128 THE MAGNETIC CIRCUIT [ART. 43 



and that a different value of && must be used for each harmonic. 

 The value of M is calculated so as to produce the required revolv- 

 ing flux, as is explained in Chapters IV, V, and VI. From eq. (63) 

 either n or i, or their product can be determined. 



Prob. 1. A single-phase four-pole induction motor has 24 stator slots, 

 two-thirds of which are occupied by the winding; there are 18 con- 

 ductors per slot. The average reluctance of the active layer is 0.09 rel. 

 per square centimeter. What current is necessary to produce a pulsating 

 flux of such a value that the maximum flux density due to the first 

 harmonic is 5 kl./sq.cm., when the secondary circuit is open? 



Ans. 8.3 amp. 



Prob. 2. Show that in the preceding problem the difference between 

 the actual flux per pole and its fundamental is less than 2 per cent. 



Prob. 3. Show that, if in Fig. 34 the angle x is counted from the 

 crest of the first harmonic, the expansion into the Fourier series is similar 

 to eq. (61), except that cosines take place of the sines, and the terms 

 are alternately positive and negative. 



43. The M.M.F. of Polyphase Windings. Consider a two- 

 phase winding of the stator of an induction motor (Fig. 35a) ; let 



2 Armature f 



[661 [fi~ 1631 l*j 



-$-*- ^ Slots ''i ~L- 

 FIG. 35a. A two-phase winding. 



the current in phase 1 lead that in phase 2 by J7 7 , or by 90 electrical 

 degrees. A little reflection will show that the resultant m.m.f. of 

 the two phases glides from right to left : Let the current in phase 1 

 reach its maximum at the instant =0; at this instant the current 

 in the coil 2 is zero, and the m.m.f. wave is distributed uniformly 

 under the coil 1 ; at the instant t= \T the current in phase 1 is zero, 

 and the m.m.f. is distributed under the coil 2. At intermediate 

 instants both coils contribute to the resultant m.m.f., so that its 

 maximum occupies a position intermediate between the centers 

 0\ and 2 of the coils 1 and 2. 



The actual rectangular distribution of the m.m.f. due to each 

 phase can be replaced by a fundamental sinusoidal one and its 

 higher harmonics, as in Fig. 34. The pulsating fundamental m.m.f. 

 of each phase can be replaced by two waves of half the ampli- 

 tude, gliding synchronously in opposite directions. Let the wave 



