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

[Magnus Danielson]

Hi Tom,

In addition to what Don has already said

On 12/19/2015 01:32 AM, Tom Wiltshire wrote:
> Hi All,
>
> I've been designing 4-pole Butterworth filters using 2-pole Sallen-Key sections.
>
> The usual way to do this is to set the Q of the first stage to 0.541, and the Q of the second to 1.307. Multiplying one by the other gives an overall Q of 0.707, which is our Butterworth response.
>
> In the interests of simpler circuits, I wondered if it would be possible to design a 4-pole filter with two identical sections (less individual component values). Such a circuit would need each stage to have a Q of sqrt(0.707) = 0.841. So I tried it in LTspice.
>
> 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

Cheers,
Magnus