[sdiy] Simple discrete Unity-Gain Follower ?

[Magnus Danielson]

From: Don Tillman <don at till.com>
Subject: Re: [sdiy] Simple discrete Unity-Gain Follower ?
Date: Mon, 5 May 2003 01:35:48 -0700

Dear Don,

>    > > There are a number of serious problems with feedback loops,
>    > > even when they're working properly.  The most blatent is that
>    > > the output is the amplifier gain times the error.  That's just
>    > > philosophically wrong at a deep fundamental level.
>    > 
>    > Don, you didn't really mean to say that, did you?
> 
> Yes, absolutely.

Well, if we agree that we disagree on that one, then we at least have some
agreement. ;O)

>    > Again, it is about applying the technique adequately.
> 
> I disagree.  It's about the process that's going on.  The
> low-feedback-amplifier process is completely different than the
> high-feedback-opamp process.  I wouldn't even call the opamp circuit
> an amplifier -- it's really a servo.
> 
> Servos work very differently than amplifiers.  

I disagree, but there is a point in what you're saying. Let's sort out what I
agree and disagree on.

First of all, and amplifier is some device which increases one or more of the
common strength properties of a signal, i.e. it increases voltage, current
and power. Buffers is a special kind of amplifiers which maintains one of these
properties (most commonly the voltage, but could be the current). Anything
which satisfy these properties classifies as an amplifier.

But I think you really meant to say gain-element (it is easy to use them
interchangeably most of the times, but let's keep them distinct this time).
You mean to say that a feedback system (a.k.a. servo system) does not amplify
by pure a gain-element, but amplifies just the error signal until equilibrium
is acheived. This is true, but to do that you have a gain-element in the
feedback loop, somewhere prior to the output.

A simple such system would look like this:

The error term is the difference between the input Vin (positive input on
op-amp) and the feedback Vfeedback (negative input on op-amp):

Verr = Vin - Vfeedback

We amplify the error through a gain-element having the voltage gain of G, to
get the output voltage:

Vout = G * Verr

We then feedback the output through a voltage divider (two resistors) having
the feedback gain of F (where F <= 1):

Vfeedback = F * Vout

Mold these together you get

Vout = G * (Vin - F * Vout)

solving for Vout

Vout + G * F * Vout = G * Vin

Vout = Vin * G / (1 + G * F)

solving for Vout/Vin

Vout/Vin = G / (1 + G * F)

You can get almost any steady state gain by setting the feedback term F
suitably. However, things isn't as easy as this, as we all know. If we let
F = 1, i.e. if we hook it up as a classic buffer/"emitter follower" we get:

Vout/Vin = G / (1 + G) = 1 / (1 + 1/G)

which only approximates the gain of 1 and the lower raw gain G you have, the
bigger error. This comes to support you thesis. However, if you really wished
to have the overall gain of 1, you should have forced Vout/Vin to be 1, so
let's try that:

1 = G / (1 + G * F)

1 + G * F = G

G * F = G - 1

F = (G - 1) / G = 1 - 1/G

so we see that for perfect unity gain we need to actually feedback just
slightly less than we thought to hit unity, but it is doable. For a raw gain of
1 000 000 you would have a 1 ohm resistor over a 1 MOhm resistor, which looks
really silly, but would be theoretically almost correct.

Which proves my point for the DC case, you need to apply the technique
appropriately. The naive assumption is that G is infinitively high, so I don't
have to care. An insightfull design can do with much less raw gain, and still
make good designs. Feedback systems doesn't necessarilly make design efforts
easier by a stroke of magic, it just makes it different and as with all other
tools it needs to be applied correctly and with insight or it may fail you in
small or in great.

For the non-steady state case the analysis becomes more complex, but there is
also treatments for them, which all needs to be applied with due insight.

> The major source of distortion in a low-feedback amplifier is device
> nonlinearity.  The major source of distortion in high-feedback-opamp
> units are, well, a whole collection of weird side effects of the
> feedback loop dynamically responding to input, load, device
> nonlinearites, frequency compensation, and so forth.  It's not a clean
> process.

I don't say it is an easy design process, I say it is a different design
process and the concerns you voice is the concerns that I too see, but I also
see that you _can_ handle them if you take care to analyse them and find a
suitable design for the total system. I am by no means saying that it is just
tossing a feedback loop around something to make it behave well, absolutely
not, but what I am saying is that if you do it _properly_ you can get it behave
very very well. It is a design technique and you have to evaluate it's pros and
cons against any other design technique when both are well applied.

I think you basically dismiss a class of solutions because experience say that
devices built with them doesn't behave as you like, but then comes the question
if they really where designed with the full insight of how to reduce those
flaws for which you judge your devices. My point here is that it is a minefield
wherever you turn, feedback or not, and even if you follow good tradition in a
certain type of design and does that work well doesn't necessarilly mean you
have done a good design in a more objective fashion, it may be a good design in
comparision with others, but that's about it.

So, I say that your concerns about reproduction quality is valid, but I don't
agree with your conclusions on which type of systems is "best". This doesn't
mean I am unwilling to listen to what experiences there are, there are in my
opinion many usefull hints on what is going on in people experiences, but that
doesn't mean they have their analysis right. Also, when I do an analysis
someone else might naturally come up and point out my errors (and I am sure
people will). While my approach isn't as easy as yours (which is indeed
motivated by a massive accumulated experience), it also identifies very well
my state of mind as a researcher. I simply want to investigate and find out
other new ways of doing things. They may or may not improve things, but they
may at least increase the knowledge of why we do things the way we do them.
People rarely know really _why_ they don't do it another way. In my experience
thinking differently like this can be greatly rewarding and new solutions which
is much neater than others do come out of this way of thinking. So, from a
mental point of view I just have to dismiss your statements as really true, but
beyond that I see that I also can make a case which makes your statement not as
clearly true as you try to make it. Be sure, I am convinced that you think it
is true based on all the experience you and your fellow friends collected, but
this doesn't make my point of view incorrect, but rather much more that "we
don't have any experience to support that".

You could make an analogy between the theoretical physics and practical physics
if you like. What the theoretical physics side have done is to look at
discrepanceis of practical physics and find out the logic behind it. The whole
quantum physics where found out by discrepancies in otherwise seemingly sound
theory.

> For instance...  Slew limiting and TIM distortion don't exist in low
> feedback amps.  And simple nonlinearity doesn't exist (as such) in
> servos.

Actually, they do exist, the question is if you hit onto them. Your point about
feedback systems I would rather formulate as "the way feedback systems usually
is applied things get worse", which I think we _both_ agree on. You make the
conclusion feedback systems is just waste of time where as I look at the
feedback systems and tries to find ways to resolve things.

>    > > And, of all the truly cherished pieces of audio equipment (hifi amps,
>    > > guitar amps, signal processors, etc.) it's really hard to name any
>    > > that use large amounts of feedback in the audio chain.  And of all the
>    > > truly despised pieces of audio equipment, most use a lot of feedback.
>    > > The correlation is pretty significant.
>    > 
>    > The problem here is that "truly cherished pieces of audio
>    > equipment" may be apparent to you, but is surely not apparent to
>    > me and many others. The objectivity in the truth of the statement
>    > remains to be prooved.
> 
> Nahhh.  Some amplifiers rule and some amplifiers suck.  I don't know
> of too many cases where one person thinks a given amp rules while
> another claims it sucks, but you can throw out those cases and do just
> fine.
> 
> I mean, no one's claiming that a McIntosh MC75 sucks, right?

This may be apparent to you, but since I don't have a McIntosh MC75 or have one
around to listen too, how could I judge it?

Also, you still miss my point, I only point out that even if your collected
experience tells you one thing, it might not be true. It's all circumstantial.
However, as we know this does not automatically lead to a conclusion that it
really is true, just that from the given evidence it is reasnoble to conclude
that. I then make the point that there is more to feedback loops than what is
usually even considered in audio-gear. There is much more analysis done of
feedback systems than what is ever taken into considerations. Since there is
alot of such analysis, we can take lessons from that and see what a more
refined analysis would give. Could we avoid TIM and IMD in a feedback system?
Possibly. Could we acheive lower THDs in a feedback system if we avoid other
distorsion factors? Possibly. Since we have not been able to rule out these
possibilities, your statement doesn't hasn't been proved correct, since we not
found the counter-exampel. We simply disagree on wiether the counter-exampel is
possibel or not. I am willing to investigate that to some degree.

>    > But don't fall for the normal audiophile ways of prooving it to
>    > me, I won't care.  I stopped listen to that in the 80thies.
> 
> Don't need proof; the goodness should just be obvious.  Follow the
> zen. 
> 
> I know it sounds like I'm being silly, but I'm completely serious.
> Most proofs involve one or more weird abstractions, a measurement or a
> metric that can be used to rank an amplifier.  But that abstraction
> requires a set of assumptions, and those assumptions won't be relevant
> to life in the real world.  Total harmonic distortion measurements,
> for instance.

Your point here is about relevance, and I agree. Most of the science (not only
in the field of audio) succseeds in abstract themself sufficiently away from
the real world problems so that their conclusions of having an optimum system
doesn't apply in real life terms. Those are flawed scientists, and the world is
full of them. Regardless of whatever measure, or model or whatever else mean of
evaluation and analysis you take, you must be able to show the relevance of
your methods. This is certainly on my agenda. I've done my homework on that
many times and make it a pride to make sure my own analysis makes sense in a
real world and point out flaws in others. So, we agree that theoretical
evidence must verify in real life observations, great. Do I have an amp for you
to listen to? No, not yeat, but that was actually not the topic, it was on
weither your experiences really implied a statement being true, and I tend to
say "well, it may seem like it, but I tend to think that's not true".

Having different experiences and conclusions isn't necessarilly bad to start
with. It is the discussions where people really doesn't agree but are able to
rub their arguments against each other which could be very giving for all
parties, even if it is hairpulling for all of them while it is in the heat of
the argument.

>    > As for ways to design things, it is *really* hard to judge a
>    > design. You must quantify aspects, and the human hearing and
>    > perception is as such not very easy to grasp. There is many
>    > concepts you got to get really right, and I've found that some of
>    > the concepts is not even well understood in the audio world at
>    > large. What I've learned was that there where really few
>    > authorities that really grasped all the peculiarities and could
>    > rationally explain them, and I mean *REALLY FEW* (approx to 0).
> 
> I agree completely.  I have a lot to say about this, but it will have
> to wait until I write up properly.

I would be eager to read what you have to say and discuss on different topics.
You have to excuse me for being such a hard-headed guy that just don't buy
certain things straight off even if you take them for granted, but then I think
you will find that we agree very much on other things, so the discussion could
be a very interesting one.

I just want to stress that I understand why things becomes so difficult since
non-linear systems is not easy to analyse compared to linear systems. Toss in
one or more feedback loops and you have to start thinking really clear.

My advice about such systems is to do as good design as possible, and use the
feedback loop to finsih it off. Care needs to be taken such that you don't run
into excess non-linearities such as clipping, saturation/slew-rate limits,
through-zero non-linearities etc. I will look some on how to avoid some of
these flaws, since it seems it is needed for the discussion, but I already
see a number of possible ideas to try, so it doesn't seem hopeless.

PS. Don, don't think I don't like discussing this with you, I do and I
certainly not mean to make it personal. I just don't really agree with you and
I think that is a great basis for a discussion from which both may gain
insight. Why isn't this done with a whiteboard and interviening dinner????

Cheers,
Magnus - too far away from Palo Alto