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  4. In a series LCR circuit having L = 30 mH, R = 8 Ω and the resonant frequency is 50 Hz. The quality actor of the circuit is

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Q. In a series $LCR$ circuit having $L = 30\,mH$, $R = 8\,\Omega$ and the resonant frequency is $50\,Hz$. The quality actor of the circuit is

Alternating Current

Solution:

$L =30\, MH$
$R =8 \,\Omega$
resonant freq $=50\, Hz$
$\omega=2 \pi f =100\, \pi$
Quantity factor $= Q =\frac{1}{ R } \sqrt{\frac{ L }{ C }}$
$=\frac{1}{ R } \sqrt{\frac{ L }{ C }} \longrightarrow (1)$
At resonance condition.
$\Rightarrow (100 \pi)^{2}=\frac{1}{30 \times 10^{-3} C }$
$\Rightarrow C =\frac{1}{30 \times 10^{-3} \times(100 \pi)^{2}}$
Now in (1) -
$Q =\frac{1}{8} \sqrt{\frac{30 \times 10^{-3} \times 30 \times 10^{-3} \times(100 \pi)^{2}}{1}} $
$=\frac{1}{8} \sqrt{900 \times 10^{-6} \times 10^{4} \pi^{2}} $
$=\frac{1}{8} \times 30 \times 10^{-1} \times \pi$
$=\frac{3 \pi}{8}$
$=1.1775$
$\approx 1.18$.