Problem 22-2    Source:   Problem 10.11 - page 98 -    MARKUS, Otávio - Book: Circuitos Elétricos - CC - CA - Ed. Érica - 9ª edição - 2004.

In the circuit shown in the Figure 22-02.1, V= 12.0 volts , R₁ = R₂ = 10 kΩ , R₃ = 20 kΩ , C₁ = 10 µF , C₂ = C₃ = 20 µF and all capacitors are initially discharged. When the S key is closed calculate:

a) the time constant of the circuit;

b) the minimum time required for the voltage between points a-b to be equal to power supply voltage;

c) the voltage between points a-b for a time equal to half time found in the previous item;

d) the moment the voltage between the points a-b reaches the half of the voltage of the power supply.

Solution of the Problem 22.2

Item a

As in the circuit we have more than one resistor and more than one capacitor, so to calculate the time constant we have to reduce the circuit to an equivalent resistance and an equivalent capacitor.

To do so, let's calculate the equivalent resistance of the association of three resistors that appear in the circuit. Note that R₁ and R₂ are in series. Adding your values, we find 20 kΩ. We also know that this series is parallel to R₃. Now, calculating the parallel we found:

Let's calculate the equivalent capacitor of the association shown in the circuit. We note that C₂ and C₃ are in series. For two capacitors we can calculate the series by the following equation:

Making substitutions for the values provided by the problem, and doing the calculation we find C_res = 10 µF. And C_res is parallel to C₁. To calculate the parallel of capacitors just add their values. Soon:

See in the Figure 22-2.2 how the circuit was reduced to a resistance equivalent and an equivalent capacitor. Therefore, having the values of C_eq = 20 uF and R_eq = 10 kΩ, we can calculate the time constant.

Item b

We know that after five time constants the circuit goes into steady state, that is, the capacitor gets the voltage from the power supply. Like this:

Item c

Since we know the time constant of the circuit, let's use the eq. 22-03 below to find the mathematical expression that defines the voltage between the terminals of the capacitor, that is:

So, we have:

Efetuando-se o cálculo:

Note that if t = 0, we find V_c = 0 volt (initial condition), and if t → ∞ , we find V_c = 12 volts (final condition).

With this equation, we can now calculate the capacitor voltage for any given time. Just replace the value of t in the equation and do the calculation.

But the problem asks to calculate the tension between the points a-b at the instant equal to half of the time found in the previous item, that is, t = 1/2 = 0.5 segundo.

Item d

In this item it is requested to calculate the time in which V_c = V / 2 = 12/2 = 6 volts. To do this, simply apply the value of V_c = 6 in the equation we found, and isolate the value of t.

Let's apply the ln function to both members. Remembering that t / 0.2 = 5 t and multiplying the two members by - 1, we have:

After these algebraic transformations we were able to isolate the value of t, or: