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Old 15th April 2011, 12:11 AM   #1
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Arrow Cauer / Elliptic Filter design program

Recently I have been developing some PC boards for analog filters used in loudspeaker crossovers. Two types of filter functions that the PCB can implement are the "High Pass + Notch" and "Low Pass + Notch" filters. These are typically used in Cauer / Elliptic filters, and now that I have the capability of making them, I wanted to nail down the specifics of Elliptic filter design. This has turned out to be a bit of a steep hill and I have learned some new things, one of which I wanted to share in this post - a good place to turn for getting poles and zeros (or corner frequencies, Qs and zero frequencies) for an Elliptic filter having specifications of my choosing.

I found two very good references for this information:
1) a book of Elliptic Filter Tables, very well presented:
"Handbook of tables for elliptic-function filters" By Kendall L. Su (link is to the google book limited version)
2) a DOS program written by Hercules G. Dimopoulos, Department of Electronics Engineering, Technological Education Institute of Piraeus, Greece
Program for Optimal Elliptic (Cauer) Filter Design

(1) is a very nice Elliptic filter reference book but it is very expensive!
(2) seems to provide all the information that you can manually look up in (1), plus it will optimize the design for a given filter order for passband, stopband, or transition band properties. The program is available at the web page link given above, and there is a nice overview of it shown there. Although it's a DOS program, it gives all the necessary information (normalized to 1 radian/s) for implementing a low-pass elliptic filter using analog building blocks. A high-pass filter just flips the frequencies about w=1. This should make designing and building elliptic filters and crossovers a snap!

I thought I would share this nice find here, since others might be interested. I know that there were a few threads here and elsewhere back in about 2006 on what were then called "elliptic" filters, but which were really some filter function (e.g. Butterworth or Linkwitz-Riley) plus a couple of zeros (notches) thrown in for fun, to increase attenuation in the stop band. That's not really an elliptic filter, but it's in the same spirit I guess. I have not seen much discussion on this topic in more recent times. I'd like to hear about people's experiences with this kind of filter for loudspeaker crossovers. If that's you please post a follow up. Thanks.

-Charlie
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Old 15th April 2011, 03:20 AM   #2
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Here's an example of an Elliptic filter that I just I worked up in an Excel spreadsheet loudspeaker/crossover design spreadsheet that I am developing. I obtained all the info - the frequencies of the corner frequency and Q for each second order stage, and the first order stage corner frequency - from the DOS program I mention above.

This is a 5th order filter with a 1kHz corner frequency and a 1 dB passband ripple. It produces a stop band depth of 65dB above 2k Hz. I like the fact that the slope is just about 60dB/octave between 1k Hz and 2k Hz with little or no "corner". Of course this will come at the price of a peak in group delay at 1k Hz and some ringing in the transient response. I'm hoping to find out whether this is really all that audible or not, when I can actually get around to building and testing something like this with the "right" project.


Click the image to open in full size.

-Charlie
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Old 15th April 2011, 05:37 AM   #3
infinia is offline infinia  United States
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Hi
I think most of these filters assume an equal source and load impedance to realize the lumped element values. For passive filters in between amplifiers to speakers the source impedance is extremely low compared to the speaker, so the transfer function response is different than theory? next load the values in Spice to check it out.
FWIW 1 dB passband ripple is higher than I would choose for 5th order,

The best filter synthesis book I've used, but also very expensive.
Handbook of Filter Synthesis Anatol I. Zverev copyright 1967 see table of contents chapter 4 deals with elliptic functions
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Old 15th April 2011, 06:28 AM   #4
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Quote:
Originally Posted by infinia View Post
Hi
I think most of these filters assume an equal source and load impedance to realize the lumped element values. For passive filters in between amplifiers to speakers the source impedance is extremely low compared to the speaker, so the transfer function response is different than theory? next load the values in Spice to check it out.
FWIW 1 dB passband ripple is higher than I would choose for 5th order,

The best filter synthesis book I've used, but also very expensive.
Handbook of Filter Synthesis Anatol I. Zverev copyright 1967 see table of contents chapter 4 deals with elliptic functions
Zverev is pretty good, and I have a copy. I should have mentioned it. But it is not as useful for filters as Su's book, because Su lists the information in a much easier to comprehend format, IMHO. I'm only using active circuits, thus source and load impedance really are not important. Likewise, values of L and C in tables such as in Zverev aren't useful for active realizations (at least the ones that I am employing), although I suppose you could use GICs for a direct passive-to-active conversion, but then you might have the termination impedance to worry about, like you mentioned. OTOH, the information in Su's book is very easy to directly convert in to an active filter - I'm using cascade realizations to achieve the overall transfer function of the filter, and Su's book nicely spells out the parameters I need to design each stage of the filter using this approach.

I'm not sure why 1dB of passband ripple not acceptable - this is +/- 0.5 dB, which is flatter than many "acoustically flat" speaker designs are in the first place. Maybe you can comment on that some more? The passband ripple can easily be reduced using a different filter specification that has less ripple.

-Charlie

Last edited by CharlieLaub; 15th April 2011 at 06:34 AM.
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Old 15th April 2011, 07:16 AM   #5
infinia is offline infinia  United States
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uh ok active filters, forgot this isn't the loudspeaker forum. haha
ripple errors are additive why not choose less ripple in the passband on the parts you can control?

Does Su just gives table of coefficients, and using an active filter cookbook you come up with and op amp topology / bi-quads? maybe you can post an example using your design above? Don't you need to combine this with a high pass filter to sum correctly acoustically? have you thought of using a line level non-active or minimum inductor design for ultimate performance ie low noise etc?
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Old 15th April 2011, 03:13 PM   #6
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Quote:
Originally Posted by infinia View Post
uh ok active filters, forgot this isn't the loudspeaker forum. haha
ripple errors are additive why not choose less ripple in the passband on the parts you can control?

Does Su just gives table of coefficients, and using an active filter cookbook you come up with and op amp topology / bi-quads? maybe you can post an example using your design above? Don't you need to combine this with a high pass filter to sum correctly acoustically? have you thought of using a line level non-active or minimum inductor design for ultimate performance ie low noise etc?
In my first post, first sentence, I mentioned that this is intended for active crossovers. The idea is to use these filters to create crossovers (two complementary filters working together), using active line level circuits, so I am posting here, where it seems more appropriate, and not in the loudspeaker forum, although it could fit there too. I'm hoping that some of those people actually look in this forum from time to time...

If you want "less ripple" then you can always use a maximally flat filter type. Allowing ripple to increase so that you gain in other areas of the crossover is part of the Elliptic filter concept, so decreasing ripple towards zero is really not the right strategy. The idea is to make the parameters only as tight as you really need to in terms of passband ripple, stopband depth, etc. Anyway, additive ripple error should only be a problem if you have more than one or two filters acting on any frequency band. Since at most two filters would be doing this, to create a bandpass function for a driver's passband (HP filter...driver passband...LP filter) this really is not an issue IMHO.

While you could create an Elliptic filter using all passive components, I'm not a big fan of inductors, so I am not planning on using them in any filter that I build, which kind of excludes passive types. I can get sufficiently low noise and distortion from modern op amps. I only need one amplifier per biquad to implement the filter stages, so I don't have to go overboard. I don't use a "cook book" per se, but the design is pretty straightforward to do, and to analyze for accuracy due to component values/tolerances. I do a Monte Carlo analysis of the circuit design first, to see if it will be able to produce the function I need. I talked about this in another thread in this forum if you want more details on that, or the circuits.

-Charlie
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Old 15th April 2011, 05:16 PM   #7
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I recently used a passive implementation of an eliptic filter as a LP for a woofer. For the application it was quite useful and successful. There are issues of course, not the least of which is expense and the DCR of the inductor(s). There was no attempt to HP the rest of the system the same way, fyi. The idea was merely to get the LF stuff OUT as fast as possible "at all costs". Which it did very nicely.

Anyhow I recently began to think about an active implementation. A quick look around, just a week or so ago did not turn up much practical information or circuit examples. The one example that comes to mind was literally a standard filter (bessel? don't recall) with a series of notch filters following. The resulting response was as desired, but it set me scratching my head a little about this way of doing things.

I do not have any of the reference texts mentioned so far on my shelf, and what I found so far on the web is a bit thin... a nice software design too would be nice.

For the passive version I built, I merely did a by hand iterative approach in PSPICE until I found a series of values around which tweaking seemed to result in a stable filter with approximately the desired characteristics. Followed of course with a real world implementation and again an iterative process of tweaking values and testing for best results. The results were quite good... especially since minimum ripple in the passband was not a major factor in the filter design... even so I didn't measure major deviations from the no filter passband freq response...

Anyhow, in my design philosophy I don't mind solid state or op-amp filters for LP functions that are low enough in frequency (lower mid bass and down) so that their effect upon the "main portion" of the sonic presentation is minimized. Otoh, I think that it is VERY VERY difficult to get opamps in particular (super low distortion ones or not) to be truly neutral. Therefore I try to avoid them above the lower mid bass, if I use them at all.

For example in a very simple 4th order LP (something between a butterworth and cheby) it is/was very very easy to hear tonal changes in a subwoofer running with the xover set to -3dB @ @70Hz merely by substituting different (high quality) opamps into the circuit. Changing even one, altered the audible timbre without any doubt.

Anyhow, despite these issues and even if others do not have the same experiences and philosophies I am quite interested in the active implementation of the eliptic (Cauer) filter, and what people come up with for topology...

Speaking of which dropping a GIC into the midst of the active filter seems - at least on the surface - to be an interesting approach.

Oh, let me add that I am a fan of using passive filter elements inbetween active stages over filter elements that are feedback around an opamp... but that's my personal preference, ymmv.

Regards,

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Last edited by bear; 15th April 2011 at 05:22 PM.
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Old 15th April 2011, 10:20 PM   #8
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Here are some screen shots of an example crossover put together based on 5th order Elliptic HP and LP filters. I reduced the Qs of the high Q stages in order to get a better match in the crossover region.

The first plot shows the notches in the HP and LP filters. The lines show the response of the tweeter + HP filter, the woofer + LP filter, and the combined response. The HP looks like it continues to roll off as frequency decreases, but this is not the case. The stopband attenuation flattens out for Elliptic filters, in contrast to all pole filters that one is used to seeing. The additional "roll off" is caused by the tweeter's response falling at low frequencies, contributing an extra 12dB/octave attenuation there. For the woofer, the stop band attenuation should be better, but because the breakup region contains peaks, these cause the stop band attenuation to degrade around 7k-9k Hz.

To generate these curves, I started with some FRD files that I found posted on the web for these drivers.

Click the image to open in full size.


This plot shows a nice flat response, about +/- 1dB 200Hz-10k Hz (on-axis). I'm not sure what the off axis response looks like, but because of the high slope the interference between drivers should be minimal.
Click the image to open in full size.

-Charlie
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Old 16th April 2011, 08:15 AM   #9
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The thing to watch with these filters is the phase tracking in the crossover region which Charlie appears to have achieved with the last post
See my blog for one approach to an elliptic filter

Consort3′s Blog

Also for students of filter design this series of articles is useful

EETimes Europe - Filter Wizard
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Old 16th April 2011, 08:35 AM   #10
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Btw in the same co-operative spirit of disseminating information I think you'll find Neville THIELE (of bass-reflex fame) has published a paper on Cauer high level passive cross-over designs in the JAES.

It would be many moons ago now. I can go through the "archives" here and get a specific reference if anyone is keen to follow it up.

Cheers,
Jonathan
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