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LC ladder filter questions

Discussion in 'Electronic Design' started by Tom Bruhns, Apr 7, 2004.

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  1. Tom Bruhns

    Tom Bruhns Guest

    Hi all,

    I have an LC filter, ladder topology. I wish to change the source
    resistance from which it is driven from the original finite value to a
    different finite value, but keep the poles and zeros in the same
    places by changing the values of the ladder components appropriately.
    The load resistance at the output is different from either the
    origianl or the new driving-source resistance. Is there some
    mechanical way to arrive at the required component value changes?

    Alternatively, is there a reasonably simple closed form for the poles
    and zeros of an LC ladder network? In this case, there are shunt
    capacitances and the series elements are parallel LCs. This
    particular filter implements just some of an elliptic-type filter, but
    is not itself a complete standard elliptical filter, so I can't just
    use a standard elliptical filter design algorithm to re-work things
    for a different input impedance.

    I feel like this should be common knowledge and I should have learned
    it somewhere along the line long ago, but it turns out to be very
    unfamiliar ground for me. And so far my attempts to work out a
    closed-form solution result in things too messy to deal with and have
    any confidence I haven't missed a term or factor somewhere. (There
    are four shunt arms, and four series including the input resistor and
    three LC parallel tanks, with associated Q-lowering resistors in my

  2. John Larkin

    John Larkin Guest

    You can scale the filter impedance easily, but it requires that both
    the driving impedance and the load impedance be scaled by the same
    ratio. If the impedance ratio (new/old) is K, just scale inductors by
    K and capacitors by 1/K. If you have internal resistors (unusual!)
    scale them by K.

    You could select a K to satisfy one end of the filter, and pad the
    impedance at the other end to essentially keep the ratio that the
    filters sees as K on both ends. If you must scale the generator and
    load impedances unequally, you've got to design a whole new filter.

    Yes, "messy" is an appropriate description. The best way to design an
    LC filter is to look up a normalized prototype in a book and scale.
    Williams' book is excellent; it has a bunch of ellipticals, too.

  3. For a general, arbitrary source and load impedance, no.
    Depends on what the filter type is. For a butterworth, yes. For a Bessel
    filter, no. That is, the element values of an LC butterworth are
    directly obtainable from a simply closed form solution. For Bessel, no
    such luck.
    If its not a standard filter, there is no hope, unless you an expert in
    filter design, even, probably impossible. A fundamental point of
    standard filters is that they are mathematically described, but *proven*
    to be physically realisable for any order. It takes a lot of work to
    discover filter responses that actually satisfy the laws of physics.

    A pure elliptic filter, of even order, can not, technically, have equal
    source and load termination resistances. There is a standard fudge
    Your feelings would be incorrect.
    Very doubtfull.
    Of course it will be. Filter design is very specialised. Only a few % of
    EE's have in depth filter design knowledge.
    I don't see how can you work out a closed form solution at all without
    the design basis of the filter? Where are the poles and zeros of the

    This don't help much.

    Kevin Aylward
    SuperSpice, a very affordable Mixed-Mode
    Windows Simulator with Schematic Capture,
    Waveform Display, FFT's and Filter Design.

    "quotes with no meaning, are meaningless" - Kevin Aylward.
  4. Tom:

    You didn't say, but if the filter is a band-pass or high-pass
    you can achieve all of this by means of "transformers".

    I put "transformers" in quotes because you can usually accomplish
    this just by "tapping" the existing coils.

    Even if the filter itself is low pass but does not have to pass
    frequencies all the way down to DC you may be abl
  5. Tom:

    You didn't say, but if your LC filter is a band-pass or high-pass
    design you can achieve what you need by means of "transformers".

    I put "transformers" in quotes because you can usually accomplish
    this just by "tapping" the existing coils.

    Even if the filter itself is low pass but does not have to pass
    frequencies all the way down to DC you may still be able
    to use transformer techniques.
  6. Bill Sloman

    Bill Sloman Guest

    I think he means "Electronic Filter Design Handbook" by Arthur B.
    Williams and Fred J. Taylor, ISBN: 0070704414 (third edition 1995). It
    is out of print, but the H.P. library ought to have a copy. The book
    got recommended here pretty regularly.
  7. Tom:

    There is one "other" way to accomplish what you want but
    your results will depend upon how different your new source and load
    resistances are from the originals.

    This is accomplished using a technique called "predistorted" design.

    Usually one has to do the pre-distortion using computer algorithms, but...

    Weinberg [see below] actually has tabulated a few predistorted

    You will find that you have "extra" degrees of freedom with pre-distorted
    designs since *you* get to choose the half-plane in which you want each
    of the reflection coefficient zeros, or return loss poles, to lie. Choice
    always nice!

    One down side of a predistorted design is that you must then
    ccept the attendant insertion loss due to the miss-match and
    the increase in element value sensitivities. The more you "offset" the
    terminations from their nominal values the more the element value
    sensitivities increase.

    A properly terminated "matched" lossless LC filter designed
    by insertion loss methods has zero insertion loss in the passband
    and so is at the absloute minimum element value sensitivity. Nice
    property, eh?

    [Changing any element of a matched design can only cause the insertion l
    oss to increase it cannot decrease, hence a matched lossless design is at
    minimum sensitivity by definition. What a wonderful property! All design
    techniques should have this property. But if the terminations are not
    i.e. in a pre-distorted design, then changing an element value can
    cause insertion loss to decrease hence a pre-distorted design may not be at
    minimum sensitivity.]

    You don't get this sensitivity increasing effect if you do the termination
    impedance leveling with transformers or coil tapping rather than
    since then it is only the turns ratio that can affect the loss at reflection

    See tables of pre-distorted LC ladder designs in Chapter 13, pp. 637 - ...

    Louis Weinberg, "Network Analysis and Synthesis", McGraw-Hill,
    New York, 1962.
  8. Russell Shaw

    Russell Shaw Guest

    No simple/easy way i know of.
    A closed equation for the transfer function is had just by deriving
    the T/F using circuit equations of the Ls, Cs, and terminations.
    It will be a rational polynomial with only LHP poles and some
    imaginary zeros.
    With your partial circuit and known load impedance, derive the
    transfer function. Now scale it by a loss factor equal to the
    extra loss of connecting unequal Rs/RL, in comparison to that
    of maximum power (equal Rs/RL). Now use the scaled transfer
    function to get the reflected incident voltage polynomial
    for an input line of Zo=Rs. With many long division steps,
    you can decompose the driving point polynomial into the
    sequence of Ls and Cs. At the right points, you'll need
    to synthesize the LC trap elements.
  9. Tom Bruhns

    Tom Bruhns Guest

    Good...then I probably didn't sleep through coverage of that, at

    The poles and zeros are, of course, at the roots of the denominator
    and numerator of the transfer function of the problem,
    it's just that the expression for the TF gets messy enough that I
    don't trust myself to keep track of all the terms. Mathcad may help.
    Actually, I don't need the poles and zeros, specifically, to solve the
    problem. I just need the TF in terms of the Ls, Cs and Rs in the
    model. Then I can (theoretically) just adjust the R I want to change
    and set up simultaneous equations for each of the coefficients in the
    numerator and denominator polynomials for the "before" and "after"
    component values. I've done this sort of thing before several times,
    with paper and pen, but for somewhat simpler networks. Conceptually
    simple; very messy in practice as the filter order gets large. And
    those simultaneous equations may not be linear and may require an
    iterative approach to a solution, but that shouldn't be too difficult
    in any event.

    It's even slightly easier than that: the zeros are, of course, at the
    resonances of the parallel tanks which are the series elements in the
    ladder, so if one of those Ls changes, the C which is shunt across it
    must change to keep LC constant. That does simplify the problem, and
    gets rid of three variables.

    Thanks much for the comments. It's good to know that I'm not off in
    the weeds unnecessarily. :)

  10. Tom Bruhns

    Tom Bruhns Guest

    Thanks for your comments, Peter, in this post and the other. I do
    need to maintain response to DC, so there will be no shunt inductive
    components in the ladder. After reading the posts so far, my
    inclination is to simply plow ahead with "writing out" (perhaps in
    Mathcad) the TF expression in terms of the Ls, Cs and Rs, evaluating
    the coefficients for the existing design values, changing the input
    resistance, and solving for the set of new Ls and Cs and Rs that yield
    the same coefficients (accounting for the DC gain change that results
    from changing the input resistance). The sixth-order numerator and
    seventh-order denominator are messy to deal with, but not impossible.
    I had hopes of a "quick fix."

    I'll see if I can find Weinberg and look for further enlightenment
    there. I also have Zverev in front of me, and he certainly has a lot
    of interesting things to say, but I'm finding most of them to be
    tangential to what I'm specifically trying to do.


  11. Tom Bruhns

    Tom Bruhns Guest

    Thanks, John and Bill. I'll have a look at that reference if I can
    find it easily, but in this case, "standard" designs aren't
    necessarily going to help, since I want to replicate the poles and
    zeros of an existing design, just with a different ratio of input and
    output impedances.

  12. Tom Bruhns

    Tom Bruhns Guest

    Yep. It's just messy. I was hoping the mess I can get in a
    straightforward but tedious way would reduce to some elegant form
    that's well-known to all but me. Seems to not be the case. It's
    probably time for me stop avoiding it and just dive in and do it.

  13. Nope. Filter synthesis might not have even been in the curriculum.
    Ahmm. Thats like saying the length is twice its half.

    I meant, what are poles and zeros *supposed* to be by *design*. If its
    not a standard filter, there must be some rational for choosing them.
    Oh dear... monkeys at a typewriter...

    I cant see this as practical with so many components.
    Filter synthesis was invented specifically to do this. That what's it
    dose, systematically solves the equations for a given response.
    Indeed not. Filter synthesis is quite involved.

    I'm not an expert, but I know enough to have been able to implement the
    relevant stuff, i.e. butterworth, tchebychev, and Bessel LP and HP
    facility in SS. Some of it was hard code typing of *lots* of poles and

    If I were you, I would try and stick to the well known filter types.
    However, if you can stand the insertion loss, here is a trivial
    solution. If the load is less than what you want it to be, stick a
    resister in series, if less put one in ¦¦, but do normal scaling for
    either the input of output load first.

    Kevin Aylward
    SuperSpice, a very affordable Mixed-Mode
    Windows Simulator with Schematic Capture,
    Waveform Display, FFT's and Filter Design.

    "quotes with no meaning, are meaningless" - Kevin Aylward.
  14. Roy McCammon

    Roy McCammon Guest

    I seem to recall there is an easy method if the filter is
    symmetric about the middle and the termination's are equal,
    you can use Bartlett's Bisection Theorem. If you can't
    find it, I can fax the info. A little to complex to
    describe here, but easy once you see it.
  15. Reg Edwards

    Reg Edwards Guest

    To change the terminating impedances simply change all the L/C ratios.

    i.e., to double both terminating impedances, whatever values they are,
    double the value of all L's and halve the value of all C's, and the job's

    For dissimilar terminating impedances, incorporate a basic matching
    1/2-section, or use a simple transformer, or tap down the coil at one end.

    What's the matter with just designing another filter?

    Overcomplicated books are written on filter design to provide an income from
    publishing and in the hope of making a name for one'self. But they are very
    seldom read in anger.
  16. maxfoo

    maxfoo Guest

    Do a search for the free proggy rfsim99.exe, has everything you need and then

    Remove "HeadFromButt", before replying by email.
  17. Tom:

    Well in all of the "textbook" literaturethat I know of, including Zverev
    and most others [BTW... I have most of of the best filter references in
    my personal library.] the only one that I ever found who discussed, let
    alone tabulated, pre-distorted design was/is Louis Weinberg.

    I have personally done quite a few predistored low pass LC ladder
    designs myself over the years, but using my "homebrew" Fortran
    based computer algorithms. I use Fortran 95 these days.

    Basically for a predistorted design the procedure is that after an exact
    approximation you subsequently shift the natural modes [poles] of that
    "perfect" approximation towards the right half plane, by an amount "d"
    and then synthesize a lossless LC ladder [can be low pass or whatever]
    for that shifted or so-called predistorted case. You then synthesize a
    lossless LC ladder for this shifted, or pre-distorted design.

    You will then find, by comparing a front end synthesis with a back end
    synthesis for the pre-distorted design, the new termination ratio that is
    no longer equal terminations on either end. This predistorted [Shifted
    by the translation "d" in the s-plane] design will usually only now have
    one double reflection zero on the jw axis and the remaaining reflection
    zeros thenmay be chosen arbitrarily by you before sysnthesis.

    i.e. if the filter is of order N, then the predistorted design will have N-1
    reflection zeros for which you are free to choose location between LHP
    or RHP.

    Once you make your choice [Choice is nice!] of reflection zero distribution
    then you can synthesize your network. Clearly you will have 2^(N-1)
    zero choices, each of which results in a somewhat different set of element
    You will have 2^(N-1) different sets of network element values to choose

    Next you synthesize your chosen set of reflection zeros into the LC
    target ladder network. Then when you realize the LC ladder you have to
    pad your LC elements with resistance to bring all the reflection zeros
    back onto the jw axis.

    I have found that this is best done by assigning "all" of the shift "d" to
    the inductors
    and then doing a "semi-uniform" dissipation pad. i.e. slightly lowering the
    Q of all of
    the inductors by padding them wth R's and leaving the C's alone. This is
    semi-unform dissipation.

    The end result is an LC ladder filter that has all refelection zeros on the
    axis, and the
    original target transfer function, but with different front end and back end
    resistances. I have found that you can "predict" the termination ratio by
    first doing this
    using computer algorithms for a "test" value of "d", and then finding
    the resulting termination ratio from synthesis. Then you can either
    or interpolate from that guessed d <-> termination ratio to "hit" the value
    termination ratio that you want. i.e. I usually need one or two synthesis
    to hit my desired ratio.

    I do this myself using my own extensive library of LC ladder approximation
    synthesis routines. Fortran stuff developed by myself over several decades
    of filter
    design. Fewer folks are doing this kinda stuff anymore, it's a dying art,
    to say
    the least!

    The beautiful thing about Wienberg's book is that he has tabulated quite a
    few of
    these predistorted low pass designs, and to the best of my knowledge,
    Weinberg is the *only* one who has ever tabulated these pre-distorted

    Weinberg tells you how to do all of this in his really great textbook. And
    then he
    tabulated quite a few pre-distorted low pass LC ladder filter designs.
    It's out of print now, but you can still buy used copies. IMHO... Weinberg
    is one, if not the best, practical textbook on the whole subject of filter
    If I had to own just one book on filter design it would be Weinberg!

    Otherwise... if you can't find what you want in Weinberg's pre-distortion
    you pretty well gotta do this whole process yourself, and for anything
    higher than say
    third or fourth order, you pretty well need a bunch of computer routines, or
    hire a
    consultant. I'm available... :)

    I've done predistorted designs myself using my own software for LC ladders
    up to
    around 20th order LC ladders with dissipation factors "d" very nearly equal
    to 1/Q
    of the highest Q inductors available in the frequency range I was working
    with. That's
    when I first discovered the "sensitivity problems with pre-distorted

    Unfortunately few applications today require such sophisticated LC ladder
    techniques, and... each year there are fewer of us who understand how to do
    kinda stuff.

    But for optimal and feasible filtering it's pretty hard to beat LC ladder
    filters, they
    are optimal in so many beautiful ways. Only digital filters can beat them in
    terms of
    stability, sesitivity, and economy... and digital filters are ranging
    upwards in frequency
    with each passing year...

    When will the "last" LC ladder filter be designed? I hope I'm there to
    design it!


    Happy filtering.

  18. The Phantom

    The Phantom Guest

    How about posting the schematic so we can all play with it?

    Including, of course, the new source and load impedances.
  19. John Larkin

    John Larkin Guest

    OK, we need at least one goofy idea per day:

    An active filter is easy to design, since you can just plop down the
    poles and zeroes. LC filters are nasty, because all the sections
    interact. So how about chopping up an LC filter into managable chunks,
    and isolate them with buffers? That would bridge the gap between
    active filter territory (10s of MHz maybe) and the range where the
    buffers get wimpy, 1 GHz maybe.

    You could call it a half-passive filter.

  20. Roy McCammon

    Roy McCammon Guest

    there is another easy way, get your employer to buy
    a copy of FILSYN from ALK engineering
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