Derivation of Transfer Functions of Second Order Active Filters Resource?

We have covered the transfer functions and operation of some second order active filters (using op amps) in my electronic circuits course. However, the professor presented us with the transfer functions rather than derive them. I like to see a derivation of transfer functions so as to gain a better intuition for them when I first learn them and, strangely, my book (Sedra and Smith's Microelectronic Circuits 6th ed.) doesn't offer a derivation/proof either.

Does anyone have a resource that can show me the derivation of one or more of the standard filter types? Or is it possible someone could walk me through a derivation? I would really appreciate it!

For example, the Low Pass Filter's transfer function was given as:

$$\frac{Vo}{Vi} = \frac{H_o}{1 + \frac{s}{w_o \cdot Q} + \frac{s^2}{w_o^2}}$$

where $w_o$ is the fundamental frequency and Q is (I assume?) a quality factor of some sort.

• If you drew out a circuit with an inductor, resistor and capacitor forming a low pass filter, can you derive an equation for Vout that looks to have the right format? – Andy aka Feb 23 '14 at 14:26

• I think you have a typo $2\zeta = RC\omega_n$ – John Zhao Feb 3 '19 at 12:18