THE SYNCHRONOUS MOTOR. 



163 



This curve is a hyperbola inasmuch as j9cos (BI) X I P"* or B cos 

 X XI = XP" = a constant. This curve is plotted in Fig. 137 with the values of B cos 

 (j9/) laid off to the left inasmuch as P tf , being an intake, is negative. The .r-axis of 

 reference Of, represents the current in the circuit ; and the line OE, making with 



Of the known angle 6, but of unknown length, represents the resultant electro- 

 motive force of machines A and B. 



Draw a horizontal line cutting OE in / and cutting the hyperbola in q. Then, 

 for that particular value of the current which corresponds to the chosen ordinate aq 

 ( = Jff) the line Op represents the actual resultant electromotive force inasmuch as 

 the vertical component of E is equal to XI. Describe about the point p a circle 

 of which the radius represents A. Then the lines Or and Os represent the two 

 possible values of B for the chosen value of XI. From this diagram it is evident 

 that for the shortest possible value of Op (smallest possible current) the circle 

 described about p just touches the ordinate aq, in which case the line A is hori. 

 zontal and parallel to Of, as shown in Fig. 138. For any shorter value of Op 

 (smaller value of E or I] the circle does not reach to the ordinate aq, which means 

 that for so small a value of the current the given value of A is too small to supply the 

 line losses RI* and the given motor intake P rf . 



3. When P ff is adjustable, maximum efficiency occurs when the differential coeffi- 

 cient of P" 'I P f with respect to is zero. From equations (15) and (16) we have 



P_ _ ^ cos (0 0) -f^ 2 cos0 

 ~P* ~ AB cos (0+0)-f A 2 cos 

 whence, applying the condition 



