Three-phase single-way diode rectifier: average and rms

A three-phase single-way (half-wave, three-pulse) uncontrolled rectifier uses three diodes, one in each phase of a star-connected supply (or transformer secondary), with the load connected between the common cathode point and the neutral.

          D1
 a o------>|-----+
          D2     |
 b o------>|-----+------o P (+)
          D3     |        |
 c o------>|-----+       Load
                          |
 n o----------------------o N (-)
 a, b, c, n: star-connected secondary
 va = Vm sin wt, vb lags 120, vc lags 240

Operation and waveforms

• Each diode's anode is connected to one phase; the cathodes are joined to form the positive output. The diode connected to the most positive phase conducts and the other two are reverse biased.
• Measured from the zero crossing of $v_a$, $v_a$ is the highest from $30^\circ$ to $150^\circ$, $v_b$ from $150^\circ$ to $270^\circ$ and $v_c$ from $270^\circ$ to $390^\circ$. These crossover points ($30^\circ$, $150^\circ$, $270^\circ$) are the points of natural commutation.
• Each diode conducts for $120^\circ$; the output is the upper envelope of the three phase voltages, with three pulses per cycle (ripple frequency $3f = 150$ Hz), never falling below $V_m/2$.
• Load current returns through the neutral; each diode must block a peak inverse voltage of $\sqrt3 V_m$ (peak line voltage).

 vo     va        vb        vc
      _----_    _----_    _----_
     /      \__/      \__/      \
    /  D1     ^   D2    ^   D3    \
   30        150       270       390  (deg)
 vo = upper envelope; minimum Vm/2 at
 crossover points, peak Vm

Average output voltage

The output consists of three identical pulses per cycle; taking D1's pulse ($v_a$ from $30^\circ$ to $150^\circ$):

$$\begin{aligned} V_{dc} &= \frac{3}{2\pi}\int_{\pi/6}^{5\pi/6}V_m\sin\omega t\,d(\omega t) = \frac{3V_m}{2\pi}\left[\cos\frac{\pi}{6} - \cos\frac{5\pi}{6}\right] \\ &= \frac{3V_m}{2\pi}\left[\frac{\sqrt3}{2} + \frac{\sqrt3}{2}\right] = \frac{3\sqrt3\,V_m}{2\pi} = 0.827\,V_m \end{aligned}$$

RMS output voltage

$$\begin{aligned} V_{rms}^2 &= \frac{3}{2\pi}\int_{\pi/6}^{5\pi/6}V_m^2\sin^2\omega t\,d(\omega t) = \frac{3V_m^2}{4\pi}\left[\omega t - \frac{\sin2\omega t}{2}\right]_{\pi/6}^{5\pi/6} \\ &= \frac{3V_m^2}{4\pi}\left[\frac{2\pi}{3} - \frac{1}{2}\left(-\frac{\sqrt3}{2} - \frac{\sqrt3}{2}\right)\right] = \frac{3V_m^2}{4\pi}\left[\frac{2\pi}{3} + \frac{\sqrt3}{2}\right] \\ &= V_m^2\left[\frac{1}{2} + \frac{3\sqrt3}{8\pi}\right] \end{aligned}$$ $$V_{rms} = V_m\left[\frac{1}{2} + \frac{3\sqrt3}{8\pi}\right]^{1/2} = 0.8407\,V_m$$

Here $V_m$ is the peak phase voltage. Form factor $= 0.8407/0.8270 = 1.017$ and ripple factor $= \sqrt{FF^2-1} = 18.3\%$, much better than single-phase rectifiers (48% for full wave).

Example: with 230 V per phase, $V_m = 325.27$ V, $V_{dc} = 268.99$ V and $V_{rms} = 273.45$ V.