CHAP. VII] M.M.F. OF DISTRIBUTED WINDINGS 125 



x = TT. All the terms on the right-hand side vanish, except the one 

 containing sin 2 nx, and we have 



from which 



A n =4/i/(n*) ........ (60) 



Thus, the required series is 



h = 4h/x (sin a; + J sin 3z -f J sin 5z+etc.) . . (61) 



This means that the amplitude of the fundamental wave is 4/x 

 times larger than the height h of the original rectangle; the ampli- 

 tude of the third harmonic is equal to one-third of that of the fun- 

 damental wave ; the amplitude of the fifth harmonic is one-fifth of 

 that of the fundamental wave, etc. In practical applications the 

 fundamental wave is usually all we desire to follow, but in some 

 special cases a few of the harmonics are important. 1 



Let now the winding of a phase be distributed in S slots per 

 pole (Figs. 15 and 16), the distance between the adjacent slots 

 being a electrical degrees. The conductors in every pair of slots 

 distant by a pole pitch produce a rectangular distribution of the 

 m.m.f . like the one shown in Fig.34, or, what is the same, an equiv- 

 alent series of sine-wave distributions. The m.m.fs. produc <! 

 by the different coils are superimposed, and, since a sum of sine 

 waves having equal bases is also a sine wave, the resultant m.in.f. 

 also consists of a fundamental sine wave and of higher harmonics. 

 Tin* fundamental waves of the m.m.fs. of the several coils are dis- 

 placed by an angle of a electrical degrees with respect to one 

 another, so that the amplitude of the resultant wave is not quite 

 S times larger than that of each component wave. The reduction 

 coefficient, or the slot factor, k $ , is the same as that for the imlu<T<l 

 e.m.f. (Art. 28), because in both cases we have an addition of sine 

 waves displaced by a electrical degrees, (see also prob. 20 ii 



1 This method of treating the m.m.fn. of distributed windings by reaolv- 

 iiit' the rectangular curve into ita higher harmonica ia due to A. Blondcl. 

 See his articl. entitled " Quelquea proprietea generates dea champs magnd- 

 tiquea tournanta," L'Ectairage Elcctrique, Vol. 4 (1895), p. 248. Some nut hors 

 consider the actual " stepped " curves of the in m.f. or flux di*tril>u 

 procedure rath.-r rurnberaome, and in the end teas accurate, in view of the 

 fart that the higher harmonics are to a considerable extent wiped out by the 

 currents in the rotor. 



