16 



SYNCHRONOUS MOTORS 



R and Z.=the resistance and inductance, respectively, of the total 



circuit, of the two machines; 

 j=the instantaneous value of the current; 

 7 = the effective value of the current, equal to the maxi- 

 mum value divided by \/2. 



Let us suppose the conditions of stability to be unknown and let 

 us seek to ascertain how two alternators connected in series will operate. 



The two sine-functions of the E.M.F. represented by the curves 

 e\ and e% in Fig- II , m ay be formulated by the equations, 



in (u>t-\ ); 

 \ 2 / 



sn 



sin cot -- 



2 



in which 6 designates the angular distance between the actual position 

 of e\ and the position of opposition of e%. 



The E.M.F. which is acting in the circuit is equal to the algebraical 

 sum of the opposing E.M.F. 's. 



\ 



inltot -- ). 

 V 2 / 



From this the current, i, may be deduced, by the well-known 

 differential equation, 



- 



at 



sinajt+-)-E 2 V~2 sinlut -- ). (A) 

 \ 27 \ 27 



In this equation let i=X sin a)t+ Y cos cut. 



If this value be substituted in the equation, the values of X and Y can 

 be determined by making the coefficients of the sine-terms and of the 

 cosine-terms successively equal to zero. We can then obtain, by 

 differentiation, substitution, etc., 1 the following value for i: 



\sm(u>t--\-cos( --}] 



L S \ *) R * M \ 2/J 



1 See Appendix A. 



