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Derivation of an Equation for the Cut-Off Diameter of a Frustum-Shaped Cavity at a Specific Resonant Frequency

I am scheduled to speak at the SmallSat Conference 2017 in Logan, UT this August. It is just one of the Swifties, where you get 3 minutes to talk about a subject, but they are letting me include a paper on the subject into the proceedings, which I have never done before. It is essentially a summary of the state of the investigation of the EMDrive as I know it, and what I believe are testable predictions that would differentiate between possible explanations for the phenomenon. I cannot put the paper here, but I am going to link to a paper I wrote that is referenced by it.

One of the important features of a waveguide that some of the proposed explanations rely upon is the cut-off frequency of the waveguide, or the cut-off diameter of the frequency. The problem is that, because the cavities being used are in a shape that has not been considered useful before this, the cut-off diameter doesn't have a reliable equation in order to predict it. This is my attempt at deriving one, and I hope that it proves valuable. I don't have the tools to test it in a simulation, but some people on the EMDrive thread at the NASA Spaceflight Forum gave it a quick go and it looks promising.

AIAA Paper

The long-awaited paper from Eagleworks on their EMDrive experiments has come out and the short version is: they measured a thrust of about 2mN / kW. This is a huge step forward, and I'm struggling to understand it all. I have only had a chance to skim it, but here are my thoughts right now. 

The reference to the Pilot-Wave understanding of quantum dynamics is fascinating this is the second time in a few weeks something about this has been published; the first time was in Wired here. (Apparently this was published in 2014, but I only read it a week ago) This idea says that a photon isn't a single thing that is both a particle and a wave, but a literal pairing of a particle and a wave. I'm looking forward to better understanding how this ties into the EMDrive, but I don't really understand the explanation yet. I asked Dr. March the following two questions: 1. Is it fair to summarize the findings as "the EMDrive is not a reactionless thruster, but more like a propeller through the zero-point field, and 2. Does this mean that a second EMDrive would react to another's wake? His response to the first was a gracious and complicated cantata scored around the theme of "No", but I'll have to read a lot more to understand it (including a dictionary, in some cases). He did recommend Dr. White's papers on the Quantum Vaccuum, so I will be reading that, as well. His answer to the second was "Yes", so I find that a very tempting experiment to try, if I ever get any farther. 

I also contacted Dr. McCulloch at Plymouth University. The question was around Pilot-Wave theory in general. He has said that his modified Casimir affect theory of inertia (MiHsC) would require light to travel faster than the speed of light when in a wave guide. He very graciously corrected my understanding that it did, pointing out that what travels faster than c was the phase velocity, not the group velocity, of the wave. Having gone back and tried to understand the difference, I see he's right, but my question today was: if the particle and the wave are separate, wouldn't the individual particles start riding the phase velocity but arrive sporadically, while the average flux of particles would be consistent with them traveling the group velocity? It seems to me that the particles, being the parts with mass, would be the relevant part to his theories, so maybe it's an answer. I look forward to hearing what he thinks. 

I genuinely don't know what to think. The possibility that there are multiple theories, and possible experiments to find out which it could be, is very exciting. What do you think?


Tuesday, the hard work of the last four months finally produced some results. As you may recall, the last chamber I made was not able to resonate at the expected frequency. I originally thought it was, but then I discovered that it was actually a resonance in my measuring equipment and not in the chamber itself. I discovered this right before I presented my project at the International Space Development Conference in San Juan (it wasn't meant to be a scientific presentation, but it would have been pretty embarrassing to present something that turned out to be incorrect). Comparing the measurements of the cavity that the Tajmar team used in their model, I found that my chamber was 5 mm shorter. I've had trouble understanding why they made their model to those measurements. I didn't have the parts on hand to make a second chamber, and didn't have the money to buy new materials. Luckily, I did have the parts to try and make the variable-sized chamber that would be the next generation model. I've pressed on to try and finish that, and have made some progress on it, but most importantly I got much better at polishing. With a few new tools and learning new techniques that were better than just hand-polishing with sandpaper, I decided a couple of weeks ago to try polishing the original chamber with the new techniques and see if the problem with the resonance was actually due to incorrect measurements. 

And amazingly changing the polishing fixed the resonance problem! I got a resonance at about 2.40 GHz and at about 2.46 GHz. I didn't test it with the computer recording software plugged in, and when I attempted to show Mike Beach, the electronics expert that has been helping me at Artisan's Asylum, after the class he was teaching ended an hour and a half later it didn't work any more. I believe that this is because the copper oxidized enough in that hour and a half to reduce the reflectivity enough that it no longer resonated, but now I know how to fix that. (Really, Dr. Paul March gave me the hint about it months ago, but I thought I could get around it so I didn't buy the correct materials. Eight hours soaking in lemon juice, he told me, but I thought polishing right before the test was enough. I now keep all parts in a bath of 5% ascorbic acid whenever it's not in use).

The Q I measured yesterday was about 50, comparable to what the Tajmar experiments had, and I'm sure I can do better. Now that something is moving the needle, I am laying out a series of tests that I will be conducting over the next few days or weeks.

I am going to run tests, recording a control sweep and then recording frequency sweeps every 15 minutes to see the degradation of the resonance over time after at least a day of soaking in the ascorbic acid. The next step will be to see whether using galinstin as a liquid "gasket" between the copper parts of the chamber increases the Q factor; it may or it may not. The next step after that will be to incorporate the silicone spray Dr. March recommended to me to see how well it protects against oxidation. I didn't want to incorporate this spray until I knew the chamber worked in case a misapplication of the spray caused a problem that prevented the chamber from working.

Testing the chamber under full expected power will take a lot of preparation. I have partially constructed a faraday cage in which to test it that is big enough for some levers to amplify the measured force. It's a bit complicated, and getting power into the cage without it leaking the powerful microwaves out has been tricky.

I also have been starting to construct the experiment controller, with the foundation for control circuitry for the satellite inside it. That part is designed roughly in my head, and I've only just started to get it out into code.

I will keep you posted. 


Yesterday, I was able to reproduce every resonance mode with just the antennae, not in a cavity. Today, I was not able to, but I think it's best to treat the modes I found as suspect until I get another set of equipment to verify the behavior of the cavity better.

International Space Development Conference

I recently presented the Build an EM Drive project to the International Space Development Conference in Puerto Rico. Puerto Rico was fun; the conference was sort of a bust for us. I got on the schedule late and, while they were great, it was so chaotic that I accidentally wasn't on the official schedule. A few people came. The presentation is included below. It's part of a change in the way we are promoting the project, as well. I will have more about that soon.


I made a plaster cast of the cavity to measure the actual inner diameter of the cavity at the small plate. I had trouble getting the plaster out. I tried chipping it out and heating it for 15 minutes at 450o in a toaster and quenching it to shock it out. It did not come out until I heated it with the oxyacetylene torch. I took care not to anneal it by accident, but the flange is more bendable now. The finish is ruined.

The measured OD of the mandrel is 73.20 mm at the small plate. The measured OD of the plaster cast is 73.90 mm. This implies I lost approx. 0.70 mm between forming imperfections and polishing. This is an interesting benchmark for future fabrication.

I am interested that the diameter at the small plate is lower than dc @ the lower resonant frequency, 2351.999 MHz. This should have cut off that frequency. Feynman II suggests that, after the cutoff diameter, the signal does not die immediately, but dies out exponentially quickly so that it is a source of losses and is essentially gone within a distance of a (the radius). The chamber may hhave been past cut-off for such little distance that it wasn't visibly affected, but I would have expected it to affect the peak more than it seems to. This may speak to my theory that the cutoff frequency equation may be affected by the slope of the walls of the chamber.

I was thinking last night about the use of Raspberry Pis in a Hadoop cluster and I found that it is "difficult" to translate FFT to a MapReduce paradigm. However, I read more about DFT and found that it is possible to see whether a specific frequency is present in a signal with a technique called "correlation." This technique translates very well to MapReduce. Considering tracking the resonant frequency and feature recognition to determine if the resonant frequency is near the fc of the small plate radius (especially as the small plate expands in the heat) is essentially a search of the frequency domain, and a full characterization of the frequency domain probably isn't necessary, this could be a useful insight. Large transistor, slow speed processors running in a distributed MapReduce fashion may overcome processing limitations we could face.

2016-05-07 (2)

I have run the experiment with better resolution on the Spectrum Analyzer. The settings are as follows:

Signal Generator    
Freq Step 0500.000 MHz
Start Freq. 2300.000 MHz
Stop Freq. 2500.000 MHz
Step Delay 00.100 s
Attenuator Off  
Power +3 dBm
Spectrum Analyzer    
Freq Span 002.000 MHz
Start Freq. 2403.000 MHz
Stop Freq. 2405.000 MHz
Module 2.3 - 2.5 G  
Top dBm -065 dBm
Bottom dBm -107 dBm
Iterations 016  
Offset dB +000  
Units dBm  

The results are stored in RFExplorer_SweepData_2016_05_07_17_04_04.rfe. The samples at the edge of the peak around 2404 GHz were:

  • 2403.956 MHz@-99dBm
  • 2404.036 MHz@-100.41 dBm

This implies a Q factor of 26,710. I was skeptical of such a high rating on my first attempt, so I verified my approach with a physicist/engineer Richard Driver. He couldn't comment on my approach or experiment, but he believed I was calculating Q correctly.

I am also seeing a resonant mode at 2351.999 that I wasn't expecting,, and one at 2507.967 MHz, and maybe another at 2534.063 MHz. The analyzer doesn't got beyond that apparently. At least, I can't make it do so.

The wavelength in air of 2403.991 MHz (the average of the peak) is 124.7925 mm (according to www.wavelengthcalculator.com), so the half-wavelength is 62.3962 mm. The measured height of the cavity is 62.70 mm - 1.36 mm = 61.34 mm (the plunge depth from teh calipers from the top of the base plate, minus the thickness of the base plate). I'm off by about 1 mm from matching the half-wavelength, and I can't tell if the that's close. Neither of the other nearly peaks (59.8094 mm and 63.7755) are any closer.

The cut-off radius of the target frequency is:

fc(mn) = X'mn/(2 * pi * a *sqrt(mu * epsilon)) c = 3 * 108
ac = X'mnc/(2 * pi * f) X'mn = 1.8412
ac = 0.03656870 m f = 2403.991 GHz
dc = 73.14 mm  

At the lower frequency:

ac = X'mn * c /(2 * pi * f) c = 3 * 108
ac = 0.03737707 m X'mn = 1.8412
dc = 74.75 mm f = 2351.999


The settings from 5/5 were:

Signal Generator  
Freq. Step 0500.000
Start Freq. 2300.000
Stop Freq. 2500.000
Step Delay 00.100
Attenuator Off
Pwr +3dBm
Spectrum Analyzer  
Freq. Span 060.000
Start Freq. 2367.000
Stop Freq. 2427.000
Module 2.3 - 2.5G


I have constructed a loop antenna from a SMA female-to-female adapter and a lenght of thin wire. I secured teh bottom plate to the cavity with 4 wing bolts and wing nuts. I inserted a straight antenna through a 3/8" hold about midway up the chamber, and inserted the loop antenna below the midpoint through a 1/4" hole. I prefer the 1/4" hold. Neither antenna was strongly secured, and the female-to-female adapter was connected physically (resting) to teh chamber wall. The straight antenna was connected to the RFExplorer Signal Generator. The loop antenna was cnnected to the RFExplorer Spectrum Analyzer and arranged approximately horizontal, parallel with the bottom plate. The settings of the devices are recorded later. The data recorded by the RFExplorer software is in file RFExplorer_SweepData_2016-05-05-22-28-52.rfe. The data show a steady peak at 2403.964 MHz that corresponds to turning the Signal Generator on and off. Using the contiguous data points 2403.429 MHz @ -91 dBm, 2403.964@-74dBm, and 2404.500@-90dBm to calculate Q, I get Q=2,244.598. I find this suspicious for several reasons:

  1. this is using contiguous data points, so it is more likely a commentary on the resolution of the equipment, meaning it is AT MINIMUM 2,244, and
  2. this seems suspiciously high considering I'm basically a mook and this is my first attempt.

This was done with:

  1. insufficient polishing,
  2. no silicone coating,
  3. no special geometry such as spherical endplates.

There remains in this experiment:

  1. tie the resonant frequency back to the actual dimensions of the cavity and
  2. resolve the resonant frequency with the ac of that frequency, and compare that to teh actual a of the cavity.


A number of problems have been bothering me that may be solved. I couldn't describe why the height of the chamber had to be dictated by the wavelength of the wave, or pinpoint where I read that. I didn't fully understand what a mode was. And, I didn't know whether the lambdag equation would work in a frustum as it did in a waveguide. This came from failing to understand that E is not oriented as in Feynmann II 23-11(a), but 23-11(b). It was very helpful to understand what was happening, but in a waveguide E goes across teh guide to allow the EM wave to propogate down it. (Feynmann II 24-3(a), Balanis Fig. 1, attached figure 1). So, figure 2 is wrong, figure 3 is more correct. This orients the propagation between the endplates, prevents the sloping sides from being near-tangential from E (meaning we are not at a non-zero portion of the Bessel function, preserving the lambeg equation) and changes the way I understand the current flow. It also explains why teh resonance modes in Tajmar were so clear; the sloped sides dampened resonance when the propagation was oriented sideways. Only TE resonance modes were appearing. This changes the way I will orient the antennas, at least. It also changes how I understand the connectivity of the loop antennas. I was so fixated on Feynmann II 23-15 that I missed Feynmann II 23-8, which shows a loop antenna and the obvious implication that the coaxial sheath shoudl be connected to the interior wall.

Journal Diagram 1


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