Design Methods

Given specification, we know

\[
\begin{aligned}
&\text{At } \omega = \omega_p, \quad \alpha = \alpha_p = \alpha_{\max} \\
&\text{At } \omega = \omega_s, \quad \alpha = \alpha_s = \alpha_{\min}
\end{aligned}
\]

1.Find the order of Butterworth Response:

\[
n = \frac{\log_{10} \left( \frac{10^{\alpha_{\min}/10} - 1}{10^{\alpha_{\max}/10} - 1} \right)}{2 \log_{10} \left( \frac{\omega_s}{\omega_p} \right)}
\]

2. Determine the cut-off frequency

\[
\omega_0 = \omega_p \left(10^{\alpha_{\max}/10} - 1 \right)^{-\frac{1}{2n}}
\]

3. Find the poles based on order:

\[
B_n = 1 \quad \text{or} \quad (s+1) \prod \left( s^2 + 2 \cos \left( \frac{k \pi}{n} \right) s + 1 \right)
\]

\[
\text{If } n \text{ is odd, there is a pole at } 0^\circ
\]

\[
\text{If } n \text{ is even, there are poles at } \pm \frac{90^\circ}{n}
\]

\[
\text{The poles are separated by } \frac{180^\circ}{n}
\]

\[
\text{The poles are located at } P_k = \cos \phi_k \pm j \sin \phi_k
\]

\[
\text{Transfer function:} \quad T_n(s) = \frac{1}{B_n}
\]

\[
\text{Alternatively, magnitude squared:} \quad |T_n(s)|^2 = \frac{1}{1 + (-1)^n s^{2n}}
\]