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Solve OpAmp Circuit Using Differential Equations

  1. Nov 17, 2016 #1
    1. The problem statement, all variables and given/known data
    upload_2016-11-17_12-22-21.png

    2. Relevant equations


    3. The attempt at a solution
    Nodal Equations
    By property of OpAmp, V2=Vo

    eq1:[tex]\frac{V_{1}-V_{in}}{R_1}+\frac{V_{1}-V_{o}}{R_2}+C_2*(\dot{V_1}-\dot{Vo})[/tex]
    eq2: [tex]V_1=C_1R_2\dot{V_o}+V_o[/tex]

    eq3:[tex] \dot{V_1}=C_1R_2\ddot{V_o}+\dot{V_o}[/tex]

    Sub 2 & 3 into 1
    [tex]\frac{C_1R_2\dot{V_o}+V_o-V_{in}}{R_1}+\frac{C_1R_2\dot{V_o}+V_o-V_o}{R_2}+C_2(C_1R_2\ddot{V_o}+\dot{V_o}-\dot{V_o})[/tex]

    Simplify
    [tex]\frac{C_1R_2\dot{V_o}+V_o-V_{in}}{R_1}+ C_1\dot{V_o}+C_2(C_1R_2\ddot{V_o})[/tex]

    [tex]\frac{C_1R_2\dot{V_o}}{R_1}+\frac{V_o}{R_1}-\frac{V_{in}}{R_1}+C_1\dot{V_o}+C_1C_2R_2\ddot{V_o}[/tex]

    [tex]\frac{C_1R_2\dot{V_o}}{R_1}+\frac{V_o}{R_1}+C_1\dot{V_o}+C_1C_2R_2\ddot{V_o}=\frac{V_{in}}{R_1}[/tex]

    Divide everything by C1C2R2 to single out Vo''
    [tex]\frac{C_1R_2\dot{V_o}}{C_1C_2R_1R_2}+\frac{V_o}{C_1C_2R_1R_2}+\frac{C_1\dot{V_o}}{C_1C_2R_2}+\frac{C_1C_2R_2\ddot{V_o}}{C_1C_2R_2}=\frac{V_{in}}{R_1C_1C_2R_2}[/tex]

    Simplify
    [tex]\frac{\dot{V_o}}{R_1C_2}+\frac{V_o}{C_1C_2R_1R_2}+\frac{\dot{V_o}}{C_2R_2}+\ddot{V_o}=\frac{V_{in}}{R_1C_1C_2R_2}[/tex]
    Rearrange
    [tex]\ddot{V_o}+\frac{\dot{V_o}}{R_1C_2}+\frac{\dot{V_o}}{C_2R_2}+\frac{V_o}{C_1C_2R_1R_2}=\frac{V_{in}}{R_1C_1C_2R_2}
    [/tex]

    This is the correct solution:
    [tex]\ddot{V_o}+\frac{\dot{V_o}}{R_1R_2}+\frac{V_o}{R_1R_2C_1C_2}=\frac{V_{in}}{R_1}[/tex]

    What am I doing wrong?
     
  2. jcsd
  3. Nov 17, 2016 #2

    gneill

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    Staff: Mentor

    I did a quick check and I have to say I'm liking your solution better than the "correct" solution. The units don't look right in their solution:

    $$\frac{[V]}{[ s ]^2} + \frac{[V]}{[Ω]^2[ s ]} + \frac{[V]}{[ s ]^2} = \frac{[V]}{[Ω]}$$
     
  4. Nov 17, 2016 #3
    I typed it wrong, correct (given) solution should be this:
    [tex]
    \ddot{V_o}+\frac{\dot{V_o}}{R_1C_2}+\frac{V_o}{R_1R_2C_1C_2}=\frac{V_{in}}{R_1}[/tex]
     
  5. Nov 17, 2016 #4

    gneill

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    Staff: Mentor

    Still don't like their solution. The RHS has V/Ω for units (so a current). The LHS is all V/s2.
     
  6. Nov 17, 2016 #5
    So mine looks right?
     
  7. Nov 17, 2016 #6

    gneill

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    Staff: Mentor

    I believe so, yes.
     
  8. Nov 17, 2016 #7
    If I want to find characteristic roots, I set LHS = 0 and solve, correct?
    So it'd look something like:
    d^2 + 2/(R1C2)*d+1/(R1R2C1C2)=0
    D1=...., D2=....
     
  9. Nov 17, 2016 #8

    gneill

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    Staff: Mentor

    Something like that, yes. You probably want to collect your to Vo' terms into a single term. Unless R1 = R2 they have different denominators.
     
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