Chapter 1   Perception and Theory of Distortion.

The instruments which are used to measure THD (total harmonic distortion) are made up of two basic elements. 1) a sharp and deep notch filter capable of attenuating a single frequency by at least 90 dB while leaving all other frequencies unchanged in amplitude. 2) a voltmeter with a dynamic range of 90 dB and an upper frequency limit of approximately 10 times the highest frequency at which distortion is to be measured. A typical instrument also includes an input attenuator, an input buffer, and a notch filter output.

The input buffer provides a high input impedance so the instrument will not load single stage circuits that are being tested in a standalone condition.

Where H₁ is the amplitude of the fundamental frequency, H₂ the amplitude of the second harmonic, H₃ is the amplitude of the third harmonic, H_n is the amplitude of the nth harmonic, and sqrt denotes taking the square root of the value inside the following parentheses. This system is quite cumbersome but, for instance, there may be occasions when the design engineer wants to know the relative amplitudes of even and odd numbered harmonics. We will revisit this question in a later section of this chapter. Also taking the square root of the sum of the squares is what a true RMS voltmeter does.

Note: I do not know what frequencies are used for IMD measurement in the rest of the world. Perhaps someone who lives there will enlighten me.

The DC level at the output of the low pass filter is 0.899723 volts. Note that this so called DC level is capable of changing at a 60 Hz rate. If the "carrier" were to be 100% modulated its peak value would change from 0 to twice the static peak so the level at the output of the low pass filter would change from 0 to 1.79945 volts. Thus for 100 % modulation the peak to peak sinusoidal voltage at the output is 1.79945 volts. Converting this to RMS gives 0.63620 volts. Remembering that the unmodulated DC output is 0.899723 volts the ratio of the unmodulated DC to the RMS AC for 100% modulation is 1.41421. We shouldn't be surprised by that. So, if we want 100% to equal 1 volt RMS we should set the DC level to 1.41421 volts DC.

This sounds good in theory but the practical limitation is that the low pass filter has a slightly different gain for AC and DC. There may be some difference between different builds of the detector and filter circuit so another means must be found to calibrate the instrument.

We are up to the seventh harmonic and we have yet to create any really dissonant notes. We don't get any notes that are seriously out of tune until the 9th harmonic which is 5 Hz higher than a D7*.

* This does not mean the D7 chord which is often played as the fourth chord in the key of C, but the d in the seventh octave.

The higher the harmonic number the lower the amplitude and the truth is the high order harmonics are below the noise floor of most amplifiers and so aren't perceived by the listener. So if all this is true why does distortion sound so bad?

If we could make an amplifier that would produce only harmonic distortion and no intermodulation distortion we wouldn't have to worry about it as much as we do. Distortion is produced by nonlinearity in the amplifier. I will discuss this in the next section. Any nonlinearity will produce both harmonic and intermodulation distortion and it's the IM distortion that sounds so bad.

So we conclude that IMD is much more objectionable than THD. As stated earlier the two go together like love and marriage, a horse and carriage, or any other pair you might care to name. I am not advocating abandonment of THD measurements. I think that we audio hobbyists should measure IMD alongside THD for a complete picture of the behavior of an amplifier.

where V is the peak value of the signal, ω is the angular frequency in radians per second, ω = 2 π f where f is the input frequency in Hz, and t is time in seconds.

The A₀ term is a DC offset that may be inherent to the amplifier. Ideally it would be zero. Other DC offsets are created by even order terms. The constants in the identities below create these DC offsets. In tube amplifiers these are not passed on to the speakers because the output transformer will not pass DC to its load. It would be passed to the speaker from a silicon based amplifier but in general it is too small to be worried about.

I just can't resist. I hope that those who put iron blocks on top of their amplifiers to trap magnetic fields hear about this. It will give them something else to worry about and could start a whole new industry making DC absorbing modules to be connected between the amplifier and speaker. Don't tell them it's just a big capacitor. End of snide remark.

The A₁ term is the voltage gain figure we all use to characterize an amplifier without giving it a second thought. There is a little more to it than I had imagined.

Now we come to those nonlinear terms. We have assumed the input to the amplifier to be Vin = V cos ωt, where V is the peak value of the signal, ω is the angular frequency of the signal measured in radians per second, and t is time in seconds. The input signal is a pure sinusoid. In the equations below we are not altering the input signal although it looks like it. The output signal is the input sinusoid multiplied by the coefficient A and squared, cubed, or raised to the 4th power.

Now we employ the trigonometric identities,

Making the appropriate substitutions we have,

In the above equation terms are grouped by the subscript number of the coefficient A. We would rather have them grouped by harmonic numbers, that is all fundamentals together, all second harmonics together, and so on. First we will expand the terms and remove the brackets.

Now we will sort them out by harmonic number.

Now that we have things in the order we want let's recombine terms.

This tells us that a squared term gives us a second harmonic plus a DC offset, a cubed term gives us a third harmonic plus a bit more fundamental, and a 4th power term gives us a 4th harmonic plus a little second harmonic and a little more DC offset.

In most listenable amplifiers the distortion is not only small but pretty much confined to the second, third, and fourth harmonics. This is another reason to limit our discussion to these conditions.

The term f(Vin) is just fancy mathematics talk for Vo (output voltage). We can further simplify and clarify by making that substitution. And while we are at it let's get rid of those useless DC terms.

I used cosines for a good reason. The identities come out in cosines and all waves begin at unity at zero radians. Vo / Vin becomes a ratio and it doesn't matter whether we use peak, average, or RMS. In fact we can do away with those cos ()ωt) terms and simplify everything.

Where Vo₁ is the measured amplitude of the fundamental frequency, Vo₂ is the measured amplitude of the second harmonic, Vo₃ is the measured amplitude of the third harmonic, and Vo₄ is the measured amplitude of the fourth harmonic. These measurements must be made with a spectrum analyzer or wave analyzer.

The purpose of all this algebraic crank turning is to make a few measurements and then come up with the values of the A coefficients. Solving each equation above for its coefficient value and remembering that we know the numeric values of V and each Vo value gives,

Data and coefficients from Newcomb D-10
Harmonic #	Vo#	Coefficient A#
1	4.00	9.93
2	2.20e-3	-57.5e-3
3	9.70e-3	606e-3
4	1.70e-3	531e-3

Don't be concerned about one of the coefficients coming out negative. All that means is that the phase of the signal is reversed. Also you will notice that the values of the coefficients are counter intuitive. This is to be expected when numbers raised to powers get involved. If you do as I did and put the equations into a spreadsheet and experiment with numbers you will find that the THD increases as the input level increases while the coefficients remain unchanged. This is to be expected. A larger input signal is swinging the nonlinearities over a wider range and they are more nonlinear at higher levels.

If you calculate the THD from the equation in section 1.1.5 you will get 0.252%. The measured THD with an HP 334A was 0.265%. The calculation is -4.49% off from the measured value. Since the "calculation" is based on measurements of the harmonics made using an HP 3581A wave analyzer which has an amplitude accuracy specification of +/- 4%, and I have not tried to calibrate it since buying it in a well-used condition, the error is easy to account for.

Where V is the peak value of the signal in volts, and t is time in seconds.

Now we will plug Vin into the gain polynomial. Binomial theorem here we come.

I refuse to deal with terms any higher than the third power. If you want to work them out for yourself I can't stop you.

In the binomial equation B = V cos (ω₁t),
and C = V/4 cos (ω₂t)

So as we apply the binomial equations the terms
are, B = V cos (ω₁t),
and C = V/4 cos (ω₂t)

Let's do away with that DC term and throw away any constants as they turn up, and they will. We will break down the big job into a series of smaller jobs. We will begin with the A₂ term.

In the equation above the first term gives us the second harmonic of the low frequency, and the third term gives the second harmonic of the high frequency. These will not be very strong in a push pull output circuit but they are there because the balance is not perfect and second order distortion is produced in earlier stages of the amplifier. The middle term is what gives us the sums and differences.

It appears that we have what we want from the second order term. What does the third order term give us? There's only one way to find out.

When this result came out I could hardly believe it. This tells us that there will be frequency components at 6880, and 7120 in addition to the previously expected 6940 and 7060 sidebands on the 7000 Hz carrier frequency. Not only that but there will also be components at 1340 and 1460 Hz. This information sent me right back to the workbench where I observed the following results.

After completing the derivations that come after the table below I added a third column to it.

IM Components from D-10
Frequency (Hz)	Amplitude (V)	Calculated Components (v)
60	4.0 V	4.09
120	0.216 V	-4.60 mV
180	26.6 mV	38.8 mV
6820	2.6 mV
6880	8.95 mV	7.28 mV
6940	2.05 mV	-2.3 mV
7000	0.88 V	1.00
7060	1.65 mV	2.3 mV
7120	7.7 mV	7.28 mV
7180	3.0 mV
13880	0.6 mV
13940	2.35 mV	1.82 mV
1400	2.7 mV	-0.288 mV
14060	2.3 mV	1.82 mV
14120	0.7 mV

Confirmation of the theoretical results caused me to remove the "Linear Rectifier and Low Pass Filter" module from the Analyzer to rework the low pass filter. I replaced the 0.022 μf capacitors with 0.01 μf. Note that the sidebands caused by the third order term are stronger than those caused by the second order. At the second harmonic of the high frequency the situation is reversed as the equations predict.

I was almost afraid to go further. The results for a single signal had come out pretty good and I had doubts that my luck would hold out. But after some consideration I decided to bite the bullet and type out the entire expression for Vo. Then I will sort out the terms by the frequencies they produce as done for the single frequency input. For better or worse, here we go.

The measured IMD using my instrument is 1.7%. Calculated from the data taken on the amplifier it is 1.45%. Calculated from the equations above and the coefficients of the amplifier which were determined earlier the IMD is 1.11%. These are some pretty large errors but at least they are within the same order of magnitude.

Anyone who knows HTML programming will understand just how much typing has been involved in all of this. If you don't know HTML, it takes 12 characters just to make one subscript or superscript. The greek characters must be spelled out plus two more characters to indicate a special symbol.

Also, this exercise was not in vain because the discovery of the contribution of third order terms caused a design change which improved the accuracy of the instrument.