--- title: "Three-phase single-way controlled rectifier, α = 60°" subject: "Power Electronics" chapter: 3 key: "semester_7/power_electronics:three-phase-single-way-60deg" source: https://www.bctengineer.com/q/semester_7/power_electronics/three-phase-single-way-60deg site: BCT Engineering --- # Three-phase single-way controlled rectifier, α = 60° Power Electronics · Chapter 3: Three phase AC to DC conversion - Asked in: 2076 Chaitra, 8 marks; 2072 Kartik, 8 marks (asked 2 times) - Video: [3 Phase Half Wave Controlled Rectifier with RL load | Power Electronics | Lecture 55](https://www.youtube.com/watch?v=hAm_lquzunc) — 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/three-phase-single-way-60deg - Whole chapter: https://www.bctengineer.com/md/semester_7/power_electronics/chapter-3 ## Question Explain the operation of a three phase single way controlled rectifier circuit with highly inductive load. Derive the expression for average value of the output voltage. Draw the wave form of output voltage for firing angle of 60°. ## Short answer Each thyristor conducts for 120° after firing at α from its natural commutation point. Average output voltage Vdc = (3√3 Vm)/(2π) cos α = 0.827 Vm cos α; for α=60°, Vdc = 0.4135 Vm. ## 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. ```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. | Interval ($\omega t$) | Conducting | $v_o$ | |---|---|---| | $90^\circ$ – $210^\circ$ | T1 | $v_a$ | | $210^\circ$ – $330^\circ$ | T2 | $v_b$ | | $330^\circ$ – $450^\circ$ | T3 | $v_c$ | #### Waveform for α = 60° ```text phase: va vb vc fired at: 90 210 330 (deg) ends at: 210 330 450 each phase is used from 90 to 210 deg of its own sine; vo goes negative for the last 30 deg of every 120 deg pulse ``` ```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). For $\alpha = 60^\circ$: $V_{dc} = 0.827V_m\cos60^\circ = 0.4135\,V_m$. (E.g. with 230 V per phase, $V_m = 325.27$ V and $V_{dc} = 134.50$ V.) ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. From where is the firing angle alpha measured in the three-phase single-way controlled rectifier? A. The zero crossing of the phase voltage B. The peak of the phase voltage C. The natural commutation point, 30 degrees after the phase zero crossing D. The point where the line voltage is zero Answer: C. Alpha is measured from the instant a diode would start conducting, so T1 fires at 30 degrees + alpha. (Key point: Firing angle reference) 2. With a highly inductive load, how long does each thyristor conduct? A. 120 degrees, even if its phase voltage becomes negative B. 180 degrees C. Only while its phase voltage is positive D. 120 - alpha degrees Answer: A. The load current is continuous, so each thyristor carries it for exactly 120 degrees until the next one is fired. (Key point: Conduction with inductive load) 3. What is Vdc of this rectifier? A. (3 sqrt(3) Vm/pi) cos(alpha) B. (3 sqrt(3) Vm/(2pi)) cos(alpha) = 0.827 Vm cos(alpha) C. 0.827 Vm sin(alpha) D. (3 Vm/(2pi)) cos(alpha) Answer: B. Integrating from pi/6 + alpha to 5pi/6 + alpha gives (3 sqrt(3) Vm/(2 pi)) cos(alpha). (Key point: Average voltage expression) 4. For alpha = 60 degrees, what is Vdc and how does the waveform look? A. 0.827 Vm; no negative part B. 0.4135 Vm; each pulse goes negative in its last 30 degrees C. 0; equal positive and negative areas D. 0.716 Vm; pulses start at zero Answer: B. Vdc = 0.827 Vm cos 60 = 0.4135 Vm; each phase is used from 90 to 210 degrees of its sine, so the end dips below zero. (Key point: Output at alpha = 60) ## Close variants set in other exams - [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× ## Related in Power Electronics - [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, 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× - [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 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 thyristor rectifier: average and RMS](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3A3ph-single-way-thyristor-avg-rms) — Power Electronics, chapter 3, asked 5× ## Related in other subjects - [3-phase half controlled bridge: no-load speed, firing angle](https://www.bctengineer.com/question.md?key=semester_7%2Futilization_of_electrical_energy%3Ahalf-bridge-motor-no-load-speed) — Utilization of Electrical Energy, chapter 3, asked 3× - [SCR analogy; half-wave rectifier for 1 A average](https://www.bctengineer.com/question.md?key=semester_4%2Fadvanced_electronics%3Ascr-analogy-hwr-120v) — Advanced Electronics, chapter 7 - [3-phase thyristor bridge: no-load speed and firing angle](https://www.bctengineer.com/question.md?key=semester_7%2Futilization_of_electrical_energy%3Afull-bridge-motor-no-load-speed) — Utilization of Electrical Energy, chapter 3, asked 3× - [Voltage regulation expression for inductive load](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Avoltage-regulation-inductive-load) — Electrical Machines, chapter 2