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UBC Theses and Dissertations

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

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• • • y j CAT. m. k£*f>±-. i.m Ap.L*X>. THEOHY OF THE PERFORMANCE OF THE INDUCTION TWOT'OT? UNDER UNBALANCED CONDITIONS by Edward 0 , Lunn A T h e s i s s u b m i t t e d for t h e Degree of MASTER OF APPLIED SCIENCE i n t h e Department of ELECTRICAL ENGINEERING The U n i v e r s i t y of B r i t i s h e & l u a b i a A p r i l , 1933 e » • TABLE OP CONTENTS I 1 . I n t r o d u c t o r y . 2 . The Magnetomotive F o r c e o f a S y m m e t r i c a l P o l y p h a s e W i n d i n g . I I 3. I n d u c t i o n s M o t o r R e a c t i o n s and M a g n e t i c C o u p l i n g . 4. F r e q u e n c y o f R e a c t i o n s i n I n d u c t i o n M o t o r s . I l l 3. G e n e r a l T h e o r y f o r U n b a l a n c e d C o n d i t i o n s . 6. B a l a n c e d A p p l i e d V o l t a g e , A l l Impedances B a l a n c e d . 7. U n b a l a n c e d A p p l i e d V o l t a g e , A l l Impedances B a l a n c e d . 8. U n b a l a n c e d A p p l i e d V o l t a g e , Primai-y Impedances U n B a l a n c e d , S e c o n d a r y Impedances B a l a n c e d . 9. U n b a l a n c e d A p p l i e d V o l t a g e , P r i m a r y Impedances B a l a n c e d , x S e c o n d a r y Impedances U n b a l a n c e d . IV^ ' TO. B l o c k e d Torque T e s t s . 11. C a l c u l a t i o n s f o r S l i p - T o r q u e Curves f o r B a l a n c e d M o t o r H a v i n g U n b a l a n c e d A p p l i e d V o l t a g e . 12. C a l c u a t i o n H S f o r S l i p - T o r q u e C u r v e s , U n b a l a n c e d P r i m a r y B a l a n c e d A p p l i e d V o l t a g e . 13. C o n c l u s i o n . A p p e n d i x . 1 THEORY OF THE PERFORMANCE 07 THE INDUC T I OH" MO TOR  UNDER UNBALANCED CONDITIONS . 1 . 1. I n t r o d u c t o r y . The g e n e r a l 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 o p e r a t i n g u n d e r "balanced c o n d i t i o n s a r e q u i t e g e n e r a l l y known. I t i s a l s o known t h a t any u n b a l a n c i n g o f t h e a p p l i e d v o l t a g e r e s u l t s i n a r e d u c t i o n o f maximum t o r q u e a n d o u t p u t , and i n a r e d u c t i o n o f e f f i c i e n c y . T h i s r e d u c t i o n a l s o o c c u r s i f t h e c o n s t a n t s o f t h e e x t e r n a l c i r c u i t s o f t h e m o t o r , s u c h as s t a r t i n g and c o n t r o l c i r c u i t s , a r e u n b a l a n c e d . I n t h i s l a t t e r c a s e t h e u n b a l a n c i n g o f m o t o r t e r m i n a l v o l t a g e i s a f u n c t i o n o f t h e c u r r e n t o r s l i p . I t i s o u r p u r p o s e t o i n v e s t i g a t e t h e s e phenomena i n a q u a n t -i t a t i v e manner. F o r t h i s , t h e method o f s y m m e t r i c a l com-p o n e n t s i s v e r y e f f e c t i v e , and a p r o p e r u n d e r s t a n d i n g o f the a p p l i c a t i o n o f t h i s t h e o r y i s d e s i r a b l e . A b r i e f summary o f t h e more i m p o r t a n t p r i n c i p l e s o f t h i s method o f i o l v i n g u n b a l a n c e d p o l y p h a s e networks i s i n c l u d e d i n a n a p p e n d i x . • 2. The Magnetomotive F o r c e o f a S y m m e t r i c a l P o l y p h a s e  W i n d i n g . P r e f a t o r y t o a g e n e r a l a n a l y s i s o f u n b a l a n c e d c o n d i t i o n s , we s h a l l c o n s i d e r t h e magnetomotive f o r c e p r o d u c e d by c u r r e n t s f l o w i n g i n a t h r e e - p h a s e , s y m m e t r i c a l , d i s t r i b u t e d w i n d i n g . The development o f a s i m p l e t h r e e -phase w i n d i n g i s shown i n F i g . 1, and below i t the c u r r e n t v a l u e s i n t h e c o n d u c t o r s a r e g r a p h i c a l l y e x p r e s s e d , f o r t h e i n s t a n t c o n s i d e r e d . Below t h i s , t h r e e component m . m . f . t r a p e z o i d a l c u r v e s show the m . m . f . e x e r t e d by each J h a e e . The r e s u l t a n t m . m . f . i s i n d i c a t e d i n r e d . The l o w e s t g r a p h CURRENT o W . M . F. o M.M.F. F i g . t . i n d i c a t e s t h e m . m . f . o f a s i n g l e p h a s e . I t i s t r a p e z i o d a l i n f o r m , and a t the i n s t a n t c o n s i d e r e d , has a maximum A a m p l i t u d e e q u a l to F a . The F o u r i e r a n a l y s i s o f the t r a p -e z i o d g i v e s t h e m . m . f . , f a , e x e r t e d by i t a t the a n g l e 9 from t h e a r b i t r a r i l y c h o s e n o r i g i n , w h i c h i n t h i s case i s the c e n t e r o f phase 1 : f - ±3- f a i n a s i n 9 + I s i n 3a s i n 3 9 + i s i n ^ a s i n ^ e + • • • • ] O ) I t w i l l be n o t e d t h a t a = 2£ f o r the t h r e e - p h a s e w i n d i n g . I f the c u r r e n t s i n the w i n d i n g a r e s i n u s o i d a l , F a i s a h a r m o n i c f u n c t i o n o f t i m e , A F a = F a cos c o t , so t h a t , e x p r e s s i n g f a a s a harmonic f u n c t i o n o f t i m e , fV= 2^ F Q cos wt ^ \ s i n n£ s i n n9 . . . . . . . (2) S i m i l a r e x p r e s s i o n s h o l d for t h e m.m.f. of t h e o t h e r two p h a s e s , where t h e t i m e - p h a s e of t h e c u r r e n t s and s p a c e -phase of the w i n d i n g s a r e d i s p l a c e d — I T and 4 i r r a d i a n s . 5 3 A O 1 1 5 0 0 f b = 24 F b cos (cot - -|tr)V Is in n£ sin n(9 - l i r ) (5) •ir1, y Z _ n a b 3 f c = 24 F c cos (cot « 4 10 > - s i n n T r s i n n(9 - 1-iT) . . . . . . (4) ^ 3 4 l t f b 3 The r e s u l t a n t m.m.f. o f the three phases a t any a n g l e 9 , at time t, is t h e sum f a + f b+ f c . I f t h e c u r r e n t s i n t h e t h r e e phases a r e b a l a n c e d so t h a t F„ - F b = F c = F, , the sum i s g i v e n by f, = 2 iF, V cosjcot - ( U - 1 )-§frf\" Is i n n TLsin n<9 -01-1 (5) -ft2 £7 L 3 'Z. 6 ( 3J) P u t t i n g n=1,3,5, . . . , we have f, = J l F , f s i n ( e - o ) t ) + —lsin(59+wt) Lsin(79-«t) T T 1 L ' 2 5 49 _ 1- slnO 19+cot) + •*» • 121 1 ft •• « A ( 6 ) I f t h e c u r r e n t s i n t h e t h r e e phases a r e u n b a l a n c e d , t h e y may be r e s o l v e d i n t o t h r e e balanced, s e t s o f c u r r e n t s a c c o r d i n g t o t h e f u n d a m e n t a l p r i n c i p l e o f t h e t h e o r y o f s y m m e t r i c a l components. We a r e n o t c o n c e r n e d w i t h the z e r o phase sequence o f c u r r e n t s , s i n c e t h i s s e t does n o t f l o w o r d i n a r i l y i n i n d u c t i o n motor w i n d i n g s , and i n any case w o u l d p r o d u c e no r e s u l t a n t m . m . f . as may be s e e n from a s t u d y o f F i g . 1. The p o s i t i v e phase sequence o f c u r r e n t s , I, , g i v e s r i s e t o f, as e x p r e s s e d by e q u a t i o n (5), b u t t h e n e g a t i v e phase sequence o f c u r r e n t s , Iz, has t i m e - p h a s e d i s p l a c e m e n t s t h e r e v e r s e o f I , . T h a t i s , i n e q u a t i o n (Jj), t h e s i g n p r e f i x e d to t h e term (N-1 )•§*!", i s r e v e r s e d . Thus t h e m . m . f . due t o I 2 i s g i v e n by f 2 = ^ | F 2 V cos wt + )^irJ\T i s i n n I L s i n n f e - (N-T )-|ir) (7) o r ^ ^ F - s i n j e +(cot-<x)) ...+ __Lsinf59 - ' ( c o t - * ) ) ct b e i n g a c o n s t a n t a n g l e . I n e q u a t i o n s ( 6 ) and (8) we see t h a t t h e a m p l i t u d e o f t h e h i g h e r , h a r m o n i c t e r m s ^ f a l l s o f f r a p i d l y . S o , f o r the p r e s e n t , we may f i x o u r a t t e n t i o n on t h e fundamental t e r m s . We see t h a t t h e m . m . f . f 2 * due to I a j has the same a n g u l a r v e l -o c i t y a s f t h u t i n t h e o p p o s i t p d i r e c t i o n . I t becomes c l e a r , t h e n , t h a t I , p r o d u c e s a " p o s i t i v e r o t a t i o n a l " m . m . f . r e v o l v i n g a t a n a n g u l a r v e l o c i t y o f co e l e c t r i c a l r a d i a n s p e r s e c o n d , w h i l e I 2 p r o d u c e s a " n e g a t i v e r o t a t i o n a l " m . m . f . r e v o l v i n g a t t h e same speed I n t h e o p p o s i t e d i r e c t i o n . The r e s u l t a n t m . m . f . i s g i v e n by the sum, f, + f a . I f we A A r e p r e s e n t f, and £ 2 ^7 v e c t o r s o f l e n g t h F, and F 2 ( c a n s i d e r i n g f u n d a m e n t a l terms o n l y ) r e v o l v i n g w i t h t h e i r c o r r e s p o n d i n g v e l o c i t i e s , we may o b s e r v e how t h e i r r e l a t i v e m a g n i t u d e s a f f e c t t h e r e s u l t a n t m . m . f . • 5 O b v i o u s l y , the l o c u s o f F R = F, + F 2 i s , i n g e r e r a l , an e l l i p s e o f s e m i - m a j o r a x i s F, •+ F 2 , and s e m i - m i n o r a x i s F, - F z . As l o n g as F , > F 2 , the r e s u l t a n t , F R , has a d i r -e c t i o n o f r o t a t i o n the same as F, , a l t h o u g h w i t h v a r i a b l e a n g u l a r v e l o c i t y . When F z z : 0 , t h e l o c u s , i s , o f c o u r s e , c i r c u l a r , and t h e m . m . f , wave had c o n s t a n t a n g u l a r v e l o c i t y . When F 2 = F, , t h e l o c u s i s a s t r a i g h t l i n e , the p e r i o d b e i n g ~ T h i s i l l u s t r a t e s t h e c h a r a c t e r o f the f i e l d i n a s i n g l e -phase w i n d i n g . The d e d u c t i o n w h i c h s h o u l d be e m p h a s i z e d , however, i s t h a t t h e p o s i t i v e phase sequence c u r r e n t s p r o d u c e a n a . m . f . wave, ahek hence f i e l d , r e v o l v i n g i n the o p p o s i t e d i r e c t i o n to t h e f i e l d p r o d u c e d b y the n e g a t i v e sequence c u r r e n t s . I I 3. I n d u c t i o n M o t o r R e a c t i o n s and M a g n e t i c C o u p l i n g . H a y i n g s e e n some o f the e f f e c t s o f u n b a l a n c e d c u r r e n t s f l o w i n g i n a t h r e e - p h a s e w i n d i n g , we s h a l l p a s s on to a c l o s e r a n a l y s i s . When a n a l t e r n a t i n g e . m . f , E , i s impressed, on a n i n -d u c t i o n motor a t s t a n d s t i l l , the r e a c t i o n s a r e the same as f o r s t a t i c m a c h i n e r y , and e x p r e s s e d by E + (R + jo)]S)i = 0 , where i:R = ohmic r e s i s t a n c e If = i n d u c t a n c e ( s e l f p l u s m u t u a l ) = 27fx( f r e q u e n c y ) I = c u r r e n t I f one w i n d i n g moves c o n t i n u o u s l y w i t h r e s p e c t to the o t h e r , the m u t u a l i n d u c t a n c e w i l l undergo p e r i o d i c c h a n g e s , dtue to changes i n m a g n e t i c c o u p l i n g between the w i n d i n g s . Thus the f r e q u e n c y o f the the r e a c t i o n s depends upon t h e speed o f t h e m o t o r . The s e l f - and m u t u a l - i n d u c t a n c e s r e f e r r e d to. a r e t h e e f f e c t i v e p o l y p h a s e v a l u e s . That i s , the p r i m a r y s e l f - i n d u c t a n c e L has s u c h a v a l u e t h a t when a b a l a n c e d p o l y p h a s e s e t o f v o l t a g e s , E , i s i m p r e s s e d on the p r i m a r y , the s e c o n d a r y b e i n g o p e n , t h e n i + (E + > L ) I = 0 , The m u t u a l i n d u c t a n c e , M , has s u c h a v a l u e t h a t , when a b a l a n c e d s e t o f c u r r e n t s , I p , axe f l o w i n g i n t h e p r i m a r y , t h e v o l t a g e ( b a l a n c e d ) a c r o s s t h e open s e c o n d a r y i s E s = j o M P . When t h e w i n d i n g s a r e s y m m e t r i c a l l y d i s p o s e d w i t h r e s p e c t to one a n o t h e r , as i s the ease w i t h a l l p r a c t i c a l m a c h i n e s , t h e m u t u a l e f f e c t s between w i n d i n g s o f d i f f e r e n t phases does n o t s h i f t t h e phase p o s i t i o n o f the v a l u e s , b u t m e r e l y a f f e c t s t h e i r m a g n i t u d e s , a l t h o u g h the c u r r e n t s i n t h e d i f f -e r e n t phases have d i f f e r e n t t i m e - p h a s e p o s i t i o n s . T h i s i s owing t o the f a c t t h a t i n the t o t a l m u t u a l e f f e c t s between a l l w i n d i n g s , the o u t - o f - p h a s e r e l a t i o n s a r e c a n c e l l e d * and by d e f i n \ t i o n , the i n d u c t a n c e c o n s t a n t s , 1 and M , a r e d e t e r m i n e d u n d e r b a l a n c e d c o n d i t i o n s . 4 . "Frequency o f R e a c t i o n s i n I n d u c t i o n M o t o r s . I f a n u n b a l a n c e d v o l t a g e o f f r e q u e n c y ~ I s a p p l i e d to a n i n d u c t -i o n motor whose r o t o r i s r e v o l v i n g a t a v e l o c i t y o f G O 0 - G V e l e c t r i c a l r a d i a n s p e r s e c o n d , the p o s i t i v e r o t a t i o n a l f i e l d p r o d u c e d b y the c u r r e n t s I p , i n the p r i m a r y w i n d i n g w i l l c u t the r o t o r a t the a n g u l a r v e l o c i t y 6),, and c o n -s e q u e n t l y i n d u c e r o t o r c u r r e n t s o f f r e q u e n c y We s h a l l show l a t e r t h a t i f t h e c o n s t a n t s o f the r o t o r c i r c u i t a r e b a l a n c e d , t h e r o t o r c u r r e n t s , I s , o f f r e q u e n c y 4£± a r e b a l a n c e d . B u t i f t h e c o n s t a n t s a r e u n b a l a n c e d , t h e s e c u r r e n t w i l l be u n b a l a n c e d , and may be r e s o l v e d i n t o the components I 5 1 and L j 2 , h a v i n g the r o t o r f r e q u e n c y ^ I S I p r o d u c e s an m . m . f . w h i c h r e v o l v e s i n the same d i r e c t i o n as the r o t o r , a t a v e l o c i t y co, r e l a t i v e to i t , and hence a t a v e l o c i t y (J0,+(Do —a>t = cOo r e l a t i v e to the s t a t o r ; t h a t i s , i t i s i n s t e p w i t h the s t a t o r m . m . f . produced, b y I P 1 . B u t I 5 2 e x e r t s a n m . m . f . r e v o l v i n g a t v e l o c i t y -<t>, r e l a t i v e t o t h e s t a t o r , . ~ r -and hence a t v e l o c i t y -©^(cu.-co,) = co0~ilr6Pr-. r e l a t i v e to the s t a t o r . Now, coming b a c k to I P Z : the m . m . f . w h i c h i t e x e r t s ; V .... ' r e v o l v e s s a t v e l o c i t y -co, r e l a t i v e to s t a t o r , and hence a t v e l o c i t y - u>0 - ((L>e-co,) = - ( 2 ^ . - 0 ) , ) r e l a t i v e to t h e r o t o r , thus c u r r e n t s o f f r e q u e n c y 2 0 J o a r e p r o d u c e d i n the r o t o r , w h i c h may be u n b a l a n c e d , t h u s c a u s i n g f u r t h e r r e f -l e c t i o n s . I f t h e r o t o r c i r c u i t i s u n b a l a n c e d , t h e r e i s a number o f t h e s e r e f l e c t i o n s , l i m i t e d o n l y by the h i g h e r r e a c t a n c e o f t h e w i n d i n g s to the c u r r e n t s o f the h i g h e r f r e q u e n c i e s . We s h a l l c o n f i n e o u r s e l v e s to c u r r e n t s whose f r e q u e n c i e s s h a l l n o t exceed 2to 4 -W,, s i n c e c u r r e n t s o f h i g h e r f r e q u e n c i e s w i l l be o f n e g l i g i b l e a m p l i t u d e , due to t h e h i g h e r r e a c t a n c e s o f f e r e d by t h e w i n d i n g s t o them. We may t a b u l a t e t h e r e l a t i o n s between the c u r r e n t r e a c t i o n s , i n o r d e r t o be a b l e to c o r r e l a t e them more r e a d i l y when we come to w r i t e the e q u a t i o n s to d e t e r m i n e the perfocmance o f the m o t o r . The s u b s c r i p t s 1 and 2 r e f e r 8 p o s i t i v e phase sequence and negative phase sequence currents r e s p e c t i v e l y . Current Frequency V e l o c i t y of corresponding xu.iu.f. R e l a t i v e to s t u t o r R e l a t i v e to r o t o r I P , ) I p2. ) <0o 2TT + co0 - COo + to, -( 2OJd-co,) Isi ) Isz ) co, + OJo +(to 0- 2o).) + CO, - CO, i * ) 6 ) 0 - 2o>, 2TT + 6 ) 0 - 2co, - (0O0 - 2co,) -60, -(2co 0- 3 0 ) , ) Is. > 2CO„-C0, 2lT +3co0- 2co, -00o -2 co0- co, _ ( iCOo - CO, ) In f i g u r e 3 the d i r e c t i o n s and r e l a t i v e v e l o c i t i e s of the var i o u s m.m.fs. are shown.. Those shown braeketted keep step. Fp2 Fp, (<Og) FA (aJ.-2LO.) F£ Qio.-^tQ..) 3 _ j f o r aW, F i g . 3 «• >- -« «• Rotor Fs /2-(2a30-co,') Fs,(co.) F« (-co,) Fk1 (2oJ.-«x>,) t a b u l a t i n g the frequency r e a c t i o n s f o r the currents we are c o n s i d e r i n g (those of frequency not exceeding(2co e >-co,)), we have the t a b l e i n Figure 4. REACTION FREQUENCY OF This i s to be read from the top Ipz I51 I* I's, r down to the l e f t , Thus the IP, % frequency of r e a c t i o n of I I* CO. 00o on I i s given i n the t h i r d OS Isi CO, % square down i n the l e f t - hand I* CO | Y^ column. Blank squares i n d i c a t e K CO' no r e a c t i o n , Tnus there i s no 6 / CO' r e a c t i o n of I on I . f o t e CO '=2CO 0 -OJ 1 F i g . 4 The accompanying o s c i l l o g r a m s o f r o t o r c u r r e n t s show c l e a r l y how t h e r o t o r c u r r e n t s a r e a f f e c t e d "by u n b a l a n c e d c o n d i t i o n s . The f i r s t s e t i s f o r b a l a n c e d a p p l i e d v o l t a g e and b a l a n c e d s t a t o r and r o t o r c i r c u i t s . There i s a s m a l l r i p p l e due t o t o o t h h a r m o n i c s , b u t o t h e r w i s e , the c u r r e n t I s , i s the o n l y one p r e s e n t i n the r o t o r . I n t h e s e c o n d s e t , the s t a t o r c i r c u i t has b e e n u n b a l -a n c e d b y added e x t e r n a l r e s i s t a n c e . T h i s i n t r o d u c e s 1^ o f f r e q u e n c y , i n the r o t o r . T h i s c u r r e n t has the same a m p l i t u d e i n a l l t h r e e p h a s e s , as has 1^ ; i . e . t h e r o t o r c u r r e n t s a r e b a l a n c e d , a n d the o n l y component o f I's i s 1^ * and o f I» I 5 ( . I n the t h i r d s e t , the r o t o r c i r c u i t , a d w e l l as the s t a t o r , has been u n b a l a n c e d by e x t e r n a l r e s i s t a n c e , w i t h the r e s u l t t h a t t h e r o t o r c u r r e n t s a r e u n b a l a n c e d , Ist t Is* » l i t .and I s 2 a l l b e i n g p r e s e n t . I t may seem t h a t I i 2 i s w r o n g l y d e s i g n a t e d b y the s u b -s c r i p t 2 , i n d i c a t i n g a n e g a t i v e sequence c u r r e n t . We h a v e , however, c h o s e n to c a l l a c u r r e n t o f n e g a t i v e phase s e q u e n c e , one w h i c h p r o d u c e s a f i e l d r e v o l v i n g i n a d i r e c t i o n o p p o s i t e to the d i r e c t i o n o f the r o t o r . H e n c e , when I i i s b a l a n c e d , Is', i s a b s e n t * • 1 o I I I 5. G e n e r a l T h e o r y f o r U n b a l a n c e d C o n d i t i o n s . The f o r e g o i n g c o n s i d e r a t i o n s w i l l e n a b l e u s to b e g i n a m a t h e m a t i c a l a n a l y s i s o f t h e p r o b l e m o f a s y m m e t r i c a l l y c o n s t r u c t e d i n d u c t i o n motor h a v i n g u n b a l a n c e d c o n s t a n t s i n i t s s t a t o r and r o t o r c i r c u i t s , and h a v i n g an u n b a l a n c e d v o l t a g e s u p p l y . L e t E P 1 , E P 2 , , be the p o s i t i v e and n e g a t i v e sequence com-p o n e n t s o f a p p l i e d v o l t a g e , ( E p 6 , the z e r o sequence component, i s d i s r e g a r d e d , s i n c e i t p r o d u c e s no z e r o sequence c u r r e n t ) E S | , E S 2 , be the p o s i t i v e and n e g a t i v e sequence compon-e n t s o f r o t o r f r e q u e n c y CO,/2TT , ( s i n c e we s h a l l c o n s i d e r t h e s e as t h e t o t a l r o t o r v o l t a g e s f o r the r o t o r c i r c u i t , t h e y a r e z e r o ) Ei, , Esz , be the p o s i t i v e and n e g a t i v e sequence compon-e n t s o f r o t o r v o l t a g e o f f r e q u e n c y co'/2-rr , ( a l s o z e r o ) Rpo » Rpi » Rpz » be -the s y m m e t r i c a l components o f r e s i s t a n e o f the p r i m a r y c i r c u i t , ( e x t e r n a l r e s i s t -ance i n c l u d e d ) Rso » R Si » RSz t "be the c o r r e s p o n d i n g q u a n t i t i e s f o r the s e c o n d a r y c i r c u i t * L p o , L P I , L p z , be the s y m m e t r i c a l components o f s e l f -i n d u c t a n c e o f the p r i m a r y c i r c u i t , I»so » ^si » ^»S2"»' oe the c o r r e s p o n d i n g q u a n t i t i e s f o r t h e • s e c o n d a r y c i r c u i t , M be the c o e f f i c i e n t o f m u t u a l i n d u c t a n c e f o r the motor as d e f i n e d i n s e c t i o n 3 , p be the number o f p o l e s , 3W be the power o u t p u t o f m o t o r , ( w a t t s ) 3T be the t o r q u e i n l b s . f t . , 3@? + jQ,)be the v o l t - a m p e r e i n p u t . E q u i v a l e n t l i n e - t o - n e u t r a l v a l u e s f o r a l l v o l t a g e s , c u r r e n t s , and c o n s t a n t s , a r e to be u s e d . 11 i We may now w r i t e the v o l t a g e e q u a t i o n s f o r the p r i m a r y and s e c o n d a r y c i r c u i t s , u s i n g the p r i n c i p l e s o f the t h e o r y o f s y m m e t r i c a l components, and r e f e r r i n g to the t a b l e o f F i g . 4 as a g u i d e , i f n e c e s s a r y . E P I = R P O Ip, + R P 2 I P 1 + j w.(I P ( ) I P , + L P 2 Ipa-i- M I S , ).. .., . ( T O ) ^ Ep^™ ^PI Ipi + Rpo Ipa +• j co0(Lpi Ipj -t L polp2 + MI. S ^.) • * © (T1 ) f. E £ | - : R s o I A I + B 5 J S i + jo), ( L S O I s , * tsz I S 2 . - f MI P l ) = 0 . (12) f E S 2 = I S , + Rso Isz + j O ) , ( L a i Is, + Lso I S i + 0 ) = 0 . (13) f E ^ , = R S o Is. + $ S 2 I « + j c o ' ( L s o i ; , + L « U a + 0 ) =0 . (14) ! E s 2 = Rsi 1st + Rso Isz + j ^ ' ^ L s , Isi + l s o Isz + M I P 2 . ) = 0 . (1 5) T h e r e i s no m u t u a l i n d u c t a n c e term i n e q u a t i o n s ( 1 3 ) and ( 1 4 ) , as may be s e e n by r e f e r e n c e to F i g . 4 . M u l t i p l y e q u a t i o n ( 10 ) by , (Ig , b e i n g the c o n j u g a t e o f I P , ) ( 1 1 ) by Ish (12) by L,, (13) by L*. (14 ) by IL, and (15) 630 co, co, ay b y Isz. The r e s u l t s a r e EP' Jei = lEgIeiJ[pi4. Rp^Ipglpiu. W T T T j . T T T . i l T T 1 ! * t /: \ E P Z - I P Z . - RPI Iwlre . Rpalpy.Ip?. j ( I P 1 I P , IPZ±I'PoIp2lpi+ M I s 2 I P 2 ) . . . . (17) "Clio-""' '•'" COo •'• C0 0 ~' • 1 -. A - • - RsoLsiIsi j Rszlszlsi ^ j (Itfplsi I51 + Ife2 I s i 4 M I P | I S | ) . . . . (18) 63, CO 1 0 Rgilsilszr Rsolsalsz. j ( L S i I 5 2 + L 5 a I s l I 5 2 .+ 0 ).-,.,. (T9) ' CO, T cu, • ^ "•' 0 0 Rsalsilsi , Rsilszlsi . j (L 5 0Is,Is', + L f f i l i 2 I s ,4 - 0 ) . • • • (20) OS' CO' ^ = Rsi Isi l a • Rsnlszls^ . j ( L s , I s i Isz+ LSJIS-JIS2+ M I P 2 I S z ) . . . . { 21 ) CO' ^ Co' ^ A d d i n g e q u a t i o n s (16) to (21) i n c l u s i v e , we o b t a i n :•' EPI'IPI . E P 2 In = P + jQ, , . . . . . . . (22) where 12 Q = ( L p j p j p , ) + ( L ^ I ^ ) + ( L ^ . I ^ ) +v(I.soIs 2l5 2) + ( L 5 j ; , I s f ) Is2.Ipz+ Ipz^z ) • • • • • • • • • • (23) The sum i n each "brackett i s p u r e l y r e a l , s i n c e each sum i s o f t h e f o r m AA o r AB 4 BA , e a c h o f w h i c h i s p u r e l y r e a l , { ( s e e a p p e n d i x ) , T h e r e f o r e Q, i s p u r e l y r e a l and hence «)Q, p u r e l y i m a g i n a r y - the r e a c t i v e v o l t - a m p e r e i n p u t t o t h e m o t o r . and t h i s i s a l s o p u r e l y r e a l - t h e w a t t s i n p u t to the m o t o r d i v i d e d by G J 0 , from e q u a t i o n (22). The c o p p e r l o s s i n t h e p r i m a r y c i r c u i t i s and s i m i l a r e x p r e s s i o n s h o l d f o r the c o p p e r l o s s e s i n the s e c o n d a r y c i r c u i t due to c u r r e n t s I S 1 , I S 2 . and Isi , I s ' i , The o u t p u t o f t h e motor i s , p e r p h a s e , W = ( i n p u t ) - ( c o p p e r l o s s e s ) = a>0P - ( c o p p e r l o s s e s ) , n e g l e c t i n g the i r o n l o s s e s . Thus (24) Hi W 1 j (Ksoisi Tsi + ^SoXs2^-S2 + Rsilsi Is2+ R52.I52I51 ) ( 25) d ) o - C O , CO, ls + t 0 o - 6 D (26) where l s and I s a r e the r o t o r c o p p e r l o s s e s c o r r e s p o n d i n g t o the c u r r e n t s I s , , I s z and Is, , 1 4 . r e s p e c t i v e l y . 13 .. I n the e q u a t i o n 5 2coo-£0, &>e-co, i s p o s i t i v e ' for a l l motor speeds from s y n c h r o n i s m 2 C O . - CO, / t o s t a n d s t i l l . T h e r e f o r e t h e q u a n t i t y C O o ~ C O r i 7 i s the power 2co.-oo, f e d h a c k i n t o t h e s u p p l y c i r c u i t . T h i s power i s drawn f r o m t h e l i n e t h r o u g h t h e p o s i t i v e sequence component. The t o r q u e i n l b s . f t . i s g i v e n by T « 33000 TT o W 746 x 2TT x 6Q(oJo-co,) = ? $ 9 P w ( t 0 o - C 0 , ) = 0.369 P { «^ s_ Is ) . . . . . . . . . . (28) T h i s i s a more g e n e r a l s t a t e m e n t o f t h e law w h i c h i s w e l l known f o r t h e t o r q u e o f i n d u c t i o n m o t o r s u n d e r b a l a n c e d c o n d i t i o n s : I T Torque i s p r o p o r t i o n a l t o r o t o r c o p p e r l o s s d i v i d e d b y s l i p " . E q u a t i o n (28) s t a t e s t h a t t h e t o r q u e i s p r o p o r t i o n a l to ( p o s i t i v e sequence l o s s e s ) _ ( n e g a t i v e r o t a t i o n a l l o s s e s ) . ( c o r r e s p o n d i n g s l i p ) ( c o r r e s p o n d i n g s l i p ) I f the f o r m o f t h e t o r q u e - s l i p c u r v e i s d e s i r e d , we must s o l v e e q u a t i o n s (TO) t o ( T 5 ) f o r the s e v e r a l c u r r e n t s and s u b s t i t u t e i n ( 2 8 ) . T h i s i s a t e d i o u s p r o c e d u r e i n the g e n e r a l e a s e ; we s h a l l , however, c o n s i d e r the s i m p l i f i c a t i o n s i n t r o d u c e d b y s p e c i a l c o n d i t i o n s . « (\. T r a n c e d A p p l i e d V o l t a g e . A l l Impedances B a l a n c e d . I n t h i s c a s e , E P 2 = 0, R P, - 0, RP Z,= E ^ , ^ R^2= L P | = L P 2 ™ L s ) =• L s z - 0, and the s i x f u n d a m e n t a l e q u a t i o n s r e d u c e t o 14 B P I = Rpo I P i+ j ^ ( L p o I P 1 + MI S | ) . . . . . . . . . . . . . - ( 2 9 ) E P Z = 0 ~ Rpo Ip2+ j <^o (Lpo 1^ + MIsi) • • • • » « « . . . (30) E S I = 0 = R ^ I S , + j <4»(Ii^ o I 5 | + M Ip) . . . . . . . . . . . . ( 3 D Bsj= 0 = . R s o - I « + J ' ^ i d s o Ia'+'-'O ) V . . . . . . . . . . . (32) E s ' ,= 0 = R s o jo>'{L S o I s ' * 0 ) (33) E ^ - ' O = R S 0 Isz+ j ( L s o • I^+-MS r a -) . i . . . . . . . . . . (54) Prom (32) I S 2 = 0 , and from (33) Is'.= 0 . C o m b i n i n g (30) and ( 3 4 ) , we f i n d t h a t IPSL= 0 and %z- 0. T h e r e f o r e , i n e q u a t i o n s ( 3 0 ) , ( 3 2 ) , (331, and ( 3 4 ) , e v e r y term i s z e r o , l e a v i n g (29) and (31) f o r c o n s i d e r a t i o n . M u l t i p l y i n g (29) by I P i _ 6)0 and ( 3 D by Isi and a d d i n g t h e two e q u a t i o n s so f o r m e d , we 6)0 o b t a i n BELIPI = Rp^l lp i - i . R?fll?'lsi-i. J [Lfolpilpi + I»soIsiIiM* ^(isr^pi + Ipi Isi ^1 • (35) C 0 o COa COo whence the r e a l power i n p u t to the motor i s Rpo Jpi Ipi + Rsolsi I« i » • » • • « » . . . (36) The t o t a l c o p p e r l o s s e s are Rpalp.Ip,* R ^ L ^ I s , . S u b t r a c t i n g t h e s e from t h e i n p u t , we o b t a i n ^ O J , Rsolsi I s i 5 • • • • • • * • * * • • (37) and the t o r q u e i s T, = 0 .369 p w O J o - C O , ' = 0.369 P R S A L . I S . . . . . . . . . . . . (38) S o l v i n g (29) and ( 3 D f o r I s , , we have I S ( = - .jco.MEp. = E P I ( a + j b ) Rpo-c^co.M2"* jcOoLpo Hence I s , = E p , ( a - j b ) , and Is ,T s l = E p ^ a ^ b 2 ) , where a and b ane f u n c t i o n s o f the motor c o n s t a n t s and the s l i p . l i U n b a l a n c e d A p p l i e d v o l t a g e . A l l Impedances "balanced. I n t h i s c a s e , e q u a t i o n s (29) to (34) a p p l y , w i t h the e x c e p t i o n t h a t E P i has a v a l u e i n (30). isz. = 0* an^- ^sr" ° i h u t Ipz and I'sz a r e n o t z e r o . From (29) and (31 ) we o b t a i n t h e v a l u e f o r p o s i t i v e power and t o r q u e g i v e n by e q u a t i o n s (37) and {38K B u t t h e r e i s , i n t h i s c a s e , a n e g a t i v e term a l s o , i n the power and t o r q u e e q u a t i o n s . F o r , from (30) and ( 3 4 ) , the r e a l p a r t o f E p ^ i s (Epii; z ) r f t j l - R P D I R i T p 2 + | £ Fsolsz l^ (39) The c o p p e r l o s s e s a r e , f o r the n e g a t i v e sequence components o f c u r r e n t , RP0Ip2lp2+ R^I^zI^. S u b t r a c t i n g t h e s e l o s s e s from the i n p u t we have = - ^ 7 L 7 ^ ^ l - ( 4 0 ) as t h e n e g a t i v e m e c h a n i c a l power. The complete, e x p r e s s i o n f o r m e c h a n i c a l power i s W - W , + W2 = ( O J o - 6 J 0 ( R ^ ) I S , _ . ( 4 1 ) and T = O.369 P (Rsok.Is, R s o & l k ) (42) ( co, to' ) As p o i n t e d out b e f o r e , the n e g a t i v e s i g n i n ( 4 T ) i n d i c a t e s t h a t power i s b e i n g f e d back i n t o the s u p p l y c i r c u i t . S i n c e the n e g a t i v e sequence c u r r e n t does work to f e e d power b a c k t o t h e s u p p l y , t h e motor f u n c t i o n s as a phase b a l a n c e r and an a means o f s u p p l y i n g m e c h a n i c a l power s i m u l t a n e o u s l y . I n the e q u a t i o n f o r t o r q u e , ( 4 2 ) , I S i m a y be d e t e r m i n e d from e q u a t i o n s (30) and (34). Isi was g i v e n as E P l ( a + j b ) . From (30) and (34), I S 2 = E P 2 ( g + j f ) . T h e r e f o r e s i n c e Is.Isi = Ep, (a* + b* )•,.• and i ; 2 i ; 2 = E P a { g " + f z ) , t h e n T, =. E P l 4>(s,c) and I z . - Ep2 4>|(2 - s ) , e] . where s and e r e p r e s e n t s l i p and motor c o n s t a n t s r e s p e c t -i v e l y . The f u n c t i o n s a r e n o t the same, as may "be s e e n , so i n g e n e r a l , JDi i s n o t e q u a l to E P I , a l t h o u g h t h i s i s t r u e a t s t a n d s t i l l . The c u r v e s i n F i g . 5 i l l u s t r a t e the f o r e g o i n g c l e a r l y . They show T, f o r p o s i t i v e sequence v o l t a g e E P I , and T z f o r n e g a t i v e sequence v o l t a g e E P 2 . The b r o k e n l i n e c u r v e , T , i s f o r the case when E P £ = - E P I , so t h a t E O A = ( E P I - E n ) ~ 0 , E O B = ( -0 .5 - j O.8660Epl+ ( - 0 , 3 + j o.366 ) ( - E P ( ) = - j 1 . 7 3 2 E P E e c = <-0,5 + i 0.866)Ep,+ ( - 0 . 5 - j 0.866 ) ( - E P I ) = +j 1.732EP That i s , E O B = - E o - , c o r r e s p o n d i n g to a s i n g l e - p h a s e v o l t a g e 2\/3 E p , a l l i e d between t e r m i n a l s B and C . The c u r v e i s , o f c o u r s e , o f t h e form u s u a l l y shown f o r s i n g l e - p h a s e o p e r a t i o These c u r v e s were c a l c u l a t e d from t e s t s on a t h r e e - p h a s e m o t o r ; the c a l c u l a t i o n s a r e shown i n a l a t e r s e c t i o n . I t w i l l be o b s e r v e d t h a t any u n b a l a n c i n g o f a p p l i e d v o l t a g e has a more s e r i o u s e f f e c t on the s t a r t i n g t o r q u e t h a n on the p u l l - o u t t o r q u e . T h i s f a c t i s i l l u s t r a t e d by means o f the c u r v e s o f F i g . 6, c a l c u l a t e d f o r the same m o t o r . I t i s e v i d e n t , a l s o , t h a t i f a l t e r n a t i n g c u r r e n t were u s e d f o r b r a k i n g p u r p o s e s , the r e s u l t s might be f a r from s a t i s f a c t o r y i f the a p p l i e d v o l t a g e were u n b a l a n c e d . The i n c r e a s e d h e a t i n g o f the motor on l o a d i s a more i m p o r t a n t c o n s i d e r a t i o n t h a n the t o r q u e r e d u c t i o n . The v a l u e 17 o f t h e n e g a t i v e sequence c u r r e n t f l o w i n g c a n he o b t a i n e d by m u l t i p l y i n g t h e n e g a t i v e sequence v o l t a g e by the s t a n d -s t i l l a d m i t t a n c e ( c o n s i d e r e d c o n s t a n t o v e r t h e motor s p e e d r a n g e t o n e g a t i v e sequence c u r r e n t s ) . F u l l v o l t a g e i m p r e s s e d on m o t o r s o f n o r m a l d e s i g n , when l o c k e d , w i l l cause 6 to 8 t i m e s f u l l - l o a d c u r r e n t to f l o w . That i s , t h e y have a n a d m i t t a n c e o f 6 t o 8 a t 100?& s l i p . I f t h e s e m o t o r s were t o have u n b a l a n c e d v o l t a g e s i m p r e s s e d on them s u c h t h a t t h e n e g a t i v e sequence component o f v o l t a g e amounted to about •1 5T6 o f t h e p o s i t i v e sequence component, ( i . e . a n u n b a l a n c e f a c t o r o f 0.15) t h e r e would r e s u l t a n e g a t i v e sequence c u r r e n t e q u a l to f u l l l o a d c u r r e n t . T h u s , on l o a d , t h e t o r q u e o f o r d i n a r y i n d u c t i o n motors i s n o t s e n s i t i v e t o a moderate degree o f v o l t a g e u n b a l a n c e , b u t t h e i r h e a t i n g c h a r a c t e r i s t i c s a r e v e r y s e n s i t i v e t o s u c h u n b a l a n c e . 8 . U n b a l a n c e d A p p l i e d V o l t a g e . P r i m a r y Impedances  U n b a l a n c e d . S e c o n d a r y Impedances B a l a n c e d . R e f e r r i n g t o e q u a t i o n s (10) to (15), i n w h i c h R S I , R^, L S | , and L 5 1 w i l l be z e r o f o r t h i s c a s e , we f i n d t h a t Isz.= 0 from (13) and t h a t I'SI= 0 from ( 1 4 ) . That i s , the o n l y c u r r e n t s i n . t h e r o t o r a r e I s , and ; o r i n o t h e r w o r d s , the r o t o r c u r r e n t s a r e b a l a n c e d , b u t , i n g e n e r a l , have d i f f e r e n t f r e q u e n c i e s . The t o r q u e i s g i v e n by e q u a t i o n (2 8) where I ^ - l / . ^ O ; t h a t i s , T = 0.369 p Rs*I SI Isi — Rsol CO, CO' (43) I t i s worthy o f n o t i c e t h a t I 3 ( i s no l o n g e r a f u n c t i o n o f E P ( , s and c o n l y , b u t a l s o o f E P 2_ ; and s i m i l a r l y I'S7. i s 18 now a f u n c t i o n o f E p , , as w e l l as o f E P 2 , s and c . T h i s may be s e e n from e q u a t i o n s {10) to (15); f o r , w r i t i n g t h e n e c e s s a r y e q u a t i o n s i n t h e f o l l o w i n g f o r m , Epj — Zpolp, + ZiptIP2_— M 0 I i ( + 0 . • • . . • . • * . . . . . ( 4 4 ) E p 2 = Z p . I p , * Z p o l p i t 0 - M.'IU . . . . (45) 0 = - M , IP, + 0 + Z s IS[ + 0 . . . « . . . . . . . . . . (46) 0 = 0 - M' 1^+ 0 + Z U l z . . . . . . . . . . . . . . . (47) where Z p o ~ R P O + j c o 0 L P O , Z P ( = R P I + j a>oLPI , Z R P 2 . + «j <A>I< P 2. » Z S = R S + j t o . L g , Z i - R s + jco 'L, , M 0 - - J « * M , M, ~ -jco.M, and M ' = - j c o ' M , we o b t a i n 2p 0 Zpz E P 1 0 where A = ZPO Zpz - M 0 0 Z P i Zpo Ep^ -Mo Zpi Zpo 0 - M ' - M , 0 , 0 0 -Jtt, 0 Z £ 0 0 - 0 2s 0 - M ' 0 A and I' = Zpj, - M TP D -^ pi Z pi Zpo 0 - M , 0 z, » 0 0 - M ' 0 0 , 4 \ E v a l u a t i n g , I S f = - E P , M , ( M o M ' - ZfoZs ) - EP,M, Zo a Z' s . . . . . . . . ( 4 9 ) ( M o M ' - ZpoZi ) ( M 0 M , - Z P 0 Z £ ) - Zp.Z^Zs Z^ I « = - E P . M ' Z P , Z S - EraM' (MOM, - ZreZs ) (50) : ( M 0 M ' - Z P 0 Z i ) ( M 0 M ( - Z P 0 Z s ) - Z M Zp,.Zs Z £ T h i s i s i n agreement w i t h the s t a t e m e n t commonly made i n t r e a t i s e s on s y m m e t r i c a l components, n a m e l y , t h a t f o r a b a l a n c e d n e t w o r k , an a p p l i e d v o l t a g e o f one sequence w i l l p r o d u c e c u r r e n t o f t h a t sequence o n l y , b u t f o r a n u n b a l a n c e d n e t w o r k , i n g e r n r a l , a v l o t a g e component o f e i t h e r sequence 19 w i l l g i v e r i s e t o c u r r e n t components o f b o t h s e q u e n c e s . 9 . Unbala.nr>.flfl A p p l i e d V o l t a g e . P r i m a r y Impedances  B a l a n c e d , s e c o n d a r y Imparl amies U n b a l a n c e d . I n t h i s case RP|= RPz= Lp,^ L p ^ 0. From e q u a t i o n s (10) to (1.5), we see t h a t c u r r e n t s I P I , I P 2 , I S i , I 5 2 ) Is,, Isz a l l have v a l u e s i and t h e t o r q u e i s g i v e n by t h e g e n e r a l t o r q u e e q u a t i o n , (28). I t s h o u l d be n o t e d t h a t the t o r q u e w h i c h we speak o f i s , i n a l l c a s e s , t h e e f f e c t i v e v a l u e . I n r e a l i t y , t h e t o r q u e i s p u l s a t i n g whenever the r o t o r c u r r e n t s a r e u n -b a l a n c e d , t h a t i s , whenever the r o t o r c o n s t a n t s a r e u n -b a l a n c e d . One may t h i n k o f a component o f s i n g l e - p h a s e t o r q u e j due t o t h e u n b a l a n c e , i n t h i s c o n n e c t i o n . U e i t h e r w i l l the motor be as q u i e t when the s e c o n d a r y c u r r e n t s a r e u n b a l a n c e d i . T h e r e i s a g r a d u a l i n c r e a s e i n t h e i n t e n s i t y o f the hum, farom the c a s e o f p e r f e c t b a l a n c e to the c o n d i t i o n where one phase o f the r o t o r i s o p e n - c i r c u i t e d . 10. B l o c k e d t o r q u e T e s t s . The t e s t s were c a r r i e d out o n a l O h . p . , 60 c y c l e , 6pole, wound r o t o r i n d u c t i o n m o t o r , h a v i n g a d e l t a - c o n n e c t e d p r i m a r y and s t a r - c o n n e c t e d s e c -o n d a r y . The c o n s t a n t s , r e d u c e d t o e q u i v a l e n t l i n e - t o - n e u t r a v a l u e s , were R p = 0.146 R5 = 0.275 co0Lp =11.1 o j 0 L s = 4.67 OJ.M. = 6.45 (co0=377) - B l o c k e d t o r q u e t e s t s were t a k e n w i t h b a l a n c e d v o l t a g e a p p l i e d and t h e p r i m a r y c i r c u i t u n b a l a n c e d b y added r e s -i s t a n c e . The c u r r e n t s were c a l c u l a t e d from e q u a t i o n s (49) and ( 5 0 ) , w h i c h , f o r s t a n d s t i l l and b a l a n c e d a p p l i e d v o l t a g e , r e d u c e to IS I= - E»,K0 ; , ( M 0 M , - Z r o Z s ) - Z p . z ^ z ; I',= - SKMOZPTZ* . (M 0M, r Z^Zs ) - Zp.Zp.ZsZi t h e r e s u l t s were as f o l l o w s : V o l t s A p p l i e d E x t e r n a l • L i n e R e s . P r i m a r y Impedance Components C u r r e n t i n fflec. Torque l b s . f t . C a l c . Obs. Is. 62 A. . 2 .03 B 2 .03 0 0 ^ 1 .5+31 L I 2W0; 34+jO; 58 Zpf0.34-j0.58 2 8 . 4 10. 3.42 5.25 62.3 A 1 .37 B 1.55 C 0 Z f t * 1 .12+J11 .1 Zpr 0 92 +J0.45 Zpz-0.2 -jO.45 30.9 5.54 4.48 4.62 61.3 A , d-i'8j" B 0.91 c 0 Z ^ 0 . 7 3 + j 1 1 . 1 Z P i-0.13+j0.26 Z P i = 0 . 1 3 - j 0 . 2 6 33.2 3.72 5.27 V* r 126 A 6.34 B 6.71 0 0 Z F 3-4 , .5 +J11.1 Z P , - 1 . +J1.93 Z»*-1. -31 .93 31 . 2 37.5 13.1 3.25 126 A 4.78 B 5 .34 C 0 Z P O - 3 . 5 2 + j l 1 . 1 zP1=0.7 t j L 5 4 z ^ o . 7 - j 1 . 5 4 13 .7 5.5 114 Z 5,w2 B 3.37 C 0 Z r „ 2 . 34 + 31 T . t - 43.2 Zfi-o.51+31. Z ^ O i 5 T - 3 l . I 13.7. 8.13 7.78 The o b s e r v e d and c a l c u l a t e d v a l u e s f o r t o r q u e a r e i n f a i r l y c l o s e agreement. The n e g l e c t o f i r o n l o s s e s t e n d to make the c a l c u l a t e d v a l u e s h i g h , but t h i s i s compensated f o r , to some e x t e n t , by the s a t u r a t i o n a t h i g h e r v o l t a g e s , w h i c h makes the a c t u a l c u r r e n t s g r e a t e r t h a n t h o s e c a l c u l a t e d . F I G U R E 5 T O R Q U E - S U P C U R V E S Showing t h e e f f e c t o f u n b a l a n c e d v o l + a g e on. b a l a n c e d m a c h i a e . . T O R Q U E - S L I P C U R V E S S h o w i n g e f f e c t of- u n b a l a n c e d V o l t a g e on a m o t o r w i t h u n b a l a n c e d p r i m a r y . • 21 11. C a l c u l a t i o n s f o r s l i p - t o r q u e c u r v e s f o r b a l a n c e d m o t o r h a v i n g u n b a l a n c e d a p p l i e d v o l t a g e . I n t h i s case e q u a t i o n s (49) and (50) r e d u c e to I ?= - M i BPI •'• • •. M 6 M ( - Z P 4 Z S I ' = - M' E P * , M Q M ' . z ^ z ; w h i c h becomej f o r t h e p a r t i c u l a r motor u n d e r c o n s i d e r a t i o n , I « - 0.62/266° 15' s B p , 0.29 5/266° 1 5* + s i ; = 0.62726g_1_j ( 2 - s ) 0.293/266° 1 5' <2-s) The d a t a f o r t h e c u r v e s a r e s z s 0.1 18.65 1 ;3 13.3 0.2 . 28. 1.5 11 i 6 0.5 30. 1.7 1 0 i 3 0;5,: 26.4 1.8 9-? 0.7 \ 21.8 1.9 9.4-t . o 16.6 K d e f i n e d , b y T = K' Ep, j : . They a r e shown p l o t t e d i n F i g . 5. 12. C a l c u l a t i o n s f o r s l i p - t o r q u e c u r v e s , u n b a l a n c e d p r i m - a r y , u n b a l a n c e d a p p l i e d v o l t a g e . The c u r r e n t and t o r q u e v a l u e s were c a l c u l a t e d from e q u a t i o n s (49), (50) and (430. The r e s u l t s a r e as f o l l o w s : s Isi Is*  ~0~ 0 • ' 0 i 4 2 3 E P z + 0 i 3 2 4 E P j 0.T 0.t62Ep, + 0 i 0 4 5 5 E P Z 0 i 4 3 3 E P z + 0 . 2 7 9 E P | 0.2 0 i 2 5 4 E P | + 0. 0 7 T E p 2 0,A31^n 4 0;2.*3EP | 0;5 0 ; 3 0 7 E P I + 0.08t E P z 0 . 4 3 2 E P t +O0 .217E, , 0.5 0 . 3 6 l E P ( + 0.097 Ew 0 . 4 2 9 ^ P 2 . + 0.t87EP, 0;7 0.386EP1 + 0.102-E« 0-.422Ep«. + 0.1 7 E P I 1 . 0 0 . 4 0 9 E P 1 + 0.105 Epj 0 . 4 0 9 E p Z + 0.105EP| g i v i n g the c u r r e n t s I 6 ( and I'st i n terms o f the v o l t a g e 22 components. The s l i p - t o r q u e c u r v e s o f F i g . 6 were o b -t a i n e d b y s u b s t i t u t i n g t h e s e v a l u e s i n e q u a t i o n (43). 13. C o n c l u s i o n . The method o u t l i n e d i s q u i t e g e n e r a l i n i t s a p p l i c a t i o n , i t s u t i l i t y b e i n g l i m i t e d o n l y by the t e d i o u s a l g e b r i s a t i o n i n v o l v e d . I t does n o t take i n t o a c c o u n t the e f f e c t o f u n s y m m e t r i c a l w i n d i n g s , o r t h e mod-i f i c a t i o n s i n t r o d u c e d by s l o t r a t i o s , s u c h c o n s i d e r a t i o n s b e i n g o u t s i d e t h e scope o f t h i s p a p e r . I t c o u l d be a p p l i e d t o s q u i r r e l - c a g e machines by u s i n g an e q u i v a l e n t t h r e e -phase network f o r the r o t o r c i r c u i t . I n c o n c l u s i o n , the w r i t e r w i s h e s to e x p r e s s h i s a p p r e c -i a t i o n o f t h e a s s i s t a n c e g i v e n by P r o f , W . B . C o u l t h a r d , i n the l a b o r a t o r y , and o f the a d v i c e o f D r . H . T i c k e r s , Head o f the Department o f M e c h a n i c a l and E l e c t r i c a l E n g i n e e r i n g o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a . A p p e n d i x . B r i e f S t a t e m e n t s o f some o f the : More Important Theorems  U s e d i n the A p p l i c a t i o n o f t h e Method o f S y m m e t r i c a l  Components. 1. Any u n b a l a n c e d system o f p o l y p h a s e c u r r e n t s o r v o l t a g e s may he r e p r e s e n t e d by two s y m m e t r i c a l systems o f p o s i t i v e and n e g a t i v e phase s e q u e n c e , r e s p e c t i v e l y , and a r e s i d u a l o r z e r o sequence v e c t o r . T h u s , i f E O A ) E O B a n d Eo C a r e the t h r e e v o l t a g e v e c t o r s i n a n u n b a l a n c e d t h r e e - p h a s e s y s t e m , t h e y may be r e p r e s e n t e d by E O A = E Q + E, 4 E ^ E o e = E 0 + a 2 E,+• a Ea. E o c = E Q 4 a E, 4 a 2 E 2 where E 0 , E , , and E z a r e the s y m m e t r i c a l components o f v o l t a g e , and a i s the v e c t o r _ 1 + j/1 , ( a z - i _ j ^ ) 2 2 "* 2 2 a = 1, a = a , ar = a , ar - a , e t c . C o n v e r s e l y : Eo = i ( E o f t + E ^ E j E, - i (Ea t )+ a V « f V E i «1 ( E M + a E ^ a JUj E x a c t l y s i m i l a r e q u a t i o n s may be w r i t t e n f o r c u r r e n t s i n a n u n b a l a n c e d s y s t e m . , There i s no z e r o hhase component o f l i n e v o l t a g e , o r o f l i n e c u r r e n t where t h e n e u t r a l i s i s o l a t e d . There may be z e r o phase components o f l i n e - t o -n e u t r a l v o l t a g e , o r o f l i n e - t o - f t i n f c . c u r r e n t . . 2. Any p o l y p h a s e system o f a s y m m e t r i c a l impedances may he r e s o l v e d i n t o two s y m m e t r i a a l systems and a r e s i d u a l v e c t o r ; Z A - Z 0 + Z, +• Z z Z B = Z 0 + a*Z, 4 a Z x Z c = Z 0 4 a Z, 4 a z Z a and Zo= 1 ( Z A 4 Z B + Z c ) Z. = 1(Z A 4 a Z & + a z Z t ) 3 Z z - 1 ( Z A 4 a z Z 6 4 a Z c ) 3 3 . I n w r i t i n g v o l t a g e o r c u r r e n t e q u a t i o n s i n terms o f s y m m e t r i c a l components; t h e sum o f the s u b s c r i p t s o f e a c h t e r m must be c o n s t a n t , o r d i f f e r b y m u l t i p l e s o f 3 , e . g . : E 0 = Z 0 I 0 4 Z 2 I , 4 Z, I 2 s u b s c r i p t s B . 0 , 3 , 3 . E , = Z, I a 4 Z e I , 4 Z , I 2 « .. 1 , 1 , - T - i 4 . • B 2 - Z 2 I 0 + Z, I , 4 Z 6 I 2 . » 2 , 2 , 2 , 2. Thus s u c h e q u a t i o n s as E , -1 , Z,, E e = I o Zr-;f e t c . , a r e w r o n g . 4. F o r a b a l a n c e d n e t w o r k , a p p l i e d v o l t a g e s o f one phase sequence may p r o d u c e c u r r e n t s o f t h a t sequence o n l y . F o r a n u n b a l a n c e d n e t w o r k , an\^ a p p l i e d v o l t a g e o f any sequence may p r o d u c e c u r r e n t s o f a l l s e q u e n c e d . 5. F o r a n u n b a l a n c e d t h r e e phase s y s t e m , the power i s g i v e n i n terms o f t h e s y m m e t r i c a l components, b y the r e a l p a r t o f 3 ( E 0 I 0 + E , S, + E Z I 2 ) , where I o j l , and I z a r e the c o n j u g a t e s o f I c , I | » I-z r e s p e c t i v e l y . 6. Terms o f the f o r m AA , AB 4 BA , ABC 4 ABC , ABC 4 ABC e t c . i a r e p u r e l y r e a l * as may.be v e r i f i e d by e x p a n d i n g i n terms o f the complex q u a n t i t i e s . 25 7. F o r a t h r e e - p h a s e s y s t e m h a v i n g component r e s i s t a n c e s , R 0 ? R , , R 2 , and component c u r r e n t s I, I% ± t h e r e s i s t a n c e l o s s I s g i v e n b y 3 ( R 0I, I t + R^.I 2I Z+ R, I, I z + R 2 I z I, ). The atoove a r e mere s t a t e m e n t s ; f o r a l o g i c a l d e v e l o p -ment and p r o o f s , the r e a d e r i s r e f e r r e d to the l i t e r a t u r e . BIBLIOGRAPHY F o r t e s c u B S y m m e t r i c a l C o o r d i n a t e s T r a n s . A . I . E . S . 1913 p . 1027 Wagner and Evans S y m m e t r i c a l Components p u b l i c a t i o n o f Westinghouse E l e c t r i c & M f g . C o . 

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