[sdiy] important questions!

[Magnus Danielson]

From: "Arun Bohm" <arun_bohm at hotmail.com>
Subject: [sdiy] important questions!
Date: Mon, 23 Dec 2002 10:27:32 +0000

> Hi

Hi Arun,

>    Why do so many DIY-ers seem to be into the TL082? What are it's 
> characteristics and what is it's pin-out. In terms of studies I've only just 
> got to the part where we discuss (in books) how the gain is affected by the 
> frequency of the input.

It's dual FET input (high input impedance) with fairly good noise and
offset properties. It's pretty cheap and available about everywhere you
usually go. It's certainly not best in class, but a fair compromise.
Some favor the TL-07x family over the TL-08x family, since they have
slightly better performance. The TL-082 now fills the default-slot for
many in the same way as the uA-741 was the default for a long time and
the NE-5532 and NE-5534 did.

> Also, another question. What is it about non-linear devices which
> allows for the sums and differences across it's junctions? I understand
> that it has to do with some fourier interpretaion of the seemingly
> parrabolic curve which describes it's voltage current relationship. But
> it has been some time since I did a fourier transform, and even then
> only for homework.

If you look into analysis you find the McLauren and Taylor series allows
you to model virtually any curve by a sum over raising powers of the
input variable x (or x-a where a is some constant around which you wish
the resolution to be good). You will also recall that their power series
exists for a number of our popular functions, like that for exp, sin, cos
etc. I.e. if we have a number of integer raised powers of the input we
can mimic all non-linear properties. With this as background it will not
be too hard to conclude that all non-linear properties can be equalently
well modeled in this raised power model. I am a bit over-generalizing,
but it is for the best of presenting the end-point here, so I give you a
schetch of the mathematical truth, not the mathematical truth itself.

Now, if we toss in a sine into a raised power of 2:

           2                            2       2
y(t) = x(t)  x(t) = Asin(wt) => y(t) = A sin(wt)

classical trigonometry gives us

                 1
[sin a][sin b] = - [cos(a-b) - cos(a+b)]
                 2

which for the special case of a = b becomes

       2                    1                         1
[sin a]  = [sin a][sin a] = - [cos(a-a) - cos(a+a)] = - [1 - cos(2a)]
                            2                         2

Making the endresult

                              2    2
        2 1                  A    A
y(t) = A  - [1 - cos(2wt)] = -  - - cos(2wt)
          2                  2    2

I.e. we've got us a DC offset of a half A square and we got a cosine of
twice the frequency!

I recommend that you do the exercise of drawing this on a paper to help
convince yourself. Having done that can help the mental process many more
times later in your engineering life if you ever come accross a similar
process! (This is really one of the secret trades, you understand what it
"does" not how it "functions"!)

Naturally, as you raises the power you get to use a few more variants of
the trigonometry rules. However, we will take another basic case, namely
that of tossing two different sines into the square-curcuit.

           2
y(t) = x(t)  x(t) = A sin(w t) + A sin(w t)
                     1     1      2     2

                                2
y(t) = (A sin(w t) + A sin(w t))
         1     1      2     2

        2        2                            2        2
y(t) = A sin(w t)  + 2A A sin(w t)sin(w t) + A sin(w t)
        1     1        1 2     1       2      2     2

The first and last terms we already know how to solve, they are obvious
enought from the initial test. The middle term is however interesting.
We can now again use the classical trigonometry case and we get:

              1
y  (t) = 2A A - [cos(w t - w t) - cos(w t + w t)]
 12        1 22       1     2          1     2

y  (t) = A A cos([w -w ]t) - A A cos([w +w ]t)
 12       1 2      1  2       1 2      1  2

So here you got the delta and sum frequencies!

The totat response from the squarer would naturally be

        2    2
       A    A
y(t) =  1 -  1 cos(2w t) + A A cos([w -w ]t) - A A cos([w +w ]t)
       --   --       1      1 2      1  2       1 2      1  2
       2    2

        2    2
       A    A
     +  2 -  2 cos(2w t)
       --   --       2
       2    2

I.e. we got two DC contributions, we got the frequency doubled input
frequencies and we got the delta and sum frequencies.

If we got some DC offset on the input, then we would also get leakage of
the input signals to the output, and in all practical curcuits we do.

For all higher powers, you go about in the same fashion. You need to add
the rules for sin times cos and cos times cos, but those are classical
textbook formulas, so I woun't bother to give them.

There are tricks to help to cancel out parts of the output by combining
the result of two, four or more non-linear devices. Such balanced systems
works by sign reversal tricks. This is how the diode ring came about. The
diode ring is also called (diode) ring-modulator by radio folks, because
a modulator modulates the amplitude (complete with sign reversals) and
thus manifest a 4-quadrant multiplier or a "mixer" in a radio-terms.

Ring-modulators is known to have existed at least back in the 40thies (I
have a radio-designers book from 1941 having it, using copper oxide
rectifiers - that nearly predates the "invention" of the semiconductor!)

Ring-modulators is used at higher frequencies today, you can get them of
the shelf for many GHz and they basically still work on the same
principle as back in the early days. For lower frequencies we tend to use
Gilbert cells which has benefits when well trimmed. Those also reach
quite high in frequency (I've seen 600 MHz but I haven't actually looked
very carefully, I'm not into radio stuff).

Anyway, with the tools that you now have you should be able to
extrapolate over to most other situations.

There is one case not covered worth mentioning, and that is the double
mixer setup. When you want to cancel either the difference or the sum
frequency, you can run two multipliers having the inputs differentiated
by a phase of 90 degrees. This brings up the full set of sin and cos
of both a and b signals and the output summing/difference network can now
form signals of either sum or difference, or any mix of both.

This is naturally what is being used in frequency shifters.

> Was this first discovered theoretically and then applied experimentally. 
> Had to have been. Not anything like observing forces between current 
> carrying wires, a totally experimental discovery.

The "theory" is classic in all senses. It had been in place way before it
could be used in much practical experiments. It was part of the basic
training for engineers when electrons came into play. Surely it took some
time to figure out the connection between know properties and observed
properties, but when you've got the basic properties (like the
exponential behaviour of a semiconductor junction (you may not even have
to know it is a semiconductor!!!)) then you can turn back to math
exercises and figuring out how to engineer a better solution and later
try the new topology in practice. This is really what we have math for,
it's a tool for us to become a better engineer. For a few souls, math as
such have a beutty, but the main power of math is that it helps us to
model, calculate and even estimate future behaviour of designs.

Oh, did you have to do any Fourier transforms in the above? No... but
once you understand it, the Fourier way of handling it becomes quite
obvious. However, Fourier transforms and analysis normally assume linear
properties, and you need to go back to the non-linear aspects first
before attempting to model it using Fourier methods. Fourier transforms
is amongst the set of linear transforms that we can use, it has its
benefits and drawbacks. Few things is actually "natural" to Fourier
transforms, but it's so powerfull that it is being used anyway with one
or a few modifications. The LaPlace transform is actually more powerfull,
but you can acheive the same result as the LaPlace transform by
postprocessing the data before tossing it into the Fourier transform.

Fourier and LaPlace where not best of friends due to scientific
disagreements. Fourier being of the younger generation and LaPlace of the
older. This helped to delay Fouriers recognition many years actually.
Luckilly we don't have to bother that much about those issues these days.
We just use the outcome of these great minds that broke the barrier for
us.

Cheers,
Magnus