# Another capacitor Laplace transform problem

1. Feb 14, 2010

### bitrex

I'm trying to use the Laplace transform to work out another capacitor problem, the voltage as a function of time on a capacitor that's discharging into another capacitor through a resistor. It's the classic two capacitor problem, but I'd like to actually find an expression for the voltage as a function of time across the capacitor that's discharging and the capacitor that's charging. I've tried setting up a coupled differential equation, like this:

$$\frac{Vc_1 - Vc_2}{R} = C_2\frac{dV_{C2}}{dt}$$
$$\frac{Vc_2 - Vc_1}{R} = C_1\frac{dV_{C1}}{dt}$$

but of course when I take the Laplace transform and try to solve it algebraically I get a system of equations the equivalent of something like A = 5B and B = 4A, which is useless. Any tips on a better way of setting this up would be appreciated.

2. Feb 14, 2010

### rsa58

I don't know about the laplace transform, but the current in the circuit is the same. so the right side of both equations is equal. integrate both sides and plug in any initial conditions? the problem maybe has to do with the ratio of capacitances.

3. Feb 14, 2010

### elibj123

The only way you could get an homogenous system of equations is by letting the initial conditions (voltage of the capacitors) be zero, then it's not a surprise you will get a trivial solutions.

Use

$$L(\frac{d}{dt}y)(s)=sY(s)-y(0)$$

4. Feb 14, 2010

### bitrex

Yes, that's what went wrong. I forgot to put in the initial conditions properly! Thank you.