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DEVELOPMENT OF A STEADY STATE SIMULATION CODE FOR KLYSTRONS

DEVELOPMENT OF A STEADY STATE SIMULATION CODE FOR KLYSTRONS. Chiara Marrelli 22/06/2011. KLYSTRON STUCTURE AND DESIGN. Klystron electronic design. I 0 – V 0 ( μ Perveance ) Cavity parameters (f 0 , R/Q, Q 0 , Q ext …) Distances between cavities Beam focusing field.

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DEVELOPMENT OF A STEADY STATE SIMULATION CODE FOR KLYSTRONS

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  1. DEVELOPMENT OF A STEADY STATE SIMULATION CODE FOR KLYSTRONS Chiara Marrelli 22/06/2011

  2. KLYSTRON STUCTURE AND DESIGN Klystron electronic design • I0– V0 (μPerveance) • Cavity parameters (f0, R/Q, Q0 , Qext …) • Distances between cavities • Beam focusing field Beam – cavitiesinteraction Inter – particlesinteraction (high current)

  3. Design methodology Intensive useofsimulationcodes: • “Disk” codes (e.g. SLAC AJDisk): • 1D – onlylongitudinalmotionallowed • Cavitiesrepresentedbytheirimpedence • Steady state simulations (no transient) • No information on the total cavity field • No information on beamfocusing • No gun simulation • QUICK EXECUTION TIME design tool • Particle In Cell (PIC) codes (e.g. Magic): • 2-3D simulation • Calculatefields and transient • Focusing system and gunsimulation • TIME EXPENSIVE verify tool

  4. New code development Development of a new klystron design code in collaboration with SLAC - S. Tantawi • 2D – steady state simulations • complete description of the interaction beam-field • focusingfieldsimulation • no gunsimulation • simulation of multiple beams interaction • Fast execution time

  5. Beam-cavityinteraction Step 1: field on beamaction Cavityfield Electron Beam Cavityfieldguessvalue Step 2: beam on fieldaction Particlestracking Currentcalculation Evaluationofcavityfieldwith the beam NO SPACE CHARGE FORCES CONSIDERED (particle-particleinteraction) Checkforconvergence

  6. Beam-cavityinteraction Step 1 (beam on field): calculation of electromagnetic field in the cavity in the presence of beam • Expansionof the total field in termsof the cavitynaturalmodes: (1,2) (3) With: • Sinceonlyonetermisconsidered in the expansions: (4,5) The field in the cavity is then equal to the field of the design mode multiplied by a complex coefficient, α, to be determined in amplitude and phase.

  7. Beam-cavityinteraction S=S1+S2 Determining α: From the cavitypowerbalance: (6) The first integral can be written as: (7) And, by using Leontovich-Schelkunoff: S2 V+ S1 (8) V-

  8. Beam-cavityinteraction On S2 we can approximate: And, by introducing the quality factor Q we get, for the integral over S2 : (9) Ve Vrefl With: V+ The balance equation becomes: (10) V- With:

  9. Beam-cavityinteraction The left side of equation (10) contains the power flowing across the waveguide aperture. The incoming wave is given by: (11) While the outcoming wave is: Ve (12) Vrefl If the aperture is small we can assume Γ = -1 and obtain: V+ So that we have: V- (13) With:

  10. Beam-cavityinteraction The amplitude of the emitted wave Ve depends only on the stored energy in the cavity |α|2u. Its phase can be obtained directly from the balance equation in the case without beam and then adding the phase of the field inside the cavity (phase of α): (14) (15) We have an expression to obtain the cavity field in amplitude and phase in the presence of the beam current density provided the knowledge of the frequency shift (driving frequency).

  11. Beam-cavityinteraction Input power from the waveguide Beam-field interaction The current density due to the electron beam can be represented by the sum of N individual electron currents: (16) The fundamental harmonic of this current density is then: (17) (18) We get: To get the velocity and position vectors we have to solve the relativistic equations of motion for every particle in the cavity (Step 2)

  12. Beam-cavityinteraction Simulationalgorithm: Assume a set of electrons distribuited uniformly in transverse position, time and initialmomenta (for first cavity); Assume an initial value α0for the field coefficient; Integrate the equations of motion for each particle through the length of the cavity with the fields: (plus focusing magnetic field); the cavity modes are obtained from a 2-D electromagnetic code, like SUPERFISH or the FEM code developed by Sami Tantawi. Calculate the integral Calculate the new value α1of the field coefficient using eq. (15); If α1=α0(within a certain tolerance), then α=α1 ; otherwise assume a new value for α0 and go to step 3; Step 2: beam on fieldaction with

  13. Beam-cavityinteraction Step 2 (field on beam): integrationof the equationsofmotion (19) Time dependent relativistic Hamiltonian (non-autonomous system): Pseudo time-independent problem by introducing two more variables, τand Pτ: (20) Set of 8 equations: The Hamilton’s equations have to be integrated numerically Symplectic method (implicit for not separable Hamiltonian) in order to preserve the phase space structure

  14. Example: pillbox input cavity (no space charge) • Test of the code self-consistency: • initially uniform beam • no space charge • cavity without beam pipe (analytic field) Outcomingpower (W) as a functionof the cavityresonantfrequency Pin=1 kW BeamCurrent=100 A BeamVoltage=100 kV ω=11.424 GHz (drivingfrequency) ω0=11.445 GHz L=0.5 cm Q0=5588 Qext=115 FocusingField=0.093 T (Brillouin field) Outcomingpower (W) as a functionof the external Q The cavity resonant frequency ω0 and the external quality factor Qext are chosen to minimize the outcoming power from the cavity when the beam is in.

  15. Example: two cavity klystron (no space charge) • Test of the code self-consistency when applied to a cavity with PIN=0 • velocity modulated beam coming from the input cavity • no space charge • cavity without beam pipe (analytic field) Pin=1 kW BeamCurrent=100 A BeamVoltage=100 kV ω=11.424 GHz (drivingfrequency) ω01=11.445 GHz (inputcavity - pillbox) L1 =0.5 cm Q01=5588 Qext1=115 ω02=11.424 GHz (outputcavity - pillbox) L2 =0.5 cm Q02=5576 Qext2=55 FocusingField=0.093 T (Brillouin field) Ldrift=30 cm (spacebetweencavities) The output cavity resonant frequency ω02 and external quality factor Qext2 are chosen to maximize the output power.

  16. Comparisonwith klystron kinematictheory • No space charge • Input cavity represented by V1 modulating the particles momenta • passive cavities represented by their equivalent parallel circuit • M = cavity coupling coefficient due to finite transit time V1 V2 t0, p0 t0, p1 t1, p1 t1, p2 I1

  17. Comparisonwith klystron kinematictheory Pin=250 W BeamVoltage=100 kV BeamCurrent=10 A ω=11.424 GHz L drift=0.1 m FocusingField=0.093 T Qext2=∞ pzn Particles normalized longitudinal momentum after the drift space and after the second cavity zn pzn new code Differences due to the fact that the kinematic theory does not take in account the effect of the beam back to the cavity kinematic theory zn

  18. Comparisonwith 1-D simulation code (AJDisk) • 1 D code currently used at SLAC for the design of round and sheet beem klystrons • beam splitted into a series of disks of charge moving only in the longitudinal direction • the disks are acted by both the cavity fields and the space charge fields Outputs: • cavity voltages • beam current in the cavities • particles minimum β • gain • efficiency • maximum output electric field

  19. Comparisonwith 1-D simulation code Low current simulations to minimize the space charge effects Pin=250 W BeamVoltage=100 kV BeamCurrent=0.5 A ω=11.424 GHz L drift=5 cm Qext2=∞ AJDisk Particles normalized longitudinal velocity (vz/c) – function of z new code

  20. Comparisonofresults The klystron kinematictheory & AJDisk Two cavity klystron Pin=250 W BeamVoltage=100 kV ω=11.424 GHz (drivingfrequency) L drift=0.1 m FocusingField=0.093 T (Brillouin field) Voltage in the 1st cavity vs beam current AJDisk new code Kin. Theory Maximum δβz after the 2nd cavity vs beam current Voltage in the 2nd cavity vs beam current High current simulations require to take in account the repulsive forces between particles: SPACE CHARGE FIELDS

  21. Inter-particlesinteraction We search a steady state solution by taking in account the Coulomb repulsion between macroparticles Simulation algorithm based on two main steps : • Calculation of the total space charge field inside the drift tube as a function of time t ; • Particles tracking inside the drift tube in presence of this field.

  22. Inter-particlesinteraction Approximations: Calculation of the space charge field only inside the drift tubes (cavity gaps small with respect to the drift lengths); Free space solution for the Laplace partial differential equation when calculating the potential due to a point charge in the particle frame Solution in the circular pipe leads to the Green function: (21) Where the ξsl are the zeros of the Bessel functions And: The evaluation of expression (10) has to be performed for every particle at every iteration; this can be very slow in case of a big number of particles To speed up the simulation n=1 n=2,3,4,… Free space fields

  23. Inter-particlesinteraction From the potentials produced by particle iin free space (particle frame) Electromagnetic fields (particle frame) Electromagnetic fields (laboratory frame) The total field (lab frame) is then given by the sum if all the particles fields Once that we have the space charge field at the location of particle j due to the presence of all the other particles in the laboratory frame, we can integrate the equations of motion for the considered particle in presence of the total (space charge and focusing) field.

  24. Inter-particlesinteraction Main development issues: 1) Since we want to perform steady state simulations only the evolution of one set of particles distributed over an RF period is evaluated, BUT To calculate the space charge field at time t* we need to take in account ALL the particles that are in the region of space around the considered particle (± 0.5 of the beam wavelength) at that time (and not only particles of the set), i.e. all the particles that satisfy the condition: (22) Where the contributions for k>2 can be neglected 2) Particles in the cavities and pipes before and after the drift space give a contribution to the space charge field; They will be used as sources for the space charge fields but their trajectories will not be modified during the iterative procedure.

  25. Inter-particlesinteraction First (partial) results: Drift space after input cavity Z normalized momentum for on axis particles pzn Pin=250 W BeamCurrent=15 A BeamVoltage=100 kV ω=11.424 GHz (drivingfrequency) ω0=11.445 GHz Ldrift= 10 cm Q0=5588 Qext=95 FocusingField=0.093 T (Brillouin field) 0.5 plasma wavelength zn • Further work to be done: • test of the results for the plasma frequency • optimization (and speeding up) of the space charge routine

  26. Work on klystrons at Cern High efficiency klystrons for the CLIC study: Efficiency goal: 80% Very low μperveance (<=0.25) Use of higher harmonic cavities (not only second but also 3rd and maybe 4th ) More systematic optimization methods required (Evolutionary Algorithms)

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