The induction motor and its engineering capabilities

Sept., x9o3.]

The

The [,nduction Motor.

18 3

Induction Motor and Its Engineering Capabilities.*

[,4 Thesis Presented to the University Faculty of Cornell University for the Degree of Docfor of Philosophy. ] BY GEORGE L. H o x I E . PART

ONE.

T H E I N D U C T I O N MOTOR.

Definition.--The induction motor, as its n a m e implies, is a motor that d e p e n d s for its operation u p o n the principles of " i n d u c t i o n . " As generally used, this term m e a n s the production of an effect at some point b y a cause located elsewhere, no a p p a r e n t mechanical connection existing between the cause and the effect. Thus, given an electric charge on a c o n d u c t i n g body, if a second c o n d u c t o r be b r o u g h t into the neighborhood, induced charges, of opposite signs, will a p p e a r on those surfaces of the second b o d y w h i c h are n e a r e s t to, and most remote from, the originally c h a r g e d body. Also, if a c u r r e n t be e s t a b l i s h e d in one of two neighboring closed c o n d u c t i n g circuits, a current, called an induced current, will flow m o m e n t a r i l y in the second circuit. Or, if a m a g n e t be created in the vicinity of a m a s s of soft iron, induced m a g n e t i c poles will appear on the latter. M a n y similar e x a m p l e s of induction m i g h t be given. A n y m o t o r t h a t d e p e n d s primarily upon such p h e n o m e n a for its operation m a y b e properly t e r m e d an induction motor. Practical Form.--Several varieties of purely induction motors h a v e been devised, b u t only one possesses any practical interest. In this class, the m a g n e t i c forces p r o d u c e d b y the field due to an induced, or secondary, c u r r e n t r e a c t i n g T h e writer wishes to t h a n k Prof. Harris J. R y a n for suggestions and advice received prior to t h e inception of this thesis, and from time to time during the progress of t h e work. Also to acknowledge his indebtedness to several friends, t h r o u g h whose kindness it was possible to obtain the data of construction and performance which form a part of this paper.

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[J. F. 1.,

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upon the field produced by an inducing, or primary, current, are m a d e to do work. Both p r i m a r y and secondary c u r r e n t s are of necessity alternating, since it is only by a c h a n g e in one c u r r e n t that a n o t h e r m a y be induced. It js also necessary, in order to get a c o n t i n u o u s r o t a r y motion, t h a t the currents present should be polyphase currents. T h i s does not necessarily imply t h a t polyphase currents should be supplied to the motor, a l t h o u g h it is u s u a l l y best t h a t this should be done. T h e e l e m e n t a r y t h e o r y of i n d u c t i o n motors is quite simple, b u t an e x t e n d e d s t u d y of their b e h a v i o r involves the consideration of some very complex relationships, and is correspondingly difficult. In fact, such a s t u d y cannot be m a d e at all w i t h o u t the use of a p p r o x i m a t i o n s which affect to a g r e a t e r or less degree the accuracy of the result. This state of affairs is not indeed peculiar to the i n d u c t i o n motor. H a r d l y a n y of N a t u r e ' s p h e n o m e n a are so simple t h a t a m a t h e m a t i c a l s t u d y of t h e m can be e n t i r e l y complete. Obviously the results of such a m a t h e m a t i c a l investigation are only correct to the same degree as were the original assumptions. It is therefore of the u t m o s t i m p o r t a n c e that one should d e t e r m i n e as a c c u r a t e l y as possible the degree to which such a p p r o x i m a t i o n s as are necessary are likely to affect the final result. S t u d y o f a P a r t i c u l a r M o t o r . - - B e f o r e t a k i n g up a n y theoretical m a t t e r s connected w i t h the i n d u c t i o n motor, it will be well to make a very careful s t u d y of a p a r t i c u l a r motor, not from a m a t h e m a t i c a l standpoint, b u t with a view of a c t u a l l y d e t e r m i n i n g w h a t goes on in the iron and copper of which it is composed. Such a s t u d y will serve to make clear the effect of some of the c u s t o m a r y approximations used in the m a t h e m a t i c a l t h e o r y of the motor, a n d will enable one to form a fairly correct idea as to t h e i r propriety. T h e m o t o r chosen for this detailed s t u d y is one for which the writer was able to obtain n e a r l y all of the electrical dimensions. Its specifications are as follows: Horse-power . . . . . . . . . . . . . . . . . . . . . Revolutions per minute . . . . . . . . . . . . . . . . Outside diameter of primary core . . . . . . . . . . .

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T h e m a g n e t i c circuit or flux-carrying parts of this motor are d r a w n to scale in Fig. s. Since there are three phases and twelve poles, there are four slots per p h a s e per pole, the phases b e i n g g r o u p e d as indicated b y the letters aaaa, bbbb,

Sept., t9o3.]

The Induction Motor.

187

cccc, anna, etc., in Fig. r. ~'~h~ m a g n e t i c field p r o d u c e d by the three c u r r e n t s at a p a r t i c u l a r ixamaxntin:.the ~pe.xation of the m o t o r is shown graphically in Fig. i by a d o t t e d line. T h e bore of t h e p r i m a r y is t a k e n as the axis of abscissas, and since this is a circle the r e s u l t i n g curve of flux distribution is a little distorted. It will be noted t h a t six poles are positive and six negative, a total of twelve. Dtstribution o f Magnetism.--In Fig. 2 an e n l a r g e d view of a section of this m a g n e t i c circuit is shown. For convenience the d i a g r a m is developed so t h a t the p r i m a r y bore becomes a s t r a i g h t line. T h e slots are l e t t e r e d as before for the separate phases, and in one slot of both p r i m a r y and :secondary the a r r a n g e m e n t of conductors is shown. In the upper part of Fig. 2 is given the conventional sinewave d i a g r a m for a three-phase circuit. Selecting the i n s t a n t w h e n the c u r r e n t A is zero, c u r r e n t s B and C are "866 of their m a x i m u m . If the reluctance of the air-gap be very large as compared with t h a t of the iron part of the circuit, so t h a t t h e total M.M.F. of these c u r r e n t s acts at the air-gap, this M.M.F., as f o u n d at each of the t w e n t y - t w o t e e t h shown in t h e figure, m a y be readily plotted. Such a construction is found at the lower p a r t of Fig. 2. T h e conventional signs b e t w e e n the positions of the teeth in this figure show the directions of the currents B and C. It is obvious t h a t the phase relations of the sine curves give no indication as to the m a n n e r in w h i c h the currents are connected to the machine. In this i n s t a n c e the scheme of connection is such that, s t a r t i n g at the left of the figure and m o v i n g to the right, each new g r o u p of slots reached is c a r r y i n g such a c u r r e n t as was carried by the preceding group a little earlier in time. T h e r e s u l t a n t effect is similar to t h a t of a m o t i o n from r i g h t to left of both currents and magnetism. Method o f Plotting.--In the case chosen for Fig. 2, where one c u r r e n t is zero and the other two are equal, it is a very simple m a t t e r to c o m p u t e the M.M.F. a c t i n g t h r o u g h each tooth since all the flux p a t h s are s y m m e t r i c a l . It is not always so obvious, however, how this s h o u l d be done, and it is perhaps desirable to show t h a t the m e t h o d followed is a proper one for a n y c u r r e n t distribution.

i88

Hoxie :

[J. F. I.

It will be a s s u m e d that in any m a g n e t i c circuit under consideration the conditions are a p p r o x i m a t e l y as follows : (I) T h e m a g n e t i c circuits are formed in two concentric rings of l a m i n a t e d metal, as in Fig. 2. (2) T h e total reluctance of any circuit is located in the air-gap b e t w e e n the rings. (3) All m a g n e t i c circuits are parallel to the paper. (4) T h e m a g n e t i c fields are set up b y currents in conductors which lie in the slots of the o u t e r ring, and which are p e r p e n d i c u l a r to the plane of the paper. (5) For each current in a single slot there are present similar currents at e q u i d i s t a n t points around the circumference, the total n u m b e r in each case b e i n g equal to the n u m b e r of m a g n e t i c circuits. A general case fulfilling these conditions is illustrated in F i g . 3 . In this figure the current in conductors A~, ,-/2, A3 and A, is 200 a m p e r e s ; that in conductors B~, B~, B3 and B 4 is Ioo a m p e r e s ; and in conductors C~, 42, C3 and C4 is 50 amperes. It is required to find the m a n n e r in which the flux is d i s t r i b u t e d a r o u n d the air-gap in this figure. Since the reluctance of the iron is a s s u m e d to be zero, t h a t of the air is the only item limiting the flux established b y a given M.M.F. As the p e r m e a b i l i t y of air is unity, a given change in M.M.F. will always p r o d u c e the same change in the flux present, no m a t t e r w h a t m a y be the flux density. U n d e r these c i r c u m s t a n c e s we m a y m a k e use of the principle of " s u p e r p o s i t i o n ;" that is, the field due to a n y combination of currents is equal to t h e g e o m e t r i c a l s u m of the fields which w o u l d be p r o d u c e d b y each current taken separately. Since all fields are radial at the air-gap in the p r e s e n t instance the g e o m e t r i c a l s u m is also the algebraic sum.

Selecting the four conductors A1, ,4 2, As, ,q4; it is evident that, taken alone, equal fields would s u r r o u n d each, and that these fields would lie in the q u a d r a n t s formed b y the lines APA I and A ' A ' . Since the flux in a n y case m u s t cross the air-gap twice, it will e n c o u n t e r an equal relt~ctance b y any path. T h e flux d e n s i t y is therefore u n i f o r m over the entire circumference, and is such as w o u l d be produced by Ioo

Sept., t9o3. ]

189

The Induction Motor.

a m p e r e t u r n s of M.M.F. a p p l i e d a t e a c h p o i n t of t h e a i r - g a p . Let u s call flux l e a v i n g t h e i n n e r r i n g p o s i t i v e , a n d flux e n t e r i n g t h e i n n e r r i n g n e g a t i v e . T h e n in Ftff. 3 t h e r e are due to c u r r e n t s in t h e A c o n d u c t o r s o n l y : From " " "

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T h e d o t t e d lines w i t h a r r o w - h e a d s , w h i c h are d r a w n in Fig. 3, i n d i c a t e g r a p h i c a l l y this a i r - g a p d i s t r i b u t i o n , o n e line c r o s s i n g e a c h i5 ° f o r e a c h 25 a m p e r e t u r n s . N o a t t e m p t has b e e n m a d e to s h o w t h e e x a c t d i s t r i b u t i o n in t h e iron. T h e f o l l o w i n g n o t e s m a y be m a d e u p o n t h i s f i g u r e : (I) N o flux c i r c u i t is f o r m e d t h a t d o e s n o t i n c l u d e o n e of the 3 c o n d u c t o r s , t h e s e c a r r y i n g a l a r g e r c u r r e n t t h a n t h e others.

Hox ie :

19 °

[J. F. I.,.

(2) Some flux travels b y a r o u t e that includes all the conductors, A, B, C, of one group. (3) T h e flux does not divide s y m m e t r i c a l l y b e t w e e n one group, A,, B,, C~, and the a d j a c e n t groups, as A2, B~, C~, b u t the flux b e l o n g i n g to one g r o u p extends almost over to the c o n d u c t o r C of the next group. It should, of course, be b o r n e in m i n d that Fig. 3 represents a very special ease of flux distribution, and particularly that the conditions practically preclude leakage. ~"

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Actual Distribution in Matar.--In the actual m o t o r the reluctance of the iron part of the m a g n e t i c circuit is not zero, b u t except for the teeth such reluctance is relatively small. It m a y be assumed, w i t h o u t g r e a t error, t h a t taking the t e e t h and air-gap together, nearly all of the reluctance of any m a g n e t i c p a t h is c o n t a i n e d therein. T h e flux, or M.M.F., at each tooth m a y then be d e t e r m i n e d b y the m e t h o d

Sept., 19o3. ]

The Induction Motor.

I91

of the preceding p a r a g r a p h w i t h o u t g r e a t labor. T h e only error of any i m p o r t a n c e in such a d e t e r m i n a t i o n lies in the assumption t h a t t h e flux is proportional to the M.M.F. The m a g n i t u d e of this error will be e s t i m a t e d later. In order to m a k e the problem as concrete as possible, it is a s s u m e d t h a t each c o n d u c t o r will carry a sine-wave of current, h a v i n g an effective value of 45 amperes. T h e results obtained, b e i n g proportional to the current, will serve equally well for a n y c u r r e n t value. A c u r r e n t of 45 a m p e r e s was t a k e n because, as n e a r l y as could be e s t i m a t e d from experimental d a t a on other motors, 45 amperes was t h e most probable value for the m a g n e t i z i n g current of this motor. It is a s s u m e d t h a t the secondary bars are present, but are u n c o n n e c t e d at the ends, so t h a t no secondary c u r r e n t may flow. T h e flux d i s t r i b u t i o n will be d e t e r m i n e d for intervals of 15 °, m e a s u r e d on sine curve A, Fig. 2, and starting w i t h the phase angle of A equal to zero. T h e successive currents in conductors l y i n g in the A slots will, therefore, be, o, "259, "5, "707, "866, "966, "i, "966, etc., w h e r e the m a x i m u m value is u n i t y and the interval is i5 ° . W i t h four conductors per slot, each c a r r y i n g 45 effective amperes, the total c u r r e n t per slot is i8o effective, or 254 m a x i m u m , amperes, and the various m o m e n t a r y c u r r e n t s are as follows : PHASE ANGLE OF PHASE A

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T h e net M.M.F., applied to the teeth and air gap by these currents, is set down in the table on the following page for the t e e t h positions, s h o w n in Fig'. 2. It m a y be noticed from this table t h a t there is a r e g u l a r progression, from t o o t h to tooth, of the zero line of flux. The position of this line is d o t t e d across t h e table.

i92

Hoxie :

[J. V. J.,

T h e point of m a x i m u m flux moves in the same direction, b u t not regularly, and is noted on the table by a j a g g e d line

of dots. 2k line m a d e up of dashes, and d i v i d i n g each flux g r o u p into equal parts, is also drawn across the table. This line is straight, and indicates a u n i f o r m progression of the

Sept.. t9o3.]

7he [ncluction Motor.

I93

flux. T h e total q u a n t i t y of flux varies s o m e w h a t , and this variation is indicated b y the n u m b e r s 704 o, 7to4, 71 i2, 7io4, 7040, etc. T h e s e n u m b e r s are c o m p u t e d u p o n the assumption that the flux and M.M.F. are proportional, and t h e y are more nearly equal to each other than this a s s u m p t i o n is to the truth, so that for all practical purposes the following conclusions m a y be r e g a r d e d as reached for this m o t o r : (I) T h e total q u a n t i t y of flux is constant. (2) T h e points of zero flux m o v e around the circumference at a uniform velocity. (3) T h e points of mid-flux m o v e a r o u n d the circumference with a uniform velocity. (4) T h e points of m a x i m u m flux m o v e around the circumference, b u t not with a uniform velocity. T h e points ()~ m a x i m u m flux m o v e with twice the velocity of the points of zero flux, b u t are m o v i n g only half of the time, so t h a t the net rate of progression is the s a m e for either. F r o m facts Nos. I, 2 and 3, we m a y state" the following - - w h i c h is true at least for this particular motor. The effect of the c h a n g i n g m a g n e t i s m upon the p r i m a r y conductors i s practically the s a m e as w o u l d be produced if ' the p r i m a r y were on open circuit and the s e c o n d a r y were replaced b y a r o t a t i n g field h a v i n g twelve poles, each pole having a flux distribution v a r y i n g from a m a x i m u m at its center to zero at a point m i d w a y b e t w e e n poles. T h a t is to say, there has been p r o d u c e d a changing" field similar to, and practically the same as, a rotating flehl. Graphical Constructian.--In Fig. 4 the flux distribution, as indicated b y the a c c o m p a n y i n g table, is s h o w n graphically. The d o t t e d parallelograms, m a r k e d I to 22, represent the position of the t e & h of Fig. a. T h e short horizontal lines ~across these indicate b y their distances from the axis, the isuccessive M.M.F.'s a c t i n g at each tooth, and the d o t t e d !lines j o i n i n g these horizontals s h o w the distribution over a portion of the air gap, for e i g h t different instants, ~ of a cycle apart. A t the lower part of F,g. 4 are drawn curves s h o w i n g the m a g n e t i c cycles t h r o u g h which typical teeth, Nos. 9, IO VoL. CLVi. No. 933. 13

i94

[J. F. I.,

Hoxfe :

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m a g n e t i c density, and the h y s t e r e s i s losses, are not the same in different teeth. T h e three sine-waves shown represent, however, the entire v a r i a t i o n in the m a c h i n e u n d e r discussion.

Sept., 19o3 .]

The Induction Motor.

[95

The counter E.M.P. induced in a single conductor of the primary circuit should, under the a s s u m e d conditions, be a sine-wave. It is of interest, however, to derive this from the determined fluctuations of the field, and to determine independently the phase and quantity relationships existing between the E.M.F.'s induced in conductors lying in four adjacent slots of a single-phase group. This was done in the following manner: Referring to Fig'. 2 and to the table of M.M.F. distribution, quantities :-"

"re

,4~'-,~

t

I

i

i I I

i

,

i

:', \

L

' I

-i )$*

",1

,

,

', '

r i

,

P, , \ I " I ~

•

~---'X--~ i • j i~, t \ ', ",

i~*, L_'__] r

,

....... ~3-

~io.

"

7~.

,

fal

I

~'-.

•

, i,~a-

•

",,

L ',,

'\

-

l~,O-

I

:

'\

-.

1

". ........

r--'~VJi "--.,.:..'.'.v

Fxo. 5.

proportional to the flux actually s u r r o u n d i n g the slot between teeth Nos. t3 and [4 were set down for phase angles of phase A, of o °, t5 °, 3 °° . . . . . . ~8o °Since these values occur ~ of a cycle apart, the difference b e t w e e n two successive values is proportional to the average E.M.P. induced in a conductor lying in such a slot during that part of a cycle. T h e s e differences were carefully determined, and in Fi~. 5 they are plotted as horizontal lines, ~5 ° in length, and

I96

[-[oa'ie :

[J. F. I,

at a p p r o p r i a t e i nt er va l s . T h e c u r v e A was sketched t h r o u g h t h e s e h o r i z o n t a l s in s uc h a m a n n e r t h a t each horizontal s h o u l d be an a v e r a g e o r d i n a t e for t h a t p a r t of the c u r v e l y i n g b e t w e e n t h e s am e v e r t i c a l lines. Curves.q2, .43 a n d A o w e r e p l o t t e d in a s i m i l ar m a n n e r , t h o u g h their c o n s t r u c t i o n lines are n o t g i v e n in Fig. 4. /ks n e a r l y as could be d e t e r m i n e d b y this graphical m e t h o d all t h e c u r v e s are sine c ur ves , and are equal. This, of course, was to be e x p e c t e d , a nd serves as a check upon t h e n u m e r i c a l work. Total Flux Present.--Since t h e p r i m a r y r e s i s t a n c e of the m o t o r is low, t h e c o u n t e r E.M.F., w i t h a m a g n e t i z i n g curr e n t of 45 a m p e r e s per c o n d u c t o r , is a l m o s t e x a c t l y t he same

//

FiG. 6. as t h a t w h i c h is i m pr es s ed, or is 220 volts, effective pressure. Since eac h p h a s e has an equal n u m b e r of conductors in e o r r e s p o n d i n g slots of each g r o u p , it follows t h a t the E.M.F. i n d u c e d in t he c o n d u c t o r s of one slot of each group m u s t b e a r to 220 vol t s t he s am e r e l a t i o n t h a t t h e E . M . F . c u r v e A, for ex am pl e , hear s to t h e s u m of c u r v e s A~, As, A3, Ai. B u t th e E.M.F.'s of Fiff. 5"~re n o t in phase, and their s u m is less t h a n f o u r t i m es either. In o t h e r words, there are times w h e n t he E.M.F. g e n e r a t e d in c e r t a i n conductors of one p h a s e is oppos e d to t h a t g e n e r a t e d in o t h e r s of the s a m e phase. T h e r e is t hen a di~erential action, as is t h e case in m a n y A. C. d y n a m o s .

The Induction Motor.

Sept., t9o3.]

I97

O b v i o u s l y t h e m a g n i t u d e of such a differential a c t i o n will d e p e n d u p o n t he space o c c u p i e d a l o n g t he p r i m a r y bore, by a g r o u p of c o n d u c t o r s b e l o n g i n g to a single phase, or i t will be g r e a t e r as t he n u m b e r of slots p e r phase and pole is. increased. In this case t he s i t u a t i o n is i l l u s t r a t e d in Fig. 6, where A,, A s, A3, and A4, r e p r e s e n t t h e E.M.F.'s in each set of slots, an d A r e p r e s e n t s t h e E.M.F.'s of t h e e n t i r e phase, Each c o m p o n e n t f a c t o r m u s t be 51"8 a v e r a g e volts, if t h e sum is to be I98 a ve r age, or 220 effective volts, or t h e r e are lost 9"2 a v e r a g e volts, or 4"65 p e r cent. of the total pressure. T h e a c t u a l flux per pole m us t , t her e f or e , be 465 per cent. larger t h a n if d i f f er e nt i a l action were absent. T h e r e are I92 c o n d u c t o r s in series in each phase, and t he m a g n e t i s m is e q u i v a l e n t to a r o t a t i n g field m a k i n g 25o r e v o l u t i o n s p er m i n u t e . T o g e n e r a t e I98 a v e r a g e volts, t h e flux m u s t be as follows: 25o ¢ x t92 x ~ - x I2 ~- ~98oooooooo x I"O46 or,

(P x 2, x56,ooo m a xw el l s per pole ; or with t h e a s s u m e d m a g n e t i z i n g c u r r e n t of 45 amperes, one a m p e r e t u r n a ppl i e d at a t o o t h sends t h r o u g h it 304 maxwells. In this discussion it has so far b e e n a s s u m e d t h at no flux s u r r o u n d s p r i m a r y c o n d u c t o r s locally. T h e total flux p a s s i n g t h r o u g h each p r i m a r y t o o t h and t he corr e s p o n d i n g flux d e n s i t y are set dow n in t he s u c c e e d i n g tables, t h e c u r r e n t v a l u e s a nd flux d i s t r i b u t i o n b e i n g as prev i ou s ly d e t e r m i n e d : F L U X D E N S I T I E S I N T]~ETH. Tooth

oo

No.

6 ............... 7 ............... 8 ...............

P h a s e Angle of P h a s e A. x5 °

3 °0

628o0 418oo 20800

4520o

~41oo

21700 I60o

O 2410o

248co 420o0 591co

48400 6o4oo 72200

9 Io

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

o 20800

II 12

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4t8oo 628o0

13

. . . . . . . . . . . . . .

836oo

14

. . . . . . . . . . . . . . .

15

. . . . . . . . . . . . . . .

762oo

84500

836o0

9320o 87000

965o0 84500

836oo

80800

72200

45°

i6oo 2J7oo 45200

68500 74600 80800 870o0 93200 762o0 591oo

I98

Hoxie :

Tooth

oo 83600 836O0

NO.

16 17

[J. F. I.,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i8 . . . . . . . . . . . . . .

62800

I9 20 2I

418oo 20809

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

P h a s e A n g l e o f P h a s e A. 15°

74600 68500 45200 217oo I600 24800

O

3° 0

45 °

60400 484o0 24100 o 24t00 484O0

42000 24800 16oo 217o0 452O0 6850O

F L U X AT T E E T H . Tooth No.

Phase Angle of Phase

oo

6

. . . . . . . . . . . . .

20060o

7 8

. . . . . . . . . . . . . . . . . . . . . . . . . .

133700 6680o

9 zo

. • . . . . . . . .... . . . . . . . . . . . . .

o 6680o

II

. . . . . . . . . . . . . . . . . . . . . . . . . .

1337o0 200600

13

..........

26750o

14 15 z6

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

267500 267500 267500

17 18

. . . . . . . . . . . . . . . . . . . . . . . . . .

267500 200600

19 2o 2I

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1337o0 66800

12

0

x5° 14440o 69600 5200 79600 1344O0 189O0O 243800 29850o 27870o 258700 2386o0 2 x8 8 0 0 144400 69600 52co

79600

3°0 772O0 o 77200 154400 193OO0 2316OO 270200 3088o0 2702oo 2316oo x930oo 154400 77200 o 77200

A.

45° 5200 696OO 144400 218800 238600 2587OO 27870o

2985oo 2438o0 t 89o0o I344oo 79600 52oo 69600 1444oo

i544oo 218800

F l u x Distribution in Primary Core.--In order to determine the effect of the reluctance a c t u a l l y present in the iron core, a n d to e s t i m a t e the error t h a t is involved in n e g l e c t i n g it, we will select t h e i n s t a n t w h e n the phase angle of p h a s e A is 3o°, and will m a k e use of t h e q u a n t i t i e s and densities f o u n d in the preceding tables for t h a t condition. T h i s part i c u l a r i n s t a n t is chosen for s t u d y because at t h a t t i m e the g r e a t e s t d e n s i t y of flux is f o u n d in a single t o o t h of the p r i m a r y , a n d the total q u a n t i t y of flux is a m a x i m u m . The v a l u e s of flux g i v e n - i n the tables are n o t to be considered as final, b u t m a y be t a k e n as first approximations, subject to later corrections s h o u l d f u r t h e r s t u d y prove such corrections necessary. A t t h e i n s t a n t chosen the flux is zero in tooth No. 7, and is m o s t dense at t e e t h Nos. i and I3. In Fig. 7 is given a d r a w i n g of t h e p r i m a r y part of the m a g n e t i c circuit w h i c h includes half of t e e t h Nos. I and I3. F r o m the fact

Sept., 19o3.]

The [nductzon Motor.

[99

that the flux d i s t r i b u t i o n in the p r i m a r y teeth is symmetrical a b o u t tooth No. 7, it follows that, n e g l e c t i n g a n y local or leakage flux, t h e d i s t r i b u t i o n in the p r i m a r y core will also be s y m m e t r i c a l , or half the flux of tooth No. 1 goes to tooth No. x3; flux from tooth No. 2 goes to t o o t h No. I2; No. 3 to No. II, and so on. T h e probable p a t h of the flux in each case is indicated by d o t t e d lines in the figure. T h e position of these lines was d e t e r m i n e d b y a m e t h o d of trial and error. F i r s t the lines were s k e t c h e d in as it was t h o u g h t t h e y should be drawn. T h e n e q u i p o t e n t i a l lines were d r a w n crossing these at right-angles. T a k i n g one section b o u n d e d by two equipotential lines, the flux of each p a t h should occupy a w i d t h

'

I

i

~.."~ .-", ..-\ ~ ..... .~--!;>-~-::..... :-~ ",. ', ~\-~" ~"--I ~ .' ~ .

~

/

,

FTG,7. proportional to its a m o u n t and to the distance b e t w e e n the two e q u i p o t e n t i a l s at t h a t point. T h e new or corrected positions of the b o u n d a r y lines of the flux p a t h s were t h u s located, new e q u i p o t e n t i a l lines s k e t c h e d in, and a second c o m p u t a t i o n made. T h i s process was repeated until the corrections b e c a m e negligible. T h e flux densities, in kilomaxwells per square inch, are m a r k e d on Fig. 7 at a n u m b e r of points. T h e flux d e n s i t y and d i s t r i b u t i o n in the p r i m a r y core being t h u s d e t e r m i n e d w i t h fair accuracy, a curve s h o w i n g the relation b e t w e e n such densities and the M.M.F. per inch of p a t h r e q u i r e d to produce t h e m was c o n s t r u c t e d . T h i s curve (Fig. 8) was t a k e n from a curve given by P a r s h a l l and

200

Hoxie

[J. F. 1.,

."

H o b a r t as t h e m a g n e t i z a t i o n c u r v e for l a m i n a t i o n s of " e l e c t r i c s t e e l " of g o o d a v e r a g e q u a l i t y . U s i n g t h i s curve, c a r e f u l d e t e r m i n a t i o n s w e r e m a d e f o r t h e M.M.E.'s p e r inch of p a t h , a t e q u a l i n t e r v a l s a l o n g e a c h p a t h in t h e core. T h e final a v e r a g e of t h e s e r e s u l t s is g i v e n b e l o w . In this a n d s i m i l a r tables, p a t h No. i m e a n s t h e p a t h b e t w e e n t e e t h I00~

/ /

mc~

f

/

/

if-

/

l

..------ - - " - - F

I

f

. f

- : ' f •y,, .~ ~ ' ~ / ¢

!,v~/ 60 i K[~ ~A

221 !2] !J

AJ 7op.r~ 7;zJ ~2,~ 1 ~ar ,~, r_~,

1

"7 ~ 9 20

~0

4~

50

)

79

,~0 FIG.

Nos.

i

9D

i(,o

~J3

J o

li;o

b0

t: o

8.

a n d I3 ; p a t h No. 2 is b e t w e e n t e e t h Nos. 2 a n d I2,

etc. : Path No. I . . . . . . . . . . . .

]Length iu I n c h e s . " . . . . .

2 ................. 3 ................. 4 ................. 5 . . . . . . . . . . . . . . . . . 6 . . . . . . . . . . . . . . . . .

13" 3

m.35 7'25 5"I 3 2"5

Average Ampere Turns Per Inch.

9"1 9'9 lO'8 ]1"6 12 io

Ampere Turtas in Core.

121 lO2 78 59 36 25

T h e flux d e n s i t i e s in t h e p r i m a r y t e e t h are in g e n e r a l h i g h e r t h a n in t h e core. T h e c o m p l e t e p a t h in a n y c a s e i n c l u d e s t w o t e e t h or a t o t a l l e n g h of 3"2 i n c h e s . T h e sect i o n of t h e t o o t h is n o t u n i f o r m , b e i n g s m a l l e r at t h e f a c e

The Induction Motor.

Sept., I9o3.j

20I

t h a n a t t h e r o o t . A l s o c o n s i d e r a b l e flux l e a v e s t h e t o o t h b e f o r e r e a c h i n g t h e face, a n d t a k e s a l o n g e r r o u t e t h r o u g h the air gap. O n e of t h e s e f a c t s t e n d s to c o u n t e r b a l a n c e t h e other. I t will b e q u i t e s u f f i c i e n t l y a c c u r a t e to m a k e u s e of the a v e r a g e s e c t i o n a n d of t h e t o t a l l e n g t h . A g a i n , m a k i n g u s e of t h e c u r v e in Fi#. & t h e M . M . F . used u p in t h e d i f f e r e n t t e e t h w a s d e t e r m i n e d , as f o l l o w s :

,

/

\/

r

\

,,

\'

.. /

/

,

/

!

, : : ........::.......::....ii?i::::;;;;!il.....> ' V / i ,

',

/ FIG. 9. AMPERE

TURNS

USED

UP

IN

PRIMARY

Path NO. I

2

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . .

3 ..... ". . . . . . . . . . . . . . . . . 4 ........................ 5 ....................... 6 . . . . . . . . . . . . . . . . . . . . . . . .

.

T~ETH. Average A m p e r e Turns Per Inch.

in Teeth.

52

I66

31

ioo

21 14 ii 7

Atllpere

Turlls

67 45 35 22

F l u x Distribution in Secondary ]ron.--In t h e d i s c u s s i o n of the flux d i s t r i b u t i o n of t h e p r i m a r y , it w a s t a c i t l y a s s u m e d t h a t t h e q u e s t i o n of s e c o n d a r y r e l u c t a n c e d i d n o t a f f e c t such d i s t r i b u t i o n . T h i s is n o t a b s o l u t e l y t r u e , as a n i n s p e c t i o n of Fz~g. 9 will i n d i c a t e . I n Fig-. 9 we h a v e a d r a w i n g of t h e s e c o n d a r y p a r t of t h e s a m e m a g n e t i c c i r c u i t t h a t was in p a r t s h o w n in Fig'. 7. I t m a y b e

2o2

Hoxie :

[J. F. I.,

noted that the symmetrical distribution of the primary is not repeated in the secondary. This is, of course, due to the differing numbers of teeth in primary and secondary. This difference does not, however, have as much effect as m i g h t at first sight be supposed, since the tops of the secondary teeth nearly join, so that the secondary presents a nearly continuous magnetic surface. In order to estimate as nearly as possible the air gap distribution to the tops of the secondary teeth, the large scale

F I G . 1o.

drawings of F i g s . z o and zz were constructed, and curves of flux density were made for each primary tooth of these figures. To determine these curves the method of Professor Goldsborough (Trans. Am. Inst. Elect. Engrs., i898) was employed. The wavy dotted line in FiEs. z o and zz is the sum of the separate density curves, and indicates graphically the actual flux density over the surface of the secondary. In estimating the quantity of flux entering a given see-

Sept.. t9o3.]

The Induction Motor.

203

ondary tooth from a particular p r i m a r y tooth it was a s s u m e d that the p r i m a r y t o o t h supplied that part of the total w a v e lying b e t w e e n two points of m i n i m u m flux on either side of the tooth. T h i s area, and the part of the area lying over the s e c o n d a r y t o o t h in question, were measured, and the ratio of one to t h e o t h e r taken to be the ratio of the total flux t h r o u g h the p r i m a r y tooth, to the part of it w h i c h entered the given secondary. M.M.F. Used up in A i r G a p . w T h e m e t h o d used for the d e t e r m i n a t i o n of the M.M.F. e x p e n d e d in s e n d i n g the flux ;2 7 o , o O Q

2 j I, ooc2

FIG. ti.

across the air g a p d e p e n d s u p o n the m a g n e t i c d e n s i t y curves of Figs. zo and zi. T h e accuracy of this d e t e r m i n a t i o n d e p e n d s therefore u p o n the a c c u r a c y of those curves. T h e m e t h o d m a y be i l l u s t r a t e d b y a p p l y i n g it to p a t h No. I. T a k i n g the flux w a v e b e l o n g i n g to t o o t h No. i, Fig. zo, the ratio of its m a x i m u m ordinate to its a v e r a g e ordinate is I'z3. T h a t is to say, if t h e r e were no t e e t h or slots, and flux were d i s t r i b u t e d u n i f o r m l y over the entire air g a p at the a v e r a g e rate at w h i c h it occurs in the n e i g h b o r h o o d

204

Hoxie

."

[J. F. I.,

of t o o t h No. I, t he d e n s i t y w o u l d be less t h a n t h e p r e s e n t m a x i m u m d e n s i t y in t he r a t i o of I" to 1"23. T h e M.M.F. w h i c h w o u l d , in s uc h an event , be necessary at t h e air g a p is M.M.F.---~ "045 X "313 X SI,3OO" ~ 723 ampere turns. W h e r e "045 is t h e l e n g t h of the air-gap, "313 is the n u m b e r of a m p e r e t u r n s r e q u i r e d to set up one maxwell p e r s q u a r e inch, t h r o u g h a d i s t a n c e of I i nch in air, and 51,3oo is t h e a v e r a g e d e n s i t y u n d e r t o o t h No. I, and half of each of t h e a d j a c e n t slots. U n d e r th e a c t u a l c o n d i t i o n s we find a flux d e n s i t y of 5 [,3 °o X 1"23 = 63,200 m a x w e l l s p e r s q u a r e inch u n d e r the

19 ~

I

//\\

i~ ~ 6.#

/

"] i

/

/

\

f----__

/ FIG.

I2.

c e n t e r of t o o t h No. i. T h e M.M.F. r e q u i r e d to set it up is M.M.F. = ' o 4 5 X 313 X 63,200 = 890 a m p e r e turns. Cons i d e r i n g all p o i nt s w i t h i n half an inch of t h e face of prim a r y t o o t h No. I to be at t he s am e m a g n e t i c p o t e n t i a l , and all p o i n t s on t h e face of a g i v e n s e c o n d a r y t o o t h to h a v e a c o n s t a n t p o t e n t i a l , it follows t h a t t h e M.M.F. by any flux p a t h across t h e air gap, f r o m p r i m a r y t o o t h No. I to the s e c o n d a r y t o o t h i m m e d i a t e l y u n d e r it, is 890 a m p e r e turns. F o r t h e c o m p l e t e p a t h No. i, w h i c h crosses t he air gap a t t e e t h Nos. i and 13, t he M.M.F. us e d up in t h e air is therefore twice 890 , or I78O a m p e r e turns.

The Induction Motor.

Sept., I9O3.]

205

T h e same m e t h o d applied to o t h e r portions of the air gap gives the M.M.F. n e e d e d for the o t h e r routes, as set down in the following table : M . M . F . USED I N DOUBLE A I R G A P . A m p e r e Tttrns Path M a x i m u m Density for No. iu Air. Double Air Gap. I . . . . . . . . . . . . . . . . . . . . . 63,OOO 1,780 2 . . . . . . . . . . . . . . . . . . . . .

55,500

1,56o

3 . . . . . . . . . . . . . . . . . . . . . 4 . . . . . . . . . . . . . . . . . . . . .

47,5 ° o 40,OOO

1,340 1,13o

5 . . . . . . . . . . . . . . . . . . . . . 6 . . . . . . . . . . . . . . . . . . . . .

31,7 ° 0 I5,9OO

89O 45 °

For the s e c o n d a r y t e e t h and core the m e t h o d is the same as was used in the primary, and the results follow. M.M.F.

USED I N SECONDARY I R O N .

Path Ampere Turns No. in Teeth. I . . . . . . . . . . . . . . . IO3 2 . . . . . . . . . . . . . . . 84

Ampere Turns tn Core. 86 80

Ampere Turns Total.

I89 I64

3 . . . . . . . . . . . . . . .

36

67

4 . . . . . . . . . . . . . . .

2O

57

77

5 . . . . . . . . . . . . . . .

X~

55

47

7

~7

24

6

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"

"

"

•

"

•

IO5

General Summary.--The final a d d i t i o n of the M.M.F.'s used up in various parts of the m a g n e t i c circuit, and for various paths, is given in the succeeding tables. A comparison is made in the second table b e t w e e n the m a g n e t o m o t i v e forces used near the air gap and in t h e cores, and a comparison is also m a d e b e t w e e n the total M.M.F. as now c o m p u t e d and the original estimate, which was m a d e on the basis of 45 effective amperes per p r i m a r y c o n d u c t o r : T O T A L 1V[AGNETOMOTIVE FORC ft. Path No. 1 . . . . . . . . . . . 2 . . . . . . . . . . . .

A, T. in Cores. 186 172

A.T. in Teeth.

A.T. in Air Gap.

Total

269 186

1,78o 1,56o

~,235 r,918

3 . . . . . . . . . . . . 4 . . . . . . . . . .

151 126

5 . . . . . . . . . . . . 6 . . . . . . . . . . . .

83 43

IO3 65 47

1,34o I,I3O 890

1,594 1,32r I,O2O

26

45 °

519

A.T.

206

Hoxie ."

[J. F. I.,

COMPARISON. Path No.

i ............ 2 ............ 3 ............ 4 ............ 5 ............ 6 ............

A.T. Cores.

I86 r72 x5I I26 83 43

A.T. in Teeth Plus Gap.

2,049 ~,746 1,443 ~,I95 937 476

Total. A.T.

2,235 1,918 ~,594 1,32I I,O2O 5~9

Previously Estimated.

3,032 1,778 1,524 1,27o I,m6 5o8

T h e r es u lts of t he c o m p u t a t i o n s are in r e m a r k a b l y close a g r e e m e n t with t h o s e w h i c h w e r e e s t i m a t e d for a magnetizing c u r r e n t of 45 a m p e r e s , t h a t b e i n g t he best v a l u e tha.t could be o b t a i n e d f r o m t h e t e s t r e p o r t s of a line of similar motors. It seems probable that the variation between d i ff er en t i n d i v i d u a l m o t o r s b u i l t f r o m t h e s a m e d r a w i n g s would be as m u c h as t h e difference b e t w e e n t h e s e results a n d t h e a v e r a g e of e x p e r i m e n t s . Since no a c c o u n t has so f a r b e e n t a k e n of l e a k a g e or of t h e small a m o u n t of flux p r e s e n t in t he slots, v e n t i l a t i n g spaces, etc., t h e r e s ul t s of t h e c o m p u t a t i o n show a slightly l a r g e r a m o u n t of flux in t h e iron, a nd t h e r e f o r e call for a l a r g e r m a g n e t i z i n g force t h a n w o u l d be a c t u a l l y needed. A c o r r e c t i o n for this w o u l d b r i n g t h e e s t i m a t e d and comp u t e d v alu es in t o still closer a g r e e m e n t . Effect o3' Correcting Certain Apflraximatzbns.--The p a r t i c u l a r v a l u e of s u c h a s t u d y as t he p r e c e d i n g lies in t he possibility t h a t th e r e s u l t s o b t a i n e d m a y assist one in f o r m i n g an idea of th e r e a s o n a b l e n e s s of cer t a i n c u s t o m a r y a p p r o x i m a t i o n s , and of t h e places w h e r e a p p r o x i m a t i o n s m a y be freel y used, a n d w h e r e only w i t h g r e a t c a u t i o n . W e are now in a position to e x a m i n e t he effect u p o n t h e p r e c e d i n g s t u d y of v a r i o u s m i n o r corrections. T h e p r o p o s i t i o n s w h i c h h a v e been a s s u m e d , t a c i t l y or o t h e r w i s e , a r e : (x) All th e r e l u c t a n c e of t h e m a g n e t i c circuits is cont a i n e d in th e t e e t h a nd air gap. (z) All p a r t s of t h e m a g n e t i c circuits h a v e a c o n s t a n t permeability. (3) N o flux s u r r o u n d s i n d i v i d u a l c o n d u c t o r s , or c o n d u c t o r g ro u p s , locally.

207

T~e Induction Motor.

Sept., 1 9 o 3 . ]

(4) T h e flux is e n t i r e l y confined to iron, except at t h e air gap. (5) T h e conditions of the m o m e n t chosen are constant. None of these s t a t e m e n t s are strictly true. It is desired to d e t e r m i n e t h e probable error involved in t a k i n g t h e m as accurate. (i) F r o m the a c c o m p a n y i n g table, it is seen t h a t the reluctance of a n y m a g n e t i c circuit is principally t h a t of the air gap and teeth. T h e per cent. of total r e l u c t a n c e f o u n d in that part of t h e circuit is as follows: Path Path Path Path Path Path

No. No. No. No. No. No

Average

I 2 3 4 5 6

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P e r Cent. 92 91"3 90"7 9o'$ 91"7 91.8 9i'3

T h e error in distribution, due to the i n a c c u r a c y of assumption No. i, c a n n o t therefore be g r e a t e r t h a n one due to a total c h a n g e in the reluctance of a b o u t IO per cent. A s shown above, this c h a n g e is practically the s a m e by a n y path. T h e alteration in the distribution is therefore e n t i r e l y negligible. (2) T h e p e r m e a b i l i t y of the air gap is of course constant. The p e r m e a b i l i t y of the iron is nearly c o n s t a n t w i t h i n a limited range, b u t is far from c o n s t a n t over a wider one. A s s u m p t i o n No. 2 m a y be quite incorrect in some instances and negligible in others. In Fig. 8, a s t r a i g h t line is drawn t h r o u g h the origin in such a w a y as to become an average of the m a g n e t i z a t i o n curve for a portion of its length. If the m a g n e t i z a t i o n curve and the s t r a i g h t line coincided, it would m e a n t h a t flux d e n s i t y in the iron was proportional to the m a g n e t o m o t i v e force applied to it. F o r any given density, the error involved in a s s u m i n g the above to be true, is indicated by the horizontal distance b e t w e e n the two curves. It m a y be n o t e d t h a t the two curves practically coincide b e t w e e n the flux densities of 25,ooo and 65,0oo maxwells per square inch.

208

Hoxie :

[J. v. I.,

In referring to Figs. 7 and 9 it is seen that, w i t h trifling exceptions, the flux densities in the p r i m a r y and s e c o n d a r y cores lie within those limits. T h e p r i m a r y and secondary teeth b e l o n g i n g to the three o u t e r p a t h s carry flux at a h i g h e r d e n s i t y than 65,ooo. T h e d i m i n u t i o n in t h e actual flux carried b y t h e s e t h r e e p a t h s m a y be n o t e d b y a trial and error method, first a p p l y i n g a eorreetion to the original value, then d e t e r m i n i n g a correction for t h a t result, and so on. T h e process is given in full for p a t h No. i, and the results are given for p a t h s Nos. 2 and 3. P A T H No. i.

FIRST APPROXI ~/IATION. D e n s i t y in p r i m a r y t e e t h . . . . . . . . . . . . . . . . . 96,0o0 A m p e r e t u r n s p e r i n c h for t h i s d e n s i t y ' 52 A m p e r e t u r n s as a s s u m e d (from s t r a i g h t line) . . . . . . . . 23 Difference p e r i n c h . . . . . . . . . . . . . . . . . 29 Difference for p a t h i n c l u d i n g t w o t e e t h . . . . . . . . . . 93 D e n s i t y in s e c o n d a r y t e e t h . . . . . . . . . . IO4,O0o a n d I io, coo Average a m p e r e t u r n s p e r i n c h . . . . . . . . . . . . . . I15 A m p e r e t u r n s as a s s u m e d . . . . . . . . . . . . . . . . . 26 Difference p e r i n c h . . . . . . . . . . . . . . . . . . . . 89 Difference for p a t h . . . . . . . . . . . . . . . . . . . . 8o Total loss in flux o w i n g to decreased p e r m e a b i l i t y of h i g h d e n s i t y teeth, e q u i v a l e n t to a loss in p a t h No. t of 173 a m p e r e t u r n s , or 7"75 p e r cent. of t h e total.

SECOND APPROXIMATION. D e n s i t y in p r i m a r y t e e t h (from preceding) . . . . . . . . . 88,6o0 D e n s i t y in s e c o n d a r y t e e t h . . . . . • . 96,9o0 a n d lOl,5OO A m p e r e t u r n s p e r inch, p r i m a r y . . . . . . . . . . . . . . 36 A m p e r e t u r n s p e r inch, as a s s u m e d . . . . . . . . . . . . 2r Difference p e r i n c h . . . . . . . . . . . . . . . . . . . 15 Difference for p a t h . . . . . . . . . . . . . . . . . . 48 A m p e r e t u r n s p e r inch, s e c o n d a r y . . . . . . . . . . . . . 66 A m p e r e t u r n s as a s s u m e d . . . . . . . . . . . . . . . . 24 Difference p e r i n c h . . . . . . . . . . . . . . . . . . . . 42 Difference for p a t h . . . . . . . . . . . . . . . . . . . . 38 T o t a l loss in flux e q u i v a l e n t to a loss of 86 a m p e r e t u r n s , or 3"9 p e r cent. of t h e total.

THIRD APPROXIMATION. D e n s i t y in p r i m a r y teeth ( f r o m p r e c e d i n g ) . . . . . . . . . 92,3oo D e n s i t y in s e c o n d a r y t e e t h " " . . . Io0,ooo and lO5,OOO A m p e r e t u r n s p e r inch, p r i m a r y . . . . . . . . . . . . . . 444

T/w [nduction Afotor.

Sept., i9o3. ]

20 9

Ampere turns per inch, as assumed . . . . . . . . . . . Difference per ineh . . . . . . . . . . . . . . . . . . . Difference for path . . . . . . . . . . . . . . . . . . . Ampere turns per inch, secondary . . . . . . . . . . . . Ampere turns, as assumed . . . . . . . . . . . . . . . . Difference per inch . . . . . . . . . . . . . . . . . . . Difference for p a t h . . . . . . . ". . . . . . . . . . . . Total loss in flux, equivalent to a loss of 5"4 per cent. of total, or to 121 ampere turns.

. . . . . . . the

23 2 t

67 85 25 6o 54

FOURTH APPROXIMATION. Density in primary teeth (from preceding) . . . . . . . . . 91,ooo Density in secondary teeth . . . . . . . . . . . 98,30o and lO4,Ooo Ampere turns per inch, primary . . . . . . . . . . . . . . 40 Ampere turns per inch, as assumed . . . . . . . . . . . . 22 Difference per inch . . . . . . . . . . . . . . . . . . . . 18 Difference for path . . . . . . . . . . . . . . . . . . . . 58 Ampere turns per inch, secondary . . . . . . . . . . . . . 75 Ampere turns, as assum}d . . . . . . . . . • ' . ..... 25 Difference per inch . . . . . . . . . . . . . . . . . . . . 50 Difference for path . . . . . . . . . . . . . . . . . . . . 45 Total loss in flux, equivalent to a loss of 4'6 per cent. of the total, or to lO3 ampere turns. FIFTH APPROXIMATION. Density in primary teeth (from preceding) . . . . . . . . . 9t,5oo Density in secondary t e e t h . . . . . . . . . . . 99,000 and IO5,OOO Ampere turns per inch, primary . . . . . . . . . . . . . . 41 Ampere turns, as assumed . . . . . . . . . . . . . . . . . ~2 Difference per inch . . . . . . . . . . . . . . . . . . . . 19 Difference for path . . . . . . . . . . . . . . . . . . . . 6t Ampere turns per inch, secondary . . . . . . . . . . . . . 80 Ampere turns, as assumed . . . . . . . . . . . . . . . . . 25 Difference per inch . . . . . . . . . . . . . . . . . . . . 55 Difference for path . . . . . . . . . . . . . . . . . . . . 5o Total loss in flux, equivalent to a loss of 5 per cent. of the total, or t I I ampere turns. So many

as five approximations

for illustration.

The

closely

enough

manner

in which

graphically

in

approximation

after the

final value two

or

three

final value

Fig. z 3. is seen

The to be

are not necessary, may

usually

be

approximations.

is approached

change

half

The

is indicated

due to each

about

except

predicted

of the

successive preceding

change. The

immediate

,of t h e h i g h

density

VOL. CLVI.

No. 933.

effect teeth

therefore

of the

is to reduce

the

low permeability flux about

a s fo1I4

2xo

[J. F. I.,

H o x i e ."

lows, the values for p a t h s Nos. 2 and 3 b e i n g o b t a i n e d as above : P e r Cent. Path

No.

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Such a c h a n g e in the flux of t h e s e teeth w o u l d cause other slight c h a n g e s which in turn w o u l d react upon these values, some increasing and some d i m i n i s h i n g them. Ex. cept as h e r e i n a f t e r noted, variations of t h a t character are negligible.

l\

_5

8.

F I G . 13 .

(3) LeaX'age F / u x . - - U n d e r the specified conditions, i.e., with the s e c o n d a r y on open circuit, the flux surrounding c o n d u c t o r s locally, or s u r r o u n d i n g g r o u p s of p r i m a r y conductors, w i t h o u t e n t e r i n g the secondary, is an extremely small per cent. of the total a m o u n t . U n d e r o t h e r conditions, however, such flux m a y b e c o m e important, and it is desirable t h a t its a m o u n t and distribution, w i t h varying p r i m a r y currents, s h o u l d b e carefully studied. T h e flux conditions t h a t h a v e heretofore been chosen for discussion are those w h e n the phase angle of c u r r e n t in phase A is

Sept., 19o3.]

T/ze [ncluction 3~otor.

21 I

3o°. T h e c u r r e n t s in the slots of Fz~. 7 are t h e r e indicated, the l a r g e r c u r r e n t s b e i n g f o u n d in t h e c e n t r a l slots. Referring to Fz~. z3, we find an e n l a r g e d d r a w i n g of. t e e t h Nos. 6, 7 and 8, w i t h th e c o n d u c t o r s l y i n g b e t w e e n t hem . L e a k a g e flux m a y be di vi ded into two parts. (a) C o m p l e t e flux p a t h in n o n - m a g n e t i c media. (b) F l u x p a t h p a r t l y in iron. F l u x as in (a) occurs along t h e total l e n g t h of each primary co n d u cto r, and al ong t h e e nd c o n n e c t i o n s as well. Flux as in (b) is f o u n d only for a c o n d u c t o r l e n g t h e q u a l ' t o the n et l e n g t h of l a m i n a t i o n s parallel to the shaft. T o o b t a i n a r o u g h idea of t he o r d e r of m a g n i t u d e of class (a), n e g l e c t i n g end c onne c t i ons , we will a s s u m e flux paths c o i n c i d e n t w i t h i n s u l a t i o n sheaths, as i n d i c a t e d in Aig. z 3. A c o m p u t a t i o n on this basis gives t he following results : Total flux per slot surrounding a single conductor only. 64 maxwells Total flux per slot surrounding a pair of conductors . . i34 " Total flux per slot surrounding all four conductors . . I65 " Total flux in insulation surrounding any single conductor of Fig..'3 . . . . . . . . . . . . . . . . 248 " T h e s e figures are n o t g i v e n as at all accurate, since t he a s s u m p t i o n on w h i c h t h e y are based Js a v e r y r o u g h one, but t h e y serve to show t h a t t h e l e a k a g e flux which occurs entirely in n o n - m a g n e t i c p a t h s is q u i t e u n i m p o r t a n t as c o mp ar ed w i t h t he total. Leakage F l u x P a r t l y in l r o n . - - T h i s is a v e r y c o m p l e x matter, an d one wh i ch is difficult of analysis. Obviously, so far as affects th e p r i m a r y a l o n e - - a n d we h a v e as y e t only considered p r i m a r y E . M . F ' . ' s - - t h e r e will be no l e a k a g e flux unless a g i v e n t o o t h carries flux at t he s a m e t i m e in two directions. F s r example, in Fzff. i3, w i t h t he condi t i ons of the m o m e n t , a n d no leakage, flux t r a ve l s d o w n w a r d t h r o u g h tooth No. 6, an d u p w a r d t h r o u g h t o o t h No. 8 ; no flux existing in t o o t h No. 7. W i t h leakage, flux w oul d t r a v e l d o w n w a r d in t o o t h No. 6, and u p w a r d p a r t l y in t o o t h No. 7, and p a r t l y in t o o t h No. 8, while s om e flux t r a v e l i n g u p w a r d in t o o t h No. 8 would r e t u r n d o w n w a r d t h r o u g h t oot h No. 7, c o m p l e t i n g a

212

Hoxie :

[J. F. I.,

circuit w i t h o u t reaching tooth No. 6. T h a t is, flux would be traveling u p w a r d on the left-hand side of t o o t h No. 7, and d o w n w a r d on its right-hand side. S u c h a flux w o u l d be m o s t dense along either edge of this tooth, and would be set up b y the difference b e t w e e n that part of the M.M.F. of the currents in slot 6- 7 acting b e t w e e n the points A and B, Fig. zd, and the p a r t of the M.M.F. of the currents in slot 7-8 acting b e t w e e n the same points. F r o m s y m m e t r y , n e g l e c t i n g any n o n - s y m m e t r y of the secondary, the M.M.F. 6-7, b e t w e e n d and B is equal to the M.M.F. 7-8 b e t w e e n C and D. T h e M.M.F. of 1-8 b e t w e e n d and ]? is less than this b y the a m o u n t used up in f o r c i n g flux from C to A and from B to D t h r o u g h the iron, or by a trifling amount. T h i s trifling difference acting along the tooth from d to B and across the air-gap at B would indeed set up a v e r y slight flux, b u t not e n o u g h to b e w o r t h considering. L e a k a g e t h a t m a y occur w i t h r e s p e c t to the secondary conductors, i. e., that which m a y circle p r i m a r y conductors, and not those of the secondary, is of more importance. S u c h flux m a y be set up along the ends of the p r i m a r y teeth and across the i n t e r v e n i n g slots; or across the tops of the s e c o n d a r y teeth and b e t w e e n the ends of the secondary projections, or along the air gap. T h e a m o u n t of this leakage m i g h t be o b t a i n e d if it were possible to c o m p u t e the reluctarme of t h e c o m b i n e d path and the M.M.F. applied to it. In so far as we m a y a p p r o x i m a t e to such determinations we m a y a p p r o x i m a t e to a d e t e r m i n a t i o n of the leakage bet w e e n p r i m a r y and secondary. F o r present purposes, with s e c o n d a r y conductors carrying no current, the M.M.F.'s are not large, and t h e distances t h r o u g h air are large, so t h a t this flux is not very important. T h e m a t t e r of l e a k a g e w i t h the m o t o r r u n n i n g normally will be discussed later, t h o u g h no v e r y satisfactory method for p r e d e t e r m i n i n g this q u a n t i t y s e e m s to be possible, at l least w i t h o u t an a m o u n t of labor d i s p r o p o r t i o n a t e to the! p r o b a b l e results. (4) Effect o f F l u x in tile S l o t s . - - T h e a m o u n t of flux that crosses t h r o u g h the n o n m a g n e t i c s u b s t a n c e s lying in a slot

Sept., 19o3. ]

T/le Induction l~Iotor,

z 13

depends principally upon the degree to which the iron of the neighboring teeth is saturated. In the case of the present motor the teeth densities are rather moderate and a very small proportion of the flux crosses through any slot. For iIlustration we will compute the probable flux found on either side of primary tooth No. I and present in the ventilating and lamination spaces. We have: Ampere turns per inch in tooth No. I . . . . . . . 52 Ampere turns per inch through slot therefore are not more than . . . . . . . . . . . . . . . . . 52 3Iaxwells per square inch in slot not more than , . I66 Area for flux in slot . . . . . . . . . . . . . . 3"78 sq. in. Area fbr flux in ventilation spaces, etc . . . . . . . '73 " Total area . . . . . . . . . . . . . . . . . . . . 4'51 " Total flux, maxwells . . . . . . . . . . . . . . . 75o Flux in tooth No. I . . . . . . . . . . . . . . . 3o8,ooo Proportion of flux in neighborhood of tooth No. ~, but not in iron . . . . . . . . . . . . . . . . "24 p. c. This is the worst case for the primary. In the secondary t h e m a x i m u m t o o t h d e n s i t y w a s e s t i m a t e d a t I I0,000 m a x wells per square inch. Making a similar computation for this case we find the following : Ampere turns per inch . . . . . . . . . . . . . . Corresponding flux density in air . . . . . . . . . Total area for this density . . . . . . . . . . . Total flux not in tooth . . . . . . . . . . . . . . Proportion of flux in neighborhood of tooth but not in iron . . . . . . . . . . . . . . . . . . . .

f4o 45o 6"67 sq. in. 3,ooo "96 p. c.

(5) Comtitions o] the Moment.--At t h e m o m e n t c h o s e n f o r investigation the flux per pole has the greatest possible value and the flux in a single tooth at the center of a pole is a t m a x i m u m density. The approximations that have been made are therefore less close than at any other moment. The only other item which may change is the position of the secondary structure. Since the primary and secondary are unsymmetrical and the number of secondary teeth is'not a multiple of the number of poles, it follows that the cond i t i o n s o f t h e m o m e n t , a s s t u d i e d i n Figs. 7 a n d ' 9 , a r e n o t precisely those of any other magnetic circuit, even at this instant. The changes which may be produced by a shifting of t h e s e c o n d a r y a r e c h a n g e s i n s e c o n d a r y - t o o t h density.

214

Notes and Comments.

[J. F. I.,

T h e conditions of Fio~s. 7 and 9 probably represent a fair average, since t h e y are favorable to a . l o w d e n s i t y in the • teeth at the left of the figures and to a h i g h d e n s i t y in those at the right, b o t h b e i n g for the same m a g n e t i c circuit. Operation.--So far n o t h i n g has been said a b o u t the theory of the m o t o r or of the principles of its operation. W e have simply e x a m i n e d the b e h a v i o r of the m a g n e t i c circuit of a p a r t i c u l a r m a c h i n e w h e n an appropriate polyphase E.M.F. was applied to it. It remains to note q u a l i t a t i v e l y the b e h a v i o r of this m o t o r w h e n r u n n i n g , and t h e n to discuss the p e r f o r m a n c e of i n d u c t i o n motors in general. [ To be continued.] M O N E Y SAVED BY U N I T E D STATES GEOLOGICAL S U R V E Y MAP. In its issue of May 2I, I9o3, the Engineering News" has an interesting article on the very expensive construction of the Pittsburg, Carnegie and Western Railroad near Jewett, Ohio, in which the following paragraph occurs: " Perhaps the most instructive location problem solved in this work was one involving the use of an atlas sheet of the United States Geological Survey with a contour interval of 3o feet. Before the issue of this particular atlas sheet the preliminary surveys and first location of the proposed line had been completed, and work was about to begin when a new resident engineer, Mr. T. H. Loomis, was appointed. Happening to be personally acquainted with the Government topographer, Mr. W. T. Griswold, who was then engaged in mapping this part of Ohio, Mr. Loomis secured from trim a p h o t o graph of the contour map then almost finished. From this photographic copy Mr. Loomis projected a new location between stations 65o and 836, effecting thereby a saving of 2,2oo feet in distance and 75 ° of curvature, be. sides eliminating 60o feet of tunnel. This change measured in dollars and cents resulted in reducing the first cost of that portion of the road by at least $85,coo. Turning to the [last annual report of Mr. Charles D. Wolcott, Director United States Geological Survey, we find that 1864 square miles of the State of Ohio have been mapped thus far, at a cost of $~2,oov; yet one small atlas sheet, covering some 220 square miles, has been the means of saving a sum more than seven limes what it has cost to map the whole ~864 square miles. As a tribute to the accuracy of the Government topographer, Mr. Griswold, it may be well to add that when the railway engineers ran their profile levels along the paper-projected line, at no place did the actual elevation differ more than i6 feet from the elevation scaled off the contour map. To any engineer acquainted with the roughness of Eastern Ohio and with the approximate methods necessarily employed in Government topographical surve) s of such vast areas, these figures speak forciby enough. And it is well that just such facts as these should become generally known, for there has been altogether too great an apathy in the past in furthering the work of the United States Geological S u r v e y - - a work that is destined, to be of incalculable value to every citizen of the United States."