LC circuit
From Academic Kids
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 <math>\sqrt{1 \over LC}<math>, 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, <math>V _{C}<math> plus the voltage across the inductor, <math>V _{L}<math> equals 0.
We also know that <math>V _{L}(t) = L \frac{di(t)}{dt}<math> and <math>i(t) _{C} = C \frac{dV(t)}{dt}<math>
After rearranging and substituting, we obtain the second order differential equation
<math>\frac{d ^{2}i(t)}{dt^{2}} + \frac{1}{LC} i(t) = 0<math>
The associated polynomial is <math>r ^{2} + \frac{1}{LC} = 0<math>, thus <math>r = j \sqrt{\frac{1}{LC}}<math> and the complete solution to the differential equation is
<math>i(t) = Ae ^{j \sqrt{\frac{1}{LC}} t}<math>
and can be solved for <math>A<math> 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 <math>\sqrt{\frac{1}{LC}}<math>.
