Single-phase cycloconverter with mathematical expression

A single-phase cycloconverter is a direct AC-to-AC frequency changer: it builds a lower-frequency output from selected portions of the input sine wave, using a positive thyristor group (for positive load current) and a negative group (for negative load current).

Circuit

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

Operation ($f_o = f_i/2$, R load)

 vo  /\  /\
    /  \/  \
 --/--------\--------/---
             \  /\  /
              \/  \/
    | P1 | P2 | N2 | N1 |
    |<--- To = 2 Ti --->|

Mathematical expressions

1. Frequency relation. If each group conducts for $n$ input half cycles,

$$T_o = n\,T_i \quad\Rightarrow\quad f_o = \frac{f_i}{n}$$

2. Output voltage (R load, constant firing angle $\alpha$). With $v_s = V_m\sin\omega t$, in every input half cycle the load sees $\pm V_m\sin\omega t$ from $\alpha$ to $\pi$:

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

3. Average voltage of a group. Each group is a phase-controlled converter; for continuous current its mean voltage is

$$V_{d} = V_{do}\cos\alpha, \qquad V_{do} = \frac{2V_m}{\pi}$$

4. Sinusoidal output (firing-angle modulation). To get a sinusoidal fundamental $v_o = V_{om}\sin\omega_o t$, the firing angle is changed during the output cycle so that

$$V_{do}\cos\alpha_P = r V_{do}\sin\omega_o t \;\Rightarrow\; \alpha_P = \cos^{-1}(r\sin\omega_o t)$$

and for the negative group $\alpha_N = \pi - \alpha_P$, so that $V_{do}\cos\alpha_P = -V_{do}\cos\alpha_N$. Here $r = V_{om}/V_{do}$ ($0 \le r \le 1$) is the voltage ratio. This is called cosine-wave crossing control.

Notes

• With R load the SCRs turn off at supply zeros (natural commutation).
• With inductive load, current lags; each group works as rectifier while $v_o$ and $i_o$ have the same sign and as inverter while they differ; groups are switched only at current zero (or a reactor is used for circulating-current mode).
• Step-up operation ($f_o > f_i$) needs forced commutation.