Old spec 5.4 example, Nyquist stability and Hurwitz limit for a third order loop (ENGR5001)

Old spec 5.4 example, Nyquist stability and Hurwitz limit for a third order loopTutorialOld spec ENGR2022:115 min

A unity feedback system has open-loop Transfer Function $KG(s) = \frac{K}{(s + 4)(s^2 + 4s + 8)}$

Formulas you may need

Critical point for neutral stability on the Nyquist diagram: $-1 + 0j$ (learn this)
Third order Hurwitz: stable if all coefficients positive and $a_1a_2 > a_0a_3$ (learn this)

(a)
Find the closed-loop characteristic equation and use the Hurwitz criterion to find the range of $K > 0$ for stability.
critical K =
(b)
Find where the Nyquist plot of $KG(j\omega)$ crosses the negative real axis (frequency and value in terms of $K$ ), and confirm the critical gain. Describe the Nyquist plots for $K = 80$ , 160 and 200.
phase crossover frequency =rad/s
(c)
For $K = 80$ , find the gain margin (dB) and the phase margin.
gain margin =dB
phase margin =deg