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

There is no connection between the maximum power output of an amplifier and its gain. They are two unrelated design values.

For a given input signal, the higher gain amplifier (or setting) will produce a higher output. But the maximum rated output of a Bryston amplifier does not change between gain settings.

Gain is how much it's going to amplify the incoming signal. Less gain is usually quieter but if you have low gain signal going in, you may need more amplification to reach peak power.  

The relationship is logarithmic. For every increase of 3dB power doubles. For 10dB power increases ten times. 

In your case switching between 23 and 29dB of gain is 6dB. So whatever input it takes to drive the amp to full power at 23dB, it will reach full power with 1/4 that level when set at 29dB.

@millercarbon   
Good answer

That gain switch is like a gas pedal. If the driver has a strong foot, then the pedal is set to 23 to take into account the heavier push to get full power. However, a driver with a weaker foot will set the gas pedal to 29, making it easier to push for full power. The engine does not know the difference nor cares.

+1 millercarbon, you're spot on.

Yes, and thanks.

Awesome question and one I’ve been contemplating myself.

and awesome answer MC, thanks to the op and the og

None. :) 

Amps tend to, by convention but not requirement, have around 28 dB of gain, regardless of their power rating. :)  This is about 20x input voltage.

That is, put in 0.1V peak to peak input signal and you should get 2V peak to peak on the output.

Of course, gain only works so long as you don't exceed the output limits, whose absolute limit in a linear amp is by the power supply rails.  So, if your rails are +- 20V, that's your peak output.  With 20x gain (around 28 dB) this means your maximum input voltage is 1v peak to peak before clipping.

Thanks everyone.  I like the gas pedal vs engine analogy.  But does a higher initial gain ultimately mean anything for the output of the amplifier (like higher possible SPL in dB for a given speaker)?

Yep, but actually your max usable output is about 0.71 of your rail voltages.

Gain is the amount of gain the amplier has usually between 20 to 30 db and power is the amount of watts it will put out into an impedance in ohms. The higher the gain the more loud it will be at a certain volume setting on your preamp or amp but with the higher gain more noise will be introduced but if you have efficient speakers  you will always want less gain to lower the noise and use more of your volume control. With less efficient speakers you may need the extra gain because you will run out of volume control too quickly so it is for system matching. Power is a relative thing that also depends on how much current you have in the circuit, tube amps are high voltage low current, and solid state amps are low voltage high current, so different types of speakers will like one or the other.

For me the easiest analogy would be....  With higher gain,  it takes less of a turn of the volume knob before you hit the amplifiers full output. 
With the lower gain setting,  you need to twist it a bit further.  

Thanks everyone. I like the gas pedal vs engine analogy. But does a higher initial gain ultimately mean anything for the output of the amplifier (like higher possible SPL in dB for a given speaker)?

Yeah I was afraid of that, as it is a horrible analogy. So awful in fact I am gonna leave it alone and just explain what gain actually is, in the hopes the truth will replace all the false ideas generated by that awful example.

Gain is nothing more than another word for multiplier. Gain is the amount by which the incoming signal is multiplied. Because gain is a multiplier it is expressed in dB. So in your case, say you have selected 23dB gain. Whatever voltage level is input, is increased by 23dB.

Remember, dB is a log function. Remember, every 3dB is twice the power. Every 10 dB is ten times the power. So your 23dB equates to 200 times. (The math in this case is straightforward. But if you need help, ask!) Therefore, say your input voltage is .1mV. With 23dB gain your output voltage will be 20mV.

Flip the switch to increase the gain to 29 and since 29 is 6dB more than 23, that is two 3dB doublings, the output will be 4X the previous example or 80mV.

That is gain. It is like a calculator that no matter what number you enter all it does is multiply by 8. Or 16. Or whatever.

However, what if the calculator has a display with only 2 digits, and you input 100? Well clearly the gain does not change but the output cannot go that high, you are not going to magically get three digits no matter what the gain.

That is the difference between power and gain. Gain is a multiplier. Power is a ceiling. You cannot get any more power from your amp no matter what gain you set it on.

millercarbon has it correct.  This gets confusing (for example, your first reply) because you confused gain (amplification factor - a multiple) and dB - a measurement of sound pressure as perceived by the human ear.
So the real question is: "what are you trying to understand?"
IN the end MC's point is one all should take to heart.  Doubling the volume  -- all things equal - requires 10X the power. Think about thta next time you consider the value of, say 60W vs 30W.
G

I'm wondering if the question is about the switch values or how it should be set?

IN the end MC’s point is one all should take to heart. Doubling the volume -- all things equal - requires 10X the power.

Don’t mean to be difficult but no, that’s not quite right. Sound volume is how loud we call something we perceive or hear. That’s why I resist measuring and object to people saying they listen at so and so many dB. Instead I prefer to say I listen at a satisfying level, or if I want to get across that it is good and loud I like to say Supertramp Loud.

It sounds goofy but at least it is clear I know the difference between volume level and SPL. Sound pressure level is what we measure with decibels. Literally the pressure difference between the compression and rarefactions of the sound wave.

The difference is that we can say a 10dB increase requires ten times the power. Because both decibels and watts are quantifiable. Volume however is totally unquantifiable. Is 10 dB twice as loud? According to who? Some can barely hear a 2dB change. For others that is a big jump and they want a volume control with finer increments. For others anything past a certain point is simply "too loud".

But your main point is definitely one people will do well to learn. To play even just 3dB louder requires twice the power. 3dB is not that much of a difference. But it requires twice the amp to do it. This more than anything else is why people will do themselves such a big favor to ignore low sensitivity speakers. You can get them. They can be made to work. But the math is stacked against you.

If the amp is rated 300Watt at 8Ω, gain set to 23dB, the amplifier need 3.5Volt input signal in order to output 300Watt at 8Ω,
If the gain set to 29dB, the amplifier need 1.75Volt input to output 300Watt at 8Ω.
Its depends on the max. output voltage of your preamp, set the gain on your power amp accordingly.

@millercarbon - thank you.  I like this:

That is the difference between power and gain. Gain is a multiplier. Power is a ceiling. You cannot get any more power from your amp no matter what gain you set it on.

@timlub - thank you. This is helpful and simple:

With higher gain, it takes less of a turn of the volume knob before you hit the amplifiers full output.
With the lower gain setting, you need to twist it a bit further.

@erik_squires - now this seems relevant to me:

That is, put in 0.1V peak to peak input signal and you should get 2V peak to peak on the output. 

Of course, gain only works so long as you don't exceed the output limits, whose absolute limit in a linear amp is by the power supply rails. So, if your rails are +- 20V, that's your peak output. With 20x gain (around 28 dB) this means your maximum input voltage is 1v peak to peak before clipping.

This is where my technical ignorance seems significant.  Maybe I’d need a textbook to understand this:

Of course, gain only works so long as you don't exceed the output limits, whose absolute limit in a linear amp is by the power supply rails.

My Sanders sound pre has a gain setting I think of 8DB, or close.

   I set it at 3 DB, for a bit of boost, but the McCormack amp does the rest

SPL in your room, in your sitting position is also largely contributed to by the room itself.
There is a phenomenon called cabin gain or the "transfer function", I am going to quote from car audio information.

Have you ever wondered why car audio systems can have so much bass as compared to a home stereo system? The reason has to do with space. Your home has a lot of it and your car doesn’t. Because there is so little space in a car the bass notes (which are long pressure waves) build up inside the passenger area. To calculate the length of a sound wave you divide the speed of sound (in feet per second) by the frequency. At sea level, the speed of sound is approximately 1,127 ft/sec. For example, a 40 Hz note has a wavelength of approximately 28 ft (at sea level).

(speed of sound)/(frequency) = wavelength
OR
(1,127 ft/sec)/(40Hz) = 28.175 ft

Since the length of the average car interior (including trunk) is in the 12 ft range the 40 Hz note will be longer then the car’s interior.

This is why notes below 70-90 Hz (depending on the vehicle) will have a greater output than the rest of the frequencies. Once this magic frequency is reached, bass output will increase by about 12 dB/octave below that frequency. This phenomenon is called cabin gain or the "transfer function". So a smaller vehicle will have a greater cabin gain and should be able to have greater low bass than a larger vehicle. This is true for identical subwoofer systems with identical power.

The physics involved are true for listening rooms, and so SPL isn’t just about the amount of energy the drivers are feed or are producing, measured at the listening position. Many use control devices to attenuate and clean up low frequency nodes which can be attributed to cabin gain, or "transfer function". Boundary reinforcement is another interesting topic and is related.

I am sure it’s very well known by most of us, but often not hardly thought about. I may be wrong but it appears, MC takes this also into consideration?

It sounds goofy but at least it is clear I know the difference between volume level and SPL. Sound pressure level is what we measure with decibels. Literally the pressure difference between the compression and rarefactions of the sound wave.

With all the math and physics measurements being tossed around, you’d think that everyone was equally gifted with identical hearing and processing by their brains. Unfortunately, this is not true and varies again by the condition and training of the listener and by their age. This is what makes "audiophiles" out of "just another person, but gifted by the Lord with perfection and the ability to communicate perfectly." I hope you are laughing at how arbitrary the listening experience can be. The most important aspect of the whole thing is how much you enjoy the experience of recorded sound and music.
I have a degree in experimental psychology and have been an electronic design engineer and owner of a graphics/media company with a recording studio for over a decade. This is irrelevant as I ENJOY music and have for a long, long time.

I am interested in this because I have just upgraded from one power amp to two monoblocks - the gain has not changed but the available power has gone from 200W per channel to 600W...

... and yes, the monoblocks are no louder than the stereo unit. 

Here is my attempt at understanding the issue. Typed out here more to get feedback, to get my working checked, than to tell anyone else what to think.

OK, so gain tells you how much bigger the output voltage is to the input voltage, i.e. the ratio of input to output. It think (guess) that typically power amps have an input to output voltage ratio around 100 times. 

Power considerations appear to come into play when you connect the amplifier to something, like a resistor, or a speaker. In order to hold the correct voltage, the amp has to pump out current. If it runs out of current, the voltage will decrease, the relationship between output voltage and input voltage will falter. 

With something simple like a resistor, one could easily work out how much power one needed from power = voltage squared divided by resistance. But speakers are not resistors, and music is not a steady state. I think that if one suddenly wants a speaker's driver to move, its resistance momentarily collapses, and so only an amplifier with a lot of power can hold its output voltage steady. As the speaker resistance collapses due to a rim shot or square wave, the amplifier has to dump a tonne of current into it.

In my mind big power amps sound powerful because they are capable of making the speaker follow the music. They sound more exciting, not louder. 

Wouldn't an increase in the gain setting be the equivalent of turning up the volume knob, as a default setting? (using more power by default?)

I have an amp with independently adjustable gain for each of 4 inputs. I don't use it, but my understanding is that it will simply adjust (offset) the default volume (power) level for a given input. This makes sense if you wanted to level out some inputs that might be varying in volume, but why the global gain switch? Gives you a better/preferred starting point for volume.. due to input or speaker behavior?

@rols

In my mind big power amps sound powerful because they are capable of making the speaker follow the music. They sound more exciting, not louder.

I thought that increased damping factor was part of the reason that higher-power amps have more speaker control (grip), but I'm probably missing other factors; interesting topic.

This is where my technical ignorance seems significant.  Maybe I’d need a textbook to understand this:

Of course, gain only works so long as you don't exceed the output limits, whose absolute limit in a linear amp is by the power supply rails.

This is nothing more than an unnecessarily techno-jargon laced way of saying exactly what I said: Gain is a multiplier, power is a ceiling.   

This discussion is getting out of hand as simple DC models are being compared with exotic AC ones. The math is being dumped out of a need for a model of understanding which doesn't match the challenge of the questions. The answer to all of them is... the questions are too difficult to separate into little pieces. The basics are just that, basics. The answers are beyond the understanding of most people because they encounter ideas such as statistics and psychology, which some people think do not "fit" into any discussion of music. However, they do... as well as other areas of mathematics and exotic theory, both electronic and as apparently "simple" as component design (so sorry, that is ALSO extremely math oriented). The bottom line is, most people simply cannot, will not, and don't either believe or know the answers. That means if they are right or wrong. Good luck.

@millercarbon

Gain is a multiplier, power is a ceiling.

a ceiling of what, though?  Current?  Voltage?  

Power.  

Seriously. Helps to think things through. What is power? Current? Voltage?  
Here I'll give you a hint: Watt is power? That's not a typo. 

They're not paying me enough. You could mention that.....

The gain switch is a nice feature on the power amp, 23dB of gain can give most active preamp better range of volume control, but passive preamp might need 29dB gain to drive the amp to full power.
Why not call Bryston to get the answer?

Very simplified electrical info following:
One can think of voltage as electrical pressure (or potential force), resistance somewhat like a valve that restricts flow, and amperage as the physical amount of electron flow, for the sake of a simplified analogy.

Electrical work (measured in Watts) can be calculated many ways, a simple one is the amount of electrical pressure (Volts) multiplied by the amount of electron flow (Amperes).

As nitroxpro states, that it is an exotic alternating current, I would say for simplification that it also meets an exotic resistance (impedance curve).

Amplification can be somewhat described as if you had a small electrical signal controlling a valve on a larger electrical signal, as the smaller signal swings from positive to negative, in higher and lower amplitude from zero, it opens and closes the valve proportionally, positive and negative.

The amount higher than the source signal the amplifier potential has, could be considered gain.
Now it takes a certain amount of power to meter the valve (like in a tap), there is the impedance and the voltage of the AC coming in, if it's too little it won't open the amplifiers (tap so to speak) to realise it's maximum output.

The maximum output potential of the second amplified signal can be considered it's available working power.
Even if you hold the tap wide open, it can't possibly feed more power than is available, than the potential electrical pressure the amplifier can produce, through that valve.

I say there is exotic resistance because the components of the crossovers, the windings on the drivers are reactive to different frequencies, this reactive quality can be measured as impedance curve.

https://www.audioholics.com/loudspeaker-design/understanding-impedance-electrical-phase/page-2

Resistance is the term used to denote the opposition to the flow of current in a circuit or material. It depends on the resistivity and dimensions of the material. The resistance offered by a material to the flow of direct current and alternating current remains the same. It is denoted by the letter R.

Impedance

Impedance is the opposition to alternating current. It is a combination of resistance and reactance. It is represented by a complex number and has a real and an imaginary part. The real part represents resistance and the imaginary part represents the reactance. Reactance can be capacitive, reactive or a combination of both.

source: https://www.electricalclassroom.com/resistance-vs-impedance/

@rixthetrick - thank you sir!  Love your posts and will refer to them again as needed.

Amplification can be somewhat described as if you had a small electrical signal controlling a valve on a larger electrical signal, as the smaller signal swings from positive to negative, in higher and lower amplitude from zero, it opens and closes the valve proportionally, positive and negative. 

This is where my technical ignorance seems significant.  Maybe I’d need a textbook to understand this:


Well, think about how much an amp can do.  Amps can't have infinite power, or output voltage, right?  They are limited devices.  Every one of them. What limits them?  Among many things, the voltage available to the output circuits. That is the absolute limit of output.

Your gain is 20x (28 dB) for instance.  OK, but your maximum voltage out is 10 Volts peak to peak.

1 volt pk-pk in x 20 = 20 Volts pk-pk. 

With this amp, 20 volts out is not possible.   You need a bigger amp.  What you'll get instead is a clipped signal that goes up to 10 and hangs there until the input voltage goes back down.

@erik_squires - Thank you. That’s helpful.

Well, think about how much an amp can do. Amps can't have infinite power, or output voltage, right? They are limited devices. Every one of them. What limits them? Among many things, the voltage available to the output circuits. That is the absolute limit of output.

Your gain is 20x (28 dB) for instance. OK, but your maximum voltage out is 10 Volts peak to peak.

1 volt pk-pk in x 20 = 20 Volts pk-pk.

With this amp, 20 volts out is not possible.   You need a bigger amp. What you'll get instead is a clipped signal that goes up to 10 and hangs there until the input voltage goes back down.

Look at it this way.

A Designer has design requirements.

Design an amp that produces 250 watts per channel with an input of x mV into an 8 ohm load that will produce the 250 watts per channel max without clipping.  an input impedance of y Ohms, output impedance of P Ohms, frequency range of 20 Hz to 20 kHz, etc.

This gives you the gain of the amp.  As MillerCarbon and many other have told you, it then is a straight forward calculation.

There are typical standards in many amp designs.   They call for an input voltage of x MV in order to produce max (before clipping) output of xxx watts per channel into an 8 ohm load.  So,  most audio amps have a set gain with the input sensitivity voltage of x mV to produce that max output.

This is electronics engineering 101.

The amp specs will tell you the amps gain based on the designed for input voltage to get that max power output.

The amp in question, has a gain switch that changes the amps gain.  They are either adding another gain stage in the amp via the switch to allow for lower input sensitivity or they are adjusting the input sensitivity and adding or removing a gain stage or just adjusting the input sensitivity.

either way, what MillerCarbon and some others have described is pretty accurate.

Its not rocket science, but it does require some knowledge of electronic design.

If you start with the required input voltage sensitivity, the required output power rating, the required load, input impedance requirement, output impedance requirements, the amp will have a calculated gain.

If you start with a set gain requirement and power output, into a set load, you still need to know what the input sensitivity is.

Pre-amps are pretty similar, except that there are standard pre-amp input voltage sensitivities, and max output voltage requirements (so you don't blow up your amp and speakers). 

anyway, I hope this didn't confuse you further.

enjoy 

To further confuse the issue,

Changes in amp output, measured in dB volts are directly related to SPL dB.
That is, increase amp output by 3dB and the SPL will increase the same.  Like with amps though, speakers have physical limits which cannot be exceeded

The practical difference is about reducing noise thats generated from the pre - amp feeding the amp. A pre-amp with a high output signal may deliver unwanted hum or hiss when the input sensitivity switch is at the higher setting. Along the same lines, if really efficient speakers are being used residual noise from the pre-amp may be audible; flipping the amp input switch to the lower setting will tend to reduce the noise. 

My Question:
Setup is active speakers and pre-amp.
Speakers have input gain knob for -10 to +10 db.
Which is best, crank the volume on the pre amp upto 80% or add gain on the speakers and less on the pre-amp.

Btw. Pass XP-22 and Adam Tensor Delta  speakers

That Bryston gain switch would be to match the volume of two disparate amps on the same preamp.

MC running a clinic here. Nice. 

A master clinic involves the recognition that in power amps variable gain is just variable negative feedback. See Roger Modjeski for an EE w ears and many successful designs the variable NFB RM-9 being a sonic master class. The cool thing about variable NFB w time and phase accurate speakers is that the intrepid listener ( given a decent recording ) can discern how NFB effects image depth....
carry on....
see also Burning Amp

Often just variable NFB but sometimes just R....

I find that speakers that has a low db rating benefits from a higher gain amp.
speakers that has a higher db rating benefits from a low gain amp.
wattage remains the same.
Also don’t forget your preamp gain.

The relationship between dB and power is logarithmic.
End of story.

This site is cries out for a Audio 101 list of stuff every audiophile should know. This would come right after W=VA.

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.