0
\$\begingroup\$

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

Question Answer

\$\endgroup\$
1
\$\begingroup\$

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.

\$\endgroup\$
  • \$\begingroup\$ Reasoning behind the critical damping I understand, however I can't see why K will equal 4/27? \$\endgroup\$ – Arsenal123 Jul 4 '16 at 11:02
  • \$\begingroup\$ It should be the value for which the imag. part of the pair of roots disappears. Did you try this? \$\endgroup\$ – LvW Jul 4 '16 at 11:39
  • \$\begingroup\$ Sorry I'm still not understanding. Apologies, can you please clarify by showing the steps in order please. May thanks \$\endgroup\$ – Arsenal123 Jul 4 '16 at 12:05
  • \$\begingroup\$ 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. \$\endgroup\$ – LvW Jul 4 '16 at 12:47
  • \$\begingroup\$ 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"). \$\endgroup\$ – LvW Jul 4 '16 at 20:25
1
\$\begingroup\$

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

\$\endgroup\$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.