Analytical Method for Predicting the Air-Gap Flux Density of Dual-Rotor Permanent- Magnet (DRPM) Machine

Transcription

1 Analytical Method for Predicting the Air-Gap Flux Density of Dual-Rotor Permanent- Magnet (DRPM) Machine W.Xie, G.Dajaku*, D.Gerling Institute for Electrical Drives and Actuators, University of Federal Defense, Munich Werner-Heisenberg-Weg Neubiberg, Germany Tel: (+49) , fax: (+49) Abstract-- In this paper an analytical method for designing DRPM machine is presented. As well known, DRPM machine has high efficiency, high torque density and can be used as Electric Vehicle Drive and so on, thus developing a simple and accurate analytical method is necessary. The aim of this paper is to calculate the air-gap flux density taking into account the harmonics of the winding current, the winding type and the influence of radial and parallel magnetized magnets. A magnetic equivalent circuit of DRPM machine is developed, and both inner and outer magnet circuits are analyzed separately. Compared with finite-element method (FEM), the analytical method gets an accurate result. Keywords DRPM Machine, Analytical Method, Air-Gap Flux Density, Slotting Effect I. INTRODUCTION HIS paper introduces an analytical method of Dual- T Rotor Permanent-Magnet Machines (DRPM) design. DRPM machines can substantially improve the efficiency due to the greatly shortened end windings and sizably boost the torque density by doubling the working portion of the air gap as well as optimizing the machine aspect ratio [1]. Fig.1 shows the geometry of DRPM machine. Due to the advantages, many applications can be envisaged in the future. Several research work on designing and optimizing DRPM machines have already been done [1]-[3]. However, there is no accurate and simple analytical method to design or calculate the DRPM performance before going to the step of finite element method (FEM) until now, so this paper introduces a mathematical method to predict the air-gap flux density of DRPM machine. Airgap flux density distribution has an important position for the machine design process. In order to improve the accuracy of analytical method, many earlier works have taken into account the leakage flux, the influence of different magnetization, the slotting effect, the saturation in stator and rotor, the winding harmonics and so on [3]-[5]. But considering about the accuracy, the influence of the W.Xie is with the Institue of Electical Drives, University of Federal Defense Munich, Germany ( Xie.Wei@unibw.de). G.Dajaku is Senior Scientist with FEAAM GmbH, D Neubiberg, Germany ( Gurakup.Dajaku@unibw.de). D.Gerling is Head of the Institute of Electrical Drives, University of Federal Defense Munich, Germany ( Dieter.Gerling@unibw.de). harmonics and slotting effect were analyzed, the leakage flux is represented by the leakage coefficient. This paper presents the principle and process of building equivalent magnetic circuit. Based on the geometry and path of magnetic field, DRPM machine works like two conventional machines in series [1], [6]. In section three, finite-element method (FEM) also verifies the independence of the separate flux path of inner part and outer part. Through the separate principle, the equivalent magnetic circuits can be analyzed separately. Reference [3] has built equivalent magnetic circuit which takes into account leakage flux based on single surface-mounted permanent-magnet (SMPM) machines. Reference [5] has analyzed the equivalent magnetic circuit taking account slotting effect and harmonic of interior-type permanentmagnet (IPM) machines. However, for the equivalent magnetic circuit of DRPM machine, there is nearly no research on the equivalent magnetic circuit. This paper introduces the complete equivalent magnetic circuit of DRPM machine. Compared with the accuracy, the easy implementation is also important, thus the analytical method has ignored the saturation of iron. And finally, the analytical model was verified by comparison with FEM results. TABLE I REQUIREMENT AND ASSUMPTIONS Parameters value units Number of pole pair 2 Diameter/Length 1.3 Outer rotor outside radius 87.5 mm Outer rotor inside radius 78.9 mm Outer rotor magnet inside radius 71.9 mm Outside radius of stator 71.2 mm Inside radius of stator 3.2 mm Inner rotor magnet outside radius 29.6 mm Inner rotor magnet inside radius 22.1 mm Inside radius of inner iron 1.6 mm /12/$ IEEE 2758

2 Fig.1. Cross-section of the dual-rotor PM machine II. ANALYTICAL METHOD In this section the analytical method to predict the airgap flux density of DRPM is introduced. Since the topology has two separate air gaps and two different flux paths, thus the design equations can be separated into two portions: one is for the inner rotor, inner permanent magnet, and the stator; the other one is for the outer rotor, outer permanent magnet, and the stator [1]. As shown in Fig.2, the green lines represent the main flux path. Two flux loops use different paths: loop 1 represents the flux path of outer part; loop 2 represents the flux path of inner part; and the black circuits present the leakage flux. Fig.2. Flux path of DRPM. Fig.3. Equivalent magnetic circuit of DRPM machine. Fig.4. Equivalent magnetic circuit without leakage flux. According to Fig.2, the inner portion can be assumed as a complete single rotor surface mounted PM (SMPM) machine. And according to reference [3], [4], the equivalent magnetic circuit (EMC) was build, as shown in Fig.3. In order to simplify the model, the leakage flux can be represented by the leakage coefficient. Thus the circuit can be simplified to Fig.4. Based on the basic theory, Equation (1) is used, and it is simple, easy and widely used in many literatures for the analysis of electric machines. The air-gap flux density is produced by the magneto motive force (MMF) and the air-gap permeance. Here B( xt, ) is the flux density, Λ( x) is permeance and Θ( xt, ) is the magneto motive force (MMF). In this paper, the improved analytical method taking into account the permeance based on EMC and MMF based on stator winding and magnet was analyzed [4]. And it is denoted that for the inner portion, the air-gap flux density depends on the equation (2). B( xt, ) = 2 Λ( x) Θ ( xt, ) (1) B ( φ, t) =Λ( φ ) [ Θ ( φ, t) +Θ ( φ, t)] (2) g1 s1 1 s1 s_1 s1 r_1 s1 Where φ is the displacement along the stator circumference, s1 Λ1( φ s 1) is the function of inner flux path permeance, Θ s_1( φs 1, t) is inner MMF due to inner stator winding, Θ ( φ, ) is inner MMF based on inner rotor magnet. r_1 s1 t 2759

3 In order to enable the analytical solution and simplify the analytical model, the following assumptions are made. End effects are ignored. The permanent magnets have linear demagnetization characteristic. Stator and rotor back-iron has infinite permeability, the saturation effects of the back iron are neglected. The airspace between the magnets and iron teeth has the same permeability as the magnets. The eddy current effect is neglected. A. Permeance The flux permeance depends on the path of flux. As the rotor rotates, the flux permeance of some type of machine will change cyclically, for example inset PM machine. But in this paper for DRPM, because of the rotor magnet is on the surface of the rotor iron, and the permeance of magnet is nearly equal to air, the permeance can be assumed as a constant. Fig.5 shows the flux path of inner part. R is the magnetic resistance of inner air gap, R 1 is the magnetic resistance of inner stator teeth, R is the magnetic 2 resistance of inner magnet, R is the magnetic resistance of 3 inner rotor yoke, R 4 is the magnetic resistance of inner stator yoke. The magnetic resistance can be computed from the relationship of length of the magnetic path, the permeability and the cross-sectional area. Equation (3) is used to calculate the average value of inner flux permeance but ignores the cross-sectional area, since the destination is just to determine the distribution of the air-gap flux density [2]. For the outer EMC, the component of the outer circuit is nearly the same with inner one, thus in this paper, only the inner part of DRPM machine is analyzed. lgap1 is length of inner air gap, H is length of inner stator teeth, s1_ teeth l is length of flux path in inner rotor yoke, r1_ Yoke l is length of flux path in stator yoke. s1_ Yoke From the geometry data Table, the result of inner permeance has been analyzed. According to the mathematical model, the permeance is a constant with the different rotor position. As well known the permeance of inner flux path in Fig.5 does not consider the stator slotting effect, but in fact, the stator slots not only influence the flux amplitude but also influence the distribution of the permeance. As shown in Fig.6, it can be seen that the detail of flux path of circle 1 (C1), and s R represents the magnetic resistance of flux path in slot. Because of the topology, both right and left slots have the flux path (line 1) in Fig.6. As a result of slotting effect the total permeance will decrease (which means the magnetic resistance will increase) and the permeance in the slot position will be much smaller than average as shown in Fig.7 [9]. A new analytical method was introduced as shown in equation (4), which is used to calculate the value of permeance produced by only slotting effect [4]. It considers the harmonic of slot permeance. Fig.7 shows the result of inner permeance taking into account slotting effect. It can be seen that in slot position, the permeance has a clear downward trend which means the magnetic reluctance amplitude becomes higher. Fig.6. Inner flux path taking into account the slotting effect. Λ = Λ + Λ cos( k N P φ ) (4) s s, s, k s s1 k = 1 Fig.5. Flux path and EMC of inner part. 1 Hr1_ teeth lgap1 Hs1_ teeth lr1_ Yoke ls1_ Yoke = 2 ( + + ) + + Λ μ μ μ * μ μ μ μ μ 1 r r r Where H is thickness of inner magnet, r1_ teeth (3) 276

4 7.4 x Inner flux path permeance [H] MMF based on inner stator winding [At] Angle of Inner Rotor position [rad.elc.deg] Fig.7. Distribution of inner flux permeance taking into account the slotting effect. B. Magnetomotive Force of Winding After calculating the permeance, before predicting the flux density, the MMF based on stator winding should also be considered. Depending on the winding type, there will be different MMF characteristics [4]. As well known, this distribution of winding for DRPM machine can decrease the end winding, and then increase the efficiency [1]. Now there is a general expression for the MMF distribution presented in equation (5) and (6) [4]. Here one layer distributed winding topology and simulation result were analyzed, as shown in Fig.8 and Fig Angle of Inner stator position [rad.elc.deg] Fig.9. MMF distribution of inner distributed winding topology with q=1. C. Magnetomotive Force of Magnet For any types of PM machines, knowledge of magnetic field distribution is a prerequisite for predicting performance parameters of PM machines, for example torque, back-emf, losses and so on [7]. For analytical method, the distribution of different magnetization will influence the waveform and accuracy of the calculation. Fig.1 introduces radial and parallel magnetization distributions. Fig.1-(a) represents parallel magnetization distributions and waveform of radial magnetization is shown in Fig.1-(b). Compared with radial magnetization, parallel magnetization is much easier to be obtained [8]. In this paper, parallel magnetization distribution is used. Θ pm 2π Fig.8. Distribution of stator wingding [6]. π sin( ) y1 π ξ 2 w, p, d, sin( ) m = ξ ξ = τ p 2 π q sin( ) 2mq m 2 w ξ Θ ( φ, t) = i cos( wt pφ + δ ) (6) w, s_1 s1 s1 2 π Where: ξ is pitch factor, p, y 1 is pitch, is harmonic times, τ is pole pitch, p ξ d, is distributed factor, ξ is winding factor. w, (5) Θ pm β mag Fig.1. Parallel and radial magnetization and waveform [7]. As it is mentioned above, the distribution of rotor MMF not only depends on the magnet topology but also on the magnetization direction. For inner portion, the inner rotor is nearly the same as the rotor of surface mounted PM (SMPM) machine. After choosing the magnet type as shown in Fig.1-(a), a general expression for the rotor MMF of magnet is presented in equation (7). ξ and Θ PM1 denote the space harmonics and corresponding amplitudes. The maximum of the MMF is based on the coercive field intensity and thickness of the magnet, here βmag is magnet pole width (pole arc). r_1 φr 1 t PM 1 ξ p φr1 ξ Θ (, ) = Θ cos( ) (7) 2π 2761

5 β β β Θ 2 2 PM1 = + π βmag βmag ( ξ + 1) ( ξ 1) 2 2 mag mag sin(( 1) ) sin(( 1) ) mag Θ ξ + ξ PM 1 Based on the geometry data, Fig.11 shows the distribution of MMF based on inner magnet. (8) flux density due to winding flux density due to magnet outer air-gap flux density Outer air-gap flux density [T] Angle of Outer rotor position [rad.elc.deg] Fig.13. Distribution of outer air-gap flux density. Fig.11. Distribution of MMF based on inner magnet. According to equation (2), both of Θ _1 ( φ r R 1, t ) and Θs_1( φs 1, t) affect the air-gap flux density. In order to increase the accuracy, not only the slotting effect, but also the harmonic and magnetization type was considered. Fig.12 and Fig.13 show the calculation result of inner part air-gap flux distribution and outer part air-gap flux distribution. flux density due to winding flux density due to magnet inner air-gap flux density.5 III. SIMULATION RESULT In the following, the Finite Element Method (FEM) is employed to verify the effectiveness of the analytical model, redefine the geometry factors, find the suitable performance, and verify if the machine can survive under the stator saturation and the limitation of current density. Thus the studied DRPM machine is analyzed using 2D- Maxwell finite element software. Fig.14-(a) shows the flux lines of DRPM machine on 2D-Maxwell. Outer flux loops are constituted with stator yoke, stator teeth, outer air-gap, outer magnet and outer rotor; inner flux loops contain stator yoke, stator teeth, inner air-gap, inner magnet, and inner rotor. The flux loops denoted the effectiveness of the assumption of Fig.2. And Fig.14-(b) represents the flux vector for DRPM machine. Outer flux loop Inner flux loop.4.3 Outer air-gap flux density [T] Angle of Outer rotor position [rad.elc.deg] Fig.12. Distribution of inner air-gap flux density. Outer flux vector Inner flux vector Fig.14. Distribution of flux of DRPM machine; a). flux lines, b). flux vector. 2762

6 Powered by TCPDF ( With the same exciting current and magnetization of magnet, Fig.15 shows the distribution of flux density of DRPM machine. Compared with the simulation of FEM, the implementation of analytical method is much easier. The obtained result shows the agreement between analytical and FEM method which also takes into account the slotting effect, magnetization and harmonics. machines," IECON th Annual Conference on IEEE Industrial Electronics Society, vol., no., pp , 7-1 Nov [9] Andriamalala, R.N.; Razik, H.; Baghli, L.; Sargos, F.-M.;, "Eccentricity Fault Diagnosis of a Dual-Stator Winding Induction Machine Drive Considering the Slotting Effects," Industrial Electronics, IEEE Transactions on, vol.55, no.12, pp , Dec. 28. Wei Xie was born in China, He graduated from the Northwestern Polytechnical University in Xi an, China in 21 and is now at the Institutes of Electrical Drives, University of Federal Defense, Munich, Germany as a PhD student. His technical interest includes electromagnetic field analysis and switched reluctance machines. Fig.15. Air-gap flux density due to one pole outer magnet of studied DRPM machine. IV. CONCLUSION Air-gap flux density in PM machines is an important parameter for predicting the performance of PM machines. In this paper, for the distribution of air-gap flux density of DRPM machine, the new analytical method takes into account the harmonics of permeance, stator winding, magnet and the slotting effect, and improves the accuracy. The inner part and outer part are analyzed separately, and the air-gap MMF components are also considered separately for the stator winding and magnets. This paper has denoted that this method is simple and effective for DRPM machine design. The analytical method has been verified by FEM calculations. Gurakuq Dajaku; Dr.-Ing. Gurakuq Dajaku is with FEAAM GmbH, Werner-Heisenberg-Weg 39, D Neubiberg, Germany, phone: , fax: -3718, Gurakuq.Dajaku@unibw.de. Born in 1974 (Skenderaj, Kosova), got his diploma degree in Electrical Engineering from the University of Pristina, Kosova, in 1997 and his Ph.D. degree from the University of Federal Defense Munich in 26. Since27 he is Senior Scientist with FEAAM GmbH, an engineering company in the field of electric drives. From 28 he is a Lecturer at the University of Federal Defense Munich. His research interests include the design, modelling, and control of permanent-magnet machines for automotive application. Dr. Dajaku received the Rheinmetall Foundation Award 26 and the ITIS (Institute for Technical Intelligent Systems) Research Award 26. Dieter Gerling; Prof. Dr.-Ing. Dieter Gerling is head of the Institute of Electrical Drives at the University of Federal Defense Munich, Werner- Heisenberg-Weg 39, D Neubiberg, Germany (phone: ; fax: -3718; Dieter.Gerling@unibw.de). Born in 1961, Prof. Gerling got his diploma and Ph.D. degrees in Electrical Engineering from the Technical University of Aachen, Germany in 1986 and 1992, respectively. From 1986 to 1999 he was with Philips Research Laboratories in Aachen, Germany as Research Scientist and later as Senior Scientist. In 1999 Dr. Gerling joined Robert Bosch GmbH in Bühl, Germany as Director. Since 21 he is Full Professor at the University of Federal Defense Munich, Germany ( REFERENCES [1] R. Qu and T. A. Lipo, Dual-rotor, radial-flux, toroidally-wound, permanent-magnet machines, IEEE-IAS Annual meeting, Pittsburgh,PA, Oct. 22, Vol. 2, pp [2] R. Qu and T. A. Lipo, Design and Optimization of Dual-Rotor, Radial-Flux, Toroidally-Wound, Permanent-Magnet Machines, IEEE 38th IAS Annual Meeting, 23, Vol. 2, pp [3] Gerber, S.; Strauss, J.M.; Randewijk, P.J.;, "Evaluation of a hybrid finite element analysis package featuring dual air-gap elements," Electrical Machines (ICEM), 21 XIX International Conference on, vol., no., pp.1-6, 6-8 Sept. 21. [4] Dajaku G.: Stator Slotting Effect on the Magnetic Field Distribution of Salient Pole Synchronous Permanent-Magnet Machines, IEEE Transactions on magnetics, September 21, Vol. 46, NO. 9, pp [5] Dajaku, G.; Gerling, D.;, "Determination of air-gap flux density due to magnets using the new analytical model," Electrical Machines (ICEM), 21 XIX International Conference on, vol., no., pp.1-6, 6-8 Sept. 21. [6] Zhenping Wang, The Research on Radial-flux, Dual-rotor PM Machine master degree paper, Shandong University, China, 26. [7] Z.Q.Zhu, David Howe, Improved Analytical Model for Predicting the Megnet Field Distribution in Brushless Permanent-Magnet Machines, Magnetics, IEEE Transactions on magnetics, Jan 22. Vol.38, No.1, pp [8] Ferraris, L.; Ferraris, P.; Poskovic, E.; Tenconi, A., "Comparison between parallel and radial magnetization in PM fractional 2763