Science and Math for Audio Humans – No Free Lunch

by Danny Maland

It's disclaimer-time again: Everything that I set before you should be read with the idea that “this is how I've come to understand it.” If somebody catches something that's flat-out wrong, or if you just think that an idea is debatable, please take the time to start a discussion via the comments.

At the end of the last installment, I “cliff-hangered” all of you. We had just arrived at the threshold of discussing what goes on when more than one loudspeaker gets connected in a circuit with a power amplifier - but I stopped before we could get in the door. I halted our collective progress because I didn't just want to expand the math and run. Rather, I wanted to go a little bit more in depth.

Returning to an old metaphor, let's say that we have a water pump, some pipes, and a couple of water wheels or turbines. The pump is like an amplifier, the pipes are like interconnection cables, and the turbines are like speakers. Just like an electrical system, there is pressure (voltage), a flow amount (current), and opposition to that flow (impedance/ resistance).

Figure 1 shows an arrangement of these items where two turbines are connected in series, that is, the second turbine is fed from the output of the first. In this case, the pump creates a certain water pressure at the input to the first turbine. The opposition to flow that the pump experiences is simply the addition of the resistance of the two turbines.

  • This kind of connection scheme is possible in the world of audio, but it's rather rare (in the case of loudspeakers, at any rate). It is far more common to see a parallel connection, like that depicted in Figure 2.

In a parallel connection, both turbine inputs are driven from the output of the pump. Each turbine experiences the same pressure, and the flow opposition experienced by the pump is the reciprocal of the sum of the reciprocals of the turbine resistances.


Assuming that the pump can provide an adequate amount of flow for either system at a pressure of, say, 1 pressure unit, two situations can develop. In Figure 1, the pump has to create a total system pressure of 1 where the flow of water has only a single path. The flow is restricted once, and then restricted again. Each turbine added to the path makes the system harder for the pump to push against, because the pump has to push against one turbine, and then the next, and then the next. Assuming that the system isn't leaking, and the pump isn't letting water go the wrong way, then each turbine has to see the same amount of flow. That is, if one liter of water is going down the pipe every second, then each turbine has to pass one liter of water every second – no ifs, ands, or buts. The pump creates a total system pressure necessary for one liter of water to go through through every second, and the pressure across each turbine will be enough to keep the flow constant. That pressure will not necessarily be the same for each turbine, and each turbine only gets part of the total system pressure. This makes sense, because if the first turbine is very hard to push, and the second turns very readily, we would expect the second turbine to see less pressure than the first – though still enough to keep the flow through the system constant.

If that can be said to be fair enough, then the happenings for Figure 2 go as follows. (Remember that the system isn't leaking, and the pump is such that water can't flow the wrong way through.) Just as in Figure 1, the pump works to get the total system pressure up to 1 pressure unit. Because that pressure can flow directly to each individual turbine, each turbine sees the same pressure. The pump experiences the opposition to flow from both turbines, but it pushes across both turbines simultaneously. It doesn't have to get across turbine one before it can push against turbine two. While there is a total system flow (as you would expect), a very resistant turbine outputs a smaller amount of flow into the total, whereas a very free turbine contributes a larger flow output. If we keep adding branches with turbines to the pump connection, the pump effectively sees a larger and larger “pipe” connected to its output. As the pipe gets bigger, the flow opposition drops – but it also gets more and more difficult to pressurize. (You can try this with a kitchen sink. Take out anything that reduces flow through the drain, and you can run the faucet at maximum pressure without filling the sink. Put a food catch in the drain hole, and the faucet can now pressurize the sink to the point that it starts to fill.)

So, why is this post titled “No Free Lunch?”

Some folks get all excited when they realize that they get more total power out of a power amplifier as the total system impedance is reduced. A power amplifier, as we experience them, is a device that is meant to create a requested voltage across its output terminals into whatever is connected. Effectively, it's an electrical pump. Ask it for 50 Vrms (Volts root mean square), and it will do its darndest to give you exactly that as a total system voltage. Assuming that you're within the design limits of the amplifier, this total system pressure is created whether the amperage (water flow) is small or large.

If we take that amplifier, and connect a single 8Ω loudspeaker to the terminals, we get the following:

If we connect another 8Ω speaker in parallel to the first, the total system impedance looks like this (the reciprocal of the sum of the reciprocals of the impedances) :

Now, this theoretical power amplifier I've conjured up can definitely put 50 Volts into 4Ω, so we get this:

“Cool! More power!” Well, yes...except power amplifiers won't quite give you twice the power for half the impedance, if you want the distortion created to be constant. Further, there's this whole issue of the current involved:

The current that the amplifier has to provide has doubled. The amplifier is working harder, generating more waste heat, and is probably creating more distortion artifacts (whether they're audible or not is another thing).

This is why amplifiers are given a minimum impedance rating by the manufacturers. At some point, the poor amplifier just can't fill an “enormous pipe” with the necessary pressure – not without melting itself down, at any rate. Drop the impedance to 2Ω, and the amplifier has to generate a flow of 25 Amperes to keep up a “pressure” of 50 Vrms. There are certainly a good number of power amplifiers “out in the wild” that can do 25 Amperes, and more, but you can't just assume that the amplifier you have on hand is one of them.

Also, please remember that what we've been calculating are totals for the whole system. Even in the best-case scenario, each 8Ω enclosure connected to our power amplifier is still getting 312.5 Watts. The amplifier has not become some sort of magical creature that puts more power into each box – rather, it has supplied more power to the system as a whole, and each component in the system still has to share that power as per the demands of physics. There's no free lunch!

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