How to Simplify Laplace Transformed Op-Amp Circuit Equations?

[trickae]

Homework Statement

http://img504.imageshack.us/img504/7714/assn2dn1.jpg
Find H(s) = V2(S) / V1(s)

Homework 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

The Attempt at a Solution

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?

 

[trickae]

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.

 

[trickae]

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 :'(