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Torque in induction disc-type instruments Goumeniouk, Gleb Ivanovitch 1936

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TORQUE M INDUCTION DISC-TYPE INSTRUMENTS hy Gle"b I . Goumeniouk Submitted f o r the Degree of LSTER OP APPLIED SCIENCE i n the department of ELECTRICAL ENGINEERING. THE UNIVERSITY OP BRITISH COLIMBIA A p r i l , 1936. IHDEX. Page 1 I n t r o d u c t i o n 2 The Media i n v o l v e d 4 Torque 13 P a r t i c u l a r expres s ions f o r , 27- 29 P r a c t i c a l c o n s i d e r a t i o n s 30 Apparatus 37 E x p e r i m e n t a l Work 41 R e s u l t s " 48 Appendix " 52 Acknowledgement 61 B i b l i o g r a p h y 6 2 NOTATION S c r i p t c a p i t a l s r e p r e s e n t v e c t o r s , "block l e t t e r s r e p r e -sent s c a l a r s » Y e c t o r product y ~ a."1 + -1 ^  ^ -S c a l a r product Radius of c u r r e n t f low (a l so r e s i s t a n c e ) t? Radius of d i s c (sometimes r e s i s t a n c e ) <P . Maximum f l u x (£> R . M . S . ( e f f e c t i v e ) f l u x £ E l e c t r o m o t i v e f o r c e , Energy ^ Thicknes s of d i s c | J 'J ^Admit tance of d i s c , a l s o c a l l e d p e r m i t t i v i t y of medium <J Conductance ^ - a^jj/e af cj',sc b Susceptance 6 F l u x d e n s i t y B = j 6J ^ C u r r e n t d e n s i t y Log to "base "e" Angle between magnets, a t the centre of the d i s c A A Phase angle "between cp„ and cj)e. Frequency of A . C . y c y c l e s per second. H Magnet ic f o r c e E l e c t r i c f o r c e ^ E l e c t r o m a g n e t i c f o r c e Host of the symbols are e x p l a i n e d i n the diagrams to which they a p p l y . TORQ.UE IJT IRDUGTIQit DISC-TYPE' IflSTBDMEHTS• Contrary to general opinion, much theoremtical work has "been done on the subject of torque i n in d u c t i o n disc-type instruments. Most of i t , however, has "been very shallow, and a l l of i t e n t a i l s assumptions of one form or another. ..•As a subject i n mathematical physics, the work should hot have any assumptions; but as a work of engineering, where the main idea i s to obtain a means of e a s i l y e x p l a i n i n g a phenomenon by the use of mathematics, s i m p l i f y i n g assumptions-are h i g h l y d e s i r a b l e . By "assumptions" we do not mean presumptions which make a v a r i a b l e into a constant: merely such ideas which, though reducing mathematical argument, do not tamper ?/ith the p h y s i c a l p r i n c i p l e s involved. I t i s quite an a r t to make such assumptions, and u n t i l that a r t i s p e r f e c t l y mastered, the r e s u l t s w i l l he l i m i t e d i n t h e i r a p p l i c a t i o n s j and u n t i l work i s turned out i n which no assumptions are present, there w i l l "be ample room f o r c r t i c i s m . In view of the above statements t h i s subject was chosen, not so much v/ith the i n t e n t i o n of g e t t i n g a complete agreement between theory and measurement, as wit h an aim to lay. down c e r t a i n fundamentals i n terms of which, no matter how complicated i n space, the phenomena connected with the apparat-us may he expressed. N a t u r a l l y some sort of experimental check i s i n order; but an excessive amount of care has not been paid to the p e r f e c t i o n of apparatus: to be s p e c i f i c , of the 25,000 l i n e s which l i n k e d the e x c i t i n g coil,.7,500 only passed through the search c o i l at the p o l e . t i p . This .was due mainly to- the f a c t that the c o i l s were bulky and concentrated; i t may be eliminated to a large extent. Assumptions w i l l be excluded wherever p o s s i b l e , but an attempt w i l l be made to give the work an engineering cha-r a c t e r . ' Mathematical a n a l y s i s , where i t gets bulky, w i l l be c a r r i e d only to a point from which ordinary, and at times laborious routine work w i l l obtain the desired expressions. In some eases, at that p o i n t , a -change w i l l be made to some approximately equivalent representation. Loss of g e n e r a l i t y w i l l be avoided. "Where a d i s c u s s i o n may be too p a r t i c u l a r , i t w i l l be omitted altogether. TEE MEDIA Let us consider f i r s t the media i n which the phenom-ena are taking place.. ..The currents w i l l flow i n a non-magnetizable conducting d i s c i n which they w i l l he r e s t r i c t e d by the p h y s i c a l boundaries ;and governed by the s p e c i f i c r e-sistance of the m a t e r i a l , the self-induetance of the current f i l a m e n t s , and by the actions of the current filaments on each, other. These ac t i o n s w i l l comprise the mutual inductance be-tween filaments of current and an e l e c t r o s t a t i c force of r e -p u l s i o n . This l a s t force w i l l not be worked w i t h d i r e c t l y , as i t s e f f e c t s are taken care of when considering the e f f e c t s of the others. The force generating the c u r r e n t s A s u p p l i e d by the time v a r i a t i o n of the magnetic f l u x which w i l l pass through •two media;-(i) a permeable core, and ( i i ) a i r . This l a s t statement at once brings to our a t t e n t i o n the f a c t that the core f l u x and the a i r f l u x w i l l not be quite i n phase, nor w i l l they be s i n u s o i d a l ; a f a c t which i s n e g l i -g i b l e or not, depending on circumstances i n which they are considered. In order to get acquainted w i t h the p e c u l i a r i t i e s of the medium i n vtfiich the currents flow, l e t us consider an i n -f i n i t e plane sheet of u n i t thickness, which i s tiireaded at some point by a p e r i o d i c a l l y varying filament of magnetic f l u x . I t i s obvious that the currents induced w i l l flow i n concentric 5 . c i r c l e s w i t h t h e i r centres at the "centre of dens i t y " of the "bundle of f l u x . . • • Any one of these current filaments of radius £?, w i l l have a c e r t a i n impedance i n i t s path. This impedance w i l l con-s i s t of a hack E.M. F. due to ohmic re s i s t a n c e j A a b a c i E . I J ' . /^rrw^j-generated "by ( i ) the f l u x through the c i r c u i t due to the ^ L'c*~w filament of current, and ( i i ) the f l u x through the c u r c u i t due to a l l other filaments i n the p l a t e . F i r s t consider the problem q u a l i t a t i v e l y , l e t a fila m e n t of radius carry a current of / wni/ ( a l l the work i s i n absolute G.G.S. u n i t s ) and l e t the r e s t of the p l a t e carry a.current^the density of which i s i n v e r s e l y p r o p o r t i o n a l to the radius. There w i l l be a c e r t a i n f l u x through that c i r c u i t of r a d i u s ^ , Change the scale of l i n e a r measurement, and from the change i n the u n i t of E.M.P. we see that the f l u x i n s i d e the circuityj>er u n i t current i n the fi l a m e n t , i s pro-p o r t i o n a l to£>/. Hence the t o t a l reactance^and also the impedance of the c i r c u i t ; i s p r o p o r t i o n a l to -Since the phase angle of the current behind the E.M.P. i s Ian ' Ljp e we see that the phase angle i s constant over the e n t i r e p l a t e . With the above f a c t s i n mind, we may c a l c u l a t e the t o t a l reactance of any r i n g of radius ,^.,. and obtain the phase angle, which i n turn we may use to f i n d the r e a c t i v e component d \ of !y i n any system of currents, where that component v a r i e s i n v e r s e l y as the current density. ' Let us take a p l a t e / cnn- t h i G k ; a n d consider a r i n g of mean radius, say 10 cms. extending from 7-5 eras, to 10-5 cms-. Let the. current density at the centre of the r i n g he /scj.cm. Then over the r e s t of the p l a t e the current density w i l l 'be (} - LP In view of the f a c t that a vast amount of work has given .by-been done on the estimation of self-inductance of c i r c u l a r c o i l s , we'-shall merely quote one of the expressions f o r a c i r c u l a r c o i l of square cross s e c t i o n . Thus we have f o r L (Weinstein 1s formula) -2i?,-L - 4-n P S u b s t i t u t i n g , we get L = 275-0 centimetre.? In order to obtain the mutual e f f e c t of the 'whole plate" on the r i n g ; we are considering, we must f i r s t o btain the mutual c o e f f i c i e n t between two coplanar ;concentric c i r c l e s . We have f o r the f i e l d at any point P i n s i d e of a c i r c u l a r current of radius £?, O P H A. 6 r de Then f-f w h e r e Y -- o casB +. d e R* - a sin^e and r a t i o n a l i z i n g ; H = t P " - a 1 Sin > d 9 P/- a*-e de This i s of e l l i p t i c form <_ ^ ,^ out f o r our purpose i t may D e most convenient to have i t i n the form of an i n f i n i t e s e r i e s . This w i l l give us ^£ =4 2 ft -L (?, 1/1,3 3 2,4/ 7 2o-/ 2 77 -c F /• 3 2.4 2o +/ /. 3 . S 2n-/ 2.4. 6 . : . 2. 2 « -c P.* - a 7 -' 2~ -/ /. 3. ? ^ - / r /_£ 2.4 .... 2„ / 2 r, -t 2n - / 2 . 4 . £ Then i f /7 i s the magnetic force per u n i t current M , 2 / / H' Aide .d*. Jo 'h 7 £ With the a i d of the above expression, we may set up an i n -t e g r a l f o r the magnetic f l u x through our r i n g of radius P - 10 cms. Thus, t h e i l u x I -r 3. 5,7 - . 2 n * / I 4 . C. & 2n I o 2TT P / + n -i- I In + I 3. S. 7 4 - 6 . 12, dr r 2 o n ^ e , 4-n + I '°2™ f y] 3. 5 , 7 . 4. •2o 2o -f / r 1.061?, 2Q ft' 2 3, 0", 7 4. 2 i + 7. 1 ' n f , 2o P, / • O S ^ 3. 0", 7 2^-+ 20 ff 2 P, • 7 ^ 2o _*// (3, 5, 7 I t should De no tea that a l l the inductance terms are p r o p o r t i o n a l to £?, . E v a l u a t i o n of the above gives cf) roughly 3 5' 2 5Hence, i n c l u d i n g the self-inductance texm$, the t o t a l f l u x i s approximately 3 75"0 / m e i . Ho?/ we can obtain tiie phase angle of the current, thus 7\ - fan f £?eQchvc drojD \ \ H'CSli'riVe drab / I f the density i s kept constant and the thickness i s reduced to % cms-. f then the f l u x through the r i n g i s reduced to £ $ approximately, ana.the new angle i s Hence the impedance angle f o r an unbounded p l a t s of thickness $ cm? and s p e c i f i c r e s i s t a n c e ^ m i c r o h m s c u ^ e is > .. . k L 2 o Tl p to1. r where - L i s the una-uctance i n centimetres and i s the angular frequency of the a l t e r n a t i o n s . For an unbounded Copper p l a t e / cn-,, t h i c k ; <^  = 6 5 i , For aluminium 7) = 8 2 5 ° . For a p l a t e 'Ooms radius n "becomes 7 7° f o r copper and ^ cj° fox aluminium. I f the d i s c s are / T " . t h i c k , we have 7). -~-.'2'46 and /S". For a brass p l a t e I mm, t h i c k , .20 cms diameter >i i s approximately £ . I t may he w e l l to discuss at t h i s point some of the laws of electromagnetism which we s h a l l have occasion to use i n what f o l l o w s . We have seen, i n our elementary f i e l d , that the impedance was p r o p o r t i o n a l to the radius of the flow c i r c l e , and that the phase angle was constant a l l over the d i s c . Hence •-we. may use the conventional method of representing our current f i e l d i n terms of the vector of current d e n s i t y ^ . Having done so, we see that the equations governing the d i s t r i b u t i o n of current are:-T'.rn e cii a I n t e g r a t i o n of the above equations w i l l give the d i s -t r i b u t i o n ?/hich we seek. .Direct i n t e g r a t i o n i s d i f f i c u l t , but since we f i n d that, i n - s o - f a r as our problem i s -concerned, a l i n e a r d i f f e r e n t i a l equation defines the voltage drop across an element of the medium, we may apply the p r i n c i p l e of super-p o s i t i o n . This w i l l enable us to combine elementary f i e l d s of current flow i n t o a f i e l d of f i r s t order complexity, and then to t r e a t the phenomena i n that complex f i e l d i n terms of e f f e c t s i n the c o n t r i b u t i n g elementary f i e l d s . One of the e f f e c t s which we s h a l l have to consider, i s that of the mutual force between a current and a magnetic f i e l d , expressed by or - / / / « x dv This i n t e g r a l i s gen e r a l l y given by . F - 5 1 * when G> i s the f l u x density at the centre of a conductor of lengt h / , c a r r y i n g a uniformly d i s t r i b u t e d current 'I In the problem under co n s i d e r a t i o n , however, the i n t e g r a l i s best obtained by taking the element of volume with, the' <& Instead of with the ^ . Thus,' we ..shall obtain where <p i s the amount of f l u x passing through a current f i e l d of density J at the centre of the f l u x , the length of the i h t e r l i n k a g e being ^ , which i s the thickness of the p l a t e c a r r y i n g the current. The above expression holds i n inverse square f i e l d s of current density and s i m p l i f i e s the work con-s i d e r a b l y , since i t eliminates the n e c e s s i t y of finding, the length of the current paths through the bundle of f l u x . I t i s ; thus independent of the shape of the' boundary of .the bundle of magnetic f l u x . The expression J<$>^ applies to f i n i t e l y bounded regions of uniformly d i s t r i b u t e d f l u x density (£> . We may allow f o r f r i n g i n g by a summation such as i l l u s t r a t e d i n the diagram, -there- the expression a p p l i e s to forces between conductors, i t may be evaluated i n terms of the expression which postulates that the electromagnetic force tending to i n -crease a v a r i a b l e x i s ' given by the increase i n e l e c t r o k i n e t i c energy per u n i t increase of the v a r i a b l e , the currents being maintained constant. I t has been stated that we may combine elementary f i e l d s , so also we may resolve f i e l d s of f i r s t order complexity. Thus, when considering the currents i n the d i s c produced "by an e c c e n t r i c a l l y - s i tuated, p e r i o d i c a l l y - v a r y i n g "bundle of f l u x , we use the p r i n c i p l e of images, which places a s i m i l a r period-i c a l l y v a rying "bundle of f l u x a t a distance rj ^ from the centre of the d i s c , F "being the radius of the disc,and r and /j distances from the centre of the d i s c to the centres of the r e a l and image "bundles r e s p e c t i v e l y . The image f l u x i s h a l f a period "behind the r e a l , as required. The d i s t r i b u t i o n of magnetic :density i s characterized by the equations V x M - 4 * r V • B = o Bote that there i s much analogy between the equation governing the d i s t r i b u t i o n s of the q u a n t i t i e s jj and & The above fundamental p r i n c i p l e s , v/ith t h e i r c o r o l l a r i e s w i l l fo rm the basis of t h i s work. TORQUE F i r s t we s n a i l consider, i n a general way, trie f a c t o r s which produce torque i n the type of apparatus with which we are concerned. the various out-of-phase magnets w i l l pass through the cores of those magnets, nearly a l l the torque w i l l be due to the e f f e c t s of these f l u x e s . There w i l l a h o be a torque due to the a i r - , or leakage-flux l i n k i n g the e x c i t i n g c o i l of a mag-net, and the currents due to the cure f l u x e s of other magnets; and yet another small torque caused by the i n t e r l i n k a g e of currents produced by the a i r - f l u x of a magnet,and the a i r - f l u x of another magnet. Snould trie cores not be symmetrically d i s -t r i b u t e d with respect to the d i s c , there w i l l be a torque due Since most of the f l u x l i n k i n g the e x c i t i n g c o i l s of to the f a c t that the leakage-flux i s not m phase with the core f l u x . two d i s c r e t e , out-of-phase bundles of p r e s s i o n f o r the torque produced by We s n a i l now a r r i v e at an f l u s i t u a t e a at d i f f e r e n t parts of the d i s c , the d i s c being a t r e s t , 14 Let the rlux e s at P\ arid B be <$„ ^ <'o(W]^  cf>B (W -The image f l u x e s at ^ ' and 6 ' ' w i l l then be - cf> n s ' ° ( ^ J and - ^s (^oJ ~c^J rji^ g current .densities at S clue to the-i'luxes at p> and &' are Q , cr CP*, oo c^ro (u>f - 7) A- f\ ° ~ 5 :— —4= i _ The forces due to the i n t e r l i n k a g e of these components of current w i t h thu f l u x at S> w i l l c l e a r l y be along J, and ; but as we are i n t e r e s t e d i n the t a n g e n t i a l components of those f o r c e s , we s h a l l only consider the r a d i a l components of the current d e n s i t i e s which are 2 ^ , ( J in -j-> (u>l- - 7) J f. <T (fin Cos J , 0 The electromagnetic force i s obtained from /6. where C6 e i s the vector of f l u x density at B j <pe, the f l u x at & and <-T the thickness of the d i s c . Hence the t o t a l t a n g e n t i a l force under & i s f, Sin ~j> _ R' N / ' o -j> ~5r r- 3 * . cr 0 5 I t-0 and the torque .S i m i l a r l y the torque due to the forces under cr d d ^ 0 / - /)J c o r ^ 0 / - (^J c/j r> e c m r These expressions may he w r i t t e n (expressing j, , -?= and J . as functions of R .-, ^  and cr 7 r, * + - 2r, r„ Cos 4 n e and F r, 2 F /-^_ ecu r, 6 X 0 2r, r , M I r f^. i We notice that the negative term Inside the square "brackets Is i n each ease equal to r^rj- + p * _ 2 P V * - cos ana so we w r i t e Torque -r, r*. P *" r/W r , r * + P - 2 P V , r, cor-7J Cor cr "/ *" + - 2 r, r_ cos -Jj r, r. 2- •' i r. r . + P 4 - 2 P V , r L KJA c o ; Since the time-mean of each of the sVo1" u>f } co - )ijf a c t o r s i s a h a l f , we have f o r the f i n a l expression Torque -2 v r, /7_ 57 o 7 'P •+ r," - 2r,r, cos-jj >PC + P 4- 2^r,r^f\ r, r , r C O J ^ * 18. R r, s, o Cos tP cos /\ In the case of p r a c t i c a l metres, the discs give 7^  about 4 ° | hence, a t normal power f a c t o r s 7s) may be neglected altogether. In that case we have f o r the torque an expression of the form J^f - f~l cos'1'oor /3 $,n LO/ which we see i s a b s o l u t e l y independent of time. Thus we have cr r, which may be generalized to C o s dune cms. R' ^ r, r 2 s/n^L r + r Cos T/Here ( 8 , • ^ (/, J I ; «« - £ and ^ and | are angles measured to elements of f l u x from some reference l i n e . The i n t e g r a l Is taken over the regions of d i s t r i b u t i o n of and 63 s , and w i l l give the t o t a l torque a c t i n g on the 20 d i s c . I t necessitates the' knowledge of the d i s t r i b u t i o n of the qu a n t i t i e s 03 „ and ©« . With the a i d of the above expression, we may obtain p a r t i c u l a r r e s u l t s , some of which are given i n the next few pages, together with diagrammatic explanations. So f a r we have considered the disc at s t a n d s t i l l . The d i s c i n motion o f f e r s an i n t e r e s t i n g problem, which w i l l be considered i n terras of the braking effect' of a permanent magnet. We shall.assume that the d i s t r i b u t i o n of the' f l u x density vector i s uniform over a f i n i t e l y bounded rectangular region; to allow f o r f r i n g i n g we resolve Into sections,and sum the e f f e c t s . The conditions of the problem are shown i n the accom-panying diagrams. The developed model'is used, since, when the magnet pole face i s small as compared w i t h the d i s c , and i s close to the edge of the d i s c , no appreciable e r r o r i s incurred. C o n s i d e r i n g the equat ions V x 1 = _ \§ v.- I V o * bh 2.77 2&, 2<f. which, when in t e g r a t e d , w i l l describe the flow of current, we make f u r t h e r s p e c i f i c a t i o n s by saying V v x far * <& where & i s the vector of e l e c t r i c f o r c e . Then i f we wri t e V x 4 n ( V 4-r J we may think of the d i s t r i b u t i o n of X i n a medium of per-m i t t i v i t y cr , as being a d i s t r i b u t i o n of f l u x density of ^ f l l ^ l i H J ^ o f c u r l of e l e c t r i c force, flowing along c e r t a i n geometrical f i g u r e s i n space. The analogous expression 4 n may help to convey t h i s idea. We may now proceed to f i n d out more about t h i s current of \ An Due to the f a c t that an electromotive force i s II generated only i n the s e c t i o n of the conducting p l a t e which i s immediately underneath the pole, we have that region occupied by a u n i -formly d i s t r i b u t e d vector of e l e c t r i c f o r c e ; . Our assumptions give plane f i n i t e boundaries to t h i s rectangular region, and from the f a c t that CuzL i s the l i n e i n t e g r a l per u n i t area, we see that V*jS> w i l l have a value only when the l i n e i n t e g r a l i s taken around a j j o r t i o n of the plane boundaries which are p a r a l l e l to tne vector ,£ . Evaluating V * ^ from the l i n e -7 i n t e g r a l standpoint, we see that 22. per cm. di f f e r e n c e i n the compon-ents of the path i n and out of trie region. I f the l i n e i n t e g r a l i s PoZ-A alonJ wn,'ch taicen as shown i n the sketch, hhe lime, • in J-e qral /alt en . The d i r e c t i o n of the vector a a 6 V t o t a l current of 4 t i s i s also shown a B V 4 n Hence the flowing i n the plane faces which are p a r a l l e l to the vector. Ed L /nes af el-ecfrtc current l/ec/or curren r a I 4 n j This vector current w i l l produce a f l u x of e l e c t r i c current, the nature of the d i s t r i b u t i o n of which, i s also roughly i n -dicated . I f we assume 0 , which i s p e r f e c t l y j u s t i f i a b l e f o r small values of S" , then the d i s t r i b u t i o n of e l e c t r i c current i s only a f f e c t e d by the vector current flowing i n the X Z boundaries of the region of \o ; and again, since — f - =• O , we may consider trie d i s t r i b u t i o n of 2 l terms of the f i e l d arounu i n f i n i t e l y long, t h i n riboons ca r r y i n g the vector current a. \7* & A- % The p e r m i t t i v i t y of the medium i s cr = J , the conductance. The problem now. 'is to f i n d the d i s t r i b u t i o n of the f l u x of e l e c t r i c current around the t h i n ribbons c a r r y i n g the vector current. To do t h i s , we again use the p r i n c i p l e of images, and as we are using the developed model, we have the fo l l o w i n g p i c t u r e : -r - - - - i -4-(\f the. velocity of the disc, at the Cenrrc of density of moyne He. flux Line r-ep.rcsentih the centre a f hhe d'S~c . The e l e c t r i c current density at any point w i l l be due to the vector current a -~~flowing i n the A ribbon conductors. 4 if Y/e make ^ =• o outside the di s c by making cr =o to the l e f t of the edge of, the d i s c . Since the energy absorbed by ohmic l o s s due to the currents i n the di s c i s supplied by the work done to rotate the d i s c against the braking force created by the permanent magnet, we may express that braking force i n terms of the energy i n the. f i e l d of j- j which w i l l be the i n t e g r a l being taken over the volume of the d i s c . Bow the energy o f the two c i r c u i t s d i a g r a m m a t i c a l l y d e s c r i b e d b e l o w - 9 P« r nn'i fi i vi I- <j <T b "2nc* C / rcutl 2 c Sh o v\/iny Sec f-ion jz>er jz> rhe -flo\N of current-en d i cu lor ro i s , f r o m the g e n e r a l e x p r e s s i o n 4 u S V/ 4 IT g i v e n by l_ ~ PSfj per cm . /'o ct'irechon f l where Z_ r e f e r s to the u n i t s of e l e c t r i c c u r r e n t l i n k i n g e i t h e r o f the e q u a l c i r c u i t s p e r u n i t o f V * S, f l o w i n g i n t h a t c i r c u i t A. rr and ^ r e f e r s t o the u n i t s o f e l e c t r i c c u r r e n t w h i c h l i n k one c i r c u i t due to a u n i t o f v — f l o w i n g i n the o t h e r c i r c u t . Sow, c o n s i d e r i n g the f a c t t h a t where $ 1 r e p r e s e n t s the f l u x l i n k a g e s i n the f i e l d ; i.e., the l i n k a g e s between the f l u x of e l e c t r i c c u r r e n t s i n the d i s c and the vector current a y x ^ - , we see that 4 t <r£ a l 5 V /_ - M 4 where,.as before, £* represents the thickness of the a i s c i n centimetres. The constants, L and M , are obtained from the geometrical c o n f i g u r a t i o n of the ribbons* The work i s quite standard and i s g i v e n ' i n the appendix. Prom t h i s expression f o r the energy of the f i e l d of we may obtain the force tending to decrease the v e l o c i t y . I t w i l l be F - . fL 4 m e r f o 1 3 V 4 e v L - M) dynes Assuming that t h i s force acts at the centre of density of the f i e l d of the magnet, the r e t a r d i n g torque w i l l be Z_ — M ) d\f ne cms I f the magnet c a r r i e s an a l t e r n a t i n g f l u x density given by 5 . 5 then the R . M . 3 . braking torque i s c r A — M l cp/a ne cms . We are now i n a p o s i t i o n to Inter p r e t the equation of motion of the d i s c , as expressed by 2 6 J7 E)rakir->cj torque due to H~>e jbermanenf nnaanck )Q hgrq + Orakinq torque cfuc to hh e n.C. maqnchs. So l e t us turn to some of the engineering aspects of the problem. Jjs T£i?ms of the Developed Model Ed of E fear romqqneh'c force, uncler 15 •disc PRACTICAL CONSIDERATIONS In t h i s s e c t i o n we s h a l l consider, i n terms of the developed model, some of the important points i n connection with the design of the disc-type apparatus. With regard to the ohmic l o s s i n the d i s c , we see that i t i s given by -an i n t e g r a l which may be expressed as the i n t e g r a l being taken over the volume of the d i s c . Since / • dJ expresses a p o t e n t i a l d i f f e r e n c e be-tween ft and & and since J Jjj'^ d a expresses a current through an area expresses .what might be c a l l e d "curgent Linkages ) c l e a r l y these Current- Il/nkojes are a fu n c t i o n of• the geometrical configur-a t i o n of the problem. Where we have a magnetic c i r c u i t i n the form of round bundles of magnetic f l u x l i n k i n g an i n f i n i t e conducting p l a t e of u n i t thickness ana p e r m i t t i v i t y cr Sn fin i /e jo lale the Cu/eeertT L e r e a d i l y evaluated to t>e = L t r -4 » 4 + per u n i t of E-.M.F. threading the c i r c u i t . Or, per u n i t of magnetic f l u x i n the c i r c u i t L <TuJ 4 n 4 C OS 7) L stands f o r the amount of e l e c t r i c current i n an i n f i n i t e p l a t e of u n i t tnickness passing tnrough the magnetic c i r c u i t . Where we have two such c i r c u i t s l i n k i n g the p l a t e , define the quantity frt as the amount of current l i n k i n g one c i r c u i t aue to u n i t magnetic f l u x i n the other. Por two c i r c u i t s j^n f i n l l c /slate f?adius of m n e h c conductor = A-3 2 tvl' i s found to be CJjxJ 4n 4 -in 1+ ^ Cos Clnd hence 4 ? 4 In 1 i + d1' cos r ' uri'i I of E.*4,F In p r a c t i c a l cases C o f I t w i l l be omitted. Gonsiaer now a common t r i p o l a r magnet Poles of radius /c O - "a  1 — f — L - f - — -Circuit Circuit 5 Circuit C We have seen that the i n t e g r a l d • cr may oe expressed i n terms of current l i n k a g e , which i n turn may be expressed i n terms of the constants L and M . I f the p l a t e were i n f i n i t e and were l i n k e d by the magnetic c i r c u i fcs shown above <d v L„ <fin Em +(LB i - ) _ r i — j cr c H-s e i-a Ec - r M CK U->C J F -+ 2 A//A8 E» Et where E*,£_,ie are trie R.M.S, Thus, E.M.Fs threading the c i r c u i t . If) ( unit 5, k% • dJ n d 0 where 2 i s the e l e c t r i c f o r c e , dj t h e T e c t o r element of path, a n a ^ A t l i e vector element of the surface bounded by the path. Remembering that and that we w r i t e IVU& • / W e c j I „ -LB - Lc - L E« = ~ Ec , by v i r t u e of the f a c t that f \ *• -d v = 2 L - M L Ef cr f o r the unbounded p l a t e of u n i t thickness. Since we have r e s t r i c t e d cr fo 0 outside the di s c and the thickness to cr J 1 S u b s t i t u l i n g f o r L , M , E „ and ^ s , we have f o r the ohmic l o s s r / \ f i r loz 2 -f-2 ST 2-2</ L I/L \ d -h 4 rj + Js~r\ rld\ A. 4-4 With regard to the design of the e x c i t i n g windings, we reduce the problem to a transformer problem by f i n d i n g the e f f e c t i v e current flowing around the core under consideration. Taking the p o t e n t i a l magnet, we have /. - 4 L, the ~ appearing since cr - o outside the disc, This gives (TcO 2 rr 7J A I 4 r lO S i m i l a r l y , the current l i n k i n g the f l u x of the current c o i l of the watt meter i s T l o 5 <r uo J /=? + c 2J r Ivfow the ohmic l o s s due to tiie currents i n the disc generated by the f l u x of tiie p o t e n t i a l magnet i s cr go 7rs tt s . Hence, as f a r as the p o t e n t i a l magnet i s concerned, the effec-t i v e r e s i s t a n c e of the d i s c , which may be considered as a s h o r t - c i r c u i t e d , s i n g l e turn, secondary winding, i s i s Rb 50 Z (T I o A. Ohms- . + — 4 S i m i l a r l y , the e f f e c t i v e r e s i s t a n c e of the d i s c , considered as the secondary of the current c o i l transformer, i s if i o I OO ^ 1 d A* i I + — 4 J I f the current winding i s s p l i t into two p a r t s , then, considering the two cores separately,- we have f o r the r e s i s t -ance i n the secondary of each £?f* - P c TT / O -7 5o S*cr \ 2 n 2d 2L J I -+ d +• — 4 By means of the above expressions, the e l e c t r i c a l design may be c a r r i e d out by standard methods. 37 3 8 . 4o. Pales found I " diome l-cr Con s is/ of if Or fa r lened hard i v o o d. " O t (3 auqe Id L Orn m ohc d I" S<juare CZoppcr F'od • ri ce d/cs TorS(C<~, SprmCj 3 dyne Cm? J per a ecj 1 ra ss 3 O c /-i-> -r diam. 0.O4-Q. cms. hhick Specific res, s fence f • (V • f 7J 1 0 nhrnr / 3 / Cm, 1' IV) casu red F lc.es ar,d rheir phases cr. ob *d fronn mca^renn^r of isc/rajes (and hheir yecrar sum and d;ffcrenCeJ across 4o Jurn search coils EXPERIMENTAL WORE As lias already been stated, the experimental work i s not the..essence of t h i s t h e s i s . In f a c t , I t i s a very minor part , required only so as to give some confirmation of the correctness of the theory developed. The apparatus used i s amply described by the sketches and the photographs. I t may be severely c r i t i c i z e d : the very f a c t that the leakage f l u x i s twice the.core f l u x i s reason enough to suggest that the apparatus be discarded altogether. On the other hand, that f a c t has been made use of to i l l u s t r a t e q u a l i t a t i v e l y the e f f e c t of leakage f l u x . I t may be w e l l to state that the expressions obtained to account f o r that e f f e c t -are.quite u n s a t i s f a c t o r y . They are based oh l o g i c a l reasoning of a type, and oh "physical i n t u i t i o n , " but they make abso-l u t e l y no attempt at r i g o u r . The p r i n c i p l e s on the strength of which the e f f e c t of leakage^.!lux v.ou!d be i n v e s t i g a t e d t h e o r e t i c a l l y , have already actual been l a i d down; but the . a r r i v a l work i s extremely laborious and involves a knowledge of the d i s t r i b u t i o n of the f l u x . Furthermore, there i s no advantage i n g e n e r a l i z i n g the work, since the d i s t r i b u t i o n depends on the p h y s i c a l , e l e c t r i c a l , and magnetic dimensions of the magnets. So the "guess method" has been used. A short explanation w i l l nov» be given of how the ex-pressions were a r r i v e d a t . We s n a i l consider the e f f e c t of the leakage f l u x of magnet 6 on the currents produced by tne f l u x of core PI . the leakage f l u x i s zero, because tne torques produced i n the two s e m i c i r c l e s are equal and opposite. At o ^ " f 1 ^ " the torque w i l l be p r o p o r t i o n a l to the d i f f e r e n c e between tne f l u x e s i n tiie two s e m i c i r c l e s , and a f u n c t i o n of tne p o s i t i o n of the point ..here that d i f f e r e n c e between the f l u x e s may be considered to act. That point w i l l be i n the v i c i n i t y of the one at which the f l u x i n the more densely f i l l e d h a l f a c t s . Again, the torque w i l l l a r g e l y depend on the leakage f l u x density due to magnet S> i n the v i c i n i t y of the pole of magnet R , since the current density is. highest i n that region. L a s t l y , i f we place tne poles i n the region where the rate of change of torque with ramus i s zero, estimating f o r that p o s i t i o n the point of a c t i o n of the e f x e c t i v e leakage f l u x , and express the e f f e c t s i n otner p o s i t i o n s i n terms of the assumed e f i e c t i v e point of a c t i o n , tnen we s n a i l obtain the c o r r e c t order of magnitude f o r the torque. When we estimate the point of a c t i o n , we should keep i n mind the f a c t that that point w i l l t r y to approach the one at which trie s c a l a r moment about the a i s c a ^ i s of the gradient of the current density i s a maximum; una at the same time, i t w i l l t r y to remain i n the region of large leakage i l u x density. With the above i n mind, the e f i e c t i v e radius of a c t i o n was assumed to be constant at 10 c < n » . } (region of maximum torque f o r constant separation) the angular parameter 9 re-l a t i n g the point of a c t i o n to trie core & was taken to be e •4/ 3 The d i f f e r e n c e i n f l u x between the two halves i s ob-tained, assuming a d i s t r i b u t i o n shown below the d i f i e r e n c e i n . f l u x i s r e a d i l y found -to be 2 i ecOS f /an "jj - 2 1 k I R cos \j) e -Fcos <j> 21 i f ? ~j) J fan ^ dy Cos •+ I V sin 3 + c o s I 5 n S ^ ] where &j i s the leakage f l u x density at J , the inner edge of the d i s c , as shown. I t i s c a l c u l a t e d from the approximate € x jj re ss i o n H = * Ri*r where J i s the current (B.M.U^)in the equivalent (^radius i?,J f s i n g l e turn representation of the e x c i t i n g c o i l , and a i s the distance of -J from the centre of the c o i l . Since f o r constant separation, the leakage f l u x density m the v i c i n i t y of pole P i s approximately constant, we should have i n the region of the v a r i a t i o n of torque due to the v a r i a t i o n of 8 j equal to zero. Hence, f o r displacements i n ' t h a t region, we should m u l t i -p l y the new Bj 'by a f a c t o r f which makes (p'j)(f) = ej ; that i s , we make — = O . Hence, we s h a l l w r i t e f o r the torque I S ; due to the leakage f l u x of core 4- 1 s > n s yL /5 10 r, sir IP P * r, Sirt & loo -f- r 2 O COJ & P + / oo r 2o I? r cos 9 v/here the two poles are s i t u a t e d at the radius , and where £>,0 i s the value of 5 j when r = to . (Region of l^"" =, o ,) To obtain the t o t a l leakage torque, we f i n d also the torque due to the 'leakage f l u x of the magnet R and the currents due to core B . The method i s h i g h l y u n s a t i s f a c t o r y ana may have "been replaced by the f a i r l y accurate expression f o r the torque due to the leakage f l u x of magnet 6 , which i s mentioned below. P T W fl P, 1 T COS + r>. - 2 r, r^ cios r P r r_ r / o v P* + r, V - 2 R \ r^ cos -V r dr^ d -v The i n t e g r a t i o n of the above may be done i n a ser i e s of se r i e s In the statement of experimental r e s u l t s , diagram-matic explanation i s given wherever considered necessary. In the c a l c u l a t i o n of E> • , f?, was estimated as follows Since the expression H a - F, / +- '• + 1 0", i s , f o r large values of a , p r o p o r t i o n a l to P'i \ the mean P,1 was estimated to be To allow f o r the f a c t that there were less turns i n the outer h a l f of the c o i l than i n the inner h a l f , R, was taken to oe 2. L cn^s t The choice of 2 .6 c m s was governed by the reason that the mean r a a i u s , considering the v a r i a t i o n i n the number of turns i n the geometrical halves of the c o i l , was about 2. "2 instead of 2. & 3 c m s t N a t u r a l l y one i s not ready to give much weight to the above method of c a l c u l a t i o n . For that reason, torque due to the leakage f i e l d of one magnet and the currents i n the disc due to trie f l u x of the other magnet was d i r e c t l y measured, as shown below. In a l l the work the core f l u x e s were Kept constant, the currents being approximately constant at 0. 37 amf* i/icl 52. A P P E N D I X Here we s h a l l he c o n c e r n e d o n l y w i t h the e v a l u a t i o n o f the c o n s t a n t s , L and /W , f r o m the g e o m e t r i c a l c o n f i g u r -a t i o n o f the c i r c u i t s , Ci'rouir 2 2d - 9 , Circuit I 2c a s e c t i o n , p e r p e n d i c u l a r to the a x i a l l e n g t h o f w h i c h , i s shown above . The c o n d u c t i o n s a r e l o n g , p a r a l l e l , t h i n r i b b o n s of w i d t h « . The work w i l l be c a r r i e d out i n terms of the l o g a r i t h -m i c d i s t a n c e between the s e c t i o n s of the c o n d u c t o r s c a r r y i n g the c u r r e n t . By the l o g a r i t h m i c d i s t a n c e , between two s e c t i o n s /3 *9 ' F we- mean d a * dnL1 d\, da - dx & r cl\i dm' d The logarithmic distance of a s e c t i o n from i t s e l f i s then In terms of the logarithmic distances, the self-inductance per u n i t a x i a l length of p a r a l l e l , c y l i n d r i c a l conductors having the sections shown i s I f & and £> are i d e n t i c a l I f we have two c i r c u i t s then the mutual inductance between them 5 4 . I f tiie c i r c u i t s are i d e n t i c a l and symmetrically placed, M = 4 We s h a l l now derive a general expression f o r the logarithmic distance between the sections of two long p a r a l l e l riboons s i t u a t e d as shown i n the accompanying diagram. We have ^ _____ a d fdf" Talcing the 2d ft 8 f i r s t , we rewrite the expression d * a 2n a d and changing the v a r i a b l e •d, fF as = — / <df x. d-a d J • d d f I In ( ^ + S ) d - * lad d f 4n -h 2 b /an ' 2L b -I F S u b s t i t u t i n g the l i m i t s i n terms of the o r i g i n a l v a r i a b l e s , .we get:-we obtain 57, ( lad i 1 4. -f 2 c ) 4 b l \ ( d . 2 c ) 1 - ^ 2 L 2 c y -+ k ' d + a -t °2 c -t- 4 b c /_?o ' / 2c V b J - 2 (a + 2c. b - 2{ d + i.c) h / _ 3 a d r Q 4 - 2 c" d + 2 c From tlae above we obtain the desired logarithmic dis-tances, making the s u b s t i t u t i o n s shown below. -i i d-_____ J d A d = a b = o -f 2c - 5" Q * c/ Q + 2c -- S 2 c = - c? a = d 2c -. -a a *• 2 c - S - Q The logarithmic distance of a l i n e from i t s e l f i s To f a c i l i t a t e computation, Rogowski (Elektrotechnik und Maschinenbau, 1 9 1 2 ) expands the integrand functions into uniformly coriTerging s e r i e s and then in t e g r a t e s . The values of L and M have been computed and are given i n the next few pages. 51-Values of L a e L 0 • / /£ • 2 / • 2 / 2 .4 2 • 5 /0 .7 2 • 4- o • 3 2 .as • f, 7. <?/3 • 7 7.244 • 8 •<7 fo . 0 i 7 ^ /• o 5 . 5 4 5 V a lues of iVl . J2 -5 . 0 a _ 0 . 5 5 = / b & 0 0 O O O • ( . 0 1 cjq 0 .0 1 Cjcj V . 0 / 7 £ . 6 . O / 9 S 2 • 2 .07Z4- .07% 4-1 , Oil £ I . 0 7 ^ 0 •3 . 1 7 2 A • 1 7 2 2 • 1 <o <=l 1 . / 6 5 £ •d • 27 U • • 2%-SO • 27?/ • 5 • 4-4-43 • 4 1 <={i • 4 O 73 • £ . • C 1 5 O . 5<f 7 • 57 2 2 • 3 4 S £ •7 • 77 3 • 7 2 5 2 • £</44 • ? • ct%°i4 • 7 5 / 5 * • ' <? I. ill /, / 3 5 /. <0 5 0 • 9 ^ 6 /. 0 I. 3 S 4 /•SO? / . 2 0^ I - 1 lo Values 0 f M / r o m RoaovMski if. «. M . / <? / 2 I s r Circ-uif b . '2nd Circot'ir a 2c a Values of L k a L 2, b L o 0 0 <=-<o •I /. / 04 •1 / 5\ 2/ • ? Z. ov 5" • 2 12. 4^ •3 2- I°l4 • 3 l(b. 84 • 4 5.4 72 • 4 9. 7/_ •i> 4,o6 8 • 5 •6 4 . 6 o4- • 4 <?, /S6 • 7 S < o 84 • 7 7, •& s. si 2 • 2 7, oE4 • 7 5 . <j 2 O • 9 6. 656 ho _, 2 6 5 ho 6, 285 _ = -£T 2 - <? _ - / € £. • _ e 0 o _> O , 0 • 1 . 0 / 7 7 0 , 0 2 31 .03 134 .0 6 2. 2 Z • 2 ,olE 4-4 \ •//<=}£ . / 9 37 •3 o, n 7.4 • 2 3-33 -3£ 43 • 4 .55 28 . 4 / 6 3 - 5 5 S 7 •3 •44&3 .4°l4Z • b a 2 2 • 76 70 . Q . 6/SO • ^ 7 3 7 • Bo 2 2. • 9 <? 3 5 •7 . 7 7 7 - - 8L(>7 1. Oo 3 1. 2 0 3 . -g 1.0 6 7 /. 206 /. 4 23 /. J?7 /, 2-*7 I . 4- 3 3 h . 2 3 /, 0 /. 3 3 6 / . 4 7- I. &4- 2 /._?57 6 / ACKNOWLEDGEMENT The experimental work was done i n the l a b o r a t o r i e s of the U n i v e r s i t y of B r i t i s h Columbia. The subject was suggested by Dr. V i c k e r s , head of the E l e c t r i c a l and Mechanical Engi-neering Departments. The l i t e r a t u r e which was a v a i l a b l e , was supplied by the Un i v e r s i t y L i b r a r y . Volumes of the ."Elektrotechnik und Maschinenbau" were obtained, through Miss Smith of the L i b r a r y s t a f f , from the U n i v e r s i t y of Washington. The work of W. Rogowski (Elektrotechnik und Maschin-enbau, 1911, 1912) i s worthy of a l l p r a i s e . One might, however, f i n d f a u l t w i t h the f a c t tnat he makes too d i r e c t a use of the analogy implied between /~) r~J O (3 disregarding the physics of the phenomena; and with the f a c t that he assumes the edge of the pole face to define a l i n e of current flow. He does no experimental work. 4z BIBLIOGRAPHY K. Bauer , A r c l i i v f u r E l e k t r o t e c h n i k , V o l . 16 pp . 58-72 Bruger , E . T . Z . 1895, p.677 S c h r o t t k e , E . T . Z . 1901 p „ 675 S t e r n , E . T . Z . 1902 p . 774 Dav id & Simons E . T . Z . 1907 p . 942 ¥ . R o g o w s k i , E l e k t r o t e c h n i k und Maschinenbau, 1912, p.205-232 W.Rogowski, E . T . Z . 1910 p . 1070 Drysda l e & J o l l e y , Measuring Instruments , V o l . 1 & 2. R.M.Mor ton , t h e s i s f o r M . A . S c . a t U . of B . C . 

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