RLC Circuit (2-11-2020)

Let the currents through the branches be as such:

• Source current: $I_S$
• Through $R_1$ : $I_{R_1}$
• Through $R_2$ : $I_{R_2}$
• Through $C_1$ : $I_{C_1}$
• Through $C_2$ : $I_{C_2}$

• Through $L$ : $I_{L}$

• The charge on $C_1$ : $Q_1$

• The charge on $C_2$ : $Q_2$

Therefore, according to Kirchoff's current law:

$I_S = I_{C_1} +I_{R_1}$ $I_S = I_{L} +I_{R_2}$ $I_L = I_{C_1} + I_{C_2}$

When viewing the circuit as it is, $I_{C_2}$ is assumed to be moving from right to left. From the junction of the resistors to that between $C_1$ and the inductor.

According to Kirchoff's voltage law:

$-V_S + \frac{Q_1}{C_1} + L \dot{I}_L=0$ $I_{R_1}R_1 + \frac{Q_2}{C_2} - \frac{Q_1}{C_1}=0$ $\frac{Q_2}{C_2} + L \dot{I}_L = I_{R_2}R_2$

Initial conditions:

$I_L(0) = Q_1(0) = Q_2(0) = 0$

This system of equations can obviously be re-arranged into a state-space form but I simply numerically integrated them using a script of code. The variation of source current with time is as follows:

The answer is $\boxed{1.3618}$ . The source currents initially and after a long time can be simply verified by inspecting the circuit. The initial source current is $5A$ and the final source current is $3.3333A$ .