# Voltage divider with filter cap gain derivation

I am a mechanical engineer by training and I'm self-studying circuits. I'm trying to understand how to derive the gain and cut-off frequency for the following circuit:

I think that I should use the generalized voltage divider approach, where $$Z_1 = R_1$$ $$Z_2 = \left(\frac{1}{R_2}+\frac{1}{j\omega C}\right)^{-1}$$ and $$\frac{V_{out}}{V_{in}}=\frac{Z_2}{Z_1+Z_2}$$

but the complex algebra is really tripping me up. I also tried brute forcing the calculation using numpy complex numbers, but I don't think I did it right. Any help would be appreciated.

The voltage divider is correct to where left off, just need to slug through the algebra.

Starting where the OP left off:

$$\frac{V_{out}}{V_{in}}=\frac{Z_2}{Z_1+Z_2}$$ $$\frac{V_{out}}{V_{in}}=\frac{\frac{1}{\frac{1}{R2}+jωC}}{R_1+\frac{1}{\frac{1}{R2}+jωC}}$$

Multiply top and bottom by $$\\left( \frac{1}{R2}+jωC\right) \$$

$$\frac{V_{out}}{V_{in}}=\frac{1}{R_1\left( \frac{1}{R_2}+jωC\right)+1}=\frac{R_2}{R_1+R_2}\frac{1}{1+jω(R_1||R_2)C}$$

So the the gain can be seen to be $$A_V=\frac{R_2}{R_1+R_2}$$.

The cutoff frequency $$2\pi f_C=\frac{1}{(R_1||R_2)C}$$