# Determining energy stored in capacitor and inductor in RLC circuit

I've been stuck on the following homework problem for a few hours now with little progress.  As you can see, it's a relatively simple RLC circuit with a couple independent sources and a voltage-controlled voltage source. Despite this, I have been unable to solve for mesh currents and nodal voltages despite repeated attempts at tackling the problem.

From what I understand, the energy stored by a capacitor over $[t_0,t]$ follows from $i=C\frac{dV}{dt}$ since $W=\int^t_{t_0} Pdt=\int iVdt = \int_{t_0}^t CV\frac{dV}{dt}dt=C\int_{V_0}^V V\ dV=\frac12C(V^2-V_0^2)$.

Similarly, for an inductor, $V=L\frac{di}{dt}$ so $W=\int_{t_0}^tLi\frac{di}{dt}dt=L\int_{i_0}^i i\ di=\frac12L(i^2-i_0^2)$.

Unfortunately, I haven't been able to find the above $V(t),i(t)$. $V_0,i_0$ were however trivial to solve for using a DC-equivalent circuit.

How does one go about solving for these functions?

A couple of suggestions:

(1) the EE stackexchange site a better home for this question

(2) simply solve for the voltage across the capacitor and the current through the inductor. Once you have those, the energies stored, as a function of time are just

$$W_L(t) = \frac{L}{2}i^2_L$$

and

$$W_C(t) = \frac{C}{2}v^2_C$$

Since this is evidently a DC circuit in steady state (big hint here), the voltages and currents are constant so the stored energies are constant.

• In steady state it is indeed a DC circuit which is how I determined $V_0,i_0$ -- treating the capacitor as an open circuit and the inductor as a closed one. Are you suggesting the voltages and currents are constant with respect to time? Is this problem not a transient one? Have I been extremely overthinking this problem? :-p – oldrinb Oct 22 '14 at 17:58
• There are no switched elements and the sources are DC sources so, as far as I can tell, this is a DC steady state problem. That's certainly how I would approach it. You're correct, assume $v_L = 0, i_C = 0$ and solve for the inductor current and capacitor voltage. If you have that, use the equations I give above to find the stored (magnetic and electric) energies. – Alfred Centauri Oct 22 '14 at 21:25
• I am such a fool! I thought that was too easy to be true and convinced myself $V_L,i_c$ varied. Thank you. – oldrinb Oct 23 '14 at 0:58