Editing LC circuit (section)

=== Impedance ===

The same analysis may be applied to the parallel LC circuit. The total impedance is then given by

: <math>Z = \frac{\ Z_\mathsf{L} Z_\mathsf{C}\ }{ Z_\mathsf{L} + Z_\mathsf{C} }\ ,</math>

and after substitution of {{mvar|Z}}{{sub|L}} {{math|{{=}} ''j ω L''}} and {{mvar|Z}}{{sub|C}} {{math|{{=}} {{sfrac|1| ''j ω C'' }}}} and simplification, gives

: <math>Z(\omega) = -j \cdot \frac{ \omega L }{\ \omega^2 L C - 1\ } ~.</math>

Using

: <math>\omega_0 = \frac{1}{\ \sqrt{ L C\;}\ }\ ,</math>

it further simplifies to

: <math>Z(\omega) = -j\ \left(\frac{1}{\ C\ } \right) \left( \frac{\omega}{\ \omega^2 - \omega_0^2\ } \right) = + j\ \frac{ 1 }{\ \omega_0 C \left( \tfrac{\omega_0}{\omega} - \tfrac{\omega}{\omega_0} \right)\ }  = + j\ \frac{ \omega_0 L }{\ \left( \tfrac{\omega_0}{\omega} - \tfrac{\omega}{\omega_0} \right)\ } ~.</math>

Note that

: <math>\lim_{\omega \to \omega_0} Z(\omega) = \infty\ ,</math>

but for all other values of {{mvar|ω}} the impedance is finite. 

Thus, the parallel LC circuit connected in series with a load will act as [[band-stop filter]] having infinite impedance at the resonant frequency of the LC circuit, while the parallel LC circuit connected in parallel with a load will act as [[band-pass filter]].