rc_circuit_1

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 — rc_circuit_1 [2015/05/07 19:54] (current)104.228.198.109 created 2015/05/07 19:54 created 2015/05/07 19:54 created Line 1: Line 1: + ====== RC Circuit Time Domain Analysis ====== + {{:​rc_circuit.png?​200|}} + ---- + ===== 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 + + $v_{OUT}=v_{IN} \left( 1- e^{\frac{-t}{RC}} \right)$