Analytical design of a bandstop Butterworth filter

Question

I am trying to analytically design a bandstop Butterworth filter. I am trying to use: $$|H(s)|^2 = \frac{1}{1+\big(\frac{s}{j\omega_1}\big)^{2N}}+\frac{1}{1+\big(\frac{s}{j\omega_2}\big)^{-2N}}$$ to derive a transfer function. I would like to know if I'm correct and how to proceed from here.

Answer 1

it looks like you are designing a parallel highpass and lowpass filter. i don't think that's the best way to do it. instead, take your butterworth LPF prototype

$$\Big|H(j\Omega)\Big|^2 = \frac{1}{1+\Omega^{2N}}$$

and make this substitution for normalized $\Omega$:

$$ j\Omega = s \leftarrow \frac{Q}{\frac{s}{\Omega_0} + \frac{\Omega_0}{s}} $$

$Q$ is inversely proportional to the bandwidth of the bandstop filter (if $N=1$, then $Q$ is the actual "Q" of the second-order filter. $\Omega_0$ is the angular frequency of where the bandstop is tuned to.