So time for a little background research.
Mixing Consoles I'd expect that most people know what a mixer/mixing console is, but for those who don't: a mixer takes in all the sound sources (microphones, guitars, keyboards, pre-recorded or streamed audio, etc.) of a performance (music, theater, radio show, etc.) and puts it all together into a either a single mono or stereo feed for the house speakers or for a broadcast (they are used in studios for all parts of recording too, but I'm pretending we're just considering the live case). The mixer allows you to set the volumes of each part, pan the sound to the left or right speaker, and usually allow some basic EQ. Most mixers beyond your basic ones will also have a whole host of auxiliary inputs and outputs for adding extra effects, sending out monitor mixes (so the musicians can hear what they're playing or singing), and pretty much any other reason you could think to send different mixes around. It's these auxiliary outputs that will be of interests to us. If you really want to know more, just google "mixing consoles" and you'll find plenty of info.
Just to get a feel for the console outputs, I looked at the specs for a number of mixer manufacturers including Mackie, Peavey, Yamaha, and Behringer and found that typical outputs are around 22dBu max output level with an source impedance of 120R. I'll go into what that means later.
Amplifier Or perhaps more specifically, a power amplifier. Amplifiers do pretty much what you expect; make something bigger. But what actually is that "something"? Well often within electrical circuits it will simply be a voltage. However, when you want to make that voltage do something in the real world you need power, and that is where things get interesting. You will need more than voltage alone. Take the starter motor in your car, for instance. The vehicle battery is only 12V but can pump out the 1.5HP (746Watts/HP) or so to get the engine started. The 48V phantom power from a mixing console? It can usually illuminate a few LED's. That source impedance I mentioned earlier will come into play....
What I am showing above is the scenario I just presented. You should notice that there is a series resistance internal to both the car battery and the mixing console phantom power supply. In the case of the former, the internal resistance is simply a fact of life due to battery chemistry and physical interconnects and ideally is as low as possible. For the latter, the internal resistance is intentionally added to protect the internal circuitry from supplying larger currents than intended (including short circuit loads) as well as possibly protect musicians from handling a faulty microphone that might now expose them to 48V. I've been zapped in the face by 120V due to a combination of bad guitar amp wiring and a microphone. It sucked.
Anyhow, for the car battery we have a low internal resistance compared to the load. And we see that most of the power is delivered to the starter motor. Looking at the values, the battery is pumping out 200W + 1000W for a total of 1200W. That's quite a bit. If you were to take a multimeter and probe the battery terminals, you would see 10V instead of 12V as shown in the diagram. When you first turn on the starter its resistance is closer to a dead short meaning that you could measure something like 4V across that battery. That's why all your dash lights dim. Another thing to note is that the 200W internally absorbed turns into heat.... so don't crank on it too long. But you get the idea, voltage sources have internal resistances that absorb some of the total power and drop some of the voltage.
The phantom power example just demonstrates the opposite. In this case, the internal resistance dominates, leaving little for the actual load (the LED). R_internal limits the available current and in turn limits the deliverable power.
This brings us to the "Maximum Power Transfer Theorem": to deliver the most power, the load needs to be equal to the source impedance. However, that doesn't mean that is the goal of a power amp design. Let's look back to the battery. If R_Starter was set equal to R_Battery we'd see 1800W delivered to the starter. But we'd also see 1800W delivered to the battery's internal resistance. Not so good. At these power levels this is woefully inefficient. You can afford to throw away a few mW lighting an LED, but 1800W? You can run a microwave off of less. And here we are just letting it go to waste. Not to mention that it's all turning into heat.... Something to consider for design later.
Our amplifier will need to be able to deliver enough current at a given voltage to power the load: headphones.
Headphones Seems kinda self explanatory. The headphones convert the electrical signal to physical vibration. Sound. While the mechanism is actually rather simple, I won't go into how they work here as it's not critical. Feel free to investigate on your own. But what kind of voltage levels do you throw at headphones? What kind of impedance do they have? What is loud?
Well I own a pair of Sennheiser EH-150's. Nothing fancy, but a starting point. Fortunately, Sennheiser has specs on their website. These headphones have a 32R impedance and a sensitivity of 115dB. That's loud. Even if you don't understand the math.
Let's look at the impedance first. The mixer has an internal impedance of 120R, and from the previous discussion, we know that the smaller impedance of the headphones will mean that it'll be difficult to deliver power to them. At least we've got a feel for that side of things.
Now let's consider the sensitivity spec. Often this spec is given as the output in dB @ 1mW, but Sennheiser actually specs it as the output dB @ 1Vrms. This will actually be convenient for later analysis. So we know that a 1Vrms signal will output 115dB. I want to stop here as this post is getting long and further discussion may need more math primers. But, we've got a better understanding of real world values and considerations for the problem at hand.
