38 TREATISE ON ALTERNATING CURRENTS. 



The energy given to the circuit during a small interval of time 

 dt is therefore 



El sin pt sin (pt 0)dt 



If we divide a complete period of the current into an infinitely 

 large number of infinitely small times dt t and take the sum of 

 the energies given to the circuit during those times, we shall 

 obtain the total energy given to the circuit during a time equal 

 to a periodic time of the current, and if, further, we divide 

 the expression thus obtained by the periodic time,, we shall 

 have the average or mean power given to the circuit. 



Denoting the mean power by F, we therefore have 



n r' 2 " 

 P = - \ * El sin pt sin (pt - 0)dt 



**J o 



J-JJ- /I / I \ 



That is the power given to an alternating- 

 current circuit is half the product of the maxi- 

 mum current and the maximum P.D. multiplied 

 by the cosine of their phase difference. 



The expression for the mean power may be written 



E I 



P = 7= . =. cos 

 \/2 \/2 



= VA cos 6 (2) 



where E and / are the R.M.S. values of P.D. and current 

 respectively. 



Thus the power given to an alternating'-eur- 

 rent circuit is the product of the R.M.S. values 

 of the current and P.D. multiplied by the cosine 

 of their phase difference. 



The E.M.S. values E and / are the quantities measured 

 respectively by a voltmeter placed between the terminals of the 

 circuit and an ammeter placed in the circuit. Since cos is always 

 less than unity, we see that the product of amperes and volts 

 is, unless 6 = 0, greater than the power given to the circuit. 



In contradistinction to the true power, El cos 6, the 

 product El is called the apparent power, and the ratio 



