INFLUENCE OF A NUMBER OF ROTOR SLOTS ON ELECTROMAGNETIC TORQUE OF THE SINGLE-PHASE SHADED POLE INDUCTION MOTOR

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1 Krzysztof MAKOWSKI INFLUENCE OF A NUMBER OF ROTOR SLOTS ON ELECTROMAGNETIC TORQUE OF THE SINGLE-PHASE SHADED POLE INDUCTION MOTOR ABSTRACT The simulation study deals with models of the shaded pole motor of different numbers of rotor slots but the same stator structure. Steady-state currents of motor windings computed by circuit harmonic model were used as input currents in 2D FE models. Flux density distribution in the air-gap and torque/speed characteristics of the tested models were presented. The simulations have shown the influence of the number of rotor slots on the starting, pull-out and rated torque of the motor. Keywords: 1-phase induction motor, electromagnetic torque, FEM, electromagnetic field 1. INTRODUCTION For steady-state simulation of single-phase shaded pole induction motors circuit-fe combined methods more often are used. Application of the FEM make possible to calculate magnetic field in the whole of magnetic circuit of the Krzysztof MAKOWSKI, D.Sc., Ph.D. krzysztof.makowski@pwr.wroc.pl Institute of Electrical Machines, Drives and Measurements, Smoluchowskiego 19, Wroclaw, POLAND PROCEEDINGS OF ELECTROTECHNICAL INSTITUTE, Issue 24, 29

2 96 K. Makowski motor. The FE models with movable grids in the air-gap enable modeling movement of the rotor with respect to the stator when the motor operation is considered. Such models usually require considerable computational time on account of reconstruction of the FE grid and renumbering of nodes at each position of the rotor. Modeling the fixed (stationary) FE grid in the whole crosssection area of the motor simplifies the problem but it permits to reduce the time of computation. The results of study of the influence of rotor slots on flux density distribution in the air-gap presented in [1] showed that the average value of the normal magnetic flux density subject to change in the range of (.5-1)% which have a little effect on simulation of steady-state operation of the motor. In the paper which deals with investigation of the influence of a number of rotor slots on electromagnetic torque of the shaded pole induction motor the harmonic circuit model together with FE models were used for the simulations. On the basis of the original model of the tested motor of 15 rotor slots, several circuit-field models were built up with different numbers of the rotor slots, i.e. of less and more rotor slots than the nominal one and the same stator structure. 2. DESCRIPTION OF THE MOTOR For study of the problem, the single-phase shaded pole induction motor of ratings listed in Table 1 was used, which cross-section diagram is shown in Fig. 1. Applying the d-q reference harmonic model described in details in the paper [2] the currents of the stator windings and rotors bars are determined for a given rotor speed, which are treated as input currents in the 2D FE model. For calculation of steady-state magnetic field in the cross-section of the motor with taking into account magnetization characteristics of magnetic core of the motor, a quasi-stationary magnetic field equation (the Poisson equation) with complex variables was applied which is formulated in the paper [3]. The procedure of the calculation includes: computation of stator and rotor winding currents for given rotational speeds using the d-q harmonic model determination of reluctivity of the magnetic circuit using B-H characteristic of iron and FEM evaluation of torque versus rotational speed characteristics of the motor for different numbers of rotor slots

3 Influence of a number of rotor slots on electromagnetic torque of the single-phase Fig. 1. Cross-section diagram of the tested motor TABLE 1 Ratings and data of the tested motor Rated power Rated voltage Rated current Rated speed Frequency 1.6 W 22 V.16 A 24 rpm 5 Hz Over-torque factor 1.3 Number of poles 2 Type of main stator winding Type of auxiliary stator winding Type of rotor winding 2 coils /178 turns 1 coil per pole squirrel cage /15 bars Material of laminated core M 6-5A

4 98 K. Makowski The voltage equations of the d-q harmonic model for steady-state operation may be written in the complex form as follows: where: U U M A = [ ] I I I I I I M A rd rq 1 (1) 1 rdν rqν [ ] = M AM r1dm r1qm rνdm rνqm A r1da r1qa rνda rνqa MA - kx r1d Mr1d Ar1d r1 kx Ar1q r1 r1q Mr1q Mrνd Arνd rνd νkx rν νkx Mrνq rν rνq Arνq In the above equation M, A represent self impedances of the main (M) and auxiliary (A) stator windings, X Mrν, X Arν are mutual reactances between the main, auxiliary stator windings and the rotor cage for the ν - space harmonic, I M, I A are currents of the main and auxiliary stator windings, I rdν, I rqν are rotor currents in d and q axis for the ν - harmonic, U M, U A are voltages of the main and auxiliary stator windings, p is the number of pole pairs, s the slip of the rotor and k = (1-s)/p. Magnitudes of the rotor bar currents may be determined by the relation: I rk = { I rdν cos[ ϕ ν + ( k 1) να] + I rqν sin [ ϕν + (k 1) να]} (2) ν where I rk denotes the total current in the k -th rotor bar, I rdν, I rqν stand for the ν harmonic components of the rotor currents in the d-q reference, α is the

5 Influence of a number of rotor slots on electromagnetic torque of the single-phase angle between two neighboring rotor bars and harmonic with respect to the reference bar. ϕ ν is the angle of the ν space Magnetic field computation was curried out for the assumption, that: 2D magnetic field is uniform along axial length of the motor reluctivity of iron only depends on magnitude of flux density eddy currents are neglected temperature effect on conductivity and permeability of materials is neglected A quasi-stationary electromagnetic field in the cross-section of the motor is described by the equation with complex variables [4]: 1 A 1 A ( ) + ( ) = J (3) x μ x y μ y jωt jωt where A = Az( x, y ) e, J = J z ( x, y) e represent the z-component of the magnetic vector potential and the total surface current density of windings directed along length of the rotor, μ is permeability of materials, and ω is the pulsation of the electromagnetic field. 3. SIMULATION RESULTS Applying the circuit-fe combined method, steady-state operation of the original shaded pole induction motor of 15 rotor slots and modified models of less and more numbers of the rotor slots than the original one were modeled at the same geometrical structure of the stator and dimensions of the rotor slots. The obtained magnetic field distributions in cross-section of the motor for the rated speed are represented by normal component of the flux density in the airgap and its harmonic space resolution, which are shown in Figs The numerical analysis of the magnetic field in the air-gap have shown that increase of the number of the rotor slots over the nominal value (i.e. more than the original motor) produces decrease of the 3 rd harmonic amplitude of the flux density and at the same time increasing electromagnetic torque at rated speed. It is also observed that increasing the 3 rd and 5 th harmonic amplitudes for the 13 rotor slots bring about a reduction of the rated torque with insignificant growth of the starting torque of the motor.

6 1 K. Makowski,8,6,4,2 Bn [ T ] -, ,4 -,6 -,8 L [mm] 13 rotor slots 14 rotor slots 15 rotor slots 16 rotor slots 17 rotor slots Fig. 2. Normal component of the magnetic flux density in the air-gap for different number of rotor slots at rated speed Amplitudes of flux density [ T ],7,6,5,4,3,2, Order of space harmonic 13 slots 14 slots 15 slots 16 slots 17 slots Fig. 3. Harmonic amplitudes of flux density in the air-gap for different number of rotor slots at rated speed

7 Influence of a number of rotor slots on electromagnetic torque of the single-phase Electromagnetic torque Te [mnm] ,2,4,6,8 Relative rotational speed n/n s [-] 15 slots 13 slots 17 slots Fig. 4. Torque versus rotational speed characteristics of the motor for different number of rotor slots TABLE 2 Values of starting, pull-out and rated torques of the tested models Number of rotor slots T s [mnm] T m [mnm] T n [mnm] LITERATURE 1. Makowski K.: Determination of magnetic field changes in single-phase shaded pole induction motors due to slotting of the rotor, Sixth International Conference on Computational Electromagnetics, CEM 26, 4-6 April, Aachen, Germany, pp Makowski K., Schoepp K.: An analysis of shaded pole induction motors with increased starting torque, Electric Machines and Power Systems, vol.8, no.6, pp , Hemisphere Publishing Corporation, 1983.

8 12 K. Makowski 3. Makowski K.: Determination of performance characteristics of a single-phase shaded pole induction motor by circuit-field method, Electrical Engineering, Springer-Verlag, 22, vol. 84, no. 5, pp Lowther D.A., Silvester P.P.: Computer aided design in magnetics, Springer-Verlag, Berlin Heidelberg New York Tokyo, Manuscript submitted Reviewed by Andrzej Pochanke WPŁYW LICBY ŻŁOBKÓW WIRNIKA NA MOMENT ELEKTROMAGNETYCNY JEDNOFAOWEGO SILNIKA INDUKCYJNEGO POMOCNICYM UWOJENIEM WARTYM K. MAKOWSKI STRESCENIE W obliczeniach symulacyjnych stanów ustalonych silników indukcyjnych z pomocniczym uzwojeniem zwartym wykorzystuje się coraz częściej metodę elementów skończonych w połączeniu z metodą obwodową. astosowanie MES pozwala na obliczenie pola magnetycznego w całym obszarze obwodu magnetycznego silnika. Modele z ruchomą dyskretną siecią elementów skończonych w szczelinie powietrznej pozwalają na modelowanie ruchu wirnika względem stojana, co jednak wiąże się ze znacznie wydłużonym czasem obliczeń charakterystyk silnika. astosowanie stałej (nieruchomej) sieci elementów skończonych dyskretyzującej cały obszar przekroju poprzecznego silnika upraszcza zagadnienie, jednocześnie jednak pozwala na znaczne skrócenie czasu obliczeń symulacyjnych. Wykonane badania [1] dotyczące wpływu położenia żłobków wirnika względem stojana w stanie ustalonym pracy silnika na rozkład pola magnetycznego w szczelinie powietrznej wykazały zmiany wartości średniej indukcji w szczelinie silnika w granicach (.5-1)%. W przedstawianej pracy dotyczącej badania wpływu żłobków wirnika na moment elektromagnetyczny jednofazowego sinika indukcyjnego z pomocniczym uzwojeniem wykorzystano harmoniczny model obwodowy w połączeniu z dwuwymiarową quasi-stacjonarną analizą rozkładu pola magnetycznego w silniku. Bazując na modelu wyjściowym silnika zawierającym 15 żłobków w wirniku zbudowano modele obwodowo-polowe silników w zakresie od 13. do 17. żłobków wirnika, przy zachowaniu pozostałej struktury budowy silnika. Otrzymane w wyniku symulacji rozkłady przestrzenne pola magnetycznego w szcze-

9 Influence of a number of rotor slots on electromagnetic torque of the single-phase linie powietrznej z analizą harmoniczną oraz charakterystyki mechaniczne badanych modeli przedstawiono na rysunkach 1-3. Numeryczna analiza rozkładu pola magnetycznego wykazała, że wzrost liczby żłobków wirnika ponad wartość nominalną powoduje zmniejszenie amplitudy 3. harmonicznej indukcji w szczelinie oraz wzrost momentu elektromagnetycznego przy prędkości znamionowej. Jednocześnie zaobserwowano, że wzrost 3. i 5. harmonicznej pola w szczelinie powietrznej przy 13. żłobkach wirnika powoduje spadek momentu znamionowego silnika przy nieznacznym wzroście momentu rozruchowego.