Sources of Harmonics(Odd and Even)
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.