[sdiy] Designing 4-pole filters with identical 2-pole stages - why not?

[Donald Tillman]

> On Dec 19, 2015, at 2:26 AM, Tom Wiltshire <tom at electricdruid.net> wrote:
> 
> Ok, so what I did is a Linkwitz-Riley filter, and whilst it ought to have a roll-off of 24dB/oct like I wanted, it should also have an attenuation of -6dB at the cutoff point, -3dB worse than the standard 4-pole Butterworth. I'd have thought that'd be visible on the LTSpice frequency response graph, but if anything, the Linkwitz filter looks like it has a bit of a peak there. Any ideas what might be going on there?

A L-R filter would require two Butterworth stages, each with a Q of .707.  Your cascaded filters have a Q of 0.842, so it's not exactly a L-R.

Let's plot the math:


(I did these in a couple minutes with the Grapher program that comes with the Mac.  Highly recommended for this sort of thing.)

The green curve is a Butterworth: Q's of .54, 1.31.
The blue curve is your cascaded pair: Q's of 0.842, 0.842.
The red curve is an L-R: Q's of 0.707, 0.707.

So your fake Butterworth filter is remarkably close.  

Your simulation curves seem off.  The Butterworth curve shouldn't peak and I have no idea what the deal is with the other curve, it doesn't look remotely right.  Do your opamp models introduce any poles?

  -- Don

--
Don Tillman
Palo Alto, California
don at till.com
http://www.till.com

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