---
title: "Single-phase cycloconverter with mathematical expression"
subject: "Power Electronics"
chapter: 6
key: "semester_7/power_electronics:single-phase-cyclo-math"
source: https://www.bctengineer.com/q/semester_7/power_electronics/single-phase-cyclo-math
site: BCT Engineering
---

# Single-phase cycloconverter with mathematical expression

Power Electronics · Chapter 6: AC voltage controller

- Asked in: 2070 Asar, 8 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/power_electronics/single-phase-cyclo-math
- Whole chapter: https://www.bctengineer.com/md/semester_7/power_electronics/chapter-6

## Question

Explain the operation of single phase cyclo-converter with required mathematical expression and waveform.

## Short answer

A single-phase cycloconverter directly changes AC frequency using positive and negative thyristor groups. For n input half-cycles per output half-cycle, fo = fi/n. RMS output voltage with firing angle alpha is Vo = (Vm/sqrt2) sqrt[(1/pi)(pi - alpha + sin2alpha/2)]; average group voltage Vd = Vdo cos alpha, and sinusoidal output uses alpha = cos^-1(r sin omega_o t).

## Answer

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

```text
          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)

| Input half | $v_{aO}$ | SCR | $v_o$ |
|---|---|---|---|
| 1 | + | P1 | $+|v_s|$ |
| 2 | − | P2 | $+|v_s|$ |
| 3 | + | N2 | $-|v_s|$ |
| 4 | − | N1 | $-|v_s|$ |

```text
 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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. If each group conducts for n input half cycles, what is the output frequency?
   A. fo = n*fi
   B. fo = fi/n
   C. fo = fi/(2n)
   D. fo = fi*n/2
   Answer: B. The output period is To = n*Ti, so fo = fi/n. (Key point: Frequency relation)

2. What are the positive and negative thyristor groups used for?
   A. P conducts positive load current and N conducts negative load current
   B. P for the positive supply only and N for the negative supply only
   C. P for rectifying and N for inverting only
   D. P supplies R and N supplies L
   Answer: A. The positive group carries positive load current and the negative group negative load current, building an alternating output. (Key point: Role of P and N groups)

3. To get a sinusoidal fundamental output, how is the firing angle of the positive group varied?
   A. alpha_P = constant
   B. alpha_P = cos^-1(r sin wo t)
   C. alpha_P = sin^-1(r cos wo t)
   D. alpha_P = pi - alpha_N always equal to 90°
   Answer: B. Setting Vdo cos(alpha_P) = r Vdo sin(wo t) gives alpha_P = cos^-1(r sin wo t); this is cosine-wave crossing control. (Key point: Firing-angle modulation)

4. What is the firing angle of the negative group relative to the positive group in the cosine-crossing scheme?
   A. alpha_N = alpha_P
   B. alpha_N = pi - alpha_P
   C. alpha_N = pi + alpha_P
   D. alpha_N = 90° - alpha_P
   Answer: B. alpha_N = pi - alpha_P makes the two groups' mean voltages equal and opposite in sign (cos alpha_N = -cos alpha_P). (Key point: Negative group angle)

## Related in Power Electronics

- [Single-phase cycloconverter operation](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Asingle-phase-cycloconverter) — Power Electronics, chapter 6
- [Single-phase step-down cycloconverter](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Asingle-phase-step-down-cyclo) — Power Electronics, chapter 6, asked 2×
- [Step-up and step-down single-phase cycloconverter](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Astep-up-step-down-cyclo) — Power Electronics, chapter 6
- [Single-phase cycloconverter and applications](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Asingle-phase-cyclo-applications) — Power Electronics, chapter 6
- [Bridge single-phase cycloconverter conduction sequence](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Abridge-cyclo-conduction-sequence) — Power Electronics, chapter 6, asked 3×
- [Three-phase to single-phase cycloconverter](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Athree-to-single-phase-cyclo) — Power Electronics, chapter 6, asked 2×

## Related in other subjects

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