We've recently been talking about mixers, and it has become apparent to me that there is a big gap about what we mean by "understanding" mixers. Is it enough to embrace at the trig formulas? Or is it possible to understand these key devices at an intuitive level? You know, as a boy, when James Clerk Maxwell was trying to understand a device, he used to ask, "What's the go of it?" If he was dissatisfied with the answer he would ask, "But what's the particular go of it?" I struggled for a long time to understand mixers (the struggle continues!). Here is an excerpt on my efforts to understand mixers from my 2009 book "SolderSmoke -- Global Adventures in Wireless Electronics."
UNDERSTANDING:
THE MIXER
That
dual gate 40673 was relatively easy to understand, but over the years I
frequently became aware of the fact that I didn’t really understand what was
happening in that mixer circuit. At times I thought that I understood it, but
then I’d dig a bit deeper and find that my understanding was incorrect, or at
least incomplete. Mixers are absolutely
key stages in almost all amateur transmitters and receivers, so I knew that as
a radical fundamentalist I’d eventually have to really understand how these
circuits work.
There
are many paths to confusion in this area.
You can be misled by graphical explanations and by “hand waving” verbal
descriptions. And I think that purely
mathematical explanations fail to provide the kind of intuitive understanding
we are looking for. Let me describe some
of the pitfalls.
When
they get to mixers, some books show three nice graphs of sine waves. They are stacked one over the other. The top two are input signals, each of a
different frequency. The third graph is
the arithmetic sum of the top two.
Moment by moment the signal strengths presented by the top two are added
together, and the result is shown on the bottom.
Soon
it becomes clear that the shape of the envelope of the resulting graph is
varying at a different frequency. A third frequency. What is happening is that because the two
input signals are of different
frequencies, they are periodically going in and out of phase: one moment both signals are at a positive
peak, and they reinforce each other.
Later, one is at a positive and the other is at a negative peak; here
they cancel each other. It is so simple
and easy to see! You can even count the
number of cycles that this new signal is going through. And—amazingly—the
frequency of this new signal is the arithmetic
difference of the first two. Voila! This is il
terzo suono of Giuseppe Tartini! Suddenly it
seems that you understand mixers and superhets.
You might think that you now understand how Major Armstrong hoped to
convert the electrical engine noise from the German bombers down to a frequency
at which he could amplify them.
But you’d be wrong.
Sorry about that. It’s just not
that simple.
It
took me a long time to realize that this explanation of mixer action was a
kind of children’s fable for mixer theory, a
misleading fairy tale with sufficient connection to the related field of
acoustics to take on an aura of legitimacy.
The
first indication that something was amiss came when I looked for the sum
frequency. I knew that mixers produce
new frequencies at BOTH the difference of the inputs AND at the sum of the
inputs. The neat little three-graph
presentation seemed to explain the difference frequencies, but what about the
sum output? How do we explain that
output using these charts? It took me a
while to realize that you can’t. Because
this is not really the explanation of how mixers work.
In
the course of writing this book (in 2007), when I got this point I was reminded
that my struggle to understand mixers has been a long battle. I was reminded of this when I turned to
Google in search for insights. Along
with the many learned and highly technical articles that popped up in the
search results, I found my own pleas for help going back some ten years. Here is a typical exchange on this subject
posted to sci.electonics.basic USENET group in 1997:
September 5,
1997 Bill Meara (wme...@erols.com) wrote:
> Here's a nagging little question that has been bothering me for some
> time:
> I have several Physics and Radio
books that give very clear
> explanations of how "beat" frequencies are generated in mixer
> circuits. These books have nice
little charts showing how the two
> waves combine to produce a third frequency that is the difference
> between the two. Great! Very
illuminating.
> But these same books are oddly silent
on how the "sum" frequency is
> developed. Can this frequency be
explained in a similarly graphic
> manner? Any hints?
An excellent
question.
It relies on nonlinear circuit elements
and high school trigonometry
(trig) identities. Ideal mixers have square-law or Vout = Vin^2
characteristics. This means that if you have two signal of different
frequencies, vin = s1 + s2 where s1 = cos (2*pi*f1*t) and s2 = cos
(2*pi*f2*t), you have vin equal to the sum of two cos, which by trig
identity gives vout equal to terms of cos(f1+f1)/2 and cos(f1-f2)/2.
The
response in this exchange is typical of what you get when you ask these kinds
of “how do mixers really work” questions.
Most experts will immediately come back at you with two things:
non-linear elements (like a diode) and trigonometry.
The equations seem to be saying that the sum and difference frequencies
that we see coming out of mixer circuits are
caused by the multiplication of the two input signals.
What? How does that work? At this point many of the books seemed to
chicken out on providing non-mathematical explanations. But for me, the math seemed to cry out for
some explanation. The equation seemed to
be saying that some very simple devices—one diode, for example—are somehow able to take two input
signals, multiply them together, and spit out new frequencies that are the
arithmetic sum and differences of the two inputs. I found myself thinking, “Diodes are good,
but are they that good? Who taught them the multiplication
tables?” And if we are seeking sum and
difference frequencies, why do we eschew addition and subtraction? Why do we use multiplication?
The
simple explanation using the three charts and Giuseppe Tartini’s Terzo Suono
explanation kept putting me on the wrong path.
I kept coming across examples, mostly from acoustics, that showed two
frequencies coming together this way to produce a third frequency. There was, of course, a common experiment in
the high school physics lab in which two tuning forks of slightly different
frequency are brought together. You can
hear the “beat,” the difference frequency that results. Where is the “non-linear” element in this
case? Displaying what I thought was a
somewhat unquestioning acceptance of what they’d learned in engineering school,
some folks told me that for this kind of beating to take place there had to be a non-linear element. Some suggested that the mixing took place in
a non-linear portion of the human ear.
Others hinted that the air itself might have non-linear qualities.
This
didn’t sound right to me. So I built a
little circuit that would electrically combine two audio signals. And I would watch the results on my
oscilloscope. There’d be no air (or
ears) involved. Sure enough, on the
scope I could see the beats as the two frequencies came closer together. There were no non-linear diodes doing
multiplication. I thought I was getting
closer to understanding.
But I wasn’t.
The
problem with this kind of mixing or combining is that the resulting beat is not
“extractable.” When I first started
seeing that word—extractable—in incoming e-mail messages, I didn’t really
understand what it meant. Tom Holden, VE3MEO, made it clear: “The beat note that you hear between the two
tuning forks is not a new signal—it’s just the period between the constructive
and destructive interference due to the superposition or addition of the two
signals… You can’t separate the beat frequency signals from the source signals
because subtracting one of the source signals from the waveform leaves you with
merely the other. You can’t hear the beat without hearing both forks singing.”
Real
mixing is obviously different from this kind of terzo suono beating. In a
real mixer you want to be
able to separate the new frequency from the old ones. You want to be able to extract it so that you
can better filter it and amplify it. And
you want to leave the input signals behind.
OK, back to the drawing boards.
Back
in 1999, I think I kind of came close to a limited understanding of the
phenomenon. Here is another USENET
exchange:
To: ianpur...@integritynet.com.au
Ian: I
really like your pages.
I have a question about the theory behind
the mixer stage: As was done
on your page, explanations of this stage are usually limited to stating
that the active device is operated in the non-linear portion of the
curve and this results in its operation as a mixer. Given that this
is
the heart of superheterodyne operation, I've always wished that the
explanations would go a bit deeper.
We recently had a very lengthy and
interesting discussion on this in the
sci.electronics.basics newsgroup. I came to some conclusions about
mixer operation that (I hope) may provide the kind
of explanation that I
think is needed for beginners and non-engineers to understand mixers:
--When we say that the active device is
operated in the non-linear
portion of its operating curve, we are really saying that we are biasing
it so that each of the two input signals will—in effect—vary the
amount of amplification that the other receives from the device.
-- When this happens, the output of the
device is a waveform that
contains sum and difference frequencies. If we ask WHY this happens, we
have to be satisfied with an answer that points to mathematics: If you
combine two signals in the manner described above, mathematical
principles dictate that the resulting waveform contains sum and
difference frequencies.
Please let me know what you think of this
explanation—does it make
sense, is it consistent with accepted theory?
Again, thanks for
the great web site. I will be visiting often.
73 Bill
N2CQR
Bill Meara
<wme...@erols.com> wrote: receives from the device.
Excellent.
This is a very good definition of non-linearity. Sometimes
"amplification" isn't involved, as when we use a diode mixer, but in that
case each signal varies the amount of _attenuation_ that the other
receives from the device. I like it, and
I think that the explanation is about as useful as any.
For a mathematical
analysis, you might want to consider that a
non-linear mixer actually _multiplies_ the two signals, rather
than adding
them. I think I've got this right, anyway...
M Kinsler
So,
back in 1999 I seem to have sort of accepted that if you take two signals of
different frequency and feed them into a non-linear device, the math tells us
that in the output you will get sum and difference frequencies. I also seem to have been coming close to
understanding the need for non-linearity: in order for the signals to really
“mix” one signal has to affect how the other signal passes through the
device. My thinking was that if you have
one signal in effect varying the bias on a transistor as that signal goes
through its cycle, another signal going through that device will see the device
as being extremely non-linear. It will
get mixed with the first signal. (In retrospect, my understanding of the role of
non-linearity was still quite flaky.)
Still,
I was not satisfied with my understanding of the mixers. I thought it was a bit of a cop-out to just
say, “Well, the math tells us that if you multiply two sine waves, the output
will contain sum and difference products.”
Math-oriented scientists and engineers often pour scorn on what they
call “arm waving” non-mathematical descriptions. But I think there is some room for scorn in the
opposite direction: I don’t think that memorizing a trig formula means that you
really understand how a mixer works. In “Empire of the Air,” Tom Lewis writes: “At Columbia,
[Edwin Howard] Armstrong developed
another trait that displeased some of the staff and would annoy others later in
life: his distrust of mathematical explanations to account for phenomena of the
physical world. All too often he found his professors taking refuge in such
abstractions when faced with a difficult and seemingly intractable conundrum…
Time and again as an undergraduate at Columbia,
Armstrong had refused to
seek in mathematics a refuge from physical realities.”
I
guess I still yearned for the clarity and intuitive understanding that had been
(falsely) promised by those three nice beat frequency charts. Time and time again, as I dug into old
textbooks and ARRL Handbooks and
promising web sites served up by Google, I was disappointed.
Then I found it.
It
was in the Summer 1999 issue of SPRAT, the quarterly journal of the G-QRP Club. Leon
Williams, VK2DOB, of Australia
had written an article entitled “CMOS Mixer Experiments.” In it he wrote, “Generally, mixer theory is
explained with the use of complicated maths, but with switching type mixers it
can be very intuitive to study them with simple waveform diagrams.”
Eureka! Finally I had
found someone else who was dissatisfied with trigonometry, someone else who
yearned for the clarity of diagrams.
Leon’s article had waveform diagrams that showed, clearly, BOTH sum and
difference output frequencies.
Switching
mixers apply the same principles used in other kinds of mixers. As the name
implies, they switch the mixing device on and off. This is non-linearity in the extreme.
Not
all mixers operate this way. In
non-switching mixers the device is not switched on and off, instead one of the
signals varies the amount of gain or attenuation that the other signal will
face. And (as we will see) it does this in a non-linear way. But the basic principles are the same in both
switching and non-switching mixers, and as Leon points out, the switching
circuits provide an opportunity for an intuitive understanding of how mixers
work.
Let’s
take a look at Leon’s
circuit. On the left we have a signal
coming in from the antenna. It goes
through a transformer and is then applied to two gate devices. Pins 5 and 13 of these gates determine
whether the signals at pins 4 and 1 will be passed on to pins 3 and 2
respectively. Whenever there is a positive signal on gate 5 or on gate 13,
signals on those gaps can pass through the device. If there is no positive signal on these
gates, no signals pass. Don’t worry
about pins 6-12.
RF A
is the signal going to pin 4, RF B is the “flip side” of the same signal going
to pin 1. VFO A is a square wave
Variable Frequency Oscillator signal at Pin 5. It is going from zero to some
positive voltage. VFO B is the flip
side. It too goes from zero to some
positive voltage.
Look
at the schematic. Imagine pins 5 and 13
descending to bridge the gaps whenever they are given a positive voltage. That square wave signal from the VFO is going
to chop up that signal coming in from the antenna. It is the result of this chopping that gives
us the sum and difference frequencies.
Take a ruler, place it vertically across the waveforms, and follow the
progress of the VFO and RF signals as they mix in the gates. You will see that whenever pin 5 is positive,
the RF signal that is on pin 4 at that moment will be passed to the
output. The same process takes place on
the lower gate. The results show up on
the bottom “AUDIO OUTPUT” curve.
Now,
count up the number of cycles in the RF, and the number of cycles in the
VFO. Take a look at the output. You will
find that that long lazy curve traces the overall rise and fall of the output
signal. You will notice that its
frequency equals RF frequency minus VFO frequency. Count up the number of peaks in the choppy
wave form contained within that lazy curve.
You will find that that equals RF frequency plus VFO frequency.
Thanks Leon!
Back
to the math for a second. Why do they say that those diodes
multiply? And what do trigonometric
sines have to do with all this?
First
the sines. Most of the signals we are
dealing with are the result of some sort of circular or oscillating
motion—coils that are being spun around magnets, resonant circuits that behave
like a playground swing. For this
reason, the trigonometry of circles can be used to determine the amplitude of a
signal at any given instant. Take the
peak value of a sine wave signal, and multiply it by sin[2Ï€(freq)(time)] and
you will get the instantaneous value of that signal.
When
we say that mixers multiply, it is important to realize that we are NOT saying
they multiply frequencies. We are saying
they multiply the instantaneous amplitudes of the input signals. And it is that multiplication that results in
the generation of the sum and difference frequencies.
Why
multiplication? Again, by looking at
switching mixers, it is easier to understand.
Consider one of the gates in Leon’s mixer. The RF input
is a sine wave. Its instantaneous value
varies according to Peak*sin(2Ï€ft). The
VFO signal is either positive or 0. If
it is positive, the RF signal passes through the gate. We can say that if it is on, it will have a
value of 1. If it is 0, no signal passes
through the gate. Mathematically we can
say that the output is a multiplication product of the two inputs. If RF is at 1.2, and VFO is positive (1), the
output from that gate will be 1.2x1=1.2.
If RF is at 1.2 and VFO is at 0, the output will be 1.2x0=0
Note
that this is very different from the simple summation in the “children’s fable”
presented at the beginning. If addition
were at work here, we’d expect the outputs to look like 1.2+1=2.2 or 1.2+0=1.2 But that is clearly NOT what we’d get with a
switching mixer. Clearly
multiplication is the operation that best models this circuit.
Now
this doesn’t mean that in every mixer circuit one
input with an instantaneous input of 2 volts and another with an instantaneous
input of 3 volts will result in an instantaneous output of 6 volts. After all,
some mixers are made up of transistors that are capable of amplification, but
others use simple diodes, and these diodes can’t amplify. Different mixing circuits use different kinds
of devices, different input levels, and different biasing voltages. So the outputs will vary quite a bit. But the shape of the output waveform will
resemble the waveform that results when you multiply the instantaneous values
of two input waveforms. We can say that
in addition to the multiplication that is the heart of the process, there are
also mathematical constants and offsets that result from the particular
characteristics of individual circuits.
You
can use a simple spreadsheet program to get a feel for this. Set up two columns each with the formula
Peak*sin(2Ï€ft). Assign different values
of frequency to each column. Set up
another column for time—make it 1-100 and think of each division as a block of
time. Graph the results. Then run a third column that multiplies the
first two. And put this third column on
the same graph. You’ll see the mixer action.
Leon’s switching mixer circuit helped
me get a bit more of the kind of intuitive understanding that I’m always
looking for. Later on, through a more
careful reading of Experimental Methods
in RF Design’s mixer chapter, I
think I started to understand how non-switching mixers work, and why
non-linearity is an essential element of a mixer circuit.
Jean
Baptiste Joseph Fourier (1768-1830)
discovered that any complex periodic waveform can be shown to be the result of
the combination of a set of sine waves of different frequencies. Here’s a great illustration of this
principle. It is from ON7YD’s web
site. The darkest line is the
complex signal that results from the sine
waves that are shown around it. A
picture is worth a thousand words.
The
key idea here is that if you see a complex periodic (repeating) waveform, you
should realize that “beneath” that waveform, there are a number of nice clean
sine waves. And here is where
non-linearity as an essential element in mixing comes in.
Let’s
consider two devices, both with dual inputs.
One is set up to be very linear.
The other is set up to be non-linear.
Let’s put two signals of different frequencies into each input. The first input is 1 volt peak at 1 MHz, the
second input is .1 volt peak at 10 MHz.
In
the linear circuit, we can think of the stronger 1 volt signal as moving the
operating point of the device up and down, up and down along the very straight
line that describes the relationship between input and output in this circuit. As it does so, the weaker 10 MHz signal just
sort of rides along.
If
we look at the output we can clearly see the two signals, one riding along with
the other. The output waveform is not
complicated, and it seems clear that there are only two signals that you could
get out of that via filtering: the two input signals. This is just like the acoustic situation that
caused me so much confusion. The key
thing to remember here is that the two signals are not really mixing.
Now
let’s look at the non-linear circuit.
Now the operating curve really is curved. The weaker 10 MHz signal will
once again, in a sense, be riding along on the stronger 1 MHz signal, but that
1 MHz signal is no longer moving up and down on that nice straight line. Now it is on that curve. Now the two signals really “mix”, mixing
almost to the same extent that liquids of two different colors mix in a
blender. You can see how the curved
operating characteristic—the non-linearity— causes the two signals to mix.
Out
of the non-linear circuit a very complex periodic waveform emerges. It is a complicated mess, but Fourier tells us that
any complex periodic waveform can be seen as being composed of sine waves of
many different frequencies. If we were
to dissect this output waveform of this device, we’d find the two original
signals, harmonics of these signals, and, most importantly, new signals at the
sum and difference frequencies of the input frequencies. And this complex signal CAN be dissected. To do this, we make use of “balanced” devices
to cancel out the input signals, and filters to shave away the harmonics and perhaps
either the sum or the difference output.
We can set things up so that only one frequency emerges from the
mix. That is extremely useful.
I
think (hope!) I’ve made progress in my effort to understand mixing; I think
I’ve moved far beyond both hand waving acoustics-based fairy tales, and the
almost equally unsatisfactory approach that equates understanding with the
ability to regurgitate trig formulas. I
now understand the difference between mixing and adding. I know why multiplication (and not addition)
is the math operation that describes what happens in a mixer. Most
importantly I think, I now know why
you need a non-linear device to have true mixing. Fourier provides the
answer: That bend in the operating curve of a non-linear device causes the
output to be the kind of complex periodic waveform that contains many different
sine waves. And among those waves are
sum and difference frequencies.
Now
I must admit that how it is that
among those sine waves there are the exact sum and difference frequencies of
the inputs, well, for me that remains a bit of a mystery. But it kind of makes sense…
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