Generalization of Pole locations
\[
B_n = 1 \quad \text{or} \quad (s+1) \prod (s^2 + 2\cos(k) s + 1)
\]
\[
\text{If } n \text{ is odd, there is a pole at } 0^\circ
\]
\[
\text{If } n \text{ is even, there are poles at } \pm \frac{90^\circ}{n}
\]
\[
\text{The poles are separated by } \frac{180^\circ}{n}
\]
\[
\text{The poles are located at } P_k = \cos(k) \pm j \sin(\phi(k))
\]
---
\[
\text{For } n = 2 \quad (\text{even})
\]
\[
\pm \frac{90^\circ}{n} = \pm \frac{90^\circ}{2} = \pm 45^\circ
\]
\[
B_2 = 1 \times \left(s^2 + 2 \cos(45^\circ) s + 1\right) = s^2 + 2 \times \frac{1}{\sqrt{2}} s + 1 = s^2 + s \sqrt{2} + 1
\]
---
\[
\text{For } n = 3 \quad (\text{odd})
\]
\[
\frac{180^\circ}{n} = \frac{180^\circ}{3} = 60^\circ
\]
\[
B_3 = (s+1) \times \left(s^2 + 2 \cos(60^\circ) s + 1\right) = (s+1) \times \left(s^2 + 2 \times \frac{1}{2} s + 1\right) = (s+1)(s^2 + s + 1)
\]
---
\[
\text{For } n = 4 \quad (\text{even})
\]
\[
\text{Poles are at } \pm \frac{90^\circ}{n} = \pm \frac{90^\circ}{4} = \pm 22.5^\circ
\]
\[
\text{Poles separation is given by } \frac{180^\circ}{n} = \frac{180^\circ}{4} = 45^\circ
\]
\[
\text{Poles are located at } \pm 22.5^\circ, \pm 67.5^\circ
\]
\[
B_4 = \left(s^2 + 2 \cos(22.5^\circ) s + 1\right) \times \left(s^2 + 2 \cos(67.5^\circ) s + 1\right)
\]
\[
B_4 = (s^2 + 1.85 s + 1)(s^2 + 0.765 s + 1)
\]
\[
\text{The transfer function is given by}
\]
\[
T_4(s) = \frac{1}{B_4} = \frac{1}{(s^2 + 1.85 s + 1)(s^2 + 0.765 s + 1)}
\]
---
\[
\text{For } n = 5 \quad (\text{odd})
\]
\[
\text{There is a pole at } 0^\circ
\]
\[
\text{Poles are at } \frac{180^\circ}{n} = \frac{180^\circ}{5} = 36^\circ
\]
\[
\text{Poles are located at } 0^\circ, \pm 36^\circ, \pm 72^\circ
\]
\[
B_5 = (s+1) \times \left(s^2 + 2 \cos(36^\circ) s + 1\right) \times \left(s^2 + 2 \cos(72^\circ) s + 1\right)
\]
\[
B_5 = (s+1)(s^2 + 1.62 s + 1)(s^2 + 0.62 s + 1)
\]
\[
\text{The transfer function is given by}
\]
\[
T_5(s) = \frac{1}{B_5} = \frac{1}{(s+1)(s^2 + 1.62 s + 1)(s^2 + 0.62 s + 1)}
\]