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The electrical characteristics of the power supply describe the quality of the power supply's outputs, and its ability to handle special situations such as disruptions or disturbances to its input power, or variations in the loads the power supply drives. While important to know, and definitely indicative of the power supply's quality level, I must concede that in most cases the average power supply purchaser doesn't need to sweat most of these details all that much. You should ensure that the power supply's figures for these characteristics are not wildly off from those of other, similar supplies. For the newer form factors, you should also check the power supply's specs against the requirements listed in the form factor specification document. Beyond that, however, don't use small differences in these numbers between "power supply A" and "power supply B" to draw excessively grand conclusions.
Hold (or Hold-up) Time: Probably the most important electrical characteristic, this is the amount of time the power supply will keep producing its output, if it loses its input. A typical figure is about 20 milliseconds (the energy-storing components within the power supply are what allow this number to exceed zero.) This value indicates the length of a blackout that the power supply may be able to tolerate before dropping the Power Good signal. It is also important to compare against the switch time of a UPS you are considering for use with your PC. The hold time should be considerably greater than the switch time to reduce the chances of problems.
Load Regulation: Sometimes called voltage load regulation. This specification refers to the ability of the power supply to control the output voltage level as the load on the power supply increases or decreases. The voltage of a DC power source tends to decrease as its load increases, and vice-versa. Better power supplies do a better job of smoothing out these variations. Load regulation is usually expressed as a "+/-" percentage value for each of the voltages the power supply delivers. 3% to 5% are typical; 1% is quite good. (The -5 V and -12 V signals usually are no better than +/- 5% even on very good units; there's no point bothering getting them better than that since they are low-current and mostly unused anyway.)
Line Regulation: The complement of load regulation, this parameter describes the ability of the power supply to control its output levels as the level of the AC input voltage varies from its minimum acceptable level to its maximum acceptable level. Again, a value for each output level is usually specified as a "+/-" percentage. +/- 1% to 2% is typical.
Ripple: Also sometimes called "AC Ripple" or "Periodic and Random Deviation (PARD)" or simply "Noise". The power supply of course produces DC outputs from AC input. However, the output isn't "pure" DC. There will be some AC components in each signal, some of which are conveyed through from the input signal, and some of which are picked up from the components in the power supply. Typically these values are very small, and most power supplies will keep them within the specification for the power supply form factor. Ripple values are usually given in terms of millivolts, peak-to-peak (mVp-p). "Peak-to peak" refers to measuring the AC voltage from its negative maximum to its positive maximum (see here for an illustration of what this means.) Lower numbers are better.
Transient Response: As shown in the diagram here, a switching power supply uses a closed feedback loop to allow measurements of the output of the supply to control the way the supply is operating. This is analogous to how a thermometer and thermostat work together to control the temperature of a house. As mentioned in the description of load regulation above, the output voltage of a signal varies as the load on it varies. In particular, when the load is drastically changed--either increased or decreased a great deal, suddenly--the voltage level may shift drastically. Such a sudden change is called a transient. If one of the voltages is under heavy load from several demanding components and suddenly all but one stops drawing current, the voltage to the remaining current may temporarily surge. This is called a voltage overshoot.
Transient response measures how quickly and effectively the power supply can adjust to these sudden changes. Here's an actual transient response specification that we can work together to decode: "+5V,+12V outputs return to within 5% in less than 1ms for 20% load change." What this means is the following: "for either the +5 V or +12 V outputs, if the output is at a certain level (call it V1) and the current load on that signal either increases or decreases by up to 20%, the voltage on that output will return to a value within 5% of V1 within 1 millisecond". Obviously, faster responses closer to the original voltage are best.
Peak Inrush Current / Input Surge Current: The absolute maximum amount of current that the power supply will draw in the moment after it is initially turned on. This is sometimes used to indicate how much "shock" the power supply is subjected to when it is turned on. Lower values are better.
Overvoltage Protection: In addition to specifying a normal maximum voltage level, good power supplies will include protection against the output voltage exceeding a certain critical level. If for some reason the voltage of the +3.3 V, +5 V or +12 V lines goes above a certain value, the power supply will shut down that output. The number is usually expressed as a "trip point voltage" (for example, +6.25 V for the +5 V line) or a percentage (which for a trip voltage of +6.25 V would be 125%). The specification will also say what the power supply does when an overvoltage is detected; usually, it will reset.
Overcurrent Protection: If the power supply's outputs exceed their maximum ratings, some power supplies will detect this condition and reset the unit. The supply will specify what percentage over the maximum rating for each voltage output will cause this to occur.