# How can AC be listed as a single voltage when it constantly varies?

How can AC be listed as a single voltage (e.g 240V AC) when it constantly varies? And what does this have to do with RMS Voltage?

A 240V AC means that it has a peak voltage nearly equal to $$240 \times \sqrt{2} \approx 340 ~ \rm{V}$$ Where the given values for the voltages for AC source is usually the RMS voltage $$V_{rms}$$, which is defined as the square root of the mean square of the value of the voltage across one periodic time Wiki, and it is related to the peak voltage $$V_p$$ through the following equation $$V_p = V_{rms} \times \sqrt{2}$$

Thus, knowing the root mean square voltage of an AC source, we can find the peak voltage using the above formula.

If you graph the voltage as a function of time you'll get a curve of the form:

$$V = V_p \sin ( 2\pi f t )$$

Where $f$ is the frequency (e.g. 50Hz in the UK or 60Hz in the USA). The only unknown constants in the equation are the frequency, $f$, and the peak voltage, $V_p$, so you just need to give these two values to completely describe the voltage.

However we often want to calculate the (average) power consumed by some piece of equipment, and the peak voltage is not ideal for this. To get an average power we need an average voltage and we get this by integrating the voltage over one cycle and dividing by the period. The trouble is that because the voltage is alternately positive and negative it integrates to zero. So instead we square the voltage, so it's always positive, integrate the square then take the square root. The result is the root mean square voltage:

$$\left(V_{RMS}\right)^2 = f\int_0^{t=1/f} V^2(t)dt$$

The 240V you mention is the root mean square voltage. When we do the integral we find the simple relationship:

$$V_{RMS} = \frac{V_p}{\sqrt{2}}$$

Using the root mean square makes calculating the electrical power straightforward.

In short, the single value for AC voltage will match the DC voltage that will do the same work.

For example: Go out to your car and remove the battery and both of the brake lights. Hook up one light to the battery. Now go get yourself a 12VAC transformer, plug it into the wall and hook up the other light to the output. Observe that both lights are of equal brightness.

Now get a good quality oscilloscope and check the output from the battery ( pure DC, 12V potential ) and the transformer ( 60hz sine, 16.9v peak voltage ).

The difference is of very little value or interest to anyone other than electrical engineers and power company employees. Everything most people come across that runs on AC (or DC) already factors this difference into the design labels, connectors etc.

And yes, we have used DC power in common residences. It's exceptionally rare these days, but you can easily find older appliances with "universal" motors that will run on either AC or DC. Malcom and Angus Young saw such a label on a sewing machine back around 1970 and decided it was a neat-sounding name for a band.