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One important factor is your choice of L:C ratio. How much current do you want flowing in your oscillating loop?
High A goes with high C, low L.
Low A goes with low C, high L.
A low value of emitter resistor has the effect of increasing gain. This is needed if your inductor has high DC resistance.
In a Colpitt's oscillator where two capacitors and one inductor determine the frequency of oscillation. Since Colpitt’s oscillator is a high frequency oscillator, its inductor value will be in µH and capacitor in pF. The circuit oscillates when the components are suitably selected to satisfy the Barkhausen criteria (βA=+1(feedback factor must be unity)) "A" is the gain of the amplifying element and "β" is the transfer function of the feedback path, so βA is loop gain around the feedback loop of the circuit. Also there must be a positive feedback. i.e., the phase shift around the loop is zero or an integer multiple of 2π.
An op amp can automatically detect the resonant frequency of a series LC, and drive it as an oscillator. It's not the same as a Colpitts, but I have tested hardware versions of this at low power.
If your inductor is a small Henry value then it tends to draw higher current. Add a half-bridge if the op amp cannot provide sufficient current.
It is true that the formula yields 11.25 kHz, and it theoretically should not be affected by series resistance. However with a real circuit I have seen the frequency slow slightly, as series resistance is increased. My simulation 'sort of' duplicates the same frequency drop.
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I only made the resistance 1k because that makes a reasonable load for a 741 op amp (5-10 mA). Better results can be gotten with a power op amp, or half-bridge, or full H-bridge.
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Or this long-tail pair arrangement:
Original circuit is one of several oscillators found at page 97 of this article:
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