As hinted at above, on the Harbeth User Group forum there is an extensive discussion about amps. Alan Shaw, the designer of Harbeth speakers and owner of the company, has long propagated that all well-designed amps sound the same. However, he acknowledges that in fact it is very difficult to have a really well-designed amp, as making a good amp is surprisingly difficult. What I've understood is the following (but bear in mind any misinterpretation is mine). Please excuse my mistakes in English, as this is not my native language.
An audio signal is measured in volt and ampere (multiplied becoming watt). Amplifying the signal would entail increasing voltage and/or current. An amp can go up to a limited voltage and can deliver a limited amount of amperage; both may in fact be related in that there is a limit to the total power (wattage) the amp can deliver. If you would attach a simple 8 Ohm resistor you could easily measure this, and I believe this may be the way in which simple measurements about amps power can be made. The amount of amplification (gain) is controlled by the volume pot.
Limitations in power lead immediately to differences in sound: the signal gets clipped, which may cause damage to speakers and amp. But if you do not come within reach of the limits, you would expect that two amps with the same gain would sound exactly the same, as that electronically speaking is the job that the amp has to do.
However, the gain and the limitations of power may not be completely constant. It is possible that an amp (for example) intentionally or unintentionally has different gain or different limits at different frequencies. If an amp has higher gain in the bass region, it will sound differently than a completely neutral amp.
Now the complicated part is that an amp doesn't drive a constant resistor but actual speakers which never offer a constant load. Several hifi journals offer impedance (=Ohm) plots of speakers over the frequency range. While some are quite constant over the range, other speakers may have quite extreme dips, starting from 8 Ohm but going down as deep as 2 Ohm or even 1 Ohm. You may wonder why this matters. If the impedance halves, the amount of power that the speaker sucks out of the amp (or needs to be driven) doubles. Hence an amp that would only need to provide a modest 10 Watt at 8 Ohm, suddenly needs to be able to provide 80 Ohm if the speaker dips to 1 Ohm at a specific frequency. Whether the speaker does go so low depends on the content of the music at that moment. If that frequency is no longer present, the amp has a far easier job.
Finally, there is something called 'damping factor'. This was discussed here recently as well. As far as I understand, a speaker 'stores' the energy (which is the effect of a coil) and then releases a current back to the amp. The amp has to be able to cope with this. The swings in impedance and required power described above have the effect that the amp afterwards has to deal with the current being returned.
All these effects may be easy to control when talking about very small amounts of power, for example with headphones or low listening levels (approx 1 watt). However, if you want to play louder you may easily need 50-100 Watt or even more. It is hard to make an amp that is able to cope with that amount of power and still offer a constant gain during dips, while being able to absorb the current returned to the amp.
As far as I understand it is possible to design an amp that is able to do all this fairly well, but such amps are not as common as one might think and may be very costly, in particular if they offer large amounts of power. Of course you may not need an extremely capable amp if your speaker is a so-called easy load (=doesn't have extreme impedance diips) or if you listen at modest levels.
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