CHAP. VII] M.M.F. OF DISTRIBUTED WINDINGS 137 



produce cut the secondary conductors at comparatively high rela- 

 tive speeds ; thus, secondary currents are induced which wipe out 

 these harmonics to a considerable degree. There are practical cases, 

 however, in which some one particular harmonic becomes of some 

 importance, and affects the operation of the machine, particularly 

 at starting. For this reason the following general outline of the 

 properties of the higher harmonics in the m.m.f. is given. 1 



In a single-phase machine (Fig. 34) all the higher harmonics of 

 the m.m.f. are pulsating at the same frequency as the fundamental 

 wave, but the width of the nth harmonic is only I/nth of that of 

 the fundamental wave. Each pulsating harmonic can be replaced 

 by two gliding harmonics of half the amplitude, one left-going, the 

 other right-going. The linear velocity of these gliding m.m.fs. is 

 only I/nth of that of the fundamental gliding waves, because they 

 cover in the time %T a distance equal only to their own base, PQ/n 

 (180 electrical degrees) . With one slot per pole, the amplitudes of 

 the higher harmonics decrease according to eq. (61), but with more 

 than one slot, or with a fractional-pitch winding they decrease 

 more rapidly, because different values of k b must be taken for each 

 harmonic (see Art. 30 above). 



In a two-phase machine, consider (Fig. 35a) the gliding waves 

 L n and R n , of the nth harmonic. For this harmonic, the distance 

 between 0\ and 0% is equal to \nn electrical degrees. At the 

 instant t=0 the crest of the wave L nl is at the point Oi; at the 

 instant t= \T the crest of the wave L n2 is at the point 2 . There- 

 fore, the two waves travel at a relative distance of $i:(n 1) < K < - 

 trical degrees, considering the base of the wth harmonic as equal 

 to its own 180 electrical degrees. In a similar manner, i lie distance 

 between the crests of the two right-going waves is found to be 

 equal to i7r(n + 1) electrical degrees. We thus obtain the following 

 table of the angular distances between the waves due to the t\\o 

 phases: 



Order of the harmonic 1 3 5 7 9 11 13 



Distance between the two Ln waves it In 3;r 4* 5* 6* 

 Distance between the two Rn waves * 2* SB 4* 6* 6* 7 - 



The waves which travel at a distance 0, 2r, 4^, etc., are simply 

 added together, while those at a distance -, 3;r, 5rr, etc., cancel each 



For a more detailed treatment sec Arn<>M. Wcchsdrtromtochnik, Vol. 

 3 (1904), Chapter 13, and Vol. 5, part I (1909), Chapt* . 



