ΦAbstract -- Magnetic gears, which can transmit torque without mechanical contact, have low vibration and acoustic noise, and are low maintenance in comparison with conventional mechanical gears. Various types of the magnetic gears have been introduced. Among them, a planetary type magnetic gear has attracted interest recently because its transmission torque is higher than the other types. This paper presents a high–speed and accurate optimal design method of the planetary type magnetic gear based on reluctance network analysis (RNA). The RNA has several features including a simple analytical model, high calculation accuracy with high speed, and ease of coupled analysis with rotational motion of rotors. The validity of the proposed method is proved by comparing with a finite element analysis (FEA) and experiment.

Index Terms—magnetic gear, optimum design, reluctance network analysis (RNA), finite element analysis (FEA)

I. INTRODUCTION agnetic gears can transmit torque without mechanical contact. Therefore, they have low vibration and

acoustic noise, and are low maintenance in comparison with conventional mechanical gears. Various types of magnetic gears have been introduced in previous papers [1]–[3]. Among them, a planetary type magnetic gear [4] has attracted interest recently.

Fig. 1 shows the basic structure of a planetary type magnetic gear used in the consideration. The magnetic gear consists of an inner and outer rotors with surface mounted permanent magnet (SPM), and ferromagnetic stationary parts which are called pole pieces. It works as a gear by modulating the magnet flux due to the pole pieces. Torque density of the planetary type magnetic gear is higher than the other types because all the magnets on the inner and outer rotors contribute to generate and transmit torque [5], [6].

To be put the magnetic gear into practical use, establishment of an optimum design method for the magnetic gear is necessary. Nowadays, a number of general purpose programs of finite element analysis (FEA) have been placed on the market. However, FEA–based optimum design is difficult because its analytical model is complicated and calculation time tends to be long.

On the other hand, reluctance network analysis (RNA) is one of the practical solutions, because the analytical model is simple, calculation accuracy is relatively high, and it is easy to combine with motion dynamics [7], [8].

This work was supported by JSPS Grant–in–Aid for Young Scientists

(A) (22686028) , and Grant–in–Aid for JSPS Fellows (24•4456). M. Fukuoka, K. Nakamura, and O. Ichinokura are with the Depeartment

of Electrical and Communication Engineering, Graduate School of Engineering, Tohoku University, 6-6-05, Aoba, Aramaki, Aoba-ku, Sendai, 980-8579, Japan (e-mail: power20@ec.ecei.tohoku.ac.jp).

In this paper, first the RNA model of an SPM type magnetic gear is described. The calculated torque is compared to FEA and experiment. Next, the RNA–based dynamic analysis method of the magnetic gear is presented. Finally, the RNA–based optimum design for an SPM type magnetic gear is presented. The proposed optimum method has high calculation accuracy with high speed.

II. ANALYSIS OF SPM TYPE MAGNETIC GEAR BASED ON RNA

A. RNA model of the SPM type magnetic gear Fig. 1 shows the structure and specifications of the SPM

type magnetic gear used in the consideration. The inner and outer rotors have permanent magnets on their surfaces. The numbers of pole–pairs of the inner and outer rotors are 3 and 31, respectively. Hence, the gear ratio is 1: 10.333 which is given by the ratio of the inner and outer pole–pairs [5]. The pole pieces are placed between the inner and outer rotors. The number of pole pieces is 34 which is given by the sum of the numbers of the inner and outer pole–pairs. The material of the permanent magnet is Nd–Fe–B of which residual flux density Br is 1.25 T and coercive force Hc is 975 kA/m, respectively. Core material of the pole pieces and the rotor back yokes are non–oriented silicon steels.

Fig. 1. Structure and specifications of an SPM type magnetic gear.

118

mm

156

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M. Fukuoka＊, K. Nakamura, O. Ichinokura

A Method for Optimizing the Design of SPM Type Magnetic Gear Based on

Reluctance Network Analysis

M

978-1-4673-0142-8/12/$26.00 ©2012 IEEE 30

In order to derive the RNA model, first of all, the

magnetic gear is divided into multiple elements as shown in Fig. 2. The number of divisions in a circumferential direction is determined based on the number of the pole pieces so that the pole pieces are divided equally into an integral number. On the other hand, it is not necessary to divide the rotor magnets equally into an integral number because the magnetomotive forces (MMFs) of permanent magnets are given by a continuous function of a rotor position angle θ in the RNA model.

Each divided element is expressed in a two–dimensional unit magnetic circuit as shown in Fig. 3. The reluctances Rr and Rθ in the circuit are given by only the dimensions of the divided element and the magnetic property of materials.

In case the unit magnetic circuit is in the core region, magnetic nonlinearity of the reluctance of the core Rm must be considered. Fig. 4 shows the B–H curve of core material. To take the magnetic nonlinearity into consideration, the B–H curve of core material is approximately expressed by the following nonlinear function [7]: , 1 where the coefficients α1 and α13 are 58 and 5, respectively. The order 13 is determined by the strength of nonlinearity. Using Eq. (1), the MMF fm can be written as follows: . 2

Consequently, α α , 3

where the cross section is S, and the magnetic path length is l, respectively. In Eq. (3), the nonlinear formula enclosed in the parenthesis indicates the reluctance Rm(φ).

Fig. 2. Divisions of the magnetic gear based on RNA.

Fig. 3. Unit magnetic circuit.

Fig. 4. B–H curve of core material and its approximate curve. On the other hand, the reluctance of the air Rair is defined

by , 4

where the permeability of vacuum is μ0. The MMF of permanent magnets fc is defined by , 5

where the length of the magnet is lm. Thus, the MMFs distributions of the inner and outer rotors Fh(θ) and Fl(θ) are given by the following function so that the rotary motion can be expressed in the RNA model:

2 tan sin , , 6 where the coefficient b is 5, the number of the inner and outer rotor pole–pairs are ph and pl, the length of inner and outer rotor magnets are lmh and lml, respectively [8].

Fig. 5 shows a part of the RNA model obtained in the way described above. Fig. 5(a) illustrates the model on the r–θ plane. The number of division in the circumferential direction is 272, that is, each pole piece is divided equally into 4 parts. Fig. 5(b) indicates the model on the r–z plane. The RNA model takes the leakage flux in the axial direction into account since the magnetic gear has a flat structure.

The number of division in a radial direction is 6. Therefore the magnetic gear is divided into 2,176 elements.

B. Steady state analysis Fig. 6 shows the flow diagram of steady state analysis

using the RNA model. When the angular velocities of the inner and outer rotors ωh, ωl are given first, both rotor positions θh, θl are obtained from , . 7

Next, the flux distribution in the RNA model is calculated, and then the generated torque of the both rotors is calculated by the following function based on magnetic energy [9]:

2 , 8

where the number of rotor division in the circumferential direction is nθ. The fluxes and the difference of MMFs in the permanent magnets are φmj and ∆fmj, respectively. All the above calculation can be performed simultaneously on SPICE which is a general purpose circuit simulator.

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(a) RNA model on the r–θ plane.

(b) RNA model.on the r–z plane.

Fig. 5. Expanded view of the RNA model of the magnetic gear.

Fig. 6. Flow diagram of steady state analysis based on RNA.

Fig. 7 shows the three–dimensional FEA model used for the comparison. The number of elements is 466,184.

Fig. 8 shows the transmission torque characteristic calculated by RNA and FEA, respectively. The internal phase angle is defined as the difference between d–axes of the inner and the outer rotors. It is understood that the maximum transmission torque is obtained when an internal phase angle is 90 deg. in the same manner as conventional synchronous machines. The figure reveals that both calculated values are in good agreement. The calculation time of the RNA is few minutes, while the FEA is about 34 hours.

Fig. 9 indicates the calculated radial component of flux density in the inner and outer air gaps at the maximum output point where the internal phase angle is 90 deg. It is understood that the complicated distribution of the gap flux

can be well predicted by RNA. Fig. 10 shows the torque waveforms at the maximum

output point obtained from RNA and FEA. It is clear that the torque ripple is very small.

Fig. 7. There–dimensional FEA model of the magnetic gear.

Fig. 8. Transmission torque characteristic calculated by RNA and FEA.

(a) The gap between inner rotor and pole pieces.

(b) The gap between outer rotor and pole pieces.

Fig. 9. Radial component of flux density in the air gaps calculated by RNA and FEA.

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Fig. 10. Calculated torque waveforms at the maximum output point obtained from RNA and FEA.

C. Dynamic analysis Fig. 11 illustrates the flow diagram of dynamic analysis

using the RNA model. First of all, when the angular velocity of the inner rotor ωh is given, the inner rotor position θh is calculated by Eq. (7). Next, the flux distribution in the RNA model is calculated, and then the generated torques of the both rotors τh and τl are calculated by Eq. (8). After that, the angular velocity of the outer rotor ωl is obtained from the motion equation: , 9

where the moment of inertia of the outer rotor is Jl, and the load torque is τLoad. Finally, the outer rotor position θl is calculated by Eq. (7) using ωl.

Fig. 12 shows the rotational speed behavior obtained from the RNA–based dynamic analysis, when the inner rotor speed is changed with no load. The figure reveals that the outer rotor is rotated following the rotational speed of the inner rotor while keeping a gear ratio of 1: 10.333.

Fig. 13 shows the torque and speed behavior when the load torque is gradually increased and inner rotor rotates with constant speed 1,000 r/min. The figures indicate that the gear can transmit the required speed and torque of the ratio of 1: 10.333 below the maximum torque, and that the gear loses synchronism when the load torque is exceeded the

Fig. 11. Flow diagram of dynamic analysis based on RNA.

Fig. 12. Rotational speed behavior with no load torque obtained from the RNA–based dynamic analysis.

(a) Rotational speed behavior

(b) Torque behaviior

Fig. 13. Rotational speed and torque behavior with load torque

maximum torque of the magnetic gear. Using RNA, it is possible to consider the loss of synchronism.

III. COMPARISON WITH THE EXPERIMENTAL RESULTS On the basis of the above results, a trial SPM type

magnetic gear was made. The structure of the magnetic gear has the same specifications shown in Fig. 1.

Fig. 14 shows the general view of the experimental set up. The trial magnetic gear operates as a reduction gear on this system. The rotational speed of inner rotor is regulated an arbitrary speed by the servomotor. The load torque is controlled by the hysteresis brake.

Fig. 15 shows the input rotational speed versus output rotational speed obtained from no–load test. The measured values are compared to the calculated ones obtained from the proposed RNA–based dynamic analysis. It is understood from the figure that the ratio of the inner and outer rotor speeds is 1: 10.333, and that the measured and calculated values are in good agreement.

Fig. 16(a) shows the transmission torque characteristics calculated by RNA and FEA. The calculated maximum

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torque of the magnetic gear is 9.51 N·m. Fig. 16(b) indicates the observed torque at load test when a speed of the inner rotor is 300 r/min. It is understood that the measured maximum torque of 9.47 N·m is almost agree well with the calculated one, and that the magnetic gear loses synchronization when the load torque exceeds the maximum torque.

IV. RNA–BASED OPTIMUM DESIGN FOR THE SPM TYPE MAGNETIC GEAR

A. Optimum design method for magnetic gear Since the magnetic gear has many design parameters such

as lengths of the magnets, air gaps of the inner and outer rotors, etc., it is necessary for the establishment of an optimum design method with high calculation accuracy and high speed. Therefore, the RNA–based optimum design of the magnetic gear is presented.

Fig. 17 shows optimized parameters and fixed parameters of the magnetic gear considered in this paper. The radial lengths of inner and outer magnets, lmh, lml, and the radial length of the pole piece lp, are optimized to maximize the transmission torque, while the diameter and the stack length of the gear are fixed 156 mm and 10 mm, respectively. The two gap lengths are fixed 1 mm each. The numbers of pole–pairs of the inner and outer rotors are fixed 3 and 31, respectively.

In this consideration, the design parameters lmh, lml, and lp are changed 1 mm to 15 mm with 1 mm step, respectively. The total combination of the design parameters is 153 = 3,375. Hence, the optimum combination of the parameters is searched efficiently by combining the RNA with SIMPLEX method [10], which is one of the optimization algorithms. Here, the magnetic gear is optimized using two–dimensional RNA model in this paper, that is, influence of the leakage flux in the axial direction is neglected.

Fig. 14. General view of experimental system with the trial magnetic gear.

Fig. 15. Speed characteristics obtained from RNA and experiment.

(a) Calculated torque. (b) Observed torque.

Fig. 16. Comparison of the calculated and measured torque.

Fig. 18 indicates the flow diagram of the RNA–based

optimum design. When the parameters are given first, the RNA model of the magnetic gear of a certain design is derived, and the torque is calculated. Next, the calculated torque of the design is compared to that of the other designs. Finally, if the toque is maximized, the optimization procedure is finished. If not, the other combination of the parameters is selected based on SIMPLEX method.

Fig. 19 shows the progress of the torque and parameters. It is understood that the optimum values can be found for the 45th time by the proposed method. The maximum transmission torque is 25.9 N·m, and lmh, lml, and lp are 15 mm, 7 mm, and 4 mm, respectively.

Fig. 17. Optimized parameters and fixed parameters.

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Fig. 18. Work flow diagram based on SIMPLEX.

(a) Transmission torque.

(b) PM radial lengths of inner rotor.

(c) PM radial lengths of outer rotor.

(d) Pole pieces radial lengths.

Fig. 19. Progress of the torque and dimensions.

B. Comparison with the FEA–based optimum design To validate the usefulness of the proposed optimum

method, it is compared to the FEA–based optimum design. The combination of lmh, lml, and lp is searched to maximize the transmission torque using two–dimensional FEA and SIMPLEX method under the same condition of the above using RNA.

Table 1 shows the comparison of the results obtained from RNA and FEA. The total calculation time of the proposed method is about 7 min, while it takes about 6 hours when the FEA is employed. The main reason of the difference between the both results is that the numbers of elements are different. It has been proved that the proposed method has high calculation accuracy with high–speed.

V. CONCLUSION This paper presented a method for optimizing the design

of the SPM type magnetic gear based on RNA. The calculated values obtained from RNA agree well with ones obtained from FEA and experiment. Using the proposed RNA–based dynamic analysis, it is possible to calculate the dynamic behavior of the magnetic gear including the loss of synchronism. It reveals that the RNA–based optimum design for the magnetic gear has high calculation accuracy with high–speed.

TABLE I

Comparison of the results obtained from RNA and FEA.

VI. REFERENCES [1] D. E. Hesmondhalgh and D. Tipping, “A multielement magnetic gear,”

IEE Proc. B, Elect. Power Appl., vol. 127, pp. 129-138, May. 1980. [2] K. Tsurumoto and S. Kikushi, “A new magnetic gear using permanent

magnet,” IEEE Trans. Magn., vol. 23, pp. 3622-3624, Sep. 1987. [3] K. Ikuta, S. Makita, and S. Arimoto, “Non-contact magnetic gear for

micro transmission mechanism,” Proc. IEEE Conf. on Micro Electromechanical Systems (MEMS ‘91), pp. 125-130, Jan. 30, 1991.

[4] T. B. Martin, Jr., “Magnetic transmission,” U.S. Patent 3 378 710, Apr. 16, 1968.

[5] K. Atallah and D. Howe, “A Novel High-Performance Magnetic Gear,” IEEE Trans. Magn., vol. 37, pp. 2844-2846, Jul. 2001.

[6] K. Atallah, S. D. Calverley, and D. Howe, “Design, analysis and realisation of a high-performance magnetic gear,” IEE Proc., Elect. Power Appl., vol. 151, pp. 135-143, Mar. 2004.

[7] K. Tajima, Y. Anazawa, T. Komukai, and 0. Ichinokura: "An analytical method for characteristics of orthogonal-core under consideration of magnetic saturation and hysteresis," EPE '97, vol. 2, pp. 6-11, 1997.

[8] K. Nakamura and O. Ichinokura, “Dynamic Simulation of PM Motor Drive System Based on Reluctance Network Analysis,” Power Electronics and Motion Control Conference (EPE-PEMC 2008), pp. 758-762, Sep. 2008.

[9] K. Nakamura, M. Ishihara, and O. Ichinokura, “Reluctance Network Analysis Model of a Permanent Magnet Generator Considering an Overhang Structure and Iron loss,” Proc. 17th Int. Conf. on Electrical Machines (ICEM 2006), PSA1-16, 2006.

[10] J. A. Nelder and R. Mead, “A simplex method for function minimization,” Computer Journal, vol. 7, pp. 308-313, 1965.

VII. BIOGRAPHIES Michinari Fukuoka received the B.E. and M.E. degrees from Tohoku

University in 2010 and 2012 in electrical engineering, respectively. Now, he is a doctor student of Tohoku University. His current research interests include design and analysis of magnetic gears. Mr. Fukuoka is a student member of the Magnetic Society of Japan (MSJ), the Institute of Electrical Engineers of Japan (IEEJ).

Kenji Nakamura received the B.E. and M.E. degrees from Tohoku University in 1998 and 2000, respectively. He was with Tohoku University as a research associate in the Graduate School of Engineering from 2000 to 2007. In 2006, he received the Ph.D. degree from Tohoku University, where he is currently an associate professor. His current research interests include design and analysis of reluctance machines and permanent magnet machines. Dr. Nakamura is a member of the Magnetic Society of Japan (MSJ), the Institute of Electrical Engineers of Japan (IEEJ), and IEEE.

Osamu Ichinokura received his B.S., M.S. and Ph.D. degrees in electrical engineering from Tohoku University in 1975, 1977 and 1980, respectively. Since 1980, he has been with the Electrical Engineering, Tohoku University. He is now a professor of the Graduate School of Engineering, Tohoku University. His current research interests are in the areas of power electronics and power magnetics. Prof. Ichinokura is a member of the Magnetic Society of Japan (MSJ), the Society of Instrument and Control Engineers (SICE), the Institute of Electrical Installation Engineers of Japan, the Institute of Electrical Engineers of Japan (IEEJ), and IEEE

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