132 TREATISE ON ALTERNATING CURRENTS. 



Draw through T the line PSTQ parallel to the line of current, 

 and draw PM, SO, TR, and QN through e, 0, T, and Q respectively, 

 at right angles to the line of current. We then have 



n Si S 



tan =-.= - = - . . . . (10) 



n r r 



that is, is independent of the current, and OT is a fixed direction 

 relative to Oi, so long as the speeds of the machines are kept 

 constant, and L is considered constant. 



In Fig. 43, OS (or PM) [= ZirnLi] is proportional to the 

 current i. 



OM is the component of e directly opposing i. OR is the 

 E.M.F. required to overcome resistance, and ON is the component 

 of E in the direction of i ; hence rectangle PSOM is proportional 

 to the output of the motor (w). 



The rectangle OSTR is proportional to the fir losses, and the 

 rectangle OSQN is proportional to the output \iE cos t//] of the 

 generator. 



From this and the equation 



w + i 2 r = iE cos ;// 



it follows that the efficiency of transformation 



OM OM 



If the output of the motor is kept constant, we have- 

 rectangle PSOM = constant 



and the locus of P is a rectangular hyperbola having OM and OS 

 as asymptotes (Fig. 44). 



Take any point P on this hyperbola. We have seen that OT 

 has a fixed direction relative to Oi ; and the point T (Fig. 44) ou 

 this direction is found by drawing through P a line parallel to Oi. 

 Again, e lies on the line through P parallel to OS, and eT = E in 

 magnitude. Let the E.M.F. of the generator be kept constant and 

 equal to E. With centre T, and radius E, describe a circle cutting 

 PM in e and e ; then the corresponding counter E.M.F. of the 

 motor may be either Oe or Oe, and the current is represented in 

 magnitude by PM\ that is, corresponding to given values of E and 

 i, there are two values of e. The relative phases in the two cases 

 are shown in the parallelograms OeTE and Oe'TEJ (Fig. 44). 



