# Passive Bessel Low Pass Filter

Discussion in 'Homework Help' started by farah_727rash, Dec 27, 2007.

1. ### farah_727rash Thread Starter New Member

May 1, 2007
6
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Hi All,

I am trying to design a Bessel LPF with a cut-off of 48.8 MHz.

I chose passive cos of the very high freq. involved. My text gives equations gives equations to find transfer function of Butterworth second order filter given our desired specs and to realize it using a LCR circuit. I designed and simulated it in Pspice but could not get exact cut-off and had quite a non-linear phase response.

I want some help to do the same for a Bessel topology cos linear phase is of the most importance in my project. But, I have no idea about the equations and such for Bessel filters. Does anyone know of any resource online that would help me design a Bessel LPF of a higher order, say 5 or so?

Thanks a lot,

Feb 24, 2006
10,340
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3. ### farah_727rash Thread Starter New Member

May 1, 2007
6
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Thanks, but I checked the book, and it does not contain anything about filters...

4. ### scubasteve_911 Senior Member

Dec 27, 2007
1,202
1
Farah,

Check out Wikipedia's information on Bessel Filters.
http://en.wikipedia.org/wiki/Bessel_filter

I'm on holidays from school and don't really want to reteach myself the math needed to analyze the filter. Most decent libraries will have books from the 60-70s purely on passive filter design and formulation. Have fun!

5. ### farah_727rash Thread Starter New Member

May 1, 2007
6
0
Thanks, just one more thing

I got my transfer function of the bessel low pass filter (fifth order):
945
--------------------------------------------------------------------
3.05e-041 s^5 + 5.802e-032 s^4 + 5.15e-023 s^3 + 2.612e-014 s^2

+ 7.452e-006 s + 945

I plotted its freq and phase response in matlab and checked to see if it met my specs.
Now, I need to realize the circuit. I know how to do it for a second order filter cos the standard form of the transfer function would be:
(1/LC) / (s^2 + s (1/CR) + (1/LC)) which could easily be designed for a R and L value after selecting a C value.

But, how to do it for a fifth order transfer function. Is there any such equation for a low pass filter from which I could find the values of the components?

Thanks a lot,
Farah.

6. ### Ron H AAC Fanatic!

Apr 14, 2005
7,050
657
I designed a filter for you and posted it here briefly, before I remembered that this was a homework assignment. I used "Handbook of Filter Synthesis" by Zverev, which is one of the most useful books I own. Any good University library should have a copy.
I simulated the design (5 pole Bessel, F3db=48.4MHz), and it worked just fine (with ideal components, of course).
Good luck.

7. ### farah_727rash Thread Starter New Member

May 1, 2007
6
0
So, my question is that given my transfer function, how do I get the component values? Could you please give me the equations so that I could calculate the component values. Looking on the net for the resources does not turn up anything and since I don't have any text dedicated solely to filter design to help me with, the equations from that book you mentioned would be a great help.

Thanks,
Farah.

8. ### The Electrician AAC Fanatic!

Oct 9, 2007
2,301
339
Realization of high order networks from a transfer function is a non-trivial problem in network synthesis. Consult the books by Tuttle or Guillemin. There are no "equations" for the general high-order problem. There are "methods", such as the use of continued fractions.

Or...

It would be easier to get a trial version of a filter design program (such
to your specs. The program will give you the realization.

9. ### Ron H AAC Fanatic!

Apr 14, 2005
7,050
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The values for the components came from a table, which was undoubtedly derived from equations. The process doesn't seem to be described in Zverev.
The book "Network Analysis and Synthesis" by Kuo has the procedure for synthesizing the filter from the transfer impedance Z21. It involves a continued fraction expansion. I have long forgotten how to do that.

Does you assignment require that you derive the filter from the transfer function, or is the filter merely part of a larger assignment? In other words, does it matter how you come up with the filter design?

10. ### farah_727rash Thread Starter New Member

May 1, 2007
6
0
The LPF is a component in the wireless communications base station I am designing for my project actually...so, no it doesn't matter how I design it as in it isn't a "homework", but I wanted to know how to do it myself instead of using software and calculators online...

But, I've it figured that component values have been calculated by mathematicians, and the values have to be denormalized to suit our specs...So, could you verify the equations:
R_new = source/load impedance * R_normalized
L_new = source/load impedance * L_normalized / (2*pi*cut-off_freq)
C_new = C_normalized / (source/load impedance * (2*pi*cut-off_freq) )

Thanks,
Farah.

11. ### Papabravo Expert

Feb 24, 2006
10,340
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Which edition? The one I have has extensive tables for orders up to 8.

12. ### Papabravo Expert

Feb 24, 2006
10,340
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Did you mean "continued fraction" or "partial fraction". If you menat partial fraction then you start by finding the roots. Once you have the roots you can use a set of linear equations to find the coefficients.

13. ### Ron H AAC Fanatic!

Apr 14, 2005
7,050
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OK, here's the design and the results. Farah, I think it answers your most recent questions.
The .ASC file will run in LTSpice, if you have it.
Papabravo, my book says "continued fraction expansion".

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14. ### scubasteve_911 Senior Member

Dec 27, 2007
1,202
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I'm starting to recall passive network synthesis via Foster I/II and Cauer I/II synthesis. This is going back to 2nd year, so I am a bit foggy. Do a search for Foster or Cauer synthesis.

Steve

Dec 27, 2007
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16. ### hgmjr Moderator

Jan 28, 2005
9,030
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Greeting Farah,

If you have the time, patience, and desire to tackle the calculation by hand, a technique I have used with success involves the following steps.

1. Derive or look-up the general expression for the transfer function of the passive RLC fifth-order. The coefficients will be expressed as a function of the R's, L's and C's of your filter circuit.

2. Find the coefficients for the fifth-order transfer function normallized to 1 rad/sec. The tables for these coefficients are fairly easy to track down.

3. Set each of the coefficients from your general case from step 1 above to the corresponding value taken from the tables from step 2. This will give you a non-linear system of n-equations with n-unknowns.

4. Use whatever numerical analysis technique you are comfortable with to solve for the unknowns. I use Newton-Raphson's Method.

5. The values for the R's, L's and C's from step 4 are the normallized values which will then need to be de-normallized as you have indicated.

You can take this approach and grow very old in the process or you can use a simulation program and solve for the values in a matter of minutes.

hgmjr

17. ### scubasteve_911 Senior Member

Dec 27, 2007
1,202
1
"4. Use whatever numerical analysis technique you are comfortable with to solve for the unknowns. I use Newton-Raphson's Method."

I'd stick to the roots function in Matlab lest your hair fall out

Steve

18. ### Ron H AAC Fanatic!

Apr 14, 2005
7,050
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I just now downloaded and ran AADE Filter Design and Analysis, which is free. I input our OP's requirements, and the program came up with exactly the same component values that I got using Zverev. The program will allow you to specify the Q of the inductors, which is a nice feature.

19. ### Papabravo Expert

Feb 24, 2006
10,340
1,850
Right. Partial fraction expansion is for doing inverse Lapalce transforms

20. ### Ron H AAC Fanatic!

Apr 14, 2005
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I hate it when the OP vanishes...