124 THE MAGNETIC CIRCUIT [ART. 42 



sinusoidal distribution. As long as the coil PQ acts alone, the 

 m.m.f. has a " rectangular " distribution in space, and, if the cur- 

 rent in the coil varies with the time according to the sine law, the 

 height of the rectangle, or the m.m.f. across the active layer, also 

 varies according to the sine law. In what follows it is important 

 to distinguish between variations of the m.m.f s. in space, i.e., along 

 t he air-gap, and those occurring in time, as the current in a winding 

 varies. 



For the purposes of analysis the rectangular distribution of the 

 m.m.f. can be replaced by an infinite number of sinusoidal distribu- 

 tions (Fig. 34), according to Fourier's series. 1 The advantages of 

 such a development over the orginal rectangle PP'Q'Q are as fol- 

 lows: 



(a) The sine wave is a familiar standard by which all other 

 shapes of periodic curves are judged. 



(b) When adding the m.m.fs. due to the coils in different slots, 

 or belonging to different phases, it is much more convenient to add 

 sine waves than to add rectangles displaced in space and varying 

 with the time. 



(c) In the actual operation of an induction motor or generator 

 the higher harmonics in the m.m.f. wave are to a considerable 

 extent wiped out by the corresponding currents in the rotor, so that 

 the rectangular distribution is actually changed to a nearly sinu- 

 soidal one (see Art. 45 below). 



Let h be the height of the rectangle; we assume that for all the 

 points along the air-gap the sum of the ordinates of all the sine 

 waves is equal to h\ or 



h = Ai sin x + Az sin 3x + A 5 sm 5z+etc. . . (59) 



Here x is the angle in electrical degrees, counted along the air-gap, 

 and A\, A s , A 5 , . . . are the amplitudes of the waves, to be deter- 

 mined as functions of h. No cosine harmonies enter into this for- 

 mula, because the m.m.f. distribution is symmetrical with respect 

 to the center line 00' of the exciting coil. To determine the ampli- 

 tude of the nth harmonic A n , multiply both sides of eq. (59) by 

 sin nx d(nx), and integrate both sides between the limits x=0 and 



1 For the general method of expanding a periodic function into a series of 

 sines and cosines, see the author's Experimental Electrical Engineering, Vol. 2, 

 pp. 222 to 227. 



