B (x) 0. Region 1. v x. Region 2 NNSS 0.5. abs(an + i Bn) [T] n NSNS 0.5. abs(an + i Bn) [T] n

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1 Analytical Calculation of Field Force and Loe of a Radial Magnetic Bearing With a Rotating Rotor Conidering Eddy Current Marku Ahren International Center for Magnetic Bearing ETH urich Switzerland Ladilav Kucera International Center for Magnetic Bearing ETH urich Switzerland Abtract: Thi paper preent an analytical model of a radial magnetic bearing with which magnetic eld force and loe are calculated. When an unlaminated rotor rotate in a radial magnetic bearing eddy current are caued to ow inide the conducting material of the rotor. Thee eddy current change the magnetic eld of the radial bearing and therefore the force on the rotor depend on the eddy current. Additionally to the levitation force a tangetial force act on the rotor. Thi tangential force caue a retardation torque which ha been meaured for many magnetic bearing ytem. When the rotor i excited the tangential force additionally lead to a cro coupling between the x and y axi which may detabilize the ytem. It will be hown that the force and loe of a radial bearing depend on the pole con- guration. It can be een that the conguration NSNS ha maller loe than NNSS. The analytical reult preented in thi paper correpond with meaurement [8]. Introduction Normally aerodynamic drag caue the larget lo in a rotating rotor but under vacuum condition eddy current loe are dominant. Due to the reduced heat tranfer in a vacuum an adequate model of eddy current loe i deirable. An analytical olution ha ome advantage compared with numerical reult achieved with Finite Element analyi becaue it give a better inight into the problem. Furthermore numerical calculation for radial bearing often do not ucceed. In order to achieve a correct and table numerical reult a mall grid width and a large number of element i neceary. For higher velocitie (rotational peed) the number of element increae rapidly and it may reach computational limit [] [3]. b S B N Nomenclature b length of a radial bearing B magnetic ux denity E electric eld trength F force H magnetic eld trength J current denity R radiu of the radial bearing U circumference of the rotor v velocity x; y; z coordinate direction penetration depth air gap rotational peed conductivity permeability J Ω R Figure : Cro-ection of a radial magnetic bearing Changing magnetic eld inide conducting material caue eddy current to ow. When a rotor rotate in a radial magnetic bearing the direction of magnetic ux denity in the air gap change from + ^B (north pole N) to ^B (outh pole S). Outide of the pole region the magnetic ux denity i zero. Thi changing magnetic eld therefore induce eddy current in the ferromagnetic and electrically conductive rotor. Thee current repectively induce magnetic eld which change the original z* y* x*

2 eld. The model in thi paper i two dimenional and therefore it can only be ued for unlaminated rotor. The magnetic ux denity ditribution i approximated with Fourier erie. With thi model the magnetic eld force and loe are calculated. The reult achieved in thi paper can be veried with meaurement of the retardation torque and the eddy current loe. The model for a radial magnetic bearing with eddy current i imilar to model ued for the analyi of magnetically levitated vehicle (MAGLEV) but the boundary condition are dierent [3]. A more adequate approach which i alo ued in thi paper i given in [9] and []. 3 Fourier Approximation In the rt tep the ux denity B will be approximated by a Fourier erie (ee [6]). B = = = A n co(k n x)+b n in(k n x) () A n + ib n with k n =n=u = n=r. e iknx + A n ib n e iknx (3) C n e iknx + Cn e iknx (4) Model.5 NNSS Region σ= µ=µ Region σ µ=µ µ r B (x) U y x v x ab(an + i Bn) [T] ab(an + i Bn) [T] n.5.5 NSNS Figure : A moving conducting medium under the applied ux denity B. For the following calculation all magnetic pole of the radial bearing are aumed to be identical. In order to olve Maxwell' equation it i neceary to dene appropriate boundary condition. The magnetic ux denity at the boundary between the tator and the air gap i aumed to be contant (^B ) and the applied ux denity i periodic. Conductivity and permeability are aumed to be contant. Saturation and hyterei eect are neglected. Another aumption i that the magnetic eld and the induced current tend to concentrate at the urface of the rotor. When the penetration depth of the magnetic eld i mall compared with the diameter of the rotor the error uing carteian coordinate (x; y) intead of cylindrical coordinate (r;') i mall. = r () The rotational peed can then be replaced by the velocity v x ==R. Hence the problem of a rotor rotating in radial bearing can be approximated by a emi-innite conducting plate moving under magnetic pole n Figure 3: Fourier erie of the two radial magnetic bearing conguration (NNSS and NSNS). B [T] B [T] NNSS x/u [ ] NSNS x/u [ ] Figure 4: Approximated ux denity B (x) of a NNSS and a NSNS radial magnetic bearing conguration. Dahed line: approximated ux denity uing 3 harmonic olid line: original ux denity.

3 Comparing the two Fourier erie in gure 3 it can be een that the conguration NNSS and NSNS (ee gure 4) have dierent harmonic ux denity component A n and B n. The rt component of the Fourier erie for the conguration NSNS i of higher order than for the conguration NNSS. Therefore it i often uppoed that NNSS lead to maller loe than NSNS but meaurement [8] how the oppoite behaviour. In [7] a imple model i preented which conrm the common meaning without calculating the magnetic eld and force. On the other hand the Fourier erie for the conguration NNSS ha more harmonic component than the conguration NSNS. It i clear that the quetion a to which conguration lead to higher loe cannot be anwered without an adequate model. 4 Magnetic Field In thi chapter all calculation will be hown for one term Ce ikx of the harmonic component of the boundary condition (ee equation 4). Maxwell' equation without time variant eld = = ) can be written a re = (5) rh = J (6) The material equation i: B = H = r H (7) Ohm' law i given by: J = (E + v B) (8) Equation 5-8 can be combined to econd-order partial dierential equation. r B + r(vb)= (9) The ux denity ha component in x and y direction the eddy current are owing in z direction and the conductive part i moving in x direction. B B x B y A ; J J z A ; v v x B x B y and J z are function of x and y. Combining equation 9 and lead to: A () When the olution for one direction of the ux denity eld i found the ux denity of the other direction can be derived from the relation rb= (conervation of ux) which = (3) The magnetic ux denity eld i driven by the applied magnetic eld denity B. Hence the olution with the ame travelling wave dependence on x are aumed and the ux denity take the form: B(x; y) = ~ B(y)e ikx The equation and can be tranformed ~ Bx q ~ Bx = ~ q ~ By = (6) p with q = k + kv x and with the general olution: ~B x = q ae qy be qy (7) ik ~B y = ae qy + be qy (8) The olution domain i divided into two region: region correponding to the air gap where = and therefore q = k and region the moving conducting medium. For region the olution are: B x = ie ikx a e ky b e ky (9) B y = e ikx a e ky + b e ky () The depth of region i aumed to be very large compared with the penetration depth (ee equation ). Therefore it can be aumed that the eld denity i zero for y!. B x = b q ik eikx e qy () B y = b e ikx e qy () The contant a b and b can be calculated with the boundary condition at y = and y =. Aty= the boundary condition depend on the applied ux denity and at y = the magnetic eld value for both region mut be B x y = () = () B y(y =) = Ce ikx (3) B y(y =) = B y(y=) (4) B x(y =) = r B x(y=) (5)

4 From equation 3-5 a b and b can be calculated: a = C b = C b = C eq q + q r k r k with = coh k+ e k e k q rk inh k. (6) (7) (8) ab(fx) ab(fy) [N] olid: NNSS dahed: NSNS 5 Force and Loe To calculate the force acting on the conducting medium Maxwell' tre tenor i ued [4] [9]. F x = F y = B x B y da (9) B y B xda (3) v [m/] Figure 5: Upper curve: levitation force F y lower curve: drag force F x x 4 3 olid: NNSS dahed: NSNS i any cloed urface urrounding the conducting body. It i ueful to chooe the urface at y = and to integrate x from to and z from to b where b i the length of the magnetic bearing (ee gure ). ab(p) [W].5.5 F x = ibu F y = bu C n C n n n C n C n n n q n q n (3) q n q n (3) C n q n and n are the conjugate complex value of C n q n and. The ubcript n refer to the n-th harmonic of the applied ux denity erie. The loe P can be calculated with: P = F x v x (33) Figure 5 how the increae of the drag force F x and the decreae of the levitational force F y with increaing velocity v x. Loe can be reduced with rotor material which havealow conductivity and a high relative permeability r. For mall value (e.g. r < ) the drag force increae dratically. In gure 5 it can be een that the conguration NSNS lead to a reduced increae of the drag force compared with the conguration NNSS. The relative dierence of the loe of thee two conguration i about 5% - % v [m/] Figure 6: Loe P due to eddy current inide the conducting and moving body. 6 Cro Coupling In the previou chapter the magnetic force have been calculated for a implied rectangular geometry of the rotor (ee gure ). Thi model i ucient for the calculation of the force of radial magnetic bearing when the ux denity ditribution in the bearing i ymmetrical. The force calculated for the rectangular model can be tranformed to the cylindrical model. The reulting levitation and drag force acting on the rotor are zero but the drag force lead to a reulting drag torque. When the rotor i excited the air gap and the ux denity ditribution are no longer ymmetrical (ee gure 7). When for example the ux denity increae in one direction the ux denity in the oppoite direction mut decreae due to the conervation of the ux. An excited rotor caue an aymmetrical ux denity ditribution which lead to dierential force. Thee force cannot be calculated with equation 9-3 and therefore it i neceary to

5 tranform the equation into the coordinate ytem of the rotor (ee gure ). ab(an + i Bn) [T] B [T] n x/u [ ] Figure 7: Dierential ux excitation. Upper plot: Fourier erie lower plot: applied ux denity. F x = + F y = B x B y in (')da + B y B x co (')da (34) B x B y co (')da B y B x in (')da (35) with ' = x=u = x=r. The integral of equation 34 and 35 can be olved in aimilar fahion to the calculation of the previou chapter. A mentioned before an excitation of the rotor caue an aymmetrical ux denity ditribution. For correct calculation the variable air gap mut be conidered. The excited rotor geometry can be tranformed with the Moebiu-tranformation into a new imaginary geometry ytem with two concentric circle. The partial dierential equation 5 and 6 have to be tranformed alo. The additional ditortion factor lead to equation which can only be olved by numerical method. Numerical calculation how that the force caued by the excited rotor are mainly caued by achange of the ux denity ditribution. The inuence of the variable air gap i mall and therefore it can be neglected (auming U=6). The implied geometry hown in gure will repreent a ytem with a medium air gap. F x = bu Cn C n+ n n+ + C n+c n n+ n + q n+ + + q n + q n+ + + q n (36) F y = ibu Cn C n+ n n+ C n+c n n+ n + q n+ + + q n + q n+ + + q n (37) Figure 8 how the baic behaviour of the force F x and F y depending on the velocity v x. The levitational force of a magnetic bearing decreae lowly with increaing peed. Contrary to the levitational force the cro coupling force which i zero at tandtill increae with increaing peed. Fx* Fy* [N] Fx* Fy* [N] v [m/] v [m/] Figure 8: Dierential magnetic bearing force F x (olid line) and F (dahed line). Upper plot: changing ux denity ditribution in x-direction lower plot: y changing ux denity ditribution in y-direction tep : ^B ; :5 ^B ; ^B 7 Summary Thi paper how a model for eddy current in an unlaminated rotor rotating inide a radial magnetic bearing. Additionally to the levitational force a drag force i acting on the rotor which lead to a retardation torque. When the rotor i excited the aymmetrical ux denity ditribution caue reulting force in the x and y direction and therefore lead to a cro coupling. From the analytical olution for the ux denity all of thee force and the loe can be calculated. It i hown that the conguration NSNS lead to lower loe than NNSS which correpond with meaurement in [8]. 8 Outlook Mot rotor for magnetic bearing application are laminated in order to reduce eddy current. For ome application unlaminated rotor are ued becaue the production cot are lower and the material trength i higher. The choice of the lamination thickne a well a it in- uence on the eddy current loe i an important deign problem. It i atonihing that the imple geometry of laminated heet lead to a dicult electromagnetic

6 problem (ee alo [9]). One approach to model the magnetic eld of radial magnetic bearing with laminated heet on the rotor i to ue a time variant magnetic eld. Thi approach i ued for tranformer but it cannot produce a drag force becaue the model conit neither of an eddy current ow nor a ux denity ditribution in the z direction. Another approach i a two dimenional Fourier approximation in the x and z direction. Thi model decribe only the problem of an unlaminated rotor which i not innite in length. Conidering the edge eect of radial bearing the calculated eddy current loe increae lightly wherea laminated rotor have much maller loe. The auption of thi model are only valid for large lamination thicknee. Hence thi model cannot be ued for laminated rotor. Both approache cannot achieve an adequate model for laminated rotor when eddy current are caued bymoving conducting material under an applied time invariant magnetic eld. It i unclear whether analytical olution can be found for thi problem. [] A. Sabni. Analyi of Force in Rectangular-Pole Geometrie Uing Numerical Integration Technique. IEEE Tranaction on Magnetic [] R. Stoll. The analyi of eddy current. Clarendon Pre Oxford 974. [] H. Vulling. Analytiche Berekening van Elektromagnetiche Velden en Krachten. Technical report TU Delft 99. [3] S. Yamamura and T. Ito. Analyi of Speed Characteritic of Attracting Magnet for Magnetic Levitation of Vehicle. IEEE Tranaction on Magnetic September 975. [4] T. Yohimoto. Eddy Current Eect in a Magnetic Bearing Model. IEEE Tranaction on Magnetic 9 September 983. Reference [] P. Allaire R. Rockwell and M. Kaarda. Magnetic and Electric Field Equation for Magnetic Bearing Application. In MAG `95 Magnetic Bearing Magnetic Drive and Dry Ga Seal Conference & Exhibition Alexandria 995. [] K. Hebbale. A Theoretical Model for the Study of Nonlinear Magnetic Dynamic. PhD thei Cornell Univerity 985. [3] Intitut fur Elektriche Machinen ETH Eidgenoiche Techniche Hochchule urich. FEMAG Benutzeranleitung 994. [4] J. Jackon. Klaiche Elektrodynamik. de Gruyter Berlin 983. [5] M. Kaarda P. Allaire E. Malen and G. Gillie. Deign of a High Speed Rotating Lo Tet Rig for Radial Magnetic Bearing. In Fourth International Sympoium on Magnetic Bearing urich 994. [6] H. Marko. Methoden der Sytemtheorie. Springer- Verlag Berlin 986. [7] F. Matumura and K. Hatake. Relation between Magnetic Pole Arrangement and Magnetic Lo in Magnetic Bearing. In 3rd International Sympoium on Magnetic Bearing Wahington 99. [8] T. Mizuno and T. Higuchi. Experimental Meaurement of Rotational Loe in Magnetic Bearing. In Fourth International Sympoium on Magnetic Bearing urich 994. [9] F. Moon. Magneto-olid Mechanic. John Wiley & Son New York 984.