HARMONIC ELECTROMOTIVE FORCE AND CURRENT. 6 1 



area, and 



FT-/ rw i 

 ~ /f 



is the height of the rectangle t'cdt" 



which has an area equal to the shaded area, the average value 

 of y during the interval /' to t" is the height of this rec- 



Axis of y 



r 



Fig. 53. 



tangle, and the height of this rectangle is equal to the area 

 under the curve divided by the length of base. 



(b) Average value of the sum of a number of variable 

 quantities. Let ;r, y, z be a number of variable quanti- 

 ties. Then the average value of (x -f y -f z -f ) is equal to 

 Av. ;r -{- Av. y-\- Av. z -f- This is evident when we consider that, 



Av. (x+y+z ) is by definition equal to f - - f I (x-\-y +z ^ 



i *Jtf 



i c*" i r v> i r l " 



which is equal to ^ ~, I xdt ^-fff I J^+^ // _ // I 



but this latter expression is by definition equal to 

 Av. x + Av. y + Av. z+ -. 



Proposition. The average value of a harmonic electromotive 

 force (or current) during a half cycle is equal to two times the 

 maximum value E (or I) divided by TT. 



The average value during one or more complete cycles is of 

 course equal to zero. 



Proof. Let e E sin ut be a given harmonic electromotive force. The average 

 value of e is by definition equal to f - ^ / edt t or - y I sin utdt. A 



