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Simple Op-amp/Laplace question

  1. May 6, 2007 #1
    1. The problem statement, all variables and given/known data

    [​IMG]
    Find H(s) = V2(S) / V1(s)


    2. Relevant equations

    Laplace transformed ckt is topologically equivalent to the standard circuit - except all dynamic elements are replaced by either s (L) or 1/s (C) and then treated as resistances. inverse laplce of the solution will give the solution

    3. The attempt at a solution
    Code (Text):

    Laplace transform of the circuit shown above
    find H(s) = V2(S) / V1(s)

    [B][U]i) KCL @ Va[/U][/B]

    (Va - V1 )/ R + VaSC1 + (Va - Vb)/R = 0

    or  Va-V1 + RVaSC1 + Va-Vb = 0
    or  2Va-V1-Vb +RSC1Va = 0
    or  (2+RSC1)Va - Vb = V1 -------------------------(1)

    [B][U]ii)KCL @ Vb[/U][/B]

    (Vb-Va)/R + (Vb-V2)SC3 + (Vb-V2)SC2 = 0
    or   Vb - Va + RSC3(Vb-V2) + R(Vb-V2)SC2 = 0
    or   (1 + RSC3 + RSC2)Vb - Va - (RSC3+RSC2)V2 = 0
    or   [(1 + RSC3 + RSC2)Vb - Va] / (RSC3+RSC2) = V2 -------------------(2)


    divide V2/v1 = H(S)

    H(S) = V2(s)/V1(s)
    =       [(1 + RSC3 + RSC2)Vb - Va]
            ----------------------------
                       (RSC3+RSC2)
    ---------------------------------------
                    [(2+RSC1)Va - Vb]


    =         [(1 + RSC3 + RSC2)Vb - Va]
             ----------------------------
           [(RSC3+RSC2)[(2+RSC1)Va - Vb]
     
    Problem is that I need Va and Vb to cancel for the next three parts of the question where we determine plots etc.

    Any help? Should i have substituted Va and Vb with each other and get them in terms of v2/v1 then?
     
  2. jcsd
  3. May 6, 2007 #2
    i found my mistake in the second KCL equation :

    it should be Vb - V2/R
    and i'll get another KCL equation at the third node v2 directly above the C2 capacitor.
     
    Last edited: May 7, 2007
  4. May 6, 2007 #3
    Code (Text):

    I think i messed up again - when trying to cancel Va and Vb
    I end up getting a cubic in the denominator and that can't
    be simplified that easily with inv laplace transforms.

    So I got
    KCL @ Va

        (2 + RSC1)Va - Vb = V1 -----------------------------(1)

    KCL @ Vb

        (Vb-Va)/R  +  (Vb-V2)SC3 + (Vb-V2)/R = 0
        or ....
        (2 + RSC3)Vb - Va - (RSC3+1)V2 = 0 ---------------(2)

    KCL @ V+

        (V2-Vb ) / R = V2SC2
        or ..
        (1-RSC2)V2 = Vb --------------------------------------(3)

    Sub (3) into (2)

        (2 + RSC3)(1-RSC2)V2 - Va - (RSC3 + 1)V2 = 0
        or ..
        [-(RS)^2 . C3C2 - 2(RS)C2 + 1 ] V2 = Va ------------------------(4)

    Sub (4),(3) into (1)

        [(2 + RSC1)(-(RS)^2 . C3C2   - 2RSC2 + 1)  - (1-RSC2)]V2 = V1
        V2/V1 = 1 / [(2 + RSC1)(-(RS)^2 . C3C2   - 2RSC2 + 1)  - (1-RSC2)]


        or better visually

        V2                              1
        ----   = --------------------------------------------------------------
        V1           (2 + RSC1)(-(RS)^2 . C3C2   - 2RSC2 + 1)  - (1-RSC2)

         
        If i expand the last term i get a cubic. Help :'(
     
     
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