butterworth filter transfer function decompostion question

[yefj]

Hello, If i understand correctly we are approximating a general response with multisection transfer function shown bellow.
In the pozar book they give me a complete result.
but i wat to develop such table by myself.
1.How do i convert a single line of the table below into a mathematical transfer function?
2.Is there some example where they take a transfer function and aproximate it and creating such line.
I know its being stuied more in digital filters.if you could please help me lwith such rf example guidance.

 

[tepalia02]

You can get some clues here:

 

[FvM]

Do you understand that the upper and lower half of the posted screenshot are unrelated because they are dealing with different filter prototypes? The upper is about maximum flat delay (Bessel) and the lower about Butterworth filter.

You are asking how to convert the Bessel circuit element table into a transfer function. The purpose of the table is however opposite, it has been derived by synthesizing a given transfer function (n-th order Bessel lowpass) with a specific circuit topology (LC ladder).

--- Updated Dec 9, 2023 ---

I reviewed the respective chapter around shown table 8.5 in Pozar Micropwave Engineering. It's not derived in the book, just mentioned, neither the characteristic of maximum flat delay filter is explained at all. If you want to understand the respective filter characteristics and design methods, you better get a serious filter design textbook. My personal favorite is Harry Y-F. Lam, Analog and Digital Filters, Design and Realization.

 

[D.A.(Tony)Stewart]

Analog and digital filters ; design and realization : Lam, Harry Y. F., 1944- : Free Download, Borrow, and Streaming : Internet Archive

xv, 632 p. ; 24 cm

archive.org