development of a steady state simulation code for klystrons
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
DEVELOPMENT OF A STEADY STATE SIMULATION CODE FOR KLYSTRONS
Chiara Marrelli22/06/2011
KLYSTRON STUCTURE AND DESIGN
Klystron electronic design
• I0– V0 (μPerveance)• Cavity parameters (f0, R/Q, Q0 , Qext
…)• Distances between cavities• Beam focusing field
Beam – cavities interaction
Inter – particles interaction
(high current)
Design methodologyIntensive use of simulation codes:
“Disk” codes (e.g. SLAC AJDisk):• 1D – only longitudinal motion allowed• Cavities represented by their impedence• Steady state simulations (no transient)• No information on the total cavity field• No information on beam focusing• No gun simulation• QUICK EXECUTION TIME design tool
Particle In Cell (PIC) codes (e.g. Magic):• 2-3D simulation• Calculate fields and transient• Focusing system and gun simulation• TIME EXPENSIVE verify tool
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 focusing field simulation no gun simulation simulation of multiple beams interaction Fast execution time
Beam-cavity interaction
Cavity field
Electron Beam
Cavity field guess value
Particles tracking
Evaluation of cavity field
with the beam
Current calculation
Check for converge
nce
NO SPACE CHARGE FORCES CONSIDERED (particle-particle
interaction)
Step 2: beam on
field action
Step 1: field on beam action
Beam-cavity interaction
0hH
0eE
• Since only one term is considered in the expansions:
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.
i
iieE
i
iihH
With: ii
i
• Expansion of the total field in terms of the cavity natural
modes:
Step 1 (beam on field): calculation of electromagnetic field in the cavity in the presence of
beam
0
(1,2)(3)
(4,5)
From the cavity power balance:
V
beam
VVS
dVJEdVHHdVEEjdSnHE ****
2
1
2
1
2
1
2
1
Beam-cavity interaction
2
21
**
***
2
1
2
1
2
1
2
1
2
1
S
SSS
dSnHEIIVV
dSnHEdSnHEdSnHE
S=S1+S2
S1
S2
V-
V+
nHnd
jEn ˆˆ
1ˆ
And, by using Leontovich-Schelkunoff:
The first integral can be written as:
222
2** ˆ
1
2
1ˆ
2
1
2
1
SSS
dSHnd
jdSEnHdSnHE
(6)
(7)
(8)
Determining α:
Beam-cavity interaction
On S2 we can approximate: ohHn
ˆ
And, by introducing the quality factor Q we get, for the integral over S2 : u
QjdSh
d
jdSnHE
SS
2
0
02
0* 1
1
2
1
2
1
22
dVeuV 2
02
1 With:
V
beam dVJeujuQ
jIIVV *0
2
0
2
0
0*
2
11
2
1
2
1
The balance equation becomes:
0
0
With: V+
V-
VeVrefl
(9)
(10)
The left side of equation (10) contains the power flowing across the waveguide aperture. The incoming wave is given by: j
IN ePZV 02
V+
V-
VeVrefl
Beam-cavity interaction
eerefl VVVVV While the outcoming wave is:
If the aperture is small we can assume Γ = -1 and obtain:
ee VVVVV
So that we have:
0
*
0
**
0
*
2
1
2
1
Z
VVP
Z
VVVV
ZIIVV e
ee
ee
extexte Q
u
Q
UP
2
00
With:
(11)(12)
(13)
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):
20
0
0
20
arctan
002
extQQ
Q
j
exte e
Q
uZV
2
0
0
2
0
00
*0
arctan
0
*
111
2
12
20
0
0
20
Qj
QQu
dVJeeQ
uP
e
V
beam
Q
j
e
INext
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).
Beam-cavity interaction
(14)
(15)
2
0
0
2
0
00
*0
arctan
0
*
111
2
12
20
0
0
20
Qj
QQu
dVJeeQ
uP
e
V
beam
Q
j
e
INext
Beam-field interaction
The current density due to the electron beam can be
represented by the sum of N individual electron currents:
Input power from the
waveguide
N
iiiibeamTOT rrqvJ
1
T
tjjbeamTOTbeam edeJ
TJ
0
2 The fundamental harmonic of
this current density is then:
*
1 0
*0
*0 )(
)()),(),((
2
dzzv
zvzztzeq
TdVJe
N
i zi
iL
iii
V
beam
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)
(16)(17)(18)
Beam-cavity interaction
1. Assume a set of electrons distribuited uniformly in transverse position, time and initial momenta (for first cavity);
2. Assume an initial value α0 for the field coefficient;3. 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.
4. Calculate the integral 5. Calculate the new value α1 of the field coefficient
using eq. (15);6. If α1=α0 (within a certain tolerance), then α=α1 ;
otherwise assume a new value for α0 and go to step 3;
0with00hH
00eE
V
beamdVJe *0
Beam-cavity interactionSimulation algorithm:
Step 2:
beam on
field action
Time dependent relativistic Hamiltonian (non-autonomous
system):Pseudo time-independent problem by introducing two more variables, τ
and Pτ : Set of 8 equations:
tqecmtqAePLPqH ii ,, 422
PqecmqAePH ,,~ 42
2
1
1
1
1
2
2
2
dt
d
AP
AP
dt
dz
AP
AP
dt
dy
AP
AP
dt
dx
zz
yy
xx
A
AP
AP
dt
dP
zz
A
AP
AP
dt
dP
yy
A
AP
AP
dt
dP
xx
A
AP
AP
dt
dP
z
y
x
2
2
2
2
1
1
1
1
(19)(20
)
The Hamilton’s equations have to be integrated numericallySymplectic method (implicit for not separable Hamiltonian)
in order to preserve the phase space structure
Beam-cavity interactionStep 2 (field on beam):
integration of the equations of motion
Example: pillbox input cavity (no space charge)
Pin=1 kWBeamCurrent=100 ABeamVoltage=100 kVω=11.424 GHz (driving frequency)ω0=11.445 GHzL=0.5 cmQ0=5588Qext=115FocusingField=0.093 T (Brillouin field)
Test of the code self-consistency:• initially uniform beam• no space charge• cavity without beam pipe (analytic field)
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.2
02
1 VVZ
P e
115
445.11
0008.0
0
extQ
GHzf
WP
Outcoming power (W) as a function of the
cavity resonant frequency
Outcoming power (W) as a function of the
external Q
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 ABeamVoltage=100 kVω=11.424 GHz (driving frequency)
ω01=11.445 GHz (input cavity - pillbox)L1 =0.5 cm Q01=5588 Qext1=115
ω02=11.424 GHz (output cavity - pillbox)L2 =0.5 cm Q02=5576 Qext2=55 FocusingField=0.093 T (Brillouin field)
Ldrift=30 cm (space between cavities)
The output cavity resonant frequency ω02 and external
quality factor Qext2 are chosen to maximize the output
power.
%3000
IV
POUT
Comparison with klystron kinematic theory
• 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
11
21
01
0010
20
101
0
1
cos1
lp
ptt
tVp
pMpp
idtt
i
iii
ii
iii
i
21
21
212
12
1
01
1
1
1
Vp
pMpp
eZIV
eN
II
jQ
Q
R
Z
i
iii
tj
N
i
tj i
V2
V1t0, p0 t0,
p1
t1, p1
t1, p2I1
d
z
d zjz
dzzE
dzezEM
e
0
0
Particles normalized longitudinal
momentum after the drift space and after the second
cavity
Pin=250 WBeamVoltage=100 kVBeamCurrent=10 Aω=11.424 GHz L drift=0.1 mFocusingField=0.093 TQext2=∞
Comparison with klystron kinematic theory
zn
pz
n
zn
pz
nDifferences due to the fact that the
kinematic theory does not take in
account the effect of the beam back to the
cavity
kinematic theory
new code
Comparison with 1-D simulation code
• 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
• cavity voltages• beam current in the cavities• particles minimum β• gain• efficiency• maximum output electric field
Outputs:
(AJDisk)
Particles normalized longitudinal
velocity (vz/c) – function of z
AJDisk
new code
Pin=250 WBeamVoltage=100 kVBeamCurrent=0.5 Aω=11.424 GHz L drift=5 cmQext2=∞
Low current simulations to minimize the space charge effects
Comparison with 1-D simulation code
Comparison of resultsThe klystron kinematic theory & AJDisk
Two cavity klystronPin=250 WBeamVoltage=100 kVω=11.424 GHz (driving frequency)L drift=0.1 mFocusingField=0.093 T (Brillouin field)
AJDisknew codeKin. Theory
High current simulations require to take in account the repulsive forces between particles:
SPACE CHARGE FIELDS
Voltage in the 1st cavity vs beam
current
Voltage in the 2nd cavity vs beam current
Maximum δβz after the 2nd
cavity vs beam current
We search a steady state solution by taking in account the Coulomb repulsion between macroparticles
Simulation algorithm based on two main steps :
1. Calculation of the total space charge field inside the drift tube as a function of time t ;
2. Particles tracking inside the drift tube in presence of this field.
Inter-particles interaction
Approximations:1. Calculation of the space charge field only inside the
drift tubes (cavity gaps small with respect to the drift lengths);
2. Free space solution for the Laplace partial differential equation when calculating the potential due to a point charge in the particle frame
Inter-particles interaction
b
zz
n k nknk
nknnknn nk
enJ
brJbrJ
brrG
''
00 1 1
0
00
0
''cos'
/'/'
2','
xJ nWhere the ξsl are the zeros of the Bessel functions
(21)
And:
2
1n
n=1n=2,3,4,…
Solution in the circular pipe leads to the Green function:
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
Free space fields
Electromagnetic fields (particle frame)
From the potentials produced by particle i in free space (particle frame)
Inter-particles interaction
Electromagnetic fields (laboratory frame)
The total field (lab frame) is then given by the sum if all the particles fieldsOnce 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.
N
iiii
iiz
N
iiii
iiy
N
iiii
iix
zzyyxx
zzqzyxE
zzyyxx
yyqzyxE
zzyyxx
xxqzyxE
1 2
32222
0
1 2
32222
0
1 2
32222
0
4
,,
4
,,
4
,,
0
1 2
32222
0
1 2
32222
0
,,
4
,,
4
,,
BzyxB
zzyyxx
xxq
czyxB
zzyyxx
yyq
czyxB
z
N
iiii
iiy
N
iiii
iix
Inter-particles interactionMain development issues:
Since we want to perform steady state simulations only the evolution of one set of particles distributed over an RF period is evaluated, BUTTo 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: out
iini tkTtt * ...2,1,0k
1)
Where the contributions for k>2 can be neglected
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.
2)
(22)
First (partial) results:
Z normalized momentum for on axis particles
Pin=250 WBeamCurrent=15 ABeamVoltage=100 kVω=11.424 GHz (driving frequency)ω0=11.445 GHzLdrift= 10 cmQ0=5588Qext=95FocusingField=0.093 T (Brillouin field)
0.5 plasma wavelength
Inter-particles interaction
Drift space after input cavity
zn
pz
n
Further work to be done:• test of the results for the plasma frequency
• optimization (and speeding up) of the space charge routine
Work on klystrons at CernHigh 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)