Krzysztof BIEŃKOWSKI 1 Andrzej POCHANKE 2 Influence of geometrical parameters on static torque characteristics in double-phase switched reluctance motor Switched reluctance motors find wide applications in controllable rotational speed drives as well as actuators in automatic systems and robotics. The static torque characteristics and control algorithms have decisive influence on exploational proprieties of drive as energy converter. The static torque profiles could be shaped by geometrical parameters of magnetic core. The influence of widths of stator s and rotor s poles on the static torque was introduced in this paper as well as width of airgap influence. The results were obtained by means of field solution in finite element models of the motor. 1. Introduction Switched Reluctance Motors are distinguished by simple and robust construction (Fig. 1.). Stator and rotor cores are made of sheets of electromagnetic steel. Stator and rotor poles are uniformly distributed on the periphery. The number of poles has to be even and moreover number of rotor poles has to be differ from number of stator poles [1,2]. The concentrated coils are situated on each stator pole. Coils on opposite poles are series connected and makes phase windings. Phase windings are connecting to the voltage source in suitable sequence by the power converter with transistor switches. In the initial period of SR machines development they reveals many of disadvantages high torque ripples, high level of vibrations and noises, low power density. However the progresses in CAD/CAM technology and material engineering as well as progresses in power electronics and microprocessor technology, they caused that SR machines was successfully going to gradually elimination their drawback or minimalization consequences of their disadvantages [2,3]. SR motors have high possibility to operate in very difficult conditions - high moisture, temperature, pollination and they are easy controllable by means of power converter. Take such properties into account SR motors are applicated onto whirling pumps, ventilators and traction drives [4]. With regard to low manufacturing costs, the large durability and maintenance free duty, SR motors are more convenient to the customer use drives than traditional commutator motors [2,4]. Low cost and high reliability of SRMs arises from the simple and robust construction [1,2]. The absence of mechanical commutator, rotor windings and permanent magnets makes they proof on mechanical stresses and makes possible to operate in high temperatures. The number of phases of the machine is the basic parameter, which determines the exploitational properties and applications of SRM. The enlarging of the number of phases is 1 Ph.D. Ing., Institute of Electrical Machines, Warsaw University of Technology. Plac Politechniki 1, 00-661 Warsaw, tel. +(48-22)6607490, fax: +(48-22)8127535, e-mail: k.bienkowski@ime.pw.edu.pl 2 D.Sc. Ing., Institute of Electrical Machines, Warsaw University of Technology. Plac Politechniki 1, 00-661 Warsaw, tel. +(48-22)6607435, fax: +(48-22)8127535, e-mail: pochanke@ime.pw.edu.pl
profitable with regard on decreasing of torque ripples, but it entail the enlargement of number of semiconductor switches and decreasing of the factor of copper utilization [1]. In high-speed drives of small power, which do not need low torque ripples as ventilators and pump, the number of phases is equal m = 1 or 2. [4]. In more responsible drives with controlled rotational speed, it is profitable to enlarge the number of phases up to three or four, and in some application even to seven [7]. stator core rotor core b pr b ps stator winding 2. Static torque calculations Fig.1. Schematic construction of double phase SRM. The electromagnetic torque produced by switched reluctance motor is hard to calculate by analytical method, because of nonlinearity of magnetic material and of the strong dependence of flux path from the rotor movement [2]. More precise results could be obtained by magnetic field solution. On presented researches two-dimensional finite elements (FE) model of the motor was prepared [5]. The magnetostatic solution of FE model deliver the information of magnetic potential in the modelling area, which could be used to calculation of flux density. The static electromagnetic torque could be calculated by integration of Maxwell stress tensor along the circle in air gap, as well as by the differentiation of the coenergy of modelling area. In the case of two-dimensional static flux excited by ampere-turns θ and given angular position α of rotor, the torque can be expressed: Where: 2π 2 Bn ( α, r) Bt ( α, r) T( θ, α) = r l dα (1) µ r radius of circle in the airgap, l effective length of core, 0 0 B ( α, r), B ( α, r) normal and tangential components of the flux density to the circle in n the point (α, r). t
If numerous static problems were being solved at different current excitation and rotor position, it is possible to obtain static torque-angle characteristics of the motor [5, 6]. During design process of the motor, it is convenient to use the torque per unit length. Precisely designed static torque could be archived by choosing effective length of the core. The examples of angular characteristics of static torque for different values of excitation are presented on fig. 2. 0,035 0,030 7 A 0,025 T 0,020 [N m/mm] 0,015 4 A 0,010 0,005 1 A 0,000 0 10 20 30 40 50 60 70 80 90 α [ ] Fig. 2. Static torque (per unit length) versus rotor position angle for three different excitation currents. 3. Influence of geometrical parameters of the core on the static torque characteristics. The calculations of influence of chosen parameters on the angular profiles of torque were executed for motor with geometrical structure introduced on fig. 1. The parameters of the motor are following: - number of stator poles / rotor poles p s / p r = 4 / 2, - length of the side of stator core b s = 80 mm, - height of stator yoke h ys = 8 mm, - inner stator diameter d si = 40 mm, - number of turns per phase n ph = 2 x 100 turns. For the motor with parameters mentioned above, the torque calculations were executed for following variables: - width of stator pole b ps, - width of rotor pole b pr, - thickness of air gap δ. In the author s opinion such parameters have the most influence on static torque produced by the motor. During calculations the others dimensions of the motor did not be changed.
3.1. The influence of width of stator pole on static torque The torque characteristics were calculated for three values of width of stator pole: b ps = 15,0 16,3 and 17,5mm. For each calculation the other parameters were constant: the air gap δ = 0,3mm; the width of rotor pole b pr = 16mm; the excitation current I = 4A. 0,020 T 0,015 [N m/mm] b ps = 17,5 0,010 b ps = 16,3 b ps = 15,0 0,005 0,000 0 10 20 30 40 50 60 70 80 90 α [ ] Fig. 3. Static torque (per unit length) versus rotor position angle for three different values of width of stator pole. 3.2. The influence of width of stator pole on static torque The torque characteristics were calculated for three values of width of rotor pole: b pr = 14,8 16,0 and 17,3mm. For each calculation the other parameters were constant: the air gap δ = 0,3mm; the width of rotor pole b pr = 16,3mm; the excitation current I = 4A. 3.3. The influence of air gap on static torque The torque characteristics were calculated for three values of air gap: δ = 0,2 0,3 and 0,4 mm. For each calculation the other parameters were constant: the width of stator pole b ps = 16,3mm; the width of rotor pole b pr = 16mm; the excitation current I = 4A. 4. Assortment of constructional parameters for the motor On the base of characteristics mentioned above, it is quite difficult to choose geometrical parameters of the motor, which are adequate for most torque generating. The better gauge could be average torque calculated for chosen set of parameters. The average torque could be calculated for full angle range between non-aligned and aligned position of the rotor as well as, for the conduction angle of the phase.
0,020 b dr = 17,3 0,015 T [N m/mm] b dr = 16,0 0,010 b dr = 14,8 0,005 0,000 0 10 20 30 40 50 60 70 80 90 α [ ] Fig. 4. Static torque (per unit length) versus rotor position angle for three different values of width of rotor pole. 0,025 δ = 0,2 0,020 T [N m/mm] δ = 0,3 0,015 0,010 δ = 0,4 0,005 0,000 0 10 20 30 40 50 60 70 80 90 α [ ] Fig. 5. Static torque (per unit length) versus rotor position angle for three different values of air gap.
In the field of presented researches the average torque for full angle range was calculated for nine variants of construction of the motor, with following parameters: variant 11 - b ps = 15 mm, b pr = 14,8 mm, variant 12 - b ps = 15 mm, b pr = 16 mm, variant 13 - b ps = 15 mm, b pr = 17,3 mm, variant 21 - b ps = 16,3 mm, b pr = 14,8 mm, variant 22 - b ps = 16,3 mm, b pr = 16 mm, variant 23 - b ps = 16,3 mm, b pr = 17,3 mm, variant 31 - b ps = 17,5 mm, b pr = 14,8 mm, variant 32 - b ps = 17,5 mm, b pr = 16 mm, variant 33 - b ps = 17,5 mm, b pr = 17,3 mm. The air gap was set at 0,2 mm and the phase current at 4A. The average torque for the first variant T 11 = 0,00796 Nm/mm, was used as the reference value for remaining variants. Comparison of relative torque values for nine alternative design of the motor is presented on fig. 6. 114% 112% T avnn /T av11 110% [%] 108% 106% 104% 102% 100% 1 2 3 1 2 3 5. Conclusions Fig. 6. Relative torques comparison for alternative design of the motor Double phase switched reluctance motor, with four stator poles and two rotor poles produces the torque with high level of torque ripples. Four angular sectors about 40 of no torque accrues for one rotor turn. The rotor has to cross those sectors stagnantly, to retain continuous movement. The torque of the motor strongly depends on the excitation current. Control of the motor is realised by control of the average current level and the conduction period of the phase, whereat the dependence of instantaneous torque on current is strongly non-linear due to magnetic saturation of the core material. The saturation depends on current level as well as rotor position. Torque ripples escalate with the saturation of the core. The widths of stator and rotor poles do not influence on maximum torque level, if excitation current is constant, but the shape of torque versus angle profiles varying with them.
On the base of presented researches, it should be affirmed, that width of stator pole has stronger influence on torque characteristics. The air gap has the most essential influence on torque level, but only in the range of partial covering of the poles. The lowest possible values of air gap should be applied in SR motors. Only mechanical limitations of the air gap have to be fulfilled. The motor already produces quite large torque at little poles covering the poles, then at quit small flux [2, 6]. Therefore, the magnetisation curve of core material should, first of all make possible to obtain flux already at little excitation. The level of saturation has minor importance. The suitable selection of widths of poles makes possible enlargement of average torque about a dozen or so percentage in relation to initial design. Presented researches do not be an attempt to optimization of the construction. The maim aim of the work is to show on the character of influences of geometrical parameters of the core on electromagnetic torque. Acknowledgement: This research project is supported by The State Committee for Scientific Research of Poland within years 2003-2006. References [1] Miller, T.J.E.: Switched Reluctance Motors and Their Control. Magna Physics Publishing, Clarendon Press, Oxford, 1993. [2] Krishan R.: Switched Reluctance Motor Drives. CRC Press London, 2001. [3] Holling, G. H.: Design of Switched Reluctance Motor Drives. Advanced Motion Controls, Inc, November, 1999 [4] J.Y. Lim et all: Single Phase Switched Reluctance Motor for Vacuum Cleaner. Proc. ISIE 2001, Pusan, Korea, 2001. [5] Bieńkowski K., Bucki B.: Model polowy przełączalnego silnika reluktancyjnego. XL Międzynarodowe Sympozjum Maszyn Elektrycznych, 15 18 czerwca 2004, Hajnowka [6] Bieńkowski K., Szypior J., Bucki B., Biernat A., Rogalski A.: Influence of Geometrical Parameters of Switched Reluctance Motor on Electromagnetic Torque. Procedings of XVI International Conference of Electrical Machines - Kraków, 5-8 września 2004, [7] Bieńkowski K., Bucki B.: Trójfazowe reluktancyjne silniki przełączalne. XXXIX Miedzynarodowe Sympozjum Maszyn Elektrycznych, Gdańsk-Jurata 9-11 czerwca 2003
Krzysztof BIEŃKOWSKI 3 Andrzej POCHANKE 4 Wpływ parametrów geometrycznych na charakterystyki statyczne momentu dwupasmowego silnika reluktancyjnego Streszczenie Silniki reluktancyjne komutowane elektronicznie znajdują coraz szersze zastosowania w napędach o regulowanej prędkości obrotowej oraz jako elementy wykonawcze w układach automatyki i robotyki. Charakterystyki statyczne momentu oraz sposób sterowania maja decydujący wpływ na właściwości eksploatacyjne napędu jako przetwornika energii. Charakterystyki statyczne momentu można kształtować przez dobór parametrów geometrycznych rdzeni stojana i wirnika. W referacie przedstawiono wpływ szerokości biegunów stojana i wirnika oraz szerokości szczeliny powietrznej na statyczny moment reluktancyjny silnika małej mocy. Obliczenia pola magnetycznego dokonano w przekroju maszyny prostopadłym do osi wału wirnika. Przy znanym rozkładzie pola w obszarze modelu silnika statyczny moment obrotowy można obliczyć przez całkowanie tensora naprężeń Maxwella wzdłuż szczeliny powietrznej lub w drodze obliczania pochodnej koenergii układu względem kąta obrotu wirnika. W przypadku dwuwymiarowego rozkładu pola magnetycznego wyznaczonego przy określonym przepływie θ zasilanego pasma i przy danym położeniu kątowym α wirnika względem stojana, moment można wyrazić zależnością (1). Obliczenia wpływu wybranych parametrów na charakterystyki kątowe momentu zostały wykonane dla silnika o strukturze geometrycznej przedstawionej na Fig.1. Wyznaczone zostały charakterystyki statyczne momentu dla trzech wartości szerokości bieguna stojana (Fig.3) i wirnika (Fig.4) oraz trzech wartości szczeliny powietrznej (Fig.5). Wyznaczono także moment średni dla dziewięciu wariantów konstrukcyjnych silnika (Fig.6). Przy stałym przepływie uzwojenia, wartości szerokości biegunów stojana i wirnika nie wpływają na wartość maksymalną momentu, lecz powodują zmianę kształtu charakterystyki kątowej momentu oraz zmianę wartości średniej momentu obrotowego. Wartość szczeliny powietrznej ma najistotniejszy wpływ na wartość momentu, ale jedynie w zakresie częściowego pokrywania się biegunów. W silnikach reluktancyjnych powinny być stosowane jak najmniejsze wartości szczeliny powietrznej, dopuszczalne ze względów mechanicznych. Odpowiedni dobór szerokości biegunów stojana i wirnika umożliwia zwiększenie średniego momentu o kilkanaście procent względem konstrukcji początkowej, przy niezmienionych innych wymiarach i stałej wartości prądu pasma fazowego. Prezentowane wyniki nie stanowią próby optymalizacji przełączalnego silnika reluktancyjnego. Wskazują jedynie na charakter zmian momentu przy zmianach parametrów geometrycznych rdzenia magnetycznego maszyny i mogą być przydatne przy optymalizacji metodą przesiewową. 3 dr inż., Instytut Maszyn Elektrycznych Politechniki Warszawskiej, Plac Politechniki 1, 00-661 Warszawa, tel. +(48-22)6607490, fax: +(48-22)8127535, e-mail: k.bienkowski@ime.pw.edu.pl 4 dr hab. inż., Instytut Maszyn Elektrycznych Politechniki Warszawskiej, Plac Politechniki 1, 00-661 Warszawa, tel. +(48-22)6607435, fax: +(48-22)8127535, e-mail: pochanke@ime.pw.edu.pl