How to use basic Second order filter to implement the Linkwitz-Riley Filter

Hi,

I am trying to implement the Linkwitz-Riley Filter,

as I refer this website (https://www.rane.com/note160.html#ref)

from this section:

"Taken to its most general extremes, cascading any order Butterworth filter produces 2x that order Linkwitz-Riley. Hence, cascading two 1st-order circuits produces a 2nd-order Linkwitz-Riley (LR-2); cascading two 2nd-order Butterworth filters creates a LR-4 design; cascading two 3rd-order Butterworth filters gives a LR-6, and so on. (Starting with LR-2, every other solution requires inverting one output. That is, LR-2 and LR-6 need inverting, while LR-4 and LR-8 do not.)"

and I try the basic High or Low pass filter(second-order) to simulate the butterworth filter.

I use second-order filter and Q-factor = 0.707 (Sigmastudio will change to 0.71, but is the same result).

and the next I try to use the simulated butterworth filter to make one Linkwitz-Riley 12 (as I know it is a 2nd-order Linkwitz-Riley)

Using Crossover I take the result will like this:

If I want to take the Flat result, the Crossover block need to invert the polarity.

if not invert the polarity will like this:

As the above idea, I use the simulated butterworth filter and trying to make the crossover,

but however I invert the signal phase, I can not get the correct answer, here is how I do:

but no invert the polarity will same as the crossover tool:

Anything that I missunderstand?

Please help mine question

BR,

Alvis

sigma studio

Second-order filter

filter

Linkwitz-Riley filter