Laplace transforms: R series with RC parallel circuit: Difference between revisions
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Find the Voltage across the capacitor for t>=0: |
Find the Voltage across the capacitor for t>=0: |
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Voltage across capacitor at t(0-)=0 |
Voltage across capacitor at t({0-})=0 |
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: <math>C=.1 Farad</math> |
: <math>C=.1 Farad</math> |
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Solve equations (1) and (2) simultaneously |
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Laplace Transform S-domain |
Take the Laplace Transform to move to the S-domain |
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:<math>\mathcal{L}\left\{\int_{0-}^\infty{(i_2) dt}\right\}=I_2/s</math> |
:<math>\mathcal{L}\left\{\int_{0-}^\infty{(i_2) dt}\right\}=I_2/s</math> |
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Take the inverse Laplace transform to move back into the t-domain |
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:<math>i_2=(1/2)(e^{-(5/6)t})</math> |
:<math>i_2=(1/2)(e^{-(5/6)t})</math> |
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:substitute this equation back into equation (1) |
:substitute this equation back into equation (1) |
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:<math>i_1=(10-20(.5e^{-( |
:<math>i_1=(10-20(.5e^{-(5t/6)}))/50</math> |
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:<math>i_1=(1/5)(1-e^{-( |
:<math>i_1=(1/5)(1-e^{-(5t/6)})</math> |
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:<math>v_{capacitor}=10/20(i_1+i_2)</math> |
:<math>v_{capacitor}=10/20(i_1+i_2)</math> |
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:<math>v_{capacitor}=10-20((1/5)(1-e^{-( |
:<math>v_{capacitor}=10-20((1/5)(1-e^{-(5t/6)})+(1/2)(e^{-(5t/6)}))</math> |
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Answer: |
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Written by: Andrew Hellie |
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Checked by: |
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Revision as of 00:41, 22 October 2009
Problem Statement
Find the Voltage across the capacitor for t>=0:
Voltage across capacitor at t({0-})=0
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
Written by: Andrew Hellie
Checked by: