RMS and Peak Voltage

An alternating voltage spends most of its time somewhere below its peak and averages zero over a cycle, so "the" voltage of an AC supply needs defining before it means anything. The definition that earns its keep is the heating-equivalent one: the DC voltage that would deliver the same power to a resistor. Since power goes as $V^2$, you square the waveform, take the mean of the square over a cycle, and take the root of that — root-mean-square, done in that order. For a sine wave the mean of $\sin^2$ over a cycle is exactly $1/2$, so the RMS value is $V_{peak}/\sqrt{2}$, about 0.707 of the peak. The $\sqrt{2}$ is not a convention or a fudge; it is the square root of that one-half, and it belongs to the sine and to nothing else.

North American mains at 120 V RMS actually swings to $120 \times 1.414 = 170\ \text{V}$ either side of neutral, 340 V peak to peak. A 230 V European supply reaches 325 V peak, 650 V peak to peak. A 24 V control transformer delivers 34 V peaks. Those peak numbers are the ones that matter when you choose an insulation rating, a rectifier's reverse voltage, or the working voltage of a filter capacitor — a 200 V capacitor across a 120 V circuit is not a comfortable margin, it is already an underrating.

The whole reason for the convention is that it makes the DC power formulas keep working. Feed 120 V RMS into $P = V^2/R$ and the answer is the true average power in watts, no correction needed, which is precisely the property RMS was constructed to have. That is why meters, nameplates and every distribution standard quote RMS, and it is why the RMS value is what a thermal instrument naturally measures — an old thermocouple-type meter reads the heating effect directly and is right by construction.

Now the traps, and there are several worth knowing. The $\sqrt{2}$ applies to a sine wave only. A square wave's RMS equals its peak; a triangle wave's is peak over $\sqrt{3}$; a rectified or chopped waveform is something else again. Second, and this catches working electricians: an ordinary averaging multimeter does not measure RMS at all. It rectifies, takes the average, and multiplies by 1.11 — a factor that assumes a sine. On the distorted current drawn by LED drivers, variable-frequency drives and switch-mode supplies, that meter can read 20 to 40% low, and only an instrument marked true RMS will tell you the truth. Third, keep peak and peak-to-peak straight: an oscilloscope shows peak-to-peak, which is twice the peak and $2\sqrt{2}$ times the RMS. Fourth, RMS current squared times resistance gives real heating, but RMS volts times RMS amps gives volt-amperes, not watts — the power factor still has to be applied. And never mix them within one calculation: peak volts with RMS amps produces an answer that is wrong by 41% and looks entirely reasonable.