Yes, the energy in the compressed air is used to spin the turbine. I know all this. I had just sort of assumed that the compression stage could be electrically driven.
When you compress air, you store some energy in it, but it also heats up, and some of that heat transfers, so you lose some of that energy. So if all you're doing is compressing air, you're getting out less than you put in when you let it expand again. In order to get more out, you need combustion. The combustion is what gives you the net energy output.
But if you're doing combustion, there's no reason to add electricity to the equation in a jet engine. Making it hybrid by adding an electric generator to the output and using that electricity to do the initial compression just adds two conversions (mechanical to electrical and electrical back to mechanical) between the output and the compression, which is going to add weight and inefficiency. There's absolutely no point to doing that rather than just powering the compression straight from the combustion output.
Now you might ask, if we don't do that for jets, why would we do that for cars? Why are hybrid car engines more efficient than straight combustion engine cars? And the answer is that car engines don't operate like turbine plane engines. A turbine plane engine (be it jet, turbofan, or turboprop) spends most of its time operating near its peak efficiency, under fairly constant load. That's the way you design it, so that normal flight conditions match peak engine efficiency.
But cars don't do that. You're operating cars under wildly different engine outputs on a regular basis, so it's not possible to keep a combustion-only car operating at peak efficiency. If you're driving in the city, you may spend a lot of time burning fuel at zero efficiency as you idle at a stop light. A hybrid car engine doesn't have the electric motor take over
part of the engine cycle, it substitutes for the entire engine cycle. It allows you to only run the combustion engine only at peak efficiency, and use the electric motor for the rest. You do lose some energy in the conversion process, but because most driving involves so much inefficient engine use that you can eliminate, there's still a net gain to be had. But if you were to run at peak engine efficiency constantly, a hybrid car would actually lose out compared to a combustion only engine.
