CHAP. IV] INDUCED E.M.F. 73 



the pole-pieces must be worked out very careiully in order to have 

 an e.m.f . approaching the true sine wave. With a larger number 

 of slots this is not so necessary because the. em.fs. induced in differ- 

 ent slots are added out of phase with each other, and the undesir- 

 able higher harmonics partly cancel each other. The voltage 

 wave is further improved by a judicious use of a fractional-pitch 

 winding. These facts are made clearer in the solution of the prob- 

 lems that follow. 1 



Prob. 30. The flux density in the air-gap under the poles of an alterna- 

 tor is constant for 50 per cent of the pole pitch, and then it drops to zero, 



ling to the straight-line law, on each side in a space of 15 pc: 

 of the pole pitch. Draw to scale the curves of induced e.m.f. for the fol- 

 lowing windings: (a) Single-phase, one slot per pole; (b) Single-phase, 

 nine slots per pole, five slots being occupied by a one-hundred per cent 

 pitch winding; (c) The same as in (b) only the winding-pitch is equal to 

 7/9; (d) Three-phase, Y-connected full-pitch winding, two slots per pole 

 per phase ; in the latter case give curves of both the phase voltage and the 

 line voltage. On all the curves indicate roughly the equivalent sine wave, 

 in order to see the influence of the number of slots and of the fractional 

 pitch in improving the wave form. 



Prob. 31. A three-phase, Y-connected alternator has three slots per 

 pole per phase, and a full-pitch winding. The field curve has an 8 per 

 cent fifth harmonic, that is to say, the amplitude of the fifth harmonic is 

 0.08 of that of the fundamental sine wave. What is the magnitude of the 

 fifth harmonic in the phase voltage and in the line voltage. Solution: In 

 formula (33) the angle a between the adjacent slots is 20 electrical degrees 

 for the fundamental wave. For the fifth harmonic the same distance 

 between the slots corresponds to 100 electrical degrees. Hence, for the 

 fundamental wave 



fc,sin 30/(3 sin 10) -0.96; 



\\hilc for the fifth harmonic 



Jfe-dn 150/(3 sin 50) -0.217. 



'\'\\\< n KM i is that, due to the distribution in three slots, the fundamental 

 of the voltage is reduced to 0.96 of its value in a unislot machine. 

 while the lil'th harmonic is reduced to only 0.217 of its corresponding value. 

 Thm-fore, the relative magnitude of the fifth harmonic in the ph.i>r 

 voltage is 8X21 .7/96 -1.8 per cent, which means that the fifth harmonic 

 is reduced to less than one-fourth of its value in the field run 

 calculating the line voltage the vectors of the fundamental waves in a 

 three-phase machine are combined at an angle of 120 degrees. Conae- 



1 For f urther details sec Professor C. A. Adams' paper on " Electron) 

 Force Wave-shape in Alternators," Trans. Amer. In*. Elec. Engt., Vol. 

 28 (1909), Part 1 1, p. 1053. 



