--- title: "Three-phase single-way controlled rectifier, α = 30°" subject: "Power Electronics" chapter: 3 key: "semester_7/power_electronics:three-phase-single-way-30deg" source: https://www.bctengineer.com/q/semester_7/power_electronics/three-phase-single-way-30deg site: BCT Engineering --- # Three-phase single-way controlled rectifier, α = 30° Power Electronics · Chapter 3: Three phase AC to DC conversion - Asked in: 2076 Asoj, 8 marks - 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-30deg - 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 highly inductive load. Derive the expression of average and RMS value of output voltage. Draw the waveforms of output voltage for α = 30° and calculate average value of output voltage. ## Short answer With a highly inductive load current is continuous and constant; each thyristor conducts 120° and commutates when the next phase is fired. Vdc=0.827Vm cosα, so at α=30° Vdc=0.716Vm; Vrms=Vm[1/2+3√3cos2α/(8π)]^{1/2}. The α=30° waveform is continuous pulses just touching zero, from 60° to 180° for T1. ## 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. The load is highly inductive, so the load current is continuous and constant. ```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 | $v_o$ | |---|---|---| | $60^\circ$ – $180^\circ$ | T1 | $v_a$ | | $180^\circ$ – $300^\circ$ | T2 | $v_b$ | | $300^\circ$ – $420^\circ$ | T3 | $v_c$ | #### Waveform for α = 30° ```text phase: va vb vc fired at: 60 180 300 (deg) ends at: 180 300 420 each phase is used from 60 to 180 deg of its own sine; vo just touches zero at the end of each pulse (boundary of continuity) ``` ```text vo .-. .-. .-. / \ / \ / \ | \ | \ | \ -----+------+-----+------+-----+------+-- 60 180 180 300 300 420 each pulse: 0.866 Vm -> Vm (at 90) -> 0 ``` The thyristor currents are $120^\circ$ blocks of the constant load current $I_o$. #### 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 $V_{dc} = V_{dm}\cos\alpha$. #### 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 $\sin\left(\frac{5\pi}{3}+2\alpha\right) - \sin\left(\frac{\pi}{3}+2\alpha\right) = 2\cos(\pi+2\alpha)\sin\frac{2\pi}{3} = -\sqrt3\cos2\alpha$) $$ V_{rms} = V_m\left[\frac{1}{2} + \frac{3\sqrt3}{8\pi}\cos2\alpha\right]^{1/2} $$ #### Calculation for α = 30° *Assumption:* phase voltage 230 V (400 V line), $V_m = \sqrt2 \times 230 = 325.27$ V. $$ \begin{aligned} V_{dc} &= \frac{3\sqrt3 \times 325.27}{2\pi}\cos30^\circ = 268.99 \times 0.866 = 232.96\ \text{V} \\ V_{rms} &= 325.27\left[0.5 + \frac{3\sqrt3}{8\pi}\cos60^\circ\right]^{1/2} = 325.27 \times 0.7768 = 252.66\ \text{V} \end{aligned} $$ **Answer: $V_{dc} = 0.716V_m$ = 232.96 V (for 230 V per phase).** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In the three-phase single-way rectifier, the firing angle alpha is measured from: A. The zero crossing of the phase voltage B. The natural commutation point, 30° after the phase zero crossing C. The peak of the phase voltage D. The zero crossing of the line voltage Answer: B. Alpha is counted from the point where a diode would start conducting, so T1 is fired at 30° + alpha. (Key point: Firing angle reference) 2. With a highly inductive load, each thyristor conducts for: A. Only while its phase voltage is positive B. 120°, even if its phase voltage becomes negative C. 180° D. (150° - alpha) Answer: B. The load current is continuous and constant, so each thyristor carries it for exactly 120° until the next one is fired. (Key point: Highly inductive load conduction) 3. The average output voltage for the inductive-load three-phase single-way rectifier is: A. 0.827 Vm cos(alpha) B. 1.35 Vm cos(alpha) C. 0.827 Vm sin(alpha) D. 0.477 Vm cos(alpha) Answer: A. Integrating over T1's 120° conduction gives Vdc = (3 sqrt3 Vm / 2pi) cos(alpha) = 0.827 Vm cos(alpha). (Key point: Average voltage formula) 4. At alpha = 30°, the average output voltage in terms of the peak phase voltage Vm is: A. 0.827 Vm B. 0.716 Vm C. 0.414 Vm D. 0.7768 Vm Answer: B. Vdc = 0.827 Vm cos 30° = 0.716 Vm; 0.7768 Vm is the rms value. (Key point: Value at alpha = 30°) ## Close variants set in other exams - [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 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× - [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 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 = 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 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× - [Full controlled rectifier, 25 A, α = 30°, derivation](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Afull-controlled-rectifier-25a-30deg-derive) — Power Electronics, chapter 2 ## Related in other subjects - [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