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

[Magnus Danielson]

Tom,

On 12/19/2015 11:26 AM, Tom Wiltshire wrote:
>
> On 19 Dec 2015, at 02:14, Magnus Danielson <magnus at rubidium.dyndns.org> wrote:
>>>
>>> The circuit, identical-stages filter above, typical 4-pole butter worth below:
>>>
>>> http://www.tomwiltshire.co.uk/Butterworth.png
>>>
>>> And the responses:
>>>
>>> http://www.tomwiltshire.co.uk/ButterworthResponse.png
>>>
>>> Now, my question is "What am I missing?" or "Why is my simulation lying to me again?" since I find it difficult to believe that people have been building unnecessarily complicated filters since 1930 without spotting that they could make life much simpler.
>>> Is it that the identical-stages version would be very sensitive, or does it have some other flaw?
>>
>> Say hello to your Linkwitz Riley filter:
>> https://en.wikipedia.org/wiki/Linkwitz%E2%80%93Riley_filter
>>
>> Don is correct in his description of the Butterworth filter:
>> https://en.wikipedia.org/wiki/Butterworth_filter
>
> 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? I'll have to have a look more zoomed in and see if I can't see the difference.
>
> Thanks Magnus

The Linkwitz-Riley is designed to have a cross-over at -6 dB, because 
when you have a low-pass and high-pass with the same signal, you want 
them to add perfectly at the cross-over rather than getting a "hump" in 
the response. What you need to do is to "de-tune" the filters so that 
you shift the individual filters -3 dB point away from your intended -3 
dB point, in fact you want your -3 dB point be the individual filters 
-1.5 dB point, which sums to -3 dB, so a slight frequency multiplication 
constant and you have it. To lazy to calculate it for you, but it is doable.

Cheers,
Magnus