Question

An AC circuit containing 800 mH inductor and a 60 $\mu$F capacitor is in series with 15$\Omega$ resistance. They are connected to 230 V, 50 Hz AC supply. Obtain total power consumed near to :-

• A. 20 watt
• B. 5 watt
• C. 40 watt
• D. 10 watt

Answer 1

Answer: (A)

20 watt

Answer 2

1. Calculate the angular frequency ($\omega$): $\omega = 2\pi f = 2\pi (50 \, \text{Hz}) = 100\pi \, \text{rad/s}$

2. Calculate the inductive reactance ($X_L$) and capacitive reactance ($X_C$): Inductive reactance: $X_L = \omega L = (100\pi \, \text{rad/s})(0.8 \, \text{H}) = 80\pi \, \Omega \approx 251.33 \, \Omega$

   Capacitive reactance: $X_C = \frac{1}{\omega C} = \frac{1}{(100\pi \, \text{rad/s})(60 \times 10^{-6} \, \text{F})} = \frac{10^6}{6000\pi} \, \Omega = \frac{500}{3\pi} \, \Omega \approx 53.05 \, \Omega$

3. Calculate the net reactance ($X$): $X = X_L - X_C = 80\pi - \frac{500}{3\pi} \approx 251.33 \, \Omega - 53.05 \, \Omega = 198.28 \, \Omega$

4. Calculate the impedance ($Z$): $Z = \sqrt{R^2 + X^2} = \sqrt{(15 \, \Omega)^2 + (198.28 \, \Omega)^2} = \sqrt{225 + 39314.27} = \sqrt{39539.27} \, \Omega \approx 198.84 \, \Omega$

5. Calculate the total power consumed ($P$): The power consumed in an AC circuit is dissipated only by the resistor. $P = \frac{V_{rms}^2 R}{Z^2} = \frac{(230 \, \text{V})^2 \times 15 \, \Omega}{(198.84 \, \Omega)^2} = \frac{52900 \times 15}{39538.38} \, \text{W} = \frac{793500}{39538.38} \, \text{W} \approx 20.07 \, \text{W}$