Calculating Stability of Phaser Feedback Loop

Hi, I have been doing some experiments developing a phaser program. I really like setting the phaser feedback to really high levels, such that the effect is really metallic and resonant, almost like a physical modeling instrument. The problem with this is that it’s easy to go overboard and get runaway feedback which gets harsh extremely quickly.

I am wondering if there is a way to calculate the point at which self oscillation will occur, given the number of phaser poles, the cutoff frequency of the allpass filters, and the feedback level.

The phasor I am using uses a variable number of 1 pole allpass filters in series, each defined by the equation:

y[n]= A⋅(x[n] + y[n−1]) − x[n−1]


A=(1−(π⋅cutoff / SampleRate)) / (1+(π⋅cutoff / SampleRate))

Here is a signal flow diagram:

Ultimately what I am looking for is an equation I can use to calculate the following:

Given a phaser with x poles with y cutoff, self oscillation will being at feedback level z

Can anyone recommend a good method or some resources I could use to figure this out?