Wait a minute. Let’s see if I understand this. (Here I’m channeling you, the puzzled reader, after you’ve grabbed me in the coffee break of a Filter Wizard lecture morning). The frequency of the n^{th} harmonic of a sinusoid is n times that of the fundamental. The n^{th} harmonic distortion is defined as the ratio of the amplitude of the n^{th} harmonic to the amplitude of the fundamental. So the 1^{st} harmonic distortion is… the ratio of the amplitude of the fundamental to the amplitude of the fundamental and that’s… unity. I. Do. Not. Understand.

You’re sweating. People are starting to look, keep their distance maybe. Cognitive surrender is just around the corner. You need help.

— and there’s a flash, the snap of freshly-pressed satin, and Equation Man is there! My trusty sidekick. He’s going to make it better. I step back, and let my algebraic alter ego take over. He picks up a cosine of angular frequency ω and…

… let’s assume we take our signal of x=v_{0} cos( ω t) and apply a mild s-shaped compression of the form y = x – kx^{3} over the normalized range -1≤x ≤ +1, with k small, because there’s not very much of this non-linearity. Because

Now, you’re right about that definition of harmonic distortion. The only extra frequency that has popped up is the 3^{rd} harmonic, which forms all of the total harmonic distortion:

We can write down that simplification in [4] because the denominator term in [3] is pretty close to unity. That’s a clue… because the fundamental amplitude is *slightly changed* from v_{0} cos( ω t) to:

Look at that. If the THD – which here is only the 3^{rd} harmonic – is *not* zero, then the amplitude of the fundamental is *not* what it was before we applied the non-linearity. Depending on the sign of k (positive for compression, negative for expansion), it could be larger or smaller.

This is **first harmonic distortion** – the inevitable change in the amplitude of the fundamental caused by a gain non-linearity. And because the harmonic distortion is a function of signal level, so* is the gain of your system at the fundamental frequency* .

So, for example, in a system with 0.01% (-80dB) 3^{rd} harmonic distortion at full scale, the absolute gain linearity error will be approximately 0.03%, compared to when measured at a much lower signal amplitude where the distortion is negligible.

If you need to calibrate the gain of your system very precisely, then it’s clearly a problem if the gain actually depends on the signal level!

Corresponding calculations can naturally be done for higher order non-linearities, with some simplifications. To briefly show the result for y = x – kx^{7} rather than the equation-rich workings:

where n is an rms composite of the 3^{rd} , 5^{th} and 7^{th} harmonics you get from this higher-order bendiness. We get:

And that ends up giving us:

In other words (OK, other letters, numbers and symbols) only about half the linearity effect of the pure 3^{rd} order non-linearity for the same amount of measured harmonic distortion. See reference 1 for the typical THD plot of this 7^{th} order non-linearity.

We can also extend this approach (reference 2) to non-linearities with a weak hyperbolic tangent form, the characteristic of the classic degenerated bipolar transistor differential pair which is typical of many high speed analog circuits. Quite useful for all you amplifier designers out there.

As an interesting (well, it is to me) aside, note that if the non-linearity is present at the output of a circuit with feedback applied around dominant pole stabilization, then there will also be a *phase* non-linearity which, just conceivably, could have an effect in power metrology applications should extremely low supply current analog techniques be used (op-amps which are too slow). See reference 3 for an introduction to the origin of this mechanism.

Well, there, Equation Man has done his bit, and you look, well, a little better. Another coffee and you’ll be fine. That afternoon session on elliptic function filters is going to fly by…

What’s the moral of this particular story? Mainly that an impairment in your system transfer function (e.g. non-linearity) may not just cause the errors that you’re expecting (e.g. harmonic distortion) but also errors that you weren’t expecting, *and therefore don’t look for * (e.g. variation of gain with signal level) until it’s too late.

In other words, don’t just look at the defects in your solution whose presence you can prove; look also for the defects whose absence you must ensure. Because that… is fundamental! / Kendall

**References:**

The 3rd armonic of equation 1 may be filtered through an low pass filter to reduce the THD, this would allow to process the signal with less distorsion.

Yes, you could use a filter to eliminate the 3rd harmonic. But you'd still have the 1st harmonic distortion! This is not a problem that can be fxed with a filter (and you just know how much it pains me to say that!).

That's right, and in addition the contribute of first harmonic is often not negligible when compared with the input signal amplitude.

maths maths

Nice post, Kendall, enjojed that. Good reminder to us all about the impact of nonlinearity in a system.

nice post

Really nice post, indeed.

However, while this equations are useful to quantify the amount of change in magnitude for the fundamental component, all this math is not required to understand or deduce the phenomenon.

From a purely heuristic point of view, considering that all energy you have inside your signal is preserved, when you create new harmonics (aka distortion) it is easy to guess that the fundamental's energy should be its source, being somehow depleted to provide the new (probably undesired) signals/harmonics.

Also, technically speaking, the changing in the fundamental's magnitude is not a distortion, but only a linear transformation. We generally call as distortion the wave changes over the whole signal. By nature and definition, the fundamental is a sine wave and will still be even after donating whichever energy it has to new created harmonics.

“Also, technically speaking, the changing in the fundamental's magnitude is not a distortion, but only a

linear transformation “Wrong. It would be linear transformation if the gain factor is independent of the input signal. A function of formThis is not the case here.

y = kxis linear only if k is constant (i.e., independent of variable x).Good point. I did forget that whatever is distorting the signal it must depend on the signal itself. I was thinking on a fixed (and immutable) one time change of the original signal. As the signal changes, so does its harmonics and the amount of amplitude of the fundamental.