Question

In an L-C-R circuit the value of ${X_L}$ , ${X_C}$ and $R$ are $300\Omega $ , $200\Omega $ and $100\Omega $ respectively. The total impedance of the circuit will be
(A) $600\Omega $
(B) $200\Omega $
(C) $141\Omega $
(D) $310\Omega $

Answer 1

Hint: - At resonance the capacitive reactance and inductive reactance are equal. At above the resonant frequency the inductive reactance will be greater than the capacitive reactance. So the RLC circuit operating above resonant frequency behaves as a purely inductive circuit. The phase difference between the current and voltage will be equal in this circuit as the inductive circuit.
Formula used:
The formula for the impedance in the LCR circuit,
$Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} $
where $Z$ = impedance
$R$ = resistance
${X_L}$ = Inductive reactance
${X_C}$ = Capacitive Reactance

Complete step-by-step solution:
Given,
The value of the resistor in the RLC circuit is $100\Omega $ ,
The value of Inductive reactance in the RLC circuit is $300\Omega $ ,
The value of Capacitive reactance in the RLC circuit is $200\Omega $ .
Since, For a series LCR circuit, the impedance is given by the formula,
$Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} $
Substitute the value of $R$ , ${X_L}$ and ${X_C}$ in the above equation we get,
$Z = \sqrt {{{\left( {100} \right)}^2} + {{\left( {300 - 200} \right)}^2}} $
$ \Rightarrow Z = \sqrt {{{\left( {100} \right)}^2} + {{\left( {100} \right)}^2}} $
On further solving the equation we get,
$Z = \sqrt {2{{\left( {100} \right)}^2}} = 100\sqrt 2 $
$ \Rightarrow Z = 141.42 \simeq 141\Omega $
And so the total impedance of the circuit is $141\Omega $ .

Hence, the correct answer is option is (C).

Additional information: In the LCR circuits when the ${X_L} \succ {X_C}$ , then the circuit is termed as Inductive circuit and similarly when the ${X_C} \succ {X_L}$ then the corresponding circuits are termed as Capacitive circuit. Always remember that the voltages in a series RLC circuit are actually phasors which are treated as vectors, so the net emf is obtained as a vector addition of three voltages given.

Note: The series resonance or the series LCR circuits are one of the most significant circuits. They have a vast number of practical uses starting from AC mains filters, radios, and also in television circuits. We have to remember all the formulas used and understand the meaning of the terms such as impedance and reactance, then this type of question will be solved easily.