ELECTRIC CIRCUITS 135 



Substituting values obtained from (1) and (2) in the equations 

 (3) and (4) the magnitude and phase relation of I can be obtained. 

 (II) Using the constants G and B: 



(1) Ii = E V(?! 2 + B,\ tan 0i = ^; 



(2) h = E VG 2 2 + 2 2 , tan fr = . 



tf2 



From the vector diagram 



and tan <t> = 



85. Rectangular Coordinates. The simplest method of deal- 

 ing with alternating-current phenomena is to express the e.m.f .'s, 

 currents, etc., as the sum of two components, one along a chosen 

 axis and the other perpendicular to it. 



In Fig. 103 which represents the e.m.f. and current in a circuit 

 of impedance z = V r 2 -f- x 2 the axis is chosen in the direction of 



T r 



*-* WW - 1 



61 I 



E fl* ^ L =ESin ^ =Ia; \^ |i r lSln* 



7 ^r ^^ I =E'6 



ei=ECos 0=Ir 



FIG. 103. 



the current and the e.m.f. is resolved into two components, e\ in 

 phase with the current and e 2 in quadrature ahead of the current. 

 The absolute value of the e.m.f. is 



E = Vei 2 + e 2 2 , 

 and it leads the current which was chosen as axis by an angle 0, 



where tan < = 



Thus when the e.m.f. is expressed as the sum of two components 

 at right angles both its magnitude and its phase are known. 



