Laplace transforms: R series with RC parallel circuit

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Problem Statement

Find the Voltage across the capacitor for t>=0:
Voltage across capacitor at t({0-})=0
Fig (1)














Use Loop Equations to solve for the currents in and


Loop 1
_______________________________________equation (1)


Loop 2
_______________________equation (2)


Solve equations (1) and (2) simultaneously


Substituting equation (1) into equation (2) gives...
simplifies to...


Take the Laplace Transform to move to the S-domain





Take the inverse Laplace transform to move back into the t-domain


substitute this equation back into equation (1)


Voltage on Capacitor

Answer

Volts


Apply the Initial and Final Value Theorems to find the initial and final values

Initial Value Theorem
Final Value Theorem



Initial Value:
Initial Value = 0 Volts


Final Value:


Final Value = 6 Volts


Volts
Volts


Bode Plot

simplified to...



Fig (1)






















Written by: Andrew Hellie

Checked by: