Question

Obtain the resonance angular frequency for the circuit shown in the figure.

Answer 1

$Z_{RL}=R+X_L=R+jwL$

$Y_{RL}=\frac{1}{Z_{RL}}=\frac{1}{R+jwL}$

$Z_c=X_c=\frac{1}{jwc}$

$Y_c=\frac{1}{X_c}=jwc$

$Y=Y_c+Y_{RL}=jwc+\frac{1}{R+jwL}=jwc+\frac{R-jwL}{R^2+w^2{L}^2}$

$Y=jwc+\frac{R}{R^2+w^2{l}^2}-\frac{jwL}{R^2+w^2{l}^2}=\frac{R}{R+w^2{l}^2}+j(wc-\frac{wL}{R^2+w^2{L}^2})$

for resonance frequency $wc-\frac{wL}{R^2+w^2{l}^2}\Rightarrow c=\frac{L}{R^2+w^2{l}^2}$

$w=\sqrt{\frac{1}{CL}-\frac{R^2}{l^2}}$