# Root mean square values

The reason for using what seems a rather complicated definition is as follows. The power P used in a resistor R is proportional to the square of the current:

P = i2R

But with alternating current the value of i and therefore of P changes, and so:

mean value of P = (mean value of i2) x R = I2R where

I = (mean value of i2)1/2 = root mean square (r.m.s) current

We can therefore define the r.m.s. value as that current that would dissipate power at the same rate as a d.c. current of the same value.

The relation between the peak and r.m.s. values for a sinusoidal wave can be seen in Figure 1. It can be shown that the r.m.s. value of current I is related to the peak value io by the equation:

Root mean square current:    I = io/(21/2) = 0.707io)

Similarly, for the voltage we have:

Root mean square voltage:    V = vo/(21/2) = 0.707vo)

In Britain the voltage supply is 230 V; this is the r.m.s. value, and so the peak value is 230/0.707 or in the region of 325 V.

## Proof of the value of the r.m.s. current

Let the current vary with time in the following way:

i= Iosin(w)t

where w is a constant related to the frequency f by the equation w = 2pf. By definition the r.m.s. current (I) is

I = (mean value of i2)1/2 = (mean value of sin2(wt))1/2

But sin2(wt) = ½ – ½ cos (2wt), and the mean value of cos (2wt) is 0.

Therefore the mean value of sin2 (wt) = ½ , and therefore I = Io(½)1/2 = Io/(2)1/2

It is important to realise that the general definition of r.m.s. value applies to any type of varying signal and not simply to one that varies sinusoidally. For example, it is quite possible for a square wave to have an r.m.s. value. The specific equation above however applies to sinusoidal variations only.

The variation of both sin(wt) and sin2(wt) are shown in Figure 2.