 ( 2 , / - l l d ) 8 + . . . ] V = i I R 2 P 1 0 h _ o P 1 0 . P™' V r e a c t i v e 3 1 R d c 2 3 L 1 8 3 1 5 (45) 25 FIGURE 6 The e f f e c t i v e r e s i s t a n c e i s c a l c u l a t e d t o be 1.01 R ^ and the r e a c t a n c e i s .165 R ^ a t 60 Hz. F o r t h i s case s k i n e f f e c t produces o n l y a 1% i n c r e a s e i n the r e s i s t a n c e and t h i s would be c o n s i d e r e d i n s i g n i f i c a n t . The same p r o c e d u r e i s now used f o r the r o t o r shown i n F i g u r e 7. For t h i s r o t o r the parameters a r e : P . = 3.44 x 10" 6 r/cm a l d = 2.01 cm f = 60 Hz max 26 I N C H FIGURE 7 and the e f f e c t i v e r e s i s t a n c e i s c a l c u l a t e d t o be 1.172 R ^ and the r e a c t a n c e i s .88 R d c « The s k i n e f f e c t can then be seen t o produce s i g n i f i c a n t v a r i a t i o n s i n both r e s i s t a n c e and r e a c t a n c e o f the r o t o r c o n d u c t o r s . The e f f e c t on the s t a t o r w i l l now be c o n s i d e r e d . The s t a t o r w i n d i n g can have many more v a r i a t i o n s i n d e s i g n than the r o t o r . Windings f o r d i f f e r e n t phases may be l o c a t e d i n the same s l o t . In a d d i t i o n , the phase p a i r v a r i e s from one s l o t t o a n o t h e r . Furthermore, c o n d u c t o r s may be e i t h e r b a r s o r s m a l l diameter w i r e s . The a n a l y s i s of each d i f f e r e n t c o n d i t i o n i s d i f f i c u l t t o do and i n a l l cases the v a r i a t i o n s i n parameters w i l l be l e s s than f o r the r o t o r due t o 27 the m u l t i - c o n d u c t o r paths used i n a l l s t a t o r s . In p r a c t i c e t h e s k i n e f f e c t v a r i a t i o n s a r e not determined by t h e c a l c u l a t i o n s d e s c r i b e d a t t h e b e g i n n i n g of t h i s c h a p t e r , f o r two r e a s o n s . F i r s t , i t i s n e c e s s a r y t o d e s t r o y t h e r o t o r o f the machine t o determine t h e geometry o f the r o t o r b a r s . Secondly, the p r e v i o u s a n a l y s i s does not take i n t o account leakage i n d u c t a n c e v a r i a t i o n due t o c u r r e n t d i s p l a c e m e n t . T h i s i s d e s c r i b e d below. 3.4 Leakage Inductance V a r i a t i o n Due t o C u r r e n t Displacement The s k i n e f f e c t as d e s c r i b e d i n S e c t i o n 3.2 w i l l f o r c e the c u r r e n t s f l o w i n g i n t h e s t a t o r and r o t o r t o t h e s u r f a c e and t h i s w i l l cause a d e c r e a s e i n t h e leakage r e a c t a n c e . F i g u r e 8 shows the c i r c u m s t a n c e s under which the leakage f l u x i s a l t e r e d - [ 11]. The leakage i n d u c t a n c e v a r i a t i o n d e s c r i b e d i n t h i s s e c t i o n i s more d i f f i c u l t t o a n a l y s e t h an t h e v a r i a t i o n i n r e s i s t a n c e and r e a c t a n c e d e s c r i b e d e a r l i e r . When the s t a t o r c u r r e n t s i n c r e a s e i n f r e q u e n c y , the leakage f l u x w i l l d e c r e a s e . T h i s i s due t o the h i g h f r e q u e n c y f o r c i n g t h e s t a t o r c u r r e n t t o t h e s u r f a c e , which produces g r e a t e r magnetic c o u p l i n g between the r o t o r and s t a t o r . The s t a t o r leakage f l u x i s u s u a l l y measured a t f u l l l i n e f r e q u e n c y so t h a t when the motor i s run on lower f r e q u e n c i e s the leakage f l u x w i l l be h i g h e r . The b l o c k e d - r o t o r t e s t , which i s used t o measure the r o t o r and s t a t o r l e a k a g e i n d u c t a n c e , w i l l g i v e a c o r r e c t v a l u e o n l y when the r o t o r i s stopped and the s t a t o r c u r r e n t i s a t l i n e f r e q u e n c y . To measure the v a r i a t i o n i n leakage i n d u c t a n c e and r o t o r r e s i s t a n c e , a v a r i a b l e f r e q u e n c y t e s t i s c o n d u c t e d . The r o t o r i s d r i v e n by a synchronous machine which i s connected t o the s o u r c e a p p l i e d t o the i n d u c t i o n motor. In t h i s 28 Low Stator Frequency STATOR High Stator Frequency //ft* " ~ ^ N 1 \*> W O \ \ N. // / / / / Y'/l I II l l i i I i ) ROTOR Low Rotor Frequency High Rotor Frequency \ fa \ N FIGURE 8 29 way s m a l l r o t o r speed v a r i a t i o n s can be e l i m i n a t e d . A c o n s t a n t v o l t s p e r h e r t z must be m a i n t a i n e d d u r i n g t h e t e s t t o p r o v i d e c o n s t a n t magnetic f l u x i n t h e motor. The v a r i a t i o n i n the s t a t o r r e s i s t a n c e and i n d u c t a n c e w i t h f r e q u e n c y can be c a l c u l a t e d . A b l o c k e d r o t o r v a r i a b l e f r e q u e n c y t e s t i s t h e n conducted t o determine t h e r o t o r v a r i a t i o n i n leakage i n d u c t a n c e and r e s i s t a n c e w i t h f r e q u e n c y . The i n c l u s i o n o f parameter v a r i a t i o n s i n the computer s i m u l a t i o n r e q u i r e s t h a t t h e most a c c u r a t e v a l u e s f o r the parameters be used a t each time s t e p . The most a c c u r a t e v a l u e s can be e s t i m a t e d knowing the r e s u l t s from the motor t e s t s and the t e r m i n a l c o n d i t i o n s a t the time s t e p i n t h e program. The motor t r a n s f o r m a t i o n as d e r i v e d i n c h a p t e r two uses c o n s t a n t v a l u e s o f i n d u c t a n c e but m u l t i p l i e s the r e s i s t a n c e v a l u e s by the speed o f t h e r o t o r a t each time s t e p . T h i s means t h a t i t w i l l t ake almost no e x t r a time t o i n c l u d e r e s i s t a n c e v a r i a t i o n s . However, i n d u c t a n c e v a r i a t i o n s w i l l slow the computer program down c o n s i d e r a b l y . The r e a s o n f o r u s i n g a t r a n s f o r m a t i o n i s t o a v o i d i n v e r t i n g a s i x by s i x m a t r i x a t e v e r y time s t e p . To i n c l u d e t h e leakage f l u x v a r i a t i o n i n t h e computer program r e q u i r e s t h a t new v a l u e s f o r t h e i n d u c t a n c e m a t r i x be used f o r each time s t e p . T h i s r e q u i r e s i n v e r s i o n o f t h e m a t r i x , which i s a v e r y time consuming o p e r a t i o n and s h o u l d be a v o i d e d i f p o s s i b l e . However, i n most c a s e s where the i n d u c t i o n motor i s c o n n e c t e d t o an i n v e r t e r the r o t o r c u r r e n t f r e q u e n c y never v a r i e s by more than f i v e h e r t z so t h a t the r o t o r leakage v a r i a t i o n does not have t o be i n c l u d e d . The s t a t o r l eakage v a r i a t i o n i s u s u a l l y l e s s s i g n i f i c a n t than t h a t of the r o t o r and can a l s o be i g n o r e d f o r most c a s e s . D u r i n g f u l l l i n e v o l t a g e s t a r t i n g , however, the r o t o r f r e q u e n c y w i l l v a r y s i g n i f i c a n t l y . The v a r i a t i o n i n t h e r o t o r leakage f l u x was found by L i w s h i t z - G a r i k and Whipple [17] t o be g r e a t e r than 6% f o r motors l e s s than f i f t y 30 horsepower. T h i s i s a s i g n i f i c a n t v a r i a t i o n and s h o u l d be c o n s i d e r e d i f a c c u r a t e s i m u l a t i o n r e s u l t s a r e r e q u i r e d under a l l o p e r a t i n g c o n d i t i o n s . 3.5 S a t u r a t i o n o f Leakage Reactance The magnetic p a t h of the leakage f l u x e s l i e s , f o r a l a r g e p a r t , i n a i r f o r which the p e r m e a b i l i t y i s a c o n s t a n t . T h i s means t h a t t h e leakage f l u x e s s h o u l d not n o r m a l l y v a r y w i t h the amplitude of t h e motor c u r r e n t s between n o - l o a d and f u l l l o a d . However, a t h i g h s l i p w i t h f u l l v o l t a g e t h e motor c u r r e n t s become v e r y l a r g e , a p p r o x i m a t e l y s i x times g r e a t e r than d u r i n g f u l l l o a d , and the i r o n i n t h e leakage f l u x p a t h s w i l l s a t u r a t e . Due t o t h i s s a t u r a t i o n t h e s t a t o r and r o t o r leakage i n d u c t a n c e may become as low as 75% t o 85% o f t h e normal v a l u e . Because the l o c k e d r o t o r t e s t i s r u n a t lower v o l t a g e and c u r r e n t s than d u r i n g f u l l l i n e v o l t a g e s t a r t i n g , the s a t u r a t i o n i n the leakage does not o c c u r . The measured v a l u e f o r t h i s t e s t i s t h e r e f o r e t h e u n s a t u r a t e d leakage i n d u c t a n c e . In the model the l o c k e d r o t o r c u r r e n t a t f u l l v o l t a g e would be 35% lower u s i n g t h e u n s a t u r a t e d leakage i n d u c t a n c e than w i t h s a t u r a t i o n . T h i s t h e n can be a s i g n i f i c a n t e f f e c t . The leakage r e a c t a n c e s have e s s e n t i a l l y t h r e e d i f f e r e n t paths and thus t h r e e d i f f e r e n t t h r e s h o l d s o f c u r r e n t which cause s a t u r a t i o n [18]. These p a t h s a r e shown i n F i g u r e 9 and t h e e x p e c t e d f l u x c u r r e n t graph i s shown i n F i g u r e 10. The l i n e a r a p p r o x i m a t i o n i s a l s o shown on the graph. A l t h o u g h i t i s i n f a c t non-l i n e a r , the e f f e c t o f s a t u r a t i o n i s . o f t e n r e p r e s e n t e d as a p i e c e - w i s e l i n e a r f u n c t i o n [19] . S i n c e t h e i n f l u e n c e of s a t u r a t i o n i n the leakage p a t h o c c u r s o n l y a t e x t r e m e l y h i g h c u r r e n t , t h e u n s a t u r a t e d leakage v a l u e s a r e n o r m a l l y used. The s a t u r a t e d v a l u e i s i n s e r t e d i n t o the model whenever motor c u r r e n t s exceed a s e t t h r e s h o l d determined by a f u l l v o l t a g e b l o c k e d r o t o r t e s t . The s a t u r a t e d leakage r e a c t a n c e v a l u e i t s e l f i s a l s o determined by t h i s t e s t . 31 primary slot leakage zigzag and belt leakage secondary slot leakage useful flux FIGURE 9 B •s FIGURE 10 3.6 S a t u r a t i o n o f Main Reactance In c o n t r a s t w i t h the p a t h of the leakage f l u x , most of the p a t h of the main f l u x i s i n magnetic m a t e r i a l w i t h o n l y a s m a l l p a r t b e i n g i n t h e a i r gap. I f the main f l u x p a t h does s a t u r a t e then the m a g n e t i z i n g branch c u r r e n t w i l l 32 i n c r e a s e by a l a r g e amount. The a n a l y s i s f o r s a t u r a t i o n of the m a g n e t i z i n g branch i s s i m i l a r t o t h a t f o r the t r a n s f o r m e r except f o r th e l a r g e r m a g n e t i z i n g c u r r e n t . High c u r r e n t s w i l l f l o w due t o s a t u r a t i o n whenever the a l t e r n a t i n g v o l t a g e a p p l i e d i s g r e a t e r t h a n the number o f t u r n s (N) times t h e s a t u r a t e d v a l u e f o r f l u x time d e r i v a t i v e dij; s/dt f o r the magnetic m a t e r i a l of the s t a t o r . S a t u r a t i o n o f the m a g n e t i z i n g b r a n c h i s i l l u s t r a t e d i n F i g u r e 11. Under no l o a d c o n d i t i o n s the magnetomotive f o r c e (mmf) causes a f l u x which i n d u c e s a back emf. The s m a l l d i f f e r e n c e between the a p p l i e d and i n d u c e d v o l t a g e s e q u a l s the IR drop and the h y s t e r e s i s drop. High c u r r e n t s w i l l f l o w whenever N d\Ji g/dt i s l e s s than the a p p l i e d v o l t a g e r e g a r d l e s s of any secondary l o a d . applied primary secondary FIGURE 11 A r e a s o n a b l e way o f d e t e r m i n i n g the amount o f s a t u r a t i o n t h a t w i l l o c c u r i n the m a g n e t i z i n g branch of a motor i s by comparing the a p p l i e d v o l t a g e d i v i d e d by the f r e q u e n c y t o th e m a g n e t i z i n g c u r r e n t . T h i s can be found by i n c r e a s i n g the a p p l i e d v o l t a g e above the r a t e d v o l t a g e d u r i n g the no l o a d t e s t . 33 The s a t u r a t e d v a l u e o f t h e m a g n e t i z i n g i n d u c t a n c e can be determined by t h i s t e s t . A l s o determined i s the a p p l i e d v o l t a g e t o f r e q u e n c y r a t i o a t which s a t u r a t i o n o f t h e m a g n e t i z i n g b r a n c h becomes s i g n i f i c a n t . Whenever t h i s r a t i o i s exceeded d u r i n g s i m u l a t i o n s , t h e s a t u r a t e d v a l u e f o r t h e m a g n e t i z i n g i n d u c t a n c e i s used. 3.7 Other E f f e c t s The u s u a l assumptions t h a t have been made i n p r e v i o u s models a r e : 1) t h a t t h e machine i s s y m m e t r i c a l 2) t h a t t h e e f f e c t s o f space harmonics can be i g n o r e d 3) t h a t a l l i n d u c t a n c e s a r e f r e q u e n c y independent 4) t h a t a l l r e s i s t a n c e s a r e c o n s t a n t 5) t h a t s a t u r a t i o n does not o c c u r The l a s t t h r e e assumptions a r e not used i n the p r e s e n t model. They were examined i n p r e v i o u s s e c t i o n s and w i l l n o t be d e a l t w i t h f u r t h e r h e r e . The proposed model m a i n t a i n s t h e assumption o f symmetry, which seems r e a s o n a b l e as almost a l l motors a r e f a c t o r y assembled u s i n g stamped and machined p a r t s so t h a t m e c h a n i c a l symmetry and thus e l e c t r i c a l symmetry s h o u l d be a s s u r e d . The e f f e c t o f space harmonics due t o the i n t e g r a l - s l o t windings i s d i f f i c u l t t o model, and s i n c e i t s e f f e c t i s p a r t l y a c c o u n t e d f o r i n the leakage i n d u c t a n c e term, i t w i l l not be c o n s i d e r e d s e p a r a t e l y . The v a r i a t i o n o f harmonic m a g n e t i z i n g i n d u c t a n c e has been a n a l y z e d by Slemon, Ismailow and Dewan [20,21] . They found t h e m a g n e t i z i n g i n d u c t a n c e t o harmonic c u r r e n t s t o be f a r l e s s than the fundamental c u r r e n t s . In v o l t a g e source i n v e r t e r s the c u r r e n t s have low harmonic c o n t e n t so t h i s e f f e c t can be i g n o r e d . However, b e f o r e m o d e l l i n g a c u r r e n t source i n v e r t e r , which has h i g h e r harmonic c o n t e n t , the e f f e c t s h o u l d be g i v e n f u r t h e r c o n s i d e r a t i o n . A l l s i g n i f i c a n t e f f o r t s mentioned by p r e v i o u s a u t h o r s have now been c o n s i d e r e d f o r implementation i n t h e computer model. For most s i t u a t i o n s t h e secondary e f f e c t s can be i g n o r e d w i t h no s i g n i f i c a n t l o s s o f a c c u r a c y but f o r s i t u a t i o n s where extremes from t h e normal o p e r a t i o n o f t h e motor a r e r e q u i r e d , then secondary e f f e c t s s h o u l d be i n c l u d e d . 35 4. VOLTAGE SOURCE REPRESENTATION 4.1 I n t r o d u c t i o n The p r e v i o u s two c h a p t e r s d e s c r i b e the r e p r e s e n t a t i o n o f t h e motor i n the computer model. T h i s c h a p t e r w i l l c o n s i d e r the r e p r e s e n t a t i o n o f t h e v o l t a g e s o u r c e . There a r e many d i f f e r e n t t y p e s o f v o l t a g e s o u r c e s t h a t must be r e p r e s e n t e d i n o r d e r t h a t the computer model can r e p r e s e n t the many d i f f e r e n t s i t u a t i o n s which a d e s i g n e n g i n e e r w i l l e n counter. A l t h o u g h t h e r e a r e many d i f f e r e n t t y p e s , the v o l t a g e waveforms produced by the v o l t a g e s o u r c e s can be r e p r e s e n t e d by a combination o f a few b a s i c s i n u s o i d a l and s t e p f u n c t i o n s . The v o l t a g e s o u r c e impedance must a l s o be c o n s i d e r e d f o r the r e p r e s e n t a t i o n t o be a c c u r a t e . T h i s c h a p t e r w i l l d e s c r i b e t h e a n a l y s i s o f t h e b a s i c t y p e s o f s o u r c e s and then develop the combinations o f f u n c t i o n s which can be used t o model them. 4.2 F u l l V o l t a g e L i n e S t a r t i n g F u l l v o l t a g e l i n e s t a r t i n g o c c u r s when the i n d u c t i o n motor i s d i r e c t l y connected t o a v o l t a g e bus. The c o n d i t i o n s which have t o be r e p r e s e n t e d a r e source impedance and non-simultaneous c o n n e c t i o n o f a l l t h r e e - p h a s e s . Source impedance i n the form of r e s i s t a n c e and i n d u c t a n c e can be i n s e r t e d i n t o t h e i n d u c t a n c e and r e s i s t a n c e m a t r i x o f the i n d u c t i o n motor model (assuming z = z f o r t h e s o u r c e ) . The source i n d u c t a n c e i s added t o the v a l u e s of the per-phase s t a t o r s e l f i n d u c t a n c e . The source r e s i s t a n c e i s added t o the s t a t o r r e s i s t a n c e . The non-simultaneous c o n n e c t i o n of the motor phases can be r e p r e s e n t e d by d e l a y s i n the v o l t a g e f u n c t i o n s . 36 The v o l t a g e s s u p p l i e d t o the motor which must be r e p r e s e n t e d i n the computer model a r e simple s i n u s o i d a l f u n c t i o n s which a r e e a s i l y implemented i n the program. 4.3 C y c l o c o n v e r t e r s One type o f v a r i a b l e f r e q u e n c y v o l t a g e s u p p l y f o r i n d u c t i o n motors i s the c y c l o c o n v e r t e r , which c o n v e r t s an i n p u t v o l t a g e o f one f r e q u e n c y t o an o u t p u t v o l t a g e a t a lower f r e q u e n c y . T h i s i s a c h i e v e d by t u r n i n g on the t h y r i s t o r s i n an a p p r o p r i a t e sequence. The s t a n d a r d 3-<i> c y c l o c o n v e r t e r c i r c u i t i s shown i n F i g u r e 12. PHASE A P H A S E B P H A S E C FIGURE 12 37 A t y p i c a l lower f r e q u e n c y v o l t a g e waveform t h a t can be produced i s shown i n F i g u r e 13. The average v o l t a g e can a l s o be e f f e c t i v e l y r e d u c e d t o g i v e a c o n s t a n t v o l t s p er h e r t z r a t i o . FIGURE 13 As shown i n F i g u r e 12 each phase v o l t a g e a p p l i e d t o t h e motor c o n s i s t s o f a combination of v o l t a g e s o b t a i n e d from a l l t h r e e phases of the i n p u t v o l t a g e . T h i s p r o c e d u r e can be s i m i l a r l y r e p r e s e n t e d i n t h e computer model. Al t h o u g h t h e r e a r e many d i f f e r e n t m o d u l a t i o n schemes used i n c y c l o c o n v e r t e r s , the one most commonly used i s the c o s i n e wave c r o s s i n g c o n t r o l method [22]. T h i s i s the o n l y c y c l o c o n v e r t e r m o d u l a t i o n t e c h n i q u e t h a t i s b u i l t i n t o the model; any o t h e r scheme t h a t i s r e q u i r e d may be programmed by the u s e r . The c o s i n e wave c r o s s i n g c o n t r o l scheme compares each phase v o l t a g e t o the r e q u i r e d v o l t a g e i n o r d e r t o determine the f i r i n g s i g n a l f o r a p a r t i c u l a r t h y r i s t o r . When a phase v o l t a g e i s d e c r e a s i n g and i s e q u a l t o the r e q u i r e d v o l t a g e then t h a t phase t h y r i s t o r i s t u r n e d on. T h i s i s shown i n F i g u r e 14 f o r one s e t of t h y r i s t o r s , but i t can e a s i l y be extended t o the c y c l o c o n v e r t e r of Figme 12. positive voltage zero voltage negative voltage control signal for A control signal for B control signal for C FIGURE 14 39 The c y c l o c o n v e r t e r has the d i s a d v a n t a g e t h a t the f r e q u e n c y can be v a r i e d over o n l y a narrow range; a t low f r e q u e n c i e s t h e power f a c t o r i s poor and a t h i g h f r e q u e n c y the harmonics become l a r g e . For t h i s reason i n v e r t e r s are o f t e n used, which do a l l o w f u l l f r e q u e n c y and v o l t a g e v a r i a t i o n . 4.4 I n v e r t e r s The i n v e r t e r a c t s as a s e t o f s w i t c h e s which connect a D.C. v o l t a g e t o a 3-<(> motor i n such a way t h a t a 3-<|> v a r i a b l e f r e q u e n c y v a r i a b l e v o l t a g e waveform i s p r o d u c e d . The b a s i c i n v e r t e r c i r c u i t i s shown i n F i g u r e 15 a l o n g w i t h the s w i t c h i n g p a t t e r n 0 S A B I V c 1 2 3 4 5 6 1 s r s s S 5 " S 3 S3" S«» S<T S2 S 2 " S 6 S 6 " S 1 s r s 5 V AB 100 50 -50 -100 -50 +50 100 V AN 50 0 -50 -50 0 +50 +50 FIGURE 15 40 The v o l t a g e waveform produced by t h i s i n v e r t e r i s shown i n F i g u r e 16. F i g u r e 17 shows how t h e b a s i c i n v e r t e r waveform i s s i m p l y produced by a s i n u s o i d f u n c t i o n w i t h a t h r e s h o l d l i m i t s e t so t h a t when the f u n c t i o n i s l e s s than one h a l f o f i t s maximum v a l u e t h e n t h e ou t p u t f u n c t i o n i s z e r o . When t h e s i n u s o i d f u n c t i o n i s g r e a t e r than one h a l f i t s maximum v a l u e then the output f u n c t i o n i s o n e - h a l f . V, AN V BN V CN FIGURE 16 41 signal li miter result 0 _ l I ^ L _ J r / t FIGURE 17 The v a r i o u s m o d u l a t i o n t e c h n i q u e s which a r e used t o improve t h e performance o f the i n v e r t e r by d e c r e a s i n g harmonics can be added as s i m p l e e n a b l e / d i s a b l e f u n c t i o n s t o t h e output waveforms. The m o d u l a t i o n f u n c t i o n can be computed from the v a l u e s o f the r o t o r p o s i t i o n , r o t o r speed and motor c u r r e n t s which are a v a i l a b l e f o r use a t e v e r y time s t e p . I f a m o d u l a t i o n scheme t h a t i s t o be s i m u l a t e d i s not a v a i l a b l e i n the program then i t can be programmed by the u s e r from t h e f u n c t i o n s a v a i l a b l e . The two most p o p u l a r m o d u l a t i o n s t r a t e g i e s and the way i n which t h e y w i l l be implemented i n th e program w i l l now be examined. The most simple m o d u l a t i o n scheme i s t o have the b a s i c square-wave v o l t a g e source o f F i g u r e 17 modulated i n p r o p o r t i o n t o r o t o r speed. With t h i s type of system the e f f e c t i v e r o t o r f r e q u e n c y i s measured and then the i n v e r t e r f r equency i s m a i n t a i n e d a t a s l i g h t l y g r e a t e r f r e q u e n c y , u s u a l l y t h r e e h e r t z . A 42 c o n s t a n t v o l t s p e r h e r t z r a t i o i s m a i n t a i n e d by u s i n g a f i x e d p u l s e width w i t h a v a r y i n g f r e q u e n c y . T h i s i s shown by the f o r m u l a f o r t h e p u l s e w i d t h b f b where f i s the i n v e r t e r f r e q u e n c y and f ^ i s the motor base f r e q u e n c y . The r a t i o of ^ / f ^ 1 S the term t o make the v o l t a g e p r o p o r t i o n a l t o t h e f r e q u e n c y . Whenever the i n v e r t e r i s r u n n i n g a t g r e a t e r than base f r e q u e n c y , t h e p u l s e w i d t h i s l e f t as o r i g i n a l l y shown i n F i g u r e 17. The d i s a d v a n t a g e o f the square-wave i n v e r t e r waveform, as j u s t d e s c r i b e d , i s t h a t i t has a h i g h harmonic c o n t e n t . For t h i s reason t h e t r i a n g l e i n t e r c e p t m o d u l a t i o n method [23] as develo p e d by Schonung and Stemmler i s o f t e n used. T h i s method compares a s i n e wave of the i n v e r t e r ' s f r e q u e n c y w i t h a t r i a n g u l a r waveform a t a m u l t i p l e o f t h e i n v e r t e r f r e q u e n c y . T h i s i s shown i n F i g u r e 18. Whenever the s i n e wave v a l u e i s g r e a t e r than the t r i a n g u l a r waveform f o r t h a t phase (see p o i n t 1, F i g . 18) th e n the i n v e r t e r connects t h e t e r m i n a l t o the p o s i t i v e bus. Whenever t h e s i n e wave i s l e s s than t h e t r i a n g l e wave the t e r m i n a l s i n e waves a r e of co u r s e a t 120° d i f f e r e n c e i n phase but i t i s found t h a t i f the t r i a n g l e r e f e r e n c e waveform i s the same f o r a l l t h r e e phases and the motor i s co n n e c t e d and not c e n t e r tapped t h e n a s i g n i f i c a n t d e c r e a s e i n v o l t a g e harmonics o c c u r s . T h i s i s the s i t u a t i o n f o r almost a l l i n v e r t e r i n d u c t i o n motor systems• 43 FIGURE 18 I t i s a l s o p o s s i b l e t o r e p r e s e n t t h e t r i a n g l e i n t e r c e p t method o r any o t h e r method, by i t s fundamental and harmonic components s i n c e t h e s e v a l u e s a r e w e l l documented i n the d e s c r i p t i o n o f new m o d u l a t i o n t e c h n i q u e s . The d e s i g n e n g i n e e r can choose between the exact r e p r e s e n t a t i o n as d e s c r i b e d i n t h i s c h a p t e r f o r f u l l l i n e s t a r t i n g , t h e c y c l o c o n v e r t e r and the i n v e r t e r o r e l s e use t h e approximate harmonic r e p r e s e n t a t i o n . The two components of an i n d u c t i o n motor d r i v e , the v o l t a g e s o u r c e and the i n d u c t i o n motor, have now been examined and t h e i r r e p r e s e n t a t i o n i n t h e computer program d e s c r i b e d . 44 5. NUMERICAL METHODS 5.1 I n t r o d u c t i o n T h i s c h a p t e r w i l l d i s c u s s the n u m e r i c a l t e c h n i q u e s which may be used t o s o l v e the systems of e q u a t i o n s d e v e l o p e d i n Chapter 2. A d e t a i l e d a n a l y s i s o f the e q u a t i o n s i n o r d e r t o o p t i m i z e t h e n u m e r i c a l s o l u t i o n method i s f a r beyond the scope of t h i s t h e s i s . R a ther, t h i s c h a p t e r attempts t o e x p l a i n why the n u m e r i c a l method used i n the program works so w e l l . The t r a p e z o i d a l r u l e i s i n v e s t i g a t e d and t h e advantages of u s i n g a v a r i a b l e time s t e p are d i s c u s s e d . 5.2 N u m e r i c a l Methods B e f o r e c o n s i d e r i n g the v a r i o u s n u m e r i c a l methods t h a t may be used t o s o l v e t h e n o n - l i n e a r d i f f e r e n t i a l e q u a t i o n , i t i s important t o c o n s i d e r what a c c u r a c y i s r e q u i r e d of t h e s o l u t i o n method. I t may seem o b v i o u s t h a t the s o l u t i o n method which always comes c l o s e s t t o the c o r r e c t v a l u e i s the b e s t , but when n u m e r i c a l methods a r e a n a l y z e d t h e "always" i s o f t e n l e f t o u t . There are many s o l u t i o n methods which can c l a i m g r e a t a c c u r a c y o v e r a l i m i t e d time of t h e s o l u t i o n but then become u n s t a b l e and d i v e r g e from the s o l u t i o n . Such a n u m e r i c a l method c o u l d not be r e l i e d upon. On the o t h e r hand the s o l u t i o n must always be a c c u r a t e enough so t h a t the results a r e u s e f u l f o r making d e s i g n d e c i s i o n s . The Runge-Kutta types of n u m e r i c a l s o l u t i o n s a r e g e n e r a l l y a v o i d e d by n u m e r i c a l a n a l y s t s as b e i n g t o o s i m p l i s t i c and i n e f f i c i e n t f o r use i n l a r g e s c a l e computer programs [24, p. 194] but they do have the advantages of h a v i n g good s t a b i l i t y , ease of s t e p s i z e change and a r e s e l f - s t a r t i n g . 45 The t r a p e z o i d a l r u l e , one type o f Newton Cotes method, has i n a d d i t i o n i d e a l r o u n d o f f c h a r a c t e r i s t i c s s i n c e the c o e f f i c i e n t s i n t h e f o r m u l a have t h e same v a l u e [25, p. 119]. P r e d i c t o r c o r r e c t o r methods a r e a v e r y e f f i c i e n t n u m e r i c a l method but s u f f e r from s t a b i l i t y problems. The i n s t a b i l i t y i s caused by a p a r a s i t i c s o l u t i o n which grows i n magnitude from the c o r r e c t o r computation [25, p. 183]. From w i t h i n the above two t y p e s o f methods t h e r e a r e an almost u n l i m i t e d number of s o l u t i o n f o r m u l a s a v a i l a b l e but t h e r e i s not one f o r m u l a which i s c l e a r l y s u p e r i o r i n a c c u r a c y and s t a b i l i t y . Because the s t a b i l i t y o f a n u m e r i c a l method i s v e r y d i f f i c u l t t o a n a l y s e , most comparisons o f t h e n u m e r i c a l method f o c u s on a c c u r a c y a t a c e r t a i n s o l u t i o n p o i n t o n l y . However, f o r the more common s i t u a t i o n o f p r e d i c t i n g many f u t u r e v a l u e s b a s e d on p r e v i o u s v a l u e s , t h e s t a b i l i t y o f t h e s o l u t i o n method i s the most im p o r t a n t f a c t o r t o be c o n s i d e r e d . The a b s o l u t e n u m e r i c a l s t a b i l i t y o f the t r a p e z o i d a l r u l e i s t h e f a c t o r which makes i t so u s e f u l and f o r t h i s r e a s o n the t r a p e z o i d a l r u l e w i l l be f u r t h e r examined. 5.3 The T r a p e z o i d a l R u l e The t r a p e z o i d a l r u l e i s one of t h e most commonly used n u m e r i c a l methods d e s p i t e i t s r a t h e r poor n u m e r i c a l e f f i c i e n c y [25]. There a r e two reasons f o r t h i s . The f i r s t i s t h a t o t h e r n u m e r i c a l methods w i l l use almost as much computer time when t h e r e i s a l a r g e r a t i o between the s m a l l e s t and l a r g e s t e i g e n v a l u e s (a l a r g e s p r e a d i n the time c o n s t a n t s ) [26]. The second, a l r e a d y mentioned, i s t h a t the t r a p e z o i d a l r u l e i s a b s o l u t e l y n u m e r i c a l l y s t a b l e f o r l i n e a r systems. The amount o f computer time r e q u i r e d by u s i n g the t r a p e z o i d a l r u l e can be e s t i m a t e d by l o o k i n g a t t h e fo r m u l a f o r the t r a p e z o i d a l r u l e . 46 . h rdy . dy y = y + — — + —-n+1 n 2 Kdx yn dx y n + J ( 4 7 ) T h i s i s a second o r d e r f o r m u l a and has an e r r o r term g i v e n by: e r r o r < k f 3 ( £ ) h 3 (48) The e r r o r term i s g r e a t e r than f o r almost a l l o t h e r n u m e r i c a l methods which have e r r o r terms t y p i c a l l y o f the o r d e r o f e r r o r < k f 5 ( £ ) h 5 (49) I t i s f o r t h i s r e a s o n t h a t t h e t r a p e z o i d a l r u l e w i l l g e n e r a l l y r e q u i r e a s m a l l e r s t e p s i z e and thus use more computer time f o r the same l e v e l of a c c u r a c y . The s t u d y o f the s t a b i l i t y o f a n u m e r i c a l method i s r e a l l y j u s t the study o f the a c c u m u l a t i o n o f e r r o r s i n the s o l u t i o n method. The r o u n d o f f e r r o r , which i s one type of e r r o r which accumulates, i s m i n i m i z e d because the c o e f f i c i e n t s i n the t r a p e z o i d a l r u l e a r e e q u a l . T h i s i s i n c o n t r a s t t o some n u m e r i c a l formulas which have n e g a t i v e c o e f f i c i e n t s and thus q u i t e poor r o u n d o f f p r o p e r t i e s . The a b s o l u t e n u m e r i c a l s t a b i l i t y i s due t o the t r a p e z o i d a l r u l e h a v i n g a c h a r a c t e r i s t i c e q u a t i o n o f t h e f i r s t degree and thus h a v i n g no p a r a s i t i c r o o t s i n the s o l u t i o n [26, p. 1798]. 5.4 T r a p e z o i d a l Method With V a r i a b l e Time" Step The t r a p e z o i d a l method as d e s c r i b e d i n S e c t i o n 5.3 has a c c u r a c y t h a t i s dependent on the s t e p s i z e . In s i m u l a t i o n s where t h e r e a r e no s w i t c h i n g or s t e p e f f e c t s and the h i g h e s t harmonic i s known, then i t i s q u i t e p o s s i b l e t o choose a s t e p s i z e which w i l l g i v e a c c u r a t e r e s u l t s a t a l l p o i n t s i n t h e s i m u l a t i o n . However, where s w i t c h i n g or s t e p i n p u t s o c c u r i t w i l l u s u a l l y be 47 r e q u i r e d t o choose a s t e p s i z e which i s v e r y s m a l l i n o r d e r t o p r o v i d e a c c u r a t e r e s u l t s around the s w i t c h i n g i n p u t , but t h e s m a l l s t e p s i z e i s s i m p l y a waste o f computer time d u r i n g the r e s t o f the s i m u l a t i o n . T h i s i s i l l u s t r a t e d i n F i g u r e 19 where the s m a l l e r s t e p s i z e produces more a c c u r a t e r e s u l t s o n l y a t t h i s s w i t c h i n g p o i n t but does not c o n t r i b u t e a p p r e c i a b l y t o the a c c u r a c y a f t e r the s w i t c h i n g change. V V i i i n | i i i i | n i l | -time, s tep l o / u n i t o f time. • t ime step I / u n i t of t ime FIGURE 19 A s i m i l a r phenomenon i s i l l u s t r a t e d by the sampling theorem i n communications which s t a t e s t h a t a s i g n a l can be r e p r e s e n t e d a c c u r a t e l y o n l y when i t i s sampled a t t w i c e t h e f r e q u e n c y o f i t s h i g h e s t harmonic. At lower sampling r a t e s the s i g n a l cannot be r e c r e a t e d a c c u r a t e l y . In comparison the t r a p e z o i d a l r u l e can p r e d i c t a waveform o n l y when the harmonics of the waveform are f a r l e s s than l / 2 a t . T h i s i s i l l u s t r a t e d i n Figure 20. 48 V it-•it-FIGURE 20 From 0 t o t j the s i g n a l c o n t a i n s no harmonics and no D.C. v a l u e . From t 2 t o to, the s i g n a l has no harmonics and a D.C. v a l u e . In both of these cases a v e r y l a r g e time s t e p c o u l d be u sed t o r e p r e s e n t t h e s i g n a l . However, from t j t o t 2 t h e s i g n a l c o n t a i n s a v e r y l a r g e number of harmonics. I t would thus r e q u i r e a v e r y s m a l l time s t e p i n o r d e r t h a t t h e s i g n a l be a c c u r a t e l y r e p r e s e n t e d . T h i s i s of c o u r s e i n t u i t i v e l y obvious because o n l y a v e r y s m a l l time s t e p would a c c u r a t e l y p o s i t i o n t h e s w i t c h change w i t h r e s p e c t t o time. The e r r o r i n the t r a p e z o i d a l r u l e has been g i v e n i n E q u a t i o n (48) as b e i n g dependent on the time s t e p cubed. T y p i c a l l y t h e r e w i l l be a s w i t c h i n g change i n a s i m u l a t i o n e v e r y 100 s t e p s . I f the time s t e p i s d e c r e a s e d by a f a c t o r o f t e n d u r i n g the time s t e p t h a t t h e s w i t c h i n g o c c u r s t h e n t h e program r u n n i n g time w i l l be i n c r e a s e d by a p p r o x i m a t e l y 10%. However, the program a c c u r a c y a t t h a t p o i n t w i l l be i n c r e a s e d by a f a c t o r o f a p p r o x i m a t e l y 1000. As 49 the s w i t c h i n g p o i n t i s the o n l y p o i n t where l a r g e i n a c c u r a c y i s e x p e c t e d t h i s d e c r e a s e i n time s t e p w i l l i n c r e a s e the program a c c u r a c y d r a m a t i c a l l y . I t has been shown how the decrease i n the time s t e p about s w i t c h i n g p o i n t s w i l l i n c r e a s e the a c c u r a c y o f the computer program w i t h o u t a p p r e c i a b l y i n c r e a s i n g the computer time r e q u i r e d . For t h i s r e a s o n the program i s w r i t t e n so t h a t i t checks f o r s w i t c h i n g changes as t h e s i m u l a t i o n p r o g r e s s e s and when one o c c u r s t h e s t e p s i z e i s r e d u c e d by a f a c t o r s e t by the program o p e r a t o r . T h i s c o n c l u d e s the i n v e s t i g a t i o n o f t h e r e q u i r e m e n t s f o r a computer program model i n d u c t i o n motor d r i v e s . The f o l l o w i n g c h a p t e r w i l l d i s c u s s the r e s u l t s o f s i m u l a t i o n s made w i t h t h e program d e v e l o p e d . 50 6. RESULTS AND DISCUSSION 6.1 I n t r o d u c t i o n An i n d u c t i o n motor s i m u l a t i o n program was w r i t t e n (Appendix 1) and the. r e s u l t s o f the s i m u l a t i o n s were compared a g a i n s t measured v a l u e s . The program was found t o produce c o n s i s t e n t l y a c c u r a t e r e s u l t s which were found t o be v e r y u s e f u l as an a i d i n the d e s i g n o f an i n v e r t e r i n d u c t i o n motor d r i v e . The r e s u l t s o f t h e s i m u l a t i o n s a r e d e s c r i b e d i n t h e f o l l o w i n g s e c t i o n s . O s c i l l o g r a m s f o r some of the measured waveforms a r e i n c l u d e d . In a l l s i m u l a t i o n s the i n d u c t i o n motor had t h e parameters g i v e n i n Ta b l e 1. These v a l u e s were o b t a i n e d u s i n g the s t a n d a r d motor t e s t s as d e s c r i b e d i n C hapter 2. T a b l e 1 - I n d u c t i o n Motor Parameters s t a t o r r e s i s t a n c e 9.00 ft r o t o r r e s i s t a n c e 6.68 ft s t a t o r s e l f i n d u c t a n c e 0.243 H mutual i n d u c t a n c e between s t a t o r phases -0.116 H r o t o r s e l f i n d u c t a n c e 0.243 H mutual i n d u c t a n c e between r o t o r phases -0.116 H mutual s t a t o r t o r o t o r i n d u c t a n c e 0.2276 H number of p o l e s 4 r o t o r i n e r t i a 0.00084 k i l o g r a m m e t e r s 2 r a t e d v o l t a g e 208. V r a t e d power 186.5 W 6.2 F u l l V o l t a g e L i n e S t a r t i n g F u l l v o l t a g e s t a r t i n g o f the motor was s i m u l a t e d f o r t h e v o l t a g e waveform o f F i g . 21. The motor was con n e c t e d t o t h e l i n e u s i n g a s i m p l e t r i p l e p o l e s i n g l e throw s w i t c h . The c u r r e n t p r e d i c t i o n i s shown i n F i g . 22 and the measured v a l u e s f o r a c o l d and f o r a h o t motor a r e shown i n F i g u r e s 23, 24, and 25. The v a l u e s o b t a i n e d f o r the f i r s t t h r e e c u r r e n t peaks are summarized i n Table I I . These v a l u e s agree f a v o u r a b l y w i t h t h e measured v a l u e s . T a b l e I I - Comparison Between S i m u l a t i o n and A c t u a l Peak C u r r e n t s Measured For F u l l V o l t a g e L i n e S t a r t i n g . 1 s t peak 2nd peak 3rd peak S i m u l a t i o n r e s u l t 6.125A -7.875A 8.0 A Measured c o l d 7.18 A -7.86 A 8.46A Measured hot 5.74 A -7.38 A 7.18A 6.3 Other V o l t a g e Sources A c y c l o c o n v e r t e r was s i m u l a t e d as shown i n F i g . 26. The shape of t h e c u r r e n t waveform i s as e x p e c t e d [22] a l t h o u g h no a c t u a l measurements c o u l d be made f o r comparison purposes s i n c e no c y c l o c o n v e r t e r was a v a i l a b l e t o t h e r e s e a r c h e r . A square p u l s e i n v e r t e r s i m u l a t i o n i s shown i n F i g u r e s 27 and 28. I t can be o b s e r v e d t h a t t h e width o f t h e v o l t a g e p u l s e determines t h e peak c u r r e n t d u r i n g t h e i n i t i a l s t a r t - u p of t h e motor. A simple r u l e o f thumb can be e a s i l y determined from t h i s o b s e r v a t i o n . The s t a r t i n g c u r r e n t o f almost a l l i n d u c t i o n motors i s a p p r o x i m a t e l y s i x times f u l l l o a d c u r r e n t . I f the s o l i d - s t a t e d e v i c e s o f an i n v e r t e r a r e t o be s i z e d t o c a r r y f u l l l o a d c u r r e n t , then t h e v o l t - s e c o n d s of any p u l s e s h o u l d be l e s s than o n e - s i x t h the v o l t - s e c o n d s of t h e o n e - h a l f s i n e -wave l i n e v o l t a g e . . 51a For s i m u l a t i o n of F i g . 28 a .0012 second p u l s e width was u s e d. A narrower p u l s e w i d t h o f .0533 second was used i n the s i m u l a t i o n shown i n F i g . 29 and the r e s u l t i n g e f f e c t of reduced torque can be o b s e r v e d . The t r i a n g u l a r modulated i n v e r t e r waveform i s shown i n F i g . 31 and t h e r e s u l t a n t c u r r e n t waveform i s shown i n F i g . 32. 6.4 Time Step M o d i f i c a t i o n The program i s w r i t t e n i n such a way as t o look ahead t o the next time s t e p t o see i f any changes t o the v o l t a g e s a t the t e r m i n a l s w i l l o c c u r . I f changes w i l l o c c u r , then the time s t e p i s reduced by a f a c t o r o f t e n and t h e s i m u l a t i o n p r o g r e s s e s u s i n g the s m a l l e r time s t e p f o r the next t e n s t e p s . At t h e l a s t o f t h e s e s t e p s the time s t e p i s i n c r e a s e d by a f a c t o r o f t e n . The v o l t a g e t e r m i n a l c o n d i t i o n s a r e a g a i n checked and i f no v o l t a g e changes w i l l o c c u r , t h e time s t e p remains the same. I f v o l t a g e c o n d i t i o n s w i l l change, then the time s t e p i s once a g a i n r e d u c e d . The time s t e p v a r i a t i o n r o u t i n e was run f o r t h e f i x e d p u l s e width waveform o f F i g . 33. The r e s u l t s o f the c u r r e n t s i m u l a t i o n f o r phase A a r e shown i n F i g . 34. As was e x p e c t e d t h e time s t e p m o d i f i c a t i o n (shown by the d o t t e d l i n e ) p r o v i d e d as a c c u r a t e r e s u l t s u s i n g a p p r o x i m a t e l y o n e - t h i r d as much computer time as the 50 /usee p e r s t e p s i m u l a t i o n ( s o l i d l i n e ) . The .5 m i l l i s e c o n d per s t e p s i m u l a t i o n (dashed l i n e ) was s i g n i f i c a n t l y l e s s a c c u r a t e when the time s t e p was not m o d i f i e d . Time s t e p m o d i f i c a t i o n i s an e x c e l l e n t method f o r r e d u c i n g the computer time r e q u i r e d t o run s i m u l a t i o n programs. For a f r e q u e n t l y used commerical program the c o s t of m o d i f i c a t i o n t o i n c l u d e time s t e p v a r i a t i o n would be s m a l l i n comparison t o the s a v i n g s i n computer c o s t s . 52 6.5 V a r i a t i o n s i n R e s i s t a n c e and Inductance S i m u l a t i o n s were made f o r f u l l v o l t a g e l i n e s t a r t i n g o f an i n d u c t i o n motor ( F i g . 35) w i t h v a r i a t i o n s i n leakage i n d u c t a n c e and r o t o r r e s i s t a n c e a t no l o a d . These s i m u l a t i o n r e s u l t s a r e shown i n F i g . 36 f o r phase B. The r o t o r leakage i n d u c t a n c e was s e t t o d e c r e a s e i n p r o p o r t i o n t o t h e s l i p f r e q u e n c y from 114% t o 100% (SL=.88) and the r o t o r r e s i s t a n c e was i n c r e a s e d i n p r o p o r t i o n t o the square o f t h e s l i p f r e q u e n c y from 85.5% t o 100%. (SR=1.17). (These v a l u e s a r e t y p i c a l f o r s m a l l motors.) As shown i n F i g . 36 the c u r r e n t i s almost i d e n t i c a l f o r t h e t h r e e d i f f e r e n t c a s e s . T h i s i s not s u r p r i s i n g s i n c e t h e v a l u e s f o r t h e r e s i s t a n c e s and i n d u c t a n c e s a r e the same f o r a l l t h r e e c a s e s , when the motor i s f i r s t c o n nected, and o n l y v a r y once t h e motor i s r o t a t i n g and t h e c u r r e n t s a r e lower. The same v a l u e s f o r v a r i a t i o n i n the r e s i s t a n c e and i n d u c t a n c e were used i n t h e s i m u l a t i o n o f F i g . 37 but t h i s time t h e motor was i n i t i a l l y r o t a t i n g a t 1800 rpm and f u l l l o a d was a p p l i e d . The s i m u l a t i o n shows t h a t f o r a s m a l l motor t h e v a r i a t i o n i n r e s i s t a n c e as a f u n c t i o n o f f r e q u e n c y do produce a s l i g h t l y h i g h e r c u r r e n t , b u t not so g r e a t as t o be important f o r most s i m u l a t i o n s . However, f o r l a r g e motors t h e f r e q u e n c y dependence o f t h e i n d u c t a n c e s and r e s i s t a n c e s i s f a r g r e a t e r and would produce l a r g e d i s c r e p a n c i e s between s i m u l a t e d and a c t u a l c u r r e n t s i f not acco u n t e d f o r . 6.6 P r a c t i c a l A p p l i c a t i o n o f Program While the s i m u l a t i o n program was b e i n g w r i t t e n and t e s t e d the d e s i g n and t e s t i n g o f an i n v e r t e r i n d u c t i o n motor d r i v e was c o n c u r r e n t l y b e i n g u n d e r t a k e n . A r o u t i n e was devel o p e d t o s i m u l a t e a v o l t a g e waveform w i t h reduced harmonic c o n t e n t [27] t h a t was b e i n g used w i t h t h e i n v e r t e r . V a r i o u s s i m u l a t i o n s were run and compared t o the a c t u a l waveforms measured. 53 The v o l t a g e waveforms a t 4, 30 and 60 Hz a r e shown i n F i g . 38, 43, 46. The s i m u l a t i o n s were compared w i t h the a c t u a l c u r r e n t waveforms as shown i n F i g . 39 and 40 f o r 4 Hz, F i g . 44 and 45 f o r 30 Hz and F i g . 47 and 48 f o r 60 Hz. The s i m u l a t i o n r e s u l t s were a c c u r a t e f o r both the 30 and 60 Hz s i m u l a t i o n s ( w i t h i n 5%), b u t p r e d i c t e d t o o g r e a t a c u r r e n t peak f o r t h e 4 h e r t z s i m u l a t i o n . The 4 Hz s i m u l a t i o n shown i n F i g . 39 p r e d i c t s a c u r r e n t peak of a p p r o x i m a t e l y 3.2 Amps whereas t h e measured v a l u e o f F i g . 40 shows a c u r r e n t peak o f 2.05 Amp. However, the g e n e r a l shape of t h e waveform i s the same. The s i m u l a t e d c u r r e n t peak may be too h i g h due t o s k i n e f f e c t a t t e n u a t i n g the v e r y h i g h f r e q u e n c y harmonics of t h i s p a r t i c u l a r waveform. The s i m u l a t i o n s were used t o v e r i f y the c o r r e c t o p e r a t i o n o f t h e i n v e r t e r . In a d d i t i o n t h e y were used t o i n v e s t i g a t e problems w i t h t o r q u e p u l s a t i o n s a t low f r e q u e n c y . The s t r o n g t o r q u e p u l s a t i o n s p r e d i c t e d i n t h e s i m u l a t i o n f o r t h e 4 Hz waveform ( F i g . 41) were o b s e r v e d d u r i n g i n v e r t e r t e s t i n g . A 4 Hz t r i a n g u l a r modulated waveform ( F i g . 42) was then compared t o t h e harmonic e l i m i n a t i o n waveform and found t o have more a c c e p t a b l e l e v e l s o f t o r q u e p u l s a t i o n s . The s i m u l a t i o n s t u d i e s have p r o v i d e d v a l u a b l e i n f o r m a t i o n t hroughout the development of t h e i n v e r t e r i n d u c t i o n motor d r i v e . S p e c i f i c a l l y , from t h e s i m u l a t i o n r e s u l t s i t was d e c i d e d t o use a d i f f e r e n t m o d u l a t i o n s t r a t e g y a t low f r e q u e n c y than the o r i g i n a l l y used harmonic e l i m i n a t i o n method. A l s o the power t r a n s i s t o r s f o r t h e i n v e r t e r were chosen f o r c u r r e n t c a r r y i n g c a p a c i t y by e n s u r i n g t h a t the t r a n s i s t o r s c o u l d s a f e l y c a r r y the c u r r e n t p r e d i c t e d by t h e s i m u l a t i o n s . 54 FIGURE 21 - Phase v o l t a g e a p p l i e d d u r i n g s i m u l a t i o n . 55 A ,A A A A V V V \J \J V o o £ 8 3 » S o28.7o»y FIGURE 23 - Measured v a l u e s o f c u r r e n t f o r f u l l v o l t a g e l i n e s t a r t i n g , f i r s t maximum. 56 C o l d 7.86 Amp FIGURE 24 Hot 7.38 Amp Measured v a l u e s of c u r r e n t f o r f u l l v o l t a g e l i n e s t a r t i n g , second maximum. \ f V - 1 \l \J\I\J A r\ -o 1 8.7 3mS V / ^ V o4 2.3 omW C o l d 8.46 Amp Hot 7.18 Amp FIGURE 25 - Measured v a l u e s of c u r r e n t f o r f u l l v o l t a g e l i n e s t a r t i n g , t h i r d maximum. 57 0 1 i 1 1 1 : — i 1 1 0 5.005 10.01 15.015 20.02 25.025 30.03 35.035 10. TIME (MILLISECONDS E+01) FIGURE 27 - F i x e d p u l s e w idth v a r i a b l e f r e q u e n c y s i m u l a t i o n p u l s e width=.05333 seconds. FIGURE 28 - F i x e d p u l s e width v a r i a b l e f r e q u e n c y s i m u l a t i o n p u l s e width=.0012 seconds. 60 5/7 INER&FRIC=.005 VRR F i n to n i n 1 I 1 1 1 1 1 1 .0.0 5.005 10.0] 15.015 20.02 25.025 30.03 35.035 40.04 TIME (MILLISECONDS E+011 FIGURE 29 - Torque and speed f o r v a r i a b l e f r e q u e n c y f i x e d p u l s e w idth p u l s e width=.00533 seconds. 61 FIGURE 30 - Torque and speed f o r v a r i a b l e f r e q u e n c y f i x e d p u l s e w idth p u l s e width=.0012 seconds. 62 7/16' REAL-4111 :V'VPHR1 T T T 0.0 5.256 12.513 13.763 , 25.025 33.281 TIME (MILLISECONDS) 37.537 43,79-1 50.05 FIGURE 31 - T r i a n g u l a r m o d u l a t i o n 60 Hz v o l t a g e waveform. 63 7/16 REAL •41-11 FIGURE 32 - T r i a n g u l a r m o d u l a t i o n 60 Hz c u r r e n t wave.form. 64 1 1 mm 4 I* rem + -!- m i rH±i m hOi-rf J3 : TFT! f 3-256 i T! - - • n &.513 I . : . : •,.|.J.L. •i :J6B I j~t |2|5lOJS-|-i-r4# . u r n , jI:SECpNQs):f L-281 tth i l b l j •H-fjj 13 j igai. bs !•!•-: ••ITTTI FIGURE 33 - F i x e d p u l s e w i d t h f i x e d f r e q u e n c y v o l t a g e waveform. 65 FIGURE 34 - S i m u l a t i o n o f c u r r e n t waveform u s i n g v a r i a b l e t i m e s t e p . 5 x 1 0 " 5 seconds p e r s t e p 5 x 1 0 _ l + seconds p e r s t e p 5 x 1 0 " s e c o n d s p e r s t e p w i t h automatic s t e p r e d u c t i o n f e a t u r e 66 FIGURE 35 - V o l t a g e s o u r c e used f o r s i m u l a t i o n of v a r i a b l e i n d u c t a n c e s and r e s i s t a n c e s . 67 FIGURE 36 - C u r r e n t s i m u l a t i o n f o r no l o a d f u l l v o l t a g e w i t h parameter v a r i a t i o n s . no v a r i a t i o n SR = 1.17 SL = .88 FIGURE 37 - C u r r e n t s i m u l a t i o n f o r f u l l l o a d f u l l v o l t a g e and i n i t i a l speed of 1800 rpm. no v a r i a t i o n SR = 1.17 SL = .88 FIGURE 38 - V o l t a g e waveform f o r 4 Hz harmonic e l i m i n a t i o n method. FIGURE 40 - Measured c u r r e n t o f 4 Hz harmonic e l i m i n a t i o n waveform 2.05 Amps peak. 71 FIGURE 41 - Torque p u l s a t i o n s f o r 4 Hz harmonic e l i m i n a t i o n m o d u l a t i o n method. FIGURE 42 - Torque p u l s a t i o n s f o r 4 Hz t r i a n g u l a r m o d u l a t i o n method. FIGURE 43 - V o l t a g e waveform of 30 Hz harmonic e l i m i n a t i o n method. 74 FIGURE 45 - Measured c u r r e n t o f 30 Hz harmonic e l i m i n a t i o n waveform 1.18A peak. U-15:REf\L 5111 K L = . l 8.759 | i - l -H- | - t -H-TIME [MILLISECONDS E+01) FIGURE 46 - V o l t a g e waveform o f 60 Hz harmonic e l i m i n a t i o n method. 76 FIGURE 47 - C u r r e n t s i m u l a t i o n o f 60 Hz harmonic e l i m i n a t i o n method. I U I t « t • H M J t V FIGURE 48 - Measured c u r r e n t of 60 Hz harmonic e l i m i n a t i o n method .965 A peak. 77 7. CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER RESEARCH 7.1 Summary T h i s t h e s i s has a n a l y z e d t h e v a r i o u s components which must be r e p r e s e n t e d i n a s i m u l a t i o n program o f an i n d u c t i o n motor d r i v e . The i n d u c t i o n motor e q u a t i o n s have been c o n s i d e r e d and a s i m p l i f y i n g t r a n s f o r m a t i o n has* been implemented. The second o r d e r e f f e c t s o f s a t u r a t i o n , leakage i n d u c t a n c e v a r i a t i o n and r e s i s t a n c e v a r i a t i o n under d i f f e r e n t o p e r a t i n g c o n d i t i o n s have been a n a l y z e d . The e f f e c t s o f leakage i n d u c t a n c e and r e s i s t a n c e v a r i a t i o n have been implemented i n the s i m u l a t i o n program. The v a r i o u s v o l t a g e s o u r c e s t h a t a r e used w i t h i n d u c t i o n motor d r i v e s have been c o n s i d e r e d and implemented i n t o t h e computer program. I t was no t e d i n Chapter 6 t h a t t h e r e s u l t s o f the s i m u l a t i o n s were e x t r e m e l y a c c u r a t e i n almost a l l cases where a c t u a l r e s u l t s were a v a i l a b l e f o r comparison. The second o r d e r e f f e c t s model was not checked f o r a c c u r a c y s i n c e t h e s e e f f e c t s o c c u r o n l y f o r v e r y l a r g e machines which were not a v a i l a b l e i n the l a b o r a t o r y . However, the modeling method r e l i e s on measured parameters f o r the i n d u c t i o n motor d u r i n g t h e c o n d i t i o n s which cause second o r d e r e f f e c t s and s h o u l d thus produce r e l i a b l e s i m u l a t i o n r e s u l t s . As f a r as can be determined t h i s program i s the f i r s t t o i n c l u d e an a c c u r a t e model o f t h e second o r d e r e f f e c t s o f v a r i a t i o n i n motor i n d u c t a n c e and r e s i s t a n c e . . I t i s a l s o t h e f i r s t program t o have a s w i t c h i n g change c h e c k i n g r o u t i n e which a u t o m a t i c a l l y reduces t h e s t e p s i z e about a s w i t c h i n g p o i n t . T h i s t h e s i s i s the f i r s t known time t h a t the r u l e o f thumb developed i n Chapter 6 which g i v e s the maximum v o l t a g e p u l s e w i d t h t h a t s h o u l d be used i n a m o d u l a t i o n scheme has been g i v e n . 78 7.2 Recommendations f o r f u r t h e r r e s e a r c h The second o r d e r e f f e c t s i n l a r g e i n d u c t i o n machines are p o o r l y c h a r a c t e r i z e d and f u r t h e r r e s e a r c h s h o u l d be made i n t h i s a r e a . T h i s use of the t r a p e z o i d a l r u l e s h o u l d be f u r t h e r i n v e s t i g a t e d as i t seems t o have b e t t e r p r o p e r t i e s than the g e n e r a l l i t e r a t u r e would suggest, p a r t i c u l a r l y w i t h r e s p e c t t o systems w i t h l a r g e time c o n s t a n t s such as the one d e s c r i b e d i n t h i s t h e s i s . The s i m u l a t i o n program d e v e l o p e d f o r t h i s t h e s i s has been shown t o be a c c u r a t e and e x t r e m e l y u s e f u l . I t would seem t o be a n a t u r a l e x t e n s i o n t o i n c l u d e the v a r i o u s s w i t c h i n g components o f the v o l t a g e s o u r c e s i n t h e s i m u l a t i o n program. 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B r a n i n , "Computer Methods of Network A n a l y s i s " , P r o c e e d i n g of th e IEEE V o l . 55, No. 11, Nov. 1967, pp. 1787-1801. [27] H.S. P a t e l , " T h y r i s t o r I n v e r t e r Harmonic E l i m i n a t i o n U s i n g O p t i m i z a t i o n T e chniques", U n i v e r s i t y o f M i s s o u r i - C o l u m b i a , Ph.D. T h e s i s , 1971. APPENDIX I - PROGRAM LISTING 82 C INDUCTION MOTOR MODEL B; BASED ON ROBERTSONS PAPER C REVISED JULY 12,1981 C THIS PROGRAM COMPUTES THE CURRENT AND SPEED OF C AN INDUCTION MOTOR WHEN FED BY A VOLTAGE SOURCE. C THE SOLUTION METHOD IS SUMMARIZED IN THE PAPER "A C DIGITAL MODEL FOR THREE-PHASE INDUCTION MACHINES" C BY STUART ROBERTSON AND KATTINGERI HEBBAR, VOL. C PAS-88, IEEE,NOVEMBER 1969, P.1624. THE OUTPUT IS IN THE C IN THE FORM FOR USE BY H.W.DOMMEL'S PLOTTING ROU-C TINE OR CAN BE OUTPUT IN THE FORM OF A TABLE OF C VALUES.(USUALLY TOO VOLUMINOUS TO BE OF REAL VALUE) C THE PROGRAM CAN MODEL THE VOLTAGE SOURCE AS A C VARIABLE FREQUENCY AND AMPLITUDE SINE WAVE INCLUDING C HARMONICS UP TO THE 22ND SO AS TO ACCURATELY MODEL C A VOLTAGE SOURCE INVERTER. C THE PLOTTING OUTPUT IS SENT TO UNIT 4 AND THE PARA-' C METERS REQUIRED BY THE, PROGRAM ARE WRITTEN ON FIVE C LINES AND READ FROM UNIT 5. THE FORMAT CODE IS F10.6 C FOR THE FIRST, F10.5 FOR THE SECOND AND F10.6 FOR THE THIRD. C THE FIVE LINES ARE READ IN AS FOLLOWS: C C VTYPE,PTYPE,RTYPE,ITYPE C DT,TMAX,VONE, VFIV, VSEV, FREQK, AMPK C VTHRE,VNINE,VELEV,VTHIR,VFIFT,VSEVT,VNINT,VTWON C RS, RR, LSS, LSM, LRR, LRM, MSR C POLEN, KF, JAY, KLOAD C C THE DEFINITION OF THESE PARAMETERS IS AS FOLLOWS: C VTYPE-VOLTAGE TYPE; THE TYPE OF VOLTAGE SOURCE TO BE REPRESENTED C 1-SINUSOIDAL VOLTAGE SOURCE WITH HARMONICS UP TO THE 21 ST C 2-CYCLOCONVERTER C 3-INVERTER WITH FIXED PULSE WIDTHS C 4-INVERTER WITH COSINE WAVE MODULATION C PTYPE-PROGRAM TYPE; WHETHER OR NOT THE TIME STEP IS REDUCED C 1-THE TIME STEP IS NOT AUTOMATICALLY REDUCED AT SWITCHING C 2-THE TIME STEP IS AUTOMATICALLY REDUCED AT INVERTER SWITHING C RTYPE-RESI STANCE TYPE;THE ROTOR RESISTANCE IS VARIED AT HIGH SLIP C 1-ROTOR RESISTANCE IS NOT VARIED C 2-ROTOR RESISTANCE IS VARIED C ITYPE-INDUCTANCE TYPE;INDUCTANCE IS VARIED AT VARIOUS FREQUENCIES C . 1-INDUCTANCE IS NOT VARIED C 2-INDUCTANCE IS VARIED WITH FREQUENCY C C C DT- TIME STEP IN UNITS OF SECONDS C TMAX- MAXIMUM TIME OF SIMULATION IN UNITS OF SECONDS C VONE- MAXIMUM AMPLITUDE OF FUNDAMENTAL VOLTAGE SINE-C WAVE IN VOLTS C VFIV- MAXIMUM AMPLITUDE OF FIFTH HARMONIC VOLTAGE IN C VOLTS C VSEV- MAXIMUM AMPLITUDE OF SEVENTH HARMONIC VOLTAGE C IN VOLTS C FREQK-FREQUENCY VARYING COEFFICIENT, IF EQUAL ZERO C THE FUNDAMENTAL VOLTAGE FREQUENCY IS SET AT C 60 HZ. FOR ALL OTHER VALUES THE FREQUENCY C AT ANY TIME IS EQUAL TO 60 HZ*FREQK*(THE TIME C IN SECS) C AMPK- AMPLITUDE VARYING COEFFICIENT, IF EQUAL ZERO C THE AMPLITUDE OF THE FUNDAMENTAL, FIFTH AND C SEVENTH VOLTAGE SINE WAVES ARE AS SET B Y VONE, 83 C VFIV AND VSEV OTHERWISE THEY ARE MULTITUDE* BY C AMPK*(THE TIME IN SECONDS) C C THE SECOND LINE IS FOR THE MOTOR PARAMETERS: C C RS- PER PHASE RESISTANCE OF THE STATOR C R R - PER PHASE EQUIVALENT RESISTANCE OF THE ROTOR C LSS-STATOR PER PHASE SELF INDUCTANCE IN HENRYS C LSM- STATOR TO STATOR PER PHASE MUTUAL INDUCTANC IN HENRYS C LRM-ROTOR PER PHASE EQUIVALENT MUTUAL INDUCTANCE IN HENRYS C LRM-ROTOR PER PHASE EQUIVALENT MUTUAL INDUCTANC IN HENRYS C MSR-MAXIMUM EQUIVALENT PER PHASE STATOR TO ROTOR C MUTUAL INDUCTANCE C C THE THIRD LINE IS: C C POLEN-THE NUMBER OF MOTOR POLES, USUALLY 2,4 OR 6 C KF- THE COEFFICIENT OF FRICTION THAT VARYS LINEARLY C WITH SPEED IN UNITS OF NEUTONS METERS/RADS PER SECOND C JAY- ROTOR INERTIA IN UNITS OF KILOGRAM*METERS*METER C LOAD- CONSTANT LOAD TORQUE IN NEUTON METERS C C DEFINE REAL DOUBLE PRECISION VARIABLES REAL* 8 TEMPI ( 5 )',TEMP2 ( 5) ,TEMP3(5) , TEMP 11 (5,5) , TEMP 1V3 ( 5, 5) , &VNEW(5),VOLD(5),INEW(5),IOLD(5),LOLD(5,5),LINV(5,5), &DINV(5,5),DTV(5,5),RES(5,5),HALFI( 5,5),HI ST(1, 9) ,RNEW(5,5), &TEMP12(5,5),IND(5,5),ISTAT(6),VSTAT(6),VSTATO(6), &DT,TMAX,RS,RR,THETA,THETAM,TEL,ROTORP, &THETM0,THETM1,DET1,DET2,TH500, &LSS, LSM,LRR,LRM,MSR,POLEN,KF,JAY, ' J &DCOND2,DCOND1,VFREQ,VAMP,FREQK,AMPK,VONE,VFIV,VSEV, &DSIN,DCOS,DABS,KLOAD,AVARY,FVARY,VAVAR,VBVAR,VCVAR, &VAVARA,VBVARA,VCVARA,VOUT,V(1441), &VTHRE,VNINE,VELEV,VTHIR,VFIFT,VSEVT,VNINT,VTWON, &ATIMEO,BTIMEO,CTIMEO,LRRF,RRF,SIGR,SIGL, &SIGA,SIGB,SIGC,VSIGA,VSIGB,VSIGC,VSINA,VSINB,VSINC, &SIGD,SIGE,SIGF,VSIGD,VSIGE,VSIGF,VSIND,VSINE,VSINF, &FREQ,FREQV,SIGTRI,SIGAB,SIGBC,SIGCA,SLOPE C C DEFINE REAL SINGLE PRECISION VARIABLES REAL STEP,LINK,SWITCH,TEST,VTESTA,VTESTB,VESTC, &ATIME,BTIME,CTIME,APUTIM,BPUTIM,CPUTIM C C DEFINE INTEGER VARIABLES INTEGER JEXP1,JEXP 2,VTYPE,PTYPE,RTYPE,ITYPE,ISWITH, &ITESTS,IPREV,I500,L500,IA500,IB500,IDINT ;, &TESTA1,TESTB1,TESTC1,TESTA2,TESTB2,TESTC2, &TESTA3,TESTB3,TESTC3,TNBLA1,TNBLB1,TNBLC1, • &TNBLA2,TNBLB2,TNBLC2,TNBLA3,TNBLB3,TNBLC3, &PNBLA1,PNBLB1,PNBLC1,PNBLA2,PNBLB2,PNBLC2, &PNBLA3,PNBLB3,PNBLC3,V1,V2,V3, &TNBLD1,TNBLE1,TNBLF1,TNBLD2,TNBLE2,TNBLF2, &TNBLD3,TNBLE3,TNBLF3,PNBLD1,PNBLE1,PNBLF1, &PNBLD2,PNBLE2,PNBLF2,PNBLD3,PNBLE3,PNBLF3 C C DEFINE SINGLE PRECISION MATRICES DIMENSION IPERM1(12),GRAPH(1002,11),IPERM2(12) C C DEFINE CHARACTER VARIABLES CHARACTER*8 XOUT(5)/* VPHA','VPHB','VPHC','TORQ','SPED'/ 84 CHARACTER*8 F(12)/'IA','SA', ' IB','SB','IC,'SC , Sc'IRA' ,'RA' ,'IRB' ,'RB' ,'IRC ,'RC'/ CHARACTER*68 TEXT/'INDUCTION MOTOR MODEL B'/ C READ IN PROGRAM CONTROL,PARAMETERS READ(5,10) VTYPE,PTYPE,RTYPE,ITYPE 10 FORMAT (411) READ(5,15) DT,TMAX,VONE,VFIV,VSEV,FREQK,AMPK 15 FORMAT (7F10.5) . READ(5,20) VTHRE,VNINE,VELEV,VTHIR,VFIFT,VSEVT,VNINT,VTWON 20 FORMAT (8F10.5) READ(5,25) RS,RR,LSS,LSM,LRR,LRM,MSR 25 FORMAT (7F10.5) READ(5,30) POLEN,KF,JAY,KLOAD,SPED 30 FORMAT (4F10.8,F10.2) C PRESET VARIABLES TO INITIAL VALUES TEE=0.0 STEP=0.0 LINK=0. L500=0 THETM0=0.0 THETM1=0.0 C THIS WILL GIVE AN INITIAL SPEED TO THE MOTOR THETM1=0.0 THETM0=SPED*DT/9.5493 GRAPH(1,1)=0.0 GRAPH(1,2)=0.0 GRAPH(1,3)= 0.0 GRAPH(1,4)=0.0 GRAPH(1,5)=0.0 GRAPH(1,6)=0.0 GRAPH(1,7)=0.0 GRAPH(1,8)=0.0 GRAPH(1,9)=0.0 GRAPH(1,10)=0. 0 GRAPH(1,11)=0 . 0 IAG=0 IBG=5 ICG=12 IDG=0 IEG=6 IFG=11 IGG=0 IONE=1 ITESTS=0 IPREV=0 DDT=DT VFREQ=1.0 VAMP=1.0 ATIME=0.0 BTIME=0.0 CTIME=0.0 TEST=0.0 SWITCH=0.0 ISWITH=0 RRF=RR LRRF=LRR DO 35 1 = 1 ,5 DO 35 J=1,5 35 RES(I,J)=0.0 DO 40 1=1,5 DO 40 J=1,5 40 DTV(I,J)=0.0 DO 45 1=1,5 DO 45 J=1,5 45 HALFI(I,J)=0.0 DO 50 1=1,5 50 H A L F I ( I , 1 ) = 0 . 5 IND(1 ,1)=3.0*(1.0+2.0*LSM/LSS) IND(2,1)=0.0 IND(3,1)=0.0 IND(4 , 1)=0.0 IND(5,1)=0.0 IND(1,2)=0.0 IND(2,2)=2.0*(LSS-LSM) IND(3,2)=LSS-LSM IND(4,2)=3.0*MSR IND(5,2)=1.5*MSR IND(1,3)=0.0 IND(2,3)=LSS-LSM IND(3,3)=2.0*(LSS-LSM) IND(4,3)=1.5*MSR IND(5,3)=3.0*MSR IND(1,4)=0.0 IND(2,4)=3.0*MSR IND(3,4)=1.5*MSR IND(4,4)=2.Q*(LRR-LRM) IND(5,4)=LRR-LRM IND(1,5)=0.0 IND(2,5)=1.5*MSR IND(3,5)=3.0*MSR IND(4,5)=LRR-LRM IND(5,5)=2.0*(LRR-LRM) RNEW(1,1)=3.0*RS RNEW(2,1)=0.0 RNEW(3,1)=0.0 RNEW(4,1)=0.0 RNEW(5,1)=0.0 RNEW(1,2)=0.0 RNEW(2,2)=2.0*RS RNEW(3,2)=RS RNEW(4,2)=0.0 RNEW(5,2)=8.162097*POLEN*MSR*SPED/60.0 RNEW(1,3)=0.0 RNEW(2,3)=RS RNEW(3,3)=2.0*RS RNEW(4,3)=-8.162097*POLEN*MSR*SPED/60.0 RNEW(5,3)=0.0 RNEW(1,4) = 0.0 RNEW(2,4)=0.0 • RNEW(3,4)=0.0 RNEW(4,4)=2.0*RR RNEW(5,4)=RR+ 5.4 413 98 *(LRR-LRM)*POLEN*SPED/60.0 RNEW(1,5)=0.0 ' . RNEW(2,5)=0.0 RNEW(3,5)=0.0 RNEW(4,5)=RR-5.441398*(LRR-LRM)*POLEN*SPED/60.0 RNEW(5,5)=2.0*RR DO 55 1=1,5 55 DTV(I,1)=DT THETA=0.0 VSTAT(1)=0. VSTAT(2)=180.*DCOS(-2.0943951) VSTAT(3) = 180.*DCOS(2.0943951 ) VSTAT(4)=0. VSTAT{5)=0. VSTAT(6)=0. VNEW(1)=VSTAT(1)+VSTAT(2)+VSTAT(3) VNEW(2)=VSTAT(1)-VSTAT(2) VNEW(3)=VSTAT(3)-VSTAT(2) VNEW(4)=0.0 . VNEW(5)=0.0 DO 60 1=1,5 60 INEW(I)=0.0 PRESET VALUES FOR THE CYCLOVONVERTER ROUTI TESTA 1=0 TESTB1=0 TESTC1=0 TESTA2=0 TESTB2=0 TESTC2=0 TESTA3=0 TESTB3=0 TESTC3=0 TNBLA1= 0 TNBLB1=0 TNBLC1=0 TNBLA2=0 TNBLB2=0 TNBLC2=0 TNBLA3=0 TNBLB3=0 TNBLC3=0 PNBLA1=0 PNBLB1=0 PNBLC1=0 PNBLA2=0 PNBLB2=0 PNBLC2=0 PNBLA3=0 PNBLB3=0 PNBLC3=0 TESTD1=0 TESTE1=0 TESTF1=0 TESTD2=0 TESTE2=0 TESTF2=0 TESTD3=0 TESTE3=0 TESTF3=0 TNBLD1= 0 TNBLE1=0 TNBLF1=0 TNBLD2=0 TNBLE2=0 TNBLF2=0 TNBLD3=0 TNBLE3=0 TNBLF3=0 PNBLD1=0 87 PNBLE1=0 PNBLF1=0 PNBLD2=0 PNBLE2=0 PNBLF2=0 PNBLD3=0 PNBLE3=0 PNBLF3=0 V1=0 V2=0 V3=0 C PRESET VALUES FOR TYPE 4 VOLTAGE SOURCE SIGTRI=0.0 SLOPE=112. C SLOPE=1680@60HZ,SLOPE=112§4HZ C LOAD IN VALUES FOR RNEW WHICH THEN BECOMES ROLD RNEW(1,1)=3.0*RS RNEW(5,2)=8.162097*POLEN*MSR*SPED/60.0 ' RNEW(4,3)=-8.162097*POLEN*MSR*SPED/60.0 RNEW(5,4)=RR+5.441398*(LRR-LRM)*POLEN*SPED/60.0 RNEW(4,5)=RR-5.441398*(LRR-LRM)*POLEN*SPED/60.0 CALL INV (5,5,IND,IPERM1,5,LINV,DET1,JEXP1,DCOND1) C ***START OF MAIN PROGRAM*** 70 STEP=STEP+1.0 ISTEP=STEP JSTEP=ISTEP+1 TEE=TEE+DT GO TO 75 C IF SWITCHING HAS OCCURRED; DIVIDE DT BY TEN AND SET C RELATED PARAMATERS 80 SWITCH=1.0 TEE=TEE-DT DT=DT/10.0 DO 88 1=1,5 88 DTV(I,1)=DT 85 TEE=TEE+DT ISWITH=ISWITH+1 75 CALL DGCOPY (VNEW,VOLD,5,1,5,5) CALL DGCOPY (VSTAT,VSTATO,5,1,5,5) CALL DGCOPY (INEW,IOLD,5,1,5,5) IF(VTYPE.EQ.1) GO TO 100 IF(VTYPE.EQ.2) GO TO 200 IF(VTYPE.EQ.3) GO TO 300 IF(VTYPE.EQ.4) GO TO 400 IF(VTYPE.EQ.5) GO TO 500 C DETERMINE THE VALUES OF VOLTAGES FOR THE SOURCE C C CALCULATE INSTANTANEOUS FREQUENCY AND VOLTAGE OF HARMONIC SOURCE C 100 IF(FREQK.NE.0.0) VFREQ=TEE*FREQK IF(AMPK.NE.O.O) VAMP=TEE*AMPK C REMOVE THE ABOVE AFTER SIMULATION C THIS ROUTINE SIMULATES A VOLTAGE SOURCE WITH HARMONIC C FREQUENCIES UP TO THE NINETEENTH. I F FREQK AND AMPK ARE C EQUAL TO ZERO THAN THE FREQUENCY IS A CONSTANT 60 HZ C AND THE VOLTAGE AMPLITUDES ARE AS SET BY VONE ETC. . C IF VAMPK AND FREQK ARE NOT EQUAL TO ZERO THAN THE VOLTAGE C AND FREQUENCY ARE RAMPED UP. VSTAT(1)=VAMP*(VONE*DCOS(VFREQ*TEE*376.99112) &+VFIV*DCOS(5.0*VFREQ*TEE*376.99112) i 88 S>+VSEV*DCOS(7.0*VFREQ*TEE*376.99112) S.+VTHRE*DCOS(3.0*VFREQ*TEE*376.99112) &+VNINE*DCOS(9.0*VFREQ*TEE*376.99l12) / &+VELEV*DCOS(11.0*VFREQ*TEE*376.99l12) ' S.+VTHI R*DCOS ( 13.0*VFREQ*TEE*376.991 12) &+VFIFT*DCOS(15.0*VFREQ*TEE*376.99i12) &+VSEVT*DCOS(17.0*VFREQ*TEE*376.99112) S,+VNINT*DCOS( 19.0*VFREQ*TEE*376.991 12) &+VTWON*DCOS(21.0 *VFREQ*TEE* 376.991 1 2 ) ) VSTAT(2)=VAMP*(VONE*DCOS(VFREQ*T£E*376.99112-2.094395) S+VFIV*DCOS(5.0*(VFREQ*TEE*376.991 12-2.094395) ) S,+VSEV*DCOS(7.0*(VFREQ*TEE*376 .991 12-2.094395) ) &+VTHRE*DCOS(3.0*(VFREQ*TEE*376.99112-2.094395)) &+VNINE*DCOS(9.0*(VFREQ*TEE*376.991l2-2.094395)) &+VELEV*DCOS(11.0*(VFREQ*TEE*376.9S112-2.094395)) &+VTHIR*DCOS(13.0*(VFREQ*TEE*376.9S112-2.094395)) &+VFIFT*DCOS(15.0*(VFREQ*TEE*376 . 99112-2.094395)) &+VSEVT*DCOS(17.0*(VFREQ*TEE*376.99112-2.094395)) £+VNINT*DCOS(19.0*(VFREQ*TEE*376.99112-2.094395)) &+VTWON*DCOS(21.0*(VFREQ*TEE*376.99112-2.094395))) VSTAT(3)=VAMP*(VONE*DCOS(VFREQ*TEE*376.99112+2.094395) &+VFIV*DCOS(5.0*(VFREQ*TEE*376 . 991 1 2+2. 094395)) &+VSEV*DCOS(7.0*(VFREQ*TEE*376.991 12+2.094395)) &+VTHRE*DCOS(3.0*(VFREQ*TEE*376.99l12+2.094395)) &+VNINE*DCOS(9.0*(VFREQ*TEE*376.99112+2.094395)) &+VELEV*DCOS(11.0*(VFREQ*TEE*376.99112+2.094395)) &+VTHIR*DCOS(13.0*(VFREQ*TEE*376.99l12+2.094395)) &+VFIFT*DCOS(15.0*(VFREQ*TEE*376.99112+2.094 395)) t+VSEVT*DCOS(17.0*(VFREQ*TEE*376.99112+2.094395)) &+VNINT*DCOS(19.0*(VFREQ*TEE*376.99112+2.094395)) &+VTWON*DCOS(21.0*(VFREQ*TEE*376.99112+2.094395))) VLINAB=VSTAT(2)-VSTAT( 1 ) VLINBC=VSTAT(3)-VSTAT(2) VLINCA=VSTAT(1)-VSTAT(3) VSTAT(1)=.3333*(VLINCA-VLINAB) VSTAT(2)=.3333*(VLINAB-VLINBC) VSTAT(3)=.3333*(VLINBC-VLINCA) C PRINT, VSTAT(1),VSTAT(2),VSTAT(3) C THIS MANIPULATION OF PHASE VOLTAGES DISCONECTS THE NEUTRAL C FROM GROUND VSTAT(1)=.3333333*(2.0*VSTAT(1)-VSTAT(2)-VSTAT(3)) C PRINT, VSTAT(1) VSTAT(2)=.333333*(2.0*VSTAT(2)-VSTAT(1)-VSTAT(3)) VSTAT(3)=.333333*(2.0*VSTAT(3)-VSTAT*1)-VSTAT(2)) GO TO 16 C THIS ROUTINE WILL SIMULATE A CYCLOCONVERTER 200 CONTINUE FREQ=377.0 FREQV=94.25 VSINA=360.0*DSIN(FREQ*TEE) VSINB=360.0*DSIN(FREQ*TEE~2.094395) . VSINC=360.0*DSIN(FREQ*TEE-4.18879) VSIND=360.0*DSIN(FREQ*TEE-1.047197) . . : VSINE=360.0*DSIN(FREQ*TEE-3.141593) . VSINF=360.0*DSIN(FREQ*TEE-5.235988) VSIGA=DCOS(FREQ*TEE-1.0471975) VSIGB=DCOS(FREQ*TEE-1.0471975-2.094395) VSIGC=DCOS(FREQ*TEE-1.0471975-4.18879) VSIGD=DCOS(FREQ*TEE-1.0471975-1.047197) VSIGE=DCOS(FREQ*TEE-1.0471975-3.141593) VSIGF=DCOS(FREQ*TEE-1.0471975-5.235988) SIGA=0.5*DSIN(FREQV*TEE) SIGB=0.5*DSIN(FREQV*TEE-2.094395) SIGC=0.5*DSIN(FREQV*TEE-4.18879) SIGD=.5*DSIN(FREQV*TEE-1.047197) SIGE=.5*DSIN(FREQV*TEE-3.141593) SIGF=.5*DSIN(FREQV*TEE~5.235988) IF(SIGA.GT.VSIGA) TNBLA1=0 IF(SIGA.GT.VSIGB) TNBLB1=0 IF(SIGA.GT.VSIGC) TNBLC1=0 IF(SIGA.GT.VSIGD) TNBLD1=0 IF(SIGA.GT.VSIGE) TNBLE1=0 IF(SIGA.GT.VSIGF) TNBLF1=0 IF(SIGA.LE.VSIGA) TNBLA1= IF(SIGA.LE.VSIGB) TNBLB1= IF(SIGA.LE.VSIGC) TNBLC1= IF(SIGA.LE.VSIGD) TNBLD1= IF(SIGA.LE.VSIGE) TNBLE1= IF(SIGA.LE.VSIGF) TNBLF1= TESTA1=TNBLA1-PNBLA 1 TESTB1=TNBLB1-PNBLB1 TESTC1=TNBLC1-PNBLC1 TESTD1=TNBLD1-PNBLD1 TESTE 1=TNBLE1-PNBLE1 TESTF1=TNBLF1-PNBLF 1 ) IF(TESTA 1.EQ.-IF(TESTB1. EQ. -IF(TESTC1.EQ.-IF(TESTD1.EQ.-IF(TESTE 1.EQ. -IF(TESTF1.EQ.-IF(V1.EQ.1) IF(V1.EQ.2) IF(V1.EQ.3) IF(V1.EQ.4) IF(V1.EQ.5) IF(V1.EQ.6) IF(SIGB.GT.VSIGA) IF(SIGB.GT.VSIGB) IF(SIGB.GT.VSIGC) IF(SIGB.GT.VSIGD) IF(SIGB.GT.VSIGE) IF(SIGB.GT.VSIGF) IF(SIGB.LE.VSIGA) IF(SIGB.LE.VSIGB) IF(SIGB.LE.VSIGC) IF(SIGB.LE.VSIGD) IF(SIGB.LE.VSIGE) IF(SIGB.LE.VSIGF) TESTA2=TNBLA2-PNBLA2 TESTB2=TNBLB2-PNBLB2 TESTC2=TNBLC2-PNBLC2 TESTD2=TNBLD2-PNBLD2 TESTE2=TNBLE2-PNBLE2 TESTF2=TNBLF2-PNBLF2 V1 = 1 V1=2 V1=3 V1=4 V1=5 V1=6 VSTAT(1)=VSINA VSTAT(1)=VSINB VSTAT(1)=VSINC VSTAT(1)=VSIND VSTAT(1)=VSINE VSTAT(1)=VSINF TNBLA2=0 TNBLB2=0 TNBLC2=0 TNBLD2=0 TNBLE2=0 TNBLF2=0 TNBLA2=1 TNBLB2=1 TNBLC2=1 TNBLD2=1 TNBLE2=1 TNBLF2=1 IF(TESTA2.EQ . - 1 IF(TESTB2.EQ . - 1 IF{TESTC2.EQ.-IF(TESTD2.EQ.-IF(TESTE2.EQ.-) V2 = V2 = V2 = V2 = V2 = 90 IF(TESTF2.EQ.-1) V2=6 IF(V2.EQ.1) VSTAT(2)=VSINA IF(V2.EQ.2) VSTAT(2)=VSINB I F(V2.EQ.3) VSTAT(2)=VSINC IF(V2.EQ.4) VSTAT(2)=VSIND IF(V2.EQ.5) VSTAT(2)=VSINE IF(V2.EQ.6) VSTAT(2)=VSINF IF(SIGC.GT.VSIGA) TNBLA3=0. IF(SIGC.GT.VSIGB) TNBLB3=0 IF(SIGC.GT.VSIGC) TNBLC3=0 IF(SIGC.GT.VSIGD) TNBLD3=0 IF(SIGC.GT.VSIGE) TNBLE3=0 IF(SIGC.GT.VSIGF) TNBLF3=0 IF(SIGC.LE.VSIGA) TNBLA3=1 IF(SIGC.LE.VSIGB) TNBLB3=1 IF(SIGC.LE.VSIGC) TNBLC3=1 I F(SIGC.LE.VSIGD) TNBLD3=1 IF(SIGC.LE.VSIGE) TNBLE3=1 IF(SIGC.LE.VSIGF) TNBLF3=1 TESTA3=TNBLA3-PNBLA3 TESTB3=TNBLB3-PNBLB3 TESTC3=TNBLC3-PNBLC3 TESTD3=TNBLD3-PNBLD3 TESTE3=TNBLE3-PNBLE3 TESTF3=TNBLF3-PNBLF3 IF(TESTA3.EQ.-1) V3=1 IF(TESTB3.EQ.-1) V3=2 IF(TESTC3.EQ.-1) V3=3 IF(TESTD3.EQ.-1) V3=4 IF(TESTE3.EQ.-1) V3=5 IF(TESTF3.EQ.-1) V3=6 IF(V3.EQ.1) VSTAT(3)=VSINA I F(V3.EQ.2) VSTAT(3)=VSINB IF(V3.EQ.3) VSTAT(3)=VSINC IF(V3.EQ.4) VSTAT(3)=VSIND IF(V3.EQ.5) VSTAT(3)=VSINE IF(V3.EQ.6) VSTAT(3)=VSINF PNBLA1=TNBLA1 PNBLB1=TNBLB1 PNBLC1=TNBLC1 PNBLD1=TNBLD1 PNBLE1=TNBLE1 PNBLF1=TNBLF1 PNBLA2=TNBLA2 PNBLB2=TNBLB2 PNBLC2=TNBLC2 PNBLD2=TNBLD2 PNBLE2=TNBLE2 PNBLF2=TNBLF2 PNBLA3=TNBLA3 PNBLB3=TNBLB3 PNBLC3=TNBLC3 PNBLD3=TNBLD3 PNBLE3=TNBLE3 PNBLF3=TNBLF3 GO TO 16 C VARIABLE FREQUENCY WITH ROTOR FEEDBACK ROUTINE C THIS ROUTINE SIMULATES A VOLTAGE SOURCE INVERTER WITH.CONSTANT C VOLTAGE AND FIXED PULSE WIDTHS. THE FREQUENCY IS VARIED C ACCORDING TO VREQK IF NOT EQUAL TO ZERO BUT USES ROTOR FEEDBACK 91 C IF VFREQK IS EQUAL TQ ZERO. 300 FVARY=18.8495+DABS(376.991 1 2*SPED*POLEN/2.0/3600.) IF (FREQK.NE.0.0) FVARY=VFREQ* 3 7 6 . 9:91 AVARY=5.0+DABS(311.*SPED*POLEN/2.0/3600.) FVARY=376.99112 VAVAR=DCOS(FVARY*TEE) VBVAR=DCOS(FVARY*TEE-2.09439) VCVAR=DCOS(FVARY*TEE+2.09439) VAVARA=DABS(VAVAR) VBVARA=DABS(VBVAR) VCVARA=DABS(VCVAR) IF(VAVARA.LE.0.5) VSTAT(1)=0.0 IF(VBVARA.LE.0.5) VSTAT(2)=0.0 IF(VCVARA.LE.0.5) VSTAT(3)=0.0 IF(VAVAR.GT.0.5) VSTAT(1)=50.0 IF(VBVAR.GT.0.5) VSTAT(2)=50.0 IF(VCVAR.GT.0.5) VSTAT(3)=50.0 IF(VAVAR.LT.-0.5) VSTAT(1)=-50.0 IF(VBVAR.LT.-0.5) VSTAT(2)=-50.0 IF(VCVAR.LT.-0.5) VSTAT(3)=-50.0 GO TO 16 C THIS PRODUCES AN ENABLE FUNCTION TO GIVE CONSTANT TIME PULSES C IF FREQK NOT EQUAL 0.0 THEN FREQ I S RAMPED UP AS DETERMINED C BY THE VALUE OF FREQK C SET TIMER FOR EACH PHASE C THIS MAKES SURE THAT THE TIMER DOES NOT GET SET RESET DURING C DURING A TIMING STEP CHANGE IF(VSTAT(1).EQ.0.0) ATIME=TEE IF(VSTAT(2).EQ.0.0) BTIME=TEE IF(VSTAT(3).EQ.0.0) CTIME=TEE C IF(ATIME.GT.TEE) ATIME=ATIMEO C IF(BTIME.GT.TEE) BTIME=BTIMEO C IF(CTIME.GT.TEE) CTIME=CTIMEO C ATIMEO=ATIME C BTIMEO=BTIME C CTIMEO=CTIME C THE LENGTH OF TIME THE PULSE HAS BEEN ON APUTIM=TEE-ATIME BPUTIM=TEE-BTIME CPUTIM=TEE-CTIME C THIS MAKES THE FIRST PULSE ONE HALF THE USUAL FOR A PULSE C THAT STARTED BEFORE THE SIMULATION PULSEL=.0012 IF(TEE.EQ.APUTIM) PULSEL=.0006 C IF THE PULSE HAS BEEN ON LONG ENOUGH THEN TURN IT OFF IF(APUTIM.GT.PULSED VSTAT(1)=0.0 IF(BPUTIM.GT.PULSEL) VSTAT(2)=0.0 IF(CPUTIM.GT..0012) VSTAT(3)=0.0 IF(APUTIM.LT.O.O) VSTAT(1)=0.0 IF(BPUTIM.LT.O.O) VSTAT(2)=0.0 IF(CPUTIM.LT.O.O) VSTAT(3)=0.0 GO TO 861 C THIS ROUTINE WILL SIMULATE A COSINE CROSSING MODULATED INVERTER 400 CONTINUE FREQ=25.1327 IF(SIGTRI.GT.1.0) SLOPE=-SLOPE IF(SIGTRI.LT.-1.0) SLOPE=-SLOPE SIGTRI=SIGTRI+SLOPE*DT SIGAB=.066667*DCOS(FREQ*TEE-.5236} SIGBC=.066667*DCOS(FREQ*TEE-2.6180) SIGCA=.066667*DCOS(FREQ*TEE-4.71239) PRINT,SIGTRI,SIGAB LINEAB=117.6 LINEBC=117.6 LINECA=117.6 IF(SIGTRI.GT.SIGAB) LINEAB=-117.6 IF(SIGTRI.GT.SIGBC) LINEBC=-117.6 IF(SIGTRI.GT.SIGCA) LINECA=-117.6 VSTAT(1)=.3333*(LINECA-LINEAB) VSTAT(2)=.3333*(LINEAB-LINEBC) VSTAT(3)=.3333*(LINEBC-LINECA) VSTATO(1)=VSTAT(1) VSTATO(2)=VSTAT(2) VSTATO(3)=VSTAT(3) VSTAT(1)=.333333*(2.0*VSTATO(1)-VSTATO(2)-VSTATO(3)) VSTAT(2)=.333333*(2.0*VSTATO(2)-VSTATO(1)-VSTATO(3)) VSTAT(3)=.333333*(2.0*VSTATO(3)-VSTATO(1)-VSTATO(2)) GO TO 861 THIS ROUTINE WILL SIMULATE A WAVEFORM THAT CAN BE SPECIFIED BY SWITCHING ANGLES 500 IF(L500.EQ.1) GO TO 550 1500=0 510 1500=1500+1 VOUT=311.0 IF(I500.GT.126) VOUT=-311.0 IF(I500.GT.126) VOUT=311.0 IF(I500.GT.234) VOUT=-311.0 IF(I500.GT.234) VOUT=311.0 V(I500)=VOUT IF(I500.LT.361) GO TO 510 DO 520 1500=361,720 520 V(I500)=V(721-I500) DO 540 1500=721,1440 540 V(I500)=-V(1441-1500) L500=1 550 TH500=1.0*377.0*TEE IA500=IDINT(TH500*1440./6.28318) IB500=IDINT{(TH500+4.1887)*1440./6.28318) IC500=IDINT((TH500+2.0944)*1440./6.28318) IF (IA500.LT.1) IA500=1 561 IF(IA500.LT.1441) GO TO 562 IA500=IA500-1440 GO TO 561 562 IF(IB500.LT.1441) GO TO 563 IB500=IB500-1440 GO TO 562 563 IF(IC500.LT.1441) GO TO 570 IC500=IC500-1440 GO TO 563 Av 570 VSTATO(1)=V(IA500) VSTATO(2)=V(IB500) VSTATO(3)=V(IC500) VSTAT(1)=.333333*(2.0*VSTATO(1)-VSTATO(2)-VSTATO(3)) VSTAT(2)=.333333*(2.0*VSTATO(2)-VSTATO(1)-VSTATO(3)) VSTAT(3)=.333333*(2.0*VSTATO(3)-VSTATO(1)-VSTATO(2)) GO TO 16 END OF FIXED VOLTAGE AND PULSE LENGTH ROUTINE END OF VOLTAGE DETERMINATION ROUTINE 93 C C FIND OUT IF SWITCHING HAS OCCURRED 861 VTESTA=VSTATO(1)-VSTAT(1) VTESTB=VSTATO(2)-VSTAT(2) • VTESTC=VSTATO(3)-VSTAT(3) TEST=0.0 IF(VTESTA.NE.O.O) TEST=1.0 IF(VTESTB.NE.O.O) TEST=1.0 IF(VTESTC.NE.O.O) TEST=1.0 IF(PTYPE.EQ.1) GO TO 16 IF(SWITCH.EQ.1 . 0 ) TEST=0.0 IF(TEST.EQ.1.0) GO TO 80 16 VNEW(1)=VSTAT(1)+VSTAT(2)+VSTAT(3) VNEW(2)=VSTAT(1)-VSTAT(2) VNEW(3)=VSTAT(3)-VSTAT(2) C THIS ROUTINE ADJUSTS THE VALUES OF THE ROTOR RESISTANCE C TO ACCOUNT FOR SKIN EFFECT IF (RTYPE.EQ.1) GO TO 90 SIGR=1.17 . SIGL=.88 RR=RRF*(1.0-((SPED*POLEN/120./VFREQ/376.991)**2)*( 1 .0-1 .C/SIGR) ) RNEW(4,4)=2.0*RR RNEW(5,5)=2.0*RR 90 RNEW(1,1)=3.0*RS RNEW(5,2)=8.162097*POLEN*MSR*SPED/60.0 RNEW(4,3)=-8.162097*POLEN*MSR*SPED/60.0 N RNEW(5,4)=RR+5.441398*(LRR-LRM)*POLEN*SPED/60.0 RNEW(4,5)=RR-5.441398*(LRR-LRM)*POLEN*SPED/60.0 C THIS ROUTINE ADJUSTS THE LEAKAGE INDUCTANCE TO ACCOUNT FOR C THE SKIN EFFECT IF(ITYPE.EQ.1) GO TO 95 SIGL=.88 LRR=LRRF*(1 . 0 -(SPED*P0LEN/120./VFREQ/376.991)*(1.0-1.0/SIGL)) IND(1,1)=3.0*(1.0+2.0*LSM/LSS) IND(2,2)=2.0*(LSS-LSM) IND(3,2)=LSS-LSM IND(4,2)=3.0*MSR IND(5,2)=1.5*MSR IND(2,3)=LSS-LSM IND(3,3)=2.0*(LSS-LSM) IND(4,3)=1.5*MSR IND(5,3)=3.0*MSR IND(2,4)=3.0*MSR IND(3,4)=1.5*MSR IND(4,4)=2.0*(LRR-LRM) IND(5,4)=LRR-LRM IND(2,5)=1.5*MSR . . IND(3,5)=3.0*MSR IND(4,5)=LRR-LRM IND(5,5)=2.0*(LRR-LRM) CALL INV (5,5,IND,IPERM1,5,LINV,DET1,JEXP1,DCOND1) C C START OF CURRENT CALCULATION ROUTINE C PRINT, 'ISTEP=',I STEP,'VSTAT( 1 ) = ' , VSTAT(1),'TEE=',TEE 95 CALL DGADD (VNEW,VOLD,TEMP1,5,1,5,5,5) CALL DGMATV (HALFI,TEMP 1,TEMP2,5,5,5) CALL DGMATV (RNEW,IOLD,TEMP3,5,5,5) CALL DGSUB (TEMP2,TEMP3,TEMP 1,5,1,5,5,5) CALL DGMATV (LINV,TEMP 1,TEMP2,5,5,5) CALL DGMATV (DTV,TEMP2,TEMP3,5,5,5) 94 CALL DGADD (TEMP3,IOLD,INEW,5,1,5,5,5) CALL DGCOPY (INEW,IOLD,5,1,5,5) ISTAT(1)=INEW(1)+INEW(2) ISTAT(2)=INEW(1)-INEW(2)-INEW(3) ISTAT(3)=INEW(1)+INEW(3) ISTAT(4)=1.1547*(DCOS(THETA+.52359)*INEW(4)-DSIN(THETA)*INEW(5)) ISTAT(6)=1.1547*(DSIN(THETA)*INEW(4)+DCOS(THETA-.52359)*INEW(5)) ISTAT(5) = 0.0-1STAT(4)-I STAT(6) C C CALCULATE TORQUE DEVELOPED TEL=-MSR*(((ISTAT(1)*ISTAT(4)+ 1STAT(2)*ISTAT(5)+1 STAT(3)*ISTAT + (6))* +DSIN (THETA) ) + ( (ISTAT( 1 ) *I STAT( 5 ) +ISTAT( 2 ) *I STAT ( 6 ).+I STAT ( 3 ) +*ISTAT(4))*DSIN(THETA ++2.0943951)) + ( ( I STAT(1)*ISTAT(6)+1 STAT(2)*ISTAT(4)+1 STAT(3)*IS +TAT(5))* +DSIN(THETA-2.0943951)))* POLEN/2.0 TEL=1.299*MSR*POLEN*(INEW(2)*INEW(5)-INEW(3)*INEW(4)) C THIS ROUTINE ADJUSTS THE ROTOR SPEED TO THE CORRECT VALUE C WHEN THE PROGRAM IS ADJUSTING THE STEP VALUE ROTORP=THETM0-THETM1 ITESTS=IPREV+ISWITH IF(ISWITH.GT.1) ITESTS=0 IF(ITESTS.EQ.O) ROTORP=ROTORP IF(ITESTS.EQ.1) ROTORP=ROTORP/10.0 IF(ITESTS.EQ.-I) ROTORP=ROTORP*10.0 IF(ISWITH.EQ.IO) IPREV=-1 IFdSWITH.NE. 1 0) IPREV=0 C . C NEW ROTOR POSITION CALCULATION THETAM=ROTORP+THETM0+(1.0*DT*DT*((TEL-((ROTORP)/DT)* &KF)-KLOAD)/JAY) C PRINT, 'ISTEP=',ISTEP,'IPREV=',IPREV,'ITESTS=',ITESTS SPED=(THETAM-THETMO)/DT*9.54S3 THETA=POLEN * THETAM/2.0 THETM1=THETM0 THETM0=THETAM ' v" IF(PTYPE.EQ.1) GO TO 900 IF(ISWITH.EQ.10) GO TO 840 '/ IF(SWITCH.EQ.1.0) GO TO 85 840 IF(ISWITH.EQ.10) DT=DT*10.0 IF(ISWITH.EQ.10)SWITCH= 0.0 DO 860 1=1,5 860 DTV(I,I)=DT IF(ISWITH.EQ.IO) GO TO 900 900 GRAPH(JSTEP,1)=VSTAT(1) GRAPH(JSTEP,2)=VSTAT(2) GRAPH(JSTEP,3)=VSTAT(3) GRAPH(JSTEP,4)=TEL GRAPH(JSTEP,5)=SPED GRAPH(JSTEP,6)=ISTAT(1) GRAPH(JSTEP,7) = I STAT(2) GRAPH(JSTEP,8)=ISTAT(3) GRAPH(JSTEP,9)=I STAT(4) GRAPH(JSTEP,10)=ISTAT(5) GRAPH(JSTEP,11)=ISTAT(6) IFdSWITH.EQ. 10) PRINT, ' REDUCED STEP AT I STEP= ' , I STEP IFdSWITH.EQ.10) ISWITH=0.0 IF (TEE.LT.TMAX) GO TO 70 C ***END OF MAIN PROGRAM*** 95 C PRINT 910,DT,TMAX C 910 FORMAT(' ','DT=',F10.8,'TMAX=*,5X,F10.8) PRINT 920,RS,RR,LSS,LSM,LRRRLRM,MSR 920 FORMAT(' ','RS=',F8.4,2X,'RR=',F8.4R2X,'LSS=*,F8.4,2X,'LSM=' +,F8.3,2X,'LRR=',F8.4,2X,'LRM=',F8.3,2X,'MSR=',F8.4) C PRINT 930 C 930 FORMATC ',1X,'TIME STEP',4X,'CURRENT:PHASE A',12X,'TORQUE', C +12X,'ROTOR ANGLE RAD.') C DO 940 1 = 1,1 STEP C 940 PRINT 950, HI ST (I , 1 ) ,HI ST( I , 2) , HI ST (I,8 ) ,HI ST (I , 9 ) C 950 FORMAT(' ',3X,F4.0,5X,E18.11,5X,E18.11,5X,E18.11) c: C ROUTINE TO OUTPUT DATA FOR PLOTTING WRITE(4) TEXT,IAG WRITE(4) IBG,DDT,ISTEP,XOUT PRINT, "THIS IS I STEP' ,I STEP WRITE(4) ICG,IDG,IEG,F DO 960 K=1,JSTEP KG=K-1 C PRINT, GRAPH(K,6) 960 WRITE(4) IFG,IGG,KG,(GRAPH(K,J),J=1,IFG) WRITEU) IONE,IONE,IONE, (GRAPH( 1 ,J) ,J=1 ,IFG) C PRINT, LINV STOP END $DATA 

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