Question

An LCR series circuit with 100$\Omega$ resistance is connected to an ac source of 200 V and angular frequency 300 rad/s. When only the capacitance is removed, the current lags behind the voltage by 60°. And when only inductance is removed the current leads the voltage by 60°. Find the value of current (in Ampere) through the LCR circuit)

Answer 1

2

Answer 2

The problem describes an LCR series circuit and provides information about its behavior under two specific conditions, allowing us to determine the reactances and then the current in the full LCR circuit.

1. Analyze the given information:

• Resistance (R) = 100 $\Omega$
• AC source voltage (V) = 200 V
• Angular frequency ($\omega$) = 300 rad/s

2. Condition 1: Only capacitance is removed (RL circuit)

• When only the capacitance is removed, the circuit consists of resistance (R) and inductance (L).
• The current lags behind the voltage by $60^\circ$.
• For an RL circuit, the phase angle ($\phi_1$) is given by: \tan \phi_1 = \frac{\text{Inductive Reactance (X_L)}}{\text{Resistance (R)}} $\tan 60^\circ = \frac{X_L}{100}$ $\sqrt{3} = \frac{X_L}{100}$ $X_L = 100\sqrt{3} \text{ } \Omega$

3. Condition 2: Only inductance is removed (RC circuit)

• When only the inductance is removed, the circuit consists of resistance (R) and capacitance (C).
• The current leads the voltage by $60^\circ$.
• For an RC circuit, the phase angle ($\phi_2$) is given by: \tan \phi_2 = \frac{\text{Capacitive Reactance (X_C)}}{\text{Resistance (R)}} $\tan 60^\circ = \frac{X_C}{100}$ $\sqrt{3} = \frac{X_C}{100}$ $X_C = 100\sqrt{3} \text{ } \Omega$

4. Determine the impedance of the LCR circuit:

• For the full LCR series circuit, the impedance (Z) is given by: $Z = \sqrt{R^2 + (X_L - X_C)^2}$
• From our calculations, we found $X_L = 100\sqrt{3} \text{ } \Omega$ and $X_C = 100\sqrt{3} \text{ } \Omega$.
• Therefore, $X_L - X_C = 100\sqrt{3} - 100\sqrt{3} = 0$.
• This indicates that the circuit is in resonance.
• At resonance, the impedance is equal to the resistance: $Z = \sqrt{R^2 + 0^2} = R$ $Z = 100 \text{ } \Omega$

5. Calculate the current in the LCR circuit:

• The current (I) in the LCR circuit is given by Ohm's Law: $I = \frac{V}{Z}$ $I = \frac{200 \text{ V}}{100 \text{ } \Omega}$ $I = 2 \text{ A}$

The value of the current through the LCR circuit is 2 Ampere.