Robinson Instability Criteria for MEIC e Ring Shaoheng Wang, Haipeng Wang, Robert Rimmer

The Definitions of Synchronous Phase Below transition SY Lee, etc 𝜓 𝑠 𝑉 𝑎𝑐𝑐 = 𝑉 0 sin 𝜓 𝑠 Above transition Wiedemann, Robinson, etc SY Lee, etc Merminga, etc 𝜓 𝑠 𝜓 𝑠 𝜓 𝑠 𝑉 𝑎𝑐𝑐 = 𝑉 0 sin 𝜓 𝑠 𝑉 𝑎𝑐𝑐 = 𝑉 0 sin 𝜓 𝑠 𝑉 𝑎𝑐𝑐 = 𝑉 0 cos 𝜓 𝑠

Synchronous phase is used in several physical relations Acceleration SR equilibrium bunch length Synchrotron tune Beam loading optimization Robinson Instability 𝑉 𝑎𝑐𝑐 = 𝑉 0 sin 𝜓 𝑠 𝑉 𝑝𝑒𝑎𝑘 cos⁡( 𝜓 𝑠 )= −2𝜋 𝑐 2 𝜂𝐸 𝜔 2 𝜎 2 𝐻𝑒 𝛿𝐸 𝐸 2 𝜐 𝑠 = 𝐻𝑒 𝑉 𝑝𝑒𝑎𝑘 𝜂cos⁡( 𝜓 𝑠 ) 2𝜋𝐸 𝛽 𝑜𝑝𝑡 =1+ 2𝐼 𝑅 𝑠ℎ𝑢𝑛𝑡 sin⁡( 𝜓 𝑠 ) 𝑉 𝑔𝑎𝑝 tan⁡(𝜓)= − 2𝐼 0 𝑅 𝑠 𝑉 𝑔𝑎𝑝 (𝛽+1) co𝑠 𝜑 𝑠 0 <sin −2𝜓 < 2 𝑉 𝑔𝑎𝑝 cos( 𝜓 𝑠 ) 𝑅 𝐿 𝐼 𝑎𝑣𝑒

Synchronous phase and acceleration 𝑉 𝑎𝑐𝑐 = 𝑉 0 sin 𝜓 𝑠 𝜓 𝑠 𝜓 𝑠 Fit both cases

Synchronous phase and synchrotron tune 𝜐 𝑠 = 𝐻𝑒 𝑉 𝑝𝑒𝑎𝑘 𝜂cos⁡( 𝜓 𝑠 ) 2𝜋𝐸 𝜓 𝑠 𝜓 𝑠 Fit both cases

Synchronous phase and SR equilibrium bunch length 𝜎= 𝑐 𝜂 𝜔 𝑠 𝛿𝐸 𝐸 = 2𝜋 𝑐 𝜔 0 𝐸 𝐻𝑒 𝑉 𝑝𝑒𝑎𝑘 𝜂 cos 𝜓 𝑠 𝛿𝐸 𝐸 𝜓 𝑠 𝜓 𝑠 Fit both cases

Synchronous phase and matching beam loading 𝛽 𝑜𝑝𝑡 =1+ 2𝐼 𝑅 𝑠ℎ𝑢𝑛𝑡 sin⁡( 𝜓 𝑠 ) 𝑉 𝑔𝑎𝑝 tan⁡( 𝜓 𝑇 )= −2𝐼 0 𝑅 𝑠 𝑉 𝑔𝑎𝑝 (𝛽+1) co𝑠 𝜑 𝑠 𝜓 𝑠

Synchronous phase and Robinson Instability 0 <sin −2 𝜓 𝑇 < 2 𝑉 𝑔𝑎𝑝 cos( 𝜓 𝑠 ) 𝑅 𝐿 𝐼 𝑎𝑣𝑒 𝜓 𝑠

Beam Loading and Phasor Diagram Vcavity IG VG VB YL YS YT -IB YT

Phasor Diagram Parameters yT: Tuning angle of impedance yT < 0 for above transition loaded impedance: 1 𝑍 = 1 𝑅 𝐿 1+𝑖 𝑄 𝐿 𝜔 𝑅𝐹 2 − 𝜔 0 2 𝜔 𝑅𝐹 𝜔 0 tan 𝜓 𝑇 =− 𝑄 𝐿 𝜔 𝑅𝐹 2 − 𝜔 0 2 𝜔 𝑅𝐹 𝜔 0 𝑉 𝐼 = 𝑅 𝐿 1−𝑖tan 𝜓 𝑇 = 𝑅 𝐿 cos 𝜓 𝑇 𝑒 𝑖𝜓 ; 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, 𝜔 0 = 1 𝐿𝐶 wRF: generator RF frequency, Synchronous beam revolution frequency times harmonic number

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. 𝑉 𝐺 = 𝑅 𝐿 𝐼 𝐺 cos 𝜓 𝑇 𝑒 𝑖 𝜓 𝑇 RL 𝑉 𝐵 = 𝑅 𝐿 𝐼 𝐵 cos 𝜓 𝑇 𝑒 𝑖 𝜓 𝑇 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.

YS in Phasor Diagram Vcavity 𝜓 𝑠 𝑉 𝑎𝑐𝑐 = 𝑉 0 sin 𝜓 𝑠 YS -IB IB YS

YS in Phasor Diagram Vcavity 𝜓 𝑠 𝑉 𝑎𝑐𝑐 = 𝑉 0 sin 𝜓 𝑠 YS -IB IB

YS in Phasor Diagram Vcavity 𝜓 𝑠 𝑉 𝑎𝑐𝑐 = 𝑉 0 cos 𝜓 𝑠 -IB YS IB

Phasor Diagram, above transition, YT<0 Vcavity IG VG VB YL YS YT -IB YT

Robinson Instability Criteria --- CEA-11

Equations of Cavity and Beam System

Stable Condition

Further clarified in CEA-1010

YS in Phasor Diagram in CEA-1010 Not impedance angle =- 𝜓 𝑇 =𝜓 𝑠 𝜓 𝑠 𝑉 𝑎𝑐𝑐 = 𝑉 0 sin 𝜓 𝑠

@ 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

Robinson work point @ 12 GeV Qext@12GeV = 1.23e5

Robinson work point @ 5 GeV Qext@5GeV = 0.01* Qext@12GeV

Robinson work point @ 5 GeV Qext@5GeV = 0.002* Qext@12GeV

Robinson work point @ 3 GeV Y = 268.2, tuning angle = -88.8 degrees 20 cavities Qext@5GeV = Qext@12GeV

Robinson work point @ 3 GeV 1 cavity Qext@5GeV = Qext@12GeV

Robinson work point @ 3 GeV 1 cavity Qext@5GeV = 0.15*Qext@12GeV