Butterworth filter via Cauer topology

[Frank-95]

Hi all.

I would like to ask some explanation regarding how to obtain L and C values for a butterworth filter using the first Cauer topology. I will consider a normalized low pass filter.

(Link added by Mentor)
https://en.wikipedia.org/wiki/Butterworth_filter

What I don't understand is this: how to pass from transfer function to L and C values?

For example: let's consider a second order filter, with 0 dB gain at DC (for simplicity). I have:

|H(jω)|² = 1 / [1 + (ω/ω_c)²ⁿ] ⇒ H(s)H(-s) = 1 /[1 + (-s²/ω_c²)ⁿ]

We find poles by putting the deniminator equal to 0. Hence it yields:

s_k = jω_ce^{j(2k - 1)π/(2n)}

For n = 2 and k = 1, 2

s1 = jω_ce^jπ/4
s2 = jω_ce^j3π/4

I thus have:

H(s) = ω_c / [(s - s1) (s - s2)]

As shown also on wikipedia.

What I don't understand now is this: how can I calculate from here the values of capacitors and inductors of the LC ladder. There are formulas at wikipedia but they don't explain how they are obtained.

Any tip?

Thank you

 

[LvW]

For a second order filter it is realtively easy: You find the transfer function H1(s) of the circuit (expresssed in terms of L and C) and - in a second step - you compare the corresponding parts of thhis function with the general second-order function Ho(s) (expressed in pole data Qp and wp). For a Butterworth filter we have Qp=0.7071 and wp=wc (desired 3dB frequency).
However, for filter orders n>2 this calculation is increasingly involved and you can (must) use corresponding tables with normalized element values (filter tables are available in corresponding filter books).

 

[Frank-95]

Okay, but I notice an analogy between:

s_k = jω_ce^{j(2k - 1)π/(2n)}

and

g_k = 2sin((2k - 1)π/(2n))

In fact, if I develop the first equation I get:

s_k = -sin[(2k - 1)π/(2n)] + jcos[(2k - 1)π/(2n)]

The normalized models I am referring about is this (R1 = R2):

So I feel pretty close to understand how normalized values are achieved, without using tables

 

[LvW]

You have shown how to find the poles of a Butterworth response - expressed in polar coordinates.
However, does it help you to find the corresponding parts values?
From the pole positions, you could, of course, derive the pole data for each complex pole pair (wp and Qp).
However, these values can be found in each book dealing with filters