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rc_circuit_1 [2015/05/07 19:54] (current)
104.228.198.109 created
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 +====== 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
 +
 +\[v_{OUT}=v_{IN} \left( 1- e^{\frac{-t}{RC}} \right) \]
  
rc_circuit_1.txt ยท Last modified: 2015/05/07 19:54 by 104.228.198.109