The transistor in the amplifier circuit shown in the figure is biased at $${{\rm I}_C} = mA.$$ Use $${V_T}\left( { = {{kT} \over q}} \right) = 26\,mV,$$ $${\beta _0} = 2000,\,{r_b} = 0$$ and $${r_0} \Rightarrow \infty $$
$$(a)$$ Determine the AC small signal midband voltage gain $$\left( {{\raise0.5ex\hbox{$\scriptstyle {{V_o}}$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle {{V_i}}$}}} \right)$$ of the circuit
$$(b)$$ Determine the required value of $${C_E}$$ for the circuit to have a lower cutoff frequency of $$10Hz$$

Given the characteristic equation $${s^3} + 2{s^2} + Ks + K = 0.$$ Sketch the root focus as $$K$$ varies from zero to infinity. Find the angle and real axis intercept of the asymptotes, break-away / break-in points, and imaginary axis crossing points, if any

The polar plot of a type-$$1, 3$$-pole, open-loop system is shown in Fig. below. The closed loop system is

The asymptotic approximation of the log-magnitude versus frequency plot of a minimum phase system with real poles and one zero is shown in Fig. Its transfer functions is