# Science and Math for Audio Humans – Putting It Together: Loudspeakers

by Danny Maland

The disclaimer, one final time:

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.

To close up this series, we're going to look at how concepts come together in the context of loudspeakers. The last time around, the focus was on microphones. Today, we'll be transducing in the opposite direction; we'll be turning electricity into sound pressure waves. To do that, we'll be using a device that looks remarkably like a dynamic moving-coil microphone. The difference is that this device is meant to handle much bigger excursions of its diaphragm, is built much larger, and (of course) is meant to produce sounds of its own.

As an aside, yes, you can wire up a garden-variety loudspeaker to a microphone input and get sound into a system with it. The frequency response won't be much to write home about, of course, but you can do it. As a general-purpose input transducer, this sort of "kludge" doesn't make the grade. However, it can work handily as a single-purpose or special-effect trick. Indeed, on the market today is a mostly-meant-for-low-frequencies input transducer that is a loudspeaker mounted in a drum shell.

Figure 1 shows the essential innards of a loudspeaker.

• A loudspeaker's diaphragm is the “piston” used to impart a push to air molecules. The piston has to be suspended in some sort of frame to work properly, which is where the surround and spider come in. The suspension of a loudspeaker also has the important task of ensuring a consistent “return to zero” (or, “return to start”) when the assemblage stops moving. Much like a dynamic microphone, the loudspeaker's coil is suspended in a magnetic field. The difference in operation is that a relatively large electrical input is presented to the coil, which turns the magnet/ coil pairing into an electric motor.
• Let's assume that our loudspeaker has been loaded into an appropriate enclosure by an equipment manufacturer. The manufacturer claims that this particular driver can handle 500 watts of continuous input. They call it “RMS power” on the spec sheet, in defiance of the folks who will complain that this is the wrong terminology. (It IS the wrong terminology, but as mentioned previously in the series, you will be okay if you know what that term is really meant to symbolize.) The manufacturer also has a frequency response chart that shows the unweighted SPL produced by the assemblage when driven with 1 watt of input – and measured 1 meter away. At another spot in the spec sheet, they've reduced that measurement to a shorthand “sensitivity” number of 99 dB/1w/1m. That number is an average that they arrived at in some way that may or may not be clear to us.
• Figure 2 shows the frequency response chart from the spec sheet. The passband we're interested in for our application is highlighted in blue.

Question: What “worst case” unweighted SPL measurement would we expect to see if the loudspeaker was driven with the maximum recommended continuous power, and measured at 1 meter?

To answer this question, we first have to determine where, in our intended passband for this device, we get the least SPL. Taking a look at the chart, we can see that the lower limit of our passband, 50 Hz, is where the loudspeaker produces 93 dB SPL-Z at 1 watt/ 1 meter.

To figure out the theoretical SPL at 500 watts, we need to get out our logarithmic math. The good news is that this calculation is very easy. We're dealing directly in power, and our reference power is 1 watt, so there is effectively no calculation required for the division.

In theory, the device can produce a continuous level of 119.99 dB SPL-Z when driven at full power, measured at one meter. Again, this is assuming the worst case in our passband, as the loudspeaker system in question is noticeably less sensitive at 50 Hz than at 100 Hz.

Question: Why might the actual SPL measurement be higher than we expect? Why might it be lower?

It is quite possible that our theoretical numbers will not match up with what we actually experience from this device. For instance, it is possible that the manufacturer measured their device in an anechoic chamber, and it is unlikely that we are in an anechoic environment. Depending on the manufacturer's test setup, their anechoic measurement may have effectively been “full-space,” where all sonic energy not directly traveling to the measurement point is lost to the measurement.

As you might remember from the beginning of this series, such an environment for actually listening to a loudspeaker's output is very rare. If we're indoors, it's far more likely that we're in a situation that is somewhere at or between “half-space” and “quarter-space.” Half-space is called that because it assumes that sound is being forced to radiate into half of a sphere, instead of a full sphere. Quarter-space and eighth-space follow on from this. In theory, each halving of the radiation space allows the observed SPL to increase by 6 dB, although this is dependent on the walls involved being large enough to reflect the wavelengths being emitted, and also on the walls being perfect reflectors.

The main reason for us to see less than the maximum output we expect is that of power compression. Power compression is caused by voice coil heating. In metals like copper, which is the most common material used for voice coils, resistance to current flow increases as temperature increases. If we run our loudspeaker with the maximum continuous input power that it can withstand, the voice coil will inevitably heat up. As it does so, it becomes more difficult for the amplifier to drive the loudspeaker assembly. As a result, we do not appear to get the full “theoretical” benefit from the power delivered. (This is not strictly true, of course – our problem is that our theoretical numbers don't take power compression into consideration.)

Another part of power compression is that a driver undergoing large excursions may very well have larger-than-intended portions of its voice coil leaving the gap in the magnet. As a result, the influence of the magnetic field on the coil is reduced. This mostly results in distortion products, but it can also contribute to voice coil heating. Thermal management for loudspeaker voice coils loses efficacy when the voice coil has traveled beyond where a designer expects it to be.

Question: What is the maximum distance from the loudspeaker for 85 dB SPL unweighted, assuming full continuous power is applied, and assuming a full-space environment?

To answer this question, we first start with our “maximum output at 1 meter” determination. In this case, our starting number is 119.99 dB SPL-Z. The difference between 85 and 119.99 dB SPL-Z is 34.99 dB. Theoretically, each time we double our distance to the loudspeaker, we lose 6 dB of measured SPL. Ultimately, we have to figure out how many doublings of distance it takes to get to a 34.99 dB loss.

Each time we double the distance, we can represent it as a power of 2. For instance, 2 raised to the first power is 2 meters, or double the distance from 1 meter. The double of that distance is 4 meters, or 2 raised to the power of 2. The double of that is 8 meters, or 2 raised to the power of 3. That 6 dB loss, then, becomes a coefficient that we can multiply with our exponent to find the total theoretical loss at a particular distance. What this turns into is dead-simple algebra.

So, 2 raised to the power of 5.8312 should give us our distance for 85 dB SPL-Z continuous. In theory, we should be able to go out 56.95 meters and still have 85 dB SPL-Z to work with. Of course, we do have to remember that we won't be running the loudspeaker at full throttle all the time, but at least we have some sort of idea about what is “in the ballpark” of expectation for the moments when we are running at full tilt.