CHAP. VII] M.M.F. OF DISTRIBUTED WINDINGS 131 



Prob. 7. Plot the actual " stepped " curves of m.m.f. distribution 

 for a two-phase winding with three slots per pole per phase, for the fol- 

 lowing instants: t = Q, t = ^T, and t = \T. Compare the maximum and 

 the average m.m.f. of the actual distribution with those of the first 

 harmonic. 1 



Prob. 8. Solve the preceding problem for a three-phase winding 



with 2 slots per pole per phase, and a winding pitch of J. Take two 



instants, t=0 and t=^T, and show that for the instants t-fgT^ 



he m.m.f. distribution is the same as for < = 0, while for ^f 



etc., the m.m.f. distribution is the same as for t = -faT. 



Prob. 9. Prove directly that two equal pulsating sine waves of m.m S. 

 or flux, displaced by 90 electrical degrees in space and in time relatively 

 to each other, give a gliding sine wave, the amplitude of which is equal 

 to that of each pulsating wave. Solution: The left-hand side of eq. 

 (62) gives the value of the m.m.f. at a point x and at an instant t, due 

 to phase 1 ; the m.m.f. produced at the same point and at the same instant 

 by the phase 2 is A sin (x + fa) cos (2nft- fa). Adding the two expressions 

 gives A sin (x +2xft), which is a left-going wave of amplitude A. 



Prob. 10. Prove, as in the preceding problem, that the three pulsating, 

 sine waves of m.m.f. produced by a three-phase winding, give together 

 a gliding m.m.f., the amplitude of which is 50 per cent larger than that 

 of each pulsating wave. 



Prob. 11. Prove by the method given in problem 9 above that m 

 pulsating m.m.f. waves displaced in space and in time by an electrical 

 angle 2^/m produce a gliding m.m.f. the amplitude of which is $m times 

 larger than that of each pulsating wave. See Arnold, W echselstromr 

 technik, Vol. 3 (1908) p. 302. 



44. The M.M.Fs. in a Loaded Induction Machine. 2 Eq. (64) 

 gives the magnetizing current i of an induction motor at no-load, 

 i.e., when the rotor is running at practically synchronous speed, so 

 that the secondary currents are negligible. When the motor is 

 loaded, the useful flux which crosses the air-gap is due to the com- 

 bined action of the primary and the secondary currents. In com- 

 mercial motors the flux at full load is but a few per cent below that 

 at no load, the difference being due to the impedance drop in the 



1 Problems 7 and 8 are intended to acquaint the student with the usual 

 method of calculation of the m.m.fs. of distributed windings and to show the 

 advantage of Blondcl's method used in the text. For numerous s 

 curves and calculations, see Boy de la Tour, The Induction Motor, f'haptrr I V. 



'The treatment in this article presupposes a general knowledge of the 

 equivalent performance diagram of induction machines; the purpose of 

 the article being to deduce the exact numerical relations. This article and 

 the one following can be omitted without impairing the continuity of trcat- 

 im Tit in the rest of the text. 



