1. RF Parameters Calculation for JLEIC Colliders (e Ring)
Shaoheng Wang
7/19/2018
CASA

2. RF System needs to …… Provide energy compensation for SR
desired bunch length
To operate in stable region
Avoid Robinson instability
Avoid coupled bunch instabilities
To operate with limited RF power and power feed-in capability

3. In Electron Ring RF system
provide energy compensation and desired bunch length
RF system provide large enough RF bucket to avoid beam loss
Beam current determination, ensuring
Total SR power < Total SR power limit:10 MW
Linear SR power < Linear SR power limit:10 kW/m
𝝈= 𝑐 𝜂 𝜔 𝑠 𝛿𝐸 𝐸 = 2𝜋 𝑐 𝜔 𝐸 𝑯𝑒 𝑉 𝑝𝑒𝑎𝑘 𝜼 cos 𝜑 𝑠 𝜹𝑬 𝑬
𝐸 𝑙𝑜𝑠𝑠 𝑝𝑒𝑟 𝑡𝑢𝑟𝑛 = 𝐶 𝑆𝑅 𝐸 4 𝑰 𝟐 = 𝑉 𝑝𝑒𝑎𝑘 sin 𝜑 𝑠
𝜹𝑬 𝑬 =𝛾 𝐶 𝑞 𝑰 𝟑 2 𝑰 𝟐 + 𝑰 𝟒
buncket height 𝛿𝐸 𝐸 = 2 𝜐 𝑠 𝛿𝐸 𝐸 𝐻𝜂 1− 𝜋−2 𝜑 𝑠 2 tan 𝜑 𝑠 = 2𝑐 𝐻𝜎 𝜔 − 𝜋−2 𝜑 𝑠 2 tan 𝜑 𝑠 >10
𝑃 𝑡𝑜𝑡𝑎𝑙 𝑆𝑅 = 𝐸 𝑙𝑜𝑠𝑠 𝑝𝑒𝑟 𝑡𝑢𝑟𝑛 𝑰 𝒂𝒗𝒆
𝑃 𝑙𝑖𝑛𝑒𝑎𝑟 𝑆𝑅 = 𝑰 𝒂𝒗𝒆 𝐶 𝑆𝑅 𝐸 4 2𝜋 𝒓 𝒅𝒊𝒑𝒐𝒍𝒆 𝟐

4. RF Power RF Forward power Cavity circuit model
𝑃 𝑓𝑤𝑑 = 𝑉 𝑔𝑎𝑝 𝜷 4 𝑹 𝑳 𝛽 𝐼 𝑎𝑣𝑒 𝑹 𝑳 𝑉 𝑔𝑎𝑝 sin 𝜑 𝑠 tan 𝝍 𝑳 2
𝑅 𝐿 = 𝑹 𝒔 1+𝜷
Beta is fixed at 3.6 from PEP II cavity, while required optimal beta varies from 5 to 60

5. Coupled bunch instability
𝑍 || 𝑡ℎ𝑟𝑒𝑠ℎ = 2𝐸 𝜐 𝑠 𝑁 𝑐𝑎𝑣 𝐼 𝑏 𝑓 HOM 𝛼 𝜏 𝑠

6. 3 GeV
 2 RF cavities is enough
But,
 Robinson instability
Open loop

7. At 6 GeV, current is limited by cavity total impedance
28 cavities
8 cavities

8. Change loading angle to move working point
 Working point is stable
But,
 Too sensitive to tuning angle

9. With direct feedback
A = 7.4

10. Impedance and Nyquist plot with Direct feedback
The Nyquist plot
shows the stability of the feedback system
The maximum gain is limited by the phase margin

11. Open Loop, RF Cavity Driven by Ig and Ib
𝑰 𝒃
Loaded RF cavity
𝑰 𝒈,𝑶𝑳
𝑹 𝑳
𝑽 𝒔 𝑲
𝑽 𝒊𝒏
The cavity gap voltage is driven by two currents
generator current: 𝐼 𝑔,𝑂𝐿 =𝐾 𝑉 𝑖𝑛
beam image current: 𝐼 𝑏,𝑖𝑚𝑎𝑔𝑒 =− 𝐼 𝑏
with
𝑉 𝑖𝑛 = 𝑉 𝑠
then
𝑉 𝑐 = 𝑉 𝑔 + 𝑉 𝑏 = 𝐾 𝑉 𝑠 − 𝐼 𝑏 𝑍 𝐿
Where the frequency dependent cavity impedance is
𝑍 𝐿 = 𝑅 𝐿 1+𝑖 𝑄 𝐿 𝜉
𝜉= 𝜔 2 − 𝜔 𝜔 0 𝜔
𝜔 0 is cavity resonance frequency
Note, Ib is the RF frequency component of beam current, Ib = 2Iave Vc Vg Ig
-Ib js YL
Above transition, open loop Vb 𝝍
tan 𝜓 =− 𝑄 𝐿 𝜉

12. Robinson Instability, 3 GeV case
Vgap Vg Ig
-Ib Vb
Vgap Vg Ig
-Ib Vb

13. Closed Loop: a two-input system
The input voltage of the generator combines the RF input and the loop feedback,
𝑉 𝑖𝑛 = 𝑉 𝑠 − 𝑉 𝑐 𝛼 𝑒 −𝑖∆𝜔 𝜏 1 = 𝑉 𝑠 − 𝐼 𝐺 − 𝐼 𝑏 𝑍 𝑐 𝛼 𝑒 −𝑖∆𝜔 𝜏 1
with ∆𝜔= 𝜔 𝑅𝐹 − 𝜔 𝑟𝑒𝑠 , and the sensor delay t1
The total klystron generator current is
𝐼 𝐺 =𝐾 𝑉 𝑖𝑛 𝑒 −𝑖∆𝜔 𝜏 2
with the controllor delay t2. The loop delay is t = t1 + t2, then from the two above euqtions, we can have
𝑉 𝑖𝑛 = 𝑉 𝑠 +𝛼 𝑍 𝑐 𝐼 𝑏 𝑒 −𝑖Δ𝜔 𝜏 𝛼 𝑍 𝑐 𝐾 𝑒 −𝑖Δ𝜔𝜏
the cavity is driven by 𝐼 𝐺 − 𝐼 𝑏 , so the cavity gap voltage can be derived
𝑉 𝑐 = 𝑍 𝑐 𝐼 𝐺 − 𝐼 𝑏 = 𝑍 𝑐 𝐾 𝑒 −𝑖∆𝜔 𝜏 𝑉 𝑠 +𝛼 𝑍 𝑐 𝐼 𝑏 𝑒 −𝑖Δ𝜔 𝜏 𝛼 𝑍 𝑐 𝐾 𝑒 −𝑖Δ𝜔𝜏 − 𝐼 𝑏
Or simplified as,
𝑉 𝑐 = 𝑍 𝑐 1+𝛼 𝑍 𝑐 𝐾 𝑒 −𝑖Δ𝜔𝜏 𝑒 −𝑖∆𝜔 𝜏 2 𝐾 𝑉 𝑠 − 𝐼 𝑏
𝐼 𝐺 =𝐾 𝑒 −𝑖∆𝜔 𝜏 𝑉 𝑠 +𝛼 𝑍 𝑐 𝐼 𝑏 𝑒 −𝑖Δ𝜔 𝜏 𝛼 𝑍 𝑐 𝐾 𝑒 −𝑖Δ𝜔𝜏
𝑉 𝑐 a
𝑉 𝑐 a𝑒 −𝑖∆𝜔 𝜏 1
𝜏 1 _
𝐼 𝐺 K Vs +
Vin
𝜏 2 Ib
𝑉 𝑐
𝐾= 𝐼 𝑔 𝑉 𝑖𝑛
Note:
Stability
Power
Loaded RF cavity

14. the effective voltage produced by the beam
The cavity voltage can be written as
where, the RF source and beam current induced voltages are
Take the beam induced contribution to cavity voltage as example, plug in the cavity impedance,
Then the beam induced contribution to cavity voltage can be expressed as
Note:
When loop is open, a = 0 , then Aw = 1, then the voltage and detuning angle expressions reduce to open loop cases.
When loop is closed, 𝑅 𝐿 is reduced by a factor of Aw
𝑉 𝑐 = 𝑉 𝑔 + 𝑉 𝑏
𝑉 𝑔 = 𝑍 𝑐 1+𝛼 𝑍 𝑐 𝐾 𝑒 −𝑖Δ𝜔𝜏 𝑒 −𝑖∆𝜔 𝜏 2 𝐾 𝑉 𝑠
𝑉 𝑏 = − 𝑍 𝑐 1+𝛼 𝑍 𝑐 𝐾 𝑒 −𝑖Δ𝜔𝜏 𝐼 𝑏 = 𝑍 𝑙𝑜𝑜𝑝 𝐼 𝑏
𝑍 𝑐 = 𝑅 𝐿 1+𝑖 𝑄 𝐿 𝜉 = 𝑅 𝐿 1−𝑖tan 𝜓 𝑐𝑎𝑣
tan 𝜓 𝑐𝑎𝑣 =− 𝑄 𝐿 𝜉
𝑍 𝑙𝑜𝑜𝑝 = 𝑅 𝐿 1+𝑖 𝑄 𝐿 𝜉 1+𝛼 𝑅 𝐿 1+𝑖 𝑄 𝐿 𝜉 𝐾 𝑒 −𝑖Δ𝜔𝜏 = 𝑅 𝐿 1+𝑖 𝑄 𝐿 𝜉+ 𝛼𝑅 𝐿 𝐾 𝑒 −𝑖Δ𝜔𝜏 = 𝑅 𝐿 1+ 𝛼𝑅 𝐿 𝐾cos Δ𝜔𝜏 +𝑖 𝑄 𝐿 𝜉+ 𝛼𝑅 𝐿 𝐾sin Δ𝜔𝜏
= 𝑅 𝐿 1+ 𝛼𝑅 𝐿 𝐾cos Δ𝜔𝜏 1−𝑖 −𝑄 𝐿 𝜉− 𝛼𝑅 𝐿 𝐾sin Δ𝜔𝜏 1+ 𝛼𝑅 𝐿 𝐾cos Δ𝜔𝜏 = 𝑅 𝐿 𝐴 𝑤 1−𝑖 tan 𝜓 𝑙𝑜𝑜𝑝
Loop Gain: 𝐴=𝛼𝑅 𝐿 𝐾
𝐴 𝑤 =1+𝛼 𝑅 𝐿 𝐾cos Δ𝜔𝜏
=1+𝐴cos Δ𝜔𝜏
tan 𝜓 𝑙𝑜𝑜𝑝 = tan 𝜓 𝑐𝑎𝑣 − 𝛼𝑅 𝐿 𝐾sin Δ𝜔𝜏 𝐴 𝑤
𝑉 𝑏 = − 𝑅 𝐿 𝐴 𝑤 1−𝑖 tan 𝜓 𝑙𝑜𝑜𝑝 𝐼 𝑏 =− 𝑅 𝐿 𝐴 𝑤 𝐼 𝑏 cos 𝜓 𝑙𝑜𝑜𝑝 𝑒 𝑖 𝜓 𝑙𝑜𝑜𝑝

15. the effective voltage induced by RF generator source
The RF source induced voltages is
where
Note,
Here we can see that the controller delay t2, contributes an additional phase shift, it can be combined into loading angle, so same loog detuning angle will be used for both beam and RF induced cavity voltage.
𝑉 𝑔 = 𝑍 𝑐 1+𝛼 𝑍 𝑐 𝐾 𝑒 −𝑖Δ𝜔 𝜏 1 + 𝜏 𝑒 −𝑖∆𝜔 𝜏 2 𝐾 𝑉 𝑠 = 𝑅 𝐿 𝐴 𝑤 1−𝑖 tan 𝜓 𝑙𝑜𝑜𝑝 𝐼 𝑔 = 𝑅 𝐿 𝐴 𝑤 𝐼 𝑔 cos 𝜓 𝑙𝑜𝑜𝑝 𝑒 𝑖 𝜓 𝑙𝑜𝑜𝑝
𝐼 𝑔 = 𝑒 −𝑖∆𝜔 𝜏 2 𝐾 𝑉 𝑠
𝐼 𝐺 = 𝑒 −𝑖∆𝜔 𝜏 2 𝐾 𝑉 𝑖𝑛
𝑉 𝑐 = 𝑍 𝑐 1+𝛼 𝑍 𝑐 𝐾 𝑒 −𝑖Δ𝜔𝜏 𝑒 −𝑖∆𝜔 𝜏 2 𝐾 𝑉 𝑠 − 𝐼 𝑏
𝑉 𝑐 a
𝑉 𝑐 a𝑒 −𝑖∆𝜔 𝜏 1
𝜏 1 _
𝐼 𝐺 K Vs +
Vin
𝜏 2
𝑉 𝑐
𝐾= 𝐼 𝑔 𝑉 𝑖𝑛
Note:
Loaded RF cavity

16. Compare Closed Loop Open Loop
-Ib
Vgap Vg Ig
-Ib Vb Vb Ig
Vgap Vg
Closed Loop
Open Loop

17. Detuning Frequency Determination
From the phasor diagram, we can see that,
With given beam current and generator loading angle 𝜓 𝐿 , in order to get desired 𝑉 𝑐 and 𝜑 𝑠 , 𝑉 𝑔 and 𝜓 𝑙𝑜𝑜𝑝 need to be determined from the phasor diagram by balance 𝑉 𝑏 + 𝑉 𝑔 = 𝑉 𝑐
From second equation, we can get an expression for Vg, and plug
it in to the first equation, then we have
Then we have
From earlier slide, we know that
then we get the transendental equation, which can be used to calculate the detuning frequency Δ𝜔,
𝑉 𝑐 = 𝑉 𝑏 cos 𝜋 2 + 𝜑 𝑠 + 𝜓 𝑙𝑜𝑜𝑝 + 𝑉 𝑔 cos Ψ 𝐿 − 𝜓 𝑙𝑜𝑜𝑝 0 = 𝑉 𝑏 sin 𝜋 2 + 𝜑 𝑠 + 𝜓 𝑙𝑜𝑜𝑝 − 𝑉 𝑔 sin Ψ 𝐿 − 𝜓 𝑙𝑜𝑜𝑝
𝑉 𝑐 =− 𝑉 𝑏 sin 𝜑 𝑠 + 𝜓 𝑙𝑜𝑜𝑝 + 𝑉 𝑏 cos 𝜑 𝑠 + 𝜓 𝑙𝑜𝑜𝑝 sin Ψ 𝐿 − 𝜓 𝑙𝑜𝑜𝑝 cos Ψ 𝐿 − 𝜓 𝑙𝑜𝑜𝑝
sin Ψ 𝐿 − 𝜓 𝑙𝑜𝑜𝑝 =− 𝑉 𝑏 𝑉 𝑐 sin 𝜑 𝑠 + 𝜓 𝑙𝑜𝑜𝑝 sin Ψ 𝐿 − 𝜓 𝑙𝑜𝑜𝑝 + 𝑉 𝑏 𝑉 𝑐 cos 𝜑 𝑠 + 𝜓 𝑙𝑜𝑜𝑝 cos Ψ 𝐿 − 𝜓 𝑙𝑜𝑜𝑝
cos Ψ 𝐿 sin 𝜓 𝑙𝑜𝑜𝑝 = sin Ψ 𝐿 cos 𝜓 𝑙𝑜𝑜𝑝 − 𝑉 𝑏 𝑉 𝑐 cos 𝜑 𝑠 + Ψ 𝐿
tan 𝜓 𝑙𝑜𝑜𝑝 = tan Ψ 𝐿 − 𝑉 𝑏 𝑉 𝑐 cos 𝜓 𝑙𝑜𝑜𝑝 cos Ψ 𝐿 cos 𝜑 𝑠 + Ψ 𝐿 = tan Ψ 𝐿 + 𝑅 𝐿 𝐼 𝑏 𝐴 𝑤 𝑉 𝑐 cos 𝜑 𝑠 + Ψ 𝐿 cos Ψ 𝐿
tan 𝜓 𝑙𝑜𝑜𝑝 = tan Ψ 𝐿 + 𝑅 𝐿 𝐼 𝑏 1+𝐴cos Δ𝜔𝜏 𝑉 𝑐 cos 𝜑 𝑠 + Ψ 𝐿 cos Ψ 𝐿
𝜉= 𝜔 2 − 𝜔 𝜔 0 𝜔
tan 𝜓 𝑙𝑜𝑜𝑝 = − 𝑄 𝐿 𝜉−𝐴sin Δ𝜔𝜏 1+𝐴cos Δ𝜔𝜏
𝑄 𝐿 𝜔 0 +Δ𝜔 𝜔 0 − 𝜔 0 𝜔 0 +Δ𝜔 +Asin Δ𝜔𝜏 + tan Ψ 𝐿 1+𝐴cos Δ𝜔𝜏 + 𝑅 𝐿 𝐼 𝑏 𝑉 𝑐 cos 𝜑 𝑠 + Ψ 𝐿 cos Ψ 𝐿 =0

18. Chain of Parameters
𝑉 𝑏 =− 𝑅 𝐿 𝐴 𝑤 𝐼 𝑏 cos 𝜓 𝑙𝑜𝑜𝑝 𝑒 𝑖 𝜓 𝑙𝑜𝑜𝑝
𝑉 𝑔 = 𝑅 𝐿 𝐴 𝑤 𝐼 𝑔 cos 𝜓 𝑙𝑜𝑜𝑝 𝑒 𝑖 𝜓 𝑙𝑜𝑜𝑝
tan 𝜓 𝑐𝑎𝑣 =− 𝑄 𝐿 𝜔 2 − 𝜔 𝜔 0 𝜔
tan 𝜓 𝑙𝑜𝑜𝑝 = tan 𝜓 𝑐𝑎𝑣 −Asin Δ𝜔𝜏 𝐴 𝑤
𝐼 𝐺 = 1−𝑖tan 𝜓 𝑐𝑎𝑣 1−𝑖tan 𝜓 𝑐𝑎𝑣 +𝐴 𝑒 −𝑖Δ𝜔𝜏 𝐼 𝑔 + 𝐴 𝑒 −𝑖Δ𝜔𝜏 1−𝑖tan 𝜓 𝑐𝑎𝑣 +𝐴 𝑒 −𝑖Δ𝜔𝜏 𝐼 𝑏
Δ𝜔 𝑡𝑜 𝜓 𝑐𝑎𝑣 𝑡𝑜 𝜓 𝑙𝑜𝑜𝑝 𝑡𝑜 𝑉 𝑏 𝑉 𝑐 𝑉 𝑔 𝑡𝑜 𝐼 𝑔 𝐼 𝑏 𝐼 𝐺 𝑡𝑜 𝑃 𝑓𝑤𝑑 𝑤𝑖𝑡ℎ 𝐷𝐹𝐵
𝑃 𝑓𝑤𝑑 = 𝑅 𝑠 𝐼 𝐺 2 8𝛽

19. Generator Current for Forward Power CalculationK Ib
𝑉 𝑐 a𝑒 −𝑖∆𝜔 𝜏 1
𝑉 𝑐 a Vs
Vin
Loaded RF cavity
𝐼 𝐺 _ +
𝐼 𝑔 ′
𝐼′ 𝑏
Follow earlier slice, we can see that
𝐼 𝑔 ′ , determined by Klystron input and feedback loop
𝐼 𝑏 ′ , determined by beam current and feedback loop
So, together with beam current, the cavity sees ( 𝐼 𝑔 ′ + 𝐼 𝑏 ′ − 𝐼 𝑏 )
𝐼 𝐺 = 𝐾 𝑒 −𝑖∆𝜔 𝜏 2 𝑉 𝑠 +𝛼 𝑍 𝑐 𝐾 𝐼 𝑏 𝑒 −𝑖Δ𝜔 𝜏 1 + 𝜏 𝛼 𝑍 𝑐 𝐾 𝑒 −𝑖Δ𝜔 𝜏 1 + 𝜏 2
= 𝐼 𝑔 1+𝛼 𝑍 𝑐 𝐾 𝑒 −𝑖Δ𝜔 𝜏 1 + 𝜏 𝛼 𝑍 𝑐 𝐾 𝑒 −𝑖Δ𝜔 𝜏 1 + 𝜏 𝛼 𝑍 𝑐 𝐾 𝑒 −𝑖Δ𝜔 𝜏 1 + 𝜏 𝐼 𝑏
= 𝐼 𝑔 ′ + 𝐼 𝑏 ′

20. Conclusion
so far, we got the formulars, euqations and plots we need, we will re-tune the working points in the parameter space