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

1. ### Tom BruhnsGuest

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
model).

Cheers,
Tom

2. ### John LarkinGuest

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.

John

3. ### Kevin AylwardGuest

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
though.
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
filter?

(There
This don't help much.

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.

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

4. ### Peter O. BrackettGuest

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. ### Peter O. BrackettGuest

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 SlomanGuest

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. ### Peter O. BrackettGuest

Tom:

There is one "other" way to accomplish what you want but
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
designs.

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
is
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
equal,
i.e. in a pre-distorted design, then changing an element value can
potentially
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
pre-distortion,
since then it is only the turns ratio that can affect the loss at reflection
zeros.

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

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

8. ### Russell ShawGuest

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.
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 BruhnsGuest

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

....
The poles and zeros are, of course, at the roots of the denominator
and numerator of the transfer function of the network...no 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.

Cheers,
Tom

10. ### Tom BruhnsGuest

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
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.

Cheers,
Tom

....

11. ### Tom BruhnsGuest

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.

Cheers,
Tom

12. ### Tom BruhnsGuest

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.

Cheers,
Tom

13. ### Kevin AylwardGuest

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
zeros.

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

http://www.anasoft.co.uk
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 McCammonGuest

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 EdwardsGuest

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
done.

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

16. ### maxfooGuest

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

17. ### Peter O. BrackettGuest

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)
reflection
zero choices, each of which results in a somewhat different set of element
values.
You will have 2^(N-1) different sets of network element values to choose
from!

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
called
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
termination
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
extrapolate
or interpolate from that guessed d <-> termination ratio to "hit" the value
of
termination ratio that you want. i.e. I usually need one or two synthesis
passes
to hit my desired ratio.

I do this myself using my own extensive library of LC ladder approximation
and
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
designs.

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.
BTW...
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
design.
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
tablulation,
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
designs.

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

Regards,

18. ### The PhantomGuest

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

Including, of course, the new source and load impedances.

19. ### John LarkinGuest

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.

John