RMS, Peak and Peak-to-Peak Converter
RMS, Peak and Peak-to-Peak Converter
Enter any one of RMS, peak, peak-to-peak or rectified average and get the other three, for a sine, a square, a triangle, a rectified sine or DC — with each shape’s crest factor, form factor, power into a load, and how far a cheap average-responding meter will be out.
RMS, peak and peak-to-peak
A 10 V peak sine into an 8 Ω load
RMS, peak, peak-to-peak and average
sine: Vrms = Vpk/√2 = Vpp/(2√2), Vavg = 2Vpk/π square: Vrms = Vpk triangle: Vrms = Vpk/√3
P = Vrms² ÷ R average-responding meter reading = 1.1107 × Vavg
- RMS
- root mean square — the DC value that would heat a resistor equally
- peak
- zero to the top of the waveform (the amplitude)
- pk-pk
- the whole vertical extent, top to bottom — what a scope’s Vpp reads
- average
- the mean of the absolute value over one cycle, not the mean of the signal, which is zero for anything symmetrical
- 1.1107
- the form factor of a sine, π/(2√2) — the number a cheap meter multiplies its rectified average by
Worked example
A 10 V peak sine into an 8 Ω load
A sine's RMS is the peak divided by √2: 10 ÷ 1.4142 = 7.071 V
Peak-to-peak is twice the peak: 20 V
The rectified average is 2/π of the peak: 0.6366 × 10 = 6.366 V
Power in the load is Vrms² ÷ R = 7.0711² ÷ 8 = 6.25 W — half the 12.5 W the load would take at the instant of the peak
Crest factor 1.4142, form factor 1.1107: on a sine an average-responding meter is exactly right, which is the only shape it is right on
Every shape’s ratios, and what a cheap meter does to it
| Waveform | RMS ÷ peak | Average ÷ peak | Crest factor | Form factor | Average-responding meter error |
|---|---|---|---|---|---|
| Sine | 0.7071 | 0.6366 | 1.4142 | 1.1107 | 0.00% |
| Square (symmetrical) | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 11.07% |
| Triangle or sawtooth | 0.5774 | 0.5000 | 1.7321 | 1.1547 | -3.81% |
| Half-wave rectified sine | 0.5000 | 0.3183 | 2.0000 | 1.5708 | -29.29% |
| Full-wave rectified sine | 0.7071 | 0.6366 | 1.4142 | 1.1107 | 0.00% |
| Steady DC | 1.0000 | 1.0000 | 1.0000 | 1.0000 | n/a — an AC range is capacitively coupled and reads nothing on DC |
What RMS actually means, and why the shape decides everything
RMS is not an average, and it is not a fudge factor. It is defined by what it does: the RMS value of a varying voltage is the steady DC voltage that would dissipate the same power in the same resistor. Power in a resistor goes as v², so you square the waveform, take the mean of that over a whole cycle, and take the root — root, mean, square, read backwards. Once you have it, P = Vrms²/R works exactly as it does for DC, which is the entire point of the quantity.
Why a sine’s RMS is 0.707 of its peak. The mean of sin² over a cycle is exactly ½, so the RMS is 1/√2 of the amplitude. A square wave spends every instant at full amplitude, so its mean square is the amplitude squared and its RMS is its peak. A triangle spends most of the cycle nowhere near its peak, so its RMS is 1/√3 = 0.577 of the amplitude. Those three numbers are the whole of the subject; everything else is bookkeeping about peak-to-peak and rectified averages.
The warning that costs people money. A moving-coil meter, and most cheap digital multimeters, do not measure RMS at all. They rectify the signal, average it, and multiply by 1.1107 — the form factor of a sine — so the scale reads correctly on a sine and on nothing else. Feed one a square wave and it reads 11.07% high; feed it a triangle and it reads -3.81% low; feed it a half-wave rectified sine and it is -29.29% low. Fluke’s own note on the subject records errors “up to 40 percent” on the chopped current waveforms that switch-mode supplies and dimmers draw. If you are measuring anything a semiconductor has been switching — a PWM duty cycle calculator output, a phase-angle dimmer, a rectifier’s input current — you need a true-RMS meter, and even then you need to check its crest factor rating.
Crest factor is the other half of the story. It is the peak divided by the RMS: 1.414 for a sine, 1 for a square, 1.732 for a triangle, and anything up to 5 or more for the current pulses a capacitor-input rectifier draws. A high crest factor is what clips amplifiers and saturates transformers long before the RMS figure looks alarming, and it is why an amplifier rated for so many watts RMS into 8 Ω still needs headroom. Peak power in the load is Vpk²/R, which for a sine is exactly twice the average.
For the decibel arithmetic that usually sits beside these voltages, use the decibel calculator; to turn an RMS voltage into dBm across a named impedance, the dBm to watts calculator. The duty-cycle version of the same question — a rectangular wave that is not symmetrical — is on the PWM duty cycle calculator, and Ohm’s law for the load itself is the Ohm’s law calculator.
Frequently asked questions
How do I convert peak to RMS?
Divide by the crest factor of the shape. For a sine that is √2, so RMS = 0.7071 × peak. For a square wave the crest factor is 1 and the RMS equals the peak; for a triangle it is √3, so RMS = 0.5774 × peak. There is no single conversion — the shape decides.
What is peak-to-peak on an oscilloscope?
The whole vertical extent of the trace, from the bottom of the waveform to the top. For anything symmetrical about zero that is twice the peak, so a sine of 10 V peak reads 20 V peak-to-peak and 7.071 V RMS. A rectified sine never goes negative, so its peak-to-peak and its peak are the same number.
Why does my multimeter read the wrong voltage on a square wave?
Because it probably is not measuring RMS. An average-responding meter rectifies the signal, takes the mean and multiplies by 1.1107, which is correct only for a sine. On a symmetrical square wave that reading is 11.07% high. A true-RMS meter squares the waveform internally and gets it right on any shape within its crest factor rating.
Is 230 V mains the RMS or the peak voltage?
The RMS. A 230 V RMS sine has a peak of 325.3 V and a peak-to-peak of 650.5 V, which is why equipment on a 230 V supply needs parts rated well above 230 V and why a rectifier and smoothing capacitor across it charge to about 325 V.
What is the average value of a sine wave?
Over a whole cycle it is zero, because the negative half cancels the positive half. What meters and textbooks call the average is the RECTIFIED average — the mean of the absolute value — which is 2/π = 0.6366 of the peak. This page reports that one.
How do I get power from an RMS voltage?
P = Vrms² ÷ R, exactly as for DC — that is what RMS is for. A 10 V peak sine into 8 Ω gives 6.25 W. Using the peak voltage instead would give 12.5 W, which is the instantaneous power at the top of the waveform, not the average.
Related calculators
References
- Fluke Corporation. Troubleshooting the electrical distribution system, application note. Average-responding meters “capture the rectified average of an ac waveform and scale the number by 1.1 to calculate the rms value”; the note’s table gives such a meter as “10% high” on a square wave and “40% low” on a single-phase diode rectifier load.
- Horowitz P, Hill W. The Art of Electronics, 3rd ed. Cambridge University Press, 2015. §1.3.2 on the decibel and the two factors, and Appendix A on the oscilloscope — what a measured amplitude actually is.
- International Electrotechnical Commission. IEC 60027-3:2002, Letter symbols to be used in electrical technology — Part 3: Logarithmic and related quantities, and their units. The standard that defines the decibel and the neper, and the rule that a power quantity takes 10 log10 while a root-power (field) quantity takes 20 log10.
- Bureau International des Poids et Mesures. The International System of Units (SI Brochure), 9th ed., 2019 — the watt as one joule per second, which is what makes the root-mean-square the right average to take for heating.