The FEM Analysis and Torque Calculationof REPM Gear

Hong Yang1Þ, Zhiyi Yang, and Han Zhao

Mechanical and Automotive Engineering Department, Hefei University of Technology,Hefei 230009, China

(Received May 1, 2001; accepted September 30, 2001)

Subject classification: 02.70.Dh; 75.50.Ww; 85.70.Ay; 96.35.Pt; S1.4

Permanent magnetic (PM) gears are magneto-mechanical devices that are widely used to replacethe ordinary mechanical gears and to transmit torque without any mechanical contact. In this pa-per, according to the demagnetization curve of REPM material and the relationship between mag-netic density and intensity of magnetization with magnetic medium in the magnetic field, the mag-netic analysis model of PM external gear is built, and its magnetic field distribution is calculatedby the finite element method (FEM). The transmission torque is calculated by the Maxwell stresstensor method and its characteristics are studied theoretically. In the end, the torque curves of PMgear are given. From the curves, the influences of geometry size, magnet number, transmissionratio, height of yoke of the magnet on transmission torque are obtained. It cannot only forecastthe field distribution to a certain extent, estimate the transmission performance, calculate the trans-mission torque, but also can give a quantitative direction to the optical design and related para-meters of PM gear.

1. Introduction The development of permanent magnetic materials has a long history.The NdFeB permanent magnet has been successfully produced in 1983. It is called thethird permanent magnetic material because of its good magnetic performances.1. NdFeB magnet has high magnetic performance. It can attract the weight 640 timesthat of itself. So, the application of NdFeB magnet can make the mechanical devicesand equipment smaller, lighter, and thinner. 2. NdFeB magnet has relative low cost. Ituses the Nd element with fluent resources. it does not need Co and rare earth Sm withshort resource. 3. The mechanical performances of NdFeB magnet are better than thatof rare-earth Co magnet and that of AlNiCo magnet. Thus, the NdFeB magnet pos-sesses the main markets swiftly, it has wide application and development prospects inmechanical engineering.

Permanent-magnetic gears are novel transmission devices formed by a pair of cylind-rical sintered or adhered NdFeB magnets. They have unique advantages that are easyprocessed, no mechanical contact, no frictional loss and no noise, low starting torque,and over-load protection etc. So, it is necessary to study the design and torque charac-teristics of permanent-magnetic gears.

2. Structures of Permanent-Magnetic Gears Permanent-magnetic gears are magneto-mechanical devices that are widely used to replace the ordinary mechanical gears andto transmit torque without any mechanical contact. These devices consist of two sepa-rated radial polarized cylindrical sintered NdFeB magnets constrained to rotate abouttheir respective axes. The permanent magnetic gears are magnetically coupled to one

1) Corresponding author; Tel: +86-755-6731151; e-mail: yinyouyang@hotmail.com

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phys. stat. sol. (a) 189, No. 3, 1057–1061 (2002)

another, and when the primary driver ro-tates, it imparts a torque to the followerand cause it to rotate. The new deviceshave broad application prospects in thefield of robots, medical instruments, chemi-cal engineering, food industry, and so on.

Similar to normal gear transmission, thePM gear transmission has three types: ex-

ternal gear transmission, internal gear transmission and pinion-rack transmission [1],see Fig. 1.

Since the transmitted torque has one of the most important properties of magneticgears [2, 3], the torque characteristics of radial type permanent magnetic gears are in-vestigated in this paper with computer aided simulation and finite element method(FEM). The Maxwell stress tensor technique is used to calculate the torque of the mag-netic gears.

3. Finite Element Analysis FEM is regarded as one of the most efficient numericalmethods, and now is routinely employed for the analysis of magnetic field. In this sec-tion we introduce the magnetic field computation of permanent magnetic gears byusing FEM. The Maxwell stress tensor technique is used to calculate the transmittedtorque of the magnetic gears.

3.1 Simulation of magnets The mathematic model of permanent magnets is built be-fore the magnetic field of permanent magnetic gears is calculated. The characteristicrelationship of parameters B and H for permanent magnetic materials is evolved into ademagnetization curve. The mathematic model of permanent magnets is built on thedemagnetization curve.

The rare-earth permanent magnet is a magnetic medium with no supply. Suppose it iswell-distributed magnetized, it is well known that the permanent magnet with magneti-zation M is represented as a distribution of equivalent surface current density,

Km ¼ M � n and H ¼ 1m0

B � H : ð1Þ

The normal unit vector n is perpendicular to the surface of the permanent magnet. Km

is the equivalent surface current density.The demagnetization curve of rare-earth permanent magnets is represented simply

as

H ¼ 1m0

B � Hc ;

where Hc is the coercive force of the permanent magnet. So,

M ¼ Hc : ð2Þ

1058 Hong Yang et al.: Torque Calculation of REPM Gear

Fig. 1. Transmission of REPM gear

3.2 Magnetic field computation The magnetic field caused by PM gears is a kind ofquasi-static magnetic field. According to the theory of the magnetic field, in the 2Dmagnetostatic approximation, the magnetic vector potential A satisfies the Poissonequation

r2A ¼ �mJ ; ð3Þwhere m is the relative magnetic permeability and J is the current density. If the axiallength of the magnetic gear is far greater than the dimensions in the cross-sectionplane, the magnetic field is assumed to be the 2D field. In Eq. (3), A and J are sub-stituted by the component in axial direction.

In order to solve Eq. (3), the boundary conditions of the magnetic gears have to beknown the magnetic flux penetrates outside the permanent magnets. So we must built-model for a bigger air region around the magnetic gears. And the far field elements areemployed on the boundary of model, thus the magnetic vector potential at the bound-ary of the model is zero. The second-order isoparametric elements are used in the inter-nal region of the model. In this work, the variation principle is used in the discretiza-tion of the field Eq. (3). High order nonlinear equations are derived, and the Newton-Raphson iteration method is used for solving the nonlinear equations. The vectorpotential A and flux density B is calculated from the nodal values

A ¼PeNeAe ;

B ¼ r� A ;

(ð4Þ

where Ne is the element shape function, Ae is the nodal magnetic vector potential.Thereby the magnetic field distribution of permanent magnetic gears can be obtained,see Fig. 2.

3.3 Calculation of torque For the 2D application of magnetic field, the methodbased on Maxwell’s stress tensor is used to calculate torque in the finite elementanalysis with extrapolated field values in the following numerically integrated surfaceintegral:

F ¼ÐS

s � dS ð5Þ

where s is Maxwell’s stress tensor and S is the integration surface.For a 2D analysis, the corresponding electromagnetic torque about the þz axis is

given by

T ¼ Z � 1m0

ðS

r � ðn � BÞ B � 12

B2n� �

dS ; ð6Þ

phys. stat. sol. (a) 189, No. 3 (2002) 1059

Fig. 2. Magnetic field distribution of permanent magneticgears with the relative angular offset a ¼ 22.5

where Z is the unit vector along þz axis, r and n are the position vector and the unitsurface normal vector in the global Cartesian coordinate system, respectively.

4. Torque Characteristics Analysis The transmitted torque of PM gear is a function ofseveral variables, including the magnetic pole pairs, the dimensions of magnets, the airgap between two gears, the thickness of yoke, transmission ratio, and the relative angu-lar offset of the two gears, and so on.

In this paper, the outer and inner radii of the PM gear are 8 and 4 mm, respectively.The axial length of magnets is 16 mm. There are eight poles on each magnetic gear. Inthe process of analyzing the magnetic field and calculating the torque, the geometryand materials used should be accurately represented in the model, and the nonlinearB–H characteristics of iron for the inner ring of magnetic gear is considered. The tor-que is computed on the driven gear.

1. The relative angular offset and the air gap between two gears: The relative angularoffset a is defined from the center of north pole on one gear to the center of southpole on the other. Figure 3 shows the torque of the radial magnetic gears as a functionof the relative angular offset and the air gap between two gears for cylindrical magnets.

When a ¼ 0, the torque is equal to zero. This position is stable. As a is increased thetransmitted torque is increased. When a is half a pole pitch, a ¼ p=p (p is the number ofpoles), the torque is the maximum. After the pullout, the transmitted torque is decreased asa is increased. When a ¼ 2p=p, the torque is also equal to zero, but this position is unstable.For a given number of poles, the level of the magnetic induction rapidly decreases with theair gap increasing. So, the torque transmitted by two gears is decreased quickly.

2. The dimension and the number of poles: In Fig. 4, the air gap between two mag-netic gears without any iron yoke is fixed to 2 mm. For given poles p the maximumtorque is obtained for a relative angular offset of p=p.

From these curves, we observed that the peak is shifted from 4 poles to 8 poles, andto 12 poles with increasing outer radius of the magnets from 8 to 16 mm, and to 24 mm,respectively. This is because the magnetic field strength is gradually increased, the areaof the surface of each pole is decreased however for increasing the number of thepoles. Therefore, the combination of the two effects leads to the maximum peak shift tolarger number of poles with larger radius of the cylindrical magnets.

1060 Hong Yang et al.: Torque Calculation of REPM Gear

Fig. 3 Fig. 4

Fig. 3. Torque versus the relative angular offset and the air gap

Fig. 4. Torque versus the number of pole pairs and the geometrical dimension

3. The transmission ratio: Under the conditions that the number of pole pairs and outradius of the driver gear are fixed to given values, the number of pole pairs and outradius of the driven gear will be increased with increasing transmission ratio. So andonly so, a stable rotation speed ratio of the two gears can be achieved. Figure 5 showsthat the maximum torque is increased linearly with increasing transmission ratio.

4. The thickness of iron yoke: In Fig. 6, the out radius of PM gears is 32 mm, its yokethickness varied from 0 to 10 mm. It is clear that the torque is enhanced by the exist-ence of an iron yoke, and its value as a function of the number of pole pairs is satu-rated for iron yoke with thickness above 1 mm. From these curves, we observed thatthe peak is shifted from 10 pole pairs for the case without any iron yoke to 12 polepairs for the case with an iron yoke of above 1 mm.

5. Conclusions In this paper, the torque characteristics of radial type permanent mag-netic gears are studied with FEM combining with Maxwell stress tensor technique. Theresults show that the transmitted torque of permanent magnetic gears is sensitive to thefactors including the number of magnetic poles, the dimensions of the magnets, the airgap between the two gears, the thickness of the yoke, transmission ratio, and the rela-tive angular offset of the two gears.

Acknowledgement The authors would like to thank the National Nature ScienceFoundation of China (59975027) for the financial support to this research work.

References

[1] Zhao Han and Tian Jie, Mechan. Sci. Technol., 19, 207 (2000).[2] Y. D. Yao, D. R. Huang, et al., IEEE Trans. Magn. 33, 2203 (1997).[3] Y. D. Yao, D. R. Huang, et al., IEEE Trans. Magn. 33, 4236 (1997).[4] Sheng Jianni, Numerical Analysis of Engineering Electromagnetic Field, Xi’an Jiaotong Uni-

versity Press, 1991.

phys. stat. sol. (a) 189, No. 3 (2002) 1061

Fig. 5 Fig. 6

Fig. 5. Torque versus the transmission ratio

Fig. 6. Torque versus the number of pole pairs for different thicknesses of iron yoke