Particle-In-Cell Simulation Tools for Design and Optimization of High Power CW Gyrotron Oscillators

Transcription

1 Particle-In-Cell Simulation Tools for Design and Optimization of High Power CW Gyrotron Oscillators S. Illy, E. Borie, G. Dammertz, B. Piosczyk, M. Schmid Forschungszentrum Karlsruhe, Institut für Hochleistungsimpuls- und Mikrowellentechnik, Association EURATOM-FZK, Hermann-von-Helmholtz-Platz 1, Eggenstein-Leopoldshafen, Germany Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 1 / 37

2 Outline Introduction Available PIC codes at FZK Design of Magnetron Injection Guns Simulation of the Gyrotron interaction Time dependent simulation of beam instabilities Collector simulation: conventional sweeping Collector simulation: sweeping with rotating magnetic field Conclusion Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 2 / 37

3 Introduction Numerical simulation tools are indispensable for the design and optimisation of high power CW Gyrotrons Examples: 14 GHz 1MW Gyrotron for W7-X 17 GHz 2MW coaxial Gyrotron for ITER The following parts require the use of Particle-in-Cell (PIC) simulation tools: 1. Magnetron Injection Gun (MIG). Goal: exact beam parameters (position, velocity ratio), low energy- and velocity spread, no instabilities. 2. Collector. Goal: acceptable average and peak power densities on the collector wall. (e.g. 2.4 MW of dissipated power in the case of the 17 GHz 2 MW coaxial Gyrotron) Collector sweep coil Hollow electron beam Output window HF Mirrors Launcher Resonator Main coil Compression region Magnetron injection gun Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Introduction 3 / 37

4 Available PIC codes at FZK 2.5D full electromagnetic PIC for simulation of beam instabilities. 2.5D electrostatic PIC for gun & collector design (quite slow). 2.5D raytracing code for gun & collector design (quite fast). 3D raytracing for collector design (with transversal sweeping magnetic field) Additional tools: Grid generation. Calculation of applied magnetic field. Visualisation (with interactive GUI) Transformation: physical to contravariant Maxwell solver: 2D, z r plane Finite difference method, field components are centred in space & time Contravariant representation to permit usage of boundary fitted grids Open boundary conditions according to Higdon (1986) Enforce Gauss Law Transformation: contravariant to physical Forschungszentrum Karlsruhe Particle distribution charge / current density Particle localisation: Iterative scheme (fast, vectorizable) Particle pusher: 2.5D relativistic leapfrog scheme Particle emission: constant current dens., space charge emission or particle injection Interpolation: EM fields on particle positions (area weighting method) Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Available PIC codes at FZK 4 / 37

5 Boundary fitted (non-orthogonal) coordinates Physical grid (for the 17GHz, 2 MW coaxial Gyrotron): Forschungszentrum Karlsruhe Corresponding logical grid: Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Available PIC codes at FZK 5 / 37

6 Design of Magnetron Injection Guns Magnetic system of the 17GHz, 2MW coaxial Gyrotron: Forschungszentrum Karlsruhe r [m] B_za [T] z [m] Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Design of Magnetron Injection Guns 6 / 37

7 Simulation results Particle positions: Self-induced electric field (r-component): Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Design of Magnetron Injection Guns 7 / 37

8 Simulation results (continued) Forschungszentrum Karlsruhe Typical output of the ESRAY-Module (for U = 8 kv, I beam = 75 A): Convergence reached (Iteration No. 13) Statistics for 144 particle(s) of type ["Electrons"]: E_kin [kev]: / ( %), alpha: / ( %), beta_z: / ( %), beta_perp: / (2.163 %), u_perp: / ( %), u_z: / ( %), gamma: / ( %), r [m]:.136 +/ ( %), P_total= e+6W, q_total=-1.5e-11as, I_total=-75A. Dynamic array memory (1D/2D/sum): 1.281M, 34.73M, M. Total CPU: 34.13s ::34 1.% Total elapsed (wall clock): s :: % Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Design of Magnetron Injection Guns 8 / 37

9 Simulation of the Gyrotron interaction The simulation of the Gyrotron interaction is a good verification method: The Gyrotron interaction is a (wanted) time-dependent instability. We have numerical tools to simulate Gyrotron resonators. Even the transient behaviour can be calculated with SELFT, a time-dependent multimode code (S. Kern, 1996) Geometry of the TE,3 -Cavity: o 2 5 o 16.2 mm 2.19 mm 3.6 mm 2 mm 2 mm L 3 mm 2 mm Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Simulation of the Gyrotron interaction 9 / 37

10 Simulation of the Gyrotron interaction: Background Electrons with relativistic Cyclotron frequency TE 28,8 -mode: Ω c = eb m γ GHz B/T 2π28, γ γ = 1 1 ( v c ) 2 ( v ) 1 + eu kev c are accelerated / decelerated by the E ϕ -component of the oscillating TE m,n -mode. Consequence: azimuthal phase bunching and energy transfer to the RF-field, if Ω rf Ω c. (14 GHz, W7-X Gyrotron) Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Simulation of the Gyrotron interaction 1 / 37

11 Simulation of the Gyrotron interaction: Results E ϕ field profile [a.u.] Stationary state: (L = 35 mm, I b = 15A, α = 1.5) T2hPIC SELFT z [m] SELFT vs. PIC: U b = 79. kv, B R = 1.16 T ) backward-wave, transient state. ) forward-wave, transient state. P out [kw] Transient behaviour: (L = 65 mm, I b = 2A, α = 1.) T2hPIC SELFT t [ns] SELFT T2hPIC L I b α Mode P out f P out f [mm] [A] [kw] [GHz] [kw] [GHz] TE,3, TE,3, TE, TE,3, TE,3, Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Simulation of the Gyrotron interaction 11 / 37

12 Development of E ϕ in steps of T HF /8 (L = 35 mm, I b = 2 A, α = 1.5, E ϕ < 7.5 MVm 1 ): Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Simulation of the Gyrotron interaction 12 / 37

13 Time dependent simulation of beam instabilities Structure of the simulation grid (designed for an applied magnetic field of 1 T) Grid length: 9.5 cm Number of grid cells: Beam radius: 1.7 r L Beam thickness: 4r L Time step:.2 ps Number of macro particles: ( per cell) Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Time dependent simulation of beam instabilities 13 / 37

14 Results of PIC simulation Development of E r (z, r) in steps of 2 ps T c /2 I = 1 A, α = 1.5, flat beam profile, no E r,stat, no TE-polarisation graphics range: E r 1 5 V/m 5 cm Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Time dependent simulation of beam instabilities 14 / 37

15 Analytic model of the Electrostatic Cyclotron Instability Combining Ampères law, Faraday s law and j = σ E in Fourier space, we obtain the dispersion relation k ( k E) = ω 2 c 2 ( I + i ) σ E. ωǫ }{{} ǫ ǫ is the dielectric tensor of the magnetised relativistic plasma. Example: " #»sjs(k Z ǫ xx = 1 2π ω2 Z u (1 u ˆf z) + kzu ˆf xr L ) 2 X p u ωγ u z k xr L u du du z. ω k zu z ωγ + sω s= With k = (k x,, k z ) T, E x and E z and E y = we obtain k 2 x ǫ xx + k x k z (ǫ xz + ǫ zx ) + k 2 z ǫ zz ω2 c 2 (ǫ xxǫ zz ǫ xz ǫ zx ) =. Beside other restrictions this dispersion relation is only valid for TM-polarisation (E x, E z, B y ), effects of TE-polarisation are ignored. Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Time dependent simulation of beam instabilities 15 / 37

16 Comparison analytic model PIC simulation Influence of velocity spread (analytic model): Im k z r L k x r L =.5 k x r L =1. δβ z /β z =% Forschungszentrum Karlsruhe.5 δβ z /β z =1% δβ z /β z =5% ω/ω c PIC-Results (I = 1A, α = 1.5) δα/α 1 o =1% Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Time dependent simulation of beam instabilities 16 / 37.9

17 Collector simulation: conventional sweeping Proposed collector for the 17GHz, 2MW Coaxial Gyrotron r [m] r [m] Collector Geometry Grid: 17GHz Coaxial Gyrotron collector [16 x 36] Forschungszentrum Karlsruhe z [m] Magnetic System z [m] B_za [T] Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Collector simulation: conventional sweeping 17 / 37

18 FEMM setup and results More than 11 elements are used. Input Parameters: Wall thickness: 2mm (top plate), 15mm (rest) I AC = 1 A, 6 loops per coil Results show no significant difference to results obtained with ANSYS. Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Collector simulation: conventional sweeping 18 / 37

19 Comparison static vs. harmonic solution (f=7hz, I=5A) Max. magnetic flux density on axis: B z,max = mt (static) B z,max =.15 i 3.24 mt (harm.) Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Collector simulation: conventional sweeping 19 / 37

20 Modeling of the launcher mirrorbox section Forschungszentrum Karlsruhe 3 input particles from self consistent cavity simulation (mono-mode) Beam parameters: E kin = 87.7 kev, I b = 8 A Particles leaving the simulation region will be stored and injected in the collector simulation huge speed-up Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Collector simulation: conventional sweeping 2 / 37

21 <E_kin> [kev] Forschungszentrum Karlsruhe Average of E kin (red) and spread in E kin (green) vs. z Distribution of E kin at cavity exit z [m] Distribution of E kin at collector entrance Spread in E_kin [%] 1.e 11.8e 11 2.e 11 p(e_kin) [a.u.].6e 11.4e 11 p(e_kin) [a.u.] 1.5e 11 1.e 11.2e 11.5e E_kin [kev] E_kin [kev] Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Collector simulation: conventional sweeping 21 / 37

22 Simulation results (averaged power density on collector wall) Common parameters: I b = 8 A, U depr = 33.1 kv, P load = 2.4 MW Average of instantaneous power density distributions for all phases of the applied coil currents Θ < 2π f sweep = 7Hz f sweep = 2 Hz Power density / W/cm Power density / W/cm z / m I sweep,ac = [4, 4, 2, 2, 4,2] A I sweep,dc = A T peak = 28 C, P ps =.38 kw (.58 kva) z / m I sweep,ac = [25, 25, 25, 1, 1, 1] A I sweep,dc =.25 I sweep,ac T peak = 22 C, P ps = 15.5 kw (5.9 kva) Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Collector simulation: conventional sweeping 22 / 37

23 Influence of the hot collector wall T = 15 C, σ = 38MS/m (instead of 58MS/m at 2 C), f sweep = 7Hz Power density / W/cm Power density / W/cm z / m I sweep,ac = [4, 4, 2, 2, 4,2] A I sweep,dc = A z / m I sweep,ac = [24, 24, 12, 12, 24, 12] A I sweep,dc = A Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Collector simulation: conventional sweeping 23 / 37

24 Modeling of secondary emission Forschungszentrum Karlsruhe Theoretical models of secondary emission distinguish between three different types of secondaries (where E is the energy of the incident electron): So-called true-secondary electrons with low kinetic energy (up to 5 ev) and a yield factor that is relatively high. Inelastically backscattered ( rediffused ) electrons with a kinetic energy in the range from zero to E and a moderate yield factor. Elastically backscattered electrons with high kinetic energy close to E but a relative low yield factor. Main reference: M.A. Furman and M.T.F. Pivi, Probabilistic model for the simulation of secondary electron emission, Physical Review Special Topics, Accelerators and Beams, Vol. 5, 22. Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Collector simulation: conventional sweeping 24 / 37

25 Secondary Emission: Simulation results Particle plots of the incident electron beam and generations #1 to #3 of secondary electrons (from top to bottom). Parameters: 8A beam current 33 kv depression voltage 2.4MW power on collector wall Sweeping coil currents: 4/4/2/2/4/2A Phase of applied sweeping coil current: deg. Forschungszentrum Karlsruhe Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Collector simulation: conventional sweeping 25 / 37

26 Instantaneous power density on the collector wall without secondaries with secondaries Power density / W cm z / m Power density / W cm z / m Maximum instantaneous power density at a sweeping phase of 315. A non-negligible reduction of the instantaneous power density can be observed. Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Collector simulation: conventional sweeping 26 / 37

27 Maximum instantaneous power density vs. sweeping phase P max / W cm P max (w/o secondaries) P max (with rediffused el.) P max (with secondaries) Phase of sweeping coil current / deg Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Collector simulation: conventional sweeping 27 / 37

28 Collector simulation: sweeping with rotating magnetic field Uses three pairs of elliptical dipole coils (laterally mounted). Geometry of idealised single loop coils: R = 376 mm,w = 277mm, h = 21 mm. Excitation currents: sinusoidal, f = 5 Hz, phase shifted by 6, I max = 7kA. Advantages: Smaller influence of eddy currents, higher sweeping frequency, low cost power supply (3-phase transformer) Cryostat Mirror box Defocusing dipole coils (3 pairs) Top view Collector R Emitter h w Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Collector simulation: sweeping with rotating magnetic field 28 / 37

29 Instantaneous power density on the collector wall Θ = I loop = 5 ka Θ = 3 Power density on collector wall [W/cm²] Power density on collector wall [W/cm²] Maximum: W/cm² Maximum: W/cm² 4.e+3 4.e e e+3 3.6e+3 3.6e+3 3.4e+3 3.4e e e+3 3.e+3 3.e+3 2.8e+3 2.8e e e+3 2.4e+3 2.4e+3 phi [rad] 3 2.2e+3 2.e+3 1.8e+3 phi [rad] 3 2.2e+3 2.e+3 1.8e+3 1.6e+3 1.6e e+3 1.2e e+3 1.2e+3 1.e+3 1.e+3 8.e+2 8.e e e+2 4.e+2 4.e+2 2.e+2 2.e+2.e z [m] P max = 3515 W/cm 2.e z [m] P max = 3913 W/cm 2 Total power on collector wall: 1MW Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Collector simulation: sweeping with rotating magnetic field 29 / 37

30 Instantaneous power density on the collector wall Θ = I loop = 1 ka Θ = 3 Power density on collector wall [W/cm²] Power density on collector wall [W/cm²] Maximum: W/cm² Maximum: W/cm² 4.e+3 4.e e e+3 3.6e+3 3.6e+3 3.4e+3 3.4e e e+3 3.e+3 3.e+3 2.8e+3 2.8e e e+3 2.4e+3 2.4e+3 phi [rad] 3 2.2e+3 2.e+3 1.8e+3 phi [rad] 3 2.2e+3 2.e+3 1.8e+3 1.6e+3 1.6e e+3 1.2e e+3 1.2e+3 1.e+3 1.e+3 8.e+2 8.e e e+2 4.e+2 4.e+2 2.e+2 2.e+2.e z [m] P max = 3944 W/cm 2.e z [m] P max = 3819 W/cm 2 Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Collector simulation: sweeping with rotating magnetic field 3 / 37

31 Averaged power density on the collector wall I loop = 4 ka I loop = 5 ka Power density on collector wall [W/cm²] Power density on collector wall [W/cm²] Maximum: W/cm² Maximum: W/cm² 4.688e e e e e e e e e e e e e e e e e e+2 phi [rad] e e e+2 phi [rad] e e e e e e e e e e e e e e e e e e e+1.e z [m] P max = 469 W/cm 2.e z [m] P max = 498 W/cm 2 P max corresponds to the averaged peak power density obtained in the case of conventional sweeping. Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Collector simulation: sweeping with rotating magnetic field 31 / 37

32 Averaged power density on the collector wall I loop = 7 ka I loop = 1 ka Power density on collector wall [W/cm²] Power density on collector wall [W/cm²] Maximum: W/cm² Maximum: W/cm² 4.315e e e e e e e e e e e e+2 3.2e e e e e e+2 phi [rad] e e e+2 phi [rad] e e e e e e e e e e+2 1.9e e e e e e e e e+1.e z [m] P max = 431 W/cm 2.e z [m] P max = 436 W/cm 2 Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Collector simulation: sweeping with rotating magnetic field 32 / 37

33 W7-X Gyrotron with mounted sweeping coils Forschungszentrum Karlsruhe Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Collector simulation: sweeping with rotating magnetic field 33 / 37

34 Infrared camera picture (short pulse) Sweeping current: 7.9 ka turns Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Collector simulation: sweeping with rotating magnetic field 34 / 37

35 Modulated transversal collector sweeping Forschungszentrum Karlsruhe By modulating ( wobbling ) the amplitude of the applied 5Hz three phase current with a lower frequency (5 1Hz), the critical maxima of the power density distribution will be smeared out coil current / kat t / s Example: Amplitude Modulation from 4 ka to 7kA at 5Hz. Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Collector simulation: sweeping with rotating magnetic field 35 / 37

36 Modulated transversal collector sweeping: Results I loop = 5 ka const. I loop = 4 7 ka, modulated at 5Hz phi [rad] e+1.e z [m] P max = 498 W/cm 2 5.e e+2 4.5e e+2 4.e e+2 3.5e e+2 3.e e+2 2.5e e+2 2.e e+2 1.5e e+2 1.e+2 7.5e+1 5.e+1 phi [rad] e+1.e z [m] P max = 283 W/cm 2 At IPP Greifswald experiments with modulated sweeping current already started. Quantitative results obtained with the W7-X Gyrotrons (14 GHz, 1MW, cw) will be available soon. 5.e e+2 4.5e e+2 4.e e+2 3.5e e+2 3.e e+2 2.5e e+2 2.e e+2 1.5e e+2 1.e+2 7.5e+1 5.e+1 Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Collector simulation: sweeping with rotating magnetic field 36 / 37

37 Conclusion In the field of high power CW Gyrotrons Particle-in-Cell codes are an indispensable tool for the design of the Magnetron Injection Gun.... design of the collector shape and magnetic sweeping system.... simulation of beam instabilities that may strongly influence the efficiency of the tube.... verification of numerical tools for cavity design (in limited sense) Dr. Stefan Illy 7th School on Fusion Physics and Technology, April 28 Conclusion 37 / 37