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According to the solution, it says it's because the phase shift is wrong. The gain network is non-inverting, so I assume that means that the LC feedback network was not 0 degrees for that particular frequency.
So I tried to work it out by hand but I am stuck here. Is my knowledge that the LC feedback not being 0 degrees for f = 1/root(2LC) the reason why it doesn't work? If so, what is the shift exactly, and how can I calculate the phase shift from the equations below
Consider the Wien bridge oscillator circuit: -
It is very similar to the oscillator in the question except that the lower capacitor is is an inductor in the question. At this point alarm bells should possibly start ringing to inform you that replacing the cap with an inductor will not work.
However, lets look at the frequency and phase response of the RC Wien bridge network. This is the filter section of the Wien bridge oscillator circuit (C2 is replaced by L in the question): -
Here's the phase response: -
As can be seen, at resonance, the phase shift is zero as one expects - this produces oscillation at Fr. With C2 changing to an inductor, the phase shift will be 90 degrees and will therefore not oscillate. This last point presumes you understand what happens when L and C are combined at resonance. If not then look on this page for a simulation.
Because your handwriting is not easy to read I didn`t follow your calculation in detail - however, the denominator of the transfer function for the RLC feedback circuit cannot be real for finite frequencies (the imaginary part cannot be zero for s=jw). However, this is required to have a zero phase at a finite frequency - as a precondition for producing oscillations. Hence, the circuit cannot work as a n oscillator. (The imag. part of the transfer function always must contain a difference in order to become zero for a finite frequency).
EDIT 1: The transfer functionof the RLC feedback path is H(s)=N(s)/D(s) with
N(s)=LCs^2 and D(s)=1+s(L/R+RC)+2LCs^2
EDIT 2: OK - I have tried to follow your calculation. It is correct, and - as expected - it is the transfer function of a second-order highpass (phase starts at +180 deg and approaches zero for infinite frequencies).
