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

[Donald Tillman]

> On Dec 18, 2015, at 4:32 PM, Tom Wiltshire <tom at electricdruid.net> wrote:
> 
> 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.

I see a problem brewing...

> 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.

Yeah, that...

The definition of a Butterworth Filter is not that the product of the poles is 0.707, but rather that poles are located on a left-half circle, with a diameter of omega, with the pole locations equally distributed as if there were phantom poles on the right-half circle.  (There are diagrams that describe this better than my words.)  And complex numbers require poles pairs with symmetric +/- Y positions.

For even order cases, the product of the poles is .707, but that's a side effect, not a cause.

So you can't build a Butterworth Filter with identical stages.

  -- Don

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