$LC$ Oscillations

Question

I understand the mathematics behind LC and RLC oscillator circuits in terms of setting up the second order differential equations and solving them, giving us the expected sinusoidal results, but I don't quite understand how LC oscillations work qualitatively in my mind - I can't explain them well in words in terms of the changing magnetic field (Faraday's Law), capacitor charge, etc.

At the moment, this is my 'in words' understanding of the concept: "As the capacitor discharges, the current increases to a maximum and then tries to decrease, but the inductor causes this decrease in current to happen at a slower rate, which means that more charge collects on the other plate of the capacitor than would normally in the discharged state, leading to a voltage across the capacitor in the opposite direction. This voltage then causes the entire cycle to repeat, this time in the other direction, and the oscillations continue indefinitely (given zero resistance/damping)."

Is this correct? If not could someone please explain how LC oscillations actually work in words?

Answer 1

If I would to explain LC oscillations in words, I would start explaining the pendulum first.

For small angles, we know it behaves like an harmonic oscillator. During this regime, wr know that two solutions are possible: the null solution (pendulum at rest in the equilibrium position) and oscillating.

In a pendulum, the energy is divided in kinetic, related to its velocity, and potential, related to its height. When it has maximum potential energy, meaning it is its highest point, there is no velocity, so no kinetic energy; when it is in its fastest point, it is also in the lowest, so no potential energy (we should choose the zero there, but you get it). The movement keeps this balance, conserving total energy.

If you look in the phase space, the region of states with the same fixed energy $E$ is an ellipse; what the system do is to run into this ellipse forever and ever, transitioning from state to state, respecting the conservation of total energy. There is no reason for the system to stop in any specific point in the energy ellipse, and the system would only stop in a point if the ellipse would be a point (zero energy).

Now the LC circuit. The energy of the capacitor could be written in terms of the charge there $E_c = \frac {1}{2C} Q^2$. The energy of the inductor is written in terms of the current $E_I = \frac 12 L I^2$.

The energy of the capacitor is stored in the electrical field, and somehow it could be thought as being a potential energy. Analogously, the energy in the inductor is stored in the magnetic field, and can be thought as a kind of kinetic energy. This analogy is not so bad: when the potential energy is at its maximum (capacitor full charge), there is no kinetic energy (no current). When the kinetic energy is at its maximum (maximum current), the capacitor is with no charges (no potential energy).

If we remember that $I=\dot Q$, we can represent the state of the system by a point $(Q,\dot Q)$ and call the set of all the possible points by phase space. For fixed total energy, the set of points is again an ellipse and the system runs the energy ellipse forever, as in the pendulum case.

Of course, I'm not considering resistance, and also I'm not considering any external force. In order to understand the LC oscillations, we don't need to, as in the pendulum case. It is just a case of balance between potential and kinetic energy.