[Amps] Re: FW: How are IMds created

[Bill Fuqua]

         You are referring to models that are determined by empirical data. 
There you can't separate the effects of second order and third order 
distortion. The math ( I don't know if you looked at it or not) is simply 
an explanation how second and third order distortion  effects contribute to 
harmonics, frequency mixing, intermodulation distortion and cross 
modulation in a 2-port device. This was for educational purposes (my 
classes) and not for the purposes of design ( however it could be in some 
cases). Up until almost 1920 radio engineers did not understand frequency 
mixing. I do recall  Ramsey (not Norman Ramsey, but I always wondered if 
they were related) published a book on radio engineering in the 20's and 
explained that it was not a beating effect but the square law 
characteristic of tubes and diodes that produced frequency mixing.

The purpose of this is to explain how these effects are produced so that 
students will have a better understanding of  of linear, well, mostly 
linear devices.

You observation of second harmonics being produced first before IMD is 
easily predicted due to the fact that the third order term has a smaller 
coefficient than the second order in the Taylor series that describes most 
non-linear devices. That does not mean that the second harmonic generated 
is mixed with the fundamental to produce the IMD products. If you had a 
device with out a second order term you would have IMD without producing 
any second harmonic energy(see math). However, such a device would be hard 
to come by.

The math speaks for itself. Unless you can find an error. As I said earlier 
that there are effects that can not be completely modeled mathematically 
although can sometimes be approximated using empirical data. One is reverse 
recovery time in a diode.  A diode in conjunction with an inductor can be 
made to frequency divide,but it is a chaotic behavior and can't be modeled 
but can be somewhat predicted. The point at which this effect takes place 
changes depending on amplitude, frequency and the direction in frequency 
you may be sweeping. This example has been used often to demonstrate period 
doubling and bifurcation in chaotic system and an easy one to do in the 
physics lab. This takes a current sense resistor say 10 ohms, a 3 mh choke, 
a good slow 1N4004 diode and a signal generator that will go to a couple 
hundred KHz and provide up to 100 volts P-P voltage, An old HP400CD will 
work. Observe the current thru the resistor and voltage from the signal 
generator on two channels and sweep frequency slowly from low to 
high  and  see what you get. Not all 1N4004 diodes will exhibit this effect 
but it is interesting when you get it. Infact, this phenomenon presents 
problems in  some high power switching motor controllers but was not 
understood until about 20 years ago.

I thought this would be relevant due to the fact that we are talking about 
tube linear amplifiers and perhaps some would like to know how IMD and 
Cross Modulation products are produced.  Vacuum tubes are easily modeled in 
this way and the types of problems I mentioned above are not present.   An 
cross modulation that is easily seen when expanding the third order term of 
the Taylor series. You can see some of the amplitude of one of the two 
signals imposed onto the amplitude of the other.

You had many references albeit one author many times. I used 
one  ...ME!  It is important that we use the tools we pick up along the way 
in a life long education process to discover for our selves why things are 
the way they are. That is what science is about and I hope even amateur 
radio. I am sorry if you can't appreciate the math and the explanation of 
why these effects exist in nearly linear devices, but I just can't do 
anything about that.  I am sure that this analysis is nothing new. Surely 
many have done it in the past. And this analysis does not negate your 
observations or measurements either but should get you thinking of your 
conclusions and why the results turned out how they did. It is these 
smaller perturbations that make for interesting science. In fact 
perturbation theory has it own place in quantum mechanics.

I am waiting for response from some of those other individuals that asked 
for a copy. Especially those that appreciated the analysis. I mostly have 
received complaints (from two people) that did not.



73
Bill wa4lav



At 08:57 PM 1/6/2004 -0500, you wrote:
>
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>Bill,<?xml:namespace prefix = o ns = 
>"urn:schemas-microsoft-com:office:office" />
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>  you stated that intermodulation products are not related to the signal 
> harmonics.
>
>Here is a definition that summarizes what a bunch of references say on the 
>subject. I already gave you a list of references from Steven Maas. There 
>are plenty others:
>
>
>
>"Non-linear device characteristics produce both harmonics and 
>intermodulation distortion when overdriven by rf signals. Harmonics are 
>produced when the non-linearity operates on a single frequency. 
>Intermodulation distortions are produced when the non-linearity operates 
>on two or more frequencies. The harmonics are n times the fundamental 
>frequency. The intermodulation distortion components are frequencies at 
>the sum or difference frequencies of their original signals and their 
>HARMONICS! "
>
>
>
>So we set out to do another experiment on this issue to determine what IMD 
>dependencies exist from harmonics. The experimental data supports what 
>Harmonic-Balance predicts for a two tone test; that the harmonics are 
>generated by a non linear element at the lowest rf levels and the same 
>non-linearity generates the IMDs when the rf drive is further increased.
>
>Summarized:
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>
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>- 2*F1-F2 and 2*F1-F2 are the frequencies where the 3rd IMDs products are. 
>There is no misunderstanding or   fluke here on what this means.
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>- Stating that the IMD is independent of the harmonics is not supported by 
>either Harmonic-Balance model nor by the experimental data taken. Do the 
>experiments yourself.
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>- The Harmonic-Balance model is by far more accurate than any 
><?xml:namespace prefix = st1 ns = 
>"urn:schemas-microsoft-com:office:smarttags" />Taylor series expansion 
>method. Harmonic Balance has been the microwave standard for a couple 
>decades now. Use of the Taylor series is limited to continuous-and 
>analytic functions, and is valid only within a close distance to an I(V) 
>operating point. You can t deviate far from the chosen bias point. Neither 
>condition is satisfied in a mixer case.   I already mentioned the 
>discontinuity at V=0. Not only is the I(V) function non continuous, their 
>derivatives do not exist at V=0. With theTaylor series you are stuck to a 
>small part of the bias regime. Also severe charge crowding or space charge 
>effects leads to the negative differential resistance regime. This is 
>another eparture from I(V) continuity, and marks another discontinuity 
>point. I can think of at least two discontinuity points and they vary from 
>device to device.
>
>
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>- Careful calibrations in a Load-Pull microwave set up accounts for the 
>power levels of  the various spectral components and their amplitude 
>dependencies (how one changes when the other is changed).
>
>Harmonics come first, then IMDs, It is in this order , becasue IMD energy 
>is derived from the mixing of the harmonics (I agree it is a necessary but 
>not sufficient condition. If both cases existed it would support your 
>case, but we can t see it experimentally), but:
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>- Notching of (specific) harmonic energy shows an effect on the associated 
>IMD levels, but not the other way around.
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>We used a microwave FET and drove it with two rf tones in the VHF range. 
>We picked an overlapping spectral area with the second harmonic and the 
>3rd IMD products. We show that the second harmonic is generated first, 
>that the 3rd order IMD comes after, as the rf two tone drive level is 
>increased. At no time does any IMD term ever exceed the level of its 
>generating harmonic.
>
>
>
>The supporting experimental data is attached as an appendix.
>
>
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>For your info the experimental data taken was done in my lab, at the Army 
>Research Lab in Adelphi Md.
>
>The Load-Pull set up was used to assure the greatest dynamic range and the 
>broadest bandwidth to avoid any in band resonances. We trace our 
>measurement standards to NIST (Boulder). We also set government standards 
>for spurious free dynamic range measurements from HF to W band.
>
>
>
>If you wish to make your claim stick, that a Taylor series is an 
>appropriate model, as a minimum you need to support your model with 
>experimental data and publish in a refereed journal (if case you haven t).
>
>If your claim proves to be correct then the methodology for reducing IMDs 
>changes, including a lot of references and some microwave curriculum that 
>is taught.
>
>While many in the microwave community welcome new microwave knowledge and 
>understanding, there is no credible evidence to support a departure from 
>the norm on this issue yet.
>
>
>
>Regards,
>
>
>
>Chris Fazi, PhD. EE
>
>ARL Program Manager and adjunct to EE Dept Johns Hopkins University.