Why use RMS values instead of "mean of absolute" [duplicate]

Question

Why do we use rms values? I can't get my head around why these values are used. I am doing an intro course to elec eng and we get given some values (normally voltages) in rms and others not, and we are expected to calculate other values but I don't know when to calculate rms or when to calculate something else.

I realise that the root-mean-square is different from the mean-root-square or mean of absolute but it seems logical to me to use the mean-root-square not the rms. Why do we want to use the rms? And when do we want to calculate rms instead of another "interpretation" of the value?

I have seen a similar question to this that addresses only power calculations where voltage is squared but we seem to use rms even when we don't square anything and we aren't interested in calculating power...

Answer 1

We use RMS when we are interested in the power delivered by a source (into an ohmic load). For a DC voltage that power is proportional to V^2: P = I * V, and I = V / R, hence V appears twice in the formula.

So the power at any one moment is proportional to V^2. Hence the average power is proportional to mean( V^2 ). When we want to know which DC voltage would cause the same power, we have the equation

mean( DC^2 ) = mean( V^2 )

which is easily reduced to

DC = sqrt( mean( V^2 ))

in other words, the root of the mean of the square => root-mean-square.

Summary: the RMS voltage is the DC voltage that would cause the same power (in an ohmic load) as the AC voltage we are studying.

Answer 2

we seem to use rms even when we don't square anything and we aren't interested in calculating power...

Yes, we do and, we also calculate the current through a capacitor by using the RMS quantities despite there being no net power dissipated or taken (the capacitor is purely reactive) yet, we still use RMS terms. If the voltage is a sinewave at 100 V RMS and the capacitive reactance were 100 ohms, then the RMS current is 1 amp.

• If we used some other measure of voltage (say) Vpk then we would get Ipk.
• If we used the average of a sinewave the result would be zero

What does mean-root-square mean? In effect squaring then rooting a sinewave is like perfect rectification of the sinewave. Then, if you take the mean (to get mean-root-square) it is 0.63689 of Vpk. But this is meaningless to a capacitor and even if we ignored the "mean" part, feeding this "corrupted and harmonic rich" signal into a capacitor is nonsense.