Question

A series LCR circuit is designed to resonate at an angular frequency ${\omega }_{\circ }={10}^{5}rad/s$. The circuit draws $16W$power from $120V$ source at resonance. The value of resistance$‘R’$ in the circuit is _______ $\Omega$.

Answer 1

Step 1. Given data:

Resonant Angular frequency ${\omega }_{\circ }={10}^{5}rad/s$

Power ($P$)$=16W$

Voltage ($V$)$=120volt$

Step 2. Find the value of resistance$R$:

At resonance:

$X_L=X_C$

Where,

$X_L=$Inductive reactance

$X_C=$Capacitive reactance

Net impedance is given as

$Z=\sqrt{R^2+{\left(X_L-X_C\right)}^2}(\because X_L=X_C)$

Where, $Z=$impedance

$R=$resistance

$X_L=$inductive reactance

$X_C=$capacitive reactance

$\Rightarrow$$Z=R$

Now, the power in terms of the voltage and resistance of the circuit is,
$P=\frac{V^2}{R}$

$16=\frac{{120}^2}{R}$

$R=\frac{14400}{16}$

$R=900\Omega$

Hence,$R$ is$\mathbf{900}\mathbf{}\mathit{\Omega }$.