Root-mean-square (r.m.s.) value

We know that a current passing through a resistance has a heating effect, the magnitude of which we measure in watts, that is power is given by

This shows that power (or heat) dissipated in a resistance is proportional to either square of the current or the voltage.

Related: The Basics of Electric Circuits

Consider say an ac current of a peak value of I amperes. Draw its waveform over half cycle and then square  the value of the current at each 30° or 15° spacing (instantaneous values), then draw a waveform using these squared values. The resulting wave will represent the power dissipated. The average or mean heating effect will therefore be the sum of all the instantaneous values divided by the number of instantaneous values.

To put it another way; the root of the mean of the squares, this can be shown mathematically to be 0.7071 of the peak value or maximum value for sinusoidal waveforms.

Where I_max is the peak value of the waveform. If we connect a resistance to a dc supply and draw a dc current equal to the value of the ac r.m.s. value, the heating effect will be the same in both cases. Therefore we can define the r.m.s. value of alternating current or voltage as the value of alternating current or voltage which will give the same heating effect as the same value of direct or voltage, i.e.

15 A (r.m.s.) = 15 A (dc)

Note that, unless otherwise stated, all the values of voltage and current that are quoted on ac equipment are given as r.m.s. values. For example, the peak value of our 240 V domestic supply is

You can also read: Power Measurement in DC Circuits