heavy beam loading and collective effects€¦ · i two important effects in storage rings: i...
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Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
1/41
Heavy Beam Loading and Collective Effects
D. Teytelman
Dimtel, Inc., San Jose, CA, USA
LLRF Workshop 2019
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
2/41
Outline
IntroductionThe focus of this tutorialCoupled-bunch Instabilities
Beam Loading EffectsBeam Loading in Storage RingsRing Circumference and Fundamental Impedance
Transient LoadingHow Not To Fix Transient LoadingRealistic Mitigation
Fundamental Impedance and InstabilitiesBunch-by-bunch FeedbackImpedance Control Loops
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
3/41
Outline
IntroductionThe focus of this tutorialCoupled-bunch Instabilities
Beam Loading EffectsBeam Loading in Storage RingsRing Circumference and Fundamental Impedance
Transient LoadingHow Not To Fix Transient LoadingRealistic Mitigation
Fundamental Impedance and InstabilitiesBunch-by-bunch FeedbackImpedance Control Loops
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
4/41
Coupled-bunch Instabilities and Beam Loading
I Looking at heavy beam loading in electron/positron storage rings;I Two important effects in storage rings:
I Coupled-bunch instabilities in the longitudinal plane;I Transient beam loading due to non-uniform fill patterns.
I In heavily loaded rings both of these will be driven by the fundamentalimpedance of the RF cavities:
I Instabilities: beam interacts with the impedances at synchrotronsidebands of revolution harmonics;
I Transient beam loading: driven by the impedance at revolution harmonics.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
4/41
Coupled-bunch Instabilities and Beam Loading
I Looking at heavy beam loading in electron/positron storage rings;I Two important effects in storage rings:
I Coupled-bunch instabilities in the longitudinal plane;I Transient beam loading due to non-uniform fill patterns.
I In heavily loaded rings both of these will be driven by the fundamentalimpedance of the RF cavities:
I Instabilities: beam interacts with the impedances at synchrotronsidebands of revolution harmonics;
I Transient beam loading: driven by the impedance at revolution harmonics.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
4/41
Coupled-bunch Instabilities and Beam Loading
I Looking at heavy beam loading in electron/positron storage rings;I Two important effects in storage rings:
I Coupled-bunch instabilities in the longitudinal plane;I Transient beam loading due to non-uniform fill patterns.
I In heavily loaded rings both of these will be driven by the fundamentalimpedance of the RF cavities:
I Instabilities: beam interacts with the impedances at synchrotronsidebands of revolution harmonics;
I Transient beam loading: driven by the impedance at revolution harmonics.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
4/41
Coupled-bunch Instabilities and Beam Loading
I Looking at heavy beam loading in electron/positron storage rings;I Two important effects in storage rings:
I Coupled-bunch instabilities in the longitudinal plane;I Transient beam loading due to non-uniform fill patterns.
I In heavily loaded rings both of these will be driven by the fundamentalimpedance of the RF cavities:
I Instabilities: beam interacts with the impedances at synchrotronsidebands of revolution harmonics;
I Transient beam loading: driven by the impedance at revolution harmonics.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
4/41
Coupled-bunch Instabilities and Beam Loading
I Looking at heavy beam loading in electron/positron storage rings;I Two important effects in storage rings:
I Coupled-bunch instabilities in the longitudinal plane;I Transient beam loading due to non-uniform fill patterns.
I In heavily loaded rings both of these will be driven by the fundamentalimpedance of the RF cavities:
I Instabilities: beam interacts with the impedances at synchrotronsidebands of revolution harmonics;
I Transient beam loading: driven by the impedance at revolution harmonics.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
5/41
Outline
IntroductionThe focus of this tutorialCoupled-bunch Instabilities
Beam Loading EffectsBeam Loading in Storage RingsRing Circumference and Fundamental Impedance
Transient LoadingHow Not To Fix Transient LoadingRealistic Mitigation
Fundamental Impedance and InstabilitiesBunch-by-bunch FeedbackImpedance Control Loops
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
6/41
Coupled-bunch Instabilities
Resonant structure
Vacuum chamber
nn+1n+2
bunch n bunch n+2n+1bunch
Time
I Bunch passing through a resonantstructure excites a wakefield which issampled by the following bunches — acoupling mechanism;
I In practice the wakefields have muchlonger damping times than illustratedhere;
I Longitudinal bunch oscillation→ phasemodulation of the wakefield→ slope ofthe wake voltage sampled by thefollowing bunches determines thecoupling.
I For certain combinations of wakefieldamplitudes and frequencies the overallsystem becomes unstable.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
6/41
Coupled-bunch Instabilities
Resonant structure
Vacuum chamber
nn+1n+2
bunch n bunch n+2n+1bunch
Time
I Bunch passing through a resonantstructure excites a wakefield which issampled by the following bunches — acoupling mechanism;
I In practice the wakefields have muchlonger damping times than illustratedhere;
I Longitudinal bunch oscillation→ phasemodulation of the wakefield→ slope ofthe wake voltage sampled by thefollowing bunches determines thecoupling.
I For certain combinations of wakefieldamplitudes and frequencies the overallsystem becomes unstable.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
6/41
Coupled-bunch Instabilities
Resonant structure
Vacuum chamber
nn+1n+2
bunch n bunch n+2n+1bunch
Time
I Bunch passing through a resonantstructure excites a wakefield which issampled by the following bunches — acoupling mechanism;
I In practice the wakefields have muchlonger damping times than illustratedhere;
I Longitudinal bunch oscillation→ phasemodulation of the wakefield→ slope ofthe wake voltage sampled by thefollowing bunches determines thecoupling.
I For certain combinations of wakefieldamplitudes and frequencies the overallsystem becomes unstable.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
6/41
Coupled-bunch Instabilities
Resonant structure
Vacuum chamber
nn+1n+2
bunch n bunch n+2n+1bunch
Time
I Bunch passing through a resonantstructure excites a wakefield which issampled by the following bunches — acoupling mechanism;
I In practice the wakefields have muchlonger damping times than illustratedhere;
I Longitudinal bunch oscillation→ phasemodulation of the wakefield→ slope ofthe wake voltage sampled by thefollowing bunches determines thecoupling.
I For certain combinations of wakefieldamplitudes and frequencies the overallsystem becomes unstable.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
7/41
Coupled-bunch Instabilities: Eigenmodes and Eigenvalues
I A system of N bunches (coupled harmonic oscillators) has Neigenmodes;
I From symmetry considerations we find that the eigenmodes correspondto Fourier vectors;
I Mode number m describes the number of oscillation periods over oneturn;
I Motion of bunch k oscillating in mode m is given by: Ame2πkm/NeΛmt
I Am — modal amplitude;I Λm — complex modal eigenvalue.
I Wakefields affect the modal eigenvalues in both real (growth rate)and imaginary (oscillation frequency) parts;
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
7/41
Coupled-bunch Instabilities: Eigenmodes and Eigenvalues
I A system of N bunches (coupled harmonic oscillators) has Neigenmodes;
I From symmetry considerations we find that the eigenmodes correspondto Fourier vectors;
I Mode number m describes the number of oscillation periods over oneturn;
I Motion of bunch k oscillating in mode m is given by: Ame2πkm/NeΛmt
I Am — modal amplitude;I Λm — complex modal eigenvalue.
I Wakefields affect the modal eigenvalues in both real (growth rate)and imaginary (oscillation frequency) parts;
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
7/41
Coupled-bunch Instabilities: Eigenmodes and Eigenvalues
I A system of N bunches (coupled harmonic oscillators) has Neigenmodes;
I From symmetry considerations we find that the eigenmodes correspondto Fourier vectors;
I Mode number m describes the number of oscillation periods over oneturn;
I Motion of bunch k oscillating in mode m is given by: Ame2πkm/NeΛmt
I Am — modal amplitude;I Λm — complex modal eigenvalue.
I Wakefields affect the modal eigenvalues in both real (growth rate)and imaginary (oscillation frequency) parts;
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
7/41
Coupled-bunch Instabilities: Eigenmodes and Eigenvalues
I A system of N bunches (coupled harmonic oscillators) has Neigenmodes;
I From symmetry considerations we find that the eigenmodes correspondto Fourier vectors;
I Mode number m describes the number of oscillation periods over oneturn;
I Motion of bunch k oscillating in mode m is given by: Ame2πkm/NeΛmt
I Am — modal amplitude;I Λm — complex modal eigenvalue.
I Wakefields affect the modal eigenvalues in both real (growth rate)and imaginary (oscillation frequency) parts;
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
7/41
Coupled-bunch Instabilities: Eigenmodes and Eigenvalues
I A system of N bunches (coupled harmonic oscillators) has Neigenmodes;
I From symmetry considerations we find that the eigenmodes correspondto Fourier vectors;
I Mode number m describes the number of oscillation periods over oneturn;
I Motion of bunch k oscillating in mode m is given by: Ame2πkm/NeΛmt
I Am — modal amplitude;I Λm — complex modal eigenvalue.
I Wakefields affect the modal eigenvalues in both real (growth rate)and imaginary (oscillation frequency) parts;
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
8/41
Modal Oscillation Example
I Harmonic number of 8;I Top plot — mode 1;I Bottom — mode 7;I All bunches oscillate at the same
amplitude and frequency, butdifferent phases;
I Cannot distinguish modes m andN −m (or −m) from a single turnsnapshot.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
9/41
Modal Oscillation With Damping
I Same modes with damping.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
10/41
Coupled-bunch Instabilities: Eigenvalues and Impedances
I Beam interacts with wakefields (impedances in frequency domain) atsynchrotron sidebands of revolution harmonics;
I Impedance functions are aliased, since they are sampled by the beam;
I Λm = (−λ‖rad + iωs) +παef 2
rf I0E0hωs
Z ‖eff(mω0 + ωs);
I Effective impedance: Z ‖eff(ω) =∑∞
p=−∞pωrf+ωωrf
Z ‖(pωrf + ω)
I Normally, instabilities in the longitudinal plane are driven by higher ordermodes in RF cavities and other resonances;
I In case of heavy beam loading in machines with large circumference,situation is anything, but normal.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
10/41
Coupled-bunch Instabilities: Eigenvalues and Impedances
I Beam interacts with wakefields (impedances in frequency domain) atsynchrotron sidebands of revolution harmonics;
I Impedance functions are aliased, since they are sampled by the beam;
I Λm = (−λ‖rad + iωs) +παef 2
rf I0E0hωs
Z ‖eff(mω0 + ωs);
I Effective impedance: Z ‖eff(ω) =∑∞
p=−∞pωrf+ωωrf
Z ‖(pωrf + ω)
I Normally, instabilities in the longitudinal plane are driven by higher ordermodes in RF cavities and other resonances;
I In case of heavy beam loading in machines with large circumference,situation is anything, but normal.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
10/41
Coupled-bunch Instabilities: Eigenvalues and Impedances
I Beam interacts with wakefields (impedances in frequency domain) atsynchrotron sidebands of revolution harmonics;
I Impedance functions are aliased, since they are sampled by the beam;
I Λm = (−λ‖rad + iωs) +παef 2
rf I0E0hωs
Z ‖eff(mω0 + ωs);
I Effective impedance: Z ‖eff(ω) =∑∞
p=−∞pωrf+ωωrf
Z ‖(pωrf + ω)
I Normally, instabilities in the longitudinal plane are driven by higher ordermodes in RF cavities and other resonances;
I In case of heavy beam loading in machines with large circumference,situation is anything, but normal.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
10/41
Coupled-bunch Instabilities: Eigenvalues and Impedances
I Beam interacts with wakefields (impedances in frequency domain) atsynchrotron sidebands of revolution harmonics;
I Impedance functions are aliased, since they are sampled by the beam;
I Λm = (−λ‖rad + iωs) +παef 2
rf I0E0hωs
Z ‖eff(mω0 + ωs);
I Effective impedance: Z ‖eff(ω) =∑∞
p=−∞pωrf+ωωrf
Z ‖(pωrf + ω)
I Normally, instabilities in the longitudinal plane are driven by higher ordermodes in RF cavities and other resonances;
I In case of heavy beam loading in machines with large circumference,situation is anything, but normal.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
10/41
Coupled-bunch Instabilities: Eigenvalues and Impedances
I Beam interacts with wakefields (impedances in frequency domain) atsynchrotron sidebands of revolution harmonics;
I Impedance functions are aliased, since they are sampled by the beam;
I Λm = (−λ‖rad + iωs) +παef 2
rf I0E0hωs
Z ‖eff(mω0 + ωs);
I Effective impedance: Z ‖eff(ω) =∑∞
p=−∞pωrf+ωωrf
Z ‖(pωrf + ω)
I Normally, instabilities in the longitudinal plane are driven by higher ordermodes in RF cavities and other resonances;
I In case of heavy beam loading in machines with large circumference,situation is anything, but normal.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
11/41
Outline
IntroductionThe focus of this tutorialCoupled-bunch Instabilities
Beam Loading EffectsBeam Loading in Storage RingsRing Circumference and Fundamental Impedance
Transient LoadingHow Not To Fix Transient LoadingRealistic Mitigation
Fundamental Impedance and InstabilitiesBunch-by-bunch FeedbackImpedance Control Loops
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
12/41
Beam/Cavity Interaction
loops
RF feedback
dynamics
Longitudinal
Generator
C R L
Beam
~IG
~VC
~IB
I RLC model of the accelerating cavitywith two input currents: generatorand beam;
I Cavity voltage ~VC is defined by thesum current;
I Low loading (~IB ~IG) — cavityvoltage is mostly defined by thegenerator current;
I High loading — cavity voltage isstrongly affected by beam current;
I “Feedback loop” from cavity voltageto beam current and back to cavityvoltage.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
12/41
Beam/Cavity Interaction
loops
RF feedback
dynamics
Longitudinal
Generator
C R L
Beam
~IG
~VC
~IB
I RLC model of the accelerating cavitywith two input currents: generatorand beam;
I Cavity voltage ~VC is defined by thesum current;
I Low loading (~IB ~IG) — cavityvoltage is mostly defined by thegenerator current;
I High loading — cavity voltage isstrongly affected by beam current;
I “Feedback loop” from cavity voltageto beam current and back to cavityvoltage.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
12/41
Beam/Cavity Interaction
loops
RF feedback
dynamics
Longitudinal
Generator
C R L
Beam
~IG
~VC
~IB
I RLC model of the accelerating cavitywith two input currents: generatorand beam;
I Cavity voltage ~VC is defined by thesum current;
I Low loading (~IB ~IG) — cavityvoltage is mostly defined by thegenerator current;
I High loading — cavity voltage isstrongly affected by beam current;
I “Feedback loop” from cavity voltageto beam current and back to cavityvoltage.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
12/41
Beam/Cavity Interaction
loops
RF feedback
dynamics
Longitudinal
Generator
C R L
Beam
~IG
~VC
~IB
I RLC model of the accelerating cavitywith two input currents: generatorand beam;
I Cavity voltage ~VC is defined by thesum current;
I Low loading (~IB ~IG) — cavityvoltage is mostly defined by thegenerator current;
I High loading — cavity voltage isstrongly affected by beam current;
I “Feedback loop” from cavity voltageto beam current and back to cavityvoltage.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
12/41
Beam/Cavity Interaction
loops
RF feedback
dynamics
Longitudinal
Generator
C R L
Beam
~IG
~VC
~IB
I RLC model of the accelerating cavitywith two input currents: generatorand beam;
I Cavity voltage ~VC is defined by thesum current;
I Low loading (~IB ~IG) — cavityvoltage is mostly defined by thegenerator current;
I High loading — cavity voltage isstrongly affected by beam current;
I “Feedback loop” from cavity voltageto beam current and back to cavityvoltage.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
13/41
Phasor Diagram
~IB
~Vc
φL
~IGφZ
~IL
~Itot
~Itot = ~IG + ~IB
φB
I Phasors at RF frequency, cavityvoltage on X axis;
I Synchronous phase φB is determinedby RF voltage, energy loss per turn;
I For minimum generator power keeploading angle φL = 0;
I Cavity is detuned to maintain properphase angle φZ between the totalcurrent and the cavity voltage;
I The larger is~IB, the higher is thedetuning.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
13/41
Phasor Diagram
~IB
~Vc
φL
~IGφZ
~IL
~Itot
~Itot = ~IG + ~IB
φB
I Phasors at RF frequency, cavityvoltage on X axis;
I Synchronous phase φB is determinedby RF voltage, energy loss per turn;
I For minimum generator power keeploading angle φL = 0;
I Cavity is detuned to maintain properphase angle φZ between the totalcurrent and the cavity voltage;
I The larger is~IB, the higher is thedetuning.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
13/41
Phasor Diagram
~IB
~Vc
φL
~IGφZ
~IL
~Itot
~Itot = ~IG + ~IB
φB
I Phasors at RF frequency, cavityvoltage on X axis;
I Synchronous phase φB is determinedby RF voltage, energy loss per turn;
I For minimum generator power keeploading angle φL = 0;
I Cavity is detuned to maintain properphase angle φZ between the totalcurrent and the cavity voltage;
I The larger is~IB, the higher is thedetuning.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
13/41
Phasor Diagram
~IB
~Vc
φL
~IGφZ
~IL
~Itot
~Itot = ~IG + ~IB
φB
I Phasors at RF frequency, cavityvoltage on X axis;
I Synchronous phase φB is determinedby RF voltage, energy loss per turn;
I For minimum generator power keeploading angle φL = 0;
I Cavity is detuned to maintain properphase angle φZ between the totalcurrent and the cavity voltage;
I The larger is~IB, the higher is thedetuning.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
13/41
Phasor Diagram
~IB
~Vc
φL
~IGφZ
~IL
~Itot
~Itot = ~IG + ~IB
φB
I Phasors at RF frequency, cavityvoltage on X axis;
I Synchronous phase φB is determinedby RF voltage, energy loss per turn;
I For minimum generator power keeploading angle φL = 0;
I Cavity is detuned to maintain properphase angle φZ between the totalcurrent and the cavity voltage;
I The larger is~IB, the higher is thedetuning.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
14/41
Outline
IntroductionThe focus of this tutorialCoupled-bunch Instabilities
Beam Loading EffectsBeam Loading in Storage RingsRing Circumference and Fundamental Impedance
Transient LoadingHow Not To Fix Transient LoadingRealistic Mitigation
Fundamental Impedance and InstabilitiesBunch-by-bunch FeedbackImpedance Control Loops
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
15/41
Resonant Modes and Revolution Harmonics
−400 −300 −200 −100 0 100 200 300 4000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (kHz)
Impedance (
arb
. units)
I A 20 kHz resonance ideally “hidden”between two revolution harmonics.
I 500 m ring;I 1.5 km ring;I 3 km ring;I 10 km ring;I 100 km ring;I Narrow resonances cannot be
“hidden” from the beam in large rings.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
15/41
Resonant Modes and Revolution Harmonics
−400 −300 −200 −100 0 100 200 300 4000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (kHz)
Impedance (
arb
. units)
I A 20 kHz resonance ideally “hidden”between two revolution harmonics.
I 500 m ring;I 1.5 km ring;I 3 km ring;I 10 km ring;I 100 km ring;I Narrow resonances cannot be
“hidden” from the beam in large rings.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
15/41
Resonant Modes and Revolution Harmonics
−400 −300 −200 −100 0 100 200 300 4000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (kHz)
Impedance (
arb
. units)
I A 20 kHz resonance ideally “hidden”between two revolution harmonics.
I 500 m ring;I 1.5 km ring;I 3 km ring;I 10 km ring;I 100 km ring;I Narrow resonances cannot be
“hidden” from the beam in large rings.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
15/41
Resonant Modes and Revolution Harmonics
−400 −300 −200 −100 0 100 200 300 4000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (kHz)
Impedance (
arb
. units)
I A 20 kHz resonance ideally “hidden”between two revolution harmonics.
I 500 m ring;I 1.5 km ring;I 3 km ring;I 10 km ring;I 100 km ring;I Narrow resonances cannot be
“hidden” from the beam in large rings.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
15/41
Resonant Modes and Revolution Harmonics
−400 −300 −200 −100 0 100 200 300 4000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (kHz)
Impedance (
arb
. units)
I A 20 kHz resonance ideally “hidden”between two revolution harmonics.
I 500 m ring;I 1.5 km ring;I 3 km ring;I 10 km ring;I 100 km ring;I Narrow resonances cannot be
“hidden” from the beam in large rings.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
15/41
Resonant Modes and Revolution Harmonics
−400 −300 −200 −100 0 100 200 300 4000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (kHz)
Impedance (
arb
. units)
I A 20 kHz resonance ideally “hidden”between two revolution harmonics.
I 500 m ring;I 1.5 km ring;I 3 km ring;I 10 km ring;I 100 km ring;I Narrow resonances cannot be
“hidden” from the beam in large rings.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
16/41
Ring Circumference and Beam Loading
Photo/image credit: CERN, SLAC
I People don’t build multi-kilometerrings just to spend money;
I Large circumference — very highenergy;
I Or very high current;I Or both.I Large circumference means heavy
beam loading of the RF system.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
16/41
Ring Circumference and Beam Loading
Photo/image credit: CERN, SLAC
I People don’t build multi-kilometerrings just to spend money;
I Large circumference — very highenergy;
I Or very high current;I Or both.I Large circumference means heavy
beam loading of the RF system.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
16/41
Ring Circumference and Beam Loading
Photo/image credit: CERN, SLAC
I People don’t build multi-kilometerrings just to spend money;
I Large circumference — very highenergy;
I Or very high current;I Or both.I Large circumference means heavy
beam loading of the RF system.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
16/41
Ring Circumference and Beam Loading
Photo/image credit: CERN, SLAC
I People don’t build multi-kilometerrings just to spend money;
I Large circumference — very highenergy;
I Or very high current;I Or both.I Large circumference means heavy
beam loading of the RF system.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
16/41
Ring Circumference and Beam Loading
Photo/image credit: CERN, SLAC
I People don’t build multi-kilometerrings just to spend money;
I Large circumference — very highenergy;
I Or very high current;I Or both.I Large circumference means heavy
beam loading of the RF system.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
17/41
Cavity Detuning and Longitudinal Stability
−1.5 −1 −0.5 0 0.5 1 1.50
500
1000
1500
2000
Frequency offset from ωrf, revolution harmonics
ℜ(Z
), k
Ω
−1.5 −1 −0.5 0 0.5 1 1.5−2000
−1000
0
1000
2000
Eigenmode numberEff
ective
drivin
g im
pe
da
nce
, kΩ
I Growth rate for mode -1 is ∝Z (ωrf−ωrev+ωs)−Z (ωrf+ωrev−ωs);
I Symmetric on resonance;I Growth rates peak when
fundamental crosses uppersynchrotron sidebands ofrevolution harmonics;
I Instability growth times very smallrelative to synchrotron period.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
17/41
Cavity Detuning and Longitudinal Stability
−1.5 −1 −0.5 0 0.5 1 1.50
500
1000
1500
2000
Frequency offset from ωrf, revolution harmonics
ℜ(Z
), k
Ω
−1.5 −1 −0.5 0 0.5 1 1.5−2000
−1000
0
1000
2000
Eigenmode numberEff
ective
drivin
g im
pe
da
nce
, kΩ
I Growth rate for mode -1 is ∝Z (ωrf−ωrev+ωs)−Z (ωrf+ωrev−ωs);
I Symmetric on resonance;I Growth rates peak when
fundamental crosses uppersynchrotron sidebands ofrevolution harmonics;
I Instability growth times very smallrelative to synchrotron period.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
17/41
Cavity Detuning and Longitudinal Stability
−1.5 −1 −0.5 0 0.5 1 1.50
500
1000
1500
2000
Frequency offset from ωrf, revolution harmonics
ℜ(Z
), k
Ω
−1.5 −1 −0.5 0 0.5 1 1.5−2000
−1000
0
1000
2000
Eigenmode numberEff
ective
drivin
g im
pe
da
nce
, kΩ
I Growth rate for mode -1 is ∝Z (ωrf−ωrev+ωs)−Z (ωrf+ωrev−ωs);
I Symmetric on resonance;I Growth rates peak when
fundamental crosses uppersynchrotron sidebands ofrevolution harmonics;
I Instability growth times very smallrelative to synchrotron period.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
17/41
Cavity Detuning and Longitudinal Stability
−1.5 −1 −0.5 0 0.5 1 1.50
500
1000
1500
2000
Frequency offset from ωrf, revolution harmonics
ℜ(Z
), k
Ω
−1.5 −1 −0.5 0 0.5 1 1.5−2000
−1000
0
1000
2000
Eigenmode numberEff
ective
drivin
g im
pe
da
nce
, kΩ
I Growth rate for mode -1 is ∝Z (ωrf−ωrev+ωs)−Z (ωrf+ωrev−ωs);
I Symmetric on resonance;I Growth rates peak when
fundamental crosses uppersynchrotron sidebands ofrevolution harmonics;
I Instability growth times very smallrelative to synchrotron period.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
17/41
Cavity Detuning and Longitudinal Stability
−1.5 −1 −0.5 0 0.5 1 1.50
500
1000
1500
2000
Frequency offset from ωrf, revolution harmonics
ℜ(Z
), k
Ω
−1.5 −1 −0.5 0 0.5 1 1.5−2000
−1000
0
1000
2000
Eigenmode numberEff
ective
drivin
g im
pe
da
nce
, kΩ
I Growth rate for mode -1 is ∝Z (ωrf−ωrev+ωs)−Z (ωrf+ωrev−ωs);
I Symmetric on resonance;I Growth rates peak when
fundamental crosses uppersynchrotron sidebands ofrevolution harmonics;
I Instability growth times very smallrelative to synchrotron period.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
17/41
Cavity Detuning and Longitudinal Stability
−1.5 −1 −0.5 0 0.5 1 1.50
500
1000
1500
2000
Frequency offset from ωrf, revolution harmonics
ℜ(Z
), k
Ω
−1.5 −1 −0.5 0 0.5 1 1.5−2000
−1000
0
1000
2000
Eigenmode numberEff
ective
drivin
g im
pe
da
nce
, kΩ
I Growth rate for mode -1 is ∝Z (ωrf−ωrev+ωs)−Z (ωrf+ωrev−ωs);
I Symmetric on resonance;I Growth rates peak when
fundamental crosses uppersynchrotron sidebands ofrevolution harmonics;
I Instability growth times very smallrelative to synchrotron period.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
18/41
Single Bunch Train in FCC-ee (Z)
0 50 100 150 200 250 300 350−300
−200
−100
0
100
200
Time (µs)
Phase (
deg@
RF
)
Transient is 392.0891 degrees peak−to−peak
0 50 100 150 200 250 300 3500
0.005
0.01
0.015
0.02
Time (µs)
Bunch c
urr
ent (m
A)
FCC−ee; 88/0 powered/parked cavities; Vgap
= 255 MV; I0 = 1.39 A; 70760by1 fill
I Non-uniform fill pattern puts power atrevolution harmonics, modulatescavity field;
I Single train is unphysical;I At 300 mA it is slightly more realistic;I Bunch length is all over the place;I As is the synchrotron frequency.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
18/41
Single Bunch Train in FCC-ee (Z)
0 50 100 150 200 250 300 350−300
−200
−100
0
100
200
Time (µs)
Phase (
deg@
RF
)
Transient is 392.0891 degrees peak−to−peak
0 50 100 150 200 250 300 3500
0.005
0.01
0.015
0.02
Time (µs)
Bunch c
urr
ent (m
A)
FCC−ee; 88/0 powered/parked cavities; Vgap
= 255 MV; I0 = 1.39 A; 70760by1 fill
I Non-uniform fill pattern puts power atrevolution harmonics, modulatescavity field;
I Single train is unphysical;I At 300 mA it is slightly more realistic;I Bunch length is all over the place;I As is the synchrotron frequency.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
18/41
Single Bunch Train in FCC-ee (Z)
0 50 100 150 200 250 300 350−50
0
50
Time (µs)
Phase (
deg@
RF
)
Transient is 87.6171 degrees peak−to−peak
0 50 100 150 200 250 300 3500
1
2
3
4
5x 10
−3
Time (µs)
Bunch c
urr
ent (m
A)
FCC−ee; 88/0 powered/parked cavities; Vgap
= 255 MV; I0 = 0.3 A; 70760by1 fill
I Non-uniform fill pattern puts power atrevolution harmonics, modulatescavity field;
I Single train is unphysical;I At 300 mA it is slightly more realistic;I Bunch length is all over the place;I As is the synchrotron frequency.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
18/41
Single Bunch Train in FCC-ee (Z)
0 20 40 60 80 100 120 140 160 1802
2.2
2.4
2.6
2.8
3
3.2
3.4
Time (µs)
Bunch length
(m
m)
Peak−to−peak bunch length spead 40.69%
I Non-uniform fill pattern puts power atrevolution harmonics, modulatescavity field;
I Single train is unphysical;I At 300 mA it is slightly more realistic;I Bunch length is all over the place;I As is the synchrotron frequency.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
18/41
Single Bunch Train in FCC-ee (Z)
0 20 40 60 80 100 120 140 160 18085
90
95
100
105
110
115
120
125
130
135
Time (µs)
Syn
ch
rotr
on
fre
qu
en
cy (
Hz)
Peak−to−peak synchrotron frequency spead 39.90%
I Non-uniform fill pattern puts power atrevolution harmonics, modulatescavity field;
I Single train is unphysical;I At 300 mA it is slightly more realistic;I Bunch length is all over the place;I As is the synchrotron frequency.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
19/41
Dealing With Beam Loading
I Two main effects of heavy beam loading in large rings:I Longitudinal coupled-bunch instabilities driven by the RF cavity
fundamental impedance;I Synchronous phase transients.
I Transient effects depend onI Total beam current;I Fill pattern.
I Fill patterns can be designed to mitigate transient effects;I But longitudinal instabilities due to the fundamental impedance remain
an issue even with completely uniform fills;I Reducing beam loading in the RF system design helps both issues.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
19/41
Dealing With Beam Loading
I Two main effects of heavy beam loading in large rings:I Longitudinal coupled-bunch instabilities driven by the RF cavity
fundamental impedance;I Transient modulation of longitudinal optics.
I Transient effects depend onI Total beam current;I Fill pattern.
I Fill patterns can be designed to mitigate transient effects;I But longitudinal instabilities due to the fundamental impedance remain
an issue even with completely uniform fills;I Reducing beam loading in the RF system design helps both issues.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
19/41
Dealing With Beam Loading
I Two main effects of heavy beam loading in large rings:I Longitudinal coupled-bunch instabilities driven by the RF cavity
fundamental impedance;I Transient modulation of longitudinal optics.
I Transient effects depend onI Total beam current;I Fill pattern.
I Fill patterns can be designed to mitigate transient effects;I But longitudinal instabilities due to the fundamental impedance remain
an issue even with completely uniform fills;I Reducing beam loading in the RF system design helps both issues.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
19/41
Dealing With Beam Loading
I Two main effects of heavy beam loading in large rings:I Longitudinal coupled-bunch instabilities driven by the RF cavity
fundamental impedance;I Transient modulation of longitudinal optics.
I Transient effects depend onI Total beam current;I Fill pattern.
I Fill patterns can be designed to mitigate transient effects;I But longitudinal instabilities due to the fundamental impedance remain
an issue even with completely uniform fills;I Reducing beam loading in the RF system design helps both issues.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
19/41
Dealing With Beam Loading
I Two main effects of heavy beam loading in large rings:I Longitudinal coupled-bunch instabilities driven by the RF cavity
fundamental impedance;I Transient modulation of longitudinal optics.
I Transient effects depend onI Total beam current;I Fill pattern.
I Fill patterns can be designed to mitigate transient effects;I But longitudinal instabilities due to the fundamental impedance remain
an issue even with completely uniform fills;I Reducing beam loading in the RF system design helps both issues.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
19/41
Dealing With Beam Loading
I Two main effects of heavy beam loading in large rings:I Longitudinal coupled-bunch instabilities driven by the RF cavity
fundamental impedance;I Transient modulation of longitudinal optics.
I Transient effects depend onI Total beam current;I Fill pattern.
I Fill patterns can be designed to mitigate transient effects;I But longitudinal instabilities due to the fundamental impedance remain
an issue even with completely uniform fills;I Reducing beam loading in the RF system design helps both issues.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
20/41
Mitigating Beam Loading in Design Stage
Cavity detuning
ωd =∣∣∣ωrfI0
Vc
RQ cosφb
∣∣∣I Minimize the number of cavities:
I Reduces fundamental impedance interacting with the beam;I Limited by the maximum coupler power and/or the maximum cavity
voltage.I Minimize detuning:
I Cavities with low R/Q;I Lower RF frequencies are preferable, especially when coupler limited;I Low R/Q favors superconducting cavities.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
20/41
Mitigating Beam Loading in Design Stage
Cavity detuning
ωd =∣∣∣ωrfI0
Vc
RQ cosφb
∣∣∣I Minimize the number of cavities:
I Reduces fundamental impedance interacting with the beam;I Limited by the maximum coupler power and/or the maximum cavity
voltage.I Minimize detuning:
I Cavities with low R/Q;I Lower RF frequencies are preferable, especially when coupler limited;I Low R/Q favors superconducting cavities.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
21/41
Outline
IntroductionThe focus of this tutorialCoupled-bunch Instabilities
Beam Loading EffectsBeam Loading in Storage RingsRing Circumference and Fundamental Impedance
Transient LoadingHow Not To Fix Transient LoadingRealistic Mitigation
Fundamental Impedance and InstabilitiesBunch-by-bunch FeedbackImpedance Control Loops
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
22/41
Phasor Argument
~IB
~Vc
φL
~IGφZ
~IL
~Itot
~Itot = ~IG + ~IB
φB
I First idea — phase modulate thegenerator to suppress the transients;
I PEP-II example: IB = 6 A, IG = 1.7 A;I To compensate fill pattern
modulation, when IB goes to 0 in thegap, IG would need to match IT !
I Factor of 10 in peak power.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
22/41
Phasor Argument
2
4
6
30
210
60
240
90
270
120
300
150
330
180 0
LER; 8/0 powered/parked cavities; Vgap
= 4.5 MV; I0 = 3 A; 1722by2 fill
IG
IB
IT
I First idea — phase modulate thegenerator to suppress the transients;
I PEP-II example: IB = 6 A, IG = 1.7 A;I To compensate fill pattern
modulation, when IB goes to 0 in thegap, IG would need to match IT !
I Factor of 10 in peak power.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
22/41
Phasor Argument
2
4
6
30
210
60
240
90
270
120
300
150
330
180 0
LER; 8/0 powered/parked cavities; Vgap
= 4.5 MV; I0 = 3 A; 1722by2 fill
IG
IB
IT
I First idea — phase modulate thegenerator to suppress the transients;
I PEP-II example: IB = 6 A, IG = 1.7 A;I To compensate fill pattern
modulation, when IB goes to 0 in thegap, IG would need to match IT !
I Factor of 10 in peak power.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
22/41
Phasor Argument
2
4
6
30
210
60
240
90
270
120
300
150
330
180 0
LER; 8/0 powered/parked cavities; Vgap
= 4.5 MV; I0 = 3 A; 1722by2 fill
IG
IB
IT
I First idea — phase modulate thegenerator to suppress the transients;
I PEP-II example: IB = 6 A, IG = 1.7 A;I To compensate fill pattern
modulation, when IB goes to 0 in thegap, IG would need to match IT !
I Factor of 10 in peak power.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
23/41
Outline
IntroductionThe focus of this tutorialCoupled-bunch Instabilities
Beam Loading EffectsBeam Loading in Storage RingsRing Circumference and Fundamental Impedance
Transient LoadingHow Not To Fix Transient LoadingRealistic Mitigation
Fundamental Impedance and InstabilitiesBunch-by-bunch FeedbackImpedance Control Loops
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
24/41
Single Bunch Train
0 50 100 150 200 250 300 350−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
Time (µs)
Phase (
deg@
RF
)
Transient is 3.0494 degrees peak−to−peak
0 50 100 150 200 250 300 3500
0.005
0.01
0.015
0.02
0.025
Time (µs)
Bunch c
urr
ent (m
A)
FCC−ee; 88/0 powered/parked cavities; Vgap
= 255 MV; I0 = 1.39 A; 65140by2 fill
I 0.3% gap (400 RF buckets, 1 µs);I Uniform train of 65140 bunches with
5 ns spacing;I Bunch length moves around by 3.4%
(peak-to-peak).
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
24/41
Single Bunch Train
0 50 100 150 200 250 300 350−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
Time (µs)
Phase (
deg@
RF
)
Transient is 3.0494 degrees peak−to−peak
0 50 100 150 200 250 300 3500
0.005
0.01
0.015
0.02
0.025
Time (µs)
Bunch c
urr
ent (m
A)
FCC−ee; 88/0 powered/parked cavities; Vgap
= 255 MV; I0 = 1.39 A; 65140by2 fill
I 0.3% gap (400 RF buckets, 1 µs);I Uniform train of 65140 bunches with
5 ns spacing;I Bunch length moves around by 3.4%
(peak-to-peak).
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
24/41
Single Bunch Train
0 50 100 150 200 250 300 3502.08
2.09
2.1
2.11
2.12
2.13
2.14
2.15
2.16
2.17
Time (µs)
Bu
nch
le
ng
th (
mm
)
Peak−to−peak bunch length spead 3.42%
I 0.3% gap (400 RF buckets, 1 µs);I Uniform train of 65140 bunches with
5 ns spacing;I Bunch length moves around by 3.4%
(peak-to-peak).
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
25/41
Fill Pattern Density Modulation
0 50 100 150 200 250 300 350−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Time (µs)
Phase (
deg@
RF
)
Transient is 3.0291 degrees peak−to−peak
0 50 100 150 200 250 300 3500
0.005
0.01
0.015
0.02
0.025
Time (µs)
Bunch c
urr
ent (m
A)
FCC−ee; 88/0 powered/parked cavities; Vgap
= 255 MV; I0 = 1.39 A; 65340 density mod fill
I Idea from J. Byrd et al., Phys. Rev.ST Accel. Beams 5, 092001 (2002):
I Charge removed from the gap isadded symmetrically to both endsof the train;
I 200 bunches removed from the gap;I Rather than double the charge, fill
200 buckets at the ends of the train inevery bucket (2.5 ns) pattern;
I Phase transient peak-to-peakamplitude is unchanged.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
25/41
Fill Pattern Density Modulation
0 50 100 150 200 250 300 350−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Time (µs)
Phase (
deg@
RF
)
Transient is 3.0291 degrees peak−to−peak
0 50 100 150 200 250 300 3500
0.005
0.01
0.015
0.02
0.025
Time (µs)
Bunch c
urr
ent (m
A)
FCC−ee; 88/0 powered/parked cavities; Vgap
= 255 MV; I0 = 1.39 A; 65340 density mod fill
I Idea from J. Byrd et al., Phys. Rev.ST Accel. Beams 5, 092001 (2002):
I Charge removed from the gap isadded symmetrically to both endsof the train;
I 200 bunches removed from the gap;I Rather than double the charge, fill
200 buckets at the ends of the train inevery bucket (2.5 ns) pattern;
I Phase transient peak-to-peakamplitude is unchanged.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
25/41
Fill Pattern Density Modulation
0 0.2 0.4 0.6 0.8 1−1.6
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
Time (µs)
Phase (
deg@
RF
)
Transient is 3.0291 degrees peak−to−peak
0 0.2 0.4 0.6 0.8 10
0.005
0.01
0.015
0.02
0.025
Time (µs)
Bunch c
urr
ent (m
A)
FCC−ee; 88/0 powered/parked cavities; Vgap
= 255 MV; I0 = 1.39 A; 65340 density mod fill
I Idea from J. Byrd et al., Phys. Rev.ST Accel. Beams 5, 092001 (2002):
I Charge removed from the gap isadded symmetrically to both endsof the train;
I 200 bunches removed from the gap;I Rather than double the charge, fill
200 buckets at the ends of the train inevery bucket (2.5 ns) pattern;
I Phase transient peak-to-peakamplitude is unchanged.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
25/41
Fill Pattern Density Modulation
0 50 100 150 200 250 300 350−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Time (µs)
Phase (
deg@
RF
)
65140by2 (3.05°)
65340 density mod (3.03°)
I Idea from J. Byrd et al., Phys. Rev.ST Accel. Beams 5, 092001 (2002):
I Charge removed from the gap isadded symmetrically to both endsof the train;
I 200 bunches removed from the gap;I Rather than double the charge, fill
200 buckets at the ends of the train inevery bucket (2.5 ns) pattern;
I Phase transient peak-to-peakamplitude is unchanged.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
26/41
How Does Fill Pattern Modulation Work?
0 200 400 600 800 100010
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
101
Revolution harmonic
Ma
gn
itu
de
(A
)
65140by2
65340 density mod
I Two fill patterns used earlier:I 65140by2: one long train of
65140 bunches every other RFbucket and 400 bucket gap;
I 65340 density mod: long trainwith density modulation.
I Both fill pattern spectra shownotches at multiples ofh/400 ≈ 327 revolutionharmonics due to identical 400bucket gaps;
I Density modulation suppresseslow-frequency revolutionharmonics where cavityimpedance is large.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
26/41
How Does Fill Pattern Modulation Work?
0 200 400 600 800 100010
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
101
Revolution harmonic
Ma
gn
itu
de
(A
)
65140by2
65340 density mod
I Two fill patterns used earlier:I 65140by2: one long train of
65140 bunches every other RFbucket and 400 bucket gap;
I 65340 density mod: long trainwith density modulation.
I Both fill pattern spectra shownotches at multiples ofh/400 ≈ 327 revolutionharmonics due to identical 400bucket gaps;
I Density modulation suppresseslow-frequency revolutionharmonics where cavityimpedance is large.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
26/41
How Does Fill Pattern Modulation Work?
0 5 10 15 2010
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
101
Revolution harmonic
Ma
gn
itu
de
(A
)
65140by2
65340 density mod
I Two fill patterns used earlier:I 65140by2: one long train of
65140 bunches every other RFbucket and 400 bucket gap;
I 65340 density mod: long trainwith density modulation.
I Both fill pattern spectra shownotches at multiples ofh/400 ≈ 327 revolutionharmonics due to identical 400bucket gaps;
I Density modulation suppresseslow-frequency revolutionharmonics where cavityimpedance is large.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
27/41
Does Fill Pattern Modulation Work?
0 50 100 150 200 250 300 3500
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Bunch number
Curr
ent (m
A)
0 50 100 150 200 250 300 350−1
−0.5
0
0.5
1
1.5
2
2.5
3
Bunch number
Phase (
deg@
RF
)
200 mA
236 mA
3.05° p2p
2.76° p2p
I Measurements from the Advanced LightSource in Berkeley:
I A train of 296 buckets, 32 bucket gap;I Buckets 1–16 and 281–296 filled to
twice the charge.I A bit of first revolution harmonic due to
the detuned harmonic cavities.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
27/41
Does Fill Pattern Modulation Work?
0 50 100 150 200 250 300 3500
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Bunch number
Curr
ent (m
A)
0 50 100 150 200 250 300 350−1
−0.5
0
0.5
1
1.5
2
2.5
3
Bunch number
Phase (
deg@
RF
)
200 mA
236 mA
3.05° p2p
2.76° p2p
I Measurements from the Advanced LightSource in Berkeley:
I A train of 296 buckets, 32 bucket gap;I Buckets 1–16 and 281–296 filled to
twice the charge.I A bit of first revolution harmonic due to
the detuned harmonic cavities.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
28/41
Outline
IntroductionThe focus of this tutorialCoupled-bunch Instabilities
Beam Loading EffectsBeam Loading in Storage RingsRing Circumference and Fundamental Impedance
Transient LoadingHow Not To Fix Transient LoadingRealistic Mitigation
Fundamental Impedance and InstabilitiesBunch-by-bunch FeedbackImpedance Control Loops
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
29/41
Bunch-by-bunch Feedback
DefinitionIn bunch-by-bunch feedback approach the actuator signal for a given bunchdepends only on the past motion of that bunch.
Controller
Beam Kicker structure
Back−endFront−end
SensorBPM Actuator
I Bunches are processed sequentially.I Correction kicks are applied one turn later.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
30/41
Feedback Control Limits: Longitudinal
I Measure longitudinal position (time of arrival);I Correct energy;I To generate required 90 phase shift the feedback must observe at least
half a synchrotron period;I Fastest controllable growth times on the order of 1–2 synchrotron
periods.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
30/41
Feedback Control Limits: Longitudinal
I Measure longitudinal position (time of arrival);I Correct energy;I To generate required 90 phase shift the feedback must observe at least
half a synchrotron period;I Fastest controllable growth times on the order of 1–2 synchrotron
periods.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
30/41
Feedback Control Limits: Longitudinal
I Measure longitudinal position (time of arrival);I Correct energy;I To generate required 90 phase shift the feedback must observe at least
half a synchrotron period;I Fastest controllable growth times on the order of 1–2 synchrotron
periods.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
30/41
Feedback Control Limits: Longitudinal
I Measure longitudinal position (time of arrival);I Correct energy;I To generate required 90 phase shift the feedback must observe at least
half a synchrotron period;I Fastest controllable growth times on the order of 1–2 synchrotron
periods.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
31/41
Longitudinal Example from ANKA
I Measured while cavity tuning walks anHOM onto a synchrotron sideband;
I Growth time is 2.3Ts, damping time is Ts;I Actual modal oscillation trajectory;I Filter is 2/3 of a synchrotron period.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
31/41
Longitudinal Example from ANKA
0 0.2 0.4 0.6 0.8 1 1.2 1.40
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Time (ms)
de
g@
RF
mar0516/143812 Data, Fit and Error for Mode #45
Data
Fit
Error
I Measured while cavity tuning walks anHOM onto a synchrotron sideband;
I Growth time is 2.3Ts, damping time is Ts;I Actual modal oscillation trajectory;I Filter is 2/3 of a synchrotron period.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
31/41
Longitudinal Example from ANKA
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
Time (ms)
Phase (
deg@
RF
)
mar0516/143812: mode 45
I Measured while cavity tuning walks anHOM onto a synchrotron sideband;
I Growth time is 2.3Ts, damping time is Ts;I Actual modal oscillation trajectory;I Filter is 2/3 of a synchrotron period.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
31/41
Longitudinal Example from ANKA
0 10 20 30 40 50−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Time (turns)
Coeffic
ient
I Measured while cavity tuning walks anHOM onto a synchrotron sideband;
I Growth time is 2.3Ts, damping time is Ts;I Actual modal oscillation trajectory;I Filter is 2/3 of a synchrotron period.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
32/41
Outline
IntroductionThe focus of this tutorialCoupled-bunch Instabilities
Beam Loading EffectsBeam Loading in Storage RingsRing Circumference and Fundamental Impedance
Transient LoadingHow Not To Fix Transient LoadingRealistic Mitigation
Fundamental Impedance and InstabilitiesBunch-by-bunch FeedbackImpedance Control Loops
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
33/41
Low Level RF to the Rescue
loops
RF feedback
dynamics
Longitudinal
Generator
C R L
Beam
~IG
~VC
~IB
I Fundamental impedances at asynchrotron sideband — instabilitygrowth times below Ts/10;
I Beam feedback cannot control suchinstabilities;
I RF feedback stabilizes cavity field —low effective impedance as seen bythe beam;
I dVCdIB≈ 0;
I Use wideband loops to lower theimpedance at multiple revolutionharmonics around the RF.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
33/41
Low Level RF to the Rescue
loops
RF feedback
dynamics
Longitudinal
Generator
C R L
Beam
~IG
~VC
~IB
I Fundamental impedances at asynchrotron sideband — instabilitygrowth times below Ts/10;
I Beam feedback cannot control suchinstabilities;
I RF feedback stabilizes cavity field —low effective impedance as seen bythe beam;
I dVCdIB≈ 0;
I Use wideband loops to lower theimpedance at multiple revolutionharmonics around the RF.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
33/41
Low Level RF to the Rescue
loops
RF feedback
dynamics
Longitudinal
Generator
C R L
Beam
~IG
~VC
~IB
I Fundamental impedances at asynchrotron sideband — instabilitygrowth times below Ts/10;
I Beam feedback cannot control suchinstabilities;
I RF feedback stabilizes cavity field —low effective impedance as seen bythe beam;
I dVCdIB≈ 0;
I Use wideband loops to lower theimpedance at multiple revolutionharmonics around the RF.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
33/41
Low Level RF to the Rescue
loops
RF feedback
dynamics
Longitudinal
Generator
C R L
Beam
~IG
~VC
~IB
I Fundamental impedances at asynchrotron sideband — instabilitygrowth times below Ts/10;
I Beam feedback cannot control suchinstabilities;
I RF feedback stabilizes cavity field —low effective impedance as seen bythe beam;
I dVCdIB≈ 0;
I Use wideband loops to lower theimpedance at multiple revolutionharmonics around the RF.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
34/41
PEP-II Collider
Parameter HER LERCircumference 2.2 kmEnergy 9 GeV 3.1 GeVBeam current 2.1 A 3.2 ACavities 28 8RF power 11 MW 4 MW
I Copper HOM damped cavity;I Cavity with the HOM loads;I Two and four cavity stations,
vector sum control, 1 MWklystrons.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
34/41
PEP-II Collider
Parameter HER LERCircumference 2.2 kmEnergy 9 GeV 3.1 GeVBeam current 2.1 A 3.2 ACavities 28 8RF power 11 MW 4 MW
I Copper HOM damped cavity;I Cavity with the HOM loads;I Two and four cavity stations,
vector sum control, 1 MWklystrons.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
34/41
PEP-II Collider
Parameter HER LERCircumference 2.2 kmEnergy 9 GeV 3.1 GeVBeam current 2.1 A 3.2 ACavities 28 8RF power 11 MW 4 MW
I Copper HOM damped cavity;I Cavity with the HOM loads;I Two and four cavity stations,
vector sum control, 1 MWklystrons.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
35/41
PEP-II Fast Impedance Control
Comb loop
Cavity
Cavity
Cavity
Cavity
Direct loop gain and phase
Direct loop outputVtotal
Klystron
Comb loop gain and phase
Station reference
∑∑
Beam
I Two feedback loops: directand comb;
I Open loop: 33 µs growthtime;
I Direct loop only: 333 µsgrowth time;
I Direct and comb: 3.3 msgrowth time.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
35/41
PEP-II Fast Impedance Control
−1500 −1000 −500 0 500 1000 1500
−20
−10
0
10
20
30
Frequency (kHz)
Loop g
ain
(dB
)
−1500 −1000 −500 0 500 1000 150010
1
102
103
Frequency (kHz)
Impedance (
kΩ
)
−10 −5 0 5 1010
−2
100
Mode
Gro
wth
rate
(m
s−1)
GrowthDamping
I Two feedback loops: directand comb;
I Open loop: 33 µs growthtime;
I Direct loop only: 333 µsgrowth time;
I Direct and comb: 3.3 msgrowth time.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
35/41
PEP-II Fast Impedance Control
−1500 −1000 −500 0 500 1000 1500
−20
−10
0
10
20
30
Frequency (kHz)
Loop g
ain
(dB
)
−1500 −1000 −500 0 500 1000 150010
1
102
103
Frequency (kHz)
Impedance (
kΩ
)
−10 −5 0 5 1010
−2
100
Mode
Gro
wth
rate
(m
s−1)
GrowthDamping
I Two feedback loops: directand comb;
I Open loop: 33 µs growthtime;
I Direct loop only: 333 µsgrowth time;
I Direct and comb: 3.3 msgrowth time.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
35/41
PEP-II Fast Impedance Control
−1500 −1000 −500 0 500 1000 1500
−20
−10
0
10
20
30
Frequency (kHz)
Loop g
ain
(dB
)
−1500 −1000 −500 0 500 1000 150010
1
102
103
Frequency (kHz)
Impedance (
kΩ
)
−10 −5 0 5 1010
−2
100
Mode
Gro
wth
rate
(m
s−1)
GrowthDamping
I Two feedback loops: directand comb;
I Open loop: 33 µs growthtime;
I Direct loop only: 333 µsgrowth time;
I Direct and comb: 3.3 msgrowth time.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
36/41
Why Two Loops
−1500 −1000 −500 0 500 1000 1500−20
−10
0
10
20
Frequency (kHz)
Loop g
ain
(dB
)
Direct loop gain 8.0 dB
−1500 −1000 −500 0 500 1000 1500−200
−100
0
100
200
Frequency (kHz)
Loop p
hase (
deg)
−1500 −1000 −500 0 500 1000 150020
30
40
50
Frequency (kHz)
Impedance (
kΩ
)
I Direct loop gain is limited by delay;I OK at 11 dB;I and 14 dB;I At 17 dB we are stop impedance
reduction;I Worse at 20 dB.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
36/41
Why Two Loops
−1500 −1000 −500 0 500 1000 1500−20
−10
0
10
20
Frequency (kHz)
Loop g
ain
(dB
)
Direct loop gain 11.0 dB
−1500 −1000 −500 0 500 1000 1500−200
−100
0
100
200
Frequency (kHz)
Loop p
hase (
deg)
−1500 −1000 −500 0 500 1000 150020
30
40
50
Frequency (kHz)
Impedance (
kΩ
)
I Direct loop gain is limited by delay;I OK at 11 dB;I and 14 dB;I At 17 dB we are stop impedance
reduction;I Worse at 20 dB.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
36/41
Why Two Loops
−1500 −1000 −500 0 500 1000 1500−20
−10
0
10
20
Frequency (kHz)
Loop g
ain
(dB
)
Direct loop gain 14.0 dB
−1500 −1000 −500 0 500 1000 1500−200
−100
0
100
200
Frequency (kHz)
Loop p
hase (
deg)
−1500 −1000 −500 0 500 1000 150020
30
40
50
Frequency (kHz)
Impedance (
kΩ
)
I Direct loop gain is limited by delay;I OK at 11 dB;I and 14 dB;I At 17 dB we are stop impedance
reduction;I Worse at 20 dB.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
36/41
Why Two Loops
−1500 −1000 −500 0 500 1000 1500−20
−10
0
10
20
Frequency (kHz)
Loop g
ain
(dB
)
Direct loop gain 17.0 dB
−1500 −1000 −500 0 500 1000 1500−200
−100
0
100
200
Frequency (kHz)
Loop p
hase (
deg)
−1500 −1000 −500 0 500 1000 150020
30
40
50
Frequency (kHz)
Impedance (
kΩ
)
I Direct loop gain is limited by delay;I OK at 11 dB;I and 14 dB;I At 17 dB we are stop impedance
reduction;I Worse at 20 dB.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
36/41
Why Two Loops
−1500 −1000 −500 0 500 1000 1500−20
−10
0
10
20
Frequency (kHz)
Loop g
ain
(dB
)
Direct loop gain 20.0 dB
−1500 −1000 −500 0 500 1000 1500−200
−100
0
100
200
Frequency (kHz)
Loop p
hase (
deg)
−1500 −1000 −500 0 500 1000 150020
30
40
50
Frequency (kHz)
Impedance (
kΩ
)
I Direct loop gain is limited by delay;I OK at 11 dB;I and 14 dB;I At 17 dB we are stop impedance
reduction;I Worse at 20 dB.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
37/41
Trade Bandwidth for Gain
−20 −15 −10 −5 0 5 10 15 20−20
−10
0
10
20
Frequency (kHz)
Ga
in (
dB
)
−20 −15 −10 −5 0 5 10 15 20−200
−100
0
100
200
Frequency (kHz)
Ph
ase
(d
eg
)
I Double peaked comb filter atsynchrotron sidebands;
I No response at revolutionharmonics;
I Almost 20 dB of gain at synchrotronsidebands.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
37/41
Trade Bandwidth for Gain
−1000 −500 0 500 1000−20
−10
0
10
20
Frequency (kHz)
Ga
in (
dB
)
−1000 −500 0 500 1000−200
−100
0
100
200
Frequency (kHz)
Ph
ase
(d
eg
)
I Double peaked comb filter atsynchrotron sidebands;
I No response at revolutionharmonics;
I Almost 20 dB of gain at synchrotronsidebands.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
37/41
Trade Bandwidth for Gain
I Double peaked comb filter atsynchrotron sidebands;
I No response at revolutionharmonics;
I Almost 20 dB of gain at synchrotronsidebands.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
38/41
A Word About Technology: RF Processing Module
I Analog direct loop: I/Q demodulation/modulation, op-amp feedbackprocessing;
I 86 ns delay, 3 MHz bandwidth, 450 ns total loop delay;I Vector sum, multiple gain/phase blocks, lead/lag compensation, ripple
loop DSP.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
38/41
A Word About Technology: RF Processing Module
Comb loop
Cavity
Cavity
Cavity
Cavity
Direct loop gain and phase
Direct loop outputVtotal
Klystron
Comb loop gain and phase
Station reference
∑∑
Beam
RF processing module
I Analog direct loop: I/Q demodulation/modulation, op-amp feedbackprocessing;
I 86 ns delay, 3 MHz bandwidth, 450 ns total loop delay;I Vector sum, multiple gain/phase blocks, lead/lag compensation, ripple
loop DSP.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
39/41
A Word About Technology: Comb Module
I Baseband digital processor clocked at 9.8 MHz (72frev);I Identical I/Q channels;I Second order IIR for comb response, FIR group delay equalizer and
lowpass;
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
39/41
A Word About Technology: Comb Module
ADC DACOne turn delay32-tap FIR filtera1z
−72+a0b2z
−144+b1z−72+1
10 MHz 10 MHz
2nd order IIR comb filterand low-pass filter
Group delay equalizer
Iin Iout
ADC DACOne turn delay32-tap FIR filtera1z
−72+a0b2z
−144+b1z−72+1
10 MHz 10 MHz
2nd order IIR comb filterand low-pass filter
Group delay equalizer
Qin Qout
I Baseband digital processor clocked at 9.8 MHz (72frev);I Identical I/Q channels;I Second order IIR for comb response, FIR group delay equalizer and
lowpass;
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
40/41
SSB Comb Filters
Hcomb(jω)
Hcomb(jω)Iout
QoutQin
Iin
−20 −15 −10 −5 0 5 10 15 20−40
−30
−20
−10
0
10
20
Frequency (kHz)
Ga
in (
dB
)
−20 −15 −10 −5 0 5 10 15 20−200
−100
0
100
200
Frequency (kHz)
Ph
ase
(d
eg
ree
s)
I Upper sideband of comb filterreduces growth rates;
I Lower sideband removes dampingimpedance;
I Single sideband comb idea;I Never implemented, only simulated;I Used comb phase offset to reduce
growth rates;I Trade-off between LLRF and beam
stability.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
40/41
SSB Comb Filters
Hap(jω)
−Hap(jω)
Hcomb(jω)
Hcomb(jω)
∑
∑ Iout
QoutQin
Iin
−20 −15 −10 −5 0 5 10 15 20−40
−30
−20
−10
0
10
20
Frequency (kHz)
Ga
in (
dB
)
−20 −15 −10 −5 0 5 10 15 20−200
−100
0
100
200
Frequency (kHz)
Ph
ase
(d
eg
ree
s)
I Upper sideband of comb filterreduces growth rates;
I Lower sideband removes dampingimpedance;
I Single sideband comb idea;I Never implemented, only simulated;I Used comb phase offset to reduce
growth rates;I Trade-off between LLRF and beam
stability.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
41/41
Summary
I Large ring circumference and high beam currents make for achallenging combination;
I RF system design should be driven by the beam loading andlongitudinal stability considerations;
I Fundamental impedance is large, but very tightly controlled, so drivingimpedance reduction is feasible;
I Cavity HOMs are relatively unpredictable, need to be damped to levelsmanageable by the bunch-by-bunch feedback;
I Gap transient response cannot be controlled by RF feedback (high peakpower), need to manage fill pattern gaps.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
41/41
Summary
I Large ring circumference and high beam currents make for achallenging combination;
I RF system design should be driven by the beam loading andlongitudinal stability considerations;
I Fundamental impedance is large, but very tightly controlled, so drivingimpedance reduction is feasible;
I Cavity HOMs are relatively unpredictable, need to be damped to levelsmanageable by the bunch-by-bunch feedback;
I Gap transient response cannot be controlled by RF feedback (high peakpower), need to manage fill pattern gaps.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
41/41
Summary
I Large ring circumference and high beam currents make for achallenging combination;
I RF system design should be driven by the beam loading andlongitudinal stability considerations;
I Fundamental impedance is large, but very tightly controlled, so drivingimpedance reduction is feasible;
I Cavity HOMs are relatively unpredictable, need to be damped to levelsmanageable by the bunch-by-bunch feedback;
I Gap transient response cannot be controlled by RF feedback (high peakpower), need to manage fill pattern gaps.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
41/41
Summary
I Large ring circumference and high beam currents make for achallenging combination;
I RF system design should be driven by the beam loading andlongitudinal stability considerations;
I Fundamental impedance is large, but very tightly controlled, so drivingimpedance reduction is feasible;
I Cavity HOMs are relatively unpredictable, need to be damped to levelsmanageable by the bunch-by-bunch feedback;
I Gap transient response cannot be controlled by RF feedback (high peakpower), need to manage fill pattern gaps.
Beam loading andinstabilities
IntroductionThe focus of this tutorial
Coupled-bunch Instabilities
Beam LoadingEffectsBeam Loading in StorageRings
Ring Circumference andFundamental Impedance
Transient LoadingHow Not To Fix TransientLoading
Realistic Mitigation
FundamentalImpedance andInstabilitiesBunch-by-bunch Feedback
Impedance Control Loops
Summary
41/41
Summary
I Large ring circumference and high beam currents make for achallenging combination;
I RF system design should be driven by the beam loading andlongitudinal stability considerations;
I Fundamental impedance is large, but very tightly controlled, so drivingimpedance reduction is feasible;
I Cavity HOMs are relatively unpredictable, need to be damped to levelsmanageable by the bunch-by-bunch feedback;
I Gap transient response cannot be controlled by RF feedback (high peakpower), need to manage fill pattern gaps.
top related