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AC-DC Voltage Conversion
The electricity you get from your utility company is in the form of alternating current (AC), while the electricity your PC requires is direct current (DC). (Read this page of the electrical basics section for more details on AC and DC.) Therefore, the primary function of your power supply is to convert your wall AC power into a DC form that your PC can use. In fact, the supply normally provides several different voltage levels, to meet the demands of different components in the machine.
In fact, while almost everything in your home runs off standard AC power, many devices actually use DC internally. Just a few examples of this are telephone answering machines, many types of audio equipment, some kinds of battery chargers, and in the PC world, certain types of printers, external modems and other peripherals. Two indicators that a device actually uses DC inside are: the ability of the device to run on batteries, and the presence of a device outside the unit that powers it. These small "bricks" with one plug for the wall and another for the device are often called AC adapters (sometimes the "term of endearment" wall wart is used, in honor of the way they are usually designed to hang right out of a wall socket. :^) ). AC adaptors are really DC power supplies that converts the AC of the wall into DC for the device.
The difference between this sort of DC power supply and the kind in your PC is the design. AC adapters are linear power supplies. These supplies are cheap and simple to make. The main problem with them is that they are tremendously wasteful; typically, 50% or more of the energy supplied to one of these adapters is wasted as heat. You can feel this readily--just touch the adapter when the unit is working; many become quite hot to the touch. A hot AC adapter means electricity is being wasted. This is an acceptable compromise for small appliances, but unacceptable for a PC power supply.
Instead of the linear design, PCs use switching power supplies. (The full name for this sort of design is actually "constant-voltage, half-bridge forward-converting switching power supply", but don't worry, that won't be on the final exam. :^) ) Explaining in detail how the design works would take many paragraphs and make your eyes gloss over unnecessarily. In a nutshell, the switching power supply uses a transistor switch and a closed feedback loop to produce DC output that is properly regulated regardless of the load on it, with only the amount of AC power required to draw the DC load being taken from the utility.
The main advantage of a switching power supply is that it is far more efficient than a linear design. When you are dealing with hundreds of watts of power, this is a more serious issue than when you are talking about an answering machine. The second advantage is that all the energy wasted in the power supply as heat has to be removed by the PC's cooling system. Therefore, more efficient power supplies produce less heat that the system has to exhaust. The main disadvantage of a switching supply is that it generates high-frequency signals within it as part of its conversion process, which can radiate out of the unit and cause interference to other electronic devices (inside or outside the PC). For this reason, you will always see PC power supplies encased in metal boxes for shielding.
The power factor of a device refers to the ratio of the actual power used by the device to the product of the current and voltage supplied to it. Traditional power supplies have a power factor of about 0.6 to 0.7. The power factor is important especially for determining the sizing of UPSes as well as circuits that supply larger units. Some newer power supplies, especially larger ones for servers, have additional circuitry added to correct the power factor of the supply. These are, unsurprisingly, called power-factor-corrected supplies, and have a power factor of (or near) 1.0. They make UPS sizing either easier or more confusing, depending on how you look at it :^) Another reason why power-factor correction is being added to some supplies is that low power factor devices, if used in sufficient quantities, cause problems for electrical utilities. In some parts of the world the utility companies are starting to impose surcharges on companies with excessive loads at low power factors, though this is not really a concern for a home PC user.
Next: Standard Output Voltages