CHAP. VII] M.M.F. OF DISTRIBUTED WINDINGS 127 



wave at the same point. Or else, represent the two gliding waves 

 by two vectors of equal magnitude Af , revolving in opposite direc- 

 tions. The resultant vector is a pulsating one in a constant direc- 

 tion, and varies harmonically between the values 2Af . 



The analytical proof is as follows: Let the exciting current 

 reach its maximum at the moment t=0. Then, if the amplitude of 

 the m.m.f. wave at this instant is equal to A, the amplitude at any 

 other instant t is equal to A cos 2nft. Therefore, the m.m.f. cor- 

 responding to a point distant x from P and at a time t is equal to 

 A cos 2xft sin x. By a familiar trigonometrical transformation 

 we have 



A sin x cos 2nft =\A sin (x+ Zrjt) +$A sin (x -2-/0. (62) 



The right-hand side of this equation represents two sine waves, of 

 the amplitude $A, gliding synchronously along the air-gap, that is, 

 covering one pole pitch during each alternation of the current. 

 The wave %A sin (x+2nft) glides to the left, because, with increas- 

 ing t, the value of x must be reduced in order to get the same phase 

 of the m.m.f. wave, that is, to keep the value of (x+2xft) constant . 

 The other wave glides to the right, because, with increasing /, the 

 value of x must be increased in order to obtain any constant value 

 of (x2xft). A similar resolution into two gliding waves can be 

 made for each higher harmonic of the pulsating m.m.f. wave; the 

 higher the order of a harmonic the lower the linear speed of its two 

 gliding wave components. 



In practice it is usually required to know the relationship 

 between the effective value i of the ma.uneti/in^ current, the num- 

 ber of turns n per pole per phase, and the eivst value of one of the 

 gliding m.m.f. waves. From the preceding explanation this rela- 

 tionship for the fundamental wave is 



(63) 



where M is the amplitude of each of t he two gliding m.in.fs.. ni\''2 

 represents the maximum height // of the original ivetan^le. and the 

 factor i IH introduced because the amplitude of each gliding \\ave 

 -half of that of t he corresponding pulsat ingwavo. The breadth 

 factor kb is the same a 1 1 iat used for the induced e.n .27 to 



29). Similar r\pre>> inns can 1 >< \\ritten foi her harmonic. 



remembering that their amplitudes decrease according to eq 



