Solve OpAmp Circuit Using Differential Equations

1. Nov 17, 2016

eehelp150

1. The problem statement, all variables and given/known data

2. Relevant equations

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

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

eq3:$$\dot{V_1}=C_1R_2\ddot{V_o}+\dot{V_o}$$

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

Simplify
$$\frac{C_1R_2\dot{V_o}+V_o-V_{in}}{R_1}+ C_1\dot{V_o}+C_2(C_1R_2\ddot{V_o})$$

$$\frac{C_1R_2\dot{V_o}}{R_1}+\frac{V_o}{R_1}-\frac{V_{in}}{R_1}+C_1\dot{V_o}+C_1C_2R_2\ddot{V_o}$$

$$\frac{C_1R_2\dot{V_o}}{R_1}+\frac{V_o}{R_1}+C_1\dot{V_o}+C_1C_2R_2\ddot{V_o}=\frac{V_{in}}{R_1}$$

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

Simplify
$$\frac{\dot{V_o}}{R_1C_2}+\frac{V_o}{C_1C_2R_1R_2}+\frac{\dot{V_o}}{C_2R_2}+\ddot{V_o}=\frac{V_{in}}{R_1C_1C_2R_2}$$
Rearrange
$$\ddot{V_o}+\frac{\dot{V_o}}{R_1C_2}+\frac{\dot{V_o}}{C_2R_2}+\frac{V_o}{C_1C_2R_1R_2}=\frac{V_{in}}{R_1C_1C_2R_2}$$

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

What am I doing wrong?

2. Nov 17, 2016

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:

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

3. Nov 17, 2016

eehelp150

I typed it wrong, correct (given) solution should be this:
$$\ddot{V_o}+\frac{\dot{V_o}}{R_1C_2}+\frac{V_o}{R_1R_2C_1C_2}=\frac{V_{in}}{R_1}$$

4. Nov 17, 2016

Staff: Mentor

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

5. Nov 17, 2016

eehelp150

So mine looks right?

6. Nov 17, 2016

Staff: Mentor

I believe so, yes.

7. Nov 17, 2016

eehelp150

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=....

8. Nov 17, 2016

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.