Downloaded fom obt.dtu.dk on: Feb 19, 218 Analyss of the magnetc feld, foce, and toque fo two-dmensonal Halbach cylndes Bjøk, Rasmus; Smth, Andes; Bahl, Chstan Publshed n: Jounal of Magnetsm and Magnetc Mateals Lnk to atcle, DOI: 1.116/j.jmmm.29.8.44 Publcaton date: 21 Lnk back to DTU Obt Ctaton APA: Bjøk, R., Smth, A., & Bahl, C. R. H. 21. Analyss of the magnetc feld, foce, and toque fo twodmensonal Halbach cylndes. Jounal of Magnetsm and Magnetc Mateals, 3221, 133-141. DOI: 1.116/j.jmmm.29.8.44 Geneal ghts Copyght and moal ghts fo the publcatons made accessble n the publc potal ae etaned by the authos and/o othe copyght ownes and t s a condton of accessng publcatons that uses ecognse and abde by the legal equements assocated wth these ghts. Uses may download and pnt one copy of any publcaton fom the publc potal fo the pupose of pvate study o eseach. You may not futhe dstbute the mateal o use t fo any poft-makng actvty o commecal gan You may feely dstbute the URL dentfyng the publcaton n the publc potal If you beleve that ths document beaches copyght please contact us povdng detals, and we wll emove access to the wok mmedately and nvestgate you clam.
Publshed n Jounal of Magnetsm and Magnetc Mateals, Vol. 322 1, 133 141, 214 DOI: 1.116/j.jmmm.29.8.44 Analyss of the magnetc feld, foce, and toque fo two-dmensonal Halbach cylndes R. Bjøk, C. R. H. Bahl, A. Smth and N. Pyds Abstact The Halbach cylnde s a constucton of pemanent magnets used n applcatons such as nuclea magnetc esonance appaatus, acceleato magnets and magnetc coolng devces. In ths pape the analytcal expesson fo the magnetc vecto potental, magnetc flux densty and magnetc feld fo a two dmensonal Halbach cylnde ae deved. The emanent flux densty of a Halbach magnet s chaactezed by the ntege p. Fo a numbe of applcatons the foce and toque between two concentc Halbach cylndes ae mpotant. These quanttes ae calculated and the foce s shown to be zeo except fo the case whee p fo the nne magnet s one mnus p fo the oute magnet. Also the foce s shown neve to be balancng. The toque s shown to be zeo unless the nne magnet p s equal to mnus the oute magnet p. Thus thee can neve be a foce and a toque n the same system. Depatment of Enegy Conveson and Stoage, Techncal Unvesty of Denmak - DTU, Fedeksbogvej 399, DK-4 Rosklde, Denmak *Coespondng autho: abj@dtu.dk 1. Intoducton The Halbach cylnde 1; 2 also known as a hole cylnde pemanent magnet aay HCPMA s a hollow pemanent magnet cylnde wth a emanent flux densty at any pont that vaes contnuously as, n pola coodnates, B em, = B em cospφ B em,φ = B em snpφ, 1 whee B em s the magntude of the emanent flux densty and p s an ntege. Subscpt denotes the adal component of the emanence and subscpt φ the tangental component. A postve value of p poduces a feld that s dected nto the cylnde boe, called an ntenal feld, and a negatve value poduces a feld that s dected outwads fom the cylnde, called an extenal feld. A emanence as gven n Eq. 1 can, dependng on the value of p, poduce a completely shelded multpole feld n the cylnde boe o a multpole feld on the outsde of the cylnde. In Fg. 1 Halbach cylndes wth dffeent values of p ae shown. The Halbach cylnde has pevously been used n a numbe of applcatons 3; 4, such as nuclea magnetc esonance NMR appaatus 5, acceleato magnets 6 and magnetc coolng devces 7. In these applcatons t s vey mpotant to accuately calculate the magnetc flux densty geneated by the Halbach cylnde. Thee exst seveal papes whee the magnetc feld and flux densty fo some pats of a Halbach cylnde ae calculated 8; 9; 1; 11, but a complete spatal calculaton as well as a detaled devaton of the magnetc vecto potental has pevously not been publshed. p = 1 p = 2 p = 2 ϕ p = 3 Fgue 1. The emanence of a p = 1, p = 2, p = 2 and p = 3 Halbach cylnde. The angle φ fom Eq. 1 s also shown.
Analyss of the magnetc feld, foce, and toque fo two-dmensonal Halbach cylndes 2/11 In ths pape we wsh to calculate the magnetc vecto potental and subsequently the magnetc flux densty at any pont n a two dmensonal space esultng fom a Halbach cylnde. Once the analytcal soluton fo the magnetc flux densty has been obtaned we wll poceed to calculate the foce and toque between two concentc Halbach cylndes. Fo p = 1 and a elatve pemeablty of 1 the moe complcated poblem of computng the toque between two fnte length concentc Halbach cylndes has been consdeed 12, and t s shown that a toque ases due to end effects. Howeve, nethe the feld no the toque s evaluated explctly. Below we show that fo specal values of p a nonzeo foce and toque may ase even n the two dmensonal case. Regon III Regon II µ = 8 Magnet Regon I R c µ = 8 R R o 2. Defnng the magnetostatc poblem The poblem of fndng the magnetc vecto potental and the magnetc flux densty fo a Halbach cylnde s defned n tems of the magnetc vecto potental equaton though the elaton between the magnetc flux densty, B, and the magnetc vecto potental, A, B = A. 2 If thee ae no cuents pesent t s possble to expess the magnetc vecto potental as 2 A = B em. 3 Fo the two dmensonal case consdeed hee the vecto potental only has a z-component, A z, and the above equaton, usng Eq. 1, s educed to 2 A z,φ = B em p + 1 snpφ. 4 Ths dffeental equaton consttutes the magnetc vecto potental poblem and must be solved. In the a egon of the poblem the ght hand sde educes to zeo as hee B em =. Once A z has been detemned Eq. 2 can be used to fnd the magnetc flux densty. Aftewads the magnetc feld, H, can be found though the elaton B = µ µ H + B em, 5 whee µ s the elatve pemeablty assumed to be sotopc and ndependent of B and H. Ths s geneally the case fo had pemanent magnetc mateals. 2.1 Geomety of the poblem Havng found the equaton govenng the magnetostatc poblem of the Halbach cylnde we now take a close look at the geomety of the poblem. Followng the appoach of Xa et al. 11 we wll stat by solvng the poblem of a Halbach cylnde enclosng a cylnde of an nfntely pemeable soft magnetc mateal, whle at the same tme tself beng enclosed by anothe such cylnde. Ths s the stuaton depcted n Fg. 2. Ths confguaton s mpotant fo e.g. moto applcatons. Fgue 2. A Halbach cylnde wth nne adus R and oute adus R o enclosng an nfntely pemeable cylnde wth adus R c whle tself beng enclosed by anothe nfntely pemeable cylnde wth nne adus R e and nfnte oute adus. The egons maked I and III ae a gaps. The Halbach cylnde has an nne adus of R and an oute adus of R o and the nne nfntely pemeable cylnde has a adus of R c whle the oute enclosng cylnde has a nne adus of R e and an nfnte oute adus. Late n ths pape we wll solve the magnetostatc poblem of the Halbach cylnde n a by lettng R c and R e. The use of the soft magnetc cylndes esults n a well defned set of bounday equatons as wll be shown late. Of couse one can also solve dectly fo the Halbach cylnde n a usng the bounday condtons specfc fo ths case. When solvng the magnetostatc poblem thee dffeent expessons fo the magnetc vecto potental, feld and flux densty wll be obtaned, one fo each of the thee dffeent egons shown n Fg. 2. The geomety of the poblem esults n sx bounday condtons. The equement s that the adal component of B and the paallel component of H ae contnuous acoss boundaes,.e. H I φ = = R c B I = Hφ I = Hφ II I = Hφ III = Hφ II = R = R = R o = R o R e H III φ = = R e. 6 The two equatons fo H φ = come fom the fact that the soft magnetc mateal has an nfnte pemeablty.
Analyss of the magnetc feld, foce, and toque fo two-dmensonal Halbach cylndes 3/11 2.2 Soluton fo the vecto potental The soluton to the vecto potental equaton, Eq. 4, s the sum of the soluton to the homogenous equaton and a patcula soluton. The soluton s A z,φ = n=1 A n n +B n n snnφ+b em snpφ, whee A n and B n ae constants that dffe fo each dffeent egon and that ae dffeent fo each n. Usng the bounday condtons fo the geomety defned above one can show that these ae only nonzeo fo n = p. Thus the soluton fo the defned geomety becomes A z,φ = A p +B p snpφ+b em snpφ, 8 whee A and B ae constants that dffe fo each dffeent egon and that ae detemned by bounday condtons. The soluton s not vald fo p = 1. Fo ths specal case the soluton to Eq. 4 s nstead 7 A z,φ = A + B 1 snφ B em lnsnφ, 9 whee A and B ae defned lke fo Eq. 8. Note that fo p = we have that B em, = B em and B em,φ = n Eq. 1. Ths means that A z = and consequently B s zeo eveywhee. The magnetc feld, H, howeve, wll be nonzeo nsde the magnetc mateal tself,.e. n egon II, but wll be zeo eveywhee else. We now deve the constants n Eq. 8 and 9 dectly fom the bounday condtons. 3. Devng the vecto potental constants The constants of the vecto potental equaton can be deved fom the bounday condtons specfed n Eq. 6. We fst deve the constants fo the case of p 1. Fst we note that the magnetc flux densty and the magnetc feld can be calculated fom the magnetc vecto potental B = 1 A z φ B φ = A z 1 H = B B em, µ µ H φ = 1 µ µ B φ B em,φ. 1 Pefomng the dffeentaton gves [ ] B = pa p 1 + pb p 1 p + B em cospφ [ ] B φ = pa p 1 + pb p 1 1 B em snpφ [ p H = A p 1 + B p 1 µ µ + B ] em p µ µ 1 cospφ [ p H φ = A p 1 + B p 1 µ µ B ] em 1 µ µ 1 snpφ. 11 Usng the adal component of the magnetc flux densty and the tangental component of the magnetc feld n the set of bounday equatons we get a set of sx equatons contanng the sx unknown constants, two fo each egon. The constants A and B wll be temed A I and B I n egon I, A II and n egon II, and A III and I n egon III. Intoducng the followng new constants a = R2p e R 2p o R 2p e + R 2p o b = R2p R 2p c R 2p + R 2p c the constants ae detemned to be and = A I = µ a 1 µ a+1 R 2p o R 1 p o R 1 p R 2p + R 2p c, 12 µ b 1 µ b+1 R 2p 1 µ b 1 µ b + 1 B I = A I R 2p c A II = µ a 1 µ a + 1 R 2p o A III = R 2p o + R 2p e 1 µ a 1 µ a + 1 B em, 13 B em R1 p o I = A III R 2p e. 14 Usng these constants n Eq. 8 and 11 allows one to calculate the magnetc vecto potental, the magnetc flux densty and the magnetc feld espectvely. The constants ae not vald fo p = 1. The soluton fo ths case wll be deved n a late secton. 3.1 Halbach cylnde n a We can fnd the soluton fo a Halbach cylnde n a f we look at the soluton fo R e and R c. Lookng at the
Analyss of the magnetc feld, foce, and toque fo two-dmensonal Halbach cylndes 4/11 pevous expesson fo the constants a and b we see that fo p > 1 : fo p < : a 1 b 1 a 1 b 1 n the lmt defned above. Ths means that the constant now becomes R 1 p o R 1 p Bem µ 1 µ+1 = R 2p o µ+1 µ 1 R 2p p 1 p > 1 R 1 p o R 1 p Bem µ+1 p 1 p < µ 1 R 2p o µ 1 µ+1 R 2p and the emanng constants fo p > 1 become A I R 2p 1 µ +1 µ 1 B I A II A III = µ 1 µ +1 R 2p o I whle fo p < they become A I B I B A II II A III = I B II R 2p o 1 µ 1 µ +1 1 µ 1 B em p 1 R1 p o µ +1 µ +1 µ 1 R 2p o B em p 1 R1 p o 1 µ +1 µ 1 15 16 17 18 Ths s the soluton fo a Halbach cylnde n a. Note that the soluton s only vald fo µ 1. In the specal case of µ = 1 the constants can be educed even futhe. 3.2 Halbach cylnde n a and µ = 1 We now look at the specal case of a Halbach cylnde n a wth µ = 1. Ths s a elevant case as e.g. the hghest enegy densty type of pemanent magnet poduced today, the so-called neodymum-on-boon NdFeB magnets, have a elatve pemeablty vey close to one: µ = 1.5 13. Usng the appoxmaton of µ 1 fo a Halbach cylnde n a educes the constant to =. 19 The emanng constants depend on whethe the Halbach cylnde poduces an ntenal o extenal feld. Fo the ntenal feld case, p > 1, the constant A II wll be gven by A II = B em R1 p o. 2 The constant A I detemnng the feld n the nne a egon s equal to A I = B em R 1 p R 1 p o. 21 The emanng constants, B I, A III and I ae zeo. Usng Eq. 11 the two components of the magnetc flux densty n both the cylnde boe, egon I, and n the magnet, egon II, can be found. B I = B p 1 emp R 1 R o p 1 cospφ R B I φ = B p 1 emp R 1 R o p 1 snpφ R = B emp p 1 1 cospφ R o φ = B em p 1 1 p snpφ. 22 Consdeng now the extenal feld case, p <, the constant A II s gven by A II = B em R1 p. 23 The constant A III detemnng the feld n the oute a egon s gven by A III = B em Ro p 1 R p 1. 24 The emanng constants, A I, B I and I ae zeo. Agan usng Eq. 11 we fnd the two components of the magnetc flux densty n egon II and III to be R R o I = B emp 1 R o p+1 Ro cospφ R I φ = B em p 1 R o p+1 Ro snpφ = B emp φ = B em p+1 p+1 p+1 R 1 cospφ p+1 R 1 p snpφ. 25 The equatons fo I and I φ ae dentcal to the expessons fo B I and B I φ n Eq. 22 except fo a mnus sgn n both equatons.
Analyss of the magnetc feld, foce, and toque fo two-dmensonal Halbach cylndes 5/11 3.3 The constants fo a p = 1 Halbach cylnde Havng detemned the soluton to the vecto potental equaton and found the constants n the expesson fo the magnetc flux densty and the magnetc vecto potental fo a Halbach cylnde both n a and enclosed by a soft magnetc cylnde fo all cases except p = 1 we now tun to ths specfc case. Ths case s shown n Fg 1. We have aleady shown that the soluton to the vecto potental poblem fo ths case s gven by Eq. 9. The bounday condtons ae the same as pevous,.e. they ae gven by Eq. 6. In ode to fnd the constants the components of the magnetc feld and the magnetc flux densty must be calculated fo p = 1 as the bounday condtons elate to these felds. Usng Eq. 1 we obtan B = [A + B 2 B em ln]cosφ B φ = [ A + B 2 + B em ln + 1]snφ H = 1 [ A + B 2 B em ln + 1 ] cosφ µ µ H φ = 1 [ A + B 2 + B em ln ] snφ.26 µ µ Usng these expessons fo the magnetc flux densty and the magnetc feld we can agan wte a set of sx equatons though whch we can detemne the sx constants, two fo each egon. Rentoducng the two constants fom Eq. 12 a = R2 e R 2 o R 2 e + R 2 o b = R2 R2 c R 2 + R2 c, 27 the followng equatons fo the constants ae obtaned: A I = R 2 + R2 c 1 µ b 1 µ b + 1 B I = A I R 2 c A II = µ a 1 µ a + 1 R 2 o + B em lnr o aµ 1 = aµ + 1 R 2 o µ b 1 1 µ b + 1 R 2 R B em ln R o A III = R 2 e + R 2 1 µ a 1 o µ a + 1 I = A III R 2 e. 28 We see that the constants A I, B I, A III and I ae dentcal to the constants n Eq. 14. The magnetc flux densty and the magnetc feld can now be found though the use of Eq. 26. 3.4 Halbach cylnde n a, p = 1 We can fnd the soluton fo a p = 1 Halbach cylnde n a f we look at the soluton fo R e and R c. In ths lmt the pevously ntoduced constants ae educed to a 1 b 1. 29 The expessons fo the constants can then be educed to A I = R 2 1 µ + 1 µ 1 B I = A II = µ 1 µ + 1 R 2 o + B em lnr o µ 1 = µ + 1 R 2 o µ + 1 1 µ 1 R 2 R B em ln R o A III = I = 1 µ 1. 3 µ + 1 Agan we see that the constants A I, B I, A III and I ae equal to the constants n Eq. 17. Ths soluton s vald fo all µ except µ = 1. Combnng the above constants wth Eq. 26 we see that the magnetc flux densty n the cylnde boe s a constant, and that ts magntude s gven by B I µ 1 = µ + 1 R 2 o µ + 1 1 µ 1 R 2 µ + 1 µ 1 1 R 2 R B em ln, 31 fo µ 1. 3.5 Halbach cylnde n a, p = 1 and µ = 1 Fo the specal case of µ = 1 fo a p = 1 Halbach cylnde n a the constants can be educed futhe to A I Ro = B em ln R A II = B em lnr o B I,,A III,I =. 32 Combnng the above constants wth Eq. 26 one can fnd the magnetc flux densty n the boe, egon I, and n the magnet, egon II, B I Ro = B em ln cosφ R B I Ro φ = B em ln snφ R Ro = B em ln cosφ φ = B em ln Ro 1 R o snφ. 33
Analyss of the magnetc feld, foce, and toque fo two-dmensonal Halbach cylndes 6/11 As fo the case of µ 1 the magnetc flux densty n the cylnde boe s a constant. The magntude of the magnetc flux densty n the boe s gven by B I Ro = B em ln. 34 R 4 Analytcal soluton whch we ecognze as the well known Halbach fomula 2. 3.6 Valdty of the solutons To show the valdty of the analytcal solutons we compae these wth a numecal calculaton of the vecto potental and the magnetc flux densty. We have chosen to show a compason between the expessons deved n ths pape and numecal calculatons fo two selected cases. These ae shown n Fg. 3 and 4. In Fg. 3 the magntude of the magnetc flux densty s shown fo a enclosed Halbach cylnde. Also shown n Fg. 3 s a numecal calculaton done usng the commecally avalable fnte element multphyscs pogam, Comsol Multphyscs 14. The Comsol Multphyscs code has pevously been valdated though a numbe of NAFEMS Natonal Agency fo Fnte Element Methods and Standads benchmak studes 15. As can be seen the analytcal soluton closely matches the numecal soluton. In Fg. 4 we show the magnetc vecto potental, A z, as calculated usng Eqs. 8 and 18 compaed wth a numecal Comsol smulaton. As can be seen the analytcal soluton agan closely matches the numecal soluton. We have also tested the expessons fo the magnetc flux densty gven by Xa et. al. 24 11 and compaed them wth those deved n ths pape and wth numecal calculatons. Unfotunately the equatons gven by Xa et. al. 24 11 contan eoneous expessons fo the magnetc flux densty of a Halbach cylnde n a wth µ = 1 as well as fo the expesson fo a Halbach cylnde wth ntenal feld enclosed by soft magnetc mateal. 4. Foce between two concentc Halbach cylndes Havng found the expessons fo the magnetc vecto potental and the magnetc flux densty fo a Halbach cylnde we now tun to the poblem of calculatng the foce between two concentc Halbach cylndes, e.g. a stuaton as shown n Fg. 5. In a late secton we wll calculate the toque fo the same confguaton. Ths confguaton s nteestng fo e.g. moto applcatons and dves as well as applcatons whee the magnetc flux densty must be tuned on and off wthout the magnet beng dsplaced n space 7. The foce between the two Halbach cylndes can be calculated by usng the Maxwell stess tenso, T, fomulaton. The foce pe unt length s gven by F = 1 µ S T ds. 35 y [mm] y [mm] 2 2 4 4 2 2 4 x [mm] 4 2 2 Numecal soluton 4 4 2 2 4 x [mm] Fgue 3. Colo onlne Compang the analytcal soluton as gven by Eq. 11 and 14 wth a numecal soluton computed usng Comsol. Shown ae contous of B = [.3,.5,.7,.9] T fo an ntenal feld p = 2 enclosed Halbach cylnde wth dmensons R c = 1 mm, R = 2 mm, R o = 3 mm, R e = 4 mm, and B em = 1.4 T, µ = 1.5. The solutons ae seen to be dentcal. The shaded aeas n the fgues coespond to the smla shaded aeas n Fg. 2.
Analyss of the magnetc feld, foce, and toque fo two-dmensonal Halbach cylndes 7/11 4 Analytcal soluton n a 2.6.8.6.4.2.8.6 φ.6.4 y [mm] 2.8.6.4.2.2.2.4.6.8.6 y [mm].6 4 4 2 2 4 x [mm] 4 2 2.6.8.6 Numecal soluton.6.6.4.4.6.4.8.4.6.8 4 4 2 2 4 x [mm] Fgue 4. Colo onlne Compang the analytcal soluton as gven by Eqs. 8 and 18 wth a numecal soluton computed usng Comsol. Shown ae contous of A z = ±[.2,.4,.6,.8]V s m 1 fo an extenal feld p = 2 Halbach cylnde n a wth dmensons R = 2 mm, R o = 3 mm and B em = 1.4 T, µ = 1.5. The ed contous ae postve values of A z whle the blue ae negatve values. As wth Fg. 3 the solutons ae seen to be dentcal..6.6 Fgue 5. An example of a concentc Halbach cylnde confguaton fo whch the foce and toque s calculated. The oute magnet has p = 2 whle the nne magnet s a p = 2. The nne magnet has also been otated an angle of φ = 45. The dotted ccle ndcates a possble ntegaton path. The Catesan components of the foce ae gven by F x = 1 T xx n x + T xy n y ds µ S F y = 1 T yy n y + T yx n x ds, 36 µ S whee n x and n y ae the Catesan components of the outwads nomal to the ntegaton suface and whee T xx, T yy and T xy ae the components of the Maxwell stess tenso whch ae gven by T xx = B 2 x 1 2 B2 x + B 2 y T yy = B 2 y 1 2 B2 x + B 2 y T xy, T yx = B x B y. 37 When usng the above fomulaton to calculate the foce a closed ntegaton suface n fee space that suounds the object must be chosen. As ths s a two dmensonal poblem the suface ntegal s educed to a lne ntegal along the a gap between the magnets. If a ccle of adus s taken as the ntegaton path, the Catesan components of the outwads nomal ae gven by n x = cosφ n y = snφ. 38 Expessng the Catesan components though the pola com-
Analyss of the magnetc feld, foce, and toque fo two-dmensonal Halbach cylndes 8/11 ponents as B x = B cosφ B φ snφ B y = B snφ + B φ cosφ, 39 the elaton fo computng the foce pe unt length becomes 2π 1 F x = µ 2 B2 B 2 φ cosφ B B φ snφ dφ F y = 2π 1 µ 2 B2 B 2 φ snφ + B B φ cosφ dφ, 4 whee s some adus n the a gap. The computed foce wll tun out to be ndependent of the adus as expected. We consde the scenao whee the oute magnet s kept fxed and the ntenal magnet s otated by an angle φ, as shown n Fg. 5. Both cylndes ae centeed on the same axs. Both of the cylndes ae consdeed to be n a and have a elatve pemeablty of one, µ = 1, so that the magnetc flux densty s gven by Eqs. 22 and 25 fo p 1. Fo p = 1 Eq. 33 apples nstead. As µ = 1 the magnetc flux densty n the a gap between the magnets wll be a sum of two tems, namely a tem fom the oute magnet and a tem fom the nne magnet. If the elatve pemeablty wee dffeent fom one the magnetc flux densty of one of the magnets would nfluence the magnetc flux densty of the othe, and we would have to solve the vecto potental equaton fo both magnets at the same tme n ode to fnd the magnetc flux densty n the a gap. Assumng the above equements the flux densty n the a gap s thus gven by B = I,1 + B I,2 B φ = I φ,1 + B I φ,2, 41 whee the second subscpt efes to ethe of the two magnets. The nne magnet s temed 1 and the oute magnet temed 2, e.g. R o,1 s the nne magnets oute adus. The ntege p 1 thus efes to the nne magnet and p 2 to the oute magnet. Thee can only be a foce between the cylndes f the nne cylnde poduces an extenal feld and the oute cylnde poduces an ntenal feld. Othewse the flux densty n the gap between the magnets wll be poduced solely by one of the magnets and the foce wll be zeo. Pefomng the ntegals n Eq. 4 one only obtans a nonzeo soluton fo p 1 = 1 p 2 and p 2 > 1. In ths case the soluton s F x = 2π µ Kcosp 1 φ F y = 2π µ Ksnp 1 φ, 42 whee K s a constant gven by K = B em,1 B em,2 R p 1,2 Rp 1 o,2 Rp 2 o,1 Rp 2,1. 43 F x and F y [N m 1 ].8.6.4.2.2.4.6.8 1 x 15 F x,ana F y,ana F x,num F y,num 1 6 12 18 24 3 36 φ [degee] Fgue 6. The two catesan components of the foce pe unt length gven by Eq. 42 compaed wth a Comsol calculaton fo a system whee the oute magnet has p 2 = 2, R,2 = 45 mm, R o,2 = 75 mm and B em,2 = 1.4 T and the nne magnet has p 1 = 1, R,1 = 15 mm, R o,1 = 35 mm and B em,1 = 1.4 T. The analytcal expesson s n excellent ageement wth the numecal data. The foce s pe unt length as we consde a two dmensonal system. Notce that the foce s ndependent of, as expected. In Fg. 6 we compae the above equaton wth a numecal calculaton of the foce. The esults ae seen to be n excellent ageement. Notce that the foces neve balance the magnets,.e. when F x s zeo, F y s nonzeo and vce vesa. If p 2 = 1 the magnetc flux densty poduced by the oute magnet s not gven by Eq. 22 but s nstead gven by Eq. 33. Howeve ths equaton has the same angula dependence as Eq. 22 and thus the foce wll also be zeo fo ths case. 5. Toque between two concentc nested Halbach cylndes Havng calculated the foce between two concentc Halbach cylndes we now focus on calculatng the toque fo the same system. The toque can also be calculated by usng the Maxwell stess tenso, T, fomulaton. The toque pe unt length s gven by τ = 1 T ds µ S = 1 B nb 1 µ 2 B2 n ds, 44 S whee agan the ntegaton suface s a closed loop n fee space that suounds the object. Agan choosng a ccle of adus as the ntegaton path, the elaton fo computng the
Analyss of the magnetc feld, foce, and toque fo two-dmensonal Halbach cylndes 9/11 toque pe unt length aound the cental axs becomes τ = 1 2π 2 B B φ dφ, 45 µ whee B and B φ ae the adal and tangental components of the magnetc flux densty n the a gap and s some adus n the a gap. Agan the computed toque wll be shown to be ndependent of the adus when vaes between the nne and oute ad of the a gap. We consde the same case as wth the foce calculaton,.e. the oute magnet s kept fxed, both magnets have the same axs, the ntenal magnet s otated by an angle φ and both of the cylndes ae consdeed to be n a and have a elatve pemeablty of one. Agan thee can only be a toque between the cylndes f the nne cylnde poduces an extenal feld and the oute cylnde poduces an ntenal feld. To fnd the toque pe unt length we must thus ntegate τ = 1 µ 2π 2 I,1 + B I,2I φ,1 + B I φ,2dφ. 46 Ths ntegaton wll be zeo except when p 1 = p 2. Fo ths specal case the ntegal gves τ = 2π p 2 2 µ 1 p 2 K 1 K 2 snp 2 φ, 47 2 whee the constants K 1 and K 2 ae gven by K 1 = B em,2 R 1 p 2,2 R 1 p 2 o,2 K 2 = B em,1 R p 2+1 o,1 R p 2+1,1. 48 The valdty of ths expesson wll be shown n the next secton. It s seen that thee ae p 2 peods pe otaton. Fo p 2 = 1 the expesson fo the magnetc flux densty poduced by the oute magnet s not gven by Eq. 22 but nstead by Eq. 33, and so we must look at ths specal case sepaately. 5.1 The specal case of p 2 = 1 Fo the specal case of a p 2 = 1 oute magnet the flux densty poduced by ths magnet n the a gap wll be gven by Eq. 33. The extenal feld poduced by the nne magnet s stll gven by Eq. 25. Pefomng the ntegaton defned n Eq. 46 agan gves zeo except when p 2 = 1 and p 1 = 1. The expesson fo the toque becomes τ = π µ K 2 K 3 snφ 49 whee the two constants K 2 and K 3 ae gven by K 2 = B em,1 R 2 o,1 R 2,1 Ro,2 K 3 = B em,2 ln. R,2 5 Note that K 2 s dentcal to the constant K 2 n Eq. 48 fo p 2 = 1. We also see that Eq. 49 s n fact just τ = m B fo a dpole n a unfom feld tmes the aea of the magnet. Table 1. The paametes fo the two cases shown n Fg. 7 and 8. Magnet R R o p B em [mm] [mm] [T] Case 1: nne 5 15-2 1.4 oute 2 3 2 1.4 Case 2: nne 1 35-1 1.4 oute 45 75 1 1.4 5.2 Valdatng the expessons fo the toque We have shown that thee s only a toque between two Halbach cylndes f p 1 = p 2 fo p 2 >, wth the toque beng gven by Eq. 47 fo p 2 1 and Eq. 49 fo p 2 = 1. To vefy the expessons gven n Eq. 47 and Eq. 49 we have computed the toque as a functon of the angle of dsplacement, φ, fo the two cases gven n Table 1, and compaed ths wth a numecal calculaton pefomed usng Comsol. The esults can be seen n Fg. 7 and 8. τ [N] 8 6 4 2 2 4 6 Analytcal Numecal 8 6 12 18 24 3 36 φ [degee] Fgue 7. A numecal calculaton of the toque pe unt length between two concentc Halbach cylndes compaed wth the expesson gven n Eq. 47 fo the physcal popetes gven fo Case 1 n Table 1. The analytcal expesson s n excellent ageement wth the numecal data. τ s pe unt length as we consde a two dmensonal system. As can be seen fom the fgues the toque as gven by Eq. 47 and Eq. 49 ae n excellent ageement wth the numecal esults. 6. Foce and Toque fo fnte length cylndes The foce and toque fo fnte length cylndes wll be dffeent than the analytcal expessons deved above, because of flux leakage though the ends of the cylnde boe. To nvestgate the sgnfcance of ths effect thee dmensonal numecal smulatons of a fnte length system coespondng to the system shown n Fg. 6 has been pefomed
Analyss of the magnetc feld, foce, and toque fo two-dmensonal Halbach cylndes 1/11 τ [N] 3 2 1 1 2 Analytcal Numecal 3 6 12 18 24 3 36 φ [degee] Fgue 8. The toque pe unt length gven by Eq. 49 compaed wth a numecal calculaton fo the physcal popetes gven fo Case 2 n Table 1. As wth the case fo p 2 1,.e. Fg. 7, the analytcal expesson s n excellent ageement wth the numecal data. τ s pe unt length as we consde a two dmensonal system. usng Comsol. Fo ths system the foce has been calculated pe unt length fo dffeent lengths. The esults of these calculatons ae shown n Fg. 9. Fom ths fgue t can be seen that as the length of the system s nceased the foce becomes bette appoxmated by the analytcal expesson of Eq. 42. A shot system poduces a lowe foce due to the leakage of flux though the ends of the cylnde. Howeve, even fo elatvely shot systems the two-dmensonal esults gve the ght ode of magntude and the coect angula dependence of the foce. Smlaly, the toque fo a thee dmensonal system has been consdeed. Hee the system gven as Case 1 n Table 1 was consdeed. Numecal smulatons calculatng the toque wee pefomed, smla to the foce calculatons, and the esults ae shown n Fg. 1. The esults ae seen to be smla to Fg. 9. The toque appoaches the analytcal expesson as the length of the system s nceased. As befoe the two dmensonal esults ae stll qualtatvely coect. Above we have consdeed cases whee the two dmensonal esults pedct a foce p 1 = 1 p 2 o a toque p 1 = p 2. Howeve, fo fnte length systems a foce o a toque can be pesent n othe cases. One such case s gven by Mhochan et al. 12 who epot a maxmum toque of 12 Nm fo a system whee both magnets have p = 1, ae segmented nto 8 peces and whee the oute magnet has R,2 = 52.5 mm, R o,2 = 11 mm, L 2 = 1 mm and B em,2 = 1.17 T and the nne magnet has R,1 = 47.5 mm, R o,1 = 26 mm, L 1 = 1 mm and B em,1 = 1.8 T. Ths toque s poduced manly by the effect of fnte length and to a lesse degee by segmentaton. The toque poduced by ths system s 12 N pe unt length, whch s sgnfcant compaed to the expected analytcal value of zeo. The toque fo fnte F x and F y [N m 1 ].8.6.4.2.2.4.6.8 1 x 15 1 6 12 18 24 3 36 φ [degee] F x,ana F y,ana L =.5*R o,2 L = 2*R o,2 L =.25*R o,2 L = 1*R o,2 L = 1*R o,2 Fgue 9. The two catesan components of the foce pe unt length fo a thee dmensonal system wth dmensons as those gven n Fg. 6. The analytcal expessons as well as the esults of a thee dmensonal numecal smulaton ae shown. τ [N] 8 6 4 2 2 4 6 8 6 12 18 24 3 36 φ [degee] Analytcal L =.25*R o,2 L =.5*R o,2 L = 1*R o,2 L = 2*R o,2 L = 1*R o,2 Fgue 1. The toque pe unt length fo a thee dmensonal system wth dmensons as those gven as Case 1 n Table 1. The analytcal expessons as well as the esults of a thee dmensonal numecal smulaton ae shown. length systems wth p 1 p 2 s, as noted above, a hghe ode effect. Ths makes t sgnfcantly smalle pe unt length than fo the coespondng system wth p 1 = p 2. 1 ths system, whch s desgned to have a toque, poduce a lage toque even though the system s much smalle. The end effects due to a fnte length of the system can
Analyss of the magnetc feld, foce, and toque fo two-dmensonal Halbach cylndes 11/11 be emeded by seveal dffeent technques. By coveng the ends of the concentc cylnde wth magnet blocks n the shape of an equpotental suface, all of the flux can be confned nsde the Halbach cylnde 16. Unfotunately ths also blocks access to the cylnde boe. The homogenety of the flux densty can also be mpoved by shmmng,.e. placng small magnets o soft magnetc mateal to mpove the homogenety 17; 18; 19. Fnally by slopng the cylnde boe o by placng stategc cuts n the magnet the homogenety can also be mpoved 2. Howeve, especally the last two methods can lowe the flux densty n the boe sgnfcantly. 7. Dscusson and concluson We have deved expessons fo the magnetc vecto potental, magnetc flux densty and magnetc feld fo a two dmensonal Halbach cylnde and compaed these wth numecal esults. The foce between two concentc Halbach cylndes was calculated and t was found that the esult depends on the ntege p n the expesson fo the emanence. If p fo the nne and oute magnet s temed p 1 and p 2 espectvely t was shown that unless p 1 = 1 p 2 thee s no foce. The toque was also calculated fo a smla system and t was shown that unless p 1 = p 2 thee s no toque. We compaed the analytcal expessons fo the foce and toque to numecal calculatons and found an excellent ageement. Note that ethe thee can be a foce o a toque, but not both. The deved expessons fo the magnetc vecto potental, flux densty and feld can be used to do e.g. quck paamete vaaton studes of Halbach cylndes, as they ae much moe smple than the coespondng thee dmensonal expessons. An nteestng use fo the deved expessons fo the magnetc flux densty would be to deve expessons fo the foce between two concentc Halbach cylndes, whee one of the cylndes has been slghtly dsplaced. One could also consde the effect of segmentaton of the Halbach cylnde, and of couse the effect of a fnte length n geate detal. Both effects wll n geneal esult n a nonzeo foce and toque fo othe choces of p 1 and p 2, but as shown these wll n geneal be smalle than fo the p 1 = 1 p 2 and p 1 = p 2 cases. It s also woth consdeng computng the foce and toque fo Halbach cylndes wth µ 1. Hee one would have to solve the complete magnetostatc poblem of the two concentc Halbach cylndes to fnd the magnetc flux densty n the gap between the cylndes. Acknowledgements The authos would lke to acknowledge the suppot of the Pogamme Commsson on Enegy and Envonment EnM Contact No. 214-6-32 whch s pat of the Dansh Councl fo Stategc Reseach. [2] K. Halbach, Nucl. Instum. Methods 169 198. [3] Z.Q. Zhu and D. Howe, IEE Poc. Elec. Powe. Appl. 148 4 21, 299. [4] J. M. D. Coey, J. Magn. Magn. Mate. 248 22, 441. [5] S. Appelt, H. Kühn, F. W Häsng, and B. Blümch, Nat. Phys. 2 26, 15. [6] J.K. Lm, P. Fgola, G. Tavsh, J.B. Rosenzweg, S.G. Andeson, W. J. Bown, J. S. Jacob, C. L. Robbns, and A.M. Temane, Phys. Rev. ST - Accel. Beams 8 25, 7241. [7] A. Tua and A. Rowe, Poc. 2nd Int. Conf. on Magn. Refg. at Room Temp., Potooz, Solvena, IIF/IIR:363 27. [8] Z.Q. Zhu, D. Howe, E. Bolte, and B. Ackemann, IEEE Tans. Magn. 29 12 1993, 124. [9] K. Atallah, D. Howe, and P.H. Mello, Eghth Int. Conf. on Elec. Mach. and Dv. Conf. Publ. No.444 1997, 376. [1] Q. Peng, S. M. McMuy, and J.M.D. Coey, IEEE Tans. Magn. 39 42 23, 1983. [11] Z.P. Xa, Z.Q. Zhu, and D. Howe, IEEE Tans. Magn. 4 24, 1864. [12] T. R. N Mhochan, D. Weae, S. M. McMuy, and J. M. D. Coey, J. Appl. Phys. 86 1999, 6412. [13] Standad Specfcatons fo Pemanent Magnet Mateals, Magn. Mate. Pod. Assoc., Chcago, USA 2. [14] COMSOL AB, Tegnègatan 23, SE-111 4 Stockholm, Sweden. [15] Comsol, Comsol Multphyscs Model Lbay, thd ed. COMSOL AB, Chalmes Teknkpak 412 88 G 25. [16] E. Potenzan, J. P. Clake, and H. A. Leupold, J. Appl. Phys. 61 1987, 3466. [17] M. G. Abele, H. Rusnek, and W. Tsu, J. Appl. Phys. 99 8 26, 93 [18] R. Bjøk, C. R. H. Bahl, A. Smth, and N. Pyds, J. Appl. Phys., 14 28, 1391 [19] A. Rowe, and A. Tua, J. Magn. Magn. Mate. 32 28, 1357. [2] J. E. Hlton, and S.M. McMuy, IEEE Tans. Magn., 43 5 27, 1898 Refeences [1] J. C. Mallnson, IEEE Tans. Magn. 9 4 1973, 678.