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# Maximum average output of 3 phase and 1 phase controlled rectifier

### The maximum average output of 3 phase and 1 phase controlled rectifier

Question: Show that the maximum average output of 3 phase-controlled rectifiers is greater than 1 phase-controlled rectifier?

Solution

For a single phase full wave converter, the average output voltage can be found from

\begin{array}{l}V_{dc}=\frac2{2\pi}\int_\alpha^{\mathrm\pi+\mathrm\alpha}V_m\sin\left(\omega t\right)d(\omega t)\\\\\;\;\;\;\;=\frac{2V_m}{2\pi}\lbrack-\cos\left(\omega t\right)\rbrack_\alpha^{\mathrm\pi+\mathrm\alpha}\\\\\;\;\;\;\;=\frac{2V_m}{2\pi}\lbrack-\cos\left(\mathrm\pi+\alpha\right)-(-\cos\alpha)\rbrack\\\\\;\;\;\;\;=\frac{2V_m}{2\pi}\lbrack\cos\alpha+\cos\alpha\rbrack\\\\\;\;\;\;\;=\frac{2V_m}{2\pi}.2\cos\alpha\\\\\;\;\;\;\;=\frac{2V_m}\pi.\cos\alpha\\\\Maximum\;average\;output\;occur\;at\;\alpha=0^o\\\\\;\;\;\;\;V_{dcmax}=\frac{2V_m}\pi.\cos0\\\\\;\;\;\;\;\;V_{dcmax}=\frac{2V_m}\pi\\\end{array}

For a single phase full converter, the average output voltage can be found from

\begin{array}{l}V_{dc}=\frac3\pi\int_{\frac{\mathrm\pi}6+\alpha}^{\frac{\mathrm\pi}2+\mathrm\alpha}V_{ab}d(\omega t)\\\\\;\;\;\;\;=\frac3\pi\int_{\frac{\mathrm\pi}6+\alpha}^{\frac{\mathrm\pi}2+\mathrm\alpha}\sqrt3V_m\sin\left(\omega t+\frac{\mathrm\pi}6\right)d(\omega t)\\\\\;\;\;\;\;=\frac{3\sqrt3V_m}\pi\int_{\frac{\mathrm\pi}6+\alpha}^{\frac{\mathrm\pi}2+\mathrm\alpha}\sin\left(\omega t+\frac{\mathrm\pi}6\right)d(\omega t)\\\\\;\;\;\;\;=\frac{3\sqrt3V_m}\pi.\cos\alpha\\\\The\;Maximum\;average\;output\;voltage\;occur\;for\;delay\;angle\;\alpha=0^o\\\\\;\;\;\;\;V_{dcmax}=\frac{3\sqrt3V_m}\pi.\cos0\\\\\;\;\;\;\;\;V_{dcmax}=\frac{3\sqrt3V_m}\pi\\\end{array}

From the output voltage equation of single-phase and three-phase controlled rectifier, we see that

\begin{array}{l}\frac{\mathbf3\sqrt{\mathbf3}{\mathbf V}_{\mathbf m}}{\mathbf\pi}\boldsymbol>\frac{\mathbf2{\mathbf V}_{\mathbf m}}{\mathbf\pi}\\\end{array}

Thus we say that the maximum average output of 3 phase-controlled rectifiers is greater than 1 phase-controlled rectifier.

Maximum average output of 3 phase and 1 phase controlled rectifier

Read: What are an ac voltage controller and its applications?

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