Amplifier Distortion Sound Demos
In this section a music sample (a .wav file) is processed to mimic the effects of amplifier distortion, and a new wav file of the distorted music is created. The goal is to compare amplifiers at high power levels where audible distortion occurs. The amplifier models are idealized, but they are realistic regarding clipping behavior, which is a primary cause of objectionable amplifier distortion. Both tube and solid-state amplifiers are modeled. Some of the wav files are available for downloading, so you can make your own comparisons. I do not claim that this exercise answers all of the questions regarding amplifiers, but it answered some major questions for me.
The advantages of this approach are:
The disadvantage is:
A typical engineering approach is employed: modeling the dominant effects (the "long poles in the tent"), and idealizing less important details. The models are validated by comparing results with measured data from real amplifiers, shown below.
My conclusions from my own listening tests are:
The comparisons provide some basis for judging between a tube amp rated at 5% THD at X-Watts, vs. a solid-state amp rated at .001% THD at Y-Watts. The audio world is inundated with snake oil, and I have been suspicious about the claims regarding the sound from single-ended triode amps at high levels of THD, but I am now a believer.
I am very fortunate to have two excellent reference sources for this section: the first is Norman Koren, who has a very informative discussion of amplifiers in his section on Feedback and Fidelity, and who provides SPICE tube models which I use. The second is Daniel Cheever, who wrote a Masters Thesis that is chock-full of valuable information. The methodology used here is similar to that used by GedLee (see their Powerpoint presentation). However they apparently have not made their wav files available to the public, and I didn't see a comparison between their models and measured amplifier responses. Their primary objective is to develop a new metric to replace THD and intermodulation distortion as quality criteria, whereas my objective is to compare amplifiers.
Doug Self describes the design of a class B solid-state amplifier with very low distortion, at power levels where no clipping occurs. Class A and AB tube amplifiers are very linear for low signal levels, when the output power is well within the limits of the final amplification stage. I am convinced that amplifiers are totally transparent under these conditions. For higher power levels response becomes non-linear, and clipping occurs, meaning that the amplifier is putting out all the power it can, and it still isn't enough. This power limitation lies essentially in the final amplifier stage, and it is a major source of amplifier distortion. This is the region of audible distortion that is modeled here.
Amplifiers are rated based on average power. It is not unusual for me to play music on my system (quite loud) at 10 Watts average power (based on an eyeball estimate from an oscilloscope display of the voltage at the speaker terminals). One of the music samples used here played at an average power of 10 Watts would require a peak power of over 800 Watts! Clipping is not at all uncommon.
An amplifier can be modeled mathematically by a curve of output voltage vs. input voltage. This isn't totally correct, because there can be transient deviations in the response, but it is a good approximation.
The amplifier model is idealized as follows:
An example of an input/output curve for a class AB push-pull tube amplifier is shown in this figure [38 kb] by the dark curve. This curve was derived using the 6550 pentode tube model developed by Norman Koren, and feedback is not used. As discussed by Norman, the effect of feedback is to make the transition from the linear region to the power saturated region more abrupt. Solid-state amplifiers in particular typically use a lot of feedback. Rather than trying to model feedback in detail, the solid-state amplifier is modeled by the abrupt transition shown by the red segment of the input/output curve. Measurements supporting this model are discussed below. The red circle is the "reference voltage" referred to below.
The single-ended triode curve shown in this figure [41 kb] is based on Norman's 12AX7 triode model. The middle blue dot is at the grid bias point, and the other two blue dots represent the class A operating range. Of course when clipping occurs the voltage exceeds this range. The red line terminating in the reference voltage shows what a perfect linear response curve would look like.
These input/output curves are represented mathematically in Matlab. An input wav file is read in, amplified, and a new (re-normalized) wav file is written out.
Distortion Harmonic Components
The three amplifiers produce different spectra of distortion products, and that is the key point in understanding the difference between the amps. THD lumps together all of the harmonics, but in reality some harmonics sound a lot worse than others. Many people have pointed to this difference, which is illustrated in this graph of harmonics produced at different volume levels [54kb], for a pure 440Hz sine wave input. This is a busy figure, but it contains the essential information for understanding this difference. The x-axis voltage scale along the bottom is defined with respect to the reference voltage shown in the previous graphs. At the maximum value of 1.0 the RMS output power is very close to the maximum possible output power, and a lot of clipping is occurring. The z-axis scale is the magnitude of each harmonic, in dB relative to the fundamental. The numbers on the curves show the ratio of the harmonic frequency to the fundamental.
It is seen that a SE Triode amp only produces three significant harmonics. Based on listening tests of the fundamental tone plus a single harmonic, the audibility thresholds for the 2nd, 3rd, 4th, 5th, and 7th harmonics are approximately -20 dB, -20 dB, -40 dB, -35 dB, and -50 dB respectively. In general, the higher order harmonics are audible at much lower levels. For the SE Triode amp the 3rd and 4th harmonics are not audible over the range of this figure, and the 2nd harmonic is only audible at a value of alpha of 0.8 or more.
There are no even harmonics for the other two amps.The solid state amp doesn't produce any distortion until the amp clips, but then the levels of the audible distortion products rise very quickly, and the 3rd, 5th, and 7th harmonics are audible for the higher values of alpha. For the push-pull pentode only the 3rd harmonic is audible.
This discussion only deals with a single 440 Hz tone, and the audibility thresholds for harmonics of tones at other frequencies will be different. In addition to simply being audible, some harmonics are more unpleasant to the ear than others. Finally, with real music there is also intermodulation distortion. But the differences in this simple case clearly show that there are gross differences between the amps that cannot be captured simply by measuring THD.
I am an engineer, and I would love to have one simple measurement that accurately quantifies amplifier quality. Unfortunately that measurement does not exist. (GedLee presents a metric that they claim represents quality much better than THD, but I would not call it "simple"). Therefore we have to trust our ears, and that is my motivation for generating distortion samples using real music.
The three models are intended to be generic representatives of common amplifier designs. Here results from the models are compared to real amplifiers, which in general use different components, and/or operate at different power levels. Perfect agreement is not expected; the issue is if the models reflect the generic characteristics of the designs.
The first figure [41 kb] compares harmonics generated by the single-ended triode model (SE) to the measured harmonics of a Cary CAD-300 SEI, given in Figure 8 of Cheever's thesis. The red dots are for the Cary, and the blue dots and lines for the model. I am satisfied with the similarity. The THD in this case is high, 9.4%. A second comparison is for a different SE design shown in Fig 2-8 of Cheever's thesis, for a THD of 1.35%. The measured 2nd and 3rd harmonics are -37.4 dB and -62.8 dB relative to the fundamental; the model generates 2nd and 3rd harmonics at -37.4 dB and -67.5 dB, demonstrating excellent agreement.
The second figure [42 kb] shows THD vs. output power, red measured for the CAD-300 SEI, blue curve from the model. Since the amp powers vary, in this figure and the next figure the power is relative to a reference that is scaled to make the curves coincide at one point, but the slopes of the curves are not changed. The agreement is not perfect, but both curves exhibit.the most notable characteristic: a relatively gentle growth of THD with increasing power. In contrast, the third figure [44 kb] shows THD vs. output power for a solid-state (SS) amplifier. The growth in THD is downright explosive! The green curve is for a Royston 3B-ST from Cheever's Figure 16; the red curve is for a Rotel RB-980BX from an article in High Performance Review, Summer 1992. The blue curve is for the SS model.
This agreement alone is convincing evidence that the SS simplified feedback model is realistic. However since I own several SS amps, I decided to bench test my Rotel RB-981. The test signal, a 440 Hz tone, had an acceptable THD of 0.0055% at the amp input. The black curve in the previous figure shows the result for THD vs. power, which is similar to the other data. The dark curve in the next graph shows a sample of measured clipped output [40 kb]. A perfect sine wave input voltage, scaled to overlay the output, is shown by the dashed red curve. For each time sample the output is plotted vs. the scaled input as shown by the collection of blue dots in this figure [40 lb]. The (barely visible) cyan curve represents a perfect linear response. The blue curve turns abruptly at saturation, confirming the SS amp model. The spectral content at a THD level of 1.713% is shown in this figure [40 kb]. The dominant distortion components of the measured data and the model are in near-perfect agreement. This last figure clearly validates the SS model.
Early SS amps had problems with "crossover distortion" as the current shifts between NPN and PNP transistors used in a push-pull design. I searched in great detail for any such effects, at various power levels, and found nothing. The amp was beautifully linear up to the specified power. There was a minor asymmetry in the clipping behavior, which started a little earlier at the wave bottom vs. the wave top, but obviously it had little effect on the harmonics as shown in the previous figure. The amplifier is specified at .03% THD at 130 Watts into 8 Ohms; I measured .0296% THD at 160 Watts into 6 Ohms. I did measure a "THD" as high as 0.24% at very low power levels, but the spectral content showed that it was due to noise rather than distortion, and I am pretty sure the noise source was external to the amp.
Sample Files and Listening Results
THD for a Pure tone
For these initial tests a pure 440 Hz sine wave is input to the amplifier model, and the THD of the output signal is computed. THD is defined by
Where a1 is the magnitude of the fundamental tone, and the other terms are magnitudes of the harmonic tones. THD is defined for a pure sine wave input, where all of the harmonics in the output are nominally integral multiples of the fundamental frequency. However in my calculations the magnitudes of any line in the spectrum are included, so my "THD" includes noise as well as distortion.
The distortion of a pure tone is quite audible at a THD level of 0.5% for the SS model. There are nine harmonics in the range of -50 to -60 dB, and it is the higher harmonics that are the most audible For the SE model distortion is only audible at a THD of 10%, where the 1st harmonic is 20 dB below the fundamental! Cheever explains that this huge difference is due to the fact that the 2nd harmonic (which is absent for the SS) is masked by the natural non-linearity of the ear itself (for a discussion see Non-linear Behavior of the Ear). For the SE triode the 2nd harmonic is dominant, and the higher order harmonics drop off far more rapidly than for the SS, as shown in this figure [43 kb]. The THD is 5% in both cases shown. I have posted pure tone distortion wav files with 0.5% distortion for the SS model, and 5% for the SE model. I can't hear the distortion in the SE file even though it is 10 times higher than the SS model.
I originally intended to define a music file having a "THD of 1%" by a file that has the same average root-mean-square (RMS) input voltage as a pure sine wave that results in a THD of 1%. The problem is that the ratio of peak to average power for the sine wave is 2:1. For the two music samples used here the ratios of peak to average power are 38:1 and 82:1. Since the power peaks of the music samples are much higher than the sine wave, the clipping at the music peaks is much more severe. For the SS model, for any THD between .001% and 1% the clipping of a music file is essentially the same, and quite bad. The same problem exists for the push-pull (PP) tube amp, albeit less severe.
The alternative I have chosen is to define a parameter "alpha" equal to the RMS input voltage divided by the reference voltage defined above. As noted previously, a value of 1 means that the average power output is very close to saturation, and there is a lot of clipping. Note that this parameter washes out any power difference between the tube and SS amps, since each amp is being compared to its own maximum power. The relation between alpha and THD is shown in this figure [42 kb]. It is seen that the SS amp operates with zero THD until the value of alpha is .71, where the peak of the sine wave just reaches the reference voltage. The THD of the SE triode is always high, and the PP tube amp is in between. This figure again illustrates the major difference between the three designs. But the difference has more to do with using THD as the distortion criterion than with real differences in distortion. At the same value of alpha the audible distortion is similar for all three amps. Cheever, among others, points out the inadequacies, and perverse results, of using THD as a standard, and I totally agree.
The listening test results quoted here were obtained using Army's double-blind tester. For a jazz selection, with a value of alpha equal to 0.5, I was 100% accurate in selecting between the reference and distorted files for the SS and PP models. I could not reliably distinguish the distorted SE file, and I could not reliably distinguish between the SS and PP models. I was 100% accurate in selecting the SE file as less distorted than the SS file. Referring to the previous figure, this value of alpha corresponds to a sine wave THD of about 5%, 0.2%, and 0%, for the SE, PP, and SS models respectively. So if the output powers were really equal, a SE design that would be rated at 5% THD sounds better than a SS rated at 0% THD!
For a 10 second sample of music there are 441,000 time samples; 5.4% of the samples exceeded the reference voltage for the SS and PP models, 4.2% for the SE model for the above tests.
For a Vivaldi selection, with a value of alpha equal to 0.4 I was 100% percent accurate in selecting between the reference and all three distorted files, and in selecting the SE file as less distorted than the PP and SS files. In this case approximately 3% of the samples exceeded the reference voltage.
Effect of Amplifier Power
Suppose the SS amp puts out 4 times the power as the SE tube amp. If you make the output volumes equal, the alpha value for the SE amp is twice as large as for the SS amp. Compared to a SS with alpha = 0.4, the SE amp with alpha = 0.8 sounds much worse. A SS with alpha = 0.2 compared to a SE with alpha = 0.4 is also better, but the difference is less dramatic. So in this situation the advantage of higher SS power outweighs the advantage of less discordant SE distortion products.
So far I have mainly tested myself on one short music segment, the Vivaldi sample. Based on this, my detection threshold is close to a value of alpha = 0.3 for the SS case, where 1% of the samples are clipped, and I am 90% accurate in selecting the distorted file. This corresponds to an average power output equal to 9% of the maximum power output. Further testing can only result in a lower threshold. I am guessing that if I run my amps at 5% of their rated power that clipping will not be audible.
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This section originally posted August 15, 2006; revised to add my measured Rotel amplifier data September 7, 2006. The section on distortion harmonic components was added September 22, and I want to thank Patrick Turner for suggesting the figure discussed in that section. I added info regarding the GedLee site when I learned about it in March 2007.