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Harmonic and Intermodulation Distortion

pma

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I bought an actual valid issue of the IEC 60268-3 standard (year 2018), the one that is linked for example below:

IEC 60268-3:2018 | IEC Webstore

IEC 60268-3:2018 Standard | Sound system equipment - Part 3: Amplifiers
webstore.iec.ch

For those who might be interested (@restorer-john , maybe others as well) - let's try to suggest a concept of power amplifier testing that would be in conformance with current valid standards. PM me if you are interested.
 

RCAguy

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That's because the test frequencies and numbers are different. If you test at 60 Hz and then 7 kHz, then you have the same information as the SMPTE IMD except now they are separated and easier to analyze.
Don't 60Hz and 7kHz need to be injected together for intermodulation to take place?
 

fpitas

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fpitas

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An interesting real-world example of IMD:


It seems that some people can hear the Doppler distortion (like me) but others can't. Maybe depends on the ear construction (?)
 

MarnixM

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HD and IMD are different aspects of the same non-linearities. Their relationship is derived from the transfer function of the device in question. And drivers would have different transfer functions from amplifiers. @pma said above that the transfer function for drivers would be much more complex in order to be accurate enough to be useful.
So the answer to your question: "yes" in theory, but "no" in practice because the transfer function would be so complex it would be infeasible to derive it.
So rather than attempt to predict the HD and IMD theoretically, we can take an easier approach: measure the HD and IMD that the device (speaker) actually produces.

Harmonic distortion
Harmonic distortion is a frequency related type of non-linear distortion and deals with “extra” harmonic tones in the produced signal on both side of the ground tone which are not existing in the input signal. Although 2nd and 3th are most prominent, harmonics residues can go up very high (12th to 15th level). The total of these extra harmonic tones in the measured audio spectrum is called Total Harmonic Distortion (THD).

Intermodulation distortion
Intermodulation distortion (IMD) is a non-linear distortion by which the audio waveform is distorted in a way that has no relation to the frequency. IMD is mainly a result of imperfection of the sound producing devices, e.g. cartridge, CD-player, amplifier and of much more important the speaker. When the speaker produces “artefacts” as result of modulation between separate wave frequencies (that are not present in the original signal) in speaker terminology this type of non-linear distortion is called intermodulation distortion (IMD).

In 2003 Geddes and Lee (1) concluded that both the hearing of IMD as THD is frequency, SPL and “masking” dependent and that test results on hearing these types of distortion are not reliable and reproducible. Based on enhanced research they concluded in 2006 that “non-linear distortion in a loudspeaker does not appear to be a strong factor in its perception” and in 2008 “recent studies have clearly shown the human PREFERENCE for THD distortion of low order or at higher amplitudes. These are viewed subjectively as enriching the sound”.

 
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Hexspa

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This post provides a quick comparison of how a distortion percentage relates to dynamic range in dB. That is, if an amplifier has 1% distortion, what does that mean? How soft (or loud) is 1%?


Figure 1 shows a 1 kHz voltage signal with 1% second harmonic distortion (HD2) added. The spectral analysis (FFT, fast Fourier transform) shows the second harmonic at 40 dB below (-40 dB) the (0 dB) signal. So, 1% distortion is 40 dB below the signal level. Recall that dB = 20log(x), 1% is 1/100 = 0.01, and 20log(0.01) = -40 dB. Aside: Power goes as 10log(x), so 1% distortion in power (Watts) is only -20 dB. In SPL (dB/W), 20 dB is about the difference between say a pretty loud sound (80 dB) and a normal conversational volume (60 dB). Since we are in voltage (maybe a preamplifier output before the power amp), 40 dB takes you from that loud sound down to a fairly quiet room, say a library.

View attachment 145896

The lower plot shows an ideal sine wave, the signal with 1% distortion, and the difference (error) signal multiplied by 10 to make it easier to see. By eye, I cannot see any difference in the ideal input signal and output with 1% distortion (Vout). Because the error is purely second harmonic distortion, notice there are twice as many cycles in the error as in the input signal.

The figure below shows the same 1% distortion but now it is third harmonic distortion (HD3). The HD3 spur is -40 dB, but notice there are now three error cycles for each full cycle (0-up-0-down-o) of the signal. And again, I can’t see anything by eye (maybe you can).

View attachment 145897

Just for grins, and for something to see, the following two pictures show 10% HD2 and HD3. As expected the harmonic terms are now only 20 dB below the signal. What is interesting is how the high level of distortion changes the look of the waveforms. Second harmonic distortion does not change the picture too much, but third harmonic distortion causes the peaks to flatten, sort of like a soft square wave. Even harmonics tend to sound more pleasant than odd harmonics, adding a bit of edge to the sound while odd harmonics create a raspy buzz.

View attachment 145898

View attachment 145899

Last is a figure showing 0.1% HD3, with a spur 60 dB below the signal. A factor of ten in distortion changes the level by 20 dB (in voltage; a 10x change in power is 10 dB).


View attachment 145900

Finally, here’s a table showing the relationship among distortion in percent and dB in voltage (or current) and power. The dominant source of distortion in most systems is the speakers, which often run 1% or more in the midrange even for good ones, and 10% or more for large low-frequency signals. Amplifiers usually run in the 0.1 – 1% range, and preamps often 0.01% or better.

View attachment 145902

Hopefully this will help relate those distortion percentages to the dB specifications used for SNR, SPL, and so forth.

--- On to IMD ---

A previous post discussed harmonic distortion (HD), but to most people intermodulation distortion (IMD) is far more objectionable. Why? Because IMD generates tones that are not harmonically related to the input signals, and these tones “stick out” more than tones that are harmonics of the input tones. They sound “off” to our ears.

As an example, I created two tones at around 523 Hz (C above middle C on a piano) and 661 Hz (the E above, the root and third of a major chord). I added 0.1 % second- and third-order distortion terms (x^2 and x^3) and plotted the spectrum.

View attachment 144910

You can see the two signal tones at -6 dB, and a group of distortion spurs (tones) around the signal. I used -6 dB (0.5 full-scale) because two signals added will peak at 0 dB when they are each at one-half full-scale. You can see that in the next picture, which shows a few cycles of the signal. The funny-looking waveform is a result of the two signals mixing together, adding and subtracting from each other at different times. Even though each signal alone would only reach 0.5, together they reach 1.0 when they are in phase so both signals add. The error signal is multiple by 100 to make it easier to see; otherwise, the 0.1 % error would be essentially invisible by eye.

View attachment 144911

Below is a zoom showing the bunched signals, and a table of all the frequencies present. Not only do we have harmonics of each signal, at 2x and 3x the fundamental tones, but also intermodulation sum and difference terms. That means that we get C and E, and their second and third octaves, but also a tone a little over high D, a little below a low C# from the second-order IMD. The third-order IMD gives back the fundamental tones (but in general out of phase, causing amplitude modulation), and tones around a couple of (lower and higher) G’s (not too bad), Bb (oops), and a strange tone about half-way between Ab and A (what?) As a matter of fact, none of the IMD tones lie exactly on a piano note. These added tones add dissonance not in the original chord structure, and our ears are pretty good at picking out this dissonance. Yuck.

View attachment 144912

View attachment 144913

So, IMD creates non-harmonic tones, tones that do not lie in the chord, and that sounds bad. One other note: IMD tones are actually higher in magnitude than HD tones. Theoretically, IMD-2 tones are 6.021 dB higher than HD-2 tones, and IMD-3 tones are 9.542 dB above the corresponding HD-3 tones. Adding insult to injury… This is another reason IMD is more important than HD in the real world.

The derivation of all this is straight-forward but tedious. Take a couple of signals represented by Vin=sin(2*pi*f1*t)+sin(2*pi*f2*t), then build a function using

Vout = Vin + HD2*Vin^2 + HD3*Vin^3

After expanding out the terms and plugging in various trig identities you will find the terms in the table above, and the relation between HD and IMD amplitudes.

Hope this is useful - Don
Here are some more visualizations of how a sine wave changes when additional harmonics are added. We have fundamental, 10 and 100% h2 and h3. These were created using a meldaproduction modulator which lets you manipulate a variety of waveform shapes in many ways including adding harmonics to a sine.

I'm not really a distortion expert but it's interesting that the 2nd harmonic keeps stretching the sine until the periods double but cross zero whereas the 3rd harmonic folds the wave and remains above or below zero. I forgot what this is called; some kind of symmetrical or asymmetrical behavior.
 

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Holmz

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…. Based on enhanced research they concluded in 2006 that “non-linear distortion in a loudspeaker does not appear to be a strong factor in its perception” and in 2008 “recent studies have clearly shown the human PREFERENCE for THD distortion of low order or at higher amplitudes. These are viewed subjectively as enriching the sound”.


Lower distortion speaker do have a quieter sound, and have a somewhat boring neutrality to how they handle the loud parts.
When people mention “dynamic”, it can oft mean this added loudness of distortion.
 
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