Editing Rectifier (section)

=== Three-phase rectifiers ===
Single-phase rectifiers are commonly used for power supplies for domestic equipment. However, for most industrial and high-power applications, [[three-phase]] rectifier circuits are the norm. As with single-phase rectifiers, three-phase rectifiers can take the form of a half-wave circuit, a full-wave circuit using a center-tapped transformer, or a full-wave bridge circuit.

[[Thyristor]]s are commonly used in place of diodes to create a circuit that can regulate the output voltage. Many devices that provide direct current actually 'generate' three-phase AC. For example, an [[automobile alternator]] contains nine diodes, six of which function as a full-wave rectifier for battery charging.

==== 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>

==== Three-phase, full-wave circuit using center-tapped transformer ====
[[File:6 phase half wave rectifier.png|thumb|300px|Controlled three-phase full-wave rectifier circuit using [[thyristor]]s as the switching elements, with a center-tapped transformer, ignoring supply inductance]]
If the AC supply is fed via a transformer with a center tap, a rectifier circuit with improved harmonic performance can be obtained. This rectifier now requires six diodes, one connected to each end of each transformer secondary [[winding]]. This circuit has a pulse-number of six, and in effect, can be thought of as a six-phase, half-wave circuit.

Before [[solid state (electronics)|solid state]] devices became available, the half-wave circuit, and the full-wave circuit using a center-tapped transformer, were very commonly used in industrial rectifiers using [[mercury-arc valve]]s.<ref name="Rissik1941">{{cite book|author=Hendrik Rissik|title=Mercury-arc current convertors [sic] : an introduction to the theory and practice of vapour-arc discharge devices and to the study of rectification phenomena|url=https://books.google.com/books?id=S4MhAAAAMAAJ|year=1941|publisher=Sir I. Pitman & sons, ltd.}}</ref> This was because the three or six AC supply inputs could be fed to a corresponding number of anode electrodes on a single tank, sharing a common cathode.

With the advent of diodes and thyristors, these circuits have become less popular and the three-phase bridge circuit has become the most common circuit.

==== Three-phase bridge rectifier uncontrolled ====
[[File:Getting behind the tridge rectifier.jpg|thumb|left|180px|Disassembled automobile [[alternator (auto)|alternator]], showing the six [[diode]]s that comprise a full-wave three-phase bridge rectifier.]]

For an uncontrolled [[three-phase]] bridge rectifier, six [[diode]]s are used, and the circuit again has a pulse number of six. For this reason, it is also commonly referred to as a ''six-pulse bridge''. The B6 circuit can be seen simplified as a series connection of two three-pulse center circuits.

For low-power applications, double diodes in series, with the anode of the first diode connected to the cathode of the second, are manufactured as a single component for this purpose. Some commercially available double diodes have all four terminals available so the user can configure them for single-phase split supply use, half a bridge, or three-phase rectifier.

For higher-power applications, a single discrete device is usually used for each of the six arms of the bridge. For the very highest powers, each arm of the bridge may consist of tens or hundreds of separate devices in parallel (where very high current is needed, for example in [[aluminium smelting]]) or in series (where very high voltages are needed, for example in [[high-voltage direct current]] power transmission).
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[[File:6 pulse bridge without inductance.png|thumb|300px|Controlled three-phase full-wave bridge rectifier circuit (B6C) using [[thyristor]]s as the switching elements, ignoring supply inductance. The thyristors pulse in order V1–V6.]]

The pulsating DC voltage results from the differences of the instantaneous positive and negative phase voltages <math>V_\mathrm{LN}</math>, phase-shifted by 30°:

[[File:DC voltage profile of B6 three-phase full-wave rectifier.jpg||500px]]

The ideal, no-load average output voltage <math>V_\mathrm{av}</math> of the B6 circuit results from the integral under the graph of a DC voltage pulse with the period duration of <math>\frac{1}{3} \pi</math> (from 60° to 120°) with the peak value <math>\hat v_{\mathrm{DC}} = \sqrt 3 \cdot V_\mathrm{peak}</math>:

: <math>
\begin{align}
V_\mathrm{dc} = {} & V_\mathrm {av} = \frac{1}{\frac{1}{3} \pi} \int_{60^\circ}^{120^\circ} \sqrt 3 \cdot V_\mathrm{peak} \cdot \sin\varphi \cdot \mathrm d\varphi \\[8pt]
= {} & \frac{3 \sqrt 3 \cdot V_\mathrm{peak}}{\pi} \cdot \left(-\cos 120^\circ + \cos 60^\circ \right) \\[8pt]
= {} & \frac{3\sqrt 3 \cdot V_\mathrm{peak}}{\pi} \cdot \Biggl[-\left(-\frac{1}{2} \right)+\frac{1}{2} \Biggl] = \frac{3\sqrt3 \cdot V_\mathrm{peak}}{\pi}
\end{align}
</math>

: <math> \Longrightarrow V_\mathrm{dc}=V_\mathrm {av}= \frac{3\sqrt{3} \cdot \sqrt 2 \cdot V_\mathrm{LN}}{\pi} \Longrightarrow V_\mathrm {av} = \frac{3\sqrt 6 \cdot V_\mathrm{LN}}{\pi} \approx 2.34 V_\mathrm{LN}</math>

[[File:3 phase rectification 2.svg|thumb|200px|3-phase AC input, half-wave and full-wave rectified DC output waveforms]]

If the three-phase bridge rectifier is operated symmetrically (as positive and negative supply voltage), the center point of the rectifier on the output side (or the so-called isolated reference potential) opposite the center point of the transformer (or the neutral conductor) has a potential difference in the form of a triangular [[Common-mode signal|common-mode voltage]]. For this reason, these two centers must never be connected to each other, otherwise short-circuit currents would flow. The [[Ground (electricity)|ground]] of the three-phase bridge rectifier in symmetrical operation is thus decoupled from the neutral conductor or the [[Ground and neutral|earth]] of the mains voltage. Powered by a transformer, earthing of the center point of the bridge is possible, provided that the secondary winding of the transformer is electrically isolated from the mains voltage and the star point of the secondary winding is not on earth. In this case, however, (negligible) leakage currents are flowing over the transformer windings.

The common-mode voltage is formed out of the respective average values of the differences between the positive and negative phase voltages, which form the pulsating DC voltage. The peak value of the delta voltage <math>\hat v_{\mathrm{common-mode}}</math> amounts {{sfrac|1|4}} of the peak value of the phase input voltage <math>V_\mathrm{peak}</math> and is calculated with <math>V_\mathrm{peak}</math> minus half of the DC voltage at 60° of the period:

: <math>
\begin{align}
\hat v_{\text{common mode}} = {} & V_\mathrm{peak} - \frac{\sqrt 3 \cdot V_\mathrm{peak} \cdot \sin 60^\circ}{2} \\[8pt]
= {} & V_\mathrm{peak} \cdot \Biggl( 1- \frac{\sqrt 3 \cdot \sin 60^\circ}{2} \Biggl) = V_\mathrm{peak} \cdot 0.25
\end{align}
</math>

The RMS value of the common-mode voltage is calculated from the form factor for triangular oscillations:

: <math>V_{\text{common mode}} = \frac{\hat v_{\text{common mode}}}{\sqrt 3}</math>

If the circuit is operated asymmetrically (as a simple supply voltage with just one positive pole), both the positive and negative poles (or the isolated reference potential) are pulsating opposite the center (or the ground) of the input voltage analogously to the positive and negative waveforms of the phase voltages. However, the differences in the phase voltages result in the six-pulse DC voltage (over the duration of a period). The strict separation of the transformer center from the negative pole (otherwise short-circuit currents will flow) or a possible grounding of the negative pole when powered by an isolating transformer apply correspondingly to the symmetrical operation.

==== Three-phase bridge rectifier controlled ====
The controlled three-phase bridge rectifier uses thyristors in place of diodes. The output voltage is reduced by the factor cos(α):

: <math>V_\mathrm{dc}=V_\mathrm {av}=\frac{3\sqrt 3 \cdot V_\mathrm{peak}}{\pi} \cdot \cos \alpha</math>

Or, expressed in terms of the line to line input voltage:<ref>{{cite book|last=Kimbark|first=Edward Wilson|title=Direct current transmission.|url=https://archive.org/details/directcurrenttra0000kimb|url-access=limited|year=1971|publisher=Wiley-Interscience|location=New York|isbn=978-0-471-47580-4|pages=[https://archive.org/details/directcurrenttra0000kimb/page/508 508]|edition=4. printing.}}</ref>

: <math>V_\mathrm{dc}=V_\mathrm {av}=\frac{3 V_\mathrm {LLpeak}}{\pi} \cdot \cos \alpha</math>

where:
: ''V''<sub>LLpeak</sub> is the peak value of the line to line input voltages,
: ''V''<sub>peak</sub> is the peak value of the phase (line to neutral) input voltages, and
: ''α'' is the firing angle of the thyristor (0 if diodes are used to perform rectification)

The above equations are only valid when no current is drawn from the AC supply or in the theoretical case when the AC supply connections have no inductance. In practice, the supply inductance causes a reduction of DC output voltage with increasing load, typically in the range 10–20% at full load.

The effect of supply inductance is to slow down the transfer process (called commutation) from one phase to the next. As result of this is that at each transition between a pair of devices, there is a period of overlap during which three (rather than two) devices in the bridge are conducting simultaneously. The overlap angle is usually referred to by the symbol μ (or u), and may be 20&nbsp;30° at full load.

With supply inductance taken into account, the output voltage of the rectifier is reduced to

: <math>V_\mathrm{dc} = V_\mathrm {av}=\frac{3 V_\mathrm {LLpeak}}{\pi} \cdot \cos (\alpha + \mu). </math>

The overlap angle ''μ'' is directly related to the DC current, and the above equation may be re-expressed as

: <math> V_\mathrm{dc}=V_\mathrm {av}=\frac{3 V_\mathrm {LLpeak}}{\pi} \cdot \cos(\alpha) - 6 f L_\mathrm {c} I_\mathrm {d} </math>

where:
: ''L''<sub>c</sub> is the commutating inductance per phase, and
: ''I''<sub>d</sub> is the direct current.

{|
|-
|[[File:Bridge rectifier at alpha=0 u=0.png|thumb|300px|Three-phase bridge rectifier at alpha=0° without overlap]]
|[[File:Bridge rectifier at alpha=0 u=20.png|thumb|300px|Three-phase bridge rectifier at alpha=0° with overlap angle of 20°]]
|}

{|
|-
|[[File:Bridge rectifier at alpha=20 u=20.png|thumb|300px|Three-phase controlled bridge rectifier at alpha=20° with overlap angle of 20°]]
|[[File:Bridge rectifier at alpha=40 u=20.png|thumb|300px|Three-phase controlled bridge rectifier at alpha=40° with overlap angle of 20°]]
|}

==== Twelve-pulse bridge ====
[[File:12 pulse bridge.png|thumb|left|350px|Twelve pulse bridge rectifier using [[thyristor]]s as the switching elements. One six-pulse bridge consists of the even-numbered thyristors, the other is the odd-numbered set.]]
Although better than single-phase rectifiers or three-phase half-wave rectifiers, six-pulse rectifier circuits still produce considerable harmonic distortion on both the AC and DC connections. For very high-power rectifiers the twelve-pulse bridge connection is usually used. A twelve-pulse bridge consists of two six-pulse bridge circuits connected in series, with their AC connections fed from a supply transformer that produces a 30° phase shift between the two bridges. This cancels many of the characteristic harmonics the six-pulse bridges produce.

The 30-degree phase shift is usually achieved by using a transformer with two sets of secondary windings, one in star (wye) connection and one in delta connection.
{{clear}}