IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 15, NO. 2, JUNE 2005 2253

Halbach Array Superconducting Magnetic Bearingfor 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 aflywheel energy storage prototype, it was designed and simulatedsome configurations of Permanent Magnetic Bearings (PMB) andSuperconducting Magnetic Bearings (SMB). The bearings wereassembled with Nd-Fe-B permanent magnets and the simulationswere carried out with the Finite Element Method (FEM). ThePMB was designed to reduce the load on SMB and provideradial positioning of the whole set. SMB were designed withYBCO superconductors and an assembly of permanent magnets.Several configurations of permanent magnets were simulated,trying to maximize the magnetic flux gradient in direction or-thogonal to the movement and flux density in the surface of thesuperconductors. Early experiments have shown an increasingstiffness and levitation force with increasing field gradient andintensity. It was also a goal to reduce the stray field outside thebearing. The levitation force of the SMB using a flux shapersconfiguration was measured and compared with FEM simula-tion, showing very good agreement. The simulation of a SMBusing Halbach array configuration shows that it increases thelevitation force and reduces the stray field.

Index Terms—Flywheel, magnetic bearing, superconductinglevitation.

I. INTRODUCTION

I N this work two types of passive magnetic bearings are pre-sented: a Superconducting Magnetic Bearing (SMB) and a

Permanent Magnetic Bearing (PMB). These passive magneticbearings have been developed at the Federal University of Riode Janeiro, to operate in a flywheel energy storage system [1].The purpose of this equipment is store energy in the flywheeland recover this mechanical energy whenever necessary. Theconversion of the electrical energy in mechanical one and vice-versa, is done by a switched reluctance machine [2], [3].

The PMB is suggested here to position the shaft radially andto reduce the load on the SMB. It makes possible to reducethe total number of superconductors blocks used in the axialbearing, that are still very expensive (about US$ 300/piece). ThePMB has low cost and no need to be cooled in , but it has thedisadvantage of being unstable. The axial force and the mechan-ical stiffness of two topologies of PMB were investigated using3D Finite Element Method (FEM) simulations. The main con-cern 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: sotelo@coe.ufrj.br; ferreira@coep.ufrj.br).

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:randrade@dee.ufrj.br).

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 samepole at the center [4]. The other topology uses a Halbach arrayto increase and concentrate the magnetic induction in the de-sired direction [5], [6]. Both SMB have the same dimension, butin the Halbach array the ferromagnetic material is replaced bypermanent magnets in order to reduce the stray field. The FiniteElement Method was used in the SMB design to calculate thefield distribution. The critical state model [7]–[9] was applied in2D FEM simulations to obtain the levitation force. Both FEMresults were compared with measurements showing good agree-ment. As expected, the stray field in the Halbach array is reducedand the levitation force can be increased over 50% compared tothe flux shapers SMB.

II. DEVELOPED BEARINGS

Two different magnetic bearings were designed for this fly-wheel prototype: a PMB that uses only Nd-Fe-B magnets and aSMB with YBCO superconductors and Nd-Fe-B magnets. Theinvestigation of these magnetic bearings are presented in the fol-lowing sections.

A. Permanent Magnetic Bearing

The main reason of using an axial PMB is reduce the loadweight of the rotor and the flywheel, above the thrust bearing. Itmakes possible to reduce significantly the quantity of supercon-ductor blocks in the SMB, bringing down the overall cost. Otheradvantage of this bearing is its stiffness, which helps to positionthe system radially. The proposed PMBs are composed of 2 per-manent magnet rings of Nd-Fe-B (N35) with the following di-mensions: 100 mm inner diameter, 120 mm outer diameter and10 mm height, as presented in Fig. 1(a).

The coercivity force and remanent field of N35 are, re-spectively, 918 kA/m and 1.198 T. To calculate the radialand axial forces of PMB, 3D Finite Element Method (FEM)simulations were used. These magnetic forces in FEM weredetermined applying the virtual work method. The radial force

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2254 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 15, NO. 2, JUNE 2005

Fig. 2. 3D FEM simulated levitation force in the PMB.

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

is important to help to restore the operational position of theflywheel when the shaft is displaced radially from the operatingposition. In order to provide a damping mechanism due tohysteresis losses, a small shim is introduced between the ringsmagnets (Fig. 1(b)). It was known that these shims reducethe axial force [10], because the magnetic induction becomesweaker as the gap increases. Force results for different gapsare presented in Fig. 2. The maximum radial displacement islimited by the airgap length of the electrical machine used inthe flywheel system. The switched reluctance machine usedin this work has an airgap of 2.5 mm. Therefore we are in-terested in the restoring forces for radial displacements whichare smaller than 2.5 mm. Fig. 3 shows that the radial restoringforce is also reduced when the shim is introduced. When thePMB 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 in35.7% and the stiffness in 63.3%, the shim is still necessary

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

Fig. 5. Flux shapers configuration using Nd-Fe-B magnets and steel.

to damp the amplitude vibration by the currents induced intothe shim by the asymmetric rotational magnetic field.

B. Thrust SMB

Two topologies of thrust superconducting magnetic bearingare presented in this section: one concentrating the magneticfield radially in steel rings and another where these steel ringsare replaced by permanent magnet rings (magnetized axially) inorder to obtain a Halbach array configuration. The magnetic ra-dial orientation was reached by gluing several ring segments thatwere magnetized as shown in Fig. 4. To concentrate the mag-netic flux in the axial direction, two consecutive rings (that aremagnetized radially) must have the same polarity in the interfaceregion (which is made of steel). This configuration is detailed inFig. 5.

A picture of the constructed bearing is shown in Fig. 4. The di-ameter of this bearing is almost the same of a compact disc. Thesteel used in the construction was SAE1020. SAE1020 mechan-ical resistance is superior to other steels because it has highercarbon level. Another advantage is its availability in the do-mestic market and its very low cost. Nevertheless this steel isnot commonly used in magnetic applications and it has as disad-vantage the fact of its magnetic properties may vary from sampleto sample of SAE1020. Other ferromagnetic materials, such asSAE1010 or SAE1008, could be used with expected better mag-netic results due to their lower carbon levels, but, its importantto remember that their mechanical properties are inferior.

The magnets arrangement of Fig. 4 is able to produce a ra-dial magnetic induction peak to peak variation of 0.68 T (in aradial region of 15 mm) for an axial distance of 4 mm above

SOTELO et al.: HALBACH ARRAY SMB FOR A FLYWHEEL ENERGY STORAGE SYSTEM 2255

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

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

the magnets, as presented in Fig. 6. This figure shows the agree-ment between measurements and 2D FEM simulated results. Inspite of the magnetic induction gradient value has a significantvalue, this magnetic flux density has symmetry above and underthe magnet assembly disc. As the blocks superconductors areplaced only under this disc all the flux in the upper part is notused to produce any force.

To increase the magnetic induction in the desired direction(the lower part, where the superconductors are situated), aHalbach array configuration can be used. This configurationis commonly applied in electric machines using permanentmagnets [6], and sometimes in PMB [5]. Some Nd-Fe-Bmagnets rings can replace the steel ring, and their magneticorientation can determine where the flux will be concentrated.The flux lines in the flux shapers bearing and in the Halbacharray bearing were obtained by 2D FEM simulations, and cangive a qualitative idea of the stray field, as presented in Fig. 7.As showed in this figure, its is possible to see that the strayfield in the Halbach configuration is lower than in the fluxshaper one, and the magnetic induction is expected to increasein the direction into the superconductors.

To visualize quantitatively these results the radial and axialmagnetic induction of flux shaper and Halbach SMBs were cal-culated from the center to the radius of the disc. These resultsare presented in Fig. 8 (in the upper part of the magnets) and in

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

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. Itis possible to see in Fig. 8 that both magnetic induction are de-creased significantly in the Halbach configuration in the upperpart of the magnets.

In the lower part of the magnets, as shown in Fig. 9, peak topeak variation of the magnetic induction is increased 34% and35% 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 celland a linear actuator. This result was compared to simulations inorder to validate the implemented model. To calculate the mag-netic force between the magnets and superconductors, the crit-ical state model was used [8], [9]. The implementation of Bean’smodel using the FEM was suggested by Sugiura et al. [7]. Thismodel solves electromagnetic equations in terms of the mag-netic vector potential in the axisymmetric coordinate system. Itassumes that the macroscopic field distribution in type-II super-conductor can be determined from the balance between Lorentz

2256 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 15, NO. 2, JUNE 2005

Fig. 10. Magnetic levitation force for zero field cooling.

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

(1)

where is the electrical potential and is the magnetic vectorpotential.

Applying Maxwell equations, the Poisson equation can beobtained as follow:

(2)

where is the shielding current density in the superconductor.The nonlinearity in the superconductor is determined by the fol-lowing 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 wascalculated using 2D FEM simulation of the flux shapers andHalbach bearings. The parameters used in these simulationwere: and . Fig. 10 shows

the simulated results for the 2 proposed topologies and thelevitation force measured from the flux shapers configuration.This figure shows an excellent agreement between calculatedand measured levitation forces for the flux shapers SMB. It isexpected that the Halbach array configuration will increase thelevitation force over 50% for values of gaps between 7 mm and1 mm. Other advantage of this new magnetic configuration isthe possibility of obtaining the same magnetic force at highergaps. For example, a Halbach array produces a levitationforce of 382 N with a gap of 4 mm, while for the flux shapersconfiguration this force is obtained with a gap of 1.1 mm.

IV. CONCLUSION

This paper presented a PMB and a SMB bearing for a fly-wheel energy storage system. In spite of the PMB being an axialbearing it presented a considerable stiffness (24 N/mm for anaxial gap of 4 mm). Two topologies of thrust SMB were ana-lyzed: a flux shaper and a Halbach array. The Halbach array isable to reduce the stray field and increase the magnetic induc-tion, which makes possible to increase the levitation force over50% for the operational region.

ACKNOWLEDGMENT

The authors would like to thank: R. de A. Abreu, N. F. B. deMello, A. da S. P. C. Real, G. C. Bordin, and S. L. P. C. Valinhofor the experimental support.

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