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 = (mean value of V²)/R where

We can therefore define the r.m.s. value as that voltage 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 voltage V is related to the peak value V_o by the equation:



Similarly, for the current we have:


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= I_osin(w)t

where ω 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(ωt))^1/2

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

Therefore the mean value of sin² (ωt) = ½ , and therefore I = I_o(½)^1/2 = I_o/(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(ωt) and sin²(ωt) are shown in Figure 2.