102 ELECTRICAL ENGINEERING 



tegrating over one-half wave; it is 

 o sin e d6 



1 C* 

 =- I Io si 



TTjO 



= - [1 + 1] = ?/ = 0.637 J , ..... (144) 



7T 7T 



2 



that is, the average value of the ordinate of a sine curve is - 



7T 



times the maximum ordinate. 



73. The Effective Value of a Sine Wave. The effective value 

 of an alternating current is the value of continuous current which 

 would have the same heating effect. 



When an alternating current i = I sin 6 flows through a 

 resistance R, energy is transformed into heat at the instanta- 

 neous rate i 2 R watts; the average rate of transformation of energy 

 is 



= - r 



TTjo 



I -cos20)d0 

 o 



sin 2 



(145) 



W2/ 



thus the alternating current i = I sin has the same average 

 heating effect or consumes the same average power as a con- 

 tinuous current^; the value / = ~ is therefore called the 



V2 V2 



effective value of the alternating current i = J sin 0. 



The effective value of the alternating e.m.f . e = E Q sin is 



E = ^= (146) 



V2 



The effective value of any alternating quantity is the square 

 root of the mean of the squared instantaneous values taken over 



