Problem + Hard 22-1 Source:
Exercise 7.8 - page 266 Book: Análise de Circuitos em Engenharia - J. David Irwin - 4ª edição - Ed. Pearson.
In the circuit shown in the figure below, the switch S has been open for a long time.
At t = 0 the switch is closed. Determine the answer of Vo (t).
Solution of the Problem + Hard 22-1
With the S switch open for a long time we know that the capacitor acts as an open circuit.
Therefore, there will be no current flow through it and the two voltage sources. So the whole current of
3 A source will circulate through the two 4 ohm resistors. Calculate the value of
Vx it's very easy, just apply the Ohm's Law, or:
Vx = 4 x 3 = 12 volts
To determine the value of Vo (0-),just make the composite mesh
by the two voltage sources and the 4 ohm resistor. Soon:
Vo(0-) = Vx + 24 + 2 Vx = 60 volts
From this moment on we will consider the switch S closed and therefore the current source of
3 A is out of circuit. As the capacitor cannot sharply vary its voltage, we conclude that:
Vo(0+) = Vo(0-) = 60 volts
On the other hand, when t → ∞ the capacitor is an open circuit
and there will be no current flow through the circuit. Soon, Vx = 0. Stay clear
that the source voltage of 24 volts will be parallel to the capacitor. So:
Vo(∞) = 24 volts
All that remains to be calculated is the time constant of the circuit. To do so, let's remove the capacitor of the circuit and determine the equivalent Thévenin of the remainder of the circuit. Let's short circuit the independent source of 24 volts and we should note that with closing the
S key the two resistors of 4 ohms were in parallel. So let's replace them with one
single resistor of 2 ohms. So we get the circuit shown in the picture below, where
We introduce a I current source in place of the capacitor. To find Rth
we must calculate the ratio V / I.
Note that the current flowing through the circuit is I. Then Vx = 2 I. Soon,
making the mesh we found:
V = 3 Vx = 6 I
So, we find Rth, or:
Rth = V / I = 6 Ω
With the value of Rth we can calculate the value of the time constant of the circuit. Like this:
τ = Rth C = 12 s
Using the eq. 22-03 , we can write the solution equation for the problem circuit.
