Nonlinear Distortion Testing in Loudspeakers
Distortion testing in loudspeakers seems to be overlooked in the DIY community, or at least, it is not emphasized as much. Which is unfortunate, because a loudspeaker must be capable of reproducing the music, not only with a flat frequency response, but also with a high degree of dynamic range at both the lowest and higher frequencies.
This paper deals only with nonlinear distortion products. Linear distortion will be covered separately. Non-linear distortion refers to the generation of different frequencies measured in a loudspeaker response that are not present in the original stimulus. There are essentially two types of nonlinear distortion, harmonic and intermodulation distortion. Harmonic distortion refers to the generation of tones that occur at multiples of the original tone (See the figure immediately below). This is contrasted with intermodulation distortion, which is the distortion that occurs when two or more frequencies are used as the stimulus and the result contains frequencies not at integer multiples of the fundamental, or stimulus frequency.
So, how is this figure generated and what does it mean? Look at it this way. First, a stimulus is applied to the unit. Then, the response is measured. Lastly, some graphical representation of the results is generated. In the graph above, what’s the stimulus? It’s a single tone, at 50 Hz. Marker 1 denotes this. That is, on the electrical input side we have a 50 Hz sine wave being applied to the drive unit. Measuring the electrical input directly is shown in the next figure.
So this is the stimulus. On the x-axis we have frequency and on the y-axis, amplitude level, in the same way that we look at a frequency response graph. The difference is that instead of graphing the frequency response as a continuous line, we have discrete vertical lines representing the amplitude of the applied signal. So, we have a single line at 50 Hz. The response is again shown on the next page.
This seems simple enough. But remember, that was a graph of the stimulus, not the response. Going back to the response, in this case using a GR Research M130 woofer in an ML TQWT alignment, is again duplicated below. Yes, it’s the same as the first graph, reprinted here for convenience.
Now, the axes are the same, but the amplitude is measured in SPL, as we are measuring the acoustic response of the loudspeaker unit. Note that we have a 50 Hz tone, the same as the input electrical stimulus present. This is the fundamental. But we also have other discrete bars, most prominently at 100, 150, 200…and so on. These are integer multiples of the original signal and represent harmonic distortion products. In general, these are referenced to the original signal. So, looking at figure three, the original is at 86dB, and the 2nd order HD product at 100 Hz is at 52dB. Or, we say the 2nd order HD product is 86-52=34 dB down. As a quick rule of thumb, -20 dB represents a signal that is 10% of the original, -30 dB around 3%, and –40 dB at 1%.
There is another way to represent the harmonic distortion products.
Here, a frequency sweep has been done as the stimulus and measured. This is the black line just over 80 dB. The response in red is chosen as the second order harmonic (I know it says yellow-that didn’t show up well). That is, as function of the fundamental frequency, the system is graphing the amplitude of the second order harmonic. So, at 1k, the fundamental level is just over 80 dB and the level of the second harmonic is just over 20 dB, or, -60 dB if you like, or, 0.1%, if you like.
This may or may not be obvious, but remember that the frequency of the second harmonic is twice that of the fundamental. So, at 1k, the driving signal is 1k and its level is ~80 dB, and the level of the 2nd harmonic is just over 20 dB, but the frequency itself is 2k. This is why you will see graphs like this stop at 10k for the second harmonic, around 7k for the third harmonic, etc. If you have a system that can only sample to 48k (i.e. a frequency of up to 20-22k), your system can’t sample a 2nd order harmonic with a stimulus of over 10k.
This graph and the prior graph both tell us similar information. The first graph can show multiple harmonic distortion products for a single frequency, while the figure above shows a relatively continuous sweep, but can usually only show a limited number of products before the graph becomes cluttered with data.
Contrast this with the following graph.
What do we see? Well, the stimulus is a set of two sine waves at frequencies of 200 Hz and 2.2kHz. Now something interesting occurs. We have harmonic distortion products, but we also have products that don’t fall on integer multiples of either stimulus frequency. This spectrum of new frequencies is termed intermodulation distortion. Note that they can be generated in a number of ways with many different stimuli. Generally speaking, IM distortion has more frequency products, and can be higher in amplitude. Note the rich spectrum. Compare this with the exact same setup, but the 200 Hz tone removed.
Note the marked reduction in the level and number of distortion products when we remove one of the two tones. Basically, more complex signals have a larger number of distortion products and higher levels. Simple harmonic distortion testing is useful, and can test the limits of excursion of a driver, but more complex stimulus can be helpful for making more detailed analysis of distortion.
Another form of stimuli for nonlinear tone testing uses a waveform that a frequency that is 100% amplitude modulated at 1/10th its frequency. See SL’s site link at the beginning for details. However, an equivalent representation of this signal is that it’s a set of 3 sine waves defined in the following way. F, F-10%, F+10% and the amplitude of the sidebands is one-half the center frequency. Still confused? An example makes it clear. Instead of a single frequency at 50 Hz, you have a center frequency at 50 Hz, and two sidebands at 50 Hz+/-5 Hz (ten percent of 50). Or, to make it really obvious, a single 50 Hz tone at say, 86 dB and two sidebands at 45 and 55 Hz with amplitude of 80 dB. A typical response, in this case for a Morel MDT55 dome midrange, is shown below.
Why is this interesting? Well, for a couple of reasons. First, it’s a more complex signal with a richer spectrum of distortion products. You can actually hear audible changes in the waveform as the amplitude is being increased, i.e. distortion and correlate it to actual numbers. Second, it’s easy to see the harmonics products, the IM products, and whether they are 2nd, 3rd, or higher order.
Now I’ll throw out a couple of examples.
This is the spectrum of distortion products that occur when a loudspeaker based on the 4" Vifa TC11 midwoofer is driven by a 35 Hz signal. When you hear this, you hear a signal that sounds quite different than it should, even though some people would be used to hearing this, and wouldn’t even call it distortion.
What do I mean? Well, first off, the second order harmonic and IM distortion products are around 27 dB down from the fundamental, or, 3-5%. The third order products are only 15 dB down, or greater than 10%.What’s really bad though, is that the ear has a decreasing sensitivity as you go down the spectrum. If you look at the Fletcher-Munson curves you can see that, compared with a 100 Hz signal, our ears’ sensitivity is down 10 dB at 70 and 20 dB at 35 Hz. So, a 70 dB 35 Hz signal sounds subjectively as loud as a 50 dB signal at 100 Hz.
Uh, oh…that’s a major problem. That means that the third order distortion products, which are down 15 dB at 105 Hz, will "sound" 20 dB louder if you look at it in phons. So, the above signal doesn’t really sound like a 35 Hz tone, it sounds like a 105 Hz tone, mixed in with some 35 Hz, and less so 70 Hz tones. There are some higher order products, but not bad.
I know what you’re thinking; a 4" midwoofer can’t be expected to produce 35 Hz bass. That’s not the problem. The problem is, any full range music, reproduced at even modest levels (70 dB at 35 Hz) will be transformed into higher distortion products. It doesn’t sound bad in this case, but it’s clearly not true to the original. It’s "fake" bass that comes out of so many small speakers.
If you didn’t quite follow it, read it again. I don’t mean to club your head with this, but it’s the reason why you will never, ever get accurate bass at virtually any level out of a 4" driver. Even if you doubled or tripled the xmax, you still won’t get there.
Things get better with a larger driver, but it appears generally unappreciated how much bigger you need to go to get undistorted bass, even at moderate levels. Consider the following two graphs. They are both ML TL’s. The first is based on the GR Research M130 5" driver and the other is based on the SS8554 8" driver. Both drive levels are adjusted to get 86 dB at 0.5m. Consider the distortion spectrums with the Fletcher-Munson curves in mind. Make the adjustments in your head and you can see the significant difference in the reproduced signal. Mind you, the M130 does not sound bad, but there is some distortion, audible as "fake" bass, even at this moderate level. The SS8554 isn’t perfect, but it’s measurably and audibly better.
The ear’s decreasing sensitivity at low frequencies is a significant problem and lead to the need for quite low distortion at the low end. To reiterate, if xmax is defined by a manufacturer as 10% harmonic distortion, then say, at 20 Hz, when the 10% figure is reached, the amplitude of the 40 Hz 2nd harmonic may be down only 20 dB. The problem is, the relative sensitivity of the ear has increased by 20 dB from 20 to 40 Hz. So what are heard are two tones, 20 and 40 Hz of equal magnitude. So, even though the driver is still operating in its linear range, obvious audible distortion is occurring. And this doesn’t take into account problems with the 3rd order product or higher. The 3rd order product has to be down almost 40dB just to be perceived as equal in magnitude to the 20 Hz fundamental. Wow!
It’s hard not to draw the conclusion that, even for moderate levels in moderate sized rooms, the smallest driver needed for undistorted low frequency extension is an 8" driver. 10-12" drivers are required for true low distortion bass at realistic levels. In addition, transient levels can be 15-20 dB higher than the average SPL level. If this weren’t taken into account, while the average levels would be relatively undistorted, distortion would occur at the peaks.
You can see why the xmax/excursion-limited prediction by modeling software that use xmax provided by the manufacturer could be poor. The builder who thinks that he can get undistorted bass from a 7" driver at 105 dBSPL half space because that’s what Unibox says may very likely be wrong, and by a significant margin. (I’m not picking on Unibox-I think it works well. But distortion and SPL predictions should always be tested-all modeling software has limitations.) Remember, higher order products can be very audibly offensive and these generally will not be accounted for in these simple xmax limited excursion predictions.
Higher order products are bad, even at levels of –40 dB and lower. IM distortion is bad. The higher the order, or the more numerous the products, the worse the perceived distortion.
Even if you don’t care about reproducing frequencies below, say 40-50 Hz, you still can’t get away unscathed. If there is low frequency information in the program material, while it won’t be reproduced by your loudspeaker, the distortion products will. You will suffer from the "fake bass syndrome."
Distortion occurs across a spectrum. There is a range of linear, undistorted bass. As the SPL is increased, generally 2nd and 3rd order harmonic distortion products rise. These are audible, but not necessarily all that offensive, especially 2nd order. It’s perceived as "richness." I suspect many people like it. It’s still "fake bass," i.e. distortion. As the level is increased, 3rd order and higher products rise, and multiple IM products occur. Now more edginess and what is typically perceived as distortion starts to occur.
Doing these measurements yourself and listening to the tones is quite revealing and I recommend that you do this to your designs, just like you would test frequency response or impedance.