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Homework Help: Laplace transform IVP

  1. Feb 14, 2010 #1
    1. The problem statement, all variables and given/known data
    I'm trying to use the Laplace transform to find the charging equation of a capacitor with an initial voltage already on the capacitor.


    2. Relevant equations

    [tex]V_i = RC\frac{dVc}{dt} + V_c[/tex]

    The initial condition of Vc will be [tex]\lambda[/tex].

    3. The attempt at a solution

    [tex]V_i(s) = sRC*V_c(s) + V_c(s) - \lambda[/tex]

    Vi will be a step function of magnitude V, so

    [tex]\frac{V_i}{s} + \lambda = (sRC + 1)V_c(s)[/tex]

    [tex]V_c(s) = \frac{V}{s(sRC+1)} + \frac{\lambda}{sRC+1} [/tex]

    So using a table of Laplace transforms I get:

    [tex]V_c(t) = V_i(1-e^{\frac{-t}{RC}}) + \frac{\lambda}{RC}(e^{\frac{-t}{RC}})[/tex]

    The problem is that this doesn't seem to agree with the formula I've seen, it's close but lambda shouldn't be divided by RC. I'm wondering how I went wrong? Thanks for any advice.
     
  2. jcsd
  3. Feb 14, 2010 #2

    vela

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    Your mistake when you took the Laplace transform of dVc/dt:

    [tex]L[RC V']=RC L[V']=RC(sV(s)-\lambda)[/tex]
     
  4. Feb 14, 2010 #3
    Whoops! Thanks for the catch...:redface:
     
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