--- title: "AC voltage controller rms at α = 90°" subject: "Power Electronics" chapter: 6 key: "semester_7/power_electronics:acvc-r-load-220v-90deg" source: https://www.bctengineer.com/q/semester_7/power_electronics/acvc-r-load-220v-90deg site: BCT Engineering --- # AC voltage controller rms at α = 90° Power Electronics · Chapter 6: AC voltage controller - Asked in: 2076 Chaitra, 8 marks; 2072 Kartik, 8 marks (asked 2 times) - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/power_electronics/acvc-r-load-220v-90deg - Whole chapter: https://www.bctengineer.com/md/semester_7/power_electronics/chapter-6 ## Question Explain the operation of single phase ac voltage controller with resistive load. If the input voltage is 220V, 50Hz, calculate the rms value of output voltage for firing angle of 90°. ## Short answer Phase control with anti-parallel thyristors: output voltage is zero until 90°, then follows supply until 180° each half-cycle. The rms output voltage is V_o = V_s√[(π-α+sin2α/2)/π]; for V_s=220 V, α=90°, V_o = 220√0.5 = 155.56 V. ## Answer A single-phase AC voltage controller controls the rms voltage across a load by phase-delayed firing of two anti-parallel thyristors, T1 for the positive half and T2 for the negative half. #### Operation with resistive load ```text T1 --->|--- +---| |----+ | ---|<--- | | | T2 | ( ~ ) 220 V, 50 Hz R vo | | +-------------------+ ``` - $0$ to $\alpha$: no thyristor on, $v_o = 0$. - At $\alpha$, T1 is fired; $v_o = v_s$ until $\pi$, where current becomes zero and T1 turns off naturally. - $\pi$ to $\pi+\alpha$: $v_o = 0$. - At $\pi + \alpha$, T2 is fired; $v_o = v_s$ until $2\pi$. ```text vo (alpha = 90 deg) |\ | \ -------+--\-------+------- 0 90 180 | /360 | / | / 270 ``` Each half cycle the load gets only the second quarter of the sine wave. The output frequency stays 50 Hz. #### RMS output voltage formula $$ \begin{aligned} V_o &= \left[\frac{1}{\pi}\int_{\alpha}^{\pi} (\sqrt{2}V_s\sin\omega t)^2\, d(\omega t)\right]^{1/2} \\ &= V_s\sqrt{\frac{1}{\pi}\left(\pi - \alpha + \frac{\sin 2\alpha}{2}\right)} \end{aligned} $$ #### Calculation Given $V_s = 220$ V, $\alpha = 90° = \pi/2$, so $\sin 2\alpha = \sin 180° = 0$: $$ \begin{aligned} V_o &= 220\sqrt{\frac{1}{\pi}\left(\pi - \frac{\pi}{2} + 0\right)} \\ &= 220\sqrt{0.5} = 220 \times 0.7071 \\ &= 155.56\ \text{V} \end{aligned} $$ **Answer: $V_{o,rms} = 155.56$ V** (half the power of full conduction, since $V_o^2/V_s^2 = 0.5$). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In the single-phase AC voltage controller with R load, what is the load voltage between 0 and alpha? A. Equal to the supply voltage B. Zero, because no thyristor is conducting C. Equal to the peak voltage D. Negative of the supply voltage Answer: B. Neither thyristor has been fired before alpha, so vo = 0 until T1 is fired. (Key point: Operation) 2. How does T1 turn off in the resistive load case? A. Forced commutation circuit B. Gate reverse pulse C. Naturally, when the current falls to zero at pi D. When T2 is fired at alpha Answer: C. With R load the current follows the voltage and becomes zero at pi, so T1 commutates naturally. (Key point: Turn-off of T1) 3. What is the rms output voltage formula for R load? A. Vs sqrt((1/pi)(pi - alpha + sin(2 alpha)/2)) B. Vs (1 + cos alpha)/2 C. Vs sqrt(alpha/pi) D. Vs cos(alpha) Answer: A. Integrating (Vm sin wt)^2 from alpha to pi over a half cycle gives this expression. (Key point: RMS formula) 4. For Vs = 220 V and alpha = 90 degrees, what is the rms output voltage? A. 220 V B. 110 V C. 311 V D. 155.56 V Answer: D. sin 180 = 0, so Vo = 220 sqrt(0.5) = 155.56 V, i.e. half the full power. 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