Guessing, but I'd assume that headphones uses some kind of equalizing filter at the input to tweak the signature. That filter might not behave as intended if you replace the voltage driver with a current driver. In normal filters for instance the first pole will be removed by doing this.
I'm not aware of any headphones that have any response shaping equalization built into them. I'm pretty sure there's probably something out there that does, but everything I've seen has the driver driven directly.
Nor am I aware of any loudspeaker or headphone driver that's designed to be driven by a current source. Everything assumes it will be driven by a voltage source.
If you look at the impedance plot of a typical dynamic driver, whether loudspeaker or headphone, you'll see that it peaks at the driver's resonant frequency. This is where the driver is working most efficiently compared to above and below this frequency. And as was mentioned previously, it's current through the voice coil that produces the magnetic field that interacts with the driver's magnets and causes the diaphragm to move.
So now when you drive it with a voltage source, because Ohm's Law says that current is equal to voltage divided by impedance, and since impedance is highest at the resonant frequency, there will be less current flowing in the voice coil at that frequency. Which makes sense because where it's most efficient is where you need less power to get the same amount of output.
But drive that same driver from a current source, and everything gets turned upside down. Because a current source wants to drive the same amount of current through the load regardless of the impedance, when you get to the peak of the impedance curve, which again is where the driver is most efficient, there will be much more current flowing through the voice coil compared to driving it with a voltage source and as a consequence, the output of the driver will be much greater. It's the same as if you used a voltage source and used EQ to give it a peak in the response at the resonant frequency.
In other words, if you start out with a driver that has a flat response when driven from a voltage source, as you transition to a current source, the frequency response starts to look like the impedance curve with a peak at the resonant frequency (as well as at high frequencies where the voice coil inductance starts coming into play). When Nelson Pass began experimenting with current source amplifiers to drive single driver loudspeakers, he had to add networks to compensate for this and avoid peaks in the low frequency response.
This isn't an issue with orthos though due to their flat impedance. But with dynamic drivers, they're designed to be driven from a voltage source.
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