Butterworth filter transfer function calculation

Hey guys, I am looking for some help in understanding the derivation for an n order butterworth filter.

Taken from wikipedia.org:

The transfer function

Plot of the gain of Butterworth low-pass filters of orders 1 through 5. Note that the slope is 20n dB/decade where n is the filter order.


Like all filters, the typical prototype is the low-pass filter, which can be modified into a high-pass filter, or placed in series with others to form band-pass and band-stop filters, and higher order versions of these.
The gain G(ω) of an n-order Butterworth low pass filter is given in terms of the transfer function H(s) as:

where

• n = order of filter
• ωc = cutoff frequency (approximately the -3dB frequency)
• G0 is the DC gain (gain at zero frequency)

It can be seen that as n approaches infinity, the gain becomes a rectangle function and frequencies below ωc will be passed with gain G0, while frequencies above ωc will be suppressed. For smaller values of n, the cutoff will be less sharp.
We wish to determine the transfer function H(s) where s = σ + jω. Since H(s)H(-s) evaluated at s = jω is simply equal to |H(jω)|2, it follows that:

The poles of this expression occur on a circle of radius ωc at equally spaced points. The transfer function itself will be specified by just the poles in the negative real half-plane of s. The k-th pole is specified by:

and hence,

The transfer function may be written in terms of these poles as:

The denominator is a Butterworth polynomial in s.

Click to expand...

From above, it says that the poles of |H(jw)|^2 are evaluated. How is this correct? I dont understand the progression from magnitude squared expression to the actual transfer function. I guess the real prolem i have with it is this part and following:
and hence,


I am sorry about the links not visible, if anyone could tell me how to make them visible i am all ears

Have you read this application note on active filters?

hgmjr

Can you provide an example filter circuit similar to the one you would like to evaluate/build?

Perhaps a concrete example would be useful in a couple of ways. It would provide the various members who are knowledgeable on the topic with a circuit to work through while at the same time it would advance your understanding of the various design strategies that are available to tackle this problem.

I for one would be interested to learn the motivation behind your curiosity about filter design. It is a subject that is neither for the faint of heart nor the mathematically shy.

That said, I admire your pluck.

hgmjr

Hi and thank you for your replies, well to answer your question I didnt find it acceptable that we just magically have these different order equations which we put in to get the transfer function. I think it should be possible to obtain the tranfer function for the filter by looking at the magnitude tranfer function formula. I guess I am just trying to understand it to some extent.

In the article mentioned from wikipedia, the author attemps to explain just that and to be honest I have spent a good few hours now sifting through different sources which seem to readily give the mathematical equations but not to justify them. I have not read the application note you provided but will try to read it when I get home from work.

I am very impressed with the amount of information Texas instruments have released. I have read some of their stuff on opamp stability etc and find it to be quite good.

I'll let you know how i get on and I will also consult my lecturer as well on this as he is frankly a legend.

Cheers!

P.s. I havent been trying to work on a particular example as such, I would just like to know the system of how the polynomials result.

There are a number of techniques that are available. The easiest approach is to use filter design software. You have made it clear that you want to know the underlying theory. Without sufficient detail knowledge of what is going on behind the curtain, it is difficult to appreciate the implecations of design decisions being made for you by the software.

For each viable filter topology, be it active or a passive filter, there is a transfer function that describes the output signal as a function of the input signal. This transfer function is most often expressed in terms of the resistors, inductors, capacitors and the frequency of the applied signal. I am sure you have seen these in your text books and on the internet.

You will also have noticed that the transfer functions contain the complex variable "s" raised to various powers. The exponent of these "s" terms convey the number poles when they appear in the denominator of the transfer function and they convey the number of zeroes when they appear in the numerator of the transfer function.

Each of these "s" terms has a coefficient that is expressed in terms of the resistors, capacitors, and/or inductors used in the filter circuit.

There are tables available that provide the normalize values for the coefficients of these various "s" terms corresponding to the filter type you wish to design. Some of the filter types are Butterworth, Bessel, and Chebyshev to name a few. Each of these filter types possess its own performance characteristics that make is suited to a given application.

You will note that I used the term "normalized". These cooefficients are often based on a filter designed for a frequency of 1 rad/sec (expressed in radians).

To tackle the calculation of the normalized values for the resistors, capacitors and inductors is mathematically challenging since it involves a non-linear system of n-equations with n-unknowns. This requires the use of some fairly sophistocated math such as Newton-Raphson's. The process is labor intensive and can get really messy for filters of order 4 and greater.

Once you obtain the normalized values for all of the resistors, capacitors, and inductors, you can then use the method referred to as scaling to get to the frequency you need for your specific design.

This explanation is not intended to be comprehensive but to simply give you a few details about what goes on in the design of a filter.

hgmjr

Here is an example of the table I was referring to in my previous post.

hgmjr

Yup indeed i am familiar with the majority of what you have told me. We have been shown in class how to scale the normalised butterworth equations to an arbitrary frequency which I am fine with. We have also been shown how (at least with a second order filter ) to translate the 2nd order butterworth equation into a realisable filter with R,L and C parameters. As mentioned, the underlying issue here is how the poles of the magnitude squared function are utilised to arrive at the transfer function for the filter and by that I mean the poles of the transfer function itself. Or is it that the poles of the magnitude squared function are the same poles of the transfer function itself? It just doesnt seem obvious to me how this is the case.

I have still not had a chance to read your link but I am sure that it will shed some light on the topic.

Thanks for your help thus far

How about posting a transfer function for an active filter (lets say a second order lowpass filter) as an example of a filter you might be interested in analyzing?

hgmjr

Respect is due to Mentaaal for being bothered to work this out. The answer you seek may be found in the excellent "Analog Filter Design" by M.E. van Valkenburg, chapter 6 "Butterworth Pole Locations". Here is the Reader's Digest condensed version; you'll notice a different mathematical method to the Wiki page. I'd like to take credit for the working, but the sad truth is I copied it (almost) verbatim from the fine van Valkenburg tome. It's the filter bible.

Thanks for the help with this question guys, that link was quite good, my lecturer had also referred me to that book.

Well doing a particular filter is not really the issue for me, perhaps my question was a stupid one. What was bothering me what I was not satisfied that the poles of h(s)*h(-s) matched the ones for h(s). I can see now that if we take the poles on the positive real axis that this is indeed true. It took a bit of explaining by my legendary electric circuits lecturer but eventually he drummed it into my skull.

Thanks for the help again!

hgmjr said:

Here is an example of the table I was referring to in my previous post.

hgmjr

Click to expand...

hi, I appreciate ur approach to this question. In fact, the table has been a relief to my headache. Tnx

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