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Old 27th April 2013, 08:43 AM   #41
system7 is offline system7  United Kingdom
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Yup, you ran into the problems I was having with series crossovers. The low order (near first-order there) looks promising, but falls on its backside with treating the drivers nicely. It's one of Lynn Olson's detested "Little Girl with a Guitar" speakers, that plain distorts with complex music. Trying to equalise out the tweeter Fs and woofer resonance then leads to too much complexity.

Let's recap what a time-aligned speaker looks like:

Click the image to open in full size.

Seems better to me to get well-behaved drivers and give them second order filters leading to a higher order acoustic response. Might as well get an easy impedance too. Then it sounds good with any amp.

There is a nice theorem by Sidney Darlington that you can simplify the resistances in a filter down to one or two: Equivalent impedance transforms - Wikipedia, the free encyclopedia. No don't spend time on it, it all gets quite deep. The story of filters giant Wilhelm Cauer, who has the elliptical filter named after him, is more accessible and shocking.

Also interesting is the pattern that appears in the complex maths of filters where you realise that all the various solutions are just placing the poles and zeroes on ellipses, circles or anything else you fancy. This, for instance is the lovely hexagonal symmetry that lurks in the 3rd. Order Butterworth filter:

Click the image to open in full size.

It is, in fact, a phase or all-pass network that is quite doable with opamps. What it does is produce pure group delay. No surprise you find neat ratios like 3:1 then. It becomes a geometry problem in the end. Which is neat.
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Well, there it is! Best regards from Steve in Portsmouth, UK.

Last edited by system7; 27th April 2013 at 08:48 AM.
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Old 27th April 2013, 01:31 PM   #42
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Quote:
Originally Posted by system7 View Post
Also interesting is the pattern that appears in the complex maths of filters where you realise that all the various solutions are just placing the poles and zeroes on ellipses, circles or anything else you fancy. This, for instance is the lovely hexagonal symmetry that lurks in the 3rd. Order Butterworth filter:

Click the image to open in full size.

No surprise you find neat ratios like 3:1 then. It becomes a geometry problem in the end. Which is neat.
Very cool - NOW I see where all your geometry inferences were pointing! I had heard about elliptical filters and had no idea what they were. You have put this in perspective for me. Clearly a lot to learn, but it really helps to have this framework. - Thanks again
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Old 29th April 2013, 07:35 PM   #43
system7 is offline system7  United Kingdom
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I'm glad you enjoyed that. Not for bricklayers, eh? You either crawl round on your belly in the ditch, or fly like an eagle over the entire landscape.

I'll give you the final piece of filtering from a higher mathematical point of view. Just to look at and enjoy without frying your brain too much.

This (correct) pole/zero view of filters:

Click the image to open in full size.

Suppose you wrap that plane around a Riemann sphere, so the origin (low frequencies) and infinity (high frequencies) just go to the South and North poles respectively, and the poles and zeroes are on the equator.

Click the image to open in full size.

It's a 1:1 mapping that starts to look extremely elegant. In fact it's also related to the Mobius Transformation and Conformal mapping. With these tools, filters and room and cabinet properties become rather simple. It's worth remembering that a mechanical filter like a speaker driver and an electrical filter share the same overall mathematical properties, we are just combining them at this forum.

Interestingly, Cauer started as a specialist in General Relativity, which means he knew ALL this stuff.
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Well, there it is! Best regards from Steve in Portsmouth, UK.

Last edited by system7; 29th April 2013 at 07:45 PM.
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