62 ELEMENTS OF ELECTRICAL ENGINEERING. 



half cycle (positive values of e) lies between the limits ut = Q, and ut = 7r, so that 

 t f = o and \f f = TT/U. Therefore the average value of e during a half cycle is 



equal to I sin ut dt. Let x = o>/, then dx = ucff ; also when t = o, 



TT Jt=o 



x = o } and when t = TT/W, x ^= TT. Therefore 



E /**= 2E 



average value of <? I sin JR* = 

 7T/a;=0 7T 



Proposition. 77^ square root of the average square of a har- 

 monic electromotive force (or current) during one or more whole 

 cycles is equal to the maximum value E (or I) divided by the 

 square root of two. That is 



These relations may be established by reducing the expression - f J / e l dt', 



but the following considerations lead to the desired result more easily. Let 

 e = TL sin ut, then e z = TL* sin 2 ut, and Av. e z = Av. (E 2 sin 2 uf) =E 2 Av. (sin 2 ut) ; 

 but we have the general relation : 



sin 2 ut -\- cos 2 ot = I (a) 



so that 



Av. (sin 2 w/) -f Av. (cos* /) = I (^) 



and, since the cosine of a uniformly variable angle passes similarly through the same 

 set of values during a cycle as the sine, we have 



Av. (sin 2 6>/) = Av. (cos z w^) (c) 



Hence, from equation (3) we have 



2 Av. 



or 



Av. (sin 2 w/)= (d) 



Therefore 



Av. e* = E 2 Av. (sin 2 6tf) = 

 or 



T/Xv7^=- 



1/2 



The use of effective values of electromotive force and current in 

 the clock diagram. According to Art. 21 the lines in a clock 

 diagram are supposed to represent maximum values of electro- 



