# How to derive the transfer function of this op-amp? For the 1st question, is this a Sallen-Key filter? What criteria must a circuit possess in order for it to classified as a Sallen-Key filter? I searched the internet but found no clear answer.

For finding the frequency: my attempt, see below, defines node voltages Va and Vb and uses nodal analysis to find the transfer function, treating vo = vb since the opamp will regulate the voltage at the inverting input the same as the non-inverting input.

After finding the transfer function, I differentiated with respect to f to find the value in terms of R and C. $$\frac{v_i-v_a}{R}=\frac{v_a-v_b}{R}+\frac{v_a-v_b}{\frac{1}{jwc}}$$ $$\frac{v_a-v_b}{\frac{1}{jwc}}=\frac{v_b}{R||\frac{1}{jwc}}$$

After substituting Va in terms of Vb,A = Vb/Vi, I got the equation below:

$$gain=\frac{jwc}{3+w^2Rc-j(6wRc+wc)}$$

This is as far as I got, I am stuck on how I can get the above answer. Is the nodal analysis equation I derived correct?

• Assuming that something is Sallen & Key if they described it in their paper, then yes as it's described by Sallen & Key in TR-50, 6 May 1954 on page 6. Or in their later paper in March of 1955 on page 77 of IRE Transactions. (Of course, anyone can make arbitrary distinctions about what is and isn't.)
– jonk
Dec 2, 2021 at 19:39

Nodal analysis is the way to go. You can work on s-domain to make the life easier.

According to your schematic, $$\\mathrm{v_b=V_o}\$$. Using the properties of a closed loop operational amplifier, you'll get

$$\mathrm{ V_i+(1+s\ RC)V_o=(2+s\ RC)V_a }$$

where

$$\mathrm{ V_a=V_o\frac{1+2s\ RC}{sRC} }$$

So the gain function can be written as

$$\mathrm{ G=\frac{V_o}{V_i}=\frac{u-1}{u^2+2u-1} }$$

where $$\mathrm{ u=1+s\ RC }$$

Before you ask, this $$\u\$$ comes from using the paralleled impedances R and C across $$\\mathrm{v_b}\$$ and GND.

After the substitution, you'll get $$\mathrm{ G=\frac{s/(RC)}{s^2+\frac{4}{RC}s+\frac{2}{R^2C^2}} }$$

It's obvious that this is a transfer function of a second order system which has a transfer function of

$$\mathrm{ G=A\frac{\omega_n^2}{s^2+2\zeta \omega_n+\omega_n^2} }$$

but with an extra zero. So, re-arranging the final equation so that it forms like the one above gave me

$$\mathrm{ G=\frac{s\ RC}{2}\cdot \frac{\frac{2}{R^2C^2}}{s^2+\frac{4}{RC}s+\frac{2}{R^2C^2}} }$$

From here, you can go for either substituting $$\s\$$ with $$\j\omega\$$, or differentiating, or any other technique. But eventually, you'll find that the maximum gain occurs at $$\\mathrm{\omega=\sqrt{2/(R^2C^2)}=\sqrt2/(RC)}\$$.

• Your 5th formula is the one. The last one has a gain larger by 1/(RC)^2. Dec 2, 2021 at 16:26
• @aconcernedcitizen wow I noticed this a quarter a year later. Thanks. Fixed. Mar 14, 2022 at 14:23