Cut-off of Resonant Modes in Truncated Conical Cavities

Transcription

1 Cut-off of Resonant Modes in Truncated Conical Cavities José J. Rodal, Ph.D. June 2015

2 Although the fact that a truncated conical cavity displays an absence of sharp cut-off frequencies has been remarked before: Due to the absence of sharp cut-off frequencies, the interior of the frustum can support nontrivial field amplitudes, even in regions of relatively small electrical cross section, Davis, A.M.J., and Scharstein, R.W.; Electromagnetic plane wave excitation of an open-ended conducting frustum, Antennas and Propagation, IEEE Transactions (Volume:42, Issue: 5), May 1994 this fact appears to be still relatively unknown, particularly to those conducting experiments on the so-called EM Drive, a truncated conical cavity excited at microwave frequencies; as all researchers up to now have designed their cavities such that the small-end diameter is above cut-off based on cylindrical formulas. I hereby show that as remarked by Davis and Scharstein, the truncated cone does not present sharp cut-off frequencies that can be calculated based on cylindrical formulas. On the contrary, continuing the cone beyond the small diameter at which cut-off would occur (according to the cylindrical formula which is inapplicable to the cone) leads to significantly higher amplitude of the electromagnetic field. It turns out that continuing the cone up to distances much closer to the apex of the cone also results in lower phase shift and higher geometrical attenuation of the electromagnetic field in the longitudinal direction. Thus, continuing the cone beyond the cylindrical cut-off frequency may result in very interesting behavior.

3 Let s take a look at the difference between a cylindrical cavity and a truncated cone. The intrinsic coordinate system for a cylinder is a cylindrical (polar coordinate) coordinate system. In a cylinder with flat plane ends one may have plane waves travel smoothly from end to end, and it is straightforward to satisfy the boundary conditions. It turns out that the intrinsic system for a truncated cone is a spherical coordinate system because spherical waves can travel such a cone smoothly if the ends are spherical sections. The truncated cone boundary conditions can be readily satisfied in such a coordinate system. Let s start by looking at the Hemholtz equation (*) expressed in spherical coordinates: r = spherical radial coordinate, along the length of the cone, from the vertex to the big base of the truncated cone, θ = spherical polar angle: the angle between radial lines and the longitudinal axis of symmetry of the cone, φ = azimuthal angle, along the circumference of a circumferential cross-section of the truncated cone. This geometry becomes clear by looking at the geometry of the truncated cone in the next image, showing a flat cross section of the truncated cone in the shape of a trapezium, with spherical coordinates r and θ: (*) where k is the wavenumber k = ω /c = 2 π f /c = 2 π / λ and where f=frequency ψ stands for the time-independent space-variable part of either the electric E field or of the magnetic B fields

4 Mode shapes TM mnp TE mnp Solution function index Spherical Bessel p Asc. Legendre n Sine m Equations above from: Weile, Daniel S., Spherical Waves

5 Very different functions govern the behavior of the solution for truncated cones than for cylinders!!!! Transverse direction n : functions describing field variation: Cylinder Cylindrical Bessel Truncated Cone Associated Legendre Longitudinal direction p : functions describing field variation: Cylinder Sine, Cosine Truncated Cone Spherical Bessel Function

6 Longitudinal direction p : functions describing field variation: Cylinder Sine, Cosine Truncated Cone Spherical Bessel Function j 1 = (Sin[k r]/(k r) Cos[k r] )/(k r) j 2 = (3/(k r) 2-1 )Sin[k r]/(k r) 3 Cos[k r] )/(k r) 2 Spherical Bessel functions j and their derivatives can show very different behavior from circular (sine, cosine) functions: notice their dependence on the longitudinal distance k r (from the cone vertex) in the denominator. The smaller k r (the closer to the vertex) the bigger the difference with the circular functions. We show above, for illustration purposes, the behavior for integer values of the subscript, however, for a truncated cone, the subscript is, in general a noninteger, and the Bessel functions are related to generalized hypergeometric series. Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 9", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover.

7 The cut-off frequency of an electromagnetic cavity is the lowest frequency for which a given mode will resonate in it. It is incorrect to use a cut-off condition for cylinder resonance as a cut-off condition for a truncated cone resonance because the cut-off condition for cylinder resonance does not take into account the fact that the electromagnetic wave field inside a truncated cone is given by spherical Bessel functions instead of harmonic (sine or cosine) functions that govern resonance for cylinders. The spherical Bessel functions can display very interesting behavior not present in circular (sine and cosine) functions: for example they can display variable wavelength along the truncated cone and display geometric attenuation in the longitudinal direction (both features not present in harmonic functions). Thus, mode shapes that would be suddenly cut-off in cylinders are not cut-off in truncated cones (unless the small base of the truncated cone is very close to the apex of the cone). The cutoff frequency of an electromagnetic waveguide is the lowest frequency for which a mode will propagate in it. It is the same frequency as the cut-off frequency for an electromagnetic cavity with identical dimensions, with longitudinal mode shape number p=0, corresponding to a constant field in the longitudinal direction. Truncated cones cannot have constant electromagnetic fields in the longitudinal direction, thus the use of cylindrical waveguide formulas for truncated cone cavities is nonsense. Here, I show that cut-off of mode shapes in truncated cones does not occur at the geometries predicted by cylindrical formulas, on the contrary, the amplitude of the resonant mode increases with truncated conical shapes that would have been cut-off according to cylindrical formulas. Furthermore the cut-off does not occur suddenly but is gradual and it only occurs at distances close to the cone s vertex.

8 We will vary this radius r1 (distance from apex of cone to small base) keeping constant cone angle. As we shrink r1, the small diameter decreases and the cavity length increases.

9 Example geometry: Baby EM Drive, Paul Kocyla ) and Jo Hinchliffe, Aachen, Germany Description Data We will vary this Cavity Length (m) Big diameter (m) Small diameter (m) Spherical radius r1 (m) Spherical radius r2 (m) Cone 1/2 angle (deg) Shawyer Design Factor Frequency (GHz) 24.1 Mode shape TE013 Dielectric None We will vary this radius r1 (distance from apex of cone to small base) keeping constant cone angle. As we shrink r1, the small diameter decreases and the cavity length increases.

10 This spreadsheet shows the different geometries studied in this report, keeping constant cone angle and diameter of the big base, while varying the diameter of the small base (hence, reducing the radial distance between the small base and the apex of the cone). I start with a truncated cone with the small base above cut-off according to the cylindrical waveguide formula, and reduce the diameter of the small base by successive increments, to 90% of the original diameter (at which point it is already below cylindrical waveguide cutoff) until it is only 5% of the original diameter. As the small base diameter is reduced, the length of the truncated cone (between bases) increases. Observe that the natural frequency decreases a little (due to the longer length of the cone). Perhaps surprisingly, the amplitude of mode TE013 actually increases by a factor greater than 2.5 from its original amplitude. While mode shape TE013 has the smallest amplitude compared to TE011 and TE012 at the initial dimensions, as we reduce the small base it becomes the mode with the highest amplitude. cone 1/2 small big TE013 Cyl.Wvg. Cut-Off angle diameter diameter length frequency TE01 TE013 TE012 TE011 r1(mm) r1/r1original r2(mm) (deg) (mm) Ds/DsOriginal (mm) (mm) L/LOriginal (GHz) (GHz) ampltd. ampltd. ampltd % % % % " " % " % % " " % " % % " " % " % % " " % " % % " " % " % " " % " " % " % " " cut-off % " " % " % " " cut-off % " " % " % " " cut-off % " " % " % cut-off cut-off cut-off % " " % " % cut-off cut-off cut-off % " " % " % cut-off cut-off cut-off

11 r1/r1 original = Ds/Ds original = 1 Each slide shows the ratio of the distance to the cone apex to the original distance, which is the same as the ratio of the diameter of the small base to the original diameter. The graphs show the variation of the electromagnetic field in the longitudinal direction, from r 1 to r 2 as if the truncated cone were turned in the horizontal direction The graphs also show the ratio of the longest halfwavelength of the electromagnetic field in the longitudinal direction for mode TE013, to the length it would have if governed by circular functions. The graphs also show the highest amplitude of mode TE013 l /((r 2 r 1 )/3) =

12 r1/r1 original = Ds/Ds original = 1 TE012 and TE013 look more like sine curves while TE011 already looks different we see TE011 being attenuated near the small end l /((r 2 r 1 )/3) = Amplitude: l TE013 < TE012 < TE011

13 r1/r1 original = Ds/Ds original = 1

14 r1/r1 original = Ds/Ds original = 98.79% l /((r 2 r 1 )/3) = l Amplitude: TE013 < TE012 < TE011

15 r1/r1 original = Ds/Ds original = 98.79%

16 r1/r1 original = Ds/Ds original = 96.38% l /((r 2 r 1 )/3) = l Amplitude: TE013 < TE012 < TE011

17 r1/r1 original = Ds/Ds original = 96.38%

18 r1/r1 original = Ds/Ds original = 90% l /((r 2 r 1 )/3) = l Amplitude: TE013 < TE011 < TE012

19 r1/r1 original = Ds/Ds original = 90%

20 r1/r1 original = Ds/Ds original = 50% All three mode shapes TE011, TE012, TE013 are now clearly attenuated near the small end. A sine does not describe well the shape. The wavelength gets longer as it approaches the apex. l /((r 2 r 1 )/3) = l Amplitude: TE011 < TE012< TE013 amplitudes switched: TE013 is now highest

21 r1/r1 original = Ds/Ds original = 50% Standing wave attenuates to zero here

22 r1/r1 original = Ds/Ds original = 35% l /((r 2 r 1 )/3) = l

23 r1/r1 original = Ds/Ds original = 35% Standing wave attenuates to zero here

24 The next image shows the electromagnetic field variation for the small base reduced to only ¼ of its original dimension. Although one can still plot the solution for mode TE011, Mathematica warns that the solution does not converge at r 1, indicating possible cut-off of mode TE011. For this geometry, for the small base reduced to only ¼ of its original dimension, the cut-off frequency (91 GHz) based on a cylindrical waveguide incorrectly predicts that mode shape TE011 should have been cut-off at a much larger base diameter (at 0.90 the original dimension instead of 0.25 the original dimension).

25 r1/r1 original = Ds/Ds original = 25% Mathematica indicates that the solution for mode TE011 has not converged at the small end, indicating cut-off of mode TE011 l /((r 2 r 1 )/3) = l

26 r1/r1 original = Ds/Ds original = 25% Standing wave attenuates to zero here

27 r1/r1 original = Ds/Ds original = 22% Mathematica indicates that the solution for mode TE011 has not converged at the small end, indicating cut-off of mode TE011 l /((r 2 r 1 )/3) = l

28 r1/r1 original = Ds/Ds original = 22% Standing wave attenuates to zero here

29 r1/r1 original = Ds/Ds original = 21% Mathematica indicates that the solution for mode TE011 has not converged at the small end, indicating cut-off of mode TE011 l /((r 2 r 1 )/3) = l

30 r1/r1 original = Ds/Ds original = 21% Standing wave attenuates to zero here

31 The next image shows the electromagnetic field variation for the small base reduced to only 1 / 5 of its original dimension. Although one can still plot the solution for mode TE011, TE012 and TE013, Mathematica warns that the solution at r 1 does not converge for any of these modes, indicating possible cut-off of all these modes TE011, TE012 and TE013. For this geometry, for the small base reduced to only 1 / 5 of its original dimension, the cutoff frequency (113 GHz) based on a cylindrical waveguide incorrectly predicts that mode shapes TE012 and TE013 should have been cut-off at a much larger base diameter (at 0.90 the original dimension instead of 0.20 the original dimension).

32 r1/r1 original = Ds/Ds original = 20% Mathematica indicates that the solution has not converged at the small end, indicating cut-off of TE011, TE012, TE013 modes

33 r1/r1 original = Ds/Ds original = 20% Mathematica indicates that the solution has not converged at the small end, indicating cut-off of TE013 Standing wave attenuates to zero here

34 r1/r1 original = Ds/Ds original = 15% Mathematica indicates that the solution has not converged at the small end, indicating cut-off of TE011, TE012, TE013 modes

35 r1/r1 original = Ds/Ds original = 15% Mathematica indicates that the solution has not converged at the small end, indicating cut-off of TE013 Standing wave attenuates to zero here

36 The next image shows the electromagnetic field variation for the small base reduced to only 1 / 20 of its original dimension. One can no longer plot the solution for mode TE011, TE012 and TE013, clearly all these modes are definitely cut-off at this small distance from the vertex of the cone. Notice that for this geometry, for the small base reduced to only 1 / 20 of its original dimension, the cut-off frequency based on a cylindrical waveguide is much larger: 454 GHz.

37 r1/r1 original = Ds/Ds original = 5% Singularity at small end: clearly cannot satisfy boundary condition at small end: cut-off of TE011, TE012, TE013 modes

38 r1/r1 original = Ds/Ds original = 5% Standing wave zero everywhere

39 Xiahui Zeng and Dianyuan Fan have recently published a paper on electromagnetic fields and transmission properties in truncated conical waveguides. They present data for the geometrical attenuation α and the phase constant β (divided by the wavenumber k, to render them dimensionless) in the vertical axis, vs. the dimensionless distance (k r) to the apex of the cone for various mode shapes. For illustration purposes, we choose (from their figures) the waveguide mode TE01 (that corresponds to the mode shapes shown in the previous figures), and identify the curve corresponding to the same cone half-angle (π/12 radians) as the truncated cone geometry previously explored in the previous figures. We also identify with a circle the original distance from the small base to the vertex of the cone, as well as a circle to identify the distance from the small base to the vertex of the cone with the small base reduced to ½ the original size. We observe, when comparing these two geometries, that when the small base is reduced to ½ the original size: 1) the (dimensionless) geometrical attenuation is increased by a factor of 28 times from 0.1 to ) the phase constant is increased by a factor of 2 from 0.5 to 1. Hence it looks like very large changes can be accomplished by simply reducing a truncated cone s small base so that it is much closer to the cone s apex, and this can be done without incurring cut-off, and achieving a higher amplitude to boot.

40 r1(mm) r1/r1original small diameter (mm) Ds/DsOriginal k r TE013 ampltd. length (mm) L/LOriginal % % % % % % % % % % % % % % % % % % % % " % % % " % % % " % % % cut-off % % % cut-off % % % cut-off % Baby EM Drive θ = π / Proposed 1/2 smaller base Baby EM Drive as tested

41 Conclusions 1. It is nonsense to use a cylindrical waveguide cut-off formula to predict cut-off of mode shapes in a truncated cone. Truncated cones cannot have constant electromagnetic fields in the longitudinal direction, thus the use of cylindrical waveguide formulas (which assume constant cross-sections) for truncated cone cavities is nonsense. 2. Truncated cones show an absence of sharp cut-off frequencies. Cut-off occurs at geometries that are close to a pointy cone, at small base dimensions that are much smaller than what is predicted by cylindrical waveguide cut-off formulas. 3. On the contrary, continuing the cone beyond the small diameter at which cut-off would occur (according to the cylindrical formula which is inapplicable to the cone) leads to significantly higher amplitudes of the electromagnetic fields. The amplitude of mode TE013 actually increases by a factor greater than 2.5 from its original amplitude. While mode shape TE013 has the smallest amplitude compared to TE011 and TE012 at the initial dimensions, as we reduce the small base it becomes the mode with the highest amplitude 4. The half-wavelength nearest the apex gets longer as it approaches the apex.

42 Conclusions 5. For the particular geometry in the examples in this report, cut-off of mode shape TE011 occurs when the small base is reduced to only ¼ of its original dimension. The cut-off condition based on a cylindrical waveguide incorrectly shows that TE011 should have been cut-off at a much larger base diameter (at 0.90 the original dimension instead of 0.25 the original dimension). 6. Cut-off of mode shape TE012 and TE013 occurs when the small base is reduced to only 1 / 5 of its original dimension. The cut-off condition based on a cylindrical waveguide incorrectly shows that TE012 and TE013 should have been cut-off at a much larger base diameter (at 0.90 the original dimension instead of 0.20 the original dimension). 7. Continuing the cone up to distances much closer to the apex also results in lower phase shift and higher geometrical attenuation of the electromagnetic field in the longitudinal direction. When the small base is reduced to ½ the original size: a) the (dimensionless) geometrical attenuation is increased by a factor of 28 times from 0.1 to 2.8, and b) the phase constant is increased by a factor of 2 from 0.5 to 1. Hence it looks like very large changes can be accomplished by simply reducing a truncated cone s small base so that it is much closer to the cone s apex, and this can be done without incurring cut-off, and achieving a higher amplitude to boot. Thus continuing the cone beyond the cylindrical cut-off frequency may result in very interesting behavior.

43 Conclusions 8. All the EM Drive formulas (McCulloch s, and Shawyer s) predict greater thrust with a larger difference between the diameters of the big and the small bases of the truncated cone. Therefore these formulas point towards the direction that the ideal geometry would be one with a small based diameter. Yet such a geometry has not been explored up to now, apparently due to Shawyer s constraining the small base diameter to be larger than the diameter that results in cut-off according to the cylindrical waveguide formula. This report shows that this constraint is nonsensical, as truncated cones resonate (and at higher amplitude) with significantly smaller base diameters. This report shows that the small based diameter could be reduced to at least ½ of its present size, and perhaps to 1 / 5 of its present size.

44 Appendix The following image was posted by Paul March (NASA Eagleworks) at the Nasa Spaceflight Forum s EM Drive thread #2. It shows several analyses done by Frank Davis at NASA using COMSOL s Finite Element Analysis to model the resonant frequencies of Shawyer s Demonstration EM Drive. As Shawyer only reported the height of the truncated cone and the diameter of the big (bottom) base, Frank Davis conducted all these studies with different small based diameters to ascertain the likely dimensions of the small base based on knowledge of the excitation frequency and the mode shape excited in Shawyer s experiment. Frank Davis in this analysis keeps the height of the truncated cone constant as he changes the small base diameter (thereby effectively changing the cone s angle) while I have kept the cone s angle constant as I changed the small base diameter (thereby effectively changing the cone s height). NASA s analysis fully confirms the results of this report. Frank Davis (using a completely different method: Finite Element Analysis instead of the exact solution I am using) finds that as the geometry of the truncated cone becomes closer to a pointy cone the mode shapes are not cut-off. Hence NASA s COMSOL Finite Element Analysis confirms that it is nonsense to use a cylindrical waveguide cut-off formula to predict cut-off of mode shapes in a truncated cone.

45 Appendix

46 References 1. Davis, A.M.J., and Scharstein, R.W.; Electromagnetic plane wave excitation of an open-ended conducting frustum, Antennas and Propagation, IEEE Transactions (Volume:42, Issue: 5 ), May 1994, 2. de Villiers, D., Analysis and Design of Conical Transmission Line Power Combiners, Ph.D. thesis dissertation, University of Stellenbosch, Engineering Department, Thesis Advisers: P. Meyer and P.W. van der Walt, December 2007, 3. de Villiers, D., and Meyer, P., Numerical calculation of analytic solutions for higher order modes in conical lines, International Journal of RF and Microwave Computer-Aided Engineering, Volume 19, Issue 1, pages , January 2009, er_modes_in_conical_lines 4. Egan, Greg, Resonant Modes of a Conical Cavity, created 24 October 2006, revised 5 August 2014; 5. Hildebrand, F., Advanced Calculus for Applications, Prentice-Hall, N.J., Mayer, B., and Reccius, A.; Conical cavity for surface resistance measurements of high temperature superconductors, Microwave Theory and Techniques, IEEE Transactions (Volume:40, Issue: 2), Feb Morse, P., and Feshbach, H., Methods of Theoretical Physics, Part II: Chapters 9 to 13, McGraw Hill, N.Y., Schelkunoff, S.A., Electromagnetic Waves, D. Van Nostrand, N.Y., Weile, Daniel S., Spherical Waves, ELEG 648 Advanced Engineering Electromagnetics, Spring 2006, Zeng, Xiahui, and Fan, Dianyuan, Electromagnetic fields and transmission properties in tapered hollow metallic waveguides, Optics Express Vol. 17, Issue 1, pp (2009), doi: /OE ,