ILC Damping Ring Wolski (2007) Dynamics with Transverse Coupled-Bunch Wake Fields in the ILC Damping Rings Kai Hock and Andy Wolski.

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Presentation transcript:

ILC Damping Ring Wolski (2007) Dynamics with Transverse Coupled-Bunch Wake Fields in the ILC Damping Rings Kai Hock and Andy Wolski

Objective: 1. Effect of varying beta function in macroparticle model 2. Analytic solution for constant beta function. Previous work – for constant beta function : 1. Equation of motion decouple into Fourier modes (Chao 1993). 2. Modes decay / grown exponentially, analytic formula available. 3. Analytic formula for bunch trajectories available but incomplete (Thompson and Ruth 1991). Current work: 1.Varying beta function – modes remain coupled, decay modes grow. 2.Analytic formula for bunch trajectories completed with error bounds.

Example of an application

v A Macroparticle Model y No wake fieldResistive wake force n n+1 n+2 Equation of motion Sands (1969), Thompson and Ruth (1991), … History

Normal Modes where coupled decoupled (Fourier modes) DFT Chao (1993)

If beta function is constant OCS6 Damping Ring: Uniform Fill

A few surprises …

Mode Frequency Spectra Turn 90 Parametric force High frequency oscillation comes from …

Mode Coupling Decay modes grow because … Growth mode Decay mode Decay mode ?

Bunch Trajectories Equation of motion Analytic solution (Thompson, Ruth 1991) Incomplete – Zero wake field limit is SHM, need – Else cannot have arbitrary

Characteristic Equation Elementary solution Contour plot General solution If beta function is constant:

Integration of Depends on initial history

If wake field very small Conjecture Depends on initial condition at t = 0 Depends on initial history Just like SHM

An analytic solution Conjecture validated

H(  ) Rogers (2006)  Feedback system bandwidth Frequency  mode number Bandwidth must cover growth modes If decay modes grow unexpectedly …