Lessons · Engineering · RMS of a sine wave
RMS: the DC voltage that would heat the same wire the same amount
For a sine wave, V_rms = V_peak / √2, about 0.707 of the peak. Power and heating use rms, so meters and ratings are in rms.
Hone is a place to practise a career, one idea a day. This is one of its lessons, written out in full and free to read without an account.
What it is for
An oscilloscope shows 339 V on a wire the label calls 240 V. Both are right: one is the peak of the wave, the other is what a heater feels. Put the peak into a power formula and every heater in the building is sized double.
How to think about it
Peak to rms: divide by 1.414. Rms to peak: multiply by 1.414. Anything about power, heating or a rating wants rms; anything about insulation or a capacitor's voltage limit wants peak.
Worked example
V_rms = V_peak / √2 = 0.707 × V_peakThe rule for a sine wave, and only for a sine wave.
A sine wave with a 100 V peak: V_rms = 100 / 1.414 = 70.7 VPeak to rms.
The other way: 240 V rms mains has a peak of 240 × 1.414 = 339.4 VRms to peak. That is the number the insulation has to survive.
Power in a 24 Ω heater: P = V_rms² / R = 240² / 24 = 2400 WRms in the power formula, because rms is defined as the DC voltage that heats the same.
Your turn
Peak 50 V. Write the rms.
V_rms = 50 / 1.414 = V
Solve one, graded on the server
The trap
Using the peak in a power calculation. Peak squared is twice rms squared, so the power comes out double. Wiring and heaters are sized on rms; only the insulation cares about the peak.