RMS in AC Circuits: Why Use Square for Mean Values?

[joel amos]

Why do we use the square of the graph as opposed to the absolute value of the graph to find the mean values?

 

[Svein]

Why do we use the square of the graph as opposed to the absolute value of the graph to find the mean values?

The definition of electrical heat in a resistor is $E=\int R\cdot I^{2}dt$. We want this expression to be the same for DC and AC. Since the AC waveform repeats itself after the period T, we want $E=\int_{0}^{T} R\cdot I_{dc}^{2}dt=\int_{0}^{T}R\cdot I_{ac}^{2}dt$. Since R and I_dc are constants, this is equivalent to $R\cdot I_{dc}^{2}\cdot T= R\cdot\int_{0}^{T}I_{ac}^{2}dt$. After some reordering, we get $I_{dc}^{2}= \frac{1}{T}\int_{0}^{T}I_{ac}^{2}dt$.
So - the equivalent DC current of an AC current is the square Root of the Mean of the Squares.