How can I deduce the damping characteristics formula?

[chrisjrollin]

Dear members,

I've got the interest arranging and deducing the formulas defining the damping characteristics of RC lag lead filter.

This is the circuit I've designed:


And this is sort of the arrangements to deduce the damping characteristics formula.


And the problem is that, I'm trying to make the arrangements into the deduced form below:



But this is the arrangement having the deduced form at its end and I could make as much far as possible:


Is there any missing steps or wrong arrangment that I've made in the calculations above?
I would gratefully take your advice and help to improve my knowledge.

Thanks for your interest to my question.

* References:
http://old.iihr.uiowa.edu/~hml/people/kruger/Teaching/ece_55141_2012/Articles/PLLExtract.pdf
http://mars.uta.edu/mae4310/handouts/compensator_realization.pdf
http://www.ece.ucsb.edu/~long/ece594a/PLL_intro_594a_s05.pdf
http://www.ti.com.cn/cn/lit/an/scha003b/scha003b.pdf

 

[LvW]

I think, you must fail if you are trying to find a second-order function for a first-order circuit.
The first part of your calculation is, of course, correct.

Note that on page 632 of the 1st document we have two different finctions: First-order F(s) and second-order T(jw).