Science and Math for Audio Humans – Power and Loudspeakers

by Danny Maland

Once again, my disclaimer. Again. Some more: 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.

In the last go-around, I tied up the loose ends of how decibels work. An important piece of that process was talking about how voltage relates to power. This is a big thing for audio folks, because pretty much everything we do comes down (eventually) to our output transducers. That is to say, loudspeakers. If we didn't have some way of converting electrical signals into air pressure, audio dudes and dudettes wouldn't have much of a job – at least, not as we recognize the job today.

Loudspeakers are an enormous topic in and of themselves, but the thing that seems best to tackle right now is the idea of how loudspeakers and power come together.

The first thing to get at is the idea of impedance. Impedance, like resistance, is opposition to current flow. Impedance is more complex than resistance, however, and is meant to describe opposition to flow in circuits where voltage varies over time. Circuits that utilize alternating current are not only subject to “straight line” DC (Direct Current) resistance, but also feel the effects of shifting magnetic fields and capacitance. (As an aside, capacitance is the ability to store electrical energy by means of electrical fields, as opposed to, say, chemical means.) For audio folks, the big thing to realize is that impedance changes with the frequency of voltage being put across a circuit.

A visualization of this can be seen in Figure 1. Figure 1 is the generalized representation of a loudspeaker driver's opposition to current flow when graphed against frequency. (Please note that this plot assumes a loudspeaker driver that uses a voice coil and a magnet.)

  • There's this marked impedance peak down at the resonant frequency of the driver, because that's where the driver vibrates the most readily. You would think that this would cause an impedance drop instead, but the driver being able to cycle the most freely means that it can also move the most through the magnetic field provided by the magnet. This being the case, the driver at its resonant frequency can “push back” to the amplifier with the most “force.”
  • If you think this is wild and wooly, it gets even more interesting when you stick that loudspeaker in a ported box, or mate it to a horn, or add a passive crossover, get the idea.
  • Now, the reason that manufacturers rate drivers and loudspeaker systems with a single number for “nominal impedance” is that it provides a “pretty close enough” shorthand for what a loudspeaker's impedance looks like through the frequency range that the manufacturer expects to be in play. An audio human ought to be aware that impedance doesn't come down to a constant number, but there's no reason to be overly concerned about the curve – unless you're designing drivers or loudspeaker enclosures, of course.
  • So, having it in our heads that loudspeakers oppose current flow as a kind of exotic resistor, we can begin to understand what happens when we start connecting loudspeakers and the amplifiers that love them. Figure 2 shows a very simple setup.

We've got one amplifier, and one loudspeaker system. We're going to take it on faith that the loudspeaker system's nominal impedance really is what the manufacturer says it is, 8Ω. (The Omega is the symbol for Ohms.) Our power amplifier is giving us nice, steady output of 2.83 V RMS.

Hey – what's this “RMS” business?

The thing is that the voltage coming from the amplifier, being an audio signal, is time variant. We could talk about the voltage at some particular instant in time if we wanted to, but that wouldn't really help us much, especially if we chose the particular instant where the voltage was at zero. (In order to go from positive voltage to negative, and then back again, you have to make “zero crossings.”) We could take an arithmetic average of the voltage, but if we're talking about a nicely symmetrical sine wave, that simple mean of the voltage values is zero. What we need is an RMS (Root Mean Square) average. RMS uses the root of the mean of the squares of the values involved.

Let's say we have measured values of 0, 1, 3, 1, 0, -1, -3, -1, and 0. If we square each of those numbers, we get 0, 1, 9, 1, 0, 1, 9, 1, and 0. What's ever so handy is that multiplying a negative number by a negative number gives us a positive result, and so the negative-going half of the sine wave no longer poses an arithmetic problem. We add up our new values and divide by the number of values (9) to get a simple mean of about 2.44. The square root of 2.44 gives us our RMS average of 1.56.

For sine waves, the RMS voltage quite handily translates into what the voltage would be if the circuit were DC. I should also mention that amplifiers are usually rated in RMS, or continuous power. Some folks rail against this as being meaningless for reasons of purity in the discipline of physics. I don't particularly see it as a problem, as long as you realize that what it is really meant by “RMS power” is “the continuous power derived from the RMS voltage provided by this amplifier into a given load.”

Enough of that, though. From last time, here's the equation for power...followed right behind by the equation for current:

I'm showing you both of these again so that I can prove a version of the power equation that lets us work only with voltage and resistance. Here it is in Figure 5.

This lets us just plug in the numbers from Figure 2.

So, 2.83 V RMS into an 8Ω load is just about 1 watt of continuous output. (It's actually 1.0011125 watts, but referenced to an even 1 watt, the decibel difference is 0.004 dB. Not worth losing sleep over.) Incidentally, this is why manufacturers use 2.83 V RMS so often when talking about loudspeaker sensitivity – put that into an 8Ω loudspeaker, measure the SPL from 1 meter away, and there's your 1watt/ 1 meter sensitivity figure. (You can also measure from 10 meters and then mathematically derive the 1 meter SPL...)

Well, that's neat and all, but what happens when you have more than one loudspeaker involved? I'd love to tell you right now, but I'm out of space.

I'll show you that whole business next time.

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