I was always dissatisfied with inverters because it seems like an LC tank and transistors would be all you need. However, after much time spent simulating and a few false positives experimenting, I am beginning to think it is flat out impossible. I have no way to prove it, but it seems that you need at least 4 passive components to create AC from DC. It seems as though it is often that there are a pair of passive components for each phase of the duty cycle.
Is this indeed the case?
A LC tank circuit is just a inductor and capacitor connected to resonate at a particular frequency. These are not active components. They can not make a frequency that doesn't already exist. They can filter, sometimes with quite high Q, a mix of frequencies to pick out or reject a narrow range.
In other words, A L-C tank can be used as the governing elements that set the frequency, and also to filter out spurious frequencies to leave you largely with a sine, but the original signal with the desired frequency content can only be made with something that consumes power. This requires at least one active device somewhere.
If it were a desing criterion for some reason, it should be possible to create a circuit that oscillates and drives a L-C tank with a single active element. It will be easier with multiple active elements or multiple windings around the same core. If you are looking to have it produce significant power, I wouldn't worry about the number of active elements and think about drive level and efficiency instead.
A blocking oscillator is able to generate a 'dirty' sine wave with only one transistor, a capacitor and an inductor. However this isn't an inverter - it produces a free running signal, but that signal is tuned by the LC time constant.
It seems that one of the most simple configurations for what you're aiming at is a Royer Converter. However, with a Royer, it is not only the LC resonance that determines the circuit's behavior. The saturation characteristics of the transformer's core play also a role. In this application note by Zetex/Diodes, Fig. 3 and the text around it have a lot of information about a Royer's secrets, in a nutshell.
Related: What is the use of transformers with 3 pairs of wires?
Astable multiviberators used for induction heating convert directly from DC to AC at resonant frequency. The capacitance at the required frequency can be calculated.
