---
title: "Three-phase single-way thyristor rectifier: average and RMS"
subject: "Power Electronics"
chapter: 3
key: "semester_7/power_electronics:3ph-single-way-thyristor-avg-rms"
source: https://www.bctengineer.com/q/semester_7/power_electronics/3ph-single-way-thyristor-avg-rms
site: BCT Engineering
---

# Three-phase single-way thyristor rectifier: average and RMS

Power Electronics · Chapter 3: Three phase AC to DC conversion

- Asked in: 2073 Shrawan, 8 marks
- Video: [3 Phase Half Wave Controlled Rectifier with R load | Power Electronics | Lecture 53](https://www.youtube.com/watch?v=zIns_hBFNWs) — Electronics Maddy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/power_electronics/3ph-single-way-thyristor-avg-rms
- Whole chapter: https://www.bctengineer.com/md/semester_7/power_electronics/chapter-3

## Question

Explain the operation of three phase single way controlled rectifier with thyristor with neat diagram and waveforms. Derive the expression for average and RMS value of the output voltage.

## Short answer

Three thyristors conduct sequentially for 120 degrees each with a highly inductive load, and alpha is measured from natural commutation points. Average output is Vdc = 3 sqrt3 Vm cos(alpha)/(2 pi), maximum 0.827 Vm. RMS output is Vrms = Vm sqrt(1/2 + 3 sqrt3 cos(2 alpha)/(8 pi)); alpha below 90 degrees gives rectifier mode and above 90 gives inverter mode.

## Answer

A **three-phase single-way (half-wave, three-pulse) controlled rectifier** uses three thyristors, one per phase, connected to a star-connected supply (or transformer secondary) with the load returned to the neutral. The output voltage is controlled by delaying the firing of each thyristor by $\alpha$ from its natural commutation point. A highly inductive load is assumed (continuous current).

```text
          T1
 a o------>|-----+
          T2     |
 b o------>|-----+------o P (+)
          T3     |        |
 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
- Each thyristor's anode goes to one phase and the cathodes are joined to the positive output. A thyristor can conduct only when its phase is the most positive **and** it has been fired.
- The firing angle $\alpha$ is measured from the **natural commutation point** (the instant a diode would start conducting), i.e. from $30^\circ$ after the zero crossing of its phase voltage. So T1 is fired at $\omega t = 30^\circ + \alpha$, T2 at $150^\circ + \alpha$, T3 at $270^\circ + \alpha$.
- When T2 is fired, $v_b > v_a$, so T1 becomes reverse biased and turns off (**natural/line commutation**).
- With a **highly inductive load**, the load current is continuous and constant, so each thyristor conducts for exactly $120^\circ$, even if its phase voltage becomes negative. The output follows the phase of the conducting thyristor.

#### Conduction intervals and waveforms
| Thyristor | Fired at | Conducts till | $v_o$ |
|---|---|---|---|
| T1 | $30^\circ+\alpha$ | $150^\circ+\alpha$ | $v_a$ |
| T2 | $150^\circ+\alpha$ | $270^\circ+\alpha$ | $v_b$ |
| T3 | $270^\circ+\alpha$ | $390^\circ+\alpha$ | $v_c$ |

Waveform for $\alpha = 60^\circ$:

```text
 vo
      |`-.           |`-.           |`-.
      |   `-.        |   `-.        |   `-.
 -----+------`-.-----+------`-.-----+------`-.
                `-.  |         `-.  |         `-.
      90       180 210         330           450
 each pulse: Vm (at 90) -> 0 (at 180)
 -> -0.5 Vm (at 210), then jumps to next phase
```

#### Average output voltage
Taking the conduction of T1 (the waveform repeats every $2\pi/3$):

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

$$
V_{dc} = \frac{3\sqrt3\,V_m}{2\pi}\cos\alpha = 0.827\,V_m\cos\alpha
$$

where $V_m$ is the peak phase voltage. Maximum value (at $\alpha = 0$) is $V_{dm} = 0.827V_m$, so $V_{dc} = V_{dm}\cos\alpha$.

- $0 \le \alpha < 90^\circ$: $V_{dc} > 0$, rectifier mode.
- $\alpha = 90^\circ$: $V_{dc} = 0$.
- $90^\circ < \alpha < 180^\circ$: $V_{dc} < 0$, inverter mode (needs a dc source in the load).

#### RMS output voltage

$$
\begin{aligned}
V_{rms}^2 &= \frac{3}{2\pi}\int_{\pi/6+\alpha}^{5\pi/6+\alpha}V_m^2\sin^2\omega t\,d(\omega t) = \frac{3V_m^2}{4\pi}\left[\omega t - \frac{\sin2\omega t}{2}\right]_{\pi/6+\alpha}^{5\pi/6+\alpha} \\
&= \frac{3V_m^2}{4\pi}\left[\frac{2\pi}{3} - \frac{\sin\left(\frac{5\pi}{3}+2\alpha\right) - \sin\left(\frac{\pi}{3}+2\alpha\right)}{2}\right] = \frac{3V_m^2}{4\pi}\left[\frac{2\pi}{3} + \frac{\sqrt3}{2}\cos2\alpha\right]
\end{aligned}
$$

(using $\sin\left(\frac{5\pi}{3}+2\alpha\right) - \sin\left(\frac{\pi}{3}+2\alpha\right) = 2\cos(\pi+2\alpha)\sin\frac{2\pi}{3} = -\sqrt3\cos2\alpha$)

$$
V_{rms} = V_m\left[\frac{1}{2} + \frac{3\sqrt3}{8\pi}\cos2\alpha\right]^{1/2}
$$

#### Example values (230 V per phase, $V_m = 325.27$ V)
| $\alpha$ | $V_{dc}$ | $V_{rms}$ |
|---|---|---|
| $0^\circ$ | 268.99 V | 273.45 V |
| $30^\circ$ | 232.96 V | 252.66 V |
| $60^\circ$ | 134.50 V | 204.85 V |
| $90^\circ$ | 0 V | 176.14 V |

## Check yourself

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

1. According to the conduction table, T2 is fired at:
   A. 30° + alpha
   B. 90° + alpha
   C. 150° + alpha
   D. 270° + alpha
   Answer: C. The firing points are 120° apart: T1 at 30° + alpha, T2 at 150° + alpha, T3 at 270° + alpha. (Key point: Firing instants)

2. The average output voltage is found by integrating Vm sin(wt) over T1's conduction interval, which is:
   A. 0 to pi
   B. pi/6 + alpha to 5pi/6 + alpha, multiplied by 3/2pi
   C. pi/6 + alpha to 5pi/6 + alpha, multiplied by 1/2pi
   D. alpha to pi, multiplied by 3/2pi
   Answer: B. T1 conducts for 120° (2pi/3), so the integral over that interval is averaged with factor 3/2pi. (Key point: Integration limits)

3. With Vm = 325.27 V, the average output voltage at alpha = 60° (inductive load) is:
   A. 232.96 V
   B. 268.99 V
   C. 155.30 V
   D. 134.50 V
   Answer: D. Vdc = 0.827 x 325.27 x cos 60° = 134.5 V. (Key point: Average value at alpha = 60°)

4. At alpha = 90° with inductive load, the average output voltage is:
   A. Maximum
   B. Zero
   C. Negative
   D. 0.5 of maximum
   Answer: B. cos 90° = 0, so Vdc = 0; positive and negative areas of the pulse cancel. (Key point: Mode at alpha = 90°)

## Close variants set in other exams

- [Three-phase single-way thyristor rectifier: average](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Athree-phase-single-way-thyristor-avg) — Power Electronics, chapter 3, asked 3×
- [Three-phase single-way controlled rectifier](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Athree-phase-single-way-controlled-avg) — Power Electronics, chapter 3, asked 3×
- [Three-phase single-way controlled rectifier, R load](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Athree-phase-single-way-r-load) — Power Electronics, chapter 3, asked 3×
- [Three-phase single-way controlled rectifier, α = 30°](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Athree-phase-single-way-30deg) — Power Electronics, chapter 3, asked 4×

## Related in Power Electronics

- [Three-phase single-way diode rectifier: average and rms](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3A3ph-single-way-diode-avg-rms) — Power Electronics, chapter 3, asked 2×
- [Three-phase single-way controlled rectifier, α = π/6](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Athree-phase-single-way-pi-6) — Power Electronics, chapter 3
- [Three-phase single-way controlled rectifier, α = 90°](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Athree-phase-single-way-90deg) — Power Electronics, chapter 3, asked 4×
- [Three-phase single-way controlled rectifier, α = 60°](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Athree-phase-single-way-60deg) — Power Electronics, chapter 3, asked 3×
- [Three-phase single-way uncontrolled rectifier](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Athree-phase-single-way-diode-avg) — Power Electronics, chapter 3
- [Three-phase controlled rectifier with three thyristors](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Athree-phase-three-thyristor-avg-rms) — Power Electronics, chapter 3