Circuit analysis problem - Laplace

Discussion in 'Homework Help' started by ineedmunchies, Apr 3, 2008.

  1. ineedmunchies

    Thread Starter New Member

    Apr 3, 2008
    The question is as shown in the first picture. Question1.jpg

    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:
    \frac{5}{s} - \frac{2}{s}=\frac{V_{1}}{1}+\frac{V_{1}-V_{2}}{2s}

    Node 2:

    These can then be rearraged to give




    Which I then put into a matrix and solved for V_{1} and V_{2}

    V_{1} = \frac{3}{s} - \frac{2}{1+\frac{1}{2s}}
    which can be simplified to
    V_{1} = \frac{3}{s} - \frac{4}{s+2}

    V_{2} = (\frac{\frac{-3}{2}}{1+\frac{1}{2s}})+\frac{4}{s}

    (*Note the 4/s is not part of the denominator, I couldn't get the brackets to work properly.)

    Which can be simplified to
    V_{2} = \frac{4}{s}-\frac{3}{s+2}

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

    and V_{2}(t) = 4-3e^{-2t}

    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.
  2. Mark44

    Well-Known Member

    Nov 26, 2007
    I think your algebra is off in your solutions for V1 and V2. Starting with your original equations for nodes 1 and 2, I get:
    V1 = (3s + 7/2)/(s(s + 1))
    V2 = (4s + 7/2)/(s(s + 1))

    I've checked, and these values satisfy your original equations, so I'm pretty confident about them.
    To do the inverse Laplace transforms, you'll need to break each of these apart into a sum of two rational expressions. The denominators will be s and s + 1, not s and s + 2 that you show.
  3. Mark44

    Well-Known Member

    Nov 26, 2007
    In case the bit about breaking up the expressions for V1 and V2 is not clear enough, you'll need to do a partial fraction decomposition for each of them.

    For V1, you need to solve for A and B in this equation:
    (3s + 7/2)/(s(s+1)) = A/s + B/(s + 1)

    When you do the inverse Laplace transform, you'll end up with v1(t) = A + B*e^(-t)

    Do the same thing to find v2(t).