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Predictions of microwave Predictions of microwave
breakdown in rf structuresbreakdown in rf structures
I. Nefedov, I. Shereshevskii
D. Dorozhkina, E. Rakova, V. Semenov
D. Anderson, M. Lisak, U. Jordan, R. Udiljak
J. Puech
T. Olsson
Institute of Physics for Microstructures RAS, Nizhny Novgorod, Russia
Institute of Applied Physics RAS,
Nizhny Novgorod, Russia
Chalmers University of Technology, Göteborg, Sweden
Centre National d'Etudes Spatiales, Toulouse, France
Powerwave Technologies, Täby, Sweden
Predictions of microwave breakdown Predictions of microwave breakdown
in rf structuresin rf structures
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Outline of the PresentationOutline of the Presentation
1. Introduction (motivation and brief description of the breakdown phenomenon).
2. Electron interaction with microwave field (rough estimates of parameters which are dangerous from the breakdown point of view).
3. Basic theories of multipactor and corona breakdown.
4. Simplified models and their limitations.
Predictions of microwave breakdown Predictions of microwave breakdown
in rf structuresin rf structures
Outline of the PresentationOutline of the Presentation
(continuation)(continuation)
5. Detailed numerical simulations are necessary for accurate predictions of the breakdown threshold.
6. Demonstration of the software developed to simulate the corona and the multipactor effects inside rfrf filters.
7. Conclusions.
Predictions of microwave breakdown Predictions of microwave breakdown
in rf structuresin rf structures
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Increase in power
and frequency bandwidth
Progress in communication systems
Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
IntroductionIntroduction..
Design of more compact devices
More complicated geometries
&& Higher intensity of rf field
Higher risk of breakdown &&
More complicated prediction of breakdown threshold
Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Microwave breakdown phenomenaMicrowave breakdown phenomena
vacuum conditions:
Ll >>0
gas environment:
Ll <<0
[ ][Torr]
102cm
3
0p
l−⋅
≈length of electron
free path gas pressure
size of device
Vacuum discharge (multipactor effect)
secondary electron emission from the
walls
Gas discharge (corona effect)
electron impact ionization of neutral
gas molecules
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Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Microwave breakdown phenomenaMicrowave breakdown phenomena
Vacuum discharge (multipactor effect)
Gas discharge (corona effect)
Necessary conditionNecessary condition
There should be electrons with energy eW
eV100201 ÷≈> WWe eV2010 ÷≈> ie WW
first cross-over
point of SEY curveionization energy
Study of electron interaction with Study of electron interaction with
microwave fieldmicrowave field
Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Electron acceleration in microwave Electron acceleration in microwave
field under vacuum conditions field under vacuum conditions
2
4max
∝⋅≤ωωωω
ωωωωωωωω
EWWe
2
2ωωωω
ωωωωmV
W =ωωωωωωωω
ωωωωm
eEV =
electric field amplitude
circular frequency
of the field
electron oscillatory velocity
Operation of a vacuum rf device is potentially dangerous from the multipactor point of view when Λ>
>
L
WW 14 ωωωω
ωωωωππππ ωωωωV=ΛNecessary
distance:
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Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Electron heating in microwave field in the Electron heating in microwave field in the
presence of collisions with neutral moleculespresence of collisions with neutral molecules
In gaseous environment the interaction of electrons with microwave field depends on 2 extra parameters:
Collision frequency νννν Energy loss factor δδδδ
p∝ννννgas pressure
in air:
νννν [s-1] = 5⋅109⋅⋅⋅⋅p[Torr]
δδδδ depends on the sort of gas
δδδδ ≈≈≈≈ (1÷3)⋅10-3
δδδδ ≈≈≈≈ (1÷3)⋅10-2
for gases with single-atom molecules
for gases with multi-atom molecules
rarefied gas: ωωωωνννν <<
average gain of electron energy between two collisions:
ωωωωWWe ≈∆
dense gas: ωωωωνννν >>
( )22 ννννωωωωωωωω ⋅≈∆ WWe
Operation of rf device under gas
environment is potentially dangerous
from the breakdown point of view if δδδδ0
eV52
lL
We
>
÷>
22
2
ννννωωωω
ωωωω
δδδδωωωω
+≈
WWe
δδδδωωωωWWe ≈ ( ) ( )22 ννννωωωωδδδδωωωω ⋅≈ WWe
Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Electron heating in microwave field in the Electron heating in microwave field in the
presence of collisions with neutral moleculespresence of collisions with neutral molecules
average
electron energy
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Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Basic theory of microwave gas breakdownBasic theory of microwave gas breakdown
Evolution of electron density n is governed by
ionization, attachment and diffusion processes:
( ) ( )nDnt
nai νννννννν −+∇=
∂
∂ 2
and depend on average electron energy and are proportional to gas pressure when is fixed
iνννν aνννν
eW
aνννν is the frequency of electron
attachment to neutral particles
iνννν is the frequency of impact
ionization of neutral particles
D is diffusivity of free
electrons in a gas
pD 1∝
eWwhen is fixed
Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Basic theory of microwave gas breakdownBasic theory of microwave gas breakdown
( ) ( )nDnt
nai νννννννν −+∇=
∂
∂ 2
Boundary conditions: at solid surface0=n
Breakdown criterion: an exponential increase of
electron density in time ( )tn γγγγexp∝
0>γγγγ in case of CW operation
t∆> 20γγγγt∆
in pulse operation regime with pulse duration
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Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Simplified model of microwave gas breakdownSimplified model of microwave gas breakdown
Breakdown criterion: t∆> 20γγγγ
The simplified model of gas breakdown is basedon the approximation of spatially uniform distribution of microwave field intensity
2, LDddai =−−= ννννννννννννννννγγγγ
estimate of
electron loss
rate due to
diffusion
p[Torr] ~ f [GHz] p
sincepE ∝ωωωω
( )2ννννωωωωEWe ∝
increase in
diffusion
losses since
pD 1∝
ωωωωE Paschen curve
The accuracy is not high enough especially in
systems with complicated geometry.
The only way for reliable predictions of the
breakdown threshold is to solve numerically the
diffusion problem for the electron density.
The main limitation of the simplified model
Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Simplified model of microwave gas breakdownSimplified model of microwave gas breakdown
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Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Basic theory of multipactor discharge Basic theory of multipactor discharge
In contrast to the gas discharge theory the theory of multipactor consider the motion of separate electrons in the microwave field:
×+= Hr
Er
dt
d
c
ee
dt
dm
2
2
electron
mass
electric field magnetic fieldelectron charge
Boundary conditions:
Collisions of an electron with a solid surface is accompanied by emission of new secondary electrons. The number of secondaries depends on the energy of the primary electrons.
( )
0
exp
>
∝
γγγγ
γγγγ tN
Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Basic theory of multipactor discharge Basic theory of multipactor discharge
Secondary emission yield curveSecondary emission yield curve
1
1W 2W eW
Multipactor criterion: an exponential increase of average electron number in time
Necessary Necessary
condition:condition:
There should be electrons with energy
1WWe >
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Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Simplified model of multipactor dischargeSimplified model of multipactor discharge
The simplified model of multipactor is based on the resonance concept
The multipactor is related to such electron trajectories which are repeated periodically in time.
Main requirements:the resonance trajectory
must be stable with respect to small deviations of initial time and position.
cross-section of
rectangular waveguide
trajectory with
small deviation of
initial position
resonance
trajectory
A spread in initial velocities of the secondary electrons is neglected
Considerable reduction of prediction accuracy for resonances of high
orderA typical situation in systems
with complicated geometry is spatial instability of resonance trajectories
The resonance trajectory completely
changes character
The only way to obtain reliable predictions of the multipactor threshold is extensive
numerical simulations
The main limitations of the model
Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Simplified model of multipactor dischargeSimplified model of multipactor discharge
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Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Requirements on numerical algorithmsRequirements on numerical algorithms
Main requirements on software
for breakdown simulations
Corona
• Accurate calculations of the spatial distribution
of the rf electric field intensity
• Accurate solution of the diffusion problem
for the electron density
Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Requirements on numerical algorithms Requirements on numerical algorithms
Main requirements on software
for breakdown simulations
Multipactor
• Accurate calculations of the rf electric and
magnetic field strengths
• Calculations of electron trajectories taking into
account a spread of electron initial velocity
and the action of the rf magnetic field on the
electron motion
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Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Software SEMA.Software SEMA. Electromagnetic calculations.Electromagnetic calculations.
Model space configuration:
• rectangular configuration;
• the filter is symmetric
relative to the plane YZ;
Assumptions:
The filter walls are assumed to be perfectly conducting (i.e. the tangential component of the
electric field is assumed to be zero on the walls).
The entrance and exit cross-sections of the filter are supposed to be nonreflecting.
y
x
z
Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Software SEMA.Software SEMA. Variable parametersVariable parameters..
Variable filter parametersVariable filter parameters:
• number of sections;
• height X, length Zof each section;
• width Y (the filter dimension along y axis)
Variable field parametersVariable field parameters:
• field frequency f [GHz];
• input wave polarization
set by a user
length Z
x
he
igh
t X
z y
set by a user
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Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Software SEMA. Software SEMA. Input wave polarizationInput wave polarization..
Quasi two-dimensional cases: the structure of
the electromagnetic field is fixed in the y-direction
TEn0
-mode (n = 1, 3, 5,…)
=a
xnEE yy
πcos
~00
x
y
a=
heig
ht X
b = width Y
independently of the relation between the
values of a and b
inside the filter: 0, =zx EE
does not depend on
y coordinate
yE
filter cross-section
Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Software SEMA. Software SEMA. Input wave polarizationInput wave polarization..
TEn1
-mode (n = 0, 2, 4,…)
=b
y
a
xnEE xx
ππcoscos
~00
=b
y
a
xnE
a
nbE xy
ππππππππsinsin
~00
a=
heig
ht X
b = width Y
x
y
transversal structure of electric field
inside the filter:
( )byE y πsin∝
( )byEE zx πcos, ∝
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Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Simulation of the diffusion problemSimulation of the diffusion problem
Variable parametersVariable parameters:
• Input Power [W]
• Gas Pressure p [Torr]
• Sort of gas:
(i) Air
set by a user
;106pD = ;106 4
pa ⋅⋅=ν316
7
100104
⋅⋅⋅=
p
Ep
effiν
22
222
2 ων
ν
+⋅=
EEeff
p⋅⋅= 9105ν
[ ] [ ] [ ][ ] [ ]1-
,
1-2
s,cmV
s,Torr,scm
aieffE
pD
ν
ν
(ii) He, Ne, Ar, Kr, Xe,
N2, O2, H2, Cl2, F2, HCl, CF4, SiH4, CH4, SF6
( ) ( )nDnt
nai νννννννν −+∇=
∂
∂ 2
n = 0 on the metal walls;
∂n/∂z = 0 in the entrance and exit filter cross sections.
Initial and boundary conditions:
n = 1 in the whole volume
of the filter at t = 0;
Singlo Data base, CPAT and Kinema Software,
http://singlo-kinema.com
Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Simulation of the diffusion problemSimulation of the diffusion problem
He, Ne, Ar, Kr, Xe,
N2, O2, H2, Cl2, F2, HCl,
CF4, SiH4, CH4, SF6
FDTD scheme is used for
simulation of the diffusion problem
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Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Calculation of electromagnetic problem Calculation of electromagnetic problem
The electromagnetic field in a filter section can be
presented as a sum of Eigen modes of this section.
( ) ( ){ }∑ −+⊥ +=
nnnn zAzA EE
vector function En
describes the
transverse structure
of the eigen modes
for the filter section
1+≤≤ ss zzz
amplitudes of the waves:
( ) ( )( )snsoutnn zzikzAA −+=+ exp
( ) ( )( )zzikzAA snsinnn −+=− exp
1+≤≤ ss zzzare propagation
constantsnk
Numerical calculations of electromagnetic field inside a filter are based on mode-expansion algorithm.
( ) ( ) ( )
( ) ...
ˆˆ
=−
−++=+ ++
sout
sin
sin
sout
zA
zATzARzA
The complete set of equations for the mode
amplitudes An includes:
1) propagation equations:
2) transmission-reflection equations:
( ) ( )
( ) ( )−=+
+=−
+−
++
1
1
ˆ
ˆ
sout
sin
sout
sin
zAPzA
zAPzA
szz =
( )−+1sin
zA
( )+sin
zA
( )+sout
zA
( )−sout
zA
( )−sin
zA ( )−+1sout
zA
Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
ModeMode--expansion algorithmexpansion algorithm
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Matrices of propagation are known.
In particular,
−+ PP ˆ,ˆ
( )( ){ }ssn zzihP −↔ ++ 1expˆ
Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
ModeMode--expansion algorithmexpansion algorithm
Matrices of reflection and transmission
can be calculated using:
1) the continuity conditions for electromagnetic field
components in the junction-planes of filter
sections;
2) the boundary conditions for the mode
amplitudes:
−+ RR ˆ,ˆ−+ TT ˆ,ˆ
( ) ( ) waveinputoutin
AzAzA _11 =+=−
( ) ( ) 011 =−=+ ++ Nout
Nin
zAzA
Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Advantages of the modeAdvantages of the mode--expansion algorithm expansion algorithm
The advantages of the mode-expansion algorithm
compared with grid-methods of electromagnetic
calculations:
1) no problem of scale matching of grids in the
junction-planes of the filter sections;
2) no need for numerical calculations of
electromagnetic field inside filter sections since
analytical solutions are available there.
The code based on the mode-expansion algorithm demonstrates high computational speed
& high accuracy of calculations
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Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Software SEMA.Software SEMA.
Additional advantages of the software SEMA:
1) Visualization of electric field distribution
2) The possibility to chose one (or several) filter section for calculations of the diffusion problem.
It significantly reduces the computing time for calculating the electron avalanche growth (if any) and increases the accuracy of these calculations.
Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Software SEMA. Software SEMA. Electrodynamics.
The software has been tested in different manners.
1) Electrodynamic part.
Calculation of dispersion
matrix: parameters S11, S12
SEMA
ideal
circuit simulation
(CNES)
Courtesy of
Alcatel Alenia Space
SPAINSPAIN
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Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Software SEMA. Software SEMA. Breakdown simulations.
2) Breakdown simulations. Comparison with experimentally observed data for breakdown thresholds.
In particular, measurements of corona breakdown was
made in a waveguide switch used at CNES:
the threshold at the pressure 12 hPa (~ 9Torr)was found to be 410 W
Courtesy ofAlcatel Alenia Space
FRANCEFRANCE
The simulation result:
10 Torr, 410 W – breakdown!breakdown!
10 Torr, 400 W – no breakdown
Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Software Software MulSymMulSym. . Multipactor simulations.
A software MulSymMulSym for simulations of multipactor problem in a rectangular filter has been developed.
It is based on the Monte-Carlo algorithm.
The electric field distribution in the filter is computed by SEMA.
MulSymMulSym makes it possible the study the development of multipactor discharges in real time in different filter sections. The software takes into account:
1) a spread of initial velocities of secondary electrons;
2) the action of the rf magnetic field on the electronmotion.
The software MulSymMulSym is currently being tested.
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Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Software SEMA. Demonstration.Software SEMA. Demonstration.
Setting of filter parameters. Example.
Courtesy of
Alcatel Alenia Space
SPAINSPAIN
Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Software SEMA. Demonstration.Software SEMA. Demonstration.
Test of “CNESCNES--filterfilter””
Frequency = 11.7 GHz
Simulation of diffusion problem in the fourth resonator
Air:Power = 32 W;Pressure = 8 Torr;
Breakdown!Breakdown!
30 W & 8 Torr;No breakdown!
Experimental result:threshold level -- 30 W
Courtesy of
Alcatel Alenia Space
SPAINSPAIN
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Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
Conclusions.Conclusions.
• Classical understanding and modelling of corona and multipactor breakdown is not sufficient for accurate prediction of breakdown thresholds in many modern rf structures and communication scenarios.
• Main complications are caused by:(i) electric field inhomogeneity (complicated
geometries);
(ii) time varying electric field (multi-carrier operation).
• Breakdown predictions require:Finding the electric & magnetic fields in the rf device.Solving for the evolution of the electron density inthese fields.
Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
PresentPresent workwork..
• Numerical code have been developed for:
(i) finding the electric and magnetic fields in
wave guide configurations built on
“rectangular” elements;
(ii) determining the evolution of the electron
density in the corresponding fields.
Corona discharges (the diffusion equation)
Multipactor discharges (electron trajectories)
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Predictions of microwave breakdown in rf structuresPredictions of microwave breakdown in rf structures
PresentPresent workwork..
• The codes have been applied to study:
(i) corona breakdown in different filters and
around sharp corners and wedges
(ii) multipactor in rectangular wave guides,
wave guides with irises, coaxial lines, etc.
(iii) multipactor in multi-carrier operation
The presented work represents a significant step
forward in the predictive modelling of breakdown
in geometrically complicated rf devices and for
multi-carrier operation scenarios.