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Once the turbo is on, more air can participate to the combustion, which also requires more fuel. Then, it's clear that the turbo can give more power (at the price of more consumption).
Question 1: Can this be used as well to improves economy ? (i.e. less fuel for same RPM than without turbo)
Question 2: How does the gain behave with RPM, passed the initial threshold ? Does it make the total efficiency curve of the engine more flat, or less flat ?
The key thing here is that a turbo extracts energy from the exhaust which would otherwise be lost to the environment, this means that it has the potential to improve the thermal efficiency of an engine.
At its simplest a turbocharger uses a paired turbine and compressor to pressurise the air inducted into the cylinders. Normally aspirated engines just use atmospheric pressure to 'fill' the cylinder which costs a certain amount of engine power as the descending piston needs to 'suck' air through the inlet port via ducting and filters, not a lot of power but a bit.
In addition having a higher pressure air charge in the cylinder means a greater mass of oxygen is available so a greater mass of fuel can be burned or the same mass of fuel can be burned more completely so you can get more energy per cycle or extract more energy from the fuel. This is especially the case at higher RPM where the ability to get enough air into the combustion chamber in the time available becomes a limiting factor.
This also means that you can get more power out of a smaller capacity engine block for relatively small overheads in terms of extra ancillary equipment especially for moderate boost pressures.
It is also worth noting that turbos can have a significant role to play in hybrid power trains, particularly as they can substantially increase the power to weight ratio of a given capacity engine at relatively constant RPM. It is also possible to use a turbo for electrical power input/output when connected to motor/generator here the turbo can output surplus power to the electrical storage system and similarity electrical power can be imputed to keep the turbine spinning in its optimum range to reduce or eliminate turbo lag.
a real world example of this is current generation F1 engines which are now getting close to 1000bhp peak output with a thermal efficiency of around 50% (better than any IC road car by some margin) with a capacity of 1.6 litres (from which typical consumer engines generate about 100bhp) albeit at the expense of a service life measured in tens of hours.
The turbo increases the amount of air that is introduced to the engine. Mass airflow sensors see this increase in air and signal the fuel injectors to add proportionally more fuel.
This gives the net result of having effectively larger cylinders - the engine with the turbo can "process" the same amount of fuel per cycle as a larger engine with no turbo.
The weight of the turbo is very small relative to the weight of the engine, so this means there's an increase in the power-to-weight ratio of the engine.
This might be the answer you're after, but it's not clear because "efficiency" isn't well-defined when referring to automotive applications. There's "fuel economy" - how far you can go per unit of fuel, and there's "fuel efficiency" - how much mechanical power you get per unit of fuel.
The most fuel efficient point of an engine is generally wide-open throttle under maximum or near-maximum load (think accelerating from a stop). Fuel economy is dictated more by driving style, which includes minimizing acceleration rates, putting fuel economy at odds with fuel efficiency.
So, from that standpoint, if you were to drive a turbo-charged vehicle the same way you would drive the non-turbo counterpart, you would get the same accelerations at a lower throttle percentage. This would seem to imply better fuel economy at the cost of lower fuel efficiency.
