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I'm talking in the context of guitar amps, but I assume that this question is relevant for any type of audio amplifier.

Very often in amplifier schematics I see two stages of amplification -- first, the signal is amplified a smaller amount by a preamp circuit and then amplified again by a power amp circuit.

This seems redundant to me. What's the point in amplifying a signal in two small steps rather than just one greater-gain amplification?

My first thought was: does this multi-stage amplification help to reduce unwanted noise from the signal? But the more I think about that, the less it makes sense, since surely the second stage would be amplifying any noise as well.

Quick and dirty answer:

Buffering is one reason. Interconnects between things can have a lot of capacitance and require a lot (comparatively) of current to drive.

Noise immunity is another. Think about this scenario: Send a signal through a wire where it picks up, say, 10mV noise, then amplify it by 100x: total noise, 1000mV. But if you instead amplify it by 10x, then send it through the wire where it gets 10mV noise, then amplify by another 10x, your total signal amplification is still 100x, but your total noise is only 100mV.

In audio gear, it is useful to do most of the signal manipulation at a standard level, known as "line level". This includes mixing, equalization, compression, etc.

Some signal sources (microphones, guitar pickups, etc.) do not inherently produce line level outputs, so a preamplifier is used to boost the signal to that level. Some signal sources (record players) require not only a boost, but also a special equalization to flatten the frequency response.

Then, after all of the signal processing is done, a second, "power" amplifier is used to drive the speaker(s).

This kind of modularity allows signal sources, processing stages, and different kinds of speakers to be mixed and matched freely.

Let me ask you this...how big would the amplifier hardware have to be to do its job properly? If it was this size, could it go right next to the pickup where the noise would the lowest and signal integrity is highest?

If it can't, then it has to be farther away. But if it's farther away, can the pickup natively produce a signal that can be reliably and accurately transmitted all the way to the single-stage amp?

If not, then you need another amplifier that is physically small enough to go right next to the signal source and reinforce the the signal just enough so that it can be accurately transmitted to where there is more physical space for the rest of the hardware. This reinforcement can come in the form of re-transmitting the pickup's high impedance output as a low impedance output, or boosting the signal levels just a bit so that the transmission noise won't drown out the signal.

A major reason for separate boxes for preamps and poweramps is the GROUND currents.

Suppose you built the preamp and the poweramp on the same PCB. Why not?

Some of the loudspeaker current will be flowing around on the GROUND, and end up combining with the input signal.

To minimize this "combining", make that PCB long and thin, so the PowerAmp Grounds are far away from the PreAmp Grounds.

How to improve on this? use long thing regions between the Preamp and the Poweramp.

In the extreme, a coax cable provides a long-thin-region, to ensure very small combining of input and output.

Give low millivolt signals from a vinyl record Moving Magnet cartridge, or even 0.5 millivolt from Moving Coil cartridges, that become near-100-volt audio outputs, the entire system needs 100,000:1 isolation.

^{simulate this circuit – Schematic created using CircuitLab}