Output Impedance

Who dat?

One of the important factors of a headphone source output quality is its output impedance, yet this spec is often not published by manufacturers.

Just because a headphone jack fits into an amplifier doesn't mean that those two devices will play along nicely. Don't worry, nothing will break, but sound can get affected:

TL;DR: Amplifier output impedance should be lower than 1/8 of the headphone's impedance

Why does Output Impedance matter?

We generally want our source output impedance to be significantly lower than the impedance of the headphones. This is due to 3 main reasons:

A large output impedance can attenuate the signal reaching the headphones, resulting in not enough volume output.
Headphones have different impedances at different frequencies. This can result in dramatic changes in frequency response. Headphones with more varied impedance curves are affected more by this effect. This is because the headphone impedance and the output impedance of the amp form a voltage divider which varies by frequency.
Electric damping factor is reduced as output impedance increases. A common rule of thumb is that the output impedance should not exceed 1/8th of the nominal impedance of the headphones. So, for 300 Ohm headphones, you'd want no more than 37.5 Ohms output impedance; for 16 Ohm headphones, no more than 2. This is just a rough guideline, as they are exceptions to the rule.

A technical explanation

Zo = Amplifier Output Impedance

ZL= Headphone Impedance (aka. Load)

Uout = Total output voltage of the amplifier

Uo = Voltage drop over output impedance

UL = Voltage drop over load impedance

As the output impedance (Zo) gets closer to the load impedance (ZL) more voltage gets dropped over the output impedance and the voltage that drives the headphones (UL) gets smaller.

You can see in the right image above that the impedance of most dynamic headphones is not linear. That means the voltage reaching the headphone drivers is also not a constant, but a function of the impedance.

Example: Assume you have an amplifier with that outputs 1V at 100 Ohm and an output impedance of 80 Ohm. You connect a headphone with the impedance shown on the right image above

Uout = 1V
Zo = 80 Ohm

The Voltage drop over the output impedance can be calculated with the formula

Uo = Uout * (Zo / Zo + ZL))

Now because ZL is the impedance load and changes from ~300 Ohm at 55Hz to ~80 Ohm from 1000Hz - 10000Hz we get

Uo = 0.2V @ 55Hz given ZL= 300 Ohm

Uo = 0.5V @ 1000Hz given ZL = 80 Ohm

so the voltage driving our headphones is

UL = Uout - Uo

UL = 0.8V @ 55Hz

UL = 0.5V @ 1000Hz-10000Hz

That shows that the amplifier's high output impedance results in a higher voltage supply at 55Hz than above 1000Hz. Supplying more power to the headphone driver at lower frequencies will result in an attenuation of the frequency region above 1000Hz.

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