How important is the power rating with headphone amps?

The power that the amp can deliver is important, the amount of power available must be somewhat more than enough to play the 'phones at the volume level you want.

Most headphones need very little power, in general.  A tenth of a watt is much more than enough for most headphones.  HOWEVER:  there are some notable exceptions.

• HifiMan HE-6 needs quite a lot of power.  At least a watt, some would say 4 watts.
• Audeze LCD-2 needs a fair amount of power- half a watt at least, I'd say.
• AKG K-1000 need a couple of watts at least.

Look at the headphone's SENSITIVITY specification.  This tells you how much power is needed. 

The sensitivity is rated by measuring how loud the headphone is with 1 milliwatt of power.  1 milliwatt is one one-thousandth of a watt, written as 1 mW, or 0.001 watt.

The AKG K-701 has a rated sensitivity of 105 dB.  That means, if you feed the k-701 0.001 watts of audio power, it produces sound at a loudness of 105 dB.  Listening for any length of time at 105 dB will damage your hearing.  It is suggested that 85 dB be used as a limit for long term listening.  90 dB might be OK too.  But 105 dB is definitely REAL loud.  

Loudness and power have a logarithmic relationship.  That is to say, every time you DOUBLE the POWER, the LOUDNESS goes up by 3 dB.

If you want to DOUBLE the LOUDNESS that you perceive, you need to increase the sound level  by 10 dB.  An increase of 10 dB will be perceived by most people as a doubling of loudness. (This is how they came up with the idea of the DECIBEL in the first place- they played two sounds, one louder the first. They asked their test subjects, "tell me when the second sound is TWICE as loud." They then called this doubling increase in volume 10 dB.  Then they asked people, "Tell me when the second sound is JUST BARELY louder than the first." They called this barely-perceptible  increase in sound 1 dB.  That's where the dB comes from- it's based on human hearing. As it turns out, to increase loudness by 10 dB (double the volume) you need about TEN TIMES the power.

So:

• Doubling the amp power (from 50 watts to 100 watts, or from 3 to 6 milliwatts,for example)  gives you 3 dB more loudness
• If you apply TEN TIMES the power, the sound will be TWICE as loud.  (for example, if you have an amp rated at 100 watts for your speakers, and want something twice as loud, you need to buy a 1,000 watt amplifier.)

Therefore, if 1 milliwatt of power produces 105 dB from the AKG K-701, then to listen at 90 dB .... do the math.  

.001 watts = 105 dB so therefore .000316  watts (0.3 milliwatts) will give you 90 dB from your AKG K-701's.

Impedance also plays a role.  The impedance is a measure of how readily the headphone will draw power from the amplifier- how much of a load it is. The relationship between power and impedance is governed by ohms law, within the limits of the amplifier.

An amplifier that produces 100 watts into an 8 ohm speaker can produce about 25 watts into the 32 ohm impedance of the AKG K-701.

This same 100 watt / 8 ohm amplifier would deliver 2.66 watts into a 300 ohm pair of Sennheiser HD800's

HOWEVER!

The lower the impedance, the more CURRENT is needed from the amplifier.  According to Ohms Law, an amplifier that produces 100 watts into 8 ohm speakers OUGHT to be able to produce 200 watts into 4 ohm speakers, and 400 watts into 2 ohm speakers.... and 800 watts into  1 ohm speakers... and, in fact, into a DEAD SHORT which is ZERO ohms, the amp should deliver INFINITE watts of power.

But we know that's not true.  Amplifiers have a limit to their current output-  at some level of watts they just CAN'T anymore.  Try running a 1 ohm car subwoofer array on a cheap amplifier and what you will get is either a BLOWN FUSE, or SMOKE.

This also happens with headphone amps. A headphone amp might be able to deliver 0.01 watts (10 milliwatts) into 300 ohms, so it will play dandy loud on your Sennheiser HD800's.  That amp SHOULD be producing about .05 watts (50 milliwatts) into a 50 ohm pair of HiFiMan HE-6's, but many amps- OTL tube amps in particular-  just CAN'T produce the needed current so in fact they cannot develop 0.5 watts into 50 ohms, they will probably TRY to produce this level of power but they won't be able to do it, and the result will be massive distortion if you crank it up as the amplifier hits the wall in terms of it's electrical limits.

See http://gilmore2.chem.northwestern.edu/tech/dbohn1_tech.htm  a tutorial on Understanding Headphone Power Requirements by HeadWize

Great post milosz.  I tried to answer a similar question the other day, but your understanding is a bit better than mine.

Informative as always milosz. Thanks for the insights.

I have a question somewhat related to power.  I've read numerous times that gain is as important as power output.  Please correct me if I'm wrong about what I say here. 

The power rating of an amp is the maximum power output before clipping occurs.  It's determined by the components that the amp is made from.  Gain is the ratio of the input power and output power and is unrelated to the power rating other than components that increase the power rating also tend to increase the gain.  Gain however can be lowered with a resistor network.  An amp can have a very high power rating, but if the gain is low enough, the amp will never really reach the maximum power with a realistic line level input.

I really wish more amps stated their gain.  Almost every amp I've tried has way too much gain for the levels I listen at.  Portable amps are the worst offenders since most portable headphones/IEMs are low impedance and low sensitivity. 

except that some of the numbers are wrong - K701 have a little lower than average sensitivity @ 93 dB re 1 mW - manufacturers are changing to dB re 1V for which the 62 Ohm K701 is sucking down 16 mA, for 16 mWrms

amp power rating must include load R and you need some math if they haven't speced your headphone's Z

and the power should be 10x (20 dB) the amount required for your max average listening level - live acoustic musical performances can average 90+ dB and peak at over 120 dB (hopefully just for seconds)

http://gilmore2.chem.northwestern.edu/articles/hearing_art.htm

a number of discrete or chip buffered op amp desktop amps can happily power the K701 to 120 dB dynamic peaks without clipping for occasional listening sessions at live levels

for all day listening the average SPL should be kept below 80 dB and most portable players will be more capable than many cheap portable amplifiers - the players use purpose designed op amps to put out ~ 1 Vrms into lower Z loads than the K701 - many aftermarket portables use standard op amps that don't work well at the required currents and low battery voltages - they really are objectively worse than just plugging into your DAP

Thanks for the info...just what I was looking for.