Question

In a series $LCR$ circuit, half power frequencies (frequency at which power dissipated is half the maximum power dissipated in the circuit) are found to be $90rad/s$ and $40rad/s$. Find the value of $\frac{{\omega }_0}{10}$, where ${\omega }_0$ is the resonance frequency (in rad / s ).

Answer 1

Answer: 6

At half power frequencies,
$Z=R\sqrt{2}$
$\Rightarrow \left|X_L-X_C\right|=R$
${\left(\omega L-\frac{1}{\omega C}\right)}^2=R^2$
$\Rightarrow {\omega }^2{L}^2+\frac{1}{{\omega }^2{C}^2}-\frac{2L}{C}=R^2$
${\omega }^4{L}^2{C}^2-{\omega }^2\left(\frac{2L}{C}+R^2\right)C^2+1=0$
The product of roots is
${\omega }_1^2{\omega }_2^2=\frac{1}{L^2{C}^2}={\omega }_1{\omega }_2=\frac{1}{LC}={\omega }_0^2$
$\Rightarrow {\omega }_0=\sqrt{{\omega }_1{\omega }_2}=60$ rad / s