What is RMS voltage?
RMS (root mean square) voltage is the square root of the mean of v² over a period: the DC voltage that dissipates the same power in a resistor. The 120 V on a US outlet is an RMS value.
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RMS voltage calculator
This RMS voltage calculator converts between the peak, peak-to-peak, RMS and average values of sine, square, triangle, sawtooth, PWM and rectified waveforms, with an optional DC offset. It also shows the crest factor, the form factor and what an average-responding multimeter would read.
I know the
Examples
V
Amplitude before any DC offset: the highest point above 0 V with no offset (the high level for PWM).
V
Optional: a DC level added to the whole waveform.
Ω
Optional: average and peak power in a resistor.
RMS voltage
3.536 V
- Peak-to-peak
- 10 V
- Average of |v| (rectified)
- 3.183 V the plain mean is 0
- Crest factor (peak ÷ RMS)
- 1.414 3.01 dB
- Form factor (RMS ÷ average)
- 1.111
- Average-responding meter, AC-coupled
- 3.536 V same as true RMS
For a waveform centered on 0 V the plain average is zero, so “average” here means the average of the rectified waveform, as meters use it.
How do you calculate RMS voltage?
RMS is the square root of the average of v² over one period. For a sine, Vrms = Vpeak ÷ √2 = 0.7071 × Vpeak, so 120 V rms mains peaks at 169.7 V. A ±Vp square wave has Vrms = Vp, a triangle or sawtooth Vp ÷ √3 (0.5774 × Vp), and a 0-to-Vp PWM signal Vp × √D.
RMS formula for common waveforms
The RMS value of any periodic voltage is the square root of the mean of its square. For the standard waveforms the integral has a closed form in terms of the peak Vp (the amplitude, or the high level of a pulse).
Vrms = √( (1 ÷ T) × ∫ v(t)² dt ) Vavg = (1 ÷ T) × ∫ |v(t)| dt
- T
- period
- Vp
- peak (amplitude) voltage
- D
- duty cycle of a pulse, 0 to 1
Sine: Vp ÷ √2 Square (±Vp): Vp Triangle, sawtooth: Vp ÷ √3
PWM (0 to Vp): Vp × √D Half-wave rectified: Vp ÷ 2 Full-wave rectified: Vp ÷ √2
With a DC offset, and the two ratios that describe the shape:
Vrms = √(Vdc² + Vac,rms²) crest factor = Vpeak ÷ Vrms form factor = Vrms ÷ Vavg
Worked examples
Example 1: 120 V mains
The 120 V on a US outlet is an RMS value. The peak is 120 × √2 = 169.7 V, the peak-to-peak swing is 339.4 V, and the rectified average is 108 V. Capacitors and semiconductors across the line must be rated for the peak, not the RMS value.
Example 2: a 12 V PWM heater at 25 % duty
Vrms = 12 × √0.25 = 6 V and Vavg = 12 × 0.25 = 3 V. In a 10 Ω heater the power is Vrms² ÷ R = 3.6 W, which is also 25 % of the 14.4 W drawn while the switch is on. Using the average voltage would give 900 mW, a quarter of the real figure.
Example 3: a sine on a DC bias
A 1 V peak sine centered on 2.5 V (a typical single-supply ADC input) swings from 1.5 V to 3.5 V. Its AC part is 707.1 mV rms, and the total is √(2.5² + 0.7071²) = 2.598 V. A meter on AC reads the first number; a DC-coupled true-RMS measurement reads the second.
RMS, average, form factor and crest factor by waveform
Multiply the peak voltage by the ratio to get the RMS or rectified average value. PWM rows are pulses from 0 V to Vp.
| Waveform | RMS ÷ peak | Average ÷ peak | Form factor | Crest factor |
|---|---|---|---|---|
| Sine | 0.7071 | 0.6366 | 1.111 | 1.414 |
| Square (±Vp) | 1 | 1 | 1 | 1 |
| Triangle | 0.5774 | 0.5 | 1.155 | 1.732 |
| Sawtooth | 0.5774 | 0.5 | 1.155 | 1.732 |
| Half-wave rectified sine | 0.5 | 0.3183 | 1.571 | 2 |
| Full-wave rectified sine | 0.7071 | 0.6366 | 1.111 | 1.414 |
| PWM, 50 % (0 to Vp) | 0.7071 | 0.5 | 1.414 | 1.414 |
| PWM, 25 % | 0.5 | 0.25 | 2 | 2 |
| PWM, 10 % | 0.3162 | 0.1 | 3.162 | 3.162 |
What an average-responding meter reads
An AC-coupled, average-responding multimeter measures the rectified average of the AC part and multiplies by 1.111, the sine form factor. Values are for a 1 V peak waveform; the true RMS is of the AC part, which is what an AC-coupled true-RMS meter shows.
| Waveform (1 V peak) | True RMS (AC) | Average-responding meter | Error |
|---|---|---|---|
| Sine | 707.1 mV | 707.1 mV | 0 % |
| Square (±Vp) | 1 V | 1.111 V | +11.1 % |
| Triangle | 577.4 mV | 555.4 mV | −3.81 % |
| Sawtooth | 577.4 mV | 555.4 mV | −3.81 % |
| Half-wave rectified sine | 385.6 mV | 389.7 mV | +1.06 % |
| Full-wave rectified sine | 307.8 mV | 297.7 mV | −3.26 % |
| PWM, 50 % (0 to Vp) | 500 mV | 555.4 mV | +11.1 % |
| PWM, 25 % | 433 mV | 416.5 mV | −3.81 % |
| PWM, 10 % | 300 mV | 199.9 mV | −33.4 % |
Measuring RMS in practice
The formulas assume ideal edges and a stable period. Real PWM has finite rise and fall times and ringing, which change the RMS slightly; a narrow pulse train also has a high crest factor, and many meters and RMS converter ICs lose accuracy above their rated crest factor, so check that figure in the datasheet. Bandwidth matters as well: a meter that cannot follow the harmonics of a fast edge reads low. For anything that is not a clean sine, use a true-RMS meter or an oscilloscope’s RMS measurement over a whole number of periods, and decide whether you want the AC-coupled reading or AC + DC. For power in a resistor, always use the RMS value; for the voltage rating of a part, use the peak.
Glossary
- RMS (root mean square)
- The square root of the mean of v² over a period: the DC voltage that dissipates the same power in a resistor.
- Peak (Vp)
- The largest value of the waveform above 0 V; the amplitude for waveforms centered on zero.
- Peak-to-peak (Vpp)
- The span from the lowest to the highest point. Twice the peak for waveforms centered on zero.
- Rectified average
- The mean of |v| over a period. Average-responding meters measure this.
- Crest factor
- Peak ÷ RMS. 1.414 for a sine; large for narrow pulses and noise.
- Form factor
- RMS ÷ rectified average. 1.111 for a sine, which is the scale factor an average-responding meter applies.
- Duty cycle
- The fraction of each period a pulse waveform spends at its high level.
- True RMS meter
- A meter that computes the RMS value directly instead of scaling a peak or average reading.
Related guides
- dBm to watts converter: RMS and peak-to-peak voltage of an RF power level.
Frequently asked questions
What is the RMS voltage formula?
Square the waveform, average the squares over one period, and take the square root. For a sine that gives Vrms = Vpeak ÷ √2 = 0.7071 × Vpeak, so a 10 V peak sine is 7.071 V rms.
How do you convert peak to RMS and RMS to peak?
For a sine, multiply the peak by 0.7071 to get RMS, and multiply RMS by 1.414 to get the peak. Other waveforms use the RMS ÷ peak ratios in the table on this page.
How do you convert Vpp to Vrms and Vrms to Vpp?
For a sine, Vrms = Vpp × 0.3536, because the peak is half the swing. Going the other way, Vpp = Vrms × 2.828, so 120 V rms swings 339.4 V peak to peak.
What is the RMS value of a 10 V peak-to-peak sine wave?
3.536 V. The peak is half the swing (5 V), and 5 V ÷ √2 = 3.536 V.
What is the RMS of a square wave?
A square wave that swings ±Vp has an RMS value equal to Vp, because v² is Vp² all the time. A logic-level square wave from 0 to Vp at 50 % duty is Vp × √0.5 = 0.7071 × Vp, so a 3.3 V clock is 2.333 V rms.
How do you calculate the RMS of a PWM signal?
For a pulse from 0 to Vp with duty cycle D, Vrms = Vp × √D and the average is Vp × D. A 12 V PWM at 25 % is 6 V rms but only 3 V average, and it is the RMS value that sets the heating in a resistive load.
What is the difference between RMS and average voltage?
RMS is the DC voltage that would heat a resistor equally; average is the mean of the waveform. For a sine centered on zero the plain average is 0, so meters use the average of the rectified wave instead: 0.6366 × Vpeak, against 0.7071 × Vpeak for RMS.
What is crest factor?
Crest factor is peak ÷ RMS. It is 1.414 for a sine, 1 for a ±Vp square wave, 1.732 for a triangle, and 1 ÷ √D for a pulse train, so a 1 % duty pulse has a crest factor of 10. High crest factors are hard for RMS meters and converters to measure accurately.
Why does my multimeter read a square wave wrong?
An average-responding meter measures the rectified average and multiplies it by 1.111, the ratio that is correct for a sine. On a ±Vp square wave it reads 11.1 % high and on a triangle 3.81 % low. A true-RMS meter reads both correctly within its crest-factor and bandwidth limits.
How do you add a DC offset to an RMS value?
The AC and DC parts add as squares: Vrms = √(Vdc² + Vac,rms²). A 1 V peak sine on a 2.5 V bias is √(2.5² + 0.7071²) = 2.598 V rms.
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Sources
- Keysight 5988-6916EN, Make Better AC RMS Measurements with Your Digital Multimeter (crest factor, pulse-train RMS, AC + DC)
- Dataforth AN128, RMS Revisited (Table 1: RMS of square, pulse, triangle, sine and rectified waveforms)
- Rohde & Schwarz Application Note 1MA98, dB or not dB? (§5.10, crest factor of sine and noise-like signals)
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