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

2. Contents
Robinson Instability
Cavity Impedance
Direct Feedback

3. Robinson Instability

4. 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 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

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

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

7. Direct Feedback and Lowered Effective Impedance

8. 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 ∗ = 𝑰 𝒇 2 𝑍 0 = 𝑰 𝒈 2 𝑅 𝐿 1+𝛽 𝛽
𝑃 𝑓 = 1+𝛽 8 𝑅 𝐿 𝛽 𝑉 𝑐 𝑅 𝐿 𝐼 𝑏 𝑉 𝑐 sin 𝜙 𝑠 tan 𝜓 + 𝑅 𝐿 𝐼 𝑏 𝑉 𝑐 cos 𝜙 𝑠 2

9. With Feedback and Coupler
Cavity K a
circulator
Damper If Ir
Iin
IFB
-Ib
1:N
Coupler
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

10. 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

11. 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 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
𝑃 𝑓 = 1 2 𝑅𝑒 𝑉 𝑓 ∙ 𝐼 𝑓 ∗ = 𝐼 𝑓 2 𝑍 0 = 𝐼 𝑔 2 𝑅 𝐿 1+𝛽 𝛽
𝑃 𝑓 = 1+𝛽 8 𝑅 𝐿 𝛽 𝑉 𝑐 𝑅 𝐿 𝐼 𝑏 𝑉 𝑐 sin 𝜙 𝑠 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

12. 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

13. 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

14. 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

15. 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.