Design and Application of a Two Pole Spherical Permanent Magnet Motor

Transcription

1 TEM Journl. Volume 7, Issue, Pges 5-58, ISSN 7-89, DOI:.8/TEM7-7, Februr 8. Desgn nd Applcton of Two Pole Sphercl Permnent Mgnet Motor Sbel Akk O, Osmn Gürdl Fts Fcult of Mrne Scences, Ordu Unverst,Ordu,5 Turke Fcult of Technolog, G Unverst, Ankr, Turke Abstrct In ths stud, n r core sphercl motor comprsng two pole permnent mgnet rotor wth 8 mm dmeter nd multpole sttor s desgned b consderng the ese of mnufcturng prortes. Sttor wndng dmensons re clculted n w to gve mmum torque to the rotor nltcll. Mgnetc feld dstrbuton nd torque vlues re clculted nltcll. Sttor cols n the oke mde from Delrn re postoned n proper vlues. Sphercl motor wth three poles sttor nd two poles rotor cn move up nd down n 6 muthl ngles wthout drect rottonl movement n n open loop sstem nd t s sutble to use n cscde n robotc sstem. Kewords Sphercl Motor; Permnent Mgnet; Geodesc Arc Length; Torque.. Introducton In the ndustrled world tod, mn sstems used from mnufcturng to the other sub-sectors re shftng to robotc pplctons v utomton sstems. Due to these developng nd rpdl growng technologes, trend n whch robotc pplctons show progress hs formed. In order to meet these needs of the ndustr, ctutors used n these robotc sstems re needed to be desgned ccordng to the needs. Desgnng sphercl motors tht cn rotte wth nd degrees of freedom offers gret dvntges n ths feld. The mjor dvntges of sphercl motors re fst recton tmes, hgh effcences nd hvng hgh precson moton [,]. DOI:.8/TEM7-7 Correspondng uthor: Sbel Akk O, Fts Fcult of Mrne Scences, Ordu Unverst, Ordu, Turke Eml: sbelkko@gml.com Receved: December 7. Accepted: Februr 8. Publshed: Februr 8. 8 Sbel Akk O; publshed b UIKTEN. Ths work s lcensed under the Cretve Commons Attrbuton-NonCommercl-NoDervs. Lcense. The rtcle s publshed wth Open Access t When the development of sphercl motors n tme s nvestgted, Wllms nd Lthwte s stud on nducton motors wth vrous speeds n 95s hs enbled nducton motor to be developed wth sphercl geometr []. Bsed on Wllms nd Lthwte s studes, Vchtsevnos et l. desgned sphercl motor wth degrees of freedom []. Then the performed electromgnetc nlss of ths motor b usng del dstrbuted current model [5]. Kneko et l. developed sphercl DC servo motor tht cn move wth three degrees of freedom [6]. Ths rotor ssembl s performed round the center shft. Becuse of the lmted movement re of DC sphercl motor nd comple mechncl structure of sphercl nducton motor nd ts mnufcturng dffcultes, Lee et l. ntroduced sphercl step motor concept nd desgned vrble sphercl step motor tkng the step motor s bss [7,8]. Lee et l. ntroduced dnmc model of sphercl motor wth vrng reluctnce n terms of torque model nd rotor dnmcs [9]. In order to plce more thn twent poles equll onto the sem sphercl sttor, Lee nd Kwn developed progrm nd desgned the rotor poles wth two permnent mgnets to provde three degrees of freedom movement []. Lee nd Kwn lso developed theor bsed on locl ntercton between the rotor nd the sttor whch re close to ech other. B nlng the knemtc reltonshp between the rotor nd the sttor poles close to ech other, Lee et l. stted tht severl equll spced sttor poles re requred for moton control []. Lee nd Kwn suggested tht the sustnblt model s n mportnt concept n order to predct the torque produced b the vrble reluctnce sphercl motor []. On ths concept, Pe mde more theoretcl reserch nd Roth nd Lee eplned ths model epermentll [,]. B et l. presented new method, referred here s bck- EMF method, for sensng the mult-dof moton of permnent mgnet sphercl motor []. In ths stud, n r core sphercl motor s desgned. The rotor of the sphercl motor s two poles NdFeB permnent mgnet. The sttor s composed of three cols. The stud s mostl focused on determnng prtculrl the sttor concl col ngles nltcll. In order to determne ζ nd ζ ngles determnng the optmum geometrcl shpe of the cols, n equton s defned. To crete the TEM Journl Volume 7 / Number / 8. 5

2 TEM Journl. Volume 7, Issue, Pges 5-58, ISSN 7-89, DOI:.8/TEM7-7, Februr 8. torque model of the desgned motor, torque clcultons re performed.. Mgnetc feld dstrbuton Mgnetc feld dstrbuton generted b the permnent mgnet n sphercl motor desgn forms the bss n the motor desgn optmton nd developng the control secton []. Mgnetc feld s dvded nto two ones. The frst one s the r gp outsde the rotor. The second one s the poles of the permnent mgnet. Mgnetc flu denst n the r gp outsde the rotor []: B µ o H I () where μo s the mgnetc permeblt of the r, B s the mgnetc flu denst nd H s the mgnetc feld strength. Mgnetc flu denst n the permnent mgnet poles s []: B µ o µ r H II + µ o M o () where, μr s the reltve permeblt of the mgnet. M o B rem s the permnent mgnetton. µ o For the mgnetc feld, H () Mgnetc feld strength s defned s the grdent of sclr potentl of Ф nd shown s n Eq. (). H Ф () As result of the mplementton of Lplce equton, the followng feld equtons re obtned []. In permnent mgnet; Ф (. M o )/µᵣ (5) In r; Ф I (6) The rdl component of the mgnetton vector s epressed now s follows []. M r ( )ᴾ ¹ M o cos ϕ α o π (p ) snθ P (7) where p s the totl pole number of the permnent mgnet. Mgnetc flu denst n the r gp one cn be epressed s follows b consderng the boundr condtons. B Ir µ o dф I dr where the sclr potentl s []: m ( n+ ) { } + ξ r { Υ ( θ φ) } ( n+ ) m Υ ( θ, φ) m n nl n Φ ξ r, I m nl (8) (9) Y n m (θ, ϕ)where the sphercl hrmonc functon [] nd defned s follows. Y m n (θ, ϕ) n+ (n m)! [P π (n+m)! n m ()]e mϕ () where P m n () shows the mth degree Legendre functon n nth row nd n m n whle n nd m m re ntegers. ξ ni Epresson n Eq. (9) s constnt whch wll be determned b the boundr condtons [] The boundr condtons re s follows. m Epnson of the coeffcent ξ ni s gven n Eq. (). B Ir r R o B IIr r R o H Iϕ r R o H IIϕ r R o nd H Iθ r R o H IIθ r R o () ξ m ni C n,m d n ξ m ni C n, m d n () where dn d n /d n. Cnm s determned b the surfce ntegrl, s shown below []. C nm M o where π π (n m)! f(ϕ)]e mϕ dϕ n+ [P (n+m)! n m ]e mϕ sn θdθ () π f(φ) ( )ᴾ cos φ α π (p ) dφ () ± b π f(ϕ)]e mϕ dϕ (5) where the boundr condtons of ϕ re π (p ) + α α < φ < π (p ) + α + α. The vlue s not seen to be n the vlues of m, ±, ±8, ±, ±6,.. π (n m)! c/ π n+ [P π (n+m)! n m ()]e mϕ sn θdθ (6) where the boundr condtons of θ re π β < θ < π + β. For the vlues of n ±, ±, ±6,.., the result of c s found. For odd numbers c. Wheren,, b nd c re rel numbers. 5 TEM Journl Volume 7 / Number / 8.

3 TEM Journl. Volume 7, Issue, Pges 5-58, ISSN 7-89, DOI:.8/TEM7-7, Februr 8.. Torque model Whle the torque model s desgned, Lorent force lw s utled. Due to the ntercton between the rotor mgnetc feld nd the sttor wndngs n whch current flows, torque s generted on the rotor []. T r (J B)dv v (7) where J s the current denst. If t s ntegrted over the whole col volume, the torque equton cn be wrtten s below. The smbol c represents the crculr loop n the wndng []. R ζ ζ T c J rb Ir (r, θ, ϕ)dl rdrdζ (8) R c If the torque produced b th wndng s nmed s T, θ nd ϕ represent the s poston of the wndng. R s the dstnce between the nner surfce of the wndng nd the center, R s the dstnce from the outer surfce of the wndng to the center, ζ s the ngulr dmeter of the r core, ζ s the ngulr dmeter of the wndngs. Output torque n sngle col s epressed s the followng equton [5]. Tble. The propertes of ths prototpe Sttor rdus 5.7 (mm) Rotor rdus (mm) Polr ngle of the α8, permnent mgnet 9 The number of sttor poles The number of rotor poles Mmum tlt ngle ± Mmum torque (unm) In Fgure, smple structure of the motor on the drwng edtor n Ansoft Mwell D cn be seen. A prototpe of the sphercl motor desgned s shown n Fgure. There re three sttor poles nd ech hs three cols (col, col nd col ). These cols generte electromgnetc forces long the sttor. The rotor of the motor lso hs two poles. T T c G(θ, ф )J (9) where: T [T, T, T ] T where, G(θ, ф ) g (θ, ф ), g (θ, ф ), g (θ, ф ) T Fgure. Bsc structure of the motor g Τ Τc g g ( θ, φ ) g ( θ, φ )... g ( θ N, φn ) J ( ) ( ) ( ) ( ) ( ) ( ) θ, φ g θ, φ... g θ N, φn J θ, φ g θ, φ... g θ, φ J N N N (). Prototpe development Snce the sphercl motor n ths performed desgn hs n r core, sttor prt of the desgned sphercl motor s mde b Delrn mterl. Sttor poles re plced wth 5 ngle to the s nd wth ngle to the s. There s. mm r gp between the sttor poles nd rotor poles. The propertes of ths prototpe re shown n Tble. () TEM Journl Volume 7 / Number / 8. 55

4 TEM Journl. Volume 7, Issue, Pges 5-58, ISSN 7-89, DOI:.8/TEM7-7, Februr 8. end rd of the wndngs (r-r). Snce D s multpler n torque equton, the mmum vlue of the torque cn be clculted wth the second dervtve of D n terms of ζ nd ζ. D s epressed s n the followng equton []. { ζ ζ +.5( sn ζ sn ζ )} D πβrem J () B keepng ζ constnt, the frst nd the second dervtves of D re tken s shown below. (b) Fgure. Sphercl motor prototpe: () The prototpe of motor, (b) Sphercl motor control crcut The mgnetc crcut would be short ccordng to the mnmum reluctnce prncple. The flu denst of the motor s shown n Fgure. The drectons of the mgnetc flu denst re s shown n the Fgure (), the sttor s strght from the rotor. D ζ D ζ ζ + sn ζ cosζ () ζ + sn ζ + snζ () The grphs drwn ccordng to the bove equtons re gven n Fgure. nd Fgure 5. For the mmum torque, ζ nd ζ must be ζ nd ζ π/. Snce ζ s not possble, ζ should be kept s smll s possble. Becuse of the phscl condtons t s tken s ζ7 Fgure. D '' ζ grph () (b) Fgure. The flu denst of the motor: () Vector representton, (b) Mgnetc representton Determnton of col concl ngle Mmum torque (Tm) s relted wth the concl ngles of sttor wndngs (ζ - ζ) nd the strt nd '' Fgure 5. D ζ grph Snce ζ π/ s not possble, to fnd the vlue of ζ ngle whch cn generte mmum torque, geodesc rc length formuls re used n sphercl coordnte sstem. In the sphercl coordnte sstem; r r snθ cosφ + r snθ sn φ + r () r To fnd the ngle (φ) between two rdl vectors s, n rdus r defned s bove, dot product s 56 TEM Journl Volume 7 / Number / 8.

5 TEM Journl. Volume 7, Issue, Pges 5-58, ISSN 7-89, DOI:.8/TEM7-7, Februr 8. used. r ( r snθ cosφ + r snθ snφ + r ) ( r snθ cosφ + r snθ snφ + r ) cosϕ r ( r ) ( r (snθ cosφ snθ cosφ + snθ snφ snθ snφ + ) r ] cosϕ r r [snθ snθ (cosφ cosφ + snφ snφ ) + ) cos( φ φ ) r [snθ snθ cos( φ φ ) + ] (5) Fgure 6. Col prmeter ζ vs Sζ (ζ 7 ) A result of the bove equton, rc length (s) cn be found s below: s r r cos [snθ snθ cos( φ φ ) + ] (6) Arc lengths for three cols re found b usng the bove equton. Ths found rc length lso gves ζ ngle necessr to produce mmum torque n three wndngs. s cos [cos 5 cos 5 + sn 5 sn 5 cos( )] 5,76 5, ,56 π (7) where ζ 7,76. Becuse of the phscl condtons t s tken s ζ7. S chnges dependng on the col ngles of ζ nd ζ nd S s gven s n equton below. S (sn ζ sn ζ ) (sn5 ζ sn 5 ζ ) (8) S vlue n Eq. (8) chnges the torque output. Accordngl, how the col concl ngles we wll use n ths motor desgn wll chnge the torque output cn be seen n the grphs. In Fgure 6, how S vlue nd thus the torque output decrese wth the ncrese of ζ ngle cn be seen. In Fgure 7, how S vlue nd thus the torque output ncrese wth the ncrese of ζ ngle cn be seen. Fgure 7. Col prmeter ζ vs Sζ (ζ ) Solvng the torque ntegrl For n, m, torque clculton s mde b usng the bove equtons. Where p s tken s p. For n nd m vlues,, b, 5 c 5 π, Y (θ, φ) 8 5 8π sn θe φ re found. When the vlues re substtuted n Eq.(8). B Ir(,) 75 M 56 d r 6 sn θcosφ s found. Substtutng these obtned results n Eq. (8): Τ 75 ζ Jµ Μ drc 56 ζ.( cosψ e snψ e )} dζ φ θ π sn θ cos φr sn ζdψ (9) cn be wrtten. Where, R c R R dr. d R 6. The epresson n the ntegrl cn (µᵣ+5 ) be defned s D (θ, ϕ) []. The ntegrted verson of ths epresson wll become s follows []. D e ( θ φ) π θ( ζ 7 5, sn sn + sn ζ )[ + snφe θ ] 5 cosφe Sζ sn ζ sn ζ sn5 ζ sn 5 ζ () φ () TEM Journl Volume 7 / Number / 8. 57

6 TEM Journl. Volume 7, Issue, Pges 5-58, ISSN 7-89, DOI:.8/TEM7-7, Februr 8. e Dc ( θ, φ ) π S( sn θ snφe + sn θ cosφe sn θ sn φe () T sn θsnφ T I T T c sn θcosφ J () T sn θsnφ where the epresson T c s gven s n equton below []. T c 75 5 Jµ M d R c Sπ () In Fgure 8, wth nltcl nd smulton results for the rotor when ϕ, θ nd θ re compred. Fgure 8. Torque vs. θr Torque vlue mesured s result of the pplcton s ppromtel µnm. In the clcultons, these vlues of μ o π -7 H/m, μ r,99 H/m, B rem, T re consdered. 5. Conclusons In ths stud, n r core sphercl motor composed of two poles permnent mgnet rotor nd multpole sttor s optmed nd desgned b consderng the producton ese. Mgnetc feld dstrbuton generted b the rotor poles of the sphercl motor nd the torque model of the sphercl motor re epressed. Another mportnt fctor ffectng the output torque s the col concl ngle determnng the col geometr. Col concl ngle of the desgned sttor wndngs re found n w to obtn mmum torque wth nltcl clculton b usng the geodesc rc length formuls b consderng the del nd phscl lmttons whch s the frst pproch n the lterture. References [] Gürdl, O. & Öner, Y. (5). Three Dmensonl Mgnetosttc Anlss nd Applcton of Iron Cored Permnent Mgnet Sphercl Actutor. J. Fc. Eng. Arch. G Unv.,, -. [] Öner, Y. (7). A permnent mgnet sphercl rotor desgn nd three dmensonl sttc mgnetc nlss. Sensors nd Actutors A: Phscl, 7(), -8. [] Wng, J., Jewell, G.W., Howe, D. (998). Anlss, desgn nd control of novel sphercl permnentmgnet ctutor. Electrc Power Applctons, IEE Proceedngs, 5(), 6-7. [] Vchtsevnos, G., Dve, K., & Lee, K. M. (987). Development of novel ntellgent robotc mnpultor. IEEE Control Sstems Mgne, 7(), 9-5. [5] Dve, K., Vchtsevnos, G. & Power, R. (987). The nlss of felds nd torques n sphercl nducton motors. IEEE Trnsctons on Mgnetcs, (), 7-8. [6] Kneko, K., Ymd, I., & Ito, K. (989). A sphercl DC servo motor wth three degrees of freedom. Journl of dnmc sstems, mesurement, nd control, (), 98-. [7] Lee, K. M., Vchtsevnos, G., & Kwn, C. K. (988, Aprl). Development of sphercl wrst stepper motor. In Proceedngs of the 988 IEEE lntemtonl Conference on Robotcs nd Automton, Phldelph, PA. Aprl (pp. 6-9). [8] Lee, K. M., Pe, J., & Roth, R. (99). Knemtc nlss of three-degrees-of-freedom sphercl wrst ctutor. Mechtroncs, (6), [9] Lee, K.M., Wng, X. & Wng, N.H. (99). Dnmc modelng nd control of bll-jont-lke vrblereluctnce sphercl motor. Smposum on Mechtroncs, ASME Wnter Annul Meetng, New Orlens, [] Lee, K.M. & Kwn, C.E. (99). Desgn concept development of sphercl stepper for robotc pplctons. IEEE Trnscton on Robotcs nd Automton, 7(), [] Lee, K. M., Pe, J., & Roth, R. (99). Knemtc nlss of three-degrees-of-freedom sphercl wrst ctutor. Mechtroncs, (6), [] Pe, J. (99). Methodolog of desgn nd nlss of vrble-relüktnce sphercl motors. Ph.D. thess, Dept. Mechncl engneerng, Georg Tech, USA. [] Roth, R. & Lee, K.M. (995). Desgn optmston of three degrees of freedom vrble reluctnce sphercl wrst motor. Trnsctons of the ASME Journl of Engneerng for Industr, 7(), [] Yn, L., Chen, I. M., Lm, C. K., Yng, G. & Lee, K. M. (). Desgn, Modelng nd Eperments of - DOF Electromgnetc Sphercl Actutors. Mechnsms nd Mchne, Sprnger Scence+Busness Med B. V., -5. [5] Yn, L., Chen, I. M., Lm, C. K., Yng, G., Ln, W., & Lee, K. M. (8). Desgn nd nlss of permnent mgnet sphercl ctutor. IEEE/ASME Trnsctons on mechtroncs, (), TEM Journl Volume 7 / Number / 8.