RMS Voltage Calculator

V_rms = V_p × k_rms, V_avg = V_p × k_avg, form factor = V_rms / V_avg, crest factor = V_p / V_rms

V_p

The peak voltage, measured from the zero line to the top of the wave, in volts. It is the amplitude, and it is what a component's voltage rating has to survive — the insulation on a wire carrying a quoted 230 volts is being asked to hold off 325. The field will not accept a negative number, because a symmetric wave's negative excursion is the same amplitude described by phase rather than a second quantity (V)

k_rms

The shape factor for the RMS value: 1/√2 for a sine, 1 for a square wave, 1/√3 for a triangle. This is the coefficient that makes the answer depend on the waveform rather than on a single number, so 0.7071 is correct for one shape and wrong for the other two. It comes from the integral of the square of the wave over one period, which is why a flat-topped wave keeps more of its amplitude in the RMS figure and a peaky one keeps less

k_avg

The shape factor for the average: 2/π for a sine, 1 for a square, 1/2 for a triangle. It applies to the rectified waveform — the absolute value of the wave — because the plain average of a symmetric alternating signal is zero. This factor is what meters that are calibrated for sines are built on, and it is why such a meter reads low on anything with a peakier shape

FF

The form factor, RMS divided by the rectified average: 1.1107 for a sine, 1 for a square, 1.1547 for a triangle. It measures how much the shape departs from flat, and it is a constant of the shape alone — it carries no unit and it does not change when you change the amplitude. It is also the ratio a cheap meter is quietly assuming when it measures an average and displays an RMS

CF

The crest factor, peak divided by RMS: 1.4142 for a sine — the square root of two, and the single most useful number on the page — 1 for a square wave and 1.7321 for a triangle. This is the ratio that tells you whether a reading is a peak or an RMS value, because a peak-reading meter on a sine displays exactly √2 times what a true-RMS meter displays on the same signal. It is likewise a shape constant, unaffected by amplitude

Use it when you have one of the four voltages and need another: a peak figure from a datasheet, an RMS figure an instrument reported, a peak-to-peak from an oscilloscope trace, an average from an older meter. It matters most where the waveform is not a sine — a dimmer, a variable-speed drive, a switched-mode supply, an inverter or a UPS all put out shapes whose peak and RMS are further apart than √2, and where a meter expecting a sine will be wrong. Use it whenever two instruments disagree about the same circuit, because the crest factor is the number that resolves the argument: if the ratio between them is not 1.4142, they are measuring different things rather than one of them being broken.

1. The defaults: 325 V peak, a sine

   1. Peak 325 V, sine
   2. RMS is 325 / √2 = 229.810 V
   3. Peak to peak is 2 × 325 = 650 V
   4. Rectified average is 325 × 2/π = 206.901 V
   5. Form factor is 229.810 / 206.901 = 1.1107, crest factor is 325 / 229.810 = 1.4142

   This is a mains supply: the 230 volts on the socket is the RMS figure, and 325 is what the voltage actually reaches, twice in every cycle, in both directions. The last line is the one worth keeping — 1.4142 is the square root of two, and it is the factor between what a peak-reading instrument and a true-RMS instrument will show on this same waveform. A wire rated for 230 volts is being asked to hold off 650 volts peak to peak.

2. The same 325 V peak as a square wave

   1. Peak 325 V, square wave
   2. The wave spends all its time at full amplitude, so RMS is 325 V
   3. Rectified average is also 325 V — the wave never leaves the top
   4. Form factor and crest factor are both 1

   A square wave is the shape that carries the most power for a given peak, because it never spends time partway up: three of the five outputs collapse onto the same number. It also shows why there is no universal peak-to-RMS factor — the answer is 1 here and 0.577 for a triangle at the same peak, a ratio of 1.73 between two waveforms that both reach 325 volts. Note that the peak-to-peak figure did not change: all three shapes in this page are symmetric about zero, so it is always twice the peak.

3. The same 325 V peak as a triangle

   1. Peak 325 V, triangle
   2. RMS is 325 / √3 = 187.639 V
   3. Peak to peak is still 650 V
   4. Rectified average is 325 / 2 = 162.5 V
   5. Form factor is 1.1547, crest factor is √3 = 1.7321

   The other extreme: a triangle spends almost all its time away from the peak, so it delivers the least power of the three shapes. Its crest factor of 1.7321 is the highest here, and that is the number a true-RMS meter needs to be able to handle — instruments are usually specified to a maximum crest factor, and a reading above it is simply not trustworthy even on a meter that claims true RMS.

4. A 12 V peak signal from an adapter

   1. Peak 12 V, sine
   2. RMS is 12 / √2 = 8.485 V
   3. Peak to peak is 24 V
   4. Rectified average is 12 × 2/π = 7.639 V
   5. The two ratios are unchanged: 1.1107 and 1.4142

   Compare the last line with the first example: the amplitude dropped by a factor of twenty-seven and both ratios are identical to four decimals, because they are properties of the waveform and not of the signal. That is what makes them useful as a check — a crest factor that moves when the amplitude moves means the waveform changed shape, not that the signal got bigger. The 8.485 V RMS here is what a meter would show for a 12 V peak sine, and it is also the figure that would be used in a power calculation.

The average voltage this page reports is the rectified average — the mean of the absolute value of the wave. The plain average of a symmetric alternating signal is zero, since the positive and negative halves cancel exactly, and that is a correct answer that is useless; the engineering convention is to rectify first and average second, and that is what is computed here. Every shape is assumed to be symmetric about zero, which matters most for the square wave: a square wave that switches between zero and a positive peak instead of between a negative and a positive one has a different RMS and a different average for the same peak, and it is not modelled. The waveforms are the three ideal shapes, so a real signal with ringing, flat tops or asymmetry will not match any of them exactly — a crest factor that comes out near but not at 1.4142 is telling you the shape is close to a sine rather than that the arithmetic failed. No distortion, offset or noise is modelled.

How do I convert a peak voltage to RMS?

Multiply by 0.7071 — which is 1/√2 — but only if the waveform is a sine, and that qualification is the whole subject of this page. The factor is 1 for a square wave and 0.5774 for a triangle, so the same 325 volt peak gives 229.81 volts RMS as a sine, 325 as a square and 187.639 as a triangle. There is no conversion from peak to RMS that does not depend on the waveform, and any table that gives you one number is silently assuming a sine.

What is the peak voltage of 230 V mains?

325 volts, and 650 volts peak to peak. The 230 is already an RMS figure — it is defined to deliver the same power into a resistive load as 230 volts of DC — so multiplying by √2 rather than dividing gives the amplitude. This is the single most useful thing on the page for practical work: insulation, creepage distances and component voltage ratings are set by the peak and the peak-to-peak, not by the number printed on the supply, which is why a part rated at 250 volts is not adequate for a 230 volt circuit.

Why is the average voltage not zero?

Because it is the rectified average: the wave is folded up to its absolute value before the mean is taken. The true mean of a symmetric alternating wave is exactly zero, since the positive half cancels the negative one, and reporting that would be both correct and pointless. The convention in electrical work is rectify first, average second, which gives 2/π of the peak for a sine — 206.901 volts on a 325 volt peak. Meters labelled average-responding are built on this quantity, and that is the source of the error discussed in the next question.

Why does my meter read lower than this page says?

Most inexpensive meters are average-responding: they measure the rectified average and multiply by 1.1107, the form factor of a sine, to display an RMS figure. That calibration is exactly right for a sine and wrong for everything else, and it always reads low on a peakier waveform — a switched-mode supply, a dimmer, a variable-speed drive. The crest factor is how you catch it: on a sine it comes out at 1.4142, and anything noticeably higher means the shape is peakier than the meter assumes. A meter specified as true RMS does the squaring and averaging properly, but only up to a stated maximum crest factor.

Does this work for a square wave?

For a bipolar square wave, one that swings between a negative and a positive peak with equal time at each, which is what this page computes: RMS and average both equal the peak and both ratios come out at 1. A square wave that switches between zero and a positive peak is a different signal with the same amplitude, and at a 50% duty cycle its RMS is 0.7071 of the peak rather than the peak itself, its average is half the peak, and its peak-to-peak is the peak rather than twice it. The page does not model that variant, so check which kind you have before using the outputs.

Why is there no sawtooth option?

Because a sawtooth and a triangle give the same five numbers. They differ only in which edge is the ramp — a slow rise with a fast fall, or the reverse — and the RMS and the average both depend on the shape of the excursion, not on its direction in time. Adding it would put two options in the list that can never disagree, and would add a fourth row to the table identical to the third. The distinction matters for some things, but not for any quantity on this page.

• NIST Guide for the Use of the International System of Units (SI), §8.6 — root-mean-square values and the convention of quoting an alternating quantity by its RMS value — NIST
• Root mean square — the definition as the square root of the mean of the square, and the 1/√2, 1 and 1/√3 factors for the standard waveforms — Wikipedia
• Crest factor — why a meter calibrated to a sine reads low on peakier waveforms, and what the crest factor specification on a true-RMS instrument means — Wikipedia