eli5: how does a class D amplifier work?

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is it possible to put one in a usb powered microphone

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All the class A to C amplifiers use transistors in their linear region. This is a current and voltage region of the transistor where a change in the input will be simply multiplied at the output. So for example a 0.01A change in the input signal may give a 1A change in the output signal without any distortion. This is a very good thing for a pre-amp or a signal amp where you have a very low volume input and just needs to bring the signal level up to high enough levels that you can transfer them thorugh a cable, send it through analog signal processing equipment, capture it with an Analog to Digital Converter or use the signal as an input signal to a powerful amplifier. The disadvantage to this however is that transistors use a lot of power in this mode and generates a lot of heat instead of a lot of signal output. Transistors work much better when they are either fully on or fully off.

A class D amplifier does this. First the input signal is transformed to this fully on or fully off signal by converting it to pulse width modulation. The frequency of the pulses is usually much higher then the frequency of the sound and the input signal determines the length of each pulse. This way it is the same amount of power in the signal if you average it out. This pulse width moduletad signal can then be sent through a high power transistor which since it is fully on or fully off is producing a lot less heat for the power you get then with the other amplifiers. After the transistor you use a low pass filter to filter out the high pitch pulses and average out the signal so that it matches the input signal but at much higher power.

I am not sure what you are asking about with the USB powered microphone. Most USB microphones consists of a microphone, a class A pre-amplifier and an ADC. The output of this is digital through the USB chord and the amplifier is just to get the weak signal from the microphone up to the level that the ADC expects. But since you say USB powered I assume some sort of analog output. A lot of microphones are powered as they need a pre-amplifier to get a signal across the cable. Not all microphones that requires power have a pre-amplifier but it is not a bad idea to have one built inn. In general you can not use a class D amplifier as a pre-amp because it requires a decent input signal to power the pulse width modulation and the transistor. It is not uncommon to see smaller amplifiers to process the signals of the big power amplifiers for this reason.

A normal microphone would not need a very strong output signal as it is usually not powering any big speakers by its self. But if you needed an amplifier to power some speakers and only had the power from a USB plug available then you would defenatly go for the much more power efficient class D amplifier rather then a class AB amplifier which you often see in high end audio equipment.