Electronic – Why harmonic distortion and not at other frequencies

acharmonics

Despite searching I can't seem to find the answer to this. In understanding harmonic distortion, one piece of the puzzle that is missing for me is:
– why does the distortion manifest at harmonic frequencies and not other frequencies.

Feel free to simply point me at a textbook / reading that explains it, if that's easier!

  • I understand that non-linear loads can create distortion. Intuitively, this is because the load can change it's characteristics (resistance, capacitance, etc etc.) in complex (difficult to predict…) fashion.
  • I understand enough of Fourier analysis to understand harmonics, 1st (fundamental), 2nd, etc etc.
  • I don't see why the distortion would tend to be at harmonic frequencies.

E.g., if I put a 1KHz sine wave through a non-linear load, why would the distortion show up at harmonics and not something a little less friendly, say, at 1.8KHz, depending on the circuit design?

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…

Thanks!!

Update: thanks to all the great quick answers below, I think I figured it out.

  • non-linear loads are still predicable: "A nonlinear impedance effects every cycle of the waveform in the same way" (Charles Cowie)

  • other impacts that do not effect every cycle the same way are transient, or interharmonic. These can be highly unpredictable, due to external forces, etc. They can change the fundamental frequency (e.g., a sharp cutoff)

  • any periodic (distorted?) waveform "can be represented by their fundamental
    component and a Fourier series of harmonics of various magnitudes, frequencies and angles.(this cites another source)"     (from relayman357)

  • there are some great math workthroughs below that illustrate this.

So the piece that I was missing was that the distorted waveform still sits on the fundamental frequency in a periodic fashion, so by definition the distortions are harmonics (different phases/amplitudes,etc.).

Non harmonic distortion (interharmonic) isn't periodic.

Loads like amplifiers don't typically change the fundamental frequency but "give it hair", so it's still periodic.

Best Answer

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.

Related Topic