If I put my notes here, I might be able to find them again later!
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Posted 2nd April 2012 at 08:45 AM by rjm
Updated 2nd April 2012 at 11:43 AM by rjm

I posted this earlier today, but I think it deserves to be put in the blog - if nothing else so I can find it again next time ... and there always seems to be a next time when it comes to calculating headphone amplifier gains.

Starting at the beginning, the encoded data on a CD goes from 0 to 1 in 2^16 steps, but in a typical CD player or soundcard, the DAC output is -2.8 V to 2.8 V or 2 V rms or 6 dB. Many sources, such as phono stages and portable audio, are lower, perhaps as low as 250 mV.

How loud the sound is depends on the source signal amplitude, the position of the volume control, the circuit gain, and the impedance and sensitivity of the headphones.

As a practical matter, most people would want the volume control at the 9-10 o'clock position for "normal" listening.

For standard "line level" source, the gain required to keep the volume control at a 9-10 o'clock varies depending on the impedance of the headphones roughly as follows:

Impedance____Gain
16 ohms______9 dB 3x
32 ohms______12 dB 4x
65 ohms______15 dB 6x
120 ohms_____18 dB 8x
300 ohms_____22 dB 12x

That's for headphones of average sensitivity of near 100dB/mW. Most over-ear models are within 6 dB of that, independent of impedance. For in-ear types, reduce the estimate above by a 6-10 dB.

The short story is if the gain is adjusted to be just right with 16 ohm headphones, you'll have to turn up the volume control way past 12 oclock with 300 ohm headphones. If the gain is just right for 300 ohm headphones, the volume control will be will be barely off -inf, stuck in the non-linear tracking region, with 16 ohm 'phones.

When designing a headphone amplifier, therefore, you have four options:
1. Take a stand: The amp is designed either for high impedance or low impedance headphones, but not both.
2. Split the difference: Around 5-6x gain 14-15 dB is good enough to be workable with all headphones, but will be increasingly non-ideal at the extremes.
3. Make the switch: Use a dip-switch or front panel control to change a feedback resistance or otherwise act to alter the circuit gain. Two settings LO (6 dB) and HI (20 dB) are sufficient.
4. Linearize: Add a series resistance of 33-120 ohm to form a voltage divider with the load. The gain of the circuit now automatically self-adjusts to lower the gain with falling headphone impedance.

There are pros and cons to all four options. After careful consideration I generally favor method 1. My main headphones are 300 ohms, and to get the best out of them I like to optimize the voltage rails and bias currents as well as adjust the gain. My amplifier does not work well with 16 ohm earbuds - well, I can live with that.

An amplifier optimized for the middle ground is poor for either extreme. The Jack-of-all-trades truly is a master of none. If your headphones typically lie in the middle values, 30-150 ohms, then you are fine going this route.

If the circuit is an op amp or similar design such that the gain is very easy to adjust, then 3. has merit. Just be careful that the switching circuit, since it is normally part of the feedback loop, doesn't introduce instabilities.

Method 4. is on paper the most elegant approach - a simple resistor to automatically keep the output power constant over variable loads - but it raises the output impedance of the circuit to extremely high values, worst case many times the load impedance. Unfortunately, it is my experience that this results in unacceptable colorations, even with high impedance headphones. The often-stated excuse "oh, headphone impedance is fairly constant over audio frequencies" is not sufficiently true.