24 SYNCHRONOUS MOTORS 



(2) These quantities are added or subtracted the same as real 

 quantities; the resultant quantities indicate immediately, by their 

 real and imaginary portions, the magnitude and the phase of the 

 resultant vector. 



(3) The multiplication of a complex quantity by a real quantity 

 only changes the dimension of the vector, without changing its phase. 

 On the contrary, the multiplication by an imaginary portion, such as 

 jb, modifies not only the magnitude of the vector, which thereby becomes 

 multiplied by b, but it also modifies its phase, which is then made to 



advance by the amount . In fact, we have 



i.e., the vector OA is then replaced by the vector OA'. 



(4) From this, it follows that the E.M.F. absorbed by an imped- 

 ance, z, composed of a resistance, r, and of a reactance, s, in series, 

 through which passes a current 



i=x+jy, 

 is obtained, in magnitude and phase, by forming the product 



(5) The work done by an E.M.F. OB having the imaginary value 



a+jb, 

 and a current OA having the imaginary value 



*+jy, 



is easily obtained by decomposing it into the work of each of the com- 

 ponents a and b of this E.M.F. The first component, a, does no work 

 on the component y of the current with which it is " in quadrature,'" 

 but it is " in phase " with the component x; likewise b is in quadrature 

 with x but in phase with y. The power required is therefore the sum of 

 the products of the analogous coefficients of E and of I. 1 We will have 



ax+by. 



1 Use can also be made of other equivalent rules. A rule is given by M. Janet 

 (Echirage Electriquc, 18. Dec., 1897, p. 530) under the following form: To find 

 the power, j is changed into j in either the expression for the current strength 



