3D EM Simulations

3D EM Simulations 

1 ube / v2.0 / 2008 

www.cst.com

• CCavities iti 

– Eigenmodes 

–Q Factor 

Overview 

– Time Domain and Resonances 

• Cavities and Particles 

• Electron Guns 

• Collector 

• Wakefields 

2 www.cst.com

f 

Eigenmode Solver 

Theoretical Background 

PEC 

f f 

f=c/L f=c/L 

f=2c/L f=2c/L 

PMC 

In free space travelling waves can 

exist i t ffor any frequencies. 

f i 

If such a plane wave is reflected at a 

perfect f wall ll ( (electric l i or magnetic) i ) 

there will be a standing wave. This 

standing wave exists independently 

from the frequency. 

The insertion of a second wall does 

not affect those standing wave as 

long g as the distance L between the 

walls fits perfectly to the wave-length. 

The standing wave of this closed 

structure is called an eigenmode. 

The frequency and the shape of the 

next eigenmode fitting between those 

two walls is predictable. 

3 www.cst.com

Eigenmode g Computation 

r r 

rot E r r= 

− jωμ 

r rH 

r σσ 

r 

rot H = jωε 

E + σ E = jω( 

ε + ) E = 

jω 

rot 

1 

μμ 

rot 

r 

E 

= 

ω 

2 

ε 

r 

E 

Eigenvalue equation for the resonant structure modes 

and resonance frequencies. q 

Eigenvalues ω and eigenvectors E r 

4 www.cst.com 

j 

ω ε 

r 

E

Eigenmodes g 

f1=34.7 GHz f2=52.0 GHz 

f3=54.3 GHz f4=57.9 GHz f5=67.7 GHz 

5 www.cst.com

1 

2 

Eigenmode g Solver 

Main user input: 

1 

2 

Which eigenmode method should be used? 

(AKS might be faster for well behaved examples, 

JD is more robust and might even find good 

solutions for bad conditioned problems) 

How many modes are required? 

The AKS method always calculates internally at 

least 10 modes. (Therefore ( nearly y no difference in 

cpu time between 1 and 10 modes). 

The JD method calculates one mode after 

another, therefore 10 modes need roughly 

10times the simulation time of 1 mode. 

Therefore the number of modes should be 

decreased when using the JD method. 

Th 

6 www.cst.com

A simple lossfree Cavity 

3 symmetry y y pplanes only y 1/8 

of the volume needs to be calculated 

Mode1 

E H 

Mode2 

Mode3 

7 www.cst.com 

cavity_03.zip

electric energy density 

0.5 ε E_peak^2 

Energy gy Densities 

magnetic energy density 

0.5 μ H_peak^2 

Volume Integration of Energy Density is possible via Result Template 

0D / Evaluate Field in arbitrary Coordinates (0D, 1D, 2D, 3D) 

Note: for a lossfree eigenmode both integrals (el. + mag.energy) will be 1 Joule, 

since i th the energy iis oscillating ill ti bbetween t electric l t i and d magnetic ti fi field. 

ld 

8 www.cst.com

Plotting Fields and getting Field Values 

All modes are normed 

to 1 Joule stored energy. 

E / H / surface current 

are stored as peak values. 

9 www.cst.com

Benchmark Pillbox 

h1 r11 Influence of Meshing 

E H 

surface current J 

QQ_theory=50 theory=50 940 

Q_simul=50 990 

cpu-time 

pass 1 = 16sec 

pass 5 = 2 min 

f_theory=229.485MHz 

f_simul=229.444MHz 

10 pillbox_01.zip 

www.cst.com

The Quality y Factor Q 

2π 

• StoredEnergy 

2π 

⋅ f ⋅W 

Q = 

= 

Energy _ consumed _ per _ period 

the higher Q, the longer the energy is kept 

• for a lossfree structure P=0 infinite Q 

• different kind of losses exist: 

• skin effect (surface) losses: Qwall • dielectric (volume) losses: Q diel 

• losses due to connected feeding lines: Qext • losses due to energy transfer between beam and mode: Qbeam 1 

Q 

tot 

1 

= 

Q 

As a circuit model, all losses can be seen as a 

parallel circuit, acting on the same mode. 

wall 

Prms 

1 

+ 

Q 

diel 

1 

+ 

Q 

ext 

1 

+ 

Q 

11 www.cst.com 

beam

Calculation of Q and Q wall diel 

Results -> Loss and Q-Calculation performs p a loss calculation in the 

postprocessing based on perturbation theory. It handles metallic losses 

due to finite conductivity (skin effect) as well as dielectric losses. 

Q=2*Pi*frq * Energy / Loss_rms 

frq = 2.96289815e+010 Hz 

Loss Loss_rms rms = 00.5 5 Loss Loss_peak peak 

Energy = 1 Joule 

12 www.cst.com

Integration of Voltage, 

Calculation of Shunt Impedance & R/Q 

Shunt Impedance Rs=Vo^2/(2W) 

with Vo : voltage „seen“ by a charged 

interacting particle. 

For voltage integration also the Transit Time Factor 

can be specified, which defines the speed of 

the particle (beta=v/c) ( ) and gguarantees 

a 

phase-correct integration of electr. field. 

R/Q (R over Q) only depends on the geometry 

(not on the loss mechanism) and is therefore 

often used to compare different cavity structures. 

13 www.cst.com

Example Superconducting Cavity 

easy construction via Macros -> 

Construct -> Superconducting p g Cavity y 

ellipt_cav_01.zip 

14 www.cst.com

Eigenmode Solver 

Calc Calculation lation of Q Qext Langer filter 

Results of the Q-Factor calculation: Log File 

EE.g. g external QQ-factor factor comparison for the Langer filter 

f [GHz] 

4.546 

4.572 

7.080 

77.185 185 

CST Q_ext 

332 

305 

26200 

9070 

Steiglitz-McBride 

332 

305 

24132 

8472 

15 www.cst.com • Oct-09

Other methods for loaded Q 

calc calculation lation 

• Using the transient analysis 

• Amplitude E-field inside a resonator is given: 

) t ( E 

= 

E 

0 t 

2 Q 

0 e 

ω ⋅ 

− 

⋅ 

⋅ 

load 

or 

ω0⋅t 

E( t) 

− 

2⋅Q 

• MMonitored it d using i an EE-field fi ld probe b 

• Measure the time difference Δt in which the E-field is 

damped by a factor of 1/e then Q load is given by 

load 

E 

Q = π⋅ 

Δt 

⋅f 

16 CST Microwave Studio • www.cst-world.com • Oct-09 

0 

0 

= 

e 

load

Other methods for loaded Q 

calc calculation lation 

lloadd = π π⋅ 

Δ t f 0 

Signal: 503.7 at t1=200 ns 

Signal: 503 503.7/e 7/e = at t2 t2= 423 423.6ns 6ns 

Δt = t2 –t1 = 223.6ns 

f0 = 2.4615 GHz 

Qload = 1729 

Q Δ ⋅ 

1/e 

17 CST Microwave Studio • www.cst-world.com • Oct-09

Time Domain Simulation and 

Resonant SStructures 

Slow energy decay since energy is kept in the resonance 

Long Simulation Time 

Prediction of signal by Autoregressive Filter 

Usage of Frequency Domain Solver 

Transient Activity 

Energy Decay 

18 www.cst.com | Oct-09

Time Domain Simulation 

AR Filter 

Stop at 31.0 ns 

Without online-AR cpu=100 sec 

Stop at 1.4 ns 

With online-AR-Filter cpu=15 sec (1port) 

Original SS-Parameter Parameter 

from Time Signals 

S-Parameter 

predicted by 

AR Filter 


Frequency q y Domain Solver 

• Simulation performed at single frequencies 

• Simulation of Steady State 

• Broadband Frequency Sweep 

20 www.cst.com • Oct-09

Thermal Calculation 

Surface and volume losses 

from previous eigenmode 

simulation can be used to perform 

a thermal analysis 

21 www.cst.com



–Q Factor 

Overview 




• Collector 


22 www.cst.com

Workflow: 

Tracking Algorithm 

1. Calculate electro- and magnetostatic fields 

2. Calculate force on charged particles 

3. Move particles according to the previously calculated 

force Trajectory 

d 

dt 

r 

r 

r 

r 

( m v ) ) 

= q( 

E + v × B 

Velocity 

( ) 

n+ 

1 n+ 

1 n n 

n+ 

1/ 

2 n+ 

1 n+ 

1/ 

2 

= q( 

E + v × B 

y 

update 

+ Δt( 

E + × B ) 

n+ 

1 n+ 

1 n n 

n+ 

1/ 

2 n+ 

1 n+ 

1/ 

2 

m v = m v + qΔt 

E + v × B 

dr 

v 

ddt 

Position update 

t = n+ 3/2 n+ 1/2 n+ 

1 

r 

r 

r r r 

r = r +Δtv 

23 www.cst.com | Oct-09 

r

Tesla-Type 9-Cell Cavity 

Electric Field 

calculated by 

MWS-E 


Tesla-Type 9-Cell Cavity 

Particle Trajectory 

(color indicates the 

energy of the particles) 


TESLA – Two-Point-Multipacting 

• Eigenmode at 1.3 GHz 

•Emax = 45 MV/m 


time 

TESLA – Two-Point-Multipacting 

half hf period 




–Q Factor 

Overview 




• Collector 

• Wakfields 

28 www.cst.com

Emission Models 

Space Charge Limited Emission: 

Assumption: 

Φ (d d d 

) 

•Unlimited number of particles 

•Particle extraction depends on 

field close to emitting surface 

Childs Law: 

J 

s 

( ) 3/2 

Φ( d) 

−Φ(0) 

4 q 

ε 2 

9 m d 

= rent 

2 

Curr 

PEC 

( 0) 

Φ 

Voltage 

Child-Langmuir Model 

d 

Virtual Cathode 


Thermionic Emission: 

Assumption: 

•Limited number of particles 

Emission Models 

•Particle extraction depends on 

field close to emitting surface 

until ntil all particles are emitted 

Richardson Richardson-Dushman Dushman Equation: 

J = AT 

s 

2 

e 

eΦ 

- 

kT 

Currrent 

Temperature Limited 

Space Charge 

Limited 

Voltage 


Gridded Gun 

control grid 

permanent magnets 


Potential 

B-Field

Gridded Gun - Mesh 

32 www.cst.com

33 

S-DALINAC Electron Source 

-100kV 

Photocathode 

Diagnosis and 

Manipulation 

Vacuum 

See also Bastian Steiner, PhD Thesis, TEMF, TU Darmstadt 

0V


Electron trajectory 

34 See also Bastian Steiner, PhD Thesis, TEMF, TU Darmstadt


Electric potential 

35 See also Bastian Steiner, PhD Thesis, TEMF, TU Darmstadt



–Q Factor 

Overview 




• Collector 


36 www.cst.com

Collector 

Potential 


Collector, including secondary electron 

emission 

E0 = 50 keV E0 = 125 keV 

E0 = 200 keV E0 = 275 keV 


E0 = initial energy of the electrons



–Q Factor 

Overview 




• Collector 


40 www.cst.com

Wakefields 

What does the Wakefield Solver do? 

simple explanation: SSpecial current excitation of f CST CS MWS S T-Solver. S 

more complex explanation: 

- Moving charged particles are represented as Gaussian current density 

- At structure discontinuities the intrinsic electromagnetic fields of the moving 

charged c a ged pa particles t c es causes tthe e appea appearance a ce o of „ „Wakefields“ a e e ds 

- These Wakefields can act back on the particles which is expressed in terms 

of a Wakepotential 


Beam Position Monitor 

Wakefields 

Note: Beta smaller 

than 1 is possible. 

detailed deta ed 

view 


Beam Position Monitor 

150 

0 

Wakefields 

Port 1 

Port 2 

-150 0 7 

Time/ns 

Output Signals at the electrode 

ports excited by the beam 

Electric field vs. time 


Wakefields 

Normalized 

output at the 

electrode 

See also: P. Raabe, VDI Fortschrittsberichte, Reihe 21, Nr. 128, 1993 


Pillbox Cavity 

Structure 

Wakefields 

Beam definition 


Wl(s)/ /(V/C) 

Wakefields 

RReference f PPulse l 

Wl(s) 


Wakefields 


Wakefields 




–Q Factor 

SUMMARY 




• Collector 


49 www.cst.com

3D EM Simulations

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