The Woodward Effect: Math Modeling and Continued ...

The Woodward Effect: Math Modeling and Continued

Experimental Verifications at 2 to 4 MHz

Paul March1 and Andrew Palfreyman

2

1309 Sedora Drive, Friendswood, TX 77546 USA 25294 Southbridge Place, San Jose, CA 95118 USA

1281-996-9265, [email protected]

Abstract. The Woodward Effect (W-E), the supposition that energy-storing ions experience a transient mass fluctuation near

their rest mass when accelerated, has been tentatively verified using linear electrical thrusters based on the Heaviside-Lorentz

force transformation. This type of electromagnetic field thruster, or Mach-Lorentz Thruster (MLT), purports to create a

transient mass differential that is expressed in a working medium to produce a net thrust in the dielectric material contained

in several capacitors. These mass differentials are hypothesized to result from gravity/inertia–based Wheeler-Feynman

radiation reactions with the rest of the mass in the universe (per Mach’s Principle) in order to conserve momentum. Thus if a

net unidirectional force is produced in such a device, then mass fluctuations in the working media should be present. A net

unidirectional and reversible force on the order of ± 3.14 milli-Newton or 0.069% of the suspended test article mass was

recorded by us in our first high frequency 2.2 MHz test article. The authors also developed a W-E model that integrates the

various engineering parameters affecting the design, construction, and performance of W-E based MLTs for the next

generation of systems. When Woodward’s (2004a, 2004b, 2005) and our test results were compared with the model’s

predictions, the test results exceeded predictions by one to two orders of magnitude. Efforts are underway to understand the

discrepancies and update the model. The test results imply that these devices, when fully developed, could be competitive

with ion engines intended for use on satellite station keeping and/or orbital transfers.

Keywords: Gravity, Inertia, Mach-Lorentz Thruster, Mass Fluctuations, Recycled Propellant, Woodward Effect.

PACS: 04.80.Cc.

INTRODUCTION

Why do we need field propulsion and power technologies? March’s previous STAIF-2004 paper (March, 2004)

described in detail the need for such, but to summarize this case we need only reflect on that paper’s presented

mission example, where the propulsion needs for a 0.01c max velocity interstellar robotic mission to our nearest

stellar neighbor Alpha Centauri was presented. This robotic mission would take 430 years for a one-way trip and it

would require a rocket engine with a specific impulse (Isp) of at least 100,000 seconds to obtain a propellant mass

fraction of 21.3-to-1. If we examine what a relativistic velocity (~0.9c) manned mission would require, we find that

the engine’s exhaust velocity has to be very close to c to be practical. Needless to say, no currently known rocket

technology comes even close to supplying a specific impulse of ~30,000,000 seconds at the power levels required to

accelerate a manned interstellar vehicle at 1-gee constant acceleration. Thus we have to look elsewhere if we are to

ever journey to the stars, or make travel in our solar system affordable for large numbers of people and their

supporting goods and services. So where can we look for a solution to this problem? We believe that the

Woodward Effect (W-E) can provide such a solution and in this paper we shall lay out experimental evidence to

support that position and a W-E math model that will allow us to make estimates of the W-E derived vxB forces.

W-E THEORY

Woodward and Mahood have provided a theoretical explanation and experimental data in several papers over the

last fifteen years on the W-E (Woodward 1990, 1996, 2003, 2004a, 2004b, 2005; Mahood 1999a, 1999b; Mahood

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http://dx.doi.org/10.1063/1.2169317

http://dx.doi.org/10.1063/1.2169317

and Woodward, 2000; Mahood, March and Woodward, 2001; and VanDeventer and Woodward, 2006.) Other

researchers such as Brito (2003, 2005) and March (2004) have also been able to independently generate forces using

similar Lorentz vxB force rectification approaches. Woodward’s papers explain in detail his ideas on the origin of

inertia, mass fluctuations and recycled propellant propulsion including several STAIF papers (Woodward 2004b,

2005, 2006), so only a summary of his theoretical treatise will be provided here for reference. The authors started

their investigations into the W-E in 1998 with March’s first formal W-E report at STAIF-2004 (March, 2004).

The W-E is based on a theory of inertia derived from a Machian interpretation of General Relativity which, when

followed to its logical conclusion, predicts that when a mass is accelerated relative to the distant stars through a

field’s local potential gradient, then its local rest mass is transiently perturbed. The resulting acceleration-induced

“mass fluctuations” can then be used to generate an unbalanced force in a local system, which can be used for local

propulsion or energy extraction. Global energy and momentum conservation is maintained by interactions with the

rest of the mass in the universe via gravito-inertial (G/I)-based Wheeler-Feynman radiation reaction forces.

Therefore to locally accelerate a spacecraft, the Machian interpretation of inertial reaction forces implies that each

star or other distant matter in the universe will move in the opposite direction to the locally accelerated mass in

response – even if only on a nano-nano-meter scale. Conservation of energy and momentum must be maintained,

but Nature doesn’t say how big the associated system box has to be or when the accounting has to be done.

A derivation from first principles of the W-E’s controlling equation was performed by Woodward (1990, 1996) and

amplified with mathematical details by Mahood (1999b). The expression for ∇2φ is provided in equation (1), which

consists of the driven 4πGρ0 G/I term (1st term) that excites the transient impulse (2nd

term) and always negative

wormhole terms (3rd

term). Typical results of this expression are displayed in Figure 1 using a numerical integration

approximation with arbitrarily chosen control parameters. This figure shows the normalized relationship between

the W-E transient impulse and wormhole terms when driven by moderate amplitude sinusoidal mass/energy density

excitations. Note that the magnitude of the combined impulse and wormhole terms has two positive and four

negative going mass density peaks per applied 4πGρ0 cycle when modeled on a barium titanate dielectric.

FIGURE 1. Excel W-E Based Numerical Solution Using Sine Driving Function with Y-Axis = +/-∆Mass Density. [Red Curve:

1st Term in Eqn. (1); Green Curve: 2nd Term in Eqn. (1); Purple Curve: 3rd Term in Eqn. (1)]

Woodward has also developed and executed a large number of “table-top” experiments that apparently confirm the

existence of these mass fluctuations and their potential for use in the propulsion arena.

Mach-Lorentz Thruster (MLT) Basics

The Mach-Lorentz Thruster (MLT) is based on the Heaviside-Lorentz force relation, which defines what happens to

an electrically charged ion when externally applied electric (E) and magnetic (B) fields interact with ion(s) that have

a non-zero velocity relative to the E- and B-field generators. The electrical engineering “right hand rule” states that

2

0

2

0

2

0

2

0

0

114 �

�

�

�

�

�

∂∂

��

�

�

��

�

�

−��

�

�

��

�

�

∂∂

+tt

Gρ

ρρ

ρρπ (1) .

1322


when the applied E- and B-fields are at right angles to each other and there is an ionic current present, then the

resulting net force on the ions will be at right angles to both of the applied fields (Figure 2).

FIGURE 2. Mach Lorentz Thruster (MLT) Lorentz q*[E + (Net Ion’s Velocity - v x B)] Force Rectification.

The W-E impulse term’s cyclic mass fluctuations and any forces resulting from them will time average to zero

unless some external force rectification modulation (FRM) is applied to the dielectric material undergoing the mass

fluctuations. This FRM must push the active dielectric when it’s heavier and pull when it’s lighter, thus generating a

net useful force on the MLT structure. In the case of a dielectric material like barium titanate, (BaTiO3) which has a

tetragonal unit crystal cell (UCC) at room temperature, and is composed of eight Ba corner ions, six face centered

oxygen ions and one center located titanium ion (Cook, Jaffe and Jaffe, 1971), the peak mass fluctuations should

occur for the highest accelerated ion(s). In the case of the BaTiO3 UCC, due to its interatomic bonds and relative

atomic masses, the center Ti ion has the highest accelerations and internal energy state changes when an external E-

field is applied to the material (Herbert and Moulson, 2003).

FIGURE 3. Mach Lorentz q*[E + (vxB)] Force Rectification Modulation in BaTiO3 UCC.

The peak mass fluctuations will then occur at the end of each of the Ti ion’s transits in the UCC’s electrostatic

potential wells where the ion’s acceleration and stored energy levels peak and the ion(s) reaches a minimum

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negative delta mass. And as they go through the center of the UCC, the Ti ions then reach their peak velocity and

minimum stored potential energy where they obtain their maximum positive delta mass. Therefore to implement the

FRM in the BaTiO3 dielectric, one has to apply a force to the Ti ions two times during each E-field cycle. But what

is the optimum time to apply these push and pull FRM impulses? We need to apply them at the extrema of the mass

fluctuations of the Ti ion, which correspond to the extrema of position, of acceleration and of energy storage (due to

the UCC lattice restoring forces) and also at the center of the Ti ion travel path where velocity is maximum and

energy storage is minimum. See Figure 3 for a summary of the Lorentz vxB FRM process using BaTiO3 ceramics.

Having selected the FRM type - a B-field/UCC restoring force pair - we can push the Ti ions when they’re more

massive with the UCC restoring forces, and pull them when they’re less massive with the externally applied B-field.

MLT ENGINEERING TOOLS

Now that we have a qualitative idea of our goal, it would be productive to generate a derivation of the MLT thrust

equation that could be used to predict the MLT’s performance. Refer to the Nomenclature for symbol semantics.

The basic equation (VanDeventer and Woodward, 2006, eqn. (3)) for the transient mass fluctuation due to the

Woodward Effect, as a function of the applied power is:

2)

02

1(

0)24( P

mct

PmGc −

∂∂=δρπ

The first term on the RHS of the equation is designated the “impulse” mass transient term and the second term is the

always-negative “wormhole” mass transient term. The basic force equation (VanDeventer and Woodward, 2006),

eqn. (5) and (8) for the Mach-Lorenz Thruster variant of the W-E is:

><=>< mBIm

dF δ

0

2 .

We model the power P input to the capacitor as:

)2sin(0 tPP ω= ,

where ω is the excitation frequency, and thus:

)2cos(2 0 tPt

P ωω=∂∂

.

Substituting (4) and (5) into (2) yields:

0

2

2

0

0

)2(sin)2cos(20

)24(

mc

tPtm

P

Gc ωωωδρπ

−= .

The instantaneous energy E of the capacitor is:

2/2CVE = .

So:

t

VCV

t

EP

∂∂=

∂∂= .

(2)

(3)

(8)

(7)

(5)

(6)

(4)

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We model the voltage V input to the capacitor as:

)sin(0 tVV ω= .

Therefore,

)cos(0 tVt

V ωω=∂∂

.

Substituting (9) and (10) into (8) gives:

)2sin()2/()cos()sin( 2

0

2

0 tCVttCVP ωωωωω == .

We can make the identification from (4) that:

2/2

00 ωCVP = .

Substituting (12) into (6) gives:

0

2

22

0

22

0

0

2

2

)2(sin)2cos(2

)8(

mc

tCVtm

CV

Gc ωωδω

ρπ−= .

From (9) and by setting the capacitor current at 90 degrees relative to the applied voltage, we define

)cos(0 tII ω= [and thus 2/000 IVP = ].

From (9) again, we know that the magnetic B-field is in phase with the inductor current, which in turn needs to be at

90 degrees relative to the capacitor voltage in order to maximize the W-E. We therefore define (unconcerned with

sign):

)cos(0 tBB ω= .

We now make the assumption that the frequency remains constant with time. Note that a superior optimization is

possible, but not explored here, whereby the frequency is allowed to vary in some optimal manner so as to maximize

the resultant thrust as derived below. Note also that the proper mass density, which appears in the coefficients of the

transient terms in (1), is assumed to be time-invariant in this particular derivation.

Multiplying both sides of (13) by (14) and (15), and integrating to get a time averaged expression similar to equation

(3); we obtain with a little rearrangement:

�� −=><ωπ

ωπ

ωωπ

ωωωπ

ωδωρπ

2

0

22

0

2

2

0

2

0

2

2

00

2

0

0

2

)(cos)2(sin)2

(2

)(cos)2cos()2

(2)8(

dtttmc

CVdtttmBI

BICV

Gc .

The integral expressions for both the impulse and wormhole terms turn out to give a nonzero, i.e. force “rectified”

result. Using these results we can then write:

)4

1(2

1)8(

0

2

2

0

2

00

2

0

0

2

mc

CVmBI

BICV

Gc−=>< δ

ωρπ . (17)

(10)

(15)

(14)

(13)

(12)

(11)

(16)

(9)

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Now we combine this expression with (3) for the rectified force due to the W-E, yielding the thrust equation for

impulse and wormhole terms when the ρ mass density is held constant in the coefficients of the time-derivative

factors in (1):

)4

1()8( 0

2

2

0

00

2

2

00

2

0

mc

CV

mGc

dBICVF −=><

ρπω

.

We can immediately eliminate the capacitor’s displacement current using the well-known reactance formula

(assuming a zero-loss component),

0 CVI ω= .

which converts equation (18) to the basic vxB thrust equation:

)4

1()8( 0

2

2

0

00

2

233

00

mc

CV

mGc

dCVBF −=><

ρπω

.

Using the Thrust Equation

Due to the large degree of dependency between the parameters in the thrust equation, there are several different

combinations to choose from in order to find the optimal design configuration for an experimental MLT apparatus.

The interdependencies are further constrained by assuming that the inductor and the capacitor form components of

the same circuit, most commonly due to using a resonance condition to optimize input excitation energy. Here we

consider only the case where the components are driven independently and non-resonantly. Note however that both

inductor and capacitor may separately be resonated, and in fact this affords the best of all possible worlds for

minimization of input power. Note also that theoretically this is a lossless system, since all power is pure reactive.

The higher the Q, or “quality factor”, of the circuits, the better will be the performance, and the lower will be the

real (i.e. dissipative) input power requirement.

We will leave the wormhole term standing in order to estimate its relative magnitude relative to the impulse term

(unity inside parentheses) for a variety of configurations. In general terms, however, it is expected (by inspection) to

be many orders of magnitude smaller, unless we allow the proper mass density to be reduced very close to zero –

and since we have assumed it to be time-invariant in the first place, this is not an option using thrust equation 20.

But note that the added complexity of allowing the proper mass density to vary with time is left as an open issue, and

may well turn out to be a key flaw in the entire analysis – not solely for the wormhole term, but for the impulse term

as well, since the proper mass density appears in the denominator of both terms. This may explain why the

experimental MLT results to date are larger than the calculated predictions by one to two orders of magnitude due to

this simplified constant mass density analysis. An effort to clarify this issue is in progress.

Independent Capacitor and Inductor Drives

We introduce the dielectric permittivity, ε, using the well-known capacitance formula, where A is the capacitor’s

area and d is the capacitor’s dielectric thickness:

dAC r /0εε= .

Using the identity for dielectric volume:

Adv = ,

and the mass density relation to eliminate density in the impulse term:

(22)

(21)

(20)

(19)

(18)

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vm 00 ρ= ,

we then get a variety of possibilities for the term in (20):

...32

0

2

22

0

2

22

0

32

2

0

32

2

0

22

00

2

======d

v

d

A

v

A

m

A

dm

vA

m

dC

ρε

ρε

ρεεε

ρ

Taking the expression with the most easily estimable parameters we get:

22

0

2

00

2

d

A

m

dC

ρε

ρ= .

Then (20) becomes:

)4

1()8(

)(

0

2

2

0

22

0

2

2

0

33

00

mc

CV

dGc

AVBF r −=><

ρπεεω .

Thus we predict the vxB derived thrust to have a cube-law scaling with frequency and a square-law scaling with

dielectric permittivity. The output thrust relative to the input capacitor voltage is predicted to follow a cube-law for

the impulse term, whereas for the wormhole term we predict instead a capacitor voltage scaling to the 5th power.

HIGH FREQUENCY MLT EXPERIMENTAL RESULTS

Two different W-E vxB based MLT test articles were built and tested during this time period. The first is the MLT-

2004 test article built and tested by March and the second unit was the Mach-2MHz vxB MLT test article built by

Woodward but mounted, shielded and tested by March. Both units were operated in the 2.0 - 4.0 MHz frequency

range where each test article generated apparent thrust levels in the 0 - ±500 milligram-force (±4.90 mN) range dependent on their input voltage, frequency, E- and B-field phase relationship, and the elapsed operating times.

MLT-2004 Testing

The construction and typical B-field plot for the MLT-2004 based vxB test article is shown in Figure 4. This MLT

consisted of eight (8), Vishay/Cera-Mite 1,000pF, 10kV, Y5R, Dissipation Factor (D.F.) = 2.0%; 5.0 gram active

dielectric ceramic capacitors (cap) that were wired in series, with eight (8), 10.0 Meg-ohm, ½ watt voltage

equalizing resistors wired across each capacitor. This series cap circuit was then wired in series with two over-

wrapped and paralleled 148-turn toroidal B-field coils with the inner coil made from #18 AWG Thermaleze magnet

wire and the outer coil made from 660/46 strand, nylon fabric covered Litz wire that provided B-field strengths through the caps of 8.5 Gauss/Amp-peak calculated and ~10.0 Gauss/Amp-peak measured. This 463.94 gram

resonant inductor / capacitor (L-C) circuit assembly became the MLT-2004 vxB test article.

As shown in Figure 5, the vertical axis of the MLT-2004 test article, as installed in its as-tested configuration, is

defined as the “Z-axis” with +Z-axis being above the test article and –Z-axis being below the test article. Thus a

MLT vxB generated force with its vector pointing down towards the Earth (making the test article heavier) would be considered a –Z-axis (green) force and a vxB force vector pointing up (making the test article lighter) would be

defined as a +Z-axis force (red).

The cap voltages were developed by injecting ~20 Watts rms of 2.20 MHz RF into this MLT’s series resonant L-C

circuit via an open twisted wire transmission line as shown in Figure 5. Typical test data from the MLT-2004 is

shown in Figure 6. The E- to B-field ±90 degree phase reversals were accomplished by manually re-wiring the

power leads of the MLT’s capacitor and inductor feed wires via brass terminals and jumpers on top of the vacuum chamber into the proper configuration.

(23)

(25)

(24)

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FIGURE 4. MLT-2004 Test Article Construction.

FIGURE 5. MLT-2004 vxB Test Article Load Cell Mounting and RF Test Set-Up.

This 20 Watts of RF input power developed a 500V-p vxB signal with a 7.4 gauss B-field at +90 degrees (+/–; +/–

connection), and 535V-p with a 7.9 gauss B-field that was –90 degrees (+/–; –/+ connection) relative to the applied

E-field. Predicted thrust values for this test article using (25) predicted a force output of –0.41 milligram-force

(– 4.12 µN) at 62.5V-p per cap in the –Z-axis and +0.54 milligram-force (+5.30 µN) at 66.9V-p per cap in the +Z-axis direction.

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FIGURE 6. MLT-2004 Experimental +/-90o Thrust Data - Predicted Thrust = +0.54 / -0.41 Milligram Force.

Using the Woodward’s (2006) “Direct” calculation approach, these calculated values increased by a factor of 4 to

–16.4 µN and +21.6 µN respectively. A check for single drive channel electromagnetic interference (EMI) artifacts was performed by lifting the MLT off its load cell support post until it unloaded the load cell and then repeating the

experiment. With the same 20 Watts of RF power applied to the MLT-2004, an approximate 0.15 gram weight decrease (–) was found for both phase conditions, which is attributed to EMI injected into the test set-up’s DI-60E

weight meter and SCAIME AG-1kg load cell. The EMI adjusted maximum measured thrust for the MLT-2004 as

displayed in Figure 6 then becomes +0.36 gram-force (+3.53 mN) for the +90o –Z green trace and –0.28 gram-force

(–2.75 mN) for the –90o red +Z trace (Figure 6). The blue temperature trace in Figure 6 came from the MLT-2004’s

LM34H temperature sensor, which was glued to one of the eight caps in the MLT-2004’s core. There was also a

large RF-induced noise signal riding on the temperature trace when the RF was applied, but the general temperature trend could still be obtained from the end-points of this data and it rose from 85F to 98F during a typical 8.0-second

data run. Vacuum testing of this test article was attempted, but vacuum pump, manifold, and chamber air leaks

prevented data collection at operating pressures at or below the desired 0.01 Torr pressure.

Mach-2MHz MLT Testing

We now turn our attention to the Mach-2MHz test article. The seismic noise platform for the new shielded

Transducer Techniques GSO 500-gram load cell was in the 0.010 – 0.080 gram peak range. As shown in Figure 7,

the test article consisted of a T106-2 powdered iron core, two Vishay/Cera-Mite 500pF, 15kV, Y5U, D.F. = 1.5%,

2.6 gram active dielectric caps wired in parallel, which were then wired in series with two paralleled 30-turn (#22

AWG) B-field coils that provided a B-field strength through the caps on the order of 30.0 Gauss/Amp-peak. The

assembly was then mounted in a Faraday Shield made out of an 8.0 oz capacity steel/tin can with the electrically isolated power leads brought out of the can’s lid. This Faraday Shield / MLT assembly weighed in at 144.6 grams.

The assembly was then suspended from the GSO 500-gram load cell over nine In/Ga/Sn alloy liquid metal electrical

contact pots made from electronic banana jacks that provided a low friction, (when not binding due to pin-to-pot

misalignments), connection to the MLT’s inductors and capacitors. These contact pots were in turn connected to the

RF source via a pair of phase reversal G41C SPDT vacuum relays in the Secondary Faraday shielded base.

During the initial “first light” test of the Mach-2MHz MLT operating at 3.8 MHz, where applied voltage levels of

122 V-peak to each of the two 500 pF caps was reached, per cap currents of 1.5 Amp-peak was obtained, along with

a B-field of 44 Gauss, yielded an apparent weight increase and reduction on the Texmate weight meter of +0.5 / −0.2

gram-force (+4.9 / –2.0 mN) as the phase between the E- and B-fields was changed from +90 to −90 degrees while the pair of G41C vacuum relays were switched.

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FIGURE 7. Woodward’s Mach-2MHz MLT Test Article, vxB core & March’s Test Stand.

A smaller EMI signal was also detected in this new configuration even with the improved grounding and shielding

system installed between the MLT-2004 and the Mach 2MHz tests, but it was less than −0.020 grams indicated with

60 Watts going to the Mach-2MHz MLT while operating at 2.13 MHz.

FIGURE 8. Mach-2MHz MLT Test Results - Predicted Thrust is 1.3 / 5.0 Milligram-Force.

After each Mach-2MHz test, a 1.00 gram lab weight was lowered onto the test article’s Faraday shield and then

lifted off to provide a secondary calibration of the displayed force signal, (see right hand side of Figure 8).

This first-light Mach-2MHz test result provided thrust that equaled 0.24% of this MLT’s suspended mass with

approximately 50 Watts rms of 3.8 MHz RF power supplied to the test article with a VSWR of approximately 3-to-

1. As test time at the fundamental resonant frequency of the Mach-2MHz MLT (2.13 MHz) on the order of minutes

accumulated on this test article though, the thrust output decreased and stabilized between the ranges of ± 0.14 to ±0.060 gram-force (±1.37 to ±0.59 mN) dependent on the applied voltage for several tests with a typical test shown in

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Figure 8. But this reduction in thrust continued until no recorded vxB force output could be detected with March’s

lab equipment.

This decrease in vxB force output verses operating time has been seen by Woodward in his ~50 kHz ceramic

capacitor based vxB test articles as well, and it is considered to be an ageing effect inherent in the BaTiO3 based

ceramic dielectrics used in these vxB devices. It can be reversed by letting the test article rest for several days, or if it is baked for several hours above the Curie temperature of the dielectric in use, provided the vxB test article can

handle without damage the higher temperatures involved in this process.

CONCLUSION

This paper presented the development of a MLT mathematical model and the tentative substantiation of

Woodward’s observed mass fluctuations. Discrepancies of at least an order of magnitude were noted between the

predicted (low) and recorded (high) data from the two HF test articles examined. The two reviewed MLT tests used

different load sensors and test set-ups which lends weight to their results and adds to the existing work of Brito

(2003, 2005), Mahood (1999a), Mahood, March and Woodward, (2001), VanDeventer and Woodward, (2006) and

Woodward (2003, 2004a; 2004b, 2005). If one looks at all these test results together, they make a strong case for the existence of mass fluctuations and their possible application in implementing recycled propellant propulsion.

More laboratory work will be performed in the future to explore the developed math model’s predictions for MLTs

operating at higher operating frequencies (4 – 108 MHz) and voltages, (> 1 kV). In this regard, one of us (March) is

working on the Mach-10MHz resonant L-C test article that is based on the High Energy Corp.’s HT50 high current,

transmitter duty, N750 (TiO2) ceramic capacitors to see if its performance falls within the range predicted by the current W-E performance math model (27.52 gram-force with 1,600 V-peak across the caps), or whether it falls

above or below the vxB thrust results obtained from these first two high frequency BaTiO3 ceramic cap based vxB

test articles. Since the Mach-10MHz MLT uses untried TiO2 ceramic dielectric material due to its much lower

dissipation factor and ageing qualities, (0.1% verses 2.0% for the Y5R dielectric), the actual thrust production from

the Mach-10MHz MLT is uncertain and could be much lower than predicted due to its much stiffer crystal lattice.

Palfreyman is also readying a test at VHF frequencies (~100 MHz) using high-K (εr = ~15,000) microwave BaTiO3

based capacitors, which is predicted to achieve MLT thrusts that are much larger than the MLT-2004’s.

NOMENCLATURE

<…> = time averaged value over one complete cycle

L = length in meters (m)

M = mass in kilograms (kg) T = time in seconds (s)

G = Newton’s gravitational constant = 6.6742(10) x 10-11 m3 * kg-1 * s-2

c = speed of light in vacuum = 2.999792458 x 108 m * s-1 = 1 / (µ0 * ε0)1/2

ε0 = permittivity of free space = 8.854187187 x 10-12 Farad * m-1

µ0 = permeability of free space = 4π x 10-7 Henry * m-1

φ = scalar potential of the gravitational field (m2 * s-2)

ρ0 = proper mass density of capacitor dielectric (kg * m-3)

m0 = proper mass of capacitor dielectric (kg)

δm = proper mass variation of the capacitor dielectric (kg) P = power applied to capacitors (Watt = Joule * s-1)F = force or thrust (Newton = N = 1.0 kg * m * s-2)

mN = force in milli-Newton (10-3 * N)

µN = force in micro-Newton (10-6 * N) d = thickness of capacitor dielectric between electrodes (m)

I = displacement current through capacitor (Ampere = Amp = Coulomb * s-1

B = magnetic field going through capacitor dielectric (Tesla = T = N * Amp-1 * m-1)

ω = radian frequency of applied voltage and current (rad * s-1)

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C = capacitance of the capacitor in Farads (F = Coulomb * Volt-1)

E = energy stored in capacitor dielectric (J = Joule = Newton * m)

V = peak voltage across capacitor’s electrodes (Volt = V = 1.0 Joule * Coulomb-1)

ν = volume of capacitor dielectric (m3)

A = area of capacitor electrodes (m2)

εr = relative permittivity of capacitor dielectric

µr = relative permeability of vxB magnetic core material L = inductance of B-field coils in Henries (Henry = H = Weber * Ampere-1)

α = magnetic core’s cross sectional area (m2)t = number of turns per unit length of solenoid coil (m-1)

l = total length of solenoid coil (m) N = total number of turns of solenoid coil (= t * l)

λ = fraction of B-field delivered to the capacitor’s dielectric = 1 – (loss fraction)

ACKNOWLEDGMENTS

Many thanks go out to Dr. Jim Woodward of CSUF, Harold (Sonny) White, David Fletcher, Graham O’Neil, Tom

Mahood, Sue, Ellen and Ryan March, and Liz Palfreyman, for providing many related discussions, pointers, techniques and forbearance while preparing the data needed to write this paper.

REFERENCES

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Brito, H. H. and Elaskar, S. A., “Overview of Theories and Experiments on Electromagnetic Inertia Manipulation Propulsion,” in

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Conference Proceedings 746, Melville, New York, 2005.

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