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UBC Theses and Dissertations

A study of the electrical effects of armature eccentricity in a four-pole, lap-wound D.C. machine Smith, Donald Sinclair 1933

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STUDY OF THE ELECTRICAL EFFECTS OF AmUTUKS ECCEITTRICITY IB A 3?Oim-POIS, IAP-WOUHD D. C. HfcGHIKS  by Donald S i n c l a i r Siaith  xx CXJO  A T h e s i s s u b m i t t e d f o r the Degree o f :  ~ 7-IASTER OF APPLIED SCIENCE i n the Department of ELECTRICAL ENGINEERING  XX,  THE UNIVERSITY OF BRITISH COLUMBIA September 1933  PRE]? A G E  The w r i t e r wishes t o acknowledge the h e l p f u l and encouragement o f f e r e d d u r i n g the course  advice of h i s  work by Dr. H. V i c k e r s , head o f the Department o f Mechanical  and E l e c t r i c a l E n g i n e e r i n g .  He a l s o  wishes', t o thank Mr* C o u l t h a r d f o r the l a r g e amount o f time he g a v e , . p a r t i c u l a r l y .in h e l p i n g w i t h the o s c i l l o g r a p h - wo^k*.  September 1933 "vanecuver, B. C.  Donald S. S m i t h  L  STU3DY  jEE  ELECTRICAL  EFFECTS  OF ARKATUKE ECCEKTPJGITY  I K A FOUR-POLE, LAP-WOUIID D. C. MACHIHS  One o f t h e d i s a d v a n t a g e s o f the lap-wound B. G. machine i s due  t o the f a c t t h a t each o f t h e p a r a l l e l armature p a t h s i s  unuer the i n f l u e n c e o f o n l y two a d j a c e n t p o l e s .  Thus, i f f o r  any reason t h e f l u x i s n o t the same f o r a l l • tke^po-ibeB*- the E. M. F.•s o f t h e p a r a l l e l p a t h s w i l l n o t he e q u a l , and t h i s l e a d s t o c i r c u l a t i n g c u r r e n t s through the brushes,.  I n the wave-  wound machine t h e r e i s no such d i f f i c u l t y as each p a r a l l e l p a t h i s s u b j e c t e d to t h e i n f l u e n c e o f a l l t h e p o l e s . F o r many t y p e s o f machines, however, t h e l a p - w i n d i n g i s more s u i t a b l e , ' and i t s use h a s been made p o s s i b l e l a r g e l y as a r e s u l t of the i n t r o d u c t i o n of e q u a l i z i n g connections.  These a r e  c o n n e c t i o n s between p o i n t s on the armature w h i c h s h o u l d be a t the same p o t e n t i a l , and t h e y were d e v e l o p e d by D r . B . G. Lamme• The a c t i o n o f t h e c u r r e n t s w h i c h f l o w i n these c o n n e c t i o n s i s s i m i l a r t o t h a t of the c u r r e n t s c i r c u l a t i n g through the armatures  o f a l t e r n a t o r s r u n n i n g i n p a r a l l e l > they s e t up  M. II. F. s which, tend t o e q u a l i z e t h e p o l a r f l u x e s * I n e q u a l i t y o f t h e main p o l e f l u x e s may be due to s e v e r a l causes,, most i m p o r t a n t o f w h i c h i s unequal a i r - g a p l e n g t h s . As t h e r e l u c t a n c e o f the magnetic c i r c u i t i s l a r g e l y concent r a t e d i n t h e a i r - g a p , a n d i n any ca.se t h e l a t t e r i s v e r y s h o r t , i t t a k e s b u t l i t t l e e c c e n t r i c i t y o f t h e armature t o cause con-  s i d e r a b l e magnetic unbalance.  P e r f e c t balance  i s not l i k e l y t o  be r e a l i z e d as there i s always the p o s s i b i l i t y of s l i g h t p e r f e c t i o n s i n the c a s t i n g s . operates  Even i f a machine when  im-  new  s a t i s f a c t o r i l y there i s the wear i n the b e a r i n g s which,  though s l i g h t , may  be s u f f i c i e n t to cause bad  c i r c u l a t i n g cur-  rents. What has been s a i d rege.rding the use of e q u a l i z i n g conn e c t i o n s a p p l i e s to lap-windings out some years ago by I>r. W.  i n g e n e r a l , but i t was  pointed  L u l o f s t h a t the f o u r - p o l e machine  o f f e r e d a s p e c i a l case, and he argued that f o r i t no such conn e c t i o n s were necessary. f u r t h e r to study and without  I t has been the o b j e c t of t h i s t h e s i s  the o p e r a t i o n o f a f o u r - p o l e machine both  e q u a l i z i n g connections.  Although  r e g a r d i n g the a c t i o n of e q u a l i z i n g connections a l a r g e r number of p o l e s than f o u r may  be  with  some i n f o r m a t i o n on machines with  obtained, the g e n e r a l  case c o u l d be s t u d i e d t o much b e t t e r advantage on a machine with six  or more p o l e s . The machine used was  a 3 K. V/. 250  v o l t , d i r e c t current  g e n e r a t o r s p e c i a l l y b u i l t f o r experimental  work.  On the .shaft,,  and a t t h a t end remote from the commutator, are f i f t e e n  slip-  r i n g s which are connected t o tapping p o i n t s on the'armature back connections.  E i g h t of the s l i p - r i n g s are connected to f o u r  p a i r s of d i a m e t r i c a l l y opposite p o i n t s e q u a l l y spaced around the armature.  The  f o u r e q u a l i z i n g connections  can then  be  completed by connecting a p p r o p r i a t e c o n t a c t s which b e a r on  the  s l i p - r i n g s , and meters o r o s c i l l o g r a p h shunts c a n he i n t r o d u c e d ••as- desired» B e s i d e s the p r o v i s i o n f o r s t u d y i n g e q u a l i z i n g c u r r e n t s , the machine p r o v i d e s f a c i l i t i e s f o r i n v e s t i g a t i n g  commutation.  Three c o i l - s i d e s from one s l o t and one c o i l - a i d e from an adj a c e n t s l o t have i n each case one of t h e i r turns opened a t the hack of the armature and l e a d s are taken to s l i p - r i n g s so i t v / i l l he p o s s i b l e to complete the connections through graph  oscillo-  shunts. w i t h a view to s i m u l a t i n g the c o n d i t i o n s which o b t a i n when  the b e a r i n g s of a machine become worn, the t e s t machine was b u i l t so t h a t the gap l e n g t h c o u l d be v a r i e d .  F o r t h i s purpose  outboard b e a r i n g s were p r o v i d e d i n p l a c e o f end-bells', and t o vary the gap i t i s • m e r e l y necessary t h i c k n e s s between the pole-frame  to i n s e r t shims o f v a r y i n g  and the "bed-plate.  As a r e s u l t of e a r l y t e s t s with, f o u r e q u a l i z i n g  connections  i t was thought a d v i s a b l e to make p r o v i s i o n f o r o t h e r s i m i l a r c o n n e c t i o n s i n order to extend  the study somewhat.  Finally,  p r o v i s i o n was made f o r e i g h t more e q u a l i z e r s by tapping the back connections i n s i x t e e n p l a c e s ; a l a r g e r number was c o n s i d e r e d , but the i n c r e a s e d complexity of connections would n o t have been warranted.  F o r these new c o n n e c t i o n s w e l l i n s u l a t e d l e a d s were  taken from the t a p p i n g p o i n t s to t e r m i n a l s spaced around, the o u t e r r i m of a f i b r e d i s c , immediately nections.  behind  the back con-  Then i n s u l a t e d connectors were made to j o i n up •  o p p o s i t e t e r m i n a l s , and thus the e q u a l i z i n g connections c o u l d be  I n s e r t e d o r removed a t w i l l . Although new  i t would have been p r e f e r a b l e to so space  these  e q u a l i z i n g connections t h a t the t o t a l of twelve would have  been e x a c t l y evenly d i s t r i b u t e d about the armature winding, i t was  d e c i d e d t h a t t h i s would be extremely d i f f i c u l t , due  to the  n e c e s s i t y i n such case of tapping conductors near the bottom of the s l o t s .  I n s t e a d , a n e a r l y even s p a c i n g was  the conductor tapped was  used i n which  t h a t one l e a d i n g d i r e c t l y to a com-  mutator segment; t h i s made more r e a d i l y p o s s i b l e the l o c a t i o n of  the-proper conductors, though even so t h i s proved a, r e a l  difficulty.  Due  to the s m a l l s i z e of the conductors and  their  consequent c r o s s i n g of one a n o t h e r i n the s l o t s ^ the o n l y method o f l o c a t i n g the c o r r e c t ones was  by the use of a  duetor*  Using s p e c i a l l y developed s h a r p ' p o i n t s t h a t would not damage the c o t t o n c o v e r i n g of t h e conductors*, and s e t t i n g - o n e s e t of p o i n t s on the a p p r o p r i a t e commutator b a r the r e s i s t a n c e s were .measured t o l o c a t e the d e s i r e d conductor..  The  s p a c i n g of these  connections and the arrangement of the removable e q u a l i z e r s i s shown i n the appendix A*  • ' -  The p o s s i b i l i t y of i n t r o d u c i n g the e q u a l i z i n g at  the commutator end was  :  -  connections  c o n s i d e r e d , but as the f o u r o r i g i n a l  connections were a t the back t h i s i d e a was  discarded.  The f i r s t s t e p i n the experimental work was  to compare the  g e n e r a l o p e r a t i o n of the machine on v a r y i n g l o a d s w i t h d i f f e r e n t numbers of e q u a l i z i n g c o n n e c t i o n s .  I n o r d e r to render  d i t i o n s as severe as p o s s i b l e the pole frame was  con-  shimmed up f o r  most o f the t e s t s to g i v e the maximum e c c e n t r i c i t y p o s s i b l e , w i t h i n l i m i t s of s a f e t y .  The  normal a i r - g a p as measured with -  f e e l e r s (not t h a t g i v e n i n the s p e c i f i c a t i o n s ) was  0.068 i n c h e s ,  and w i t h maximum e c c e n t r i c i t y the average l e n g t h of two gaps was  0.096 Inches and  inches* - Without any was  t h a t o f the two bottom gaps 0.040  shims the magnetic balance  of the machine  very good; i n f a c t the maximum • c i r c u l a t i n g c u r r e n t through  the brushes was to  top  0.08  ampe., which i s n e g l i g i b l y s m a l l compared  the f u l l l o a d c u r r e n t of the machine of 16 amps.  This  was  when-,no.-equalizers.- were used. F o r a l l the t e s t s the machine was and  s e p a r a t e l y e x c i t e d from the.. 250  f i r s t tests. the. Sciirage motor was  d r i v e n as a  generator  volt batteries.  used and  For  the  i t proved to be  i d e a l f o r the purpose because of i t s speed, range; f o r most of the t e s t s , however, a three phase i n d u c t i o n motor had used,.and consequently g e n e r a t o r l o a d was  there was  increased.  to be  a s m a l l drop i n speed as  'This drop was  the  not s u f f i c i e n t  to  a f f e c t the r e s u l t s a p p r e c i a b l y so no c o r r e c t i o n f o r i t i s made In the c a l c u l a t i o n s * at  times due  frequency,  Considerable  d i f f i c u l t y was  encountered  to sudden f l u c t u a t i o n s i n the supply v o l t a g e  and c e r t a i n o f the i n c o n s i s t a n t readings may  a t t r i b u t e d to t h i s cause.  Loading of the generator was  and  be done by  means of l o a d i n g r h e o s t a t s , and b e s i d e s the ammeter .in the l o a d c i r c u i t ammeters were p l a c e d i n each of the connections brushes of l i k e p o l a r i t y .  From these readings  between •  can be c a l c u l a t e d  the c i r c u l a t i n g c u r r e n t s , these b e i n g the u s e l e s s c u r r e n t s which  f l e w t h r o u g h the b r u s h e s and b r u s h c o n n e c t i o n s .  ISumexxeally-  they are h a l f the d i f f e r e n c e o f the c u r r e n t s s u p p l i e d by p a i r s of brushes.of  like  polarity.  The accompanying graphs show the v a r i a t i o n i n the  two  c i r c u l a t i n g c u r r e n t s on a base o f l o a d c u r r e n t , u s i n g d i f f e r e n t numbers o f e q u a l i z i n g c o n n e c t i o n s .  T y p i c a l s h e e t s of t e s t r e -  s u l t s from which these c u r v e s have been p l o t t e d a r e shown i n a p p e n d i x B.  The  f i r s t s e t of c u r v e s c o v e r s  threecases;  f i r s t l y , t h a t i n w h i c h no e q u a l i z i n g c o n n e c t i o n s a r e u s e d , secondly,- t h a t i n w h i c h t h e r e a r e f o u r such c o n n e c t i o n s ,  and  l a s t l y , t h a t i n w h i c h t h e r e tire t w e l v e .  inter-  The  r e s u l t s are  e s t i n g as they i n d i c a t e p o o r e r o p e r a t i o n w i t h f o u r c o n n e c t i o n s than w i t h none a t a l l .  On the o t h e r hand t h e r e seems l i t t l e  to  choose between the use o f twelve c o n n e c t i o n s and o p e r a t i o n w i t h o u t any, a t ' l e a s t as r e g a r d s the c i r c u l a t i n g c u r r e n t s .  On  l i g h t l o a d s these c u r r e n t s a r e c o n s i d e r a b l y s m a l l e r ' w h e n u s i n g the t w e l v e c o n n e c t i o n s t h a n when t h e y a r e d i s p e n s e d - w i t h y in set  the v i c i n i t y of f u l l : l o a d the o p p o s i t e i s t r u e .  The  but second  of c u r v e s a f f o r d s a comparison o f the c i r c u l a t i n g c u r r e n t s  when u s i n g ten and  twelve c o n n e c t i o n s , and i n t h i s case  i s not much d i f f e r e n c e i n f a v o u r o f the l a r g e r number.  there With  o n l y ten c o n n e c t i o n s , the s p a c i n g i s n e c e s s a r i l y r a t h e r uneven. I t would appear from these c o n s i d e r a t i o n s t h a t a f u r t h e r i n c r e a s e i n the number of e q u a l i z e r s beyond t w e l v e would improve o p e r a t i o n somewhat, though p r o b a b l y n o t to any g r e a t e x t e n t .  I n any case the v a l u e s  o f c i r c u l a t i n g c u r r e n t s a r e n o t exces-  s i v e l y l a r g e comparing them w i t h the f u l l l o a d c u r r e n t o f t h e machine o f 16 amps.  Perhaps i t would have been b e t t e r t o p l o t  the c u r v e s on a base o f t h e normal l o a d c u r r e n t p e r b r u s h , o r h a l f the load c u r r e n t , as t h i s presents  t h e case i n a t r u e r  .light.., C e r t a i n i t i s t h a t a s t u d y o f these c u r v e s c o n v i n c e s one t h a t t h e r e i s a marked d i f f e r e n c e between t h e f o u r - p o l e machine and those w i t h s i x or more p o l e s , f o r w i t h these l a t t e r , t h e I n t r o d u c t i o n o f e q u a l i z i n g c o n n e c t i o n s I n v a r i a b l e improves operation.  I.t would appear t h a t i t i s unnecessary and i n a d -  v i s a b l e t o use e q u a l i z i n g c o n n e c t i o n s on a n o r m a l l y ed four-pole machine.  construct-  While the values of the c i r c u l a t i n g  currents are quite appreciable,  i t must be remembered t h a t t h e  case s t u d i e d i s an extreme one and c e r t a i n l y n o t o f t e n  likely  to be p a r a l l e l e d .in p r a c t i c e . I t w i l l be n o t i c e d that, i n t h e case i n which no e q u a l -  izers, are used, t h e r e i s n o t much v a r i a t i o n i n the, v a l u e s o f the two c i r c u l a t i n g c u r r e n t s as the load i s v a r i e d . e x p l a i n e d by the  This i s  f a c t t h a t t h e r e s i s t a n c e o f the. armature to  the c i r c u l a t i n g c u r r e n t s i s n o t a f f e c t e d by changing v a l u e s o f  load current.  I n any path i n t h e armature t a k e n by a c i r -  c u l a t i n g c u r r e n t , the  r e s i s t a n c e drop i n one h a l f due t o l o a d  c u r r e n t i s i n the opposite  d i r e c t i o n t o t h a t i n the o t h e r  so the n e t e f f e c t I s z e r o .  The f a c t t h a t t h e r e i s some  half,  v a r i a t i o n i n t h e c i r c u l a t i n g c u r r e n t s i s p r o b a b l y due to b o t h  a change i n b r u s h c o n t a c t r e s i s t a n c e and a change i n the p o l a r f l u x e s as a r e s u l t o f t h e o r d i n a r y armature r e a c t i o n e f f e c t ; The q u e s t i o n o f the e f f e c t on t h e c i r c u l a t i n g c u r r e n t s o f an incres.se i n t h e r e s i s t a n c e o f the e q u a l i z i n g c o n n e c t i o n s was o n l y c o n s i d e r e d f o r t h e ease i n w h i c h f o u r c o n n e c t i o n s wer used.  W i t h a 0.2 ohm. e x t e r n a l r e s i s t a n c e i n each e q u a l i z e r  t h e r e was n o t much d i f f e r e n c e i n t h e c i r c u l a t i n g c u r r e n t s , and i f a n y t h i n g , t h e r e was a s l i g h t Improvement i n t h a t r e g a r d . The  e q u a l i z i n g c u r r e n t s themselves dropped o n l y about b%  a l t h o u g h t h e r e s i s t a n c e added was l a r g e ; t h e t o t a l armature r e s i s t a n c e i s o n l y 0.4 ohm.  The e q u a l i z i n g c u r r e n t s must be  l i m i t e d l a r g e l y by i n d u c t a n c e , and hence t h e s i z e o f t h e connections i s probably not very important,  though o f c o u r s e ,  they must be o f s u f f i c i e n t c r o s s s e c t i o n t o p r e v e n t  over-  heating. C o n s i d e r a b l e d i f f i c u l t y was e n c o u n t e r e d i n the t e s t s due to t h e c o n t a c t drop between the c o n t a c t s and t h e s l i p - r i n g s , and a t t h i s j u n c t u r e i t m i g h t be w e l l t o go i n t o t h i s  matter  w i t h a v i e w t o showing t o what an e x t e n t i t may a f f e c t t h e v a l u e o f the r e s u l t s .  Although  t h e c o n t a c t drop i s n o t normal-  l y l a r g e i t may beeorae a p p r e c i a b l e when s p a r k i n g o c c u r s ; and there i s not only the i n c r e a s e d r e s i s t a n c e of the e q u a l i z i n g connections  t o c o n s i d e r b u t a l s o , and what i s p r o b a b l y more-  important,  the a d d i t i o n a l unbalance o f t h e armature c i r c u i t .  The  l a t t e r i s due t o t h e f a c t t h a t t h r e e o r e v e n , a l l . f o u r . o f  the opened c o i l s a r e i n one p a r a l l e l p a t h o f the armature, so  t h a t p r a c t i c a l l y a l l the I n c r e a s e d r e s i s t a n c e i s i n t h a t one path*  T h i s must l e a d t o a f u r t h e r change i n the: c u r r e n t d i s -  t r i b u t i o n , with the path of increased resistance .shirking i t s share o f t h e l o a d .  I n v i e w o f t h i s , i t was f i n a l l y  decided  to make s o l i d c o n n e c t i o n s between s l i p - r i n g s when p o s s i b l e and thus a v o i d t h e c o n t a c t drop c o m p l e t e l y . F o r t a k i n g o s c i l l o g r a m s , however, i t was n e c e s s a r y  to use  c e r t a i n o f the c o n t a c t s , and f o r t h e o s c i l l o g r a m s to be o f any v a l u e t h e c o n t a c t d r o p had t o be l o w . a l l y -equipped slip-rings.  The machine was o r i g i n -  w i t h s o f t copper c o n t a c t s to r u n on t h e b r a s s The c o n t a c t f i n g e r s were o f r a t h e r s o f t b r a s s and  c o n s e q u e n t l y no g r e a t p r e s s u r e , was p o s s i b l e ; moreover, any attempt  t o change t h e p r e s s u r e by t u r n i n g t h e h o l d e r f o r the  c o n t a c t f i n g e r s r e s u l t e d i n t h e c o n t a c t s r i d i n g on one edgeonly.  W i t h t h i s arrangement s p a r k i n g was v e r y common.  In  o r d e r t o i n c r e a s e t h e c o n t a c t pressure:, w i t h t h e i d e a o f r e d u c i n g the c o n t a c t drop, e x t r a p i e c e s o f s p r i n g phosphor b r o n z e were made t o r e i n f o r c e t h e c o n t a c t f i n g e r s .  The i n -  c r e a s e d p r e s s u r e d i d improve t h e c o n t a c t t e m p o r a r i l y , b u t a f t e r a s h o r t r u n t h e c o n t a c t became b l a c k e n e d bad as e v e r .  and s p a r k i n g was as  The b l a c k e n i n g was due t o copper dust and  . v a s e l i n e , the l a t t e r b e i n g n e c e s s a r y  to keep down t h e tempera-  ture a t the c o n t a c t . The n e x t attempt a t i m p r o v i n g t h e c o n t a c t s was w i t h the use o f c a r b o n b r u s h e s b u t these proved u n s a t i s f a c t o r y .  They  have t h e g r e a t advantage t h a t they q u i c k l y wear t o t h e shape o f  the B l i p - r i n g s , and moreover they a r e s e l f - l u b r i c a t i n g , b u t the c o n t a c t drop i s n o r m a l l y s e v e r a l times t h a t £or a copper brush.  The c a r b o n used was r a t h e r s o f t and i t may be t h a t a  somewhat h a r d e r c a r b o n o r a m i x t u r e o f carbon and copper would prove c o n s i d e r a b l y b e t t e r . The b e s t r e s u l t s were o b t a i n e d w i t h b r a s s c o n t a c t s .  The  c o n t a c t p i e c e i t s e l f was about h a l f an i n c h l o n g , t h r e e - , e i g h t h s wide and o n e - e i g h t h  thick.  T h i s was s o l d e r e d to a  v e r y f l e x i b l e s t r i p o f shim b r a s s which m e r e l y h e l d i t i n position.  The p r e s s u r e a t the c o n t a c t came from a p i e c e of  s p r i n g phosphor b r o n z e p l a c e d behind and b e a r i n g on t h e m i d d l e  the t h i n c o n t a c t  of the contact p i e c e .  finger  The r e a s o n  f o r t h i s arrangement was t h a t t h e t h i n s t r i p s u p p o r t i n g the c o n t a c t p i e c e would a l l o w i t t o l i e f l a t a g a i n s t t h e s l i p - r i n g , no m a t t e r how t h e s u p p o r t f o r t h e c o n t a c t s was t u r n e d t o change the pressure.  The s l i p - r i n g s were g i v e n a t h i n c o a t i n g o f  v a s e l i n e w h i c h , even w i t h l a r g e p r e s s u r e , k e p t down the temperature t o a reasonable v a l u e .  W i t h t h i s arrangement t h e  machine c o u l d be o p e r a t e d f o r q u i t e a time w i t h o u t t h e c o n t a c t s b l a c k e n i n g and consequent s p a r k i n g . I n o r d e r t o g a i n an i d e a o f t h e a c t u a l c o n t a c t d r o p , one of t h e s l i p - r i n g s was f u r n i s h e d w i t h a. second b r u s h ,  suitably  I n s u l a t e d , and a d i r e c t c u r r e n t c i r c u i t completed through the the two. W i t h t h e machine d r i v e n a t normal., speed.,* and o f course u n e x c i t e d , s i m u l t a n e o u s were t a k e n .  r e a d i n g s o f c u r r e n t and v o l t a g e  The c o n t a c t drop f o r the two b r u s h e s was o f the  - 13 o r d e r o f 0*3 v o l t s * v a r y i n g f r o m 0.24 w i t h 1.2 amps»> f l o w i n g to 0.34 w i t h 7*0 amps.  Thus, assuming the drop a t each c o n t a c t  to he t h e same, we have a v a r i a t i o n o f the- c o n t a c t drop from 0.12 v o l t s t o 0.17 v o l t s w i t h a c u r r e n t v a r i a t i o n f r o m 1.2 t o 7.0 amps.  T h i s i s s t i l l r a t h e r more than was hoped f o r . h u t  i t was f o u n d t h a t by e x e r t i n g e x t r a p r e s s u r e , s a y w i t h a p e n c i l h e l d behind the c o n t a c t , the v o l t a g e drop, was reduced volt.  t o 0.01  T h i s p r o v i d e s an easy way o f o b t a i n i n g a c c u r a c y i n the  • o s c i l l o g r a m s as t h i s e x t r a p r e s s u r e i s r e a d i l y a p p l i e d w h i l e .they a r e b e i n g t a k e n .  The b i g decrease  i n the c o n t a c t drop  w i t h i n c r e a s e d p r e s s u r e would seem t o i n d i c a t e t h a t t h e f i l m o r d i n a r i l y e x i s t i n g i s b r o k e n down. B e f o r e g o i n g on t o an e x a m i n a t i o n  o f t h e o s c i l l o g r a m s . , we  w i l l p r o c e e d w i t h a s t u d y o f the case I n which no e q u a l i z i n g c o n n e c t i o n s a r e used, i n l i g h t o f t h e t h e o r y o f Dr. L u l o f s .  In  the f i r s t p l a c e , and i n as b r i e f a manner as p o s s i b l e , h i s •general e x p r e s s i o n s f o r b r u s h c i r c u l a t i n g c u r r e n t s w i l l be developed, and then an a t t e m p t w i l l be made t o a p p l y these to the case we a r e i n v e s t i g a t i n g . C o n s i d e r the e f f e c t o f an armature e c c e n t r i c i t y w h i c h makes a l l t h e gaps unequal..  We. w i l l n o t be v e r y f a r wrong i f  we assume t h a t t h e i n c r e a s e i n f l u x f r o m H v a l u e and the decrease  x  over i t s normal  from IT, a r e e q u a l , a l t h o u g h due t o the  s a t u r a t i o n o f t h e i r o n t h i s w i l l n o t be e n t i r e l y t r u e . same h o l d s f o r the f l u x e s i n S, and S^.  The  BIAGRAMATIC REPRESEKTATIOIT <&, TYPICAL L&S-SflKDIHG. (Armature back c o n n e c t i o n s ..shown d o t t e d . ) "Making use o f t h i s assumption we may then c o n s i d e r the f l u x d i s t r i b u t i o n t h e same as w i t h e q u a l gaps, e x c e p t t h a t we now have two superimposed normal p o l a r f l u x e s , ones, as shown, n strengthen H  z  l  fluxes n  x  n, and  s, as w e l l as t h e  \7ith the t o p gaps l o n g e r than the l o w e r  n, must be d i r e c t e d from  t o IT, so as to  b y t h e same amount as IT, i s weakened; s i m i l a r ,  c o n s i d e r a t i o n s show t h a t s The armature  t  s, must go from S  x  x  t o S, .  i s then n o t o n l y r o t a t i n g i n i t s normal  b u t a l s o i n the two f i e l d s n n, and s the e f f e c t o f n  x  L  s,  .  field,  Consider f i r s t  n, ; as t h e armature c o n d u c t o r s r o t a t e i n t h i s  f i e l d t h e r e a r e 21. II, F.s i n d u c e d i n them which a r e down under 17 and up under U . v  r  As a r e s u l t , c u r r e n t s flow through t h e  ,.two p a r a l l e l p a t h s i n t h e armature., and t h e c i r c u i t i s comp l e t e d t h r o u g h t h e b r u s h e s B a n d B and t h e b r u s h c o n n e c t i o n . The d i r e c t i o n  o f t h e c u r r e n t s i s down under II, , up under 1 ^  and b o t h up a n d down under the south, p o l e s ; t h i s l a s t i s a p p a r e n t i f we f o l l o w t h e w i n d i n g t h r o u g h from one p o s i t i v e b r u s h to t h e o t h e r .  JTow examine t h e armature r e a c t i o n  effect  of these c u r r e n t s ; o b v i o u s l y t h e r e i s an II. l\. F. s e t up i n the d i r e c t i o n S, S s  x  L  and., f o r t h e p r e s e n t n e g l e c t i n g the. c r o s s f l u x  , t h i s would produce  a f l u x i n t h e same d i r e c t i o n  M. H* ,F. ••• B u t the •armature i s r o t a t i n g I t s •••.own ampere-turns* under S  z  as t h e  I n t h i s f i e l d s e t up b y  a n d E:» l . . J , s a r e induced- w h i c h a r e down  a n d up under 3,.  As b e f o r e , c u r r e n t s f l o w down under  S-x* up under B| and I n b o t h d i r e c t i o n s  under the o t h e r two p o l e s .  The M. II. F. due t o these c u r r e n t s i s i n t h e o p p o s i t e  direction  to n ^ n , ' , so t h a t due t o the unequal a i r - g a p s c i r c u l a t i n g c u r r e n t s f l o w , which t e n d p a r t i a l l y to wipe o u t the e f f e c t which caused them.  L e t us now c o n s i d e r t h e combined e f f e c t o f t h e two c r o s s fields.  H a v i n g gone t h u s f a r , i t . is•• apparent t h a t f o r t h e pur-  pose o f d e t e r m i n i n g t h e v o l t a g e s g e n e r a t e d between b r u s h e s o f l i k e p o l a r i t y . , i t i s s i m p l y n e c e s s a r y t o c o n s i d e r t h e machine as two separate two-pole machines.  I f then 0, i s the r e s u l t a n t  f l u x i n the d i r e c t i o n o f 3J IT, t h e ' I i . If. F. i s : X  ' '  M  P  • B  ~  a  • •-'•16'. - •  0 s ir  5  'I™-~^«««-~--s — — s i n c e 60 x 1 0 60 x 10 8  s  p-s.a«2  How l e t I , be t h e c i r c u l a t i n g c u r r e n t i n t h e armature  which  f l o w s down under IT , and up under H , and 1 ^ he t h e c i r c u l a t i n g z  c u r r e n t w h i c h f l o w s down under S and up under S,. x  The n e t .  f l u x i n t h e d i r e c t i o n o f t h e n o r t h p o l e s ( f r o m 12 t o IT, ) i s ; x  n ^n  r  —  M*.  F . ' due t o  11.  1'j.;  — reluctance of f l u x path  n , n,-  H, .being- the r e l u c t a n c e o f a p a t h through two o p p o s i t e p o l e s , t h e armature,and y o k e .  The r e a s o n f o r u s i n g f i s a p p a r e n t i f  we c o n s i d e r the d i r e c t i o n s o f t h e c i r c u l a t i n g c u r r e n t s i n a lap-winding*  Hue t o t h e f a c t t h a t c u r r e n t s , e q u a l t o - I f l o w z  i n b o t h d i r e c t i o n s under !•!, and IT the II* E. F.s due t o these x  p a r t s o f t h e w i n d i n g w i p e each o t h e r o u t . 0.4 ir H I . f l u x from S , t o S i s *.  Similarly, the net  x  Hence 3,  r  ( n ^ , -  R  t  JsoTlo'  10 r, 8  where r  (  i s t h e r e s i s t a n c e ' t o 1, o f t h e p a t h t h r o u g h t h e  armature, b r u s h e s , and b r u s h c o n n e c t i o n s .  60 x 10 r^ 'This g i v e s us s i m u l t a n e o u s e q u a t i o n s from which I , and I may s  be c a l c u l a t e d .  On s o l v i n g we  Ri  I -I  -  n,. n,  2  -  get:  +  0.1  .^ntrrvmUHUAU a MnOTwH ]»»'•»«• Wi»^«mir.)*]ii>.M^^  60 x 1 0 r-« R. 2 H  I, -  +  tlx n,  itx  ^^-^ 60 x 1 0  2  X  6  r *  6  iw-mr-^..••.nr.-n.ij -  1  And  s 2. s,  M  (.0,1 80 x 1 0  s  <L'l-^ 2.. K  z  60 x 1 0  II•>....  K  "m,.Int., - n. -,-mf m  K R  s  s3 S  i_  .—..  r  x  |  _  lO.l* ) 60 x 1 0  «,  9  a  _ 6  . ..  ¥  x  Ri  x,  These are the expressions developed by Dr. Lul'ofs though not in q u i t e the same form as he gave them; he used  the.,  q u a n t i t y which he c a l l e d magnetic r e s i s t a n c e , but magnetic r e l u c t a n c e has been used here as i t is. now  commonly used,  o r i g i n a l papers Br. L u l o f s d i s c u s s e d a numerical  in his  example of  e c c e n t r i c i t y i n the d i r e c t i o n of one p a i r of opposite p o l e s , but our case of e c c e n t r i c i t y a l o n g a n e u t r a l a x i s i n t r o d u c e s new  difficulties.  and s  L  I n the f i r s t p l a c e the c r o s s f i e l d s n  S | must be c a l c u l a t e d , a n d then we w i l l  mine values of the r e l u c t a n c e s R, and R r , and x^.  z  proceed  f o r 11  and  n,  to d e t e r -  and. the r e s i s t a n c e s  The f i n a l step w i l l be a s u b s t i t u t i o n o f  values In the expressions  1  these  I^.  I n the c a l c u l a t i o n of the c r o s s f i e l d s we w i l l assume them equal, as t h i s Is t h e o r e t i c a l l y the oase when the pole frame i s r a i s e d v e r t i c a l l y .  18 Suppose <j>, i s t h e f l u x i n t h e two top p o l e s and 0z t h e f l u x i n t h e two "bottom p o l e s , w i t h t h e r e l u c t ances a s shown I n t h e s k e t c h , 2.  fig*  R" I s t h e r e l u c t a n c e o f each  of the t o p gaps and R  f  the r e l u c t -  ance o f each o f the b o t t o m gaps... 2?ig. i2. DIAGRAM SHOWING KAGHBT1C  Then  R3SLUC TAKGES  jv  ijfc  2 (H' *  Rs) (See Appendix 0  Also,, a s s i i m l n g a l i n e a r open c i r c u i t c h a r a c t e r i s t i c , , we have —  0  where p i s t h e n o r m a l f l u x p e r p o l e o f t h e  ^machine-;:- •• How i n o r d e r t o d e t e r m i n e t h e v a l u e s o f the r e l u c t a n c e s of, t h e v a r i o u s parfes o f t h e magnetic c i r c u i t we make use o f the open c i r c u i t c h a r a c t e r i s t i c  o f t h e machine.  F o r the  .machine . i n q u e s t i o n t h e f i e l d - c u r r e n t i s 0...99 amps* w i t h an open c i r c u i t - v o l t a g e o f 250 v o l t s a t 1200 R*P,.Lt.  There a r e  1950 f i e l d t u r n s p e r p o l e , so t h e number o f ampere-turns p e r p o l e i s 1930,..  Mith  a l l t h e gaps on t h e machine e q u a l we have  t h e - n o r m a l f l u x p e r p o l e • 4 from t h e -general K^M,.!?*. ;equation;; _60_ P  X T  10  ** 20 —  ."  d.765 x 1 0 /lines,. 6  0,4  , iLt .we have toes 4 <+  « K  I  IT  Q.4 ^ x • 1950 0.785 x 1 0  «t R^ + 4  6  = 3170 x 1 0 u n i t s o f reluctance I t i s now n e c e s s a r y  to separate  out the v a r i o u s terms o f  The r e l u c t a n c e l\ o f the gap i s e a s i l y  this expression.  o b t a i n e d from the gap d i m e n s i o n s . R  "  •  T  0.063 x 2.54 • ' 18.3 " x 2.54  s  A  s  i  14G3 x 1 0 ^  I n t h i s . e a s e A i s t h e e f f e c t i v e a i r - g a p a r e a and i t i s o b t a i n e d from the a p p a r e n t a i r - g a p a r e a w i t h t h e use o f C a r t e r * s f a c t o r . :  Y/~e have now* f** f  +  R  B 3  1707 x 10" ' fe  VJe proceed t o c a l c u l a t e the v a l u e o f B , t h e r e l u c t a n c e 3  of a p o l e .  .The r a d i a l - l e n g t h o f the p o l e i s 3*92 i n c h e s and  the c r o s s a r e a 11.55 square Inches.  Hence the normal f l u x  d e n s i t y , n e g l e c t i n g leakage i s : ft The name o f t h i s u n i t was f o r m e r l y the o e r s t e d b u t the I n t e r n a t i o n a l E l e c t r o t e c h n i c a l Commission i n 1930 adopted o e r s t e d f o r the u n i t o f m a g n e t i z i n g f o r c e , l e a v i n g the u n i t o f magnetic r e l u c t a n c e nameless.  - 21 -  .. •  s  SJJZ§§-JLJ£ lines 6  p e r square i n c h  11.56 1G.25 k i l O ' - l i n e s p e r s q u a r e cm.  s  Prom t h e m a g n e t i s a t i o n c u r v e f o r c o m m e r c i a l s h e e t s t e e l t h e ampere-turns p e r cm. a t t h i s f l u x d e n s i t y a r e 4.1; . t h i s g i v e s a t o t a l of 4.1 x 3.92 x 2*54  Hence  K  T h i s makes  of765 x  -  3  40.3 ampere-turns p e r p o l e .  s  f ^ *f  io? * 53  5  1654 x l(T  2  1 0 _ b  .  fc  '^e can now go h a c k t o our d e t e r m i n a t i o n of the v a l u e s of  0, and 0  i i n t h e f i r s t p l a c e we w i l l assume t h a t the r e l u c t -  X  ances R'&nd R  a r e i n t h e same p r o p o r t i o n to R a s a r e the c o r -  u  responding l e n g t h s o f t h e gaps. t e r m i n e them more a c c u r a t e l y  I t would be p o s s i b l e t o de-  b y m a k i n g use of Garter's f a c t o r ,  as was done i n c a l c u l a t i n g t h e e f f e c t i v e gap a r e a w i t h no eccentricity,  b u t the refinement would p r o b a b l y n o t be i n  k e e p i n g w i t h the r e s t o f t h e c a l c u l a t i o n .  Taking t h e average  l e n g t h o f each o f t h e upper gaps a s 0.096 i n c h e s and t h a t o f e a c h o f t h e l o w e r a s 0*04 I n c h e s t h e v a l u e s o f R* and R" a r e : R'  H  H  E  N  C  A  • 0-040 x 1463 x 10" = . ..• 0.068 .  " 4L, '07.  6  "  0.068  S  4 x 1654 4 x 1654 but  x  1  4  6  3  x  1  0  ^~  860. x 10"  2066. x 10"  * 2(860 * 55) t 2(2066 + 53} ^  fe  b  s ,4221 5427  0.765 x 10  fe  22  —  1  T h i s g i v e s 0, s 0.669 x  10  , fc- 0.-861 x  10  whence n  n, =.  z  o^ s,  6  fe  0.096 x 1 0 l i n e s . f a  I t might appear t h a t In - these c a l c u l a t i o n s the q u e s t i o n of leakage  f l u x , was  n e g l e c t e d c o m p l e t e l y hut t h i s i s not the case.  S i n c e we a r e working from the 0. C. C. the v a l u e o£  will  a c t u a l l y i n c l u d e leakage paths,as w e l l as the paths through  the  yoke* There appear to he two  simple means by which we  e x p e r i m e n t a l l y t h i s t h e o r e t i c a l l y determined f l u x , and thus get some i d e a of how assumptions have l e a d us.  v a l u e of the c r o s s  f a r from the t r u t h  our  The most obvious and simple d e t e r -  m i n a t i o n of the c r o s s - f l u x i s by means of the v o l t a g e s between brushes  can check  of l i k e p o l a r i t y .  be. talien w i t h the f i e l d e x c i t e d by  open-circuit  The r e a d i n g s should  the normal no-load f i e l d  c u r r e n t , and not when a v o l t m e t e r a c r o s s the t e r m i n a l s of the machine r e g i s t e r s normal v o l t a g e .  T h i s l a t t e r i s no  as with the c o n n e c t i o n s between brushes  criterion,  of l i k e p o l a r i t y broken  there a r e f o u r d i f f e r e n t v o l t a g e s between v a r i o u s s e t s of p o s i t i v e and n e g a t i v e  brushes.  A c t u a l l y upon measurement, the v o l t a g e s between brushes l i k e p o l a r i t y , w i t h 0.96 from A t o C and 29.8  amps, i n the f i e l d , were 29.5  v o l t s from D to B.  T h i s shows' a  ence of o n l y about 1% i n the c r o s s f l u x e s and f o r our we w i l l take the average, 29.65 v o l t s .  How  of  volts differpurposes  substituting  this  v a l u e i n our genera,! E. !£. J?. e q u a t i o n we w i l l  g e t the v a l u e of  the c r o s s - f l u x :  a  60 x 1 0  2  29.65 x 60 x .10 1632 x 1200  0 = 2 *  0.0908 x 10  s  6  8  lines.  T h i s cheoks reasonably w e l l w i t h the p r e v i o u s l y value n  2  n, s  s  x  s, s  determined  0.096 x 1 0 l i n e s . f e  A second check on the value o f the c r o s s - f i e l d should he p o s s i b l e by moans o f an o s c i l l o g r a m o f generated c o l l v o l t a g e . V/ith the brushes l i f t e d o r the connections between brushes o f like  p o l a r i t y broken, there w i l l be no' c u r r e n t s f l o w i n g i n the  armature, p r o v i d e d , o f c o u r s e , t h a t t h e r e a r e no e q u a l i z i n g connections.  Hence i f an o s c i l l o g r a m can be taken of the v o l t -  age generated  i n one conductor  the o r d l n a t e s s h o u l d r e p r e s e n t  not o n l y v o l t a g e b u t a l s o f l u x d e n s i t y to a d i f f e r e n t But  the i n t e g r a l o f f l u x d e n s i t y over the ISO  scale.  electrical  degrees between brushes g i v e s the t o t a l f l u x from any p o l e . Hence the r a t i o of the areas under two p a r t s of the generated v o l t a g e o s c i l l o g r a m , c o r r e s p o n d i n g t o a top and bottom pole s h o u l d be the same as the r a t i o of the f l u x e s i n those p o l e s . Bue  to the s m a l l s c a l e o f the o s c i l l o g r a m s taken,  difficult  i t was r a t h e r  to measure the areas a c c u r a t e l y , b u t a value of the  r a t i o o f 1.33 was o b t a i n e d which i s p r o b a b l y f a i r l y That I s J ~ - = 1-33  good.  ..  •  ' '  • - 24. »• . •» . Cik .„ o.765 x 10  But as "before  fc  From these two we g e t <jh = 0.874 x 1 0 0.657 x 10 and hence  n , •=  4  fe  s e, «. 0.108 x 1 0 ' l i n e s t  T h i s v a l u e i s h i g h b u t the e r r o r i s n o t more than might be exp e c t e d c o n s i d e r i n g the s m a l l a r e a s which had t o bo measured. We can now go on t o a d e t e r m i n a t i o n o f the r e l u c t a n c e s R, and R .  Considering f i r s t  x  the r e l u c t a n c e o f a c r o s s - f l u x path  when there are no o t h e r f l u x e s present, we see t h a t i t i s equal  2 E +  to  3  or  U " + R ) + i\+ r  (See f i g . 2 . )  T  s i n c e R" + R' » 2 B R^.2"" H,* SR * ' B * T #  ' v:-  With t h e machine f u l l y e x c i t e d this, e x p r e s s i o n I n v o l v e s an ^approximation. due t o t h e s a t u r a t i o n o f the iron,, b u t we w i l l use I t f o r want o f a b e t t e r . S u b s t i t u t i n g a c t u a l v a l u e s we g e t : R , sr R a 2x53xl0"  6  x  +  2x1463x10"" 4- 4xl654xl0"  fe =  9648xl0"  fc  How l e t us proceed with a d e t e r m i n a t i o n of r , and r , x  keeping i n mind t h a t they were d e f i n e d as the r e s i s t a n c e s t o I, and !  L  o f t h e paths through the armature, brushes, and brush  connections.  I f a?* i s the -aaanaiaia?© -reeiS'tance. as.:-measured i n  the u s u a l way* the r e s i s t a n c e o f each q u a r t e r of the armature Is 4 ra- -  Then, i f r  i s the brush r e s i s t a n c e , the t o t a l r e -  s i s t a n c e to a c i r c u l a t i n g c u r r e n t I i n the brushes w i l l be 4 r  +  2 r. .  Hence the r e s i s t a n c e to a c u r r e n t I i n each  branch w i l l he 8 r ^ +  4  -. From a c t u a l measurement r ^ s  0.40 ohms.,, h u t a s u i t a b l e v a l u e o f r ^ I s n o t so e a s i l y o b t a i n able.  However we w i l l n o t be f a r wrong i f we assume a one  v o l t d r o p a t each b r u s h c o n t a c t w h i c h , f o r a c u r r e n t o f say 2 amps., g i v e s a v a l u e of r ^ of 0.5 ohms. Hence  r, s  r  z  -  8 x 0.4 + 4 x 0.5  •s  5.2 ohms.  How s u b s t i t u t i n g i n the e x p r e s s i o n f o r I , we get 9648 x 10~ x 0.096 x 1Q fe  I -  6  +  0.1  1632  -  x 1632 x 1200 x 0.096 x 10* 60  X  10  a  X  5.2  60 x 10* x 5.2 x 9648 x, 10'^ ( Q U ^ x ( 1 6 3 2 ) x 1200 • Tl632) X 1200 60 x 10'x 9648 x 1 0 x 5.2 1  i  «  A  i  q  o  p.,566 +1.893 0.094 + 1.043 T  A L J O  A  r i  -  2.16 amps.  l r 8 9 5 -,0.566 . 1 . 1 7 amps. 0.094 * 1.043  These a r e the c i r c u l a t i n g c u r r e n t s i n the armature the c o r r e s p o n d i n g c u r r e n t s i n the brushes and b r u s h w i l l bo 4.32 and 2.34 amps.  conductors; connectors  These v a l u e s do not agree v e r y  well w i t h the e x p e r i m e n t a l l y determined v a l u e s o f 2.16 and 1.40 amps, though they a r e o f the same o r d e r .  Looking f o r a p o s s i b l e  cause o f t h i s d i s c r e p a n c y we n a t u r a l l y q u e s t i o n the value of r e s i s t a n c e used, b u t an examination and I  z  o f the e x p r e s s i o n s f o r I,  r e v e a l s the f a c t t h a t an e r r o r i n the value of the r e -  s i s t a n c e w i l l n o t r e s u l t i n a n y t h i n g b u t a s m a l l e r r o r i n the currents. to  T h i s i s due to the f a c t t h a t the r e s i s t a n c e appears  the same power i n the more s i g n i f i c a n t terms o f both numer-  a t o r and denominator.  The o t h e r q u a n t i t y which c o u l d cause  - 28 t r o u b l e i s the r e l u c t a n c e . o f the c r o s s path, and an e r r o r i n t h i s case- w i l l r e s u l t I n a l a r g e e r r o r i n the f i n a l  result.  • l o o k i n g f u r t h e r t o more g e n e r a l causes of p o s s i b l e e r r o r wo have to examine the e f f e c t o f s a t u r a t i o n on the r e s u l t s . The  s i m p l e s t way to do t h i s i s by-means o f a h a l f v o l t a g e  test,  f o r a t h a l f v o l t a g e we a r e working on the s t r a i g h t p a r t o f the open c i r c u i t  characteristic.  T h e . c a l c u l a t i o n s a r e s i m i l a r to  those f o r f u l l e x c i t a t i o n so w i l l be o m i t t e d .  ITew v a l u e s of  the c r o s s f l u x and a l s o o f magnetic r e l u c t a n c e and o f r e s i s t ance a r c determined, and then s u b s t i t u t e d i n the g e n e r a l exp r e s s i o n s f o r I, and I  .  x  The r e s u l t i n g v a l u e s a r e :  ••!,.••••«  1.07 amps*  I j." -  0.54' amps.  The b r u s h c i r c u l a t i n g c u r r e n t s a r e then 2.14 and 1.08 amps, as compared w i t h e x p e r i m e n t a l l y amps.  determined v a l u e s  o f 1.15 and 0.86  A g a i n t h e r e i s r a t h e r poor agreement, and i n f a c t  I s much the same d i s c r e p a n c y  there  a s there was w i t h f u l l e x c i t a t i o n .  "This would seem to i n d i c a t e t h a t s a t u r a t i o n d i d not, p l a y an important r o l e , a t l e a s t at no-load. up,  of course,  the values  'As the machine i s loaded  o f the c i r c u l a t i n g c u r r e n t s change  somewhat,, due t o the s a t u r a t i o n o f the p o l e a t the l e a d i n g edge. So f a r no mention has been made o f the i n t e r p o l e s and their possible effects  I n g e n e r a l , when there i s no l o a d c u r -  r e n t f l o w i n g , there a r e two ways i n which the presence of: i n t e r p o l e s might a f f e c t the r e s u l t s ; f i r s t l y ature EJCPS under, the i n t e r p o l e s , and secondly  due t o the armdue to the l e a k -  age f l u x from the main p o l e s t h r o u g h the i n t e r p o l e s changing the main p o l e f l u x d i s t r i b u t i o n . c o n s i d e r e d as limited  The  f i r s t need h a r d l y be  the i n t e r p o l e f l u x i s not only s m a l l , but i s  to the commutation sone.  The  second e f f e c t would a l s o  appear to be s m a l l as, even w i t h the armature e c c e n t r i c , there w i l l never be anything but a s m a l l I I . I f . F* along the poles.  inter-  T h i s I s more obvious i f we c o n s i d e r f i r s t the case of  e q u a l gaps where the• M*  IS* I?.*- t e n d i n g t o produce l e a k a g e  flux  t h r o u g h t h e i n t e r p o l e s I s t h e o r e t i c a l l y a e r o , due  t o the f a c t  t h a t each main p o l e f i e l d c a r r i e s .just s u f f i c i e n t  ampere-turns  to  take the f l u x h a l f way S7e come now  theory.  round a f l u x path.  to the q u e s t i o n of the value of Dr. L u l o f s'  I n the f i r s t p l a c e i t does not In our case g i v e r e -  s u l t s i n c l o s e agreement w i t h experimental be t h a t the w r i t e r has advantage.  r e s u l t s , but i t may  not a p p l i e d the theory to the b e s t  On the other hand i t g i v e s a good p i c t u r e of what  a c t u a l l y takes p l a c e , and  f o r t h i s reason alone i s v a l u a b l e .  As to the p o s s i b i l i t y o f u s i n g a g r a p h i c a l method of d e t e r m i n i n g c i r c u l a t i n g c u r r e n t s , b e g i n n i n g w i t h t h e open c i r c u i t . characteristic,,...  i t i s e v i d e n t ; t h a t i t w o u l d e n t a i l a•• p r o c e s s o f  s u c c e s s i v e approximations. two  interdependent  T h i s , because we are d e a l i n g w i t h  q u a n t i t i e s , namely f l u x and c u r r e n t .  The  m e r i t of L u l o f s ' method l i e s i n the use o f simultaneous equat i o n s , the only way Thus f a r we i z i n g connections  of a v o i d i n g a p p r o x i m a t i o n s of t h i s k i n d .  have not examined the p a r t played by i n a f o u r - p o l e machine.  equal-  U n l i k e the machine  w i t h s i x o r more p o l e s i n which these connections  are respons-  i b l e e n t i r e l y f o r c o r r e c t i n g the f l u x d i s t r i b u t i o n * the f o u r p o l e machine b a s t h e two e f f e c t s a c t i n g t o g e t h e r .  The b e s t  method o f s t u d y i n g these e f f e c t s b o t h s i n g l y and t o g e t h e r i s by means o f o s c i l l o g r a m s showing t h e f l u x d i s t r i b u t i o n around the armature under d i f f e r e n t , c o n d i t i o n s . Since  the S, IT. IP. generated i n a conductor as i t r e v o l v e s  w i t h t o o armature i s p r o p o r t i o n a l to the f l u x d e n s i t y , an o s c i l l o g r a m o f 3.  II. E . would give t o another s c a l e the f l u x  d i s t r i b u t i o n i n the was considered^but troduce  gaps.; . A t of I r s t  the use o f .a -search  n o t o n l y would i t have been d i f f i c u l t  another conductor i n t h e s l o t s , b u t the E,-  coil to i n -  F. around  the c o i l would be the summation o f two E* 11, F.s out o f phase, and  these would have t o be separated  out.  Instead, a s i n g l e  c o n d u c t o r of the armature winding was u t i l i z e d ; f o r t h e purpose one  was s e l e c t e d which connected d i r e c t l y from a commutator  segment to one o f the s l i p - r i n g s a t t h e back of the armature. Then the commutator segment was grounded to the s h a f t by an e a s i l y removable c o n n e c t i o n d i r e c t l y on the shaft.. then o n l y necessary  and a c o n t a c t f i n g e r p l a c e d t o b e a r  To. g e t o s c i l l o g r a m s o f v o l t a g e i t was  to connect the two c o n t a c t f i n g e r s , one  from e i t h e r end o f t h e conductor, of the o s c i l l o g r a p h .  to a h i g h r e s i s t a n c e element  As a p r e c a u t i o n a second c o n t a c t f i n g e r  was p r o v i d e d on t h e s l i p - r i n g so t h a t t h e o s c i l l o g r a p h c u r r e n t was n o t drawn from the same c o n t a c t as the l o a d c u r r e n t . E v i d e n t l y the o s c i l l o g r a m s w i l l not r e p r e s e n t  absolutely  a c c u r a t e l y the f l u x d i s t r i b u t i o n f o r there are e r r o r s i n v o l v e d . In the case i n which there are no c u r r e n t s f l o w i n g i n the armature the only e r r o r i s t h a t due t o the drop a t the c o n t a c t s , and  i t has been shown p r e v i o u s l y t h a t by t a k i n g proper  pre-  c a u t i o n s a t the time of photographing, t h i s w i l l be n e g l i g i b l e . When there are armature c u r r e n t s flowing,. however, there i s the effect  of r e s i s t a n c e drop and i n some cases  inductance  drop to  Now the generated E . M . F. p e r conductor i s approx-  consider.  i m a t e l y 0.6 v o l t s 'and the r e s i s t a n c e drop along the conductor w i t h a c u r r e n t o f 4 amps. f l o w i n g i s only of the order of 0.015 volts  so t h i s e f f e c t may be n e g l e c t e d .  F u r t h e r , when no equal-  i z e r s are u t i l i z e d the conductor c u r r e n t i s p r a c t i c a l l y ; o f cons t a n t value between commutation p e r i o d s , so the e f f e c t of s e l f and mutual inductance izers,  need not.be c o n s i d e r e d .  When u s i n g  equal-  on the other hand, the armature c u r r e n t s a r e f a r from  constant  and as a r e s u l t  w i l l be i n e r r o r .  the v o l t a g e wave of the o s c i l l o g r a m  From an examination of a c t u a l o s c i l l o g r a m s  such as no. 6, i t i s e v i d e n t t h a t the e f f e c t  i s not very l a r g e  and f o r our purposes may be n e g l e c t e d . The  f l u x d i s t r i b u t i o n i n the gaps due s o l e l y to the f i e l d  windings i s shown by o s c i l l o g r a m no. 9, which was taken with the brushes l i f t e d Under these  and no e q u a l i z i n g connections  in circuit.  c o n d i t i o n s no c u r r e n t s can flow i n the armature;  that t h i s i s t r u e i s e v i d e n t i f we remember t h a t the t o t a l  flux  l e a v i n g the armature i s the same as t h a t e n t e r i n g , and hence the summation of the E . M. F.s around the armature w i l l be zero a t  30  -  any  instant.  Oscillogram  9 was  -  the one  used p r e v i o u s l y f o r a  c a l c u l a t i o n of the r a t i o of the p o l a r f l u x e s hut as was  pointed  out the s c a l e i s s m a l l , w i t h the r e s u l t t h a t the measurement o f a r e a i s not v e r y a c c u r a t e .  I t would appear t h a t enlargement  w i t h an o r d i n a r y p r o j e c t o r would o f f e r the b e s t means of i n g dependable r e s u l t s .  Due  t o the e c c e n t r i c i t y of the arma-  t u r e the f l u x i s not o n l y d i f f e r e n t under the top and p o l e s , but  obtain-  bottom  t h e r e i s a v a r i a t i o n i n f l u x d e n s i t y under i n d i v i d -  u a l p o l e s , and  t h i s shows up c l e a r l y i n the  oscillogram.  6  8 7 9  The  V o l t a g e per c o n d u c t o r w i t h no e q u a l i z i n g connecti o n s and a t no l o a d . C o i l c u r r e n t w i t h no e q u a l i z i n g c o n n e c t i o n s and a t no l o a d . V o l t a g e per c o n d u c t o r w i t h b r u s h e s l i f t e d and no e q u a l i z i n g connections.  two p o r t i o n s o f the wave c o r r e s p o n d i n g to the two  shorter  gaps have t h e i r maximum o r d l n a t e s a d j a c e n t to the n e u t r a l a x i s  between these p o l e s , as would be expected.  F u r t h e r , the t o o t h  harmonic o r r i p p l e on t h e c r e s t s o f t h e waves i s more pronounced f o r the s h o r t gaps than f o r the l o n g ones. How l e t us examine the e f f e c t o f b r u s h c i r c u l a t i n g c u r r e n t s on the wave form; i n t h i s case no e q u a l i z i n g are used b u t t h e c u r r e n t s I, and I  x  connections  w i l l c i r c u l a t e through the  ax-mature, b r u s h e s , and b r u s h c o n n e c t i o n s . r e p r e s e n t s these c o n d i t i o n s on no l o a d .  O s c i l l o g r a m no. 8 I n the f i r s t p l a c e i t  w i l l be n o t i c e d t h a t t h e p o l a r f l u x e s a r e now much more n e a r l y e q u a l than i n the p r e v i o u s c a s e ; a c t u a l l y the r a t i o i s o f the. o r d e r o f 1.05 t o 1, though an a c c u r a t e d e t e r m i n a t i o n has n o t been made*  Ho. 8 i s s i m i l a r t o no. 9 i n t h a t t h e s l o p e s o f the  wave tops a r e much t h e same, due t o t h e f a c t t h a t the armature K. M. The to  i s more o r l e s s e v e n l y d i s t r i b u t e d over each p o l e .  r a t h e r l a r g e S. K. F,s i n t h e i n t e r p o l e zones w i l l be due the commutation o f t h e c i r c u l a t i n g c u r r e n t s ; o s c i l l o g r a m no.  9 shows t h a t t h e leakage gibly small.  f l u x through the i n t e r p o l e s i s n e g l i -  O s c i l l o g r a m no. 7 i s one o f c u r r e n t I n an arma-  t u r e c o i l and was taken a t the same time as no. 3; the c u r r e n t s • a r e a c t u a l l y the sum o r d i f f e r e n c e o f 1 are more o r l e s s c o n s t a n t  1  and 1  under any one p o l e .  x  and o f course  Unfortunately  t h e r e i s no r e c o r d o f the commutation p e r i o d . The  e f f e c t of e q u a l i z i n g connections  on the main p o l e  f l u x e s I s shown by o s c i l l o g r a m s no. 4 and no. 17; the f i r s t o f these was taken w i t h a l l twelve e q u a l i z i n g c o n n e c t i o n s i n c i r c u i t and t h e second w i t h o n l y f o u r .  F o r these  oscillograms  - 32 the b r u s h e s were l i f t e d so t h a t the o n l y c u r r e n t s f l o w i n g were equalizing currents.  With only four connections i n c i r c u i t  the e q u a l i z a t i o n o f f l u x i s marked, but i t i s somewhat l e s s t h a n t h a t due  t o the b r u s h c i r c u l a t i n g c u r r e n t s ; w i t h twelve  e q u a l i z e r s the f l u x i s v e r y n e a r l y the same f o r a l l f o u r  2 3 4  poles.  V o l t a g e per c o n d u c t o r w i t h 12 e q u a l i z e r s and a t no l o a d . C o i l c u r r e n t w i t h b r u s h e s l i f t e d and 12 e q u a l i z e r s i n circuit. V o l t a g e per c o n d u c t o r w i t h b r u s h e s l i f t e d and 12 e q u a l izers in circuit. T h i s e q u a l i z a t i o n o f f l u x due  s o l e l y to the e q u a l i z i n g  c u r r e n t s , i s of utmost importance f o r a machine w i t h s i x o r more p o l e s , and has Recently  i n the p a s t been s t u d i e d  mathematically.  R. M. B a k e r , i n a paper e n t i t l e d " E q u a l i z i n g  Currents  i n the Armature o f a D. C. Machine" p r e s e n t e d a t an A. I . 1.  B.  1? 13 12  V o l t a g e per conductor with brushes l i f t e d and w i t h 4 equalizers. V o l t a g e per conductor a t no l o a d and with 4 e q u a l i z e r s . C o i l c u r r e n t a t no l o a d and w i t h 4 e q u a l i z e r s .  c o n v e n t i o n , a t t a c h e d the prob 1 era i n r a t h e r a d i f f e r e n t  way,  u s i n g f o r h i s t e s t s a f o u r - p o l e machine, w i t h a l a r g e number o f equalizing connections.  S t a r t i n g w i t h an o s c i l l o g r a m o f arma-  t u r e c u r r e n t taken w i t h a l l b r u s h e s l i f t e d he i n t e g r a t e d t o g e t the r e s u l t i n g c o r r e c t i v e M. M. P., and showed how the i n e q u a l i t i e s o f s a i n p o l e f l u x were thus smoothed o u t . H i s treatment  i s a p p l i c a b l e , o f c o u r s e , o n l y when a  l a r g e number o f c o n n e c t i o n s  comparatively  i s used, as he takes the o s c i l -  l o g r a m of c u r r e n t i n one c o i l as i t r o t a t e s w i t h the armature,  -.34  -  and t r e a t s i t as a s t e a d y space d i s t r i b u t i o n o f c u r r e n t i n the f o r m o f a c u r r e n t sheet.  l o r o u r c a s e , t h a t I s with a s m a l l  number o f c o n n e c t i o n s , t h i s t r e a t m e n t i s h a r d l y a p p l i c a b l e as the  space d i s t r i b u t i o n of. c u r r e n t i s not c o n s t a n t ,  with re-  s p e c t t o the a r m a t u r e , the wave I s a r e c t a n g u l a r one  with  changes i n o r d l n a t e s a t each e q u a l i s i n g p o i n t ; f u r t h e r , the o r d i n a t e a t any p o i n t i e c o n s t a n t l y v a r y i n g . the  With r e s p e c t to  f i e l d p o l e s we have then a s e r i e s of r e c t a n g u l a r , p u l s a t i n g  waves t r a v e l l i n g a t the speed of the a r m a t u r e . would appear some e x t e n t .  From t h i s , i t  t h a t the c o r r e c t i v e II. IS. 3P. would a l s o v a r y t o I t might i n our case be p o s s i b l e to a r r i v e at a  wave of average c u r r e n t d i s t r i b u t i o n around the armature, and use  t h i s t o f i n d the c o r r e c t i v e M.. M. F. b u t t h e p r o c e s s would  be t e d i o u s and o f d o u b t f u l v a l u e . To.go f u r t h e r back,we m i g h t f r o m p u r e l y t h e o r e t i c a l cons i d e r a t i o n s determine the f i n a l f l u x d i s t r i b u t i o n due c e r t a i n i n i t i a l unbalance. first,  to a  The process would be as f o l l o w s :  assuming s i n u s o i d a l d i s t r i b u t i o n of f l u x i n the gap,,  we would get e x p r e s s i o n s f o r the c u r r e n t s i n a l l s e c t i o n s of the  armature.  Then f o r each s e c t i o n another g e n e r a l e x p r e s s i o n  c o u l d he o b t a i n e d by s i m p l e i n t e g r a t i o n , g i v i n g the average v a l u e of c u r r e n t a t any p o i n t .  I n s e r t i n g numerical values  and  e v a l u a t i n g these l a s t e x p r e s s i o n s f o r a number o f p o i n t s , a wave c o u l d be p l o t t e d g i v i n g the average c u r r e n t d i s t r i b u t i o n . The  next s t e p v/ould be the i n t e g r a t i o n of the c u r r e n t wave t o  g i v e the c o r r e c t i v e 15. II. F. from w h i c h we would p r o c e e d t o  find  t h e new d i s t r i b u t i o n o f  first  approximation^and I t  peat the  process  several  obtained.  The w r i t e r  neglecting  mutual  and I n d u c t a n c e even w i t h not  of  flux.  this  times before  inductance  rents  a s o l u t i o n along  equalizing  and. a l s o  connections  to  a  re-  w o u l d be  these  the  lines,  resistance  themselves,  a* s a t i s f a c t o r y  i n our c o n s i d e r a t i o n of the  but  solution  a n d of e q u a l i z i n g  due t o b o t h  the  oscillograms,  was  effects  currents.  3?or n o r m a l o p e r a t i o n  connections"the  these c u r r e n t s .  twelve and f o u r c o n n e c t i o n s  both cases  c u r r e n t s were  circulating  corrective  t h e m a c h i n e was n o t  be n o t i c e d  the  case  pole  circulating is  fluxes  fairly  say  less  flowing;  what  the  the  f.  M13  Is  were  and i n  It  and  is  to  correction  t h a n when no b r u s h  with, t h e  difference  brush c i r c u l a t i n g currents,  the  twelve is,  connections  for both  correction  with is  complete.  The e f f e c t 6 w h i c h was full  to  of f o u r connections  apparently  c u r r e n t s were  difficult  and w i t h o u t  is  loaded.  cur-  with  t h r o u g h the b r u s h e s  but  the  M.  respectively,  brush connections, In  have  circulating  O s c i l l o g r a m s n o . 2 and n o .  taken with  that  of  we  corrective  any-number of e q u a l i s i n g  at  only  obtained.  examined s e p a r a t e l y  it  gives  good r e s u l t s  effects  these s i m p l i f i c a t i o n s  Bo f a r  of  last  w o u l d p r o b a b l y be n e c e s s a r y  attempted  the  But  of  the  taken with  load.  The f l u x  load current  is  shown i n o s c i l l o g r a m  twelve e q u a l i z i n g per pole  still  connections  remains n e a r l y  no.  i n , and equal  for  - 36 all  p o l e s as i n the p r e v i o u s c a s e , the o n l y p o i n t of i n t e r e s t  a t t a c h e d to the o s c i l l o g r a m b e i n g the change i n s l o p e o f the wave c r e s t f o r two o f the p o l e s .  T h i s i s due t o the o r d i n a r y  armature r e a c t i o n e f f e c t w h i c h i s c r o s s  6 5 16  magnetizing.  V o l t a g e p e r c o n d u c t o r w i t h 12 c o n n e c t i o n s and a t f u l l load. C o i l c u r r e n t w i t h 12 e q u a l i z i n g c o n n e c t i o n s and a t f u l l load. C o i l c u r r e n t w i t h 4 e q u a l i z i n g c o n n e c t i o n s and w i t h brushes l i f t e d .  The f o r e g o i n g a n a l y s i s o f the v o l t a g e o s c i l l o g r a m s has been almost p u r e l y a q u a l i t a t i v e one, and c o u l d p o s s i b l y be c a r r i e d f u r t h e r w i t h the a i d o f o t h e r o s c i l l o g r a m s .  However,  i t s e r v e s t o show i n a s i m p l e manner how the f l u x d i s t r i b u t i o n i s changed by v a r y i n g c o n d i t i o n s . I t had o r i g i n a l l y been one o f the aims o f t h i s t h e s i s to  study  commutation w i t h r e f e r e n c e  culating this.,  currents  all  r e c o r d c o u l d he due  to the  speed of  •speeds tive  coil  the  spot of  the  considerably  with  satisfactorily  Oscillograms  In  w i t h t h e machine  at  taken with fully  the  .mutation, but necessary  film  that  t o open out  speed before  gram n o ,  is  7,  this  c o u l d be  typical  coil  is  a faint  indicate  12, w h i c h was  no l o a d ,  the  the to  achieved  current  trace  is  in  satisfactory  w h i c h has,, a l r e a d y -been d i s c u s s e d ,  and a t  if  peak  encountered.  It  In and the com-  w o u l d be increasing  much i n f o r m a t i o n c o u l d be g a i n e d .  connections,  sensi-  amplifier.  t h e waves c o n s i d e r a b l y b y  and n o .  such  sufficiently  a l l w h i c h c a n be s a i d .  •whatever of commutation, equalizing  r  e q u a l i z i n g connections  there  c o m m u t a t i o n . p e r i o d w h i c h seems t o  U ith  obtained with  difficulty  twelve  loaded,  are  is  high  an improvement  coupled  a n d 12  This  no  would be necessary  It  but  as  consequent  a s l i g h t l y .higher  such case,  5,. 7,  be  c o u l d be  a resistance  nos.  conditions  oscillograph.  oscillograph.  oscillograph.  and i l l u s t r a t e  w h i c h was  and the  t o be n o t  results  to  were u n s a t i s f a c t o r y ,  There might  lamp v/ere o p e r a t e d  cathode ray  oscillograms  i n the  cir-  "view  commutation p e r i o d .  appears  a record.  b u t much b e t t e r  use', off-the amplify  of  film  of brush  With"a  multi-element  oscillograms  light  photographic  voltage,  5,  the  obtained  to produce  effect  c u r r e n t under d i f f e r e n t  r a p i d change o f c u r r e n t ,  oscillograph  no.  of  tolcen on the W e 3 t i n g h o u s e  Unfortunately  the  and e q u a l i z i n g c u r r e n t s .  oscillograms  were  to  Oscillo-  g i v e s no r e c o r d taken w i t h  little  better.  four It  .38  is  even  not  possible  tell  to  takes p l a c e .  whether  or not  over-commutation  '  From the  results  of  make some o b s e r v a t i o n s place,  '  the  ordinary tests  It  is  possible  on c o m m u t a t i o n i n g e n e r a l .  even w i t h the armature e c c e n t r i c i t y  I n the  the b r u s h e s ,  p r o v i d e d the  riding  commutator.  This In i t s e l f  that  commutation i s  never very bad.  On t h e  m u t a t i o n c a n n o t h e l p b u t be a d v e r s e l y netic dent  balance  Is  upon t h e  poor as  the  ?  l o a d c u r r e n t and i s  F u r t h e r , any e c c e n t r i c i t y  change  the  flux  length  of  This phase and to  of  Is  the  from  thesis the  is  of magnetic  between  the  o r more p o l e s ,  and to. examine  effects  case  the  i n the  conclusion that  of  the  the  the  of  machine.  to  the  depen-  circulating will  change  of  discussion.  in  even  one  correction, It  has  aimed  the  two  of  a four-pole  to  the  essential,  the  This  com-  confirms  four-pole  have been i n c l u d e d i n t h i s altogether  to  lap-wound  does n o t w a r r a n t  these c o n n e c t i o n s .  six  corrective  The d i s c u s s i o n p o i n t s  Lulofs with respect  though n o t  by  analysis  some e x t e n t  performance  Certain details  t a t i o n which,  remains  armature  due t o  mag-  f o u r - p o l e machine a n d one w i t h  former.  the machine w i t h  c o n c l u s i o n of I'r.  proof  when t h e  imbalance and I t s  machine w i t h e q u a l i z i n g c o n n e c t i o n s plicating  is  were  gaps.  a n e x t e n t an a b s t r a c t  differentiate  the  i n no way a c o m p l e t e  subject  to q u i t e  of  the' i n t e r - p o l e s ,  inter-pole  effect  not a f f e c t e d  currents.  latter  was  o t h e r h a n d , com-  affected  Inter-pole  first  a maximum,, t h e r e  no c a s e o f s p a r k i n g a t smoothly on the  to  might  presenpossibly  •'- 39 • be h e l p f u l t o anyone e l s e w o o i n g a l o n g p a r a l l e l l i n e s on t h e same machine*  •  BIBLIOGRAPHY  (1)  "Armature R e a c t i o n i n a D . G-* L a p Wound Machine" \T. i,ulofs,. E l e c t r i c i a n , , 1912  (2)  " E q u a l l i n g Currents  (3)  I n t h e A r m a t u r e o f a 33.-G-. Machine * R* If. B a k e r , ' A . 1* B» B, p a p e r Ho. '§2-5  -'"The D e v e l o p m e n t o f t h e D . 0, G e n e r a t o r i n A m e r i c a " 3 . G . Lamme, E l e c . J o u r . , Yol»XEl  1  -(40)  (41) Arrangement  of E q u a l i s i n g  Connections  The f o l l o w i n g t a b l e shows t h e a r r a n g e m e n t o f e q u a l i z i n g c o n n e c t i o n s on t h e m a c h i n e . Bumbers i n t h e f i r s t c o l u m n r e f e r to commutator segments, and t h e s e c o n d column g i v e s the s l i p r i n g o r t e r m i n a l t o w h i c h t h e c o m m u t a t o r s e g m e n t s a r e j o i n e d by a s i n g l e armature conductor. numbering o f the.commutator s e g ments i s a s shown I n f i g . 3 , a n d f o r l o c a t i o n on t h e m a c h i n e s e q u e n t n o . 66 h a s a d r i l l h o l e on t h e e n d . I T h i s was u s e d for g r o u n d i n g o f t h e s e g m e n t t o t h e s h a f t when t a k i n g v o l t a g e o s c j 3 'lograms^T F i g * - 4 shows t h e n u m b e r i n g ' o f t e r m i n a l s p l a c e d o n t h e f i b r e d i s c , a n d i n t h i s e a s e , nos.. 12 a n d 13 a r e c l o s e l y s p a c e d , and thus can be l o c a t e d *  (42a)  OG^oocor-iotne*#CvcO3HO5C5C0£ -CD >  <t!  ••I O  H H H H O O O O O  H  CM. w to  H  K l <^  CM CM 02 CM OJ CM  C2  w in CM OJ  tOHCQCOCMCD^lOCS SO !> O SO CM CO H H H  02 • *  W (35 W H » W t o tO 10 CO CD -•f * ' * * • ^  o wo  cq oii(• 9  Hsl**>-OHHiOlOl>-  02 tO ^« CD 2> CO OJ O H • H H  CQ  t  •  o  0.2 01 tO tO tO  K J> ^  W O  , T  •  t  o  to CO  o  O) CO ^1  to  CU  O IO 2> ^ O .^i ^  o  e  ,  • • »  O  to CD o H  >*:  to  ••*.•..  *•  O.  ..»•'•  •*•• . . . »  .*  •  W  O H , C M ^:  •  Q  • « »  O.O O I f  o  LO lO lO en O 01 to o5 • • •» * O'H Ol CM lO IO lO co  *  • » •  t o t o t o o t o 01 H CO t ~ H H H 02 tO tQ t o I> t~ l O o t o IO O CO OCDC\2C0s>00t0 02t0 « ••• m * » *. »• » •••«• O 01 ^ IO CD CO O H 02 H H H  O" 0> IO CO  O IO £>• ^ •». * e > CQ 05 tO ^ H tO ^  Cv2COtOOCOOtOZ>iO  c Q C - c o c n o c n o ^ c\2 r-lOltOtOlOD-t^CO  •  • . * - * • «  00  ».  'A  *' # * #' « • * * - ^1 LO £ - £ - 0 > t t > CM tO £CQCOCQCOCOCOfc-S-E-  t o o i o t o t o o o t o lO O ^ t ^ t O r i O M  OOMlOC-OH'vHcDGO • • - r-l rH H r-1  « o 1  H  .»  *.•  •*  O f f l ^ O S ^ O O O tO 0.2 "xji ^ 2> CO O Cv2 tO COCOCQCOCOC0C 1>-£w  O  s o o  to  Ol 01 IO O CO O CO O^COOlCPE-CDOCO  O OJ. ^ !> cn> H cc t — H H H H  »  op  <* fc {*-=»'  «  O O O O C O f f l t O ^ K ) H 02 01 tO -tf.tfl CO t>  O O O O O O O O O tfJlOlOlOtOlOiOtOlO  CM 02 02 CM 01 02-02 O l CM  •H H s5  Dl CD U  o  Q ?> tO O "J H 1  O SO  ocncnoscot^co^H H 0> tO ^ IO CD J>o o o o o o o o o  lOlOlOtOlOlOlOlOlO 02 02 02 <A2 02 W CC! 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O H t O l O £ > 0 5 0 2 * ^ i O CO H'H'HH  b£  O  Pi o o  S3 •H •N  v!  fcOC-  CO O tO O  CD tO  -a • . * « «:»'•»•*.« « Ofr-^ O CO t ~ CO CO 0 2 OJ • H 02 O l tO -NP -LO' CD  H CJ  CD  O O O O \Q IO l O IO 03 TO 03 0 2  jd  1  0J  O O O O O O O O O LO LO IO tO tO IO IO 1 0 to Qi 02 02 O l C\2 02 02 CM O i  P ol 0;  inioioiniOLOioioiOLO  (.'.'? Gv2 CQ 02 O l O i OJ 02 OJ O i rA f~U-^ r-i r-^ r~\ r-\ rH r-i  O LO 03 02 H  O LO O ' - O l O l f l O O 02 H r H O 05 CO CO 05 02 02 02 02 H H H H r ir ir ir ir ir lr i r l  £3i LO  - LO IO LO LO IO l O LO O H H'H H H O f f l 02 02 02 OJ K M r l H H r i H H H H H H  O LO O O O w w H H  (43a)  Treatment  o f F l u x D i s t r i b u t i o n b y means o f S o l u t i o n s Electric Circuit  of I  F o r t h e p u r p o s e o f d e t e r m i n i n g flux d i s t r i b u t i o n m toe . m a g n e t i c c i r c u i t " u n d e r s t a t i c c o n d i t i o n s , t h e following metnod S S ^ o p t e d . : Rather than ^ i t o ^ e q u a t i o n s ^ f ^ ^ ^ ^ ^ a n d r e l u c t a n c e s , a n analagous e l e c t r i c ° f : t h l s method h a s no p a r t i c u l a r m e r i t , the two. .djdgiara^ox i x g . 5 , shov t h e m a g n e t i c c i r c u i t I n w h i c h we a r e i n t e r e s t e d a n d a l s o the e l e c t r i c c i r c u i t which tie w i l l s o l v e . h  F~IG. ID i ag i-am  of  Analogous  * = -  (?^ (R* (R„ (R* H  + H + t-  B  R + H 4  1  + r +• r + r  •(» r  h^lagrtefic  x  + Kx i + R i is + + i^L ii ••+:  +* iij.) i<t + xi ^ I3 -  are true ¥  3  i i +  —  la  eluctancess  Circuit  equations  + JR )i| + Ko.) ix + Ri.) i  R^ 12. 1.  R  EI <3ct ric  following 2B 2E 2B 23  5  -  A-  Iff r if R,i, r i-r + Hi U r isR'x I3 "Li* « 0 = 0 3?  ' -ft •»  *  «  .'»  (2)  . . . ( 4)  (5)  Hon vie w i l l assume t h a t i , ^ I and 1* =. 6 . This i s not exactly t r u e b u t a g e n e r a l s o l u t i o n o f the f i v e o r i g i n a l e q u a t i o n s shows t h a t t h e a s s u m p t i o n i s w a r r a n t e d i n o u r c a s e . s  (43b) We h a v e  These  then:.  2E -  (R^ + R' +r * Rj.)!,  «*• R^ i  2E ~  ( R , •+ R»+r + R i j i j .  4 R, i,  three  may be  x  s o l v e d by  ,.23£ 2 r R* +  = .RJ + 3R^R + 1  2B {R* -  Rx  3, s  f c  -  (6)  + R i,  (7)  z  determinants  a n d we  x  *  get:  3 r R + r*• x  3R'i 1  I > - ' - "  2  where D sta,nds Eence  ii i-5  r  4  x  ( B ^ . R a . ^ r * ft?-. ) + 3R..RV 4 3 r R' +  ini)  i.  + R i  ¥  * *  ix ^ iif.  for  the  denominator of  we c a n p r o c e e d  i>  *  i*. ~  1,  4  LA  to  find  the  the  expression  ratio  of  for i »  i , <+ i ^ t o  k  i + 5  i^  R ^ ^ R ^ t x + R'J. » R » » RT.+ r 4- 3R'^ R,+ r ••+- R ' i +- R.+ 3R,-v r - R ' tm  z  R* ,* r . •t.-^Rji: R^ + r 4 2R2. How g o i n g b a c k w gelt/  4SJ  and a p p l y i n g  U«: .±..X i  + r  this  ftjB  result •  to  , (R',4 .R?,!  + 2 (R t  R , 3  our magnetic  circuit  

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