Letters to the editor Dear Editor, I read with considerable interest the article by Bruno Putzeys in Linear udio Vol 1, The F- word - or, why there is no such thing as too much eedback. In his excellent article, Bruno explains that increasing the amount o eedback might cause problems or the higher harmonic distortion components. He uses earlier work o Baxandall, who did his measurements on the non-linearity o a semiconductor device inside a eedback loop, while the amount o eedback was increased. Bruno uses in his igure 14 a square law non-linearity as example. I ve been wondering or some time whether this kind o non-linearity is ound in ampliiers using valves, which obey the Child-Langmuier-Compton equation. In 2010 I perormed measurements on my SPT70 valve ampliier(1) and the results are shown in Figure 1. The amount o eedback is given by 20. log(nb/o) on the horizontal axis, where nb and o are the gain o the ampliier with nb and open loop, respectively, with a 4 Ohm dummy load. This diers rom the way it is presented by Bruno s igure, but is the same as Baxandall s representation. My hesitation to accept Baxandall s and Bruno s approach seems to be right and the consequences are o importance. In my speciic valve ampliier all harmonic levels decrease when the eedback is in- Figure 1: Distortion in SPT70 as unction o the amount o eedback. The harmonic components are (top to bottom): 2, 3, 4, 5, 6 and are measured at 1kHz with 0 db = 10Vrms in a 4 Ohm load. 1
creased. Subjectively this is clearly noticeable. More eedback gives cleaner sound with more details (but harms other qualities o the sound reproduction(2)). The consequence is that the type o nonlinearity in my valve ampliier is o another nature; not semi conductor (power o e ) or square-law, but like a power o 1.5. This means that raising eedback in my amp does not create stronger higher order harmonic components. Reerences: 1 - Menno van der Veen: High-end Valve mpliiers 2, chapter 8; Elektor ISBN 978-0-905705-90-3. 2 - Menno van der Veen: Ontwerpen van Buizenversterkers, chapter 2; Elektor ISBN 978-90-5381-261-7; also available in German, not yet in English. Menno van der Veen, Zwolle The Netherlands Bruno Putzeys replies: Dear Menno, I thank you or your comments. I take it your main problem is the idea that moderate amounts o eedback are always worse than either lots o it or none at all. I agree that such a broad-brush conclusion would be unwarranted. What matters is that it can happen and certainly has happened to several experimenters, and that this is one o the actors that led to a suspicion o negative eedback. You can see a practical example in http://www.passlabs.com/pds/articles/ distortion_and_eedback.pd (igure 11, page 10). It is clear that this circuit will not appreciably improve unless at least 20dB o loop gain is deployed. One does suspect, however, that that circuit was expressly designed to illustrate a point. Practical ampliiers are more complex, creating a jumble o nonlinearities. Whatever undamental law underlies one active component is no longer going to be immediately obvious rom the behaviour o the circuit. n early drat o the article contained a paragraph to this eect: i the open loop distortion spectrum is already suiciently rich, newly created harmonics arising rom lower order terms might never become signiicant. This appears to be the case with your setup. I the variable eedback listening test is perormed on such an ampliier, subjective perormance would improve rom the word go. Your listening result conirms this. This is a urther strong argument in avour o using negative eedback, but at the end I elt that mentioning this in the article risked conusing the reader who irst had to understand that some crucial anti-eedback experiments were made on equipment that did indeed sound worse with moderate loop gains applied. 2
I won t attempt to comment directly on the evolution o the spectrum based on the Child-Langmuir-Compton equation. Partly because as I said, your ampliier is more complex than a single valve so the spectrum is not determined by just this equation, and partly because I wanted to point out an interesting generality which you can then set loose on any kind o non-linear transer (that is, I m giving you 10 0 homework). Observing 10 the Baxandall style plot rom the article, we can t help noticing that the relative distribution o harmonics settles down as loop 20 30 40 50 60 70 gain increases. The 80 closed loop nonlinearity asymptotically (or large values o loop 90 100 110 gain) approaches a 120 40 30 20 10 0 10 20 30 new unction that isn t Loop Gain (db) the original nonlinearity. 40 50 60 70 80 Harmonic Magnitude (db) So although we already igured out that the elementary explanation o eedback reducing the harmonics o the orward nonlinearity by a actor equal to loop gain was a bit oversimpliied, we see something else looming on the horizon: another unction whose harmonics are indeed being reduced by loop gain. That would be great, because it would make it easier to predict the spectrum o ampliiers with requency dependent loop gain (i.e. all o them). So what unction is it? Take the model presented in the article, but now put in any unction: x V out=(v in) y 3
Writing this down we get: y = ( ( x y ) It s not hard to see that or most unctions there s going to be no algebraic solution. That doesn t mean we can t get to see the shape o it. pply the inverse o on both sides. ( y) = ( ( ( x y ) 1 or ( y) = ( x y) Shuling a bit we get: ( y) + y = x ( y) + y = x (1) Right. Strictly speaking that brings us no urther to a closed orm solution or y. But hang on. s increases, y approaches x. Put on your numeric hat and try to solve the equation by successive approximation. First, rewrite (1) as: y = x ( y) nd substitute it into itsel: x ( y) + y = x (2) ter how many iterations would this procedure converge? Very rapidly i is large. May I suggest we don t even iterate once? The only error we re making is to neglect the distortion o the distortion. 4
For large values o, we may say that: ( x) + y x Rearranging: y x ( x) Conclusion: Negative eedback does not attenuate the harmonics o the open-loop nonlinearity, but o its inverse. 5