# Mean value & RMS value of Alternating Current & Voltage

### Mean value & RMS value of Alternating Current & Voltage

The mean value of sinusoidal current or voltage in one complete cycle is zero, for half cycle, the mean value can be found as given below.

Let  I = I0 sin ωt

$\displaystyle I_m = \frac{\int_{0}^{T/2}I dt}{\int_{0}^{T/2}dt}$

$\displaystyle I_m = \frac{\int_{0}^{T/2}I dt}{T/2}$

$\displaystyle I_m = \frac{2}{T}\int_{0}^{T/2}I_0 sin\omega t dt$

$\displaystyle I_m = \frac{2 I_0}{\pi}$

Similarly ,

$\displaystyle V_m = \frac{2 V_0}{\pi}$

### Root Mean square value of voltage & current :(Vrms & Irms)

I = I0 sin ωt

$\displaystyle I_{rms} = \sqrt{\frac{\int_{0}^{T}I^2 dt}{\int_{0}^{T}dt}}$

$\displaystyle I_{rms} = \frac{I_0}{\sqrt 2}$

Similarly ,

$\displaystyle V_{rms} = \frac{V_0}{\sqrt 2}$

Exercise : The peak value of an alternating current is 10 A. Find its rms value.