the Hartley oscillator

The Hartley oscillator is the mirror-image twin of the Colpitts: it makes radio-frequency sine waves from an LC tank, but it taps its feedback from a divided inductor instead of divided capacitors. Picture the resonant tank as a swing again. Both circuits keep the swing going by feeding energy back in step; they differ only in where on the swing they grab hold. Colpitts grabs at the junction of two capacitors; Hartley grabs at a tap part-way along the coil (or between two inductors in series).

Mechanically, the tank is one capacitor across an inductor that has a tap, splitting it into L1 and L2. The amplifier drives one end and feeds back from the tap, so the inductive divider sets the fraction returned, with the right phase to satisfy Barkhausen. The oscillation frequency is f equals 1 over (2 times pi times the square root of (L_total times C)), where L_total is the whole coil L1 plus L2 (plus their mutual coupling if they share a core). For instance a 10 microhenry total inductance with a 100 picofarad capacitor rings near 5 MHz.

Historically the Hartley was beloved in tube radios because a single tapped coil was easy to wind and made tuning across a band simple, by ganging a variable capacitor. Today the Colpitts is generally preferred at high frequency because capacitors are cheaper and more stable than tapped inductors, and stray capacitance is easier to absorb than stray inductance. The honest comparison: Hartley and Colpitts obey the same physics and the same Barkhausen condition; the choice is practical, about whether a divided inductor or a divided capacitor is easier and more stable in your particular band.