You're absolutely correct. Production of new frequency components can be
done with either nonlinear or time-variant circuits. A square-law diode
detector is an example of the first; a multiplier is an example of the
second. I stand corrected -- thanks for pointing it out.

Roy Lewallen, W7EL

Tim Wescott wrote:
Disclaimer (disflamer?): Everything that Roy says is true enough to get you
down the road of radio circuit design -- but:

To be absolutely, mathematically correct, if you hold your mouth right, a
"perfect" mixer with it's driving oscillator, is a linear device. It is
_not_ a time-invariant device. It's linear because the IF signal that
results from putting in the sum of any two RF signals is exactly equal to
the sum of the IF signals that each result from each of the RF signals. If
it were nonlinear then this would not be the case (and it wouldn't be a
useful device for mixing).

What gives a mixer it's "mixerness" is that it is linear but time-varying
(output = input * some function of time). It is very easy to confuse
time-varying linear with non-linear, and even easier in practice because in
order to get the effect you need to use componant non-linearities to get the
job done, just as you do with a class A amplifier. But it's usually harder
to get the nonlinearities out of a mixer than an amplifier, so in real
design you have to pay attention to non-linear effects like blocking and
intermodulation in a mixer to a much greater extent than you do with an
amplifier, and this reinforces the idea that a mixer is fundamentally
nonlinear.

This means that when you're analyzing a mixer (and ignoring real-mixer
things like intermodulation) you can still use all the linear circuit theory
stuff as long as you stay away from anything that depends on
time-invariance. This means that _simple_ Laplace and Fourier analysis is
out, but you can still use _careful_ Fourier analysis to figure out what the
output will be for a given input and oscillator frequency. In fact, that's
exactly what you are doing when you analyze a mixer: all of the desired
behavior of a mixer can be exactly predicted with Fourier analysis.

"Roy Lewallen" wrote in message
...

Your friend is right.

If you simply add or subtract two waveforms, no new frequencies are
created. You end up with only the frequencies you started with and no
more. (Theoretically, you could make one or more disappear if one of the
added waveforms contained a precise negative of one or more frequency
components of the other -- but you can never get any new frequencies.)
That's because addition is a linear process, with linear having a
precise definition that's appeared here a number of times before.
(Subtraction is just addition, with one waveform inverted before
adding.) Multiplication, though, is a nonlinear process by the precise
definition used in circuit analysis, and it does create additional
frequencies. Multiplying the two original signals of 1500 and 1955
generates the two new frequencies of 455 and 3455, for a total of four
frequencies after multiplication. Adding them wouldn't do it.

Most good mixers are actually more like switches than multipliers, but
they're still nonlinear -- very much so -- and don't do anything
remotely like adding the two signals. A doubly balanced mixer produces
the sum and difference frequencies while not letting the original two
frequencies get through to the output.

The generation of the new frequencies by multiplication of the two
originals is easily shown mathematically, as your friend says, with a
short derivation by means of a trig identity. I'll be glad to post the
derivation if you or other readers are interested, although it's widely
available elsewhere.

Roy Lewallen, W7EL

Joer wrote:

I'm trying to settle a debate with a friend, and my knowledge of
mixers is pretty rusty.

Say you have a receiver whose IF is 455 kHz, and it's tuned to a
station at 1500 kHz. If all's working OK, at the output of the mixer
you should have four frequencies:

1500 (original signal)
1955 (oscillator signal - osc. working above the signal freq.)
3455 (sum)
455 (difference)

My question is by what process does the mixer produce the 3455 and 455
frequencies. I say it's an add and subtract process, my friend says
(via mathematics) it's a multiplication process. Who's right?

thanks,

Joe W9TXU