> I understand how an inductor and a capacitor connected in parallel can act as an oscillator if an instantaneous voltage is applied then removed.
Same as a mass and spring. Give it a smack, and it shakes at its preferred frequency.
But everything has friction, so the shake gets smaller and stops. Try to take power out of it, and it stops faster.
Stick an amplifier on it. Take a small part of the shake, amplify it, and put signal back into the L-C tank in the right way. Now instead of the shake getting smaller, it gets larger.
If an amplifier has positive feedback, it is pretty sure to oscillate. The L-C tank just influences the frequency of oscillation.
Hartley and Colpits are just common ways to get positive feedback, when a single tube mostly wants to do negative feedback (because it is inverting). The Harley takes output at the cathode, and steps it up in the coil. The tube is unity voltage gain, the coil has voltage gain of 2 or 3, and that's all it takes.
A more obvious way is the "tickler": the L-C drives the grid, the plate feeds a small coil wrapped around the main L. If it won't oscillate, reverse the connections to the tickler coil.
How do you start it? While Dave mentions Class C oscillators, they won't self-start without a big bang. Most oscillators start in Class A. And since they are amplifying their own output, and there is always some random noise, they start-up fine.
With an ideal amplifier, the output would get bigger and bigger to infinity. In real life, if you don't do anything else, the output rises until the amplifier can't make any more power, until its effective gain is less than the losses in the L-C tank.
For very best frequency stability, you add some kind of gain control to keep the amp in class A. That's what the H-P patent does (or pretends to do).
With tubes, for many situations it works well enough to rise through Class A to the edge of either Class B or more often Class A2. Get the grid current to flow through a resistor (47K usually works) and it will drive the grid negative, amplifier gain drops without large distortion. This trick works good for a few milliwatts of output.
For high power from a single tube, just let it rise through Class A and B into Class C, and adjust the feedback so it sits near optimum Class C conditions. When you whack the L-C tank that hard, you push it off of its preferred frequency, and it will drift with voltage and tube-swaps. If you want big RF to melt metals, the exact frequency isn't critical: I know a place that melts tons of ultra-pure (no fuel soot) metals with 200KW single-tube oscillators. And years ago it was legal-enough to stick a telegraph key under a Class-C oscillator for radio transmission. Very distinctive chirp on each dit-dah. It was even done with speech, though the resulting frequency shift must have trashed-up the audio in the receiver.
Frequency modulation is usually done with a variable reactance in the L-C tank. Today the vari-cap diode is universal. Before that, we added a second tube with a small reactance from plate to grid. The input reactance, as shown by Miller, is the small reactance times the gain. The gain of a tube can be shifted by varying its current, most easily by shifting the grid bias. That was how FM radios did AFC.
FM transmitters almost universally started with phase modulation, not frequency modulation. There were even special tubes for this, but the reactance tube would work too. A classic FM transmitter modulation system was a huge affair.
A crude frequency modulator was the wobble generator. In the L-C tank, you had a fixed (or manually adjusted) C, and a tuning capacitor modified to spin 360 degrees, and a motor to spin it. Say you had 100pFd fixed and 0pFd-5pFd variable spining 3600RPM: you could sweep the IF passband of an FM radio 60 times a second, so a very simple o-scope hookup would let you trim the passband.
The Theremin is another FM oscillator. Fixed L-C plus the variable C of your hand near a metal plate or loop.
There are RF condenser mikes where the C in the L-C tank is the capacitance of the capsule. Maybe 30pFd static, varying +/-0.01pFd on loud sounds. This controls a 10MHz oscillator, and you can use a fairly long cable (no tube inside the mike). A frequency discriminator recovers the audio.