Editing LC circuit (section)

==<span class="anchor" id="LC_parallel_anchor">Parallel circuit</span>==
[[File:Parallel LC Circuit.svg|thumb|Parallel LC circuit]]

When the inductor (L) and capacitor (C) are connected in parallel as shown here, the voltage {{mvar|V}} across the open terminals is equal to both the voltage across the inductor and the voltage across the capacitor. The total current {{mvar|I}} flowing into the positive terminal of the circuit is equal to the sum of the current flowing through the inductor and the current flowing through the capacitor:

: <math>\begin{align}
 V &= V_\mathsf{L} = V_\mathsf{C}\ , \\
 I &= I_\mathsf{L} + I_\mathsf{C} ~.
\end{align}</math>

=== Resonance ===
When {{mvar|X}}{{sub|L}} equals {{mvar|X}}{{sub|C}}, the two branch currents are equal and opposite. They cancel each other out to give minimal current in the main line (in principle, for a finite voltage {{mvar|V}}, there is zero current). Since total current in the main line is minimal, in this state the total impedance is maximal. There is also a larger current circulating in the loop formed by the capacitor and inductor. For a finite voltage {{mvar|V}}, this circulating current is finite, with value given by the respective voltage-current relationships of the capacitor and inductor. However, for a finite total current {{mvar|I}} in the main line, in principle, the circulating current would be infinite. In reality, the circulating current in this case is limited by resistance in the circuit, particularly resistance in the inductor windings.

The resonant frequency is given by
: <math>f_0 = \frac{ \omega_0 }{\ 2 \pi\ } = \frac{1}{\ 2 \pi \sqrt{ L C\;}\ } ~.</math>

Any branch current is not minimal at resonance, but each is given separately by dividing source voltage ({{mvar|V}}) by reactance ({{mvar|Z}}). Hence  {{math|''I'' {{=}} ''{{sfrac| V |Z}}''}} , as per [[Ohm's law]].

* At  {{math|''f''<sub>0</sub>}} , the line current is minimal. The total impedance is maximal. In this state a circuit is called a ''rejector circuit''.<ref>{{cite web |url=https://en.oxforddictionaries.com/definition/rejector_circuit |archive-url=https://web.archive.org/web/20180920195850/https://en.oxforddictionaries.com/definition/rejector_circuit |url-status=dead |archive-date=September 20, 2018 |title=rejector circuit |website=Oxford Dictionaries. English |access-date=2018-09-20}}</ref>
* Below  {{math|''f''<sub>0</sub>}} , the circuit is inductive.
* Above  {{math|''f''<sub>0</sub>}} , the circuit is capacitive.

=== 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]].