Independent and dependent initial conditions

Question

Consider the RLC circuit below.

In fact, it is a normalized filter with Butterworth approximation of 3rd order.

Now suppose that for some reason I am interested in the time domain formulation. Using KCL and KVL I ended up with these state equations:

\begin{gather} \frac{\mathrm{d}i_L}{\mathrm{d}t} = \frac{1}{L}(v_{C1} - v_{C2})\\ \frac{\mathrm{d}v_{C1}}{\mathrm{d}t} = \frac{1}{C_1}\left(\frac{v_{C1}}{R1} + i_L\right)\\ \frac{\mathrm{d}v_{C2}}{\mathrm{d}t} = \frac{1}{C_2}\left(\frac{v_{C2}}{R2} - i_L\right) \end{gather}

But when I am trying to get a transient response of these equations using some numerical integration scheme like Runge-Kutta or implicit backward Euler, the solution blows up.

Only for an initial condition \$(0,0,0)\$ it is zero everywhere. Whenever I try to put the different initial conditions, let's say \$(0.5,1,1)\$ it blows up.

Am I using initial conditions the wrong way? How to choose them correctly? I understand that by saying that for example \$v_{C1} = 10\$, it means that \$i_L = 10/(1||1)\$. Or is it even true? What is then \$v_{C2}\$?

Answer 1

I think the problem might be that your cap currents don't follow your +/- convention.

Here I've drawn all of the +/- indicators and current arrows:

^{simulate this circuit – Schematic created using CircuitLab}

The resulting KCL and KVL equations are:

$$ \begin{align} C_1\frac{dv_{C_1}}{dt} + \frac{v_{C_1}}{R_1} + i_L &= 0 \\ C_2\frac{dv_{C_2}}{dt} + \frac{v_{C_2}}{R_2} &= i_L \\ L\frac{di_L}{dt} &= v_{C_1} - v_{C_2} \\ \end{align} $$

Translating your equations back to KCL current equations I get: $$ \begin{align} i_{C_1} &= i_{R_1} + i_L \\ i_{C_2} &= i_{R_2} - i_L \\ \end{align} $$ which means the current arrows look like:

^{simulate this circuit}