Eddy Currents in Permanent Magnets of a Multi-pole Direct Drive Motor

Transcription

1 Acta Technica Jauineni Vol. 6. No Eddy Cuent in Pemanent Magnet of a Multi-pole Diect Dive Moto G. Gotovac 1, G. Lampic 1, D. Miljavec 2 Elaphe Ltd. 1, Univeity of Ljubljana, Faculty of Electical Engineeing 2 Abtact. Eddy cuent in pemanent magnet of diect dive ynchonou moto caue pemanent magnet heating which can caue a dop in pefomance, educe duability o even caue machine failue. In thi aticle eddy cuent in pemanent magnet of multi-pole diect dive moto with inuoidal cuent ae invetigated with finite-element analyi with an emphai on the dependency of loe on diffeent deign paamete. Analytical equation ae ued fo compaion and evaluation of the poible eplacement of the moe time conuming numeical calculation. Keywod: eddy-cuent loe, pemanent magnet, finite element method, diect dive moto 1. Intoduction Diect dive moto ae ued in application whee high efficiency and eliability i needed. Thei ue ha inceaed in ecent yea in populion ytem fo electic vehicle with an aim of mitigating the enegy ouce poblem and enabling new concept like in-wheel dive. Since thee i no mechanical gea between the moto and the wheel of the ca, diect dive moto equie a high ated toque. Pemanent magnet ynchonou moto with high pole numbe have a vey favouable toque to weight atio and good efficiency, which make them a uitable candidate fo a diect dive moto. Howeve, thee moto uually have lotted tato, which can, in connection with a high pole numbe, intoduce high fequency component into both the ai-gap pemanent-magnet flux denity a well a the ai-gap tato-cuent magnetic flux. Futhemoe, the winding of thee moto have vey low inductance which enable an inceae of the cuent ipple and with thi ome additional high fequency hamonic in the ai-gap magnetic flux. The high fequency magnetic field ocillation upeimpoed on the main otational magnetic flux denity caue eddy-cuent induction in pemanent magnet, which can lead to pefomance deceae, additional loe and even to oveheating and demagnetization of the magnet. 133

2 Vol. 6. No Acta Technica Jauineni Figue 1. The geomety of one pole pai with two adially magnetied magnet The effect of the tato geomety on the magnetic flux denity hamonic content ha been tudied in [1, 2], whee analytical model have been intoduced to enable fat calculation of the induced eddy cuent denity and the loe in the pemanent magnet. In thi aticle eddy cuent in pemanent magnet of multi-pole diect dive moto with inuoidal cuent ae invetigated with finite-element analyi with an emphai on the dependency of loe on diffeent deign paamete. A a uppot, analytical olution fom liteatue [3, 4] will be calculated to evaluate whethe it can be ued intead of FEM analyi when fat eult ae needed. 2. Geomety and vaiable The geomety of one pole pai and the moto paamete ae peented in Fig. 1 and Table 1. Table 1. Paamete deciption and value fo oiginal deign Label Deciption P Rated powe 28 kw at P Roto pole pai 28 P Stato pole pai 28 N Numbe of tato lot 168 R Stato oute adiu mm R Magnet inne adiu mm m R Magnet oute adiu mm a Ai-gap length 1.0 mm g m Magnet thickne 3.45 mm d Slot opening length 2.75 mm 134

3 Acta Technica Jauineni Vol. 6. No Theoy Label Deciption g Slot depth 11.5 mm l Stato axial length 60 mm ω Rotational angula fequency ν FWM PWM fequency 10 khz γ Magnet electical conductivity S m 1 I 0 Max. cuent amplitude 565 A I Max. cuent ipple amplitude 56.5 A FWM The eddy-cuent loe ae calculated by a time and pace aveage of intantaneou loe via equation 2π Nω P N ω I = e,,t ϕ 2πγ V 0 2 J ( ϕ ) d d dt (1) In ode to obtain a ueful eult, eddy-cuent ditibution ae needed fo each undelying caue. The following aumption have alo been made: i) The oto and the tato coe have an infinite pemeability. ii) The eddy cuent flow only in the diection of the z-axi of the cylindical coodinate ytem. iii) The compenation field geneated by eddy cuent i neglected. A conequence of a lotted tato and the fact that the winding ae dicetely ditibuted aound the cicumfeence of the tato i the intoduction of highe hamonic component into the amatue ai-gap magnetic flux denity [2]. To calculate the eddycuent contibution fit the dicete moto winding i eplaced by a cuent heet on the vey top of lamination teeth (at = R ) a in [3]. The cuent heet equation i alo tanfomed into the oto coodinate ytem, which i elevant fo eddy cuent lo calculation in pemanent magnet. ( ϕ,t) J = = J co( ( 3k 1) p ϕ ( ( 3k 1) p+p) ωt) 3k 1 k=1,2,... 3k+1 k=1,2,... J co( ( 3k+1) p ϕ+ (( 3k+1) p p ) ωt) In (2) J n (n = 3k 1 o 3k+1 ) ae the cuent ditibution coefficient which can be calculated via (3) fom the geometic paamete [4]. 2N I J n= πr 0 K wn (2 (3) 135

4 Vol. 6. No Acta Technica Jauineni whee I 0 i the main cuent hamonic component, K wn i the winding facto which eflect the hape of the cuent heet [4] and depend on the pecific winding and lot ditibution; in ou cae: 2R npd nπ K wn = in in nb p 0 2R (4) 2 If the magnetic flux denity poduced by induced eddy-cuent i neglected, the magnetic vecto potential can then be calculated with the ue of Laplace equation and bounday condition, following fom the cuent heet definition on the edge of the tato and the infinite pemeability of the oto coe. ( =R ) (,t) ( =R ) H Θ =J,...B Θ =0 ϕ (5) whee R and R ae the oute tato coe and inne oto coe adiu epectively, H θ i the tangential component of the magnetic field tength and B θ i the tangential component of the magnetic flux denity. We can diectly calculate the eddy-cuent ditibution fom the magnetic vecto potential A z (,φ,t): ( ϕ ) Az,,t J e (, ϕ,t) = γ +C() t t whee J e i the eddy cuent denity and γ i the conductivity of the pemanent magnet mateial. The tem C(t) in the expeion (6) i the integation contant which i ued to enue the magnet ae inulated fom each othe. The eddy cuent denity in equation (6) thu atifie the following equation fo the aea of the magnet S: (6) e (, ϕ,t ) ϕ (7) J d d = 0 S that i equivalent to the tatement, that no cuent flow between the magnet. 3. Finite element analyi The two dimenional FEM analyi [5] wa ued to obtain accuate eult fo eddycuent denity ditibution and the loe in pemanent magnet. Finite element calculation offe an atifactoy altenative fo etimation of the effect of deign paamete on the loe in pemanent magnet. On the othe hand, etting up a meauement of eddy cuent and eddy-cuent loe i vey difficult, and calculating eult indiectly fom heat geneation i only a accuate a the aumption of the tato and winding loe and the conductivity of the heat path. The meh line of the length anging fom 0.3 mm up to 2 mm wee defined fo diffeent egion and a meh with node and 7000 uface element wa ceated. The quality of the meh element wa veified by the pogam intenal meh-quality tet. Only one magnetic pole pai wa modelled uing ymmetical peiodical condition (Fig. 2). A non-linea B(H) chaacteitic of M270-35A electical teel wa implemented to model the oto and tato coe and each of the magnet volume wa defined a a olid conducto. 136

5 Acta Technica Jauineni Vol. 6. No Figue 2. Example of the finite element meh The emanent flux denity of the magnet wa et to zeo in ode to calculate jut the contibution of the tato cuent to the loe. The magnetic-flux denity in the tato teeth wa below atuation level, i.e. below 1.6 Tela. If the tato teeth wee atuated, we would expect a lage deviation fom the analytical eult, which aume infinite pemeability of the tato coe. 4. Reult The eult of the finite element analyi wee compaed to the analytical equation decibing eddy-cuent loe fom the liteatue [3] in ode to check the aea of validity of analytical fomulation and to daw concluion about deign guideline fo eduction of pemanent magnet heating. The focu wa on vaiation of fou impotant deign paamete: Ai-gap length (a g ) Numbe of magnetic pole-pai (p ) Stato lot width (d ) Magnet thickne ( m ) Thee paamete ae the mot influential on the hamonic content of the ai-gap magnetic field. The ai-gap length and the pole numbe, have a imila effect, becaue the hamonic content of the ai-gap magnetic field depend heavily on the atio between the ai-gap length and the geometical modulation wavelength, which i tightly connected to the pole pai numbe. The lot width wa choen becaue it i eponible fo modulation of both the tato magnetic field denity and the pemanent magnetic field denity. The lat paamete that wa vaied i the pemanent magnet thickne, which modifie the MMF of the magnet a well a the effective ai gap between oto and tato (if the ai gap i kept contant). The oiginal moto model (Table 1) wa et up in uch a way that the geomety wa paameteized o the paamete could eaily be invetigated. They wee then vaied between the boundaie found in Table

6 Vol. 6. No Acta Technica Jauineni Table 2. Bounday condition fo paamete vaiation Paamete > < p = p g a 0.3 mm 1.9 mm d 1.40 mm 3.95 mm d 1.5 mm 4.5 mm m The opeating point of the machine wa taken to be at the maximum capability. The coil then cay a total cuent with an amplitude of 565A (each tun caie 100 ARMS). Even in thee exteme condition the eddy-cuent loe in the oiginal machine ae negligible in compaion to loe fom othe ouce. Figue 2: Dependence of total eddy cuent loe in all pemanent magnet of the machine on the ai-gap length fo a inuoidal ditibution of cuent in the tato winding Figue 3: Dependence of total eddy cuent loe in all pemanent magnet of the machine on the lot width fo a inuoidal ditibution of cuent in the tato winding The total lo exhibited by all magnet i 10 W, which i 0.2 W pe magnet. The loe ae alo quite inenitive to paamete vaiation including the pole-pai numbe 138

7 Acta Technica Jauineni Vol. 6. No vaiation. When educing the pole pai numbe and the ai-gap length the total loe in all magnet appoach 40 W, and ae elatively mall compaed to othe ouce of loe. Figue 4: Dependence of total eddy cuent loe in all pemanent magnet of the machine on the magnet thickne fo a inuoidal ditibution of cuent in the tato winding Figue 5: Dependence of total eddy cuent loe in all pemanent magnet of the machine on the pole pai numbe fo a inuoidal ditibution of cuent in the tato winding The analytical method of calculation ha howed it uefulne in the invetigated paamete ange, epecially fo the cae whee the effective ai gap i lage (lage a g o m ) o thee ae le highe hamonic in the ai-gap magnetic flux (mall d ). Since any two-dimenional calculation of eddy cuent i an appoximation in itelf, the analytical calculation can be taken a a fat way to obtain an etimate of loe caued by the tato cuent. If the loe exceed a cetain level, a moe accuate 3D model hould be ued in ode to have accuate input fo cooling ib dimenioning. 139

8 Vol. 6. No Acta Technica Jauineni 5. Concluion FEM analyi of loe in pemanent magnet of a diect dive multi-pole machine loaded with inuoidal cuent howed that cae hould be taken when deigning uch a machine to keep the loe in pemanent magnet a low a poible. Beide the loe caued by the inuoidal cuent, alo the loe caued by the cuent ipple and the pemeance vaiation of the ai gap contibute to total loe in the pemanent magnet of a pemanent magnet ynchonou moto and add up o they can caue poblem with oveheating and demagnetization. The dependence of loe on deign paamete can theefoe cucially influence the choice of moto paamete in the deign tage. Figue 2-5 eveal that the analytical method ued in thi pape can be ued to pedict eddycuent loe to a low level of accuacy, but in many cae it i a quick tet of the paamete combination. Fom the deign paamete point of view inceaing the pole numbe i vey effective, wheea inceaing both the magnet thickne a well a the ai-gap length will educe the eddy-cuent loe, but not a effectively. An altenative option i egmenting of magnet, o uing buied magnet deign. The final action that can be taken i inceaing the cooling capability of the oto, which can be done uing ib, ai-gap liquid cooling o imply ai diection, depending on the volume of loe. Acknowledgement Opeation patially financed by the Euopean Union, Euopean Social Fund. Refeence [1] Gotovac G., Lampic G. and Miljavec D.: Analytical model of pemeance vaiation loe in pemanent magnet of the multi-pole ynchonou machine, Submitted to IEEE Tan. Magn., (2012). [2] Wang, J., Atallah, K., Chin, R., Ahad, W. M., Lendenmann, H.: Roto eddycuent lo in pemanent-magnet buhle AC machine, IEEE Tan. Magn., Vol. 46., No.7., (2010), pp [3] Huang Y. et al: Eddy-cuent lo pediction in the oto magnet of a pemanent magnet ynchonou geneato with modula winding feeding a ectifie load, IEEE Tan. Magn., Vol. 47., No. 10., (2011), pp [4] Zhu, Z. Q., Howe, D.: Intantaneou magnetic field ditibution in buhle pemanent magnet dc moto. ii. Amatue-eaction field, IEEE Tan. Magn., Vol. 29., No. 1, (1993), pp [5] Cedat Flux 2D 10.4, Softwae fo electomagnetic deign, (2011). 140