Editing LC circuit (section)

===Impedance===
In the series configuration, resonance occurs when the complex electrical impedance of the circuit approaches zero.

First consider the [[Electrical impedance|impedance]] of the series LC circuit. The total impedance is given by the sum of the inductive and capacitive impedances:

: <math> Z = Z_\mathsf{L} + Z_\mathsf{C} ~.</math>

Writing the inductive impedance as {{mvar| Z}}{{sub|L}} {{math| {{=}} ''jωL'' }} and capacitive impedance as {{mvar| Z}}{{sub|C}} {{math| {{=}} {{sfrac|1| ''j ω C'' }} }} and substituting gives

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

Writing this expression under a common denominator gives

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

Finally, defining the natural angular frequency as

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

the impedance becomes

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

where <math>\, \omega_0 L\ \,</math> gives the reactance of the inductor at resonance.

The numerator implies that in the limit as {{math| ''ω'' → ±''ω''<sub>0</sub> }}, the total impedance {{mvar| Z }} will be zero and otherwise non-zero. Therefore the series LC circuit, when connected in series with a load, will act as a [[band-pass filter]] having zero impedance at the resonant frequency of the LC circuit.