Chapter 10, Problem 3P

A constant voltage $v_s\left(t\right)=10\text{V}$ is applied to the RC circuit shown in Fig. P10.3. Assume that the switch has been in position 1 for a long time. At t = 0, the switch is moved instantaneously to position 2. For $t\ge 0$, the voltage $v\left(t\right)$ across the capacitor satisfies the following differential equation and initial condition:

$RC\frac{dv\left(t\right)}{dt}+v\left(t\right)=0$ (a) Find the transient solution, $v_{tran}\left(t\right)$. What is the time constant of the response?

(b) Find the steady-state solution $v_{ss}\left(t\right)$.

(c) Determine the total solution $v\left(t\right)$.

(d) Sketch the total solution $v\left(t\right)$. How long does it take for the response to reach 99% of its steady-state value?

(e) Mark each of the following statements as true (T) or false (F):

___________Increasing the value of resistance R will increase the time the voltage $v\left(t\right)$ reaches 99% of its steady-state value.

___________Increasing the value of capacitance C will decrease the time the voltage $v\left(t\right)$ reaches 99% of its steady-state value.

___________Doubling the value of resistance R will double the lime constant of the response.

___________Doubling the value of capacitance C will double the time constant of the response.