Question

Obtain the resonant frequency ${\omega }_r$ of a series LCR circuit with $L=2.0H,C=32\mu F$ and $R=10\Omega$ . What is the Q-value of this circuit?

Answer 1

Step 1: Find Resonant frequency
At resonance $X_L=X_C$

${\omega }_{r}L=\frac{1}{{\omega }_{r}C}$

$\Rightarrow {\omega }_r=\frac{1}{\sqrt{LC}}$
Putting values, we get

$\Rightarrow {\omega }_r=\frac{1}{\sqrt{2\times 32\times {10}^{-6}}}$

${\omega }_r=125rad{S}^{-1}$

Step 2: Find Quality factor or $Q$ value

As Quality factor $\left(Q\right)$ is given by:

$Q=\frac{{\omega }_{r}L}{R}=\frac{1}{{\omega }_{r}CR}$

Putting ${\omega }_r=\frac{1}{\sqrt{LC}}$

$Q=\frac{1}{R}\sqrt{\frac{L}{C}}$

Putting values, we get
$Q=\frac{1}{10}\sqrt{\frac{2}{32\times {10}^{-6}}}$

$Q=25$
Final Answer: $125rad{S}^{-1};24.$