Free RMS Voltage Calculator

Understanding Root Mean Square (RMS)

The root mean square (RMS) is a statistical measure that represents the magnitude of a varying quantity. For a discrete set of numbers $y_1, y_2, \dots, y_N$, the RMS is defined as the square root of the arithmetic mean of their squares:

$$Y_{\text{RMS}} = \sqrt{\frac{1}{N}\sum_{i=1}^{N} y_i^{2}}.$$

This is distinct from the simple arithmetic average; the RMS gives more weight to larger values and is especially meaningful when dealing with alternating signals.

RMS for Periodic Continuous‑Time Functions

Most real‑world AC voltages and currents are continuous‑time functions that repeat at regular intervals. Such a periodic function $y(t)$ has a period $T$ and its RMS over one period is:

$$Y_{\text{RMS}} = \sqrt{\frac{1}{T}\int_{0}^{T} [y(t)]^{2}\, dt}.$$

Because the waveform is identical in every cycle, the RMS over a single period is the same as the RMS over the entire signal. If a waveform can be expressed as the sum of several same‑period components $y_1(t), y_2(t), \dots$, the overall RMS is obtained by combining the individual RMS values in quadrature:

$$Y_{\text{RMS}} = \sqrt{Y_{1,\text{RMS}}^{2} + Y_{2,\text{RMS}}^{2} + \cdots}.$$

This superposition property is crucial when a waveform includes a DC offset.

Why RMS Matters for AC Voltages

In alternating current (AC) circuits, the RMS voltage is the equivalent direct current (DC) voltage that would dissipate the same real power in a purely resistive load. For any periodic AC waveform $v(t)$, its RMS value $V_{\text{RMS}}$ therefore represents the heating effect of the waveform. This is why AC voltage ratings (e.g., 120 V household power) are almost always expressed as RMS values: they directly indicate the effective power‑delivering capability.

How to Use This RMS Voltage Calculator

The tool simplifies RMS computations for the most common waveforms. Follow these steps:

1. Select the waveform shape – choose from sine, square, triangle, sawtooth, half‑wave rectified sine, or full‑wave rectified sine.
2. Pick the characteristic voltage type – indicate whether you know the peak voltage ($V_p$), peak‑to‑peak voltage ($V_{pp}$), or average voltage ($V_{avg}$).
3. Enter the value – input the numerical magnitude and, if needed, a DC offset $V_0$.
4. Read the result – the calculator immediately displays the corresponding RMS voltage. You can also work backwards by supplying the RMS to find the characteristic voltage or the DC offset.

Calculating RMS Voltage for Common Waveforms

The RMS formula for a given waveform depends on its shape and on which characteristic voltage is known. The table below lists the expressions without any DC offset. If a DC offset $V_0$ is present, the total RMS becomes:

$$V_{\text{RMS,total}} = \sqrt{V_{\text{RMS,AC}}^{2} + V_{0}^{2}},$$

where $V_{\text{RMS,AC}}$ is the AC‑only value from the table.

Waveform	$V_{\text{RMS}}$ from $V_p$	$V_{\text{RMS}}$ from $V_{pp}$	$V_{\text{RMS}}$ from $V_{avg}$
Sine	$\displaystyle \frac{V_p}{\sqrt{2}}$	$\displaystyle \frac{V_{pp}}{2\sqrt{2}}$	$\displaystyle \frac{\pi V_{avg}}{2\sqrt{2}}$
Square	$V_p$	$\displaystyle \frac{V_{pp}}{2}$	$V_{avg}$
Triangle	$\displaystyle \frac{V_p}{\sqrt{3}}$	$\displaystyle \frac{V_{pp}}{2\sqrt{3}}$	$\displaystyle \frac{\pi V_{avg}}{2\sqrt{3}}$
Sawtooth	$\displaystyle \frac{V_p}{\sqrt{3}}$	$\displaystyle \frac{V_{pp}}{2\sqrt{3}}$	$\displaystyle \frac{\pi V_{avg}}{2\sqrt{3}}$
Half‑wave rectified sine	$\displaystyle \frac{V_p}{2}$	$\displaystyle \frac{V_{pp}}{4}$	$\displaystyle \frac{\pi V_{avg}}{2}$
Full‑wave rectified sine	$\displaystyle \frac{V_p}{\sqrt{2}}$	$\displaystyle \frac{V_{pp}}{2\sqrt{2}}$	$\displaystyle \frac{\pi V_{avg}}{2\sqrt{2}}$

Note that triangle and sawtooth waves share the same RMS formulas. For a half‑wave rectified sine, the RMS is exactly half the peak, while a full‑wave rectified sine has the same RMS as a pure sine wave of the same peak.

Incorporating a DC Offset

When a constant DC voltage $V_0$ is added to any AC waveform, the total RMS is found by summing the powers of the AC and DC components. For example, a sine wave with peak amplitude $V_p$ and offset $V_0$ yields:

$$V_{\text{RMS,total}} = \sqrt{\left(\frac{V_p}{\sqrt{2}}\right)^{2} + V_0^{2}}.$$

This quadrature addition holds for all waveform shapes handled by the calculator, allowing you to account for both alternating and direct parts in a single computation.