EmDrive Basic Theory
1
EmDrive Theory
Fg1
T = Q(Fg1-Fg2)
Microwave energy is fed to a specially tapered
resonator,which produces a net thrust from the large
number of EM reflections in the high Q structure.
(typically 5x104)
Fg1>Fg2 as guide velocity Vg1>Vg2
Fg2
Static Thrust given by:
Magnetron
2 P QS
T 0 0
c
 0


 0

 g1 g 2




2
Derivation of Basic Thrust Equation
From principle of operation
T  QFg1  Fg 2 
now
Fg 
2 P 0
c g
Ref: A. L. Cullen
Thus force is inversely proportional to guide wavelength
Then
0 
2 PQS 0  0
T

c  g1 g 2 
where

 02 

S 0  1 
 g1g 2 
1
S0 corrects for Einstein’s equation for the addition of relativistic velocities
Relationship between guide wavelength and waveguide geometry is non-linear
3
Axial Component of Sidewall Force
F
Area
A2
F2
1
At first sight it might appear that because
A1= A2+A3 then F1= F2+F3
F3
Area A1
However as the relationship between F and A
is highly non-linear,the cavity can be
designed to maximise F1/F2 and minimise
F3/F1
Area
A3
Detail design requires software developed
by SPR Ltd and is included in IPR
Guide Wavelength for Circular TM01 at 2GHz
5500
However intuitively it can be seen that at
cut-off :
5000
4500
Guide wavelength (mm)
4000
Guide wavelength is infinity
3500
3000
Group velocity and force is zero
2500
Area A2 is clearly not zero
2000
1500
1000
500
0
0
50
100
150
200
250
300
350
400
450
500
Waveguide Diam eter (m m )
4
Conservation of Momentum
Vg = 0.99c
Waveguide velocity = 0.5c
Vg measured inside waveguide = 0.99c
Vg measured outside waveguide = 0.99c (not 1.49c !)
This applies if the open end wall is now closed
An EM wave propagated inside a CLOSED waveguide is an OPEN system
Momentum can therefore be exchanged between the EM wave and the waveguide end walls
Momentum is therefore conserved
5
Conservation of Momentum
Assume that at velocity c, the EM wave has
momentum 
At initial propagation 1 = 0.4  and  2 = - 0.4 
Then  1 +  2 = 0
w
At end plates  1 = 0.9  and  2 = - 0.1 
1
2
Then momentum lost by EM wave at end plates
= 2(  1 +  2 ) = 1.6 
Momentum gained by waveguide
 w = - 1.6 
Then  w +2(  1 +  2 ) = 0
Vg2 = .1c
Vg0 = .4c
Vg1 = .9c
6
Conservation of Energy
Doppler Shift for Fo =4GHz, Qu = 5x107
40
35
Ff
30
Fr
25
20
Vg2
15
Vg1
Doppler Shift (Hz)
10
Vr
Vf
5
m otor
0
generator
-5
-10
-15
Cavity Acceleration
-20
-25
-30
-35
Cavity acceleration produces
unequal Doppler Shifts in Ff and Fr
during each wavefront transit.
Doppler Mathematical model
illustrates Doppler shift for both
Motor and Generator modes.
-40
0
5
10
15
20
25
30
35
40
45
50
Transit Num ber x 107
+ve acceleration gives a frequency decrease and hence an energy loss
(motor)
-ve acceleration gives a frequency increase and thus an energy increase
(generator)
7