Frank Raffaeli wrote:

gwhite wrote in message ...
[snipped long diatribes]

Dude, you are responding to one of the shorter messages.

Class A works just fine in multipliers/modulators -- "non-linearity" of
circuit elements is not required. Maybe you can analyze the old
MC1496. That would be enlightening to you. But more important and more
simple (it will save you loads of time), just apply *the* linearity test

[snip]
Hmmm .... you may be mistaking the (sometimes linear) current steering
effect for the mechanism within the transistor that makes current
steering possible: the relationship between gm and Ic ... or from
another POV, the change in rbb with respect to bias current. These
effects are non-linear.


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*. The rest of the terms are unwanted and that
includes the multipliers usage as a frequency translator, which I proved
to be a fundamentally linear operation. No one needs to take my word
for it. Anyone interested can idealize the gilbert cell to only the
linear terms and the device will still be able to produce DSB-AM -- IOW,
non-linearity is *not* required.

The *linear terms* on both the inputs of a class-A biased gilbert cell
are exactly what are anticipated in the higher level system drawing, the
non-linear terms are *not required* for DSB-AM to occur (to the extent
higher order products exist they are unwanted products). Driving the LO
port into the switch mode is notwithstanding. For example:

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

The linear terms are *exactly* what make the gilbert cells "linear
multipliers." Linear means linear.


Here is a system which uses the above linear system, and another "linear
circuit" to produce large carrier AM:


System 1 System 2
+---------+ +---------+
| | | |
in1 | /¯¯¯\ | | /¯¯¯\ | out
x(t) O----( X )------( + )----O y(t)
| \___/ | | \___/ |
| | | | | |
| | | | | |
| O | +--- | ---+
|cos(wc·t)| |
+---------+ |
|
in2 |
cos(wc·t) |
O-------------------+


That these two systems can be (and have been) made with "linear"
circuits is indisputable. The output is indistiguishable from large
carrier AM so we can just as well call it that.

Now it is interesting that as far as the x(t) - y(t) concern _alone_
goes, this "total" system is _not_ linear (apply the linearity test to
see how/why). But that has absolutely nothing to do with any circuit
issues like exp(x) and does has everything to do with the definition of
linearity as it is taught in the EE curriculum. No "non-linear" circuit
element was required to result in a non-linear system with regard to the
x(t) - y(t) transfer. The "non-linearity" is the sole consequence of
the fact that this "total system" is a Multiple-Input-Single-Output
(MISO) system. It is interesting that System 2 is the MISO system.
With regard to either of the inputs, and its respective output, the
response of System 2 is linear -- identical in manner to the simple
op-amp summer. But with regard to the total output of System 2, when
neither of inputs are zero, the system is non-linear _with respect to a
single input_ by the EE definition. That ought to make KA's blood boil:
a "linear" circuit reduced to non-linearity by definition.


In Summary:
Non-linearity in terms of circuit elements is not required to "make
AM." The concept of linearity in the EE curriculum is consistant across
courses. That includes Signals and Systems, Circuits, and Electronic
Design. That some are not up to speed on the linearity property is
notwithstanding.


Anyway, with all this talk of AM modulators I recall some papers I
read back in the early 1980's on vector modulation ... or the use of
the sum of two phase-modulated signals at high power, with a
compensating signal at low power such that the sum yielded a perfect
amplitude modulated wave ... very little power required from the
actual linear "AM" section. The technique is used to generate an AM
signal of many kilowatts, whilst using only a few hundred watts in the
required linear [c(t)] stage.

If the audio signal is expressed as a(t), and the phase-modulated
signal is limited to a p-p deviation of slightly less than pi/2, and
the compensation signal is c(t), then the crude diagram is as follows:
______
PM(-a(t)) -----| \ ______
| T1 0--------| \
PM (a(t)) -----|_____/ | T2 0----- "AM" out
c(t)--|_____/

T1, T2 = 90-degree hybrids

IIRC, many AM broadcast transmitters now employ this technique.
Hopefully, someone out there has a better recollection than I and can
provide a reference.


You might hinking of the Chirex Outphasor, or a similar idea (I don't
know how c(t) fits into the Chirex). Yes, it can be used to make an AM
signal. The thing is it *is* a non-linear method and you aren't calling
out the amplitude to phase mapping that is required for your +/- a(t)
signal.