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Influence of Geometrical Parameters of Switched Reluctance Motor on Electromagnetic Torque Krzysztof Bieńkowski, Jan Szczypior, Bogdan Bucki, Adam Biernat, Adam Rogalski Institute of Electrical Machines, Warsaw University of Technology, Poland Nowowiejska 20 A, 00-661 Warsaw, Poland, phone: (+4822) 660-73-35, email: [email protected] Abstract - Researches on torque characteristic of switched reluctance motor using Finite Elements Method are presented in this paper. Self-prepared additional software makes possible to find optimal set of constructional parameters of the motor. Characteristics of torque versus rotor position are presented for different values of some constructional parameters. Influence of internal diameter of stator, stator’s and rotor’s pole breadth, rotor’s pole height and air gap on electromagnetic torque and torque ripple coefficient are presented. An Effective design of SRM requires determination of the set of geometrical parameters, which enable producing maximum electromagnetic torque. For that reason only development of CAD procedures allows to utilize all anticipated advantages of these motors. The aim of presented researches is to formulate remarks concerning designing process and methods of determining these parameters leading to optimal SRM construction. I. INTRODUCTION II. FEM MODEL OF SWITCHED RELUCTANCE MOTOR Switched Reluctance Motor is an electric machine, which produces torque in consequence of variability of reluctance for magnetic flux excited by currents flowing in the stator windings. Reluctance changes during rotation of the rotor for the reason of properly formed stator and rotor cores [1]. To the effective operation of SR Motor commutator should to precisely determine position of the rotor. On this signal it must calculate the time of connection and disconnection of each phase band to source voltage. These moments are not constant but depend of the rotational speed and load of the machine. Only development of power electronics, CAD systems and microprocessor control methods caused fully using of advantages of those motors [2]. Concentrated coils are placed around the stator’s poles. Coils placed on reciprocal poles are series connected and creating phase band. Each band of the stator winding is connected to supply by transistor switches. Current in each band should flow at such positions of rotor, to assure continuous rotation with minimum torque ripples. It can be archived through such reconnecting phase’s bands which causing current flowing during increasing inductance of phase band. If this condition is not fulfilled some breaking torque will appear. Electromagnetic torque depends from changes of reluctance caused by the rotor rotation. Reluctance as function of the rotor position depends on flux excitation – current linkage of the winding, (because of non-linear characteristic of magnetisation of core) and on constructional parameters determining magnetic circuit. At given current, number of turns, B versus H curve and selected variant of stator construction, the reluctance depends on rotor pole breadth, it heights and thickness of air gap. Available commercial FEM processors deliver possibility for modelling electromechanical converter with any construction [3], but pre-processing of the model is arduous and time-consuming. Result of single solution is distribution of magnetic potential in the whole area of the model. To solve problem of determining torque characteristic special software based on PC-OPERA package was created. In result of the software operation the set of torque for different positions of the rotor ware obtained. The structure chart of this software is presented on Fig.1. Edition of data file Preprocesing of the first task Solution of the task Preprocesing of the next task Torque calculations, saving of the value Checking of the finish conditions Not filled Filled Finish of the calculations, preparation of the results, export to Matlab environent Figure 1. Structure diagram of the software. The software was in detail described in [6]. The problem of determination of static torque characteristic versus angular position of the rotor amounts to calculation of the set of values of electromagnetic torque for given set of angular position of the rotor. This characteristic of SRM is periodic due to periodicity of the magnetic circuit structure, so the set of angular position of the rotor could be reduced to one pole pith. If the stator and rotor poles are symmetrical the range of calculations could be reduced even to the half of pole pith. Single value of torque, at fixed position of rotor, can be calculated by integrating of Maxwell stress tensor in the air gap region. Simplifying assumption of constant magnetic flux distribution along the length of the core leads to solution of the two-dimensional magnetostatic problem. Therefore the expression on electromagnetic torque has form: 2π T =r l∫ 2 0 Where: r l Bn,Bt Bn Bt µ0 dα (1) – radius of air gap circle, – length of the core, – components of flux density. In the cross section of the motor model, there are following regions: stator and rotor cores, stator windings (coils), air regions. The motor following parameters: Number of stator/rotor pole pairs ps/pr = 3/2 External diameter of stator core dse = 80 mm Internal diameter stator d = 32 mm Effective length core lFe = 100 mm Breadth of stator pole bps = 8 mm Breadth of rotor pole bpr, = 9 mm Height of rotor pole hpr, = 5 mm Air gap δ = 0. 3 mm Researches of influence of geometrical parameters on electromagnetic torque were carried out with changing values of breadth and height of rotor pole and the air gap. During changing of one parameter the others were changing too, assuming the constant copper loses in stator windings, but external diameter of stator core was still constant. Family of torque characteristics at different values of breadth of rotor pole is presented on Fig. 3. Torque values between computed points (marked with suitable marks) were obtained by spline interpolation method. 1,6 Beside of produced torque, the torque ripples coefficient is very important parameter to assess the construction. This coefficient is defined as: t rpl = Where: Tmax Tmin Tav Tmax − Tmin 2Tav bpr =9 mm 1,4 1,2 (2) 1 – the maximum value of torque, – the minimum value of torque, – the average value of torque. bpr = 8 mm 0,8 0,6 0,4 bpr = 7 mm III. RESULTS The construction of the first researched motor is schematically represented on Fig. 2. Stator core Shaft Rotor core Air regions Stator winding Figure 2. Cross section of researched motor. Torque T [Nm] II.1 The static torque characteristics and torque ripple coefficient 0,2 0 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 Rotor position angle α [deg] Figure 3. Family of torque characteristics for different pole breadth of rotor. Change of breadth of rotor pole, in examined range, practically does not influence on the maximum value of torque. Increasing of breadth of rotor pole causes only changes of maximum torque rotor position. Worth mentioning is fact of existence of optimal value of breadth of rotor pole for which the average torque has maximum value. The average torque and torque ripples coefficient is calculated for 30° operational range of the phase band (between -35° and -5°), with the assumptions of constant phase current, what is fulfilled at motor operation with low speed (under the base speed). The average torque and torque ripple coefficient for different values of breadth of rotor pole is presented on Fig. 4. The smallest value is defined by mechanical conditions (bearing clearances, tolerance of brackets, shaft bending and stretching deformations of stator core). The torque ripples can be slightly reduced by growing air gap (Fig. 7). 1,4 2 Tav 1,6 1 0,8 1,4 δ = 0,3 [mm] 0,6 1,2 1 trpl 0,4 0,8 δ = 0,4 [mm] 0,2 0,6 0 0,4 6 7 8 9 0,2 10 0 bpr [mm] -45 -40 -35 The influence of height of rotor pole on the maximum torque is shoved on Fig. 5. There are presented the maximum torque value because height of rotor pole increasing does not influence rotor position angle with maximum of torque. -30 -25 -20 -15 -10 -5 0 Rotor position angle α [°] Figure 4. Average torque and torque ripple coefficient for different values of breadth of rotor. Figure. 6. Family of torque characteristics for different air gap. 1,5 Tav 1,2 1,6 1,4 0,9 T [Nm], trpl [-] 1,2 1 T [Nm] 1,8 T [Nm] Tav [Nm], trpl [-] 1,2 δ = 0,2 [mm] 0,8 0,6 0,6 trpl 0,3 0 0,4 0,1 0,2 0,2 0,3 0,4 0,5 bpr [mm] 0 0 1 2 3 4 5 6 hpr [mm] Figure 5. Maximum torque for different values of height of rotor pole. Height of pole increasing from 0 to 5δ caused torque increasing from 0 to 1.3 Nm but further torque increasing is very slow. Enlarging the height of rotor pole above 10÷15δ is practically not justified. The family of torque characteristics at different values of air gap is presented on Fig. 6.In SRM the air gap should be as small as possible. Figure. 7. Average torque nad torque ripple coefficient for different values of air gap. The second researched model has the same structure as presented on Fig. 2. and following geometrical parameters: Number of stator/rotor pole pairs ps/pr = 3/2 External diameter of stator core dse = 300 mm Internal diameter of stator d = 180 mm Effective length of core lFe = 150 mm Breadth of stator pole bps = 49 mm Breadth of rotor pole bpr, = 56 mm Height of rotor pole hpr, = 20 mm Air gap δ = 1 mm The family of torque characteristics at different values of internal stator diameter d is presented on Fig.8. 0,6 0,5 250 0,4 trpl [-] 300 150 T [Nm] 200 0,3 0,2 0,1 100 d 0 150 190 50 160 170 0 -50 -40 -30 -20 Rotor position angle α [°] -10 0 180 190 d [mm] 180 200 170 Figure. 10. Torque ripple coefficient for different values of internal diameter of stator. 160 The family of torque characteristics at different values of breadth of rotor pole is presented on Fig. 11. Breadth of stator pole, in examined range, considerably influence on the maximum value of torque and shape of the characteristic. The torque ripples can be slightly reduced by growing breadth of stator pole. At bps = 49 mm the average torque has maximum value. Figure. 8. Family of torque characteristics for different internal stator diameter. The change of internal stator diameter d, in examined range, influence on the maximum value of torque and shape of the characteristic. At d = 180 mm the average torque has maximum value, assuming the constant copper loses in stator windings. Torque ripple coefficient can be slightly reduced by growing internal diameter of stator. The average torque and torque ripple coefficient for different values of breadth of rotor pole are presented on Fig. 9. and 10. 300 Bps = 42 mm Bps = 49 mm 250 200 200 T [Nm] 150 Tav [Nm] Bps = 57 mm 190 100 180 50 170 0 -50 160 150 150 -40 -30 -20 -10 0 Rotor position angle α [deg] Figure.11. Family of torque characteristics for diferent breadth of stator pole. 160 170 180 190 200 d [mm] Figure. 9. Average torque for different values of internal diameter of stator. The average torque and torque ripple coefficient for different values of breadth of stator pole is presented on Fig. 12 and 13. Set of motor’s parameters which allow to produce high torque is not suitable in respect of torque ripple. Criterion of maximum efficiency is contradicted to criterion of minimum torque ripples. The influence of rotor’s pole height and air gap on electromagnetic torque are presented for switched reluctance motor with dse = 80 mm. Enlarging the height of rotor pole above 10÷15δ is practically not justified. To archive high torque the air gap should be as small as possible for mechanical reasons. The influence of internal diameter of stator, and breadth of stator pole on electromagnetic torque are presented for switched reluctance motor with dse = 300 mm. Breadth of stator pole significantly influence on the shape of torque versus angular position of rotor. 200 190 Tav [Nm] 180 170 160 150 40 45 50 55 60 bps [mm] 0,6 Acknowledgement This research project is supported by The State Committee for Scientific Research of Poland within years 2003-2006. 0,5 REFERENCES Figure. 12. Average torque for different breadth of stator pole. trpl [-] 0,4 0,3 0,2 0,1 0 40 45 50 55 60 bps [mm] Figure. 13. Torque ripple coefficient for different breadth of stator pole. IV. CONCLUSIONS Worked out software makes possible to choose set of constructional parameters of the SRM which produce maximum of electromagnetic torque or minimum of torque ripple coefficient at the constant copper losses. Breadth of rotor pole significantly influence on the average torque and torque ripple therefore, the shape of rotor core should focussing designer’s attention. [1] Miller E.T.J., Switched Reluctance Motors and Their Control, 1993; Magna Physics Publishing [2] Krishan R. “Switcher reluctance Motor Drives”, CRC Press LLC, London, 2001 [3] Wei Wu, John B. Dunlop, Stephen J. Collocott, and Bruce A. Kalan Design Optimization of a Switched Reluctance Motor by Electromagnetic and Thermal FiniteElement Analysis, 3334 IEEE Transations on Magnetics, VOL. 39, NO. 5, SEPTEMBER 2003 [4] Risse S., Henneberger, “Design and Optimization of SRM for Electric Vehicle Propulsion”, ICEM proceedings pp1526-1530, Helsinki, 28-30 August 2000 [5] K. Bieńkowski, J. Szczypior: “Influence of constructional parameters of Switched Reluctance Motor on electromagnetic torque”, Berichte und Informationen HTW Dresden 1/2002 ISSN1433-4135 [6] Bieńkowski K., Bucki B. „FEM model of switched reluctance motor”, XL International Symposium on Electrical Machines proceedings, Hajnówka, Poland, 1518 June 2004