Root mean square value of AC

Root mean square value of AC

It is defined by the root of square mean of AC current. 

It is denoted by Irms Or Ieff

We have 

I = Io sin(wt) 

Where, Io = peak value of AC also called current amplitude

   By defination

Irms = [ integration of I × I with respect to dt where upper limit is T and lower limit is 0]/T

On sq.  Both side

Irms × Irms = [ integration of I × I with respect to dt where upper limit is T and lower limit is 0]/ T

Now after putting I value in it

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

Irms × Irms = Io × Io/T [ integration of sin(wt) × sin(wt) with respect to dt where upper limit is T and lower limit is 0 ]             ........ (1) 

But 

Cos(2wt) = 1 - 2sin(wt) × sin(wt) 

2sin(wt) × sin(wt) = 1 - cos(2wt) 

Sin(wt) × sin(wt) =[1 - cos(2wt) ] / 2

From eq. (1) 

Irms × Irms = Io × Io/2 [ integration of [ 1-cos(2wt) ] / 2 with respect to dt where upper limit is T and lower limit is 0]

Irms × Irms = Io × Io/2T {[ integration of 1 with respect to dt where upper limit is T and lower limit is 0] - [ integration of cos(2wt) with respect to dt where upper limit is T and lower limit is 0]}

Irms × Irms = Io × Io /2T  [ (t) of upper limit is T and lower limit is 0  - (sin2wt/2w) of upper limit is T and lower limit is 0 ]

Irms × Irms = Io × Io/2T [ T - 1/2w (sin2wt) of upper limit is T and lower limit is 0 ]

Irms × Irms = Io × Io /2T [ T - 1/2w(sinwt - sin0)]

Irms × Irms = Io × Io/2 [ T - 1/2w (sin2wt) ]

But w = 2π/T

Irms × Irms = Io × Io/2T[T - 1/2× 2π/T  (sin(2× 2πT/T) ]

Irms × Irms = Io × Io/2T [ T - sin4π/(4π/T) ]

Irms × Irms = Io × Io/2T [ T - 0 ]

Irms × Irms =[(Io × Io) /2 ] of power of 1/2

Irms = {Io/[(2)of power of 1/2]}

So, now we can say that we have our correct value of Irms.