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

[Tom Wiltshire]

On 19 Dec 2015, at 01:09, Donald Tillman <don at till.com> wrote:

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

Aha! Thanks very much, Don. Your words gave me a good mental image. I was able to check it by plotting the pole positions given by this tool:

http://sim.okawa-denshi.jp/en/OPseikiLowkeisan.htm

So I learned that the first stage with Q=0.541 gives the "inside" pair of poles, close to the real axis and further from the imaginary axis, whereas the second stage with Q=1.307 gives the "outside" pair of poles further from the real axis and closer to the imaginary axis. Looked at like that, it's obvious that the stages can't be the same. Thanks.

Ok, so next question - does the order of the two sets of poles make any difference? Why do people put the higher-Q section second?

Tom