[sdiy] Harmonics Question

[Tom Wiltshire]

On 3 Nov 2011, at 03:51, Kyle Stephens wrote:

> Follow up question: just how do vacuum tubes tend to produce predominantly even harmonics, and likewise why do transistors rock the odd harmonics?

This is down to the transfer functions. An amp's transfer function is basically a waveshaping function, and although obviously the ideal is that it's a straight line, it won't be. Especially if you drive it really hard.

If you look up waveshaping theory, you'll see that there are two basic types of waveshaping function, even and odd. Odd functions produce odd harmonics and even functions produce even harmonics. The odd functions are rotationally symmetric, which covers a typical transistor amp's transfer function with overdrive at the top and bottom ends - hence the odd-only overdrive harmonics. Even functions are mirror symmetric, which is not something you see with amps, but rectifier circuits have this character and are used in some octave-up pedals.

There's a couple of examples on this page:

http://crca.ucsd.edu/~msp/techniques/v0.11/book-html/node78.html

You can see a classic clipping distortion (rotationally symmetric function, hence odd harmonics) and a squaring function (mirror symmetric hence even harmonics - in the example it doubles the sine frequency)

Most interesting transfer functions combine both odd and even elements, which gives functions which aren't symmetric and produce both types of harmonic. Tube amps have transfer functions which are substantially rotationally symmetric (like transistor amps), but often have significant offsets from the centre zero point, which spoils that rotational symmetry making the function non-symmetric and introducing even harmonics.

There are two elements to 'tube sound' from this point of view. One is the soft clipping. This gives a slower build up of distortion harmonics, and preserves dynamics better whilst still overdriving. The second is the non-zero offset, which stops the transfer function from producing entirely odd-order harmonics. Neither of these characteristics are unique to tubes. Transistor differential amps have the soft clipping character, as do various circuits using diodes to alter an op-amps feedback. The non-zero offset (or equivalently, unequal gain for the upper and lower halves of the waveform) is also easily added to non-tube circuits.

There's a java applet on this page for playing with some of this stuff:

http://music.columbia.edu/cmc/MusicAndComputers/applets/4_6_shapetable.php


Hope this helps!

Tom