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Vrms = Vpeak / √2
Vrms = Vpp / (2√2)
RMS (Root Mean Square) voltage is the effective voltage that would produce the same power dissipation in a resistive load as a DC voltage. For a pure sine wave, Vrms ≈ 0.707 × Vpeak.
| Peak (V) | RMS (V) | Peak-to-Peak (V) |
|---|---|---|
| 1 | 0.71 | 2 |
| 5 | 3.54 | 10 |
| 10 | 7.07 | 20 |
| 120 | 84.85 | 240 |
| 170 | 120.20 | 340 |
| 311 | 220.00 | 622 |
| 325 | 230.00 | 650 |
RMS (Root Mean Square) voltage is the effective value of an AC voltage that produces the same heating effect in a resistive load as an equivalent DC voltage. For a sinusoidal waveform, Vrms = Vpeak / √2 ≈ Vpeak × 0.7071. The RMS value is obtained by squaring the instantaneous voltage values over one complete cycle, taking the mean (average) of those squared values, and then taking the square root of that mean. This mathematical process gives RMS voltage its name and its physical significance. When someone refers to "120 V" or "230 V" mains power, they are quoting the RMS voltage, which is the standard way AC voltages are specified worldwide on equipment, in building codes, and on electrical test instruments.
Determine the peak (maximum) voltage of the AC waveform. This can be measured with an oscilloscope or derived from the peak-to-peak voltage by dividing by 2 (for symmetrical waveforms).
For a pure sine wave, divide the peak voltage by √2: Vrms = Vpeak ÷ 1.4142. Alternatively, multiply by 0.7071. This relationship is derived from integrating the square of the sine function over one complete period.
If you have the peak-to-peak voltage, use Vrms = Vpp / (2√2) = Vpp × 0.3536. This is useful when reading waveform amplitude directly from an oscilloscope display.
The √2 factor only applies to sine waves. For square waves, Vrms = Vpeak. For triangle waves, Vrms = Vpeak / √3. For complex waveforms, a true-RMS meter or numerical integration is required.
RMS voltage is essential for calculating real power in AC circuits. The power formula P = V²/R uses RMS voltage directly, giving the same result as the equivalent DC power. Using peak voltage in power calculations would overestimate the actual power by a factor of 2.
All electrical equipment, from household appliances to industrial machinery, is rated in RMS voltage and current. This standardization ensures that power ratings, fuse sizes, and wire gauges are based on the effective heating value, providing consistent safety and performance specifications.
Multimeters display RMS voltage because it is the most practical value for technicians and engineers. True-RMS meters accurately measure non-sinusoidal waveforms from variable frequency drives, dimmers, and switching power supplies where average-responding meters would give incorrect readings.
| Waveform | Vrms / Vpeak | Crest Factor | Form Factor |
|---|---|---|---|
| Sine Wave | 0.7071 (1/√2) | 1.414 | 1.111 |
| Square Wave | 1.000 | 1.000 | 1.000 |
| Triangle Wave | 0.5774 (1/√3) | 1.732 | 1.155 |
| Sawtooth Wave | 0.5774 (1/√3) | 1.732 | 1.155 |
| Half-Wave Rectified Sine | 0.500 | 2.000 | 1.571 |
| Full-Wave Rectified Sine | 0.7071 | 1.414 | 1.111 |
| PWM (50% Duty Cycle) | 0.7071 | 1.414 | 1.414 |
RMS stands for Root Mean Square. It is calculated by taking the square root of the mean (average) of the squared instantaneous values over one full cycle. Mathematically, Vrms = √(1/T × ∫ v(t)² dt) over one period T. The squaring makes all values positive, the mean averages them, and the square root brings the result back to voltage units.
RMS voltage directly relates to power dissipation. A 120 V RMS AC source delivers the same power to a resistor as a 120 V DC source. Average voltage (for a full sine cycle) is zero due to symmetry, and peak voltage would overstate the effective power delivery. RMS is the only AC measurement that equates directly to DC for power calculations.
Average-responding meters measure the rectified average voltage and multiply by 1.111 (the form factor for a sine wave) to display an RMS-equivalent reading. This is only accurate for pure sine waves. True-RMS meters actually compute the RMS value mathematically and give correct readings for any waveform shape, including distorted signals from electronic loads.
Yes, but not with the simple Vpeak / √2 formula. For complex waveforms, you need either a true-RMS measuring instrument or numerical computation. If the waveform can be expressed as a Fourier series, the total RMS is the square root of the sum of squares of each harmonic's RMS value: Vrms = √(V₁² + V₂² + V₃² + ...).
All standard AC power formulas use RMS values. Real power P = Vrms × Irms × cos(φ), apparent power S = Vrms × Irms, and reactive power Q = Vrms × Irms × sin(φ). Using RMS values ensures that AC power calculations give the correct time-averaged results that correspond to actual energy consumption and heat generation.
Convert RMS voltage to peak voltage for component selection and insulation design.
Calculate the full voltage swing of an AC waveform from negative to positive peak.
Calculate electrical power from voltage, current, and resistance using standard power formulas.