Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Circuit analysis - Laplace

  1. Apr 3, 2008 #1
    1. The problem statement, all variables and given/known data

    The question is as shown in the first picture. Question1.jpg

    2. Relevant equations
    3. The attempt at a solution
    It asks that initial condition current generators are used, what I believe models the circuit for t>0 is shown in the second picture with the switch open, and a current generator added. (I am not sure if this is correct.)

    I then used KCL to write equations at node 1 and node 2.

    Node 1:
    [tex]\frac{5}{s}[/tex] - [tex]\frac{2}{s}[/tex]=[tex]\frac{V_{1}}{1}[/tex]+[tex]\frac{V_{1}-V_{2}}{2s}[/tex]


    Node 2:
    [tex]\frac{2}{s}[/tex]+[tex]\frac{2}{s}[/tex]=[tex]\frac{V_{2}}{1}[/tex]-([tex]\frac{V_{1}-V_{2}}{2s}[/tex])


    These can then be rearraged to give

    [tex]\frac{3}{s}[/tex]=[tex]V_{1}[/tex](1+[tex]\frac{1}{2s}[/tex])-[tex]V_{2}[/tex]([tex]\frac{1}{2s}[/tex])

    and

    [tex]\frac{4}{s}[/tex]=[tex]V_{1}[/tex]([tex]\frac{-1}{2s}[/tex])+[tex]V_{2}[/tex](1+[tex]\frac{1}{2s}[/tex])

    Which I then put into a matrix and solved for [tex]V_{1}[/tex] and [tex]V_{2}[/tex]

    Giving
    [tex]V_{1}[/tex] = [tex]\frac{3}{s}[/tex]-[tex]\frac{2}{1+\frac{1}{2s}}[/tex]
    which can be simplified to
    [tex]V_{1}[/tex] = [tex]\frac{3}{s}[/tex]-[tex]\frac{4}{s+2}[/tex]

    and
    [tex]V_{2}[/tex] = [tex]\frac{\frac{-3}{2}}{1+\frac{1}{2s}}[/tex]+[tex]\frac{4}{s}[/tex]
    Which can be simplified to
    [tex]V_{2}[/tex] = [tex]\frac{4}{s}[/tex]-[tex]\frac{3}{s+2}[/tex]

    Then convert these back to the time domain to give:
    [tex]V_{1}[/tex](t)=3-4[tex]e^{-2t}[/tex]
    and [tex]V_{2}[/tex](t)=4-3[tex]e^{-2t}[/tex]

    Can anyone tell if there is a mistake here or not?
    I don't feel confident this is the correct answer. I think the 5 ohm resistor and 5A current should effect the circuit somehow but do not know how to encorporate it into my equations.
     

    Attached Files:

  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted