A single loop contains, in series, an independent DC voltage source v_s=30.0 V…

A single loop contains, in series, an independent DC voltage source \(v_s=30.0\,\mathrm{V}\) (positive terminal at node \(a\)), a resistor \(R=6.00\,\Omega\) from \(a\) to node \(b\), an inductor \(L=3.00\,\mathrm{H}\) from \(b\) to node \(c\), and a capacitor \(C=\frac{1}{12}\,\mathrm{F}\) from \(c\) to the reference node \(n\) (the negative source terminal is at \(n\)). A switch is connected directly across the capacitor, from \(c\) to \(n\). The loop current \(i(t)\) leaves the positive source terminal, passes through \(R\) and \(L\), and, when the switch is open, enters the capacitor terminal marked positive for \(v_C\) (positive of \(v_C\) at \(c\)). For \(t<0\) the switch is closed, short-circuiting the capacitor, and DC steady state exists. At \(t=0\) the switch opens and remains open, so that for \(t\ge 0\) the circuit is the driven series combination of \(v_s\), \(R\), \(L\), and \(C\). Determine \(v_C(0^-)\), \(i(0^-)\), \(v_L(0^-)\), the current through the closed switch at \(t=0^-\) (positive from \(c\) to \(n\)), and the capacitor current \(i_C(0^-)\). Then determine \(v_C(0^+)\), \(i(0^+)\), \(i_C(0^+)\), \(\mathrm{d}v_C/\mathrm{d}t\) at \(t=0^+\), \(v_L(0^+)\) from KVL, and \(\mathrm{d}i/\mathrm{d}t\) at \(t=0^+\). State which variables are continuous and which jump. For \(t\ge 0\) determine \(\alpha\), \(\omega_0\), \(\omega_d\), the characteristic roots, and the damping classification, then determine the complete responses \(v_C(t)\) and \(i(t)\).

A closed switch across the capacitor clamps \(v_C=0\) for all \(t<0\), so the DC loop is just \(v_s\) and \(R\).