About rms

Remember that the root mean square, or rms amplitude
is the DC voltage which will deliver the same average power as the AC signal. |

To explain why this is called root mean square, you need to think about how to calculate the average power delivered by the AC signal.

In Chapter 2, one formula for calculating power was introduced:

Substituting *V* = *IR* or *I* = *V*/*R* gives two
additional formulae:

With a *V*/*t *graph, the *P* = *V*^{ 2}/*R*
formula provides the most convenient method for calculating the power at any instant:

The upper graph shows an AC signal with 10 V peak amplitude. This is the AC power supply illuminating a lamp:

Assuming that the resistance of the lamp remains constant, *R* = 100 , it follows that the power developed will be
different at different times during each cycle. Since *V* is changing, *P*
will be changing also. For example, at *t* = 2 ms on the graph, *V* =
5.9 V and *P* = *V*^{ 2}/*R* = 0.35 W. At *t*
= 5 ms, *V* = 10 V and *P* = *V*^{ 2}/*R* =
1.0 W, while at *t* = 10 ms, *V* = 0 and *P* = 0 V and
*V*^{ 2}/*R* = 0 W.

Calculating a *P* value for every moment in the cycle gives the lower graph in
the diagram. This is how the power produced varies during each cycle.

The apparent brightness of the lamp does not depend on the power developed at any
instant, but upon the *average* power. Look again at the *P*/*t*
graph. You can see that the average power is equal to *half* the maximum power.

The average power, *P*_{av} is therefore given by:

where *V*_{p} is the peak amplitude

Now, reverse the process and think about the DC voltage, *V*_{DC} which
would deliver the same average power:

Combining these two formulae:

rearranging:

taking the square root of each side gives:

In other words, *V*_{DC} = *V*_{rms} as already defined.

It is now possible to explain that the rms value is the square root of the mean,
or average, of the square of the voltages during a complete cycle of the AC
signal. |

*V*_{rms} is the single DC value which will deliver the same average
power as the AC signal.

Back to Signals