EXACT SOLUTION OF TheTRAVELLER'S TEST GEOMETRYWe 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.
TheTraveller's frustum of a cone dimensions The following data from TheTraveller,
https://www.reddit.com/r/EmDrive/comments/40l1za/have_had_my_1st_build_design_data_confirmed/cz15iar, is used since it exactly matches the calculated Force/Power, and it features mode shape TE013 that user TheTraveller previously stated was recommended to him by Shawyer.
Geometry of the frustum of a cone resonant cavity:
BigDiameter: 0.259 m = 10.20 in
SmallDiameter: 0.159 m = 6.26 in
Length : 0.288 m = 11.34 in
Shape of end plates = Flat
____________________________________________________________
Predicted Eigenvalues of Resonance:
Predicted mode shape = TE013
Predicted natural frequency = 2.4053 GHz
(Rodal: both the eigenfrequency and eigenmode probably predicted using TheTraveller's spreadsheet modeling the conical frustum as the summation of a large number of cylindrical cavities with different diameters)
____________________________________________________________
Predicted quality of resonance and predicted force/power
Predicted Q (unloaded)= 86,200 (Rodal: not clear as to how TheTraveller calculated a Q, what approximate formula he used, but this predicted Q is definitely in the theoretical range for TE013, and these dimensions for copper)
Predicted Force/Power= 389 mN/kW (Rodal: probably predicted using Shawyer's formula, based on the above Q, the geometry, air as an internal medium and the predicted natural frequency and mode shape)
Therefore, this is the input geometry (in SI units:meters):
bigDiameter = 0.259;
smallDiameter = 0.159;
axialLength = 0.288;
from which one can calculate:
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 TheTraveller'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 = 0.461345 m
r2 = 0.751499 m
halfAngleConeDegrees = 9.84901 degrees
(notice that TheTraveller's cone is much closer to a cylinder than NASA's cone, which has a half cone angle of 14.8125 degrees. We suspect that the reason that TheTraveller chose this geometry, closer to a cylinder, was to meticulously obey Shawyer's prescription mandating for the small diameter base to exceed the dimensions for the cut-off frequency for a fictional open waveguide with the same diameter as the small base )
COMPARISON OF EXACT FREQUENCY with TheTraveller, for mode shape TE013measured frequency: TheTraveller has not yet provided a measured natural frequency for his test
calculated natural frequency
(exact solution, Dr. Rodal using Wolfram Mathematica): 2.38793 GHz
calculated natural frequency
(by TheTraveller, using spreadsheet approximating cone as summation of cylinders): 2.4053 GHz
difference: (2.4053 - 2.38793)/2.4053 = 0.69 %
The exact solution is very close to the frequency calculated by TheTraveller approximating the cone as the sum of a large number of cylinders of constant cross-section, because the half cone angle of his frustum, being less than 10 degrees is pretty close to the geometry of a cylinder.
This difference of 0.69% in frequency is negligible, considering the fact that the TheTraveller's cone is presently "held by gravity" and is bound to have larger geometrical defects, which will affect the experimental natural frequency.
For reference, we also give the natural frequencies of the first four TE01p modes for TheTraveller's geometry:
TE011 = 1.77026*10^9 Hz
TE012 = 2.07472*10^9 Hz
TE013 = 2.38793*10^9 Hz
TE014 = 2.75007*10^9 Hz
COMPARISON OF QUALITY OF RESONANCE Q with TheTraveller, for mode shape TE013TheTraveller does not specify what specific copper alloy he is using, so I will use, for reference, the same copper alloy used by NASA Eagleworks:
Material: 101 Copper Alloy properties (SI units):
mu = 0.999991*4*Pi*10^(-7);
resistivity = 1.71*10^(-8);
calculated quality of resonance
(exact solution, Dr. Rodal using Wolfram Mathematica): 86,552
calculated quality of resonance
(by TheTraveller, using spreadsheet approximating cone as summation of cylinders): 86,200
difference: (86,552 - 86,200)/86,200 = 0.41 %
(The difference in calculated Q is insignificant, given the fact that imperfections in geometry, surface and material will be responsible for much smaller values of experimental Q)
calculated skin depth (exact solution, Dr. Rodal using Wolfram Mathematica: 1.34682 micrometers
MAGNETIC VECTOR FIELD, for TheTraveller's experiment, mode shape TE013I attach an image of the Magnetic Vector Field on a plane parallel to the longitudinal axis for TheTraveller's experiment.
Notice that the optimal place to locate the exciting loop would have been along the longitudinal axis, closer to the small diameter base of the frustum, and if the loop was preferred to be located on the conical wall, it would have been better located near the small base, at about ~22.5% +/-2.5% of the distance measured along the lateral conical wall from the small end to the large end of the frustum. (The magnetic vector field image below shows the length of the magnetic vectors proportional to their intensity and one can therefore see that the intensity of the magnetic field is much larger near the small base than at the middle of the frustum.)
TheTraveller, instead reported that the <<Coaxial-fed current loop antenna approximately located at the middle of the conical side wall of the frustum>>
Today, TheTravellerReturns responds to questions about the exciting loop from IslandPlaya, as follows:
The 1/2 loop diameter, orientation and position is Secret Squirrel Secret Sauce stuff at this point in time. It will probably change when I get a stable 100W Rf output that will enable me to get some good tuning data at full power, where it is important and the money shot.
Initial loop diameter was derived from the equation I have posted but here it is again:
https://drive.google.com/file/d/0B7kgKijo-p0iQnUtV3JQRzhYNWc/view?usp=sharing
Can tell you that no part of the SMA connector is inside the frustum. The end of the SMA is in line with the side wall, very much like the linked image.
More like this image but not this coupler design:
https://drive.google.com/file/d/0B7kgKijo-p0idkFLYWQ5Y3l4dWc/view?usp=sharing
But that is enough Secret Squirrel Secret Sauce info for today.
______________
HEATING OF THE FRUSTUM WALLSObserve that heating of the copper frustum walls occurs by induction heating, as a result of the eddy-currents induced on the walls by the magnetic field paralell to the walls, hence one can readily see that the eddy-currents are strongest on a ring around the cone, near the small base of the frustum, and that the eddy-currents and hence the heating of the small end plate will look like an outer ring on the small base (but not touching the circumference of the small base). There is also a ring of eddy-current, and hence heating, on the large base of the frustum.
______________
POYNTING VECTOR FIELDFor reference we also attach an image for the Poynting vector field on a plane parallel to the longitudinal axis for TheTraveller's experiment.