128 TREATISE ON ALTERNATING CURRENTS. 



I = impedance of armature = \/{?' 2 4- (27rnL)- J }. 

 S = reactance of armature = 2-rrnL. 

 ^ = phase difference between i and E. 

 = * 6 - 



= * H> 



Then the input = w -f V 

 and also = ^ cos ^ 

 therefore 



w + 2 r = iEcos\t> ...... (1) 



Solving this equation for i, we get 



. = 



Since * is always real, we must have 



E* cos 2 ^ greater than, or equal to, 

 therefore the maximum output is given by 



t* = f ...... (3) 



4r 



This occurs when i// = ; that is, when the current is in phase 

 with the impressed P.D. 



The current corresponding to maximum output is then seen 

 to be 



To obtain the corresponding value of e we proceed as follows : 

 There are three E.M.F.s e, E, and Si which have a resultant 

 ri. Of these E is in phase with i, whilst Si is at right angles 

 to it, and e differs in phase with i by an angle <. The components 

 of e along and at right angles to i are e cos ^ and e sin <p ; there- 

 fore we must have 



E e cos = ri 

 and 



e sin <f> = Si 



But when the output is a maximum 



E = 2ri 



therefore 



e cos d> = n* 



