Direct Feedback Analysis for RF Cavity System Shaoheng Wang 6/18/2015

Contents Robinson Instability Cavity Impedance Direct Feedback

Robinson Instability

Beam Loading and Phasor Diagram Vcavity Beam Syn. Motion - RF coupled system IG VG VB YL YS yT: Tuning angle of impedance yT < 0 for above transition 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 YT -IB YT

Criteria for R. Instability 𝟐𝐜𝐨𝐬 𝝍 𝒔 𝒀 > sin −𝟐𝝍 >𝟎 Where, 𝑌= 𝑉 𝑏𝑟 𝑉 𝑔𝑎𝑝

Cavity Equavilent Circuit 𝑌= 𝑉 𝑏𝑟 𝑉 𝑔𝑎𝑝 = 𝐼 𝑏 ∗ 𝑅 𝑠 𝑉 𝑔𝑎𝑝

Direct Feedback and Lowered Effective Impedance

Forward Power Caculation for RF Cavity with Beam Loading In waveguide: Forware current: If Forward voltage: Vf = Z0*If Reflected current: Ir Reflected voltage: Vr = -Z0Ir At coupler: Input current: I1 = If + Ir Input voltage: V1 = Vf + Vr = Z0(If - Ir) Output current: I2 = I1/N= (If + Ir )/N Output voltage: V2 = NV1 = NZ0(If - Ir) Cavity: Cavity current: Ic = I2 - Ib = (If + Ir )/N - Ib Cavity voltage: Vc = ZIc = Z((If + Ir )/N - Ib) Since V2 = Vc, ie, NZ0(If - Ir) = Z((If + Ir )/N - Ib)  𝑰 𝒓 = 𝑁 2 𝑍 0 −𝒁 𝑰 𝒇 +𝒁𝑁 𝑰 𝒃 𝑁 2 𝑍 0 +𝒁 𝑽 𝒄 = 𝑁 2 𝑍 0 𝒁 2 𝑁 𝑰 𝒇 − 𝑰 𝒃 𝑁 2 𝑍 0 +𝒁 Use RL = (N2Z0)||R, 1 𝑍 𝐿 = 1 𝑅 𝐿 +𝑠𝐶+ 1 𝑠𝐿  𝑽 𝒄 = 𝒁 𝑳 2 𝑰 𝒈 − 𝑰 𝒃 𝐼 𝑔 = 𝐼 𝑓 𝑁 𝑃 𝑓 = 1 2 𝑅𝑒 𝑽 𝒇 ∙ 𝑰 𝒇 ∗ = 1 2 𝑰 𝒇 2 𝑍 0 = 1 2 𝑰 𝒈 2 𝑅 𝐿 1+𝛽 𝛽 𝑃 𝑓 = 1+𝛽 8 𝑅 𝐿 𝛽 𝑉 𝑐 2 1+ 𝑅 𝐿 𝐼 𝑏 𝑉 𝑐 sin 𝜙 𝑠 2 + tan 𝜓 + 𝑅 𝐿 𝐼 𝑏 𝑉 𝑐 cos 𝜙 𝑠 2

With Feedback and Coupler Cavity K a circulator Damper If Ir Iin IFB -Ib 1:N Coupler 1 2 𝒆 −𝒊∆𝝎𝜹 𝑰 𝒇 = 𝐾𝑰 𝑭𝑩 𝑰 𝒇 =− 𝑰 𝒓 𝐴=𝛼𝐾/𝑁 Due to the loop: 𝑰 𝒇 = 𝐾𝑰 𝑭𝑩 = 𝐾 𝑰 𝒊𝒏 +𝐴𝑁 𝒆 −𝒊∆𝝎𝜹 𝑰 𝒃 −𝐴 𝒆 −𝒊∆𝝎𝜹 𝑰 𝒓 1+𝐴 𝒆 −𝒊∆𝝎𝜹 𝑰 𝑭𝑩 = 𝑰 𝒊𝒏 −𝛼 𝐾 𝑰 𝑭𝑩 + 𝑰 𝒓 𝑁 − 𝑰 𝒃 𝒆 −𝒊∆𝝎𝜹 Compare with no feedback loop case: 𝑰 𝒇 = 𝐾𝑰 𝒊𝒏 𝑰 𝟏 = 𝑰 𝒇 + 𝑰 𝒓 = 𝐾 𝑰 𝒊𝒏 +𝐴𝑁 𝒆 −𝒊∆𝝎𝜹 𝑰 𝒃 + 𝑰 𝒓 1+𝐴 𝒆 −𝒊∆𝝎𝜹 On waveguide side of the coupler: 𝑽 𝟏 = 𝑍 0 𝑰 𝒇 − 𝑰 𝒓 = 𝑍 0 𝐾 𝑰 𝒊𝒏 +𝐴𝑁 𝒆 −𝒊∆𝝎𝜹 𝑰 𝒃 − 1+2𝐴 𝒆 −𝒊∆𝝎𝜹 𝑰 𝒓 1+𝐴 𝒆 −𝒊∆𝝎𝜹 𝑰 𝟐 = 𝑰 𝟏 𝑁 = 𝑰 𝒇 + 𝑰 𝒓 𝑁 = 𝐾 𝑰 𝒊𝒏 +𝐴 𝑁𝒆 −𝒊∆𝝎𝜹 𝑰 𝒃 + 𝑰 𝒓 𝑁 1+𝐴 𝒆 −𝒊∆𝝎𝜹 On cavity side of the coupler: 𝑽 𝟐 =𝑁 𝑽 𝟏 =𝑁 𝑍 0 𝑰 𝒇 − 𝑰 𝒓 = 𝑁𝑍 0 𝐾 𝑰 𝒊𝒏 +𝐴𝑁 𝒆 −𝒊∆𝝎𝜹 𝑰 𝒃 − 1+2𝐴 𝒆 −𝒊∆𝝎𝜹 𝑰 𝒓 1+𝐴 𝒆 −𝒊∆𝝎𝜹

With Feedback, use Vc and Ib as Variables 𝑰 𝒄 = 𝑰 𝟐 − 𝑰 𝒃 = 𝐾 𝑰 𝒊𝒏 −𝑁 𝑰 𝒃 + 𝑰 𝒓 𝑁+𝐴𝑁 𝒆 −𝒊∆𝝎𝜹 When a is small, probe current is ignored In cavity: 𝑽 𝒄 = 𝒁𝑰 𝒄 =𝒁 𝐾 𝑰 𝒊𝒏 −𝑁 𝑰 𝒃 + 𝑰 𝒓 𝑁+𝐴 𝑁𝒆 −𝒊∆𝝎𝜹 When Ir is zero and N=1, impedance seen by beam is droped by (1+A) 𝑽 𝟐 = 𝑽 𝒄 𝑁 2 𝑍 0 𝐾 𝑰 𝒊𝒏 +𝑁𝐴 𝒆 −𝒊∆𝝎𝜹 𝑰 𝒃 − 1+2𝐴 𝒆 −𝒊∆𝝎𝜹 𝑰 𝒓 =𝒁 𝐾 𝑰 𝒊𝒏 −𝑁 𝑰 𝒃 + 𝑰 𝒓 𝑰 𝒓 = 𝑁 2 𝑍 0 −𝒁 𝐾 𝑰 𝒊𝒏 + 𝑁 3 𝑍 0 𝐴 𝒆 −𝒊∆𝝎𝜹 +𝑁𝒁 𝑰 𝒃 𝒁+ 𝑁 2 𝑍 0 +2𝐴 𝑁 2 𝑍 0 𝒆 −𝒊∆𝝎𝜹 𝑽 𝒄 = 2𝑁 𝑍 0 𝒁𝐾 𝒁+ 𝑁 2 𝑍 0 +2𝐴 𝑁 2 𝑍 0 𝒆 −𝒊∆𝝎𝜹 𝑰 𝒊𝒏 − 𝑁 2 𝑍 0 𝒁 𝒁+ 𝑁 2 𝑍 0 +2𝐴 𝑁 2 𝑍 0 𝒆 −𝒊∆𝝎𝜹 𝑰 𝒃 𝑰 𝒊𝒏 = 𝒁+ 𝑁 2 𝑍 0 +2𝐴 𝑁 2 𝑍 0 𝒆 −𝒊∆𝝎𝜹 2𝑁 𝑍 0 𝒁𝐾 𝑽 𝒄 + 𝑁 2𝐾 𝑰 𝒃 𝐄𝐱𝐩𝐫𝐞𝐬𝐬𝐞𝐝 𝐚𝐬 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧 𝐨𝐟 𝑽𝒄 ,𝑰𝒃 𝑰 𝒓 = 𝑁 2 𝑍 0 −𝒁 2𝑁 𝑍 0 𝒁 𝑽 𝒄 + 𝑁 𝟐 𝑰 𝒃 𝑰 𝒇 = 𝐾 𝑰 𝒊𝒏 +𝑁𝐴 𝒆 −𝒊∆𝝎𝜹 𝑰 𝒃 −𝐴 𝒆 −𝒊∆𝝎𝜹 𝑰 𝒓 1+𝐴 𝒆 −𝒊∆𝝎𝜹 𝑰 𝒇 = 𝑁 2 𝑍 0 +𝒁 2𝑁 𝑍 0 𝒁 𝑽 𝒄 + 𝑁 2 𝑰 𝒃

With Feedback, Forward Power Calculation 𝑰 𝒇 = 𝑁 2 𝑍 0 +𝒁 2𝑁 𝑍 0 𝒁 𝑽 𝒄 + 𝑁 2 𝑰 𝒃 𝑰 𝒇 = 𝐾 𝑰 𝒊𝒏 +𝑁𝐴 𝒆 −𝒊∆𝝎𝜹 𝑰 𝒃 −𝐴 𝒆 −𝒊∆𝝎𝜹 𝑰 𝒓 1+𝐴 𝒆 −𝒊∆𝝎𝜹 𝐼 𝑔 = 𝐼 𝑓 𝑁 𝑰 𝒈 = 1 2 𝒁 𝑳 𝑽 𝒄 + 1 2 𝑰 𝒃 𝒁 𝑳 = 𝑁 2 𝑍 0 𝒁 𝑁 2 𝑍 0 +𝒁 𝑰 𝒈 = 𝑉 𝑐 2 𝒁 𝑳 + 𝐼 𝑏 𝑒 𝑖 𝜙 𝑠 − 𝜋 2 2 𝑃 𝑓 = 1 2 𝑅𝑒 𝑉 𝑓 ∙ 𝐼 𝑓 ∗ = 1 2 𝐼 𝑓 2 𝑍 0 = 1 2 𝐼 𝑔 2 𝑅 𝐿 1+𝛽 𝛽 𝑃 𝑓 = 1+𝛽 8 𝑅 𝐿 𝛽 𝑉 𝑐 2 1+ 𝑅 𝐿 𝐼 𝑏 𝑉 𝑐 sin 𝜙 𝑠 2 + tan 𝜓 + 𝑅 𝐿 𝐼 𝑏 𝑉 𝑐 cos 𝜙 𝑠 2 𝑍 𝐿 = 𝑅 𝐿 1−𝑖 𝑄 𝐿 𝜔 0 𝜔 − 𝜔 𝜔 0 = 𝑅 𝐿 1−𝑖 tan 𝜓

Impedance Open loop Closed loop 𝑽 𝒄 = 𝑁 2 𝑍 0 𝒁 2 𝑰 𝒈 − 𝑰 𝒃 𝑁 2 𝑍 0 +𝒁 𝑽 𝒄 = 𝑁 2 𝑍 0 𝒁 2 𝑰 𝒈 − 𝑰 𝒃 𝑁 2 𝑍 0 +𝒁 𝑽 𝒄 = 2𝑁 𝑍 0 𝒁𝐾 𝒁+ 𝑁 2 𝑍 0 +2𝐴 𝑁 2 𝑍 0 𝒆 −𝒊∆𝝎𝜹 𝑰 𝒊𝒏 − 𝑁 2 𝑍 0 𝒁 𝒁+ 𝑁 2 𝑍 0 +2𝐴 𝑁 2 𝑍 0 𝒆 −𝒊∆𝝎𝜹 𝑰 𝒃 𝑽 𝒄,𝒃 = − 𝑁 2 𝑍 0 𝒁 𝒁+ 𝑁 2 𝑍 0 +2𝐴 𝑁 2 𝑍 0 𝒆 −𝒊∆𝝎𝜹 𝑰 𝒃 = − 𝒁 𝑳 𝒁 𝒁+2𝐴 𝒁 𝑳 𝒆 −𝒊∆𝝎𝜹 𝑰 𝒃 𝑽 𝒄, 𝒃 = − 𝒁 𝑳 𝑰 𝒃 𝑽 𝒄,𝒃 = −1 1 𝑅 𝐿 + 2𝐴 𝑅 𝑰 𝒃 =− 𝑅 𝐿 ∥ 𝑅 2𝐴 𝑰 𝒃 =− 𝑅 1+𝛽+2𝐴 𝑰 𝒃 𝑽 𝒄, 𝒃 = −𝑅 𝐿 𝑰 𝒃 When on resonance 𝑽 𝒄,𝒃 = − 𝑅 𝐿 1+𝐴 𝑰 𝒃 RL = R/2 when matched 𝒁 𝑳 = 𝑁 2 𝑍 0 𝒁 𝑁 2 𝑍 0 +𝒁 𝑅 𝐿 = 𝑁 2 𝑍 0 𝑅 𝑁 2 𝑍 0 +𝑅

Robinson Instability Alleviated with Direct Feedback Energy = 5 GeV, Current = 3 A, Cavity Number = 10 A = 0, R. Instability Condition = 0.0 A = 8.5, R. Instability Condition = 0.09 Impedance Phase Angle Real Impedance A = 0 A = 8.5 A = 0 A = 8.5

Direct Feedback Loop Stability Loop transfer function Nyquist Plot 45o phase margin 𝑨 𝒎𝒂𝒙 = 𝑸 𝑳 𝟒𝒇𝜹 −𝟏 Group delay of PEP II cavity is 350 nSec  Amax = 8.78

Conclusion We need direct feedback to lower the cavity impedance seen by the beam No extra power is needed for the loop There is limitation for the direct feedback, comb filter feedback is needed for further multi-bunhch instability studies.