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A supercharger is driven (spun) by a belt that is connected to the crankshaft via a pulley. This pulley uses the rotation of the crank to drive the belt and spin the pulley on the supercharger, which is connected to a set of gears that have "stepped up" ratios. This means that they spin the impeller a certain amount faster than the pulley itself is spinning. This is how it produces boost.
A turbocharger, on the other hand, is driven off of exhaust gasses produced by the engine. This in turn spins a turbine which is connected to an impeller by a shaft, which creates boost
but they differ on how the fan works
