# LC circuit

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Lc_circuit.png
LC circuit diagram

An LC circuit consists of an inductor and a capacitor. The electrical current will alternate between them at an angular frequency of [itex]\sqrt{1 \over LC}[itex], where L is the inductance, and C is the capacitance.

An LC circuit is an idealized model since it assumes there is no dissipation of energy due to resistance. For a model incorporating resistance see RLC circuit.

By Kirchhoff's voltage law, we know that the voltage across the capacitor, [itex]V _{C}[itex] plus the voltage across the inductor, [itex]V _{L}[itex] equals 0.

We also know that [itex]V _{L}(t) = L \frac{di(t)}{dt}[itex] and [itex]i(t) _{C} = C \frac{dV(t)}{dt}[itex]

After rearranging and substituting, we obtain the second order differential equation

[itex]\frac{d ^{2}i(t)}{dt^{2}} + \frac{1}{LC} i(t) = 0[itex]

The associated polynomial is [itex]r ^{2} + \frac{1}{LC} = 0[itex], thus [itex]r = j \sqrt{\frac{1}{LC}}[itex] and the complete solution to the differential equation is

[itex]i(t) = Ae ^{j \sqrt{\frac{1}{LC}} t}[itex]

and can be solved for [itex]A[itex] with the addition of initial conditions. Since the exponential is complex, it means that it is AC. The real part is a sinusoid with amplitude A and the angular frequency is [itex]\sqrt{\frac{1}{LC}}[itex].

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