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Butterworth filters, help

  1. Mar 29, 2006 #1
    I want to design LPF using Butterworth transfer function & I have the following information
    Minimum gain attenuation Amin ,
    Maximum gain attenuation Amax ,
    Frequency of passband edge wp,
    Frequency of stopband edge ws,
    Amin,Amax,wp & ws can take any value

    (Using only Butterworth table for wp=1 & є=1)=====>which is the proplem Here :redface:
    Last edited: Mar 29, 2006
  2. jcsd
  3. Mar 29, 2006 #2
    You have to calculuate the order of the Butterworth filter. Do you not have any equations or anything?
  4. Mar 30, 2006 #3
    Hi Corneo

    I don’t know if you understand me :-)

    For example if I have Amax=1dB ,Amin=25dB ,wp=20π rad/s ,ws=30π rad/s & dc gain=1 V/V

    I can find the Butterworth transfer function
    T(jw)= (1+є² (w/wp)^2n)^-½ with є=0.5088 N=9
    Then I can find the poles

    But I want to get the poles from Butterworth table for є=1 & wp=1rad/s

    n Factors of Polynomial T(s)
    1 (s + 1)
    2 s2 + 1.414s + 1
    3 (s + 1)(s2 + s + 1)
    4 (s2 + 0.7654s + 1)(s2 + 1.8478s + 1)
    5 (s + 1)(s2 + 0.6180s + 1)(s2 + 1.6180s + 1)
    6 (s2 + 0.5176s + 1)(s2 + 1.414s + 1)(s2 + 1.9318s + 1)
    7 (s + 1)(s2 + 0.4450s + 1)(s2 + 1.247s + 1)(s2 + 1.8022s + 1)
    8 (s2 + 0.3986s + 1)(s2 + 1.111s + 1)(s2 + 1.6630s + 1)(s2 + 1.9622s + 1)

    For this table we know that Amax =3dB for all poles. This is mean even if we scale T(jw)--->T(jw/wp) in order to reach to wp we will still need Amax to be 1dB.

    So how I can scale the transfer function & choose the order such that the new function will match my bounded value.
    Last edited: Mar 30, 2006
  5. Mar 30, 2006 #4
    I'm not sure if I fully understood your problem here. Are you trying to find the final Butterworth transfer function for any arbitrary [itex]\omega_c[/itex], given that you have found the normalized Butterworth transfer function?
  6. Apr 7, 2006 #5


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    is one of your problems that you don't have butterworth polynomial for N=9? do you need a closed form expression?

    also, what you should do with your spec is determine where the -3 dB frequency would be (assuming you did that right and correctly determined a 9th order is needed.
  7. Apr 10, 2006 #6


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    Science Advisor
    Homework Helper

    You're looking for the middle term in your polynomials?

    For your odds:

    Divide 180 by n. (let it equal t)

    Your first term is (s+1).
    For the second term, in polar coordinates:
    [tex](1 \angle{t})+(1 \angle {-t})[/tex]
    Third polynomial:
    [tex](1 \angle{2t}) + (1 \angle {-2t})[/tex]

    For your fifth polynomial you have:
    [tex](1 \angle {36}) + (1 \angle {-36}) = 1.6180[/tex]
    The polynomial is [tex]s^2 + 1.6180s + 1[/tex]
    [tex](1 \angle {72}) + (1 \angle {-72}) = .6180[/tex]
    The polynomial is [tex]s^2 + .6180s + 1[/tex]

    Making your final:
    [tex](s + 1) (s^2 + .6180s + 1) (s^2 + 1.6180s + 1)[/tex]

    Edit: Technically, you should use 180-t everywhere I used t, but, as you can see.....

    Edit: Here's a graphical representation of what you're doing: http://www.crbond.com/filters.htm
    Last edited: Apr 10, 2006
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