Problem 22-3    Source:   Problem 7.13 - page 367 -    THOMAS, Roland E. & ROSA, Albert J. & TOUSSAINT, Gregory J. - Book: The Analysis & Design of Linear Circuits - Ed. John Wiley & Sons, Inc. - 6ª edição - 2009.

In the circuit shown in the Figure 22-03.1, the key has been in position A for a long time. At t = 0 the key is passed to the position B. Determine v_C (t) to t ≥ 0.

Solution of the Problem 22.3

Since the switch has been in position A for a long time, then the initial capacitor voltage is the voltage on the resistor that is parallel to it. Since the three resistors are in series and have the same values, so the initial capacitor voltage is V_i = 12 / 3 = 4 V. By passing the key to position B , we must calculate the equivalent Thévenin disregarding the capacitor. We easily realize that this equivalent is formed by the parallel of the two resistors, which results in an equivalent resistance of 5 kΩ and a voltage of Thévenin of 6 volts, since both resistors have values equals. Now we can calculate what the time constant of the circuit is. Like this:

Knowing the time constant of the circuit, the start and end value of the capacitor voltage, then let's use eq. 22-03, shown below, to find the mathematical expression that defines the voltage between the terminals of the capacitor, that is:

So, we have:

Performing the calculation, we arrive at:

Note that if we take t = 0, we will find v_c = 4 volts (initial condition), and if t → ∞ , we will find v_c = 6 volts (final condição).

With this equation, we can now calculate the capacitor voltage for any desired time. Just replace the value of t (in ms) in the equation and perform the calculations.