Powered speakers
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Powered speakers (or active speakers) are speakers that have built-in amplifiers. They can be connected directly to a mixing console or sound source, without the need for an amplifier. Such speakers are available with passive or active crossovers built into them. Powered speakers avoid the necessity of speaker to amplifier matching and limiting is built in. Another advantage of powered speakers is a reduction in the length of cable between amplifier output and speaker driver (some claim this affects maximum slew rate, the rate of change in audio signals) since cable length is very small in powered speakers. Powered speakers ease set-up issues for an audio system, at least to some extent. However since low-level signal cables have to be run to the speakers from a signal source, noise pick-up can be a problem. This can be reduced (even effectively solved) by incorporating "balanced" line inputs into the speakers. When driven by balanced signals (and cables) from the sound source, noise can be greatly reduced. Some powered speaker manufacturers, notably DB Technologies and Phonic, are now incorporating UHF wireless receivers so the speaker requires only a power lead.
Alternatively, a passive speaker (or unpowered speaker) is a speaker which does not have its own power source and draws power from somewhere else, as opposed to an active speaker which has a built-in amplifier. Active speakers have a short run of cable between the amplifier and the speaker, which reduces signal and power loss. The amplifier can also be matched to the speaker (chiefly an impedance issue) more exactly. Passive speakers are lighter and cheaper, but require longer lengths of cable to run to a separate amplifier. This can be desirable if you have amplifiers that can run multiple speakers.
[edit] servo-driven speakers
By including a negative feedback loop in the amplifier-speaker system, distortion can be substantially reduced. If mounted at the speaker cone, the sensor is usually an accelerometer. It's also possible to monitor the back EMF generated by the driver voice coil as it moves within the magnetic gap. In either case, specialist amplifier designs are needed and so servo speakers are inherently powered speakers.