Kevin Aylward wrote:
gwhite wrote:
Frank Raffaeli wrote:

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



Non-linearity is *not* required to create DSB-AM out of
transconductance type multipliers like the gilbert cell.

I have nothing more to say on this. I have better things to do.

However, what the hell...:-)

We know for the diode:

gm = 40.Id.

That is, the gm or IV slope is a fuction of I. This allows another
transister to give an output:

Iout = gm.Vi

Iout = 40.Id.Vi

Which is a multiplication of Id with Vi, or a modulator.

Now, lets pretend that the diode equation is linear:

Id = Io.(1 + k.Vd)

gm is then

gm = d(Id)/d(Vd)

therefore

gm = Io.k

Thus the gm is a constant, independent of applied current or voltage.
This
means a transistor using this as a control parameter would give an
output:

Iout = gm.Vi = Io.k.Vi

Which has no multiplication factors.

To achive muliplication one can consider adding a nonlinear term

Id = Io.(1 + k.Vd + cVd^2)

gm = Io.(k + 2c.Vd)

and subsequently

Iout = Io.(k + 2c.Vd).Vi

Which does have a multiplication term.

This can be formalised. To achieve multiplication from a gm source, we
must have

Vo = gm(V1).V2.

That is, gm must be a function of V1. However, Gm is defined by Vi as

gm = dI/dVi, therefore

I = integral(gm(Vi))

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.

Kevin Aylward

http://www.anasoft.co.uk
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