12 



ALTERNATING CURRENTS 



vector OA, Fig. 8, of length rl, on the vertical axis ; while the second 

 component, pLI cos pt, may be represented by the projection on the 

 same axis of a rotating vector OB = Lpl. The impressed e.m.f. at 

 any instant is the algebraical sum of these two projections ; but this 

 is clearly the same as the projection of the diagonal OE of the 

 rectangle constructed on OA and OB as sides. Hence OE will be 

 the rotating vector corresponding to the impressed e.m.f. But since 



OE = \/OA 2 + OB 2 = \/r 2 I 2 + jp 2 L 2 f 2 = 



we see that 



e.m.f. 

 current = 



and that the current lags behind the e.m.f. by an angle AOE such 

 that 



tan AOE = ^ 



r 



The quantity \A* 2 4- .P 2 L 2 is termed the impedance of the circuit, 

 and the angle AOE = tan" 1 is the angle of lag of the current 

 behind the impressed e.m.f. 



B 



RESISTANCE INDUCTANCE 



W\r--WraS 



FIG. 8. Vector Diagram for Inductive 

 Resistance. 



CAPACITY 



FIG. 9. Arrangement of Resistance, 

 Inductance, and Capacity in Series. 



We may now consider a circuit containing resistance, inductance, 

 and capacity. Such a circuit is diagrammatically represented in 

 Fig. 9.* The impressed e.m.f. may be regarded as made up of the 



* This equation holds good whether maximum or r.m.s. values be considered, 

 t We shall, in the diagrammatic representation of a circuit, use a coiled line to 

 indicate an inductive resistance, and a zig-zag line to indicate non-inductive resistance. 



