Levitation of superconductive cable in earth magnetic field

Le i a ion o supe conduc i e cable in ea h magne ic ield 267 LEVITATION OF SUPERCONDUCTIVE CABLE IN EARTH MAGNETIC FIELD P. Ka ban a) , I. Doležel b) , B. Ul ych a) a) Facul y o Elec ical Enginee ing UWB, Sady P a icá ník 14, 314 06 Plze, E-mail:{ ul ych, ka ban}@k e.zcu.cz b) Ins i u e o The momechanics ASCR, Dolejško a 5, 182 00 P aha 8, E-mail: [email protected] Summa y The pape ep esen s an in oduc o y s udy abou a supe conduc i e cable le i a ing in Ea h’s magne ic ield. Buil a e wo ma hema ical models o he p oblem p o iding bo h he shape o he a c o he cable and o ces ac ing along i . The heo e ical analysis is supplemen ed wi h an illus a i e example. 1. INTRODUCTION Ea h’s magne ic ield is p oduced by o a ion o liquid and elec ically conduc i e shell 3 o Ea h’s co e 2 wi h espec o ela i ely unmo able shell 1, see Fig. 1a (ano he iew is in Fig. 1b). Fig. 1a: Ea h magne ic ield This ield whose s eng h on he no he n hemisphe e eaches alues be ween 38–56 A/m (which depends on he s uc u e o he co esponding li hosphe ic pla es) ep esen s one o e y impo an a ibu es o he plane Ea h. The human popula ion employs his ield • unconsciously – he ield ep esen s an “umb ella” p o ec ing Ea h’s su ace agains he impac o elec ically cha ged pa icles a i ing o Ea h in he consequence o he sola ac i i y and also om space and • consciously – Ea h’s magne ic ield is used o o o ien a ion on Ea h’s su ace – he i s magne ic compasses we e buil in China, abou 600 yea s be o e Ch is , o geological su ey – subsu ace deposi s o e omagne ic o es p oduce local anomalies in Ea h’s magne ic ield, o de e mina ion o he ime scales in geology and also in a cheology. Nowadays, howe e , we can o en see a ious conside a ions aimed a i s non adi ional employmen . Men ioned can be, o example, „ e he s“ (pa icula s can be ound in [1] and [2]), which a e supe conduc i e cables unwound in a sui able di ec ion om a ious space objec s (sa eli es, he las sec ions o boos e ocke s, space labs e c.) ha can be used as: • Elec ic ol age sou ce – a sui ably o ien ed conduc o o leng h l mo ing a a eloci y in magne ic ield o lux densi y B induces ol age ( ) i0 d l u = × ⋅  B l [3]. T anspo o such a cable o he o bi is much cheape han anspo o classical pho o ol aic cells. • A sui ably o ien ed conduc o o leng h l ca ying cu en I and mo ing in magne ic ield o lux densi y B is a ec ed by he Lo en z o ce o alue ( ) L0 d l I= ×  F l B [3] ha deccele a es o accele a es he space objec and shi s i o a lowe o highe o bi . These non adi ional space applica ions ha a e in ensi ely s udied and now al eady also expe imen ally alida ed (see, o example, [1] – p ojec s TSS-1a Oidipus-C, NASA+I alian Space Agency) could also be ans e ed o Ea h’s su ace. The pape deals wi h one possible applica ion – possibili y o employmen o Ea h’s magne ic ield o le i a ion o a supe conduc i e cable closely abo e Ea h’s su ace, which could be used, o ins ance, in me eo ology, whe e he le i a ing cable could eplace he ial balloons, pa icula ly o in es iga ion o lowe laye s o he a mosphe e. Fig. 1b: Ea h magne ic ield wi h he indica ed posi ion o he cable (see Figs. 3a, b) Ad ances in Elec ical and Elec onic Enginee ing 268 2. FORMULATION OF THE PROBLEM Fig. 2. depic s an a angemen o a ypical su- pe conduc i e cable o ea hly condi ions. A simila cable could play a ole in he conside ed case. Fig. 2: A angemen o he supe conduc i e cable I s supe conduc i e sys em consis s o a g ea e amoun o hin Cu ubes 2 illed in wi h he ac ual supe conduc o and placed in Ke la shell 1. The shell ep esen ing he ca ying elemen is lown h ough liquid He 3 ha secu es he supe conduc i e egime o he cable. The cable o s a ing leng h 0 2 l ≡ s is loca ed on Ea h’s su ace (nea he equa o ) in he ho izon- al posi ion be ween poin s P and Q (see Fig. 1b) whose abscissa is o ien ed pe pendicula ly o he o ce lines o Ea h’s magne ic ield. Nea he equa- o we can conside Ea h’s magne ic ield app oxi- ma ely pa allel o Ea h’s su ace, so ha i s lux densi y 0 B = z B (see Fig. 3a). Fig. 3a: The in es iga ed a angemen o a supe conduc - ing cable in he domain wi h uni o m lux densi y – s a ing posi ion The poin s P and Q ep esen he eel d ums equipped wi h b akes. The cable is wound on hem and a e hei b aking o i can eely unwind o a gene al leng h 2 s l > (Fig. 3b). I he cable ca ies cu en I in he indica ed di ec ion, i begins o be a ec ed by he o al Lo en z o ce ( ) L d l l I − = ×  F l B o ien ed in di ec ion y . This o ce hen li s he cable upwa ds o a gen- e al posi ion ( ) y x = co esponding o he “b aked o ” leng h s o he cable. In his posi ion, howe e , he speci ic Lo en z o ce ac ing on he uni leng h o he cable has gene ally wo componen s ( ) L 0 L, 0 L, x y = ± +x y . Thus, he le i a ion e ec o Ea h’s magne ic ield can be expec ed o de- c ease wi h g owing leng h s o he cable.            Fig. 3b: The in es iga ed a angemen – inal posi ion o he cable The aim o he pape is o e alua e he • dependence ( ) y x = on he “b aked o ” leng h s o he cable and on he alue I o he supe conduc i e cu en , • dependence o he dis ibu ion o o ces ac ing on he cable on he same pa ame e s, • es ic ion o he conside ed le i a ing e ec by he “b aked o ” leng h s and cu en I . 3. MATHEMATICAL MODELS OF THE PROBLEM AND ITS SOLUTION The ma hema ical desc ip ion o he p oblem may be ca ied ou in wo ways: • Solu ion o a nonlinea o dina y di e en ial equa ion o mula ed as an ini ial p oblem and exp essing he o ce and o que balance in an in- ini esimal elemen o he cable. • Solu ion o a sys em o algeb aic equa ions exp essing only he balance o o ces in a ini e elemen o he cable ( he o que balance is he e sa is ied au oma ically) Each o hese wo models has i s ad an ages and d awbacks. The di e en ial model allows using o nume ical algo i hms o p o essionally sophis ica ed unc ion p ocedu es p o iding bo h con e gence and equi ed accu acy o solu ion o he co esponding equa ion. On he o he hand, he algeb aic model is mo e lexible and p o ides an easie ealiza ion o some pa ial, speci ic compu a ions such as local dis ibu ion o ex e nal o ces, supp ession o de- o ma ion o pa s o he supe conduc i e cable e c. Tha is why we used bo h models ha p o ided a good acco dance o he esul s. Le i a ion o supe conduc i e cable in ea h magne ic ield 269 3.1. Di e en ial ma hema ical model Fo ob aining he di e en ial equa ion we s a om Fig. 4a and Fig. 4b showing he li ed cable and si ua ion in i s elemen . Fig. 4a: The li ed cable wi h an elemen He e 1 2 , H H deno e he ho izon al componen s o o ces a poin s 1 and 2, 1 2 , V V he e ical com- ponen s and q is he uni weigh o he cable. Fi s le us exp ess he pa icula componen s o he Lo en z o ces. F om Fig. 4b we can easily de- i e ha L L 2 2 1 d d d d 1 1 x y y' F BI s , F BI s y' y' = − ⋅ = ⋅ + + (1) Fig. 4b: De ailed si ua ion in an elemen be ween poin s 1 and 2 (see Fig. 2a) He e 1 2 , H H deno e he ho izon al componen s o o ces a poin s 1 and 2, 1 2 , V V he e ical com- ponen s and q is he uni weigh o he cable ha can gene ally be qui e nonuni o m. Fi s le us exp ess he pa icula componen s o he Lo en z o ces. F om Fig. 4b we can easily de- i e ha L L 2 2 1 d d d d 1 1 x y y' F BI s , F BI s y' y' = − ⋅ = ⋅ + + (1) and as '2 d 1 d s y x = + we immedia ely ha e L L d d d d x y F BI y, F BI x = − = . (2) The balance equa ions o he o ces in he ele- men ead 1 L 2 d x H F H + = , 1 L 2 d d y V F q s V − − = (3) and o he o que wi h espec o poin 2 L 1 L 1 d d d d d d d d 2 2 2 y x x y x F V x F q s H y ⋅ − + ⋅ = ⋅ − . (4) As a any poin o he cable V Hy' = , we im- media ely ha e ( ) d d d d V Hy' H y' y' H = = ⋅ + (5) whe e (see (3)) 2 1 L d d x H H H F = − = , 2 1 L d d d y V V V q s F = − = − . A e subs i u ing om (2) and o d s we inally ha e 2 1 d d d d ' q y x BI x H y' y' BI y + − = ⋅ + and hence ( ) 2 2 1 1 ' q y BI y' H y'' + − + = ⋅ , (6) which ep esen s a nonlinea di e en ial equa ion desc ibing he esul an shape o he cable. Fo ce H is gi en as (see (2)) 0 H BIy H = + (7) whe e 0 H is he ho izon al ension in he cable a i s beginning ( x l = ± ). The i s ini ial condi ion ead ( ) ( ) 0 y l y l = − = , he second one is he known leng h s o he cable. 3.2. Algeb aic ma hema ical model Conside an a angemen o he supe conduc i e cable in Ca esian coo dina e sys em as is depic ed in Fig. 3a. In di ec ion x he dis ance / 2 l is di- ided in o e N uni o m elemen s x ∆ (Fig. 5a) wi h an equi alen numbe o co esponding nonuni o m elemen s i s ∆ on he cu e ( ) s y x ≈ desc ibing he shape o a c o he cable. To a gene al dis ance i x we assign he gene al leng h i s o he co esponding a c o he cable.                Fig. 5a: Dis ibu ion o o ces on he cable The pa i s o he cable is a ec ed by he ol- lowing o ces (Fig. 5b): Ad ances in Elec ical and Elec onic Enginee ing 270 • s0 F , s, i F – in e nal o ces p oducing ension in he cable, • L, i – he Lo en z o ces ac ing pe pendicula ly on pa icula elemen s i s ∆ , • g, i F – weigh o pa icula elemen s i s ∆ , • ex F – ex e nal o ce loading he cable. Fig. 5b: Fo ces in an elemen o he cable Acco ding o Figs. 5a, 5b we ha e s0 0 s0 ex 0 ex s, 0 s , 0 s , s0 s0 ex ex g, 0 g , g , g s , s, s , s, , , , , , , , cos , sin . i x i i i i i i x i i i i i i F F F F F F F F F s F F ψ ψ ψ ψ α α = − = = + = = = = ∆ = = x x F F F F F F F ψ ψψ ψ ψ ψψ ψ ψ ψψ ψ (8) A he same ime he e holds 2 2 e 1 , , sin , cos . i i i i i i i x s x N x s s ψ ψ α α ∆ = ∆ = ∆ + ∆ ∆∆ = = ∆ ∆ (9) The condi ions o balance o pa icula o ces now ead 0 L , s , 1 ex L , s , g , 1 1 : 0, : 0. i s x k x i k i i k i k k k x F F F F ψ ψ ψ ψ = = = − + + = − + + =    (10) A e subs i u ion om (8) and (9) o (10) we ob ain s0 s, 1 s0 1 s, sin cos 0 sin cos i k k i i k i k k k ii F BI s F F BI s F α α α α = = − + ∆ + =  − ∆ =   (11) Analogously o he componen s in di ec ion ψ we ha e ex g S, 1 1 s, ex g 1 1 cos sin 0 cos / sin .(12) i i k i k i i k k i i i k i k i k k F BI s s F F F BI s s α α α α = = = = − ∆ + ∆ + =    = − + ∆ − ∆           Compa ison o (11) and (12) p o ides s0 1 ex g 1 1 sin cos cos sin i k k k i i i k i k k k i F BI s F BI s s α α α α = = = − ∆ = − + ∆ − ∆    (13) The algo i hm o solu ion o his ma hema ical model may be ealized in he ollowing s eps: • Fo some alue o s0 F solu ion o (13) by, o example, he Regula Falsi me hod using (9), p o ides, by means o [4], he alue o i ψ . • Solu ion o (11) wi h espec o (8) p o ides o he alue i ψ he o ce S, i F . • The i s wo s eps a e epea ed o all alues o e 1, , i N =K ; a he same ime we calcula e he alue o e 1 N k k s = ∆  ha is necessa y o co ec ion o s0 F . • Condi ion e 1 N k k s s = ∆ =  hen p o ides (again using he Regula Falsi me hod) he alue o s0 F . 4. COMPUTER MODEL, ACHIEVED AC- CURACY OF SOLUTION Solu ion o he ma hema ical model 3.1 (equa- ion (6)) was ca ied ou by he ou h-o de Runge- Ku a me hod w i en in Ma lab. P o ed was e y good con e gence o he nume ical p ocess – o inding o ( ) y x in he in e al 0 50 x l ≤ ≤ = m wi h accu acy o 3 alid digi s we needed only 50 s eps. Solu ion o he ma hema ical model 3.2 was pe - o med by a use p og am w i en in Bo land Delphi. The con e gence o he nume ical p ocess o ind- ing bo h ( ) y x and ( ) S x F was good again. Accu- acy o 3 alid digi s was eached in abou 200 s eps. 5. ILLUSTRATIVE EXAMPLE 5.1. Technical speci ica ion Conside ed is an a angemen o he supe con- duc ing le i a ing cable acco ding o Fig. 2. I s di- mensions and all physical pa ame e s a e lis ed in Tab. 1. I is necessa y o ca y ou such a se o es - ing compu a ions ha would p o ide answe s o ques ions in pa ag aph 2. 5.2. Resul s and hei discussion Dependencies ( ) y x ≡s on he b aked-o leng h o he cable and cu en I ollow om Fig. 6. Table 2: Physical pa ame e s o he cable quan i y symbol uni alue Le i a ion o supe conduc i e cable in ea h magne ic ield 271 dis ance PQ 2 l m 100 leng h o b aked-o cable s m 120 130 140 ex e nal o ce ex F N 0 (*) speci ic weigh g N/m 2.5 Ea h’s magne ic lux densi y [6] B T 5 5 10 − ⋅ cu en in he cable I A 5 5 10 ⋅ 6 10 6 1.5 10 ⋅ * Ex e nal o ce is no conside ed a his s age o esea ch 0 10 20 30 40 50 0 10 20 30 40 50 x (m) y (m) s = 120 m s = 130 m s = 140 m Fig. 6: Dependence o he li y o he cable on leng h s ( 6 1.5 10 I = ⋅ A) We can see ha he li inc emen o he cable dec eases wi h he leng h s o he b aked-o cable. I is ob iously caused by he change o o ien a ion o ec o ( ) L s along he leng h s when his leng h changes, see Figs. 8 and 9. Inc ease o li ( ) y x could be, o cou se, achie ed by diminishmen o speci ic mass g o he cable, by g ow h o cu en I and, especially, by g ow h o dis ance PQ 2 l = . 3600 3700 3800 3900 4000 4100 4200 4300 4400 0 10 20 30 40 50 x (m) F s (N) s = 120 m s = 130 m s = 140 m Fig. 7: Dependence o he in e nal o ce s F on leng h s ( 6 1.5 10 I = ⋅ A) E alua ion o dis ibu ion o he o ces ac ing on he cable on i s b aked-o leng h s and cu en I can be ca ied ou om Figs. 7, 8 and 9. 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 x (m) Lx (N/m) s = 120 m s = 130 m s = 140 m Fig. 8: Dependence o componen L x o he Lo en z o ce L on leng h s ( 6 1.5 10 I = ⋅ A) 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 x (m) L,y (N/m) s0 = 120 m s0 = 130 m s0 = 140 m Fig. 9: Dependence o componen L, y o he Lo en z o ce L on leng h s ( 6 1.5 10 I = ⋅ A) Fig. 7 shows ha in e nal o ce s s F = F p o- ducing ension in he cable s ongly depends on he b aked-o leng h s o he cable. This is e iden ly associa ed wi h he balance o o ces ac ing on he cable. O ien a ion o he ec o o speci ic o ce L,s igu ing in he balance changes wi h s and depends on i s alue. Bu i s alue is ( om he iew- poin o s eng h o Ke la whose Young modulus is [5] 11 1.24 10 E= ⋅ N/m 2 ) qui e accep able. Figs. 8 and 9 show he quali a i e and quan i a- i e changes o ec o L,s p oduced by il ing o he cable wi h g ow h o leng h s . The ec o is always pe pendicula o he angen o cu e ( ) y x ≡s, which esul s in dec ease o he componen ( ) L,y s , i.e. he le i a ion componen o he o al Lo en z o ce L,y F . This ac ep esen s a ce ain es ic ion o le i a ion e ec s, bu his can be compensa ed – as said abo e – by p olonging o he o iginal dis- ance o he cable 2 l . Res ic ion o he conside ed le i a ion e ec by he alue o b aked-o leng h s o he cable may be e alua ed om Figs. 10, 11 and 12. Ad ances in Elec ical and Elec onic Enginee ing 272 0 10 20 30 40 100 105 110 115 120 125 130 135 s(m) y max = y ( x =0) (m ) 0 0,5 1 1,5 2 2,5 F s,max = F s (x =0) (kN) y max Fs(x=0) Fig. 10: Dependence o he maximum li max y and in e - nal o ce s,max F on leng h s o he cable ( 5 5 10 I = ⋅ A) 0 10 20 30 40 50 100 105 110 115 120 125 130 135 140 145 s (m) y max = y ( x =0) (m) 0 1 2 3 4 5 F s,max = F s (x=0) (kN) y max Fs(x=0) Fig. 11: Dependence o he maximum li max y and in e - nal o ce s,max F on leng h s o he cable ( 6 10 I= A) 0 10 20 30 40 50 100 110 120 130 140 150 s (m) y max = y(x=0) (m) 0 1 2 3 4 5 6 7 8 F s,max = F s (x=0) (kN) y max Fs(x=0) Fig. 12: Dependence o he maximum li max y and in e nal o ce s,max Fon leng h s o he cable ( 6 1.5 10 I = ⋅ A) The abo e h ee igu es show ha he le i a ion e ec (in he a angemen o pa ame e s in Tab. 1) is limi ed. F om ce ain alue o s ( o example o 5 5 10 I = ⋅ A om leng h 130 s ≈ m) he o ce ( ) s 0 x = F s ops changing and, he e o e, he bal- ance o o ces ac ing on he cable emains he same. Fu he inc ease o i s leng h s would lead o in- c ease o i s weigh , which would b eak he balance ha is he basic condi ion o success ul le i a ion. 6. CONCLUSION The pape shows ha he idea o a le i a ing su- pe conduc i e cable in Ea h’s magne ic ield is, om he iewpoin o he heo e ical p inciples, qui e eal. I s p ac ical ealiza ion would equi e, howe e , solu ion o a numbe o echnological p oblems (a su icien ly i m and ligh supe conduc- i e cable, easy gene a ion o high cu en , anspo o liquid He o he unwound cable e c.) and also p oblems o economic cha ac e . Fu he esea ch o heo e ical ques ions in he domain should be aimed a • possibili ies o inc ease o he pe pendicula componen L y F o he Lo en z o ce by inc ease o s i ness o a ce ain pa o he cable, • solu ion o he ask as a ully 3D p oblem ( e- spec ing o e ec s o wind e c.). Acknowledgemen This wo k has been inancially suppo ed om he G an Agency o he Czech Republic (p ojec No. 102/04/0095). REFERENCES [1] Lo enzini, E., Sanma in, E.: Elec odynamic e he s – opes in space. Scien i ic Ame ican, July 2005. [2] Cosmo, M.L., Lo enzini, E.: “Te he s in space, “Handbook, a ailable on add ess h p://c a- www.ha a d.edu/spg oup/handbook.h ml. [3] Cha i, M.V.K., Salon, S.J.: “Nume ical me hods in elec omagne ism,” Academic P ess, 2000. [4] Ge ald, C.F., Whea lley, P.O.: „Applied nu- me ical analysis,” Pea son Addison Wesley, NY 2004. [5] www.azom.com. [6] G oup o au ho s, „The joy o knowledge: The ana omy o Ea h,“ London 1996, Czech ansla ion Alba os P aha 1995.