7.3 Charging capacitor

We now examine the case of a capacitor being charged by a battery in the circuit shown.

The switch is open for $t<0$, and we therefore assume the capacitor is initially un-charged, meaning  v_c(t)=0 for t<0. The switch is closed at t=0 and remains closed.  It therefore behaves as a short circuit for $t\ge0$, connecting the battery in series with a resistor and a capacitor.  As we will see, current begins to flow in the circuit for $t\ge0$ and the voltage across the capacitor increases as the current flows, then asymptotically approaches the battery voltage $V_B$. This is called bounded exponential growth.

Analysis begins by applying KCL to the node shown, and proceeding as in the previous chapter to develop the first order linear differential equation.

This differential equation is similar to the one developed in the previous chapter with the addition of the forcing function, V_B.  An educated guess at the form of solution with three constants to be determined, follows:

The solution at this point is nearly complete, but the remaining constant K2 needs to be determined by applying the boundary condition:

We finally arrive at the solution for v_c(t). The capacitor voltage increases from an initial value of 0, exponentially increasing and asymptotically approaching  VB. An approximate value for the duration of the transient is T=RC seconds.