Average and effective values

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_RMS 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.

Waveforms

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

description	waveform	mean value	RMS value
DC Signal with an unchanging value apk over time.
Sine Pure sine shaped signal symmetrical around zero with an amplitude apk.
Sine with offset Sine wave with a top value apk and an offset from zero ao.
Half-wave rectified sine Signal that only contains the positive (or negative) values of a sine shaped signal with a top value apk.
Full-wave rectification sine Signal calculated from the absolute values of a sine with a top value apk.
Half sine impulse Half sine cycle with a width of δT and a top value apk.
Square wave bipolar Square wave that contains positive ap as well as negative an values and has a duty-cycle δ.
Square wave unipolar Square wave with only positive (or negative) values apk with a duty-cycle δ.
Trapezium Trapezium shaped signal with an amplitude apk and a width δw. The rising and the falling edge δf has the same value.
Triangle Triangle wave shape with a top-top value Δa whereby the mean amplitude an offset ao has. The rising edge has a width δ.
Triangle impulse Triangle impulse with a top value apk. The rising edge has a width δu and the falling edge δd.
Saw tooth impulse Saw tooth impulse with a top value apk and a width δ.
Saw tooth impulse with offset Saw tooth impulse with an amplitude of the rising edge Δa and a start amplitude am. The impulse with is 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*2/π (≈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]