How Do CLC Circuits Oscillate in Series and Parallel Configurations?

[freemind]

Hello folks,

Consider an ideal circuit with a capacitor, inductor, and another capacitor, in that order (the inductor is in between). Let's start off with the initial condition being that there is a charge $$q_0$$ on capacitor 1. Here's what I'm wondering: would this circuit behave like a regual LC circuit, in that as one capacitor discharges, the other capacitor will charge up (and this cycle will reverse repeatedly)? Or am I missing something important here?

Thanks

P.S: the point of this is to find the oscillation parameters of the circuit, but I'm not asking for that answer.

 

[gneill]

The problem statement doesn't indicate whether the components are connected in series or in parallel. We can consider both situations.

We assume that potentials are referenced from a common point, at the bottom of the circuit.

Parallel connection:
The capacitors will immediately redistribute the charge across both capacitors the instant the switch closes (we're assuming ideal components and wiring). The peak voltage of the ensuing oscillations will be lowered accordingly. The circuit will behave as a typical LC oscillator where the "C" in this case is the sum of the two capacitors: C = C1 + C2. The voltage across them will be the same at all times after the circuit is connected. So the charge won't ping-pong back and forth between the capacitors. Instead they gain and lose their charge in sync, trading potential for current with the inductor.

Series connection:
In this case the net capacitance is given by the series-capacitance formula: C = C1*C2/(C1 + C2). The capacitors can have different voltages over time, and charge will move from one to the other, driven by the oscillations of the current. Note that the potentials on the capacitors will oscillate about an offset voltage, not zero volts. This is due to the capacitors starting out with unequal charges hence unequal voltages, and the current always moving the same amount of charge for each. The offset would be equal to the voltage that would be obtained if the capacitors were simply connected in parallel. If both capacitors are of equal value the offset is half the initial voltage of the one initially charged. So for example if that voltage was 100 V, both capacitors would oscillate between 0 and 100 V potential difference with the common reference point, with their sinewaves centered on 50 V. Due to the phase difference you would see a 200 V sinewave across the inductor.