Predictions of microwave breakdown in rf structures

1

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

2

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.



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.



3

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


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

4


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


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:

5


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


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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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


Basic theory of microwave gas breakdownBasic theory of microwave gas breakdown

( ) ( )nDnt


∂

∂ 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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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


Simplified model of microwave gas breakdownSimplified model of microwave gas breakdown

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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


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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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


Simplified model of multipactor dischargeSimplified model of multipactor discharge

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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


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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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


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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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


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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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


∂

∂ 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


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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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


ModeMode--expansion algorithmexpansion algorithm

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Matrices of propagation are known.

In particular,

−+ PP ˆ,ˆ

( )( ){ }ssn zzihP −↔ ++ 1expˆ


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


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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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.


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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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


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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Software SEMA. Demonstration.Software SEMA. Demonstration.

Setting of filter parameters. Example.

Courtesy of


SPAINSPAIN


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


SPAINSPAIN

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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.


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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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.

Documents

Predictions of microwave breakdown in rf structures