GENERAL THEORY OF THE INDUCTION MOTOR. 285 



force required to overcome the reluctance of the magnetic circuit. This reluctance is 

 nearly all in the air gap, and the reluctance of the iron part of the magnetic circuit 

 may be neglected. Therefore the total magnetomotive force due to the wattless com- 

 ponent of the magnetizing currents in all of the stator phases is approximately equal 

 to the magnetomotive force required to force the flux 4> twice across the air gap, and 

 this is equal to twice the length g of the gap space multiplied by the flux density under 

 a stator tooth in the middle of a polar region, that is to 2^X 7r$/(2A/) X (t-\- s) //. 

 For the case of a two-phase stator winding let us consider the instant when the 

 current M w is at its maximum value ^2M W in one winding and at zero value in the 

 other winding. This actual current ^iM*, inflowing through the Z/(2/) stator 

 conductors in one band supplies the magnetomotive force 2g X TT*/ (2/7) X (^ ~h J ) /* 

 so that 



10 2p 



from which 



in which M w is the effective value of the wattless component of the magnetizing cur- 

 rent in each phase of a two-phase stator winding, Z is the total number of stator con- 

 ductors, / is the number of stator polar regions, g is the radial length of the air 

 gap, $ is the total flux emanating from a polar region of the stator, /I is the length 

 of the stator and rotor iron parallel to the motor shaft, / is the distance between adja- 

 cent polar regions {N and S) of the stator, / is the breadth of a stator tooth, and 

 s is the breadth of a stator slot. This assumes that the rotor slots are nearly closed 

 so that the effective area of the gap space is at all times sensibly equal to the area of 

 the ends of the stator teeth. 



For the case of a three-phase stator winding let us consider the instant when the 

 magnetizing current M w is at its maximum value ^2 M w in one phase and at half- 

 maximum M w l V2 in the other two phases. At this instant the three bands of stator 

 conductors, each having Z/(3/) conductors, work together to supply the magneto- 

 motive force 2g X K$l (2A/) X (t + s ) !* The ampere-turns in the respective bands 

 are M w j ^2 X ZJ(ZP), V^ X ^/(3/) and Af w / ^2 X ^/(3/) so that the 

 aggregate is 2 ^M^Z/ ($p). Therefore we have 



( m) 



from which the value of M w may be calculated. 



Calculation of magnetic leakage reactance per phase. The magnetic leakage 

 of an induction motor, that is, the passage of magnetic flux between the stator and 



* The effect of a concentrated stator winding, or of a stator winding distributed in 

 bands of prescribed width, in maintaining a harmonically distributed flux should in 

 strictness be formulated in a manner involving a "phase constant" of distribution of 

 the winding. The correct theory is very much like the theory of electromotive force 

 which is developed in Art. 61. It would be of no practical use, however, to develop 

 the refined theory of magnetizing action. Equation (ii) gives an over-estimate of M w , 

 and equation (iii) gives an under- estimate of M w . 



