Prove that the magnitude slope of -1 in the Bode plot corresponds to \(-20 d B\) per decade.

(a) Sketch the polar plot for an open-loop system with a transfer function \(1 / s^{2}\), that is, sketch

\[\left.\frac{1}{s^{2}}ight|_{s=C_{1}}\]

where \(C_{1}\) is a contour enclosing the entire RHP. (Hint: assume \(C_{1}\) takes a small detour around the poles at \(s=0\).)

(b) Repeat part (a) for an open-loop system whose transfer function is \[G(s)=\frac{1}{s^{2}+\omega_{o}^{2}}\]