# Sources of Harmonics(Odd and Even)

Exploring into harmonics this post explains very nicely about the odd and even harmonics with their individual importance. But what comes to my mind is:

What are the sources of harmonics? Why are they generated? Here is the document from National Semiconductor giving explanation about different terms of ADCs and DACs. And on page 19, it talks about symmetrical non-linearities are responsible for odd harmonics and half rave rectification is responsible for even harmonics(specifically ADC and DAC). But when I tried to search about half rave rectification on Google, I didn't get anything useful. Anyone please explain about the terms symmetrical non-linearities and half rave rectification and how are these responsible for harmonics generation.

• That should be half wave, not rave. – Justin Aug 30 '13 at 18:19
• Your first sentence makes no sense at all. In general, I it seems you need to read up on what Fourier said and how any continuous repeating function can be described as a series of sine waves, each wave with its own amplitude and phase shift. – Olin Lathrop Aug 30 '13 at 18:22
• @Justin So its printing mistake by National.. – AKR Aug 31 '13 at 1:43
• @OlinLathrop Can you please share some resources to read about sources of harmonics. – AKR Aug 31 '13 at 1:43
• @AKR - yes, it's a typo in the document. The term is "half wave rectification". – Pete Becker Aug 31 '13 at 20:17

What are the sources of harmonics? Why are they generated?

Assume that we have a voltage waveform of the form:

$v_s(t) = \cos(\omega t)$

In words, the source voltage waveform is composed of a single sinusoid of (angular) frequency $\omega$.

If this waveform is the input to a linear circuit, the output will also be composed of a single sinusoid of the same frequency as the input.

For example, a linear voltage amplifier scales the input signal by some constant $A_v$:

$v_o(t) = A_v v_s(t) = A_v \cos(\omega t)$

Now, consider what happens when the amplifier is non-linear. For example:

$v_o(t) = A_v v_s(t) + 2\alpha v^2_s(t)$

This amplifier has a 2nd order non-linearity. By a simple trigonometry identity, we have:

$v_o(t) = A_v \cos(\omega t) + \alpha[1 + \cos(2\omega t)]$

See what happened? The output is no longer composed of a single frequency but, due to the non-linear term, now has a DC component as well as 2nd harmonic component.

If instead of a 2nd order non-linearity, the amplifier had a 3rd order non-linearity:

$v_o(t) = A_v v_s(t) + 4\beta v^3_s(t)$

you might guess that a 3rd harmonic will be generated. Let's see:

$v_o(t) = (A_v + 3\beta)\cos(\omega t) + \beta \cos(3\omega t)$

Note that the 3rd order non-linearity creates a 3rd harmonic as well as an additional 1st order term.

Essentially, even-order nonlinearities generate even harmonics while odd-order nonlinearities generate odd harmonics.

Now, a symmetric circuit, such as a complementary push-pull circuit, generates odd-order harmonics for the reason that the even-order nonlinearities cancel.

An example of a circuit that creates 2nd order harmonics is a single-ended FET (a square-law device) amplifier.

• +1. It is worth pointing out that harmonics which distort the signal at a frequency of interest (or close to it) are the worst ones. While 2x, 3x, ... frequencies can usually be filtered out, in-band noise will propagate to the output. – Vasiliy Aug 30 '13 at 19:17