The process is quite complex, even for me, but I can try:

Headphones+amp can be modeled with this simple circuit somewhat accurately. On the left we have the output impedance of the amp (Rout). In the middle we have the voice coil (Rc + Lc). The right side is the diaphragm, or how it looks on the electric side.

The impedance of this circuit without Rout (only headphone) is Rc + jωLc + (1 / (1 / jωL + R + jωC)), where ω = 2πf

Below is the impedance curve of an fictional headphone and we see how we can get some values from it. The peak value (at 100 Hz here) is basically R + Rc, because voice coil inductance Lc kick in at 10 kHz or so, is neglectable at ~2 kHz here the minimum value (Rc) is seen. Another way to get Rc is to measure the DC resistance of the headhone (impedance at 0 Hz). Here we have Rc = 120 Ω, so R must be 122 Ω. We also have fo = 100 Hz. The next step is to iterate (for example in excel) the other values. Here L = 204 mH and C = 12.4 µF give the correct shape for the impedance bump around 100 Ω. Same with Lc.

What is the frequency response error, if we feed this headphone from an Rout = 100 Ω amp?

Lmax = 20*log(242 / (242 + 100)) = -3.00 dB

Lmin = 20*log(120 / (120 + 100)) = -5.26 dB

The error is Lmax-Lmin = 2.26 dB. Sometimes the impedance maximum is at 20 kHz (and even more beyond that), but most of the time the maximum is around 100 Hz due to the resonance frequency of the diaphragm.

What is the damping ratio of this headphone (without amp)? L = 204 mH, C = 12.4 µF and R = 122 Ω

Damping ratio = sqrt(L/C)/(2R) = sqrt(0.204/0.0000124)/(2*122) = 0.53

This means the headphone itself is underdamped and needs electrical damping. Is the 100 Ω amp providing enough electrical damping? Now, R becomes R* = R*(Rout+Rc)/(R+Rout+Rc) = 122*(100+120)/(122+100+120) = 78.5 Ω and we have:

Damping ratio = sqrt(L/C)/(2R*) = sqrt(0.204/0.0000124)/(2*78.5) = 0.82.

Still underdamped. How low output impedance do we need to have critical damping?

Zout_critical_damping = (sqrt(L / C)*(R + Rc) - 2*R*Rc) / (2*R - sqrt(L / C)) = 15 Ω

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