Problem $2.$ Motor Dynamic Response

The parameter values for a certain armature$-$controlled motor are

$K_T=K_b=0\mathrm{.}2N*\frac{m}{A}$

$b=5{10}^{-4}N*m*\frac{s}{r}ad,R_a=0\mathrm{.}8$

$L_a=4{10}^{-3}H,J=5{10}^{-4}kg*m^2$

For these parameters, discuss the nature of the poles of the transfer function $($e$.$g$.$ real, complex, stable, unstable, etc.$).$ From the four transfer functions obtained in Problem $1,$

$1$

a$.$ Obtain the step response of $i_a(t)$ and $(t)$ if the applied voltage is $v_{in}=10V$ with zero torque load, from tilde$(I{)}_a\frac{s}{t}$ilde$(V{)}_{in}(s)$ and tilde$()\frac{s}{t}$ilde$(V{)}_{in}(s).$

b$.$ Obtain the step response of $i_a(t)$ and $(t)$ if the load torque is $T_L=0\mathrm{.}2N*m,$ with zero input voltage, from tilde$(I{)}_a\frac{s}{t}$ilde$(T{)}_L(s)$ and tilde$()\frac{s}{t}$ilde$(T{)}_L(s).$