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Three-phase half-wave Controlled Rectifier



Three-phase half-wave Controlled Rectifier circuit with R load










Three-phase half-wave Controlled Rectifier circuit with R load Three-phase half-wave Controlled Rectifier circuit with R load Average Load/Output Voltage
{V_{dc}} = \frac{3}{\pi }\int_{\pi /6 + \alpha }^\pi  {\sqrt 2 V\sin \theta d\theta }
       = \frac{{3\sqrt 2 }}{\pi }V\left( {1 + \cos \left( {\frac{\pi }{6} + \alpha } \right)} \right)
Three-phase half-wave Controlled Rectifier circuit
The rms output voltage is obtained from with resistive load

{V_{rms}} = {\left[ {\frac{3}{{2\pi }}\int\limits_{\frac{\pi }{6} + \alpha }^\pi  {V_m^2{{\sin }^2}\omega td\left( {\omega t} \right)} } \right]^{\frac{1}{2}}}
{V_{rms}} = \sqrt 3 {V_m}{\left[ {\frac{5}{{24}} - \frac{\alpha }{{4\pi }} + \frac{1}{{8\pi }}\sin \left( {\frac{\pi }{3} + 2\alpha } \right)} \right]^{\frac{1}{2}}}
For a continuous load current with highly inductive load, the average output voltage is given by
{V_{dc}} = \frac{3}{{2\pi }}\int\limits_{\pi /6 + \alpha }^{5\pi /6 + \alpha } {{V_m}} \sin \omega td\left( {\omega t} \right)
{V_{dc}} = \frac{{3\sqrt 3 {V_m}}}{{2\pi }}\cos \alpha
The rms output voltage is obtained from
{V_{rms}} = {\left[ {\frac{3}{{2\pi }}\int\limits_{\pi /6 + \alpha }^{5\pi /6 + \alpha } {V_m^2{{\sin }^2}\omega t\left( {\omega t} \right)} } \right]^{\frac{1}{2}}}
{V_{rms}} = \sqrt 3 {V_m}{\left[ {\frac{1}{6} + \frac{{\sqrt 3 }}{{8\pi }}\cos 2\alpha } \right]^{\frac{1}{2}}}
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