My interpretation/explanation is a bit different from Brian Drummond`s description. For my opinion, we have no "resonant circuit" (L1, C2, C3) because this would not explain why we have positive feedback. The common node between C2 and C3 is grounded - and therefore, the working principle is as follows:
- The feedback path resembles a third-order lowpass which can produce a phase shift of -180deg at the common node between L2 and C2 for the desired oscillation frequency. Together with the inverting characteristic of a common-emitter stage (between B and C) we have a total phase shift of -360 deg (positive feedback) .
- Lowpass: It is a simple task to see that the lowpass function is realized as a classical 4-element ladder circuit: ro-C3 (grounded), L2-C2 (grounded). The resistance ro is the dynamic output resistance of the transistor.
- The task of C1 is to couple the feedback signal to the base node and L1 decouples the feedback node from Vcc - otherwise we would have no feedback at all. The capacitor C5 cancels the negative feedback effect of R3 for the oscillation frequency (and, thus, allows a larger gain).
Keeping in mind that neither L1 nor C1 (should) contribute to the feedback path the lowpass function between output (collector) and the upper end of C2 can be derived. For this purpose, the transistor is assumed to be a current source I with an internal dynamic resistance ro:
H(s)=Vout/I=ro/[1 + s*(C2+C3)*ro + s²*L2*C2 + s³*C2*C3*L2*ro)
It can be shown that this lowpass has an amplitude peak with a phase shift of exactly -180deg for w=1/SQRT(L2*Cp) with Cp=C2C3/(C2+C3).
EDIT2: With respect to Brian Drummond`s last comment - I agree that two alternative methods exist to explain the working principle of the circuit - however, finally both approaches join in a common loop gain function. Let me explain:
In my detailed answer I have, from the beginning and based on a visual inspection, the feedback path treated as a 3rd-order lowpass. This can explain why such a network produces a 180deg phase shift at one single frequency.
However, another view is possible - as proposed by Brian Drummond: The collector path contains a parallel LC tank circuit. It is well-known that at the resonant frequency wo=1/SQRT(LC) the voltage across the tank circuit assumes a maximum and the CURRENT through each of the parallel branches also is at its maximum value. If we treat the transistor as anon-ideal current source (internal resistance ro) the current through the capacitive branch is
Ic=I*[sro*C2/D(s)] with denominator D(s) as given in the function H(s) above (EDIT1). This current resembles a bandpass function.
The signal which is fed back to the base node is the voltage V2 across C2. This voltage is nothing else than V2=Ic*(1/sC2) . Therefore, we arrive again at the lowpass function H(s).
Summary: The tank circuit contains a current function which has a bandpass character. Using the voltage across a capacitor as a feedback signal we divide the current function by sC2 and arrive at a lowpass function with a peak and a phase shift of -180deg at w=wo. This is the way to explain the relation between bandpass and lowpass functions (and the role of the resonant frequency wo) in the shown circuit.