Sensors & Transducers, Vol. 75, Issue 7, Jul 4, pp. 5-57 Sensors & Transducers 4 b IFSA Publishing, S. L. http://www.sensorsportal.com Rotor s Mass nbalance Compensation Control on Bearingless Permanent Magnet Snchronous Motors for Minimizing the Control Currents Tao ZHANG, Wei NI, Xiaohui WANG, Chen ZHANG, Hongun JIA Facult of electronic and Electrical Engineering, Huaiin Institute of Technolog, Huaian, Jiangsu 35, China School of information and control engineering, Nanjing niversit of Information Science and Technolog, Nanjing, Jiangsu 44, China E-mail: zhangtaohit@6.com Received: 4 Ma 4 /Accepted: 3 June 4 /Published: 3 Jul 4 Abstract: The vibration originated from the rotor s unbalance mass impacts on the current oscillation. When the rotor rotating at high speed, power amplifier circuit will be burned down b ecessive vibration control current, and at the same time the ecessive control force will be generated. The vibrating force transfers to frame, which causes the sstem vibration and influence the dnamic properties and safe operation. So unbalance compensation on the rotating machine is necessar, especiall for the high speed bearingless motor. Based on introductions of the rotor motion equation of bearingless permanent magnet snchronous motor (BPMSM) and analsis on generation mechanism of unbalance force, feed-forward compensation controller is designed on the technolog of adaptive notch filter in this paper. The control sstem is simulated with Matlab/Simulink toolbo. The simulation results have shown that the current oscillation can be minimizing b the controller, and vibrating force is eliminated with the rotor rotating around inertia center. Copright 4 IFSA Publishing, S. L. Kewords: Bearingless motor, Bearingless permanent magnet motor, Suspended rotor, Compensation control, Mass unbalance.. Introduction The high-speed motor has some remarkable advantages, such as: ) It possesses smaller volume, less row material, higher power densit, and higher efficienc; ) It can drive the load directl without the transmission mechanism, which means less transmission losses and noise; 3) The rotor of highspeed motor has less rotational inertia and thus has higher dnamic response. For its promising application in special electrical transmission and high-speed direct-drive fields, the high-speed motor has becoming the international research topics in the electrical engineering fields [-3]. The bearingless permanent magnet schronous motor has some merits, such as no friction, no lubrication and no sealing, high speed, high precision, high power densit and long life. It has being innovated the traditional supporting forms of high speed motors [5, 6]. But the rotor vibration rejection problem has alwas been a critical issue in bearingless motor sstems. The vibration originated from the rotor s unbalance mass impacts on the current oscillation. When the rotor rotating at high speed, power amplifier circuit will be burned down b ecessive vibration control current. The vibrating force transfers to frame, which causes the sstem vibration and influence the dnamic properties and http://www.sensorsportal.com/html/digest/p_3.htm 5
Sensors & Transducers, Vol. 75, Issue 7, Jul 4, pp. 5-57 safe operation. It is desirable to eliminate the rotor vibration in high speed rotation machiner. The feedforward compensation is alwas used to reject the rotor vibration. But it is the ke technolog to estimate the unknown amplitude, frequenc and phase of vibration signals. The researches have been researched a vast kind of methods for the rotor vibration control. These methods onl fit for low speed mainl based on the minimum the current method, the minimum displacement method, and so on. But, when the rotor rotates in high speed, the above methods can not reject the sstem vibration. In this paper, the vibration producing mechanism of bearingless permanent magnet tpe motor is introduced. The rotor dnamics equations were deduced. Then, based the decouple control sstem of bearingless permanent magnet snchronous motor, the feedforward compensation control is designed to reject the rotor mass unbalance vibration using notch filter technolog. Finall, the simulation researches are carried out using Matlab/Simulink. The research results have shown that the rotor vibration rejection method proposed in this paper can reject the power amplifier current oscillations well. The rotor vibration of BPMSM was eliminated. So the rotor rotates around the center of inertia freel.. Producing Mechanism and Mathematical Model.. Producing Mechanism Fig. shows production mechanism of radial suspension force and torque in BPMSM. There are three magnetic fields in BPMSM, suspension windings magnetic fields with P B pole pair number, torque winding magnetic field and rotor permanent magnet field with same P M pole pair number. The torque is produced as same as in the traditional PMSM. Compared with the rotor permanent magnet field, the amplitude of torque winding magnetic field is ver small and thus its influence on radial suspension force can be negligible. The radial suspension force is mainl produced b the interaction between rotor magnetic field and suspension winding magnetic field. The direction of resultant radial suspension force is pointed to the increase of the magnetic field... Mathematical Model In order to suspend the rotor steadil, the BPMSM must produce the stable controllable radial suspension force in single direction when the rotor magnetic field, torque winding magnetic field and suspension winding magnetic field rotate in high speed. According to the generation principle of Mawell forces, the torque winding magnetic field and suspension winding magnetic field must rotate in the same direction and rotation mechanical angular velocit must satisf P M =P B ± and P M M =P B B =. Where, M, B are the rotation mechanical angular velocit of the torque winding magnetic and suspension winding magnetic field, respectivel. is the corresponding electrical angular velocit. In other words, the rotation electrical angular velocities of suspension winding magnetic field, rotor permanent magnet field and torque winding magnetic field must be equal. The suspension winding magnetic field and torque winding magnetic field are generated b the three phase smmetric currents in two sets of three phase smmetric stator windings. According to the relationships among the winding pole pair number, mechanical angular velocit and electrical angular velocit in motors, the necessar conditions which the BPMSM can produce the single direction stable controllable radial suspension force is the same phase sequence in two sets of stator windings and the same angular frequenc in stator windings currents. The two sets of winding distributions inserted into stator with the same phase sequence are shown in Fig.. In BPMSM with this stator, the resultant radial suspension force pointed to positive direction when the angular difference between suspension winding magnetic field and torque winding magnetic field is equal to zero. z - c - b - a - z b c - - z - q N a (a) Torque S N F 8 S N (b) Force Fig.. Production mechanism of radial suspension force and torque. S z - Fig.. Windings distribution. The three phase smmetric currents are applied to the two sets of stator windings. 5
Sensors & Transducers, Vol. 75, Issue 7, Jul 4, pp. 5-57 The currents in torque windings can be simpl written as i = IM cos( t θm) i = IM cos( t θm), () iz = IM cos( t 4 θm) The currents in radial suspension windings can be simpl written as ia = IBcos( t θb) ib = IBcos( t θb), () ic = IBcos( t 4 θb) According to the above analsis, using the Mawell's stress tensor method, the radial suspension force in θ direction can be epressed as follow Bg ( q, t) df ( q, t) = ds, (3) m where B g is the air gap flu densit, μ is the permeabilit of vacuum, ds is the infinitesimal area of the rotor surface. When the magnetic field fundamental wave is onl considered, the air-gap flu densit of the P B pole pair number suspension windings magnetic field generated b currents in suspension winding can be epressed as follow. B (, t q ) = B cos( w t-q - Pq), (4) b B B B B P M pole pair number torque windings magnetic field generated b currents in torque winding and permanent magnets mounted onto the rotor surface can be written as follow. B (, t q ) = B cos( w t-q - P q), (5) m M M M M Thus, the resultant air gap flu densit is given as follow. B (, t q ) = B (, t q) B (, t q), (6) g b m In the BPMSM with magnets surface-mounted rotor, the magnetic field generated b the currents in torque windings can be neglected compared to that generated b permanent magnets. From (3), the radial suspension force on the rotor surface can be epressed as follow. [ Bb( t, q) Bm( t, q)] rl df ( q, t) = dq, (7) m The resultant radial suspension force in and - direction on - static coordinate can be concluded b the integral along the rotor circumference. π [ Bb( t, θ) Bm( t, θ)] rl FX = cosθdθ μ, (8) π [ Bb( t, θ) Bm( t, θ)] rl FY = sinθdθ μ At the same time, considering the necessar conditions of producing the single direction stable controllable suspension force P B =P M ± and P B B = P M M =, the radial resultant suspension force can be epressed as follow. π rlbmbb FX = cos( θb θm) μ, (9) π rlbmbb FY = sin( θb θm) μ The radial suspension force onl decided b the rotor radius r, rotor length l and amplitudes of magnetic fields generated b permanent magnets and suspension windings magnetic fields B M, B B. The direction of the resultant radial suspension force depends on the electricit angular degree difference between the suspension windings magnetic field and torque windings magnetic field. In other words, the direction of the resultant radial suspension force depends on initial phase angular difference between suspension winding current and torque winding current. The torque is generated b the interaction between the torque windings magnetic field and rotor permanent magnets field. When the rotor is operating in the center of motor, the torque can be epressed as follow. T = p i, () e M PM Mq 3. Rotor Dnamic Equations The simplified model of rigid rotor in bearingless motor is shown in Fig. 3. When the rotor rotating angular velocit is, the central ais of inertia c is not coincided with the geometric central ais m due to the eistence of mass imbalance. So in the rotating coordinate sstem m, the center of inertia can be epressed as follow. c = m ρcos( β t), () c = m ρsin( β t) where ρ is the eccentricit between the center of inertia and geometric central. is the rotating angular velocit. β is the initial phase angle. For simple analzation, the influence of groscopic effect is neglected. sing Newton's law, the motion equations of rotor geometric center in and direction can be epressed as follow. 53
Sensors & Transducers, Vol. 75, Issue 7, Jul 4, pp. 5-57 N u c( c, c ) k c c ρ m Rotor π/4 k c r m m ρ χ tβ m m (a) (b) Fig. 3. Simplified model of a rigid rotor sstem suspended in BPMSM. d m dm M c km = Mg, () d m dm M c km = Mg where M is the eccentric mass of the rotor. When the rotor rotates, the centrifugal force produced b eccentric mass is decomposed to and direction as follow. Fz = Mρ cos( t β ), (3) Fz = Mρ sin( t β ) The centrifugal force is added to the equation (). The motion equations of rotor geometric center in and direction can be epressed as follow. d m dm M c km Mg M t d m dm M c km Mg M t = ρ cos( β ) = ρ sin( β) (4) Then, the m and m can be got from soluting the equation (4). w where A= r, k w 6c w M (- ) k Mk k M M w B = r. k w 6c w M ( - ) k Mk k M M From the equation (5), the rotor rotation center deviated from the geometric center the centrifugal force. The displacement signals measured b and direction displacement sensor are sine wave. The phase difference is 9º. Through the PID controller, the control current waveform is also sine waveform. The power amplifier produced the control forces to enforce the rotor rotation around the geometric center ais. According to action force and reaction force principle, unbalance vibration will be produced. Mg m = Acos( t β arctan[ ξ / ( )]) k / / M k M k, (5) Mg m Bsin( t β arctan[ ξ / ( )]) = k / k / M k M From equation (5) can be seen that the displacement signals consist of the vibration signal caused b the centrifugal force for rotor mass imbalance. the amplitude of vibration signal is proportional to the square of the speed. When the speed is ver high, the control current in power amplifier will be saturation. At the same time, the vibration will transfer to the base of motor, which caused the sstem vibration. The compensation control of rotor mass unbalance proposed in this paper is deviating the vibration signal from the rotor displacement signal and compensate to the displacement signal. In other words, the vibration components are removed from the displacement signal. The displacement close-loop does not control the vibration components. The rotor rotates around the center of inertia and the vibration is eliminated. At the same time, the control current oscillation in power amplifier is also reduced. 4. Control Sstem Design The bearingless permanent magnet snchronous motor rotor mass unbalance compensation control sstem is shown in Fig. 4, which consists of the feedforward compensation control unit and the rotor field decoupling control close loop. 54
Sensors & Transducers, Vol. 75, Issue 7, Jul 4, pp. 5-57 X s Y s Feed-forward Control X s c Y s c PID PID q PID F F i = i q θ Force/ Current VR i α i β i α i β /3 i A i B i C i A i B CRPWM /3 CRPWM i C i A i B i C i A i B i C PMSM Displacement Sensor θ Fig. 4. Simulation diagram of the control sstem on vibration control for bearingless PMSM. 4.. Displacement Control The displacements in and direction is detected b the radial displacement sensor. and are the given signals in displacement control close loop. The compensation signals c and c are etracted from the feedback signals X s and Y s. The error among the given signals, compensation signals and feedback signals pass through the PID controller. The radial forces command values F and F will be produced. Then, using coordinate transformation, the control currents command values i α, i β are got. Finall, based on the 3/ transformation, the currents command values in radial coils i A, i B and i C are got. The radial displacements are controllable through adjusting the currents in suspension coils. 4.. Torque Control The angular velocit feedback signal of rotor is detected b the velocit sensor. The error between the feedback signal and given signal is calculated. When the error signal pass through the PID controller, the current command value of q-ais is produced. Then, using the transformation from the two-phase rotating coordinate sstem to the twophase stationar coordinate sstem, the current command values of i α and i β are produced. The current command values of i A, i B and i C are produced using /3 transformation. The velocit of rotor is controllable through adjusting the currents of torque coils. 4.3. Feed-forward Control For simple analsis, it is assumed that the rotor onl have the static unbalance for inertial center misalignment with the geometr center. The bearingless permanent magnet tpe motor has the same stiffness in and direction. When the angular velocit of rotor is, the centrifugal force acting on the rotor can be epressed as equation (3). The feedforward compensation controller includes two parts. The rotor vibration is caused b the centrifugal force. And thus, the simulation sstem must consider the effect of the centrifugal force. The calculation module of centrifugal force F z and F z according to the equation (3) is shown as Fig. 5. CLOCK ρ cos sin m F z F z Fig. 5. Block diagram of centrifugal force calculation. The vibration signal etraction module is shown in Fig. 6. Its purpose is to realize the etraction of vibration signal from the displacement signal. The displacement signal of geomentric center X s and Y s detected b the radial displacement sensor transform to X r and Y r using the transformation from the rectangular coordinate to the snchronous rotating coordinate. X r = Xscost Yssint, (6) Yr = Xssint Yscost The vibration signal with the frequenc equals to the rotational frequenc is transformed to the DC signal. The other signals are transformed to high frequenc signals. Then, the X r and Y r pass through the low pass filter (LPF). The high frequenc signals are filtered. The output signals X r and Y r are onl consisting the DC signals. Finall, the inverse coordinate transformation has been done as follow. 55
Sensors & Transducers, Vol. 75, Issue 7, Jul 4, pp. 5-57 CLOCK sin cos X r LPF X r K i X c X s Y s Y r LPF Y r K i Y c Fig. 6. Block diagram of vibration signal etraction. X c = Xrcost Yrsin t, (7) Yc = Xrsint Yrcost From equation (7), the vibration signals X c and Y c etracted from the displacement signals are the compensation signals. The compensation signals have the same frequenc with the F z and F z. But the phase difference is equal to 8. The amplitudes of compensation signals can be adjusted. 5. Simulation Research In order to verif the validit of the above vibration control method, the feed-forward compensation control sstem is constructed based on the rotor magnetic field decoupling control sstem. The control sstem is shown in Fig. 5. The calculation of centrifugal force and etraction of vibration signals are following the rotational velocit. So the control sstem can realize the real-time compensation. The parameters of bearingless permanent magnet tpe motor are given as Table. The simulation sstem is constructed using Matlab/Simulink toolbo. The simulation process is from s to. s using Ode3t. Fig. 7. shows the currents waveform without feed-forward compensation control. The control currents in power amplifier are sine waveform. The amplitude is.5 A. After the compensation control is added to eliminate the rotor vibration, the amplitude of control current is. A shown in Fig. 8. Fig. 7. Power amplifier output current waves without feed-forward compensation control. Fig. 8. Power amplifier output current waves with feed-forward compensation control. The trajector of rotor geometric center is shown in Fig. 9. In (a), the eternal control force enforce the rotor rotating around the geometric center in the radial displacement closed loop sstem. The radial vibration amplitude of rotor is about to -4 mm. The trajector of rotor geometric center is shown in (b) with compensation control. The rotor rotates around the center of inertia because the vibration component in feedback signals is offset b the compensation signals. So the amplitude of radial displacement is equal to the eccentricit (. mm). But the control currents in power amplifier are decreased substantiall. 56
Sensors & Transducers, Vol. 75, Issue 7, Jul 4, pp. 5-57 is eliminated with the rotor rotating around inertia center. Acknowledgements This work was supported b The Natural Science Foundation of Jiangsu Province (BK46, BK348) and Colleges and niversities Natural Research Project of Jiangsu Province (3KJB47). References Fig. 9. The trajector of rotor geometric center. 6. Conclusions In this paper, introducing of the rotor motion equation of BPSMS and analsis on generation mechanism of unbalance force, feed-forward compensation controller is designed based on the technolog of adaptive notch filter. The control sstem is simulated with Matlab/Simulink toolbo. The simulation results have shown that the current oscillation can be minimizing b the feed-forward compensation controller, and vibrating force []. M. Ooshima, A. Chiba, T. Fukao, Characteristics of a permanent magnet tpe bearingless motor, IEEE Transactions on Industrial Application, Vol. 3, Issue, 996, pp. 363-37. []. M. Ooshima, A. Chiba, T. Fukao, Design and analsis of permanent magnet-tpe bearingless motors, IEEE Transactions on Industrial Application, Vol. 43, Issue, 996, pp. 9-99. [3]. A. Chiba, S. Onoa, T. Kikuchi, An analsis of a prototpe permanent magnet bearingless motor using finite element method, in Proceedings of the 5 th International Smposium on Magnetic Bearings, Kanazawa, Japan, 996, pp. 35-36. [4]. Raoul Herzog, nbalance compensation using generalized notch filters in the multivariable feedback of magnetic bearings, IEEE Transactions on Control Sstems Technolog, Vol. 4, Issue 5, 996, pp. 58-586. [5]. Zihe Liu, Kenzo Nonami, Adaptive non-stationar vibration control for magnetic bearing sstem from start-up to operational speed, in Proceedings of the Seventh International Smposium on Magnetic Bearings, ETH, Zurich,, pp. 567-57. [6]. Juan Shi, Ron Zmood and Li Jing Qin, The indirect adaptive feed-forward control in magnetic bearing sstems for minimizing selected vibration performance measures, in Proceedings of the 8 th International Smposium on Magnetic Bearings, Mito, Japan,, pp. 3-8. 4 Copright, International Frequenc Sensor Association (IFSA) Publishing, S. L. All rights reserved. (http://www.sensorsportal.com) 57