How this instrument works
RMS stands for root-mean-square: square every instantaneous value of the waveform over one full cycle, average those squares, then take the square root. That specific order — not a plain average — exists because instantaneous power in a resistor is proportional to voltage squared, so only the square-then-average-then-root sequence gives a steady value that dissipates the same heat as a DC source. For a pure sine wave the mean of sin²(θ) over a cycle works out to exactly one-half, and the square root of one-half is 1/√2, which is why V_rms = V_peak/√2 and nothing more exotic.
An electrician sizing insulation or a component's voltage rating cares about the peak, not the RMS, because breakdown happens at the instantaneous crest; a technician troubleshooting a wall socket cares about the RMS, because that is the number stamped on every appliance and the figure a standard voltmeter displays. The two are easy to confuse precisely because a 120 V outlet never sits at 120 V — it swings between +170 V and −170 V, spending most of the cycle away from either extreme, and 120.2 V is simply the effective level that would deliver identical heating from a steady DC source.
The 1/√2 factor is exact only for an undistorted sine wave. A switch-mode power supply, a dimmer-chopped lamp circuit, or a variable-frequency drive all draw current in short, spiky pulses rather than a smooth curve, and an averaging-type multimeter — one that measures the average and secretly multiplies by 1.11 assuming a sine — will read those waveforms wrong by a wide margin. Only a true-RMS meter, which genuinely squares, averages, and roots the actual waveform, gives a trustworthy number outside the sine-wave case this formula assumes.
- Enter Peak voltage — the crest value read off an oscilloscope trace or a transformer's peak rating, in volts or millivolts.
- Read RMS voltage — the effective value the formula returns, the same figure printed on equipment nameplates and shown by an ordinary voltmeter.
- Read Average (rectified) voltage — the mean level a full-wave rectifier's output settles near before any filter capacitor smooths it.
- Switch any field's unit between V and mV to match the scale of the signal you're measuring.
Worked example — 170 V peak on a US wall outlet
A US wall outlet is rated 120 V, but that number describes the RMS level, not what the waveform actually reaches. The sine wave riding on that circuit crests at 170 V — enter 170 into Peak voltage and the formula returns V_rms = 170 ⁄ √2 = 120.208152802 V, confirming the nameplate figure comes from dividing the true peak by √2, not from measuring some steady 120 V level directly.
The same 170 V peak, run through the average formula, gives V_avg = 2 × 170 ⁄ π = 108.225361302 V — noticeably lower than the RMS figure, because averaging the folded sine wave in a straight line counts less area than squaring and rooting it does. A technician troubleshooting a rectifier's DC output before any filter capacitor is added should expect a reading close to 108.2 V from a 170 V peak input, not 120 V — the RMS figure only applies to the raw AC side of the circuit.
Questions
Why is a US wall outlet rated 120 V if the wave peaks at 170 V?
Because the 120 V nameplate figure is always an RMS rating, the effective level that delivers the same heating as a steady 120 V DC source — never the instantaneous crest. The sine wave riding on that circuit actually swings up to about 170 V and back down to −170 V every cycle; dividing that peak by √2 gives 120.208152802 V, which rounds to the familiar 120 V rating. A component rated for the circuit must still tolerate the full 170 V peak, not just 120 V.
What does 'root-mean-square' actually describe?
It describes a three-step calculation performed on the waveform: square every instantaneous value, average those squares over one full cycle, then take the square root of that average. Squaring first matters because power dissipated in a resistor depends on voltage squared, so this sequence — not a simple average — produces the steady value that would deliver identical heating from a DC source. For a sine wave, that sequence always simplifies to peak ⁄ √2.
Why does the average (rectified) voltage come out lower than RMS?
Because it is computed differently: V_avg = 2 × V_peak ⁄ π takes the plain arithmetic mean of the folded (rectified) sine wave, while V_rms squares each point before averaging and rooting the result. Squaring weights the larger values more heavily, so RMS always exceeds the plain average for the same peak — for any sine wave, RMS is about 11% above the average (1.111 = (π ⁄ 2)/√2), a ratio called the form factor.
Does the V_peak/√2 formula work for any AC signal, like a square wave?
No — it only holds for an undistorted sine wave. A square wave's RMS equals its peak exactly, because every instant already sits at the extreme value, and a triangle wave's RMS is peak ⁄ √3, a different constant entirely. Feed a distorted or non-sinusoidal signal, such as a variable-frequency drive's chopped output, into an averaging-based meter that assumes 1/√2 and it will misreport; only a true-RMS meter measures any waveform shape correctly.
What is the average (rectified) voltage actually used for?
It predicts the DC level a full-wave rectifier delivers before a filter capacitor smooths the ripple — the number a technician checks first when troubleshooting a power supply's raw output stage. It is not the number a load 'feels' for heating purposes; that is what V_rms describes. Rectifier and power-supply datasheets quote this average figure specifically because it is what a bare bridge circuit, without filtering, actually produces.
How is RMS voltage related to the power delivered to a resistor?
Directly: average power equals V_rms squared divided by resistance, exactly as a DC calculation would use a steady voltage. That is the entire reason RMS exists as a concept — it lets an AC circuit's heating effect, motor torque, or lamp brightness be computed with the same P = V²⁄R formula used for DC, instead of integrating instantaneous power over the waveform by hand every time.