RMS, peak, peak-to-peak and average: what they mean and how to convert them
An AC waveform has no single "voltage". Its peak is the highest it reaches, its peak-to-peak the full swing a scope shows, its average the mean of the rectified wave, and its RMS the DC voltage that would heat a resistor equally, which is why the mains is called 230 V although it swings to 325 V. For a given shape the four are fixed multiples of each other: RMS = peak/√2 for a sine, peak for a square wave, peak/√3 for a triangle, peak × √duty for a pulse train. This calculator converts between them for any shape, gives the crest and form factors, shows what a cheap meter would read, and the power into a load.
How to use the RMS calculator
- Type the one figure you know and say which it is: RMS, peak, peak-to-peak or average. Switching the selector converts the value you typed.
- Pick the shape: sine, square, triangle, sawtooth or pulse train (with its duty). Volts or amps, as you like.
- Add a load resistance for the power. The drawing shows two cycles with the peak, RMS and average levels across it; the All waveforms tab compares every shape at the same figure.
What RMS is
RMS = √( mean of v² over a cycle )
Root mean square: square the waveform, average it over a cycle, take the square root. The squaring is what makes it useful: the power in a resistor is v²/R at every instant, so the average power is (mean of v²)/R = RMS²/R, exactly as if a DC voltage equal to the RMS were applied. That is why RMS is the figure on the mains, on transformers and on anything to do with heating or power, and why a 230 V RMS supply that peaks at 325 V delivers the same power to a heater as 230 V DC would.
The factors for each shape
| Waveform | RMS / peak | Average / peak | Crest factor | Form factor |
|---|---|---|---|---|
| Sine | 0.7071 | 0.6366 | 1.414 | 1.111 |
| Square | 1 | 1 | 1 | 1 |
| Triangle | 0.5774 | 0.5 | 1.732 | 1.155 |
| Sawtooth | 0.5774 | 0.5 | 1.732 | 1.155 |
| Pulse train (25 % duty) | 0.5 | 0.25 | 2 | 2 |
The sine's 1/√2 comes from sin² averaging to one half; the square wave is at full amplitude all the time; the triangle and sawtooth are straight ramps, and a ramp's square averages to a third. The crest factor (peak/RMS) says how spiky a wave is, and the form factor (RMS/average) is what calibrates a meter.
Meters and the form factor
Most inexpensive multimeters rectify the AC and measure its average, then multiply by the sine wave's form factor, 1.111, so that the display reads RMS. That works only for sine waves. On a square wave the meter reads 11 % high, on a triangle 4 % low, and on a narrow pulse train or a phase-controlled dimmer it can be wildly wrong. A true-RMS meter computes the real root mean square and is right for any shape up to its crest-factor limit (typically 3 to 5). This calculator's "meter reads" figure is the average-responding reading for the wave you entered.
Pulse trains and PWM
RMS = peak × √D DC = average = peak × D AC part = peak × √(D(1 − D))
A PWM signal from 0 to Vpeak with duty D is at the peak for the fraction D of the time, so the mean of v² is V² D and the RMS is V √D: at 25 % duty the RMS is half the peak, not a quarter. The DC component is V D, which is what a filter or a motor's inertia sees; the rest, V √(D(1 − D)), is the switching ripple. Into a resistive load the heating follows the RMS, so a 25 % duty delivers 25 % of the full power even though the RMS voltage is 50 %.
Your waveform, step by step
- Peak: 325.3 V; peak-to-peak 650.5 V.
- RMS: 325 V × 0.7071 = 230 V; average 325 V × 0.6366 = 207.1 V.
- Factors: crest 1.414, form 1.111; an average-responding meter reads 230 V; 529 W into 100 Ω.
Worked example: the 230 V mains into a 100 Ω heater
230 V is the RMS of a sine wave, so the peak is 230 × √2 = 325.3 V and the peak-to-peak 650.5 V: a capacitor after a rectifier charges towards 325 V, and the parts must be rated for it. The rectified average is 230 × 2√2/π = 207.1 V, and an average-responding meter, scaled by 1.111, shows 230 V: correct, because the wave is a sine. Into a 100 Ω heater the power is 230² / 100 = 529 W, with 2.3 A RMS flowing and 1.06 kW at the crest of each cycle. The same 230 V RMS as a square wave would have a peak of only 230 V; as a triangle a peak of 398 V.
Questions
Is the mains 230 V or 325 V?
Both: 230 V RMS, 325 V peak, 650 V peak-to-peak. Power sums use 230 V; voltage ratings must cover 325 V.
Why does my meter read wrong on a square wave?
It is average-responding and calibrated for sines: it reads the average times 1.111, which on a square wave is 11 % too high. Use a true-RMS meter.
Does RMS apply to current too?
Yes, with the same factors: P = IRMS² R. Switch the calculator to amps.
What about a wave with a DC offset?
The total RMS is √(DC² + ACRMS²). This calculator handles the pulse train, which has one; for an offset sine add the squares yourself.