Root Mean Squared

Question

The process for finding the root mean square (RMS) of a signal is:

square all of the values of the signal
find the mean value of this squared signal
take the square root of the mean

When finding the RMS, think about the time period over which you are taking the mean.

An alternating current is passed through a wire, with a value $ I(t) = I_0 \sin(\omega t) $. Show that the root mean square of this signal is given by $ \frac{I_0}{\sqrt{2} } $.

Hints

As the signal is a sine wave, it is periodic with time period $ T = \frac{2\pi}{\omega} $.

The mean square of the signal is given by: $$ \frac{1}{T} \int_{0}^{T} I(t)^2 \, dt $$ Take the square root of this expression to get the root mean square.

Answer

For further questions, look at the RMS value for some other waveforms, and derive these expressions