An Analysis of Linear Induction Motors for Propulsion and Suspension of Magnetically Levitated Vehicles Citation Yang, Fang-Chou (1976) An Analysis of Linear Induction Motors for Propulsion and Suspension of Magnetically Levitated Vehicles. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/DM0P-XW77. https://resolver.caltech.edu/CaltechTHESIS:11092017-124408265 Abstract A four-layer single-sided LIM used for propulsion and suspension of magnetically levitated vehicles is studied. The track is assumed to be made of conductors with uniaxial μ and σ. A general analysis allows us to exclude unsuitable geometries. The machine performance is given for the promising geometries. A possible way of computing the effective μ and σ for a composite track is sketched. From the analysis of an extremely simplified geometry, the conditions for the validity of the effective μ and σ concept are given. Finally, a three-dimensional correction is introduced. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Electrical Engineering) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Electrical Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): Papas, Charles Herach Thesis Committee: Unknown, Unknown Defense Date: 1976 Record Number: CaltechTHESIS:11092017-124408265 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:11092017-124408265 DOI: 10.7907/DM0P-XW77 Related URLs: URL URL Type Description http://resolver.caltech.edu/CaltechAUTHORS:20171108-144658929 Related Item Caltech Antenna Laboratory Technical Report 77 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 10553 Collection: CaltechTHESIS Deposited By: Mel Ray Deposited On: 15 Nov 2017 18:05 Last Modified: 24 Aug 2024 00:22 Thesis Files Preview PDF - Final Version See Usage Policy. 28MB Repository Staff Only: item control page

AN ANALYSIS OF LINEAR INDUCTION MOTORS FOR PROPULSION AND SUSPENSION OF MAGNETICALLY LEVITATED VEHICLES T h e sis by Fang-Chou Yang In P a r t ia l F u lf illm e n t o f the Requirements Fo r the Degree o f D octor o f P h ilo so p h y C a lif o r n ia I n s t it u t e o f Technology Pasadena, C a lif o r n ia 1976 (Subm itted -i i ACKNOWLEDGMENTS I would l i k e to express my deep g r a titu d e to my a d v is o r , P ro fe s s o r C h a rle s H. Papas, f o r h is guidance and encouragement d u rin g the course o f t h is work and throughout my s tu d ie s a t the C a lif o r n ia I n s t it u t e o f Technology. I would a ls o l i k e to express my thanks to Dr. T s ih C. Wang f o r i n i t i a t i n g my in t e r e s t s in m agnetic le v it a t io n system s, to Mr. W a lter K. DeLogi and Mr. R ich a rd T. Woo f o r c o r r e c tin g my E n g lis h . S p e c ia l thanks are extended to M rs. Ruth S tr a tto n f o r ty p in g the m a n u scrip t. T h is work is d e d ica te d to my parents and my w ife . -i i i - ABSTRACT A f o u r - la y e r s in g le - s id e d LIM used f o r p ro p u ls io n and suspens io n o f m a g n e tic a lly le v it a t e d v e h ic le s i s s tu d ie d . assumed to be made o f conductors w ith u n ia x ia l µ ~ The tra c k is and σ. ~ A general a n a ly s is a llo w s us to exclu d e u n s u ita b le g e o m e trie s. The machine perform ance is g iven f o r the prom ising g e o m e trie s. A pos­ s ib le way o f computing the e f f e c t iv e tr a c k is sketched. µ ~ and σ ~ f o r a com posite From the a n a ly s is o f an e xtre m e ly s im p lif ie d geom etry, the c o n d itio n s f o r the v a l i d i t y o f the e f f e c t iv e ~ σ concept a re g iv e n . in tro d u c e d . µ ~ and F i n a l l y , a th re e -d im e n s io n a l c o r r e c tio n is - iv TABLE OF CONTENTS I. INTRODUCTION 1 II. GENERAL FORMULATION 7 III. SEVERAL EXAMPLES 18 IV. COMPOSITE TRACKS 90 V. THREE-DIMENSIONAL CORRECTIONS 110 V I. CONCLUSION 137 REFERENCES 138 APPENDIX A. The Governing Equations 140 APPENDIX B. C o n d itio n s f o r T hin and T h ic k R eaction R a ils 143 APPENDIX C. The E v a lu a tio n o f In te g ra l S1 (α , ky o , L) 146 APPENDIX D. The E v a lu a tio n o f In te g ra l S2 (α, kyo, L) 148 APPENDIX E. The E v a lu a tio n o f In te g ra l S3 (α, kyo, L) 150 APPENDIX F. R e s u lts f o r Geometry ( A - ii- 2 ) w ith 157 -1 CHAPTER I INTRODUCTION With the in c r e a s in g need in t r a f f i c o f people and com m odities, autom obiles have c re a te d s e rio u s p o llu t io n and n o is e problem s; the a i r l i n e s have a ls o reached a high degree o f s a tu r a tio n . Under the c i r ­ cum stances, i t seems th a t the development o f a ra p id mass tr a n s p o r ta ­ tio n system i s d e s ir a b le . U n fo rtu n a te ly , the co n v e n tio n a l w heel- supported ra ilw a y s fa c e some d i f f i c u l t i e s as the t r a v e llin g speed becomes la r g e . Not o n ly does the n o is e become in t o le r a b le , but the s s io n o f power between the r o t a t i ng-type motor and the w h ee ls. As a r e s u lt , such n o n -co n ta ct systems as m agnetic le v it a t e d (MAGLEV), tr a c k a i r cushion v e h ic le s (TACV) are suggested as s u b s t it u t io n s f o r fu tu r e tr a n s p o r ta ­ t io n . A lthough the TACV is s t i l l a c o m p e titiv e ca n d id a te f o r high speed ground tr a n s p o r ta tio n (HSGT), in the fo llo w in g we w i l l m a in ly c o n s id e r the MAGLEV system. There are two fundam ental schemes f o r m agnetic l e v it a t io n , nam ely, a t t r a c t iv e and r e p u ls iv e schemes. The a t t r a c t iv e system uses the a t t r a c t iv e fo rc e between a m agnetic f i e l d source and a ferrom ag­ n e t ic m a te r ia l. From Earnshaw 's theorem, t h is k in d o f c o n fig u r a tio n is b a s ic a lly u n s ta b le , i . e . , a s m a lle r c le a ra n c e w i l l in c re a s e the a t t r a c t iv e fo rc e and make the c le a ra n c e even s m a lle r . back c o n tro l system is n e ce ssa ry . Thus, a fe e d ­ For the r e p u ls iv e scheme, the fo rc e e x e rte d on a m agnetic f i e l d source moving o ver a conducto r by the -2 f i e l d o f the induced eddy c u rre n t w ith in the co nducto r is employed. Such a system is in h e r e n tly s t a b le . E x te n siv e s tu d ie s o f these suspension systems have been made(1,2). D e f in it e ly , the d e t a ils w i l l depend on source and tra c k geom etry. C e rta in general r e s u lt s w i l l be d e sc rib e d below. And a c ­ t u a l l y , up to to d a y, there is s t i l l no c le a r e viden ce as to which one is the b e t t e r one. Both are n o is e le s s , not lim it e d by speed, and can be o perated in vacuum. In r e a l i t y the suspension system is o n ly p a rt o f a whole HSGT v e h ic le . Thus, w hether a system i s o p tim ize d o r not can o n ly be judged by a n a ly z in g the whole v e h ic le from the tr a d e o f f among the perform ances, te c h n ic a l m e r its , economic c o n d itio n s , e tc . G e n e ra lly sp e a kin g , in o rd e r to have a s u f f i c i e n t l y s u p p o rtin g f o r c e , the a t t r a c t iv e scheme can o n ly a llo w a sm all c le a ra n c e o f about 1 o r 2 cm, but the r e p u ls iv e scheme can have a c le a ra n c e o f 20 cm. T h is w i l l a ls o p la y an im p o rta n t r o le in d e c id in g what kin d s o f p ro p u ls io n systems to use. g e n e r a lly co n d u cto rs. o ver i t , A ls o , as we know, fe rro m a g n e tic m a te r ia ls are Thus, when the m agnetic f i e l d source is moving th e re alw ays are eddy c u rre n ts induced. u n d e sire d r e p u ls iv e fo rc e component. enough, sometimes t h is w i l l T h is w i l l g iv e an When the v e lo c it y becomes la rg e r e s u lt in a n e t r e p u ls iv e fo r c e . co u rse , we cannot a llo w t h is to happen. Of Thus, tra c k s and sources must be s u it a b ly designed to preven t t h is phenomenon from happening w ith in the o p e ra tin g speed range. For the r e p u ls iv e scheme, s in c e i t m a in ly depends on the eddy c u rre n ts induced, eddy c u rre n t lo s s e s are in e v i t a b le . These w i l l g iv e an a d d it io n a l drag fo r c e . An in cre a se d -3 p ro p u ls iv e fo r c e i s n e cessary to overcome i t . A cco rd in g to form er a n a ly s e s , i t is found th a t a t high speed t h is drag fo r c e w i l l decay to a very sm all v a lu e . However, a very la rg e drag fo rc e w i l l e x is t a t a very sm all speed. C e rta in methods(3) have been suggested to bypass t h is low speed drag. S in ce the proposed suspension system is a n o n -co n ta c t one, the c o n v e n tio n a l method o f using the d ir e c t r e a c tio n w ith the ground is not a v a ila b le . Thus, a new p ro p u ls io n system is needed. In the e a r ly p e rio d , j e t s , p r o p e lle r s and ro c k e ts have been suggested. However, p o l- lu t io n and n o is e problems e v e n tu a lly le d to the use o f lin e a r machines, The idea o f lin e a r machines is not new. Back in the 19th cen­ tu r y and the e a r ly 20th c e n tu ry , people have t r ie d to a p p ly i t in t r a in tr a n s p o r ta tio n , luggage h a n d lin g , e t c . ( 4 , 5 ) . However, due to economic and p r a c t ic a l re a so n s, the in t e r e s t in lin e a r machines de­ c lin e d . R e c e n tly as L a ith w a ite ( 6 , 7, 8) and P o lo u ja d o ff( 4 , 5) put i t , "e n g in e e rin g fa s h io n s " have been changing. Due to L a it h w a it e 's con­ tin u o u s e f f o r t s in prom oting them, lin e a r machines are being re e v a lu ­ ated and they begin to be e x te n s iv e ly used in HSGT, im pact e x tr u s io n , E.M. pumps, a c tu a to r s , e tc . Now, i f the r e p u ls iv e suspension scheme is used in HSGT, because o f i t s high c le a ra n c e perhaps the lin e a r synchronous(9) motor is p r e fe ra b le . A ls o , pow er-pickup and w eigh t problems are more e a s ily handled by the lin e a r synchronous m otor. However, in the f o llo w in g d is c u s s io n s we w il l m a in ly c o n s id e r lin e a r in d u c tio n m otors. synchronous motors w i l l no t be in c lu d e d . L in e a r -4 The p r in c ip le o f the lin e a r in d u c tio n motor can be e a s ily under­ stood from the c o n v e n tio n a l r o t a t in g in d u c tio n m otor. A c t u a lly , the s im p le s t form o f LIM is j u s t an u n r o lle d r o t a t in g in d u c tio n m otor. A a lt e r n a t iv e forms o f LIMs s e rv in g d if f e r e n t purposes can be c o n s tru c te d by j u s t adding o r changing the prim ary o r the secondary s tru c tu re s . L a ith w a ite and N a sa r(6) have g iven a com plete c l a s s i f i c a ­ t io n in terms o f the d if f e r e n t p o s s ib le to p o lo g ic a l c o n fig u r a tio n s . A ls o , a "m achine's good f a c t o r " (10) and a c o n s id e ra tio n o f the econ­ omic problems were in tro d u c e d to decide which kin d o f LIM i s p r e f e r ­ a b le . Among them, the "d o u b le -s id e d s h o rt p rim ary sheet r o t o r motor" (DSLIM)( 8 , 11, 1 5 , 16) seems to be the b e st ca n d id a te f o r HSGT. A c t u a lly , the DSLIM had a lre a d y been used in p ro to ty p e HSGT v e h ic le s b u i l t in Germany, the U .S ., e tc . But those v e r t i c a l l y b u i l t DSLIMs s u ffe re d from s e v e ra l severe problem s, such as la t e r a l i n s t a b i l i t y in c u rv e s , high c ro ss s e c tio n s o f the v e h ic le s , e tc . Hence, an a lt e r n a t iv e , the s in g le - s id e d LIM (S LIM )( 1 2 , 1 3 , 14) was in tro d u c e d . A lthough the a n a ly ­ ses o f these co n v e n tio n a l r o t a t in g motors are w e ll known and are not too d i f f i c u l t , the a n a ly s is o f L IMs i s much more c o m p lic a te d , due to the in e v it a b le end e f f e c t and the dissym m etry o f prim ary c u r re n ts . Those e x tra e f f e c t s w i l l in tro d u c e u n d e s ira b le phenomena. Today, r e ­ search in L IMs c o n c e n tra te s on e x p la in in g these e f f e c t s and lo o k in g f o r methods to compensate them. Yamamura and I t o (1 6 ), Wang(1 1 ) , and Dukow icz(15) co n sid e re d DSLIM w ith n o n -fe rro m a g n e tic sheet r o t o r s . N asar and Del C id , J r . (14) and Wang(13) d is c u s se d SLIMs w ith nonfe rro m a g n e tic sh ee t ro to rs backed by s t e e l. -5 - However, no m a tte r w hich LIM was c o n s id e re d , up to now a l l o f the a n a ly s e s were made under the assum ption o f sep a rated suspension and p ro p u ls io n system s. A lthough an e xp erim e n tal v e h ic le has a lre a d y been c o n s tru c te d by Rohr I n c . (17) , no s e rio u s a n a ly s is has been made to c o n s id e r a v e h ic le w ith a LIM used f o r both suspension and propuls io n purposes. Of c o u rse , t h is would in tro d u c e e x tra c o n tro l problem s, b u t i t a ls o o f f e r s advantages such as le s s w e ig h t. The SLIM a n a lyze d by N asar and Del C id , J r . ( 1 4 ) , Wang( 1 3 ) , e tc . w i l l g iv e e it h e r an a t t r a c t iv e o r a r e p u ls iv e f o r c e , depending on the v e lo c it y o f the v e h ic le . However, i f we want the LIM to be used f o r both le v it a t io n and p r o p u ls io n , a deeper a n a ly s is must be made as to w hich geometry and which m a te ria l to s e le c t so th a t o n ly e it h e r an a t t r a c t iv e o r a r e p u ls iv e fo rc e i s observed w ith in the o p e ra tin g speed range. The fe rro m a g n e tic tra c k was ru le d o u t as u n s u ita b le f o r DSLIM by L a ith w a ite and B a r w e ll(8) because o f s k in and n o n lin e a r e f f e c t s . These a u th o rs , however, were o n ly concerned about the th r u s t fo r c e . As we want to in c lu d e a l e v it a t in g f o r c e , the in h e re n t fo r c e between the m agnetic f i e l d source and fe rro m a g n e tic m a te ria l has to be re co n sid e re d s e r io u s ly . Furtherm ore, the u n d e sire d s k in e f f e c t w hich g iv e s r is e to n o n lin e a r e f f e c t s can a ls o be p a r t ly taken care o f by s u it a b ly la m in a tin g the fe rro m a g n e tic m a te ria l o r composing the tra c k both from f e r r o ­ m agnetic and n o n -fe rro m a g n e tic m a t e r ia l. The fo llo w in g d is c u s s io n w i l l begin w ith an in v e s t ig a t io n o f the p o s s i b i l i t y o f u sin g a lin e a r in d u c tio n motor as a combined p ro p u ls io n -6 and s u p p o rtin g system in a HSGT v e h ic le . L in e a r machines o f fo u r la y e r s w i l l be used as the s t a r t in g p o in t o f the a n a ly s is . Most o f the proposed c o n fig u r a tio n s can be d e sc rib e d by s u it a b ly s p e c if y in g those param eters as c o n d u c t iv it y , p e r m e a b ilit y , la y e r th ic k n e s s , e tc . f o r d if f e r e n t re g io n s . The a n a ly s is o f machine perform ance w i l l e x ­ clu d e u n s u ita b le geom etries and g iv e p o s s ib le methods to reduce the u n d e sire d end e f f e c t s . E x p re ssio n s f o r machine perform ances w i l l then be e v a lu a te d f o r those p ro m isin g geom etries. A p o s s ib le way o f computing the e f f e c t iv e c o n d u c t iv it y and p e rm e a b ilit y f o r a com posite tr a c k w i l l be sketch ed. Up to now these param eters were determ ined e x p e r im e n ta lly , and no t h e o r e t ic a l a n a ly s is has e v e r been attem pted. D e f in it e ly , i t i s d i f f i c u l t . An e x tre m e ly s im p lif ie d geometry w i l l be co n sid e re d to check the v a l i d i t y o f t h is e f f e c t iv e c o n d u c t iv it y and p e rm e a b ility concep t. P o s s ib le n e ce ssa ry c o n d itio n s f o r the v a l i d i t y w i l l then be g iv e n . S t a r t in g from t h is p o in t , an oth er p o s s ib le c o n fig u r a tio n w i l l be suggested and a n a lyze d . The t r a n s f e r m a trix method i s employed to s o lv e th e 2 -d im e n sio n al problem . A t the end o f t h is d is c u s s io n , a 3 -d im e n sio n al c o r r e c tio n w i l l be in tro d u c e d . Of c o u rse , the p r e v io u s ly n o n e x is te n t la t e r a l fo rc e w i l l now appear. w i l l then be g iv e n . A co rre sp o n d in g q u a n t it a t iv e appro xim atio n -7 CHAPTER II GENERAL FORMULATION The lin e a r in d u c tio n motors w hich w i l l be co n sid e re d c o n s is t o f fo u r la y e r s as shown in F ig . 2 .1 . The source w ith a given c u rre n t o r B - f ie ld d is t r ib u t io n (depending on how the source i s connected) i s l o ­ cated a t the in t e r f a c e o f re g io n s 3 and 4. Region 3 i s th e fr e e space. Regions 1 and 4 are any zero c o n d u c t iv it y m a t e r ia l, such as the i n ­ f i n i t e l y la m in a te d ir o n o r fr e e space. Regions 1 and 2 which make up th e tr a c k are moving a t a c o n s ta n t v e lo c it y v r e la t iv e to the so u rce . The m a te ria l p r o p e r tie s o f re g io n 2 in i t s r e s t frame are a r b i t r a r i l y s p e c i­ f ie d by c o n sta n t c o n d u c t iv it y and co n sta n t p e rm e a b ility u n ia x ia l r e s p e c t iv e ly . These te n so rs co n sta n ts can g e n e r a lly d e s c rib e the m a te ria l used in LIMs s a t is f a c ­ t o r ily . M a x w e ll's e q u a tio n s in v o lv in g moving m agnetic m a te ria l are com­ p lic a t e d . However, in t h is ca se , n e g le c tin g the d isp lace m en t c u rre n t and r e l a t i v i s t i c e f f e c t s is alw ays a good a p p ro x im a tio n . In the r e s t frame o f the so u rce , M a x w e ll's e q u atio n s can then be s im p lif ie d to g iv e us (see Appendix A ): (2 .1a) (2 .1b) (2 .1 c ) ( 2 . 1d) o r , f o r a n o n s in g u la r σ ~ ( i . e . , re g io n 2) -8 - F ig . 2.1 F ig . 2.2 -9 (2 .2a) (2.2b) (2 .2 c ) (2 .2d) and, f o r σ = 0 ( i . e . , re g io n s 1 ,3 ,4 ) ~ (2 .3 a) (2 .3b) (2 .3 c ) (2 .3d) W ith a e- i ωt tim e dependence, a F o u r ie r tra n sfo rm p a ir , g iven in (2 .4 a ) and (2 .4 b ) , can be in tro d u c e d to s o lv e the d i f f e r e n t i a l equa­ t io n s : (2 .4 a) (2 .4b) Then, by a p p ly in g s u it a b le boundary c o n d itio n s , in p r in c ip le , the problem has a unique s o lu t io n . However, g e n e r a lly t h is is te d io u s . To make the problem e a s ie r , we assume a t f i r s t th a t the source i s i n f i n i t e l y extended along the x - a x is . Thus, x-dependence can be suppressed. The 3-d im e n sio n al c o r ­ r e c t io n f o r some sim p le geom etries w i l l be made l a t e r in Chapter V. -1 0 For t h is 2 -d im e n sio n a l problem w ith the c o o rd in a te system as shown, the o n ly non-zero components o f As suggested by Freeman(1 8 ), C u lle n and B a rto n (19 ) , e t c . , the t r a n s f e r m a trix method can be used to f in d the f i e l d . A c t u a lly we have: Region 1: (2 .5 ) w ith (2 .6 ) Regions 2 ,3 : (2 .7 ) w ith (2 .8 ) Region 4: (2 .9 ) w ith A ls o , in the above (2 .1 0 ) -1 1 (2 .11 ) (2 .12 ) γk are s u it a b ly chosen such th a t they c o n ta in o n ly p o s it iv e re a l p a r ts . In these form ulas the boundary c o n d itio n s can then be in tro d u c e d . The f i e l d d is t r ib u t io n in the whole space can be o b ta in e d . Now the case w ith a s p e c if ie d c u r re n t source w i l l be co n sid e re d ( i . e . , s e r ie s connected s o u rce ): The c o n t in u it y c o n d itio n s o f g iv e (2 .13 a) (2.13b) Here ε ments o f is p o s it iv e but s m a ll, t ij = (T3( z 3 ) Τ2 ( z 2 ) ] i j z3 g iv e are the e le ­ T3 ( z 3 ) T2 ( z 2 ). The boundary c o n d itio n s a t (2.14a) (2.14b) (2 .14 c) i s the F o u r ie r component o f the s p e c if ie d c u rre n t d is t r ib u t io n . Combine (2 .1 3 ), (2 .14 ) to get: -1 2 (2 .15 a) (2.15b) The F o u r ie r components o f the f i e l d d is t r ib u t io n can be o b ta in e d from e q u a tio n s (2 .5 ) to (2 .1 2 ). The f i e l d d is t r ib u t io n in re a l space can then be o b ta in e d by u sin g the F o u rie r in v e rs e tra n sfo rm (2 .4 b ). From the above d e r iv a t io n , i t i s obvious th a t the problem can a ls o be s o lv e d by a p p ly in g tra n s m is s io n lin e th e o ry . components W ith the f i e l d s e rv in g as the v o lta g e and the c u rre n t r e s p e c t iv e ly , the c h a r a c t e r is t ic impedance can e a s ily be d e fin e d . A fte rw a rd the f i e l d d is t r ib u t io n can be o b ta in e d by s u it a b ly m atching the in p u t impedance o r by using the Sm ith c h a rt. Once we know the f i e l d d is t r ib u t io n , we can e v a lu a te the machine perform ances. Among them, the fo r c e s , e f f ic ie n c y , power in p u t, power f a c t o r a re the most im p o rta n t. As d e sc rib e d by Fano, Chu, A d le r (20) in c a lc u la t in g fo rc e s i n ­ v o lv in g m agnetic m a t e r ia l, the Minkowski fo rm u la tio n w hich we have used so f a r cannot g iv e s a t is f a c t o r y r e s u lt s . C hu's fo rm u la tio n is in tro d u c e d . Thus, an a lt e r n a t iv e c a lle d By u sin g t h is new fo r m u la tio n , the fo rc e s w it h in the m agnetic m a te ria l can be c le a r ly e x p la in e d . D e ta ils concerning the fo rc e s e x is t in g in the given LIM systems are in c lu d e d in Appendix A. In our problem , the fo rc e which has to be c a lc u la t e d is j u s t the t o t a l fo rc e e x e rte d on the s o u rce , w hich by the p r in c ip le o f r e a c t io n , is ju s t the t o t a l fo rc e a c tin g on the combined re g io n s 1 and 2. A cco rd in g to Chu's fo r m u la tio n , t h is fo rc e can be o b ta in e d sim p ly -1 3 by in t e g r a t in g the M a x w e ll's s t r e s s energy te n s o r o v e r the s u rfa c e in the fr e e space j u s t o u ts id e o f re g io n 2 , i . e . , (2.16a) w ith (2.16b) F o rtu n a te ly , in the fre e space, the M a x w e ll's s t r e s s energy te n s o rs f o r C hu's and M in k o w sk i's fo rm u la tio n s happen to be the same. Furtherm ore, in the fre e sp ace, the e l e c t r i c p a r t o f the s t r e s s energy te n s o r is always sm all compared to the m agnetic p a r t. D e f in it e ly , i t can be n e g le c te d , and a c t u a lly t h is is e q u iv a le n t to n e g le c tin g the d i s ­ placem ent c u rre n t w hich was done a t the v e ry b e g in n in g . Thus, the c a lc u ­ la t io n o f fo rc e s can be s im p lif ie d . F i n a l l y , the tim e averaged t o t a l fo rc e s are given by: (2.17) w ith (2 .18 ) Thu s, f o r the 2 -d im e n sio n al case: (2 .19 ) (2.20a) (2.20b) -14- (2.21a) (2.21b) where the F̄ i re p re s e n t the fo rc e s a c tin g on a u n it le n g th o f source in the x d ir e c t io n . Combine (2 .2 0 ), (2 .2 1 ) , (2 .1 5 ) and (2 .7 ) to get the fo rc e s f o r the case o f a s p e c if ie d c u r re n t so urce: (2 .2 2 ) (2 .2 3 ) where are the elem ents o f m a trix T2 ( z 2 ). Sometimes i t is e a s ie r to e v a lu a te the s t r e s s energy te n s o r in t e g r a l a t the s u rfa c e z 3- ε . Then we get: (2 .24 ) (2 .2 5 ) The average m echanical power in p u t is : -1 5 - (2.26) The average power in p u t is : (2.27a) (2.27b) (2 .27c) (2.27d) Power f a c t o r is given by: (2.28) E f f ic ie n c y i s d e fin e d as: (2 .29 ) F in a lly , the induced eddy c u rre n t lo s s is found from: (2.30a) or (2.30b) By in s p e c tio n o f the above fo rm u la s, an e q u iv a le n t c i r c u i t can be c o n s tru c te d as shown in F ig . 2 .2 , w ith -16- and thus P̄ can be r e w r itte n as: Now Lcan be decomposed in t o th re e p a rts : Z ~ The term c o n tr ib u te s to the m echanical power, g iv e s the induced eddy c u rre n t lo s s . w h ile o rd e r to have a high e f f ic ie n c y o r a low energy lo s s , to be co n ce n tra te d around im a gin ary p a r t Im~ L Z kyo such th a t is re q u ire d i s s m a ll. The re p re se n ts the d iffe r e n c e between the average sto re d e l e c t r i c energy and m agnetic energy. it , Thus, in However, as we understand in t h is case the s to re d e l e c t r i c energy i s r e l a t i v e l y sm all com­ pared to the sto re d m agnetic energy. So the term Im ~ZL w i l l m a in ly take care o f the average sto re d m agnetic energy o f the system . In the above, the problem o f a source w ith s p e c if ic c u rre n t d i s ­ t r ib u t io n a t z3 is an a lyze d . B a s ic a lly , t h is is the case o f the so - c a lle d s e rie s -c o n n e c te d so u rce . w ith a s p e c if ie d The case o f a paral l e l -connected source B f i e l d d is t r ib u t io n a t z 3 can be analyzed in the same way. D e f in it e ly , i f the source is i n f i n i t e l y extended such th a t i t can be d e sc rib e d as a t r a v e llin g wave, i t does not make any d iffe r e n c e w hether the source is s e r ie s o r p a r a lle l connected. However, t h e ir b e h a v io r is q u ite d if f e r e n t f o r a more r e a l i s t i c so u rce . We do not in te n d to -1 7 c o n s id e r the p a r a lle l-c o n n e c te d case here. T h is com pletes the general a n a ly s is . In the next c ha p te r we s ha l l s t a r t from t h is general fo rm u la tio n and c o n s id e r s e v e ra l exam ples. -1 8 CHAPTER I I I SEVERAL EXAMPLES Up to now, the source c i r c u i t has been n e g le cte d in a l l o f the a n a ly s e s . B u t, in the machine d e s ig n , th e re i s alw ays a source w in d in g r e s is ta n c e Zs. ~ And no m a tte r what k in d o f source co n n e ctio n i s used, i t i s q u ite c le a r th a t |~ZL| >> ~ Zs the source w inding lo s s s m a ll. the v a lu e o f β4 ≅ 0 around kyo i s n e ce ssa ry to make S u r e ly , under c e r t a in c irc u m s ta n c e s, ( i . e . , u sin g i n f i n i t e l y lam in ate d ir o n in re g io n 4 as assumed in n e a r ly a l l o f the LIMs c o n sid e re d in p r e v io u s ly pub­ lis h e d papers) w i l l ach ie ve t h is . w eig h t to the vehi c l e s . µ can a ls o be used. ~ But t h is a ls o means a d d it io n a l Thus, o t he r l i g h t e r m a te ria l w ith a d if f e r e n t As p o in te d out p r e v io u s ly , the LIMs are supposed to be used f o r both suspension and p ro p u ls io n . around kyo Thus, i f i s re q u ire d in (2 .2 5 ) to g iv e a s u f ­ f i c i e n t a t t r a c t iv e s u p p o rtin g f o r c e , β4 ≅ 0 i s suggested. o th e r hand, i f around kyo On the i s re q u ire d in (2 .2 5 ) to g iv e a s u f f i c i e n t r e p u ls iv e s u p p o rtin g f o r c e , then the e f f e c t of can be n e g le c te d . D e f in it e ly , f o r the general c a se , the tr a d e ­ o f f among the w e ig h t, c o s t o f m a te ria l in re g io n s 3, 4 , and the source w in din g lo s s , w i l l determ ine what v alu e o f machine d e sig n . β4 is s u p e r io r in the B u t, u n fo r tu n a te ly , the a r b it r a r in e s s o f w ill 4 β make the F o u rie r in v e rs e in t e g r a ls much more co m p licated to e v a lu a te . Thus, in the fo llo w in g , o n ly two extreme cases o f f i n i t e l y lam inated iro n and d is c u s s e d ; and o t he r values o f β4 = i / μo β4 β4 ≅ 0 f o r the i n ­ f o r the fre e space w i l l be ly in g between them w i l l be guessed -1 9 to g iv e in te rm e d ia te r e s u lt s . The same assum ptions can be made f o r β1. However, because reg io n 1 i s n o t on the v e h ic le , w eigh t is not a s e rio u s problem . In r e a l i t y , the e a rth (which can be co n sid e re d as a fre e space) w ith β1 = i / μ o , and i n f i n i t e l y lam inated back iro n w ith β1 ≅ 0 , are most p r a c t ic a l and w i l l be co n sid e re d in the f o llo w in g as the o n ly p o s s i b i l ­ it ie s . A n a ly t ic a lly e v a lu a tin g the in t e g r a ls (2 .24 ) to (2 .3 0 ) i s v e ry d i f f i c u l t , i f n o t im p o s s ib le . Only two extreme a p p ro x im a tio n s, nam ely, th in and t h ic k r e a c tio n r a i l s , w i l l be c o n sid e re d . a c tio n r a i l i s th in o r t h ic k is d e fin e d by around ky o where |~ Κx |2 i s l arge. |γ 2h2 | << 1 or That a r e ­ |γ2h2 | >> 1 The d e t a ils about when those ap proxim atio n s are reasonable in d e s c r ib in g the LIMs are given in Appendix B. Under these assum ptions, s im p lif ic a t io n s can d e f i n i t e l y be made in (2.24) to (2 .30 ) to e v a lu a te the perform ance o f the LIM s. However, in s te a d o f doing t h is im m ed ia tely, we w i l l f i r s t t r y to an a lyze the in te g ra n d s . I t w i l l be found th a t t h is is a co n v e n ie n t way o f lo o k in g in to t he sy s te m -o p tim izin g problem and the compensation f o r end e f f e c t s . F i r s t , l e t us c o n s id e r the case w it h (A) β1 ≅ 0: β1 ≅ 0 . (re g io n 1 is i n f i n i t e l y lam in ate d ir o n ) From ( 2 .8 ) , (2 .1 3 ), (2 .3 1 ): (3.1) -2 0 - (3 .2 ) (3 .3 ) , Here the fo r c e in t e g r a ls kz = |ky |. F̄y , F̄z And Fy , ~ ~ Fz r e s p e c t iv e ly . are the in te g ra n d s f o r h2 , h3 are r e s p e c t iv e ly the th ic k n e s s o f re g io n s 2 and 3. I t i s n o t d i f f i c u l t to show th a t f o r a re a l ky both and tanhγ2h2 ) have p o s it iv e re a l p a rts . have the same s ig n s as those o f And, t h e ir im aginary p a rts w i l l γ2. Thus, f o r a given source c u rre n t d is t r ib u t io n , a nonzero value of α w i l l r e s u lt in s m a lle r t o t a l power in p u t and fo r c e s , and thus degrade machine perform ance. e f f e c t o f t h is term of P1 and P2. α Of c o u rse , as p o in te d out b e fo re , the w i l l s tr o n g ly depend on the r e la t iv e magnitude Now, the th in and t h i ck r e a c tio n r a i l assum ption w i l l be used to s im p lif y the problem . -2 1 (i ) Thin re a c tio n r a i l s : S in ce γ2h2 |γ2h2 | can be made. i s s m a ll, the appro xim atio n th a t tanh γ2h2 ≅ A ls o , f o r L I Ms used in HSGT, h3 has to be sm all , i . e . , tanh kz h3 ≅ kz h3 is a ls o a reaso n a b le a p p ro x im a tio n . We can now begin to c o n s id e r d if f e r e n t kinds o f LIMs. ( i- 1 ) α ≅ 0: (re g io n 4 i s i n f i n i t e l y lam inated ir o n ) T h is i s the most p o p u la r SL IM co n sid e re d by N asar, Del C id , J r . (1 4 ), Wang(1 3 ), e tc . The r e s u l t can a l so be d ir e c t ly a p p lie d to o b ta in the perform ance o f DSLIM an a lyze d by Yamamura and I t o ( 1 6 ) , Wang( 1 1 ) , and Dukow icz( 1 5 ) . From ( 3 .1 ) , ( 3 .2 ) , ( 3 .3 ) , we get: (3.4a) (3.4b) (3 .4 c ) (3 .5 a) (3.5b) (3 .5 c ) -2 2 - (3 .6a) (3.6b) Here I t should be n o tic e d th a t µ2 ~ alw ays appears in the form o f µ 2 z / ( µ 2 z h 3 + µ o h 2 ) . Thus, e x ce p t in the case th a t h2 >> h3 , w hich i s g e n e r a lly not tru e in t h is th in r e a c tio n r a i l geom­ e t r y , the e f f e c t o f the p e rm e a b ility is s m a ll. A f t e r some te d io u s a lg e b r a ic m a n ip u la tio n s , are p lo tt e d as fu n c tio n s o f kyv/ ω , i t i s found th a t A ls o , equal to ay , ar , a i ky in F ig . 3 .1 . With "a" d e fin e d as at have extrema a t "a" s a t is f y in g e q u a tio n s given below , r e s p e c t iv e ly : (3 .7 a) (3.7b) (3 .7 c ) From these p lo t s , i t i s q u ite obvious th a t d if f e r e n t arrangements of |~Kx|2 can g iv e machines which can be used f o r d if f e r e n t purposes, ( e . g . , b ra k in g o r p ro p u ls io n in HSGT, o r even o th e r than HSGT system s). However, LIMs used f o r both p ro p u ls io n and le v it a t io n are our o n ly con­ cern now. -2 3 - F ig . 3.1 Integrands f o r geometry ( A - i- 1 ) w ith (1) Re (3) (2) Im (4) -2 4 In o rd e r to get a n e t p ro p u ls io n f o r c e , a l l y re q u ire d to s a t i s f y 1 > ao > 0 . t h is re g io n w i l l be c o n sid e re d . kyo = ao ω/v is gener­ Thus, in the f o llo w in g , o n ly Now, we w i l l lo o k more c a r e f u lly in to e q u a tio n s ( 3 .4 ) , ( 3 .5 ) , ( 3 .6 ) , e s p e c ia lly ( 3 .5 ) . I t can be shown th a t t here w i l l o n ly be one v a lu e o f "a" s a t i s ­ fy in g each o f the e q u a tio n s (3 .7 ) f o r 1 ≥a ≥0 . r o o ts , A ls o , a i > ay > ar are alw ays tr u e . f o r t h is t hin r e a c tio n r a i l case. ay > az From (3 .7 a ) , m o n o to n ic a lly in c r e a s in g fu n c tio n o f And, f o r t hose ay can be o b ta in e d is found to be a K1 v 2 /ω . Whether the machin e has a r e p u ls iv e o r an a t t r a c t iv e l e v it a t io n fo rc e depends m a in ly on w hether ao N e v e rth e le s s , a sm all g e n e r a lly means a la r g e r energy lo s s ao < az and thus a s m a lle r e f f ic ie n c y . sm all ao < az is s m a lle r o r la r g e r than S in ce ay > az i s alw ays tr u e , a a ls o means a sm all th r u s t f o r c e . n o tic e d th a t the maximum v alu e o f p o s it iv e la r g e r than th a t o f r e p u ls iv e az. Furtherm ore, i t F z / |~ ~ Κx |2 is is always much ~Fy/|~Κx|, ( a c t u a lly the r a t io i s about So, t h is k in d o f LIM c o n fig u r a tio n i s n o t good f o r use as a r e p u ls iv e le v it a t io n system . w ith 1 > ao > az Thu s, i t i s p r e fe ra b le to use a source such th a t a n e t a t t r a c t iv e and a la rg e p r o p u ls iv e fo rc e can be o b ta in e d . Two methods are suggested to op e ra te t he LIMs su ch t ha t t he above requirem ents o f (1) F ix in g e n t speed ranges. 1 > ao > az can be met: ky o , then t r y in g to use d if f e r e n t ω T h is i s the method most people su g g e st. a frequency c o n v e rte r i s needed. fo r d if f e r ­ And, th u s , -2 5 (2) F ix in g ω, then t r y in g to vary kyo as ν c hanges. T h is is s im ila r to an antenna a rra y to c hange t he d i r e c t i v i t y by s u i t ­ a b ly a rra n g in g the c u rre n t in each elem ent. In the id e a l case where the source is a t r a v e llin g wave w it h t he in t e g r a ls f o r perform ance are easy to e v a lu a te . (Of c o u rse , we cannot c o n s id e r t he i n f i n i t e t o t a l fo rc e s and power any lo n g e r. But q u a n t it ie s per u n it le n g t h in the y - d ir e c - tio n are not d i f f i c u l t to d e r iv e ) . The fo llo w in g r e s u lt s can be ob­ ta in e d ; (3 .8a) (3.8b) (3 .8 c ) (3 .8d) (3 .8e) E quations (3 .8 ) are p lo tt e d in F ig . 3.2 as fu n c tio n s o f The maximum th r u s t fo rc e w i l l o ccu r a t responding maximum value o f 1/4 ky o • μoμ2z/ (μ2zh3+ μoh2 ). ν. w ith a c o r ­ Both the -2 6 - F ig . 3.2 Power and fo rc e s f o r geometry ( A - i - 1) w ith id e a l source and ωΚ1 = 2000, ωμοσxh2 = 5 , kyo = 1 0 . (1) (2) (3) (4) -2 7 s u p p o rtin g fo rc e and the e f f ic ie n c y are m o n o to n ic a lly in c r e a s in g fu n c ­ t io n s o f ν. Thus, in o rd e r to have a high e f f ic ie n c y , i t is sug­ gested th a t the machine be operated in the a t t r a c t iv e s u p p o rtin g fo rc e re g io n . At vm, a la r g e r e f f ic ie n c y can be o b ta in e d by making k2 y o/ ωK1 s m a lle r. U n fo rtu n a te ly , a p u re ly t r a v e lin g wave source does not e x is t in n a tu re . G e n e r a lly , we can o n ly have a f i n i t e source w ith a |~ Κx |2 which i s an o s c i l l a t i n g fu n c tio n w ith a f a s t decaying envelop and a maximum v a lu e a t ky o . W ith the g iven in fo rm a tio n about the i n ­ te g ra n d s, i t i s no t d i f f i c u l t to understand how t h is s o - c a lle d end e f f e c t comes in t o p la y . We w i l l e x p la in t h is by assuming the source has f or ( i . e . , a source w ith o t he rw is e . |y | ≤ L, and Κx = 0 T h is i s t he most p o p u la r uncompensated source used by n e a r ly a l l o f t he r e s e a r c h e r s .) Now, l e t us go back to F ig . 3 .1 , where a t y p ic a l p lo t f o r i s shown. When becomes p ro p o r tio n a l to j u s t p ic k up the v a lu e o f i s o b ta in e d . Fy ~ L δ (ky - ky o ). at becomes i n f i n i t e l y la r g e , The in t e g r a l f o r ky o , and the id e a l source r e s u lt However, f o r the more r e a l i s t i c case where the s it u a t io n is d if f e r e n t . non-zero v a lu e s a t I t i s obvious th a t ky ≠ ky o . a c o n tr ib u tio n from t h is re g io n . end e f f e c t comes in to p la y . ¯Fy w i l l |~ Kx|2 L is f i n i t e , w i l l now have Thus, t he in t e g r a ls w i l l a ls o re c e iv e And t h is e x p la in s how t h is s o - c a lle d D e f in it e ly , the o v e r a ll r e s u lt depends -28d r a s t ic a lly on how fu n c tio n s By o b se rv in g the F ~y / |~ Kx |2 ~Fy/|~Kx|2 , |~ Kx |2 v a ry . curve more c a r e f u lly , i t can be seen th a t th e re is o n ly one lo c a l maximum lo c a te d a t those p o in ts p a rt o f ky o , kyo ± π/L, are n o t too c lo s e to |~Kx|2 in t o the la r g e r ~Fy/|~Kx|2 re g io n . happens to be equal to ky ≠ kyo in f lu e n c e . ~Fy/|~Kx|2. |~Kx|2 at i s sm all such th a t And t h is i s j u s t the s it u a t io n in w hich the end T h is a ls o e x p la in s i s suggested to reduce the u n d e sire d end e f f e c t . f o r th e case where th a t |~Kx|2 Thus, the r e s u lt a n t L e f f e c t degrades the machine perform ances most. L On the c o n tra ry , ay ω/v, those n o n -zero fo rc e w i l l decrease enorm ously, e s p e c ia lly when why a la rg e is So the o v e r a ll e f f e c t w i l l be w i l l always p ic k up s m a lle r the spread i s w ide. ~Fy/|~Kx|2 ky o , th e re i s alw ays an o the r p a rt wh i c h w i l l go s m a ll, i . e . , the end e f f e c t causes l i t t l e kyo /v. Thu s, i f ω ay ω/v, then although w i l l spread in t o t he re g io n where s m a lle r than th a t a t if ky = ay kyo i s near As ay ω/v, alth ough th e re is a p o s s i b i l i t y w i l l go to the reg io n where ~Fy/|~Kx|2 i s la r g e r , the in - cre a se is always sm all compared to the decrease coming from the spreadin g o f |~Kx|2 in t o the o p p o s ite d ir e c t io n . w i l l depend on how ~Fy/|~Kx|2 Thus, alth ough the d e t a il v a r ie s , a decrease in the r e s u lt a n t fo rc e can g e n e r a lly be observed. In the above we have m a in ly ta lk e d about the in flu e n c e o f the source fu n c tio n . Now we w i l l a ls o say something about the o th e r param­ e te r s . I t is known th a t a s m a lle r K1ν2 / ω Now, i f th e re are two cases w ith the same w i l l g iv e a s m a lle r ω/ν then ro u g h ly sp e a kin g , the slo p e to the r ig h t o f but d if f e r e n t ay ay . ay, w i l l be ste e p e r -2 9 f o r the case w ith la r g e r fe re n t K1 ay. Thus, i f we have two systems w ith d i f ­ o p e ra tin g a t the same ω , ν, then we are g o in g to get a more s e rio u s end e f f e c t f o r the case w ith la r g e r s e t to t h e ir co rre sp o n d in g v a lu e s o f ay ω/v. is suggested to g iv e a h ig h e r e f f ic ie n c y . o n ly a p p lie d to the case where K1 when ky o 's are (Remember, a la r g e r ao And the above argument is ay = ao > ½). So g e n e r a lly , we are e x p e c tin g a s m a lle r end e f f e c t f o r a system which uses a tra c k w ith a sm all σxh2. A ls o , as we suggested b e fo re , the machine can a ls o be used f o r s u p p o rtin g purposes. In t h is ca se , i t i s p r e fe ra b le to op­ e ra te in the a t t r a c t iv e re g io n . Thus, i t i s more d e s ir a b le to make σxh2 s m a ll. T h is happens to be c o n s is te n t w it h the sm all end e f f e c t requirem en t. So, i t seems th a t com posite tra c k s whi ch w i l l g iv e a r e la t iv e l y s m a lle r σx are p ro m isin g . N ote, f o r p r a c t ic a l m a te ria l suggested in MAGLEV system s, non­ fe rro m a g n e tic m a te ria l g e n e r a lly has a h ig h e r c o n d u c t iv it y . alth ough m a te ria l w ith high µ ~ Thus, does not o f f e r too many advantages; com posite tr a c k s w ith fe rro m a g n e tic m a te ria l co n ta in e d are s t i l l recommended f o r use in t h is s p e c ia l case. We w i l l postpone our d i s ­ c u ssio n about how to c o n s tru c t the com posite tra c k s u n t il Chapter IV. However, i t sh o u ld a ls o be remembered th a t low , o th e rw ise re g io n . ~Fy/|~Kx|2 K1 cannot be made too w i l l become too sm all in the low s l i p T h is w i l l e v e n tu a lly decrease the fo rc e and degrade the machine perform ance. A s u it a b le compromise i s thus n e cessary. For the geometry we are c o n s id e r in g , one th in g seems w o rth w h ile m entioning . With a l l o f the s p e c if ie d source and tra c k c o n f ig u r a t io n s , we know th a t the maximum fo rc e i s going to o c c u r a t -3 0 - f o r the id e a l source. maximum valu e o f than kyo. However, we a ls o know th a t f o r t h i s ~Fy/|~Kx|2 i s going to appear a t aym vω/ νm the which i s le s s Thus, the maximum t h r u s t fo r c e p o in t i s not the p o in t where the end e f f e c t i s most s e r io u s . L a t e r , t h i s can be shown to be d i f f e r e n t from some o f the o th e r c o n f ig u r a t io n s . In the above, we con sid ere d m a in ly the end e f f e c t on the fo r c e in the y - d i r e c t i o n . PF. S i m i l a r arguments can a ls o be a p p lie d to S in c e , f o r a given ν, the in te g ra n d f o r the lo s s in t e g r a l m o n o to n ic a lly de cre a sin g f o r is 0 ≤ ky ≤ ω/v, so the end e f f e c t w i l l not in tro d u c e too much in f lu e n c e to the lo s s . the end e f f e c t w i l l F̄ z, η, Thus, as f o r the e f f i c i e n c y , have s i m i l a r in f lu e n c e to th a t o f w i l l g e n e r a lly degrade the e f f i c i e n c y . For most o f F ky, ,y i . e . , i t ~Fz/|~Kx|2 is a ls o m o n o to n ic a lly in c r e a s in g , so t h a t the end e f f e c t i s not s e r io u s f o r the s u p p o rtin g f o r c e e i t h e r . (i- 2 ) α = 1: ( i . e . , source w ith o u t back ir o n ) S i m i l a r to the case α ≅ 0 , we can get from ( 3 . 1 ) , ( 3 . 2 ) , (3 .3 ) and the t h in r e a c t io n r a i l assumption: (3.9a) (3.9b) (3.10a) -31 - (3.10b) (3.11a) (3.11b) A l l o f the ap p ro xim a tio n s (3 .9 b ) , (3 .1 0 b ), (3.11b) are i n t r o ­ duced f o r the convenience o f in te g ra n d an a lyse s w ith kyh2 << 1 , kyh3 << 1 are v e ry a c c u ra te . 0 < ky < ω/v where However, as f a r as the r e a l i s t i c perform ance in t e g r a ls are concerned, ( 3 .9 a ) , (3 .1 0 a ), (3.11a) w i l l be more a c c u ra te . Formulas (3 .9 a ), (3 .1 0 a ), (3.11a) are p lo tte d in F ig . 3 .3 . those "a" param eters as d e fin e d in ( i - 1 ) , i f o n ly the re g io n For 0≤ a ≤ 1 is c o n s id e re d , we w i l l get: (3.12a) (3.12b) (3.12c) In t h is case a i > ay > ar is always tr u e , m o n o to n ica ll y in c r e a s in g fu n c tio n s o f And a l l o f them are γ = µo σx h2v. In o rd e r to have -3 2 - F ig . 3.3 Integrands f o r geometry ( A - i- 2 ) w ith (1) (2) (3) (4) γ = µoσxh2v = 1 0 . -3 3 - F ig . 3.4 Power and fo rc e s f o r geometry ( A - i- 2 ) w ith id e a l source and ωµoσxh2v = 2 5 , kyo = 1 0 . (1) (2) (4) (5) PF (3) -3 4 h ig h e r e f f ic ie n c y and enough s u p p o rtin g f o r c e , we suggest o p e ra tin g t hi s mach in e w ith Now, s in c e ao > ay, i . e . , an a t t r a c t iv e scheme is p r e fe r a b le . ay ≅ az, we cannot draw the la r g e s t p o s s ib le p ro p u ls io n fo rc e f o r t h is k in d o f geometry o p e ra tin g in the a t t r a c t iv e s u p p o rtin g fo rc e re g io n . and Furtherm ore, f o r the same tr a c k c o n fig u r a tio n s , ω, ν, ky , geometry ( i- 1 ) always g iv e s la r g e r v a lu e s o f ~Fz/|~Kx|2, |~ ZL | than those o f ( i- 2 ) . fo rc e s and power are re q u ire d , la r g e r geometry ( i- 2 ) . ~Fy/|~Kx|2, So, i f the same amounts o f |~ Kx|2 i s alw ays n e cessary f o r Source w in d in g s o f h ig h e r c o n d u c t iv it ie s must be used to p reven t in t o le r a b le ohmic lo s s e s . Thus, t h is geometry perhaps is not p re fe rre d in HSGT v e h ic le s where la rg e power and fo rc e s are always n e ce ssa ry . However, i f the c o s t f o r a high c o n d u c t iv it y source w in din g i s manageable o r high power i s not a req u ire m en t, t h is LIM w i l l s t i l l be a p p lic a b le . Thus, we w i l l s t ill p re se n t some a n a ly s is f o r t h is geometry. For the id e a l source w ith the machine perform ance i s easy to get: (3.13a) (3.13b) (3 .1 3 c) -3 5 - (3.13d) (3 .1 3e) Here, o f co u rse , ky o h2 << 1 and ky o h3 >> 1 have been assumed such th a t (3 .9 b ), (3 .1 0 b ), (3.11b) can be used. I t should be n o tic e d th a t machine perform ance. µ ~ does not have any e f f e c t on the In (3 .1 3 a ), f o r a given w ith a co rre sp o n d in g v alu e o f fo r c e occurs a t w hich i s j u s t p ro p o rtio n a l to th in g e ls e . At co rre sp o n d in g ky o , the maximum th r u s t νm, F̄ z K2o and is independent o f e v e ry ­ ay ω/vm happens to be equal to w i l l be 0 . kyo and thus the Thus, i f an a t t r a c t iv e fo rc e i s a ls o d e s ire d , we cannot draw the p o s s ib le maximum p ro p u ls io n fo rc e f o r t h is given geometry. kyo T h is i s q u ite d if f e r e n t from the p re v io u s case where > ay ω/vm > az ω/vm and thus a t the maximum p ro p u ls io n fo r c e p o in t a re a so n a b ly la rg e a t t r a c t iv e fo rc e can a ls o be o b ta in e d . f u r t h e r e viden ce th a t geometry ( i- 1 ) i s s u p e r io r . a given ky o , η T h is g iv e s S im ila r to ( i - 1 ) , f o r is a m o n o to n ic a lly in c r e a s in g fu n c tio n o f νm , a la r g e r e f f ic ie n c y can be o b ta in ed by making ν. At kyo /(ωµoσx h2 ) s m a lle r , For the more r e a l i s t i c case o f the end e f f e c t a ls o p la y s an im p o rta n t r o le in degrading machine perform ance. The reason is d e f i n i t e l y the same as b e fo re , i . e . , |~Kx|2 w ill in to the re g io n where the in te g ra n d is s m a lle r than th a t a t spread ky o . Now, as we m entioned, f o r an id e a l source w ith ay(vm) happens to be equal to ( kyoνm)/ω. Thus, the most s e rio u s end -3 6 e f f e c t is observed. G e n e r a lly , i t can be argued th a t the end e f f e c t can p o s s ib ly be reduced by making σxh2 s m a lle r , which i s a ls o the same c o n c lu s io n as th a t o f ( i- 1 ) . Of c o u rse , a s u it a b le low er σxh2 must be chosen to p reven t the machine from going in t o the low e f f i c i ­ ency re g io n . (i i ) T h ic k r e a c tio n r a i l s : For |γ2 |h2 >> 1 , a d d itio n to t h i s , fo rm u la s. tanh γ2 h ≅ 1 tanh kz h3 = kzh3 However, in t h is case, the a lg e b r a ic a n a ly s is i s found to be ( ii- 1 ) α ≅ 0 . Thus, in s te a d o f con­ α ≅ 0 , we w i l l take care o f the problem w ith Then s e v e ra l r e s u lt s f o r those o f In w i l l f u r t h e r s im p lif y the o r ig in a l much more co m p lica te d f o r problem s w ith s id e r in g is a p p ro x im a te ly tr u e . α ≅ 0 α = 1 f ir s t . can be o b ta in ed by comparing w ith α = 1 . α = 1: (re g io n 4 is fr e e space) From ( 3 .1 ) , ( 3 .2 ) , (3 . 3) and the given assum ptions: (3 .1 4 ) (3.15) -3 7 - (3.16) (3.17) (3.18) (3 .19 ) A lg e b ra ic a n a lyse s f o r s u it s in F ig . 3 .5 . ~Fy/|~Kx|2 and ~Fz/|~Kx|2 y ie ld the re - For the eddy c u rre n t lo s s and re a l P, r e s u lt s can be o b ta in e d s im p ly by m u lt ip ly in g (ω/ky - ν) and ω/ky w ith ~Fy/|~Kx|2 r e s p e c t iv e ly . For µ = µo~ I, a maximum v alu e f o r ~ (here K = ~Fy/|~Kx|2 is found to be a t w ith a co rre sp o n d in g v a lu e o f -3 8 - If ~μ ≠ μο~ Ι and μ2o / μ 2y μ2z << 1 are tr u e , And the co rre sp o n d in g maximum v a lu e i s μο/ (2√2( 2 + √2) ). Thus, i f the peak p ro p u ls io n fo r c e is o f most concern, the b ig g e r extreme v a lu e o f n e t ic tr a c k w i l l make the tra c k a l i t t l e n on ferrom agn etic one. ~Fz/|~Kx|2 ~Fy/|~Kx|2 f o r the ferrom ag- b i t more fa v o ra b le than the is ze ro a t az = For the n o n ferro m agn etic t r a c k , w ith K3-1 reduces to z e ro , i . e . , the o n ly zero p o in t i s a t a = 1 . m a tte r w hether has a maximum at a = 1. ~μ = μο~Ι or ~μ ≠ μο~Ι, ~Fz/|~Kx|2 And, no More in fo rm a tio n about these in te g ra n d s can be o b ta in e d from F ig . 3 .5 . I t can e a s ily be seen th a t f o r the t h ic k r e a c tio n r a i l , t h e s k in e f f e c t w i l l p reven t the medium in re g io n 1 from in f lu e n c in g the machine perform ance. im p o rta n t r o le . µ ~ begins to p la y a v e ry Fo r the no n fe rro m a gn etic t r a c k , o n ly the r e p u ls iv e fo r c e can be observed. a t a s m a lle r "a ". And the p e rm e a b ility A la r g e r r e p u ls iv e fo r c e can be o b ta in e d o n ly Thus, i f high e f f ic ie n c y i s d e s ir e d , t h is geometry is not recommended to be used as a LIM both f o r p ro p u ls io n and l e v i t a ­ t io n . However, i f i t i s m a in ly suggested f o r the p ro p u ls io n purpose, th e re w i l l always be an e x tra r e p u ls iv e fo rc e to h e lp in su p p o rtin g the v e h ic le s . As f o r the fe rro m a g n e tic t r a c k , we w i l l get a v e ry s im ila r re ­ s u lt to th a t o f the t h in re a c tio n r a i l ( A - i- 2 ) . Both r e p u ls iv e and -3 9 a t t r a c t iv e fo rc e s can be o b ta in e d . tru e f o r both cas e s. a p p lie d h ere. Fu rtherm ore, 0 ≤ az ≅ ay < 1 is Thus, most o f the arguments given th e re can be Depending on w hether ao is la r g e r o r s m a lle r than the machine can have e it h e r an a t t r a c t iv e o r a r e p u ls iv e fo r c e . az, B u t, i f high e f f ic ie n c y is a ls o re q u ir e d , the a t t r a c t iv e one is p r e fe ra b le . U n fo rtu n a te ly , the la r g e s t p ro p u ls io n fo rc e can o n ly be o b ta in e d a t ay = az. I t is a ls o n o tic e d th a t a t is a l i t t l e of ky h ay the peak in te g ra n d f o r F̄y b i t s m a lle r than th a t o f ( A - i- 2 ) which i s o n ly o f the o rd e r m u lt ip lie d by th a t o f ( A - i- 1 ) . Thus, s im ila r to ( A - i- 2 ) , a high c o n d u c t iv it y source w in d in g is re q u ire d to reduce the ohmic lo s s when s im ila r f o r c e s as in ( A - i- 1 ) are d e s ire d . And, a c t u a lly , t h is seems to be a common requirem en t f o r LIMs w ith α = 1 fo rc e s as LIMs w ith α4 = 0 when s im ila r are needed. For the id e a l s o u rce , the machine perform ance can e a s ily be o b ta in e d by s u b s t it u t in g and (3 .1 9 ). P lo ts o f ky w ith kyo in e q u atio n s (3 .1 5 ), (3.17) F̄y , F̄z , e tc . are then shown in F i g . 3 .6 . Now, the maximum th r u s t fo r c e p o in t w i l l s a t i s f y w ith K i = K o r K2. The c o r r e s ponding fo rc e w i l l be on the o rd e r o f A c t u a lly , as d e riv e d from the in te g ra n d a n a ly s e s , the id e a l source r e s u lt s f o r µ ≠ µo~I ~ w i l l be q u a lit a t iv e ly s im ila r to ( A - i- 2 ) . Most o f the o th e r p r o p e r tie s can a ls o e a s ily be observed from F ig . 3 .6 . With the same argument as in ( A - i) , i t is understood th a t the end e f f e c t can be reduced by making Ki s m a lle r. U sing a com posite tra c k g e n e r a lly can a t t a in t h is o b je c tiv e . However, the decrease o f a ls o g iv e a s m a lle r Ki w i l l ay and thus a s m a lle r e f f ic ie n c y . A s u it a b le com­ promise should be made to determ ine which v a lu e o f Ki to use. -4 0 - F ig . 3.5 Integrands f o r geometry ( A - ii- 1 ) w ith (1) (2) (3) (4) -4 1 - F ig . 3.6 Power and fo rc e s f o r geometry ( A - ii- 1 ) w ith id e a l source and ωΚ5 = 200,000, ky o = 1 0 , (1) (2) (3) (4) (5) Power f a c t o r -4 2 - ( i i -2) α ≅ 0: (re g io n 4 i s i n f i n i t e l y lam inated ir o n ) From ( 3 .1 ) , ( 3 .2 ) , (3 .3 ) and the g iv e n assum ptions: (3.20a) (3.20b) (3.21a) (3.21b) (3.22a) (3.22b) -4 3 A f t e r comparing to the co rre sp o n d in g form ulas in ( A - i i - 1 ) , the d if f e r e n t terms in (3 .2 0 b ), (3 .2 1 b ), (3.22b) are u n d e rlin e d and by changing these terms to " 1 " , form ulas f o r ( A - ii- 1 ) can be o b ta in e d . I t i s seen th a t o th e r. μ = μo~I ~ and ~μ ≠ μo~I behave q u ite d i f f e r e n t ly from each By o b se rv in g t h is f a c t and the r e s u lt s from some rough a lg e b ra ic c a lc u la t io n , the fo llo w in g remarks can be made. N onferrom agnetic tr a c k s : • ~Fz/|~Kx|2 i s always n e g a tiv e . ( A - ii- 1 ) ~Fz/|~Kx|2 is m onotonical l y in c r e a s in g f o r w ith a minimum v a lu e o f - μo/4 v a lu e equal to 0 a t • ay And s im ila r to th a t in at a = 0 0 ≤ a ≤1 and a maximum a = 1. is lo c a te d a t w ith K4-1 ly in g between and Ferrom agnetic tra c k s : w ith has a maximum ly in g between • ay is lo c a te d at and w ith Thus, ay > az a = 1 and a = az. Κ5-1 ly in g between is g e n e ra lly tru e . -4 4 - F ig . 3 .7 Integrands f o r geometry ( A - ii- 2 ) w ith (1) (2) (3) (4) -45- F ig . 3.8 Power and fo rc e s f o r geometry ( A - ii- 2 ) w ith id e a l source and ωΚ5 = 200,000 , = 0 .0 1 , h3= 0 .0 1 , (1) (2) (4) (3) (5) Power f a c t o r kyo = 1 0 . -46- A ls o , no m atter which t r a c k i s used, ~Fz/|~Kx|2 ~Fy/|~Kx|2 and w i l l always be la r g e r than those o f the corresponding values o f ( A - i i - 1 ) . Most arguments given in ( A - i i - 1 ) can be a p p lie d here f o r the nonferrom agnetic t r a c k except t h a t , due to the r e l a t i v e l y l a r g e r values o f ay , ~Fy/|~Kx|2 and duce la r g e r fo rc e s and e f f i c i e n c y . advantage over t h a t o f LIM w ith re-obs erved. ~ F z /|~ K x |2 , t h i s case w i l l For the fe rro m a g n e tic t r a c k , the α = 1 Due to the f a c t t h a t pro- which was mentioned before i s ay > az, i f the LIM i s used f o r both p r o p u ls io n and l e v i t a t i o n purposes, the LIM i s suggested to be operated in the a t t r a c t i v e re g io n . Then, a peak p r o p u ls io n fo r c e can always be o b ta in ed w h ile the a t t r a c t i v e f o r c e i s s t i l l r e l a t i v e l y la r g e . For the id e a l s o u rce , the r e s u l t s can be o b ta in ed s im p ly by s u b s t i t u t ­ ing ky t io n o f w ith ν ky o . Some p lo t s w ith the machine performance as a fu n c ­ are given in F ig . 3 .8 . For fe rro m a g n e tic t r a c k s , some r e s u l t s are q u a l i t a t i v e l y s i m i l a r to those o f ( A - i - 1 ) . not a δ- f u n c t io n , the end e f f e c t w i l l a ls o appear. When Decreasing K ~x is σx is suggested to reduce i t . (B) β1 = β3 : (re g io n 1 i s f r e e s pace) In (A), β1 ≅ 0. T h is means a l o t o f lam inated ferro m ag n etic m a te r ia l mus t be used, and thus more c o s t . So, from the economic p o in t o f view , the f r e e space, or j u s t the e arth f o r re g io n 1 may be p r e f e r ­ a b le . From ( 2 . 8 ) , (2 . 13) , (2.31): (3.23a) -4 7 - (3.23b) (3 .23 c) Here (3.23d) (3.23e) For t h ic k re a c tio n r a i l s , th e re is no d iffe r e n c e whether reg io n 1 i s t he fr e e space o r a lam in ate d ir o n , o r any o t he r m a te r ia l. in t he fo llo w in g o n ly t he th in r e a c tio n r a i l Thus, is n ecessary to be con­ s id e re d . ( i- 1 ) α ≅ 0: (re g io n 4 is i n f i ni t e l y lam inated ir o n ) Under the t h in r e a c tio n r a i l assum ption, (3.23) w i l l g iv e us: (3.24a) (3.24b) - 48- (3.25a) (3.25b) (3.26a) (3.26b) O f co u rse , (3 .2 4 b ), (3.25b) and (3.26b) are in tro d u c e d o n ly f o r the purpose o f s im p lif y in g the in te g ra n d a n a ly s is . be used f o r the in t e g r a l e v a lu a tio n . G e n e ra lly they cannot (The id e a l source w ith ky o h3 << 1 is one e x c e p tio n .) Except f o r a f a c t o r , t h is f a c t o r (3.24) i s e x a c t ly the same as ( 3 .9 ) . However, w i l l make the a lg e b r a ic a n a ly s is much more co m p lica te d . Several remarks w i l l be given: -49- N onferrom agnetic tra c k s : • S in ce [1 + kz (h2+h3 )] ~ 1, e xcept f o r the v alu e ~Fz/|~Kx|2, the d iffe r e n c e between t h is geometry and ( A - i- 2 ) i s s m a ll. Thus we w i l l get s im ila r p lo t s o f ~Fy/|~Kx|2 and ZL ~ And • f o r both cases. w ith γ = μσxh2ν. Of c o u rse , the co rre sp o n d in g maximum va lu e s w i l l be a l i t t l e b i t la r g e r than those o f ( A - i- 2 ) . ~Fz/|~Kx|2 w i l l be d if f e r e n t from the co rre sp o n d in g ( A - i- 2 ) r e ­ s u l t . A c t u a lly , i . e ., i t w i l l alw ays be r e p u ls iv e . T h is i s not s u r p r is in g because now th e re i s no b a c k -iro n to g iv e the a t t r a c t iv e fo rc e component any lo n g e r. a A ls o , th e re is a minimum v alu e o f - μo/4 maximum v alu e o f 0 at For a > 0 And f or • For a > 0 and cannot be n e g le c te d ) i t can be s hown t ha t a l l y in c r e a s in g . ky = 0 a = 1. F e rro m agnetic t r a c ks: • at At ay 0 < a < l ie s to t he r i g ht o f is mon o toni c - a = i t can be s hown t ha t i f az l i e s to the (left/right) o f And, th e re is on ly -50one v alu e o f az ly in g between 0 and 1. A ls o , at ky = ω/v, which is the maximum v a lu e of ~Fz/|~Kx|2. O ther p ro p e rtie s o f the in te g ra n d s can ro u g h ly be o b ta in e d from the p lo t s in F ig . 3 .9 . For the same source d is t r ib u t io n s , i t can be seen th a t f o r non­ fe rro m a g n e tic tra c k s the machine perform ance in terms o f Fy , η, and PF is ro u g h ly s im ila r to th a t o f ( A - i- 2 ) . The main d if f e r e n c e i s th a t now we g e t o n ly a p u re ly r e p u ls iv e fo r c e . And the sm all a b s o lu te v a lu e o f the in te g ra n d f o r F̄z a t la rg e "a" w i l l e xclu d e t h is geometry from being used as a LIM f o r both s u p p o rtin g and p ro p u ls io n purposes. n e t ic t r a c k s , the s it u a t io n i s d if f e r e n t . is s t i l l For ferrom ag­ Of c o u rse , i f s m a ll, the r e s u lt d e f i n i t e l y w i l l be n e a r ly the same as b e fo re . However, i f µ2y i s la rg e enough, then becomes im por­ ta n t; e s p e c ia lly when such th a t ay > az is tru e . Then, we can t r y to design a LIM to su p p o rt and s im u lta n e o u s ly a c c e le r a te the v e h ic le . Thus, from t h is p o in t o f view , we can conclude th a t a la r g e r v a lu e o f h2 i s more d e s ir a b le . And making la rg e i s the e a s ie s t way to reach t h is o b je c t iv e . we can a ls o t r y to make equal to 1 a t the peak v e lo c it y . σx µ2y and As f o r the f a c t o r la r g e r to get a v alu e a p p ro x im a te ly However, as f a r as the end e f f e c t is concerned, i t i s more d e s ir a b le to keep i t a t a moderate v a lu e (say 0 .8 ). Then i t seems th a t a somewhat s m a lle r σx . σx is p re fe rre d o ver a h ig h e r A ll o f t h is w i l l make the com posite ir o n the most p ro m isin g tra c k m a t e r ia l. -5 1 - F ig . 3.9 Integrands f o r geometry ( B - i- 1 ) w ith (1) (2) (4) (3) -5 2 - F ig . 3.10a Power and fo rc e s f o r geometry ( B - i- 1 ) w ith id e a l source and (2) (1) (4) (3) (5) Power f a c t o r -5 3 - F ig . 3.10b Power and fo rc e s f o r geometry ( B - i- 1 ) w ith id e a l source and (2) (1) (3) (4) (5) Power f a c t o r -5 4 ( i- 2 ) α = 1: (re g io n 4 is fr e e space) Under the th in r e a c tio n r a i l assum ption, (3 .2 3 ) w i l l g iv e us: (3.27a) (3.27b) (3.28a) (3.28b) -5 5 - (3.29a) (3.29b) Some te d io u s a lg e b r a ic a n a lyses w i l l g iv e us the fo llo w in g r e s u lt s : N onferrom agnetic tr a c k s : can be f u r t h e r approxim ated as 4ky2 . And Then (3.24b) and (3.27b) lo o k n e a rly the same; so do (3.25b) and (3 .2 8 b ). By comparing the form ulas w ith each o th e r i t fo llo w s th a t and G e n e r a lly , the c o r ­ responding maximum v alu es are s m a lle r f o r geometry ( B - i- 2 ) . A c t u a lly ~Fy/|~Kx|2 f o r geometry ( B - i- 2 ) is at o n ly about o n e -h a lf o f the v alu e o f ~Fy/|~Kx|2 at f o r the case ( B - i -1 ). T h is is s im ila r to the p re v io u s case ( B - i- 1 ) and w i l l always be n e g a tiv e . minimum value o f -uo /4 occurs a t A c t u a lly , a ky = 0, and a maximum value -5 6 of "0" occurs a t a lit t le ky = ω/v. However, f o r the same "a" i t w i l l be b i t s m a lle r f o r geometry ( B - i- 2 ) . Ferrom agnetic tr a c k s : • For 1 > a > 0 , if is s t i l l s m a ll, then the same c o n c lu s io n as f o r th a t o f no nferro m agnetic tra c k s can be used as an a p p ro xim a tio n . • For a > 0 , if is not s m a ll, th e n , except in a very sm all re g io n where s t ill kz is sm all such th a t is tru e so as to make the in te g ra n d s behave in the same man­ ner as th a t o f no nferro m agnetic t r a c k , the s it u a t io n is d i f f e r ­ e n t f o r the rem aining re g io n s . • In o rd e r to have a LIM w ith reaso nable e f f ic ie n c y , i t is sug­ gested th a t kyo be a l i t t l e kyo = 0 .8 ω /ν). Now f o r the re g io n where |~ Κχ |2 b i t s m a lle r than a > 0 , ω/ν (sa y , if th e n , in is la r g e ( i . e . , say around kyo ≅ 0 .8 ω /ν ), the appro xim atio n be made. can Then, i t can be shown th a t ~Fym/|~Kx|2 = μo/4. c re a s in g w ith Fz /| K X |2 ~ az ≅ ay. and a minimum v alu e o f and is found to be m o n o to n ic a ll y i n ­ A maximum v a lu e o f - μo/4 For sm all μ2y/μo ω/v h2, the terms at a = 0 ~Fy/|~Kx|2, μo/4 a = 1 can a ls o be observed. ~Fz/|~Kx|2 are found to behave in a s im ila r manner as in ( B - i- 1 ) . ence is th a t the most im p o rta n t f a c t o r at and ~ZL. The main d i f f e r ­ γ = μ o h 2 σ x v o f the p re v io u s case -57- F ig . 3.11 Integrands f o r geometry ( B - ii- 2 ) w ith (1) (2) (3) (4) μoh2σxv = 5 , -5 8 - F ig . 3.12 Power and fo rc e s f o r geometry ( B - ii- 2 ) w ith id e a l source and (1) (2) (4) (5) Power F a c to r (3) -5 9 - ( B - i- 1 ) is now changed to ~Fy/|~Kx|2, ~Zr , ~ Zi γ/2 and the co rre sp o n d in g maximum va lu e s o f f o r t h is case are s m a lle r. i t is a d if f e r e n t s to r y . A c t u a lly , i t For a la rg e μ2y/μoω/vh2, lo o ks more s im ila r to ( A - i i - 1 ) . In both geom etries ( B - i- 2 ) and ( A - i i - 1 ) , we have ay = az, Thus, most arguments g iv e n in ( A - ii- 1 ) can a ls o be a p p lie d here. A f t e r the above a n a ly s is , a general d is c u s s io n w i l l be g iv e n . Now th a t the LIMs are supposed to be used both f o r s u p p o rtin g and p r o p u ls io n , la r g e fo rc e s in both d ir e c t io n s and high e f f ic ie n c y are a l l re q u ire d . T h is w i l l e v e n tu a lly e lim in a te the p o s s i b i l i t y o f u sin g LIMs w hich g iv e r e p u ls iv e s u p p o rtin g fo r c e s . So cases ( A - i i ) and (B) m agnetic tr a c k s are not recommended. w ith n o n fe rro ­ For the rem aining geom etries which can be operated to g iv e a t t r a c t iv e s u p p o rtin g f o r c e s , they can be d iv id e d in t o two c a te g o r ie s , nam ely, ay > az and ay ≅ az. The p ro p e rty th a t a la r g e r p ro p u ls io n fo r c e can o n ly be o b ta in e d in a re g io n where ~Fz/|~Kx|2 ay = az is r e l a t i v e l y sm all w i l l f u r t h e r e xclu d e the cases w ith which g e n e r a lly o ccu r f o r α4 = 1 . Thus, f i n a l l y , the o n ly r e ­ m aining geom etries are ( A - ii- 2 ) and ( B - i- 1 ) w ith μ≠μ I o and ( A - i- 1 ) . I t should a ls o be n o tic e d th a t maximum v alu es o f ~Fz/|~Kx|2 are g e n e r a lly la r g e r than those o f ~Fy/|~Kx|2 f o r the rem aining ca se s. And th u s , f o r a s u it a b le | ~ K x | 2 , a la r g e r F̄z can always be o b ta in e d . T h is happens to be c o n s is t e n t w ith t he requirem ent f o r v e h ic le d e s ig n , s in c e an a c c e le r a ­ tio n o f " l g" is always n ecessary to su p p o rt the v e h ic le , w h ile f o r com­ f o r t , human beings can o n ly t o le r a t e a h o r iz o n ta l a c c e le r a tio n o f s e v e ra l te n th s o f " lg " . -6 0 - O f co u rse , i f |~ Kx |2 is s p e c if ie d , the machine perform ance can be e v a lu a te d e it h e r n u m e ric a lly o r a n a ly t ic a lly . G e n e r a lly , num erical in te g r a tio n s are s tr a ig h tfo r w a r d and a n a ly t ic a l e v a lu a tio n s are d i f f i ­ c u lt . However, f o r some o f the s im p lif ie d geom etries w ith t h in o r t h ic k r e a c tio n r a i l s , a n a ly t ic a l e v a lu a tio n s seem to be p o s s ib le . Now, we w i l l c o n s id e r the most p o p u la r source d is t r ib u t io n and t r y to e v a lu a te the machine perform ance in t e g r a ls . Of co u rse , i t would be the b e st s it u a t io n i f we c o u ld a n a ly ­ t i c a l l y e v a lu a te them f o r a l l o f the recommended g e o m e trie s. geometry ( A - ii- 2 ) i s q u ite d i f f i c u l t . However, Thus, alth o u gh geometry ( A - ii- 1 ) is not p ro m is in g , i t w i l l be a n a l y t i c a l l y examined as an a lt e r n a t iv e o f the t h ic k r e a c tio n r a i l s . And we hope th a t i t w i l l g iv e us some i n ­ fo rm a tio n about geometry ( A - i i - 2 ) . (R e s u lts f o r geometry ( A - ii- 2 ) are also given in Appendix F .) (a) Geometry ( A - i- 1 ) : In t h is case a l l o f the in te g ra n d s (w ith the source term exclud ed) can be e a s ily f a c t o r iz e d . The in t e g r a ls can be r e w r itte n as the summa­ t io n s o f in t e g r a ls s im ila r to S1 (α , ky o , L ) . Appendix C can then be used to get: (3.30a) (3.30b) -6 1 - (3 .3 0 c) (3.30d) where (3.30e) (3 .3 0 f) w ith In the above d e r iv a t io n , the th in r e a c tio n r a i l appro xim atio n is used f o r the whole range o f o b v io u s ly not very a c c u ra te . ky extended from -∞ to kyo where kz is la r g e , g e n e r a lly γ 2h2 I t is A c t u a lly , t h is th in r e a c tio n r a i l assump­ t io n is o n ly in tro d u c e d to an a lyze the in te g ra n d s w ith in o r even o n ly around ∞. 0 ≤ ky ≤ω /v |~ Kx | 2 is la r g e . In the re g io n where is a ls o la r g e , i . e . , the t h in r e a c tio n r a i l c o n d itio n cannot be s a t is f ie d . However, i f we go back to the o r ig in a l e x a ct e x p re ssio n s f o r the in te g ra n d s , we f in d th a t the con­ t r ib u t io n s from the la r g e kz re g io n are alw ays v e ry s m a ll. And, in the m eanw hile, i f we lo o k in t o the th in r e a c tio n r a i l a p p ro x im a tio n s, -6 2 then, in t he same re g io n we wi l l al s o see very sm all in te g ra n d s . F u r ther more, the source fu n c tio n ky o . |~ Κx |2 i s o n ly s i g n i f i c a n t around Thus, we can co nclude th a t the use o f t he t h in r e a c tio n r a i l ap p roxim a tio n f o r t he who le range o f ky does not in tro d u c e too much e r r o r f o r the machine perform ance in t e g r a ls . T h is argument can a ls o be a p p lie d to o th e r g e o m etries. Num erical v a lu e s can be used in (3 .30 ) to e v a lu a te the machine perform ance. However, b e fo re doing t h a t, we w i l l form ulas more c a r e f u lly . la rg e ( e . g ., > π ). For p r a c t ic a l ca se s, Thus, lo o k in t o these | Im 2αmL| i s g e n e ra lly can be n e g le cte d in I t i s q u ite obvious th a t the term p ro p o rtio n a l to L in S1 (αm, ky o , L ) . S1 w i l l g iv e a r e s u lt co rre sp o n d in g to the id e a l source case ( i . e . , w i l l g iv e form ulas (3 .8 )). The rem aining term o f the reason f o r the end e f f e c t . is e s s e n t ia lly A c t u a lly , we w i l l have (3.31a) (3.31b) (3 .3 1 c) H ere, the s u b s c rip ts "1" are in tro d u c e d to re p re s e n t the c o n tr ib u tio n from the term in S 1 (α m , k y o , L ). A ls o , (3.30d) is used -6 3 fo r F̄z which w i l l make the a n a ly s is s im p le r. Now, two extreme cases w ith 4ω/K1ν2 being e it h e r la r g e o r sm all w i l l be co n sid e re d in in v e s t ig a t in g the e f f e c t s due to the term as shown above. From (3 .3 0 ): (3.32a) (3.32b) I t i s n o tic e d th a t term |α2 - ky o | is much la r g e r than can be n e g le cte d in (3 .3 1 ). |α1 - k y o | ; th u s , the By lo o k in g in t o (3 .3 1 ) , we see th a t the p r o p e r tie s o f the e x p re ssio n s (3 .31 ) can be observed by c o n s id e rin g : (3.33a) (3.33b) -6 4 - (3 .3 3 c) I t should f i r s t be noted th a t these terms are sm all when A ls o , f o r the id e a l source case we know th a t and Im ¯P ≤ 0 w ith in the o p e ra tin g range o f (3.31) and ( 3 .3 3 ) , we r e a liz e th a t -Im ¯P1 and ¯F1 Re ¯P1 ao ≅ 1 . Re ¯P ≥ 0 , ¯ F y ≥ 0 , 0 ≤ ao ≤ 1 . Now, from is g e n e r a lly n e g a tiv e ; v ary in a s im ila r way and are n e g a tiv e o n ly when i s n e g a tiv e f o r ao > 1/3. shown th at the e f f ic ie n c y decreases f o r And, ro u g h ly , i t can be 1 > ao > 1/2. the machine perform ance i s thus observed in the high The degrading o f ao re g io n . We then have: Now, we need o n ly c o n s id e r the fo llo w in g terms: (3.34a) (3.34b) -65- F ig . 3.13a Machine performance f o r geometry ( A - i- 1 ) w ith A l l param eters are the same as in F ig . 3 .2 . L = 2.512m. Curves (1 ), (2 ), (3) and (4) re p re se n t the same q u a n ti­ t ie s as in F i g . 3 .2 . Curve (6) is the e f f ic ie n c y and Curve (5) is the power fa c t o r . -66- F ig . 3.13b A l l q u a n t it ie s are the same as in F ig . 3 .1 3a, except L = 1.256m. -6 7 - I t can be shown th a t the phase angles o f (3.34a) and (3.34b) are a l l m o n o to n ic a lly d e cre a sin g fu n c tio n s o f ao fo r 0 ≤ ao ≤ 1 . A c t u a lly , f o r (3.34a) ( o r , P̄1 ), the phase angle in c re a s e s from -π /4 at And, ao = 1 to π/2 at ao ≅ to f o r (3 .3 4 b ), i t w i l l vary from and f i n a l l y to 7 /4π at π/4 ao = 0 . at 5/4π ao ≅ 1 at to ao = 0 . π at ao = Thus, e xce p t in a v e ry sm all re g io n in c re a s e s in Re ¯P and p o s s ib le in c re a s e s in PF are expected. For a2o ≅ (K1v2)/ω , both served to d ecrease. -Im ¯P1. If and machine e f f ic ie n c y are ob- (K1v2)/ ω >> γ2 , F̄z 1 On the c o n tr a r y , i f s im ila r to F̄y behaves in the same way as (K1v2)/ω << γ2 , ¯Fz1 behaves in a way -Re ¯P1. From the above a n a ly s is , i t is not s u r p r is in g th a t most c o n c lu ­ s io n s are b a s ic a lly s im ila r to those o b ta in e d from the in te g ra n d a n a ly s is . T y p ic a l p lo t s o f machine perform ance f o r t h is r e a l i s t i c source are given in F i g . 3.1 3. The degrading in the machine perform ance can the be observed by comparing w ith the r e s u lt s shown in F ig . 3.2 f o r the id e a l source. (b) Geometry ( A - ii- 1 ) : S im ila r to ( a ) , F̄y and ¯P can be e va lu a te d by u sin g s u it a b le f a c t o r iz a t io n in (3.14a) and (3.15a) to get: (3.35a) -6 8 - (3.35b) where α3 , and α4 are ro o ts o f k2y + i ( ω- ky v)K 6= 0 w ith or: (3.36a) (3.36b) w ith D e f in it io n s f o r Appendices D and E. fo r S2 (αm, ky o , L) and S3 (αm, kyo, L) are given in However, i t should be noted th a t the e x a c t e x p re s sio n S3 (αm, kyo, L) i s d i f f i c u l t to g e t. Thus, o n ly an a sy m p to tic form ula i s d e riv e d . As f o r F̄z, the in t e g r a l is much more c o m p lic a te d . However, as we m entioned, the end e f f e c t i s g e n e r a lly sm all p ro vid e d t ha t la r g e . kyo L is Thus, we can always use the form ula f o r the id e a l source as an approxim ation f o r the more r e a l i s t i c source we are now c o n s id e rin g . I t is obvious th a t id e a l r e s u lt s can be ob ta in ed by j u s t c o n s id e r­ ing the terms p ro p o rtio n a l to L in S2 (α m , k y o , L ) and S3 (α m , k y o , L ). -6 9 - Here we w i l l begin to c o n s id e r e f f e c t s due to the rem aining term s. However, as we j u s t m entioned, the end e f f e c t s on the s u p p o rtin g fo rc e are r e l a t i v e l y s m a ll, so we are no t going to in c lu d e any f u r t h e r a n a ly ­ ses f o r the F̄z. Now, under the assum ption o f ky o L >> 1 and | Im(αmL)| >> 1, we can w r ite : (3.37a) (3.37b) S u b s c rip t "1" i s in tro d u c e d to p o in t out th a t (3.37a) and (3.37b) correspond to the term in (3 .14 ) and (3 .1 6 ). Now l e t us c o n s id e r two extreme cases: From (3 .3 6 ), we have (3.38a) (3.38b) -7 0 - (3.39a) (3.39b) When e r a l l y s m a ll. ao is not c lo s e to "1 ", v a lu e s f o r However, when 1-ao very la rg e in flu e n c e i s observed. is s m a ll, P̄1 and F̄1 are gen­ ( e . g ., 1- ao ≅ ω/K6v2) a The u n d e rlin e d terms in (3. 37a) and (3.37b) dominate and produce a phase angle o f about 0° in i s a ls o an in c re a s e in the p ro p u ls io n fo r c e . ¯P1. There Roughly sp e a kin g , e x p re s ­ s io n s (3 .37 ) w i l l help to ease the degrading o f the machine perform ance. From (3 .3 6 ), we have: (3.40a) (3.40b) (3 .4 0 c) (3.37a) and (3.37b) can then be r e w r itte n as: -7 1 - (3.41a) (3.41b) When ao e v e r, when e n t. is c lo s e to 1, (3.41a) and (3.41b) are r e l a t i v e l y s m a ll. ao is sm all ( e . g ., a2o ~ (K6v2 )/ω) How- the s it u a t io n is d i f f e r ­ ReP̄ 1 , Im ¯P1 and F̄y 1 are a l l p o s it iv e and somewhat la r g e r . As f o r the c o n tr ib u tio n due to the term the r e s u lt s in Appendix E are used. decomposed in t o two p a r ts . As we can see, S3 (am, ky o , L) is One i s due to the re s id u e s o f a l l the p o le s , w h ile the o th e r i s due to the in te g r a tio n s along the branch c u ts . However, if is tru e (Note th a t K5= σxμ2z >> K6 covers most o p e ra tin g re g io n s . in the very low speed r e g io n .) |β2 | is la r g e , thus The o n ly e x ce p tio n o ccu rs i s much la r g e r than | |. 1 β Most terms in the branch c u t p a rts are then shown to be much s m a lle r than the co rre sp o n d in g terms in the re sid u e p a r ts . |Im( F u rth e r c o n d itio n s o f m α L)| > π w i l l make i t p o s s ib le f o r us to w r ite down the f i n a l approx­ im ate r e s u lt s as: -7 2 - (3.42a) (3.42b) Here of α3 and α4 are d e fin e d in (3 .36 ) w h ile β1 and β2 are ro o ts γ22 = 0 , i . e . , (3.43a) (3.43b) wi t h -7 3 - Now, under the c o n d itio n we have: (3 .43 c) (3.43d) We w i l l begin to c o n s id e r two extreme cases: I t i s obvious th a t the v a lu e s f o r ao is not c lo s e to 1. α3 F̄y2 are sm all when We are most concerned w ith the case when where the v a lu e s are la r g e . terms w ith and 2 P̄ a ao~1 Now, under the assumption in v o lv e d can be n e g le c te d . For t hose the c o n tr ib u tio n s due to the u n d e rlin e d terms a p p ro x im a te ly g iv e us Re P̄2 ≅ 0 , Fy2 ≅ 0 and a p o s it iv e Im ¯ P 2 . For the rem aining term s, i t can be shown th a t Re ¯ P 2 , Im ¯P2 and are a l l n e g a tiv e . F̄y2 And the magnitudes are so la rg e th a t they w i l l not o n ly cancel the p o s it iv e v alu es in P̄1 and F̄y 1 , but w i l l a ls o e v e n tu a lly g iv e us la rg e n e g a tiv e v alu es o f Re P̄, Im P̄ and F̄y. Thus, we can conclude th a t la rg e de­ creases in the p ro p u ls io n fo r c e , the power f a c t o r and the e f f ic ie n c y w i l l be observed f o r T h is c o n c lu s io n i s b a s ic a lly s im ila r to th a t d e sc rib e d in the e a r l i e r in te g ra n d a n a ly s e s . In t h is case, the v alu es are r e l a t i v e l y sm all f o r an "a o" which -74- i s not too s m a ll. a2o ~ (K6v2)/ω. Thus, we are most concerned w ith the re g io n where Of co u rse , the terms co rre sp o n d in g to be n e g le cte d any lo n g e r. α3 cannot A f t e r some rough a lg e b r a ic m a n ip u la tio n s , i t can be shown th a t the u n d e rlin e d terms w i l l g iv e us a p o s it iv e value fo r Im P̄2 and n e a r ly ze ro values f o r Re ¯P2 and F̄y 2 . However, the most im p o rta n t c o n tr ib u tio n comes from the rem aining term s. F̄y2 < 0 , Re ¯P2 < 0 and Im ¯P2 < 0 are o b ta in e d . G e n e r a lly , And the v a lu e s are so la r g e th a t o v e r a ll decreases in the p ro p u ls io n fo r c e , e f f ic ie n c y and power f a c t o r w i l l be observed. Thus, the most s e rio u s degrading in machine perform ance i s seen around a 2 o (Ky v2 )/ω. T h is i s a ls o the same c o n c lu s io n as th a t o b ta in ed from the in te g ra n d a n a ly s is . A t y p i­ ca l p lo t o f the machine perform ances f o r t h is more r e a l i s t i c source i s then given in F ig . 3.1 4. And, by comparing w ith F ig . 3 .6 , the degrad­ in g o f the machine perform ances is observed. -7 5 - F ig . 3.14a Machine performance f o r geometry ( A - ii- 1 ) w ith A l l param eters are the same as in F i g . 3 .6 . L = 2.512m. Curves (1 ), (2 ), (3) and (5) re p re se n t the same q u a n t it ie s as in F ig . 3 .6 . Curve (4) is the machine e f f ic ie n c y . -7 6 - F ig . 3.14b A l l q u a n t it ie s are the same as in F ig . 3.14a except L = 1.256m. -7 7 (c) Geometry ( B - i - 1): We can now a p p ly Appendices C and D to (3.24a) and (3.25a) to get: (3.44a) (3.44b) (3 .4 4 c) Here δm are ro o ts o f 1 , 2 , 3 , 4. A ls o , δm+4 = δ*m. And -7 8 - w ith F̄z i s v e ry co m p lica te d and we w i l l not a n a ly ze i t any f u r t h e r . B u t, s im ila r to the p re v io u s case (b ), the r e s u lt f o r the id e a l so u rce can be used as a good a p p ro xim a tio n p ro vid e d th a t ky o L >> 1. The fo rm u las g iv e n in (3.44) a re c o m p lic a te d . By lo o k in g in t o Appendix D we know th a t in t e g r a l S2 (δ i, ky o , L) can be decomposed in t o two p a r t s , nam ely, I 1 Now we w i l l c o n s id e r the c o n tr ib u ­ t io n from I1 . and 2 I2. I t i s obvious th a t th e id e a l so u rce r e s u lt can be o b ta in e d by j u s t c o n s id e rin g the terms p ro p o rtio n a l to S2, and, s im ila r to geometry ( A - i- 1 ) , |Im 2δ i L| la r g e ( e . g . , ≳ π ) f o r most p r a c t ic a l ca se s. the terms in v o lv in g L in S1 and can be co n sid e re d as Thus, in can be n e g le c te d . S1 and S2 Under the above assum ption , we can then w r it e down the fo llo w in g e x p re s sio n s : (3.45a) -7 9 - (3.45b) here δ1 and δ2 a re ro o ts o f And the s u b s c r ip ts "1" are in tro d u c e d to d is t in g u is h (3 .4 5 ) from the o th e r terms where the c o n tr ib u tio n s a re due to in t e g r a ls s im ila r to I2. Now, we w i l l begin to c o n s id e r two extreme cases: In t h is case, the e q u a tio n f o r i ( ω-k y v )K 1 = 0 . A ls o , γ >> 1 δi is approxim ated as ensures th a t K1v >> 1, ω/Κν2 << 1 . Thus, s im ila r to ( A - i- 1 ) , most terms in v o lv in g δ1 ≅ α 1 , δ2 ≅ α 2 ) can be neg le c te d . la s t term in k2 y - i / [ ( δ2-k y o )2] (Note: (Perhaps the o n ly e x ce p tio n is the For the F̄ y 1 ; i t can be r e w r itte n as "1" p a r t, a v e ry sm all a d d it io n a l p ro p u ls io n fo r c e is added. m aining p a r t , the n e g le c t i s re a so n a b le . For the r e ­ A c t u a lly , the whole t hin g can be shown to be c a n c e lle d by ano ther term due to an in t e g r a l s im ila r to I 2 ). By a p p ly in g (3 .3 3 ), ro u g h ly s p e a k in g , the power f a c t o r , e f f ic ie n c y and t he p rop u ls ion f o rce decrease f or a la r g e r ao ( e . g . , ao > 2/ 3) , i . e . , a degrading of t he mach in e p e r f ormance w i l l be observed. maximum decreases general l y oc c u r a t a v alu e o f i s tru e . ao whe re And th e -8 0 - In t h is case, the e q u atio n f o r δi can be approxim ated as and the ro o ts a re found to be: w ith When γ δ1 and S in ce is sm all, δ2 i s sm all to o . However, K1v/γ Thus can be f u r t h e r s im p lif ie d as: |δ2 - kyo| >> |δ1 - ky o |, s o , s im ila r to th e p re v io u s in (3 .4 5 ), most terms in v o lv in g same argument as the p re v io u s la s t term in is la r g e . F̄y 1 . 1 / [ ( δ2 -ky o )2 ] γ >> 1 case can be n e g le c te d . (The γ >> 1 case can a ls o be a p p lie d to the From a l a t e r c o n s id e r a tio n , i t can be shown th a t t h is term w i l l be c a n c e lle d by an o th er term due to I 2.) A ls o , we have: (3.46a) (3.46b) -8 1 (3 .4 6 c) From (3 .45 ) and (3 .4 6 ) we know th a t Re P̄1 is p o s it iv e when is p o s it iv e and n e g a tiv e when Im ¯P1 A2 by changing is n e g a tiv e . A2 to Re ¯P1 is A s im ila r argument can be a p p lie d to Thus A3 = the in flu e n c e o f (3.45a) on the o v e r a ll power f a c t o r la r g e ly depends on ao . However, an in c re a s e in the power f a c t o r is expected when U n fo rtu n a te ly , a decrease in the p ro p u ls io n and fo rc e is always observed when w i l l o ccu r f o r a v a lu e o f ao ≳ γ . And, the most s e rio u s decrease ao ~ γ Now, we begin to c o n s id e r the c o n t r ib u t io n due to the term in ao >> γ S2. I2 However, in s te a d o f u sin g (3 .4 4 ) d i r e c t l y , we go back to the o r ig in a l fo r c e and in te g ra n d form ulas to get: (3.47a) -8 2 - (3.47b) Here: (3.48a) (3.48b) I4 Appendix D. can be e v a lu a te d in the same manner as A c t u a lly , f o r mos t p r a c t ic a l c a se s , i s a f a s t v a ry in g f u n c tio n . I 2 (δm, ky o , L) kyoL is r e l a t i v e l y sm all . i s la r g e . And, thus by comparing to lS 1 ( δ m , k y o , L ) | ( i.e ., an in t e g r a l whose li m i t s are |I4(δm, kyo, L)| in - ∞ and A ls o , the in t e g r a l +∞), |I4(δm, kyo, L)| can be approxim ated as: (3.49a) and (3.49b) -8 3 T h is appro xim atio n is b a s ic a lly s im ila r to the n e g le c t o f (see Appendix D). ( i) I3 in I2 Now, we can begin to c o n s id e r two extreme cases. γ >> 1: In t h is case, both e quatio ns can be approxim ated as δ2≅δ4 can be co n sid e re d as tr u e . That i s , δ1≅ δ3 and Thus, (3.50a) (3.50b) δ1 and δ2 are a p p ro x im a te ly given a t (3 .3 2 ) , i . e . , t i v e l y , so we have: α1 and α2 , re sp e c ­ -8 4 The l a s t two terms in P̄2 can be shown to have a sm all e f f e c t on the power f a c t o r ( a c t u a lly , And the l a s t th re e terms in 1/ky o L F̄y2 w i l l g iv e a drag fo r c e w hich is about o f the id e a l p ro p u ls io n fo r c e . Now, we w i l l begin to lo o k a t the e f f e c t due to the f i r s t terms in and ¯P2 ¯Fy2. The im a g in a ry p a rt in can be seen to have an o p p o s ite e f f e c t compared to the co rre sp o n d in g terms in (3 .4 5 a ), (3 .4 5 b ), i . e . , g e n e r a lly , i t w i l l ease the end e f f e c t . However, s in c e π/2 >> ω/K1v2, the in flu e n c e i s r e l a t i v e l y s m a ll. w i l l a ls o g e n e r a lly r e s u lt in p o s it iv e n e g a tiv e Im ¯P2 fo r ao Re ¯P2 The re a l p a rt in and which i s not v e ry sm all ( e . g . , ¯Fy2 and a1 > 1/3 where ℓ n 1/ao < π/2). Thus , we can a l so s ay th a t i t w i l l make the o v e r a ll end e f f e c t a lit t le b i t sm a lle r . As f o r the rem aining p a r ts , the term ℓn δ2/kyo w i l l j u s t cancel the co rre sp o n d in g term in to the la rg e n e ss o f - i ω/K1v2 in |δ2 - kyo| and P̄1 , and - i π/2 in F̄ y1. Due K1v2/ω, the e f f e c t o f the term ℓn δ2/kyo is n e g lig ib le . U n fo rtu n a te ly , f o r the re a l p a rt o f ℓn δ2/kyo, b o th the power f a c t o r and the p ro p u ls io n fo r c e w i l l be decreased. However, the c o n tr ib u tio n due to t h is term i s sm all to o . By com bining a l l o th e r r e s u lt s we can conclude th a t the most s e rio u s d e g ra d atio n in the machin e perform ance w i l l o c c u r a t ao where 1 - ao ≅ ω/K1v2 < γ ( i i ) γ << 1 Now the term i K1kyov can be n e g le cte d in the e quatio ns -8 5 And, we w i l l have: By u s in g the above r e la t io n s δ1 ≅ - δ4 and δ2 ≅ - δ 3 , form ulas (3.47a) and (3.47b) can be reduced to (3.51a) (3.51b) where -8 6 - Now, we w i l l make a v e ry rough ap p ro xim a tio n f o r the above form u­ la s and hope th a t we can g e t some idea about the e f f e c t s due to them. A f t e r some te d io u s a lg e b r a ic m a n ip u la tio n and the n e g le c t o f the s m all term s , we get: (3.52a) -8 7 - (3.52b) where (3 .52 c) The l a s t terms in P̄2 and F̄y2 ( i . e . , w ith going to be c a n c e lle d by the co rre sp o n d in g terms in p e c t iv e ly . ao . π in v o lv e d ) are ¯P1 and ¯Fy1, r e s - As f o r the rem aining term s, t h e ir v alu es la r g e ly depend on However, the maximum in flu e n c e d e f i n i t e l y w i l l be observed a t ao ≅ γ. And a t th a t p o in t , the c o n tr ib u tio n to p o s it iv e , i . e . , i t w il l ease the end e f f e c t . ¯Fy2 i s found to be E s p e c ia lly i f th e re i s even a p o s s i b i l i t y th a t the o v e r a ll end e f f e c t w i l l h e lp to in c re a s e the p ro p u ls io n fo rc e . Some o f the arguments g iven above can be observed from comparing the p lo t s , F ig . 3.15 and F ig . 3.1 0. -88- F ig . 3.15a Machine perform ance f o r geometry ( B - i- 1 ) w ith A l l param eters are the same as those o f F ig . 3.10a; L = 1.256m. Curves (1 ), (2 ), (3) and (5) re p re se n t the same q u a n t it ie s as in F ig . 3.10a. machine e f f ic ie n c y . Curve (4) is the -89- F ig . 3.15b Machine performance f o r geometry ( B - i- 1 ) w ith A l l param eters are the same as those o f F ig . 3.10b; L = 1.256m. Curves (1 ), (2 ), (3) and (5) re p re se n t the same q u a n t it ie s as in F ig . 3.10b. machine e f f ic ie n c y . Curve (4) is the -90 CHAPTER IV COMPOSITE TRACKS From the p re v io u s d is c u s s io n i t fo llo w s th a t a com posite tra c k is h ig h ly recommended. D e f in it e ly , in o rd e r to s a t is f y d if f e r e n t p u r­ poses, the LIM tra c k c o n fig u r a tio n s should be d if f e r e n t a ls o . I t is obvious th a t a com posite tr a c k w i l l make re g io n 2 inhomogeneous , and i t thus makes the problem im p o s s ib le to s o lv e e x a c t ly . However, f o r p r a c t ic a l a n a ly t ic a l purposes, some appro xim atio ns can always be made. For example, i f the tr a c k i s b u i l t l a t i v e l y long w avelength t r a v e llin g source f i e l d does not see the inhom ogeneity, the tra c k can be a p p ro x im a te ly co n sid e re d as homogeneous w ith some e f f e c t iv e u n ia x ia l ~σ, ~μ. T his idea was p ro b a b ly f i r s t sug- gested by M is h k in (2 1 ), C u lle n and B a rto n (19) in a n a ly z in g r o ta r y machines w ith s lo t t e d r o t o r s . a p p lic a b ilit y . Experim ents have a lre a d y been made to check i t s I t a ls o g iv e s the reason why we used u n ia x ia l σ ,~ ~ µ to re p re s e n t the tra c k p r o p e r tie s from the very beg in n in g o f our a n a ly ­ s is . The v a lu e s o f the e f f e c t iv e t io n . σ ,~ ~ µ depend on the tra c k c o n fig u r a ­ T h e ir v alu es are determ ined from a homogeneous tra c k f o r given u n ifo rm ly d is t r ib u t e d e .m .f. and m .m .f. in such a way th a t the same amounts o f c u rre n ts and flu x e s are o b ta in e d . For example, f o r the gen­ e ra l com posite tr a c k shown in F ig . 4 .1 , we w i l l get (assuming th a t w ith in the tr a c k the m a te ria l is u n ifo rm ly d is t r ib u t e d in the z d ir e c t io n ) : (4.1a) -9 1 - (4.1b) (4 .1 c ) To o b ta in µi µ ~ we change σa in to µa, σb in to and b µ σi in to in the above fo rm u la s. I t should a ls o be n o tic e d th a t in the l i m i t bx →0 these e x ­ p re s s io n s reduce to the f a m ilia r form ulas g iv e n by C u lle n and B a rto n ( 19 ) . As we mentioned e a r l i e r , t h is e f f e c t iv e a lre a d y been e x p e rim e n ta lly v e r if ie d . has n ever been attem pted. µ ,~ ~ σ concept has A t h e o r e t ic a l p ro o f, however, There is no doubt th a t i t a n a lyze t h is com posite tra c k geometry e x a c t ly . i s im p o s s ib le to However, in o rd e r to gain some in t u i t i o n about i t s v a l i d i t y , an e xtre m e ly s im p lif ie d geometry, s p e c if ic a lly a f in it e ly lam inated ir o n , w i l l be an a lyze d . Due to the la m in a tio n , the problem b a s ic a lly becomes a 3-dim en­ s io n a l one. The t r a n s f e r m a trix method is thus not a p p lic a b le . The most com plete method f o r a n a ly z in g problems w ith lam in ate d iro n was p r e v io u s ly given by Bondi and M ukherji( 2 2 ) who e v a lu a te d the induced eddy c u rre n t lo s s e s . Here we w i l l use a s im ila r fo rm u la tio n and extend i t to the LIM problem . The f i n i t e l y lam in ate d iro n i s supposed to be used in re g io n 2. Of c o u rse , f o r d if f e r e n t LIM g e o m e trie s, the s it u a ­ tio n w i l l be d if f e r e n t . Here we w i l l c o n s id e r the geometry ( A - i i - 2 ) . The r e s u lt s f o r some o f the o th e r c o n fig u r a tio n s are p o s s ib ly o b ta in ed by some s u it a b le s u b s t it u t io n s . The governing e q u atio n s are the same as b e fo re . now we have σ = o~ ~ I and µ = µ~ ~ I in re g io n 2. But The source i s s t i l l -92- F ig . 4.1 F ig . 4.2 -9 3 i n f i n i t e l y extended in the x - d ir e c t io n . Due to the p e r io d ic c h a ra c ­ t e r o f the t r a c k , the f i e l d d is t r ib u t io n w i l l a ls o be p e r io d ic in the x d ir e c t io n . A F o u r ie r s e r ie s expansion can be used to ta k e care o f the x v a r ia t io n . Thus, o n ly one la m in a tio n ( i . e . , | χ | ≤ h/2) sho uld be c o n s id e re d . Fo r the y v a r ia t io n , the F o u r ie r tra n sfo rm te ch n iq u e i s employed. And, in the f o llo w in g , o n ly one F o u r ie r component i s con­ s id e re d . I t can e a s ily be seen th a t fu n c tio n s o f f o r the c o o rd in a te system shown in F ig . 4 .2 . (which make and ~ Hx Jx = 0 ~ at The in s u la t io n re g io n s x = ±h/2) are assumed to be v e ry t h in . are thus co n tin u o u s a t the in t e r f a c e s . concluded th a t ~ Hx = 0 x at x = h/2 . Bx ~ Hence, i t may be U sing s im ila r arguments as in Bondi and M u k h e rji( 2 2 ) , we can decompose the f i e l d in re g io n 2 in t o th re e p a r ts , namely: (a) ~Hx = 0 everywhere but ~ J x ≠ 0 , (b) ~ J x = 0 everywhere but ~ H x ≠ 0 , (c ) ~Jx = 0, ~Hx = 0 everyw here. P a rt (c ) i s the s o - c a lle d deeply p e n e tra te d f i e l d w hich w i l l be found to be the most im p o rta n t. I t p la y s a r o le in red u cin g the u n d e sire d s k in e f f e c t . The f i e l d components in each re g io n are thus: Region 2: (a) ~Hx = 0 (4 .2a) -9 4 (4 .2b) (4 .2 c ) (4 .2d) (4 .2 e ) (4 .3 a) (4 .3b) (4 .3 c ) -9 5 (4.3d) ( 4 .3e) (4 .4a) (4.4b) (4 .4 c ) (4 .4 d ) (4 .4e) where (4 .5 a) (4 .5b) (4 .5 c ) (4.5d) -9 6 A s u it a b le Riemann s u rfa c e sheet sho uld be taken to g iv e p o s i­ t iv e re a l p a rts f o r the kz,ℓ kzm , and kz Region 3 (4 .6a) (4 .6b) (4 .6 c ) (4 .6d) Thus we have unknowns {qm1 } , {qm2}, {bm}, {αℓ } , f . can be s o lv e d by m atching the boundary c o n d itio n s a t At z = 0: (Note: are continuou s Jz = 0 at z =0 In p r in c ip le , they z =0 and z = h3 . (4 .7 a ) is a u to m a tic a lly s a t is f ie d from the boundary c o n d itio n s g iven above.) (4.7b) -9 7 i.e ., (4 .7 c ) ( 4 .7 d) ( 4 .7e) (4 .7 f) (4 .7g) Then, w ith (4 .8 a) (4.8b) -98(4 .8 c ) T h is s e t o f e q u a tio n s is m anipulated to g iv e : (4 .9 a) (4.9b) (4 .9 c ) (4.9d) -9 9 - (4 .9 e ) I f (4 .9 a ) can be so lve d f o r αℓ , a l l o f the o th e r unknowns can be o b ta in e d from the rem aining e q u a tio n s . Then we can f in d the f i e l d d is t r ib u t io n everyw here. I t is understood th a t the f i n a l r e s u lt s we want are machine p e r­ formances such as fo r c e s , power, e f f ic ie n c y , e tc . A l l o f these can be expressed in terms o f the f i e l d d is t r ib u t io n a t z = h3 as given by (2 .2 4 ), (2 .2 5 ) , (2 .2 7 ). z = h3 are concerned, As f a r as the f i e l d s a t form ulas ( 4 .6 ) , (4 .7 ) can be used to g iv e : (4.10a) (4.10b) (4 .1 0 c) For average c o n t r ib u t io n . F̄ y, P̄ the But f o r F̄z m≠0 terms in Hz do not g iv e any these terms do have some e f f e c t . However, the c o n tr ib u tio n s from due to the f a s t decay o f m≠ 0 can be co n sid e re d as n e g lig ib le . t r u e .) So, f o r Hz , we pro b ab ly need o n ly to c o n s id e r the term w ith ~ m=0 . Thus q0'1 a∞ . h ≲ h3 is is the most im p o rta n t unknown co n sta n t to e v a lu a te . Now, l e t us assume th a t common f a c t o r (N ote, g e n e r a lly αℓ Then we s e t can be so lv e d f o r , except f o r a -100- (4.11a) (4.11b) (4 .1 1 c) I t should be n o tic e d th a t A, B, C, α∞ are a l l ℓ ,m independent. From ( 4 .9 ) ,( 4 .1 1 ) : (4.12) And the H- f i e l d a t ~ z = h3 i s thus: (4.13a) (4.13b) (4 .13 c) Here f o r ponding E ~ Hz, o n ly the ~ f ie ld i s given by (2 .2 7 ) , the e x p re ssio n s f o r m= 0 E = ^ ~ term is in c lu d e d . ℓ x Hz . /ky~ o µ ω The c o r r e s - Using (2 .2 4 ), (2 .2 5 ), ¯F, ¯P are found to be e x a c tly the same as those given in ( A - i i - 2 ) , except th a t µ /ykz now should be re p la ce d by γ2 o -101- Now we come to the p o in t where we have to e v a lu a te A , B, C, i . e . , we should s o lv e (4 .9 a ) . Doing i t e x a c t ly is a h o peless ta s k . An ap­ p ro xim a tio n can be made by tr u n c a tin g t h is s e t o f e q u a tio n s , cons id e r in g o n ly a lim it e d number o f e q u a tio n s w ith the same number o f unknowns . However, t h i s w i l l le a d o n ly to s ome num erical d a ta . In the problem we are cons id e r in g i t s eems th a t an a n a ly t ic a l s o lu t io n is more des ir a b le . Thus , an a lt e r n a t iv e approxim ation w i l l be made. F i r s t , l e t us lo o k a t the term: I t can be r e w r itte n as: Now, i f kyoh/2π << 1 in the re g io n where in Kx ~ i s assumed, kzd/kz,ℓd << 1 can be shown to hold is la r g e . k z d / k z , ℓ d can thus be n e g le cte d 1 - kzd/kz,ℓd. Furtherm ore, even f o r m= 1, i f |F(m )| >> ukz d can always be o b ta in e d . G e n e r a lly , the requirem ent (|4π2d2o|)/h2 ≳ 1 c o n s id e rin g . reasonable: (|4 π2d2 |)/h 2 ≳ 1 , is no t d i f f i c u l t to meet in the LIM w hich we are now Under these c o n d itio n s , the fo llo w in g approxim ation is -1 0 2 - (4 .14 ) The n e xt ste p is then t r y in g to f in d an e x p l i c i t fu n c tio n o f fo r αℓ p o s s i b i l i t i e s we can f in d are If independent o f to make The magnitude o f the e r r o r term in A w i l l alw ays be la r g e r than 0 . αℓ The o n ly α ℓ = a∞ and α ℓ = a ∞ , then set o f m. ℓ w hich w i l l decrease f a s t e r than Thus, we must e x clu d e t h is p o s s i b i l i t y . A ls o , we are e x p e c tin g a a∞ when ℓ in c re a s e s . On the c o n tr a r y , f o r i s no lo n g e r ze ro . And the e r r o r term w i l l alw ays be s m a ll. Thus, we may conclude th a t the assum ption w i l l g iv e a q u ite good approx­ im a tio n . Now we use in (4 .11 ) to g e t: (4.15a) (4.15b) (4 .1 5 c) w ith (4 .1 5d) -103Thus: (4.16) If Now compare the above form ula w ith c ie n t ly sm all such th a t h/2do << 1 is s u f f i ­ i s tr u e , then i s thus reduced and to h µo/µ . T h is means th a t m a te ria l c o n s ta n ts . µ2y ≅ µ and σ ≅ 0 are the e f f e c t iv e And they are c o n s is t e n t w ith the v a lu e s given by e m p ir ic a l form ulas ( 4 .1 ) . However, i f h/2d << 1 above s im p lif ic a t io n i s n o t v a lid any lo n g e r. i s n o t met, then the Thus, we can conclude th a t the c o n d itio n s under w hich form ula (4 .1 ) g iv e s a c c u ra te v a lu e s f o r the e f f e c t iv e m a te ria l c o n sta n ts are ky h << 1 and h/2d Up to now we o n ly c o n s id e r one F o u r ie r component w ith v a r ia t io n . machine eikyy E v e n tu a lly we have to take the F o u r ie r in v e rs e to get the perform ances. I t i s obvious th a t the assum ptions w hich we make above cannot hold f o r a l l ky. g e n e r a lly decay when i t i s away from t io n s are reaso n a b le around However, the source fu n c tio n w i l l ky o . Thus, i f the above assump­ ky o , a good s o lu t io n can s t i l l by a p p ly in g t h is kind o f approxim ation to a l l g iv e a zero p ro p u ls io n f o r c e . be o b ta in ed ky. In the geometry we are now c o n s id e r in g , w ill Thus i t is out o f the q u e stio n to use m erely t h is k i nd o f tr a c k as the LIM in HSGT v e h ic le s . can be used f o r the le v it a t io n p u rp o se.) added. << 1. (Of c o u rse , i t E x tra c o n d u c t iv it y should be One example is shown in the LIM b u i l t by the Rohr I n c . (1 7 ) w ith -104s u it a b ly arranged co n d u ctin g bars in s e r te d in t o the lam in ate d ir o n . D e f in it e ly , i f the requirem ents kyoh << 1 e f f e c t iv e m a te ria l co n sta n t concept is s t i l l and h/2do << 1 h o ld , the v a lid . However, we are not going to g iv e any f u r t h e r a n a lyse s o f th a t k in d . In s te a d , an a l ­ t e r n a t iv e geometry is suggested. h/2do << 1 Now we w i l l t r y to l i f t and in tro d u c e a tr a c k w ith a la r g e r a p p ro x im a te ly e q u a ls 1. Then the approxim ation h the c r it e r io n such th a t h/2doπ t a n (√ih/2d) ≅ √ih/2d i s not tru e any lo n g e r, nor is the e f f e c t iv e m a te ria l c o n sta n t co n cep t. In ste a d , the d ir e c t method, as in tro d u c e d in t h is c h a p te r, should be used. However, even f o r sm all compared to "A ". h/2dπ ~ 1 , "C" can s t i l l be shown to be Thus, µo(A-B)/µ(A-C) ≅ µo/µ √ih/2d cot(√ih/2d). Now h/2doπ ~ 1 , s o , e x ce p t p o s s ib ly in a very sm all re g io n where su ch t ha t f o r most |√ih/2d| << 1, ky around h/2dπ ky o . ω/ν ~ ky is going to be o f the o rd e r o f "1" And f o r h/2dπ ~ 1 , the argument o f the cotang ent fu n c tio n w i l l be la r g e enough to a llo w us to make the approx­ im a tion th a t Thus, and: (4.17a) (4.17b) (4 .1 7 c) -105- ( 4 .1 7d) ( 4 .1 7e) ( 4 .1 7 f ) A n a ly se s can be made o f these in te g ra n d s . i s no lo n g e r equal to ze ro . Fy / ~Κx|2 ~ to "0" a t A c t u a lly , i f i s a m o n o to n ic a lly d e cre a sin g fu n c tio n o f 0 ≤ ky ≤ ω/ν . At ky fo r I t s v alu e w i l l decrease from ky = ω/ν . A ls o , in th a t same re g io n , can be shown to be p o s it iv e under the c o n d itio n th a t F o r the id e a l so u rce , (4 .17 ) can be used to e v a lu a te the machine perform ances. However, t h is ap p ro xim a tio n w i l l be good o n ly under the c o n d itio n s th a t kyoh3 << 1 and h/2πdo ~ 1 . Thus f o r the given source and tr a c k c o n f ig u r a t io n s , g e n e r a lly the approxim ation can ­ not s a t i s f a c t o r i l y co ver the e n t ir e v e lo c it y range 0 ≤ ν ≤ ω/k y o . If the necessary c o n d itio n s are s a t i s f i e d , the maximum fo r c e w i l l be found at w ith a co rre sp o n d in g fo r c e d e n s ity o f -106- S p e c ia l care should be taken i f (4 .17 ) is a p p lie d to the more r e a l i s t i c source I t is obvious th a t the c o n d itio n s ky h3 << 1 la r g e re g io n o f ky . and h/2πd ~ 1 cannot be s a t i s f i e d f o r a Thus, f o r a source d is t r ib u t io n the a d d it io n a l c o n d itio n th a t L |~ Kx|2∝ is v e ry la r g e seems to be n e ce ssa ry to ensure the use o f (4 .17 ) a good a p p ro x im a tio n . Now, i f a l l the c o n d itio n s f o r the a p p ro xim a tio n s to be v a lid are s a t is f i e d , i t can be seen th a t the examined geometry can o f f e r fo rc e s o f the same o rd e r o f magnitude as those o f geometry ( A - i- 1 ) . Thus, i f the geometry ( A - i- 1 ) can meet the fo r c e requirem en ts f o r the LIMs, the s tu d ie d geometry can a ls o do i t w ith s ource c u rre n t magnitudes o f the same o rd e r. However, i t seems th a t the most im p o rta n t advantage which t h is geometry can o f f e r is the " in flu e n c e s due to the end e f f e c t . " From a remark g iv e n e a r l i e r , we know th a t now t o n ic a lly d e cre a sin g fu n c tio n . If Fy / |~ ~ Kx |2 the source fu n c tio n is a mono~ Kx is not a δ- f u n c t io n , alth o u gh p a rt o f i t w i l l s pread in to a re g io n where ~ F y / | ~ K x | 2 s m a lle r , th e re always are o th e r p a rts w hich w i l l go in to la r g e r in te g ra n d re g io n s . Thus, the n e t e f f e c t is g e n e r a lly s m a ll. S im ila r ly , s in c e is always p o s it iv e , i t is n o t necessary to w orry about |~Kx|2 ~Fz/|~Kx|2 being pushed in to the r e p u ls iv e re g io n . In the above a n a ly s e s , a LIM o f geometry ( A - ii- 2 ) w ith re g io n 2 c o n s is t in g o f lam inated iro n is c o n sid e re d . We would e xpect th a t a s im ila r approach c o u ld be a p p lie d to o th e r c o n fig u r a tio n s . In some -107cases t h is i s found to be tru e indeed. A c t u a lly , s im ila r form ulas w i l l by e x- h o ld w ith s u it a b le s u b s t it u t io n o f p r e s s io n s A, B, C. However, g e n e r a lly th e s e A, B, C 's w i l l be s ome- what more co m p lica te d and the e q u a tio n s f o r them w i l l be more d i f f i c u l t to s o lv e . Es p e c ia lly f o r a th in re a c tio n r a i l geom etry, we s h a ll f in d th a t we cannot use the s ame approxim ation any lo n g e r w h ile s o lv in g f o r αℓ . A ls o , because le s s m a te ria l is employed, a somewhat la r g e r e f f e c ­ t iv e c o n d u c t iv it y is always n e cessary to g iv e a la rg e enough p ro p u ls io n fo r c e . Thus, lam in ate d iro n alo ne i s not good enough to b u ild a th in re a c tio n r a i l LIM ( i . e . , com posite r e a c tio n r a i l s w ith both f e r r o - and n o n fe rro -m a g n e tic m a te ria l are neces s a r y ) . Hence, we n e g le c t f u r t h e r cons id e r a t io n s o f t h i s k in d o f c o n fig u r a tio n . As f o r ( A - i i - 1 ) , we s h a ll f in d (3 .1 4 ), (3 .1 5 ) a n d (3.16) to be s t ill tru e w ith μoγ2/μ2ykz re p la ce d by μo(A-B)/μ(A-C). Here by the same fo rm u las as in the p re v io u s c a se , i . e . , However, f o r A and αℓ B ,C are d e fin e d (4 .1 1 b ), (4 .1 1 c ). they are Somewhat d if f e r e n t . Now we have: (4.18a) (4.18b) (4 .18 c) -108E quatio n s (4 .1 8 b ), (4 .1 8 c) are s im ila r to (4 .1 1 a ), (4 .9 e ). Thus, the ap p ro xim a tio n given f o r the p re v io u s case can be used d i r e c t l y . And, f o r h/2d << 1 , we s h a ll o b ta in the same c o n c lu s io n th a t the r e a c tio n r a i l works as i f i t is a nonconducting fe rro m a g n e tic m a te r ia l. Thus, i t i s n o t s u it a b le f o r use in the LIM e xcept f o r le v it a t io n p u r­ poses o n ly . However, f o r h/2πdo ~ 1 , we can use the by now f a m ilia r approach to g e t: (4.19a) (4.19b) (4 .1 9 c) (4 .1 9 d ) (4.19e) -109- (4 .1 9 f ) Again we observe th a t the magnitude o f eve ry term is r e l a t i v e l y sm all compared to the cas e w ith α ≅ 0 . I t is al s o n o tic e d th a t i s a m o n o to n ic a lly d e cre a sin g fu n c tio n o f both th a t [ ( ωµσh2 µ2o)/µ2 ] < 1 alw ays p o s it iv e . a b le . ky which i s g e n e r a lly tr u e . and Fy / |~ ~ Kx |2 ν, provided ~ F z / | ~ K x | 2 a ls o The arguments given in the p re v io u s case are a p p lic ­ The end e f f e c t i s n o t s e r io u s f o r t h is geometry. p ro p e rtie s th a t the power f a c t o r and |~ ZL | However, the are r e l a t i v e l y low ( s o th a t the same amount o f c u rre n t can o n ly draw a very sm all p ro p u ls io n fo r c e ) compared to the case w ith α ≅ 0 w i l l make t h i s geometry un­ fa v o ra b le . In the above, we d i s cuss ed the LIM w h ic h has a tr a c k w ith sm all s c a le in homogeneity s uch th a t the s ource o n ly s ees an average e f f e c t as des c rib e d above. Of co u rs e, d if f e r e n t s it u a t io n s are pos s ib le . For example, in the LIM b u i l t by Rohr I n c . , we can put the i n se rte d con­ d u cto r b a rs q u ite f a r away from each o th e r such th a t the d is ta n c e s be­ tween them are la r g e r than and σ ~ concept f a i l s . 2π/ kyo. Then d e f i n i t e l y , the e f f e c t iv e µ ~ However, in t h is case, i t seems th a t each bar o p e ra tes in d e p e n d e n tly o f the o th e rs . problem and can be e a s ily s o lv e d . T h is i s , o f c o u rse , a d if f e r e n t - 110CHAPTER V THREE DIMENSIONAL CORRECTIONS I t is obvious th a t g e n e r a lly the c u rre n t source and the re a c ­ t io n r a i l cannot be i n f i n i t e l y extended in the x - d ir e c t io n . Thus, a c o r r e c tio n should be made, i . e . , a more r e a l i s t i c geometry such as shown in F ig . 5.1a should be in v e s tig a t e d . T his is a tough th re e -d im e n sio n a l problem and th e re are no sim p le methods a v a ila b le to handle i t . Thus, in s te a d o f c o n s id e rin g th a t geom etry, we s h a ll lo o k in to an a lt e r n a t iv e geometry as shown in F ig . 5 .1 b . With a s u it a b le source arrangem ent, the boundary c o n d itio n Thus, w ith in Jx = 0 at x = ±Wo can be s a t is f ie d . |x | ≤ Wo , t h is a lt e r n a t iv e geometry is supposed to be a good approxim ation f o r the o r ig in a l one. With M a x w e ll's e quatio ns and m a te ria l p ro p e rtie s as in tro d u ce d in Chapter 2, the F o u rie r tra n sfo rm method can be used to s o lv e t h is boundary value problem . Due to the p e r io d ic it y in the x - d ir e c t io n , in a d d itio n to the F o u rie r tra n sfo rm p a ir ( 2 .4 a ) , ( 2 .4 b ) , another F o u rie r s e r ie s p a ir ( 5 . 1a ) , ( 5 . 1b) i s n ecessary to take care o f the x - v a r ia t io n : (5.1a) (5.1b) For n e it h e r F o u r ie r component can the t r a n s f e r m a trix method be a p p lie d any lo n g e r. A d ir e c t method sho uld be used. F i r s t , we can s o lv e the M a x w e ll's e q u atio n s in each re g io n , in tro d u c in g c e r t a in un­ knowns which can be determ ined by m atching the boundary c o n d itio n s . - 111- F ig . 5.1 a , b -112In o rd e r to make the problem s im p le r, o n ly the case c o n s id e re d . We have: Region (2): ( i . e . , 0 ≤ z ≤ h2 ) β1 ≅ 0 w i l l be (5.2a) (5 .2b) (5 .2 c) Here, the fo llo w in g r e la t io n a ls o h o ld s: (5 .2d) w ith (5 .2 e ) and ( 5 .2 f ) whe r e ( 5 .2 g) and are the ro o ts o f -1 1 3- (5 .2h) From ( 5 .2 ) , ( 2 .1 ) , i t fo llo w s : (5 .3a) (5 .3b) (5 .3 c ) Using the boundary c o n d itio n : we o b ta in : (5 .4 ) In re g io n ( 3 ) , ( i . e . , h2 ≤ z ≤ h2 + h3 ): -114(5.5a) (5.5b) (5 .5 c ) and where (5.5d) (5 .5e) (5 .5 f) (5 .5g) Now we match the boundary c o n d itio n s a t it y o f Hxn ~ z = h2 . The c o n tin u ­ and the r e la t io n s above g iv e us: (5 .6 a ) w ith (5 .6 b ) The c o n t in u it y o f Bzn ~ and the p re v io u s r e la t io n s a ls o g iv e us: (5 .7 a) w ith -115- (5 .7b) so, (5 .8 ) Furtherm ore, u sin g we get: (5 .9a) (5 .9 b ) (5 .9 c ) -116(i i ) β4 = β3 : (5.10a) (5.10b) (5 .1 0 c) Now a l l o f the f i e l d d is t r ib u t io n s can be o b ta in e d . The machine perform ances can be c a lc u la t e d by using (2 .1 7 ), (2 .2 7 ), ( 2 .2 8 ), (2 .2 9 ), e tc . Among these we s h a ll now c o n s id e r o n ly F̄ x , ¯Fy which are the s im p le s t and the most im p o rta n t. (5.11a) (5.11b) These form ulas are s t i l l too co m p lica te d to e v a lu a te . We s h a ll s im p lif y them even more by assuming the medium in re g io n 2 to be is o ­ t r o p ic ; then, (5.12a) w ith -117(5.12b) A f t e r some m a n ip u la tio n s , i t can be found th a t: (5.13a) (5.13b) where (5 .1 3 c) (5.13d) Here F (n) ~ i s an even fu n c tio n o f n . Now we take the source d is t r ib u t io n in t o account. make Jx = 0 at x = ±Wo In o rd e r to w ith in the r e a c tio n r a i l , the fo llo w in g e x - tended p e r io d ic source c u rre n t d is t r ib u t io n J ℓx( x , y) i s necessary: (5.14a) (5.14b) (5 .1 4 c) -118- Thus, f o r n odd (5.15a) f o r n even (5.15b) fo r (5 .1 5 c) n =0 A p p ly in g (5 .1 5 ) to (5 .1 3 ), we have: (5.16 a) (5.16b) So, i f Jx (x ,y ) i s n e it h e r even nor odd, th e re i s a p o s s i b i l i t y th a t a la t e r a l fo rc e e x is t s . Now l e t us c o n s id e r one example. The case analyzed. We s h a ll assume th a t the r e a c tio n r a i l f o r sm all sm all n: n , |γ 2nh2 | << 1 and kznh3 << 1 b4 ≅ 0 w i l l be i s th in such t h a t, are tru e . Then, f o r -119(5.17) A ls o , in o rd e r to compare t h is w ith the tw o -dim ensional c a se , we s h a ll assume th a t J x ( x ,y ) is u n ifo rm ly d is t r ib u t e d f o r |x | ≤ a. Then, n odd o th e rw ise Here Kx (ky ) i s ~ in tro d u c e d to take care o f the y - v a r ia t io n . we g e t a zero la t e r a l fo r c e . As f o r (5.18) Of co u rse , F̄y , we have: (5.19) The "n " summation can be e v a lu a te d by u sin g s e v e ra l s e r ie s sum­ mation form ulae (5.20a) where (5.20b) -120A remark sh o u ld be made here. c o n d itio n s |γ 2nh2 | << 1 , kznh3 << 1 O b v io u s ly , f o r la rg e n the are n o t s a t i s f i e d and the ap­ p ro x im a tio n (5 .17 ) is not v a lid any more. However, f o r la rg e n the fo rc e c o n t r ib u t io n as given by (5 .13 ) is sm all and so is the in v a lid a p p ro xim a tio n (5 .1 7 ). Hence, u sin g (5 .17 ) f o r la rg e n (w hich was d e riv e d u s in g the th in re a c tio n r a i l a p p ro xim a tio n ) w i l l o n ly in tro d u c e a sm all e r r o r in the e x p re s sio n o f the t o t a l fo r c e and we may conclude th a t (5 .1 7 ) i s a f a i r l y good appro xim atio n f o r a l l I f the y - v a r ia t io n o f Jx i s the same as th a t o f the i d e a l i s t i c source f o r the tw o-dim ensional ca se , √2 π δ ( ky - ky o ). n . Kx (ky ) ~ w i l l be g iv e n by Of c o u rse , (5 .2 0 ) w i l l g iv e us an i n f i n i t e fo r c e . How­ e v e r, f o r the fo rc e o f u n it le n g th in the y - d ir e c t io n , we go back to the o r ig in a l fo r c e e q u a tio n in re a l space to get: (5.21a) where (5.21b) I f , fu rth e rm o re , we assume (ω-ky o ν ) K1 << k2 y o , then (5 .21 ) r e ­ duces to : (5 .22 ) -121Comparing t h is to ( 3 .8 ) , we can say the f i n i t e w idth r e a c tio n r a i l has an e f f e c t iv e c o n d u c t iv it y o f T h is i s e x a c t ly the same c o n c lu s io n as R u s s e ll and N orsw orthy(23) made, u sin g an o th er approach. i s reduced. A ls o , i t i s observed th a t the p ro p u ls io n fo rc e The in c re a s e o f "a" w i l l g e n e r a lly reduce t h is s o - c a lle d la t e r a l end e f f e c t . Of c o u rse , the above c o n d itio n (ω-ky ov) K1 << k2 yo i s g e n e r a lly not tr u e . Then, we cannot in tro d u c e the e f f e c t iv e c o n d u c t iv it y concept any more. However, by comparing w ith ( 3 .8 ) , we can see the la t e r a l end e f f e c t w i l l change the p ro p u ls io n fo rc e by an amount: (5.23) From t h is co m p lica te d e x p re ssio n we cannot conclude whether t h is term w i l l decrease o r in c re a s e the p ro p u ls io n fo r c e . |γ20a | ≳ π and Assuming l γ2 0 (Wo -a ) l << 1, ( 5 . 23) becomes: (5.24a) O r, we can assume both |γ 2 0 a | and |γ20(Wo - a)| ≳ π to get: (5.24b) -122Thus, i f (5 .2 4 ) w i l l be p o s it iv e , i . e . , the la t e r a l end e f f e c t w i l l in c re a s e the p ro p u ls io n fo r c e . maximum fo r c e p o in t , U n fo rtu n a te ly , a t the Thus, the la t e r a l end e f f e c t w i l l reduce the p ro p u ls io n fo r c e a t th a t p o in t. Fo r the more general source can t r y to e v a lu a te the in t e g r a l o f the term we (5 .2 0 ). However, due to the presence γ 3 2 0 the in t e g r a l i s too tough to e v a lu a te . However, a change o f p ro p u ls io n fo r c e can alw ays be o b served , namely (5.25) (The main term i s j u s t given by (3.30b) e xce p t f o r a f a c t o r o f " a " ) . Now, aga in u sin g the assum ptions we s im p lif y (5 .2 5 ) and get (5 .2 6 ). |y2a| ≳π, |γ 2 (ωo - a)| << 1, The same e x p re s sio n (5 .2 6 ) can a ls o be used f o r ano ther s it u a t io n w ith both |y2a| and | ( Wo - a) | ≳ π , γ2 e xce p t th a t now a f a c t o r 1/2 sho uld be in tro d u ce d : (5.26) Note t h a t, i f the s in g u la r it y a t the source fu n c tio n is not s u it a b ly c o n s tru c te d ky = 0 w i l l p o s s ib ly g iv e an i n f i n i t e fo rc e and a ls o i n f i n i t e power in p u t, w hich is n o t p r a c t ic a l. Thus i s alw ays n e cessary to get r id o f t h is s in g u la r it y . s in ky oL = 0 -123As f o r the id e a liz e d source c a se , the in te g ra n d w i l l be p o s i­ t iv e in the re g io n where in g r e g io n , and n e g a tiv e in the rem ain­ Thus, i f ky o belongs to the nega­ t iv e r e g io n , a s m a lle r p ro p u ls io n fo r c e than in the tw o-dim ensional case w i l l be o b ta in e d . Because an odd power o f γ 20 appears in the above in t e g r a l, two e x tra branch p o in ts in the complex ky plane sh ou ld be in tro d u c e d , making the in t e g r a l e x tre m e ly hard to e v a lu a te . In o rd e r to o b ta in an e x p l i c i t fo rm ula f o r the fo r c e , we s h a ll go back to the o r ig in a l s e r ie s fo rm u la tio n (5 .1 9 ). Fo r each "n " , an in t e g r a l o f the f o llo w in g form sh o u ld be computed: (5 .27 ) We know th a t f o r the tw o-dim ensional ca se , the p o r tio n a l to the above in t e g r a l w ith n =0 . Now f o r fo rc e i s p ro ­ n ≠ 0 , suppose th a t then the in t e g r a l w i l l g e n e r a lly be s m a lle r than the tw o-dim ensional case (co rre sp o n d in g to e ffe c t w ill n =0 ) . Thus, t h is la t e r a l end reduce the p ro p u ls io n fo rc e even more. However, i f th e re is a p o s s i b i l i t y th a t the la t e r a l end e f f e c t w i l l compensate the lo n g it u d in a l end e f f e c t . A c t u a lly , i t had a lre a d y been found th a t the o v e r a ll c r i t i c a l p o in t should be around -124- The f a c t o r iz a t io n o f w i l l le a d to in t e g r a ls o f the form In Appendix C t h is in t e g r a l i s worked o u t. A ls o , in t e g r a ls o f the form w i l l appear. These in t e g r a ls need f u r t h e r c o n s id e r a tio n . g iv e an i n f i n i t e r e s u lt u n le ss s in ky oL = 0 . They w i l l I f t h is c o n d itio n is s a t is f ie d , we e a s ily get from Appendix C: (5.28) F i n a l l y , we have (5.29a) where α 1 , α2 are ro o ts o f -125o r: (5.29b) (5 .29 c) w ith Now, l e t us c o n s id e r the case th a t the o r i g i n a l l y sym m etrical source i s s h if t e d o v e r a d is ta n c e ε to the r ig h t , i . e . , we c o n s id e r a source c u rre n t d is t r ib u t io n : (5.30a) The co rre sp o n d in g F o u r ie r components a re then found to be: (5.30b) -126By u sin g (5 .1 6 ), the fo r c e components f o r the id e a liz e d source w i l l be given as: (5.31a) (5.31b) Here Fr (n ) ~ re p re se n ts the re a l p a r t o f F (n ): ~ (5 .3 1 c) Going through c o m p lica te d s e r ie s summation p ro ce d u re s, we g e t the e x p l i c i t forms f o r ~ Fy and ~Fx: (5.32a) -127- (5.32b) If γ 2 0 (Wo - a) ≳ π, then tanh γ20(Wo - a) ≅ co th γ20(Wo - a), and (5.32 a) reduces to (5 .2 1 a ); i . e . , when the re g io n w hich i s n o t covered by the source is la r g e , the in flu e n c e o f the s h i f t on F̄y s m a ll. |γ 2 (Wo - a)| << 1, L e t us now lo o k a t the more p r a c t ic a l case o f |γ20ε| << 1 . w i l l be Then, a f t e r r e t a in in g o n ly f i r s t o rd e r terms in the h y p e r b o lic f u n c t io n s , we have: (5 .3 3 ) By comparing t h is w ith form ula (5 .2 1 ), i t i s seen th a t th e re i s alw ays an in c re a s e o f the p ro p u ls io n fo rc e by the amount o f As f o r the la t e r a l f o r c e , because |γ20a| ≳ π and |γ20Wo | ≳ π g e n e r a lly h o ld , (5 .32 b) reduces to: (5.34) Thus, i f |γ20ε| i s la rg e enough, th e re is a p o s s i b i l i t y th a t a huge la t e r a l fo rc e in the same o rd e r as th a t o f the p ro p u ls io n fo rc e is o b ta in e d . Of c o u rse , f o r the p r a c t ic a l machine o p e ra tio n , t h is -128- F ig . 5.2 Curves (1) and (2) re p re se n t (5 .3 1 a )/ f o r ε = 0 and ε = 0.01m, r e s p e c t iv e ly ; Curve (3) re p re se n ts [(5 .3 1 b )/ Here Wo - 0.3m and a = 0.28m. fo r ε = 0.01m. A ll param eters e xcept are the same as those o f F ig . 3 .2 . ε -129- F ig . 5 .3 Curves (1) and (2) re p re se n t fo r ε = 0 and ε = 0 .0 5 , r e s p e c t iv e ly ; Curve (3) re p re se n ts Here fo r ε = 0 .0 5 . Wo = 0.3m , a = 0.2m. A l l param eters e xcept ε are the same as those o f F ig . 3 .2 . -130cannot happen. A sm all d isp la ce m e n t from the sym m etrical p o s itio n w i l l in tro d u c e a r e s to r in g fo r c e to push the source back to the o r ig ­ in a l p o r t i o n . T h is fo r c e i s given by I t can e a s ily be seen th a t F̄ x alw ays has the o p p o s ite s ig n o f ε. Thus, as f a r as the la t e r a l d isp lace m en t is concerned, the system is s t a b le . A ls o , f o r sm all d is p la c e m e n ts, these e x tra fo rc e s are r e la - t i v e l y s m a ll. For ¯Fy i s o f the o rd e r o f ε. i t i s o f the o rd e r o f ε2 , w h ile f o r F̄x it F o r the more general source form ula s im ila r to (5 .20 ) can be o b ta in e d by u sin g (5 .1 6 ) , (5 .3 0 ), (5 .3 1 ) , and some s e r ie s summations. A c t u a lly , we have: (5.35a) (5.35b) I t is d i f f i c u l t to e v a lu a te these in t e g r a ls . |γ20ε| << 1, |γ20a | >> 1 and |γ20(Wo -a ) | << 1 The assum ptions are then used to -131s im p lif y (5.35) to : (5.36a) (5.36b) Thus, the unsym m etrical case w i l l in tro d u c e one more term in the F̄y in t e g r a l, w hich can be e a s ily e v a lu a te d to be: (5 .3 7 ) (5 .3 7 ) i s alw ays p o s it iv e in the normal o p e ra tin g re g io n , i . e . , the unsym m etrical p o s it io n g e n e r a lly in c re a s e s the p ro p u ls io n fo r c e . A ls o , (5.36b) can be seen to have the o p p o s ite s ig n o f the p r a c t ic a l cases where The branch p o in ts a t c u lt to e v a lu a te . ky o L γ 2= 0 ε in most o f is la r g e . ( a ls o p o le s) make (5 .3 6 ) v e ry d i f f i ­ Thus, s im ila r to the p re v io u s ca se , we go back to the o r ig in a l s e r ie s fo rm u la tio n to g e t e x p l i c i t e x p re s s io n s . the f a c t o r s Due to 1 /n 2 , 1/m2 and 1/(m2 - n 2 ) , the t o t a l fo rc e s are m a in ly gen­ e ra te d by sm all n , m term s. Going through a l l the f a c t o r iz a t io n and i n t e ­ g r a tio n procedures we get: -132- (5.38a) (5.38b) whe r e (5 .3 8 c) ( 5 .3 8d) w ith α1 , α2 b e in g d e fin e d in (5 .2 9 ). -133- F ig . 5 .4 D if f e r e n t fo rc e components o f s e r ie s e x p re s sio n s (5.38a) and (5 .3 8 b ). (1) (3) (2) (4) where A l l o f the param eters d e fin e d in the same way as in F ig . 3 .2 . -134- F ig . 5.5 D if f e r e n t fo rc e components o f s e r ie s e x p re s sio n s (5.38a) and (5 .3 8 b ). (1) (3) (2) (4) where A l l o f the param eters d e fin e d in the same way as in F ig . 3 .2 . -135- F ig . 5.6 D if f e r e n t la t e r a l fo rc e components in the s e r ie s e x p re ssio n (5 .3 8 b ). (1) (2) (3) (4) where ε = 0.0/m and a l l o f the o th e r param eters are the same as those o f F ig . 5 .2 . -1 3 6The a n a ly s is is e s s e n t ia lly com pleted. some r e s u lt s are shown in F ig s . 5.2 and 5 .3 . For the id e a l source Comparing these to F ig . 3 .2 , we n o tic e the in flu e n c e o f the 3 -d im e n sio n al c o r r e c tio n s we made in t h is c h a p te r. As f o r the more r e a l i s t i c s o u rc e , we o n ly gave the s e r ie s r e p re s e n ta tio n . S e ve ra l components w ith sm all n , m are shown in F ig s . 5 .4 , 5 .5 and 5 .6 . When ε /Wo and seen th a t the s e r ie s r e p re s e n ta tio n f o r ε /a F̄y are v e ry s m a ll, i t is converges v e ry f a s t . However, i t seems th a t we need many terms to g iv e a f a i r l y good appro x­ im a tio n f o r ¯Fx. In the above, we m a in ly co n sid e re d the th re e -d im e n s io n a l c o r r e c ­ tio n s f o r geometry ( A - i- 1 ) . c o n f ig u r a t io n s . S im ila r approaches can be used to o th e r -137CONCLUSION In t h is work we s tu d ie d the f o u r - la y e r s in g le - s id e d LIM used f o r p ro p u ls io n and suspension o f m a g n e tic a lly le v it a t e d v e h ic le s . tr a c k is assumed to be made o f a co n d u cto r w ith u n ia x ia l The moving ~μ and σ ~. The source s i t t i n g a t the bottom o f the v e h ic le has a given c u rre n t d i s ­ t r ib u t io n . F o u r ie r tra n sfo rm te ch n iq u e s in c o n ju n c tio n w ith the t r a n s ­ f e r m a trix method are used to s o lv e t h is tw o -dim ensional boundary v alu e problem . The machine perform ance in terms o f fo r c e s , e f f ic ie n c y , and power f a c t o r is given in in t e g r a l form. source Kx ∝ ~ (ky -ky o ) The r e s u lt s f o r the id e a l are then o b ta in e d . component o f the given c u rre n t s o u rc e ). (Here Kx ~ is the F o u rie r In a d d it io n to t h i s , the a n a ly ­ s is o f the in te g ra n d s a llo w s us to exclud e the u n s u ita b le geom etries. The machine performance in t e g r a ls are e va lu a te d f o r the prom ising geom­ e t r ie s and source d is t r ib u t io n The com posite tr a c k seems to show a l o t o f p o t e n t ia l. way o f computing the e f f e c t iv e sketched. µ ~ and σ ~ f o r a com posite t r a ck is From the a n a ly s is o f an e xtre m e ly s im p lif ie d geom etry, the c o n d itio n s f o r the v a l i d i t y o f the e f f e c t iv e g iv e n . A p o s s ib le µ ~ and σ ~ concepts are Using the same a n a ly s is , another p o s s ib le LIM c o n fig u r a tio n is a ls o suggested. F i n a l l y , a th re e -d im e n sio n a l c o r r e c tio n i s in tro d u ce d in o rd e r to take care o f the f i n i t e w idth o f the LIM. F o u rie r tra n sfo rm and F o u rie r s e r ie s techniques are used to s o lv e t h is p e r io d ic ary v a lu e problem . A general a n a ly s is i s g iv e n . 3 -d im e n sio n al bound­ P ro p u ls io n fo rc e and la t e r a l fo rc e s are c a lc u la t e d f o r s e v e ra l s p e c ia l cases. -138REFERENCES 1. J. R. R eitz, et al , "Technical f e a s i b i l i t y of magnetic le v ita tio n as a suspension system fo r HSGT v e h ic le s ", Ford Motor Co., Tech. Rep. FRA-RT-72-40 , produced under Contract DOT-FR-10026 with the U.S. Dept. of T ransportation, Feb. 1972. 2. Η. T. Coffey et a l , "The f e a s i b i l i t y of m agnetically le v it a t in g high speed ground v e h ic le s ", SRI Tech. Rep. DOT-FR-10001, Feb. 1972. 3. Robert H. Bocherts, et a l, "B a se lin e sp e c ific a tio n s fo r a magneti­ c a lly suspended high-speed v e h ic le ", Proc. IEEE 61, 569 (1973). 4. Michel Poloujadoff, "L in e ar induction m achines: I. H is tory and theory o f op eration", IEEE Spectrum 8, 72 (1971). 5. Ib id , Part I I , 6. E. R. Laithw aite and S. A. Nas ar, "Linear-m otion e le c tric a l "A p p lic a tio n s " , IEEE Spectrum 8, 79.(1971 ). machines", Proc. IEEE 58, 531 (1970). 7. E. R. Laithw aite, Induction Machines fo r Special Purposes (London: George Newms, 1966). 8. E. R. Laithw aite and F. T. Barw ell, "A p p lication o f lin e a r induc­ tion motors to high-speed tran sport system s", Proc. IEE 116, 713 (1969). 9. D. F. W ilk ie , "Dynamics, co n tro l, and rid e q u a lity o f a magneti­ c a lly le v ita te d high-speed ground v e h ic le ", Transport Research 1972. 10. E. R. Laithw aite, "The goodness of a machine", Proc. IEE 112, 538 (1965). 11. T. C. Wang, " L i near induction motor fo r high-speed ground tra n s­ p o rta tio n ", IEEE Trans. IGA7, 632 (1971). 12. B. T. Ooi and D. C. White, "T raction and normal forces in the lin e a r induction motor", IEEE Trans. PAS-89, 638 (1970). -13913. T. C. Wang, J. W. Smylie and R. Y. Pei, "S in g le -sid e d lin e a r induction motor", in IEEE Conference Record of 1971 6th Annual Mtg. of the IEEE In d u stry & General A p p lica tio n s Group, p . 1-10. 14. S. A. Nasar, L. Del Cid, J r . , "P rop u lsion and le v it a t o r forces in a sin g le -sid e d lin e a r induction motor fo r high-speed ground tra n s­ p ortation, Proc. IEEE 61, 638 (1973). 15. J. K. Dakowicz, "A n a ly sis of lin e a r induction machines in the d iscre te windings and f in it e iron le n g th ", G.M. research p u b lica ­ tion GMR-1426 (1973). 16. S. Yamamura, H. Ito , and Y. Ishikawa, "Theories of the lin e a r in ­ duction motor and compensated lin e a r induction motor", IEEE Trans. PAS-91, 1700 (1972). 17. Η. H. Kolm and R. D. Thornton, "Electrom agnetic f l i g h t " , S c ie n t if ic American 229, 17 (1973). 18. E. M. Freeman, "T ra v e llin g waves in induction machines; input impedance and equivalent c i r c u i t s " , Proc. IEE 115, 1772 (1968). 19. A. L. Cullen and T. H. Barton, "A sim p lifie d electrom agnetic theory o f the induction motor using the concept of wave impedance", Proc. IEE 105C, 331 (1958). 20. R. M. Fano, L. Chu, and R. A dler, "Electrom agnetic F ie ld s, Energy, and Forces (John Wiley & Sons, In c ., 1960). 21. E. M ishk in, "Theory of the sq u irre l-c a g e induction motor derived d ir e c t ly from M axw ell's f ie ld e q uation s", Quant. J. Mech. A p p l. Math 7, 472 (1954). 22. H. Bondi, K. C. Mukherjee, "An a n a ly sis of tooth rip p le phenomenon in smooth laminated pole shoes", Proc. IEE (London) 104, 349 (1957). 23. R. L . R u sse ll, K. H. Norsworthy, "Eddy currents and wall lo sse s in screened-rotor induction m otors", Proc. IEE 105A, 163 (1958). -140APPENDIX A Let K and K' be the coordinate systems which are at re st with respect to the source and the reaction r a i l , re sp e c tiv e ly. Then, a fte r neglecting the r e l a t i v is t ic e ffe c t and the displacement current, M axw ell's equations w ithin the reaction r a il are given as (A -1a) K: (A -1b) (A -1c) K ': (A -1d) (A -1e) (A -1f) We a lso know: (A-2a) (A-2b) (A-2c) (A-2d) So, f in a lly , we have M axw ell's equations in K : (A-3a) (A-3b) (A-3c) (A-3d) -141This i s the fa m ilia r M inkow ski's form ulation. However, because there i s moving magnetic material present, as fa r as those q u a n titie s such as force d ensity, energy, etc. are concerned, t h is form ulation is not c le a r enough to explain everything. Thus, an a lte rn a tiv e ca lle d E-H form ulation introduced by Chu(20)is preferable, i. e . , we have: (A-4a) (A-4b) with (A-4c) (A-4d) The su b sc rip t "c " is introduced to d is t in g u is h those f ie ld s from the ordinary f ie ld s in M inkow ski's form ulation. A c tu a lly , with a su ita b le transform ation, i t can be found that these two form ulations are equivalent. (A-5a) (A-5b) (A-5c) (A-5d) In the E-H form ulation, the force d ensity can be expressed as: (A-6a) And the input power d ensity i s given as: -142(A-6b) Then the total force and the total power input can be obtained by su ita b le volume in te g ra tio n s w ithin the reaction r a i l . Further inform ation about machine performance can thus a lso be obtained. fo rtu n ate ly, those in te g ra tio n s are tedious. Un­ A sim pler in te g ra tio n can ge n e ra lly be obtained by introducing a stre s s energy tensor T to transform the volume integral into a surface in te g ra l. (A-7a) with (A-7b) I t should be noticed that except at the free space where D = εoE and B = μ o H , t h is stre s s energy tensor is d iffe re n t from the fa m ilia r M inkow ski's tensor. As fo r the power input, i t seems to be much sim pler to evaluate i t d ir e c tly at the source which is located in free space, i . e . , (A-7b) -143APPENDIX B Conditions fo r Thin and Thick Reaction R a ils Here we w in say a reaction r a il is thin when the condition that |γ2h2 | << 1 is s a t is f ie d around a region where the source function is most s ig n ific a n t . S im ila r ly , the condition fo r the thick reaction r a il . is required Now that we know that the source function i s ge n e ra lly concentrated w ithin the region f i r s t consider th is region. |γ2h2 | >> 1 /ν ≥ ky ≥ 0 , we w ill Now (B -1a) (B -1b) With ky = a ω/v , the only minimum of |γ2 | w ill occur at: (B-2a) where (B-2b) And the corresponding minimum value is (B-3a) or (B-3b) -144or (B-3c) Now le t us consider two d iffe re n t cases: then (i ) and (B-4a) then (ii) 4R/27 can be considered as sm all. Thus (B-4b) Thus, the condition fo r |γ2h2 | >> 1 ( i. e . , th ic k reaction r a i l s ) can be rew ritten as: (B-5a) sm aller of S im ila r ly , the condition fo r Iγ 2h2 | < < 1 ( i . e ., thin reaction r a i l s ) is (B-5b) la rg e r of A c tu a lly , fo r the left-handed sid e s of values o f the are ge n e ra lly good enough to give accurate approxima­ order of tio n s. However, in some cases, even r e la t iv e ly relaxed conditions are d if f i c u l t to be met. range of 0 ≤ ky And, conditions cannot be s a t is f ie d fo r the whole v.ω Under t h is s itu a tio n , ge n e rally these conditions / ≤ -145can be fu rth e r relaxed. As we understand, the source function is ge n e ra lly concentrated around consider Iγ2h2 | |γ20h2| << 1 and around ky o . I t seems that we need only to ky o , i. e . , (γ20h2 |. |γ20h2| >> determine whether a reaction 1 And, the conditions can thus be used as the c r it e r ia to r a il i s thin or thick. And d e fin it e ly , those cond itions are much e a sie r to be s a t is f ie d . For example, we can consider a case with √ ω μ2zσx ~ 60 , Then, i t can be shown that even the value o f (B-5b). ω/ν ~ 300 . h2 = 5 χ 10-3 cannot s a t is f y condition However, as we know, in most o f the p ra c tica l cases kyo about 15. Thus, the region fo r which we are most concerned c o rre s­ ponds to ky h2 = 5 χ 10-3 around 15. For those ky , |γ2h2| is is about 0.3 fo r , which is good enough to say that the plate i s thin fo r th is more relaxed c rite r io n . -146APPENDIX C The Evaluation of the In te gral with a complex α Using a su ita b le variab le change, we have (C -1a) with (C -1b) Further, can be rew ritten as: (C -1c) (C-2b) Now we w ill assume Im α > 0 . Then, by the using of Jo rd a n 's lemma and residue theorem, i t can be shown that: (C-3a) (C-3b) (C-3c) So, f in a lly , we have -147(C-4a) or (C-4b) A s im ila r procedure can be used to evaluate the in te gra l with Im α < 0 to get: (C-4c) -148APPENDIX D The Evaluation of the In te gral (D -1 ) with a complex α . can be rew ritten as: (D-2a) (D-2b) Using Appendix C, we have (D-3a) (D-3b) For I 2 , changes of v a ria b le s w ill give us: (D-4a) -149where (D-4b) with In (D.4b), the sin e and cosine in te g ra ls S i (x) and Ci (x) have been used with (D- 5a) (D-5b) I3 is complicated; however, fo r many p ra ctica l cases where i s large , I 3 is r e la t iv e ly small compared to the remaining part of (D.4a). Thus i t can ge n e ra lly be neglected. ky oL -150APPENDIX E The evaluation of the inte gral (E -1) α , β1, β2 are a ll complex. The branch cuts are chosen as shown in Fig . (E .1 ). the d e fin itio n o f -3π/2 < θ1 < π , - π/2 < θ2 < 2π = √ ℓ1 ℓ2 eχρ[ ( i / 2 )(θ 1+ θ2 )], the p o sitiv e n e ss of insured. (Note: and Thus, with √(ky - β1) (ky - β2) Re √(ky - β1)(ky - β2 ) is To obtain the branch cut I, an a rb it ra ry s t r a ig h t lin e is drawn from β1 with 0 < η1 < π/2 . Then another s t r a ig h t lin e can be drawn from β2 with η2 = π/2 - η 1 . These lin e s w ill meet at the point P1. The branch cut I is ju s t the lo c i of P1 . A s im ila r method can be used to get branch cut I I . ) Now, using a change o f v a ria b le , we have: (E-2a) where can be fu rth e r rew ritten as: -151- (E-2b) (E-2c) Here ε is introduced to take care of the pole at F ir s t , le t us consider I + (ε). residues (at iε ky = 0 . By using Jo rd a n 's lemma: and p o ssib ly at α ') (E-3a) (E-3b) I t should be noticed that in te g ra ls along and C2 , ε2 C1 and C1 + C2 . ε2 has been neglected in the contour This is j u s t if ie d because along i s always small compared to k2 y provided βi C1 is not too close to the o r ig in . Applying Jord a n 's lemma again, we have - p o ssib le residue at a' (E-4) -152Now: (E-5a) (E-5b) To derive (E-5b), changes of va ria b le s have been used. A lso , in (E-5b), In (E-5b), fa c to riz a tio n can be applied to the integrand. Thus, we need only consider in te g ra ls and "I" is s t i l l too d if f i c u l t to evaluate. the situ a tio n s we now have, β is large. However, in most of Thus, asymptotic expansions can always give us quite reasonable approxim ations, i. e . , (E-6a) (E-6b) Combining (E-3), (E -4 ), (E -5 ), and (E -6 ), f i n a l l y we have -153- (E-7a) A s im ila r procedure w ill give us: (E-7b) As fo r I o (ε ), we can follow the same procedure as e ith e r I + (ε) or I - (ε ). Now a contour s im ila r to care o f Io (ε ), i.e . I - (ε) is chosen to take (E-8a) ( E- 8b) F a cto riza tio n can be applied to the integrand in (E-8b). Thus, we need only consider in te g ra ls and Now the asymptotic expansion cannot be used any longer. try to evaluate i t exactly. We have to A fte r changing the v a ria b le s, we get: -154- (E-9a) ( E-9b) So, f i n a l l y we have (E-9c) -155Now we can go back to consider those con trib u tion s due to the resid ues. ferent. Of course, depending on where α i s , the r e s u lt w ill be d i f ­ For the case we are consid e ring, branch cuts I and I I are in the regions I and I I I re sp e c tive ly. region I I or IV. And α is found to be e ith e r in Thus, these are the regions we are going to d iscu ss. We can use the same procedure as in Appendix C to get: α in region I I : (E -10a) α in region IV: (E -10b) F in a lly , we can combine (E -3 ), (E -7 ), and (E-10) to give the expression fo r S 3(α , kyo , L). Of course, we can a lso use a contour s im ila r to that fo r I + (ε) to evaluate (E-8 ). Then, although the re s u lts fo r e ith e r the branch cut part or the residue part w ill be somewhat d iffe re n t, the total r e s u lt should be the same. However, in most o f the examples which we are consid ering, |β2 | >> |β1 | a contour s im ila r to that fo r the branch cut n e g lig ib le . cantly sim p lifie d . i s ge n e rally true. The use of I- (ε) w ill make the co ntrib ution due to And thus, fu rth e r analyses can be s i g n i f i ­ -156- Fig. Ε -1 -157APPENDIX F Results fo r Geometry ( A - ii- 2 ) with We apply Appendices C and E to (3.20a) and (3.21a) to get: Here α i. are roots of and As fo r F̄z , i t i s much more complicated. However, the ideal source r e s u lt can be used as a good approximation provided that ky oL i s large.