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Foorier transformation question

  1. Jul 13, 2009 #1
    this is the last part of a bigger question
    if you i forgot to mention some data please tell

    in the last part i got
    [itex]\ddot{v_c}+2\dot{v_c}+2v_c=v_s(t)[/itex]
    [itex]\dot{v_c(0)}=2[/itex]
    [itex]v_c(0)=0[/itex]

    i need to find out if the response for [itex]v_s(t)=u(t)+cos(t)[/itex]
    equals the sum of [itex]v_c^u[/itex](the response for spet function) and [itex]v_c^cos[/itex] the response for cosine
    i start by finding the response for cos(t)
    [itex]v_c(t)->V_c(s)[/itex]
    [itex]\dot{v_c}->sV_c(s)-v_c(0)[/itex]
    [itex]\ddot{v_c}->s(sV_c(s)-v_c(0))-\dot{v_c(0)}[/itex]
    [itex]s(sV_c(s)-v_c(0))-\dot{v_c(0)}+2sV_c(s)-v_c(0)+2V_c(s)=\frac{s}{s^2+1}[/itex]
    [itex]
    V_c(s)[s^2+2s+2]+2=\frac{s}{s^2+1}
    [/itex]
    so i get
    [itex]
    V_c(s)=\frac{s}{(s^2+2s+2)(s^2+1)}-\frac{2}{s^2+2s+2}
    [/itex]
    where [itex]\dot{v_c}=j\omega v_c[/itex]

    so now i need to break the fractures into a simpler ones
    but here its all complex and i dont know how to get
    a simpler fractures and there foorier transformation
    ??

    so this is where i got stuck
    and i even didnt get closer to the main solution of the problem

    ??
     
    Last edited: Jul 13, 2009
  2. jcsd
  3. Jul 13, 2009 #2
    First of all, unless you put in [itex]s = j\omega[/itex], the s-domain transformation is known as a Laplace Transform. The Fourier Transform is a special case of the Laplace Transform with [itex]s = j\omega[/itex].

    As for your query, do you know how to express the right hand side in terms of a partial fraction expansion, in terms of functions of s whose inverse Laplace transform (time domain function) is known to you?
     
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