rc_circuit_1

# RC Circuit Time Domain Analysis

## Nodal Equation

Writing the nodal equation at $v_{out}$ yields: $C\frac{dv_{OUT}}{dt}+\frac{v_{OUT}}{R}-\frac{v_{IN}}{R}=0$ Simplify: $RC\frac{dv_{OUT}}{dt}+v_{OUT}=v_{IN}$

## Solution using method of homogeneous and particular solutions

$RC\frac{dv_{OUT}}{dt}+v_{OUT}=0$

Guess that the solution is of the form:

$v_{OUT}=e^{st}$

Plugging the guess into the homogeneous equation yields:

$RCse^{st} + e^{st}=0$

$e^{st} \left( RCs + 1 \right) =0$

$s=\frac{-1}{RC}$

Noting that $Ae^{st}$ is also a solution we get

$v_{OUT}=Ae^{\frac{-t}{RC}}$

Find a particular solution:

$RC\frac{dv_{OUT}}{dt}+v_{OUT}=v_{IN}$

Guess that is a constant, $v_{OUT}=K$:

$RC\frac{dK}{dt}+K=v_{IN}$

$K=v_{IN}$

Sum the homogeneous and particular solutions:

$v_{OUT}=Ae^{\frac{-t}{RC}}+v_{IN}$

Use the initial condition to find $A$. For example if $v_{OUT}=0|_{t=0}$ then

$A + v_{IN}=0$

$A=-v_{IN}$

The final solution is: $v_{OUT}=-v_{IN}e^{\frac{-t}{RC}}+v_{IN}$

or

$v_{OUT}=v_{IN} \left( 1- e^{\frac{-t}{RC}} \right)$ 