Proceedngs of the th orld ongress n echansm and achne Scence Aprl 4, 4, Tanjn, hna hna achne Press, edted b Tan Huang Desgn of Lorentz orce Tpe Integrated otor-earng Sstem Usng Permanent agnets and oncentrated ndngs Sung-Ho Park, hong-on Lee enter for Nose and Vbraton ontrol (NOVI) Dept. of echancal Eng., KAIST, Scence Town, Daejeon, 35-7, Korea e-mal:shpark@cwllab.kast.ac.kr Yohj Okada Dept. of echancal Eng., Ibarak Unverst, Htach, Ibarak-Pref., 36-85, Japan Abstract: The ntegrated motor-bearng sstem ntegrates both functons of actve magnetc bearngs and electrc motor nto a unt. In ths paper, a generalzed approach to generaton of torque and radal force n ntegrated motor-bearng sstems usng Lorentz forces s presented. The constrants to arrangement of permanent magnets and dstrbuton of concentrated col wndngs are derved for the dsk and clndrcal tpe confguratons to be realzed. It s found that the dfference between the number of permanent magnets and the number of magnetc poles for motorng should be an nteger multple of the number of wndngs. The phase dfference of poston control current s determned b the rule that the number of poles for poston control should be more or less b than the number of the motorng poles. Tpcal feasble combnatons of desgn parameters satsfng the constrants are also lsted. Kewords: earngless motor, agnetc bearng, Lorentz force, A Snchronous motor Nomenclature A agntude of motorng current Peak of ar gap flu denst generated b rotor magnets agntude of radal force current k Number of turns per concentrated wndng l Effectve length of wndng part crossng the flu Number of pole pars for motorng current N Number of pole pars for radal force current N Set of natural numbers Q Number of pole pars for rotor permanent magnets r Effectve radus of stator wndngs t Tme Number of wndngs ( 5 ) ndng ptch angle η ndng angular nterval θ Angular coordnate fed at stator φ Phase shft of radal force control current ϕ Phase shft of motorng current ω A current frequenc [rad/sec] Introducton Actve magnetc bearng (A) s an electromagnetc devce that supports the rotor usng the controlled electromagnetc forces wthout an contact. There has been a wde range of A applcatons n ndustr, owng to such advantages as free of mechancal contact and lubrcaton, hgh perpheral speed and precson operaton, and adjustablt of the bearng stffness and dampng wthn the phscal lmts []. Nevertheless, the conventonal A sstem normall requres a relatvel long bearng span or shaft to accommodate an electrc motor to drve the rotor as well as As, lowerng the fleural crtcal speeds. On the other hand, the ntegrated motor-bearng sstem ntegrates both functons of A and electrc motor nto a unt and thus s capable of smultaneousl generatng torque for motorng and radal or aal forces for levtatng the rotor through the control of rotatng magnetc flu dstrbuton. It can therefore possess a smple and compact mechancal structure, et preservng nherent advantages of A. In return, however, the desgn and analss of ts magnetc feld and the controller desgn become a lttle comple because the motorng and levtaton of the rotor has to be smultaneousl controlled. Snce one and half decades ago, etensve works have been carred out n order to develop varous knds of ntegrated motor-bearng sstems [], such as nducton tpe, permanent magnet tpe, homopolar tpe, reluctance tpe, etc. ost of them have the desgn prncples based on use of the awell forces for the poston control of the rotor. In recent ears, new ntegrated motor-bearng sstems usng the Lorentz forces for the rotor poston as well as torque control were proposed [3]. awell force s an attractve force between the magnetc poles of opposte polart produced b the magnetc flues gong through the ar gap. It has a nonlnear relatonshp between the force magntude and the length of ar gap, and heav lamnated cores are necessar for most of ts applcatons. On the other hand, Lorentz force s bdrectonal and holds a good lnear propert, b vrtue of whch the analss and desgn of the sstem can be made eas. In addton, t can be obtaned even f no ferrte core s emploed. A coreless confguraton serves to reduce the sstem weght and takes awa such troubles as ron loss, magnetc saturaton, hsteress, edd current effect, etc. However, the sstem proposed b Han, et al. [3] necesstates a comple frequenc demodulaton skll wth two stator dsks to avod sngulart n calculatng control nputs, resultng manl from the fact that the sstem has the same number of poles n the stator wndngs as that of the permanent magnets on the rotor dsk. Okada, et al. [4] developed two prototpes of ntegrated motor-bearng sstem: one s of the dsk tpe and the other s of the clndrcal tpe, usng eght permanent magnets on the
rotor and s concentrated stator wndngs drven b three phase currents. Though the proposed dsk tpe sstem, equpped wth one rotor dsk and one stator dsk, accomplshed stable operaton, the capablt of force generaton was not satsfactor. The clndrcal tpe one was desgned as a remed to the defect of the dsk tpe, but the reported that ther theor was vald onl for eght-pole motors. Km, et al. [5] developed another Lorentz force tpe ntegrated motor-bearng sstem, whch has four permanent magnets nsde the rng-shape rotor and s-concentrated wndngs on the slotted stator core, but the desgn formula s also lmted to the case of four pole motors. In ths paper, a general formula for generaton of torque and radal force n ntegrated motor-bearng sstems usng Lorentz forces s presented. The formula s based on the use of common wndngs for both torque and poston control, as t s benefcal to the compactness of the sstem. In spte of the dfference n confguraton, the prncple can be applcable to the clndrcal and dsk tpes n a smlar wa; ths paper deals wth both of them. Prncples of Torque and Radal orce Generaton onsder a par of concentrated wndngs n an orthogonal magnetc flu feld. gure shows the conceptual drawng for a dsk tpe, and g. s the equvalent one for a clndrcal tpe. The dot and cross n the crcles shown n g. mean the magnetc flu comng out of and gong nto the page, respectvel, the loop n the wndngs ndcates the current flow, and the arrow on the loop shows ts drecton. In g., the dot and cross n the crcles represent the current flow drecton, and the magnetc flu runs across the wndngs n the radal drecton. gures (a) and (a) llustrate the basc prncple of torque generaton. Accordng to the Lorentz law, electromagnetc forces, whch are represented b the thck arrows n gs. and, are produced n the upper and lower sdes of wndngs for the counter-clockwse current flows n the rght and left wndngs. The reacton forces to the rotor have counter drecton to the forces generated n the wndngs, resultng n a couple moment. The contour arrow at the center ndcates the motorng torque. or radal force generaton, the current flow drecton n the left sde wndng s reversed whle the current flow n the rght wndng and all the magnetc flues reman unchanged as shown n gs. (b) and (b). The drecton of the force generated n the left sde wndng s then reversed. Hence, the resultant of the reacton force vectors generated b both wndngs wll head for the upper drecton. Reversng the current flows n both wndngs, of course, can easl change the radal force drecton. As the rotor rotates at a certan speed, the magnetc flu polart ma alternate perodcall, leadng to oscllaton or fluctuaton of the forces. Therefore, a proper combnaton of fluctuatng forces s essental n order to obtan stable and relable forces for the radal poston control of the ntegrated motor-bearng sstem. 3 Theoretcal Approach to Generalzed Sstem Let us consder the equ-spaced concentrated wndngs along the perpher of the stator as shown n g. 3. The dervaton of a new desgn formula developed here begns on the bass of the so-called P ± formula presented b Okada, et al. [6]: the dfference between the numbers of poles for the motorng and poston control current dstrbutons should be plus or mnus two n order to obtan the decoupled controllablt of torque and rotor poston. The postve sgn s assgned to the magnetc flu generated b rotor permanent magnets comng out of the page, and to the effectve current runnng from the center of the stator to the outer radal drecton n g. 3 (a). Under that sgn conventon, the stator wll delver a postve torque n the counterclockwse drecton. To keep consstenc of analss n clndrcal confguraton as shown n g. 3 (b), the magnetc flu n the outer radal drecton and the current gong nto the page should take the postve value. or smplct of analss, assume that the magnetc flu denst generated b the permanent magnets on the rotor takes a snusodal waveform of T I R I I T I (a) I (a) I I I (b) (b) gure Generaton of (a) torque and (b) radal force n dsk tpe confguraton gure Generaton of (a) torque and (b) radal force n clndrcal tpe confguraton
cos( ωt Qθ ) () where s the ampltude of the flu denst, ω s the A crcular frequenc, Q s the number of pole pars of the rotor permanent magnets and θ s the angular coordnate. Now consder the currents for motorng torque gven n the form of I Acos( ω t + ϕ,, L, () where denotes the number of pole pars for motorng current n the stator wndngs, A s the magntude of the current, ϕ s the phase shft, and the angular spacng of the wndngs η s defned as η (3) Note that the phase dfference between the currents n the two adjacent wndngs s η and equaton () forms the spatal current dstrbuton of pole pars n the stator D D D l r D D D D D D + D + (a) Dsk tpe r D (b) lndrcal tpe D η η D D gure 3 oordnates and wndng dstrbuton for a generalzed sstem θ θ wndngs. The stator wndng dstrbuton can be represented b usng Drac s delta functon as D kδ ( θ η + kδ ( θ η (4),, L, where k s the number of turns n a concentrated wndng and s the wndng ptch angle. Then the motorng torque produced b these currents can be obtaned, usng Lorentz law, as T rl { sn( ϕ + ( Q ) sn(ω t + ϕ ( Q + ) } RI D Akrl sn( Q ) Thus, the condtons for the torque to be a nonzero constant, regardless of the rotatng speed of the rotor become ( Q + ) η mπ (6) ( Q ) η nπ where m and n are arbtrar ntegers. In other words, the number of pole pars Q for the rotor permanent magnets and the number of pole pars for the motorng current should be determned so that the satsf the relaton, for the number of wndngs, gven b Q + n (5) (7) Ν Equaton (7) mples that the phase dfference, η n equaton (), should be restrcted to a proper fracton of π so that all motorng currents are not gven n phase or out of phase of the reference state correspondng to, sa,. It also means that (Q ) should be an nteger multple of. Note, however, that Q, phscall a natural number, can take a negatve nteger, snce equaton () stll holds for Q, whch s equvalent to reversal of the sgn conventon for θ. or a negatve nteger value of Q, the torque drecton wll also be reversed so that the rotor wll rotate clockwse. Substtutng equaton (7) nto equaton (5), we obtan T Akrl sn(q snϕ (8) mplng that the motorng torque s proportonal to the number of wndngs, the magntudes of the motorng current and the magnetc flu, the number of turns, and the effectve length of the wndng. It suggests that the torque attans ts mamum value when the wndng ptch angle s tuned to the ptch angle of the rotor permanent magnets. In addton, we can control the sstem lke a standard snchronous A motor b adjustng the phase shft ϕ from to 9, dependng on the ncrease of load torque. hen ϕ 9, we can operate the sstem as a servomotor controlled b the current magntude A. The relaton between the rotor speed N S n rpm and the motorng current frequenc f n Hz can be wrtten as 6 f N S [RP] (9) Q
where ω f. Smlarl to the prevous procedure for generaton of motorng torque, consder the currents for the radal control force n the form of I cos( ω t + φ N,, L, () where N denotes the number of pole pars for radal force control current n the stator wndngs, s the magntude of the current, and φ s the phase shft. Note that the phase dfference between the currents n the two adjacent wndngs s Nη. Applng these currents to the wndngs represented b equaton (4), we obtan the epressons for the radal control forces n the and drectons as l RI sn(( Q + ) + sn(( Q + ) sn(( Q ) sn(( Q ) l RI sn(( Q + ) sn(( Q + ) + sn(( Q ) sn(( Q ) D snθ kl cos( φ + ( Q N + ) cos(ω t + φ ( Q + N + ) cos( φ + ( Q N ) cos(ω t + φ ( Q + N ) D cosθ kl sn( φ + ( Q N + ) sn(ω t + φ ( Q + N + ) sn( φ + ( Q N ) sn(ω t + φ ( Q + N ) () () Elmnatng the terms ncludng ω t, we derve the condtons for generaton of a stable and controllable radal force gven b N ± N N (3) Note here that the condton for a stable radal force concdes wth the P ± formula and the control currents also should not be gven n phase or out of phase, n the same wa as for the motorng currents. Substtutng equatons (7) and (3) nto equaton (), we obtan ± kl sn(( Q ± ) cosφ kl sn(( Q ± ) snφ (4) where the double sgns are algned. Equaton (4) proves that the strength and the two dmensonal drecton of the radal force can be lnearl controlled b the magntude and the phase shft φ of the current, respectvel. The sgn of the radal force n equaton (4) depends completel upon how to locate two orthogonal coordnates of the sstem. It also reveals that the wndng ptch angle to get the mamum radal force s slghtl dfferent from that for the mamum torque,.e., the angle becomes narrower or wder accordng to the pole pars of the control current and the sgn of Q. The radal control force s also ndependent of the rotatonal speed of the rotor, whch makes t smple to desgn the radal poston controller of the sstem, not requrng a comple frequenc demodulaton step. nall, the torque generated b the radal force currents and the radal force generated b the motorng current are calculated as T rl { sn( φ + ( Q N) sn(ω t + φ ( Q + N) } l RI sn(( Q + ) + sn(( Q + ) sn(( Q ) sn(( Q ) l D krl sn( Q RI sn(( Q + ) sn(( Q + ) + sn(( Q ) sn(( Q ) RI D snθ Akl cos( ϕ + ( Q + ) cos(ω t + ϕ ( Q + + ) cos( ϕ + ( Q ) cos(ω t + ϕ ( Q + ) D cosθ Akl sn( ϕ + ( Q + ) sn(ω t + ϕ ( Q + + ) sn( ϕ + ( Q ) sn(ω t + ϕ ( Q + ) (5) (6) (7) rom equatons (5), (6), and (7), t s possble to decouple the torque and control force generaton, when the hold + N, N (8) Equatons (7), (3), and (8) suggest that the mnmum number of wndngs requred for generaton of the decoupled stable torque and control force s fve.
4 onclusons In the prevous secton, the mechansms of generatng the torque and the radal force have been descrbed for a generalzed ntegrated motor-bearng sstem usng Lorentz forces. The constrants to arrangement of permanent magnets and dstrbuton of concentrated col wndngs are derved for the dsk and clndrcal tpe confguratons to be realzed. It s found that the number of wndngs should not be less than fve f the motorng and radal force currents share common wndngs. The dfference between the number of permanent magnets on the rotor and the magnetc poles for motorng has to be an nteger multple of the number of wndngs. The phase dfference of poston control current s determned b the rule that the number of poles for poston control should be more or less b than that of the motorng poles. The radal control force s ndependent of the motorng torque and the rotatonal speed of the rotor, not requrng a comple frequenc demodulaton scheme. It s also found that the wndng ptch angle requred for the mamum radal force s slghtl dfferent from that for the mamum torque. In recent ears, a few works wth some specal arrangements of permanent magnets and wndngs have been performed n an attempt to acheve the decoupled stable torque and control forces n ntegrated motor-bearng sstems. or eample, the formula proposed b Okada, et al. [4] corresponds to the case of 6,, N, Q 4,.5, and the sstem developed b Km, et al. [5] to 6,, N, Q, 45. Tpcal feasble combnatons of desgn parameters satsfng the constrants dscussed n the prevous secton are lsted n Table. References. runet., Practcal applcatons of the magnetc bearngs to the ndustral world, Proc. st Int. Smp. agnetc earngs, ETH, Zurch, pp. 5-44, 988. Salazar A. O., hba A., and ukao T., A Revew of developments n bearngless motors, Proc. 7 th Int. Smp. agnetc earngs, ETH, Zurch, pp. 335-34, Aug. 3. Han. S., Lee.., and Okada Y., Desgn and control of a dsk-tpe ntegrated motor-bearng sstem, IEEE/ ASE Trans. on echatroncs, 7( ):5-, ar. 4. Okada Y., Konsh H., Kanebako H.,and Lee.., Lorentz force tpe self-bearng motor, Proc. 7 th Int. Smp. agnetc earngs, ETH, Zurch, pp.353-358, Aug. 5. Km S. J., Abe K., Kanebako H., Okada,Y.,and Lee.., A Lorentz force tpe self-bearng motor wth new 4-pole wndng confguraton, Proc. 8 th Int. Smp. agnetc earngs, to, Japan, pp. 35-4, Aug. 6. Okada Y., Dejma K., and Ohsh T., Analss and comparson of P snchronous motor and nducton motor tpe magnetc bearngs, IEEE Trans. on Industr Applcatons, 3( 5):473, 995 Table easble combnatons of desgn parameters Phase dfference ndng ptch angle N Q η Nη a. T a. 6 π 6 π 7 5 5 4π 5 π π -4 π 4 π 3 6 7 8 3 3 3 7 π 7 π 8 π 3 3 π π π 5 π 4 8 π 8 π 7 3 π 3 π π -4 π 4 π 5 8 π 8 π 9 7 4π 7 π π π 6 π 5 9 π 9 π 8 4π 7 7 π π π 5 π 6 9 π 9 π 4π 7 6π 7 π π 3 π 5 π 4 9 π 9 π π 4 π π π -7 π 7 π 6 π π 9 π π 4 π π π 6 π 7 π π π 3π 4 π π 3 π 6 π 5 π π 3 3π 4 π π 3 π π 5 π 6 * The optmum wndng ptch angles were calculated from equatons (8) and (4). Practcalt of desgn, however, s another ssue for dscusson, consderng the concept and performance of the desgn. In fact, the wndng ptch angle, whch ma dffer from the optmal value, should be determned consderng practcal restrctons. or eample, n case when the neghborng wndngs are allowed to overlap, the ar-gap length tends to become large and the magnetc saturaton problem ma occur. The handlng of wndng ends should be also consdered, especall n the dsk tpe confguraton.