ANSYS 2011 中国用户大会优秀论文

Transient Analysis of a Magnetic Gear Integrated Brushless Permanent Magnet Machine Using Circuit-Field-Motion Coupled Time-Stepping

Finite Element Method

S. L. Ho, Shuangxia Niu, and W. N. Fu

The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

In this paper, a novel magnetic gear integrated brushless permanent magnet machine is studied. The main merit of the machine is that the torque produced by the stator windings can be transmitted between the low-speed rotor and the high-speed rotor through the modulation of ferromagnetic pole pieces; hence it can output a large torque at a low speed. The operating principle of the machine is discussed and its steady-state and transient performances are analyzed using circuit-field-motion coupled time-stepping finite element method. Detailed analysis about its gear ratio, cogging torque, core losses, and power factor are reported. Theoretical analysis agrees well with the FEM computation results.

Index Terms—Ferromagnetic pole, gear radio, magnetic gear, permanent magnet motor, time-stepping finite element method.

I. INTRODUCTION AGNETIC GEAR (MG) has good potential for torque and speed transmission in practical engineering as MGs

do not have problems on lubrication and cooling. Besides, its intrinsic overloading protection feature can successfully avoid many mechanical problems, such as the broken teeth and backlash [1]. MG can also offer many merits, such as minimum acoustic noise, freedom from maintenance and improved reliability [2]. An electric machine combined with MG to provide motive power has also been reported [3]. However, the reported unit has three airgaps, the structure is complex, and hence is difficult to manufacture. A novel direct-drive MG integrated brushless permanent magnet (PM) machines that can directly combine MG with an electric machine mechanically and magnetically is reported [4]. Torque can be transmitted between the low-speed rotor and the high-speed rotor through the modulation of ferromagnetic pole pieces. Until now, only electric circuit method has been used to analyze its transient process. This is because there are two rotating bodies in the machine, and the analysis using finite element method (FEM) of the machine is more difficult than that for conventional machines.

Time-stepping finite element method (TS-FEM), which couples magnetic field equations with external electric circuit equations and mechanical torque balance equations, has been successfully used to analyze the transient performance of electric machines. Effects of saturation, eddy current and mechanical movement of the rotor can all be taken into account in the model [5]. The purpose of this paper is to employ a novel circuit-field-motion coupled TS-FEM with two rotating rotors to analyze this new machine. The steady-state and transient performances of machine are studied. Cogging torque, core losses and power factor of the machine

are simulated and analyzed.

II. MACHINE STRUCTURE Fig. 1 gives the configuration of the magnetic gear

integrated machine. This machine is designed with 2 pole-pair PMs in the high-speed rotor, 21 pole-pair PMs in the outer stator. Between the high-speed rotor and outer stator, 23 modulating ferromagnetic pieces are added. These ferromagnetic modulation pole pieces serve as the low-speed rotor and are used to modulate the space harmonics of the airgap flux density. Out of the 21 pole-pair PMs, the outer stator is designed with three-phase, 2 pole-pair concentrated windings.

(a)

(b)

Fig. 1. The magnetic gear integrated brushless PM machine. (a) Machine structure. (b) Front view.

M

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A. Gear Ratio Torque transmission of the MG is based on the modulation

of magnetic field using ferromagnetic pole pieces between the outer stator and the high-speed rotor. According to [6], the number of pole pairs in the space harmonics of the flux density distribution and the speed of the space harmonics produced by either the high or low speed rotor PMs, is given by

skm knmpp +=, (1)

srkm knmpmp += ωω , (2) where; , ,,5 ,3 ,1 ∞= Lm ∞±±±= , ,2 ,1 ,0 Lk ; p is the pole-pair number of the inner rotor; sn is the number of the ferromagnetic pole pieces and rω is the inner high-speed rotor speed of the machine. In order to transmit torque at different speeds, the pole pairs of the outer PMs must be equal to

kmp , and if the ferromagnetic pole pairs are stationary, the speed of the space harmonics of the outer rotor must be equal to km,ω . When 1 ,1 −== km , the largest space harmonic component is obtained. Thus, if the ferromagnetic pole pieces are stationary, the speed of the outer rotor of the gear is

sr npp −=− ωω 1,1 (3) and the gear ratio is given by

pnpG sr −= (4) If the outer PMs are stationary, the speed of the ferromagnetic pole pieces or the low-speed rotor is

sr npωω =−1,1 (5) and the gear ratio is given by

pnG sr = (6) For this novel magnetic gear integrated machine, the gear

ratio is 5.112/23 === pnG sr , meaning that the low-speed rotor can amplify the output torque from the high-speed rotor by a factor of 11.5.

B. Machine Design and Parameters The primary torque is produced by the three-phase stator

windings, and the frequency of the stator winding determines the synchronous speed of the high-speed rotor. The stator is fed by a three-phase current source at 100 Hz, and the rated speed is 3000 rpm. Upon modulation of the ferromagnetic gear, the speed becomes 3000/11.5=261 rpm and the torque is boost up accordingly for the direct drive. The design data are given in Table I.

TABLE I DESIGN DATA

Number of phases 3 Rated frequency 100 Hz Rated inner-rotor speed 3000 rpm Rated outer-rotor speed 261 rpm Number of stator slots 6 Number of inner-PM pole pairs 2

Number of outer-PM pole pairs 21 Number of ferromagnetic pole pieces 23 Stator outer radius 120.0 mm Outer PMs outer radius 85.3 mm Outer PMs inner radius 75.0 mm Inner PMs outer radius 61.0 mm Inner PMs inner radius 45.5 mm Each airgap length 0.6 mm Axial length 50 mm Stator winding turns per coil 10 Remanence of PMs 1.1 T

III. CFM-TS-FEM ANALYSIS This machine is composed of one magnetic gear and a PM

brushless machine. There are two airgaps and two rotating bodies. The circuit-field-motion coupled time-stepping finite element method (CFM-TS-FEM) is employed to analyze the steady-state and transient performances of the machine [5].

A. Determination of the Initial Positions In order to produce positive torque, the position of the

rotating magnetic field in the airgap and the rotor position should be controlled properly. In CFM-TS-FEM, it means that the initial positions of the two rotors should be predetermined properly. As two magnetic devices (MG and PM brushless machine) are included in the computation, it becomes one of the difficulties in the simulation. First of all, the static performance of the magnetic gear needs to be evaluated. The simulation is conducted with the high-speed rotor rotating at the rated speed while the other parts (ferromagnetic pole pieces and outer stator) are fixed. The corresponding static torque is obtained as shown in Fig. 2 which shows the torque-angle curve varies sinusoidally and at any rotor angle, the torque ratio of the outer rotor with respect to the inner rotor is 11.5, which agrees with the principle of MG. The torque curves intersect at the no-load equilibrium points. If the MG carries no load, it will stay at these no-load equilibrium positions. As the load increases, the magnetic torque will go up along the torque curve to the new equilibrium position. When the load is larger than the maximum static torque of the magnetic gear, the speed of the outer rotor will return to zero and the high-speed rotor will vibrate continuously and this is the intrinsic overloading protection property of MG.

Fig. 2. Static torque performance of magnetic gear.

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Fig. 3. Magnetic flux distribution at full load.

In this paper, considering the overloading factor (1.8~2.2) of PM brushless machine, the rated load is designed with half of the maximum static torque of the MG, namely 205 Nm. In order to analyze the full load performance of the machine, the initial position of high-speed rotor should be at

5.22/2/90 =p mechanical degrees from the no-load equilibrium position. For the outer stator windings, since the magnetic field produced by the stator currents is synchronous with the high-speed rotor, the initial position should be consistent with that of the high-speed rotor.

B. Magnetic Flux Density Analysis Fig. 3 shows the magnetic field distribution under full load

obtained using CFM-TS-FEM. It is shown that the majority of the flux lines can penetrate through the flux-modulation poles and both airgaps. They can be effectively used to transfer torque and power between the low-speed and high-speed rotors. Due to the outer stator PMs, the variation of the magnetic flux density and their corresponding harmonic spectra in the airgaps adjacent to the high-speed rotor and adjacent to the stator, are computed, as shown in Figs. 4 and 5, respectively. It is shown that with the modulation of the ferromagnetic pole pieces, the asynchronous harmonic component with 2 pole pairs in the airgap is greatly improved (fig. 4), compared with that before modulation (Fig.5). On the contrary, the harmonic component with 21 pole pairs is reduced. This means with the presence of the ferromagnetic pole pieces, the flux density can be modulated accordingly. For the same reason, the variation of the magnetic flux density and their harmonic spectra in the airgaps adjacent to the high-speed rotor and adjacent to the stator, due to the high-speed rotor PMs are obtained, as shown in Figs. 6 and 7, respectively.

Flux

den

sity

(T)

(a)

(b) Fig. 4. Flux density in airgap adjacent to the high-speed rotor, due to the stator PMs. (a) Waveform. (b) Spectrum.

IV. TORQUE TRANSMISSION AND LOSS ANALYSIS The torque capacity of the MG can be calculated using

Maxwell’s stress tensors in the outer and inner airgaps. When computing the cogging torque of the machine, the initial position of the high-speed rotor and low-speed rotor are at the no-load equilibrium position, and then the rotors rotate with the speed ratio 11.5/1. The simulated cogging torque is shown in Fig. 8. The torque ripple is relatively small compared with the rated output torque of the machine. The reason is that the cogging torque ripple is approximately related to the inverse of the smallest common multiple cN of the number of the inner rotor poles p2 and the number of the ferromagnetic pole pieces sn and in this paper, 92=cN .

(a)

(b)

Fig. 5. Flux density in airgap adjacent to the stator, due to the stator PMs. (a) Waveform. (b) Spectrum.

ANSYS 2011 中国用户大会优秀论文

(a)

(b)

Fig. 6. Flux density in airgap adjacent to high-speed rotor, due to the high-speed rotor PMs. (a) Waveform. (b) Spectrum.

The full-load torque transmission of the MG is simulated with TS-FEM, as shown in Fig. 9. During the simulation, the stator windings are fed with 57 A, three-phase sinusoidal currents. The corresponding induced torque by the stator windings is about 17.5 Nm as shown in Fig. 9 (b), which is used to produce the motive torque on the high-speed rotor. The output torque on the low-speed rotor is about 203 Nm and the transmission ratio agrees well with the gear ratio. The transient core loss at the full-load operation is simulated with dynamic core loss model [7], and the core loss curves are shown in Fig. 10. The core loss is about 90 W, which are mainly induced in the stator core, rotor core and ferromagnetic pole pieces. The induced voltages are simulated, as shown in Fig.11. The power factor is high and in excess of 0.9.

(a)

0 10 20 30 40 500.0

0.1

0.2

0.3

0.4

0.5

Pole-pair number (b)

Fig. 7. Flux density in airgap adjacent to the stator, due to the high-speed rotor PMs. (a) Waveform. (b) Spectrum.

Fig. 8. Cogging torque of the high-speed rotor.

(a)

Torq

ue (N

m)

(b)

Fig. 9. Full load torque waveforms. (a) Torque on the rotors. (b) Torque produced by stator windings.

Fig. 10. Simulation result showing the core loss curve versus time at full-load operation.

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Fig. 11. Input current and induced voltage waveforms.

V. CONCLUSION A novel magnetic gear integrated PM brushless machine is

designed and analyzed with circuit-field-motion coupled TS-FEM. The integrated design can effectively reduce the speed and amplify the torque of the machine by about 11.5 times. The computation and analysis of the gear ratio, cogging torque, core losses, and power factor are reported. The theoretical analysis agrees well with the CFM-TS-FEM simulation results.

ACKNOWLEDGMENT This work was supported in part by The Hong Kong

Polytechnic University under Grants 1BBZ7 and B-Q18X.

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[3] L. Jian, K.T. Chau and J.Z. Jiang, “A magnetic-geared outer-rotor permanent-magnet brushless machine for wind power generation,” IEEE Trans. Ind. Appl., vol. 45, pp. 954-962, 2009.

[4] K. Atallah, J. Rens, S. Mezani and D. Howe, “A novel ‘Pseudo’ direct-drive brushless permanent magnet machine,” IEEE Trans. Magn., vol. 44, no. 11, pp. 4349-4352, 2008.

[5] W.N. Fu, P. Zhou, D. Lin, S. Stanton and Z.J. Cendes, “Modeling of solid conductors in two-dimensional transient finite-element analysis and its application to electric machines,” IEEE Trans. Magn., vol. 40, no. 2, pp. 426-434, March 2004.

[6] K. Atallah, S.D. Calverley, D. Howe. “Design, analysis and realization of a high-performance magnetic gear,” IEE Proceedings-Electric Power Applications, vol. 151, no. 2, pp. 135-143, March 2004.

[7] D. Lin, P. Zhou, W.N. Fu, S. Stanton and Z. J. Cendes, “A dynamic core loss model for soft ferromagnetic and power ferrite materials in transient finite element analysis,” IEEE Trans. Magn., vol. 40, no. 2, 1318-1321, March 2004.