120 ELECTRICAL ENGINEERING 



where 



Z = VR 2 + X 2 is the impedance of the circuit. 



(4) Fig. 89 shows the vector diagram of a circuit containing a 

 condenser of reactance X c , when an alternating e.m.f. E is im- 



E 



pressed on its terminals and a current I = -^r flows through it 



A c 



leading the e.m.f. by 90 degrees. 



(5) Fig. 90 shows the diagram for the same circuit with a re- 

 sistance R added in series. 



The e.m.f. consumed by the resistance is OEi = EI = IR, in 

 phase with 01 = I. 



The e.m.f. consumed by the reactance is OE 2 = E 2 = IX C1 

 lagging 90 degrees behind 01. 



The impressed e.m.f. is 



OE = E = VES + # 2 2 = IVR* + X c * = 7Z, 

 lagging behind 07 by angle <f>, where 



E 2 IX C X c 

 *" l *-ll-rf"lf' 



(6) If an alternating e.m.f. E is impressed on a circuit con- 

 taining a resistance 72, an inductive reactance X 2irfL and a 



condensive reactance X c = fri connected in series, determine 



*- TTjO 



the magnitude and phase relation of the current and draw the 

 vector diagram for the circuit. (Fig. 91.) 



07 = 7 is the current taken as horizontal. 



OEi = EI = IR is the e.m.f. consumed by the resistance and is 

 in phase with 7. 



OE% = E 2 = IX is the e.m.f. consumed by the inductive re- 

 actance and leads 07 by 90 degrees. 



OEz = E s = IX c is the e.m.f. consumed by the condensive re- 

 actance and lags behind 07 by 90 degrees. 



OE = E = 7Z is the e.m.f. impressed on the circuit, or the 

 e.m.f. consumed by the impedance Z. It is the vector sum of the 

 three components EI, E 2 and E 3 and leads the current by angle 

 </>; therefore, 



E = VES + (E* - # 3 ) 2 



= 7 VR* + (X - X c ) z 



