AN ANALYSIS OF LINEAR INDUCTION MOTORS FOR PROPULSION AND SUSPENSION OF MAGNETICALLY LEVITATED VEHICLES. Fang-Chou Yang. Antenna La b o ra to ry

AN ANALYSIS OF LINEAR INDUCTION MOTORS FOR PROPULSION AND SUSPENSION OF MAGNETICALLY LEVITATED VEHICLES Fang-Chou Yang Antenna La b o ra to ry R eport No. 77

- i i - ACKNOWLEDGMENTS T his work was done under the su p e rv isio n o f P ro fe sso r Charles H. Papas and the cognizance o f Dr. Jay H. H a rris o f NSF.

-i i i - ABSTRACT A fo u r-la y e r s in g le -s id e d LIM used f o r p ro p u lsio n and suspen- sio n o f m a g n e tica lly le v ita te d v e h ic le s is stu d ie d. The tra ck is assumed to be made o f conductors w ith u n ia x ia l ~µ and ~σ. A general a n a ly s is allow s us to exclude u n su ita b le geom etries. machine perform ance is g iven f o r the prom ising g e o m e trie s. The A poss ib le way o f computing the e ffe c t iv e ~µ and ~σ f o r a composite tra c k is sketched. From the a n a ly s is o f an extrem ely s im p lifie d geometry, the c o n d itio n s fo r the v a lid it y o f the e ffe c tiv e ~µ and ~σ concept are given. F in a lly, a three-dim ensional c o rre c tio n is introduced.

- iv - TABLE OF CONTENTS I. INTRODUCTION 1 I I. GENERAL FORMULATION 7 I I I. SEVERAL EXAMPLES 18 IV. COMPOSITE TRACKS 90 V. THREE-DIMENSIONAL CORRECTIONS 110 V I. CONCLUSION 137 REFERENCES 140 APPENDIX A. The Governing E quations 142 APPENDIX B. C onditions f o r Thin and Thick Reaction R a ils 145 APPENDIX C. The E va lu a tio n o f In te g ra l S1(α, ky o, L) 148 APPENDIX D. The E va lu a tio n o f In te g ra l S2 (α, kyo, L) 150 APPENDIX E. The E va lu a tio n o f In te g ra l S3(α, ky o, L) 152 APPENDIX F. R e su lts fo r Geometry ( A - ii- 2 ) w ith 159

-ν - L is t o f Symbols: A v e cto r dyadic F o u rie r component Ā A (c) Ai tim e average value Q u antity given according to Chu's fo rm u latio n i-com ponent o f v e cto r A, i = x, y, z A* complex conjugate o f A ReA ImA re a l p a rt o f A Imaginary p a rt o f A a = = in chapters I I I, IV w idth o f the source in the x -d ir e c tio n B d E magnetic in d u ctio n vecto r skin depth (see 4.5c) e le c t r ic f ie ld in te n s ity vecto r u n it v e cto r along i- a x is. i = x, y, z F H hk h fo rce vecto r magnetic f ie ld in te n s ity vecto r th ickness o f region k, k = 1, 2, 3, 4 th ickness o f each lam ination id e n t it y m a trix J K k kyo volume cu rre n t d e n sity su rfa ce cu rre n t d e n sity wave number vecto r wave number o f the t r a v e llin g wave source

- v i - L P Pmech PLoss PF Tk Ti j t i j v h a lf length o f the s ta to r covered by the t r a v e llin g wave source. power in p u t m echanical power induced cu rre n t lo ss machine power fa c to r tra n s fe r m a trix fo r region k as defin ed in (2.8 ), k = 2, 3 i j component o f the s tre s s energy te n so r, i, j = x, y, z i j component o f T3T2 as defined in (2.1 3 ), i, j = 1, 2 v e lo c ity o f the tra ck w ith respect to the v e h ic le, g e n e ra lly in the y - d ir e c tio n. Wo ~Zs ~ZL βk γ k σki µki w idth o f the re a ctio n r a il in the x -d ir e c tio n source w in din g impedance in p u t impedance o f the machine wave number o f region k as defined in (2.1 2 ), k = 1, 2, 3, 4 wave number o f region k as defined in (2.8 ), k = 1, 2, 3, 4 i i component o f the c o n d u c tiv ity o f region k, k = 2, i = x, y, z i i component o f the p e rm e a b ility o f region k, k = 1, 2, 3, 4, i = x, y, z

-1 - CHAPTER I INTRODUCTION With the in c re a sin g need in t r a f f ic o f people and com m odities, autom obiles have created serio u s p o llu tio n and n oise problem s; the a ir lin e s have a ls o reached a high degree o f s a tu ra tio n. Under the c i r - cumstances, i t seems th a t the development o f a ra p id mass tra n s p o rta - tio n system is d e s ira b le. U n fo rtu n a te ly, the conventional wheel - supported ra ilw a y s face some d if f ic u lt ie s as the t r a v e llin g speed becomes la rg e. Not o n ly does the n o ise become in t o le r a b le, but the sm all f r ic t io n c o e f f ic ie n t a ls o in h ib it s the tra n sm issio n o f power between the r o t a t ing-type motor and the w heels. As a r e s u lt, such non-contact systems as m agnetic le v ita te d (MAGLEV), tra c k a ir cushion v e h ic le s (TACV) are suggested as s u b s titu tio n s f o r fu tu re tra n sp o rta - tio n. Although the TACV is s t i l l a co m p e titiv e candidate fo r high speed ground tra n s p o rta tio n (HSGT), in the fo llo w in g we w ill m ainly co n sid e r the MAGLEV system. There are two fundamental schemes f o r magnetic le v it a t io n, namely, a t t r a c t iv e and re 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 rce between a magnetic f ie ld source and a ferrom ag- n e tic m a te ria l. From Earnshaw's theorem, th is kind o f c o n fig u ra tio n is b a s ic a lly u n sta b le, i. e., a sm a lle r cleara n ce w ill in crease the a t t r a c t iv e fo rce and make the clearance even sm a lle r. Thus, a feedback co n tro l system is necessary. For the re p u ls iv e scheme, the fo rce exerted on a m agnetic f ie ld source moving over a conductor by the

-2 - f ie ld o f the induced eddy cu rre n t w ith in the conductor is employed. Such a system is in h e re n tly s ta b le. E xtensive stu d ie s o f these suspension systems have been made(1,2). D e fin it e ly, the d e ta ils w ill depend on source and tra ck geometry. C e rta in general r e s u lts w ill be described below. And act u a lly, up to today, there is s t i l l no c le a r evidence as to which one is the b e tte r one. Both are n o is e le s s, not lim ite d by speed, and can be operated in vacuum. o f a whole HSGT v e h ic le. In r e a lit y the suspension system is o n ly p a rt Thus, whether a system is optim ized o r not can o n ly be judged by an a lyzin g the whole v e h ic le from the tra d e o ff among the perform ances, te ch n ica l m e rits, economic c o n d itio n s, e tc. G e n e ra lly speaking, in o rd er to have a s u f f ic ie n t ly supporting fo rc e, the a t t r a c t iv e scheme can o n ly a llo w a sm all clearance o f about 1 o r 2 cm, but the re p u ls iv e scheme can have a clearance o f 20 cm. T his w ill a ls o p lay an im portant ro le in d e cid in g what kinds o f pro- p u lsio n systems to use. g e n e ra lly conductors. A ls o, as we know, ferrom agnetic m a te ria ls are Thus, when the magnetic f ie ld source is moving over i t, there always are eddy cu rre n ts induced. T his w ill g ive an undesired re p u ls iv e fo rce component. When the v e lo c it y becomes la rg e enough, sometimes th is w ill r e s u lt in a net re p u ls iv e fo rc e. Of course, we cannot a llo w th is to happen. Thus, tra cks and sources must be s u ita b ly designed to prevent th is phenomenon from happening w ith in the o p e ra tin g speed range. For the re p u ls iv e scheme, sin ce i t m ainly depends on the eddy cu rre n ts induced, eddy cu rre n t lo sse s are in e v i- ta b le. These w ill give an a d d itio n a l drag fo rce. An increased

-3 - p ro p u lsiv e fo rce is necessary to overcome i t. According to form er a n a lyse s, i t is found th a t a t high speed th is drag fo rce w ill decay to a very sm all v alu e. a t a very sm all speed. However, a very la rg e drag fo rce w ill e x is t C e rta in methods(3) have been suggested to bypass t h is low speed drag. Since the proposed suspension system is a non-contact one, the conventional method o f using the d ir e c t re a c tio n w ith the ground is not a v a ila b le. Thus, a new p ro p u lsio n system is needed. In the e a rly p e rio d, j e t s, p ro p e lle rs and rockets have been suggested. However, p o l- lu tio n and n o ise 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 centu ry and the e a rly 20th cen tu ry, people have t r ie d to apply i t in t r a in tra n s p o rta tio n, luggage h a n d lin g, e t c. ( 4, 5 ). However, due to economic and p r a c tic a l reasons, the in te r e s t in lin e a r machines dec lin e d. R ecently as L a ith w a ite ( 6, 7, 8) and P o lo u ja d o ff( 4, 5) put i t, "engineering fa sh io n s" have been changing. Due to L a ith w a ite 's continuous e ffo r t s in promoting 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, impact e x tru s io n, E.M. pumps, a c tu a to rs, e tc. Now, i f the r e p u ls iv e suspension scheme is used in HSGT, because o f it s high clearance perhaps the lin e a r synchronous(9) motor is p re fe ra b le. A ls o, power-pickup and w eight problems are more e a s ily handled by the lin e a r synchronous motor. However, in the fo llo w in g d iscu ssio n s we w il l m ainly co n sid e r lin e a r in d u ctio n m otors. L in e a r synchronous motors w ill not be in clu d e d.

-4 - The p r in c ip le o f the lin e a r in d u ctio n motor can be e a s ily understood from the conventional ro ta tin g in d u ctio n motor. A c tu a lly, the sim p le st form o f LIM is ju s t an u n ro lle d r o ta tin g in d u ctio n motor. A number o f a lt e r n a t iv e forms o f LIMs se rv in g d iffe r e n t purposes can be co n stru cted by ju 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 asar(6) have given a complete c la s s if ic a - tio n in terms o f the d iffe r e n t p o ssib le to p o lo g ic a l c o n fig u ra tio n s. A ls o, a "m achine's good f a c t o r " (10) and a co n sid e ra tio n o f the economic problems were introduced to decide which kind o f LIM is p re fe r a b le. Among them, the "d ouble-sid e d sh o rt prim ary sheet ro to r motor" (DSLIM)( 8, 11, 1 5, 16) seems to be the best candidate f o r HSGT. A c tu a lly, the DSLIM had a lre a d y been used in prototype HSGT v e h ic le s b u ilt in Germany, the U.S., e tc. But those v e r t ic a lly b u ilt DSLIMs su ffe re d from several severe problem s, such as la te r a l in s t a b ilit y in cu rve s, high cross se ctio 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 (SLIM )(1 2, 1 3, 14) was in troduced. Although the a n a ly- ses o f these conventional ro ta tin g motors are w ell known and are not too d i f f i c u l t, the a n a ly s is o f LIMs is much more com plicated, due to the in e v ita b le end e ffe c t and the dissymmetry o f prim ary cu rre n ts. Those e xtra e ffe c ts w ill in troduce u n d e sirable phenomena. Today, re search in LIMs concentrates on e x p la in in g these e ffe c ts and lo o k in g fo r methods to compensate them. Yamamura and I t o (1 6 ), Wang(1 1 ), and Dukowicz(15) considered DSLIM w ith non-ferrom agnetic sheet ro to rs. Nasar and Del C id, J r. (14) and Wang(13) discussed SLIMs w ith non- ferrom agnetic sheet ro to rs backed by s te e l.

- 5 - However, no m atter which LIM was co n sid e re d, up to now a ll o f the analyses were made under the assumption o f separated suspension and p ro p u lsio n systems. Although an experim ental v e h ic le has a lre a d y been co n stru cted by Rohr Inc.(17), no serio u s a n a ly s is has been made to co n sid e r a v e h ic le w ith a LIM used f o r both suspension and propulsio n purposes. Of course, th is would in tro d u ce e x tra c o n tro l problem s, but i t a ls o o ffe r s advantages such as le s s w eight. The SLIM analyzed by Nasar and Del C id, J r. (1 4 ), Wang(1 3 ), e tc. w ill g ive e it h e r an a t t r a c t iv e o r a re p u ls iv e fo rc 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 fo r both le v it a t io n and p ro p u lsio n, a deeper a n a ly s is must be made as to which 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 re p u ls iv e fo rce is observed w ith in the o p e ra tin g speed range. The ferrom agnetic tra ck was ru le d out as u n su ita b le f o r DSLIM by L a ith w a ite and B a rw e ll(8) because o f sk in and n o n lin e a r e ffe c ts. These auth o rs, however, were o n ly concerned about the th ru s t fo rc e. As we want to in clu d e a le v it a t in g fo rc e, the in h e re n t fo rce between the magnetic f ie ld source and ferrom agnetic m a te ria l has to be reconsidered s e r io u s ly. Furtherm ore, the undesired skin e ffe c t which g ives r is e to n o n lin e a r e ffe c ts can a lso be p a r tly taken care o f by s u ita b ly la m in a t- ing the ferrom agnetic m a te ria l o r composing the tra ck both from f e r r o magnetic and non-ferrom agnetic m a te ria l. The fo llo w in g d iscu ssio n w ill begin w ith an in v e s tig a tio n o f the p o s s ib ilit y o f using a lin e a r in d u ctio n motor as a combined p ro p u lsio n

-6- and supporting system in a HSGT v e h ic le. L in e a r machines o f fo u r la y e rs w ill be used as the s ta r tin g p o in t o f the a n a ly s is. Most o f the proposed c o n fig u ra tio n s can be d e scribed by s u ita b ly s p e c ify in g those parameters as c o n d u c tiv ity, p e rm e a b ility, la y e r th ic k n e s s, e tc. f o r d iffe r e n t re g io n s. The a n a ly s is o f machine performance w ill exclude u n su ita b le geom etries and g iv e p o s s ib le methods to reduce the undesired end e ffe c ts. Expressions f o r machine performances w ill then be evaluated f o r those prom ising geom etries. A p o s s ib le way o f computing the e ffe c t iv e c o n d u c tiv ity and per- m e a b ility f o r a composite tra c k w ill be sketched. Up to now these parameters were determ ined e x p e rim e n ta lly, and no th e o re tic a l a n a ly s is has ever been attem pted. D e f in it e ly, i t is d i f f i c u l t. An extrem ely s im p lifie d geometry w ill be considered to check the v a lid it y o f t h is e ffe c t iv e c o n d u c tiv ity and p e rm e a b ility concept. co n d itio n s f o r the v a lid it y w ill then be g iv e n. P o s sib le necessary S ta rtin g from t h is p o in t, another p o s s ib le c o n fig u ra tio n w ill be suggested and analyzed. The tra n s fe r m a trix method is employed to so lv e the 2-dim ensional problem. A t the end o f th is d is c u s s io n, a 3-dim ensional c o rre c tio n w ill be in tro d u ce d. Of course, the p re v io u s ly n o n e xiste n t la t e r a l fo rce w ill now appear. A corresponding q u a n tita tiv e approxim ation w ill then be given.

-7 - CHAPTER II GENERAL FORMULATION The lin e a r in d u ctio n motors which w ill be considered c o n s is t o f fo u r la y e rs 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 is connected) is lo cated a t the in te rfa c e o f regions 3 and 4. Region 3 is the fre e space. Regions 1 and 4 are any zero c o n d u c tiv ity m a te ria l, such as the in f i n i t e l y lam inated iro n o r fre e space. Regions 1 and 2 which make up th e tra c k are moving a t a constant v e lo c it y v r e la t iv e to the source. The m a te ria l p ro p e rtie s o f region 2 in i t s r e s t frame are a r b it r a r ily s p e c i fie d by constant c o n d u c tiv ity and constant p e rm e a b ility u n ia x ia l tensors r e s p e c tiv e ly. These constants can g e n e ra lly d e scrib e the m a te ria l used in LIMs s a t is f a c t o r ily. M axw ell's equations in v o lv in g moving magnetic m a te ria l are comp lic a te d. However, in th is case, n e g le ctin g the displacem ent cu rre n t and r e la t i v i s t i c e ffe c ts is always a good approxim ation. In the re s t frame o f the source, M axw ell's equations can then be s im p lifie d to g ive us (see Appendix A): (2.1a) (2.1b) (2.1c) ( 2.1d) o r, fo r a n o n sin g u la r ~σ ( i. e., region 2)

-8- F ig. 2.1 F ig. 2.2

-9- (2.2a) (2.2b) (2.2c) (2.2d) and, f o r ~σ = 0 ( i. e., regions 1,3,4 ) (2.3a) (2.3b) (2.3c) (2.3d) With a e- i ωt time dependence, a F o u rie r transform p a ir, given in (2.4a) and (2.4 b ), can be introduced to so lve the d if f e r e n t ia l equatio n s : (2.4a) (2.4b) Then, by a p p ly in g s u ita 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 tio n. However, g e n e ra lly th 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 is in f in it e ly extended along the x -a x is. Thus, x-dependence can be suppressed. The 3-dim ensional c o r re c tio n fo r some sim ple geom etries w ill be made la t e r in Chapter V.

-10- For t h is 2-dim ensional problem w ith the co o rd in ate 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 tr a n s fe r m a trix method can be used to fin d the f ie ld. Region 1: A c tu a lly we have: (2.5) w ith (2.6) Regions 2,3: (2.7) w ith (2.8) Region 4: (2.9) w ith (2.10) A ls o, in the above

-11- (2.11) (2.12) γ k are s u ita b ly chosen such th a t they contain o n ly p o s itiv e re a l p a rts. In these form ulas the boundary c o n d itio n s can then be in tro d u ce d. The f ie ld d is t r ib u t io n in the whole space can be obtained. Now the case w ith a s p e c ifie d c u rre n t source w ill be considered ( i. e., s e rie s connected source): The c o n tin u ity c o n d itio n s o f give (2.13a) (2.13b) Here ε is p o s itiv e but sm a ll, t i j = (T3( z 3 ) Τ2 (z2 ) ] i j are the e le ments o f T3(z 3 ) T2 (z2 ). The boundary co n d itio n s at z 3 g ive (2.14a) (2.14b) (2.14c) is the F o u rie r component o f the s p e c ifie d cu rre n t d is t r ib u t io n. Combine (2.1 3 ), (2.14) to get:

-12- (2.15a) (2.15b) The F o u rie r components o f the f ie ld d is t r ib u t io n can be obtained from equations (2.5) to (2.12). The f ie ld d is t r ib u t io n in real space can then be obtained by using the F o u rie r in verse transform (2.4b). From the above d e riv a tio n, i t is obvious th a t the problem can a ls o be solved by ap p lyin g tra n sm issio n lin e th eory. With the f ie ld components se rv in g as the voltage and the cu rre n t r e s p e c tiv 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 ed. A fterw ard the f ie ld d is t r ib u t io n can be obtained by s u ita b ly matching the in p u t impedance o r by using the Smith c h a rt. Once we know the f ie ld d is t r ib u t io n, we can evalu ate the machine perform ances. Among them, the fo rc e s, e ffic ie n c y, power in p u t, power fa c to r are the most im portant. As d escribed by Fano, Chu, A d le r (20) in c a lc u la tin g fo rce s in v o lv in g m agnetic m a te ria l, the Minkowski fo rm u la tio n which we have used so f a r cannot g ive s a tis fa c to r y r e s u lts. Thus, an a lte r n a tiv e c a lle d Chu's fo rm u la tio n is in tro d u ce d. By using th is new fo rm u la tio n, the fo rce s w ith in the magnetic m a te ria l can be c le a r ly e xp la in e d. D e ta ils concerning the fo rce s e x is tin g in the given LIM systems are in clu d e d in Appendix A. In our problem, the force which has to be c a lc u la te d is ju s t the to ta l fo rce exerted on the source, which by the p r in c ip le o f re a c tio n, is ju s t the to ta l fo rce a ctin g on the combined regions 1 and 2. According to Chu's fo rm u la tio n, th is fo rce can be obtained sim ply

-13- by in te g ra tin g the M axw ell's s tre s s energy te n so r over the su rfa ce in the fre e space ju s t o u tsid e o f region 2, i. e., (2.16a) w ith (2.16b) F o rtu n a te ly, in the fre e space, the M axw ell's s tre s s energy tensors fo r Chu's and M inkow ski's fo rm u la tio n s happen to be the same. Furtherm ore, in the fre e space, the e le c t r ic p a rt o f the s tre s s energy te n so r is always sm all compared to the m agnetic p a rt. D e f in it e ly, i t can be n e g le cte d, and a c tu a lly th is is e q u iv a le n t to n e g le ctin g the d is placement cu rre n t which was done a t the very beginning. Thus, the c a lc u la tio n o f fo rce s can be s im p lifie d. F in a lly, the time averaged to ta l fo rc e s are given by: w ith (2.17) (2.18) Thus, fo r the 2-dim ensional case: (2.19) (2.20a) (2.20b)

-14- (2.21a) (2.21b) where the F i represent the fo rce s a c tin g on a u n it length o f source in the x d ir e c tio n. Combine (2.2 0 ), (2.2 1 ), (2.15) and (2.7) to get the fo rce s f o r the case o f a s p e c ifie d c u rre n t source: (2.22) (2.23) where are the elements o f m a trix T2 (z2 ). Sometimes i t is e a s ie r to evaluate the s tre s s energy tensor in te g ra l a t the su rfa ce z 3- ε. Then we get: (2.24) (2.25) The average m echanical power in p u t is :

-15- (2.26) The average power in p u t is : (2.27a) (2.27b) (2.27c) (2.27d) Power fa c to r is given by: (2.28) E ffic ie n c y is defin ed as: (2.29) F in a lly, the induced eddy cu rre n t lo ss is found from: (2.30a) or By in sp e ctio n o f the above form ulas, an e q u iv a le n t c ir c u it can be constructed as shown in F ig. 2.2, w ith (2.30b)

-16- and thus P can be r e w r itte n as: Now ~ZLcan be decomposed in t o th re e p a rts: The term co n trib u te s to the mechanical power, w h ile g ives the induced eddy cu rre n t lo s s. Thus, in order to have a high e ffic ie n c y o r a low energy lo s s, is require d to be concentrated around kyo such th a t is s m a ll. The im aginary p a rt Im ~ZL represents the d iffe re n c e between the average stored e le c t r ic energy and magnetic energy. However, as we understand i t, in t h is case the stored e le c t r ic energy is r e la t iv e ly sm all compared to the stored magnetic energy. So the term Im ~ZL w ill m ainly 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 cu rre n t d is trib u tio n a t z 3 is analyzed. B a s ic a lly, th is is the case o f the so- c a lle d se ries-co n n ecte d source. The case o f a paral l el -connected source w ith a s p e c ifie d B f ie ld 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 in f in it e ly extended such th a t i t can be described as a t r a v e llin g wave, i t does not make any d iffe re n c e whether the source is s e rie s o r p a r a lle l connected. q u ite d iffe r e n t fo r a more r e a lis t ic source. However, t h e ir behavior is We do not intend to

-17- co n sid e r the p a ra lle l-c o n n e c te d case here. T his completes the general a n a ly s is. In the next c ha p te r we sha l l s t a r t from t h is general fo rm u la tio n and co n sid e r several examples.

-18- CHAPTER III SEVERAL EXAMPLES Up to now, the source c ir c u it has been neglected in a l l o f the analyses. But, in the machine d e sig n, there is always a source w inding re s is ta n c e ~Zs. And no m atter what kind o f source connection is used, i t is q u ite c le a r th a t ~ZL >> ~Zs around kyo is necessary to make the source w inding lo s s sm a ll. S u re ly, under c e rta in circu m stances, the value o f β4 0 ( i. e., using in f in it e ly lam inated iro n in region 4 as assumed in n e a rly a ll o f the LIMs considered in p re v io u s ly publis h e d papers) w ill achieve t h is. But th is a ls o means a d d itio n a l w eight to the vehic le s. Thus, o t he r lig h t e r m a te ria l w ith a d iffe r e n t ~µ can a lso be used. As pointed out p re v io u s ly, the LIMs are supposed to be used fo r both suspension and p ro p u lsio n. Thus, i f around kyo is require d in (2.2 5 ) to give a s u f f ic ie n t a ttra c tiv e supporting fo rc e, β4 0 is suggested. On the o th e r hand, i f around kyo is required in (2.25) to give a s u f f ic ie n t re p u ls iv e supporting fo rc e, then the e ffe c t o f can be neglected. D e f in it e ly, fo r the general case, the tra d e o f f among the w eight, co st o f m a te ria l in regions 3, 4, and the source w inding lo s s, w ill determine what value o f β4 is s u p e rio r in the machine design. But, u n fo rtu n a te ly, the a r b itr a r in e s s o f β4w ill make the F o u rie r in verse in te g ra ls much more com plicated to e valu a te. Thus, in the fo llo w in g, only two extreme cases o f β4 0 fo r the in f in it e ly lam inated iro n and β4 = i / μo f o r the fre e space w ill be d iscu sse d ; and o the r values o f β4 ly in g between them w ill be guessed

-19- to give in te rm ediate r e s u lts. The same assumptions can be made fo r β1. However, because region 1 is not on the v e h ic le, w eight is not a se rio u s problem. In r e a lit y, the e arth (which can be considered as a fre e space) w ith β1 = i / μ o, and in f in it e ly lam inated back iro n w ith β1 0, are most p r a c tic a l and w ill be considered in the fo llo w in g as the o n ly p o s s ib ilit ie s. A n a ly t ic a lly e v a lu a tin g the in te g r a ls (2.24) to (2.30) is very d i f f i c u l t, i f not im p o ssib le. Only two extreme approxim ations, namely, th in and th ic k re a ctio n r a ils, w ill be considered. That a r e a ctio n r a il is th in o r th ic k is defin ed by γ 2h2 << 1 o r γ2h2 >> 1 around ky o where ~Κx 2 is l arge. The d e ta ils about when those approxim ations are reasonable in d e sc rib in g the LIMs are given in Appendix B. Under these assum ptions, s im p lific a t io n s can d e f in it e ly be made in (2.24) to (2.30) to evaluate the performance o f the LIMs. However, in ste a d o f doing th is im m ediately, we w ill f i r s t t r y to analyze the in tegrands. I t w ill be found th a t th is is a convenient way o f lo o k in g in to t he system -optim izin g problem and the compensation fo r end e ffe c ts. F ir s t, le t us co n sid er the case w ith β1 0. (A) β 1 0 : (region 1 is in f in it e ly lam inated iro n ) From (2.8 ), (2.1 3 ), (2.31): (3.1)

-20- (3.2) (3.3) Here, kz = ky. And ~Fy, ~Fz are the integrands fo r the fo rce in te g r a ls Fy, Fz re s p e c tiv e ly. h2, h3 are re s p e c tiv e ly the th ickness o f regions 2 and 3. I t is not 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 itiv e re a l p a rts. And, t h e ir im aginary p arts w ill have the same s ig n s as those o f γ2. Thus, fo r a given source cu rre n t d is t r ib u t io n, a nonzero value o f α w ill r e s u lt in sm a lle r to ta l power in p u t and fo rc e s, and thus degrade machine performance. Of course, as pointed out b e fo re, the e ffe c t o f th is term α w ill stro n g ly depend on the r e la t iv e magnitude o f P1 and P2. Now, the th in and t h ick re a c tio n r a il assumption w ill be used to s im p lif y the problem.

-21- (i ) Thin re a ctio n r a i l s : Since γ2h2 is sm a ll, the approxim ation th a t tanh γ2h2 γ2h2 can be made. A ls o, fo r L IMs used in HSGT, h3 has to be sm all, i. e., tanh kz h3 kz h3 is a lso a reasonable approxim ation. We can now begin to co n sid e r d iffe r e n t kinds o f LIMs. ( i- 1 ) α 0: (region 4 is in f in it e ly lam inated iro n ) This is the most popular SLIM considered by Nasar, Del C id, J r. (1 4 ), Wang(1 3 ), e tc. The re su lt can a lso be d ir e c t ly a p p lie d to o b ta in the performance o f DSLIM analyzed by Yamamura and I t o (1 6 ), Wang(1 1 ), and Dukowicz(1 5 ). From (3.1 ), (3.2 ), (3.3 ), we get: (3.4a) (3.4b) (3.4c) (3.5a) (3.5b) (3.5c)

-22- (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, except in the case th a t h2 >> h3, which is g e n e ra lly not tru e in t h is th in re a c tio n r a il geome try, the e ffe c t o f the p e rm e a b ility is sm a ll. A fte r some tedious a lg e b ra ic m a n ip u la tio n s, are p lo tte d as fu n ctio n s o f ky in F ig. 3.1. With "a" defin ed as kyv/ ω, i t is found th a t a t A ls o, have extrema a t "a" equal to ay, ar, ai s a tis fy in g equations given below, re s p e c tiv e ly : (3.7a) (3.7b) (3.7c) From these p lo ts, i t is q u ite obvious th a t d iffe r e n t arrangements o f ~Kx 2 can give machines which can be used fo r d iffe r e n t purposes, ( e.g., braking o r p ro p u lsio n in HSGT, o r even o th e r than HSGT system s). However, LIMs used f o r both p ropulsion and le v it a t io n are our o n ly concern now.

-23- F ig. 3.1 Integrands fo r geometry (A -i-1 ) w ith (1) Re (2) Im (3) (4)

-24- In o rd e r to get a net p ro p u lsio n fo rc e, kyo = ao ω/v is genera lly re q u ire d to s a t is f y 1 > ao > 0. Thus, in the fo llo w in g, o n ly t h is region w ill be considered. Now, we w ill look more c a r e f u lly in to equations (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 ill o n ly be one value o f "a" s a t is fy in g each o f the equations (3.7) f o r 1 a 0. And, fo r t hose ro o ts, ai > ay > ar are always tru e. A ls o, ay > az can be obtained fo r th is t hin re a ctio n r a il case. From (3.7 a ), ay is found to be a m o n o to n ically in c re a sin g fu n ctio n o f K1 v 2 /ω. Whether the machine has a re p u ls iv e o r an a t t r a c t iv e le v it a t io n fo rce depends m ainly on whether ao is sm a lle r o r la rg e r than az. N e ve rth e le ss, a sm all ao < az g e n e ra lly means a la r g e r energy lo ss and thus a sm a lle r e ffic ie n c y. Since ay > az is always tru e, a sm all ao < az a lso means a sm all th ru s t fo rc e. Furtherm ore, i t is n o tice d th a t the maximum value o f p o s itiv e ~Fz/ ~Κx 2 is always much la rg e r than th a t o f re p u ls iv e ~Fy/ ~Κx, ( a c tu a lly the r a tio is about So, th is kin d o f LIM c o n fig u ra tio n is not good fo r use as a re p u ls iv e le v it a t io n system. Thus, i t is p re fe ra b le to use a source w ith 1 > ao > az such th a t a net a t t r a c t iv e and a la rg e p ro p u lsiv e fo rce can be obtained. Two methods are suggested to operate t he LIMs such t ha t t he above requirem ents o f 1 > ao > az can be met: (1) F ix in g ky o, then try in g to use d iffe r e n t ω fo r d if f e r ent speed ranges. This is the method most people suggest. And, th u s, a frequency co n verte r is needed.

-25- (2) F ix in g ω, then t r y in g to vary kyo as ν c hanges. T his is s im ila r to an antenna array to c hange t he d ir e c t iv it y by s u it ably arranging the cu 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 te g r a ls fo r performance are easy to e valu a te. (Of course, we cannot co n sid e r t he in f in it e to ta l fo rce s and power any lo n g e r. But q u a n titie s per u n it le n g th in the y - d ir e c - tio n are not d i f f i c u l t to d e riv e ). The fo llo w in g r e s u lts can be obta in e d ; (3.8a) (3.8b) (3.8c) (3.8d) (3.8e) Equations (3.8) are p lo tte d in F ig. 3.2 as fu n ctio n s o f ν. The maximum th ru s t fo rce w ill occur a t w ith a c o r responding maximum value o f 1/4 ky o μoμ2z/ (μ2zh3 + μoh2 ). Both the

-26- F ig. 3.2 Power and fo rce s fo r geometry ( A - i- 1) w ith id e a l source and ωκ1 = 2000, ωμοσxh2 = 5, kyo = 1 0. (1) (2) (3) (4)

-27- supporting fo rce and the e ffic ie n c y are m o n o to n ica lly in c re a sin g fu n c tio n s o f ν. Thus, in o rd er to have a high e ffic ie n c y, i t is suggested th a t the machine be operated in the a t t r a c t iv e su p p ortin g fo rce region. At vm, a la rg e r e ffic ie n c y can be obtained by making k2yo/ ωk1 sm a lle r. U n fo rtu n a te ly, a p u re ly tr a v e lin g wave source does not e x is t in nature. G e n e ra lly, we can o n ly have a f in it e source w ith a ~Κx 2 which is an o s c illa t in g fu n ctio n w ith a f a s t decaying envelop and a maximum value a t ky o. With the given in fo rm atio n about the in tegrands, i t is not d i f f i c u l t to understand how th is s o -c a lle d end e ffe c t comes in to p la y. We w ill e x p la in th is by assuming the source has ( i. e., a source w ith f o r y L, and Κx = 0 o t he rw ise. This is t he most popular uncompensated source used by n e a rly a l l o f t he re s e a rc h e rs.) Now, le t us go back to F ig. 3.1, where a ty p ic a l p lo t fo r is shown. When L becomes in f in it e ly la rg e, becomes p ro p o rtio n a l to δ(ky - ky o ). The in te g ra l fo r Fy w ill ju s t p ick up the value o f ~Fy a t ky o, and the id e a l source r e s u lt is obtained. However, fo r the more r e a lis t ic case where L is f in it e, the s itu a tio n is d iffe r e n t. I t is obvious th a t ~Kx 2 w ill now have non-zero values a t ky ky o. Thus, t he in te g r a ls w ill a lso re ce ive a c o n trib u tio n from t h is region. And t h is e x p la in s how th is s o -c a lle d end e ffe c t comes in to p la y. D e fin it e ly, the o v e ra ll r e s u lt depends

-28- d r a s t ic a lly on how fu n ctio n s ~Fy / ~Kx 2, ~Kx 2 vary. By observing the ~Fy/ ~Kx 2 curve more c a r e f u lly, i t can be seen th a t there is o n ly one lo c a l maximum lo ca te d a t ky = ay ω/v. Thus, i f those p o in ts ky o, kyo ± π/l, are not too clo se to ay ω/v, then although p a rt o f ~Kx 2 w ill spread in to t he region where ~Fy/ ~Kx 2 is s m a lle r than th a t a t ky o, there is always anothe r p a rt whi c h w ill go in to the la rg e r ~Fy/ ~Kx 2 re g io n. So the o v e ra ll e ffe c t w ill be sm a ll, i. e., the end e ffe c t causes l i t t l e in flu e n c e. On the co n tra ry, i f kyo happens to be equal to ay ω/v, those n o n-zero ~Kx 2 a t ky kyo w ill always p ick up sm a lle r ~Fy/ ~Kx 2. Thus, the re s u lta n t fo rce w ill decrease enorm ously, e s p e c ia lly when L is sm all such th a t the spread is w ide. And th is is ju s t the s itu a tio n in which the end e ffe c t degrades the machine performances most. T his a lso e x p la in s why a la rg e L is suggested to reduce the undesired end e ffe c t. As fo r the case where kyo is near ay ω/v, although there is a p o s s ib ilit y th a t ~Kx 2 w ill go to the region where ~Fy/ ~Kx 2 is la r g e r, the in - crease is always sm all compared to the decrease coming from the spreading o f ~Kx 2 in to the opposite d ir e c tio n. Thus, although the d e ta il w ill depend on how ~Fy/ ~Kx 2 v a rie s, a decrease in the re s u lta n t fo rce can g e n e ra lly be observed. In the above we have m ainly ta lk e d about the in flu e n c e o f the source fu n c tio n. Now we w ill a ls o say something about the o th e r parame te rs. I t is known th a t a s m a lle r K1ν2 /ω w ill g ive a sm a lle r ay. Now, i f there are two cases w ith the same ω/ν but d iffe r e n t ay, then roughly speaking, the slope to the r ig h t o f ay w ill be steeper

-29- fo r the case w ith la rg e r ay. Thus, i f we have two systems w ith d i f fe re n t K1 o p e ra tin g a t the same ω, ν, then we are g o in g to get a more se rio u s end e ffe c t fo r the case w ith la r g e r K1 when ky o 's are se t to t h e ir corresponding values o f ay ω/v. (Remember, a la rg e r ao is suggested to g ive a hig h e r e ffic ie n c y. And the above argument is o n ly a p p lie d to the case where ay = ao > ½). So g e n e ra lly, we are expecting a s m a lle r end e ffe c t fo r a system which uses a tra ck w ith a sm all σxh2. A ls o, as we suggested befo re, the machine can a lso be used f o r supporting purposes. In th is case, i t is p re fe ra b le to ope ra te in the a t t r a c t iv e re g io n. Thus, i t is more d e s ira b le to make σxh2 sm a ll. T his happens to be c o n s iste n t w it h the sm all end e ffe c t requirem ent. So, i t seems th a t composite tra cks whi ch w ill g ive a r e la t iv e l y s m a lle r σx are prom ising. Note, fo r p r a c tic a l m a te ria l suggested in MAGLEV system s, nonferrom agnetic m a te ria l g e n e ra lly has a hig h e r c o n d u c tiv ity. Thus, although m a te ria l w ith high ~µ does not o f f e r too many advantages; composite tra c k s w ith ferrom agnetic m a te ria l contained are s t i l l recommended f o r use in t h is s p e c ia l case. We w ill postpone our d is cussion about how to co n stru ct the composite tra cks u n t il Chapter IV. However, i t sh o u ld a ls o be remembered th a t K1 cannot be made too low, otherw ise ~Fy/ ~Kx 2 w ill become too sm all in the low s lip regio n. T his w ill e v e n tu a lly decrease the fo rce 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 co n sid e rin g, one th in g seems w orthw hile m entioning. With a l l o f the s p e c ifie d source and tra ck c o n fig u ra tio n s, we know th a t the maximum fo rce is going to occur at

- 3 0 - fo r the ideal source. However, we also know that fo r th is νm the maximum value o f ~Fy/ ~Kx 2 is going to appear at ayω/ vm which is less than kyo. Thus, the maximum thrust force point is not the point where the end e ffe ct is most serious. Later, th is can be shown to be d iffe re n t from some o f the other configurations. In the above, we considered mainly the end e ffe ct on the force in the y -d ire ctio n. S im ila r arguments can also be applied to F z, η, PF. Since, fo r a given ν, the integrand fo r the loss integral is monotonically decreasing fo r 0 ky ω/v, so the end e ffe ct w ill not introduce too much influence to the loss. Thus, as fo r the e ffic ie n c y, the end e ffe ct w ill have s im ila r influence to that of F y, i. e., i t w ill generally degrade the e ffic ie n c y. For most of ky, ~Fz/ ~Kx 2 is also monotonically increasing, so that the end e ffe ct is not serious fo r the supporting force either. (i-2) α = 1: ( i. e., source without back iron) S im ila r to the case α 0, we can get from (3.1), (3.2), (3.3) and the thin reaction r a il assumption: (3.9a) (3.9b) (3.10a)

-31 - (3.10b) (3.11a) (3.11b) A ll o f the approxim ations (3.9 b ), (3.10b), (3.11b) are in t r o duced fo r the convenience o f integrand analyses w ith 0 < ky < ω/v where kyh2 << 1, kyh3 << 1 are very accu rate. However, as fa r as the r e a lis t ic performance in te g ra ls are concerned, (3.9 a ), (3.1 0 a ), (3.11a) w ill be more accu rate. Formulas (3.9 a ), (3.10a), (3.11a) are p lo tte d in F ig. 3.3. For those "a" parameters as defin ed in ( i- 1 ), i f o n ly the region 0 a 1 is considered, we w ill get: (3.12a) (3.12b) (3.12c) In th is case a i > ay > ar is always tru e, And a ll o f them are m onotonicall y in c re a sin g fu n ctio n s o f γ = µoσx h2v. In order to have

-32- F ig. 3.3 Integrands fo r geometry ( A -i-2 ) w ith γ = µoσxh2v = 1 0. (1) (2) (3) (4)

-33- F ig. 3.4 Power and fo rce s fo r geometry (A -i-2 ) w ith id e a l source and ωµoσxh2v = 2 5, kyo = 1 0. (1) (2) (3) (4) (5) PF

-34- hig h e r e ffic ie n c y and enough supporting fo rc e, we suggest operating t his machine w ith ao > ay, i. e., an a t t r a c t iv e scheme is p re fe ra b le. Now, sin ce ay az, we cannot draw the la r g e s t p o s s ib le p ro p u lsio n fo rce fo r t h is kind o f geometry o p e ra tin g in the a t t r a c t iv e supporting fo rce re g io n. Furtherm ore, fo r the same tra c k c o n fig u ra tio n s, ω, ν, and ky, geometry ( i- 1 ) always g ives la rg e r values o f ~Fy/ ~Kx 2, ~Fz/ ~Kx 2, ~ZL than those o f ( i- 2 ). So, i f the same amounts o f fo rce s and power are re q u ire d, la r g e r ~Kx 2 is always necessary fo r geometry ( i- 2 ). Source w indings o f higher c o n d u c tiv itie s must be used to prevent in to le r a b le ohmic lo s se s. Thus, th 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 rce s are always necessary. However, i f the co st fo r a high c o n d u c tiv ity source w inding is manageable o r high power is not a requirem ent, t h is LIM w ill s t i l l be a p p lic a b le. Thus, we w ill s t i l l present some a n a ly s is fo r th is geometry. For the id e a l source w ith the machine perform ance i s easy to get: (3.13a) (3.13b) (3.13c)

-35- (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 ), (3.10b), (3.11b) can be used. I t should be n o tice d th a t ~µ does not have any e ffe c t on the machine performance. In (3.13a), fo r a given ky o, the maximum th ru s t fo rce occurs a t w ith a corresponding value o f which is ju s t p ro p o rtio n a l to K2o and is independent o f everyth in g e ls e. A t νm, ay ω/vm happens to be equal to kyo and thus the corresponding Fz w ill be 0. Thus, i f an a t t r a c t iv e fo rce is a lso d e sire d, we cannot draw the p o s s ib le maximum p ro p u lsio n fo rce fo r t h is given geometry. T h is is q u ite d iffe r e n t from the previous case where kyo > ay ω/vm > az ω/vm and thus a t the maximum p ro p u lsio n fo rce p o in t a reasonably la rg e a ttra c tiv e fo rce can a ls o be obtained. T his gives fu rth e r evidence th a t geometry ( i- 1 ) is su p e rio r. S im ila r to ( i- 1 ), fo r a given ky o, η is a m o n o to n ica lly in c re a sin g fu n c tio n o f ν. At νm, a la r g e r e ffic ie n c y can be obtained by making kyo /(ωµoσx h2 ) sm a lle r, For the more r e a lis t ic case o f the end e ffe c t a lso p lays an im portant ro le in degrading machine performance. The reason is d e f in it e ly the same as b efore, i. e., ~Kx 2 w ill spread in to the region where the integrand is sm a lle r than th a t a t ky o. Now, as we mentioned, fo 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

-36- e ffe c t is observed. G e n e ra lly, i t can be argued th a t the end e ffe c t can p o s s ib ly be reduced by making σxh2 sm a lle r, which is a lso the same co n clu sio n as th a t o f ( i- 1 ). Of course, a s u ita b le lower σxh2 must be chosen to p reven t the machine from going in t o the low e f f i c i ency regio n. (i i ) T hick re a ctio n r a ils : For γ2 h2 >> 1, tanh γ2h2 1 is approxim ately tru e. In a d d itio n to t h is, tanh kz h3 = kzh3 w ill fu rth e r s im p lify the o r ig in a l form ulas. However, in t h is case, the a lg e b ra ic a n a ly s is is found to be much more com plicated fo r problems w ith α 0. Thus, in ste ad o f cons id e rin g α 0, we w ill take care o f the problem w ith α = 1 f i r s t. Then several r e s u lts fo r α 0 can be obtained by comparing w ith those o f α = 1. ( ii- 1 ) α = 1: (region 4 is fre e space) From (3.1 ), (3.2 ), (3. 3) and the given assumptions: (3.14) (3.15)

-37- (3.16) (3.17) (3.18) (3.19) A lg e b ra ic analyses fo r ~Fy/ ~Kx 2 and ~Fz/ ~Kx 2 y ie ld the re - s u its in F ig. 3.5. For the eddy cu rre n t lo s s and re a l P, r e s u lts can be obtained sim ply by m u ltip ly in g (ω/ky - ν) and ω/ky w ith ~Fy/ ~Kx 2 re s p e c tiv e ly. For ~µ = µo~i, a maximum value f o r ~Fy/ ~Kx 2 is found to be at (here K = w ith a corresponding value o f

-38- I f ~μ μο~ι and μ2o/μ 2y μ2z << 1 are tru e, And the corresponding maximum value is μο/ (2 2( 2 + 2) ). Thus, i f the peak p ro p u lsio n fo rce is o f most concern, the b ig g e r extreme v a lu e o f ~Fy/ ~Kx 2 f o r the ferrom agn e tic tra c k w ill make the tra ck a l i t t l e b it more fa v o ra b le than the n on ferrom agn etic one. ~Fz/ ~Kx 2 is ze ro a t az = w ith For the nonferrom agnetic tra c k, K3-1 reduces to ze ro, i. e., the o n ly zero p o in t is a t a = 1. And, no m atter whether ~μ = μο~ι o r ~μ μο~ι, ~Fz/ ~Kx 2 has a maximum at a = 1. More in fo rm a tio n about these integrands can be obtained from F ig. 3.5. I t can e a s ily be seen th a t f o r the th ic k re a ctio n r a i l, th e skin e ffe c t w ill prevent the medium in region 1 from in flu e n c in g the machine performance. And the p e rm e a b ility ~µ begins to p lay a very im portant r o le. For the nonferrom agnetic tra c k, o n ly the re p u lsiv e fo rce can be observed. A la r g e r re p u ls iv e fo rce can be obtained o n ly a t a sm a lle r "a ". Thus, i f high e ffic ie n c y is d e sire d, t h is geometry is not recommended to be used as a LIM both fo r p ro p u lsio n and le v it a tio n. However, i f i t is m ainly suggested fo r the p ro p u lsio n purpose, there w ill always be an e x tra re p u ls iv e fo rce to help in su p p ortin g the v e h ic le s. As fo r the ferrom agnetic tra c k, we w ill get a very s im ila r re s u lt to th a t o f the th in re a ctio n r a il ( A - i- 2 ). Both re p u ls iv e and

-39- a t t r a c t iv e fo rce s can be obtained. Furtherm ore, 0 az ay < 1 is tru e f o r both cases. Thus, most o f the arguments given there can be a p p lie d here. Depending on whether ao is la rg e r or sm a lle r than az, the machine can have e ith e r an a t t r a c t iv e o r a re p u ls iv e fo rce. But, i f high e ffic ie n c y is a ls o re q u ire d, the a t t r a c t iv e one is p re fe ra b le. U n fo rtu n a te ly, the la rg e s t p ro p u lsio n fo rce can o n ly be obtained a t ay = az. I t is a ls o n o tice d th a t a t ay the peak integrand fo r Fy is a l i t t l e b it s m a lle r than th a t o f (A -i-2 ) which is o n ly o f the order o f ky h m u ltip 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 tiv ity source w inding is required to reduce the ohmic lo ss when sim ila r fo rc e s as in (A -i-1 ) are d e sire d. And, a c tu a lly, th is seems to be a common requirem ent fo r LIMs w ith α = 1 when s im ila r fo rc e s as LIMs w ith α4 = 0 are needed. For the id e a l source, the machine performance can e a s ily be obtained by s u b s titu tin g ky w ith kyo in equations (3.1 5 ), (3.17) and (3.19). P lo ts o f Fy, Fz, e tc. are then shown in F ig. 3.6. Now, the maximum th ru s t fo rce p o in t w ill s a t is f y w ith Ki = K o r K2. The c o rre s ponding fo rce w ill be on the order o f A c tu a lly, as d erived from the integrand analyses, the id e a l source r e s u lts fo r ~µ µo~i w ill 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 ro p e rtie s can a lso 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 ffe c t can be reduced by making Ki sm a lle r. Using a com posite tra ck g e n e ra lly can a tta in t h is o b je c tiv e. However, the decrease o f Ki w ill a lso g ive a s m a lle r ay and thus a sm a lle r e ffic ie n c y. A s u ita b le compromise should be made to determ ine which v a lu e o f Ki to use.

-40- F ig. 3.5 Integrands fo r geometry ( A - ii- 1 ) w ith (1) (2) (3) (4)

-41- F ig. 3.6 Power and fo rce s fo 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 fa c to r

-42- (i i -2) α 0: (region 4 is in f in it e ly lam inated iro n ) From (3.1 ), (3.2 ), (3.3) and the given assum ptions: (3.20a) (3.20b) (3.21a) (3.21b) (3.22a) (3.22b)

-43- A fte r comparing to the corresponding form ulas in ( A - ii- 1 ), the d iffe r e n t terms in (3.20b), (3.21b), (3.22b) are u n d e rlin ed and by changing these terms to "1 ", form ulas fo r ( A - ii- 1 ) can be obtained. I t is seen th a t ~μ = μo~i and ~μ μo~i behave q u ite d if f e r e n t ly from each o th e r. By observing th is fa c t and the r e s u lts 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. Nonferrom agnetic tra c k s: ~Fz/ ~Kx 2 is always n e g ative. And s im ila r to th a t in ( A - ii- 1 ) ~Fz/ ~Kx 2 is m onotonical ly in c re a sin g fo r 0 a 1 w ith a minimum value o f - μo/4 a t a = 0 and a maximum value equal to 0 a t a = 1. ay is lo ca te d a t w ith K4-1 ly in g between and Ferrom agnetic tra cks: w ith has a maximum ly in g between a = 1 and a = az. ay is lo ca te d at w ith Κ5-1 ly in g between and Thus, ay > az is g e n e ra lly tru e.

-44- F ig. 3.7 Integrands fo r geometry ( A - ii- 2 ) w ith (1) (2) (3) (4)

-45- F ig. 3.8 Power and fo rce s fo r geometry ( A - ii- 2 ) w ith id e a l source and ωκ5 = 200,000, = 0.01, h3= 0.01, kyo = 1 0. (1) (2) (3) (4) (5) Power fa c to r

- 4 6 - Also, no matter which track is used, ~Fy/ ~Kx 2 and ~Fz/ ~Kx 2 w ill always be larger than those of the corresponding values of (A -ii-1 ). Most arguments given in (A -ii-1 ) can be applied here fo r the nonferromagnetic track except that, due to the re la tiv e ly larger values of ay, ~Fy/ ~Kx 2 and ~ Fz/ ~ K x 2, th is case w ill pro- duce larger forces and e fficie n cy. For the ferromagnetic track, the advantage over that o f LIM with α = 1 which was mentioned before is re-observed. Due to the fa ct that ay > az, i f the LIM i s used for both propulsion and le v ita tio n purposes, the LIM is suggested to be operated in the a ttra ctiv e region. Then, a peak propulsion force can always be obtained while the a ttra ctiv e force is s t i l l re la tiv e ly large. For the ideal source, the resu lts can be obtained simply by su b stitu t ing ky with kyo. Some plots with the machine performance as a function of ν are given in Fig. 3.8. For ferromagnetic tracks, some results are q u a lita tiv e ly sim ila r to those of (A -i-1). When ~Kx is not a δ-function, the end e ffe ct w ill also appear. Decreasing σx is suggested to reduce it. (B) β1 = β3: (region 1 i s free space) In (A), β1 0. This means a lo t of laminated ferromagnetic material must be used, and thus more cost. So, from the economic point of view, the free space, or ju st the earth for region 1 may be preferable. From (2.8), (2.13), (2.31): (3.23a)

-47- (3.23b) Here (3.23c) (3.23d) (3.23e) For th ic k re a c tio n r a i l s, there is no d iffe re n c e whether region 1 is t he fre e space o r a lam inated ir o n, o r any o t he r m a te ria l. Thus, in t he fo llo w in g o n ly t he th in re a ctio n r a il is necessary to be consid e re d. ( i- 1 ) α 0: (region 4 is i n f i nit e ly lam inated iro n ) Under the th in re a ctio n r a il assum ption, (3.23) w ill give us: (3.24a) (3.24b)

- 48- (3.25a) (3.25b) (3.26a) (3.26b) Of course, (3.24b), (3.25b) and (3.26b) are introduced o n ly fo r the purpose o f s im p lify in g the integrand a n a ly s is. G e n e ra lly they cannot be used fo r the in te g ra l e v a lu a tio n. (The id e a l source w ith ky o h3 << 1 is one e x ce p tio n.) Except fo r a fa c to r, (3.24) is e x a c tly the same as (3.9 ). However, th is fa c to r w ill make the a lg e b ra ic a n a ly s is much more com plicated. Several remarks w i l l be given:

-49- Nonferrom agnetic tra cks: S in ce [1 + kz (h2 + h3)] ~ 1, e xcept f o r the v alu e ~Fz/ ~Kx 2, the d iffe re n c e between t h is geometry and (A -i-2 ) is sm a ll. Thus we w ill get s im ila r p lo ts o f ~Fy/ ~Kx 2 and ~ZL fo r both cases. And w ith γ = μσxh2ν. Of course, the corresponding maximum values w ill be a l i t t l e b it la rg e r than those o f ( A - i-2 ). ~Fz/ ~Kx 2 w ill be d iffe r e n t from the corresponding (A -i-2 ) re s u lt. A c tu a lly, w ill always be re p u ls iv e. i. e., i t This is not s u rp ris in g because now there is no b ack-iro n to give the a t t r a c t iv e fo rce component any lo n g e r. A ls o, there is a minimum value o f - μo/4 a t ky = 0 and a maximum value o f 0 a t a = 1. F e rro magnetic tr a c ks: cannot be neglected) For a > 0 i t can be s hown t ha t ay l ie s to t he r ig ht o f And f or 0 < a < is mon otoni c - al l y in c re a s in g. A t a = For a > 0 i t can be shown t ha t i f az l i e s to the (left/right) o f And, th e re is on ly

-50- one value o f a z ly in g between 0 and 1. A ls o, at ky = ω/v, which is the maximum v a lu e o f ~Fz/ ~Kx 2. Other p ro p e rtie s o f the integrands can roughly be obtained from the p lo ts 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 fo r nonferrom agnetic tra cks the machine performance in terms o f Fy, η, and PF is roughly s im ila r to th a t o f ( A - i-2 ). we get o n ly a p u re ly re p u ls iv e fo rce. The main d iffe re n c e is th a t now And the sm all absolu te value o f the integrand fo r Fz a t la rg e "a" w ill exclude t h is geometry from being used as a LIM fo r both supporting and p ro p u lsio n purposes. For ferrom agn e tic tra c k s, the s itu a t io n is d if f e r e n t. Of course, i f is s t i l l s m a ll, the r e s u lt d e f in it e ly w ill be n e a rly the same as before. However, i f µ2y i s la rg e enough, then becomes im porta 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 support and sim u lta n eously a c c e le ra 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 rg e r value o f is more d e s ira b le. And making µ2y and h2 la rg e is the e a s ie s t way to reach t h is o b je c tiv e. As fo r the fa c to r we can a lso tr y to make σx la rg e r to get a value approxim ately equal to 1 a t the peak v e lo c ity. However, as fa r as the end e ffe c t is concerned, i t is more d e sira b le to keep i t a t a moderate value (say 0.8 ). Then i t seems th a t a somewhat sm a lle r σx is p re fe rre d over a higher σx. A ll o f t h is w ill make the composite iro n the most prom ising tra ck m a te r ia l.

-51- F ig. 3.9 Integrands fo r geometry ( B -i-1 ) w ith (1) (2) (3) (4)

-52- F ig. 3.10a Power and fo rce s fo r geometry ( B -i-1 ) w ith id e a l source and (1) (2) (3) (4) (5) Power fa c to r

-53- F ig. 3.10b Power and fo rce s fo r geometry (B -i-1 ) w ith id e a l source and (1) (2) (3) (4) (5) Power fa c to r

-54- ( i- 2 ) α = 1: (region 4 is fre e space) Under the th in re a ctio n r a il assum ption, (3.23) w ill g ive us: (3.27a) (3.27b) (3.28a) (3.28b)

-55- (3.29a) (3.29b) Some tedious a lg e b ra ic analyses w ill give us the fo llo w in g r e s u lts : Nonferrom agnetic tra c k s: can be fu rth e r approximated as 4ky2. And Then (3.24b) and (3.27b) lo o k n e a rly the same; so do (3.25b) and (3.28b). By comparing the form ulas w ith each o th er i t fo llo w s th a t and G e n e ra lly, the c o r responding maximum values are s m a lle r f o r geometry ( B - i- 2 ). A c tu a lly ~Fy/ ~Kx 2 a t fo r geometry ( B -i-2 ) is o n ly about o n e -h a lf o f the value o f ~Fy/ ~Kx 2 at f o r the case ( B - i-1 ). This is s im ila r to the previous case (B -i-1 ) and w ill always be negative. A c tu a lly, a minimum value o f -uo /4 occurs a t ky = 0, and a maximum value

-56- of "0" occurs a t ky = ω/v. However, f or the same "a" i t w ill be a l i t t l e b it sm a lle r fo r geometry ( B -i- 2 ). Ferrom agnetic tra c k s: For 1 > a > 0, i f is s t i l l s m a ll, then the same co n clu sio n as fo r th a t o f nonferrom agnetic tra cks can be used as an approxim ation. For a > 0, i f is not s m a ll, then, except in a very sm all region where kz is sm all such th a t is s t i l l tru e so as to make the in te g ra n d s behave in the same manner as th a t o f nonferrom agnetic tra c k, the s itu a tio n is d if f e r ent fo r the rem aining regions. In o rd e r to have a LIM w ith reasonable e ffic ie n c y, i t is suggested th a t kyo be a l i t t l e b it sm a lle r than ω/ν (say, kyo = 0.8 ω /ν). Now fo r a > 0, i f then, in the region where ~Κχ 2 is la rg e ( i. e., say around kyo 0.8 ω /ν), the approxim ation can be made. Then, i t can be shown th a t and ~Fym/ ~Kx 2 = μo/4. ~Fz / K X 2 is found to be m onotonicall y in creasin g w ith az ay. A maximum value o f μo/4 a t a = 1 and a minimum v alu e o f - μo/4 a t a = 0 can a ls o be observed. For sm all μ2y/μo ω/v h2, the terms ~Fy/ ~Kx 2, ~Fz/ ~Kx 2 and ~ZL. are found to behave in a s im ila r manner as in ( B - i- 1 ). The main d if f e r ence is th a t the most im portant fa c to r γ = μ o h 2 σ x v o f the previous case

-57- F ig. 3.11 Integrands f o r geometry ( B - ii- 2 ) w ith μoh2σxv = 5, (1) (2) (3) (4)

-58- F ig. 3.12 Power and fo rce s fo r geometry ( B - ii- 2 ) w ith id e a l source and (1) (2) (3) (4) (5) Power Factor

-59- ( B -i-1 ) is now changed to γ/2 and the corresponding maximum values o f ~Fy/ ~Kx 2, ~Zr, ~Zi fo r t h is case are sm a lle r. For a la rg e μ2y/μoω/v h2, i t is a d iffe r e n t s to ry. A c t u a lly, i t looks more s im ila r to ( A - ii- 1 ). In both geom etries (B -i-2 ) and ( A - ii- 1 ), we have ay = az, Thus, most arguments given in ( A - ii- 1 ) can a ls o be a p p lie d here. A fte r the above a n a ly s is, a general d is c u s sio n w ill be given. Now th a t the LIMs are supposed to be used both fo r su p p ortin g and p ro p u lsio n, la rg e fo rce s in both d ir e c tio n s and high e ffic ie n c y are a l l re q u ire d. T h is w ill e v e n tu a lly e lim in a te the p o s s ib ilit y o f using LIMs which give re p u ls iv e supporting fo rc e s. So cases ( A - ii) and (B) w ith n o n fe rro magnetic tra c k s are not recommended. For the rem aining geom etries which can be operated to give a t t r a c t iv e supporting fo r c e s, they can be d iv id e d in to two c a te g o rie s, namely, ay > az and ay az. The property th a t a la r g e r p ro p u lsio n fo rce can o n ly be obtained in a region where ~Fz/ ~Kx 2 is r e la t iv e ly sm all w ill fu rth e r exclude the cases w ith ay = az which g e n e ra lly occur fo r α4 = 1. Thus, f in a lly, the o n ly re maining geom etries are ( A - ii- 2 ) and (B -i-1 ) w ith μ μoi and ( A - i- 1 ). I t should a ls o be n o tice d th a t maximum values o f ~Fz/ ~Kx 2 are g e n e ra lly la rg e r than those o f ~Fy/ ~Kx 2 fo r the rem aining cases. And th u s, fo r a s u ita b le ~ K x 2, a la rg e r Fz can always be obtained. T his happens to be c o n s iste n t w ith t he requirem ent f o r v e h ic le d e sig n, sin c e an a c c e le ra tio n o f "l g" is always necessary to support the v e h ic le, w h ile fo r comf o r t, human beings can o n ly to le ra te a h o riz o n ta l a c c e le ra tio n o f several tenths o f " lg ".

-60- Of course, i f ~Kx 2 is s p e c ifie d, the machine performance can be evaluated e ith e r n u m e rica lly o r a n a ly t ic a lly. G e n e ra lly, num erical in te g ra tio n s are stra ig h tfo rw a rd and a n a ly tic a l e v a lu a tio n s are d i f f i c u lt. However, fo r some o f the s im p lifie d geom etries w ith th in or th ic k re a c tio n r a ils, a n a ly tic a l e v a lu a tio n s seem to be p o s s ib le. Now, we w ill co n sid e r the most popular source d is t r ib u t io n and t r y to e valu a te the machine performance in te g r a ls. Of course, i t would be the best s itu a t io n i f we could a n a ly t i c a l l y evalu ate them fo r a ll o f the recommended geom etries. However, geometry ( A - ii- 2 ) is q u ite d i f f i c u l t. Thus, although geometry ( A - ii- 1 ) is not prom ising, i t w ill be a n a ly t ic a lly examined as an a lte r n a tiv e o f the th ic k re a c tio n r a ils. And we hope th a t i t w ill g ive us some in form ation about geometry ( A - ii- 2 ). (R e su lts fo r geometry ( A - ii- 2 ) are also given in Appendix F.) (a) Geometry ( A -i-1 ): In t h is case a ll o f the integrands (w ith the source term excluded) can be e a s ily fa c to r iz e d. The in te g r a ls can be re w ritte n as the summatio n s o f in te g ra ls s im ila r to S1 (α, ky o, L ). Appendix C can then be used to get: (3.30a) (3.30b)

-61- (3.30c) (3.30d) where (3.30e) ( 3.3 0 f) w ith In the above d e riv a tio n, the th in re a ctio n r a il approxim ation is used fo r the whole range o f ky extended from - to. I t is o b v io u sly not very accu rate. A c tu a lly, th is th in re a c tio n r a il assumptio n is only introduced to analyze the integrands w ith in 0 ky ω/v o r even o n ly around kyo where ~Kx 2 is la rg e. In the region where kz is la rg e, g e n e ra lly γ 2h2 is a ls o la rg e, i. e., the th in re a ctio n r a il 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 exact expressions fo r the in te g ra n d s, we fin d th a t the contr ib u tio n s from the la rg e kz region are always very sm a ll. And, in the meanwhile, i f we look in to the th in re a ctio n r a il approxim ations,

-62- then, in t he same region we wi l l al s o see very sm all in te g rands. F u rthermore, the source fu n ctio n ~Κx 2 is o n ly s ig n if ic a n t around ky o. Thus, we can conclude th a t the use o f t he t hin re a c tio n r a il approxim ation fo r t he who le range o f ky does not in tro d u ce too much e rro r fo r the machine performance in te g r a ls. T his argument can a lso be a p p lie d to o th e r geom etries. Num erical values can be used in (3.30) to e valu a te the machine performance. However, before doing th a t, we w ill look in to these form ulas more c a r e f u lly. For p r a c tic a l cases, Im 2αmL is g e n e ra lly la rg e ( e.g., > π). Thus, can be neglected in S1 (αm, ky o, L ). I t is q u ite obvious th a t the term p ro p o rtio n a l to L in S1 w ill give a r e s u lt corresponding to the id e a l source case ( i. e., w ill give form ulas (3.8)). The rem aining term o f is e s s e n t ia lly the reason f o r the end e ffe c t. A c tu a lly, we w ill have (3.31a) (3.31b) (3.31c) Here, the su b s c rip ts "1" are introduced to represent the c o n trib u tio n from the term in S 1 (α m, kyo, L). A lso, (3.30d) is used

-63- fo r Fz which w ill make the a n a ly s is sim p le r. Now, two extreme cases w ith 4ω/K1ν2 being e ith e r la rg e or sm all w ill be considered in in v e s tig a tin g the e ffe c ts due to the term as shown above. From (3.30): (3.32a) (3.32b) I t is n o tice d th a t α2 - ky o is much la rg e r than α1- k y o ; th u s, the term can be neglected in (3.31). By lo o kin g in to (3.3 1 ), we see th a t the p ro p e rtie s o f the expressions (3.31) can be observed by co n sid e rin g : (3.33a) (3.33b)

-64- (3.33c) I t should f i r s t be noted th a t these terms are sm all when ao 1. A ls o, fo r the id e a l source case we know th a t Re P 0, F y 0, and Im P 0 w ith in the operating range o f 0 ao 1. Now, from (3.31) and (3.3 3 ), we r e a liz e th a t Re P1 is g e n e ra lly negative; -Im P1 and F1 vary in a s im ila r way and are negative o n ly when is n egative fo r ao >1/3. And, roughly, i t can be shown th at the e ffic ie n c y decreases fo r 1 > ao > 1/2. The degrading o f the machine perform ance i s thus observed in the high ao re g io n. We then have: Now, we need o n ly co n sid e r the fo llo w in g terms: (3.34a) (3.34b)

-65- F ig. 3.13a Machine performance fo r geometry (A -i-1 ) w ith A ll parameters are the same as in F ig. 3.2. L = 2.512m. Curves (1 ), (2 ), (3) and (4) represent the same q u a n titie s as in F ig. 3.2. Curve (6) is the e ffic ie n c y and Curve (5) is the power fa c to r.

-66- F ig. 3.13b A ll q u a n titie s are the same as in F ig. 3.13a, except L = 1.256m.

-67- I t can be shown th a t the phase angles o f (3.34a) and (3.34b) are a ll m o n o to n ica lly decreasing fu n ctio n s o f ao fo r 0 ao 1. A c tu a lly, fo r (3.34a) (o r, P1), the phase angle in crease s from -π/4 a t ao = 1 to π/2 a t ao to 5/4π a t ao = 0. And, fo r (3.34b), i t w ill vary from π/4 a t ao 1 to π a t ao = and f in a lly to 7/4π a t ao = 0. Thus, except in a very sm all region increase s in Re P and p o s s ib le in crease s in PF are expected. For a2o (K1v2)/ω, both Fy and machine e f f ic ie n c y are observed to d ecrease. I f (K1v2)/ ω >> γ2, Fz 1 behaves in the same way as -Im P1. On the c o n tr a r y, i f (K1v2)/ω << γ2, Fz1 behaves in a way s im ila r to -Re P1. From the above a n a ly s is, i t is not s u rp ris in g th a t most co n clu sions are b a s ic a lly s im ila r to those obtained from the integrand a n a ly s is. T yp ica l p lo ts o f machine performance fo r th is r e a lis t ic source are given in F ig. 3.13. The degrading in the machine performance can the be observed by comparing w ith the r e s u lts shown in F ig. 3.2 fo r the id e a l source. (b) Geometry ( A - ii- 1 ) : S im ila r to (a ), Fy and P can be evaluated by using s u ita b le fa c to r iz a t io n in (3.14a) and (3.15a) to get: (3.35a)

-68- (3.35b) where α3, and α4 are roots o f k2y + i ( ω- ky v)k6 = 0 w ith or: (3.36a) (3.36b) w ith D e fin itio n s fo r S2 (αm, ky o, L) and S3(αm, kyo, L) are given in Appendices D and E. However, i t should be noted th a t the e xact expression fo r S3 (αm, kyo, L) is d i f f i c u l t to get. Thus, o n ly an asym ptotic form ula is d e rived. As fo r Fz, the in te g ra l is much more com plicated. However, as we mentioned, the end e ffe c t is g e n e ra lly sm all provided t ha t kyol is la rg e. Thus, we can always use the form ula fo r the id e a l source as an approxim ation fo r the more r e a lis t ic source we are now co n sid e rin g. I t is obvious th a t id e a l r e s u lts can be obtained by ju s t c o n sid e r ing the terms p ro p o rtio n a l to L in S2 (α m, kyo, L) and S3 (α m, k y o, L ).

-69- Here we w ill begin to co n sid e r e ffe c ts due to the rem aining terms. However, as we ju s t mentioned, the end e ffe c ts on the supporting fo rce are r e la t iv e ly s m a ll, so we are not going to in clu d e any fu rth e r a n a ly ses fo r the Fz. Now, under the assumption o f kyol >> 1 and Im(αmL) >> 1, we can w r ite : (3.37a) (3.37b) S u b scrip t "1" is introduced to p o in t out th a t (3.37a) and (3.37b) correspond to the term us co n sid e r two extreme cases: in (3.14) and (3.1 6 ). Now le t From (3.36), we have (3.38a) (3.38b)

-70- (3.39a) (3.39b) When ao is not clo se to "1 ", values fo r P1 and F1 are gene r a lly sm a ll. However, when 1-ao is s m a ll, ( e.g., 1 - ao ω/k6v2) a very la rg e in flu e n c e is observed. The u n d e rlin ed terms in (3. 37a) and (3.37b) dominate and produce a phase angle o f about 0 in P1. There is a ls o an in crease in the p ro p u lsio n fo rce. Roughly speaking, e xpress 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) (3.40c) (3.37a) and (3.37b) can then be r e w r itte n as:

-71- (3.41a) (3.41b) When ao is clo se to 1, (3.41a) and (3.41b) are r e la t iv e ly sm a ll. However, when ao is sm all ( e.g., a2o ~ (K6v2 )/ω) the s itu a tio n is d if f e r ent. Re P1, Im P1 and Fy 1 are a ll p o s itiv e and somewhat la r g e r. As fo r the c o n trib u tio n due to the term the r e s u lts in Appendix E are used. As we can see, S3(am, ky o, L) is decomposed in to two p a rts. One is due to the resid u e s o f a ll the p o le s, w h ile the other is due to the in te g ra tio n s along the branch cu ts. However, i f is tru e (Note th a t K5= σxμ2z >> K6 is la rg e, thus covers most operating re g io n s. The o n ly exception occurs in the very low speed re g io n.) β2 is much la rg e r than β1. Most terms in the branch cut p a rts are then shown to be much s m a lle r than the corresponding terms in the residue p a rts. Furth er co n d itio n s o f Im( αml) > π w ill make i t p o s s ib le fo r us to w rite down the f in a l approximate r e s u lts as:

-72- (3.42a) (3.42b) Here α3 and α4 are defin ed in (3.36) w h ile β1 and β2 are roots o f γ22 = 0, i. e., (3.43a) (3.43b) wi t h

-73- Now, under the c o n d itio n we have: (3.43c) (3.43d) We w ill begin to co n sid e r two extreme cases: I t is obvious th a t the values fo r P2and Fy2 are sm all when ao is not c lo s e to 1. We are most concerned w ith the case when a ao~1 where the v a lu e s are la r g e. Now, under the assumption t hose terms w ith α3 in v o lved can be neglected. For the c o n trib u tio n s due to the u n d e rlin ed terms approxim ately give us Re P2 0, Fy2 0 and a p o s itiv e Im P2. For the rem aining term s, i t can be shown th a t Re P2, Im P2 and Fy2 are a ll n e g ative. And the magnitudes are so la rg e th a t they w ill not only cancel the p o s itiv e values in P1 and Fy 1, but w ill a ls o e v e n tu a lly give us la rg e negative v alu es o f Re P, Im P and Fy. Thus, we can conclude th a t la rg e decreases in the p ro p u lsio n fo rc e, the w ill be observed fo r power fa c to r and the e ffic ie n c y T his co n clu sion is b a s ic a lly s im ila r to th a t d e scribed in the e a r lie r integrand analyses. In t h is case, the values are r e la t iv e ly sm all fo r an "ao" which

- 7 4 - is not too sm a ll. Thus, we are most concerned w ith the region where a2o ~ (K6v2)/ω. Of course, the terms corresponding to α3 cannot be neglected any longer. A fte r some rough a lg e b ra ic m a n ip u latio n s, i t can be shown th a t the u n d erlin ed terms w ill give us a p o s itiv e value fo r Im P2 and n e a rly zero values fo r Re P2 and Fy 2. However, the most im portant c o n trib u tio n comes from the rem aining terms. G e n e ra lly, Fy2 < 0, Re P2 < 0 and Im P2 < 0 are obtained. And the values are so la rg e th a t o v e ra ll decreases in the p ro p u lsio n fo rc e, e ffic ie n c y and power fa c to r w ill be observed. Thus, the most se rio u s degrading in machine performance is seen around a 2 o (Ky v2 )/ω. T his is a lso the same co n clu sio n as th a t obtained from the integrand a n a ly s is. A t y p i cal p lo t o f the machine performances fo r t h is more r e a lis t ic source is then given in F ig. 3.1 4. And, by comparing w ith F ig. 3.6, the degradin g o f the machine perform ances is observed.

-75- F ig. 3.14a Machine performance f o r geometry ( A - ii- 1 ) w ith A ll parameters are the same as in F ig. 3.6. L = 2.512m. Curves (1 ), (2 ), (3) and (5) represent the same q u a n titie s as in F ig. 3.6. Curve (4) is the machine e ffic ie n c y.

-76- F ig. 3.14b A ll q u a n titie s are the same as in F ig. 3.14a except L = 1.256m.

-77- (c) Geometry ( B - i- 1): get: We can now a p p ly Appendices C and D to (3.24a) and (3.25a) to (3.44a) (3.44b) (3.44c) Here δm are ro o ts o f 1, 2, 3, 4. A ls o, δm+4 = δ*m. And

-78- w ith Fz is very com plicated and we w ill not analyze i t any fu rth e r. But, s im ila r to the previous case (b ), the r e s u lt fo r the id e a l source can be used as a good approxim ation provided th a t kyol >> 1. The form ulas given in (3.44) are co m p licated. By lo o k in g in to Appendix D we know th a t in te g ra l S2 (δi, ky o, L) can be decomposed in to two p a rts, namely, I 1 and 2 I2. Now we w ill co n sid e r the c o n trib u tio n from I 1. I t is obvious th a t the id e a l source r e s u lt can be obtained by ju s t co n sid e rin g the terms p ro p o rtio n a l to L in S1 and S2, and, s im ila r to geometry ( A - i- 1 ), Im 2δ il can be considered as la rg e ( e.g., π ) fo r most p r a c tic a l cases. Thus, in S1 and S2 the terms in v o lv in g can be n eglected. Under the above assum ption, we can then w rite down the fo llo w in g exp re ssions: (3.45a)

-79- (3.45b) here δ1 and δ2 are ro o ts o f And the s u b s c rip ts "1" are introduced to d is tin g u is h (3.45) from the o th e r terms where the c o n trib u tio n s are due to in te g r a ls s im ila r to I 2. Now, we w ill begin to co n sid e r two extreme cases: In th is case, the equation fo r δi is approximated as k2y - i ( ω-ky v)k 1 = 0. A ls o, γ >> 1 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 i / [ ( δ2-ky o )2] (Note: δ1 α1, δ2 α2 ) can be negle cte d. (Perhaps the o n ly exception is the la s t term in F y 1 ; i t can be re w ritte n as For the "1" p a rt, a very sm all a d d itio n a l p ro p u lsio n fo rce is added. For the re m aining p a rt, the n e g le ct is reasonable. A c t u a lly, the whole t hing can be shown to be ca n ce lle d by another term due to an in te g ra l s im ila r to I 2 ). By a p p ly in g (3.3 3 ), roughly speaking, the power fa c to r, e ffic ie n c y and t he p rop u ls ion f orce decrease f or a la rg e r ao (e.g., ao > 2/ 3), i. e., a degrading of t he machin e p e rformance w ill be observed. And th e maximum decreases general l y occu r a t a value of ao where is tru e.

-80- In th is case, the equation fo r δ i can be approximated as and the ro o ts a re found to be: w ith When γ is sm all, is sm all too. However, K1v/γ is la rg e. Thus δ1 and δ2 can be fu rth e r s im p lifie d as: Since δ2 - kyo >> δ1 - ky o, so, s im ila r to the previous γ >> 1 case in (3.4 5 ), most terms in v o lv in g 1 /[ (δ2 - ky o )2 ] can be neglected. (The same argument as the previous γ >> 1 case can a ls o be ap p lie d to the la s t term in Fy 1. From a la t e r c o n s id e ra tio n, i t can be shown th a t th is term w ill be ca n ce lle d by another term due to I 2.) A ls o, we have: (3.46a) (3.46b)

-81- (3.46c) From (3.45) and (3.46) we know th a t Re P1 is p o s itiv e when is p o s itiv e and Re P1 is negative when A2 is n e g ative. A s im ila r argument can be a p p lie d to Im P1 by changing A2 to A3 = Thus the in flu e n c e o f (3.45a) on the o v e ra ll power fa c to r la r g e ly depends on ao. However, an in crease in the power fa c to r is expected when ao >> γ and U n fo rtu n a te ly, a decrease in the p ro p u lsio n fo rce is always observed when ao γ. And, the most se rio u s decrease w ill occur fo r a value o f ao ~ γ Now, we begin to co n sid e r the c o n trib u tio n due to the term I 2 in S2. However, in ste a d o f using (3.44) d ir e c t ly, we go back to the o r ig in a l fo rce and in te g rand form ulas to get: (3.47a)

-82- (3.47b) Here: (3.48a) (3.48b) I4 can be evaluated in the same manner as I2 (δm, ky o, L) in Appendix D. A c tu a lly, fo r mos t p r a c tic a l cases, kyol is la rg e. is a fa s t varyin g fu n c tio n. And, thus by comparing to ls 1 (δ m, k y o, L ) (i.e., an in te g ra l whose lim it s are - and + ), I4(δm, kyo, L) is r e la t iv e ly sm all. A ls o, the in te g ra l I4(δm, kyo, L) can be approxim ated as: (3.49a) and (3.49b)

-83- T his approxim ation is b a s ic a lly s im ila r to the n e g le ct o f I 3 in I 2 (see Appendix D). Now, we can begin to c o n s id e r two extreme cases. ( i) γ >> 1: In t h is case, both e quatio ns can be approxim ated as That i s, δ1 δ3 and δ2 δ4 can be considered as tru e. Thus, (3.50a) (3.50b) δ1 and δ2 are approxim ately given a t (3.32), i. e., α 1 and α2, respect iv e ly, so we have:

-8 4 - The la s t two terms in P2 can be shown to have a sm all e ffe c t on the power fa c to r (a c tu a lly, And the la s t three terms in Fy2 w ill give a drag fo rce which is about 1/kyoL o f the id e a l p ro p u lsio n fo rce. Now, we w ill begin to look a t the e ffe c t due to the f i r s t terms in P2 and Fy2. The im aginary p art in can be seen to have an o p p o site e ffe c t compared to the corresponding terms in (3.45a), (3.45b), i. e., g e n e ra lly, i t w ill ease the end e ffe c t. However, sin ce π/2 >> ω/k1v2, the in flu e n c e is r e la t iv e ly sm a ll. The re a l p a rt in w ill a ls o g e n e ra lly r e s u lt in p o s itiv e Re P2 and Fy2 and negative Im P2 fo r ao which is not very sm all ( e.g., a1 > 1/3 where l n 1/ao < π/2). Thus, we can a l so say th a t i t w ill make the o v e ra ll end e ffe c t a l i t t l e b it sm a lle r. As fo r the rem aining p a rts, the term - i π/2 in ln δ2/kyo w ill ju s t cancel the corresponding term in P1, and F y1. Due to the largeness o f δ2 - kyo and K1v2/ω, the e ffe c t o f the term - i ω/k1v2 in ln δ2/kyo is n e g lig ib le. U n fo rtu n a te ly, fo r the re a l p a rt o f ln δ2/kyo, b o th the power fa c to r and the p ro p u lsio n fo rce w ill be decreased. However, the c o n trib u tio n due to th is term is sm all to o. By combining a ll o th er re s u lts we can conclude th a t the most serio u s degradation in the machin e performance w ill occur 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

-85- And, we w ill have: By using 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

-86- Now, we w ill make a very rough approxim ation fo r the above formula s and hope th a t we can get some idea about the e ffe c ts due to them. A fte r some te d io u s a lg e b ra ic m anipulation and the n e g le ct o f the s mall term s, we get: (3.52a)

-87- (3.52b) where (3.52c) The la s t terms in P2 and Fy2 ( i. e., w ith π in v o lv e d ) are going to be cance lle d by the corresponding terms in P1 and Fy1, re s- p e c t iv e ly. As fo r the rem aining term s, t h e ir values la r g e ly depend on ao. However, the maximum in flu e n c e d e f in it e ly w ill be observed at ao γ. And a t th a t p o in t, the c o n trib u tio n to Fy2 is found to be p o s it iv e, i. e., i t w il l ease the end e ffe c t. E s p e c ia lly i f there is even a p o s s ib ilit y th a t the o v e ra ll end e ffe c t w ill help to increase the p ro p u lsio n fo rce. Some o f the arguments given above can be observed from comparing the p lo ts, F ig. 3.15 and F ig. 3.10.

-88- F ig. 3.15a Machine performance fo 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) represent the same q u a n titie s as in F ig. 3.10a. Curve (4) is the machine e ffic ie n c y.

-89- F ig. 3.15b Machine performance fo 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) represent the same q u a n titie s as in F ig. 3.10b. Curve (4) is the machine e ffic ie n c y.

-90- CHAPTER IV COMPOSITE TRACKS From the previous d iscu ssio n i t fo llo w s th a t a composite tra ck is h ig h ly recommended. D e f in it e ly, in order to s a t is f y d iffe r e n t purposes, the LIM tra c k c o n fig u ra 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 trade w ill make region 2 inhomogeneous, and i t thus makes the problem im p o ssib le to so lve e x a c tly. However, fo r p r a c tic a l a n a ly tic a l purposes, some approxim ations can always be made. I f the tra c k has a f in e ly d is tr ib u te d inhom ogeneity such th a t the re la t iv e ly long wavelength t r a v e llin g source f ie ld does not see t he inhom ogeneity, the t r a ck can be approxim ately considered as homogeneous w ith some e ffe c t iv e u n ia x ia l ~σ, ~µ. T his idea was probably f i r s t suggested by M is h k in (2 1 ), C u lle n and B a rto n (19 ) in a n a lyzin g ro ta ry machines w it h s lo tte d ro to r s. Experim ents have a lre a d y been made to check it s a p p lic a b ilit y. I t a ls o g ives the reason why we used u n ia x ia l ~σ, ~µ to represent the t r a c k p ro p e rtie s from the very beginning o f our a n a ly s is. The values o f the e ffe c t iv e ~σ, ~µ depend on the t r a ck c o n fig u ra tio n. T h e ir values are determined from a homogeneous tra c k fo r given u n ifo rm ly d is tr ib u te 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 obtained. For example, fo r the gene ra l com posite tra ck shown in F ig. 4.1, we w ill get (assuming th a t w ith in the tra c k the m a terial is u n ifo rm ly d is tr ib u te d in the z d ir e c tio n ) : (4.1a)

-91- (4.1b) (4.1c) To o b ta in ~µ we change σa in to µa, σb in to µband σ i in to µi in the above fo rm u la s. I t should a lso be n o tice d th a t in the lim it bx 0 these exp ressio n s reduce to the f a m ilia r form ulas given by C u lle n and B arton( 19 ). As we mentioned e a r lie r, t h is e ffe c t iv e ~µ, ~σ concept has a lre a d y been e x p e rim e n ta lly v e r if ie d. A th e o re tic a l p ro o f, however, has never been attem pted. There is no doubt th a t i t is im p o ssib le to analyze t h is composite tra ck geometry e x a c tly. However, in o rd er to gain some in t u it io n about it s v a lid it y, an extrem ely s im p lifie d geometry, s p e c if ic a lly a f i n i t e l y lam inated ir o n, w ill be analyzed. Due to the la m in a tio n, the problem b a s ic a lly becomes a 3-dimens io n a l one. The tr a n s fe r m a trix method is thus not a p p lic a b le. The most complete method fo r analyzin g problems w ith lam inated iro n was p re v io u s ly given by Bondi and M ukherji(2 2 ) who evaluated the induced eddy cu rre n t lo s se s. i t to the LIM problem. Here we w ill use a s im ila r fo rm u latio n and extend The f i n i t e l y lam inated iro n is supposed to be used in region 2. Of course, fo r d iffe r e n t LIM geom etries, the s it u a tio n w ill be d iffe r e n t. Here we w ill co n sid e r the geometry ( A - i i - 2 ). The r e s u lts fo r some o f the o th e r c o n fig u ra tio n s are p o s s ib ly obtained by some s u ita b le s u b s titu tio n s. The governing equations are the same as before. But now we have ~σ = o ~I and ~µ = µ ~I in region 2. The source is s t i l l

-92- F ig. 4.1 F ig. 4.2

-93- in f in it e ly extended in the x - d ir e c t io n. Due to the p e rio d ic charact e r o f the tra c k, the f ie ld d is t r ib u t io n w ill a ls o be p e rio d ic in the x d ir e c tio n. A F o u rie r s e r ie s expansion can be used to take care o f the x v a r ia tio n. Thus, o n ly one la m in a tio n ( i. e., χ h/2) should be considered. is employed. For the y v a r ia tio n, the F o u rie r transform technique And, in the fo llo w in g, o n ly one F o u rie r component is consid e re d. I t can e a s ily be seen th a t fu n ctio n s o f x fo r the co o rd in a te system shown in F ig. 4.2. The in s u la tio n regions (which make ~Jx = 0 a t x = ±h/2) are assumed to be very th in. ~Bx and ~Hx are thus continuous a t the in te r fa c e s. Hence, i t may be concluded th a t ~Hx = 0 a t x = h/2. U sing s im ila r arguments as in Bondi and M u k h e rji(2 2 ), we can decompose the f ie ld in region 2 in to three p a rts, namely: (a) ~Hx = 0 everywhere but ~ J x 0, (b) ~ J x = 0 everywhere but ~ H x 0, (c) ~Jx = 0, ~Hx = 0 everywhere. P a rt (c) is the s o -c a lle d deeply penetrated f ie ld which w ill be found to be the most im portant. I t plays a r o le in reducing the undesired s k in e ffe c t. The f i e l d components in each re g io n are thus: Region 2: (a) ~Hx = 0 (4.2a)

-94- (4.2b) (4.2c) (4.2d) (4.2e) (4.3a) (4.3b) (4.3c)

-95- (4.3d) (4.3e) (4.4a) (4.4b) (4.4c) (4.4d) (4.4e) where (4.5a) (4.5b) (4.5c) (4.5d)

-96- A s u ita b le Riemann su rfa ce sheet should be taken to give p o s i t iv e real p arts fo r the kz,l, kz,m and kz Region 3 (4.6a) (4.6b) (4.6c) (4.6d) Thus we have unknowns {qm1 }, {qm2}, {bm}, {αl }, f. In p r in c ip le, they can be solved by matching the boundary c o n d itio n s a t z = 0 and z =h3. A t z = 0: are continuous (4.7a) (Note: J z = 0 a t z = 0 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 given above.) (4.7b)

i. e., -97- (4.7c) (4.7d) (4.7e) ( 4. 7 f ) (4.7g) Then, w ith (4.8a) (4.8b)

-98- (4.8c) T his s e t o f equations is m anipulated to g ive: (4.9a) (4.9b) (4.9c) (4.9d)

-99- (4.9e) I f (4.9a) can be solved fo r αl, a ll o f the o th e r unknowns can be obtained from the rem aining equations. Then we can fin d the f ie ld d is t r ib u t io n everywhere. I t is understood th a t the f in a l r e s u lts we want are machine p e r formances such as fo rc e s, power, e ffic ie n c y, e tc. A ll o f these can be expressed in terms o f the f ie ld d is t r ib u t io n a t z = h3 as given by (2.2 4 ), (2.2 5 ), (2.27). As fa r as the f ie ld s a t z = h3 are concerned, form ulas (4.6 ), (4.7) can be used to g ive: (4.10a) (4.10b) (4.10c) For average F y, P the m 0 terms in Hz do not g ive any c o n trib u tio n. But fo r Fz these terms do have some e ffe c t. However, due to the fa s t decay o f the c o n trib u tio n s from m 0 can be considered as n e g lig ib le. (Note, g e n e ra lly h h3 is tru e.) So, fo r ~Hz, we probably need o n ly to co n sid e r the term w ith m = 0. Thus q0'1 is the most im portant unknown constant to e valu a te. Now, le t us assume th a t α l can be solved fo r, except fo r a common fa c to r a. Then we set

-100- (4.11a) (4.11b) (4.11c) I t should be n o tice d th a t A, B, C, α are a ll l, m independent. From (4.9 ),(4.1 1 ): (4.12) And the ~H- f ie ld a t z = h3 is thus: (4.13a) (4.13b) (4.13c) Here fo r ~Hz, o n ly the m = 0 term is in clu d e d. The c o rre s - ponding ~E f ie ld is given by ~E = ^ l x ωµo/ky~hz. Using (2.2 4 ), (2.2 5 ), (2.2 7 ), the expressions fo r F, P are found to be e x a c tly the same as those given in ( A - ii- 2 ), except th a t µoγ2/µ2ykz now should be replaced by

-101- Now we come to the p o in t where we have to e valu a te A, B, C, i. e., we should so lve (4.9 a ). Doing i t e x a c tly is a hopeless ta sk. An approxim ation can be made by tru n ca tin g t h is se t o f e q u a tio n s, cons id e rin g o n ly a lim ite d number o f equations w ith the same number o f unknowns. However, t h is w ill lead o n ly to some num erical data. In the problem we are cons id e rin g i t seems th a t an a n a ly tic a l so lu tio n is more des ir a b le. Thus, an a lte r n a tiv e approxim ation w ill be made. F ir s t, le t us look at the term: I t can be re w ritte n as: Now, i f kyoh/2π << 1 i s assumed, kzd/kz,ld << 1 can be shown to hold in the region where ~Kx is la rg e. k z d / k z, l d can thus be neglected in 1 - kzd/kz,ld. Furtherm ore, even fo r m = 1, i f ( 4π2d2 )/h 2 1, F(m ) >> ukzd can always be obtained. G e n e ra lly, the requirem ent ( 4π2d2o )/h2 1 is not d i f f i c u l t to meet in the LIM which we are now c o n sid e rin g. Under these c o n d itio n s, the fo llo w in g approxim ation is reasonable:

-102- (4.14) The next step is then try in g to fin d an e x p lic it fu n ctio n o f l fo r αl to make independent o f m. The o n ly p o s s ib ilit ie s we can fin d are αl = a and I f α l = a, then The magnitude o f the e rro r term in A w ill always be la r g e r than 0. A ls o, we are expecting a se t o f α l which w ill decrease fa s te r than a when l in cre a se s. Thus, we must exclude t h is p o s s ib ilit y. On the c o n tra ry, fo r is no longer zero. And the e rro r term w ill always be sm a ll. the assumption Thus, we may conclude th a t w ill g iv e a q u ite good approxim ation. Now we use in (4.11) to get: (4.15a) (4.15b) (4.15c) w ith (4.1 5d)

-103- Thus: (4.16) Now compare the above form ula w ith I f h is s u f f i c ie n t ly sm all such th a t h/2do << 1 is tru e, then and i s thus reduced to µo/µ. T his means th a t µ2y µ and σ 0 are the e ffe c tiv e m a te ria l co n sta n ts. And they are c o n s iste n t w ith the values given by e m p irica l form ulas (4.1 ). However, i f h/2d << 1 is not met, then the above s im p lif ic a t io n is not v a lid any lo n g e r. Thus, we can conclude th a t the c o n d itio n s under which form ula (4.1) g ives accurate values fo r the e ffe c t iv e m a te ria l constants are ky h << 1 and h/2d << 1. Up to now we o n ly co n sid e r one F o u rie r component w ith eikyy v a r ia tio n. E v e n tu a lly we have to take the F o u rie r in v e rse to get the machine perform ances. I t is obvious th a t the assumptions which we make above cannot hold fo r a ll ky. However, the source fu n ctio n w ill g e n e r a lly decay when i t i s away from ky o. Thus, i f the above assumptio n s are reasonable around kyo, a good s o lu tio n can s t i l l be obtained by applyin g t h is kind o f approxim ation to a ll ky. In the geometry we are now c o n sid e rin g, w ill give a zero p ro p u lsio n fo rc e. Thus i t is out o f the question to use m erely t h is k i nd o f tra c k as the LIM in HSGT v e h ic le s. (Of course, i t can be used fo r the le v it a t io n purpose.) Extra c o n d u c tiv ity should be added. One example is shown in the LIM b u i l t by the Rohr I n c.(1 7) w ith

-104- s u ita b ly arranged conducting bars in se rte d in to the lam inated iro n. D e f in it e ly, i f the requirem ents kyoh << 1 and h/2do << 1 h o ld, the e ffe c t iv e m a te ria l constant concept is s t i l l v a lid. However, we are not going to give any fu rth e r analyses o f th a t kin d. Instead, an a l te rn a tiv e geometry is suggested. Now we w ill t r y to l i f t the c r it e r io n h/2do << 1 and in tro d u ce a tra c k w ith a la rg e r h such th a t h/2doπ approxim ately equals 1. Then the approxim ation ta n ( ih/2d) ih/2d is not tru e any lo n g e r, nor is the e ffe c t iv e m a te ria l constant concept. Instead, the d ir e c t method, as introduced in t h is ch a p te r, should be used. However, even f o r h/2dπ ~ 1, "C" can s t i l l be shown to be sm all compared to "A". Thus, µo(a-b)/µ(a-c) µo/µ ih/2d cot( ih/2d). Now h/2doπ ~ 1, so, except p o s s ib ly in a very sm all region where ω/ν ~ ky such t ha t ih/2d << 1, h/2dπ is going to be o f the order o f "1" fo r most ky around ky o. And f o r h/2dπ ~ 1, the argument o f the cotangent fu n ctio n w ill be la rg e enough to a llo w us to make the approxim ation th a t Thus, and: (4.17a) (4.17b) (4.17c)

-105- (4.1 7d) (4.1 7e) (4.1 7 f ) Analyses can be made o f these in te g rands. At is no longer equal to zero. A c tu a lly, i f ~Fy / ~Κx 2 is a m o n o to n ica lly decreasing fu n c tio n o f ky fo r 0 ky ω/ν. Its value w ill decrease from to "0" a t ky = ω/ν. A ls o, in th a t same re g io n, can be shown to be p o s itiv e under the co n d itio n th a t For the id e a l source, (4.17) can be used to evalu ate the machine perform ances. However, th is approxim ation w ill be good only under the c o n d itio n s th a t kyoh3 << 1 and h/2πdo ~ 1. Thus fo r the given source and tra c k c o n fig u ra tio n s, g e n e ra lly the approxim ation cannot s a t is f a c t o r ily cover the e n tir e v e lo c ity range 0 ν ω/ky o. I f the necessary co n d itio n s are s a t is f ie d, the maximum fo rce w ill be found at w ith a corresponding fo rce d e n sity o f

-106- S p e cia l care should be taken i f (4.17) is a p p lie d to the more r e a lis t ic source I t is obvious th a t the co n d itio n s ky h3 << 1 and h/2πd ~ 1 cannot be s a t is f ie d fo r a la rg e region o f ky. Thus, fo r a source d is t r ib u t io n ~Kx 2 the a d d itio n a l c o n d itio n th a t L is very la rg e seems to be necessary to ensure the use o f (4.17) a good approxim ation. Now, i f a l l the c o n d itio n s fo r the approxim ations to be v a lid are s a t is f ie d, i t can be seen th a t the examined geometry can o ffe r fo rce s o f the same o rder o f magnitude as those o f geometry ( A - i- 1 ). Thus, i f the geometry (A -i-1 ) can meet the fo rce requirem ents fo r the LIMs, the stu d ie d geometry can a lso do i t w ith source cu rre n t magnitudes o f the same o rd er. However, i t seems th a t the most im portant advantage which t h is geometry can o ffe r is the "in flu e n c e s due to the end e f f e c t. " From a remark given e a r lie r, we know th a t now ~Fy / ~Kx 2 is a monot o n ic a lly decreasing fu n c tio n. I f the source fu n c tio n ~Kx is not a δ-fu n c tio n, although p a rt o f i t w ill s pread in to a region where ~ F y / ~ K x 2 s m a lle r, there always are o th er p a rts which w ill go in to la rg e r integrand re g io n s. Thus, the net e ffe c t is g e n e ra lly sm a ll. S im ila r ly, sin ce ~Fz/ ~Kx 2 is always p o s itiv e, i t is not necessary to worry about ~Kx 2 being pushed in to the re p u ls iv e region. In the above an a lyse s, a LIM o f geometry ( A - ii- 2 ) w ith region 2 c o n s is tin g o f lam inated iro n is considered. We would expect th a t a s im ila r approach could be a p p lie d to o th e r c o n fig u ra tio n s. In some

-107- cases th is is found to be tru e indeed. hold w ith s u ita b le s u b s titu tio n o f A c tu a lly, s im ila r form ulas w ill by exp re ssio n s A, B, C. However, g e n e ra lly th e se A, B, C 's w ill be some- what more com plicated and the equations f o r them w ill be more d i f f i c u l t to so lv e. Es p e c ia lly fo r a th in re a ctio n r a il geometry, we sh a ll fin d th a t we cannot use the same approxim ation any lo n g e r w h ile s o lv in g fo r αl. A ls o, because le s s m a te ria l is employed, a somewhat la rg e r e ffe c tiv e c o n d u c tiv ity is always necessary to give a la rg e enough p ro p u lsio n fo rce. Thus, lam inated iro n alone is not good enough to b u ild a th in re a c tio n r a il LIM ( i. e., composite re a c tio n r a ils w ith both fe r r o - and nonferro-m agnetic m a te ria l are necessa ry ). Hence, we n e g le ct fu rth e r cons id e ra tio n s o f t h i s kind o f c o n fig u ra tio n. As fo r ( A - ii- 1 ), we sh a ll fin d (3.1 4 ), (3.15) a n d (3.16) to be s t i l l tru e w ith μoγ2/μ2ykz replaced by μo(a-b)/μ(a-c). Here B, C are defined by the same form ulas as in the previous case, i. e., (4.11b), (4.11c). However, fo r A and α l they are Somewhat d if f e r e n t. Now we have: (4.18a) (4.18b) (4.18c)

-108- Equations (4.18b), (4.18c) are s im ila r to (4.11a), (4.9e). Thus, the approxim ation given fo r the previous case can be used d ir e c t ly. And, f o r h/2d << 1, we s h a ll o b ta in the same co n clu sio n th a t the re a c tio n r a il works as i f i t is a nonconducting ferrom agnetic m a te ria l. Thus, i t is not s u ita b le fo r use in the LIM except f o r le v it a t io n purposes o n ly. However, fo r h/2πdo ~ 1, we can use the by now f a m ilia r approach to get: (4.19a) (4.19b) (4.19c) (4.19d) (4.19e)

-109- (4.1 9 f ) Again we observe th a t the magnitude o f every term is r e la t iv e ly sm all compared to the case w ith α 0. I t is al s o n o tice d th a t ~Fy / ~Kx 2 is a m o n o to n ically decreasing fu n ctio n o f both ky and ν, provided th a t [ ( ωµσh2 µ2o)/µ2 ] < 1 which i s g e n e ra lly tru e. ~ F z / ~ K x 2 a lso always p o s itiv e. The arguments given in the p revio u s case are a p p lic a b le. The end e ffe c t is not se rio u s fo r t h is geometry. However, the p ro p e rtie s th a t the power fa c to r and ~ZL are r e la t iv e ly low (so th a t the same amount o f cu rre n t can o n ly draw a very sm all p ropulsion fo rce ) compared to the case w ith α 0 w ill make t h is geometry unfa v o ra b le. In the above, we d is cussed the LIM w h ich has a tra c k w ith sm all s ca le in homogeneity such th a t the s ource o n ly sees an average e ffe c t as des crib e d above. Of cours e, d iffe r e n t s itu a tio n s are pos s ib le. For example, in the LIM b u ilt by Rohr In c., we can put the in se rte d conductor bars q u ite fa r away from each other such th a t the d istance s between them are la rg e r than 2π/ kyo. Then d e f in it e ly, the e ffe c t iv e ~µ and ~σ concept f a ils. However, in th is case, i t seems th a t each bar operates independently o f the o th e rs. This is, o f course, a d iffe r e n t problem and can be e a s ily solved.

-110- CHAPTER V THREE-DIMENSIONAL CORRECTIONS I t is obvious th a t g e n e ra lly the cu rre n t source and the re a c tio n r a il cannot be in f in it e ly extended in the x - d ir e c t io n. Thus, a c o rre c tio n should be made, i. e., a more r e a lis t ic geometry such as shown in F ig. 5.1a should be in v e s tig a te d. This is a tough t hree-dim ensional problem and there are no sim ple methods a v a ila b le to handle i t. Thus, in ste a d o f co n sid e rin g th a t geometry, we s h a ll look in to an a lte r n a tiv e geometry as shown in F ig. 5.1b. With a s u ita b le source arrangement, the boundary c o n d itio n Jx = 0 a t x = ±Wo can be s a t is f ie d. Thus, w ith in x Wo, t h is a lte r n a tiv e geometry is supposed to be a good approxim ation f o r tile o r ig in a l one. With M axw ell's equations and m a te ria l p ro p e rtie s as introduced in Chapter 2, the F o u rie r transform method can be used to so lve th 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 transform p a ir (2.4 a ), (2.4 b ), another F o u rie r s e rie s p a ir ( 5.1a ), (5.1b) is necessary to take care o f the x -v a r ia tio n : (5.1a) (5.1b) For n e ith e r F o u rie r component can the tra n s fe r m a trix method be a p p lie d any lo n g e r. A d ir e c t method should be used. F ir s t, we can so lve the M axw ell's equations in each re g io n, in tro d u cin g c e rta in unknowns which can be determ ined by m atching the boundary c o n d itio n s.

- 111- F ig. 5.1 a, b

-112- In o rd er to make the problem s im p le r, o n ly the case β1 0 w ill be considered. We have: Region (2): ( i. e., 0 z h2) (5.2a) (5.2b) (5.2c) Here, the fo llo w in g r e la t io n a lso holds: (5.2d) w ith and (5.2e) (5.2 f) whe r e ( 5.2g) and are the ro o ts o f

-113- (5.2h) From (5.2 ), (2.1 ), i t fo llo w s: (5.3a) (5.3b) (5.3c) Using the boundary co n d itio n : we o b ta in : (5.4) In region (3 ), ( i. e., h2 z h2 + h3):

-114- (5.5a) (5.5b) (5.5c) where and (5.5d) (5.5e) ( 5. 5 f ) (5.5g) Now we match the boundary co n d itio n s a t z = h2. The co n tin u it y o f ~Hxn and the re la tio n s above g ive us: (5.6a) w ith (5.6b) The c o n tin u ity o f ~Bzn and the previous r e la tio n s a ls o g ive us: (5.7a) w ith

-115- (5.7b) so, (5.8) Furtherm ore, u sin g we get: (5.9a) (5.9b) (5.9c)

-116- ( i i ) β4 = β3 : (5.10a) (5.10b) (5.10c) Now a l l o f the f ie ld d is trib u tio n s can be obtained. The machine performances can be c a lc u la te 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 co n sid e r o n ly F x, Fy which are the sim p le st and the most im portant. (5.11a) (5.11b) These form ulas are s t i l l too com plicated 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 tro p ic ; then, (5.12a) w ith

-117- (5.12b) A fte r some m a n ip u la tio n s, i t can be found th a t: (5.13a) (5.13b) where (5.13c) (5.13d) Here ~F(n) is an even fu n ctio n o f n. Now we take the source d is t r ib u t io n in to account. In order to make Jx = 0 a t x = ±Wo w ith in the re a ctio n r a i l, the fo llo w in g extended p e rio d ic source c u rre n t d is t r ib u t io n J lx( x, y) is necessary: (5.14a) (5.14b) (5.14c)

-118- Thus, fo r n odd (5.15a) fo r n even (5.15b) fo r n = 0 (5.15c) A pplying (5.15) to (5.1 3 ), we have: (5.16 a) (5.16b) So, i f Jx (x,y ) is n e ith e r even nor odd, there is a p o s s ib ilit y th a t a la t e r a l fo rce e x is t s. Now le t us co n sid e r one example. The case b4 0 w ill be analyzed. We s h a ll assume th a t the re a ctio n r a il is th in such th a t, fo r small n, γ 2nh2 << 1 and kznh3 << 1 are tru e. Then, fo r sm all n :

-119- (5.17) A ls o, in o rd er to compare th is w ith the tw o-dim ensional case, we s h a ll assume th a t J x (x,y ) is u n ifo rm ly d is tr ib u te d fo r x a. Then, n odd otherw ise (5.18) Here ~Kx (ky ) is introduced to take care o f the y - v a r ia tio n. Of course, we get a zero la t e r a l fo rc e. As fo r Fy, we have: (5.19) The "n " summation can be evaluated by using several s e rie s summation form ulae where (5.20a) (5.20b)

-120- A remark should be made here. O b vio u sly, fo r la rg e n the c o n d itio n s γ2nh2 << 1, kznh3 << 1 are not s a t is f ie d and the approxim ation (5.17) is not v a lid any more. However, fo r la rg e n the fo rce c o n trib u tio n as given by (5.13) is sm all and so is the in v a lid approxim ation (5.17). Hence, using (5.17) fo r la rg e n (which was d e rived using the th in re a c tio n r a il approxim ation) w ill o n ly in troduce a sm all e r r o r in the e xpression o f the to ta l fo rce and we may conclude th a t (5.17) is a f a i r l y good approxim ation fo r a l l n. I f the y - v a r ia tio n o f J x is the same as th a t o f the id e a lis t ic source fo r the tw o-dim ensional case, ~Kx (ky ) w ill be given by 2 π δ (ky - ky o ). Of course, (5.20) w ill g iv e us an in f in it e fo rc e. Howe v e r, f o r the fo rce o f u n it length in the y - d ir e c t io n, we go back to the o r ig in a l fo rce equation in re a l space to get: where (5.21a) (5.21b) duces to: I f, furtherm ore, we assume (ω-kyoν ) K1 << k2yo, then (5.21) r e (5.22)

-121- Comparing th is to (3.8 ), we can say the f in it e w idth re a ctio n r a il has an e ffe c t iv e c o n d u c tiv ity o f T his is e x a c tly the same co n clu sio n as R u sse ll and Norsworthy(23) made, using another approach. A ls o, i t is observed th a t the p ro p u lsio n fo rce is reduced. The in crease o f "a" w ill g e n e ra lly reduce t h is s o -c a lle d la t e r a l end e ffe c t. Of course, the above co n d itio n (ω-ky ov) K1 << k2y o is g e n e ra lly not tru e. any more. Then, we cannot in tro d u ce the e ffe c t iv e c o n d u c tiv ity concept However, by comparing w ith (3.8 ), we can see the la t e r a l end e ffe c t w ill change the p ro p u lsio n fo rce by an amount: (5.23) From t h is com plicated expression we cannot conclude whether t h is term w ill decrease o r in cre a se the p ro p u lsio n fo rce. Assuming γ20a π and l γ20(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)

-122- Thus, i f (5.24) w ill be p o s it iv e, i. e., the la t e r a l end e ffe c t w ill in cre a se the p ro p u lsio n fo rc e. U n fo rtu n a te ly, a t the maximum fo rce p o in t, Thus, the la t e r a l end e ffe c t w ill reduce the p ro p u lsio n fo rce a t th a t p o in t. Fo r the more general source can t r y to e valu a te the in te g ra l (5.2 0 ). we However, due to the presence o f the term γ 3 2 0 the in te g ra l is too tough to e v a lu a te. However, a change o f p ro p u lsio n fo rce can always be observed, namely (5.25) (The main term is ju s t given by (3.30b) except f o r a fa c to r o f "a "). Now, again using the assumptions y2a π, γ2 (ωo - a) << 1, we s im p lify (5.25) and get (5.2 6 ). The same e xpression (5.26) can a lso be used fo r another s itu a tio n w ith both y2a and γ2(wo - a) π, except th a t now a fa c to r 1/2 should be introduced: (5.26) Note th a t, i f the source fu n ctio n is not s u ita b ly co n stru cted the s in g u la r it y at ky = 0 w ill p o s s ib ly g iv e an in f in it e fo rce and a ls o in f in it e power in p u t, which is not p r a c t ic a l. Thus s in kyol = 0 is always necessary to get r id o f th is s in g u la r it y.

-123- As fo r the id e a liz e d source case, the in te g rand w ill be p o s i t iv e in the region where and n e g ative in the rem aining re g io n, Thus, i f kyo belongs to the negat iv e re g io n, a s m a lle r p ro p u lsio n fo rce than in the two-dim ensional case w ill be o b ta in e d. Because an odd power o f γ20 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 should be in tro d u ce d, making the in te g ra l extrem ely hard to e valu a te. In o rd er to o b ta in an e x p lic it form ula f o r the fo rc e, we s h a ll go back to the o r ig in a l s e rie s fo rm u la tio n (5.19). For each "n ", an in te g ra l o f the fo llo w in g form should be computed: (5.27) We know th a t fo r the tw o-dim ensional case, the fo rce is p ro p o rtio n a l to the above in te g ra l w ith n =0. Now fo r n 0, suppose th a t then the in te g ra l w ill g e n e ra lly be s m a lle r than the two-dim ensional case (corresponding to n = 0 ). Thus, t h is la t e r a l end e ffe c t w ill reduce the p ro p u lsio n fo rce even more. However, i f there is a p o s s ib ilit y th a t the la t e r a l end e ffe c t w ill compensate the lo n g itu d in a l end e ffe c t. A c tu a lly, i t had a lre a d y been found th a t the o v e ra ll c r it i c a l p o in t should be around

-124- The fa c to r iz a t io n o f w ill lead to in te g r a ls o f the form In Appendix C th is in te g ra l is worked o u t. A ls o, in te g r a ls o f the form w ill appear. These in te g r a ls need fu r th e r c o n s id e ra tio n. They w ill g iv e an in f in it e r e s u lt unless s in ky ol = 0. 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 in a lly, we have (5.29a) where α 1, α2 are roots o f

-125- or: (5.29b) (5.29c) w ith Now, le t us co n sid e r the case th a t the o r ig in a lly sym m etrical source is s h ifte d over a d ista n ce ε to the r ig h t, i. e., we co n sid er a source cu rre n t d is trib u tio n : (5.30a) The corresponding F o u rie r components are then found to be: (5.30b)

-126- By using (5.1 6 ), the fo rce components f o r the id e a liz e d source w ill be given as: (5.31a) (5.31b) Here ~Fr (n ) represents the re a l p a rt o f ~F(n): (5.31c) Going through com plicated s e rie s summation procedures, we get the e x p lic it forms f o r ~Fy and ~Fx: (5.32a)

-127- (5.32b) I f γ20(wo - a) π, then tanh γ20(wo - a) coth γ20(wo - a), and (5.32a) reduces to (5.2 1 a ); i. e., when the region which is not covered by the source is la rg e, the in flu e n c e o f the s h if t on Fy w ill be sm a ll. Le t us now look a t the more p r a c tic a l case o f γ 2 (Wo - a) << 1, γ20ε << 1. Then, a f t e r r e ta in in g o n ly f i r s t o rd e r terms in the h y p e rb o lic fu n c tio n s, we have: (5.33) By comparing th is w ith form ula (5.2 1 ), i t is seen th a t there is always an in cre a se o f the p ro p u lsio n fo rce by the amount o f As fo r the la t e r a l fo rc e, because γ20a π and γ20wo π g e n e ra lly h o ld, (5.32b) reduces to: (5.34) Thus, i f γ20ε is la rg e enough, there is a p o s s ib ilit y th a t a huge la t e r a l fo rce in the same o rd e r as th a t o f the p ro p u lsio n fo rce is obtained. Of course, fo r the p r a c tic a l machine o p e ra tio n, th is

-128- F ig. 5.2 Curves (1) and (2) represent (5.31a)/ fo r ε = 0 and ε = 0.01m, re s p e c tiv e ly ; Curve (3) represents [(5.3 1b)/ fo r ε = 0.01m. Here Wo - 0.3m and a = 0.28m. A ll parameters except ε are the same as those o f F ig. 3.2.

-129- F ig. 5.3 Curves (1) and (2) represent fo r ε = 0 and ε = 0.0 5, re s p e c tiv e ly ; Curve (3) represents fo r ε = 0.05. Here 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.

cannot happen. -130- A sm all displacem ent from the sym m etrical p o s itio n w ill in tro d u ce a re s to rin g fo rce to push the source back to the o r ig in a l p o r t io n. T his fo rce is given by I t can e a s ily be seen th a t F x always has the opposite sig n o f ε. Thus, as f a r as the la t e r a l displacem ent is concerned, the system is s ta b le. A ls o, fo r sm all displacem en ts, these e x tra fo rce s are r e la - t iv e ly s m a ll. For Fy i t is o f the o rd er o f ε2, w h ile f o r Fx i t is o f the o rd e r o f ε. F o r the more general source form ula s im ila r to (5.20) can be obtained by using (5.1 6 ), (5.3 0 ), (5.3 1 ), and some s e rie s summations. A c tu a lly, we have: (5.35a) (5.35b) I t is d i f f i c u l t to e valu a te these in te g r a ls. The assumptions γ20ε << 1, γ20a >> 1 and γ20(wo - a ) << 1 are then used to

s im p lify (5.35) to: -131- (5.36a) (5.36b) Thus, the unsymmetrical case w ill in tro d u ce one more term in the Fy in t e g r a l, which can be e a s ily evaluated to be: (5.37) (5.37) is always p o s itiv e in the normal o p e ra tin g re g io n, i. e., the unsym m etrical p o s itio n g e n e ra lly in cre a se s the p ro p u lsio n fo rce. A ls o, (5.36b) can be seen to have the o p p o site sign o f ε in most o f the p r a c tic a l cases where ky ol is la rg e. The branch p o in ts a t γ2 = 0 (a lso p oles) make (5.36) very d i f f i c u lt to e v a lu a te. Thus, s im ila r to the previous case, we go back to the o r ig in a l s e rie s fo rm u la tio n to get e x p lic it e xp re ssio n s. Due to the fa c to rs 1 /n2, 1/m2 and 1/(m2 -n2), the to ta l fo rce s are m ainly generated by sm all n, m term s. Going through a l l the fa c to r iz a t io n and in t e g r a tio n procedures we get:

-132- (5.38a) (5.38b) whe re (5.38c) (5.3 8d) w ith α1, α2 being defin ed in (5.2 9 ).

-133- F ig. 5.4 D iffe re n t fo rce components o f s e rie s expressions (5.38a) and (5.38b). (1) (2) (3) (4) where A ll o f the parameters defined in the same way as in F ig. 3.2.

-134- F ig. 5.5 D iffe re n t fo rce components o f s e rie s e xpressions (5.38a) and (5.38b). (1) (2) (3) (4) where A ll o f the parameters defined in the same way as in F ig. 3.2.

-135- F ig. 5.6 D iffe re n t la te r a l fo rce components in the s e rie s expression (5.38b). (1) (2) (3) (4) where ε = 0.0/m those o f F ig. 5.2. and a l l o f the o th e r param eters are the same as

-136- The a n a ly s is is e s s e n t ia lly com pleted. For the id e a l source some r e s u lts are shown in F ig s. 5.2 and 5.3. Comparing these to F ig. 3.2, we n o tic e the in flu e n c e o f the 3-dim ensional c o rre c tio n s we made in th is chapter. As f o r the more r e a l i s t i c source, we o n ly gave the s e rie s re p re se n ta tio n. Several components w ith sm all n, m are shown in F ig s. 5.4, 5.5 and 5.6. When ε/wo and ε/a are very s m a ll, i t is seen th a t the s e rie s re p re se n ta tio n fo r Fy converges very fa 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 ation f o r F x. In the above, we m ainly considered the three-dim ensional c o rre c tio n s f o r geometry ( A - i- 1 ). S im ila r approaches can be used to o th er c o n fig u ra tions.

-137- CONCLUSION In t h is work we stu d ie d the fo u r -la y e r s in g le -s id e d LIM used fo r p ro p u lsio n and suspension o f m a g n e tica lly le v ita te d v e h ic le s. The moving tra c k is assumed to be made o f a conductor w ith u n ia x ia l ~µ and ~σ. The source r ig id ly attached to t he bottom o f the v e h ic le has a given cu rre n t d is tr ib u t io n. F o u rie r tra n s f orm techniques in conjunctio n w it h t he tra n sfe r m a trix method are used to so lve t h is two-dimens io n a l boundary value problem. The machine performance in terms o f fo rc e s, e ffic ie n c y, and power fa c to r is given in in te g ra l form. The r e s u lts fo r t he id e a l source ~Κχ δ(ky - ky o) are obtained. (Here ~Κx is the F o u rie r component o f the given cu rre n t source). In a d d itio n to th is the a n a ly s is o f the integrands under the assumption o f th in or th ic k re a ctio n r a ils gives us the fo llo w in g co n clu sio n s: (a) Comparing w ith the case t hat t he source has back iro n, the geometry w ith region "4" being fre e space g ives much sm a lle r fo rce s and power, more se rio u s end e ffe c t a t the v e lo c ity where t he p ro p u lsio n fo rce is maximum. (b) I f region "1" is lam inated iro n, the e ffe c t o f the permeab i l i t y o f the re a ctio n r a il is small when t he re a ctio n r a il is th in. (c) I f region "1" is fre e space o r the re a ctio n r a il is th ic k, t he in d u ctio n motor can o n ly give re p u ls iv e supporting fo rce s when t he re a ctio n r a ils are non-ferrom agnetic. And th e fo rce s are la rg e o n ly in the low e ffic ie n c y regio n. On the co n tra ry, la rg e a ttra c tiv e supporting fo rce s are obtained in the high

-138- e ffic ie n c y region When the re a ctio n r a ils are ferrom agnetic. Thus, the a t t r a c t iv e scheme is p re fe ra b le, and ferrom agnetic tr a c k s are recommended. (d ) Small σν and 1/k y ol are require d to reduce the degrading o f the machine performance due to the end e ffe c t. However, σ is not allow ed being too sm a ll, otherw ise a s u f f ic ie n t prop u lsio n fo rce can not be obtained. A s u ita b le compromise is necessary and the composite tra c k is recommended. From the above in fo rm a tio n, u n su ita b le c o n fig u ra tio n s are discard ed. The machine performance in te g r a ls are evaluated fo r the prom ising geom etries and source d is t r ib u t io n The composite tra ck seems most prom ising. A p o s s ib le way o f computing the e ffe c t iv e ~μ and ~σ fo r a composite tra c k is sketched. From the a n a ly s is o f an extrem ely s im p lifie d geometry, the c o n d itio n s fo r the v a lid it y o f the e ffe c tiv e ~μ and ~σ concepts are determined to be ky oh <<1 and h/2do<< 1. Using a p a r a lle l approach, another p o s s ib le LIM c o n fig u ra tio n w ith r e la t iv e ly th ic k lam in a tio n s a t is fy in g h/2do π is analyzed. I t is found th a t t h is geometry has a very sm all end e ffe c t. F in a lly, a three-dim ensional c o rre c tio n is introduced to ta ke care o f the f in it e w idth o f the LIM. F o u rie r transform and F o u rie r s e rie s techniques are used to so lve th is p e rio d ic 3-dim ensional boundary value problem. A general a n a ly s is is given. The p ro p u lsio n

-139- fo rce and the la t e r a l fo rce are c a lc u la te d fo r several sp e c ia l cases. For a geometry s im ila r to ( A - i- 1 ), th is s o -c a lle d la t e r a l end e ffe c t g e n e ra lly in cre a se s the p ro p u lsio n fo rce s in the sm all v e lo c ity region. In the high v e lo c it y region ( i. e., v is clo se to ω/ k y o ), an e ffe c t iv e c o n d u c tiv ity concept can be developed and a decrease in the p ro p u lsio n fo rce is observed. As fo r the la t e r a l displacem ent o f the source, i t is found to be s ta b le. In t h is th e ses, we o n ly co n sid e r the case w ith s e rie s connected source. With m inor changes, the r e s u lts fo r the p a r a lle l o r s e r ie s - p a r a lle l h yb rid connected sources can be obtained. The machine p e rfo r mance is expected to be d r a s t ic a lly d iffe r e n t when the end e ffe c t is taken in to c o n s id e ra tio n. F u rth e r a n a ly s is o f t h is kind is thus h ig h ly recommended. Here, we m ainly co n sid e r a LIM w hich has a tra ck w ith a f in e ly d is tr ib u te d inhom ogeneity ( i. e., ky o h << 1, with h being the p e rio d o f the s tru c tu re ). I t appears th a t larg e supporting fo rce s e x is t o n ly w ith in a narrow v e lo c it y range and change la r g e ly w ith the v e lo c ity. Control systems are thus d i f f i c u l t to c o n s tru c t. I t is suspected th a t the opp o site s itu a t io n w ith kyoh 1 can p o s s ib ly overcome t h is d if f ic u lt y. F u rth e r a n a ly s is o f tra cks o f th is kind is suggested.

-140- REFERENCES 1. J. R. R e itz, e t al., "T echnical f e a s ib ilit y o f magnetic le v it a t io n as a suspension system fo r HSGT v e h ic le s ", Ford Motor C o., Tech. Rep. FRA-RT-72-40, produced under C o n tract DOT-FR-10026 w ith the U.S. D ept. o f T ra n sp o rta tio n, Feb. 1972. 2. Η. T. C offey e t al., "The f e a s ib ilit y o f m a g n e tica lly 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. B ocherts, e t al., "B a se lin e s p 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. M ichel P o lo u ja d o ff, "L in e a r in d u ctio n machines: I. H is to ry and theory o f o p e ra tio n ", IEEE Spectrum 8, 72 (1971). 5. Ib id, P a rt I I, " A p p lic a tio n s ", IEEE Spectrum 8, 79 (1971). 6. E. R. L a ith w a ite and S. A. Nasar, "Linear-m otion e le c t r ic a l m ach ines", P ro c. IEEE 58, 531 (1970). 7. E. R. L a ith w a ite, Induction Machines fo r S p e cia l Purposes (London: George Newms, 1966). 8. E. R. L a ith w a ite and F. T. B a rw e ll, "A p p lic a tio n o f lin e a r in ductio n motors to high-speed tra n s p o rt system s", Proc. IEE 116, 713 (1969). 9. D. F. W ilk ie, "Dynamics, c o n tr o l, and rid e q u a lit y o f a magneti - c a lly le v ita te d high-speed ground v e h ic le ", T ransport Research 1972. 10. E. R. L a ith w a ite, "The goodness o f a m achine", Proc. IEE 112, 538 (1965). 11. T. C. Wang, "L in e a r in d u ctio n 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. W hite, "T ra ctio n and normal fo rce s in the lin e a r in d u c tio n m otor", IEEE Trans. PAS-89, 638 (1970).

-141-13. T. C. Wang, J. W. Sm ylie and R. Y. P e i, "S in g le -s id e d lin e a r in d u ctio n m otor", in IEEE Conference Record o f 1971 6th Annual Mtg. o f the IEEE In d u stry & General A p p lic a tio n s Group, p. 1-10. 14. S. A. Nasar, L. Del C id, J r., "P ro p u lsio n and le v it a t o r fo rce s in a s in g le -s id e d lin e a r in d u ctio n motor fo r high-speed ground tra n s- p o rta tio n, Proc. IEEE 61, 638 (1973). 15. J. K. Dukowicz, "A n a ly sis o f lin e a r in d u ctio n machines in the d is c re te w indings and f in it e iro n le n g th ", G.M. research p u b lic a - tio n GMR-1426 (1973). 16. S. Yamamura, H. Ito, and Y. Ishikaw a, "Theories o f the lin e a r in - d u ction motor and compensated lin e a r in d u ctio n m otor", IEEE Trans. PAS-91, 1700 (1972). 17. Η. H. Kolm and R. D. Thornton, "E le ctrom agnetic f lig 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 in d u ctio n m achines; in p u t impedance and e q u iv a le n t c ir c u it s ", Proc. IEE 115, 1772 (1968). 19. A. L. C u lle n and T. H. Barton, "A s im p lifie d e lectro m ag n etic theory o f the in d u c tio n motor u sin g the concept o f wave im pedance", P ro c. IEE 105C, 331 (1958). 20. R. M. Fano, L. Chu, and R. A d le r, "E lectrom agnetic F ie ld s, Energy, and Forces (John W iley & Sons, I n c., 1960). 21. E. M ish k in, "Theory o f the s q u irre l-c a g e in d u ctio n motor derived d ir e c t ly from M axw ell's f ie ld e q u a tio n s", Quant. J. Mech. A p p l. Math 7, 472 (1954). 22. H. Bondi, K. C. M u kh e rji, "An a n a ly s is o f tooth r ip p le phenomenon in smooth lam in ate d po le sh o e s", P ro c. IEE (London) 104, 349 (1957). 23. R. L. R u s s e ll, K. H. Norsworthy, "Eddy cu rre n ts and w a ll lo sse s in scre e n e d -ro to r in d u ctio n m otors", Proc. IEE 105A, 163 (1958).

-142- APPENDIX A Let K and K' be the co o rd in a te systems which are a t r e s t w ith re sp e ct to the source and the re a c tio n r a i l, r e s p e c tiv e ly. Then, a fte r n e g le ctin g the r e l a t i v i s t i c e ffe c t and the displacem ent c u rre n t, M axw ell's equations w ith in the re a c tio n r a il are given as K: (A -1a) (A -1b) K' : (A -1c) (A -1d) (A -1e) (A -1f) We a ls o 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)

-143- This is the f a m ilia r M inkow ski's fo rm u la tio n. However, because there is moving magnetic m a te ria l p re se n t, as f a r as those q u a n titie s such as fo rce d e n sity, energy, e tc. are concerned, t h is fo rm u la tio n is not c le a r enough to e x p la in e ve ry th in g. Thus, an a lte r n a tiv e c a lle d E-H fo rm u latio n introduced by Chu(20) is p re fe ra b le, i. e., we have: (A-4a) (A-4b) w ith (A-4c) (A-4d) The s u b s c rip t "c" is introduced to d is tin g u is h those f ie ld s from the o rd in a ry f ie ld s in M inkow ski's fo rm u la tio n. A c tu a lly, w ith a s u ita b le tra n sfo rm a tio n, i t can be found th a t these two fo rm u latio n s are e q u iv a le n t. (A-5a) (A-5b) (A-5c) (A-5d) In the E-H fo rm u la tio n, the fo rce d e n sity can be expressed as : (A-6a) And the in p u t power d e n sity is given as:

(A-6b) Then the to ta l fo rce and the to ta l power in p u t can be obtained by s u ita b le volume in te g ra tio n s w ith in the re a c tio n r a i l. F u rth er in fo rm atio n about machine performance can thus a ls o be obtained. -144- Unfo r tu n a te ly, those in te g ra tio n s are te d io u s. A sim p le r in te g ra tio n can g e n e ra lly be obtained by in tro d u c in g a s tre s s energy te n so r T to transform the volume in te g ra l in to a su rfa ce in te g r a l. (A-7a) w ith (A-7b) I t should be n o tice d th a t except a t the fre e space where D = εoeand B = μoη, t h is s tre s s energy te n so r is d iffe r e n t from the f a m ilia r M inko w ski's te n so r. As fo r the power in p u t, i t seems to be much sim p le r to evalu ate i t d ir e c t ly a t the source which is lo cate d in fre e space, i. e., (A-7b)

-145- APPENDIX B C o n d itio n s f o r Thin and T hick Reaction R a ils Here we w ill say a re a c tio n r a il is th in when the co n d itio n th a t γ2h2 << 1 is s a t is f ie d around a region where the source fu n c tio n is most s ig n if ic a n t. S im ila r ly, the co n d itio n γ2h2 >> 1 i s require d fo r the th ic k re a c tio n r a i l. Now th a t we know th a t the source fu n ctio n is g e n e ra lly concentrated w ith in the region ω/ν ky 0, we w ill f i r s t co n sid e r th is re g io n. Now (B -1a) (B -1b) With ky = a ω/ν, the o n ly minimum o f γ2 w ill occur a t: (B-2a) where (B-2b) And the corresponding minimum value is (B- 3a) or (B-3b)

o r -146- (B-3c) Now le t us co n sid e r two d iffe r e n t cases: ( i ) R 1/4, then am > 1/2, and (B-4a) ( i i ) R 1/4, then 4R/27 can be considered as sm a ll. Thus (B-4b) Thus, the co n d itio n f o r γ 2 h 2 > > 1 ( i. e., th ic k re a c tio n r a ils ) can be re w ritte n as: sm a lle r o f (B-5a) S im ila r ly, the c o n d itio n fo r γ2h2 << 1 ( i. e., th in re a c tio n r a ils ) is (B-5b) A c tu a lly, f o r the le ft-h a n d e d sid e s o f values o f the o rd er o f are g e n e ra lly good enough to g ive accurate approximatio n s. However, in some cases, even r e la t iv e ly relaxed c o n d itio n s are d i f f i c u l t to be met. And, c o n d itio n s cannot be s a t is f ie d f o r the whole range o f 0 ky ω/ν. Under t h is s it u a t io n, g e n e ra lly these co n d itio n s

-147- can be fu r th e r re la xed. As we understand, the source fu n c tio n is g e n e ra lly concentrated around ky o. I t seems th a t we need o n ly to co n sid e r γ2h2 around ky o, i. e., l γ20h2. And, the co n d itio n s γ20h2 << 1 and γ20h2 >> 1 can thus be used as the c r it e r ia to determ ine whether a re a ctio n r a il is th in o r th ic k. And d e f in it e ly, those c o n d itio n s are much e a s ie r to be s a t is f ie d. For example, we can co n sid e r a case w ith ωμ2zσx ~ 6 0, ω/ν ~ 3 0 0. Then, i t can be shown th a t even the value o f h2 = 5 x 10-3 cannot s a t is f y co n d itio n (B-5b). However, as we know, in most o f the p r a c tic a l cases kyo is about 15. Thus, the region fo r which we are mos t concerned c o rre s - ponds to ky around 15. For thos e γ 2 h 2 is a b o u t 0.3 fo r h2 = 5 x 10-3, which i s good enough to say th a t t he p la te is t hin fo r t his more relaxed c r it e r io n.

-148- APPENDIX C The E va lu a tio n o f the In te g ra l w ith a complex α Using a s u ita b le v a ria b le change, we have (C -1a) w ith (C -1b) F u rth e r, s 1(α, ky o, L) can be re w ritte n as: (C -1c) (C-2b) Now we w ill assume Im α > 0. Then, by the using o f Jo rd a n 's lemma and re s id u e theorem, i t can be shown th a t: (C-3a) (C-3b) (C-3c) So, f in a lly, we have

-149- (C-4a) or (C-4b) A s im ila r procedure can be used to e valu a te the in te g ra l w ith Im α < 0 to get: (C-4c)

-150- APPENDIX D The E valu atio n of the In te g ra l (D -1) w ith a complex α. s2 (α, ky o, L) can be re w ritte n as: (D-2a) (D-2b) U sing Appendix C, we have (D-3a) (D-3b) For I2, changes o f v a ria b le s w ill g ive us: (D-4a)