Kevin Aylward wrote:

gwhite wrote:

Non-linearity is *not* required to create DSB-AM out of
transconductance type multipliers like the gilbert cell. In fact,
*non-linearity is specifically something that designers hope to
minimize* -- just like in any linear device. The standard linear
approximation practice ensues: that is, the taylor expansion of
exp(x) is done and the linear term is the desired one and *it is all
that is required or wanted for this linear multiplier*.

Ahh... now I see where the confusion is, and I did already address this
by my comment on the non availability of real, linear voltage controlled
resisters. I stated that in principle, one might be able to find a
device that was strictly linear in order to achieve modulation. I also
stated that such devices do not appear to exist, such that in practise,
one generally has to use a non-linear device to achieve multiplication.

Once again, it is *not* the non-linear aspect of it that "produces" the
multiplication (frequency translation in this case).

Maybe this will help you:

The Circuits and Filters Handbook
0849383412
CRC Press (c) 1995
Wai Kai Chen - Editor in Chief


Transconductance Multipliers

A direct, straightforward technique to realize the multiplication
function exploits the possibility of controlling the transconductance
of transistors through an electrical variable (current or voltage).
Although this feature is exhibited also by unilateral amplifiers,
most practical realizations use differential amplifiers to reduce
offset problems and enhance linearity [25]. Figure 32.32 shows a
generic schematic for a differential amplifier, consisting of two
identical three-terminal active devices with common bias current.
The expressions on the right display its associated transconductance
characteristics for npn-BJTs and n-channel MOSTs, respectively [25].
These characteristics are approximated to a first-order model as


i_y*v_x
i_z ~ --------- (BJT), i_z ~ sqrt(Beta*i_y)*v_x (32.39)
4*U_t

[~ := "approx. equal to"]

which clearly displays the multiplication operation, although restricted
to a rather small linearity range. Practical circuits based on this idea
focus mainly on increasing this range of linearity, and follow different
design strategies. Figure 32.33 shows an example, known as the Gilbert
cell, or Gilbert multiplier [23]. Corresponding realizations using MOS
transistors are discussed in [2] and [53]. Sanchez-Sinencio et al. [61]
present circuits to realize this multiplication function using OTA blocks.
On the other hand, [17] presents a tutorial discussion of different
linearization techniques for MOS differential amplifiers.

There you have it. For the BJT, i_z is linear according to i_y or v_x.
That is, you change v_x by 1 dB and the ouput changes by 1 dB provided
the coefficient for v_x (which is i_y) is not zero (vice versa for i_y
too). Note all the talk about linearity from the circuits perspective:
'...enhance linearity," and "although restricted to a rather small
linearity range," and "Practical circuits based on this idea focus mainly
on increasing this range of linearity," and "...a tutorial discussion of
different linearization techniques." So explicitly again: *non-linearity
is not required or even wanted for this particular multiplying function*.
The circuits perspective of linearity is consistant with the Signals and
Systems perspective. That some engineers don't know the definition is
notwithstanding.

The following from Lathi may help get you up to speed on the linearity
property:

SIGNALS, SYSTEMS AND COMMUNICATION
B. P. LATHI
Copyright © 1965 by John Wiley & Sons, Inc.
Library of Congress Catalog Card Number: 65-22428
SECOND CORRECTED PBINTINC, SEPTEMBER, 1967

pp2-4
+----------+
| |
f(t) O----+ A system +----O r(t)
| |
+----------+
Figure 1.1


1.1 PROPERTIES OF LINEAR SYSTEMS

For every system there is an input signal (or driving function) and an
output signal (or response function) (Fig. 1.1). A system processes the
input signal in a certain fashion to yield the output signal. The word
linear at once suggests that the response of a linear system should change
linearly with the driving function (note that this is not the same as saying
that the response should be linearly proportional to the driving function,
although this is a special case of linearity); that is, if r(t) is the response
to f(t) then kr(t) is the response to kf(t). Symbolically, if

f(t) - r(t)
then kf(t) - kr(t) (1.1)

The linear system, however, implies more than Eq. 1.1. We define a
linear system as a system for which it is true that if r1(t) is a response to
f1(t) and r2(t) is a response to f2(t) then r1(t) + r2(t) is a response to f1(t)
+ f2(t), irrespective of the choice of f1(t) and f2(t). Symbolically, if

f1(t) - r1(t)
f2(t) - r2(t)
then f1(t) + f2(t) - r1(t) + r2(t) (1.2)

Equation 1.2 actually expresses the principle of superposition symbolically.
Thus linear systems are characterized by the property of superposition.
We may consider Eq. 1.2 as the defining equation of a linear system;
that is, a system is linear if and only if it satisfies Eq. 1.2, irrespective
of the choice of f1(t) and f2(t).
Sometimes the condition is stated in the form,

a·f1(t) + b·f2(t) - a·r1(t) + b·r2(t) (1.3)

irrespective of the choice of f1(t), f2(t) and constants a and b. This will be
seen to be exactly equivalent to Eq. 1.2. Note that Eq. 1.2 is stronger
than and implies Eq. 1.1.


1.2 CLASSIFICATION OF LINEAR SYSTEMS

Linear systems may further be classified into lumped and distributed
systems. They may also be classified as time-invariant and time-variant
systems. We shall briefly discuss these classifications.


Lumped and Distributed Systems

...

Time-Invariant and Time-Variant Systems

As already mentioned, linear systems can also be classified into time-
invariant and time-variant systems. The systems whose parameters do
not change with time are called constant-parameter or time-invariant
systems. Most of the systems observed in practice belong to this category.

O---- L -----+
------+ |
f(t) | R(t)
-i(t)-+ |
O------------+
Figure 1.2

Linear time-invariant systems are characterized by linear equations
(algebraic, differential, or difference equations) with constant coefficients.
Circuits using passive elements are an example of time-invariant systems.
On the other hand, we have systems whose parameters change with time
and are therefore called variable parameter or time-variant (also time-
dependent) systems. Linear time-variant systems are characterized by
linear equations with time-dependent coefficients in general. An example
of a simple linear time-variant system is shown in Fig. 1.2. The driving
function f(t) is a voltage source applied at the input terminals of a
series R-L circuit where the resistor R(t) is a function of time. The
response is the current i(t). Note that the principle of superposition
must apply for a system to qualify as a linear system whether time-
variant or time-invariant. The reader may convince himself that
the system shown in Fig. 1.2 is a linear system. A linear modulator is
another example of linear time-variant system. In this ease the gain of
the modulator is proportional to the modulating signal.
The system characterized by Eq. 1.4


d^2r dr
---- + a·-- + b·r = f(t) (1.4)
dt^2 dt


is a linear time-invariant system, whereas the system characterized by
Eq. 1.5


d^2r dr
---- + -- + (2·t + 1)·r = f(t) (1.5)
dt^2 dt

is a linear time-variant system. Note that both these systems satisfy the
principle of superposition. This can be easily verified from Eqs. 1.4
and 1.5.
It is evident that for a time-invariant system if a driving function f(t)
yields a response function r(t), then the same driving function delayed by
time T will yield the same response function as before, but delayed by
time T. Symbolically, if

f(t) — r(t)
then f(t-T) - r(t-T) (1.6)


This property is obvious, in view of the time invariance of the system
parameters. Time-variant systems, however, do not in general satisfy
Eq. 1.6.

Some of Lathi's key statements with regard to this thread a
1. "... note that this is not the same as saying that the response should
be linearly proportional to the driving function, although this is a
special case of linearity"
2. "Linear systems may further be classified into lumped and distributed
systems. They may also be classified as time-invariant and time-variant
systems."
3. "...linear systems can also be classified into time-invariant and time-
variant systems"
4. "Linear time-variant systems are characterized by linear equations with
time-dependent coefficients in general."
5. "A linear modulator is another example of linear time-variant system.
In this ease the gain of the modulator is proportional to the modulating
signal."

Note that these remarks are from pp2-4 of text for a BS level course in the
EE curriculum. You can't get more basic than that.

With regard to #4 & #5 specifically, the following is a linear system with
time dependent coefficients *and* is also a "linear modulator":

The System
+---------------+
| |
in | /¯¯¯\ | out
x(t) O--------( X )---------O y(t)
| \___/ |
| | |
| | |
| O |
| cos(wc·t) |
+---------------+

I used this example early on to emphasize the points of #4 & #5. I made
it as simple as I could think of for the purpose of making it simple for
you -- all to no avail. After all this work, I really hope you finally
"got it," because if you have not, I am done helping.

~~~~~~~~~~~~~~~~~~~~~~~~

"No. Its Yes. Non-linear action generates the multiplication products. I
was not drawing any real distinction between class c and b in this
context. The practical difference is minimal. They both do not amplifier
the waveform in a linear manner. I was not meaning to infer that it was
an "only" c. I was referring to the fact that you need at least some
method that generates non linarity." -- Kevin Aylward

"You can not achieve multiplication without a non-linear circuit."
-- Kevin Aylward

But the worm does turn:

It should go without comment that when one analyses the simple
transistor multiplier that one only selects the first order linear term,
and that this is term that generates the multiplication.

I see you are finally coming around, or maybe. One can never be too sure.

This is trivially obvious, and was
what I showed in my original analysis, gm is
inherently a small signal property.

Yes, and it is all you need for "multiplication." Your original claim
that "non-linearity" was "required" for multiplication was false, as
I wrote some 10,000 or so words ago.

Back and forth,... you are all over the map.

Indeed, as this is only valid for
small signals, more complex multipliers log the input signal so that in
conjunction with the exponential relation results in perfect
multiplication at all signal levels, originally due to Gilbert I might
add.

Doesn't matter. Non-linearity is not required for the modulation to occur
and that is what started you out on your march. You said it was, you
were wrong.

This confusion here appears to me to be one of semantics or x-wires, as
is often the case on strongly held, but oppositely apposed views. gwhite
claims that you don't inherently require a non-linear device to achieve
multiplication, ...

You are squirming. That "nothing is perfectly linear" is irrelevent.
Linear circuits are popular and much work goes into making non-ideal
(in the sense of being non-linear) devices appear as linear as possible.
Finally, the non-linear aspect (although ever present in some amount) is
not requisite for modulation to occur. If anything is trivally obvious
it should be that no device is a perfect ideal in *whatever* manner ideal
has been defined.

I claim that all practical devices have a non-linear
transfer function, ...

That is not now, nor has it ever been under contention.

...and it is this transfer function that results in
multiplication.

Well it can... sure. That is not now, nor has it ever been under
contention. But it is not *only* that aspect that can produce the
multplication. The actual configuration can use the first order terms
*alone* to acheive the multiplication.

To the extent the original circuit was non-linear, it was not that
non-linear aspect that was needed or even desired to produce the
modulation. In fact, a lot of work goes into linearizing the
transfer characteristic rather than the opposite. That the LO or
carrier port of the circuit can be driven Class C or D is
notwithstanding: that non-linear aspect is not inherently needed
for the modulation to occur. It exists more as an annoying non-
ideality that simply must be acknowledged and then mitigated in
many modulation applications.

The reason for Class C or D is for efficiency, as I wrote long ago.
The amp is not Class C or D "so that the modulation will occur."
In fact, the point is to find out how Class C or D can be used and
still achieve the desired effect. For tubes, that will likely be
plate modulation. The desire is for efficiency, and then to find
a way to make it work effectively. The point was not to "use Class
C or D so that modulation will occur."

I don't believe we are arguing about the same point.

Dude, it is pretty straightforward stuff. If you don't know what
is under discussion you might want to think before you write how
stupid someone else is.

I have better things to do.

Gee, thanks for making it so easy. So what if there are some
things you are shaky on? Get over it -- that is life. No one
knows everything.


~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~
If gm(Vi) is represented by a Taylor expansion, any required terms
linear in Vi will integrate to Vi^2, that is

I = aV^2 + terms...

That is, the I verses V relation must be non-linear to achieve a gm that
is a function of voltage or current.

I don't know what your point was with all the gm stuff, but if you hook
it up like this:

The System
+---------------+
| |
in | /¯¯¯\ | out
x(t) O------+-( X )---------O y(t)
| | \___/ |
| | | |
| +----+ |
| |
+---------------+


....then the system is indeed a non-linear one [y(t) = x(t)^2].
This squaring function is a perfect example of why I wrote a long
time ago: "'Multipliers' cannot be generally stated to be either
linear or non-linear. A system which includes a multiplier must
be put through the linearity test to see if the configuration is
linear or non-linear. IOW, it can be either."


It takes a lot of work for designers to produce linear characteristics
in their linear multipliers. They like to draw straight and long lines
on their z = x·y graphs. That one non-linear operation is cleverly used
to cancel another (and get those nice long straight lines) is a design
detail. The bottom line is the end linearity, the linearity they use as
a selling point.


From AN-531
The analysis of operation of the MC1496 is based on the
ability of the device to deliver an output which is proportional
to the product of the input voltages V_x and V_y. This holds true
when the magnitudes of V_x and V_y are maintained within the
limits of linear operation of the three differential amplifiers in
the device. Expressed mathematically, the output voltage
(actually output current, which is converted to an output
voltage by an external load resistance), V_o is given by

V_o = K·V_x·V_y (1)


.... The MC1595 multiplier contains the basic circuit configuration
of the MC1496 plus additional circuitry which results in linear
multiplier operation over a large input voltage range.
....
Figure 3 shows the MC1496 in a balanced modulator
circuit operating with +12 and –8 volt supplies. Excellent
gain and carrier suppression can be obtained with this circuit
by operating the upper (carrier) differential amplifiers at a
saturated level and the lower differential amplifier in a linear
mode...
Operating with a high level carrier input has the
advantages of maximizing device gain and insuring that any
amplitude variations present on the carrier do not appear on
the output sidebands. It has the disadvantage of increasing
some of the spurious signals...
[Saturation has nothing to do with the multiplication function.
In fact the extra non-linearity has the caveat of spuroius products.]
....
Operation with the carrier differential amplifiers in a
linear mode theoretically should produce only the desired
sidebands with no spurious outputs. Such linear operation is
achieved by reducing the carrier input level to 15 mV rms or
less.
This mode of operation does reduce spurious output
levels significantly...
[Again, non-linear operation is not required to produce the
modulation. In fact, it reduces spurious products that would
otherwise need filtering, if possible.]


And it goes on and on and on and on. Linear means linear.
Get over it.