SYNCHRONOUS MACHINERY 259 



This may be further simplified by combining the first two terms 

 and becomes 



e = VA 2 + 5 2 sin (0 + 7) - C cos (3 - <), . (251) 



A 



where 7 = cos * 



J5 2 



Thus the generated e.m.f. consists of a sine wave of funda- 

 mental frequency and a third harmonic. 



When <j> = 90 degrees, as on short circuit, the third harmonic is 



e^= - C cos (3 - 90) 



= - C sin 30; 

 when 6 = 90, 



e,= -Csin270 = C; 



therefore, at short circuit, the generated e.m.f. contains a third 

 harmonic, which passes through zero at the same instant as the 

 fundamental wave, and the wave of e.m.f. is symmetrical, with a 

 peak at the centre. 



Thus single-phase armature reaction produces a double-fre- 

 quency pulsation of the field and a third harmonic of e.m.f. 



Since the magnetic circuit is surrounded by a field winding, 

 consisting of a large number of turns, the pulsations will be less 

 than the above results indicate. The variation of the flux linking 

 with the field winding induces in it e.m.fs. and currents which 

 oppose the variation and limit it to a small value. 



In a machine with a large number of field turns the pulsation 

 produced by armature reaction up to full-load current is very small 

 and the armature reaction may be considered as constant in value 

 with reference to the fields and demagnetizing. 



In the case of a short circuit, however, where from three to five 

 times full-load current flows in the armature, a large pulsation of 

 flux is produced in the magnetic circuit and very large e.m.fs. 

 and currents may be induced in the field windings. 



If a short circuit occurs on one phase only of a three-phase 

 alternator, the armature reaction can be separated into the 

 ordinary three-phase armature reaction with equal currents and 

 a single-phase armature reaction due to the excess of the short- 

 circuit current over normal current acting in the turns of one phase. 

 This single-phase armature reaction produces a double-frequency 

 pulsation of the field and a third harmonic of e.m.f. in all the 

 phases. 



