Author Topic: EM Drive Developments - related to space flight applications - Thread 2  (Read 2103211 times)

Offline D_Dom

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Piping RF into the chamber is interesting, here is a link with various ports as an example.
http://www.designright.com/vacuum1.html

edited with a better source
« Last Edit: 02/16/2015 11:38 PM by D_Dom »
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Offline aero

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@Mulletron
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I think this definitively puts evanescent wave theories to bed.

It does if we credit you with knowing everything that is to be known about evanescent waves.

I recognize that you have researched the literature on this topic extensively and while I am willing to credit you with knowing everything that is known about evanescent waves, I'm not prepared to take that last step and credit that that is everything that is to be known.

We know that evanescent waves couple quite strongly with identically generated waves out of phase, if the sources are near enough to each other. There could be and probably is this type of coupling between the two ends of the cavity. That coupling should only stress the cavity material though and I don't know how it would produce thrust.

Dr. Rodel seems to have shot down my thought of evanescent wave photons escaping the cavity superluminally even though there have been several papers published claiming that evanescent waves, under the right conditions, do carry superluminal momentum. (Google it.)

So we're coming around to the thought that there may be a whole bunch of electromagnetic energy in the vacuum chamber in the form of waves, some of them of the correct wavelength and phase to couple with the evanescent waves escaping from the thruster cavity. If such waves are transient, then could the ... and so forth. Isn't this Dr. White's theory?
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Offline Star-Drive

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How hard would it be to put the 1165 inside a more vacuum resilient container?

http://www.empowerrf.com/pdfs/1165.pdf

Notice the paper says max altitude 50K feet!

It is operating beyond specs in that vacuum.....why they keep dying!

We gotta figure this out, pronto!
This appears to be a great finding, Mulletron, thanks for pointing it out.  If the Amp that Eagleworks has been using is not rated for the hard vacuum (5*10^(-4) Torr) in their tests, do you have a best suggestion on how they should proceed?
Nope, sealing things up like that is beyond my expertise. No clue how to help. Anyone?

So why does it have to be inside the chamber again? Can't rf be piped in somehow?

Folks:

The reason I thought that the EMPower unit could take vacuum is that the first page of the data sheet indicated that they are "hermetically" sealed.  At NASA hermetically sealed always means vacuum rated.  The Air Force and her contractors obviously have another definition of that word, but my bad for assuming it meant the NASA definition.  In any regards and as I stated earlier, EMPower has given us permission to just drill a hole into the top plate of their amp's chassis so it can vent to hard vacuum conditions when operating in same, since there are no components in their unit that would degrade over time in a hard vacuum such as electrolytic caps.

Best, Paul M.
Star-Drive

Offline Notsosureofit

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@ RODAL

Arrgh, Mondays !

Looked over my bleary weekend, noticed I was using diameters AGAIN !

Mode   Frequency (MHz)  Quality Factor, Q   Input Power (W)  Mean Thrust (μN)   Calculated w/o
                                                                                                                          dielectric
TE012     1880.4               22000                         2.6                55.4                   10.8
TM212   1932.6                 7320                        16.9                91.2                   38.5
TM212   1936.7               18100                        16.7                50.1                   93.5
TM212    1937.115             6726                       50                   66                    104.0

Anyway, shows it pays to rewrite everything in the same place !

@ aero

In the equivalence calc, the lower Doppler freq component, as seen in the accelerated frame, is evanescent. (that is its less than the cutoff in the rest frame)
« Last Edit: 02/17/2015 01:09 AM by Notsosureofit »

Offline Star-Drive

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@Mulletron
Quote
I think this definitively puts evanescent wave theories to bed.

It does if we credit you with knowing everything that is to be known about evanescent waves.

I recognize that you have researched the literature on this topic extensively and while I am willing to credit you with knowing everything that is known about evanescent waves, I'm not prepared to take that last step and credit that that is everything that is to be known.

We know that evanescent waves couple quite strongly with identically generated waves out of phase, if the sources are near enough to each other. There could be and probably is this type of coupling between the two ends of the cavity. That coupling should only stress the cavity material though and I don't know how it would produce thrust.

Dr. Rodel seems to have shot down my thought of evanescent wave photons escaping the cavity superluminally even though there have been several papers published claiming that evanescent waves, under the right conditions, do carry superluminal momentum. (Google it.)

So we're coming around to the thought that there may be a whole bunch of electromagnetic energy in the vacuum chamber in the form of waves, some of them of the correct wavelength and phase to couple with the evanescent waves escaping from the thruster cavity. If such waves are transient, then could the ... and so forth. Isn't this Dr. White's theory?

Aero:

What field strength are you estimating to be in between the frustum and the interior of the vacuum chamber in watts per square meter to get the forces you originally calculated for your evanescent wave solution when assuming the that there were Teflon seals in between the two end caps and the copper frustum body?  We've already measured that value at several locations in the chamber...

Best,  Paul M.
Star-Drive

Offline Mulletron

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@Mulletron
Quote
I think this definitively puts evanescent wave theories to bed.

It does if we credit you with knowing everything that is to be known about evanescent waves.

I recognize that you have researched the literature on this topic extensively and while I am willing to credit you with knowing everything that is known about evanescent waves, I'm not prepared to take that last step and credit that that is everything that is to be known.

We know that evanescent waves couple quite strongly with identically generated waves out of phase, if the sources are near enough to each other. There could be and probably is this type of coupling between the two ends of the cavity. That coupling should only stress the cavity material though and I don't know how it would produce thrust.

Dr. Rodel seems to have shot down my thought of evanescent wave photons escaping the cavity superluminally even though there have been several papers published claiming that evanescent waves, under the right conditions, do carry superluminal momentum. (Google it.)

So we're coming around to the thought that there may be a whole bunch of electromagnetic energy in the vacuum chamber in the form of waves, some of them of the correct wavelength and phase to couple with the evanescent waves escaping from the thruster cavity. If such waves are transient, then could the ... and so forth. Isn't this Dr. White's theory?

I'm not hostile to evanescent waves. I've spent months reading about them. I have lots to learn. The only analogue to have evanescent waves production like this is using light and thin metal films. Thin metals every time I look. For example, copper actually does have a refractive index if it is really thin. Thin films of metal behave like dielectrics. This would satisfy the surface plasmon polaritron idea. But the thickness of the frustum is many many times thicker than skin depth at 2ghz of 1.45um to 1.67um depending on what resource I use.
http://en.wikipedia.org/wiki/Surface_plasmon
http://en.wikipedia.org/wiki/Surface_plasmon_polariton
http://en.wikipedia.org/wiki/Surface_plasmon_resonance
http://refractiveindex.info/?shelf=main&book=Cu&page=Rakic
http://www.microwaves101.com/encyclopedias/skin-depth-calculator

I'm very sure it isn't evanescent waves. However that doesn't stop anybody from proving me wrong.
« Last Edit: 02/17/2015 12:08 AM by Mulletron »
Challenge your preconceptions, or they will challenge you. - Velik

Offline aero

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@Mulletron
Quote
I think this definitively puts evanescent wave theories to bed.

It does if we credit you with knowing everything that is to be known about evanescent waves.

I recognize that you have researched the literature on this topic extensively and while I am willing to credit you with knowing everything that is known about evanescent waves, I'm not prepared to take that last step and credit that that is everything that is to be known.

We know that evanescent waves couple quite strongly with identically generated waves out of phase, if the sources are near enough to each other. There could be and probably is this type of coupling between the two ends of the cavity. That coupling should only stress the cavity material though and I don't know how it would produce thrust.

Dr. Rodel seems to have shot down my thought of evanescent wave photons escaping the cavity superluminally even though there have been several papers published claiming that evanescent waves, under the right conditions, do carry superluminal momentum. (Google it.)

So we're coming around to the thought that there may be a whole bunch of electromagnetic energy in the vacuum chamber in the form of waves, some of them of the correct wavelength and phase to couple with the evanescent waves escaping from the thruster cavity. If such waves are transient, then could the ... and so forth. Isn't this Dr. White's theory?

I'm not hostile to evanescent waves. I've spent months reading about them. I have lots to learn. The only analogue to have evanescent waves production like this is using light and thin metal films. Thin metals every time I look. For example, copper actually does have a refractive index if it is really thin. Thin films of metal behave like dielectrics. This would satisfy the surface plasmon polaritron idea. But the thickness of the frustum is many many times thicker than skin depth at 2ghz of 1.45um to 1.67um depending on what resource I use.
http://en.wikipedia.org/wiki/Surface_plasmon
http://en.wikipedia.org/wiki/Surface_plasmon_polariton
http://en.wikipedia.org/wiki/Surface_plasmon_resonance
http://refractiveindex.info/?shelf=main&book=Cu&page=Rakic
http://www.microwaves101.com/encyclopedias/skin-depth-calculator

I'm very sure it isn't evanescent waves. However that doesn't stop anybody from proving me wrong.

And the problem with applying the thought of evanescent wave coupling to the quantum vacuum is that meep detects a force, but my meep simulation does not include anything to represent the quantum vacuum so that thought is ruled out. Resorting to the quantum vacuum to explain the force is not necessary.

I've brought this up before regarding the skin depth of copper. It seems to me that that data point (skin depth) is taken under rather benign conditions. The EM fields within the frustum are anything but benign, what with 50 Watts at a Q factor of over 6000. That is like 300 kW of radiant power bouncing around in the cavity. Perhaps under these conditions the electrons within the copper are excited at depth not normally measured by the techniques used measure skin depth. That is, maybe the skin depth is much larger at this high power?
« Last Edit: 02/17/2015 12:28 AM by aero »
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Offline Mulletron

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Evanescent waves and near field effects are virtual.
http://en.wikipedia.org/wiki/Near_and_far_field#Quantum_field_theory_view
http://en.wikipedia.org/wiki/Virtual_particle#Manifestations

Sorry to be pushing the oracle on you so much but I'm moving fast on my feet. There are other sources.

http://iopscience.iop.org/0295-5075/76/2/189;jsessionid=74CB8C68DB78D80A8FF4200696294DAC.c1

And you were probably right about evanescent waves being superluminal. There is ample evidence that virtual photons can be superluminal. I'm agnostic.
http://math.ucr.edu/home/baez/physics/Quantum/virtual_particles.html

We just met in the vacuum.

It is time to believe in the reality that this Q-thruster is as real as the QV itself. There is ample theory to support it dating back decades.

« Last Edit: 02/17/2015 12:44 AM by Mulletron »
Challenge your preconceptions, or they will challenge you. - Velik

Offline Rodal

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@ RODAL

Arrgh, Mondays !

Looked over my bleary weekend, noticed I was using diameters AGAIN !

Mode   Frequency (MHz)  Quality Factor, Q   Input Power (W)  Mean Thrust (μN)   Calculated w/o
                                                                                                                          dielectric
TE012     1880.4               22000                         2.6                55.4                   10.8
TM212   1932.6                 7320                        16.9                91.2                   38.5
TM212   1936.7               18100                        16.7                50.1                   93.5
TM212    1937.115             6726                       50                   66                    104.0

Anyway, shows it pays to rewrite everything in the same place !

....

Great !

In order to understand the above, (please correct me if I am wrong), you used in your formula the actual frequency and mode shapes that took place in the EM Drive experiment with the dielectric so in that sense you did calculate with the dielectric in a very restricted sense.

PS: Very unusual Monday in the Triangle (Raleigh/Durham/Chapel Hill) it is snowing over here  :)
« Last Edit: 02/17/2015 01:42 AM by Rodal »

Offline Notsosureofit

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@ RODAL

Correct !  Hmm, note the inverse rel between the 2nd and 3rd.  Typo ? (or just loading)

Edit:  sums are close 262.7 vs 246.8    ~6%

PS:  about 7 ft here so far (acc to the news)

« Last Edit: 02/17/2015 02:05 AM by Notsosureofit »

Offline Mulletron

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http://forum.nasaspaceflight.com/index.php?topic=36313.msg1330846#msg1330846
To drive the point home, here are some observations reported of PT symmetry breaking featuring non-reciprocity of light. Keep in mind that light also carries momentum.

http://www.nature.com/nphys/journal/v11/n1/abs/nphys3152.html
http://arxiv.org/abs/1308.4564
http://adsabs.harvard.edu/abs/2014NatPh..10..394P
http://www.nature.com/nphys/journal/v6/n3/abs/nphys1515.html
http://goo.gl/gYGJ40

Quote
One of seven types of hypothetical space drives suggested by Marc Millis of the Breakthrough Propulsion Physics Program at NASA's Glenn Research Center (see Millis drives).
The radiation pressure on one side of the induction sail would be increased by some yet undiscovered means, and the pressure on the other side lowered. The spacecraft would move toward the low-pressure region.
http://www.daviddarling.info/encyclopedia/I/induction_sail.html
http://goo.gl/Sb2rRg (Popsci source of image)
http://www.nasa.gov/centers/glenn/technology/warp/ideachev.html

Eagleworks has an induction sail sitting in that vacuum chamber.

Hopefully it will scale up and be useful in some space flight applications someday.
« Last Edit: 02/17/2015 05:46 AM by Mulletron »
Challenge your preconceptions, or they will challenge you. - Velik

Offline zen-in

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How hard would it be to put the 1165 inside a more vacuum resilient container?

http://www.empowerrf.com/pdfs/1165.pdf

Notice the paper says max altitude 50K feet!

It is operating beyond specs in that vacuum.....why they keep dying!

We gotta figure this out, pronto!
This appears to be a great finding, Mulletron, thanks for pointing it out.  If the Amp that Eagleworks has been using is not rated for the hard vacuum (5*10^(-4) Torr) in their tests, do you have a best suggestion on how they should proceed?
Nope, sealing things up like that is beyond my expertise. No clue how to help. Anyone?

So why does it have to be inside the chamber again? Can't rf be piped in somehow?

Folks:

The reason I thought that the EMPower unit could take vacuum is that the first page of the data sheet indicated that they are "hermetically" sealed.  At NASA hermetically sealed always means vacuum rated.  The Air Force and her contractors obviously have another definition of that word, but my bad for assuming it meant the NASA definition.  In any regards and as I stated earlier, EMPower has given us permission to just drill a hole into the top plate of their amp's chassis so it can vent to hard vacuum conditions when operating in same, since there are no components in their unit that would degrade over time in a hard vacuum such as electrolytic caps.

Best, Paul M.

If the 1165 amplifier is "NASA" hermetically sealed the reason for a 50,000 ft ceiling may be to limit distortion of the internal compartments of the unit, resulting in out of spec performance.   Drilling one hole may not do it because there is no guarantee every part of the amplifier will vent.   

Heat dissipation is a lot more difficult in a vacuum since just about all the heat has to escape by radiation.   Maybe pre-cooling the copper cavity will help.    A class C amplifier would be more efficient and would work just as well if a CW output was used.   But class C amplifiers are not linear amps.   The output goes from a low level to the maximum design power level with hardly any change in input level.

Getting back to the 1165 amp:   It is class AB so has an efficiency < 75% provided the load is 50 Ohms resistive.   It also can't handle an SWR > 3:1.    One way to protect the amplifier is to put a circulator between it and the em-drive.   The reflected wave from the em-drive gets dissipated as heat in the circulator instead of the amplifier.   Or worse the resulting high RF voltages at the output of the amplifier cause arcing.   

Since the em-drive has such a high Q it's next to impossible to drive it with the right frequency.   The frequency will always be off so the complex impedance at the input of the em-drive will almost never be 50 Ohms resistive.   This results in most of the power being reflected back to the amplifier, damaging it.    , Instead of using a signal generator, if  the cavity was the frequency determining element then locking to the desired frequency might be easier.   Cavity oscillators have been around for a long time.   High power cavity oscillators use tubes.  (Eimac) 
« Last Edit: 02/17/2015 06:05 AM by zen-in »

Offline Star One

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Aside from re running the reverse orientation tests. Are there any other tests that Eagleworks needs to run. I am aware that they need to get to a certain performance level before then can hand off for replication attempt. But for the life of me the only other test case I can think of wanting results for is

* Re Run frustum reverse orientation in Hard vacuum
* Run forward and reverse orientation of frustum in a null configuration

If it is handed off for a replication attempt, is this to be done in more than one other location, in other words are multiple teams to attempt this or just one?

My understanding is given what was said in the conclusion of the Brady et al paper is that they want to create a testable unit to be used by Glen Research Center and JPL and Johns Hopkins. However, I believe of the two NASA centers only Glen has signed on. No clue on whether or not Johns Hopkins has also signed on. Which is the reason they need to get the thrust levels up because the Balance at Glen has a much higher floor of detectable thrust.

Thanks. Hopefully they all sign on for the best verification.

Offline Notsosureofit

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@ RODAL

Arrgh, Mondays !

Looked over my bleary weekend, noticed I was using diameters AGAIN !

Mode   Frequency (MHz)  Quality Factor, Q   Input Power (W)  Mean Thrust (μN)   Calculated w/o
                                                                                                                          dielectric
TE012     1880.4               22000                         2.6                55.4                   10.8
TM212   1932.6                 7320                        16.9                91.2                   38.5
TM212   1936.7               18100                        16.7                50.1                   93.5
TM212    1937.115             6726                       50                   66                    104.0

Anyway, shows it pays to rewrite everything in the same place !

....

Great !

In order to understand the above, (please correct me if I am wrong), you used in your formula the actual frequency and mode shapes that took place in the EM Drive experiment with the dielectric so in that sense you did calculate with the dielectric in a very restricted sense.


FYI

Cleanup and de-typo of the take on applying the Equivalence Principle.


The proposition that dispersion caused by an accelerating frame of reference implied an accelerating frame of reference caused by a dispersive cavity resonator. (to 1st order using massless, perfectly conducting cavity, no dielectric)


Starting with the expressions for the frequency of a cylindrical RF cavity:

f = (c/(2*Pi))*((X/R)^2+((p*Pi)/L)^2)^.5

For TM modes, X = X[sub m,n] = the n-th zero of the m-th Bessel function.
[1,1]=3.83, [0,1]=2.40, [0,2]=5.52 [1,2]=7.02, [2,1]=5.14, [2,2]=8.42, [1,3]=10.17, etc.

and for TE modes, X = X'[sub m,n] = the n-th zero of the derivative of the m-th Bessel function.
[0,1]=3.83, [1,1]=1.84, [2,1]=3.05, [0,2]=7.02, [1,2]=5.33, [1,3]=8.54, [0,3]=10.17, [2,2]=6.71, etc.

Rotate the dispersion relation of the cavity into Doppler frame to get the Doppler shifts, that is to say, look at the dispersion curve intersections of constant wave number instead of constant frequency.

df = (1/(2*f))*(c/(2*Pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

and from there the expression for the acceleration g from:

g = (c^2/L)*(df/f) such that:

g = (c^2/(2*L*f^2))*(c/(2*Pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

Using the "weight" of the photon in the accelerated frame from:

"W" = (h*f/c^2)*g =>  "W" = T = (h/L)*df

gives thrust per photon:

T = (h/(2*L*f))*(c/(2*pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

If the number of photons is (P/hf)*(Q/2*pi*f) then:

NT = P*Q*(1/(4*pi*L*f^3))*(c/(2*pi))^2*X^2*((1/Rs^2)-(1/Rb^2))


This does fit (as far as I've gotten) the concept of a self-accelerating Dirac wavepacket (which does conserve momentum).

Slow goin', thanks for your patience.



« Last Edit: 02/19/2015 11:58 PM by Notsosureofit »

Offline Rodal

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How hard would it be to put the 1165 inside a more vacuum resilient container?

http://www.empowerrf.com/pdfs/1165.pdf

Notice the paper says max altitude 50K feet!

It is operating beyond specs in that vacuum.....why they keep dying!

We gotta figure this out, pronto!
This appears to be a great finding, Mulletron, thanks for pointing it out.  If the Amp that Eagleworks has been using is not rated for the hard vacuum (5*10^(-4) Torr) in their tests, do you have a best suggestion on how they should proceed?
Nope, sealing things up like that is beyond my expertise. No clue how to help. Anyone?

So why does it have to be inside the chamber again? Can't rf be piped in somehow?

Folks:

The reason I thought that the EMPower unit could take vacuum is that the first page of the data sheet indicated that they are "hermetically" sealed.  At NASA hermetically sealed always means vacuum rated.  The Air Force and her contractors obviously have another definition of that word, but my bad for assuming it meant the NASA definition.  In any regards and as I stated earlier, EMPower has given us permission to just drill a hole into the top plate of their amp's chassis so it can vent to hard vacuum conditions when operating in same, since there are no components in their unit that would degrade over time in a hard vacuum such as electrolytic caps.

Best, Paul M.

If the 1165 amplifier is "NASA" hermetically sealed the reason for a 50,000 ft ceiling may be to limit distortion of the internal compartments of the unit, resulting in out of spec performance.   Drilling one hole may not do it because there is no guarantee every part of the amplifier will vent.   

Heat dissipation is a lot more difficult in a vacuum since just about all the heat has to escape by radiation.   Maybe pre-cooling the copper cavity will help.    A class C amplifier would be more efficient and would work just as well if a CW output was used.   But class C amplifiers are not linear amps.   The output goes from a low level to the maximum design power level with hardly any change in input level.

Getting back to the 1165 amp:   It is class AB so has an efficiency < 75% provided the load is 50 Ohms resistive.   It also can't handle an SWR > 3:1.    One way to protect the amplifier is to put a circulator between it and the em-drive.   The reflected wave from the em-drive gets dissipated as heat in the circulator instead of the amplifier.   Or worse the resulting high RF voltages at the output of the amplifier cause arcing.   

Since the em-drive has such a high Q it's next to impossible to drive it with the right frequency.   The frequency will always be off so the complex impedance at the input of the em-drive will almost never be 50 Ohms resistive.   This results in most of the power being reflected back to the amplifier, damaging it.    , Instead of using a signal generator, if  the cavity was the frequency determining element then locking to the desired frequency might be easier.   Cavity oscillators have been around for a long time.   High power cavity oscillators use tubes.  (Eimac)
It will help to improve radiation of heat by adding low emissivity heat sinks to the locations that get most heated.   Lowering the emissivity of what is usually shiny aluminum,  will improve radiation and decrease the temperature. A thin coat of non reflective, flat paint on the inside and outside surfaces, and a thin coat on the hot parts will allow up to six times the amount of power to radiate for the same component temperature rise.  Anodizing works almost as well, provided the anodized surface is about .001 thick.

Material                      Emissivity (the higher the emissivity the better radiation of heat)
Silver, polished               0.01
Gold                               0.02
Aluminum, brush         0.15
Black Lacquer .24mil       0.48
Polished Steel                  0.53
Anodized Aluminum           0.81
White Flat Paint 1mil    0.94

Warning: emissivity tables have conflicting numbers.  A very extensive one is here:  http://www.engineeringtoolbox.com/emissivity-coefficients-d_447.html  while the one in the Emissivity article in Wikipedia is very restricted.  I would advise to consult several published values and place particular emphasis on published emissivity values in articles for heat radiation in electronics : experimental values used for heat radiation in electronics should rule !  Thickness (of the paint coat or thickness of anodizing layer) is critical for effective heat removal (you don't want to have a thick layer of coating which would trap the heat).
« Last Edit: 02/17/2015 02:10 PM by Rodal »

Offline Rodal

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@ RODAL

Arrgh, Mondays !

Looked over my bleary weekend, noticed I was using diameters AGAIN !

Mode   Frequency (MHz)  Quality Factor, Q   Input Power (W)  Mean Thrust (μN)   Calculated w/o
                                                                                                                          dielectric
TE012     1880.4               22000                         2.6                55.4                   10.8
TM212   1932.6                 7320                        16.9                91.2                   38.5
TM212   1936.7               18100                        16.7                50.1                   93.5
TM212    1937.115             6726                       50                   66                    104.0

Anyway, shows it pays to rewrite everything in the same place !

....

Great !

In order to understand the above, (please correct me if I am wrong), you used in your formula the actual frequency and mode shapes that took place in the EM Drive experiment with the dielectric so in that sense you did calculate with the dielectric in a very restricted sense.


FYI

Cleanup and de-typo of the take on applying the Equivalence Principle.


The proposition that dispersion caused by an accelerating frame of reference implied an accelerating frame of reference caused by a dispersive cavity resonator. (to 1st order using massless, perfectly conducting cavity, no dielectric)


Starting with the expressions for the frequency of a cylindrical RF cavity:

f = (c/(2*Pi))*((X/R)^2+((p*Pi)/L)^2)^.5

For TM modes, X = X[sub m,n] = the n-th zero of the m-th Bessel function.
[1,1]=3.83, [0,1]=2.40, [0,2]=5.52 [1,2]=7.02, [2,1]=5.14, [2,2]=8.42, [1,3]=10.17, etc.

and for TE modes, X = X'[sub m,n] = the n-th zero of the derivative of the m-th Bessel function.
[0,1]=3.83, [1,1]=1.84, [2,1]=3.05, [0,2]=7.02, [1,2]=5.33, [1,3]=8.54, [0,3]=10.17, [2,2]=6.71, etc.

Rotate the dispersion relation of the cavity into Doppler frame to get the Doppler shifts, that is to say, look at the dispersion curve intersections of constant wave number instead of constant frequency.

df = (1/(2*f))*(c/(2*Pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

and from there the expression for the acceleration g from:

g = (c^2/L)*(df/f) such that:

g = (c^2/(2*L*f^2))*(c/(2*Pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

Using the "weight" of the photon in the accelerated frame from:

"W" = (h*f/c^2)*g =>  "W" = T = (h/L)*df

gives thrust per photon:

T = (h/(2*L*f))*(c/(2*pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

If the number of photons is (P/hf)*(Q/2*pi) then:

NT = P*Q*(1/(4*pi*L*f^3))*(c/(2*pi))^2*X^2*((1/Rs^2)-(1/Rb^2))


This does fit (as far as I've gotten) the concept of a self-accelerating Dirac wavepacket (which does conserve momentum).

Slow goin', thanks for your patience.

Excellent!  Thank you for posting the complete equations.

One suggestion:  In the expression NT = P*Q*(1/(4*pi*L*f^3))*(c/(2*pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

the speed of light in vacuum "c" appears in the numerator without being divided by the SquareRoot of the relative electric permittivity and relative magnetic permeability.

Since the relative electric permittivity of the dielectric is 2.3, this would decrease the values in the table by a factor of Sqrt[2.3]=1.52 if the whole cavity would be occupied by the dielectric.  Granted that only a portion of the truncated cone contains the dielectric, which will decrease the dividing factor, but any amount will reduce the effective value of c in the medium, giving lower thrust and hence values closer to the experimental measurements. 

For example, very roughly, assuming that 1/3 of the longitudinal length is occupied by the dielectric, and using the average as a medium with those average properties, Sqrt[(2.3*1/3)+1*(2/3)]=1.20, the thrust values would be reduced by a factor of 1.20, so for the most important test (the one in recently performed in vacuum, -the other experimental values may have been affected by thermal convection effects in the air and are therefore less reliable-), instead of 104 μN you would get 87 μN, which better compares with the experimental value of 66 μN.  
« Last Edit: 02/17/2015 12:13 PM by Rodal »

Offline Notsosureofit

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Correct !  I just don't want to get into trying to get the exact dispersion relation.  Got enough on the plate looking for a mechanism.

Offline Rodal

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Mulletron was kind enough to send me personally this very interesting post that I would like to share:

http://physics.stackexchange.com/questions/130662/emdrive-cavity-modes

Quickly, so as not to abuse your attention:

1)The author of the Physics Stack Exchange post finds out that it is difficult to satisfy the Boundary Conditions and reaches the (false) conclusion:

Quote
When I apply boundary conditions, Er must be zero at the side walls so P0n(cos(θw)) = 0 can only happen for specific angles. If the wall is not at the right angle, Er must be zero every where. Same is true for Eϕ, but that has zeros for P1n(cos(θw)) so it would be a different mode.
...
In other words, if the EmDrive guys don't build the cavity to specific angles, it will simply reflect all power and won't have any RF in it at all!

This conclusion is incorrect.   Greg Egan found a clever way to satisfy all the Boundary Conditions for any arbitrary angle, by adding a parameter, and solving two eigenvalue problems (one eigenvalue problem for the Legendre Functions and another eigenvalue problem for the spherical Bessel functions and an arbitrary parameter).  Greg Egan's conclusion is correct and there are other papers in the literature as well.  Actually the basis of the exact solution goes back to the great US engineer Schelnukoff in 1938 .   Unfortunately nobody at http://physics.stackexchange.com/ set the author of the post straight regarding the Boundary Condition problem.

2) The fact that the truncated cone does resonate is also confirmed by the COMSOL Finite Element studies performed for NASA.



3) I'm making good progress with the exact solution.

4) One benefit of the exact solution is that I am able to plot the modes as a contour line to show the wave and the number of zeros, to ascertain the exact meaning of the mode shape quantum numbers m, n and p.  I have found out that while m and p can be characterized as we previously did, the meaning of n depends on whether the mode is TM or TE and whether n is even or odd.  No wonder then that Mulletron and NotSoSureOfIt found confusion in the literature concerning the meaning of the mode shape numbers.  I may have to correct the article on Wikipedia once again.
« Last Edit: 02/17/2015 01:22 PM by Rodal »

Offline Rodal

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....
This does fit (as far as I've gotten) the concept of a self-accelerating Dirac wavepacket (which does conserve momentum).
...
Please tell us more when you have a chance about the <<self-accelerating Dirac wavepacket (which does conserve momentum)>> as conservation of momentum has been the biggest problem of the scientific media (Prof. Baez and Sean Carroll for example) and with the serious science-fiction media (Greg Egan) with the EMDrive.  How does momentum get conserved in the EMDrive when there is nothing coming out of the EM Drive?

http://physics.stackexchange.com/questions/160702/self-accelerating-wavepackets-what-are-they-and-can-they-impulse-a-spaceship

http://www.nature.com/nphys/journal/vaop/ncurrent/fig_tab/nphys3196_F1.html

http://newsoffice.mit.edu/2015/self-accelerating-particles-0120

Or are you considering that evanescent waves coming out of the EM Drive in outer space (with no fields or matter nearby to interact with) are responsible for conservation of momentum with an effectiveness much greater than a photon rocket?
« Last Edit: 02/17/2015 01:37 PM by Rodal »

Offline Notsosureofit

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@ RODAL

Thanks for the refs.  I was not aware of self-acceleration when we started.  (but I spent a year as a NASA Fellow w/ S. Schwebel in the 60's on lift in a grav. grad., later did thesis on acoustic cavity resonator solutions for AF model atmosphere)

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