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In Reply to: Re: Regarding Boyk and Sussman, Cheever, etc. posted by john curl on March 14, 2007 at 13:58:39:
Many years ago I saw part 5, but not part 6, and there are many facts that are brought out in this series of articles.
In part 5, it is interesting that Baxandall proves that the parabolic approximation cannot hold up at low currents with a fet, and higher order distortion is inevitable when fets are operating well below their normal working current.I think this sequence of articles is about the best out there on this subject. Regarding the FET characteristics, I was coming at it from the angle of the Massobrio and Antognetti device modeling book I referenced above. Though it's nominally a text about SPICE modeling, a major portion of the book is concerned with deriving equations relating the terminal currents and voltages of the device based purely on considerations of device physics. The formula they get for a FET is pretty messy, involving, among other things, a square root term with Vgs cubed inside the square root. Then they show the usual quadratic approximation with a graph comparing the two. The graph shows them to be very close. What I took home from this was that the simple quadratic approximation was not based on device physics at all. Rather, it just happens to be a pretty accurate and much more convenient approximation to the formula obtained from device physics. I could scan these pages from the book and send them to you if you're interested.
This is the sort of thing that we should be talking about and even debating, rather than the usual.
Amen! Technical discussions are what this forum is all about. I don't know how it turned into the nightmare it did. I think people realized that it was essentially unmoderated, so they know they can come here and create havoc.
Follow Ups:
I was looking at part 6 with the BJT results. It seems to me that with BJT amps in particular, typical feedback levels are usually over 20db to adequately reduce the gain. Is this correct? If you look at the graph it is clear that for higher harmonics, up to about 40 db of feedback the 6th harmonic (and presummably it gets even worse for the harmonics higher than the 6th) is significantly higher than with no feedback. However; aren't there stability issues if you use so much feedback? At 20db, other than a lower 2nd and 3rd harmonic, all other harmonics are higher in level than with no feedback.The nulls created by phase of the distortion are interesting but notice that they don't occur at the same amount of feedback for each harmonic. So it doesn't seem like it would be possible to design an amp with feedback to "catch" all of these null points, thus minimizing feedback and lowering distortion.
Hi Brad,I think John answered your question above. But there are a few things I want to add.
First, these distortion percentages (and also the relationship of higher-order terms to the fundamental and lower-order terms) are not a fixed quantity for a given level of feedback! They vary with signal level. This is discussed by Basandall, but I'll summarize the results below. In order to get the numbers he got, Baxandall had to assume some value of AC current swing relative to the DC (bias) value. So "signal level" in this context represents the variation in device current relative to the DC current value. He chose those values both to simplify the theoretical distortion computation, and also to induce plenty of measured distortion so the effect of feedback could be assessed easily.
So how do the distortion components vary with signal (current) swing? Per Baxandall, and provided gross distortion is not occurring, the following relationships apply:
1) Percent second harmonic distortion is proportional to signal level
2) Percent third harmonic distortion is proportional to the square of the signal level
3) Percent fourth harmonic distortion is proportional to the cube of the signal level
4) Percent fifth harmonic distortion is proportional to the fourth power of the signal level
...and so on.These relationships describe the so-called "weakly nonlinear" behavior of class A circuits. They are a direct result of the Taylor series representation of the circuit's nonlinear transfer characteristic with a sine wave input. It should be noted that with topologies for which transistor switching occurs, such as power amp class AB output stages, these relationships do not apply. Let's temporarily ignore the class AB case so we can focus on what Baxandall's results show.
One could view the relationships listed above as a "glass half empty" or a "glass half full" situation. For the "half empty" perspective, one can see that as signal level (current) increases, the higher-order terms will rise very fast. For the "half full" perspective, you could say that the higher-order terms go down very fast as current swing is reduced.
Is there a way to take advantage of the "half full" perspective? Yes! The key (for class A stages) is that the quantity we're interested in is voltage , not current. For the type of gain stages normally used (common-base, common-emitter, common-source and common-gate), the voltage gain is proportional to the load impedance of the stage. If we make this impedance larger and larger, then for a fixed voltage swing at the output (what we want), the current swing required gets smaller and smaller. Because the required current swing for a fixed voltage swing gets smaller and smaller as the load impedance increases, the higher-order harmonics go down and down at a very fast rate. But now, because of the high load impedance, we have this high voltage gain. But the needs of the circuit are for only moderate gain. What do we do with the extra gain? It's used to provide high feedback. I hope you can see the cycle here. The increased load impedance makes the current swing less for a given voltage swing, reducing the distortion for a given voltage swing requirement. But the increased voltage gain also results in increased feedback, further reducing the distortion. So we get a double benefit here. Of course, the amount of feedback has to be limited because of stability considerations, but that's a separate topic. But this is the general idea of high-feedback amplifiers. Note that this situation does not fit the false dichotomy often mentioned in connection with the low-feedback vs. high feedback debate. That false dichotomy is "Have high distortion in the open-loop amplifier and cover it up with high feedback" vs. "Make the open-loop amplifier as linear as possible and use low feedback". The example above clearly shows the open-loop amplifier is getting more linear as the amount of feedback goes up.
So here's something to think about. Because of the points (1)-(4) above, what can you say about the applicability of Cheever's distortion formula?
Thanks for clarifying a few things for me. I already had the jist of most of what you were saying but the explanation regarding load impedance and using feedback to an advantage there is interesting.However; you said, "Of course, the amount of feedback has to be limited because of stability considerations, but that's a separate topic. "
But this is pertinent to the situation because it is a limit as to how much feedback you can apply to a given circuit. This also then would have an effect on where on these distortion curves you end up.
I see the exponential increase in high order harmonics with level as a big no no in general and that it largely negates the advantages of reducing lower order distortion. Audibility of distortion is for me one of the major issues and underplayed by many engineers.
I assume when you say Class AB is different you mean that it is more strongly non-linear? Crossover distortion being one serious issue (and not fixed with negative feedback). What are some others?
"The example above clearly shows the open-loop amplifier is getting more linear as the amount of feedback goes up."
Linear in terms of total distortion yes, but at what price to audible harmonics?
From Cheever's thesis he seems to focus not so much on the amp mechanics but on the resulting distortion PATTERNS that are coming out of them. He has derived an audibility pattern (he calls Aural harmonics) that is level dependent (he has dbA in exponential term) and gives strong weighting for audibility towards high order harmonics (the n11 term in the denominator).
Unless very large amounts of feedback are used, the higher order harmonics will increase significantly when feedback is applied. Cheever claims that these will be much more likely to be audible than relatively high levels of low order harmonics (although already the third must be a small fraction of the 2nd to be inaudible according to his aural harmonic sequence). However, stability is an issue limiting most amps in the application of feedback. I don't say his metric is right but if he is right about the audibility of these harmonics (ie. his aural harmonic series) then it seems clear to me that amp design should focus on making a pattern that is similar or lower at each harmonic than this pattern. In the real measurements I have seen, I don't see this happening.
Brad, I'll get back to you a bit later tomorrow. You raise some very interesting points, and it's clear to me that I should explain my position better. It's much easier for me to talk about mathematical things than it is to express my views about the Cheever thesis (which I also hope to do briefly in your thread at the top of the page). Stay tuned, I haven't forgotten you.
Sometimes.
One cannot easily use the nulls in the distortion, but you can see the trend, and avoid worst case.
I would very much appreciate any input on a more accurate fet model. Also, device manufacturing techniques, such as diffused or ion implantation many change the transfer function also. It is sometimes obvious when changing manufacturers of the same part number fet.
Hi John,
Do you think more modern Fets will have the same behavior as the ones measured by Baxandall? The question then would be, do they fit more closely to a quadratic relationship or do they deviate further?
FET's MUST deviate from their quadratic relationship as they drop further away from their normal operating current, or they would exceed the physics limit of Gm/I =40, because the change in Gm in fets is normally slower than that of bipolar transistors with lower current.
Okay, I'll scan the relevant pages and email them to you. Device physics is not my thing, so I couldn't follow it in detail.There may be multiple emails, as I can't zip them together because of the Mac/PC thing.
Andy, I appreciate anything that you can do. It is difficult to get real insight on these design problems. Of course, you can 'cover' them up with negative feedback, BUT if Boyk is correct, then we are not doing ourselves any favors.
I was attending engineering classes when Spice circuit analysis was first introduced. I was there to hear the active circuit models explained. However, my classmates and I knew that it was only an approximation of the real thing.
When I first worked with Mark Levinson we made complementary differential fet input stages. However, we found a different distortion spectrum when we used National Semi fets instead of Siliconix fets, yet they had the same part number. I know that the amount of higher order harmonics is very important, so I try almost anything to get lower. I don't worry too much about 2'nd or 3rd harmonic. It is just too much part of the music to make me concerned with .01% or less, BUT a little 5th, or worse: 7th or 9th harmonic, at listening levels, and I know that I have failed to make a truly successful product. This is why I measure individual harmonics down to .0001% with noise averaging to resolve xover artifacts that MUST reside in a typical IC op amp's distortion spectrum.
Based on the specifics of the manufacturing, such as layer thickness and uniformity, I could imagine the actual behavior to vary from fet to fet. That being the case, I would guess then that it would be necessary to obtain an actual transfer function for the exact transistor you are using in order to model it adequately. That would be a big PITA I guess.
We usually just build and test a prototype. Excessive reliance on modeling can give only partial results.
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