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U.Becker, T.Weiland, TH-Darmstadt, Schlossgartenstr. 8, 64289 Darmstadt, Germany M.Dohlus, DESY, Notkestr. 85, 22603 Hamburg, Germany S.L¨ tgert , Philips, Stresemannallee 101, 22529 Hamburg, Germany u D.Sprehn, SLAC, Stanford, CA 94309, USA
Abstract To facilitate the design of high power klystrons an investigation into the reliability and accuracy of three modern particle-in-cell codes was performed. A 150MW S-band klystron for which measurements were available was used for this comparison. The ?eld calculations of the particle-in-cell codes are based on a ?nite difference time domain scheme, and use a port approximation to speed up the convergence to steady state. However they differ in many details (eg. calculation of , or ; ; space charge correction; 2D or 3D modelling of output cavity). properties of numerical ?eld calculation and the space charge conservation. The ?rst problem is solved by Yu' s port approximation (PAp) [2]: only the beam pipe is simulated, the cavities are represented by their boundary ?elds, the fundamental mode of each cavity is simulated by an equivalent circuit while the higher modes are neglected. The amplitude of the circuit resonator is controled by a predictor corrector technique to reach steady state in few rf periods. The effect of noise rises with simulation time and is therefore reduced by the fast turning on process. A further reduction is obtained by spatial ?ltering of ?elds and current densities and a systematic (not randomized) particle injection. To ful?l the continuity either the mapping of the particle motions to grid currents and the ?ltering have to be charge conserving (FCI, MAFIA) or the space charge has to be corrected from time to time (CONDOR). A. CONDOR The space charge correction in CONDOR is calculated as a ~ correction potential. Poisson' s equation for divD  is solved and the correction is added every nth cycle where n is set by the user. In these calculations n was set between 1 and 3. The klystron is split into two segments (cavities 1 to 4 and 5,6,7). To transfer particles from one segment to the next, the particle data for the last rf periods are stored in a data dump and when the next segment is started the particles are reinjected periodically. To reduce the noise, the particle data were averaged over four periods. The ?elds are solved at the split-boundary using Neumann boundaries, no EM ?elds are dumped, and harmonics travelling in the drift tube are neglected. Therefore, it is best to devide the klystron at a point where the rf currents are relatively small. The output cavity is modelled directly in rz-coordinates (no PAp) whereby the dissipation of the output power is simulated by the rmax -boundary. B. FCI For the calculation [3] of the time dependent EM-?elds, the wave equation for the scalar and vector potentials ; in the Lorentz gauge are employed rather than for ?elds , , because in the potential representation the Lorentz condition is consistent with the continuity equation for charge and current which is automatically satis?ed in the simulation. The ?eld distribution of the cavity modes in the drift tube is calculated by the FD method bu using a re?ned mesh in the cavity region of the drift tube. Particle noise is controlled by a sophisticated ?ltering algorithm as well as by introducing a small damping term into Maxwell' s equations. PAp is used for all cavities. Because the output cavity is of a pill-box type, the ?eld calculation for this cavity has a reduced accuracy.


The developement of high power klystrons requires computer programs simulating the complex interaction of electrons with electromagnetic ?elds as realistic as possible. To analyse and verify the abilities of three modern codes (CONDOR, FCI and MAFIA) their calculations are compared with measurements at the 150-MW S-band klystron, which has been designed and built at SLAC [1] for the SBLC test facility at DESY. The design of the 7 cavity klystron is based on CONDOR calculations. design values 535 kV 700 A 3 s, 60 Hz 150 MW 55 dB  40 % 2998 MHz 0.21 T operation values 527 kV 680 A 3 s, 60 Hz > 150 MW 54 dB 42 % 3002 MHz 0.18 T

Beam voltage Beam current RF Pulsewidth, rep rate RF output power Saturated gain Ef?ciency Center frequency Solenoidal focusing ?eld

State of the art are two dimensional particle-in-cell (PIC) codes as CONDOR, FCI and MAFIA-TS2. These codes use the fact that with exception of input and output cavities most klystrons have a pure symmetry of revolution. In these codes the beam is simulated by macroscopic ring charges (macro particles) and their motion is integrated numerically (in 5D phase space). Therefore the electromagnetic ?elds or their potential representations are calculated in time-domain and take into account: electrostatic ?elds (eg. gun or dc beam), magnetostatic ?elds (eg. solenoid), resonant ?elds (cavities), transient processes and self-consistent ?eld particle interactions. Problems are: the long simulation time (many periods until steady state), the noise caused by macro particles (gain > 55 dB), unphysical



C. MAFIA In the 2D simulations the steady state solution for all cavities is derived with the PAp. Therefore, for every dc voltage a special run is necessary to obtain the beam-loading conductances of the cavity modes which are later needed to perform the iteration process of the PAp. Unsymmetric effects and the excitation of higher order modes in the output circuit due to the strong coupling to the external waveguides (Qext  14) are studied with a 3D PIC-simulation using MAFIA-TS3. The interface plane between the two and three dimensional simulation is located between cavity 6 and 7. There the periodic particle and electromagnetic ?eld information is stored in the 2D run and reinitialized in the 3D run. The 3D simulation takes into account the cavity with all modes and the real broadband waveguides extracting the power. A detailed description of the interface and other characteristics of klystron simulation using MAFIA is given in [5].

cavity 1 2 3 4 5 6 7

frequency MHz 3000 3009.3 3029 3176.8 3447.8 3384.8 2998

R/Q Ohm 87 89 91 107 98 127 117

Ql 127 8200 8300 8900 8700 9500 14

zcenter inch 27.775 33.195 39.665 55.765 62.695 66.015 70.015

d) Changes: due to a late change of the geometrical description, the FCI calculation are done for a slightly different setup. The geometry of the bucking coil is: zmin =13.3inch, zmax =16.63inch, rmin=7.9inch, rmax =9,975inch. The cathod center is located at 14in, the center of the ?rst cavity at 26inch while the distance between all other cavities is unchanged.

A. Measurements Only external parameters like input-, re?ected-, output-power and the intercepted current could be measured directly. A parasitic oscillation (f8.5GHz) was observed [1] and could be avoided by adjusting the solenoid ?eld (see values in coil settings table). This instability cannot be predicted or analysed by the monopole monomode PAp model. B. Comparison a) Re?ected Input Power: essentially the input power is either transfered to the beam or re?ected. Neglecting wall losses, one can relate the cavity 1 voltage to the input power, re?ection and cold cavity parameters:
2 V1 =Pin = 2(1 + r)2 Ql R=Q ;

To ensure that the klystron simulations are based on the same operating conditions and to avoid differences caused by precalc. the following parameters have been chosen: a) Coil Settings: the solenoid ?eld can be adjusted with four independent coils: coil bucking 1 2 3 current dens. A/inch2 -123.8 1656 1710 1224 zmin ,zmax inch 13.67, 17.00 21.0, 73.5 21.0, 73.5 59.55, 72.86 rmin ,rmax inch 7.97, 10.04 8.34, 9.34 9.84, 11.173 9.84, 11.173

which can be solved for r and compared: Ib/A measur. CONDOR FCI MAFIA 480 0.169 0.046 0.176 495 0.179 0.216 0.046 0.184 511 0.190 0.179 0.058 0.193 523 0.187 527 0.183 0.219 0.059 0.206

In this z scale the cathod center is located at z=14.415inch. (The ?eld is plotted in [4] and [5].) b) Beam Voltage and Current: the PIC-simulation starts right behind the gun region, where the DC-current is reinjected according to the particle properties calculated by the codes EGUN (for CONDOR and MAFIA) and SuperSAM (for FCI). The calculated perveances differ less than 2% from the measured values. For the simulations the current was scaled or extrapolated to the values printed in the Pout (Pin)-diagrams. c) Cold Cavity Parameters: the parameters for the input and output cavity are design values and have been measured. The parameters of the other cavities have been calculated in good agreement by SUPERFISH, SuperLANS and MAFIA-E. The PIC simulations are very sensitive to the cavity parameters especially of cavity 2 and 3. Therefore the PAp is applied with identical cold cavity parameters in all three PIC-codes. Only the output cavity is simulated directly by MAFIA3D and CONDOR. The frequency and Q factor of the complete 3D discretization (including coupling slots and waveguides) have been veri?ed in a separate 3D calculation.

The small re?ection of the FCI calculation indicates an inaccurate simulation of the input cavity. The effect of this error is a reduced cavity 1 voltage and therefore a reduced gain ( -1dB). b) Linear Gain: the gains calculated by all programs are too high: CONDOR +7.6dB, FCI +4.8dB, MAFIA2D +2.1dB MAFIA3D +1.8dB at operation frequency. In the MAFIA calculation the maximum of the frequency response is shifted by at least 8MHz to the measurement. c) Fourier Coef?cients of Beam Current: for Ub =511kV and identical cavity 1 voltage (3kV) the fourier components of the beam current and the cavity voltages are documented. In the range of the ?rst three cavities the ampli?cation of the cavity voltages and the 1st harmonic of the current are highest in CONDOR and lowest in MAFIA. The difference between CONDOR and FCI at the beginning is completly compensated to the end of the tube. The ampli?cation along the last drift space is signi?cantly higher in MAFIA so that the cavity 7 voltage is only 2.5% lower than by the other codes.

d) Output Cavity (Ub=511kV, U1 =3kV): even with very similar bunching and cavity 7 voltage the output power of CONDOR (155.1MW) and FCI (166.1MW) show a big difference. For these parameters MAFIA2D is in excellent agreement with CONDOR and MAFIA3D is further 10MW lower. The CONDOR/FCI differences and the MAFIA3D/2D differences of 10MW can only be explained by errors of the PAp. CONDOR (without PAp) and MAFIA2D (with PAp) give similar results because the cav. 7 voltage differs. e) Numerical Effort: CONDOR
r/mm x/mm y/mm z /mm t/ps

160 Power out (MW)
3 2

1 4 5

120 80 40 0 0 100 200 300 400 500 RF Input drive (W)
3 1 5

1: 2: 3: 4: 5: 527kV, 523kV, 511kV, 503kV, 495kV, 680A 676A 656A 644A 628A



MAFIA 2D 1.5

MAFIA 3D1 1.6..2 1.6..2.5 1.6..2 1.57 15 8520
Power out (MW) 160 120



0.88..0.91 1.85 60 1240

2.5 1.6

1.5 1.57 42 1065


80 40 0 0 200


1: 3: 5: 6: 527kV, 511kV, 495kV, 480kV, 680A 656A 628A 600A

rf-periods particles/ rf-period
1 3

45 556

output cavity, increased resolution in the cavity regions, integration of motion

100 200 RF Input drive (W)


The measured input re?ection coef?cient indicates that the beam loading of the input cavity is not simulated precisely by FCI. All codes calculate overly high gains (especially CONDOR) and saturated output powers (especially FCI). The linear frequency characteristicas are shifted upwards (especially by MAFIA). The port approximation causes signi?cant errors in the output cavity.

Power out (MW)

160 120 80 40 0 200 100

3 5 6

1: 3: 5: 6: 527kV, 511kV, 495kV, 480kV, 680A 656A 628A 600A

[1] D. Sprehn, R.M. Phillips, G. Caryotakis: “The Design and Performance of 150-MW S-Band Klystrons”, SLAC-PUB6677, Stanford Linear Accelerator Center, September 1994. [2] Simon Yu: “Particle-In-Cell Simulation of High Power Klystrons”, SLAC-AP-34, September 1984. [3] T. Shintake: “High Power Klystron Simulations using FCIField Charge Interaction Code” KEK 90-3 A/D, May 1990. [4] S. L¨ tgert: “FCI Parameter Study for the 150 MW DESY u Klystron”, SBLC meeting at DESY, Hamburg, February 1995. [5] U. Becker, M. Dohlus, T. Weiland: “Three Dimensional Klystron Simulation”, to be published in Particle Accelerators.
Power out (MW) 160 120 80 40 0 80 100

200 300 400 500 RF Input drive (W)



1 3 5 6

1: 3: 5: 6: 527kV, 511kV, 495kV, 480kV, 680A 656A 628A 600A

200 300 400 500 RF Input drive (W)



2 4

70 Gain (dB)


50 2980 2990

1: 2: 3: 4:

511kV 656A MEASUREMENT CONDOR P_in=0.5W FCI P_in=7.5W MAF 2D 3D P_in=2.0W

3000 3010 Frequency (MHz)




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