To get more horsepower out of a car it needs to get more fuel (and air so that the fuel can burn). This is done by forcefeeding the engine by means of a turbocharger.
It works like this, Starting at the exhaust: Exhaust comes out of the exhaust manifold and flows into the turbine housing of the turbocharger, there it makes the turbine spin and then exits to the exhaust system. The turbine wheel is connected to the compressor wheel with a common shaft. The compressor wheel, which is positioned in the intake of the engine, after the airfilter but before the throttle body, spins up with the turbine wheel and as it goes faster and faster it compresses the air going into the engine thus creating an increase in the pressure in the intake of the engine. This makes more air enter the cylinder each time the intake valves open, and because of that more fuel can be added too. As the engine speeds up, so does the speed of the turbocharger, and the pressure increases. When the pressure in the intake after the turbo reaches a predetermined level some of the exhaust will start to bypass the turbine via the wastegate valve, so the turbo won't reach a selfdestructive speed, and the engine won't get too much boost for it's own good.
Because: as the speed of the turbocharger increases so does the wear on the bearings. The MAP (Manifold Absolute Pressure)sensor will sense the pressure in the intake and send a signal to the ECU that will open the bypass (wastegate) for the exhaust when a certain pressure in reached (0,9 bar for the Escort RS Cosworth). Then when the pressure decreases it will start to close again.

Before the compressed air enters the combustion chamber it will go through an intercooler. The intercooler is a very important component on a turbocharged engine because of the heat generated by the compression of the air. When a gas (here the gas is the air sucked into the engine) is compressed the temperature raises. As per the ideal gas law. This is not good because hot air at a given pressure takes up more space than cold air at the same pressure. Therefore the compressed air from the turbo goes through the intercooler which basically is a radiator, only it cools air instead of water. On most cars this is positioned at the front of the car in front of the radiator, or under an airintake in the hood, to provide good airflow and cooling. When the air goes through the intercooler it gets cooled of and takes up less space than it did before the intercooler. This will make it possible the get even more air and fuel into the cylinder each time the intake valves open. The bigger and better the intercooler is the more the air will be cooled, to a certain extent of course, as the air cannot become cooler than the air on the outside. Because of the increased volume of the IC the lag time of the turbo will increase as the intercooler size increases..

The lag time of a turbocharged engine is the time it takes from you hit the throttle to the turbo spins up and creates the desired pressure in the intake of the engine. This only apply when the engine is in the rev-range where it can achieve full boost. This time will be affected by a number of parameters in the engine: The size of the turbo (see further down), the total length (in reality the volume) of the airhoses from the turbo into the engine, the diameter of the air hoses, the size of the intercooler, the size and shape of the exhaust manifold and the backpressure of the exhaust system.
The size of the turbo is vital because a larger turbo has larger inertia and will take longer to spin up, it also will take more exhaust gases to make it spin and create pressure. The length of the airhoses are a compromise between short length and having the turbo and intercooler in desirable places. More plumbing will of course add to the lag time as the compressed air will take longer to reach the engine and all the air in the hoses has to be compressed. The size of the intercooler is also a trade-off between having the air cooled very effectively and minimizing lag. The lag time will increase because it will take a bigger amount of air to create the same pressure in a bigger volume. The backpressure of the exhaust system will determine how fast the "spent" exhaust can exit from the turbine part of the turbo. So a low backpressure exhaust system is desirable. But of course lower backpressure means more noise and less cleaning of the exhaust gases... Rally cars usually have 3" exhaust systems straight through with only one rear silencer.

But: As there are conflicts between getting as little lag as possible and as much power as possible there are another way of getting less lag, and that is ASL ;Anti Lag System or bang-bang system as it also is called. This will make sure that that there are a minimum of lag (or no lag at all)at all times. It works by injecting large amounts of fuel into the combustion chamber whenever the driver lifts his foot of the throttle, i.e. when changing gears, when braking before a turn and at any other time the throttle is "off". The fuel injected into the engine will not burn as there are no ignition (no sparks from the sparkplugs) when the car is coasting like when braking or making a gearshift. The fuel/air mix will just leave the combustion chamber unburnt, but when it comes into contact with the exhaust manifold which is very hot (glowing red possibly) it will ignite and create a large volume of gas that will enter the turbo and create very high levels of boost when the car is off-throttle. Then when the driver decides to hit the throttle again he won't have to wait for the turbo to spin up, it is already at full boost so the engine will create maximum power (for the given revs it's at)instantly.
The not-so-great part of a bang-bang system is that it will destroy the exhaust system, turbocharger and exhaust manifold of a streetcar in a very short time. The reason for this is that the engine and exhaust system is designed for the ignition taking place inside the engine in the combustion chamber and not in the exhaust manifold. It is simply not strong enough to take those explosions for a sustained amount of time. Rallycars use it though, but those engines are built for the abuse (but still they seem to give in fairly often).

The size of the turbocharger is of vital importance when it comes to the characteristics of the engine it is turbocharging. The smaller the turbocharger the faster the boost will come, and it will also come at lower revs as a small turbine will need a smaller volume of air (exhaust) to get up to speed than a large turbine. But the "problem" with a small turbocharger (or small compressor wheel and housing more correctly) is that it will not create very much boost when the engine is reving high (above 4000rpm or so). This is because the engine with it's vacuum will suck air out of the intake almost as fast as the compressor will push it in and almost no boost pressure will build up. While a bigger compressor will push so much air into the intake manifold that there will still be boost available at high revs. The problem with a bigger compressor is that it will need a larger turbine on the other side of the common shaft to power it and that will increase lag. A good example of this is the turbocharger in the Ford Sierra RS500; a Garrett T4. It will not create any significant boost under 4500rpm , but above 4500 rpm it will provide huge amounts of boost (more than 2,0 bars up to 8000 rpm) right up to the redline. The Garrett T25 turbo in the latest Escort RS Cosworth on the other hand is a totally different size, it will give almost instant response compared to the T4. It will produce 1,4 bars of boost from 2000 rpm but as the engine revs increase above 5500 rpm it will not provide any significant boost. The good thing is that one can match a bigger compressor housing to a smaller turbine. This will provide the best of both a large and a small turbocharger; relatively little lag and more boost at the top. An example of this is the Garrett T35 in the YBT Escort RS Cosworth. It has a turbine comparable to the Garrett T3 and a compressor nearly the size of the T4. This still does have a bit of lag though and will not provide too much boost below 2500rpm. Some people think it has way too much lag but one just has to be aware of it and adjust ones driving style.

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