What’s the relationship between gain (dB) and power (watts)?

I think the OP has been answered but I've been trying to come up with a simple way of talking about gain structure for a while. I don't know if this will add much but here goes...

The maximum output voltage of an amplifier can be calculated from the output power spec, this cannot be exceeded and if the signal reaches this level then it will clip and distort.

Dynamic range - is the space between the noise floor and the maximum output voltage. This is the space for the signal and we want this to be big for sound quality.

When we amplify (or add gain) we reduce the dynamic range i.e. the signal and the noise get bigger but the maximum voltage remains the same.

Headroom - is the space between the signal and the maximum voltage which we can think of as a reserve for momentary peaks in signal level.

So in an ideal world we’d just have the gain set to the level that we wanted to listen at with a bit of headroom. It’d be great if it was that simple but…

Most amplifiers (for a variety of reasons) offer the user little or no control of the amount of gain applied.

Different sources have wildly different output voltages, consumer line level is defined as 0.316V RMS but most DACs will put out 2V, that’s 16dB difference!

 

That’s why we have a volume control or attenuator. For a number of reasons this is usually before the main gain stage which means it attenuates the small signal before it is amplified. And the killer is that the noise added to the small signal will be amplified by the following stages (raising the noise floor and reducing the dynamic range).

 

So the point of saying all that is to explain that if your system has excessive amounts of gain then you are sacrificing dynamic range and listening to more noise than you need to… take a look at how far down you have the volume control set. So if you can set a lower level of gain and still get the volume you want you’re best going with that… the maximum output is a constant and won’t change when you adjust the gain – which means it won’t affect your headroom.

Is it true then that the output stage of the amp 'transforms' the amplified signal from the gain stage (in voltage) to current (in amps)?

That is literally true for Valve amps in that they have an output transformer and similarly a key feature of the output stage of an SS amp is to provide the current required to drive the speakers.

The bigger the power supply that's available will allow for a bigger/wider peak voltage swing to be transformed into current to the speakers before it clips?

Not strictly speaking but I think I know what you're getting at... it is common for manufacturers to use a power supply that is rated lower than full RMS output 100% of the time. That's because music doesn't require that level of power all of the time (google 'crest factor' if you want to understand why). How much the power supply is derated and the amount of reservoir capacitance (needed for sudden peaks in the music) is a judgement call... some amps may struggle with heavy loads. I spec my power supplies for 100% RMS output, which is over engineering but means the amps deliver what they say in the specs without fail.

Thus, power equals headroom? Lot of gain + minimal headroom = a limited amplifier.

Kind of... if the ceiling in my house is 2m high then there's a risk I might bump my head if I jump in the air. 3m would eliminate that risk but there's no additional advantage in having 10m high ceilings... that's about as far as I can stretch that analogy. If your amplifier is capable of 300W but you only need 10W (you may be surprised at how little you actually need) then that extra headroom is of little use. If the gain of the amplifier is high to allow you to achieve 300W output from a low level signal then you will likely be getting a worse signal to noise ratio than if you went for a 60W amp with a well specced power supply.

I'm trying to explain without maths and jargon... hope that helps.