The circuit has a reference node 0 and node a

The circuit has a reference node 0 and node \(a\). An independent DC voltage source of \(10\,\mathrm{V}\) is connected from 0 to node \(s\) with its positive terminal at \(s\). Resistor \(R=2.0\,\mathrm{k}\Omega\) joins \(s\) to \(a\). A capacitor \(C=0.50\,\mu\mathrm{F}\) is connected from \(a\) to 0, with \(v_C=v_{a0}\) polarised positive at \(a\). An ideal voltage-controlled current source of value \(g v_x\) with \(g=0.50\,\mathrm{mS}\) is connected from \(a\) to 0, with the source arrow pointing towards 0, so that the dependent current \(g v_x\) leaves node \(a\). The controlling voltage is \(v_x=v_C\). For \(t<0\) a switch in series with the independent source has been open, and \(v_C(0^-)=4.0\,\mathrm{V}\). At \(t=0\) the switch closes and remains closed. (a) For \(t>0\), replace the capacitor by an open circuit and determine \(v_C(\infty)\). Deactivate the independent source (replace it by a short circuit) and determine the equivalent resistance seen by the capacitor, treating the dependent source as a conductance \(g\). Determine the time constant. (b) Determine \(v_C(0^+)\) by continuity and write \(v_C(t)\) for \(t>0\). (c) Write the current in \(R\) directed from \(s\) to \(a\) and the dependent-source current in its assigned direction, both for \(t>0\). Write the capacitor current under the passive sign convention and confirm KCL at node \(a\) for \(t>0\).

The VCCS is linearly controlled by \(v_C\) itself, so it acts as a conductance \(g\) in parallel with \(C\).