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 (Resistor Branch)
___________________________________equation (1)


Loop 2 (Capacitor Branch)
_______________________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

T-domain

S-domain


Transfer Function


Bode Plot


Fig (1)





















How to use break points and asymptotes to obtain the magnitude frequency response of the system...

The break points are the values of s in H(s) that make the numerator and or the denominator 0.

The location of the break points determines the magnitude frequency response of the system at that frequency.

Zeros are where the numerator is equal to zero.

Poles are when the denominator is equal to zero.

Use Convolution to find the output of the system


Written by: Andrew Hellie

Checked by: Kendrick Mensink