DETAILED STUDY OF OPERATION WITH NORMAL LOAD 43 



Power-Values as Function of the Lag-Angle 6. To express the 

 power-values PI and P 2 in terms of the angle 6, it is only necessary 

 to substitute, in the two equations (13), other values for / cos(Z 1 , 7) 

 and for I cos (E 2 , -0- 



7 cos (Ei, 7) is nothing more than the watted current i for the 

 case represented in Fig. 23. 



Let J\ = -y and /2 = -~. (See eq. 14, p. 42.) 



The diagram gives, readily, 



i=I cos (Z.EI, 7)=/i cos;- / 2 cos (d+f). . (15) 



The current 7 cos (E 2 , 1} will be obtained by analogy, by project- 

 ing the diagram of Fig. 26 upon a right line making with OA^ an 

 angle that is again equal to the angle -jr. From this we obtain the 

 absolute value 



7 cos (Z.E 2 , I)=J\ cos (Y 6} J 2 cos?- (16) 



Tf Tf 



Replacing J\, J 2 by their values, , , we will therefore have 



t 



P l =^[E 1 cos r -E 2 cos(d- r )} (17) 



P 2 =^[E 2 cos ft 0)- E 2 cos r ]. .... (18) 



These are expressions wherein only two variables, 7i 2 and d, enter. 

 The utility of these expressions will be shown later. 



Use of this Diagram for the Study of Different Loads. These loads 

 cannot be determined without imposing some condition which makes 

 the variables (i.e. E.M.F. current, power, lag, etc., )vary according to 

 some definite law. In that case, to each value of the lag there will 

 correspond a single (or several) quite definite values of the E.M.F. E 2 , 

 and vice versa. Consequently, on the diagram, the point A 2 , instead 

 of being able to range over the whole surface, can be displaced only 

 on a single curve such as CC', for example (Fig. 26), whose equation, 

 in polar co-ordinates, with respect to the point O as a pole, is noth- 

 ing more than the relation imposed between E 2 and 6. 



For each point of this curve, which may be called the locus of A 2 , 

 there correspond two radial vectors issuing from two fixed poles. 



One of these vectors, OA 2 , is drawn from the point O as a pole; 

 and it indicates the E.M.F. E 2 , in magnitude and in phase (starting 

 from OA\}\ the other, A^A 2 , is drawn from the point AI as a pole; 



