# Passive Butterworth Filter Optimization

Discussion in 'The Projects Forum' started by jml2307, Mar 19, 2012.

1. ### jml2307 Thread Starter New Member

Mar 19, 2012
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Hello all,

This is my first post here and any help would be greatly appreciated. I am writing a C++ program that optimizes a passive implementation of a Butterworth filter in low, high, and band pass and band stop. My problem deals with the circuits' transfer functions. The goal of my program is as follows:

Given a set of possible values of inductors and capacitors that can make up a network (of nth order), find the set which minimizes the error between the realizable filter function and the idealized function.

My problem is as follows:

Once I have the normalized values of each component scaled for cutoff frequency and impedance, what is the relation between these and the transfer function coefficients? eg, if a 4th order network is desired and c1, c2, l1 and l2 are the normalized values of the capacitors and inductors, respectively, what are the equations that express the coefficients as functions of the components? a_i = f(c1,c2,l1,l2)

Any help would be greatly appreciated and if you could point me toward any applicable and specific reference material I would be extremely relieved since I am not an EE and combing circuits texts for the past two weeks has been not paying off.

2. ### The Electrician AAC Fanatic!

Oct 9, 2007
2,300
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It appears that you want to derive the transfer function of the filter given the filter topology, low pass, high pass, order, etc. This isn't too hard, although for the higher order filters the mathematical expressions become complicated.

If you will post a schematic of a typical filter, I can show you how to derive the transfer function. Include a source impedance and load impedance (Rs and RL) in the schematic.

3. ### jml2307 Thread Starter New Member

Mar 19, 2012
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The filter topology is a basic Cauer ladder structure and I am searching for an optimal network over 2nd through 5th order. The source and load impedances I would like to keep generalized. ie an expression including Rs and Rl or the ratio of the two would be ideal. This way I could evaluate over a wider frequency range and not stick to the text book 1 ohm source / load impedance.

I have looked at Zverev's "Handbook...," Harper's "Passive Electronic Component Handbook," and Budak's "Passive and Active Network Synthesis," (and other papers on filters) but I am not finding the functions I need. I did find the continued fractions expansion of the Cauer form in Budak but I am still not finding the explicit link.

The mathematics does not bother me, in fact that is the primary purpose of this project. I am just having a difficult time finding the correct source material without having to learn the obvious prerequisite circuits courses. I guess I'm trying to skip a bit of the prerequisites.

Thanks again

4. ### jml2307 Thread Starter New Member

Mar 19, 2012
5
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I guess my initial post should have asked for an expression for each coefficient in the transfer function as follows:

a_i = f(Rs, Rl, C1, C2, L1, L2,...)

5. ### The Electrician AAC Fanatic!

Oct 9, 2007
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A method for finding transfer functions is to use the nodal method of analysis and form an admittance matrix (Y matrix) for the circuit. Then invert the matrix to obtain a Z matrix and the transfer function will be a ratio of two of the elements of the Z matrix.

I use Mathematica to do the symbolic matrix algebra because for the higher order circuits the algebra becomes cumbersome and prone to mistakes when done by hand.

See this thread, especially starting at post #43:

Also see this:

for help in building the admittance matrix.

I've attached 3 images; the first shows a second order and a fourth order ladder network. Each can be a Butterworth low pass filter if the correct element values are chosen.

Notice that you don't need to allow for a node between Rs and L1. You can just treat the Rs-L1 series combination as a single impedance.

The second and third images show the derivation of the transfer functions in terms of the ladder elements. I think this is what you want.

For other filters such as band pass, band reject, etc., you only need to start with a suitable modification of the ladder elements and apply the Shekel method.

If you need any more help, just ask.

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6. ### jml2307 Thread Starter New Member

Mar 19, 2012
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That is exactly what I have been looking for. I will work through this over the next couple days and post my final results. I greatly appreciate your help.

7. ### jml2307 Thread Starter New Member

Mar 19, 2012
5
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Electrician, in post #45 of the Two stage BJT... thread you referenced, you mentioned the original paper by Shekel. I assume this is the paper "Analysis of Linear Networks" by Joseph Sheckel c. 1957?

Thanks

Oct 9, 2007
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