Half-controlled bridge with R load, α = 45°

The circuit is a single-phase half-controlled (semi-converter) bridge: thyristors T1, T2 on top and diodes D1, D2 at the bottom, feeding $R = 10\ \Omega$. With a resistive load the output is a chopped sine wave from $\alpha$ to $\pi$ in each half cycle. Assumption: the supply is taken as 230 V, 50 Hz as stated in the text (the figure shows 220 V; results for 220 V are given at the end).

              P (+)
        +-------+-------+
       T1              T2
 a o----+               +----o b
       D1              D2
        +-------+-------+
              N (-)
 R = 10 ohm between P and N
 + half: T1 (a->P), D2 (N->b);  - half: T2, D1

Waveforms

 vo, io    __            __
          |  \          |  \
 ---------+---\---------+---\-----
 is        __
          |  \
 ---------+---\---------+---/-----
                        |__/
          45  180      225  360  (deg)

Average output voltage

$V_m = \sqrt2 \times 230 = 325.27$ V, $\alpha = 45^\circ$.

$$\begin{aligned} V_{dc} &= \frac{V_m}{\pi}(1+\cos\alpha) = \frac{325.27}{\pi}(1.7071) = 176.75\ \text{V} \\ I_{dc} &= \frac{V_{dc}}{R} = 17.67\ \text{A} \end{aligned}$$

Output dc power

$$P_{dc} = V_{dc}I_{dc} = \frac{V_{dc}^2}{R} = \frac{176.75^2}{10} = 3123.97\ \text{W}$$

Input power factor

RMS output voltage:

$$\begin{aligned} V_{rms} &= V_s\left[\frac{1}{\pi}\left(\pi-\alpha+\frac{\sin2\alpha}{2}\right)\right]^{1/2} = 230\left[\frac{2.3562+0.5}{\pi}\right]^{1/2} \\ &= 230 \times 0.9535 = 219.30\ \text{V} \end{aligned}$$

With R load, $i_s$ has the same rms value as $i_o$: $I_s = V_{rms}/R = 21.93$ A. All real input power is consumed in R:

$$\begin{aligned} P_{ac} &= \frac{V_{rms}^2}{R} = \frac{219.30^2}{10} = 4809.43\ \text{W} \\ PF &= \frac{P_{ac}}{V_sI_s} = \frac{4809.43}{230 \times 21.93} = \frac{V_{rms}}{V_s} = 0.9535\ \text{(lagging)} \end{aligned}$$

Answer (230 V): $V_{dc} = 176.75$ V, $P_{dc} = 3124$ W, input PF = 0.954 lagging.

With 220 V (figure value): $V_{dc} = 169.06$ V, $P_{dc} = 2858.2$ W, PF = 0.954 (PF does not depend on the supply voltage).