# Average and effective values

*Last Modification: April 16, 2013*

The average (or mean) and effective (or RMS) values, are common used terms to indicate the magnitude of a periodic signal. This can be a voltage, current, power or another quantity. This article lists the equations for the average and effective values for a number of different waveforms. You can find the background on this subject in the article Theory & definitions.

### Random waveform

The mean and RMS value of a random waveform can be calculated with the equations below:

[equ. 1]
[equ. 2]

In these equations is *a(t)* the signal function. This can be interchanged with a voltage *v(t)*, current *i(t)*, power *p(t)* or another quantity. *A _{mean}* and

*A*must than be replaced by the corresponding quantity. With emphasis must be mentioned that the RMS-value may only be calculated for the voltage and current. For other quantities is the RMS-value meaningless.

_{RMS}## Waveforms

Here below is a list of common waveforms and their derivatives for the mean and RMS values.

description | waveform | mean value | RMS value |

## DCSignal with an unchanging value |
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## SinePure sine shaped signal symmetrical around zero with an amplitude |
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## Sine with offsetSine wave with a top value a._{o} |
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## Half-wave rectified sineSignal that only contains the positive (or negative) values of a sine shaped signal with a top value |
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## Full-wave rectification sineSignal calculated from the absolute values of a sine with a top value |
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## Half sine impulseHalf sine cycle with a width of |
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## Square wave bipolarSquare wave that contains positive a values and has a duty-cycle _{n}δ. |
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## Square wave unipolarSquare wave with only positive (or negative) values δ. |
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## TrapeziumTrapezium shaped signal with an amplitude δ. The rising and the falling edge _{w}δ has the same value._{f} |
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## TriangleTriangle wave shape with a top-top value a has. The rising edge has a width _{o}δ. |
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## Triangle impulseTriangle impulse with a top value δ and the falling edge _{u}δ._{d} |
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## Saw tooth impulseSaw tooth impulse with a top value δ. |
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## Saw tooth impulse with offsetSaw tooth impulse with an amplitude of the rising edge δ |

#### Duty-cycle

De duty-cycle *δ* is expressed as coefficient and is always smaller than 1. Often a duty-cycle will be expressed as a percentage: to obtain the coefficient, the percentage-number must be divided by 100. If a waveform has more than one with declarations *δ _{x}*, is the total of widths never greater than 1.

#### Average value sine wave

The average value of a sine-shaped voltage or current is 0. But, often in literature, the value v_{pk}*π/2 (≈0,637*v_{pk}) is used. This is not the real value of the average values of a sine, but the average of the __absolute__ values of a sine. To avoid confusion, there has to be a clear indication which average value is meant.

## Form factor

The form factor is the ratio between the effective and average value:

[equ. 3]

Sometimes when the formfactor is calculated the result will be infinitive. This is the case with pure alternating voltages with an average value of 0. An exception is then made by using the absolute values. For a sine wave the form factor will become π/(2*√2) ≈1,11.

## Crest factor

The crest factor is the ratio between the (absolute) peak value and the effective value:

[equ. 4]