AC Voltage Converter: Peak to RMS

This page does one calculation, but it is the one that shows up everywhere in AC work: Vrms = Vpeak / √2, with √2 ≈ 1.4142. Enter a peak voltage and the RMS value appears immediately. The default input of 311 V returns 219.91 V, which is no accident — 220 V mains is specified by its RMS value, and the matching sine wave peaks at 220 x 1.4142 ≈ 311 V. On 120 V systems the same relationship gives a 170 V peak. RMS (root mean square) is the value that delivers the same average power into a resistive load as a DC voltage of the same figure: 120 V RMS heats a resistor exactly as much as 120 V DC.

The √2 comes out of the integral: for v(t) = Vp sin(ωt), the mean of the square over one cycle is Vp²/2, and its square root is Vp/√2 ≈ 0.7071 Vp. That derivation assumes a pure sine wave. A square wave has an RMS equal to its peak (factor 1); a triangle wave, Vp/√3 ≈ 0.577 Vp. The peak-to-RMS ratio is the crest factor: 1.414 for a sine, 1 for a square, 1.732 for a triangle. Ordinary multimeters measure the rectified average and scale it by 1.111 on the assumption that the input is sinusoidal — on distorted waveforms such as modified-sine inverter output or phase-cut dimmers they read wrong, and only a true-RMS instrument gets it right.

The direction selector handles both paths: peak to RMS divides by 1.4142 and RMS to peak multiplies by the same factor. What the tool does not do: DC offsets, non-sinusoidal waveforms, or anything involving power factor. It also accepts any number without complaint: a negative input flows straight through and returns a negative RMS, which is physically meaningless, since RMS is the root of a mean of squares and can never be negative. Use it for what it actually is — converting the amplitude of a clean sine, like mains voltage or a signal generator output, into its effective value.