# Deriving the transfer function of a notch filter from schemtic

I cam across this schematic for a notch filter online. It was part of lab session at a University. Currently, I am trying to design a circuit and I need to use a notch filter out the 60 Hz frequency. The description in the image suggests it is exactly what I need.

However, I am not really an electrical engineer and although I brushed up on my understanding of transfer functions through op amps, this is still quite challenging for me.

Would be great if someone could help derive the transfer function and help me get values for C2 and Rg1. Some explanation of how you solved it would be amazing. Many thanks

I have looked at this filter and derived its transfer function using the fast analytical techniques or FACTs as described in the book I wrote on the subject. The technique consists of looking at the time constants involving each of the capacitors in two different conditions: a zeroed excitation (the input voltage source is replaced by a short circuit) and a nulled response. As you have two capacitors in the circuit with independent state variables, this is a second-order filter.

I used SPICE to verify all my calculations in Mathcad and a dc operating point is fast and efficient as the below figure shows. In this sheet, I have determined the time constants for the denominator (the input is zeroed for this purpose) and the numerator also:

I then assembled the time constants and formed second-order polynomials in the numerator and the denominator. If the capacitors have equal values, then you can define the resonant frequency $$\\omega_0\$$ as follows:

$$\\omega_0=\frac{1}{C\sqrt{{R_2}{R_3}}}\$$ with the labels shown in the sheet below:

A quick SPICE simulation lets me test the calculations and verify the notch attenuates by 32.25 dB and is tuned at 60 Hz:

I am a power supply designer more than a filter master :-) and I've read that the circuit connected at the low-side terminal of $$\C_1\$$ could be modeled as an emulated inductance. I derived the transfer function (impedance) and if you neglect the high-frequency pole of the impedance "seen" below $$\C_1\$$ low-side terminal, you indeed have an equivalent inductance defined as $$\L_{eq}\approx R_3R_2C_2 \$$ in series with resistance $$\R_3\$$:
The new transfer function is thus that of an impedance divider involving $$\R_1\$$, $$\C_1\$$ and the equivalent lossy inductance. The transfer function is not that far from the original full-blown expression: