An LCR circuit contains resistance of $110 \Omega$
and a supply of $220 V$ at $300 rad / s$ angular
frequency. If only capacitance is removed from
the circuit, current lags behind the voltage by
$45^{\circ} .$ If on the other hand, only inductor is
removed the current leads by $45^{\circ}$ with the
applied voltage. The $rms$ current flowing in the circuit will be :
- $1 A$
- $2.5 A$
- $1.5 A$
- $2 A$
The Correct Option is D
Solution and Explanation
To determine the \(rms\) current flowing in the given LCR circuit, we need to analyze the conditions specified by the question step-by-step.
The circuit has a resistance of \(R = 110 \Omega\) and a supply voltage \(V = 220 \, \text{V}\) with an angular frequency of \(\omega = 300 \, \text{rad/s}\).
The problem states that the current lags by \(45^\circ\) when the capacitance is removed, which implies that the circuit contains only resistance and inductance. The phase angle \(\phi\) in an \(RL\) circuit is determined by:
\(\tan \phi = \frac{X_L}{R}\),
where \(X_L\) is the inductive reactance. Since \(\phi = 45^\circ\) and \(\tan 45^\circ = 1\), we have:
\(\frac{X_L}{R} = 1 \Rightarrow X_L = R = 110 \, \Omega\).
Substituting the inductive reactance formula \(X_L = \omega L\), we get:
\(110 = 300 \times L \Rightarrow L = \frac{110}{300} = \frac{11}{30} \, \text{H}\).
Similarly, when the inductor is removed, the current leads by \(45^\circ\), indicating a resistor-capacitor \((RC)\) circuit. Here, the phase angle is represented by:
\(\tan \phi = \frac{-X_C}{R}\),
where \(X_C\) is the capacitive reactance. Using \(\phi = 45^\circ\) and \(\tan 45^\circ = 1\), we have:
\(\frac{-X_C}{R} = 1 \Rightarrow X_C = -R = -110 \, \Omega\).
Therefore, \(X_C = \frac{1}{\omega C} = 110 \Rightarrow C = \frac{1}{300 \times 110} = \frac{1}{33000} \, \text{F}\).
The circuit behaves as an \({\text{RLC}}\) circuit, fully balanced because the inductive and capacitive reactances cancel each other out. In such a condition, the impedance \(Z\) of the circuit is purely resistive:
\(Z = R = 110 \, \Omega\).
Finally, we calculate the root mean square ( \(rms\)) current using the formula:
\(I_{\text{rms}} = \frac{V}{Z} = \frac{220}{110} = 2 \, \text{A}\).
Thus, the \(rms\) current flowing in the circuit is \(2 \, \text{A}\).
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Concepts Used:
LCR Circuit
An LCR circuit, also known as a resonant circuit, or an RLC circuit, is an electrical circuit consist of an inductor (L), capacitor (C) and resistor (R) connected in series or parallel.
Series LCR circuit
When a constant voltage source is connected across a resistor a current is induced in it. This current has a unique direction and flows from the negative to positive terminal. Magnitude of current remains constant.
Alternating current is the current if the direction of current through this resistor changes periodically. An AC generator or AC dynamo can be used as AC voltage source.
