Errect of Skew on Induction Motor

NDUCTION inotor designers must account for the saturation of the vari- ous components of magnetic reactances to make accurate predictions of the motor's performance. To predict the saturated value of these component re- actances is difficult because the superpo- sition principle can no longer be applied wxith accuracy. When saturation is pres- ent, to be rigorous, the net field must be studied not the component fields. To determine the net field under satu- rated conditions, it should first be dealt with as though there were no saturation. To find the net field with no saturation, it is valid to find the field of each com- ponent of reactance and superimpose them in the various areas of interest. In studying a particular motor, the author found it convenient to separate fluxes, having a portion of their path in the iron, into three components: the slot leakage flux, the air-gap space har- monic leakage flux, and the air-gap space fundamental flux. It is this latter flux with which this paper is concerned. Designers have long recognized that the space fundamental air-gap flux varies with axial position because of skewing the rotor slots with respect to the stator slots, but to the author's knowledge nothing has been published giving quan- titative data showing how much this flux may vary. It is important that designers have a knowledge of this effect because it not only affects the degree of saturation but also affects iron losses and noise. Nomenclature 2a =magnitude of major axis of ellipse 2b =tmagnitude of minor axis of ellipse B =magnetic induction I1 = stator current I,=rotor current Paper 55-270, recommended by the AIEE Rotating Machinery Committee and approved by the AIEE Committee on Technical Operations for presenta- tion at the AIEE Middle Eastern District Meeting, Columbus, Ohio, May 4-6, 1955. Manuscript submitted January 31, 1955; made available for printing March 16, 1955. C. E. LINsKOUS is with the General Electric Com- pany, Fort Wayne, Ind. The author wishes to express his gratitude to M. L. Schmidt and F. W. Suhr, General Electric Com- pany, for the many helpful ideas and suggestions contributed to this study. I,[=magnetizing current L =length of stack N =-amplitude of backward xvave of mag- netic induction P= amplitude of forward ixave of magnetic induction R2=-forwxard rotor resistance R2=backward rotor resistance s - slip t=time x =axial distance of a plane from the end of the stack which has the leading end of the rotor bars X2= rotor leakage reactance Xm= magnetizing reactance a= the skexv, electrical degrees 6=electrical angle away from center of the reference winding pole in the direc- tion of rotation N =electrical angle axvay from major axis of ellipse in the direction of rotation f= electrical angle by which the forward magnetizing current, referred to the reference windings, leads the voltage applied to the reference winding 7b=electrical angle by which the backward magnetizing current, referred to the reference wxinding, leads the applied voltage -=electrical angle by which the major axis of the ellipse is located from the reference winding. Measure in the direction of rotation c =angular velocity of applied voltage NL=no load FL= full load -lI,T =maximum torque SS= standstill or locked rotor axial plane = a plane perpendicular to the axis or, in other words, a plane paral- lel to the plane of the laminations reference mmf =the peak value of a space sine wave of magnetomotive force (mmf) which would produce the flux required to balance the line voltage for a balanced polyphase motor with no skexv or, in other words, the mmf that would produce the design flux density maximum mmf =the actual maximum mmf divided by the reference mmf de- fined in the foregoing VARIATION OF NET ROTATING FIELD IN UNBALANCED POLYPHASE AND SINGLE- PHASE MOTORS AT A PARTICULAR AXIAL LOCATION Techniques have been established for many years for resolving an unbalanced polyphase or a single-phase motor into forward and backward rotating sinusoidal fields of constant amplitude and angular velocity.'-6 Let it be assumed that the fundamental flux density waves at a particular axial location have been found by these techniques-to be Bf=P cos (cot-0+rf) Bb = N cos (cot +9+ TO) (1) (2) wxhere Bf=magnetic induction of the forward field at the angular location 0 Bb = magnetic induction of the backward field at the angular location 0 P = peak value of the forward magnetic induction wave, positive sequence N -=peak value of the backward magnetic induction wave, negative sequence c =electrical angular velocity t =time 0=angular location of a point on the stator measured from the center of a refer- ence wxinding pole, positive in the direction of rotation f= a phase angle for the forward wave for a voltage applied to the reference winding of Vm cos ct -rb=a phase angle for the backward wave for a voltage applied to the reference winding of V,m cos cwt These waves are represented by rotat- ing vectors in Fig. 1. At any instant, these two vectors may be added to get a resultant vector representing the net field. The projection of the resultant vector along the line labeled major axis in Fig. 1 is (P+-N) cos (27r-ct_f±+b)2 The projection of the resultant vector along the line labeled minor axis in Fig. 1 is (P-N) sin (27rf-wt_r)+Tb2 These are the same form as the well- known expression for an ellipse in para- metric form and, therefore, the tip of the resultant vector traces out an ellip- tical locus. The expression for the an- gular velocity of the resultant vector is MAJOR AXIS\ -REFERENCE MAIN POLE Fig. 1. Vectors representing the forward and backward rotating fields Linkous-Effect of Ske-w on Induction Motor Magnetic Fields Errect of Skew on Induction Motor Magnetic Fields C. E. LINKOUS ASSOCIATE MEMBER AIEE 760 AUGUST 1955 Authorized licensed use limited to: Zhejiang University. Downloaded on July 31,2010 at 12:55:40 UTC from IEEE Xplore. Restrictions apply. (F + B) 'i- DENSITY AT INSTANT OF 0p Fig. 3. Normalized curves for determining the maximum magnetic induction 0 I (B) Fig. 2. Locus of rotating vector representing the resultant field very interesting. It is developed in Appendix II. The locus is shown in Fig. 2(A). The angle of tilt of the major axis of the ellipse away from the reference pole is TfTrb (3) 2 Thus the ellipse shown in Fig. 2(A) gives the amplitude of the wave of mag- netic induction as the peak of the wave crosses any particular angular position. However, this is not in general the maxi- mum magnetic induction which that loca- tion experiences. This can be seen from Fig. 2(B). The plot of a sine function in polar co-ordinates is a circle. The construc- tion of Fig. 2(B) shows that the mag- netic induction at the angle \\ the angular posil of its occurrence the load is vai from no load standstill \WITlKI,.L'AD)~KCD Fig. 15 (rig Locus of resul vector A. Substituting this value of yp i equation 8 w ill then give the expression Bmax Taking the partial derivative of equat 8 wi-ith respect to yp and equating it to z gives /b Tp=tan - tan X a This is the value of j p which makes J maximum at the angle X. Substitut equation 9 into equation 8 gives Brnax a (b)(b) 4 cos2 [tan 1 tan x)] co,s [-tan-' (- tan ( a2 Substituting into equation 10 that a =P+-N (9) 6=tan- X (P sin (at+-rf) -N sin (at+rb) (15) B a VP cos (at +Art) +AN cos (ct A+Tb)J :ing Differentiating this with respect to time gives (p2 IN2 ( a=P2A+N2+2PNcos(2wt+Trf±+rb) (w) (16) At the time 1 aot =-(w --rf- rb)2 (11) the vector has its maximum angular velocity of b =P-N X = 0- = 0 Tf-Tb 2 (13) gives Brnax p2-2PXY l -2 '7 'cos' tan - L<+- )X tan (0-Tff T)]b X C T f Tb -X cs 2 Substituting equation 17 into equation 15 showxs that 7T 7f- 7~b0= + 2 2 when the angular velocity is maximum. Note that this is the angle at which the amplitude of the resultant wvave is mini- mum. Similarly, at the time ct= - (rf+rb)2 P -N \2 .f \b(6) [tan(> (14) LP N 2 (7) the angular velocity has its minimum value of The equation for the magnetic induction at the instant the peak of the sine axve passes point P is B = -.,a2 cos2 ypA+b2 sin2 yPX cos [X-tan-I(- tan yP)] (8) The problem is to find the value of yp which will make B a maximum at the angle Appendix II. Development of the Angular Velocity of the Vector Representing the Resultant Field It follows from Fig. 1 that the location of the resultant vector at any time t is rin =(PN) (a) (21) Substituting equation 21 into equation 15 shows that Tf-Tb 2 (22) when the angular velocity is minimum. Note that this is the angle at which the Linkous-Effect of Skew on Induction Motor Magnetic Fields (17) (18) y =a cos -y x= -b sin -y (19) (20) (12) omax .- P+N (w) P-N 764 AUGUST 1955 Authorized licensed use limited to: Zhejiang University. Downloaded on July 31,2010 at 12:55:40 UTC from IEEE Xplore. Restrictions apply. amplitude of the resultant wave is maxi- mum. References 1. THE REVOLVING FIELD THEORY OF THE CAPACITOR MOTOR, Wayne J. Morrill. AIEE Discussion P. L. Alger (General Electric Company, Schenectady, N. Y.): Designers have long realized that spiralling, or skewing, the slots of an induction motor increases the react- ance and affects the performance in many w-as, besides reducing the magnetic noise and locking tendency. Despite its import- ance, no one until now has made any thor- ough analysis of this subject. This paper is therefore of great interest and importance to all motor designers. Mr. Linkous has brought out very