1 88 ALTERNATING CURRENTS 



we take the primary current Ii as our vector of reference in a hori- 

 zontal direction. The resultant of OA = rili, AB = pLiii, and 

 BC = pMI 2 in the primary circuit gives us V, the primary phase 

 p.d., the angle AOC = 0i being the angle of lag of the primary 

 current behind the p.d., so that cos BI = power factor of motor. In 

 the secondary, the resultant of OD = E 2 l2 and DE = pL^Lz gives 

 us the e.m.f. ^MIi, while the angle EOD = 62 is the angle by which 

 I 2 lags behind joMIi. From C let OF be drawn parallel to OD. 

 Then angle BFC = 2 , while BCF is a right angle. 



109. Transformation of Vector Diagram. 

 Deduction of Circle Diagram 



Let us now transform our vector diagram by dividing the length 

 of each vector in it by Ii. We then obtain the diagram of Fig. 128. 

 A, B, and E now become fixed points (0 A = r\, AB = pLi, and 

 OE = pM. being constant). Hence, since, as the load varies, D 

 describes a semicircle on OE as diameter, and since BC is parallel 

 and proportional to ED, it follows that C is constrained to move along 

 a circle described on BF as diameter. 



Since is fixed with respect to the circle, it follows by a well- 

 known geometrical proposition that OP x OC = constant for all 

 possible positions of C. Hence 



constant constant T 

 OP = Qp = V~~ = constant x Ix 



since V is .by supposition maintained constant. OP is, therefore, 

 proportional to the primary current, and we have the important 

 result that as the load changes the extremity of the primary current 

 vector moves along a circle. Since the angle which this vector makes 

 with the horizontal is Oi, we may, in our new diagram, take the 

 horizontal direction to be that of the primary phase p.d. vector. 



Let the line OB be drawn, intersecting the circle at Q. Then 



we have OQ . OB = OP . 00, or Qp = QJ>- Hence, the triangles 



OP BC 

 OQP and OCB are similar, and ^p = ^g, or 



OP 



QP = TT B0 = constant x I a (4), 



OB 



since OB, p, and M are all constant. Thus, QP is proportional to 

 the secondary current. 



