A supercharger and a turbocharger are pretty much identical. The Turbo uses the exhaust gasses to spin a compressor that forces air into the cylinders. The supercharger uses the crank (basically the mechanical turning of the engine) to spin the compressor. Both are just methods of forcing more air into the cylinders, so you can combust more fuel and get more power. Remember, it is the combusting fuel that gives you power. The turbo/super-charger just gives you the air to mix with that fuel.
Since a turbocharger uses the exhaust gases to spin the compressor, you have to wait for the exhaust air to get moving to a significant velocity before the compressor begins moving. This is what is refered to as "Turbo Lag." Smaller turbos "spool up" quicker, meaning it takes less exhaust velocity to spin the compressor. This reduces the turbo lag. Larger turbos require more exhaust velocity to get moving, therefore have a larger lag associated with them. However, they can also cram much more air into the chambers allowing you to add much more fuel. And more fuel combustion means more power.
A supercharger runs of the spinning of the engine. Because of this it takes very little engine "rpm" to get air forced into the cylinders. However, since you are adding a work device to the engine, you are also using some engine power. It is estimated that 20% of the added power a supercharger generates is spent just creating that power. Because of this, the supercharger will never be able to create the high-end power of a large turbocharger.
The nice thing about a turbo is that the energy required to run the compressor is coming directly from a stream that is being wasted. A normally aspirated (no turbo, no supercharger) just dumps that stream of exhaust out the back of the car. So, by using this energy to run a compressor, you get "free" work. Obviously, its not "free" as you introduce more back pressure in your exhaust system which means the engine now has to work a little harder to force the exhaust out. However, the back pressure introduced can be minimized and is significantly lower than the drain of a supercharger.
Both systems work incredibly well. A supercharger is much easier to install as all it really requires is a longer drive belt to reach the supercharger's drive shaft and the physical installation of the supercharger. A turbocharger has to re-route the exhaust to the turbo inlet which requires some significant piping. Plus the need for a waste-gate and blow off valve (I won't go into detail here.)
Both systems require proper fuel management as just adding more air to the chamber significantly leans the air/fuel ratio. More fuel will have to be added to each cylinder to compensate.
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