rc_circuit_1

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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