Distortion spectrum vs feedback
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Alfred Centauri
diyAudio Member

Join Date: Jul 2007
Location: Metro Atlanta, Georgia
NFB per se does not generate higher order harmonics.

Quote:
 Originally Posted by janneman It's a Baxandall graph, not Linsley-Hood as SY noted. The graph is a result of feeding back a signal through a square-law device. Any harmonics in the feedback get again distorted by the square law, go through the feedback again, get distorted etc etc. So initially, when feeding back, you get a whole slew of additional harmonics before the feedback gets so powerfull that they all get supressed.
I'm going to make the argument that this simply isn't correct. I've read Baxandall's WW articles on NFB around non-linear amplifiers and, with all due respect, his explanation of the mechanism of higher order harmonic generation with negative feedback is, at best, misleading.

The fact is, higher order harmonic generation occurs in even a simple series circuit with a passive non-linear element.

For example, let G be a non-linear conductance, i.e., the current through G is a non-linear function of the voltage across G:

i_G = g(v_G) (1).

If a voltage source v_S is placed across G, the current i_G is simply g(v_S).

Now, place a resistor R in series with v_S and G. By KVL and Ohm's law, the voltage across the conductance is given by:

v_G = (v_S - i_G * R) (2).

Substitution of (2) into (1) yields:

i_G = g(v_S - i_G * R).

The attentive reader will recognize this equation as having precisely the same form as would a non-linear transconductance amplifier with negative feedback.

Importantly, this equation implies that i_G has derivatives to all orders, i.e., the spectrum of i_G has harmonics to all orders for any non-linear g(v_G).

The point is this: there is no feedback in the simple series circuit in any reasonable sense of the word and yet, for any non-linear conductance, no matter how simple, the addition of a resistor in series with this conductance changes the transfer characteristic from one that generates a limited number of harmonics to one with an unending series of harmonics.

This, for me, clearly refutes the claim that it is NFB per se that creates higher order harmonics via some sort of circulation of distortion products through the non-linear amplifier.

Bottom line: it is not NFB per se that creates higher order terms in the transfer characteristic. There is no circulation of distortion products. Baxandall's presentation does not, as I read it, imply anything of the sort.

Rather, his presentation is, I think, more of an iterative approach to finding a solution, i.e., assume the output is of a certain form, plug it in, correct as necessary and try again. Through this process, one is inescapably led to the result that, when a circuit variable is equal to a non-linear function of itself and another variable, the only admissible solutions have an unending harmonic spectrum. The very same approach can be applied to the simple series circuit above where, again, there is no NFB as currently understood, no alleged circulation of distortion products, no alleged distortion of distortion etc.

And unsurprisingly, the results are precisely the same (as they must be since the mathematical form of the output equations are the same!).

Evidently, when any non-linear element, active or passive, is combined with other circuit elements including linear ones, the resulting transfer or driving point characteristic, in general, is not a finite order polynomial but rather, an infinite one.

So, whether it's local NFB, global NFB, or simply placing a resistor in series with the diode, triode plate, cathode, emitter, source, etc., you cannot escape the generation of an unending series of higher order terms no matter how simple the essential non-linearity.

If anyone's interested, I've done a number of simulations to illustrate the points above. More to come...

 1st December 2011, 10:56 AM #62 DF96   diyAudio Member   Join Date: May 2007 Despite having been the source of this thread I seem to have missed it until now. I was on holiday in Copenhagen at the time. The point made above about higher order products arising simply from combinations of linear and low-order components is a valid one. You just need some algebra! In theory it should kill off wild claims about feedback delays etc creating distortion, but I doubt if it will because such people are rarely convinced by something as prone to personal opinion as the facts of algebra. It is a while since I read Baxandall's articles on this but I would be surprised if he was misleading, unless he was deliberately slanting his argument to fit his supposed audience. He was clever. However, I suppose that the originator of an idea does not always present it in the clearest way as he is still trying to grasp it for himself. (I found this in another context when I was looking at Class E amplifiers: invented by the Sokals, but it was Raab who gave the first clear analysis). Just to clarify my remark which gave rise to this thread, low amounts of feedback can be a problem if the original amp has significant low order and little high order distortion. It does not take much feedback to generate higher order products. If the amp already has a reasonable amount of higher order then the extra products may add to or cancel the intrinsic ones. This may be real life. If you believe, as some seem to, that triodes give 'pure second order' then 6dB of feedback would be unhelpful. If you realise that trodes have all orders, and each stage multiplies up the (infinite) orders from the other stages then it might be OK.
Alfred Centauri
diyAudio Member

Join Date: Jul 2007
Location: Metro Atlanta, Georgia
Quote:
 Originally Posted by DF96 It is a while since I read Baxandall's articles on this but I would be surprised if he was misleading, unless he was deliberately slanting his argument to fit his supposed audience.
I too do not think he would intentionally be misleading. I think it is the case, however, that he was unintentionally misleading. I should have been more clear. Thanks for pointing that out!

AC

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