$5-$ Consider the positioning servomechanism shown below where

$e_i=K_{\text{p}\text{o}\text{t}}{}_i,e_o=K_{\text{p}\text{o}\text{t}}{}_o,K_{\text{p}\text{o}\text{t}}=10\frac{V}{r}ad,K_m=$motor constant $=10N\frac{m}{A},R_a=10$

Gear ratio $=10$:$1,J_L=$load inertia $={10}^{-2}kg{m}^2,J_m=$motor inertia $={10}^{-2}kg{m}^2,C=200uF$ and $V_0=K_A(e-e_f).(40pt)$

Liput protrilismeter

a$)$ What is the range of the amplifier gain $K_A$ for which the system is stable?

b$)$ Choose gain $K_A$ that gives two dominant roots with damping $=0\mathrm{.}7$ for the system.

c$)$ Plot the step response of the system and verify your choice of $K$ by looking at the percentage of overshoot.