I see, you want to get an explicit expression of the phase-frequency relation of your filter.
I doubt that you can derive this expression directly from the transfer function of a filter.

Look, since s = iw + d, where w is frequency, i is imaginary unity, d is phase factor. The phase factor is just a parameter which were set as initial condition. It doesn't depend on the frequency again and is constant for all frequencies.

To derive such a phase-frequency relation, one needs a formula which relates the phase of the "incoming" frequency to the phase of the "outgoing" frequency. So you could think of a system where a harmonic oscillator with frequency w1 and phase d1 is coupled to another harmonic oscillator with freq. w2 and phase d2. The exact value of the coupling strength of these oscillators and the value of the inertia of each oscillator would specify, which kind of filter this system should describe.
An appropriate model for such a system is the driven harmonic oscillator with damping:

http://upload.wikimedia.org/math/a/1/0/a1030f6cb947558b4fd472723ad2b059.png
Also see:
http://en.wikipedia.org/wiki/Harmonic_oscillator#Driven_harmonic_oscillators

It's possible to derive the phase shift of the oscillation with w1 relative to the driving force with w2:

http://upload.wikimedia.org/math/d/2/0/d20d6b00980a90b649899fb6b402e04e.png
Also see the paragraph in wikipedia above.

Which would represent the wanted expression in the most general form.

The next step would be to iterate the parameter zeta, m1 and m2 in the differential equation in that way, that the amplitude-frequency-responce of the DEQ fits the transfer function of your filter of interest.

Here you go,
Ollie

PS: This was my last replay to this thread.

--- In Doepfer_a100@yahoogroups.com , "Tim" <timothy@...> wrote:
>
> OK, so, now before anyone goes sticking this on a T-shirt, believing it to be the answer to life, the universe and everything, I think it only fair to warn you that this:
>
> > > I'm guessing it is just some funny facet of the algebra concerned. The (normalized) transfer function for 'stage n', n=1 to 8 (so 30dB is n=5), appears to be
> > >
> > > H(s)=1/((s+1)^n+k*(s+1)^(2n-8))
>
> is complete bollocks (i.e. it is _wrong_)!! I finished that 'other thing' I was working on (a new webpage), and so couldn't help trying the short-cut route of banging this expression into Mathematica to see how the phase from it looked, and in the process discovered the most elementary of schoolboy mistakes in my working (bad, bad boy!). So I now think it is:
>
> H(s)=(s+1)^(8-n)/((s+1)^8+k)
>
> which at least gives the correct expression for n=8 (the 48dB output). The phase plots look similar to the simulation output, but I didn't get any insight as to what might be 'special' about n=5 (maybe some property of the odd 16th roots of unity or something - make a nice little homework for some budding student!).