SYNCHRONOUS MACHINERY 233 



The resultant e.m.f. around the closed circuit at any instant is 

 + e 2 + e 3 = E sin + E sin (0 - 120) + E Q sin (0 - 240) 

 = Eo (sin + sin cos 120 cos sin 120 

 + sin 6 cos 240 - cos sin 240) 



A/3 



/ A3 



JEJ ( sin 6 J sin -- cos 8 



V3 \ 

 + -2-COS0J = 0, 



sin 



and therefore if the three windings are exactly similar and are 

 displaced exactly 120 degrees there will be no resultant e.m.f. 

 acting around the winding and no circulating current will flow. 

 This result has been derived on the assumption that the e.m.f. 

 wave is a true sine wave. If the e.m.f. wave is made up of a 

 fundamental and a third harmonic, the third harmonics in the 

 three phases will be displaced by 3 X 120 = 360 and will there- 

 fore be in phase and will combine to a resultant third harmonic 

 of three times the magnitude of that in one phase and this will 

 cause a third harmonic of current to circulate through the closed 

 winding. This current may be of the order of full-load current 

 in the case of alternators of low reactance. The reactance of the 

 alternator winding to the triple frequency current is three times 

 that offered to the current of fundamental frequency but only 

 the true reactance and not the synchronous reactance of the 

 winding is effective in limiting the circulating current. The third 

 harmonic of e.m.f. does not appear at the terminals since it is con- 

 sumed in producing the circulating current. 



(2) If the ends Si, s 2 and s 3 are connected together and the ends 

 /i, / 2 and/s are joined to the three terminals A, B and C, Fig. 201, 

 the windings are connected "Y" or "star." 



The e.m.f. between A and B is 



CAB = e! e 2 = Eo sin E sin (0 120) 



= E Q (sin sin cos 120 + cos sin 120) 



= EQ (sin + \ sin + -y cos 0} 



V3 E Q sin (0 + 30); .... (234) 



