Alternators in Parallel. 311 



Engineers,' paragraph 189, we find that 



P ■ p _ F , (^ 2 + r 2 )(2R + r)+R(2Rr + ^-Z 2 n 2 )cos2^ 

 1+2 (^n 2 +r*){(2R + r) 2 + ZVj; ' ^ ij 



= * E ' (H + ^)%OV when * = °' ' ' ' ' (2) 

 Pi-P 8 _ 2R(R + r)sin2* 



1± 



Pi + P] (2Rr + r 2 )(lt-H'-Rcos2«) ,-- ^ R ow "^ 



The expression in (2) for Pi + P 2 , when the alternators are 

 in phase, is similar to the power given by a single alternator 



P=iE 2 



r I 



substituting ~ and - for r and I, so that the total power of 



two alternators in parallel is the same as that of a single 

 alternator haying half the internal resistance and half the 

 self-induction. 



It can be proved similarly that in the case of three alter- 

 nators in parallel, the total power is the same as that of one 

 alternator having \ the internal resistance and ^ the self- 

 induction. 



It is reasonable to suppose, by induction, that the law holds 

 good generally, so that the law for alternators in parallel is 

 the same as that for cells in parallel. 



Another point of interest is, that if any number m of alter- 

 nators are working in parallel, then when 



l 2 n 2 = (m — 1) (mRr -f r 2 ) 



the total power equals -^ , and the current follows Ohm's 



law. +r 



Examining the expression in (1) for Pj + Pg, it is seen that 

 as the difference in phase a. increases, and cos 2a diminishes, 

 the total power increases if l 2 n 2 is smaller than 2Rr + r 2 , and 

 vice versa. 



In the former case, part of the extra work done by the 

 leading machine is used to bring up the lagging machine 

 into step again, and the remainder goes into the external 

 circuit. 



In the latter case, the controlling power does not all come 

 directly from the leading machine, but the lagging machine 

 partly borrows it from the external circuit. 



