# OpAmp circuit analysis

1. Nov 28, 2016

### eehelp150

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

Find the differential equation for Vo

2. Relevant equations
KCL

3. The attempt at a solution
At node v1:
$\frac{V_1-V_{in}}{R_1}+\frac{V_1-V_p}{R_2}+C_2(\dot{V_1}-\dot{V_2})=0$
At node vp:
$C_1\dot{V_P}+\frac{V_P-V_1}{R_2}=0$
At node vn:
$\frac{V_N}{R_3}+\frac{V_N-V_o}{R_4}=0$
At node v2:
$\frac{V_2}{R_6}+\frac{V_2-V_o}{R_5}+C_2(\dot{V_2}-\dot{V_1})=0$
At node Vo:
$\frac{V_o-V_N}{R_4}+\frac{V_o-V_2}{R_5}=0$

This is my attempt at solving for Vo:
Take equation of node VP and solve for V1
$C_1\dot{V_P}+\frac{V_P-V_1}{R_2}=0$
$V_1=R_2C_1\dot{V_P}+V_P$
derivative
$\dot{V_1}=R_2C_1\ddot{V_P}+\dot{V_P}$

Take equation of node Vo and solve for V2
$\frac{V_o-V_N}{R_4}+\frac{V_o-V_2}{R_5}=0$
$V_2 = \frac{R_5}{R_4}(V_o-V_{N})+V_o$
derivative
$\dot{V_2} = \frac{R_5}{R_4}(\dot{V_o}-\dot{V_{N}})+\dot{V_o}$

Plug these four (V1, dV1, V2, dV2) equations into node V1 equation
$\frac{V_1-V_{in}}{R_1}+\frac{V_1-V_p}{R_2}+C_2(\dot{V_1}-\dot{V_2})=0$
$\frac{R_2C_1\dot{V_P}+V_P-V_{in}}{R_1}+\frac{R_2C_1\dot{V_P}+V_P-V_P}{R_2}+C_2(R_2C_1\ddot{V_P}+\dot{V_P}-\frac{R_5}{R_4}(\dot{V_o}-\dot{V_{N}})+\dot{V_o})=0$
Simplify
$\frac{R_2C_1\dot{V_P}+V_P-V_{in}}{R_1}+C_1\dot{V_P}+C_2(R_2C_1\ddot{V_P}+\dot{V_P}-\frac{R_5}{R_4}(\dot{V_o}-\dot{V_{N}})+\dot{V_o}=0$
By property of opamps: VN = VP
$\frac{R_2C_1\dot{V_P}+V_P-V_{in}}{R_1}+C_1\dot{V_P}+C_2(R_2C_1\ddot{V_P}+\dot{V_P}-\frac{R_5}{R_4}(\dot{V_o}-\dot{V_{P}})+\dot{V_o}=0$

This is where I am stuck... Are my original equations correct? Can someone give me a hint as to how to get rid of Vp?

2. Nov 28, 2016

### Staff: Mentor

This "equation" doesn't belong. Omit it, solve the others.

3. Nov 28, 2016

### eehelp150

Why does this equation not belong?

4. Nov 28, 2016

### Staff: Mentor

What rule did you use to form it?

5. Nov 28, 2016

### eehelp150

Nodal analysis
These points all have a voltage of Vo, dont they?

6. Nov 28, 2016

### Staff: Mentor

Show precisly how you applied nodal analysis here.

7. Nov 28, 2016

### eehelp150

(Vo-Vn)/R4 + (Vo-V2)/R5 = 0

8. Nov 28, 2016

### Staff: Mentor

That can't be right. There are 4 paths for current to/from that node, including the unknown current from the op-amp's output.

You already accommodated that node to the extent possible, there's nothing more to be done here.

9. Nov 28, 2016

### eehelp150

So I just ignore it? I'm not really understanding why...

10. Nov 28, 2016

### Staff: Mentor

You are forming an equation based on ∑ currents into a node = 0. That node has 4 currents, yet you are ignoring two of them and writing an "equation" based on just the other two. That's not valid.

Yes, pay no further attention to this node. You already have it covered.

Last edited: Nov 28, 2016
11. Nov 28, 2016

### eehelp150

At node v1:
$\frac{V_1-V_{in}}{R_1}+\frac{V_1-V_p}{R_2}+C_2(\dot{V_1}-\dot{V_2})=0$
At node vp:
$C_1\dot{V_P}+\frac{V_P-V_1}{R_2}=0$
At node vn:
$\frac{V_N}{R_3}+\frac{V_N-V_o}{R_4}=0$
At node v2:
$\frac{V_2}{R_6}+\frac{V_2-V_o}{R_5}+C_2(\dot{V_2}-\dot{V_1})=0$

Any tips for solving these equations?

Last edited: Nov 28, 2016
12. Nov 28, 2016

### Staff: Mentor

4 equations, 5 unknowns, so it's easy to solve to obtain the ratio $\dfrac {V_o}{V_{in}}$.

The instructions say "Find the differential equation for Vo" and all I can think that means is keep the Laplace operator, s, in your final result. Is that your understanding?

Last edited: Nov 28, 2016
13. Nov 28, 2016

### eehelp150

We are not supposed to use Laplace. Can you give any hints as to how I'd start solving it? I can't see any way to get rid of everything but Vo and Vin

14. Nov 28, 2016

### Staff: Mentor

It looks like a second-order system, so the answer will involve first and second derivatives, too.

If you were to use Laplace you would end up with both s and s2 terms. These can be substituted by the first and second derivatives, represented as either $D^2V_o$ or $\ddot V_o$ form, to give the general transfer function involving possibly all of the first- and second-order derivatives of both $V_o$ and $V_{in}$

Do you have a worked example to follow?

15. Nov 28, 2016

### eehelp150

No worked example to follow. We have to solve this using algebra. No laplace allowed.

16. Nov 28, 2016

### The Electrician

Are you allowed to use mathematical software, such as Wolfram Alpha, to help with the work, or must you do it all by hand?

17. Nov 28, 2016

### eehelp150

Everything by hand, sadly.

18. Nov 28, 2016

### The Electrician

Have you studied Gaussian elimination, or any of the related methods of systematized solution of simultaneous equations?

19. Nov 28, 2016

### eehelp150

Yes, but I'm a bit rusty. How would guassian work with differentials?

20. Nov 28, 2016

### The Electrician

You would probably have to use the differential operator D technique; look up "Operational Calculus". You don't have to use Gaussian elimination; doing so just helps avoid errors.

You can just do what you were doing in post #1. You will have to find the right substitutions to get rid of Vp in your last equation.

I will tell you that your equations are correct, and solving using the Laplace variable S gives the correct answer. Even though you're not supposed to do that, you don''t have to tell your instructor that you verified your answer that way.

I'm afraid you will just have to slog it out for what you hand in.