Author Topic: EM Drive Developments - related to space flight applications - Thread 2  (Read 2103714 times)

Offline Rodal

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....
Found some time to read over this patent. It is a gold mine of good info. Great find. A couple takeaways I found is that it confirms that TE modes are highly desirable compared to TM. Also this caught my eye:
Quote
It has been found possible to predict the resonances approximately by defining a phase shift per unit length as 21r/ \g, where Ag is given by the usual formula for circular wave-guides of diameter D, but where D and hence Ag vary along the cone. If this phase shift is integrated from the location of the cut-off diameter to the position of the plunger or movable end wall, resonances will be found when the integral has values of 11 pi."
Looks like math for predicting resonant modes for cones. It looks like some of the text got messed up in the character translation over to Google patents, see the bold part.

Glad you agree that this 1969 patent is a gold mine for people interested in EM Drives.

Please see  the attached Adobe Acrobat .pdf file of page 3 of the original patent, top of column 4, for the actual formulas and symbols

The patent states

"n * Pi" instead of "11 * Pi"

"2 * Pi / Lambdag " instead of b]21r/ \g[/b] 

"Lambdag" instead of  "Ag "  where Lambdag must mean the waveguide's wavelength

I attach the Adobe Acrobat .pdf file for the whole patent

Publication number   US3425006 A
Publication date   Jan 28, 1969
Filing date   Feb 1, 1967
Priority date   Feb 1, 1967
Inventors           Wolf James M
Original Assignee   Johnson Service Co
« Last Edit: 01/28/2015 12:59 PM by Rodal »

Offline aero

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I'm puzzled as to why you are using an excitation frequency of 2.45 GHz which does not correspond to any natural frequency of the cavity.  If you want to excite TE110 you should use an excitation frequency of  2.32677 GHz (using the speed of light in air, while if you use the speed of light in vacuum it would be 2.32745 GHz).   The natural frequency of mode shape TE110, 2.33 GHz, is independent of the length of the cavity.

It's very simple. 2.45 GHz is a given.
Cavity length and radius are the independent variables to be adjusted to establish resonance at 2.45 GHz.
And yes, I'm quite sure I want TE 1,1,1 mode.
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Offline Rodal

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I'm puzzled as to why you are using an excitation frequency of 2.45 GHz which does not correspond to any natural frequency of the cavity.  If you want to excite TE110 you should use an excitation frequency of  2.32677 GHz (using the speed of light in air, while if you use the speed of light in vacuum it would be 2.32745 GHz).   The natural frequency of mode shape TE110, 2.33 GHz, is independent of the length of the cavity.

It's very simple. 2.45 GHz is a given.
Cavity length and radius are the independent variables to be adjusted to establish resonance at 2.45 GHz.
And yes, I'm quite sure I want TE 1,1,1 mode.

If you insist in specifying the exciting frequency as 2.45GHz and the diameter of the cylindrical cavity, and having the length as the variable to be adjusted, then you are going the wrong way in reducing the length !

The inverted formula for length in terms of the other variables is simply:

length = pNr/(2 Sqrt[( fr/cMediumNr)^2 - ( xbesselNr/([Pi]*diameterNr))^2])

If you specify:

diameterNr = 7.54898 centimeter
cMediumNr = cAir = (29970500000 centimeter)/second
fr = 2.45*10^9/second

and for TE111 you must have:

xbesselNr = X'11 = 1.84118378134065
pNr = p = 1

Then, this results in a length of

length = pNr/(2 Sqrt[( fr/cMediumNr)^2 - ( xbesselNr/(\[Pi]*diameterNr))^2])
            = 1 /(2 Sqrt[(2.45*10^9/second/ (29970500000 centimeter)/second)^2 - (1.84118378134065/([Pi]* 7.54898 centimeter))^2])
             = 19.531439054546873` centimeters

instead of 12.2364 centimeter.


Conversely, the following inputs

diameter = 7.54898 centimeter
length=19.531439054546873` centimeters
cMedium = cAir = (29970500000 centimeter)/second

give the following lowest modes and frequencies

{{"TE", 1, 1, 0}, 2.32677*10^9},
{{"TE", 1, 1, 1},  2.45*10^9},
{{"TE", 1, 1, 2}, 2.7872*10^9},
{{"TM", 0, 1, 0}, 3.03906*10^9},

which fully verifies that length= 19.531439054546873` centimeters gives a frequency of 2.45 GHz and mode shape TE111, for a diameter = 7.54898 centimeter, in air.



But instead of inputing  the correct length (for TE111) that is length=19.5314 centimeters, first you inputed the incorrect length of 12.2364 centimeter, which was much smaller than the correct length, but then proceeded to make it worse by reducing it even further to  9.36 centimeter which does not make sense.  You should have input length=19.5314 centimeters to get TE111     :)



But in general, it is inadvisable to pose the problem by specifying the frequency, diameter and mode shape to calculate the length, because you are then ill-posing the problem.  If the frequency you specify is low enough, there will be no real length solution.

The lowest frequency you can specify is

fr= (cMediumNr* xbesselNr/([Pi]*diameterNr))

which will result in an infinite length !
« Last Edit: 01/28/2015 12:10 AM by Rodal »

Offline aero

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Isn't it possible to increase radius while leaving length at or around 12 cm? I would prefer that but can't find a radius that works with 2.45 GHz.
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Offline Rodal

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Isn't it possible to increase radius while leaving length at or around 12 cm? I would prefer that but can't find a radius that works with 2.45 GHz.

Yes, it's possible. 

If you insist in specifying the exciting frequency as 2.45GHz and the length of the cylindrical cavity, and having the diameter as the variable to be adjusted, then

The inverted formula for diameter in terms of the other variables is, trivially:

diameter= (2   xbesselNr)/([Pi] Sqrt[4 (fr/cMediumNr)^2 - (pNr/lengthNr)^2])

If you specify:

lengthNr = 12 centimeter (Please notice you asked 12 cm instead of the previous 12.2364 cm)
cMediumNr = cAir = (29970500000 centimeter)/second
fr = 2.45*10^9/second

and for TE111 you must have:

xbesselNr = X'11 = 1.84118378134065
pNr = p = 1

Then, this results in a diameter of

length =(2   xbesselNr)/([Pi] Sqrt[4 (fr/cMediumNr)^2 - (pNr/lengthNr)^2])
            = (2    1.84118378134065)/([Pi] Sqrt[4 (2.45*10^9/second/29970500000 centimeter)/second)^2 - (1/(12 centimeter))^2])
             = 8.332965999678832` centimeters

instead of  the diameter=7.54898 centimeter you used.


Conversely, the following inputs

diameter = 8.332965999678832`  centimeter
length=12 centimeters
cMedium = cAir = (29970500000 centimeter)/second

give the following lowest modes and frequencies

{{"TE", 1, 1, 0}, 2.10786*10^9},
 {{"TE", 1, 1, 1}, 2.45*10^9},
{{"TM", 0, 1, 0}, 2.75314*10^9},
{{"TM", 0, 1, 1}, 3.02311*10^9},

which fully verifies that diameter = 8.332965999678832`  centimeter gives a frequency of 2.45 GHz and mode shape TE111, for a length=12 centimeters, in air.

quod erat demonstrandum  :)



But in general, it is inadvisable to pose the problem by specifying the frequency, length and mode shape to calculate the diameter, because you are then ill-posing the problem. If the frequency you specify is low enough, there will be no real diameter solution.

The lowest frequency (for a specified length) you can specify is

fr= cMediumNr*(pNr/lengthNr) /2

which will result in an infinite diameter!




We have now looked at this three different ways, and all these ways show that you inputed the incorrect parameters to best excite a TE111 natural frequency.
« Last Edit: 01/28/2015 02:01 AM by Rodal »

Offline Rodal

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If you specify:

lengthNr = 12.23642686` centimeter
cMediumNr = cAir = (29970500000 centimeter)/second
fr = 2.45*10^9/second

and for TE111 you must have:

xbesselNr = X'11 = 1.84118378134065
pNr = p = 1

Then, this results in a diameter of  8.277559638179381` centimeter;

Conversely, the following inputs

diameter = 8.277559638179381`  centimeter
length=12.23642686` centimeters
cMedium = cAir = (29970500000 centimeter)/second

give the following lowest modes and frequencies

{{"TE", 1, 1, 0}, 2.12197*10^9},
{{"TE", 1, 1, 1},  2.45*10^9},
{{"TM", 0, 1, 0}, 2.77157*10^9},
{{"TM", 0, 1, 1}, 3.03007*10^9},

quod erat demonstrandum   :)
« Last Edit: 01/28/2015 02:01 AM by Rodal »

Offline Mulletron

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Well according the above posts and others, http://forum.nasaspaceflight.com/index.php?topic=36313.msg1320981#msg1320981 we're all getting pretty adept at using our skills/resources for calculating resonant modes of cylinders, which is a good first step. Well I'm learning new skills as I go along. So I can do cylinders all day, cones...not so much..yet. I certainly didn't come to the table knowing how to calculate resonant modes of anything 5-6 months ago.

I'm trying to figure out how to use the (2 * Pi ) / (Lambda * g) expression from the patent or the Volumetric Mean approach (which is better?) toward calculating resonant modes of conical frustums. I think the holy grail would be a quick and easy correction to convert from cylinder solutions to conical frustums.

So the (2 * Pi ) / (Lambda * g) above, does that mean what when I take 6.28 and divide that by x wavelength, and get a multiple of pi, that diameter or can support a resonant mode?

Trying to figure out how to convert this cylinder to a cone, by keeping the diameter as the small diameter, adjusting the length to arrive at the new large diameter along a 45 or 90 degree cone, and still maintain resonance @ 2.45ghz TE111:
2.45ghz, TE111
L=0.1224489m
D=0.08278945m
« Last Edit: 01/28/2015 11:43 AM by Mulletron »
Challenge your preconceptions, or they will challenge you. - Velik

Offline Rodal

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Well according the above posts and others, http://forum.nasaspaceflight.com/index.php?topic=36313.msg1320981#msg1320981 we're all getting pretty adept at using our skills/resources for calculating resonant modes of cylinders, which is a good first step. Well I'm learning new skills as I go along. So I can do cylinders all day, cones...not so much..yet. I certainly didn't come to the table knowing how to calculate resonant modes of anything 5-6 months ago.

I'm trying to figure out how to use the (2 * Pi ) / (Lambdag ) expression from the patent or the Volumetric Mean approach (which is better?) toward calculating resonant modes of conical frustums. I think the holy grail would be a quick and easy correction to convert from cylinder solutions to conical frustums.

So the (2 * Pi ) / (Lambdag ) above, does that mean what when I take 6.28 and divide that by x wavelength, and get a multiple of pi, that diameter or can support a resonant mode?

Trying to figure out how to convert this cylinder to a cone, by keeping the diameter as the small diameter, adjusting the length to arrive at the new large diameter along a 45 or 90 degree cone, and still maintain resonance @ 2.45ghz TE111:
2.45ghz, TE111
L=0.1224489m
D=0.08278945m

Regarding the Edit of http://forum.nasaspaceflight.com/index.php?topic=36313.msg1320981#msg1320981, glad  :) that we also agree that the diameter should be 8.28 centimeters for that length and TE111 frequency.

As to the comparison between the longitudinal integration (of D and lambdag expressed as a function of the lengthwise coordinate of the cone) as suggested by Wolf's 1969 patent in comparison with the Volumetric Mean approach I have to run the calculations when I get a chance.  But I  first observe that the patent's author (Wolf) admits that the suggested approach is unable to obtain any TM mode that is constant in the length direction: TMmn0 , (notice that he uses indices lmn instead of mnp).  I have some reservations that this integration approach offers any benefit over the Volumetric Mean approach but I will keep an open mind until I get a chance to compare.  [Note: I corrected my nomenclature to lambdag to better show that lambdag means the cylindrical-waveguide's wavelength]

Meanwhile, I would particularly appreciate @NotSoSureOfIt 's comments regarding Wolf's suggestion on how to calculate the resonances of a truncated cone cavity (lines 60 to 75 of column 3 and lines 1 to 11 of column 4) of the US patent #3,425,006 (which I attach below as an Adobe Acrobat .pdf document).

Publication number   US3425006 A
Publication date   Jan 28, 1969
Filing date   Feb 1, 1967
Priority date   Feb 1, 1967
Inventors           Wolf James M
Original Assignee   Johnson Service Co
« Last Edit: 01/28/2015 01:22 PM by Rodal »

Offline Rodal

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..

So the (2 * Pi ) / (Lambdag) above, does that mean what when I take 6.28 and divide that by x wavelength, and get a multiple of pi, that diameter or can support a resonant mode?

...

Not precisely.  Instead the patent's author (Wolf) means that resonance occurs when the definite Integral of (2 Pi / lambdag) taken along the lengthwise coordinate of the cone is an integer multiple of Pi (where the cylindrical-waveguide's wavelength lambdag and diameter D are expressed as functions of the lengthwise cone coordinate, under the integral sign).

Since Pi is a constant under the integral, it seems to me that it is simpler to express this as follows: resonance occurs when the definite Integral of (2 / lambdag) taken along the lengthwise coordinate of the cone is an integer multiple of unity: 1, 2, 3, 4, 5, ....(where the cylindrical-waveguide's wavelength lambdag and diameter D are expressed as functions of the lengthwise cone coordinate, under the integral sign).
« Last Edit: 01/28/2015 02:15 PM by Rodal »

Offline Rodal

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Isn't it possible to increase radius while leaving length at or around 12 cm? I would prefer that but can't find a radius that works...
That's another reason why to compare a numerical solution (by Finite Difference, Finite Element or any other numerical method that relies on a mesh to obtain results) to an exact solution, you should first pick dimensions, discretize the problem to a fine mesh to ensure convergence (this is what takes most of the person's time in a numerical solution) and if necessary, explore different exciting frequencies (keeping dimensions constant, instead of keeping the same excitation frequency and changing dimensions, which means changing the mesh).  For example using the exciting frequency of 2.63GHz for TE111 (for MeepDiameter = 7.54898 centimeter, MeepLength = 12.2364 centimeter) as obtained by @Mulletron and me.

To insist on a fixed excitation frequency for a given mode (like TE111) means that you will have to change dimensions (and hence discretize with a mesh again with the correct dimension that is associated with a natural frequency) if there are no natural frequencies for the frequency you are insisting on, and this only if there is a finite dimension for the frequency you are insisting to get (see previous posts for minimum natural frequency for a given diameter or a given length).
« Last Edit: 01/28/2015 02:49 PM by Rodal »

Offline Rodal

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...

What did you use for the bandwidth source around the frequency of interest (Drive frequency 2.45 E+9 Hz)?

Could you try running all these cases again, everything the same as before except with a significantly narrower bandwidth source around the frequency of interest ?  .  Reportedly harminv does a better job the narrower the source is around the frequency of interest .
Ok, I did run it again with bandwidth = 0.2 * Drive frequency, for cases up to resolution of 80, but I didn't get anything. Once I narrow the bandwidth to exclude the resonant frequency at 1.87 GHz, there are no resonances within the bandwidth.
Harminv does work better at identifying the resonant frequency with narrower bandwidth, when the frequency is within the bandwidth. I set the drive frequency to 1.873 GHz, narrowed the bandwidth to 0.07 * frequency and got this:

frequency                             Quality factor               error
1,873,339,229.3075 Hz   18,325,307.0778158    1.673972608680621e-7+0.0i

....


A bandwidth of 0.2 * Drive frequency is way too large for the cases you are trying to match.  Look at my message http://forum.nasaspaceflight.com/index.php?topic=36313.msg1322181#msg1322181 for example.

The exact solutions for the lowest modes with those cavity dimensions are:

{{"TE", 1, 1, 0}, 2.12197*10^9},
{{"TE", 1, 1, 1},  2.45*10^9},
{{"TM", 0, 1, 0}, 2.77157*10^9},
{{"TM", 0, 1, 1}, 3.03007*10^9},

Therefore the bandwidth between {{"TE", 1, 1, 1},  2.45*10^9} and {{"TM", 0, 1, 0}, 2.77157*10^9} is

( 2.77157*10^9 - 2.45*10^9) / 2.45*10^9 = 0.13

hence the bandwidth separating those modes is  0.13 * Drive frequency  which is much smaller than the bandwith of 0.2 * Drive frequency you first used (with Drive frequency = 2.45*10^9).

This means that you should first calculate the exact natural frequencies around the frequency you are interested in, and therefore the bandwidth between them.  After that input in Meep an even smaller bandwidth for the excitation frequency (as shown in the above example)  :)In this case it means that you should input in Meep a bandwidth smaller than 0.13 * Drive frequency

« Last Edit: 01/28/2015 05:47 PM by Rodal »

Offline aero

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Ok, I'll try that.
I did correct my analytical formula spread sheet while verifying your formula, and now my analytical formula calculations agree with yours in all respects.
Still need agreement from meep/Harminv however.
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Offline Notsosureofit

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Meanwhile, I would particularly appreciate @NotSoSureOfIt 's comments regarding Wolf's suggestion on how to calculate the resonances of a truncated cone cavity (lines 60 to 75 of column 3 and lines 1 to 11 of column 4) of the US patent #3,425,006 (which I attach below as an Adobe Acrobat .pdf document).

Publication number   US3425006 A
Publication date   Jan 28, 1969
Filing date   Feb 1, 1967
Priority date   Feb 1, 1967
Inventors           Wolf James M
Original Assignee   Johnson Service Co

Mmmm..  That's the argument that I used to come up w/ "volumetric".  Been otherwise occupied, but I'll take a look at it as time permits.

Edit:   Lessee, that's the "guide" wavelength, for equal phase planes.

Basically, you want to solve for the k[sub z] w/ R as a function of z.

like k[sub z]^2 = (omega/c)^2- (X[sub m,n]/R[fn z])^2    or X'

« Last Edit: 01/29/2015 01:12 AM by Notsosureofit »

Offline Rodal

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Meanwhile, I would particularly appreciate @NotSoSureOfIt 's comments regarding Wolf's suggestion on how to calculate the resonances of a truncated cone cavity (lines 60 to 75 of column 3 and lines 1 to 11 of column 4) of the US patent #3,425,006 (which I attach below as an Adobe Acrobat .pdf document).

Publication number   US3425006 A
Publication date   Jan 28, 1969
Filing date   Feb 1, 1967
Priority date   Feb 1, 1967
Inventors           Wolf James M
Original Assignee   Johnson Service Co

Mmmm..  That's the argument that I used to come up w/ "volumetric".  Been otherwise occupied, but I'll take a look at it as time permits.

Edit:   Lessee, that's the "guide" wavelength, for equal phase planes.

Basically, you want to solve for the k[sub z] w/ R as a function of z.

like k[sub z]^2 = (omega/c)^2- (X[sub m,n]/R[fn z])^2    or X'
Yes, thank you for looking at it, and formulating the eigenvalue problem :)

I get an interesting closed-form expression containing an ArcTan term for that integration, so, it looks like it should give different results than the cylindrical cavity formula based on any mean diameter measure like the Mean, Geometric Mean or the Volumetric Mean.  I will double check my closed-form integration and run some numbers when I have time available, to see what difference it makes. 

It is interesting that the patent's author (Wolf) did not actually give the resulting closed-form integral but he just hinted at how to solve the eigenvalue problem for the cone :).
« Last Edit: 01/29/2015 07:01 PM by Rodal »

Offline Stormbringer

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 ;D

Well inventors often try to deliberately hide stuff in their schematics. Even Leonardo Da Vinchi did (His tank and Orinthopter thing for example) And of course the alchemists  and later chemists did. There is an ancient time honored tradition of trying to screw your would be successors and possible competitors over by booby trapping your paper records.

« Last Edit: 01/29/2015 01:20 PM by Stormbringer »
When antigravity is outlawed only outlaws will have antigravity.

Offline Rodal

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« Last Edit: 01/29/2015 01:11 PM by Rodal »

Offline Rodal

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******EDIT: This study does not take into account the cutoff frequency condition that eliminates several of these mode shapes.  A new study incorporating the cutoff frequency will be posted *****


MODE SHAPE STUDY of NASA Brady et.al.'s EXPERIMENTS according to different assumed GEOMETRIES

1) We conduct a thorough study of the mode-shapes of the experiments in NASA Brady et.al.'s " Anomalous Thrust Production ..." report  (http://www.libertariannews.org/wp-content/uploads/2014/07/AnomalousThrustProductionFromanRFTestDevice-BradyEtAl.pdf) , taking into account @Mulletron's assumed geometry as well as the extra experiment conducted at 2168 Mhz.



2) The experimental data can be found in p .18, Table 2. Tapered Cavity Testing :  Summary of Results and in the section on p .18, F.Tapered Cavity RF Evaluation, General Findings and Lessons Learned; of (http://www.libertariannews.org/wp-content/uploads/2014/07/AnomalousThrustProductionFromanRFTestDevice-BradyEtAl.pdf )



3) We define the Volumetric  Mean as follows:

VolumetricMeanDiameter=Sqrt[(SmallDiameter^2+SmallDiameter*BigDiameter+BigDiameter^2)/3]

For a derivation of the Volumetric Mean (equating the volume of an equivalent cylinder to the volume of a truncated cone): See http://forum.nasaspaceflight.com/index.php?topic=36313.msg1319655#msg1319655



4) Let's define as "Mulletron geometry" the following definition for the NASA Brady et. al. cavity:

Mulletron Best estimate as of 11/9/2014   
http://forum.nasaspaceflight.com/index.php?topic=36313.msg1320903#msg1320903
   
cavityLength = 0.27637 m
bigDiameter = 0.30098 m
smallDiameter = 0.15875 m

then

VolumetricMeanDiameter=Sqrt[(SmallDiameter^2+SmallDiameter*BigDiameter+BigDiameter^2)/3]
                                       = 0.2335031034055008` meter



5) Let's define as "Aero geometry" the following definition for the NASA Brady et. al. cavity:

Aero Best estimate as of 11/9/2014    http://forum.nasaspaceflight.com/index.php?topic=29276.msg1285896#msg1285896
   
cavityLength = 0.24173 m
bigDiameter = 0.27246 m
smallDiameter = 0.15875 m

then

VolumetricMeanDiameter=Sqrt[(SmallDiameter^2+SmallDiameter*BigDiameter+BigDiameter^2)/3]
                                       = 0.21808946107809366` meter

6) Let's define as "Fornaro geometry" the following definition for the NASA Brady et. al. cavity:

Fornaro estimate    http://forum.nasaspaceflight.com/index.php?topic=36313.msg1302455#msg1302455
   
cavityLength = 0.332 m
bigDiameter = 0.397 m
smallDiameter = 0.244 m

then

VolumetricMeanDiameter= Sqrt[(SmallDiameter^2+SmallDiameter*BigDiameter+BigDiameter^2)/3]
                                       = 0.32352897860933574` centimeter

7) These are the experimentally reported frequencies and the COMSOL-calculated mode shapes:

frequencyBradyA =  1.9326*10^9 1/second; TM211
frequencyBradyB =  1.9367*10^9 1/second; TM211
frequencyBradyC =  1.8804*10^9 1/second; TE012
frequencyBradyD =  2.168*10^9   1/second; TE012

Notice the contradiction in NASA's report: mode shape TE012 at 1.88GHz, mode shape TM211 at a higher frequency 1.93 GHz and mode shape TE012 again at an even higher frequency 2.17 GHz (? ).  This is impossible.  Our interpretation is that NASA Brady et.al.'s made a typo and they meant to write TM211 (or another mode) for frequencyBradyD =  2.168*10^9 .



8) Given the the geometrical dimensions (using the previously defined Volumetric Mean Diameter), and the value of speed of light in air, we use the frequency equation (see: http://en.wikipedia.org/wiki/Microwave_cavity#Cylindrical_cavity ) to calculate frequencies as a function of the mode shape quantum numbers: circumferential (m), radial (n), and longitudinal (p).   Mode shapes are reported as TXmnp where "X" can stand for E= electric transverse mode or M= magnetic transverse mode.



9) Then I obtain the following mode for the three different assumed geometries:


Mulletron Geometry

{{"TE", 1, 1, 0}, 7.52226*10^8},
{{"TE", 1, 1, 1}, 9.27278*10^8},
{{"TM", 0, 1, 0}, 9.82506*10^8},
{{"TM", 0, 1, 1}, 1.12219*10^9},
{{"TE", 2, 1, 0}, 1.24783*10^9},
{{"TE", 1, 1, 2}, 1.31979*10^9},
{{"TE", 2, 1, 1}, 1.36054*10^9},
{{"TM", 0, 1, 2}, 1.46332*10^9},
{{"TM", 1, 1, 0}, 1.56547*10^9},
{{"TE", 0, 1, 0}, 1.56547*10^9},
{{"TE", 2, 1, 2}, 1.6532*10^9},
{{"TM", 1, 1, 1}, 1.65671*10^9},
{{"TE", 0, 1, 1}, 1.65671*10^9},
{{"TE", 3, 1, 0}, 1.71642*10^9},
{{"TE", 1, 1, 3}, 1.79216*10^9},
{{"TE", 3, 1, 1}, 1.80003*10^9},
{{"TM", 0, 1, 3}, 1.90034*10^9},
{{"TM", 1, 1, 2}, 1.90438*10^9},
{{"TE", 0, 1, 2}, 1.90438*10^9},
{{"TE", 3, 1, 2}, 2.03029*10^9},
{{"TE", 2, 1, 3}, 2.05014*10^9},
{{"TM", 2, 1, 0},2.09819*10^9},
{{"TM", 2, 1, 1}, 2.16712*10^9},
{{"TE", 4, 1, 0}, 2.17252*10^9},

Mode shapes bracketing NASA-reported frequencies:

frequencyBradyA = 1.9326*10^9 1/second;
{{"TE", 0, 1, 2}, 1.9043845840829124`*^9}; {{"TM", 1, 1, 2}, 1.9043845840829124`*^9};
{{"TE", 3, 1, 2}, 2.0302945503765867`*^9};
frequencyBradyB = 1.9367*10^9 1/second;
{{"TE", 0, 1, 2}, 1.9043845840829124`*^9}; {{"TM", 1, 1, 2}, 1.9043845840829124`*^9};
{{"TE", 3, 1, 2}, 2.0302945503765867`*^9};
frequencyBradyC =  1.8804*10^9 1/second;
{{"TE", 3, 1, 1}, 1.8000272999857957`*^9};
{{"TM", 0, 1, 3}, 1.9003447870125527`*^9};
frequencyBradyD =  2.168*10^9 1/second;
{{"TM", 2, 1, 1}, 2.167116717351836`*^9};
{{"TE", 4, 1, 0}, 2.17251756371995`*^9};

Frequencies of NASA-reported mode shapes:

{{"TE", 0, 1, 2}, 1.9043845840829124`*^9}
{{"TM", 2, 1, 1}, 2.167116717351836`*^9}


Aero geometry

{{"TE", 1, 1, 0}, 8.05391*10^8},
{{"TE", 1, 1, 1}, 1.01634*10^9},
{{"TM", 0, 1, 0}, 1.05194*10^9},
{{"TM", 0, 1, 1}, 1.22102*10^9},
{{"TE", 2, 1, 0}, 1.33602*10^9},
{{"TE", 2, 1, 1}, 1.47283*10^9},
{{"TE", 1, 1, 2}, 1.47846*10^9},
{{"TM", 0, 1, 2}, 1.62597*10^9},
{{"TM", 1, 1, 0}, 1.67611*10^9},
{{"TE", 0, 1, 0}, 1.67611*10^9},
{{"TM", 1, 1, 1}, 1.78707*10^9},
{{"TE", 0, 1, 1}, 1.78707*10^9},
{{"TE", 2, 1, 2}, 1.82267*10^9},
{{"TE", 3, 1, 0}, 1.83773*10^9},
{{"TE", 3, 1, 1}, 1.93947*10^9},

{{"TE", 1, 1, 3}, 2.02665*10^9},
{{"TM", 1, 1, 2}, 2.08483*10^9},
{{"TE", 0, 1, 2}, 2.08483*10^9},
{{"TM", 0, 1, 3}, 2.13665*10^9},
{{"TE", 3, 1, 2}, 2.21685*10^9}   

Mode shapes bracketing NASA-reported frequencies:

frequencyBradyA = 1.9326*10^9 1/second;
{{"TE", 3, 1, 0}, 1.8377297671354957`*^9};
{{"TE", 3, 1, 1}, 1.9394709580599272`*^9};
frequencyBradyB = 1.9367*10^9 1/second;
{{"TE", 3, 1, 0}, 1.8377297671354957`*^9};
{{"TE", 3, 1, 1}, 1.9394709580599272`*^9};
frequencyBradyC = 1.8804*10^9 1/second;
{{"TE", 3, 1, 0}, 1.8377297671354957`*^9};
{{"TE", 3, 1, 1}, 1.9394709580599272`*^9};
frequencyBradyD = 2.168*10^9 1/second;
{{"TM", 0, 1, 3}, 2.136646973612425`*^9};
{{"TE", 3, 1, 2}, 2.216853242229605`*^9};

Frequencies of NASA-reported mode shapes:

{{"TE", 0, 1, 2}, 2.0848310776817226`*^9};
{{"TM", 2, 1, 1}, 2.330443792149791`*^9}


Fornaro geometry

{{"TE", 1, 1, 0}, 5.4291*10^8},
{{"TE", 1, 1, 1}, 7.06031*10^8},
{{"TM", 0, 1, 0}, 7.09111*10^8},
{{"TM", 0, 1, 1}, 8.40576*10^8},
{{"TE", 2, 1, 0}, 9.00604*10^8},
{{"TE", 2, 1, 1}, 1.00738*10^9},
{{"TE", 1, 1, 2}, 1.05341*10^9},
{{"TM", 1, 1, 0}, 1.12986*10^9},
{{"TE", 0, 1, 0}, 1.12986*10^9},
{{"TM", 0, 1, 2}, 1.14793*10^9},
{{"TM", 1, 1, 1}, 1.21668*10^9},
{{"TE", 0, 1, 1}, 1.21668*10^9},
{{"TE", 3, 1, 0}, 1.23881*10^9},
{{"TE", 2, 1, 2}, 1.27515*10^9},
{{"TE", 3, 1, 1}, 1.31847*10^9},
{{"TM", 1, 1, 2}, 1.4462*10^9},
{{"TE", 0, 1, 2}, 1.4462*10^9},
{{"TE", 1, 1, 3}, 1.45887*10^9},
{{"TM", 2, 1, 0}, 1.51434*10^9},
{{"TM", 0, 1, 3}, 1.52853*10^9},
{{"TE", 3, 1, 2}, 1.53283*10^9},
{{"TE", 4, 1, 0}, 1.56799*10^9},
{{"TE", 1, 2, 0}, 1.57208*10^9},
{{"TM", 2, 1, 1}, 1.58018*10^9},
{{"TE", 2, 1, 3}, 1.62624*10^9},
{{"TM", 0, 2, 0}, 1.62771*10^9},
{{"TE", 4, 1, 1}, 1.63166*10^9},
{{"TE", 1, 2, 1}, 1.6356*10^9},
{{"TM", 0, 2, 1}, 1.68913*10^9},
{{"TM", 2, 1, 2}, 1.76299*10^9},
{{"TM", 1, 1, 3}, 1.76356*10^9},
{{"TE", 0, 1, 3}, 1.76356*10^9},
{{"TE", 4, 1, 2}, 1.80928*10^9},
{{"TE", 1, 2, 2}, 1.81283*10^9},
{{"TE", 3, 1, 3}, 1.83526*10^9},
{{"TM", 0, 2, 2}, 1.86127*10^9},
{{"TM", 3, 1, 0}, 1.88132*10^9},
{{"TE", 1, 1, 4}, 1.88531*10^9},
{{"TE", 5, 1, 0}, 1.89177*10^9},
{{"TM", 3, 1, 1}, 1.93471*10^9},
{{"TM", 0, 1, 4}, 1.93972*10^9},
{{"TE", 5, 1, 1}, 1.94487*10^9},
{{"TE", 2, 2, 0}, 1.97744*10^9},
{{"TE", 2, 1, 4}, 2.01761*10^9},
{{"TE", 2, 2, 1}, 2.0283*10^9},
{{"TM", 2, 1, 3}, 2.03145*10^9},
{{"TM", 1, 2, 0}, 2.06869*10^9},
{{"TE", 0, 2, 0}, 2.06869*10^9},
{{"TE", 4, 1, 3}, 2.07175*10^9},
{{"TE", 1, 2, 3}, 2.07485*10^9},
{{"TM", 3, 1, 2}, 2.08669*10^9},
{{"TE", 5, 1, 2}, 2.09612*10^9},
{{"TM", 0, 2, 3}, 2.11731*10^9},
{{"TM", 1, 2, 1}, 2.11736*10^9},
{{"TE", 0, 2, 1}, 2.11736*10^9},
{{"TM", 1, 1, 4}, 2.12984*10^9},
{{"TE", 0, 1, 4}, 2.12984*10^9},
{{"TE", 2, 2, 2}, 2.17375*10^9},

Mode shapes bracketing NASA-reported frequencies:

frequencyBradyA = 1.9326*10^9 1/second;
{{"TE", 5, 1, 0}, 1.891774278367002`*^9};
{{"TM", 3, 1, 1}, 1.9347074025015635`*^9};
frequencyBradyB = 1.9367*10^9 1/second;
{{"TM", 3, 1, 1}, 1.9347074025015635`*^9};
{{"TM", 0, 1, 4}, 1.9397152624956703`*^9};
frequencyBradyC = 1.8804*10^9 1/second;
{{"TM", 0, 2, 2}, 1.8612746091297557`*^9};
{{"TM", 3, 1, 0}, 1.8813198077493927`*^9};
frequencyBradyD = 2.168*10^9 1/second;
{{"TE", 0, 1, 4}, 2.129842918761221`*^9};
{{"TE", 2, 2, 2}, 2.173747734247802`*^9};

Frequencies of NASA-reported mode shapes:

{{"TE", 0, 1, 2}, 1.4461980111408324`*^9}
{{"TM", 2, 1, 1}, 1.5801773963619902`*^9}





CONCLUSIONS

1)The "Fornaro Assumed Geometry" gives:

1a) frequencies of NASA-reported mode shapes:

{{"TE", 0, 1, 2}, 1.4461980111408324`*^9}
{{"TM", 2, 1, 1}, 1.5801773963619902`*^9}

that are much lower (25%) than the tested frequencies.  If one takes into account that the NASA's experiments were conducted with a dielectric, whose inclusion will lower the natural frequency, it is clear that NASA's truncated cone cavity must have had dimensions significantly smaller than assumed by Fornaro.

1b)  The mode shapes obtained under the Fornaro geometrical assumptions have the incorrect transverse field:  Transverse Magnetic modes for the experiments that resulted in measured thrust (Brady a, b and c) and Transverse Electric mode for the experiment that resulted in no measured thrust (Brady d).  A correct geometry should result in the opposite: TE modes for Brady a, b and c and TM mode for Brady d. It is clear then that NASA's truncated cone could not have Fornaro's assumed geometry.

1c) The Fornaro geometry has so many natural frequencies in the experimental range, so close to each other, that it would have been extremely difficult to tune the EM Drive to a particular frequency.  What we learn from this is that if one wants to drive EM Drives with these natural frequencies and mode shapes, EM Drives should be small enough (smaller than Fornaro's dimensions) so that natural frequencies are apart enough from each other that the EM Drive can be tuned if one wants to excite some of these frequenciesTo achieve the thrust required for crewed space missions would then require a large number of small EM Drives, rather than a large single EM Drive as pictured in some of Shawyer's concepts or in Aether Drives in steampunk.
One could aim to excite the lowest natural frequency: TE110 with a large EM Drive, which for Fornaro's geometry means 0.54291 GHz (less than 1/3 the frequencies tested at NASA by Brady et.al.), but I don't think that this mode TE110 (see picture below) is likely to produce significant thrust according to any theories that claim coupling with external fields (like the Quantum Vacuum, etc.)



2) The "Mulletron assumed geometry":

2a) achieves an outstanding fit by very closely predicting the frequency of 2.168 GHz for Brady's experiment "d", mentioned in the section on p .18, F.Tapered Cavity RF Evaluation, General Findings and Lessons Learned; of (http://www.libertariannews.org/wp-content/uploads/2014/07/AnomalousThrustProductionFromanRFTestDevice-BradyEtAl.pdf ). This is a very important fit, since it is the only experiment that which we are sure was conducted without a dielectric, and hence it should be best modeled by the exact solution:

frequencyBradyD =  2.168*10^9 1/second;
{{"TM", 2, 1, 1}, 2.167116717351836`*^9};

2b) it predicts that the mode shape should be TM211, which is a Transverse Magnetic mode. Brady et.al reported no thrust measurement at this frequency.  Our explanation is that this may have been due to the fact that Transverse Electric modes are required for thrust measurements (either as an artifact due to heating of the ends by heat induction or as a legitimate propulsion for example by coupling with the Quantum Vacuum).  Hence Transverse Magnetic modes should result in no thrust, which agrees with the experimental results.

Brady et.al. write:

Quote
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.

Brady et.al. report mode shape TE012 for this frequeny. The assumption that the mode shape was TE012 may be an error in the report, because their Table 2. Tapered Cavity Testing: Summary of Results, clearly shows the mode shape TE012 occurring at a lower frequency than TM211, so it doesn't make any sense for Brady et.al. to write that the mode shape TE012 can occur both below and above TM211.  We think that they may have intended to write "TM211 mode at 2168 MHz" which would be in agreement with Mulletron's geometry.

2c) it predicts  mode shape TE012 for the frequencies of NASA Brady et.al.'s experiments a and b:

frequencyBradyA = 1.9326*10^9 1/second;
{{"TE", 0, 1, 2}, 1.9043845840829124`*^9};
frequencyBradyB = 1.9367*10^9 1/second;
{{"TE", 0, 1, 2}, 1.9043845840829124`*^9};

Brady et.al. report mode shape TM211 for these frequencies, based on their numerical calculations using the Finite Element code COMSOL.  However the data on their Figure 16.  "Predicted and Actual S21 plots" shows that their  COMSOL numerical calculations are substantially in error (probably due to an unconverged, insufficiently fine finite-element mesh), particularly at this frequency range in question: 1.9 GHz.   It makes most sense that the actual mode shape was TE012, rather than a magnetic mode (TM211) for the previously discussed reasons.

2d) it has a problem for Brady et.al.'s experiment c, where it predicts mode shape TM013, with mode shape TE311 giving a lower frequency:

frequencyBradyC =  1.8804*10^9 1/second;
{{"TE", 3, 1, 1}, 1.8000272999857957`*^9};
{{"TM", 0, 1, 3}, 1.9003447870125527`*^9};

NASA's Brady et.al. reported mode shape TE012 for this frequency, which makes more sense because this is NASA's experiment that resulted in the highest thrust/PowerInput.  It is possible that the discrepancy is that the exact solution we are using does not include the effect of the dielectric.  Mulletron's geometry predicts a frequency of 1.90 GHz for mode shape TE012 without the dielectric:

{{"TE", 0, 1, 2}, 1.9043845840829124`*^9}

It is very possible that the dielectric lowers the frequency and that the actual mode shape for Brady et.al.'s experiment c at 1.8804 GHz was TE012 (or actually TE013: two half-wave patterns in the empty section of the cavity in the longitudinal direction, like TE012, and another extra half-wave in the dielectric).



3) The "Aero assumed geometry":

3a) The strength of Aero's assumed geometry is that it is the only assumed geometry that very consistently predicts mode shape TE (transverse electric) modes for the Brady et.al. experiments that measured thrust: (experiments a, b and c), while predicting mode shape TM (transverse magnetic) for the Brady experiment that measured no thrust: Brady et.al. experiment d:


Mode shapes bracketing NASA-reported frequencies:

frequencyBradyA = 1.9326*10^9 1/second;
{{"TE", 3, 1, 1}, 1.9394709580599272`*^9};
frequencyBradyB = 1.9367*10^9 1/second;
{{"TE", 3, 1, 1}, 1.9394709580599272`*^9};
frequencyBradyC = 1.8804*10^9 1/second;
{{"TE", 3, 1, 0}, 1.8377297671354957`*^9};
frequencyBradyD = 2.168*10^9 1/second;
{{"TM", 0, 1, 3}, 2.136646973612425`*^9};

3b) The presence of a dielectric inside the cavity may have resulted lowering the frequency predicted for mode shape TE012 from 2.08GHz to 1.88 GHz:

Frequencies of NASA-reported mode shapes:

{{"TE", 0, 1, 2}, 2.0848310776817226`*^9};
{{"TM", 2, 1, 1}, 2.330443792149791`*^9}



4)  It is clear that the actual geometry of NASA's Brady et.al. experiments was smaller than as predicted by Fornaro's assumptions.  Based on the evidence discussed above, the geometry was close to Mulletron's and Aero's assumed geometry, with Mulletron's geometry having an edge for the above-discussed reasons.



Images of modes TE01p,  TM21p, and TE11p  (where p can be any p=0,1,2,3,...):

electric field ________________    solid lines

magnetic field - - - - - - - - - - - - -  dashed lines
« Last Edit: 02/07/2015 03:15 PM by Rodal »

Offline aero

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Very interesting.
Here's another little wrinkle you can play with.
Define x = r/R where
r is the small diameter of the cone,
and R is the big diameter of the cone.
Then let Rcy equal the diameter of the cylindrical cavity. I get
f(x) = (x^2+x+1)
R = sqrt(Rcy^2 /f(x)/3)
Of course the formula can be inverted to solve for Rcy.

For my geometry, (aero geometry) I get x = 0.5826543346
For Mulletron geometry, x = 0.527443684
and for Fornaro geometry, x = 0.6146095718

This is not quite the formula that we need (check it before using), we need something that relates L / Rcy, I think, because I read somewhere that cylindrical cavities won't resonate when L/Rcy < ~ 2.02. It has to do with cut-off frequency as I recall.
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Offline Rodal

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Isn't it possible to increase radius while leaving length at or around 12 cm? I would prefer that but can't find a radius that works with 2.45 GHz.

Yes, it's possible. 

If you insist in specifying the exciting frequency as 2.45GHz and the length (12 cm) of the cylindrical cavity, and having the diameter as the variable to be adjusted, then

....
             = 8.332965999678832` centimeters

instead of  the diameter=7.54898 centimeter you used.

@Aero, is the reason why you insist in keeping a frequency of 2.45GHz in your modeling because you are looking at making a small EM Drive using a kitchen's microwave's magnetron as the source (which are nominally rated to operate at ~ 2.45 GHz)  ?

« Last Edit: 01/30/2015 03:01 PM by Rodal »

Offline aero

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Quote
@Aero, is the reason why you insist in keeping a frequency of 2.45GHz in your modeling because you are looking at making a small EM Drive using a kitchen's microwave's magnetron as the source (which are nominally ~ 2.45 GHz)  ?

I certainly won't, but look around. Magnetron technology advances but there are orders of magnitude more magnetrons made for the kitchen microwave than for anything else. That means research and technology advance will and does focus on the 2.45 GHz operating point. Even now there exists all digital 1000W, 2.45 GHz microwave power generators on the market. A little pricy at $39.95 each in lots of 10,000 but price/lot size will come down. The web link, which I don't have available at the moment, indicated that they are pretty slick little devices, digitally tunable, sized for microwave application and long lived.

We are researching EM thruster theory. Its not to early to think about the price point for the manufacture of these devices. Yes, after a few 100 or few 1000 are made and work, then research money will start to flow into the development of the microwave generators at whatever frequency. Until then, as the Chinese experiments illustrate, 2.45 GHz is the most convenient frequency to use for research.
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