$1$

Consider a control circuit consisting of a PD $($proportional and derivative elements$)$ and a second order process with a transfer function that is defined by

$G_p(s)=\frac{1}{{}_p{s}^2+{}_{p}s+1}$ Process transfer function

$G_c(s)=K(1+{}_{D}s)$ Controller transfer function

$($a$)$ Develop the general characteristic equation that describes the behavior of the control circuit. Use the symbols given.

$($b$)$ Describe the behavior of the circuit where

${}_D=0\mathrm{.}20$

${}_P=3\mathrm{.}0$

$K=2\mathrm{.}0$