gwhite wrote:
Kevin Aylward wrote:

gwhite wrote:


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

Indeed it is, in the case of the bipolar or fet multiplier, I proved it
mathematically. Unfortunately it appears you cannot understand the
proof. See below for a more exact analysis.


Maybe this will help you:


{snip ramblings}

Nope.


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

"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

I retracted this in the pure technical sense. I restricted the claim to
electrical connected, real devices. However, this statement was made in
support of my fundamental claim that the Class A amp you presented
achieved multiplication specifically because of its non-linear transfer
function due to the diode equation. I stand by that claim, which is the
one that matters for this discussion. I have proved this, as an
extension to my prior diode presentation, in detail below.


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.

Nope. I was trying to give you your due on some points.


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.

You shouldn't have bothered, it was much wasted much effort. I
mathematically proved that the diode has to have a non-linear transfer
function in order to obtain a gm that varies with a control signal, and
thus achieve multiplication.

The fact that you are unable to explicitly, understand, discuss and
refute that proof on mathematical points says much about your ability
and credibility.


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.

Nope. I am correct. I prove it below. Excluding, my retraction of the
ideal, unavailable VRC, or the light bulb and light dependant resister,
where the controlled terminals are not connected to the controlling
terminals. This was a minor slip-up as I was implicitly referring to
transistor type circuits. I regret not being more precise, but nobody's
perfect.



...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.

Not in the transistor case it can't. I have proven otherwise. All you
are doing here is throwing away non-linear terms after the fact. This
dose not mean that non-linear terms are not required in order to achieve
the final linear result.

Cut the bull****, and show me an actual mathematical derivation
supporting your view, rather than quoting end results from books you
don't understand.


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.

I have mathematically proved otherwise. Refute my proof by a
mathematical argument or retract your claims.

In fact, a lot of work goes into linearizing the
transfer characteristic rather than the opposite.


Yeah... and I don't know this...get real. What part of "I have been a
successful analogue design engineer for over 20 years" do you fail to
understand.

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.


{snip more waffel}


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

Dude, it is pretty straightforward stuff.

Indeed it is. Stuff that you have missed by not doing the actual
details. You made a simple mistake. Live with it.



~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~
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,

I know you don't. Its quite sad really. You have no idea of electronic
circuits and are unable to do a derivation of the gain of an amplifier.
I honestly though you were sufficiently qualified to understand these
mathematical details, obvious I was wrong.

I am taking a guess here that you still wont understand my black and
white disproof of your claims, because you simple don't have the
mathematical competence to do so, nor the competence to understand that
you lack such competence.

but if you
hook it up like this:

Oh dear...

I showed that if gm = kVi., i.e. the required linear gm, then the only
way to achieve this was if its defining equation was non-linear. This is
because gm=dI/dv. Linear gm vith its own current controlling it,
inherently implies nonlineaity of the I/V graph. The fact that you
introduce some daft multiplier connected up as a squaring circuit shows
you just haven't got a clue.

{snip irrelevant}

Notes:
Vin1 is applied across an effective resistance re=1/gm. To get any gain
change gm,re must be a function of either the current or voltage i.e.
Vin2 or Ini2, in this case its the current though it. A resistance that
changes is value based on the current through it is necessarily a
non-linear resister. Its that simple.

{snip irelevent waffle}

Stop quoting text you don't understand. Show me real mathematical
derivations.



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)

Get over it.

Err. where is your derivation of this equation? I wager if you could,
you would.

Let me do it for you.

The small signal output current of a transister amplifier is given by:

Iout = V1*gm

Where gm is the transister gm, and is given by, gm=d(Ie)/d(Vbe)

Ie is the emitter current, Vbe = V1 is the voltage across the base
emitter diode, and V1 is the input signal.

Now, please let me know where you disagree with any exact point.

I am now going to assume that the transistors gm is controlled by
another current I.

That is, gm = k.I

so that, from the above expression for Iout:

Iout = V1.k.I

That is, a linear multiplication of I and V1. All well and good so far.

However, we noted that

gm = dI/dV, so that we must have

k.I = dI/dV

or

dI/I = k.dV

and upon solving this differential equation, we obtain

I = c.exp(V.k)

That is, to achieve linear multiplication by this transistor gm method,
it necessarily requires a non-linear relation between I and V for the
transistors gm characteristics. This forms a proof of my statement that
the class A modulator achieves such modulation by a non-linear process.

gwhite:

Please do not respond if you are going to past masses of waffle text
that you don't understand. Provide a *mathematical* *explicit* disproof
of my mathematics or present your retraction.

Kevin Aylward

http://www.anasoft.co.uk
SuperSpice, a very affordable Mixed-Mode
Windows Simulator with Schematic Capture,
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