# EmDrive

EmDrive (also Relativity Drive) is the name of a spacecraft propulsion system proposed by British aerospace engineer Roger J. Shawyer, who develops prototypes at Satellite Propulsion Research Ltd (SPR),[1] the company he created for that purpose in 2000.[2] New Scientist ran a cover story on EmDrive in its 8 September 2006 issue.[3] The device uses a magnetron producing microwaves directed inside a specially shaped, fully enclosed tapering high Q resonant cavity whose area is greater at one end, upon which radiation pressure would act differently due to a relativistic effect caused by the action of group velocity in different frames of reference. The inventor claims that the device generates a thrust even though no detectable energy leaves the device. If proven to work as claimed, the EmDrive could allow the design of spacecraft engines that would be electrically powered and would require no reaction mass. Such an engine would be a breakthrough in airflight and spaceflight.[4][5][6][7][8]

The device and its mode of operation are highly controversial. As of 2014, it is still not proven if the EmDrive is a genuinely new propulsion method; a misinterpretation of spurious effects mixed with mathematical errors; or a scam. The proposed theory immediately received virulent criticism because it seems to violate basic Newtonian laws of physics, notably conservation of momentum,[9][10] though the inventor insists on the contrary.[11] Whatever it be, peer reviewed independent replication has been provided by Chinese researchers from the Northwestern Polytechnic University on both mathematical and experimental grounds.[4][12] in 2008,[13] 2010,[14] 2012,[15] and 2013.[16]

Should the EmDrive produce a real thrust, various conjectures have been made to explain the underlying physics. Shawyer claims the thrust would be caused by radiation pressure imbalance due to group velocities of electromagnetic waves within the framework of special relativity. Dr. Yang predicts a resulting net force using classical electromagnetism.[14] A more complete theory has been proposed in 2013 by Argentine physicist Fernando Minotti from CONICET, who explains the alleged forces on asymmetric electromagnetic resonant cavities by a particular class of scalar-tensor theory of the Brans–Dicke type.[17] Dr. Harold G. "Sonny" White, a NASA mechanical engineer and physicist investigating field propulsion at Johnson Space Center, notes that such resonant cavities may operate by creating a virtual plasma toroid that would realize net thrust using magnetohydrodynamics upon quantum vacuum fluctuations.[18]

## SPR Ltd claims

The following claims are summarized from Shawyer's scientific papers, available on the LPR Ltd web site.[19][20][21][22][23][24][25][26]

### Group velocities in tapered waveguides

The EmDrive exploits an idea first suggested by Allen Cullen in the 1950s, an electrical engineer then at University College London, that involves measuring forces created by radiation pressure of microwaves against the internal walls of a resonant cavity. Cullen published a number of articles on this topic during the 1950s, notably in Nature.[27][28][29][30][31][32]

Roger Shawyer's idea is to try to design a microwave cavity as a conical frustum in such a manner that forces due to radiation pressure on one side are greater than the other.

Cullen showed the propagation rate of electromagnetic waves in space (group velocity) and the resulting force it exerts can be varied depending on the geometry of a waveguide within which it travels.[30] The increasing confinement of a narrowing waveguide (convergent) leads to a widening wavelength and a decrease of the group velocity (lower momentum transfer). Conversely, a widening waveguide (divergent) leads to a narrowing wavelength and an increase of the group velocity (higher momentum transfer).[33]

Shawyer states that if the electromagnetic wave travelling in a tapered waveguide is bounced between two reflectors, with a large group velocity difference at the end surfaces, the force difference resulting from the radiation pressure difference will give a resultant thrust to the waveguide linking the two reflectors, in the direction of the larger surface.

This imbalance between the radiation pressures can be also strengthened by the addition, near the smaller end plate, of a dielectric resonator or a ferrite material, whose relative permeability or relative permittivity, or both, are higher than unity. Such electric materials weaken the group velocity of waves travelling through them, lowering further the radiation pressure at the small end reflector.[33][34]

If the reflectors are separated by a multiple of half the effective wavelength of the electromagnetic wave, this thrust will be multiplied by the Q factor of the resulting resonant cavity. Thus looking for high Q cavities is necessary to significantly increase thrust magnitude.

### Conservation of momentum in open systems

Standard Newtonian mechanics and thus the law of conservation of momentum indicate that, no matter what shape the cavity is, the forces exerted upon it from within must balance to zero. Shawyer claims this statement ignores special relativity in which separate frames of reference have to be applied when velocities approach the speed of light. In the EmDrive, the system of electromagnetic waves and the waveguide can be regarded as an open system, both having separate frames of reference. This effect similarly explains the principle of the laser gyroscope, which is also an apparently closed system device, but where the beams act as if having an external frame of reference (which they have, since the speed of light is constant).

The following derivation is based on Cullen.[30] The forces acting on each end reflector are:

$F_{g1} = \frac{2 P_0}{c} \left(\frac{v_{g1}}{c}\right) = \frac{2 P_0}{c} \frac{\lambda_0}{\lambda_{g1}} \qquad \text{and} \qquad F_{g2} = \frac{2 P_0}{c} \left(\frac{v_{g2}}{c}\right) = \frac{2 P_0}{c} \frac{\lambda_0}{\lambda_{g2}}$

where:

• $F_{g1}$ is the force acting on the large reflector
• $F_{g2}$ is the force acting on the small reflector
• $v_{g1}$ is the group velocity of microwaves at the end of the largest cross-section
• $v_{g2}$ is the group velocity of microwaves at the end of the smallest cross-section
• $\lambda_0$ is the wavelength of the microwaves in free-space propagation
• $\lambda_{g1}$ is the wavelength of the microwaves at the end of the largest cross-section
• $\lambda_{g2}$ is the wavelength of the microwaves at the end of the smallest cross-section
• $P_0$ is the input power
• $c$ is the speed of light

The concept of the microwaves and waveguide as an open system can be illustrated in a thought experiment where the waveguide is subject to a proper acceleration in the direction of the thrust until a significant fraction of the speed of light is reached. Newtonian mechanics can't apply and is replaced with special relativity, which involves two relativistic effects on the EmDrive:

First, as the two forces $F_{g1}$ and $F_{g2}$ are dependent upon the local group velocities of microwaves $v_{g1}$ and $v_{g2}$, the thrust should be calculated according to Einstein's velocity-addition formula given by:

$v = \frac{v_1 + v_2}{1 + \left(v_1 v_2 \right) / c^2}$

Secondly, as the wave velocities are not directly dependent on any velocity of the waveguide, the waves and waveguide form an open system. Thus the reactions at the end reflectors are not constrained within a closed system of waveguide and beam, but are reactions between waveguide and waves, each operating within its own frame of reference, in an open system.

If the waveguide moves at a velocity $v_w$ then as the end reflectors are also moving with velocity $v_w$ the forces acting on each end reflector, given by the previous equations, are modified as follows:

$F_{g1} = \frac{2 P_0}{c^2} \left( \frac{ v_{g1} - v_w}{1 - v_{g1} v_w\ /c^2}\right) =\frac{2 P_0}{c^2} v_{ga} \qquad \text{and} \qquad F_{g2} = \frac{2 P_0}{c^2} \left( \frac{ v_{g2} + v_w}{1 + v_{g2} v_w\ /c^2}\right) =\frac{2 P_0}{c^2} v_{gb}$

Subtracting $F_{g1}$ - $F_{g2}$ the net thrust is then:

$T = \frac{2 P_0}{c^2} \left( \frac{ v_{ga} - v_{gb}}{1 - v_{ga} v_{gb}\ /c^2}\right)$

This equation shows that as the waveguide is accelerated in the direction of thrust, the thrust will decrease to zero. The null thrust is reached when $v_{ga}$ = $v_{gb}$.

If Einstein's velocity-addition formula is not used in the solution to the thrust equation, relative velocities and thrust would exceed the c limit, which is impossible and demonstrates that the EmDrive is an open system, where group velocities are independent of waveguide velocity, and that it is the relative velocities that give rise to the forces.

The momentum exchange is between the electromagnetic waves and the waveguide. As the vehicle accelerates, momentum is lost by the electromagnetic waves and gained by the waveguide.

### Static thrust equation

The derivation of the basic thrust equation detailed by Shawyer is based on Cullen.[30]

Assuming the vacuum permeability and the relative permittivity both equal to unity, and supposing that the waveguide is resonant at the microwave frequency, with conductive and dielectric losses such that there are Q return paths (each at power $P_0$), the total static thrust equation is:

$T = \frac{2 P_0 Q_u}{c} \left(\frac{\lambda_0}{\lambda_{g1}} - \frac{\lambda_0}{\lambda_{g2}}\right) \left(1 - \frac{\lambda_0^2}{\lambda_{g1} \lambda_{g2}} \right)^{-1}$

where:

• $T$ is the thrust
• $P_0$ is the power
• $Q_u$ is the unloaded Q factor of the cavity
• $\lambda_0$ is the wavelength of the microwaves in free-space propagation
• $\lambda_{g1}$ is the wavelength at the end of the largest cross-section
• $\lambda_{g2}$ is the wavelength at the end of the smallest cross-section

which can be simplified to:

$T = \frac{2 P_0 Q_u D_f}{c}$

where $D_f$ is the Design factor of the cavity.

### Dynamic equation and conservation of energy

The Q factor or simpler the Q of any resonant circuit is a dimensionless quantity that can be defined as the stored energy divided by the energy lost per cycle.

Any force produced by the engine and converted into kinetic energy is withdrawn from the energy stored in the cavity, through a decrease in the Q factor due to Doppler shift. In other words, during proper acceleration, the apparent force on the wider diameter of the cone lessens. Taking into account the input power, the circulating power, the output power transferred to the engine and the power losses in dynamic operation, Shawyer demonstrates the system fulfills the law of conservation of energy.[21]

The maximum Q of the engine, under static thrust conditions, is defined as the unloaded Q. Under proper acceleration, the Q of the engine at an average velocity over time is defined as the loaded Q. The relationship between the unloaded Q and the loaded Q is given by the dynamic equation:

$\left(\frac{Q_l}{Q_u}\right)^2 + \frac {2 Q_l\ \bar{v}\ D_f}{c} = 1$

where:

• $Q_u$ is the Unloaded Q
• $Q_l$ is the Loaded Q
• $\bar{v}$ is the average velocity of the device over time

Conventional microwave and resonant cavity technologies limit the maximum Q of resonators to around 50,000. According to the thrust equation, this restricts the specific thrust to a maximum threshold for every velocity. For example, at a velocity of 3 km/s, the specific thrust of a resonator at such a Q would reach a limit of 200 millinewtons per kilowatt. Shawyer states this could allow "first generation engines" where typical applications would be transfers to low Earth orbit, maintenance of communications satellites and the primary propulsion for unmanned space missions.

But superconducting microwave resonant cavities would tremendously boost the Q factor. In 1995, German superconducting resonant cavities for use in particle accelerators readily achieved Q of several billion.[35]

Superconducting resonators would be used in "second generation engines" that would change everything: the German resonator having a Q of 5×109 would allow static specific thrust of about 3 kN/kW, that is 3 tonnes of thrust per kilowatt of input power, "enough to lift a large car" according to Shawyer.[3]

However, with such high specific thrusts, those engines would be subject to the dynamic equation where the effect at these high values of unloaded Q is quite important. Thus in the example where the resonator has an unloaded Q of 5×109 and an average velocity of only 0.1 m/s, the dynamic equation quickly reduces the specific thrust from 3.15 tonne per kilowatt (static thrust at rest) to 0.93 tonne per kilowatt when Q is loaded. The dynamic equation therefore would constrain the applications of second generation engines to those where the acceleration and kinetic energy output is limited.

### Thrust limitation by Doppler shift

The Q losses and the overall thrust reduction when the cavity is accelerated, while the electromagnetic waves inside are reflected back and forth between the two end plates of the resonator, has been found to be caused by the Doppler effect.[25]

• With a positive acceleration, the overall Doppler shift inside the cavity is negative. This leads to a reduction in stored energy in the cavity, and thus a reduction in Q, and a reduction in thrust. The kinetic energy gained by the cavity is then balanced by the stored energy lost by the cavity. This is EmDrive in "motor" mode.
• With a negative acceleration, the overall Doppler shift is positive. This leads to an increase in stored energy, which is balanced by the loss of kinetic energy from the cavity. This is EmDrive in "generator" mode.

This dual mode illustrates that EmDrive works as a classic electrical machine. The "generator" mode offers a method of decelerating a vehicle.

More importantly, the Doppler shifts occuring in each transition will, under very high Q and high acceleration, cause the frequency of the electromagnetic wavefront to move outside the operating narrow resonant bandwidth of the cavity, dramatically limiting the thrust, thus the acceleration provided by the thruster.

A thruster has been designed with a compensating system where frequency offset is used as well as a dynamic modification of the axial length of the cavity over a few microns, according to the acceleration experienced by the engine. The extension results from a pulsed voltage being applied to piezoelectric elements in the sidewall of the cavity. This dynamic compensation enables the effect to be partially reduced, and allows acceleration of up to 0.5 m/s2 for a theoretical specific thrust of 1 tonne per kilowatt.[25]

This acceleration limitation, in the vertical plane only, could allow second generation superconducting EmDrives to be deployed as lift engines in a number of aerospace vehicles.[24]

The weight of such aircrafts being almost cancelled (but not their mass, hence inertia), they would have no need for wings anymore and could be designed with various shapes. The acceleration of the vehicle itself would be produced by conventional propulsion of limited power, like small propellers, gas turbines, rocket engines or even ion thrusters or plasma propulsion engines, although Shawyer suggests liquid hydrogen turbines would be the best choice as this fuel under cryogenic state could also cool down the superconducting resonators before being burnt in the small propelling turbines.

### Publications

No peer reviewed publications has been proposed by Shawyer to date, as opposed to independent work by Chinese researchers at the Northwestern Polytechnical University in Xi'an[13][14][15][16] and Argentine physicist Fernando Minotti.[17]

Shawyer initially wrote a theory paper in 2003, as required by the contract signed with U.K. government Department of Trade and Industry.[19]

He then regularly presented his work at various international conferences, in Brighton in 2005,[20] at the 59th International Astronautical Congress (IAC) in Glasgow in 2008,[21] the CEAS 2009 European Air and Space Conference held in Manchester,[22] the 2nd Conference on Disruptive Technology in Space Activities (TECHNO DIS) at CNES' Toulouse Space Show, France in 2010,[23] and at the 64th International Astronautical Congress, Beijing, China in 2013.[25][26]

Shawyer also filed four patents on the EmDrive technology.[33][34][36][37]

## SPR Ltd devices

### Feasibility Study and first prototype

In July 2001, A £45,000 Research and development grant was first awarded to SPR Ltd by the U.K. government's Department of Trade and Industry[38] under their SMART award scheme, as part of a three-year, £250,000 programme, and the work started with a mission analysis phase. The two technical objectives of the initial study, completed in 2002, were the derivation of a thrust equation and the verification of that equation by experiment.[1][39]

The first objective has been achieved with the completion of the theory paper.[19] For the second goal, an experimental prototype was designed and built with the following measured characteristics:

Big end 160 mm 100 mm 160 mm 2.45 GHz 5,900 0.497 850 W 16×10−3 N

The couple thruster + magnetron weighted 9.4 kg (15 kg with the electromagnetic shielding enclosure on). The maximum thrust measured was very close to the 16.6 mN thrust predicted from the static thrust equation of the theory paper. The thrust could be varied from zero to maximum by varying the input power, or by varying the resonant frequency of the thruster. Efforts were made to test for possible thermal and electromagnetic spurious effects: the primary method was to carry out all tests in both nominal and inverted orientations, and to take the mean of the results. The thruster was also sealed into a hermetic enclosure to eliminate coupling with electromagnetic radiation and any buoyancy effects of the cooling air. Three different types of test rig were used, two using 1 mg resolution balances in a counterbalance test rig, and another using a 100 mg resolution balance in a direct measurement of thruster weight. Comparison of the rates of increase of thrust for the different spring constants, using pulsed input power, indicated that the thrust would be produced by momentum transfer and would not be caused by any unknown spurious effect. The total test programme encompassed 450 test runs of periods up to 50 seconds, using 5 different magnetrons.[1][39]

### Demonstrator Engine

In 2003, after the positive results of the experimental prototype, a review of the study programme was examined by John Spiller, an independent space engineer hired by the U.K. government for that purpose. Spiller concluded in his report:

 “ The thruster's design is practical and could be adapted fairly easily to work in space […] The drive needs to be developed further and tested by an independent group with its own equipment. […] It certainly needs to be flown experimentally. ” —John Spiller, in his review of SPR Ltd work.[40]

## References

### Notes

1. "EmDrive.com". Satellite Propulsion Research Ltd (SPR) web site. Roger Shawyer / SPR Ltd.
2. ^ "Satellite Propulsion Research". Aerospace Member Directory. ADS Group.
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15. ^ a b c d Yang, Juan; Wang, Yu-Quan; Li, Peng-Fei; Wang, Yang; Wang, Yun-Min; Ma, Yan-Jie (2012). "Net thrust measurement of propellantless microwave thrusters" (PDF). Acta Physica Sinica (in Chinese) (Chinese Physical Society) 61 (11). doi:10.7498/aps.61.110301. edit
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21. Shawyer, Roger (29 September–3 October 2008). "Microwave Propulsion - Progress in the EmDrive Programme" (PDF). 59th International Astronautical Congress (IAC 2008). Glasgow, U.K.: International Astronautical Federation. Lay summary.
22. ^ a b c Shawyer, Roger (27 October 2009). "The Emdrive Programme – Implications for the Future of the Aerospace Industry" (Word document). CEAS 2009 European Air and Space Conference. Manchester, U.K.: Royal Aeronautical Society.
23. ^ a b c d Shawyer, Roger (10 June 2010). "The EmDrive - A New Satellite Propulsion Technology" (Word document). Toulouse Space Show'10, 2nd Conference on Disruptive Technology in Space Activities (TECHNO DIS 2010). Toulouse, France: CNES.
24. ^ a b Shawyer, Roger (16 September 2012). "Second generation EmDrive" (PDF). SPR Ltd.
25. Shawyer, Roger (23–27 September 2013). "The Dynamic Operation of a High Q EmDrive Microwave Thruster" (PDF). 64th International Astronautical Congress (IAC 2013). Beijing, China: International Astronautical Federation.
26. ^ a b Shawyer, Roger (23–27 September 2013). "The Dynamic Operation of a High Q EmDrive Microwave Thruster (poster)" (PDF). 64th International Astronautical Congress (IAC 2013). Beijing, China: International Astronautical Federation.
27. ^ Cullen, A. L. (12 March 1949). "Absolute Power Measurement at Microwave Frequencies". Nature (Nature Publishing Group) 163 (4141): 403. doi:10.1038/163403b0. edit
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29. ^ Cullen, A. L. (19 May 1951). "Absolute Measurement of Microwave Power in Terms of Mechanical Forces". Nature (Nature Publishing Group) 167 (4255): 812. doi:10.1038/167812a0. edit
30. Cullen, A. L. (April 1952). "Absolute power measurement at microwave frequencies" (PDF). Proceedings of the IEE - Part IV: Institution Monographs (Institution of Electrical Engineers) 99 (2): 100–111. doi:10.1049/pi-4.1952.0012. edit
31. ^ Cullen, A. L. (April 1952). "A general method for the absolute measurement of microwave power". Proceedings of the IEE - Part IV: Institution Monographs (Institution of Electrical Engineers) 99 (2): 112–120. doi:10.1049/pi-4.1952.0013. edit
32. ^ Cullen, A. L.; Stephenson, I. M. (December 1952). "A torque-operated wattmeter for 3-cm microwaves". Proceedings of the IEE - Part IV: Institution Monographs (Institution of Electrical Engineers) 99 (4): 294–301. doi:10.1049/pi-4.1952.0031. edit
33. ^ a b c GB application 2334761, Shawyer, Roger John, "Microwave thruster for spacecraft", published 1999-09-01, assigned to Shawyer, Roger John
34. ^ a b GB application 2229865, Shawyer, Roger John, "Electrical propulsion unit for spacecraft", published 1990-10-03, assigned to Shawyer, Roger John
35. ^ Bauer, S.; Diete, W.; Griep, B.; Pekeler, M.; Schwellenbach, J.; Vogel, H.; Vom Stein, P. (November 1999). "Production of Superconducting 9-Cell Cavities for the TESLA Test Facility" (PDF). 1999 Workshop on RF Superconductivity (SRF 1999). Santa Fe, New Mexico, USA: Los Alamos National Laboratory.
36. ^ GB application 2399601, Shawyer, Roger John, "Thrust producing device using microwaves", published 2004-09-22, assigned to Shawyer, Roger John and Satellite Propulsion Research Ltd
37. ^ a b GB application 2493361, Shawyer, Roger John, "A high Q microwave radiation thruster", published 2013-02-06, assigned to Shawyer, Roger John and Satellite Propulsion Research Ltd
38. ^ a b Margaret, Hodge (5 December 2006). Column 346W. "Answer about the Electromagnetic Relativity Drive". Daily Hansard Official Report (London: House of Commons of the United Kingdom).
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43. ^ Fisher, Richard (1 September 2006). "Microwave engine gets a boost". The Engineer (London).
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48. ^ Foster, John; Haag, Tom; Patterson, Michael; Williams, George J., Jr.; Sovey, James S.; Carpenter, Christian; Kamhawi, Hani; Malone, Shane et al. (1 September 2004). The High Power Electric Propulsion (HiPEP) Ion Thruster (PDF) (Technical report). NASA. NASA/TM-2004-213194.
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52. ^ EM Drive topic at Talk-Polywell forum
53. ^ David, Leonard (8 September 2004). "Cubesats: Tiny Spacecraft, Huge Payoffs". Space.com.