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It's actually more like a transformer coupled oscillator. See you can use a transformer in two ways to get your oscillator running. You can either use the phase shift of an antiphase secondary or you can use the step up capabilities of the transformer. In a common emitter configuration the transformer is used to phase shift the signal to 180dgs and the transistor shifts it to another 180dgs, totalling in a 360dgs full positive feedback. Since the emitter in a common collector topology doesn't have a phase shift (mostly) you can just use the transformer to step the output a little and feed it back to the input, with the same phase, turning a common bootstrap into a positive feedback. One of the minor disadvantages is that there are not many ways to implement negative feedbacks, especially amplitude dependent. One of the ways I've chosen is to add an emitter degeneration resistor equalling to the critical dampening value for the LC tank circuit, which should bring an almost perfect sinewave. I've further reduced the amplitude, with which you can feed the tank circuit with the antiparallel diode at the emitter. The tank circuit completely restores the output, and it's a perfect sinewave.
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