Question

Which of the following combinations should be selected for better tuning of an $\text{LCR}$ circuit used for communications ?

• A. $R=20\Omega ,L=1.5\text{H},C=35\mu \text{F}$
• B. $R=25\Omega ,L=2.5\text{H},C=45\mu \text{F}$
• C. $R=15\Omega ,L=3.5\text{H},C=30\mu \text{F}$
• D. $R=25\Omega ,L=1.5\text{H},C=45\mu \text{F}$

Answer 1

The correct option is C $R=15\Omega ,L=3.5\text{H},C=30\mu \text{F}$

The quality factor of an $LCR$ circuit is given by,

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

$Q_1=\frac{1}{20}\sqrt{\frac{1.5}{35\times {10}^{-6}}}$

$\Rightarrow Q_1=50\times \sqrt{\frac{3}{70}}=10.35$

$Q_2=\frac{1}{25}\sqrt{\frac{2.5}{45\times {10}^{-6}}}$

$\Rightarrow Q_2=40\times \sqrt{\frac{5}{90}}=9.43$

$Q_3=\frac{1}{15}\sqrt{\frac{3.5}{30\times {10}^{-6}}}$

$\Rightarrow Q_3=\frac{100}{15}\sqrt{\frac{35}{3}}=22.77$

$Q_4=\frac{1}{25}\sqrt{\frac{1.5}{45\times {10}^{-6}}}$

$\Rightarrow Q_4=\frac{40}{\sqrt{30}}=7.30$

Clearly $Q_3$ is maximum out of $Q_1,Q_2,Q_3$ and $Q_4.$

Hence, option $\left(\text{C}\right)$ should be selected for better tuning of an $LCR$ circuit.