# Calculating the transfer function of this op-amp circuit

The first op-amp is a differential, second one is a follower and third one is a second order low pass filter. I've been struggling with the low pass and the final output of the circuit and I am not sure whether my calculations are correct or not, they are still incomplete though(Always had troubles with these kind of circuits). Refer to the input voltage as Vi, first stage output as Vo1, second stage output as Vo2 and final output as Vo.

## My calculations so far:

• Side note: I believe Multisim has a tool where you can calculate the transfer function. But if you have to do this by hand, I would suggest you think of each stage like a block utilizing control theory. What's the math behind three blocks being fed back to the input? – KingDuken Sep 8 '18 at 18:40

Here's how I would start it: -

You can calculate the filter TF seperately but concentrate on solving the feedback problem first with the unknown TF as "TF": -

Write down what you know: -

1. V1 = Vin - V2
2. V2 = V1.TF

From (2) V1 = V2/TF and from (1) V2/TF = Vin - V2

Solve for V2 using (1): -

V2(1 + 1/TF) = Vin therefore

$\dfrac{V_{OUT}}{V_{IN}} = \dfrac{1}{1+TF}$

Then solve the transfer function TF (it looks fairly easy given that it is a sallen key 2nd order preceded by a simple 1st order filter with a buffer between them hence no impedance interactions).

Sallen key stage (help from wiki): -

That's the hard bit and H(s) just multiplies with the transfer function of the low pass filter which I'll leave you to do.

Then you'll have "TF" which you can insert into the equation I derived.