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Class D amps, how do they work?

Class D amps are usually analog and not digital. Like most power amps, they usually have a power transformer in the power supply.

They're a lot more efficient than linear amps of comparable power rating. Unlike linear amps, they use output transistors that switch totally on and totally off to reduce the power dissipation (wasted power) to very small levels.
 
It might not sound so intuitive, but transistors waste a lot less power when they are totally on or totally off than they do when they are partially on.

So the class d amps run the transistors all the way on directly to all the way off and directly back to all the way on, etc. This is efficient, but ends up injecting a bunch of high frequency energy into the signal in predictable frequency ranges, so a low-pass filter gets rid of the extraneous high frequency content, and gets you back to your original signal.
 
Class D amps convert the analog waveform from the preamp into a series of pulses that go instantly from 0-100%. Typically, the width of the pulse is varied with the audio. Then, it all runs through a big power inductor that converts everything back to the analog waveform.

Why bother?
1. When transistors are partially on, as during 99% of an analog waveform, they are just acting as a big variable resistor, so they dump the excess energy as heat (that is why they get hot!). The class D amps are only ON or OFF so there is no heat producing effect.

2. Because they are only ON or OFF, the transistors can be a much lower quality type for the amount of power that they handle. They don't need to be low distortion or quiet because they are just an ON and OFF switch, where you can't tell that they are less than perfect. The transistors are generally types that are used to drive motors, etc.

The end result is that you can get a very powerful amp that can be small because it doesn't need big heat sinks and ventilation, and it can be lower cost for the power generated.

Problems with Class D amps is that it is hard to get the pulse rate high enough that it does not interfere with high frequencies. This is why you see Class D used mostly for bass and not for full range sound.
 
For further clarification:

We can break the amp into two sections - front-end AC to DC conversion, and audio amplifier.

Traditional power amps use large heavy 50/60 Hz transformers to step the wall voltage (120VAC in the US) down to something more usable by the audio amplifier stage. The transformer typically also creates a pair of matched "voltage rails", such as +/- 50VAC with a common "center tap" that's used as the ground reference for the audio amp. The audio amp can be whatever topology you like - Class A/B, B, or D typically. Yeah - you can have a class D audio amp with a big heavy transformer!

Transformer size is largely proportional to the frequency of the AC voltage it's converting. 60 Hz transformers will be very large. 100,000 Hz transformers can be pretty small. This is the
basic idea behind a "switchmode power supply", or SMPS. You take the wall AC, turn it into high voltage DC then chop it up into a high frequency square wave which can be passed through a transformer and stepped down to get whatever voltage rails you need for your audio amp. This high frequency supply introduces high electromagnetic and radio-frequency noise which has to be carefully shielded and filtered from the audio amp. You can put an SMPS in front of any kind of audio amp you like, even linear amps.

Traditional amps are heavy because of the large transformer power supply and because of large aluminum heatsinks used to cool linear power amplifiers. SMPS supplies reduce the transformer weight significantly (at the expense of a more complicated power supply) and Class D amplifiers reduce the cooling components required.