Understanding Mixers -- A Free Book Excerpt -- Enlightenment from SPRAT

https://soldersmoke.com/book.htm

     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[2π(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(2πft).  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(2πft).  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…