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

Question

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.

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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.

Answer 1

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