Cycloconverter doubling time period, rms output

Doubling the time period means $T_o = 2T_i$, i.e. $f_o = f_i/2$ (50 Hz to 25 Hz). This is done by a single-phase step-down cycloconverter in which each output half cycle is made from two input half cycles.

Circuit (centre-tapped transformer type)

          a o----+---P1->|---+
            |    +---|<-N1---+
  ~ vs    Tr|                |
  (input) O o----[ Load ]----+  (common point K)
            |                |
          b o----+---P2->|---+
                 +---|<-N2---+

P1, P2 form the positive group (load current K to O) and N1, N2 the negative group (load current O to K).

Working (R load, firing angle $\alpha$)

• In half cycles 1 and 2 the positive group acts as a full-wave rectifier, so two positive humps appear.
• In half cycles 3 and 4 the negative group gives two negative humps.
• Each SCR turns off naturally at the supply zero (R load).

Waveforms

 vs  /\    /\
    /  \  /  \
 --/----\/----\/----\/---
             (alternate humps -ve)
 vo  /\  /\
    /  \/  \
 --/--------\--------/---
             \  /\  /
              \/  \/
    | P1 | P2 | N2 | N1 |
    |<--- To = 2 Ti --->|

With firing delay, each hump starts at $\alpha$ after the zero crossing.

RMS output voltage

Every output half cycle is made of two identical chopped half-sine humps, so the rms over one hump equals the rms over the whole output. With $v_s = \sqrt{2}V_s\sin\omega t$ (per half-winding):

$$\begin{aligned} V_o &= \left[\frac{1}{\pi}\int_{\alpha}^{\pi} 2V_s^2\sin^2\omega t\; d(\omega t)\right]^{1/2} \\ &= V_s\sqrt{\frac{1}{\pi}\left(\pi - \alpha + \frac{\sin 2\alpha}{2}\right)} \end{aligned}$$

• For $\alpha = 0$: $V_o = V_s$.
• Example (assumed values): $V_s = 230$ V, $\alpha = 30°$:

$$V_o = 230\sqrt{\frac{1}{\pi}\left(\pi - \frac{\pi}{6} + \frac{\sin 60°}{2}\right)} = 230 \times 0.9855 = 226.66\ \text{V}$$

Answer: $V_o = V_s\sqrt{\frac{1}{\pi}\left(\pi - \alpha + \frac{\sin 2\alpha}{2}\right)}$, e.g. 226.66 V for 230 V and $\alpha = 30°$; output frequency 25 Hz from 50 Hz.