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Pi-Network Transfer Function

Discussion in 'Electronic Design' started by Jumbaliah, Nov 2, 2004.

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  1. Jumbaliah

    Jumbaliah Guest

    I'm trying to come up with some component values for a pi-network low
    pass filter for my future DDS project. I started to try and figure
    out the basic transfer function for a 3rd order pi-network (two caps
    and one inductor). It doesn't look right to me so I wanted a second
    opinion.

    Vi-+--Z2--+--Vo
    | |
    Z1 Z3
    | |
    Gnd Gnd

    What I have is Vo/Vi = Z3/(Z2+Z3).

    The equations I used to reach this was the following:
    I2 is a current loop from Vi -> Z1 -> Gnd -> back to Vi
    I1 is a current loop from Gnd -> Z1 -> Z2 -> Z3 -> Gnd

    eq1 Vi = I2*(Z1) - I1*(Z1)
    eq2 Vo = I1*(Z3)
    eq3 0 = -I2*(Z1) + [Z1 + Z2 + Z3]*I1

    I arranged eq1 and eq3 to solve for I2. I solved for I1 in eq2 and
    sub'd that into eq1 and 3. I factored and cancelled and flipped the
    equation till I got Z3/(Z2+Z3). It doesn't seem right that Z1 isn't
    involved. I must have missed something here.

    Would anyone have the general transfer function for a nth degree
    pi-network?

    Thanks.
    Burnboy
     
  2. -----BEGIN PGP SIGNED MESSAGE-----
    Hash: SHA1
    There are loads of books with schematics and tables of component values for
    filters.
    What you might have overlooked is that Z1 is loading a perfect voltage
    source, and so there is no effect on voltage V1 from Z1. Simple :)

    You could put a current I1 into node 1, that would make Z1 a part of the
    trans-resistance transfer function.


    Best Regards

    Jens


    - --
    Key ID 0x09723C12,
    Analogue filtering / 5GHz RLAN / Mdk Linux / odds and ends
    http://www.tingleff.org/jensting/ +44 1223 211 585
    "I lost my heel" "Oh, don't worry about him!" 'Bringing up Baby'
    -----BEGIN PGP SIGNATURE-----
    Version: GnuPG v1.2.4 (GNU/Linux)

    iD8DBQFBhyv1imJs3AlyPBIRAm0eAJ44T/uxzgYT/GuAeoYCSfrLYm6aXwCfTvs1
    fGTV5Sg5Vr6MZlv8zFE7vyw=
    =Gb+s
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  3. You seem to be missing something, namely the input and output
    impedances.


    Vi--Za--+--Z2--+----+--Vo
    | | |
    Z1 Z3 Zb
    | | |
    Gnd Gnd Gnd


    Za is the source impedance, and Zb is the load impedance. Vi
    represents the potential of the Thevenin source, rather than the
    actual voltage across the output terminals of the driver.

    In a typical RF system, Za and Zb will both be 50 ohm, and (close to)
    resistive.

    Regards,
    Allan
     
  4. This is called "filter design". There are no *useful* completely general
    formulas for general filters. However, this is a subject extensively
    studied such that there are loads of programs out there that will
    calculate suitable values for specific filters, e.g. SuperSpice, and
    other general filter design programs. I suggest a web search.

    Kevin Aylward

    http://www.anasoft.co.uk
    SuperSpice, a very affordable Mixed-Mode
    Windows Simulator with Schematic Capture,
    Waveform Display, FFT's and Filter Design.
     
  5. Ban

    Ban Guest

    A pi filter is not usable for your application, if the input impedance is
    low impedance. Here it is better to use a 3-pole T-filter consisting of
    L-C-L. with the values 120u/27n/22u you will get around 100k bandwidth into
    50R load. The coils should have a Q of 20 or higher.
    If you put 47R in front to have also a 50R source impedance the bandwidth
    goes up to 138k, but stays almost Bessel without over- or undershoot.
    Look up the transfer function in google, find the Laplace transform like:
    1/(1+1.752S+1.23S^2+0.338S^3) ; S= j*f/f_fil ; f/f_fil is the normalized
    frequency operator with f_fil 3dB down.
     
  6. Hi,

    You cannot separate the source and load impedances from the
    rest of the transfer function. A quick way to design one though,
    without the resulting maths, is to use a table of coefficients
    for a Butterworth network. They are found in many books and just
    need scaling for cut-off frequency and impedance.

    Alternatively, split the inductor down the middle to get a
    couple of back-to-back L-networks (choosing an intermediate load
    impedance). There will be lots of possible values, only limited
    if you must have a specific loaded-Q. This latter method is
    common when designing tx tank circuits.

    If you must have the complete transfer function here is what
    my spice program says -


    W(s) = RL / [RS + RL + (RS.RL.C2 + L + RS.RL.C1)s + ...

    (RL.C2 + RS.C1)L.S^2 + RS.RL.C2.C1.L.s^3]


    where C1 is the source-end capacitor.


    Cheers - Joe
     
  7. Jumbaliah

    Jumbaliah Guest


    Thanks for the replies everyone.

    I've looked in my textbooks and a bit online but so far haven't found
    useful tables of component values for pi-networks. I agree with you
    all that I should have been using input and output impedances. I've
    updated my diagram and formulas to suit.

    I originally started using a filter design program from here:
    http://www.aade.com/

    It is simple enough however when I chose my filter parameters it
    calculated some capacitance and inductance values. I then wanted to
    change them to standard values but it doesn't recalculate the other
    components. This is why I wanted the transfer function of an nth
    degree pi-network. This way I could get approx values from the filter
    software, adjust some parts to standard values, then solve for the
    changes in remaining parts.

    I figure right now I'll shoot for a filter with passband of 0-30MHz,
    stopband at 35MHz+. Maybe a 7th order Chebyshev or Butterworth.

    How did you get that transfer function from spice, Joe? Do you happen
    to have one for the 7th order?

    With a 7th order Pi-network with input and output impedances I've got
    6 equations that would require a lot of paper to solve.... I only went
    as far as the third substitution ;).

    Thanks,
    Burnboy
     
  8. John Miles

    John Miles Guest

    Pi networks are sort of old-school. They were popularized by Collins
    and other manufacturers back in the days when vacuum-tube power
    amplifiers ruled the ham-radio roost. They could match the final
    amplifier's relatively-high plate impedance to a low-Z antenna,
    providing low-pass filtering at the same time. I don't see them
    mentioned very often in current literature.

    The ARRL Handbook would probably be a good place for further research on
    pi networks. If you can find the old "Single Sideband Principles and
    Circuits" volume by Sabin, Bruene, and Schoenike, it will probably go
    into more detail. Not sure what else to recommend as far as books you
    can actually find outside of a well-stocked university library.

    -- jm
     
  9. Ban

    Ban Guest

    Use spice to evaluate the components. Here is a 7th order Tschebycheff(0.2dB
    ripple) at 32.5MHz which will be 30dBdown at 50MHz. But beware, the
    overshoot is high with a pulsed input and unsuitable for Video or
    Oscilloscope applications.

    ___ ___ ___ ___ ___
    +-|___|--UUU-+--UUU-+--UUU-+--UUU-----o
    | 50R 0.33u | 0.56u| 0.47u| 68n |
    / \ --- --- --- .-.
    ( ~ ) --- --- --- | |50R
    \_/ |150p |150p |68p | |
    | | | | '-'
    | | | | |
    === === === === ===
    GND GND GND GND GND
    (created by AACircuit v1.28 beta 10/06/04 www.tech-chat.de)
     
  10. Reg Edwards

    Reg Edwards Guest

    For design of simple filters and frequency response, download simple
    programs -

    LPF_HPF

    and

    BANDPASS1

    from website below.
    ----
    ............................................................
    Regards from Reg, G4FGQ
    For Free Radio Design Software go to
    http://www.btinternet.com/~g4fgq.regp
    ............................................................
     
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