Circuits & Electrical Power · RMS and peak
Two numbers for the same wave
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Two numbers for the same wave

An AC voltage is never one number. It sweeps from zero up to a crest, back through zero to an equal trough, and round again, fifty or sixty times a second. So which figure goes on the nameplate?

Not the average — the average of a full sine is exactly zero, which would make every outlet in the building look dead. Instead we ask a better question: what steady DC voltage would heat the same element at the same rate? That equivalent is the root-mean-square value, and for a sine it is Vrms=Vpeak2V_{rms} = \dfrac{V_{peak}}{\sqrt{2}} — read aloud V-R-M-S equals V-peak over root two. VrmsV_{rms} is the effective or heating-equivalent voltage in volts, VpeakV_{peak} is the crest of the wave above zero, also in volts, and 2\sqrt{2} is a bare number, 1.414, carrying no unit at all — it comes from the SHAPE of a sine and holds at any frequency.

Here is the fact to carry out of this lesson. The 120 V at the wall is RMS. The wave itself crests at about 170 V, every cycle, and the insulation, the rectifier diodes and the surge arrester all have to survive that 170 — not the 120. Meters, nameplates and code tables quote RMS unless they explicitly say otherwise; oscilloscopes draw the peak because they draw the wave.

The named trap is crossing the √2 backwards, and it is worth building one reflex against it: the peak is always the larger number. If your answer for a peak came out below the meter reading, you divided where you should have multiplied, and no amount of re-checking the arithmetic will fix a relation used upside down.