# Calculating Stability of Phaser Feedback Loop

I have been doing some experiments with phaser programs. 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:

$$y[n] = A \cdot ( x[n] + y[n-1] ) - x[n-1]$$

where:

$$A = \frac{1 - \pi \cdot \frac{cutoff}{SampleRate}}{1 + \pi \cdot \frac{cutoff}{SampleRate}}$$

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?

So it's $|z|<1$, x and y don't really matter.