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Robinson Instability Criteria for MEIC e Ring
Shaoheng Wang, Haipeng Wang, Robert Rimmer
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The Definitions of Synchronous Phase
Below transition SY Lee, etc π π π πππ = π 0 sin π π Above transition Wiedemann, Robinson, etc SY Lee, etc Merminga, etc π π π π π π π πππ = π 0 sin π π π πππ = π 0 sin π π π πππ = π 0 cos π π
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Synchronous phase is used in several physical relations
Acceleration SR equilibrium bunch length Synchrotron tune Beam loading optimization Robinson Instability π πππ = π 0 sin π π π ππππ cosβ‘( π π )= β2π π 2 ππΈ π 2 π 2 π»π πΏπΈ πΈ 2 π π = π»π π ππππ πcosβ‘( π π ) 2ππΈ π½ πππ‘ =1+ 2πΌ π
π βπ’ππ‘ sinβ‘( π π ) π πππ tanβ‘(π)= β 2πΌ 0 π
π π πππ (π½+1) coπ π π 0 <sin β2π < 2 π πππ cos( π π ) π
πΏ πΌ ππ£π
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Synchronous phase and acceleration
π πππ = π 0 sin π π π π π π Fit both cases
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Synchronous phase and synchrotron tune
π π = π»π π ππππ πcosβ‘( π π ) 2ππΈ π π π π Fit both cases
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Synchronous phase and SR equilibrium bunch length
π= π π π π πΏπΈ πΈ = 2π π π πΈ π»π π ππππ π cos π π πΏπΈ πΈ π π π π Fit both cases
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Synchronous phase and matching beam loading
π½ πππ‘ =1+ 2πΌ π
π βπ’ππ‘ sinβ‘( π π ) π πππ tanβ‘( π π )= β2πΌ 0 π
π π πππ (π½+1) coπ π π π π
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Synchronous phase and Robinson Instability
0 <sin β2 π π < 2 π πππ cos( π π ) π
πΏ πΌ ππ£π π π
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Beam Loading and Phasor Diagram
Vcavity IG VG VB YL YS YT -IB YT
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Phasor Diagram Parameters
yT: Tuning angle of impedance yT < 0 for above transition loaded impedance: 1 π = 1 π
πΏ 1+π π πΏ π π
πΉ 2 β π π π
πΉ π 0 tan π π =β π πΏ π π
πΉ 2 β π π π
πΉ π 0 π πΌ = π
πΏ 1βπtan π π = π
πΏ cos π π π ππ ; yS: Synchronous phase angle see definition in earlier slice; yL: Loading angle angle between the generator current and the cavity voltage ; IB: fundamental harmonic component of beam current IB = 2 I0, where I0 is the average beam current; w0: cavity resonance frequency, π 0 = 1 πΏπΆ wRF: generator RF frequency, Synchronous beam revolution frequency times harmonic number
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Loaded Generator-Cavity-Beam System
Both generator and beam are considered as current source, generator feed energy in to cavity; beam extracts energy from cavity. π πΊ = π
πΏ πΌ πΊ cos π π π π π π RL π π΅ = π
πΏ πΌ π΅ cos π π π π π π When above transition, particles with higher energy have larger revolution time. Beam current needs to lags cavity voltage to satisfy the phase stability criteria, which means the beam effective impedance is inductive. So, for the generator to see a resistive impedance, the cavity needs to be capacitive-detuned, it means yT < 0 and w0 < wRF, and voltage lags beam current by the phase |yT|, because then cavity impedance looks capacitive.
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YS in Phasor Diagram Vcavity π π π πππ = π 0 sin π π YS -IB IB YS
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YS in Phasor Diagram Vcavity π π π πππ = π 0 sin π π YS -IB IB
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YS in Phasor Diagram Vcavity π π π πππ = π 0 cos π π -IB YS IB
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Phasor Diagram, above transition, YT<0
Vcavity IG VG VB YL YS YT -IB YT
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Robinson Instability Criteria --- CEA-11
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Equations of Cavity and Beam System
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Stable Condition
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Further clarified in CEA-1010
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YS in Phasor Diagram in CEA-1010
Not impedance angle =- π π =π π π π π πππ = π 0 sin π π
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@ 12 GeV RL 12.9 MW IB 2*0.11 A YT -32.4 degree YS 57.6 YL Vcavity 2.5
Vcavity 2.5 MV VB VG 2.93 IG 0.39 @ 12 GeV Vcavity VB IG YS YT YT -IB VG
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Robinson work point @ 12 GeV
= 1.23e5
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Robinson work point @ 5 GeV
= 0.01*
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Robinson work point @ 5 GeV
= 0.002*
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Robinson work point @ 3 GeV
Y = 268.2, tuning angle = degrees 20 cavities =
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Robinson work point @ 3 GeV
1 cavity =
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Robinson work point @ 3 GeV
1 cavity =
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