Predictions of microwave breakdown in rf structures

Predictions of microwave breakdown 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 Predictions of microwave breakdown in rf structures 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 1

Predictions of microwave breakdown in rf structures Outline of the Presentation 1. Introduction (motivation and brief description of the breakdown phenomenon).. 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 in rf structures Outline of the Presentation (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 rf filters. 7. Conclusions.

Introduction. Progress in communication systems Increase in power and frequency bandwidth 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 phenomena vacuum conditions: l >> L 0 length of electron free path Vacuum discharge (multipactor effect) 0 secondary electron emission from the walls l [ cm] gas environment: l << L 0 3 10 p[torr] size of device gas pressure Gas discharge (corona effect) electron impact ionization of neutral gas molecules 3

Microwave breakdown phenomena Vacuum discharge (multipactor effect) Gas discharge (corona effect) W e Necessary condition There should be electrons with energy > W1 0 100eV W e > W i We 10 0eV first cross-over point of SEY curve ionization energy Study of electron interaction with microwave field Electron acceleration in microwave field under vacuum conditions max W e 4 W mv W ω ω = Necessary distance: ω E ω ω ee V ω ω = mω Λ = πv ω ω 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 4 ω W > W1 L > Λ 4

Electron heating in microwave field in the presence of collisions with neutral molecules In gaseous environment the interaction of electrons with microwave field depends on extra parameters: Collision frequency ν Energy loss factor δ ν p in air: gas pressure ν [s -1 ] = 5 10 9 p[torr] δ depends on the sort of gas δ (1 3) 10-3 for gases with singleatom molecules δ (1 3) 10 - for gases with multi-atom molecules Electron heating in microwave field in the presence of collisions with neutral molecules rarefied gas: ν << ω dense gas: ν >> ω average gain of electron energy between two collisions: W e W ω W e W ω ω ν ω W W ω average e electron energy ( ) δ ω + ν W e δ ( W δ ) ( ω ν ω ) W ω W e Operation of rf device under gas environment is potentially dangerous from the breakdown point of view if W e > 5 ev L > l δ 0 5

Basic theory of microwave gas breakdown Evolution of electron density n is governed by ionization, attachment and diffusion processes: n = t ( Dn) + ( ν )n i ν a ν i is the frequency of impact ionization of neutral particles ν a is the frequency of electron attachment to neutral particles D is diffusivity of free electrons in a gas D 1 when ν i and ν a depend on average electron energy and are proportional to gas pressure when is fixed p W e W e is fixed Basic theory of microwave gas breakdown n = t Boundary conditions: ( Dn) + ( ν )n n = 0 i ν a at solid surface Breakdown criterion: an exponential increase of electron density in time n exp( γ t) γ > 0 γ > 0 t in case of CW operation t in pulse operation regime with pulse duration 6

Simplified model of microwave gas breakdown The simplified model of gas breakdown is based on the approximation of spatially uniform distribution of microwave field intensity Breakdown criterion: increase in diffusion losses since D 1 γ > 0 t γ = ν i ν a ν d, ν d = D L E Paschen curve p ω p[torr] ~ f [GHz] p estimate of electron loss rate due to diffusion Eω p W e since ( E ν ) ω Simplified model of microwave gas breakdown The main limitation of the simplified model 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. 7

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: electron mass d r e dr = ee + H dt c dt m electron charge electric field magnetic field 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. Basic theory of multipactor discharge Necessary condition: There should be electrons with energy Secondary emission yield curve 1 W e > W 1 W1 W Multipactor criterion: an exponential increase of average electron number in time N exp γ > 0 We ( γ t) 8

Simplified 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. resonance trajectory cross-section of rectangular waveguide trajectory with small deviation of initial position Simplified model of multipactor discharge The main limitations of the model A spread in initial velocities of the secondary electrons is neglected A typical situation in systems with complicated geometry is spatial instability of resonance trajectories Considerable reduction of prediction accuracy for resonances of high order The resonance trajectory completely changes character The only way to obtain reliable predictions of the multipactor threshold is extensive numerical simulations 9

Requirements 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 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 10

Software SEMA. Electromagnetic calculations. Model space configuration: rectangular configuration; the filter is symmetric relative to the plane YZ; Assumptions: y 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. x z Software SEMA. Variable parameters. Variable filter parameters: number of sections; height X, length Z of each section; width Y (the filter dimension along y axis) Variable field parameters: field frequency f [GHz]; input wave polarization height X length Z set by a user x y set by a user z 11

Software SEMA. Input wave polarization. Quasi two-dimensional cases: the structure of the electromagnetic field is fixed in the y-direction TE n0 -mode (n = 1, 3, 5, ) a = height X filter cross-section x b = width Y y E 0 y = E ~ 0 y nπ x cos a independently of the relation between the values of a and b inside the filter: E x, E z = 0 E y does not depend on y coordinate Software SEMA. Input wave polarization. TE n1 -mode (n = 0,, 4, ) transversal structure of electric field x E E 0 y 0 x = = E nb a ~ 0 x ~ E 0 x nπ x π y cos cos a b nπ x π y sin sin a b a = height X y inside the filter: E y sin π y E x, E cos z ( b) ( π y b) b = width Y 1

Simulation of the diffusion problem Variable parameters: Input Power [W] Gas Pressure p [Torr] Sort of gas: (i) Air a 4 p 16 3 7 Eeff 4 10 p D = 10 6 p; ν = 6 10 ; ν i = 100p (ii) He, Ne, Ar, Kr, Xe, D ν = 5 10 E eff set by a user -1 [ cm s], p[ Torr], ν[ s ] -1 [ V cm ], ν [ s ] E eff 9 p i, a ν = E ν + ω N, O, H, Cl, F, HCl, CF 4, SiH 4, CH 4, SF 6 Simulation of the diffusion problem He, Ne, Ar, Kr, Xe, N, O, H, Cl, F, HCl, CF 4, SiH 4, CH 4, SF 6 Singlo Data base, CPAT and Kinema Software, http://singlo-kinema.com n = t ( Dn) + ( ν )n i ν a Initial and boundary conditions: n = 1 in the whole volume FDTD scheme is used for simulation of the diffusion problem of the filter at t = 0; n = 0 on the metal walls; n/ z = 0 in the entrance and exit filter cross sections. 13

Calculation of electromagnetic problem Numerical calculations of electromagnetic field inside a filter are based on mode-expansion algorithm. The electromagnetic field in a filter section can be presented as a sum of Eigen modes of this section. A A + n n E + { An ( z) An ( z) } = + n E amplitudes of the waves: out = An zs + exp ikn z zs in = An ( zs + ) exp ikn( zs z) ( ) ( ( )) z s z z n ( ) s+1 vector function E n describes the transverse structure of the eigen modes for the filter section z z z k n s s+1 are propagation constants The complete set of equations for the mode amplitudes A n includes: in = ˆ out A zs+ 1 P+ A zs + 1) propagation equations: in + = ˆ out A zs P A zs+ 1 ) transmission-reflection equations: A A Mode-expansion expansion algorithm out out ( ) ˆ in( ) ˆ in z = R A z + + T A ( z ) s + + s + ( z ) =... s A out ( z ) s z = z s A out ( ) ( ) ( ) ( ) s ( z + ) A in( zs+1 ) A in A in out ( zs ) ( zs + ) A ( z ) s s+1 14

Mode-expansion expansion algorithm Matrices of propagation In particular, Pˆ P ˆ, ˆ + P are known. { exp( ih ( z z ))} + n s+ 1 Matrices of reflection R ˆ +, R ˆ and transmission T ˆ, ˆ can be calculated using: 1) the continuity conditions for electromagnetic field components in the junction-planes of filter sections; ) the boundary conditions for the mode amplitudes: in out A z = A z + A A in s + T ( 1 ) ( 1 ) = input _ wave out ( z ) = A ( z ) 0 N + 1 + N + 1 = Advantages of the mode-expansion 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; ) 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 15

Software SEMA. Additional advantages of the software SEMA: 1) Visualization of electric field distribution ) 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. Electrodynamics. The software 1) Electrodynamic part. Calculation of dispersion matrix: parameters S 11, S 1 has been tested in different manners. SEMA Courtesy of Alcatel Alenia Space SPAIN ideal circuit simulation (CNES) 16

Software SEMA. Breakdown simulations. ) 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 1 hpa (~ 9Torr) was found to be 410 W The simulation result: 10 Torr, 410 W breakdown! 10 Torr, 400 W no breakdown Courtesy of Alcatel Alenia Space FRANCE Software MulSym. Multipactor simulations. A software MulSym 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. MulSym 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; ) the action of the rf magnetic field on the electron motion. The software MulSym is currently being tested. 17

Software SEMA. Demonstration. Setting of filter parameters. Example. Courtesy of Alcatel Alenia Space SPAIN Software SEMA. Demonstration. Test of CNES-filter Frequency = 11.7 GHz Simulation of diffusion problem in the fourth resonator Air: Power = 3 W; Pressure = 8 Torr; Breakdown! 30 W & 8 Torr; No breakdown! Experimental result: threshold level -- 30 W Courtesy of Alcatel Alenia Space SPAIN 18

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 in these fields. Present work. 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) 19

Present work. 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. 0