Halbach Array Superconducting Magnetic Bearing for a Flywheel

IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 15, NO. 2, JUNE 2005

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Halbach Array Superconducting Magnetic Bearing for a Flywheel Energy Storage System Guilherme G. Sotelo, Antonio C. Ferreira, and Rubens de Andrade, Jr.

Abstract—In order to develop a new magnetic bearing set for a flywheel energy storage prototype, it was designed and simulated some configurations of Permanent Magnetic Bearings (PMB) and Superconducting Magnetic Bearings (SMB). The bearings were assembled with Nd-Fe-B permanent magnets and the simulations were carried out with the Finite Element Method (FEM). The PMB was designed to reduce the load on SMB and provide radial positioning of the whole set. SMB were designed with YBCO superconductors and an assembly of permanent magnets. Several configurations of permanent magnets were simulated, trying to maximize the magnetic flux gradient in direction orthogonal to the movement and flux density in the surface of the superconductors. Early experiments have shown an increasing stiffness and levitation force with increasing field gradient and intensity. It was also a goal to reduce the stray field outside the bearing. The levitation force of the SMB using a flux shapers configuration was measured and compared with FEM simulation, showing very good agreement. The simulation of a SMB using Halbach array configuration shows that it increases the levitation force and reduces the stray field. Index Terms—Flywheel, magnetic bearing, superconducting levitation.

I. INTRODUCTION

I

N this work two types of passive magnetic bearings are presented: a Superconducting Magnetic Bearing (SMB) and a Permanent Magnetic Bearing (PMB). These passive magnetic bearings have been developed at the Federal University of Rio de Janeiro, to operate in a flywheel energy storage system [1]. The purpose of this equipment is store energy in the flywheel and recover this mechanical energy whenever necessary. The conversion of the electrical energy in mechanical one and viceversa, is done by a switched reluctance machine [2], [3]. The PMB is suggested here to position the shaft radially and to reduce the load on the SMB. It makes possible to reduce the total number of superconductors blocks used in the axial bearing, that are still very expensive (about US$ 300/piece). The , but it has the PMB has low cost and no need to be cooled in disadvantage of being unstable. The axial force and the mechanical stiffness of two topologies of PMB were investigated using 3D Finite Element Method (FEM) simulations. The main concern in this paper is to investigate two topologies of trust SMB. The first one is a flux shaper topology having the magnetic flux

Manuscript received October 4, 2004. This work was supported by the CNPq. G. G. Sotelo and A. C. Ferreira are with the COPPE/Department of Electrical Engineering, Federal University of Rio de Janeiro, Rio de Janeiro 21.941-972, Brazil (e-mail: [email protected]; [email protected]). R. de Andrade, Jr., is with the Graduate School of Electrical Engineering, Federal University of Rio de Janeiro, Rio de Janeiro 21.941-972, Brazil (e-mail: [email protected]). Digital Object Identifier 10.1109/TASC.2005.849624

Fig. 1. Two proposed PMBs using Nd-Fe-B magnetic rings. (a) Without shim, (b) with shim.

concentrated between two consecutive magnets with the same pole at the center [4]. The other topology uses a Halbach array to increase and concentrate the magnetic induction in the desired direction [5], [6]. Both SMB have the same dimension, but in the Halbach array the ferromagnetic material is replaced by permanent magnets in order to reduce the stray field. The Finite Element Method was used in the SMB design to calculate the field distribution. The critical state model [7]–[9] was applied in 2D FEM simulations to obtain the levitation force. Both FEM results were compared with measurements showing good agreement. As expected, the stray field in the Halbach array is reduced and the levitation force can be increased over 50% compared to the flux shapers SMB. II. DEVELOPED BEARINGS Two different magnetic bearings were designed for this flywheel prototype: a PMB that uses only Nd-Fe-B magnets and a SMB with YBCO superconductors and Nd-Fe-B magnets. The investigation of these magnetic bearings are presented in the following sections. A. Permanent Magnetic Bearing The main reason of using an axial PMB is reduce the load weight of the rotor and the flywheel, above the thrust bearing. It makes possible to reduce significantly the quantity of superconductor blocks in the SMB, bringing down the overall cost. Other advantage of this bearing is its stiffness, which helps to position the system radially. The proposed PMBs are composed of 2 permanent magnet rings of Nd-Fe-B (N35) with the following dimensions: 100 mm inner diameter, 120 mm outer diameter and 10 mm height, as presented in Fig. 1(a). The coercivity force and remanent field of N35 are, respectively, 918 kA/m and 1.198 T. To calculate the radial and axial forces of PMB, 3D Finite Element Method (FEM) simulations were used. These magnetic forces in FEM were determined applying the virtual work method. The radial force

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Fig. 4. Photo of Nd-Fe-B magnets and steel SAE1020 rings, used for constructing the prototype of superconducting thrust magnetic bearing. Fig. 2.

3D FEM simulated levitation force in the PMB.

Fig. 5.

Flux shapers configuration using Nd-Fe-B magnets and steel.

to damp the amplitude vibration by the currents induced into the shim by the asymmetric rotational magnetic field. B. Thrust SMB

Fig. 3. 3D FEM simulated restoring force in the PMB for a radial displacement with a gap of 4 mm.

is important to help to restore the operational position of the flywheel when the shaft is displaced radially from the operating position. In order to provide a damping mechanism due to hysteresis losses, a small shim is introduced between the rings magnets (Fig. 1(b)). It was known that these shims reduce the axial force [10], because the magnetic induction becomes weaker as the gap increases. Force results for different gaps are presented in Fig. 2. The maximum radial displacement is limited by the airgap length of the electrical machine used in the flywheel system. The switched reluctance machine used in this work has an airgap of 2.5 mm. Therefore we are interested in the restoring forces for radial displacements which are smaller than 2.5 mm. Fig. 3 shows that the radial restoring force is also reduced when the shim is introduced. When the PMB operates without shim it has a stiffness of 24 N/mm, whilst with the shim the stiffness is approximately 8.8 N/mm, for a 4 mm gap. Despite the reduction of the axial force in 35.7% and the stiffness in 63.3%, the shim is still necessary

Two topologies of thrust superconducting magnetic bearing are presented in this section: one concentrating the magnetic field radially in steel rings and another where these steel rings are replaced by permanent magnet rings (magnetized axially) in order to obtain a Halbach array configuration. The magnetic radial orientation was reached by gluing several ring segments that were magnetized as shown in Fig. 4. To concentrate the magnetic flux in the axial direction, two consecutive rings (that are magnetized radially) must have the same polarity in the interface region (which is made of steel). This configuration is detailed in Fig. 5. A picture of the constructed bearing is shown in Fig. 4. The diameter of this bearing is almost the same of a compact disc. The steel used in the construction was SAE1020. SAE1020 mechanical resistance is superior to other steels because it has higher carbon level. Another advantage is its availability in the domestic market and its very low cost. Nevertheless this steel is not commonly used in magnetic applications and it has as disadvantage the fact of its magnetic properties may vary from sample to sample of SAE1020. Other ferromagnetic materials, such as SAE1010 or SAE1008, could be used with expected better magnetic results due to their lower carbon levels, but, its important to remember that their mechanical properties are inferior. The magnets arrangement of Fig. 4 is able to produce a radial magnetic induction peak to peak variation of 0.68 T (in a radial region of 15 mm) for an axial distance of 4 mm above

SOTELO et al.: HALBACH ARRAY SMB FOR A FLYWHEEL ENERGY STORAGE SYSTEM

Fig. 6. Axial magnetic induction in a gap of 4 mm of the configuration presented in Figs. 4 and 5.

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Fig. 8. Magnetic induction in the thrust SMB above part.

Fig. 7. Flux lines for flux shaper configuration (left) and for Halbach configuration (right).

the magnets, as presented in Fig. 6. This figure shows the agreement between measurements and 2D FEM simulated results. In spite of the magnetic induction gradient value has a significant value, this magnetic flux density has symmetry above and under the magnet assembly disc. As the blocks superconductors are placed only under this disc all the flux in the upper part is not used to produce any force. To increase the magnetic induction in the desired direction (the lower part, where the superconductors are situated), a Halbach array configuration can be used. This configuration is commonly applied in electric machines using permanent magnets [6], and sometimes in PMB [5]. Some Nd-Fe-B magnets rings can replace the steel ring, and their magnetic orientation can determine where the flux will be concentrated. The flux lines in the flux shapers bearing and in the Halbach array bearing were obtained by 2D FEM simulations, and can give a qualitative idea of the stray field, as presented in Fig. 7. As showed in this figure, its is possible to see that the stray field in the Halbach configuration is lower than in the flux shaper one, and the magnetic induction is expected to increase in the direction into the superconductors. To visualize quantitatively these results the radial and axial magnetic induction of flux shaper and Halbach SMBs were calculated from the center to the radius of the disc. These results are presented in Fig. 8 (in the upper part of the magnets) and in

Fig. 9. Magnetic induction in the thrust SMB below part.

Fig. 9 (in the lower part), for a distance of 4 mm from the disc. It is possible to see in Fig. 8 that both magnetic induction are decreased significantly in the Halbach configuration in the upper part of the magnets. In the lower part of the magnets, as shown in Fig. 9, peak to peak variation of the magnetic induction is increased 34% and 35% for the radial and axial components, respectively. III. THE LEVITATION FORCE IN SMB The levitation force measurements obtained with the SMB, were performed with zero field cooling, using a 500 N load cell and a linear actuator. This result was compared to simulations in order to validate the implemented model. To calculate the magnetic force between the magnets and superconductors, the critical state model was used [8], [9]. The implementation of Bean’s model using the FEM was suggested by Sugiura et al. [7]. This model solves electromagnetic equations in terms of the magnetic vector potential in the axisymmetric coordinate system. It assumes that the macroscopic field distribution in type-II superconductor can be determined from the balance between Lorentz

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the simulated results for the 2 proposed topologies and the levitation force measured from the flux shapers configuration. This figure shows an excellent agreement between calculated and measured levitation forces for the flux shapers SMB. It is expected that the Halbach array configuration will increase the levitation force over 50% for values of gaps between 7 mm and 1 mm. Other advantage of this new magnetic configuration is the possibility of obtaining the same magnetic force at higher gaps. For example, a Halbach array produces a levitation force of 382 N with a gap of 4 mm, while for the flux shapers configuration this force is obtained with a gap of 1.1 mm. IV. CONCLUSION

Fig. 10.

Magnetic levitation force for zero field cooling.

and pinning forces on the fluxoids. Using Faraday’s law and the , it can definition of the magnetic vector potential be written:

This paper presented a PMB and a SMB bearing for a flywheel energy storage system. In spite of the PMB being an axial bearing it presented a considerable stiffness (24 N/mm for an axial gap of 4 mm). Two topologies of thrust SMB were analyzed: a flux shaper and a Halbach array. The Halbach array is able to reduce the stray field and increase the magnetic induction, which makes possible to increase the levitation force over 50% for the operational region. ACKNOWLEDGMENT

(1) where is the electrical potential and is the magnetic vector potential. Applying Maxwell equations, the Poisson equation can be obtained as follow: (2) where is the shielding current density in the superconductor. The nonlinearity in the superconductor is determined by the following equations: (3) (4) where the function ‘ ’ is the sign of the correspondent value. The critical current density was expressed as a function of the magnetic induction , using Kim’s model [7], [11]: (5) Applying the model presented above the magnetic force was calculated using 2D FEM simulation of the flux shapers and Halbach bearings. The parameters used in these simulation and . Fig. 10 shows were:

The authors would like to thank: R. de A. Abreu, N. F. B. de Mello, A. da S. P. C. Real, G. C. Bordin, and S. L. P. C. Valinho for the experimental support. REFERENCES [1] R. de Andrade Jr. et al., “A superconducting high-speed flywheel energy storage system,” Physica C, vol. 408–410, pp. 930–931, 2004. [2] L. G. B. Rolim et al., “Flywheel generator with switched reluctance machine,” in Proceedings of the XV International Conference on Electrical Machines, vol. 1, Bruges, 2002. [3] J. L. da Silva Neto, L. G. B. Rolim, and G. G. Sotelo, “Control of a power circuit interface of a flywheel-based energy storage system,” in Proceedings of the IEEE International Symposium on Industrial Electronics, vol. 1, Rio de Janeiro, 2003, pp. 962–967. [4] M. Zeisberger et al., “Optimization of levitation forces,” IEEE Trans. Appl. Supercond., vol. 11, no. 1, pp. 1741–1744, 2001. [5] J. P. Yonnet et al., “Stacked structures of passive magnetic bearings,” J. Appl. Phys., vol. 70, no. 10, pp. 6633–6635, 1991. [6] Z. Q. Zhu and D. Howe, “Halbach permanent magnet machines and applications: a review,” Proc. IEE—Elect. Power Appl., vol. 148, no. 4, pp. 299–308, 2001. [7] T. Sugiura, H. Hashizume, and K. Miya, “Numerical electromagnetic field analysis of type-II superconductors,” Int. J. Appl. Electromagn. Mater., vol. 2, pp. 183–196, 1991. [8] C. P. Bean, “Magnetization of hard superconductors,” Phys. Rev. Lett., vol. 8, pp. 250–253, 1962. [9] , “Magnetization of high-field superconductors,” Rev. Mod. Phys., vol. 9, pp. 31–39, 1964. [10] Y. H. Han et al., “Design a hybrid high T superconductor bearings for flywheel energy storage system,” Physica C, vol. 372–376, pp. 1457–1461, 2002. [11] P. W. Anderson and Y. B. Kim, “Hard superconductivity: theory of the motion of Abrikosov flux lines,” Rev. Mod. Phys., vol. 9, pp. 39–43, 1964.