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Power Good Signal
When the power supply first starts up, it takes some time for the components to get "up to speed" and start generating the proper DC voltages that the computer needs to operate. Before this time, if the computer were allowed to try to boot up, strange results could occur since the power might not be at the right voltage. It can take a half-second or longer for the power to stabilize, and this is an eternity to a processor that can run half a billion instructions per second! To prevent the computer from starting up prematurely, the power supply puts out a signal to the motherboard called "Power Good" (or "PowerGood", or "Power OK", or "PWR OK" and so on) after it completes its internal tests and determines that the power is ready for use. Until this signal is sent, the motherboard will refuse to start up the computer.
In addition, the power supply will turn off the Power Good signal if a power surge or glitch causes it to malfunction. It will then turn the signal back on when the power is OK again, which will reset the computer. If you've ever had a brownout where the lights flicker off for a split-second and the computer seems to keep running but resets itself, that's probably what happened. Sometimes a power supply may shut down and seem "blown" after a power problem but will reset itself if the power is turned off for 15 seconds and then turned back on.
The nominal voltage of the Power Good signal is +5 V, but in practice the allowable range is usually up to a full volt above or below that value. All power supplies will generate the Power Good signal, and most will specify the typical time until it is asserted. Some extremely el-cheapo power supplies may "fake" the Power Good signal by just tying it to another +5 V line. Such a system essentially has no Power Good functionality and will cause the motherboard to try to start the system before the power has fully stabilized. Needless to say, this type of power supply is to be avoided. Unfortunately, you cannot tell if your power supply is "faking" things unless you have test equipment. Fortunately, if you buy anything but the lowest-quality supplies you don't really need to worry about this.