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Many ICs have short circuit protection built in, but how do they stop current flow in case of a short?

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Here you see how it is done. If the temperature or current of the device becomes too high then the base current of the driver transistor (most right) position will be restricted or even cut of completely. Of course this is a very simplified representation of a stabilizer but it shows what is done for the protection against overcurrent and overtemperature.

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That's a very broad question, so a correspondingly broad answer would be that they generally reduce the drive to the transistor that is producing the current, usually by shunting the drive current elsewhere.

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The general idea is to measure the current of the output device and to reduce the current in case some preset limit is exceeded.

Measuring the current inside a chip can be done in different ways. One method would be to use a shunt resistor and measure the voltage drop across it. Instead of using a resistor the voltage drop across the output device itself could be monitored.

If an overcurrent condition is detected the driving signal is reduced to reduce the current through the output device. This could be done using some digital logic and a mono-flop or using a continuous feedback mechanisms of varying complexity. Of course feedback alone could be used to simply limit the maximum current.

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