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Well James. It looks like to me you just said you need energy to run the turbocharger in the form of burning fuel. So, the turbine side of the turbocharger you admit needs power. I understand we burn fuel in our engines, pretty basic stuff. But when it leaves the engine on a normally aspirated engine, it just goes out the exhaust pipe fairly freely. Lets for grins say that it is at 1psi at the exhaust manifold. If there is a restriction in the exhaust system like say...A Turbocharger....then the pressure is going to rise in the exhaust manifold to say 3psi. This pressure and restriction WILL place a load on the engine, requiring more power from the engine in order to operate. That is where the power is coming from, our engines are big air pumps, and if you restrict the flow on the outlet, it needs more energy to overcome that.
Something you have not thought to mention, is that the air being compressed raises the intake and cylinder pressure about as much as there is a restriction on the exhaust, therefore, it is kind of close to a zero net sum situation, so it really doesn't take much extra energy to run.
I will say this again.....Pay Attention. The Turbocharger makes the overall engine more efficient. It does not rob power and slow it down, it's benefits far outweigh its gains. If it didn't than nobody would ever put them on an engine.
My only point is, it does take some power from the engine in order to run. It is doing so by requiring more force on the piston during its exhaust stroke than it would require if there were no turbocharger in the line.
You do a thought experiment and tell me what the engine would run like if the compressor side of the turbocharger just dumped the air into free space instead of the intake manifold. The engine would deliver less power than if it were not installed due to the exhaust restriction.
This really is just a nuance I wanted to point out, it is a misconception that turbochargers run for free, as if they don't require energy to operate. They do, you admitted yourself that you need to use "a stream of hot gas generated by burning some sort of fuel" which makes my point exactly, turbochargers need energy to run.
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