Editing Pulse-width modulation (section)

=== Periodic pulse wave ===
[[Image:Duty cycle general.svg|thumb|350px|Fig. 1: a periodic [[pulse wave]], showing <math>y_\text{min}</math>, <math>y_\text{max}</math> and constant duty cycle D.]]

If we consider a periodic [[pulse wave]] <math>f(t)</math> with period <math>T</math>, low value <math>y_\text{min}</math>, a high value <math>y_\text{max}</math> and a constant duty cycle D (Figure 1), the average value of the waveform is given by:

<math display="block">\bar{y} = \frac{1}{T}\int^T_0f(t)\,dt</math>

As <math>f(t)</math> is a pulse wave, its value is <math>y_\text{max}</math> for <math>0 < t < D \cdot T</math> and <math>y_\text{min}</math> for <math>D \cdot T < t < T</math>. The above expression then becomes:

<math display="block">\begin{align}
  \bar{y} &= \frac{1}{T} \left(\int_0^{DT} y_\text{max}\,dt + \int_{DT}^T y_\text{min}\,dt\right)\\
          &= \frac{1}{T} \left(D \cdot T \cdot y_\text{max} + T\left(1 - D\right) y_\text{min}\right)\\
          &= D\cdot y_\text{max} + \left(1 - D\right) y_\text{min}
\end{align}</math>

This latter expression can be fairly simplified in many cases where <math>y_\text{min} = 0</math> as <math>\bar{y} = D \cdot y_\text{max}</math>. From this, the average value of the signal (<math>\bar{y}</math>) is directly dependent on the duty cycle D.

However, by varying (i.e. modulating) the duty cycle (and possibly also the period), the following more advanced pulse-width modulated waves allow variation of the [[average]] value of the waveform.