# 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 = i

^{2}R

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

mean value of P = (mean value of i

^{2}) x R = I

^{2}R where

I = (mean value of i^{2})^{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 i

_{o} by the equation:

Root mean square current: I = i_{o}/(2^{1/2}) = 0.707i_{o})
Similarly, for the voltage we have:

Root mean square voltage: V = v_{o}/(2^{1/2}) = 0.707v_{o})
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

_{o}sin(

w)t

where

w is a constant related to the frequency f by
the equation

w = 2

pf.
By definition the r.m.s. current (I) is

I = (mean value of
i

^{2})

^{1/2} = (mean value of sin

^{2}(

wt))

^{1/2}But sin

^{2}(

wt) = ½ – ½ cos (2

wt), and the
mean value of cos (2

wt) is 0.

Therefore the mean value of sin

^{2} (

wt) = 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(wt) and sin^{2}(wt) are shown in Figure
2.