Control Engineering - Root Locus

Please see attached Question & Answer screenshot. I have managed to answer up to part 4 of the questions however in part 5, after countless working out and re-evaluating my approach, I still can't seem to understand how K=4/27 from the gain criterion? Also I'm having trouble understanding how the time-constant is found? Please can you help me

Thanks

From filter theory we know that a second-order transfer function is overdamped for a pole quality factor Qp<0.5 (two real poles on the neg. real axis); it is underdamped for Qp>0.5 (a conjugate-complex pole pair in the left half of the s-plane).

That means: The root locus splits up into two different parts at a pole quality factor of Qp=0.5. This is - by definition - a critically damped system. This case applies to your example because we have a 3rd-order system with 3 poles including one pole pair (real or conjugate-complex, depending on the gain block K). As a consequence, the system is critically damped when the gain reaches a value K=Kc corresponding to a value Qp=0.5, which is identical to the "breakaway point".

As a simple and intuitive example: Two decoupled equal RC sections provide a quality factor of Qp=0.5 and exhibit a "critically damped" response.

• Reasoning behind the critical damping I understand, however I can't see why K will equal 4/27? – Arsenal123 Jul 4 '16 at 11:02
• It should be the value for which the imag. part of the pair of roots disappears. Did you try this? – LvW Jul 4 '16 at 11:39
• Sorry I'm still not understanding. Apologies, can you please clarify by showing the steps in order please. May thanks – Arsenal123 Jul 4 '16 at 12:05
• Solving the given 3rd-order equation results in 3 roots (poles of the closed-loop function): 1 real (negative) pole and a pair of two poles of the form: [-x +-j*SQRT(y)] with x being the real part and y is the img. part. This part y depends on K and you need the value of K which gives y=0. In this case, we have a double pole at -x. This is the case of critical damping. I have checked the solution 4/27 - and it is correct. The double pole is at x=-0.333. – LvW Jul 4 '16 at 12:47
• By the way - I forgot to mention that the damping factor d is defined as d=1/2Qp. That means the damping factor d is unity for Qp=0.5 (another intuitive explanation for the term "critical damping"). – LvW Jul 4 '16 at 20:25

Breakaway points - dk/ds = 0 => s= -1/3 substitute this value in characteristic equation to get the value of K at breakaway point.