Solve OpAmp Circuit Using Differential Equations

[eehelp150]

Homework Statement

Homework Equations

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?

 

[gneill]

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:

$$\frac{[V]}{[ s ]^2} + \frac{[V]}{[Ω]^2[ s ]} + \frac{[V]}{[ s ]^2} = \frac{[V]}{[Ω]}$$

 

[eehelp150]

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:

$$\frac{[V]}{[ s ]^2} + \frac{[V]}{[Ω]^2[ s ]} + \frac{[V]}{[ s ]^2} = \frac{[V]}{[Ω]}$$

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]

 

[gneill]

Still don't like their solution. The RHS has V/Ω for units (so a current). The LHS is all V/s².

 

[eehelp150]

Still don't like their solution. The RHS has V/Ω for units (so a current). The LHS is all V/s².

So mine looks right?

 

[gneill]

So mine looks right?

I believe so, yes.

 

[eehelp150]

I believe so, yes.

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=...

 

[gneill]

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