Editing Rectifier (section)

==== Three-phase, half-wave circuit ====
[[File:3 phase half wave rectifier.png|thumb|300px|Controlled three-phase half-wave rectifier circuit using [[thyristor]]s as the switching elements, ignoring supply inductance]]
An uncontrolled three-phase, half-wave midpoint circuit requires three diodes, one connected to each phase. This is the simplest type of three-phase rectifier but suffers from relatively high [[harmonic distortion]] on both the AC and DC connections. This type of rectifier is said to have a pulse-number of three, since the output voltage on the DC side contains three distinct pulses per cycle of the grid frequency:

[[File:DC voltage profile of M3 three-phase half-wave rectifier.jpg||500px]]

The peak values <math>V_\mathrm{peak}</math> of this three-pulse DC voltage are calculated from the RMS value <math>V_\mathrm{LN}</math> of the input phase voltage (line to neutral voltage, 120 V in North America, 230 V within Europe at mains operation): <math>V_\mathrm{peak} = \sqrt 2 \cdot V_{\mathrm{LN}}</math>. The average no-load output voltage <math>V_\mathrm {av}</math> results from the [[integral]] under the graph of a positive half-wave with the period duration of <math>\frac{2}{3} \pi</math> (from 30° to 150°):
<div style="overflow:auto;>
: <math>
\begin{align}
V_\mathrm{dc} = {} & V_\mathrm {av} = \frac{1}{\frac{2}{3} \pi} \int_{30^\circ}^{150^\circ} V_\mathrm{peak} \sin\varphi \, \mathrm d\varphi = \frac{3 V_\mathrm{peak}}{2 \pi} \cdot \left(-\cos 150^\circ + \cos 30^\circ \right) \\[8pt]
= {} & \frac{3 V_\mathrm{peak}}{2 \pi} \cdot \Biggl[ -\left(-\frac{\sqrt3}{2} \right)+\frac{\sqrt3}{2} \Biggl] = \frac{3\sqrt3 \cdot V_\mathrm{peak}}{2 \pi} \\[8pt]
\Longrightarrow {} & V_\mathrm{dc} = V_\mathrm {av} = \frac{3 \sqrt3 \cdot \sqrt 2 \cdot V_\mathrm{LN}}{2 \pi} \\[8pt]
\Longrightarrow {} & V_\mathrm{av} = \frac{3 \sqrt 6 \cdot V_\mathrm{LN}}{2 \pi} \approx 1.17 V_\mathrm{LN}
\end{align}
</math>
</div>