Resonant Cavity Space-Propulsion: institutional experiments and theory

[Rodal]

This is a thread focused on objective analysis of institutional experiments and theories concerning resonant cavity space-propulsion, and discussing possible space applications.  The emphasis is on institutional experiments: those at Universities and R&D institutions (whether federal R&D as in NASA, or R&D by private companies) as opposed to Do-It-Yourself experiments.  The term "EM Drive" is associated with Roger Shawyer's UK-patented concept (http://emdrive.com/) and it has its own devoted NSF threads (notice, though that the name EMDrive is currently trademarked in the US and in Europe for industrial applications totally unrelated to Shawyer's patent).  The emphasis on this thread is on non-Shawyer experiments and theories. 

Objective skeptical inquiry is welcomed.   Disagreements should be expressed politely, focusing on the technical, engineering and scientific aspects.   As such, the use of experimental data, mathematics, physics, engineering, drawings, spreadsheets and computer simulations are strongly encouraged, while subjective wordy statements are discouraged.  This link

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enables typing of mathematical symbols, including differentiation and integration, Greek letters, etc.

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Conspiracy discussions are discouraged.  Commercial advertisement and promotion is prohibited.   X-Prize, venture capital and public research grant funding discussions are ok.

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[Rodal]

Welcome everybody !   

There are many possible ways to start this thread.  For example

One possible discussion is the effect on experimental results of conducting experiments in partial vacuum: thermal effects as well as a discussion of any theoretical effects.

Numerical modeling (Meep, COMSOL, SuperFish, ANSYS, etc.), accuracy and relevance vs. experiments is another possible topic of discussion.

A discussion of whether and how statistical methods to analyze these experiments can be made relevant is another possible topic.

What is the best design of an experiment to minimize thermal effects, Lorentz forces and other effects such that a conclusive, convincing, statement can be made regarding these institutional experiments?

What theories could allow a resonant-cavity to actually result in useful spaceflight propulsion?  How would conservation of energy be addressed?

Has Dr. White and his group published any more papers since http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20150006842.pdf or given any presentations since this one at NASA Ames?:



with particular attention to this question from a NASA Ames scientist:



Are NASA's proposed experiments at NASA Glenn going to take place?

Paul March's discussion of the cylindrical resonant cavity with RF chokes experiment at NASA using an interferometer, and the discussions we had with user StrongGR and his paper (http://arxiv.org/abs/1505.06917v1 ) analyzing it...





Countless topics are open for discussion and still unsettled!

[Rodal]

IMPORTANCE OF DIELECTRIC INSERT IN NASA'S REPORTED RESULTS

Can the dielectric insert in NASA's tests be responsible for higher results in NASA's experiments than in Tajmar's experiments at TU Dresden (http://bit.ly/1mylt2q and http://emdrive.wiki/Experimental_Results)?  Can the dielectric insert contribute to real thrust in what Dr. White at NASA has called "Q-thrusters" or can it contribute to experimental artifacts?  Prior discussions with Notsosureofit ( http://emdrive.wiki/@notsosureofit_Hypothesis ) indicated that his theory would show thrust even in a cavity with uniform cylindrical cross-section as long as the dielectric insert was inserted asymmetrically.  Prof. Woodward has been said to be of the opinion that if NASA's experiments show real thrust useful for spaceflight that it must be a result of the Mach effect due to the dielectric insert.

It has been questioned (by others at another NSF thread) whether NASA's reported test without a dielectric insert was in resonance.

The explicit reference to NASA's reported result without a dielectric insert is the following:

page 18 of
 Anomalous Thrust Production from an RF Test Device Measured on a Low-Thrust Torsion Pendulum
David A. Brady, Harold G. White, Paul March, James T. Lawrence, and Frank J. Davies
July 28-30, 2014, Cleveland, OH
AIAA 2014-4029 Propulsion and Energy Forum
(This material was declared a work of the U.S. Government and therefore is not subject to copyright protection in the United States:
http://www.libertariannews.org/wp-content/uploads/2014/07/AnomalousThrustProductionFromanRFTestDevice-BradyEtAl.pdf
)

We performed some very early evaluations without the dielectric resonator (TE012 mode at 2168 MHz, with power levels up to ~30 watts) and measured no significant net thrust.

This is a very significant statement, as NASA reports to have measured no thrust without a dielectric insert: this NASA reported result therefore runs contrary to the claims of R. Shawyer regarding his "EM Drive" (see tabulated claimed results without a dielectric insert in http://emdrive.wiki/Experimental_Results for his "Demonstrator" and "Boeing Flight Thruster") as well as the claimed results by Yang (http://emdrive.wiki/Experimental_Results).  (It has been recently reported that China's academician's terminated Yang's resonant cavity propulsion project in 2014. )   

Up to now, reported "anomalous thrust" data by NASA has been with resonant cavities containing a dielectric insert asymmetrically placed inside the cavity.

So, was the NASA test without a dielectric insert in resonance?

Paul March provided a valuable NASA report by Frank Davis (one of the co-authors of the 2014 NASA report on the "Anomalous Thrust Production" paper linked above) [attached below as a PDF file "Frustum modes overview"], using COMSOL Finite Element Analysis that calculates the natural frequency (for mode shape TE012) without a dielectric insert to be very close to the frequency at which the test (without a dielectric insert) was reported by NASA:

measured frequency at which test was performed:       2.168 GHz
calculated natural frequency (COMSOL FEA analysis):   2.179 GHz
difference: (2.179 - 2.168)/2.168 = 0.5%

Therefore the evidence supports that the NASA test without a dielectric was indeed in resonance: as the measured frequency was extremely close to the calculated natural frequency (only 0.5% difference which is easily explainable by minor differences in dimensions between the actual tested piece and the calculated dimensions).

Furthermore, Paul March stated that they had S11 and S21 measurements during these tests and that testing was performed when the cavity was shown to be in resonance as per NASA measurements.

[Rodal]

EVIDENCE THAT NASA'S TEST WITHOUT A DIELECTRIC INSERT WAS PERFORMED AT A RESONANT NATURAL FREQUENCY OF THE CAVITY


For thoroughness, and to further address (and hopefully "put to bed") the hypothesis advanced by others "that NASA's test without a dielectric insert was not in resonance", we calculate the natural frequency of mode shape TE012 for NASA's dimensions.

We use an exact solution of Maxwell's equations for standing-wave resonance of a truncated cone, I obtained using Wolfram Mathematica.  The solution uses spherical Bessel functions and associated Legendre functions (as per Wolfram Mathematica definitions) and it also uses an intrinsic system of embedded spherical coordinates for the frustum of a cone.  The solution is similar to Greg Egan's solution (http://gregegan.customer.netspace.net.au/SCIENCE/Cavity/Cavity.html) except in its generality: the solution of Maxwell's equations obtained using Wolfram Mathematica can calculate mode shapes for arbitrarily large quantum numbers m,n,p (while Egan's as presented was restricted to low order).  I have compared my solution (using Mathematica) to the examples shown by Egan, and the comparison is excellent.

NASA's frustum of a cone dimensions

(as given by Paul March in a post as Star-Drive in thread 2 of the NSF EM Drive thread, see: https://forum.nasaspaceflight.com/index.php?topic=36313.msg1326997#msg1326997  )

The copper frustum we built and now are using has the following internal copper surface dimensions.
Large OD : 11.00 " (0.2794m)
Small OD: 6.25" (0.1588 m) &
Length : 9.00 " (0.2286m) 

and as given by Frank Davies (NASA/JSC/EP5) (see Frank Davis document https://forum.nasaspaceflight.com/index.php?action=dlattach;topic=39214.0;attach=1091650 )

Bottom diam.: 11.01 inch (279.7 mm)
Top diam.: 6.25 inch (0.1588 mm)
Height: 9.00 inch (228.6 mm)
Material: 101 Copper Alloy

The only difference being 11.01 inches for the large end as given by Frank Davis vs. 11.00 inches given by Paul March.  I will use the dimensions given by Frank Davis, for proper comparison to his COMSOL FEA analysis:


I used the following input parameters, as used by Frank Davis:

bigDiameter = (11.01 inch)*(2.54 cm/inch)*(1 m/(100 cm));
smallDiameter = (6.25 inch)*(2.54 cm/inch)*(1 m/(100 cm));
axialLength = (9 inch)*(2.54 cm/inch)*(1 m/(100 cm));

tanHalfAngleCone = (bigDiameter - smallDiameter)/(2*axialLength);

halfAngleConeRadians = ArcTan[tanHalfAngleCone];

halfAngleConeDegrees = (180/Pi)*halfAngleConeRadians;

r2 = Mean[{axialLength /(1 - ( smallDiameter /bigDiameter)), bigDiameter/(2*Sin[halfAngleConeRadians])}];

r1 = Mean[{axialLength /(( bigDiameter/ smallDiameter) - 1), smallDiameter/(2*Sin[halfAngleConeRadians])}];

Notice that, since the exact solution assumes spherical ends, while NASA's truncated cone has flat ends, the spherical radii r1 and r2 are calculated as the mean value of the radii to a) the intersection of the ends with the lateral conical walls and b) the top of the dome.  From analysis of the problem and verification using numerical analysis (comparison with COMSOL FEA solutions for a large number of examples) I have found out that this mean value is an excellent approximation to the solution of Maxwell's equations for a truncated cone with flat ends.

These input parameters result in the following values (in SI units) for the spherical radii and the cone half angle:

r1 = 0.305316 m

r2 = 0.537845 m

halfAngleConeDegrees = 14.8125 degrees





COMPARISON OF SOLUTIONS WITH NASA's experiment

measured frequency at which NASA test was performed:                                              2.168 GHz
calculated natural frequency (exact solution, Dr. Rodal using Wolfram Mathematica):2.165 GHz
calculated natural frequency (COMSOL FEA analysis by Frank Davis at NASA):            2.179 GHz

difference between NASA's COMSOL FEA and measurement: (2.179 - 2.168)/2.168 = 0.5%

difference between exact solution and measurement: (2.165 - 2.168)/2.168 = -0.1%

Therefore the evidence supports that the NASA test without a dielectric was indeed in resonance: as the measured frequency was extremely close to the calculated natural frequency

[Rodal]

....

Rodal posted this back a few pages.  It may or may not relate to other frustums and would probably depend on material thickness ect.  It is the researchgate link.  http://forum.nasaspaceflight.com/index.php?topic=39004.msg1468340#msg1468340  I think it mentioned time to buckle if I remember correctly. 

Edit: I don't think much long term thrust could be had from this after thermal equilibrium is reached.  The force should decrease with time as the thermal expansion decelerates.  It might be helpful to know the time to thermal equilibrium.

Edit2: Also, any positive thrust signal observed due to thermal expansion should be an equal and opposite signal upon powering down.  Thermal contraction would give a negative thrust. 

1) The thermal buckling force is associated with the temperature profile evolution and it is also primarily associated with the structural stability of the structure in question.  The first thing to understand is that the flatness of a membrane, plate or a simple column is unstable, like a ball on top of a hill is in an unstable form of equilibirum.  A force pushing a ball on top of a hill, once it overcomes any small obstacles on the way, or friction, will result in the ball naturally seeking a more stable configuration towards the bottom of the hill.  Similarly, a force on a column will result in buckling of the column once the buckling force limit is exceeded.





2) The thermal buckling force rapidly increases with time until buckling is reached.  My article shows that, contrary to the intuition of some people that have stated that thermal effects are too slow to be responsible for artifacts in these experiments, it shows that the rise of this thermal buckling force can occur in the time frames associated with the force vs. time of these experiments. 

3) It does not necessarily follow that the structure will return to its previous unstable equilibrium configuration once the temperature decreases and the temperature returns to the original temperature.  For the structure to return to its previous point of unstable equilibrium a number of assumptions must take place.  For example, one must assume perfect elasticity of the structure.  Any deviation from perfect elasticity will prevent the structure to return to its previous unstable configuration: for example if the metal follows an elastic-plastic material stress-strain law or if there is friction involved.  In most of our experience in the real world we are accustomed to multiple examples of buckling and the familiar experience that flatness of plates and membranes are unstable configurations and that once buckling is experienced it is difficult for the structure to perfectly return exactly to its original unstable flat configuration.



4) The example of thermal buckling I gave on my paper was not meant as the only explanation for the thermal forces that can be encountered on these experiments.  On the contrary, if one goes back to read my post, one will find that I explicitly wrote that upon further examination it was found that  the main thermal effect on NASA's experiments by Dr. White's team at Eagleworks (NASA Johnson) was instead due to thermal expansion leading to shifting of the center of mass, which produced spurious force vs. time artifacts in NASA's experimental traces.  Also, it is trivial to show that these thermal effects will remain when testing in vacuum.  As in vacuum there is no thermal convection due to fluid flow in an atmosphere, these effects associated with thermoelasticity  (1) thermal buckling, 2) thermal expansion shifting of the center of mass, etc. etc.) may become more prominent as the thermal convection effects are eliminated.

[Rodal]

EVIDENCE THAT NASA'S TEST WITHOUT A DIELECTRIC INSERT WAS PERFORMED AT A RESONANT NATURAL FREQUENCY OF THE CAVITY


For thoroughness, to further "put to bed" the hypothesis advanced by others "that NASA's test without a dielectric insert was not in resonance", we calculate the natural frequency of mode shape TE012 for NASA's dimensions, under the naive assumption that it would have had spherical ends instead of flat ends, to show that spherical ends have a negligible effect on the natural frequency. 

We use the dimensions given by Frank Davies (NASA/JSC/EP5) (see Frank Davis document https://forum.nasaspaceflight.com/index.php?action=dlattach;topic=39214.0;attach=1091650 )

Bottom diam.: 11.01 inch (279.7 mm)
Top diam.: 6.25 inch (0.1588 mm)
Height: 9.00 inch (228.6 mm)
Material: 101 Copper Alloy

I used the following input parameters, as used by Frank Davis:

bigDiameter = (11.01 inch)*(2.54 cm/inch)*(1 m/(100 cm));
smallDiameter = (6.25 inch)*(2.54 cm/inch)*(1 m/(100 cm));
axialLength = (9 inch)*(2.54 cm/inch)*(1 m/(100 cm));

tanHalfAngleCone = (bigDiameter - smallDiameter)/(2*axialLength);

halfAngleConeRadians = ArcTan[tanHalfAngleCone];

halfAngleConeDegrees = (180/Pi)*halfAngleConeRadians;

r2 = bigDiameter/(2*Sin[halfAngleConeRadians]);

r1 = smallDiameter/(2*Sin[halfAngleConeRadians]);

Notice that, for this calculation we naively calculate the natural frequency for NASA's test without a dielectric as if it would have had spherical ends instead of flat ends (the case for flat ends was calculated in my previous post).

These input parameters result in the following values (in SI units) for the spherical radii and the cone half angle:

r1 = 0.310475 m

r2 = 0.546933 m

notice that these spherical radii are longer than the ones previously calculated for the case simulating flat ends.

halfAngleConeDegrees = 14.8125 degrees

which is the same cone angle previously calculated for the case simulating flat ends.





COMPARISON OF SOLUTIONS WITH NASA's experiment

measured frequency at which NASA test was performed:                                              2.168 GHz
calculated natural frequency (exact solution, assuming  spherical ends):                     2.129 GHz
calculated natural frequency (exact solution, assuming  flat ends):                               2.165 GHz
calculated natural frequency (COMSOL FEA analysis by Frank Davis at NASA):            2.179 GHz

difference between NASA's COMSOL FEA and measurement: (2.179 - 2.168)/2.168 = 0.5%

difference between exact solution (flat ends) and measurement: (2.165 - 2.168)/2.168 = -0.1%

difference between exact solution (spherical ends) and measurement: (2.129 - 2.168)/2.168 = -1.80%


Therefore we have shown that even when assuming that the resonant cavity would have had spherical instead of flat ends, the natural frequency would have differed from the tested frequency by only 1.80%

(NASA's test was performed with a resonant cavity that had flat ends, which has a calculated natural frequency in excellent agreement with NASA's tested frequency)


Here is an updated list of calculations that verify the experimental measurement by NASA:



COMPARISON OF SOLUTIONS WITH NASA's experiment (as of Dec 19, 2016)

measured frequency at which NASA test was performed:                                              2.168 GHz
calculated natural frequency (Rodal exact solution, assuming  spherical ends):                 2.129 GHz
calculated natural frequency (Rodal exact solution, assuming  flat ends):                       2.165 GHz
calculated natural frequency (analysis by X_Ray method A):                                         2.16512 GHz
calculated natural frequency (analysis by X_Ray method B):                                         2.1653438 GHz
calculated natural frequency (COMSOL FEA analysis by Frank Davis at NASA):               2.179 GHz
calculated natural frequency (FEKO BEM analysis by Monomorphic):                              2.17895Ghz
calculated natural frequency (Keysight EMPro FEA analysis by X_Ray):                          2.17983GHz

[RotoSequence]

Is it possible to evaluate these experimental device's sensitivities to deviations from the calculated optimum resonance frequency at this time?

[Rodal]

Is it possible to evaluate these experimental device's sensitivities to deviations from the calculated optimum resonance frequency at this time?

Great question.

Using the definition of quality of resonance

we can express it as

Δf /f = 1/Q

where
f= natural frequency
Δf= bandwidth of resonance
Q= quality factor of resonance (a measure inverse to damping)

Q= 1/(2ζ)

where ζ  is the damping ratio

and hence we can express the previous difference between calculation and experimental frequency in terms of an equivalent maximum Q for the calculated frequency to be in the frequency bandwidth of the experimentally measured frequency

difference between NASA's COMSOL FEA and measurement: (2.179 - 2.168)/2.168             = 0.5%     = 1/200

difference between  (Mathematica) exact solution and measurement: (2.165 - 2.168)/2.168= -0.1% = - 1/1000

So,

1) the COMSOL Finite Element Analysis carried out by NASA would only have found the bandwidth of resonance for a Q smaller than 200

2) the exact solution with Wolfram Mathematica would only have found the bandwidth of resonance for a Q smaller than 1000

We can also restate this as:

for a Q = 10,000 one needs a numerical solution to differ from the measured frequency by less than 0.01%

for a Q = 50,000 one needs a numerical solution to differ from the measured frequency by less than 0.002%

for a Q = 100,000 one needs a numerical solution to differ from the measured frequency by less than 0.001%

________

Conclusion: no numerical solution is close to the precision needed to find the bandwidth of resonance for the high Q that one is seeking for.  The higher the Q, the more precision is needed.

The precision is unattainable because one does not know the exact geometry of the resonant cavity to that precision.

All that one can do with the numerical solutions for a resonant cavity with a high Q (>10,000) is to tell where the resonance is, to a precision less than 1%, perhaps 0.1%.  Finding the actual bandwidth of resonance and the resonance peak has to be done empirically, experimentally by S21 and S11 measurements.

So, for NASA's experiment without a dielectric we can say that the exact solution says that there was indeed a natural frequency for mode TE012 within 0.1% of the measured frequency, but as to whether the measured frequency was at the resonant peak, one has to rely on NASA's team having actually found peak resonance with S21 and S11 measurements (because we don't know the dimensions of NASA's resonant cavity to the precision required to calculate the resonant peak with all the required digits of numerical precision).

________

If I have the time, I would like to make a plot of the resonant frequency calculated by the exact solution, for variable:

1) End diameter
2) Cone angle
3) Length

[dustinthewind]

...

Thanks for clarifying that you addressed separately the buckling and expansion.  I apologize as I may have not made that clear in my statement.  I was away a day so wasn't able to respond immediately.  I take it that buckling is a one time thing right?  So if it occurs once it shouldn't occur again once the deformation is permanent?  If so than yeah, I guess that would give a single impulse without a retraction event. 

You mentioned the artifact thrust from NASA is partly due to thermal expansion or displacement of mass center?  Did they also observe the retraction of that center of mass when they shut down the power?  I have to think back to those plots and I vaguely think that maybe there was a retraction event, but I will have to go back and look. 

I was thinking that if we knew the time to thermal equilibrium, would it be advisable to at least keep the frustum powered on for a time longer than that so that we are observing more than just those events?

Edit: well to compound the problem in most cases the frustum is in air so there is the convection problem included with long term tests.

Edit2: one thing that worries me is putting the frustum in an insulated box (closed system) might eliminate thermal equilibrium (continuous heating) unless maybe an internal heat sink could reduce that to some extent. 

[Rodal]

...

Thanks for clarifying that you addressed separately the buckling and expansion.  I apologize as I may have not made that clear in my statement.  I was away a day so wasn't able to respond immediately.  I take it that buckling is a one time thing right?  So if it occurs once it shouldn't occur again once the deformation is permanent?  If so than yeah, I guess that would give a single impulse without a retraction event. 

You mentioned the artifact thrust from NASA is partly due to thermal expansion or displacement of mass center?  Did they also observe the retraction of that center of mass when they shut down the power?  I have to think back to those plots and I vaguely think that maybe there was a retraction event, but I will have to go back and look. 

I was thinking that if we knew the time to thermal equilibrium, would it be advisable to at least keep the frustum powered on for a time longer than that so that we are observing more than just those events?

Edit: well to compound the problem in most cases the frustum is in air so there is the convection problem included with long term tests.

Edit2: one thing that worries me is putting the frustum in an insulated box (closed system) might eliminate thermal equilibrium (continuous heating) unless maybe an internal heat sink could reduce that to some extent. 

First I want to thank you for your post, which gives me the opportunity to clarify a number of issues.

1) It is my present understanding that all NASA Johnson Eagleworks experiments with a resonant cavity (both those in ambient conditions as well as those in vacuum) have the issue of thermal expansion shifting the center of mass, and hence severely affecting the experimental results.  I am referring solely to thermal expansion shifting the center of mass, and I am not referring to thermal buckling.  It is my understanding (from Paul March's posts in the EM Drive threads) that Paul has accepted the importance of the effect due to thermal expansion shifting the center of mass.  Actually in his latest posts, Paul wrote that

a) NASA was working on software to automatically subtract the effect of thermal expansion shifting the center of mass from what they thought was the real anomalous force

b) NASA changed their experimental set-up to minimize this  effect of thermal expansion shifting the center of mass

2) It took some time for NASA to fully understand and be able to model the effect of thermal expansion shifting the center of mass, as its analysis is not trivial.  Early on, Paul March recognized that it was affecting the baseline.  If you look at NASA's traces of force vs. time this shifting of the baseline is seen in all traces, to a greater or lesser degree.  It changes with time and with test.  Initially NASA's team addressed the "shifting baseline" problem by subtracting the baseline shift.  However this did not address the effect on the magnitude of the force vs time itself, because the effect of thermal expansion shifting the center of mass  is registered as if it would be an anomalous force, when in effect is purely thermal.  In other words, to properly take into account this effect, one cannot do it solely by "rectifying the baseline".

3) It was only recently (late 2014, after their AIAA conference report) that NASA started to try to address the whole issue  of thermal expansion shifting the center of mass by trying to subtract this effect by modeling software that would model the effect on the magnitude of the force vs. time.  That's the situation as of the last time that Paul March was able to post in the NSF site  We are eagerly awaiting their upcoming publication of their experiments (I presume to happen in the near future in some AIAA refereed journal ?) to be able to explore their ability to fully address this effect of thermal expansion shifting the center of mass in their experiments.

[Rodal]

EXPERIMENTAL PROOF THAT NASA'S TEST WITHOUT A DIELECTRIC INSERT WAS IN RESONANCE AT THE FREQUENCY REPORTED IN NASA'S REPORT


Since we had concluded in http://forum.nasaspaceflight.com/index.php?topic=39214.msg1470613#msg1470613

that:

for NASA's experiment without a dielectric we can say that the exact solution says that there was indeed a natural frequency for mode TE012 within 0.1% of the measured frequency, but as to whether the measured frequency was at the resonant peak, one has to rely on NASA's team having actually found peak resonance with S21 and S11 measurements (because we don't know the dimensions of NASA's resonant cavity to the precision required to calculate the resonant peak with all the required digits of numerical precision).

Finally, we reproduce again the experimental data from NASA Johnson Eagleworks Laboratory that proves that their experiment without dielectric inserts in their frustum of a cone cavity was indeed in resonance.

The resonance for mode shape TE012 without dielectric inserts was measured with an Agilent Model 9923A, 4.0 GHz Field Fox Vector Network Analyzer (VNA) both in the S11 and S21 modes (as shown in the pictures below) using the frustum RF loop antenna as input and the frustum sense antenna located 180 degrees around from the loop antenna with both antennas being at the same 15% of the height from the large end of the frustum, i.e., 0.15 * 9.00” = 1.35” or 34.29mm away from the large end.   

The TE012 resonant frequency without the dielectric PE disc inserts was measured at 2.167137 GHz using either the S11 or S21 methods as shown by the two attached VNA slides. 

Thus, any claims made about this test without dielectric inserts in NASA's frustum of a cone cavity with mode shape TE012 at  2.167 GHz not being in resonance are shown to be completely baseless, false and misleading.

This, factual information shows without a doubt that indeed NASA's frustum of a cone without dielectric inserts was in resonance with mode shape TE012 at  2.167 GHz in agreement with NASA's report and in agreement with the COMSOL Finite Element Analysis calculation and in agreement with the exact solution I calculated using Wolfram Mathematica.

[X_RaY]

EXPERIMENTAL PROOF THAT NASA'S TEST WITHOUT A DIELECTRIC INSERT WAS IN RESONANCE AT THE FREQUENCY REPORTED IN NASA'S REPORT


Since we had concluded in http://forum.nasaspaceflight.com/index.php?topic=39214.msg1470613#msg1470613

that:

for NASA's experiment without a dielectric we can say that the exact solution says that there was indeed a natural frequency for mode TE012 within 0.1% of the measured frequency, but as to whether the measured frequency was at the resonant peak, one has to rely on NASA's team having actually found peak resonance with S21 and S11 measurements (because we don't know the dimensions of NASA's resonant cavity to the precision required to calculate the resonant peak with all the required digits of numerical precision).

Finally, we reproduce again the experimental data from NASA Johnson Eagleworks Laboratory that proves that their experiment without dielectric inserts in their frustum of a cone cavity was indeed in resonance.

The resonance for mode shape TE012 without dielectric inserts was measured with an Agilent Model 9923A, 4.0 GHz Field Fox Vector Network Analyzer (VNA) both in the S11 and S21 modes (as shown in the pictures below) using the frustum RF loop antenna as input and the frustum sense antenna located 180 degrees around from the loop antenna with both antennas being at the same 15% of the height from the large end of the frustum, i.e., 0.15 * 9.00” = 1.35” or 34.29mm away from the large end.   

The TE012 resonant frequency without the dielectric PE disc inserts was measured at 2.167137 GHz using either the S11 or S21 methods as shown by the two attached VNA slides. 

Thus, any claims made about this test without dielectric inserts in NASA's frustum of a cone cavity with mode shape TE012 at  2.167 GHz not being in resonance are shown to be completely baseless, false and misleading.

This, factual information shows without a doubt that indeed NASA's frustum of a cone without dielectric inserts was in resonance with mode shape TE012 at  2.167 GHz in agreement with NASA's report and in agreement with the COMSOL Finite Element Analysis calculation and in agreement with the exact solution I calculated using Wolfram Mathematica.

Based on this measurement data I've got a look to my calculated frequency for this case and find:

Mode      calculated(GHz)   Comsol(GHz)  diff Comsol(%)  diff Comsol(GHz)  measured NASA(GHz)  diff meas.(%)
TE012   2,1653438127        2,1794            -0,64                -0,014                     2,167138                      -0,08279

Maybe its based on tiny differences between the final real measured cavity and the Comsol simulation.
Of course there are much larger differences for many of the other modes in my spreadsheet*. As I wrote elsewhere
I believe more in field simulations because it works.

* I use it only for general overview.

[Rodal]

EXPERIMENTAL PROOF THAT NASA'S TEST WITHOUT A DIELECTRIC INSERT WAS IN RESONANCE AT THE FREQUENCY REPORTED IN NASA'S REPORT


Since we had concluded in http://forum.nasaspaceflight.com/index.php?topic=39214.msg1470613#msg1470613

that:

for NASA's experiment without a dielectric we can say that the exact solution says that there was indeed a natural frequency for mode TE012 within 0.1% of the measured frequency, but as to whether the measured frequency was at the resonant peak, one has to rely on NASA's team having actually found peak resonance with S21 and S11 measurements (because we don't know the dimensions of NASA's resonant cavity to the precision required to calculate the resonant peak with all the required digits of numerical precision).

Finally, we reproduce again the experimental data from NASA Johnson Eagleworks Laboratory that proves that their experiment without dielectric inserts in their frustum of a cone cavity was indeed in resonance.

The resonance for mode shape TE012 without dielectric inserts was measured with an Agilent Model 9923A, 4.0 GHz Field Fox Vector Network Analyzer (VNA) both in the S11 and S21 modes (as shown in the pictures below) using the frustum RF loop antenna as input and the frustum sense antenna located 180 degrees around from the loop antenna with both antennas being at the same 15% of the height from the large end of the frustum, i.e., 0.15 * 9.00” = 1.35” or 34.29mm away from the large end.   

The TE012 resonant frequency without the dielectric PE disc inserts was measured at 2.167137 GHz using either the S11 or S21 methods as shown by the two attached VNA slides. 

Thus, any claims made about this test without dielectric inserts in NASA's frustum of a cone cavity with mode shape TE012 at  2.167 GHz not being in resonance are shown to be completely baseless, false and misleading.

This, factual information shows without a doubt that indeed NASA's frustum of a cone without dielectric inserts was in resonance with mode shape TE012 at  2.167 GHz in agreement with NASA's report and in agreement with the COMSOL Finite Element Analysis calculation and in agreement with the exact solution I calculated using Wolfram Mathematica.

Based on this measurement data I've got a look to my calculated frequency for this case and find:

Mode      calculated(GHz)   Comsol(GHz)  diff Comsol(%)         diff Comsol(GHz)   measured NASA(GHz)       diff measured(%)
TE012   2,1653438127        2,1794              -0,64                          -0,014                     2,167138                              -0,08279

Maybe its based on tiny differences between the final real measured cavity and the Comsol simulation.
Of course there are much larger differences for many of the other modes in my spreadsheet*. As I wrote elsewhere I believe more in field simulations because it works.

* I use it only for general overview.

Excellent, your solution gives the same natural frequency (2.165 GHz) I calculated with the exact solution using Wolfram Mathematica, as reported here:  http://forum.nasaspaceflight.com/index.php?topic=39214.msg1469866#msg1469866

Further evidence that validates NASA's report that the test without dielectric insert was in TE012 mode shape resonance at the measured frequency !

I think that NASA built the truncated cone cavity to within measurement tolerances of +/-0.01” , giving internal dimensions as follows

bigDiameter = (11.00")  +/-0.01” ---> total % error = 0.18% = 1/550
smallDiameter = (6.25")  +/-0.01”--->total % error = 0.32% = 1/313
axialLength = (9")  +/-0.01” --->         total % error = 0.22% = 1/450

Therefore (taking the median total % error = 0.22% = 1/450) the dimensional tolerance of NASA's frustum is such that it is only for a Q<450 that one can hope to be within the resonant bandwidth, given the uncertainty due to dimensions ( 1/450).

Importantly, the difference between both X-Ray's solution and the exact solution using Wolfram Mathematica from the measured frequency of -0.1% is well within the geometrical tolerance uncertainty of NASA's truncated cone itself.  See NASA's design dimensions for their frustum of a cone, attached below

[Rodal]

QUALITY OF RESONANCE "Q" FOR NASA'S TEST WITHOUT A DIELECTRIC INSERT

Finally, what was the predicted Quality of Resonance ("Q") for NASA's test without a dielectric insert?

Using the following resistivity for the copper alloy used for this test:

Material: Copper alloy 101

resistivity = 1.71*10^(-8) ohm meter

Sources for this material value:
http://www.azom.com/article.aspx?ArticleID=2850#_Physical_Properties_of  http://www.husseycopper.com/production/alloys/electrical/c-101-00/

Using the following geometrical dimensions for the frustum of a cone, as used by Frank Davis:

bigDiameter = (11.01 inch)*(2.54 cm/inch)*(1 m/(100 cm));
smallDiameter = (6.25 inch)*(2.54 cm/inch)*(1 m/(100 cm));
axialLength = (9 inch)*(2.54 cm/inch)*(1 m/(100 cm));

the exact solution, using Wolfram Mathematica to solve Maxwell's equations, gives:

Q = 78642

So, a very good Q value is predicted for mode shape TE012 at the frequency:

measured frequency at which NASA test was performed:                                              2.168 GHz
calculated natural frequency (exact solution, Dr. Rodal using Wolfram Mathematica):      2.165 GHz

for NASA's test without a dielectric insert that resulted in no thrust.

The fact that this NASA test resulted in zero "anomalous force", and that Paul March at NASA had the great insight to introduce dielectric inserts at the small end to produce the anomalous force, is one of the most important data point in the history of EM Drive experiments

[X_RaY]

QUALITY OF RESONANCE "Q" FOR NASA'S TEST WITHOUT A DIELECTRIC INSERT

Finally, what was the predicted Quality of Resonance ("Q") for NASA's test without a dielectric insert?

Using the following resistivity for the copper alloy used for this test:

Material: Copper alloy 101

resistivity = 1.71*10^(-8) ohm meter

Sources for this material value:
http://www.azom.com/article.aspx?ArticleID=2850#_Physical_Properties_of  http://www.husseycopper.com/production/alloys/electrical/c-101-00/

Using the following geometrical dimensions for the frustum of a cone, as used by Frank Davis:

bigDiameter = (11.01 inch)*(2.54 cm/inch)*(1 m/(100 cm));
smallDiameter = (6.25 inch)*(2.54 cm/inch)*(1 m/(100 cm));
axialLength = (9 inch)*(2.54 cm/inch)*(1 m/(100 cm));

the exact solution, using Wolfram Mathematica to solve Maxwell's equations, gives:

Q = 78642

So, a very good Q value is predicted for mode shape TE012 at the frequency:

measured frequency at which NASA test was performed:                                              2.168 GHz
calculated natural frequency (exact solution, Dr. Rodal using Wolfram Mathematica):      2.165 GHz

for NASA's test without a dielectric insert that resulted in no thrust.

The fact that this NASA test resulted in zero "anomalous force", and that Paul March at NASA had the great insight to introduce dielectric inserts at the small end to produce the anomalous force, is one of the most important data point in the history of EM Drive experiments

These are great news. I came to nearly the same conclusion some time last year(Q=79011). I never post it, at least I am not sure about the formula (found an approximation in an cern paper about cavities if my memory is correct) and my implementation. No -3dB bandwidth needed for the calculation, its mode,volume, conductivity dependent.
Based on this the Q at larger volumes is in general (mode dependent) bigger than for smaller volume. I think more energy can be stored in larger volumes.
If I try to use to divide all dimensions by a factor of 10, I get a 10 times higher resonant frequency (good so far) but a Q of only 24985. Could you so kind to check this please, I can feel something may still wrong with this calculation although the number for the original dimensions fits yours very well.

[Rodal]

...
These are great news. I came to nearly the same conclusion some time last year(Q=79011). I never post it, at least I am not sure about the formula (found an approximation in an cern paper about cavities if my memory is correct) and my implementation. No -3dB bandwidth needed for the calculation, its mode,volume, conductivity dependent.
Based on this the Q at larger volumes is in general (mode dependent) bigger than for smaller volume. I think more energy can be stored in larger volumes.
If I try to use to divide all dimensions by a factor of 10, I get a 10 times higher resonant frequency (good so far) but a Q of only 24985. Could you so kind to check this please, I can feel something may still wrong with this calculation although the number for the original dimensions fits yours very well.

You are correct on all counts 

I will be posting further...

[X_RaY]

...
These are great news. I came to nearly the same conclusion some time last year(Q=79011). I never post it, at least I am not sure about the formula (found an approximation in an cern paper about cavities if my memory is correct) and my implementation. No -3dB bandwidth needed for the calculation, its mode,volume, conductivity dependent.
Based on this the Q at larger volumes is in general (mode dependent) bigger than for smaller volume. I think more energy can be stored in larger volumes.
If I try to use to divide all dimensions by a factor of 10, I get a 10 times higher resonant frequency (good so far) but a Q of only 24985. Could you so kind to check this please, I can feel something may still wrong with this calculation although the number for the original dimensions fits yours very well.

You are correct on all counts 

I will be posting further...

Thanks very much   

[Rodal]

Notice the significance of this:

1) Two completely independent researchers, using completely different solutions (X-Ray and Rodal) predict that the NASA cavity had a natural frequency for mode shape TE012 at the same frequency at which NASA reported their experimental finding that the cavity without dielectric inserts experienced no "anomalous force"

2) Furthermore both independent researchers predict that the cavity should have experienced strong resonance with a Q=79000

This independently confirms:

3) NASA's experimental results are of such high quality that the frequencies and resonance can be independently confirmed by independent researchers

4) Roger Shawyer's strange claim that the cut-off frequency formula for open waveguides should also apply to a closed cavity like the one used by NASA is shown to be false regarding resonance, as per point 1 through 3 above, with Q=79000

[X_RaY]

Unloaded Q (Q_0) for eigen resonance conditions is related to the statistical lifetime of any given photon within the cavity before it disappear via conversion into heat.
The effective or loaded Q is different, it depends of the external Q and the coupling factor, in general it's (much) lower than the unloaded Q. The photon may coupled out of the cavity much earlier before it's statistical life time for the cavity alone(Q_0)is reached, it's energy is lost outside the cavity.
My Questions are now :

a) What's known about the external Q (value)in general for an amplifier (with/without external load and circulator) for Z_in=50 Ohm? What's the way to compute this?

b) What was the coupling factor in the NASA experiment?

I don't believe Q_0 is THE important value for thrust generation. Would be unnatural since the antenna feed is a connection to the outside, it can't be ignored.

[Rodal]

PROOF THAT THE QUALITY FACTOR OF RESONANCE "Q" SCALES LIKE √L AND THEREFORE THAT THE FORCE/POWER ALSO SCALES LIKE √L

1) Force per power input of EMDrive and its relationship to a photon rocket

We start with the definition of Power as the time derivative of work, and therefore equal to the vector dot product of force times velocity:



For an ideal photon rocket with a perfectly collimated photon beam, the exhaust velocity (not the spaceship velocity !!!) is the speed of light c and therefore:

F*c = P_in

where P_in is the Power input into the exhaust ("Power Input" here only stands for the power at this late stage, notice that there may be further losses from the power plant, coupling factor, etc.).  Therefore, for an ideal photon rocket, the force per input power is

(F /  P_in)_photonRocket = 1/c

Side note: for rockets exhausting particles-with-mass at speeds much lower than the speed of light, for example ion thrusters, this ratio is 2/v instead of 1/c, where v is the speed of the particle-having-mass (as propellant particles-with-mass, unlike photons, need to be accelerated to the exhaust speed, see https://en.wikipedia.org/wiki/Spacecraft_propulsion#Power_to_thrust_ratio and  https://en.wikipedia.org/wiki/Specific_impulse#Energy_efficiency for the reason for the factor of 2, as E=(1/2)mv² instead of E=mc² https://en.wikipedia.org/wiki/Kinetic_energy#Relativistic_kinetic_energy_of_rigid_bodies ).  Therefore, the efficiency (F /  P_in) for ion thrusters is much larger than the one for photon rockets (since v<< c, and hence 2/v>>1/c) and that's why this type of photon rocket has not seen, and is not envisioned to have, practical use.

Interestingly the force per input power for the EM Drive, according to all three different theories (McCulloch, Shawyer and Notsosureofit) can be expressed similarly as:

(F /  P_in)_EMDrive = (1/c) Q g

where Q is the quality factor of resonance and g is a dimensionless factor due to geometry, relative magnetic permeability, relative electric permittivity and mode shape, depending on the theory.  So, the force per input power for an EM Drive is predicted to be superior to a photon rocket as follows:

(F /  P_in)_EMDrive/ (F /  P_in)_photonRocket  =  Q g

in other words, the theoretical outperformance of the EMDrive is speculated to be due to just the quality of resonance Q and the dimensionless factor g.

For the purpose of this discussion we will avoid dealing with the strange consequences of these theories regarding conservation of momentum and conservation of energy issues inherent to the concept of proposing a closed resonant electromagnetic cavity for space propulsion.


2) Geometric factor "g" for different theories

2a) McCulloch

McCulloch has presented a number of simple formulas for the EMDrive (http://www.ptep-online.com/index_files/2015/PP-40-15.PDF), all having the general form

(F /  P_in)_EMDrive = (1/c) Q g

The simplest of which has the following simple definition for the dimensionless factor "g":

g=(L/D_s - L/D_b)

where:

L = length of fustrum of a cone, measured perpendicular to the end faces
D_s=diameter of small end of the fustrum of a cone
D_b=diameter of big end of the fustrum of a cone

So, it is evident that for this formula from McCulloch, the factor "g" is a dimensionless factor that only depends on the geometrical ratios L/D_s and L/D_b:

g_McCulloch = g (L/D_s,L/D_b)

It is also obvious that if one scales the EM Drive geometry such that the geometrical ratios L/D_s and L/D_b are kept constant, that the dimensionless factor "g" will remain constant in McCulloch's equation.

2b) Shawyer

Shawyer has presented a formula ( http://www.emdrive.com/theorypaper9-4.pdf ) for the EM Drive where the dimensionless factor "g" is defined as follows:

g = 2 D_f

where D_f is a dimensionless factor called the "Design Factor" by Shawyer, and where D_f is a function of the diameters and in addition it is also a function of the relative magnetic permeability and the relative electric permittivity, as well as the natural frequency of resonance:

g_Shawyer =    g(D_b/D_s,L/D_b,μ_r,ε_r,m,n,p)

where the diameters of the fustrum of a cone appear explicitly in his formula for the "design factor" and where the length and the mode shape quantum numbers appear only implicitly because the design factor is dependent on the natural frequency at which resonance with a particular mode shape occurs.

It is simple to show that if one scales the EM Drive geometry such that the geometrical ratios L/D_s and L/D_b, and the material properties μ_r,ε_r are kept constant, and the mode shape is kept the same, that the dimensionless factor "g" will remain constant in Shawyer's equation.

2c) Notsosureofit

Notsosureofit has presented a formula for the EM Drive (http://emdrive.wiki/@notsosureofit_Hypothesis) where the dimensionless factor "g" is defined as follows:

g=(Ψ_mn²/(4π³))(c/f_mnp)³(1/L)(1/(D_s) ²-1/(D_b)²)

where

Ψ_mn= X_mn (the zeros of the cylindrical Bessel functions) for TM modes
Ψ_mn= X'_mn (the zeros of the first derivative of the cylindrical  Bessel functions) for TE modes

(Side note: This link is an excellent source for the numerical values of X_mn and of  X'_mn for m<11 and n<6: http://wwwal.kuicr.kyoto-u.ac.jp/www/accelerator/a4/besselroot.htmlx )

Therefore, it can be shown that the "g" factor in Notsosureofit's hypothesis is a function of the geometrical ratios

g_{Notsosureofit} =    g(L/D_s,L/D_b,μ_r,ε_r,m,n,p)

how this is so, will be shown in detail in the next section.


3) Natural frequency scaling

For simplicity, since the truncated cone resonant cavities tested by NASA, Shawyer, Tajmar, and others have all been close to a cylindrical cavity, we will derive the scaling relationship for the natural frequencies of a cylindrical cavity, but this can also be done with the more complicated equations for a truncated cone (which instead of cylindrical Bessel functions are expressed in terms of spherical Bessel functions, and instead of harmonic (cosine) functions in terms of Associated Legendre functions).  The reason why all EM Drive experiments have been performed up to now with EM Drive geometries close to a cylindrical cavity is because experimenters have tried to follow Shawyer's strange prescription that the small diameter of the truncated cone should be larger than the cut-off frequency for an open, constant-cross-section waveguide having the same diameter (although the EM Drive is a closed cavity, and not an open waveguide, and it is well-known that such cut-off equations are inapplicable to closed cavities).  This prescription forbids geometries of truncated cones where the small diameter is much different from the large diameter.  Therefore it turns out that one can use a mean radius

R = (D_s + D_b)/4

to model the fustrum of a cone cavity as a cylindrical cavity, having natural frequencies

f_mnp=(c/R) a_mnp

where c is the speed of light, R is the previously defined mean radius and where m,n,p are the so called "quantum numbers" defining the mode shape, where m is the integer related to the circumferential direction, n is the integer related to the polar radial direction and p is the integer related to the longitudinal axial direction.

And where

a_mnp= √((( Ψ_mn/ π)²+(p R/L)²)/(4 μ_r ε_r))

It is also trivial to show that since

R = (D_s + D_b)/4

then

R/L =(D_s/L + D_b/L)/4

hence

a_mnp= a_mnp( L/D_s,L/D_b,μ_r,ε_r,m,n,p)

and that for constant geometrical ratios, constant medium properties μ_r,ε_r, and for the same mode shape m,n,p, a_mnp will remain constant and hence that the frequency will scale like the inverse of any geometrical dimension:

f_mnp=(c/R) constant
                         =(c/L) constant
                         =(c/ D_b) constant
                          =(c/ D_s) constant

3a) Proof that Notsosureofit's dimensionless factor is constant for constant geometrical ratio, constant medium properties and constant mode shape

Returning to Notsosureofit's dimensionless factor expression in point 2c:

g=(Ψ_mn²/(4π³))(c/f_mnp)³(1/L)(1/(D_s) ²-1/(D_b)²)

and replacing the frequency expression:

f_mnp=(c/R) a_mnp

one obtains:

g=(Ψ_mn²/(4π³))(R/a_mnp)³(1/L)(1/(D_s) ²-1/(D_b)²)

and therefore,

g_{Notsosureofit} =    g(L/D_s,L/D_b,μ_r,ε_r,m,n,p)

since:

R/L =(D_s/L + D_b/L)/4
(R/D_s)²=((1/4)(1+D_b/D_s))²
(R/D_b)²=((1/4)(1+D_s/D_b))²
(D_b/D_s)= (L/D_s)/(L/D_b)

Therefore for constant geometrical ratios, constant medium properties μ_r,ε_r, and for the same mode shape m,n,p, the dimensionless factor g will remain constant.  It is trivial to show the same result for Shawyer's design factor, and hence for the dimensionless factor g in Shawyer's expression.

So, in general we can state that all theoretical expressions, McCulloch's, Shawyer's and Notsosureofit, are such that the dimensionless factor g will remain constant for constant geometrical ratios, constant medium properties μ_r,ε_r, and for the same mode shape m,n,p.


4) Quality of resonance (Q) scaling

The definition of quality of resonance factor (Q) can be stated as follows (https://en.wikipedia.org/wiki/Q_factor#Definition_of_the_quality_factor):



Q ≝ ω EnergyStored /PowerLoss

where

ω = angular frequency
EnergyStored =∫Electromagnetic Energy Density dV
PowerLoss =  ((ω δ) /2)  (∫ Electromagnetic Energy Density dA)
                 = R_s  (∫ Electromagnetic Energy Density dA)/ μ
                 = ρ  (∫ Electromagnetic Energy Density dA)/ (μ δ)

where 

R_s = "surface resistance"
   = ρ / δ
ρ = resistivity of the interior wall of the EM Drive resonant cavity
μ = magnetic permeability of the interior wall of the EM Drive resonant cavity
   = μ_oμ_r
δ =skin depth (the penetration depth of the electromagnetic energy into the interior metal wall)
     (https://en.wikipedia.org/wiki/Skin_effect)

in general, for arbitrary frequencies, the skin depth is:


where

ε = electric permittivity of the interior wall of the EM Drive resonant cavity
   = ε_oε_r

At angular frequencies ω much below 1/(ρε), for example, in the case of copper, for frequencies much below exahertz (10^9 GHz, the range of hard X-rays and Gamma rays), the skin depth can be expressed as follows:



Now, using the fact that

PowerLoss =((ω δ) /2)  (∫ ElectromagneticEnergy dA)

one immediately obtains:

Q=(2/SkinDepth)( ∫Electromagnetic Energy Density dV/ ∫ Electromagnetic Energy Density dA)

Alternatively one can arrive at the same result, using the formula for power loss that depends on the "surface resistance" R_s:

PowerLoss = R_s  (∫ Electromagnetic Energy Density dA)/ μ
PowerLoss = ρ  (∫ Electromagnetic Energy Density dA)/ (μ δ)

one gets:

Q = ω μ    (∫Electromagnetic Energy Density dV)/ (R_s ∫ Electromagnetic Energy Density dA)
Q = ω μ δ (∫Electromagnetic Energy Density dV)/ (ρ ∫ Electromagnetic Energy Density dA)

and using the fact (at angular frequencies ω  much below 1/(ρε) ) that the angular frequency ω is a function of the square of the skin depth δ:

ω = 2 ρ / (μ δ²)

it is straightforward to show that the quality of resonance Q is:

Q=(2/SkinDepth)( ∫Electromagnetic Energy Density dV/ ∫ Electromagnetic Energy Density dA)

the electromagnetic energy density integrated over the cavity volume, divided by the electromagnetic energy density integrated over the cavity surface area, divided by the skin depth.

4a) Skin depth scaling

At frequencies much below 1/(ρε) the skin depth can be expressed as

SkinDepth = √(ρ/(μ π f_mnp))

where
ρ = resistivity of the interior wall of the EM Drive resonant cavity
ε = electric permittivity of the interior wall of the EM Drive resonant cavity
   = ε_oε_r
μ = magnetic permeability of the interior wall of the EM Drive resonant cavity
   = μ_oμ_r
f_mnp = resonant frequency at mode shape m,n,p
        = ω_mnp/(2π)

Plugging in the expression for the frequency

f_mnp=(c/R) a_mnp

into the skin depth expression, results in the following expression:

SkinDepth = √R√(ρ/(μ π c a_mnp))

or, using the previously derived expressions for a_mnp one concludes that the skin depth scales like the square root of any geometrical dimension, for constant resistivity and magnetic permeability of the interior wall of the cavity and for constant geometrical ratios, constant medium properties μ_r,ε_r, and for the same mode shape m,n,p.

In other words, for increasing dimensions of the cavity, preserving all geometrical ratios, and keeping material properties constant and for the same mode shape, the skin depth will increase with the square root of the dimension, while the frequency will decrease, as the inverse of the dimension.


4b) Quality of resonance (Q) scaling

Having revealed the scaling law for the skin depth, what now remains to be shown is the scaling for the energy integral ratio in the expression for Q:

Q=(2/SkinDepth)(∫Electromagnetic Energy Density dV/ ∫ Electromagnetic Energy Density dA)

The expressions under the integrals are dependent on each mode shape, as the electromagnetic energy distribution depends on mode shape, of course.  However, we can notice that the lowest mode shapes (those with low values of m,n,p, for example TE012, TM212) have been of interest in the EM Drive experiments so far.  So, for simplification purposes we can assume that the distribution of the electromagnetic field is of low order, and hence not that much variable throughout the cavity, for low mnp number mode shapes (for example m=0 means a constant distribution in the azimuthal circumferential direction of the cavity).  Under this assumption one can (for approximation purposes) take the energy out of the integral:

(∫Electromagnetic Energy Density dV/ ∫ Electromagnetic Energy Density dA) ~
                     ~ (Electromagnetic Energy Density /Electromagnetic Energy Density) (∫dV/ ∫ dA )
                     ~ InteriorVolume/InteriorSurfaceArea
                     ~ π R²L/(2 π R (R+L) )
                     ~ R/(2(1+R/L))

and substituting this and the previously found scaling law for the skin depth, into the expression for the quality of resonance factor Q, leads to:

Q=(2/SkinDepth)(∫Electromagnetic Energy Density dV/ ∫ Electromagnetic Energy Density dA)
 ~(2/(√R√(ρ/(μ π c a_mnp)))) R/(2(1+R/L))
 ~ √R b

where the factor b is:

b = (1/((1+R/L)√(ρ/(μ π c a_mnp))))   

or, using the previously derived expressions for a_mnp one concludes that the quality of resonance (Q) scales like the square root of any geometrical dimension, for constant resistivity and magnetic permeability of the interior wall of the cavity and for constant geometrical ratios, constant medium properties μ_r,ε_r, and for the same mode shape m,n,p.

In other words, for increasing dimensions of the cavity, preserving all geometrical ratios, and keeping material properties constant and for the same mode shape, the quality of resonance (Q) will increase with the square root of the dimension, also the skin depth will increase with the square root of the dimension, while the frequency will decrease, as the inverse of the dimension.

Furthermore, we previously proved that all three theories for the EM Drive (McCulloch, Shawyer and Notsosureofit) have expressions for the force/inputPower to be proportional to the quality of factor Q times a dimensionless factor g:

(F /  P_in)_EMDrive/ (F /  P_in)_photonRocket  =  Q g
(F /  P_in)_EMDrive = (1/c) Q g

and we previously proved that the dimensionless factor g (for all three theories: McCulloch, Shawyer and Notsosureofit) remains perfectly constant for constant geometrical ratios, constant medium properties μ_r,ε_r, and for the same mode shape m,n,p.

Therefore one concludes that the force per input Power (for all three theories: McCulloch, Shawyer and Notsosureofit) scales like the square root of any geometrical dimension, for constant resistivity and magnetic permeability of the interior wall of the cavity and for constant geometrical ratios, constant medium properties μ_r,ε_r, and for the same mode shape m,n,p.

In other words, to maximize the force per input power, according to all three theories: (McCulloch, Shawyer and Notsosureofit) the most efficient EM Drive would be as large as possible, this being due to the fact that the quality of factor of resonance Q (all else being equal) scales like the square root of the geometrical dimensions.
Small cavity EM Drive's (all else being equal) are predicted to have smaller quality of resonance Q and therefore smaller force/inputPower.

It is not clear whether this has been known to EM Drive experimenters, given the fact that the recent experiments by Prof. Tajmar at TU Dresden, Germany, (under advise from Roger Shawyer according to the report) were performed with a much smaller EM Drive, and the fact that there are several EM Drive researchers discussing really tiny EM Drives (as the group in Aachen, Germany) for use in CubeSats.  Such EM Drives are predicted to be much more inefficient, having substantially lower force/inputPower.