The Cycloid Permanent Magnetic Gear Frank T

IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 44, NO. 6, NOVEMBER/DECEMBER 2008 1659 The Cycloid Permanent Magnetic Gear Frank T. Jørgensen, Torben Ole Andersen, and Peter Omand Rasmussen

Abstract—This paper presents a new permanent-magnet gear based on the cycloid gearing principle, which normally is characterized by an extreme torque density and a very high gearing ratio. An initial design of the proposed magnetic gear was designed, analyzed, and optimized with an analytical model regarding torque density. The results were promising as compared to other high-performance magnetic-gear designs. A test model was constructed to verify the analytical model. Index Terms—Analytical modeling, finite element analysis (FEA), magnetic gear, permanent magnets. Fig. 1. Permanent-magnet and mechanical spur gear. I. INTRODUCTION ECENTLY, magnetic gears have gained some attention R due to the following reasons: no mechanical fatigue, no lubrication, overload protection, reasonably high torque density, and potential for very high efficiency. Focus [1], [4], [5] have been addressed to a kind of planetary magnetic gear, probably already invented before the strong NdFeB magnets came into the market in the early 1980s [6]. An active torque density is in the range of 100 N · m/L, which is a very high torque density for a magnetic device [4]. However, there is still a need for increased torque density and a better utilization of the permanent magnets. The torque density of a magnetic coupling is in the range of 400 N · m/L, and this is, in principle, Fig. 2. (a) Inner type spur gear. (b) Inner spur gear with high magnetic a magnetic gear with a 1 : 1 gearing ratio. interaction and low gearing ratio. In this paper, a magnetic gearing topology with better utilization of the permanent magnet is presented. This gearing order to optimize the layout of the new cycloid magnetic gear, topology makes it possible to increase the torque density to a parametric analytical model to calculate the torque density is almost twice the state-of-the-art magnetic gears and, therefore, developed. might be a useful alternative in applications using traditional The construction of an initial test model is given, and mea- mechanical gears or, at least, in gearing applications where surement is performed in order to validate the analytical model. some of the other advantages, e.g., overload protection, oil-free Next, an optimization is utilized with the analytical model to construction, and separation, are vital. quantify the cycloid permanent-magnet gear capability, and This paper will first give a description of the cycloid per- finally, a conclusion is given. manent magnetic gear and how the idea is derived from the classical magnetic spur gear. Due to the fact that the cycloid permanent magnetic gear is a 2-DOF topology, description of II. DESCRIPTION AND DEVIATION OF THE CYCLOID gearing ratios with different fixed axes (1 DOF) is stated. In PERMANENT MAGNETIC GEAR A permanent-magnet version to the classical spur gear can be Paper IPCSD-07-129, presented at the 2006 Industry Applications Society made, where the teeth are substituted by permanent magnets; Annual Meeting, Tampa, FL, October 8–12, and approved for publica- see Fig. 1. tionintheIEEETRANSACTIONS ON INDUSTRY APPLICATIONS by the From Fig. 1, it is quite clear that a lot of the magnets are Electric Machines Committee of the IEEE Industry Applications Society. Manuscript submitted for review October 10, 2007 and released for publication inactive and cannot assist in transferring torque between the February 19, 2008. Current version published November 19, 2008. This work two rings. In addition, volume taken up from the gear is quite was supported in part by ELFOR, in part by Danfoss Drives A/S, and in part by high because the two rings are separated. In order to reduce the Sauer Danfoss A/S. The authors are with the Institute of Energy Technology, Aalborg University, volume and also increase the interaction, it is therefore more DK 9220 Aalborg, Denmark (e-mail: [email protected]; [email protected]; por@iet. suitable to use an inner type spur gear; see Fig. 2(a). aau.dk). From Fig. 2, it is shown that more magnets gets active if Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. the number of poles on the inner ring is getting closer to the Digital Object Identifier 10.1109/TIA.2008.2006295 number of poles on the outer ring. If the number of poles on

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Fig. 5. (a) Spur-gear coordinate system and magnets. (b) Cycloid-gear coordinate system and magnets.

III. GEARING RELATIONSHIP FOR THE CYCLOIDAL GEAR The movement principle of the permanent-magnet cycloid gear is described in the previous section, where the outer rotor is fixed from rotation. However, it is also possible to fix other Fig. 3. Cycloidal movement principle shown with nine illustrations. parts of the gear. In Fig. 4, the other possible combinations of fixed parts are shown together with equations for the gearing ratio. Gear configuration in Fig. 4(c) is the configuration already described in Section II, and the gear configuration in Fig. 4(b) is equivalent to the internal spur gear with the relatively low gearing ratio. The configuration shown on Fig. 4(a) has the largest gearing ratio, and the input and output axes are separated by an air gap. The configuration in Fig. 4(c) has similar characteristics as the one in Fig. 4(a) and may be preferred in some applications due to its layout for integration. The torque density is almost similar for all configurations.

Fig. 4. Different configurations for the cycloid-gear parts. IV. ANALYTICAL MODEL the inner ring is equal to the number of poles on the outer ring, In order to be able to design and optimize the cycloid mag- the gearing ratio will be unity, and the gear can be considered netic gear, a model is required. An analytical model is preferred as a magnetic coupling. With almost equal number of poles on when optimization has to be applied because of significant the two rings, the gearing ratio is very low. Fig. 2(b) shows an reduced computational time as compared to a finite element example where the number of poles on the outer rotor is 44 and analysis (FEA). on the inner rotor 42. This example gear has very high magnetic Due to the fact that the permanent-magnet cycloid gear is a ≈ interaction but the gearing ratio is only 44/42 1.05. variation of the permanent-magnet spur gear, it is obvious to use In order to improve the low gearing ratio for the gear shown the same theory used for this gear type. In [2], the authors have in Fig. 2(b), a cycloidal principle is considered. Cycloidal gear- derived an analytical model for permanent magnetic spur gears ing principle has significant gear-reduction capability. Move- with parallel magnetized magnets. This model is only briefly ment principle for this gear is explained with nine illustrations introduced as function expressions in this paper and, for further shown in Fig. 3. The illustrations show an outer rotor with detailed explanation, is the reader referred to [2] and [3]. 44 poles and an inner rotor with 42 poles. The outer rotor is stationary while the inner rotor is magnetically connected to outer rotor and placed eccentrically relative to outer rotor. A. Magnetic-Field Expression Clockwise angular-position change for the air gap will result The magnetic-field solutions for a parallel magnetized mag- in a small change of the inner rotor rotation in anticlockwise netic ring are written by (1) and (2). This magnetic ring is the direction. The angular position for the air gap has rotated one so-called source shown in Fig. 5(a). anticlockwise revolution from illustrations one to nine while the Equations (1) and (2) have origin in the source co- inner rotor has only rotated 1/21 revolution anticlockwise. ext ordinate system and, therefore, named Br (r ,φ, v) and This gearing principle has significant gear-reduction ability. ext B (r ,φ, v), respectively, Same example is shown in Fig. 4(c), where the outer rotor φ ∞ part C is fixed. An eccentric B is driving the inner magnetic − 1 iN +1 (2) 1 ( 2 p ) plate, and this plate will make a combined orbit and rotational B (r ,φ, v)=μ0 iNpr r 2 motion. The rotational part of this motion is transferred to the i=1,3,5,... output shaft A. The gear ratio is (−21/1) = −21, which is (2) 1 × U (v)cos N iφ (1) much higher than a simple inner spur-gear configuration. i 2 p

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∞ − 1 iN +1 B. Torque-Calculation Expression (2) 1 ( 2 p ) Bφ (r ,φ, v)=μ0 iNpr 2 The instantaneous torque on a parallel magnetized spur gear i=1,3,5,... 1 is described by × U (2)(v)sin N iφ . (2) i 2 p T (R2s,R1s,R2d,R1d,d)=Tr(v)+Tt1(v)+Tt2(v). (9) These field-solution equations are transformed into the drive- ext ext magnet coordinate system Br (r, φ, v) and Bφ (r, φ, v). Fur- The three terms are scalars and represent one radial torque ther explanation of the field transformation is explained in [3]. integration and two tangential torque integrations in the drive- The two field expressions depend on coefficient terms, and magnet coordinate system, which is tilted an angle ϕ for these terms can be expressed as (3)–(8). The field expressions calculating the torque at a operational point of maximal torque (1) and (2) are indirectly used in parts of the torque calculation between source-and-drive-magnet parts. The torque-integration expression (9) expressions are written in (10)–(12) and subexpressions for the integration are written in (13)–(15) (2) Hri(v) Hφi(v) U (v)=−2μ0 Mri−2μ0 Mφi (3) i L (v) L (v) i i π − 3 2Ms cos N L(R2d R1d) (Npi) 2 (Npi) Npi pole Hri(v)=R2sμ0 R2s Npi+2R1s R2s R2sμ Tr(v)= Nr N −1 N − (Npi) (Npi) pole p R1s R2s R2sμ0Npi p × (−1) Sr(q)r(q, v) (Npi) (Npi) p=0 q=0 −2 R μR1sR Npi 1s 2s × ext 3 cos (φedge(φ, p)) Bx (r(q),φedge(φ, p), v) (Npi) 2 (Npi) (Npi) −2R2sμ R +4 R R μ0R1s ext 2s 1s 2s +sin(φedge(φ, p)) By (r(q),φedge(φ, p), v) (Npi) (Npi) (10) + R R R2sμNpi 1s 2s 2 2π −MsLR 1d Npole (Npi) (Npi) −2R R R μ Tt1(v)= 1s 2s 2s 0 N 3 3 t 2 2 N −1 − (Npi) (Npi) pole Nt 2R2sμ0 R2s + R2sμ R2s Npi p × (−1) Sr(q)sin(θ(q)) (4) p=0 q=0 3 − (Npi) 2 − (Npi) (Npi) 2π Hφi(v)= R2sμ0 R2s Npi 4 R1s μR1sR2s × ext cos θ(q)+p + φ Bx 3 Npole − (Npi) 2 (Npi) (Npi) R2sμ R2s Npi +2R1s R2s R2sμ 2π × R1d,θ(q)+p + φ, v Npole − (Npi) (Npi) 2R1s R2s R2sμ0 2π +sin θ(q)+p + φ Bext (N i) (N i) y + R p R p R μN i Npole 1s 2s 2s p 3 2π (Npi) (Npi) (Npi) 2 × R1d,θ(q)+p + φ, v (11) +2 R1s R2s μ0NpiR1s +2R2sμ R2s N pole 3 N i 2 (N i) (N i) 2 2π p − p p MsLR2d N +2R2sμ0 R2s R1s R2s R2sμ0Npi pole Tt2(v)= (5) Nt − Npole 1 Nt Li(v)= (−2+Npi) (2 + Npi) p × (−1) Sr(q)sin(θ(q)) × − (Npi) − (Npi) p=0 q=0 2μμ0R2s 2μμ0R1s 2π (N i) (N i) × ext −μ2R p +μ2R p cos θ(q)+p + φ Bx 2s 1s Npole − 2 (Npi) 2 (Npi) μ0R2s +μ0R1s (6) 2π × R2d,θ(q)+p + φ, v Br 2Np Npole Mri = μ π 1 2 2π ext 0 1− iNp +sin θ(q)+p + φ B 2 y π π 1 π π Npole × sin cos i − iNp cos sin i 2π Np 2 2 Np 2 × R2d,θ(q)+p + φ, v (12) (7) Npole − B 2N π q 2π r p θ(q)= + (13) Mφi = 2 N N N μ0π − 1 pole t pole 1 2 iNp q r(q, v)=R + (R − R ) (14) π π 1 π π 1d N 2d 1d × cos sin i − iNp sin cos i . r Np 2 2 Np 2 π θedge(p)=φ + (1 + 2p). (15) (8) Npole

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TABLE I DIMENSIONS

Fig. 6. Analytic and FEA-calculated torque–angle curve.

In Table I, it is seen that the initial cycloid permanent- magnet gear has a calculated torque density of 141.9 N · m/L, i.e., around 40% more than the “planetary” magnetic gears developed in [4]. In order to validate the analytical model for the initial design, a static FEA was made, and the results from the two models are shown in Fig. 6. In Fig. 6, good agreement is seen between the two calculation methods. The reason for the small deviation is mainly caused by the assumption of a unity relative permeability of the permanent magnets in the derivation of the analytical model. The FEA model uses 1.05 in relative permeability for the NdFeB magnets.

V. C ONSTRUCTION OF A TEST MODEL Starting point for making a test model is the cycloidal mov- C. Modified Torque Calculation From Spur Gear to ing principle shown in Fig. 3. It was chosen to use sheet steel Cycloid Type yokes for both rings, meaning that the height of the permanent magnets can be reduced by a factor of two because the steel will The torque-calculation model for the spur gear is general, and bridge neighboring magnets (magnetic mirrors). The steel yoke thereby, it is possible to change the separation distance between has also the advantages of limiting magnetic field around the the two rings to the eccentricity of the inner ring, see Fig. 5. construction. This dimensional change will only change the equations for flux In order to make the test model simple, rectangular magnets transformations [3]. Source and drive magnet is inside of each were used instead of arc-shaped magnets, and to reduce large other, and the torque is calculated on the outer drive magnet (9). unbalanced magnetic forces, two gear sets were used. Fig. 7 The system configuration is equivalent to the inner magnetic shows a rendered view of the constructed model. Outer dimen- spur gear shown in Fig. 2(b). sions of the experimental test model are 130 × 130 mm, and Torque equations, for the gear type shown in Fig. 4(b), can the gearbox total length is 86 mm. be used to calculate the nominal corresponding output torque of The input shaft is placed at the center of the gearbox. A a gear type shown in Fig. 4(c). This torque transformation can bearing is placed on the eccentric and connected to the inner be written by steel yoke. Circular motion of the input shaft will force the inner yoke to move in orbital motion. Inner yoke magnets will PA TCyc = Tin · . (16) perform a cycloidal motion. Strong magnetic-flux paths be- P C tween the inner and outer ring will ensure torque on the output Pole-number configuration for the cycloid permanent-magnet shaft. The output shaft is connected to the inner yoke with initial design comes partly from a previous developed magnetic tree columns and eccentrics. These eccentrics can compensate gear [1]. The gear in [1] has 44 permanent magnets on the for misalignment between inner yoke and output shaft. The outer ring, and this pole number is reused for the cycloid columns will act like mechanical coupling and transfer torque permanent-magnet gear design. The inner ring is designed with from the inner yoke to the output shaft. 42 magnets. The minimum air gap is fixed at 0.5 mm, and the The rotor yokes are placed on the eccentric input shaft. The eccentricity is optimized to be 2.5 mm. The parameters for the eccentricity is made by a central eccentric bushing. Together initial design are listed in Table I. with six support eccentrics, 12 extra needle bearings are added.

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Fig. 8. Sketch of dynamic test bench.

Fig. 7. Rendered view of the cycloidal gear design.

Their functionality is to ensure a parallel motion of the inner rotors relative to the output shaft reference. The proposed Fig. 9. Measured efficiency for the cycloid magnetic gear. construction will therefore have 18 bearings in total. The total volume of the magnetic gear could have been torque on both sides of the gear was measured with torque reduced by choosing a circular design and also by choosing transducers (Fig. 8). smaller bearings. Efficiencies were calculated from the measured torque val- The cycloid gear has been tested in two situations. The first ues, and the results are shown in Fig. 9. test was a static test where the output torque was measured The efficiency were measured at 1500, 500, and 50 r/min. to be 33 N · m. This measurement is shown in Fig. 6 with a The highest efficiencies were generally obtained at low speed single measurement at peek output torque for comparison with and high torque. The best gear efficiency measured at 50 r/min the 2-D models. The measured torque of the experimental test was 94%. Efficiencies of 500 and 1500 r/min were 93% and model is lower than the results from the 2-D models. For this 92%, respectively. deviation, 3-D end effects are assumed to be the most dominant factor. VI. OPTIMIZATION It has to be noted that the length of the magnets are very small, so large 3-D effects should be expected. The length of The initial design may not give the full picture of the capabil- the magnet is based on the availability of standard magnets ity of the cycloid permanent-magnet gear. Optimizations were and, thus, not optimized. However, for future optimization, it therefore performed by formulating the analytical equations might be worth to consider the third dimension in the analytical- to a general optimization problem [7] with a cost function calculation methods, since it apparently has a large effect for (17), equality constrains (18), inequality constrains (19), and a this relatively short layout. Basically, this means that the stack number of constants from Table I. The cost function (17) is the length cannot be considered as a simple scaling parameter as it torque density of the magnetic spur gear with the torque cal- is for typical electrical machines. culated from the parallel magnetization expressions developed The constructed cycloid gear is reasonably comparable with in [2]. The volume is set by the area given by drive-magnet the gear developed in [1]. Both gears use 44 times two of the wheel radius and a constant length. It is necessary to have same type of rectangular magnets in the outer ring, and the an equality constrain h1 for limitation of the used permanent- air-gap radius is, thus, similar. In [1], 216 standard rectangular magnet material, i.e., the area, since the length is fixed. This magnets were used, while the cycloid gear in this paper only permanent-magnet area constrain were set up to be the same uses 172 of the same type of magnet. In [1], the stall torque was area as used for the analytical model of the initial magnetic- −4 2 measured to 16 N · m, which means that the cycloid gear has gear test model (Aconst =35.4 · 10 m ). Another equality doubled the torque density. constrain h2 is set to keep a separation distance of 0.5 mm The other test was a dynamic test where input and output between the magnetic wheels. Inequality constrains is set for torques were measured. The test was performed with two servo the minimization routine. The outer radius has to be somewhat drives. One of the drives was set up as load drive, and the other greater than the inner radius in g1 and g2. Largest radius of the was set up as input drive. Three input velocities were tested, drive wheel radius has to be less than a certain value in g3.The and different loads were applied by the load control drive. The smallest radius for the inner gear wheel had to be greater than

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TABLE II VII. MANUFACTURABILITY OPTIMIZATION RESULTS The proposed test-model design seems quite complicated at first sight, and other designs might be worth considering for simplifying the manufacturing process. Argumentation for the chosen design is shortly described, and possible simplifications for future designs are mentioned. The proposed test-model configuration is equivalent to the configuration shown in Fig. 4(c). The advantage of choosing this configuration is that the output shaft can be designed with relatively large distance between bearings for better support of loads on the output shaft. Two rotors are used for the test- a certain value in g4. The distance of the eccentric also has a model design, primarily to compensate for reaction forces in the limitation in g5 and g6 gearbox housing. The design with two rotors makes the system more complex regarding the number of parts, and this design might not be the best design regarding manufacturability. There f(x)=f(x1,x2,...,xn) ⇒ are also used double-needle bearings in the eccentrics that − T (R2s,R1s,R2d,R1d, d) support the inner magnetic parts. Such configuration could be f(R2s,R1s,R2d,R1d,d)= 2 Lπ (R2d) replaced by simple hardened columns. It is the general opinion that internal tolerances for keeping Np · (17) gear part fixed must have the same level of precision as other N pole mechanical cycloid gear types. However, tolerances for place- hj(x)=(x1,x2,...,xn)=0 ⇒ ment of magnets might have minor demands for precision.

h1(R2s,R1s,R2d,R1d)=Aconst

− 2 − 2 2 − 2 π R2s R1s +R2d R1d =0 VIII. CONCLUSION

h2(R2s,R1d,d)=R1d −d−g−R2s =0 (18) A new cycloid magnetic-gear configuration has been presented. This magnetic gear is characterized by having high g (x)=g (x ,x ,...,x ) ≤ 0 ⇒ i i 1 2 n torque density and a high gearing ratio. The maximum achiev- −3 g1(R2s,R1s)=R1s − R2s +4 · 10 ≤ 0 able torque of the proposed gear was calculated by analytical expressions derived in [2] and the transformation equation g (R ,R )=R − R +4 · 10−3 ≤ 0 2 2d 1d 1d 2d (16). An initial design of the cycloid gear was able to reach −3 · g3(R2d)= − 61.5 · 10 +R2d ≤ 0 141.9 N m/L, which is around 40% more than the “planetary” magnetic gears developed in [4]. An experimental test model of g (R )=48.5 · 10−3 −R ≤ 0 4 1s 1s the initial design was constructed and tested. The experimental −3 · g5(d)= − 20 · 10 + d ≤ 0 test model reached 33 N m. Optimizations for cycloid gears showed that it was possible to reach torque densities up to g (d)=1· 10−3 − d ≤ 0. (19) 6 183 N · m/L, which is almost twice the density as compared to the “planetary” magnetic-gear types [4]. A cycloid magnetic A constrained nonlinear-minimization routine was per- gear could therefore be a possible choice for future applications formed to find the optimal dimensions for the magnetic spur where, for example, a motor or generator is integrated together gear. The initial design is very close to the computer-optimized with the cycloid-gear design. The proposed configurations solution. The initial design was 141.9 kN · m/m3, and the might also be used as power-split devices for hybrid cars or computer optimized solution was 142.5 kN · m/m3. The reason wind turbines with a fixed-speed synchronous generator. why the results are so close to each other is mainly caused by tight constrains limitations. The computerized model is locked ACKNOWLEDGMENT by an area constrain h1, and radius and inner radii must also be within limited values. The authors would like to thank B. Kristensen for construct- The influence of using different amount of magnetic material ing the prototype. was also investigated. The investigation has only been made with the 42- and 44-pole configuration and wider boundary on the radius constrains. The result of this analysis is shown in REFERENCES Table II. The optimal torque density is increased to a certain [1] P. O. Rasmussen, T. O. Andersen, F. T. Jørgensen, and O. Nielsen, “De- velopment of a high-performance magnetic gear,” IEEE Trans. Ind. Appl., amount if more magnetic material is put into the magnetic vol. 41, no. 3, pp. 764–770, May/Jun. 2005. cycloid gear. [2] F. T. Jørgensen, T. O. Andersen, and P. O. Rasmussen, “Two dimensional The last result of the torque-density optimization gave model of a permanent magnet spur gear,” in Conf. Rec. 40th IEEE IAS · 3 · Annu. Meeting, 2005, vol. 1, pp. 261–265. 183 kN m/m or 183 N m/L, which is nearly twice the [3] E. P. Furlani, “Analytical analysis of magnetically coupled multipole cylin- capability of state-of-the-art permanent magnetic gears. ders,” J.Phys.D,Appl.Phys., vol. 33, no. 1, pp. 28–33, Jan. 2000.

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[4] K. Atallah and D. Howe, “A novel high-performance magnetic gear,” IEEE Torben Ole Andersen was born in 1966. He re- Trans. Magn., vol. 37, no. 4, pp. 2844–2846, Jul. 2001. ceived the B.Sc.M.E. degree from the University [5] K. Atallah, S. D. Calverley, and D. Howe, “Design, analysis and realisation of Southern Denmark, Odense, Denmark, in 1989, of a high-performance magnetic gear,” Proc. Inst. Electr. Eng.—Electr. and the M.Sc. and Ph.D. degrees in mechanical/ Power Appl., vol. 151, no. 2, pp. 135–143, Mar. 9, 2004. control engineering from the Technical University [6] T. B. Martin, Jr., “Magnetic transmission,” U.S. Patent 3 378 710, of Denmark, Lyngby, Denmark, in 1993 and 1996, Apr. 16, 1968. respectively. [7] J. S. Arora, Introduction to Optimum Design. New York: McGraw-Hill, From 1996 to 2000, he was with the R&D Depart- 1989. ment, Sauer-Danfoss A/S, where he worked in the area of ﬂuid power and control engineering. In 2000, he joined Aalborg University, Aalborg, Denmark, as an Associate Professor. Since 2005, he has been a Full Professor in ﬂuid power and mechatronics. His main research interests include design and control of mechatronic systems. Frank T. Jørgensen was born in 1973. He received the B.Sc. degree in mechanical engineering jointly from the University College of Aarhus, Aarhus, Peter Omand Rasmussen was born in Aarhus, Denmark, and from the Fachschule Kempten, Denmark, in 1971. He received the M.Sc. degree Kempten, Germany, where he spent a year and a half as an exchange student, and the M.Sc. degree in electrical engineering and the Ph.D. degree from Aalborg University, Aalborg, Denmark, in 1995 and in electromechanical system design from Aalborg 2001, respectively. University, Aalborg, Denmark, where he is currently In 1998, he joined Aalborg University as an As- working toward the Ph.D. degree with a main focus on magnetic gears. sistant Professor. Since 2002, he has been an Asso- ciate Professor. His research areas are in the design He is currently working in industry. He got work- and control of switched reluctance and permanent- ing experience from the company Grundfos A/S, where he was educated as a magnet machines. mechanic.

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