Problem $2(25$ pts$)$:

Four systems are represented by the transfer functions given below.

A$.$ G$\_($A$)($s$)=(0\mathrm{.}1)/(15$s$+3)$

B$.$ G$\_($B$)($s$)=(200)/(8$s$^(2)+160$s$+7200)$

C$.$ G$\_($C$)($s$)=(1)/($s$^(2)+2$s$+900)$

D$.$ G$\_($D$)($s$)=(1)/($s$^(2)+60$s$+900)$

Your tasks:

A$.$ Find the time constant of G$\_($A$).(3$ pts$)$

B$.$ Find the natural frequency, $\backslash$omega $\_($n$),$ and damping ratio, $\backslash$zeta $,$ of the $2^($nd $)$ order transfer functions: G$\_($B$),$G$\_($C$),$G$\_($D$)(2$ pts each$)$

C$.$ Match the Transfer Functions given in the problem statement with their respective step responses and Bode plots. Be

aware of the axis limits for the step functions, as they are unique. The Bode plots can be found on the next page. $(4$ pts

each$)$

For each of the four systems, format your answers in the following form: G$\_($x$)-$ step$\_($y$)-$ bode z

Provide justification $($no more than a sentence$)$ to explain your choice $($for each system$).$