Ideally you want the HP input impedance to be about 8 (or more) times higher than the output impedance of your amp.
As a
rule of thumb rather than
ideally, as for each headphone model what is or isn't ideal can be quite different and depends on its electro-acoustic properties. There are headphone models that can be driven from "any" output impedance without problems with sonic accuracy and there are headphone models suffering from some inaccuracies even when the 1/8 rule is being applied.
The output impedance of an amp driving headphones affects the frequency response, damping and distortion. Of these the frequency response criteria seems to be the most demanding: If you make it good enough, the other things are taken care of also. So, headphone manufacturers could calculate it as a spec data point for the customers using this equation:
Rout ≤ (0.06 * Zmax * Zmin) / (Zmax - 1.06 * Zmin),
were Zmax and Zmin are the maximum and minimum impedances respectively measured on a "head" to mimick the real life acoustic impedance seen by the headphones (headphone impedance changes a little bit depending on whether it is on head of not). This equation limits the frequency response error to 0.5 dB or less. So, if you measure Zmax = 200 Ω and Zmin = 50 Ω you get
Rout ≤ (0.06 * 200 * 50) / (200 - 1.06 * 50) = 600 / 147 = 4 Ω
If Zmax is just a little more than 6 % bigger than Zmin, the allowed amp output impedance becomes very large. If Zmax is exactly 6 % bigger than Zmin, this equation gives infinite amp output impedance and if Zmax is less than 6 % bigger than Zmin, the "allowed" output impedance is
negative. These are of course non-sensical values and in these cases the limiting factor for the amp output impedance is the voltage division between the amp output impedance and the headphone: In order to limit the voltage division attenuation to 3-4 dB at most, we want at least 2/3 of the amp voltage over the headphone and at most 1/3 over the amp output impedance:
20 * log (2/3) ≈ -3.5 dB
This limits the amp output impedance to half of the headphone impedance at most. So, if you measure Zmax = 111 Ω and Zmin = 98 Ω (Zmax is about 13 % bigger than Zmin), you get
Rout ≤ (0.06 * 111 * 98) / (111 - 1.06 * 98) = 91.7 Ω
Since this violates the voltage division rule, the largest allowed amp output impedance is limited to Zmin/2 = 49 Ω. It is easy to show that the voltage division rule steps in whenever Zmax is at most 20 % bigger than Zmin. We can put all of this together by modifying the original equation into a form that avoids issues such as negative/infinite impedances and division by zero:
Rout ≤ (0.06 * Zmax * Zmin) / MAX( 0.14 * Zmin, Zmax - 1.06 * Zmin ),
where MAX(
a,
b ) simply means picking the bigger value of
a and
b.