228 ELECTRICAL ENGINEERING 



138. Electromotive Force Equation. Fig. 190 represents a 

 two-pole, single-phase alternator. The armature winding is a 

 single coil of n turns revolving at a constant speed of N r.p.s. 

 The magnetic field is assumed to be uniform. 



The e.m.f. generated in the winding goes through one com- 

 plete cycle during each revolution, and thus the frequency in 

 cycles per second is equal to the speed in revolutions per second 

 orf = N. 



The angular velocity of the coil is co radians per second, and 

 therefore 



2vf. . . . . . . (226) 



If $ is the maximum flux inclosed by the coil, that is, the flux 

 inclosed when the coil is vertical, as shown, and time is measured 

 from this instant, then at time t, after the coil has turned through 

 an angle 6, the flux inclosed is < = $ cos 6, and the e.m.f. gener- 

 ated in the coil is 



at 

 = -n < jr(3>cos 0) 10- 8 , 



but 6 = ut = 2 irft, and thus 



e = - n-r (< cos 2irfi) 10~ 8 



= 2 7r/Vi<J> 10~ 8 sin 2 irft volts . . (227) 

 = E sin 6, 



This is a sine wave of maximum value 



#0 = 27r/n$10- 8 vo).ts ..... (228) 

 and effective value 



E = -L = 4.44 fn$ 10- 8 volts. . . . (229) 



This is the electromotive force equation for an alternator 

 which produces a sine wave of e.m.f. and has a concentrated 

 winding, that is, all the turns wound in a single coil. 



This result may also be obtained as follows. The flux cut per 

 second by each turn of the coil is 4/$ lines, and therefore the 

 average e.m.f. generated in the coil is 



E avg = 4/n$10- 8 vol'ts, .... (230) 



