Electronic – How to calculate the DC blocking capacitor at the output of a transistor amplifier

There are two "tricks" to answering those questions with transistor amplifiers.

The first trick is to understand that the current-transfer ratio (h_FE) of a transistor effectivly multiplies the resistance at its emitter. So, to find the impedance looking into the base, you calculate the effective resistance at its emitter, multiply this by h_FE, and then add the internal base resistance.

The second trick is to realize that the result of the calculation above is typically orders of magnitude larger than the other resistances connected to the base of the transistor and can therefore be ignored. In other words, the input impedance of a transistor amplifier is usually very close to the impedance of its bias network alone. Indeed, bias networks are very often designed so that this is the case.

The output impedance is a question of how much the output voltage changes with output current: ΔV/ΔI. The transistor itself is essentially a current source, and whatever current it is passing is shared between the various resistances connected to the emitter. Therefore, the output impedance is equal to the net emitter resistance, not including the resistance of the external load.

Relying on the headphone impedance for filtering is probably a bad idea. Different kinds of headphones will be significantly different. Even different instances of the same model of headphones can be different. The headphone impedance isn't just resistive but also reactive, and it will take some work to characterize it. Someone might plug your thing into a line input, instead of headphones, and expect it to work.

Instead, consider buffering the output. For headphones, an op-amp will do. You can, in fact, find tons of designs online for "headphone amplifiers". Some of them even use very expensive op-amps from Burr-Brown costing $50 or more, and, I'm told, these sound better than unobtanium flux linkages. Personally, I just use whatever op-amps I have in the parts drawer. Anyway:

^{simulate this circuit – Schematic created using CircuitLab}

Now calculating C1 is easy, because you also get to pick R1. Just make it anything significantly bigger than your source impedance. Make C1 whatever you like to get the desired frequency response. The headphone impedance is largely irrelevant, because the output impedance of OA1 is so small.