Analysis and Measurement of 3D Torque and Forces for Permanent Magnet Motors with Slotless Windings

Transcription

1 Analysis Measurement of 3D Torque Forces for Permanent Magnet Motors with Slotless Windings A. Looser, T. Baumgartner, C. Zwyssig J.W. Kolar Power Electronic Systems Laboratory ETH Zurich CH-89 Zurich, Switzerl Abstract Slotless windings, skewed rhombic, are widely used in industry. Beside the drive torque, ossibly undesired transverse torques forces are generated, which have not been analyzed reviously. An analytical derivation of the torque force comonents in all three directions is detailed in this aer for the skewed winding. It is shown that for some winding configurations alternating transverse torque comonents are generated, which may comromise stable oeration in alications where for examle magnetic or gas bearings are involved. Moreover the windings are analyzed with regard to a otential use as active magnetic radial bearings in high-seed alications. Finally, measurements are resented to verify the theoretical results. I. INTRODUCTION Slotless windings are widely used for small sized electric motors in industrial alications ranging from a hundred milliwatts to a few hundred watts []. The main advantages of a slotless design are the absence of cogging torque, the elimination of losses caused by slot sace harmonics, the low winding inductance. For the last two reasons it is also an excellent choice for high-seed motors []. An overview of different tyes of slotless windings is given in [3] []. Commonly used winding tyes are the skewed tye the rhomic winding. As can be seen from Figure both windings exhibit comlex three dimensional structures with two layers overlaing. This can result in undesirable transverse torque force comonents beside the intended drive torque in axial direction, even for erfect alignment symmetry of the winding. In some high-seed alications where rotordynamics must be considered, e.g. when magnetic bearings or gas bearings are involved, these transverse torque comonents have to be either avoided by choosing an aroriate winding tye, configuration geometry or at least exactly quantified incororated into the rotordynamic model. The magnetic field in the airga required for the analysis of the torque force comonents has been studied in [5], [6] [7]. Winding factors for the harmonic field comonents resulting in alternating drive torque have been analyzed in []. However, no analysis or measurement of the transverse torque forces comonents have been reorted in literature. Fig.. a) b) Skewed winding a) rhombic winding b) In this aer torque force calculations for the skewed tye winding are derived. The most common configuration, the two-ole, three-hase winding is analyzed in detail, including ossible force generation for an active magnetic bearing. A winding test bench is built the theoretical results are verified by measurements. II. DERIVATION OF FORCE AND TORQUE FOR THE SKEWED WINDING A. Magnetic Field Distribution According to [5] the flux density in the airga of a slotless ermanent-magnet machine can be described by Fourier series for both the radial the azimuthal field comonents as B r = B θ = n= n= B n) r cosnqθ + δ) ) B n) θ sinnqθ + δ), ) where q is the number of ole airs of the rotor, δ the load angle B r n) B n) θ are the Fourier coefficients of the n th harmonic. The field in axial direction is assumed to be B z =. It is well known that for machines driven with symmetric sinusoidal currents only the fundamental comonent contributes to a constant torque. Therefore, from a design oint of view,

2 /m / z L v A+ A- B+ A+ A- r y x Fig.. Winding scheme 3D structure of the skewed tye winding. Fig. 3. Coordinate system definition winding loo arametrization. the flux density of an otimal machine only consists of the fundamental comonent, hence for the winding analysis carried out in this aer it is assumed that that the geometry of the rotor the ermanent-magnets is chosen such that all the harmonic field comonents vanish. Indeed for the q = ole air machine a urely fundamental field distribution is achieved by a diametral magnetization direction in the cylindrical ermanent-magnet [7]. The time varying fundamental field in the airga for a rotor rotating with seed Ω is then described in cylindrical coordinates by B r = ˆB r r) cosqθ qωt + δ) 3) B θ = ˆB θ r) sinqθ qωt + δ) ) To derive the force torque comonents in cartesian coordinates is described by B r cosθ) B θ sinθ) B = B r sinθ) + B θ cosθ). 5) B. Winding Definition A winding is defined by the number of hases m 3 the number of ole airs. It is to be noted that the ole air number of the rotor q does not necessarily need to be equal to ole air number of the winding. For each ole a winding zone sreads over an angle ϕ/ = /m with ϕ = /m being the electrical hase shift from one hase to the adjacent hase Figure ). For m = m = a different winding descritions would be necessary. However, because no constant torque can be achieved with a one-hase winding, it is not considered further. The two-hase winding has a very low utilization factor when having only two windings situated / to each other, because then only half of the volume is used for the winding. However introducing windings connected in series on the oosite sides in the unused sace to overcome this drawback results then in an m = hase winding, which is then covered by the given definition. A single winding loo starting from angle φ as deicted in Figure 3 can be described by the vector r cosθ) v = r sinθ) 6) zθ) with the z comonent given by { θ φ), for < θ φ < zθ) = L +L θ + φ), for < θ φ < 7) L being the winding hight in z direction. The current density in the winding loo belonging to a hase conducting a current of hase shift ϕ is given by where J is defined as ds is given by C. 3D Force Torque J = J ds ds 8) J = Ĵ cosqωt ϕ) 9) ds = v dθ. ) The Lorentz force on a infinitesimal conductor element with volume dv calculated from the trile roduct is given by dv = rdφ dr) ds = L dθrdφdr ) df = J B dv = J v B dθrdφdr ) + r L where the factor results from the two layer structure of the winding. The torque generated by that element in resect to the origin is given by dt = v df. 3)

3 The torque the force generated by a single hase is then obtained by integration over one winding loo one hase zone summation over all ole airs as T h = J R k= F h = J R k= ϕ+k+/m ϕ+k /m ϕ+k+/m ϕ+k /m φ+ φ φ+ φ v v B dθrdφdr + v r L ) B dθrdφdr, + r L 5) where R are the winding inner the winding outer radius see Figure ). Summing u over all m hases yields the total torque the total force D. Solution for = q T = T = F = m k= m k= T h ϕ= k m 6) ϕ= F h. 7) k m Evaluating ) 6) yields the the torque T x m sin m )ĴL sinδ) R ˆB rr dr + L) r r L 8) where for = m = 3 the transverse torque comonents are T x = 3 R 3ĴL sinωt δ) ˆBr + ˆB ) θ rdr 8 9) + = 3 3ĴL cosωt δ) 8 R ˆBr + ˆB ) θ rdr + r L ). ) For any other combination of m > the transverse torque comonents vanish T x = = ). Evaluating 5) 7) yields a vanishing force F = for any combination of m. E. Two-Pole Three-Phase Winding For the two-ole three-hase winding, q = =, m = 3 the transverse torque comonents T x given by 9) ) do not vanish but alternate with a frequency of twice the rotational seed. In order to comare the amlitudes of the drive torque T z the transverse torque comonents T x the integral over the radius needs to be solved R R R 5 R Fig.. Machine cross-section symbol definitions: diametrically magnetized cylindrical ermanent-magnet rotor inside a slotless stator [7]. therefore the radial deendency of the flux density has to be considered. The equations for the flux density B for q = are derived in [7] the arameter definitions are given in Figure. Simlifying the equations for the radial azimuthal comonents of the flux density by setting the relative ermeability of the ermanent-magnet the stator back iron µ r,m resectively µ r,f e yields ˆB r r) = B remr R r R r + ) R r. ) ˆB θ r) = B remr R The torque generated by a m = 3 hase winding is obtained by evaluating 8), 9) ) to T = 3 3ĴBremL R sinωt δ) 3 K 3 K 3 3ĴBremL R sinδ) K 3 3ĴBremL R cosωt δ) where the K K are given by L + L K = log + R3 L + L ) L + R L + R L L + R3 [ 3) ) K = R R L + R L + R3 ) ] + R L R ) log + L + R. 5) + L + R 3 A high ratio of L/R /R results in windings roducing relatively high transverse torque T xy comared to the drive torque T z as can be seen from Figure 5. This is rather unfavorable since for common machines the winding length is usually larger than its outer diameter leading to a

4 L/R= L/R=.8 Txy/Tz L/R= L/R=.5.6 calculated measured R3/R.8 Fig. 5. Ratio of transverse torque Txy to drive torque Tz for the two-ole, three-hase winding in deendency of the geometry arameters active length L, winding inner radius R3 winding outer radius R. The calculated value for the exerimental winding of Table I is shown with a cross, the measured value with a circle. the transverse torque in the area of % to % of the drive torque or more. III. F ORCE G ENERATION FOR ACTIVE M AGNETIC B EARINGS The skewed tye winding is a romising cidate for an active magnetic bearing for high-seed alications due to its low inductance the relatively low negative stiffness resulting from the large magnetic air ga [8]. There are many configurations for generating a net force; e.g. for the configuration = q + the torque always vanishes: T =. The force for this configuration m 3 is obtained from 5) 7) as R Z B r + B θ rdr cosδ) mjl sin m r F = sinδ). 6) r R3 + L For q = the simlifications for the flux density B made in section II-E can be used to calculate the integral in 6) yielding cosδ) m sin m )K3 JLR Brem sinδ) F = 7) where K3 is defined as! L + L + R3 L L + R. K3 = log L + L + R L L + R3 8) Fig. 6. Winding test bench with the stator mounted on the iezoelectric load cell the ermanent-magnet fixed to the ositioning stage. IV. M EASUREMENTS A. Measurement Setu The measurements are erformed on a granite measurement latform. The ermanent-magnet is mounted on a xyztranslation stage with an additional rotational stage allowing for accurate ositioning rotation of the magnet. The stator is mounted on a multi comonent load cell to measure the winding reaction force torque comonents resulting from the injected winding currents. A hotograh of the described assembly is shown in Figure 6. A Sitzenberger&Sies DM3 ower suly is used to generate the three-hase symmetric sinusoidal currents fed to the star configured winding at a frequency of Hz. The three hase currents are measured with current sensors LEM LTS 6-NP). The current sensor outut signals force signals are acquired by a National Instruments LabVIEW card at a samle rate of khz. The motor winding to be measured is exected to deliver a nominal drive torque of 3 mnm. In order to measure such small quantities a iezoelectric multi comonent dynamometer Kistler 956C) is used, roviding six force measurements in airs Fx Fx, Fy Fy Fz Fz, thereby allowing to comute the resulting forces as well as two of the three torque comonents. The force comonents are Fx = Fx +Fx, Fy = Fy +Fy Fz = Fz +Fz the two torque comonents are Ty = a Fx Fx )+b Fz Fz ) Tz = c Fy Fy ) + d Fx, where the term d Fx accounts for the coordinate transformation from the sensor to the stator coordinate system x, y, z) x, y + d, z). a, b c result from the load cell geometry. To obtain the torque Tx the measurements are reeated for the winding rotated by an angle of 9. A multichannel charge amlifier from Kistler of tye 57 is used. For all the measurements the amlifier s internal second order low-ass filters are used the cut-off frequency is set to 3 Hz.

5 Magnitude db) 6 8. Frequency Hz) Fig. 7. Frequency resonse of the drift comensation FIR filter with an attenuation of.56 db at. Hz. To verify correct behavior of the load cell, quasi-static measurements of test forces alied by a dial gauge test weights were carried out. It was concluded that the resulting forces the torque was measured with high recision, while for T z the measured values were 3% lower. Therefore a correction factor had to be introduced for the calculation of T z. As the generated charge roduced by the small forces is low the charge amlifier oerates with high gain, signal drifting needs to be comensated. It is common for iezoelectric force measurements, that drift comensation is done manually by subtracting a linear aroximation of the drift considering the beginning the end of the measurement when no forces are alied. For symmetric sinusoidal three hase currents the resulting force torque comonents are suosed to be offset-free. This simlifies drift comensation the measured data can be rocessed off-line through a FIR high-ass filter therewith eliminating the frequency comonents below Hz. The frequency resonse of the alied drift comensation filter is shown in Figure 7. B. Two-Pole Three-Phase Winding The geometric data for the q = = test motor is listed in Table I. Sinusoidal three hase currents are injected such that the resulting stator field rotates in mathematically ositive direction. Figure 8 shows the measured force torque comonents for the rotor osition θ = after signal drift comensation. Note that for the rotor fixed at osition θ = the stator field rotating in ositive direction the load angle is δ = Ωt. Measurements are reeated for rotor ositions in stes of. Another series of measurements is furthermore required to obtain the torque comonent M x by turning the winding by 9. Figure 9 shows the combined resulting force torque comonents from these measurements for a load angle δ = /, where the maximum drive torque is generated. It can be seen that the transverse torque rotates with twice the rotational seed. For the tested winding the ratio of transverse torque to drive torque is exected to be T xy /T z =., however the measurement shows a different ratio with.3, see Figure 5. Also the force comonents do Current A) Force mn) Torque Nmm) F x F y F z t s) Fig. 8. Measurement of stator reaction torque forces for the one ole air three-hase test motor with injected stator currents at rotor osition θ =. TABLE I TWO-POLE THREE-PHASE TEST MOTOR Symbol Quantity Value R Permanent-magnet radius 7. mm Winding inner radius 8. mm R Winding outer radius. mm R 5 Iron outer radius 5. mm L Active length 7.5 mm B r Remanent flux density. T N Turn number er Winding not vanish comletely. These non-vanishing forces are u to % of the torque roducing differential forces in the sensor. It can be excluded that these discreancies result from not erfectly symmetric currents, as the current measurement error is smaller than % the amlitude deviation of the injected currents was less than %. The errors however may be caused by misalignment of rotor stator. The centering of the rotor in the axial direction is satisfactorily achieved by minimizing the resulting force in z-direction. However the resulting forces in x- y-directions are very sensitive to the ositioning of the rotor in these directions. Moreover, minimizing the forces for a secific angular rotor osition did not yield satisfying results for other angular rotor ositions. Another source for the non-vanishing forces might be the two layer structure therewith associated asymmetries of the winding which has not been accounted for in the above modeling. C. Active Magnetic Bearing Winding Measurements have also been erformed for a slotless tye active magnetic bearing winding with a = ole air skewed winding a q = ole air rotor. The geometry is chosen the same as for the one ole air test motor Table I), excet for the winding ole air number. The measurement rocedure I r I s I t T z

6 Force mn) Torque Nmm) Ωt Fig. 9. Combined measured torque forces for the one ole air threehase test motor for a constant load angle of δ = /. Current A) Force N) Torque Nmm) F x F y.5 F z T z t s) Fig.. Measurement of stator reaction torque forces for the slotless magnetic bearing at θ =. is also exactly the same. However the hase currents needed to be set much lower not to overdrive the charge amlifier, which configuration was not changed. Figure shows the measurement for a fixed rotor osition at θ = while the three hase currents are alied. It can be seen that forces can be generated in both x- y-direction with generating only very small amount of undesired torque or an axial force. Also, thecalculated force amlitude values are in very good agreement with the measured values. The calculated force 3 3 I r I s I t F x F y F z T x T z amlitude for a eak current of A resulting in a current density of Ĵ =.3 A/mm is.7 N, while the measured value range is.69 N to.7 N. V. CONCLUSION The skewed tye winding with three hases one ole air q = = is widely used in industrial motor alications. Desite its oularity, its eculiarity to roduce transverse torque comonents in addition to the drive torque seems not to be well known. However when rotor dynamics are of concern, e.g. when magnetic or gas bearings are involved, a thorough evaluation of the winding behavior is crucial. Therefore analytical exressions for the 3D force torque are resented in this aer. For the two-ole winding, the measurements verify the existence of transverse torques with twice the frequency of the rotational seed, however with even a higher transverse to axial torque ratio than calculated. For the magnetic bearing winding the theoretical results are in very good agreement with the measurements. This roves the feasibility of using such a winding for a magnetic bearing. ACKNOWLEDGMENTS The authors would like to thank Josef Stirnimann Jens Boos from insire AG for kindly roviding the iezoelectric load cell technical advice. A secial thanks goes to Mark Vohrer from ATE GmbH for manufacturing the test windings. REFERENCES [] U. Kafader J. Schulze, Similarity relations in electromagnetic motors - limitations consequences for the design of small dc motors, in Proceedings of 9th International Conference on New Actuators ACTUATOR ),, [] N. Bianchi, S. Bolognani, F. Luise, Potentials limits of highseed PM motors, vol., no. 6, , Nov. Dec.. [3] M. Nagrial, J. Rizk, A. Hellany, Design erformance of ermanent magnet slotless machines, in Electrical Machines, 8. ICEM 8. 8th International Conference on, Set. 8,. 5. [] B. Hagemann, Entwicklung von Permanentmagnet-Mikromotoren mit Luftsaltwicklung, Ph.D. dissertation, Universitt Hannover, Germany, 998. [5] K. Sridhar, Electromagnetic models for slotless m synchronous motor drives, in Alied Power Electronics Conference Exosition, 995. APEC 95. Conference Proceedings 995., Tenth Annual, no., Mar 995, vol.. [6] M. Markovic Y. Perriard, Simlified design methodology for a slotless brushless dc motor, Magnetics, IEEE Transactions on, vol., no., , Dec. 6. [7] J. Luomi, C. Zwyssig, A. Looser, J. W. Kolar, Efficiency otimization of a -W, 5 -rm ermanent-magnet machine including air friction losses, in Proceedings of IEEE nd IAS Annual Meeting Industry Alications Conference IAS 7), 7, [8] T. Baumgartner, A. Looser, C. Zwyssig, J. W. Kolar, Novel highseed, lorentz-tye, slotless self-bearing motor, to be ublished.