Average value of AC in one complete cycle

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Average value of AC in one complete cycle


We have eq. Of induced current 

     I = Io sin(wt)    .... (1) 
Where, Io = peak value of AC also called current amplitude

                       Note

 
 Iav = [ integration of I with respect to dt where T is upper limit and 0 is lower limit] / T
 Where, Iav = average current of AC in one cycle 
T = time taken by the alternating current AC to complete it's one variation Or one cycle

Now, from equation (1) 
We get

Iav = Io/T  [ integration of sin(wt) with respect to dt where T is upper limit and 0 is lower limit]

Iav = Io/T [ -cos(wt) / w]  where bracket has lower limit of 0 and upper limit of T

Iav = - Io/Tw [ coswt ] where bracket has upper limit T and lower limit 0

Iav = - Io/Tw [ coswt - cos0 ]

Iav = - Io/Tw [ coswt - 1 ]

But w = 2π/T

Iav = - Io/(T × 2π/T)  [ cos2πT/T - 1 ]

Iav = - Io/2π [ cos2π - 1 ]

Iav = - Io/2π [ 1 - 1 ]

Iav = - Io/2π [ 0 ]

Iav = 0

Now, it is prove that 
Average value of AC in one complete cycle is 0.

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