Feedback

[Magnus Danielson]

Hi folks!

I just wanted to comment some on this feedback. For some of you it
migth not be obvious on how it work, it's a kind of non-linear thing
so let's make a model just for the sake of it.

First, the linear case that we normally assume:


1 + +---+ 3 +----+   4
----| + |---| G1 |-*--
    +---+   +----+ |
    - |            |
    2 |     +----+ |
      *-----| G2 |-*
            +----+

This model is an op-amp with the diffrential mode gain G1 and common
mode gain of 0. Time delay and other frequency behaviour is rejected.
The feedbackloop provides the gain G2 (0 <= G2 <= 1) using resistor
divider. The equations we can draw from this is:

U3 = U1 - U2

U4 = G1 * U3

U2 = G2 * U4

U3 = U1 - G2 * U4 = U1 - G2 * G1 * U3 => U3 + G1 * G2 * U3 = U1 =>

U3 * (1 + G1 * G2) = U1 =>

          1
U3 = ----------- U1
     1 + G1 * G2

         G1            U4       G1            1
U4 = ----------- U1 => -- = ----------- = ---------
     1 + G1 * G2       U1   1 + G1 * G2   1/G1 + G2

So, the gain of the op-amp (U4/U1) is set mostly by G2 and the gain
error (which comes from the 1/G1) reduces as G1 becomes greater. G1 is
by the way also called the open loop gain. This can reach a gain of
over a million, so it can be great.

Now, while this model was simple it was also quite linear. Time for a
model having a non-linear output, thus:


1 + +---+ 3 +----+ 4 +------+   5
----| + |---| G1 |---| f(x) |-*--
    +---+   +----+   +------+ |
    - |                       |
    2 |        +----+         |
      *--------| G2 |---------*
               +----+

Now, from this we get:

U3 = U1 - U2

U4 = G1 * U3

U5 = f(U4)

U2 = G2 * U5

>From this we get:

U5 = f(G1 * U3) = f(G1 * U1 - G1 * U2) = f(G1 * U1 - G1 * G2 * U5)

So, from this point on it really depends on the non-linear function
f(x). So, if we let f(x) = x then we just get the same thing as above,
but by letting

                2        3
f(x) = x + a * x  + b * x

Solving this set of equations are not easy since it forms an recursive
loop, but lets just try it for the sake of it:

                                                          2
U5 = G1 * U1 - G1 * G2 * U5 + a * (G1 * U1 - G1 * G2 * U5)
                                   3
     + b * (G1 * U1 - G1 * G2 * U5)

ugly, but surely we can clean it up some:

                     2     2         3     3
U5 = G1 * U1 + a * G1  * U1  + b * G1  * U1

                               2                        2     2     2
     - G1 * G2 * U5- 2 * a * G1  * G2 * U1 * U5 + a * G1  * G2  * U5

                 3          2                  3     2          2
     - 3 * b * G1  * G2 * U1  * U5 + 3 * b * G1  * G2  * U1 * U5

             3     3     3
     + b * G1  * G2  * U5

Eh... who just said cleanup??? ;)

Now, just go looping over this by taking the U5 expression from the
round before and insert U5 at each that point and you end up with the
end with the end expression. Well, short story.... it just blows up!

As you increase gain G1 and/or G2 the overtones start to rise on you,
and now they are not only the second and third, they exist at higher
degrees as well all of a sudden. Now, just imagined how ugly this
would have been with higher degrees in the f(x) formula aswell.

Just be happy that we did not consider the propagation delay, phase
and amplitude changes due to internal topology not to speak how that
design really behaves as a non-linear curcuit.

So, where this little exercise enougth enligthening to understand why
negative feedback in high gain op-amps may not be so beneficial to
distorsion?

This is just one reason why to avoid it.

Cheers,
Magnus