Question 2014.S3.32: A series RLC circuit is observed at two frequencies. At ω1= ......
A series RLC circuit is observed at two frequencies. At \omega_1= 1 krad/s, we note that source voltage V_1= 100∠0° V results in a current I_1= 0.03∠31° A. At \omega_2= 2 krad/s, the source voltage V_2= 100∠0° V results in a current I_2= 2∠0° A. The closest values for R, L, C out of the following options are
(a) R = 50 Ω; L = 25 mH; C = 10 μF
(b) R = 50 Ω; L = 10 mH; C = 25 μF
(c) R = 50 Ω; L = 50 mH; C = 5 μF
(d) R = 50 Ω; L = 5 mH; C = 50 μF
We know that
\phi=\tan ^{-1}\left\lgroup\frac{X_{ L }-X_{ C }}{R} \right\rgroup (i)
From the data given
\begin{gathered}R=\frac{V_2}{I_2}=\frac{100}{2}=50 \Omega \\Z=\frac{V_1}{I_1}=\frac{100 \angle 0^{\circ}}{0.03 \angle 31^{\circ}}\end{gathered}
where \phi=31^{\circ} . Substituting in Eq. (i), we have
\begin{aligned}\tan 31^{\circ} & =\frac{X_{ L }-X_{ C }}{R} \\\frac{\omega_1 L-\frac{1}{\omega_1 C}}{R} & =0.600 \\\omega_1 L=\frac{1}{\omega_1 C} & =0.6 \times 50\end{aligned}
\omega_{1}L-{\frac{1}{\omega_{1}C}}=30 (ii)
\omega_{2}L-{\frac{1}{\omega_{2}C}}=0 (iii)
Given that {\omega}_{1} = 1000 rad/s, {\omega}_{2}= 2000 rad/s.
Therefore, C = 25 μF and L = 10 mH.