Why harmonic distortion and not at other frequencies?

A sinewave of 1 kHz only contains one frequency: 1 kHz. Let's describe that mathematically:

\$x = sin(2 \pi f t) \$

Where \$f\$ is the 1 kHz, \$t\$ equals time and \$X\$ is the sinusoidal signal.

If an amplifier is ideal then it would only amplify the signal, i.e. increase the amplitude:

\$y = A x = A sin(2 \pi f t)\$

Note how that still has is just a sinewave, only the amplitude (value of the minimums and maximums) have changed.

But that's a linear amplifier, it will not introduce harmonics.

Now what if the amplifier distorts.

Do you remember the Taylor series? It is a way to express any function in the form of a polynomial like this:

\$y = A x + B x^2 + C x^3 ...\$

What I wrote there is the Taylor expansion that describes the behavior of an amplifier with distortion.

If you fill in \$x = sin(2 \pi f t) \$ you will get \$sin(2 \pi f t) \$, \$x = sin^2(2 \pi f t) \$ and \$x = sin^3(2 \pi f t) \$ terms and these are the harmonic frequencies.

Note that there is no way to get terms other than \$x = sin^n(2 \pi f t) \$ making it impossible to get frequencies that are not a multiple of the base frequency of \$x\$.

Bonus question:

What would be needed to get other (non-harmonic) frequencies?

With a sinewave as input, there is no way. But if we combine two or more sinewaves of different frequencies, then we can get intermodulation products. For example, make \$x\$ a signal consisting of a 1 kHz (\$f_1\$) and a 200 Hz (\$f_2\$) tone:

\$x = sin(2 \pi f_1 t) + sin(2 \pi f_2 t)\$

Then at the output of a distorting amplifier we will find sum and difference frequencies so we would get:

• 200 Hz
• 400 Hz ( 2 x 200 Hz, the 2nd harmonic of \$f_2\$)
• 600 Hz ( 3 x 200 Hz, the 3rd harmonic of \$f_2\$)
• 800 Hz (1 kHz - 200 Hz)
• 1 kHz
• 1.2 kHz ( 1 kHz + 200 Hz)
• 1.4 kHz ( 1 kHz + 2 x 200 Hz)
• etc etc

note how they're all 200 Hz (\$f_2\$) apart!

How many frequency components are present depends on how much the amplifier distorts and the amplitudes of the signals.

Fourier was saying that any periodic wave can be made up of the fundamental and its harmonics in various ratios.

Figure 1. This fabulous illustration of the Fourier Transform by Lucas V. Barbosa on Wikipedia's Fourier transform page shows the transformation of a periodic waveform from the time domain to the frequency domain. The frequency plot shows the relative strength of the harmonics with clarity that could not be obtained from staring at the time plot.

• It should be apparent that the more square the time domain waveform is then the more harmonics you will have and these should be visible in the frequency domain.

I come across this in studying audio systems. They rely on THD as a measure of fidelity (for non-clipping signals), but it baffles me why the distortion falls into nicely behaved harmonics ...

If the harmonics are not integral multiples of the fundamental then, by definition they are not harmonics. If you mixed your fundamental f with an f × 1.8 as in your example then they would beat and only come into phase every five cycles of the fundamental (and nine cycles of the 'harmonic'). The result would be a different waveform with a new fundamental of f/5.

Figure 2. Fundamental (blue), fundamental × 1.8 (orange) and the sum (red).

A very large proportion of electronic circuits (including audio) belongs to the category of "time-invariant circuits" (linear circuit or not).

To give a idea of what are other circuits, logically called "time-variant circuits", it suffices to realize that they need to have some kind of clock, for example an internal oscillator (which could be a Larsen effect, or some auto-generated thermal noise) or external (50 Hz or 60 Hz hum). All "inert" (1) electric circuits are "time-invariant" because they are not "aware" of time (!).

Now, add the fact that the time response of a "time-invariant circuits" to a periodic signal is a also a periodic signal, and at the same frequency, add a little Mathematics "à la Fourier" and you see that distorsion can't be something else than harmonic.

(1) one could say "not alive"