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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

Start with the homogeneous equation:

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

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

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

rc_circuit_1.txt · Last modified: 2015/05/07 19:54 by 104.228.198.109