POWER 1 5 



of Fig. 10, since in this case each of the vectors OB and OC becomes 

 very large in comparison with OA. The impressed e.m.f. and the 

 p.d.'s across the two portions of the circuit are given by 



impressed e.m.f. = current X resistance 

 p.d. across inductive resistance = current x v/r 8 + jp a L a 



current 

 p.d. across condenser terminals = 



Up 



7. Power in Alternating Current Circuit 



Let the current in a circuit be given by 



i = I sin pt 

 and the impressed e.m.f. by 



e = E sin (pt + 6) 

 The power w at any instant is 



w = ei = El . sin (pt + 9) . sin pt 



. 2 sin (pt + 8) . sin pt 

 {cos 6 - cos (2pt + 9)} 

 cos B - iEI . cos (2pt + 6) 



We therefore see that the expression for the instantaneous power 

 consists of two terms, one of which, EI cos 0, is a constant, while 

 the other is a cosine (or sine) wave of frequency equal to double the 

 frequency of the e.m.f. and current waves. 



Now when we;speak, without in any way qualifying the expression, 

 of the power in an alternating current circuit, we understand by this 

 term the mean value of the power over a complete period. The mean 

 value of the second term in the expression for the instantaneous 

 power w is, however, zero over any whole number of periods. Thus 

 the mean value of the power becomes equal to the first or constant 



ft I 



term ^EI cos 0. This may be written in the form -^ /- . cos 0, 



E I 



and since -,-= = r.m.s. value of e.m.f., and - /^ = r.m.s. value of 



current, we see that 



mean power = e.m.f. X current x cos 



r.m.s. values of e.m.f. and current being understood. The power, there- 

 fore, is not simply equal to the product of the e.m.f. and current, but 

 is equal to this product multiplied by cos 6. The multiplier which 

 converts volt-amperes or apparent power into watts or true power is 



