Circuit Theory/LC Tuned Circuits

CL in Series

$Z=Z_L+Z_C$

$Z=j\omega L+\frac{1}{j\omega C}$

$Z=\frac{(j\omega {)}^{2}LC+1}{j\omega C}$

There is one frequency at which $Z_L=Z_C$

$j\omega L=\frac{1}{j\omega C}$

$(j\omega {)}^2=\frac{1}{LC}$

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

The frequency $\omega =\frac{1}{\sqrt{LC}}$ is called The Resonant Frequency is denoted as ω_o

CL in Parallel

$\frac{1}{Z}=\frac{1}{Z_L}+\frac{1}{Z_C}$

$\frac{1}{Z}=\frac{1}{j\omega L}+j\omega C$

$\frac{1}{Z}=\frac{(j\omega {)}^{2}LC+1}{j\omega L}$

$Z=\frac{j\omega L}{(j\omega {)}^{2}LC+1}$

There is one frequency at which $Z_L=Z_C$

$j\omega L=\frac{1}{j\omega C}$

$(j\omega {)}^2=\frac{1}{LC}$

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

The frequency $\omega =\frac{1}{\sqrt{LC}}$ is called The Resonant Frequency is denoted as ω_o