# Butterworth filter via Cauer topology

• Frank-95
In summary, to obtain L and C values for a Butterworth filter using the first Cauer topology, one must first find the transfer function and then compare it to the general function for a second-order filter with a Butterworth response. For higher order filters, tables with normalized element values can be used. The normalized values can be found by developing the equation for the pole positions in polar coordinates. However, for specific values, it is recommended to refer to filter design books.
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.

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/ωc)2n] ⇒ H(s)H(-s) = 1 /[1 + (-s2c2)n]

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

sk = jωcej(2k - 1)π/(2n)

For n = 2 and k = 1, 2

s1 = jωcejπ/4
s2 = jωcej3π/4

I thus have:

H(s) = ωc / [(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

Last edited by a moderator:
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).

Okay, but I notice an analogy between:

sk = jωcej(2k - 1)π/(2n)

and

gk = 2sin((2k - 1)π/(2n))

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

sk = -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

You have shown how to find the poles of a Butterworth response - expressed in polar coordinates.
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

Last edited:

## 1. What is a Butterworth filter via Cauer topology?

A Butterworth filter via Cauer topology is a type of electronic filter that is designed to have a flat magnitude response in the passband and a sharp cutoff at the cutoff frequency. It is named after British engineer Stephen Butterworth and German mathematician Wilhelm Cauer.

## 2. How does a Butterworth filter via Cauer topology work?

A Butterworth filter via Cauer topology works by using a combination of inductors and capacitors to selectively pass or block certain frequencies in an electronic circuit. The Cauer topology refers to the specific arrangement of these components, which allows for a more precise and efficient filtering process.

## 3. What are the advantages of using a Butterworth filter via Cauer topology?

One of the main advantages of using a Butterworth filter via Cauer topology is its ability to provide a flat magnitude response in the passband. This means that the filter does not introduce any additional distortion or attenuation to the signal within its desired frequency range. Additionally, the Cauer topology allows for a steeper cutoff than other filter designs, making it ideal for applications that require a sharp transition between the passband and stopband.

## 4. Are there any limitations to using a Butterworth filter via Cauer topology?

One of the limitations of a Butterworth filter via Cauer topology is its larger size and complexity compared to other filter designs. This may make it less practical for certain applications where space and simplicity are important factors. Additionally, the filter's performance may be affected by external factors such as temperature and component tolerances.

## 5. Where can I use a Butterworth filter via Cauer topology?

A Butterworth filter via Cauer topology can be used in a variety of electronic systems and applications, such as audio amplifiers, signal processing circuits, and communication systems. It is particularly useful in applications that require a precise and flat frequency response, such as in audio equalizers or radio receivers.

Replies
20
Views
2K
Replies
1
Views
2K
Replies
6
Views
2K
Replies
16
Views
1K
Replies
1
Views
3K
Replies
7
Views
3K
Replies
8
Views
1K
Replies
23
Views
5K
Replies
2
Views
6K
Replies
2
Views
4K