Calculation of Coupling Impedance for Beamline Components: Comparison of Analytical and Numerical Results for Resistive Pipe

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Calculation of Coupling Impedance for Beamline
ComponentsComparison of Analytical and
Numerical Results for Resistive Pipe

• Lukas Hänichen

GSI / CERN Beam dynamics and collective effects
collaboration meeting February 19th, 2009
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Outline

• Introduction Motivation
• Background Definitions
• Benchmark Problem
• Analytical Approach
• Numerical Approach
• Results
• Conclusion Outlook

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Introduction Simplified Synchrotron Layout and
Beamline Components
Kicker for Injection and Extraction
Dipole
Quadrupole
Beampipe
Accel. Cavity
Bellows, Transitions...
Couplers, Peripherals...
Source Synchrotron SOLEIL
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Motivation Prediction of longitudinal and
transverse beam stability
Synchrotron Oscillation
Betatron Oscillation
Dipole momentum (1st order mom.)
Current density (0th order mom.)
Modulated Current
Longitudinal Instability
Transverse Instability
Component design
Beam induced EM fields
EM field code
Stability analysis
tracking code
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Background (1) Quantitative description of beam
interaction with beamline components

• Analogy Electrical network / beamline
  components

I
Z
U
IdQ/dt

• Converts current into voltage
• Coupling quantity is typically
• frequency dependant
• Network impedance Z

• Charge / dipole momentum is converted into
• Longitudinal / Transverse EM field
• Coupling Impedances Z and Zx,y can be defined
  similarly

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Background (2) Longitudinal / Transverse
Coupling Impedance definition

• Longitudinal Coupling Impedance
• Longitudinal electric field induced by moving
  disc charge
• With finite transverse dimensions
• Integral over transverse charge distribution and
  covered path

• Transverse Coupling Impedance
• Transverse electric and magnetic field induced by
  moving charge with offset ?

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Background (3) Sources for impedance
Broadband (continuous)
Narrowband (discrete)

• Space charge impedance
• self induced fields assuming PEC environment
• Resistive wall impedance
• Finite conducting parts
• oppose resistance to wall
• current
• External impedances
• Instrumentation
• Ferrites

• Resonant Modes
• Accelerating Cavities
• Higher order resonances
• in beampipe or adjacent
• parts

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Background (4) Connection between Wake Function
and Coupling Impedance

• Wake function is the response to a Delta charge
  distribution
• Wake potential is the response to an arbitrary
  bunch shape (typ. Gaussian)
• In practice the excitation has to be of finite
  length

Arbitrary Distribution

• Delta
• Distribution

• Wake potential for delta distribution is
• the wake function (Green function)

Wake potential for arbitrary distribution
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Benchmark problem (1) Impedance Contribution of
straight circular pipe

• Relevance
• Most basic geometry
• Present in many sections of synchrotron
• Represents the lower Z limit for the whole system

• Source for analytical solution Al-khateeb et al.

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Benchmark problem (2) Impedance Contribution of
straight circular pipe

• Challenges
• Distributed not localized
• Quasi- infinitely extending structure in
  longitudinal direction
• Open boundaries due to beam enter and exit plane
• Representation of transverse beam profile
• High conductivity complicates observation of
  effect (bad noise figure expected)
• Longitudinal coupling impedance investigated
  first, assuming
• Thick wall (no field penetration)
• Wall resistance described by surface impedance

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Analytical approach (1)Wave equation and beam
representation

• Wave equation and continuity equation

• Excitation homogenous, circular beam w/ radius a

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Analytical approach (2)Wave equation in
frequency domain

• Frequency domain

• Longitudinal electric field (Monopole)

b
a
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Analytical approach (3)Obtain coupling
impedance

• Apply boundary conditions, determine coefficients
• Plug field solution into impedance definition
• Evaluate impedance for ? a (envelope of
  transverse beam profile)
• Impedance can be split into

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Numerical approach (1) Time domain vs Frequency
domain

• The discrete EM field problem can be adressed by
  either solving
• A transient field problem excited by a transiting
  bunch
• (Finite Difference Time Domain methods) or
• A time harmonic field problem for multiple
  frequency samples

• Time domain simulation with CST PARTICLE STUDIO
  has been chosen
• Whole frequency range covered with single run
• Frequency range adjustable by choice of bunch
  shape
• Reliable code at hand allowing for easy
  reproduction and exchange
• Surface impedance model included
• Z is obtained by DFT

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Numerical approach (2) Wakefield Calculation
with CST PARTICLE STUDIO
ripple
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Numerical approach (3) Simulation Overview
Coupling impedance calculation

• PEC
• Check if
• Smooth pipe
• Corrugated pipe

• Lossy
• Check if
• Smooth pipe
• Parameter studies
• Corrugated pipe

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Results (1) Resistive wall

• CST PARTICLE STUDIO to compute resistive wake
  field
• Symmetry planes apply due to center beam
• Z is obtained by applying DFT
• Surface Impedance Boundary
• (Leontovich condition)
• ?10000 Sm-1 to boost resistive
• behaviour for studies
• Limit for pencil beam

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Results (2) Resistive wall impedance

• Circular pipe R0.01m
• Conductivity ?104Sm-1
• Structure length L1m
• Wakelength Lwake5m

• Comparison with analytical formula shows very
  good agreement
• Accuracy in lower frequency range may be
  increased by using longer bunches for excitation

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Results (3) Alternate approach - Conductive
losses introduced using bulk material

• Upper frequency limit fmax determines Minimum
  Skin Depth dmin
• Lower frequency limit fmin determines Maximum
  Skin Depth dmax
• Mesh has to resolve a few dmin !
• Bulk has to be thicker than an few dmax !

Bulk material Conductivity ?104Sm-1
Bulk thickness T
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Results (4) Bulk material with refined mesh

• Comparison with analytical formula shows
  agreement
• Long calculation periods (refined mesh)
• Deviation may be related to coarse meshing
• Influence of wall thickness not fully
  investigated yet

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Results (5) Parameter studies - Wakelength

• Circular pipe R0.01m
• Conductivity ?104Sm-1
• Structure length L1m
• Wakelength Lwake5m

• Truncation of wake calculation causes ripple due
  to window effect in time domain
• Wakelength should be in the range of total length
• Ripple decreases when wakelength is a multiple of
  total length

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Results (6) Parameter studies length scaling

• Circular pipe R0.01m
• Conductivity ?104Sm-1
• Wakelength Lwake1m
• Z scaled with 1/L

• Impedance scales perfectly linear with structure
  length
• No visible boundary effects at upper and lower z
  plane
• Computational cost scales linear with length

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Results (7) Space charge impedance -
representation of beam profile

• PEC model
• Assuming homogenous current distribution

• Only single point beam excitation available at
  the moment !
• Scanning beam profile with single pencil beam and
  use superposition (computational cost increases
  linear with number of beams !)
• Manual assembly of FIT beam vector should be
  possible in future
• (no additional cost !)
• Any profile is possible

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Results (8) Space charge impedance

• Impedance does not completely vanish for
  ultrarelativistic beam
• Matches qualitatively with analytical solution
  (factor of 2p/L has to be added)

ß0.5
ß1
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Results (5) Corrugated wall resistive impedance

• Analytical description pending
• Comparison with smooth pipe shows almost constant
  increase of resistance
• (due to increase of effective conductor length)

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Conclusions

• Good agreement in upper frequency range (gt10 MHz)
  
• Application to transverse impedance should be
  straightforward
• Deviation in lower frequency range (10 kHz... 10
  MHz) may be related to
• Insufficent bunch length for representing LF
  wavelengths
• This appears to be limited by computational cost
• Inaccurate surface impedance model at LF

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Outlook / Upcoming Studies (1)

• Transverse Coupling Impedances
• Applicability Test alternate method to treat
  lower frequency range (10 kHz... 10 MHz)
• Computational Efficiency Test performance of
  code using ?-f-z FIT mesh (MAFIA)
• Application Beam Pipe, GSI Slow Extraction
  Septum (A. Plotnikov)
• Improved beam representation
• Multilayer walls (fully discretized or
  specialized boundary condition)
• Thin walls (fully discretized or specialized
  boundary condition, transmission)

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Thank you very much !

• I would like to thank
• Prof. Oliver Boine-Frankenheim and the GSI for
  supporting this work
• Ahmed Al-khateeb for helpful discussions about
  analytical formulations
• Prof. Thomas Weiland and colleagues at TEMF for
  support and helpful discussions
• The audience for your attention