AC voltage controllers, also known as AC choppers or AC regulators, are static switches used to control the AC voltage supplied to a load. They are widely used in applications like light dimming, fan speed control, and industrial heating. Solving numerical problems related to these controllers is crucial for understanding their performance and design, especially in competitive exams like GATE.

AC voltage controllers typically use thyristors (SCRs) or TRIACs as switching elements. The control is achieved by varying the firing angle ($\alpha$) of these devices. The output voltage waveform is a chopped version of the input AC voltage. Key parameters to consider include RMS output voltage, average output voltage, RMS load current, average load current, and power delivered to the load.

These controllers operate on a single-phase AC supply. They can be implemented using two SCRs connected in inverse parallel or a TRIAC. The firing angle ($\alpha$) determines the portion of each half-cycle of the AC input that is allowed to pass to the load.

Consider a single-phase AC voltage controller connected to a resistive load of 10 $\Omega$. The RMS supply voltage is 230 V. If the firing angle is 60 degrees ($\pi/3$ radians), calculate the RMS output voltage and the RMS load current.

<strong>Solution:</strong>

Given: $V_s = 230$ V (RMS) $R = 10 \Omega$ $\alpha = 60^{\circ} = \frac{\pi}{3}$ radians

1. <strong>Calculate RMS Output Voltage ($V_{o,rms}$):</strong> Using the formula: $V_{o,rms} = V_s \sqrt{1 - \frac{2\alpha}{\pi} + \frac{\sin(2\alpha)}{2\pi}}$ $V_{o,rms} = 230 \sqrt{1 - \frac{2(\pi/3)}{\pi} + \frac{\sin(2(\pi/3))}{2\pi}}$ $V_{o,rms} = 230 \sqrt{1 - \frac{2}{3} + \frac{\sin(2\pi/3)}{2\pi}}$ $V_{o,rms} = 230 \sqrt{\frac{1}{3} + \frac{\sqrt{3}/2}{2\pi}}$ $V_{o,rms} = 230 \sqrt{\frac{1}{3} + \frac{\sqrt{3}}{4\pi}}$ $V_{o,rms} \approx 230 \sqrt{0.3333 + \frac{1.732}{12.566}}$ $V_{o,rms} \approx 230 \sqrt{0.3333 + 0.1378}$ $V_{o,rms} \approx 230 \sqrt{0.4711}$ $V_{o,rms} \approx 230 \times 0.6864$ $V_{o,rms} \approx 157.87$ V

2. <strong>Calculate RMS Load Current ($I_{o,rms}$):</strong> $I_{o,rms} = \frac{V_{o,rms}}{R}$ $I_{o,rms} = \frac{157.87}{10}$ $I_{o,rms} \approx 15.79$ A

When the load is inductive, the current waveform is different from the voltage waveform due to the inductor's property. The thyristors or TRIACs may continue to conduct even after the voltage crosses zero if the current is still positive. This leads to a 'current chopping' effect and requires careful consideration of the firing angle and load parameters (resistance and inductance) to determine the output voltage and current.

Note: For inductive loads, the extinction angle $\beta$ is typically greater than $\pi$. The calculation of $\beta$ often involves solving transcendental equations based on the load impedance and firing angle.

Three-phase AC voltage controllers are used for higher power applications. They typically employ six thyristors or TRIACs to control the voltage supplied to a three-phase load. The control strategy involves adjusting the firing angles of these devices to regulate the RMS voltage and power delivered to the load. Numerical problems here involve understanding phase voltages, line voltages, and the effect of firing angles on the three-phase system.

When solving numerical problems:

• Ensure angles are in radians for trigonometric functions in calculations.
• Differentiate between RMS and peak values of voltage and current.
• Pay close attention to the load type (resistive vs. inductive) as it significantly affects the formulas.
• For inductive loads, correctly identify or calculate the extinction angle ($\beta$).
• Understand the basic waveforms of controlled AC output voltage for different firing angles.