Consider a voltage v = Vdc + vac where Vdc = a constant and the average value of vac = 0. Apply the integral definition of RMS to prove the formula RMS =√ [Vdc 2 +(RMS of vac) 2 ]. Hint: v2 = [(Vdc) 2 + 2 *vac* Vdc +(vac) 2 ] and the average value of vac* Vdc = 0.

Proof is as follows

Proof:

Given that ,

for any function f with period T, RMS is given by

In our case, function is

Now open the square term as follows

Rearranging  terms

You can see that

so

So it has been proved that given expression for root mean square (RMS) is valid

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Explanation: