# How to find the transfer function of this first order active filter?

Going through Design with Operational Amplifiers and Analog Integrated Cricuits, in Chapter 3, problem 3.4 I am asked:

The circuit of Fig. P3.4 is a noninverting differentiator.

• (a) Derive its transfer function.
• (b) Specify component values for a unity-gain frequency of 100 Hz. I know that for a differentiator such as this, H(s) = -RCs and for example, a resistor and capacitor in parallel, the impedance Z1 is equal to R/RCs +1.

The closest example I could find was that of a Deboo integrator, whose transfer function is H(s) = 1/RCs. I am still struggling to understand how to extract the transfer function for Figure3.4. Any help would be appreciated.

• How is the differentiator relevant? It has a capacitor between input and inverting input and thus different behavior. Trying to find "the closest" circuit for which you know the transfer functions is a stupid approach as only one component needs to be different for a completely different behavior. What you need to do is to learn the method that is used to derive the transfer function. That is called circuit analysis and is explained in many textbooks. I suggest you start simple, derive the transfer function of the differentiator first and work your way up. – Bimpelrekkie Jan 12 at 13:07

You need nothing than the equations of feedback theory for opamps.

The general form for the closed-loop gain is

Acl=Hf * Aol/(1+Hr * Aol)

which simplifies for infinite openloop gain Aol to

Acl=Hf/Hr.

Here, the function Hf is the forward damping and Hr is the feedback function. Both are defined as follows:

Hf=Vd/Vin for Vout=0 (grounded) with Vd=diff. voltage at the opamp input nodes.

Hr=Vd/Vout for Vin=0

This way, the problem is reduced to simple voltage dividers.

Alternative(Edit): Perhaps the following method is easier to understand:

For an ideal opamp we have Vd=Vp-Vn=0 >>>>Vp=Vn

1) It easy to find Vn=f(Vout)

2) Vp=f(Vin, Vout) needs application of the superposition rule.

From both equations it shouldn`t be a problem to isolate Acl=Vout/Vin