Christopher Marki
Marki Microwave - Director of Research
RF/Microwave component design, marketing and product development.
Interests
Web Links
Marki MicrowaveEEWeb Stats
- Karma: 9
- Followers: 8
- Following: 0
Christopher's Blog :
Return to BlogThe Taylor Expansion for RF Mixers: Pretty, and Pretty Useless
Posted Jul 13, 2011 at 11:00 pm
“I have never let my schooling interfere with my education.” -Mark Twain
In my blogging past, I have always written to a very RF savvy audience. Now that we are in a more generalized electrical engineering forum, I believe it would be most appropriate to talk about something that Marki Microwave is most well known for: RF/Microwave mixers.
For those unfamiliar with Marki Microwave, we offer the world’s largest selection of “hybrid” mixers (more on hybrid technology some other time). With a career dating back four decades, my father started Marki Microwave in 1991 with the intention of offering the broadest selection of high-performance mixers in the industry. In time, I will educate my readers about the nuances and history of mixer technology, and how Marki has fit into this grand framework. Before we can get into the details, however, I need to insult some professors.
Figure 1 Figure explaining up and down conversion. Taken from the Marki Mixer Tutorial.
Most electrical engineers do not really know what a mixer is. In short, a mixer is a 3-port electrical device whose output signal frequency is equal to the sum (or the difference) of the 2 input signals’ frequencies (see Fig. 1). In other words,
Figure 2
We include the absolute value here because we do not have negative frequencies.
In my experience, few electrical engineers actually know the above equation. More typically, people think of the mixer as a “multiplier” or two signals. This is understandable; the mixer symbol literally looks like a “times” symbol (see Fig. 1), and technically, the mixer does behave with a “multiplication-like” effect. The multiplicative notion of the mixer is further reinforced in RF courses. Often, when the mixer is introduced, the professors describe mixer operation based on the concept of the Taylor series expansion of two or more signals on a nonlinear device. It looks something like this for a single diode mixer:
Figure 3
After some smoke and mirrors (i.e., algebra and trigonometry), the college professor usually arrives at the conclusion that the second order expansion includes two important terms:
Figure 4
The critical function of frequency translation of signals, therefore, is distilled into a few crude mathematical brush strokes. Digging deeper, we can show that this approach to explaining and understanding mixers is terribly flawed and leads to many erroneous and misleading conclusions.
First, the Taylor expansion necessarily assumes that one of the signals is a “large signal” and one of the signals is a “small signal.” Basically, the math only works well if the small signal only slightly perturbs the I-V characteristics of the device (usually a diode). In practice, a mixer will work quite well, even when the large signal and small signal are within 3 dB! Certainly, the requisite small signal assumption is not a necessity to create excellent mixers in the real world.
Next, due to the tedium of the algebra, the professor will often truncate the expansion to the 3rd order and represent the higher order terms with the catchall symbol for laziness: “…” I am not suggesting that a reasonable derivation would include higher order terms (that is, after all, usually saved for some horrifyingly malicious homework problem), I am simply saying that by stopping at the 3rd order, the student is mislead into believing that 4th order and higher terms are not important. In practice, we find that higher order terms can wreak havoc on a system if not properly controlled. The Taylor expansion understanding of mixers trivializes this concept.
Lastly, and most sinfully, the Taylor expansion derivation concludes that the sum and difference frequencies are proportional to the large signal voltage. This would imply that by turning up the large signal (i.e., the LO voltage), one could increase the output level of the mixed signal. Let me put this in simple terms: Passive mixers do not provide gain, EVER! The notion that I can simply turn up the voltage swing on my large signal, and subsequently increase the power level of my converted signal, is a horrible misrepresentation of the actual physics. Yes, active mixers do exist, but the signal gain is attributed to some kind of DC bias on the nonlinear devices. The concept of taking energy from a strong AC signal, and imparting some of that energy onto a weak AC signal is extremely important and interesting, but the classic RF mixer does not do this (Incidentally, there is a class of amplifiers called Parametric Amplifiers, that do perform this kind of mixing/amplifying. The physics is extremely interesting. I even once dabbled in such research while at UCSD). As a mixer guy, I wish mixers behaved like parametric processers, but they do not. Alas, passive mixers always have loss—such is Mother Nature.
To conclude, it is important to recognize the Taylor expansion approach for what it is: a convenient mathematical technique to introduce students to the idea that “nonlinear” devices spread energy in the frequency domain. Most undergraduate electrical engineering curriculums barely touch upon the concepts of the nonlinear interaction of signals in semiconductor devices, so it is not surprising that the mixer is so often poorly explained. It is my hope that I can reprogram some of these concepts and offer a more physically intuitive understanding of what makes a good RF mixer. For more information about RF/microwave mixers, you can read our tutorial .
For what it’s worth, there is one professor who does in fact teach mixers in a way that actually makes physical sense; he is Professor Lee at Stanford. I highly recommend his text, it gives an excellent introduction to RF mixers, even if Marki-style mixers are totally ignored.
Comments on this post:
Featured Products
TechXchange General Community
Technical questions? Find your answers here! Digi-Key experts available 24-7!
Maxim Control and Automation
Broad offering of high-performance analog, robust communications and innovative application-specific ICs
Honeywell MICRO SWITCH V15W
Designed for harsh-duty wash down areas, the V15W series of basic switch offers rugged, precise on/off control
7 months ago: You make it sound incorrect to think of a mixer as a multiplier, but that is misleading at best.
cos(A)cos(B) = cos(A-B)cos(A+B)/2
You described it as the right side of this equation, which is correct. But if the right side is correct then so is the left side, which is simply the multiplication of two frequencies.
Not only is this correct, but it's a useful way to think about a mixer, particularly if you wanted to implement one. You go about looking for a way to do a multiply in analog. One such way is to take the log of a signal using a diode, add this to the log of another signal, then exponentiate (reverse the log) using a diode again. It also gives you a obvious implementation method if you wanted to perform a mixing operation digitally.
I want to make it clear that there is nothing wrong with describing a mixer as the right side of the equation, only that you shouldn't make it sound like the left side (product) is incorrect. The right side gives you insight into what the mixer does in frequency space.
As for the Taylor series rant, I agree it makes little sense, but then again I've never heard anyone attempt to describe or model a mixer that way. In college when we studied AM (also known as "product" modulation), we started with the equation above.
7 months ago: Olin,
I am glad you find the expansion useful, I think most EEs would be empowered to understand the concept that nonlinear devices spead energy in the frequency domain. As I said in my blog, for that, the expansion can be useful.
Nevertheless, the Taylor series has serious shortcoming when describing many mixer phenomena including:
1. why one cannot continue to improve conversion loss by increasing LO drive
2. why a square wave LO can actually improve nonlinear performance
3. why higher order mixing terms (i.e. beyond 3rd order) actually can make or break a system
The trick is instead to think of the mixer as a switch (aka. a "commutator"), not a multiplier. Deep mixer understanding can be achieved when understanding the commutating mixer concepts. At that point, the Taylor series concept only plays a secondary role.
I hope in time we can both agree that the Taylor expansion has its uses, but that the mixer switching concept is far more powerful. Let's save that material for a future article.
Cheers,
Christopher
7 months ago: You apparently didn't really read what I wrote. As I said, I agree the Taylor expansion makes little sense in this case. One of my points was that you seemed to be ranting against the Taylor expansion, but I hadn't heard anyone try to describe a mixer that way. It seemed kindof like ranting triangular tires. They're not a good idea, but it's also not problem that comes up a lot.
My other point was that you seem to be resisting describing a mixer as a multiplier, although it is. Unlike the Taylor expansion, this way of thinking about a mixer is both useful and illuminating.
Yes, a commutator is a poor man's mixer. To understand why that works and what it's drawbacks are, it again helps to think of a real mixer as a multiplier. Then it's easy to see that a commutator is a multiplier by a square wave instead of the real sine wave. From the Fourier expansion of a square wave we can see that it contains the sine wave, but also other harmonics. A commutator is therefore a "real" mixer with some harmonics mixed in. Now you can go back to the original equation and see what this noise does to the result in frequency space, which gives clues as to how to deal with it. With good frequency planning and filtering, a commutator can be a useful way to perform mixing, but please don't try to tell us that's what a mixer *is*.
7 months ago: For those interested in reading more about commutating mixers and their advantages, please read our T3 Primer.
http://www.markimicrowave.com/menus/appnotes/t3_primer.pdf
7 months ago: "A commutator is therefore a "real" mixer with some harmonics mixed in. Now you can go back to the original equation and see what this noise does to the result in frequency space, which gives clues as to how to deal with it."
This implies that the higher frequency components of a square wave are only noise to be filtered out. In fact they can improve, sometimes dramatically, the mixer performance. This is an unfortunate and sometimes damaging misconception.
7 months ago: That sounds interesting. Can you show some examples where these extra harmonics are useful and not just noise?
I've mostly run into mixers in RF applications where the harmonics causes aliases of the intended signal at different frequencies, or conversely, allow unintended frequencies to cause signals in the output band. The advantage of commutator-type mixers is then their simplicity and robustness (no actual arbitrary multiply needs to be performed), with the extra frequencies a artifact that must be dealt with.
I'd like to hear more about applications where these extra frequencies are actually put to good use.
7 months ago: In terms of applications, the single most important trend in mixer technology today is delivering extremely linear mixers. Therefore, any mixer that gives superior third order intercept and low levels of single tone intermodulation are the "holy grail" by today's standards. We find that commutating mixers driven by square-wave LO consistently yield the best performance.
7 months ago: It must be a frequency difference thing. At higher frequencies the square wave reduces intermodulation distortion, generally at the expense of conversion loss. The higher order terms are easier to filter out when they are outside of the band of interest anyways.
I'm interested to know more about the method for implementing the mixing function in software, however. I assumed that the standard way was to just straight multiply the frequencies together, but I imagine there are a wealth of techniques for digitally mixing a signal up and down.