RMS Voltage Calculator

RMS Voltage Calculator finds RMS from a peak, peak-to-peak or average voltage for six wave shapes including PWM, turns RMS back into peak, and gives the power in a load resistance.

RMS Voltage Calculator Formulas for Peak, Vp-p and Average

RMS stands for root mean square. To calculate RMS voltage from first principles, square the voltage at every instant, take the mean of the squares over one cycle, and take the square root. Tony Kuphaldt's Lessons in Electric Circuits describes the result as an AC quantity expressed in terms functionally equivalent to DC.

For a known wave shape the squaring is already done, and one factor is enough. The RMS Voltage Calculator takes a peak, peak-to-peak or average voltage and applies the factor for the chosen shape. Choosing RMS as the known value runs it backward and returns the peak. The peak to RMS formula for a sine wave divides by the square root of two.

$$V_{RMS}=\frac{V_P}{\sqrt{2}}$$

That is 0.7071 times the peak. The default entry of 40 V peak returns 28.28 V DC Equivalent, and the first card adds the 80 V peak-to-peak swing and the 25.46 V average.

$$V_{RMS}=\frac{V_{P-P}}{2\sqrt{2}}$$

Converting peak-to-peak to RMS multiplies by 0.3536, so a 20 V peak-to-peak sine wave is 7.071 V RMS. Starting from the average of the rectified sine multiplies by 1.111 instead.

$$V_{RMS}=\frac{\pi \times V_{AVG}}{2\sqrt{2}}$$

The Formulas With Your Numbers panel of the RMS Voltage Calculator prints the line used, such as 40 V × 0.7071 = 28.28 V. A reading of 30 volts RMS goes in with RMS Voltage as the known value. The answer is 42.43 V Peak of the Wave, with an 84.85 V peak-to-peak swing.

What RMS Voltage Means

RMS voltage is the DC voltage that would heat the same resistor by the same amount. Fluke's guide to true-RMS meters calls it the effective, or DC heating, value of any AC wave shape. That is why the main result of the RMS Voltage Calculator carries the label DC Equivalent.

RMS is used for AC because the plain average of a symmetric wave is zero, which says nothing about the power it delivers. Squaring removes the sign, so both halves of the cycle count. RMS itself is neither AC nor DC. It is a way of measuring either, and the RMS of a steady DC voltage is that same voltage.

RMS does not need converting to DC for power work, since it already is the DC equivalent. A rectifier is a different matter. The rectifier chapter of the same textbook puts the loss in a silicon bridge at two diode drops, or 1.4 V. A 120 V RMS supply with a filter capacitor therefore charges toward its 169.7 V peak less 1.4 V.

No single figure is a good RMS value. The right one matches the rating on the equipment, because nameplates, outlets and meters all quote RMS. For a speaker, it is the voltage its power rating works out to in the table further down.

RMS by Wave Shape

The 0.7071 factor is for sine waves only. The Wave and Load panel of the RMS Voltage Calculator offers six shapes, and the RMS by Wave Shape panel lists all six at the peak entered. At the 40 V default the spread runs from 20 V to 40 V.

Crest factor is peak divided by RMS, and form factor is RMS divided by average. The fourth card of the RMS Voltage Calculator shows both. Kuphaldt's textbook lists crest factors of 1.414, 1 and 1.732 and form factors of 1.11, 1 and 1.15 for sine, square and triangle waves.

The Waveform Diagram panel draws two cycles of the chosen shape. Lines mark the peak, the RMS level and the average, so the gap between them is visible. On a square wave the three lines sit on top of each other, and on a narrow pulse they spread far apart.

The form factor decides the answer when the known value is an average. A triangle wave with a 20 V average has a 40 V peak and 23.09 V RMS, which is 1.155 times the average. Using the sine factor of 1.111 on that wave would give 22.21 V and understate it.

For the two rectified shapes the peak-to-peak entry means the source sine before the diodes. A 120 V RMS supply peaks at 169.7 V. Half-wave rectified it measures 84.85 V RMS with a 54.02 V average, and full-wave rectified it stays at 120 V RMS with a 108 V average.

DC Offset and PWM RMS

A wave that rides on a DC level heats a load more than the wave alone. For sine, square and triangle shapes the two parts combine as the square root of the sum of squares.

$$V_{RMS}=\sqrt{V_{AC}^2+V_0^2}$$

A 5 V peak sine wave is 3.536 V RMS. On a 2.5 V offset the RMS Voltage Calculator returns 4.33 V, and the result note changes to DC Offset Is Included. A negative offset gives the same answer, because the offset is squared.

Rectified and pulse shapes already contain a DC level of their own, so a third term appears. The tool adds twice the wave average times the offset under the root. Its formula panel shows that step whenever an offset is entered on one of those waves.

The Pulse or PWM waveform turns the tool into a PWM RMS calculator. A pulse that switches between zero and Vp for a fraction D of each cycle has an RMS value of Vp times the square root of D. Its average is Vp times D.

$$V_{RMS}=V_P\times \sqrt{D}$$

Choosing that waveform in the RMS Voltage Calculator opens a Duty Cycle field that starts at 25%. A 12 V peak at 25% returns 6 V RMS with a 3 V average, and the result note states that the pulse runs from 0 V to the peak. Peak-to-peak equals the peak for this shape.

The RMS always sits above the average until the duty reaches 100%. A heater or lamp driven by PWM responds to the RMS figure, while a meter on its DC range shows the average. At 25% duty the heating power is a quarter of full power, not a sixteenth.

RMS Volts to Watts in a Load

RMS voltage is the figure that sets power. The second card of the RMS Voltage Calculator divides RMS voltage squared by the Load Resistance entry, which opens at 8 Ω. That makes the tool an RMS calculator for watts and an RMS current calculator in one step.

$$P=\frac{V_{RMS}^2}{R}$$

The 40 V peak default is 28.28 V RMS, which puts 100 W and 3.536 A RMS into 8 Ω. Peak Power at Crest reads 200 W, twice the average, because the peak voltage squared is twice the RMS squared for a sine wave. Changing the load to 4 Ω doubles both figures.

Amplifier ratings in watts RMS work backward from the same formula, so an amplifier RMS calculator needs the speaker impedance as well. The table gives the sine wave an amplifier must deliver for the power figures people most often ask about.

So 200 W RMS into 4 Ω is the 40 V peak default, and 1,000 W into 4 Ω needs 63.25 V RMS. Asked the other way, 800 RMS has no wattage until the load is named, since 56.57 V RMS is 800 W into 4 Ω and 400 W into 8 Ω. For any other load, the Ohm's Law Calculator returns the voltage from power and resistance. The Electrical Power Calculator covers the watts when volts and amps are known.

The same shape factors apply to current. A triangular current with a 3 A peak is 1.732 A RMS over the time it flows, the same triangle factor used for winding currents in the Flyback Switch Mode Regulator Calculator.

Is 120V, 230V or 240V RMS or Peak?

AC supply voltages are RMS. The result note in the RMS Voltage Calculator says so for a sine wave. A 120 V or 230 V outlet rating is already RMS, with peaks near 170 V and 325 V. The same holds for 220 volts and 240 volts.

The RMS value of 240 volts is 240 V, and the RMS value of 220V AC is 220 V. The maximum voltage an AC supply reaches is its peak, which is 1.414 times the RMS figure. Insulation and capacitors are chosen for that peak.

Converting RMS to peak uses the RMS Voltage entry, which makes the tool an RMS to peak calculator. Typing 120 V returns 169.7 V Peak of the Wave, with the line Vp = 1.414 × Vrms under it. The Peak to Peak Voltage Calculator starts from the same RMS figure when the full swing is the goal.

Measuring RMS Voltage

A multimeter on its AC range reports RMS, but how it gets there matters. Fluke explains that an average-responding meter is accurate on pure sine waves and can read up to 40% low or 10% high on distorted ones. A true-RMS meter handles both.

The reason is the form factor. An average-responding meter measures the rectified average and multiplies it by 1.111. A 40 V peak square wave is 40 V RMS, but that meter would display 44.4 V. The same meter on a 40 V peak triangle would show 22.2 V against a true 23.09 V.

An oscilloscope gives peak-to-peak directly. Select Peak-to-Peak as the known voltage, enter the reading and pick the shape seen on the screen. The RMS Voltage Calculator then returns the figure a true-RMS meter should agree with.

Audio gear often shows level in decibels in place of volts. The third card converts the RMS result, and 28.28 V reads 29.03 dBV and 31.25 dBu. Sound On Sound's decibel glossary sets those references at 1.0 V RMS and 0.775 V RMS. The 100 W in the load reads 50 dBm, which is power against 1 milliwatt.