Transmission Line As Oscillators: A Chaotic Question
I was reading this paper on Transmission lines with non-linear loads just when this crazy question came to my mind.
https://ieeexplore.ieee.org/document/4244619 which analyses non-linear loads termination for Transmission lines
While reading this paper, the first thing that struck my mind was to have a negative resistance at the load : to be specific, a non linear active device as load whose input impedance can be made negative real (negative resistance) for a particular frequency/ a band of frequency. I considered a lossless T-line to make calculations easier.
So, yes!!! the reflected voltage V- is now greater(opposite in phase) than the incident V+ at the load.Why? Because the load reflection coefficient < -1. So if the other end of the T-line is not matched, the voltage at the load grows indefinitely ? Obviously no, the non-linearity of the device moves the pole location and reflection coefficient becomes dependent on the incident V+ and hence saturates the load voltage. Interestingly, if you work out the mathematics, the frequency of the oscillation will be dependent on the length of the T-line (2x the Transit Time). (lower the length of the T-Line higher the frequency provided the negative resistance exists at that frequency or Barkhausen criteria is satisfied)
So here is a crazy question : if I were to load a open (source reflection coefficient = +1) T- line with a negative resistance (for example one of the commonly used 3-point Transistor oscillator or maybe a tunnel diode) will it oscillate ?As stupid as it sounds, we would expect as the oscillations will grow until the Vload saturates if source is open. But something interesting happens, contrary to what I expect.
If I were to load a open T- line with a negative resistance (for example one of the commonly used 3-point oscillator or a tunnel diode for instance) will it oscillate ?
There are Transmission line based oscillators but the above question doesnt make sense. While I do find some arguments against the assumption, they are not exhaustive and they need to be shaped by suggestions. Interestingly, the opposite happens, as the Rs decreases, we have a undamped oscillation. Thats not all, if we keep reducing the source Resistance Rs. If you did not expect this, you're not alone.
I expected the oscillation to occur if input is open and output of T-Line is loaded by negative resistance. Why? Because the reflection is maximum and same phase at source. Somehow this does not occur. I would like to get more inputs on this and I will edit this in the near-future as I find more resources and more inputs from experts in this. Please feel free to comment if you contradict any of these thoughts or if you find valuable resource that would help me with this.
It is found that as the source resistance decreases with the reflected voltage, the poles move from right half of s plane and hence oscillation occur. As the source resistance further decreases to negative,we have a chaos region which is briefly explained in the following paragraphs - the number of poles and zeros explode to chaos?At the load we have something similar to an AM (amplitude modulation)spectrum with carrier frequency as 1/2T. The Barkhausen criteria of oscillation is satisfied for many frequencies if we consider the system linear for small change in V+ .Now how does this happen -something I am not comfortable intuitively, as we have more frequencies at which the negative resistance exists and remember, these may be thought as sub- harmonics of 1/2T as Transmission line length is fixed for our analysis.
Some thoughts on the assumptions and their effects:
1) The oscillation frequencies of the non-linear devices are not in sub-THz region (consider an equivalent of transit frequency in MOSFETs ) they are in lower Microwave frequency range. So the length of the T-Line to resonate at that frequency is not small. So what ?
Well T-Lines are not lossless, exponential decay is significant as we increase the length of the T-Line and hence gain of the active device may not suffice for undamped oscillation (poles at sigma =0 line).
2) My assumption that the output should have only 2 states of oscillation seems to be wrong and this depends on the nature of the load. This has been proved with Tunnel diode as load to a T-Line. As the source resistance decreases to negative value the oscillation occurs and we have more than 2 levels - this is wonted characteristic for non-linear dynamic system. So, the system can be thought of as similar to a mixer output with high direct LO leakage(1/2T frequency) and number of the frequencies at the input changes in a chaotic manner as Rs decreased. The non-linearity of the device might lead to a chaos state depending on the V+ as reflection coefficient decreases
There are some nice resources on this, for example: loading a transmission line with Tunnel diode by biasing it in the negative resistance region. The results as stated earlier are a bit scary, at-least for me, with the load voltage exhibiting not 2 states of oscillation (single tone and its harmonics ) but going into complete chaos as the reflection coefficient increases more negatively.But most importantly ratio of these "states" follow a constant - Feigenbaum constant. I would like to update more information on the relation between the Constant which is around 4.669 and the system in the near-future. As reflection coefficient becomes more negative, more and more tones (which are not harmonics to fundamental 1/2T tone ) appear.
Also, if the reflection coefficient is further increased in magnitude, there is a blank space where the number of tones reduce again (number of levels decreases from chaos) and then again goes into chaos. - Minimum order of the system to display chaotic behaviour is 3
So this non-linearity of the device seems to discourage the very idea. Seriously who wants to inject power into the neighbouring bins (multiple of 1/2T away from the 1/2T tone). It is possible that the T-line can radiate as well forming a monopole antenna if not shielded properly.
3) This seems to be the most important argument that I can find out of everything else, although there is some fault in the argument, per se. How can I tap the oscillations out without loading the T-Line ? Unfortunately I could not find any resources regarding this nor could I find any help. One flawed solution could be to add a fanout at the load which now changes the load impedance and hence is a bad idea as oscillations may die to satisfy the boundary conditions at the load.
As already stated, the above points are just thoughts that needs to be shaped by constructive criticism and most importantly, these points are not exhaustive. Please feel free to comment if any inaccuracy found. I would like to add more points to this idea with more resources available. Please do comment your opinions and ideas on the same.
REFERENCE LINKS
2) Well, this paper is interesting, it does not use only the Transmission line for the resonance but it uses a synchronous blend between both.
https://ieeexplore.ieee.org/document/7806048
I like to think about complex issues and details, analyze intuitively and discuss topics. Please feel free to comment your thoughts. This is an article written on May 17, 2020.
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