I've spent sometime learning about step responses for series/parallel RLC circuits, but this problem takes things a step further, since C seems to be in parallel with a series combination of RL and a voltage source. I'm trying to find out what i(t) is for t≥0, since the switch has been closed for a long time before opening at t =0. My understanding is that for the step response, the solution calls for a second-order differential equation that takes into account both the forced response and the function for natural response. So when I focused on the t=0- condition, I set the inductor as a short circuit and the capacitor as a open circuit. From there, I also set i(t=0-) = i_L(t=0-). I used KVL to solve for i_L(t=0-):
i_L(t=0-) = (20V + 10V)/(5Ω + 5Ω) = 3 A = i_L(t=0+) <-- (I believe this is the initial current across inductor and R2?)
As for finding v_c(0-), I did a voltage divider to end up with: v_c(0-) = (5Ω)/(5Ω + 5Ω) = 5V = v_c(0+) <-- (I'm assuming this is the initial voltage of capacitor?) Now what I don't completely understand is the t≥0 condition. With the switch open, the left side of the circuit is ignored, leaving only the right side of the RLC circuit as the circuit reaches steady state. The problem is:
From the way I modified the circuit, it seems to represent a series RLC circuit. Which would lead to the neper frequency (α = (R/2L) = 5 and resonant radian frequency, ω = (1/sqrt(LC)) = 5, leading to a critically damped response. From here, the general solution of the step response for series RLC circuit is: x(t) = Xf + D'1te^-at + D'2e^-at, with x either being v or i, and "Xf" being either the final voltage or final current
At t=∞, it's readily known that capacitor acts as a open circuit again, and the inductor as a short circuit, and because V2 is included in the reduced circuit, my understanding is that the final voltage, or v_c(t=∞), is 10V, and that the final current across R2 is ~ 0 A at the steady state. I even verified this with the use of the Falstad online circuit simulator shown below:
I am quite lost on this issue on finding i(t). What I do know is that I am dealing with a series RLC circuit for t≥0, and that the general solution for i(t) takes the form of:
iL(t) = I_f + D'1*t*e^(-at) + D'2*e^(-at).
Does anyone have any ideas as to how to solve for i(t) for t≥0?