DEVELOPMENT OF A STEADY STATE SIMULATION CODE FOR KLYSTRONS

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


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


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:


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


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

QQ

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).


(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

QQ

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)


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


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)


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


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)

DEVELOPMENT OF A STEADY STATE SIMULATION CODE FOR KLYSTRONS

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