Average value of AC in half cycle

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Average value of AC in half cycle


We have equation 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/2 is upper limit and 0 is lower limit ] / (T/2) 

Where, Iav = average value of AC in half cycle

Now from eq. (1) 

Iav = 2Io/T [-coswt/w ]  where bracket has upper limit T/2 and lower limit 0

Iav = - 2Io/Tw [coswt] where bracket has upper limit T/2 and lower limit 0

Iav = - 2 Io/Tw [ cos(wt/2) - cos0 ]

Iav = - 2Io/Tw [ cos(wt/2) - 1 ]

But w = 2π/T

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

Iav = - 2Io/2π  [ cosπ - 1 ]

Iav = - Io/π [ -1-1 ]

Iav = - Io/π [ -2 ]

Iav = 2Io/π

So now 
We can say it is prove that average value of AC in half cycle is 2Io/π . 

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