1. Strategic Communications at TRIUMF
Lorentz force induced oscillations
Ramona Leewe,
Ken Fong,
TRIUMF
TTC meeting 2019
02/07/2019
Strategic Communications at TRIUMF

2. Overview TRIUMF’s e-linac driving configuration
First operational experiences
Mathematical formulation for Lorentz force oscillation on a single cavity
Stability analysis/ oscillation growth rate (linearized system)
Limit cycle analysis
Nonlinear Lyapunov stability
Simulations
What can we do with these information?
Conclusions

3. TRIUMF’s e-linac driving configuration
CAV 1
CAV 2
Klystron
300kW
CAV1
CAV2
EACA +
TRIUMF’s e-Linac acceleration cryomodule, consists of 2 TESLA type cavities and is operated with a single klystron in CW mode and vector sum control.
Controller

4. Operational experience
Amplitude oscillation in both cavities (operational gradient dependent)
Vector sum perfectly stable
Oscillation frequency ≈ 160 𝐻𝑧
Cavity bandwidth ≈ 300 𝐻𝑧
Time to grow oscillations ≈6 −10 𝑠𝑒𝑐
Cavity 1
Cavity 2
Pick up signals for cavity 1 and cavity 2. In this case the system is moving in a limit cycle.

5. 𝒙 +𝒙=−Λ 𝒗 𝟐 − 𝒗 𝟎 𝟐 Lorentz force on a single cavity, no feedback
Mathematical problem formulation
Electrical part
Lorentz force: 𝐹=− 𝐾 𝐿 𝑉 𝑎𝑐𝑐 2
Mechanical system: 𝑥 +𝑥=𝐹
Equation of motion:
Voltage at the cavity:
𝑑𝑖 𝑑𝑡 = 1 𝑅 𝑑𝑉 𝑑𝑡 + 𝑑 2 𝐶 𝑑 𝑡 2 𝑉+2 𝑑𝐶 𝑑𝑡 𝑑𝑉 𝑑𝑡 + 𝑑 2 𝑉 𝑑 𝑡 2 𝐶+ 1 𝐿 𝑉= 2 𝑍 0 𝑑 𝑣 𝑓 𝑑𝑡
𝜏 𝑣 = 1−𝑗𝑎 𝑣= 𝑣 𝑓
𝑣 2 − 𝑣 0 2 =−2 𝑣 𝑓 𝑎+𝑥 (1+ 𝑎 2 ) 𝑠𝜏 𝑎+𝑥 2 𝑥
High quality factor, long time constant 𝜏, voltage at capacitor does not rise instantaneously
simplified, damping coefficient = 0
𝒙 +𝒙=−Λ 𝒗 𝟐 − 𝒗 𝟎 𝟐
Voltage at the cavity is derived from a parallel resonant circuit, but it’s important to note here, that superconducting cavities have a high quality factor and long time constant, this means that the voltage does not rise or fall instantaneously, which means there is a phase lag between the RF voltage and the mechanical movement. So, the combined change of current through the circuit
Ramona Leewe, Ken Fong

6. Lorentz force on a single cavity, no feedback
Mechanical system
System linearization
No damping and no external force
𝑥 𝑦 = 0 − 𝑥 𝑦
Eigenvalues ±𝑖, circle with radius 1 in the phase space
Adding perturbation, Lorentz force
𝑥 𝑦 = 0 − 𝑥 𝑦 + 𝐹=0 −𝐺
Or more general
𝑥 =𝑓 𝑥,𝑦 =−𝑦 𝑦 =𝑔 𝑥,𝑦 =𝑥−𝐺
Calculating the Jacobian and evaluate at x,y=0
𝑥 𝑦 = 0 −1 1 − 𝑑𝐺 𝜕𝑦 𝑥,𝑦=0 𝑥 𝑦
Perturbation modifies the trajectory and becomes either a stable or unstable spiral
𝑥 𝑦 = 0 −1 1 −𝜇 𝑥 𝑦
𝜇= Λ4 𝑣 𝑓 2 𝑎 1+𝑎 𝜔 𝜔
𝜔 =𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 𝑡ℎ𝑒
𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐𝑎𝑙 𝑏𝑎𝑛𝑑𝑤𝑖𝑑𝑡ℎ
𝑡𝑜 𝑡ℎ𝑒 𝑚𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙
𝑜𝑠𝑐𝑖𝑙𝑙𝑎𝑡𝑖𝑜𝑛 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
Ramona Leewe, Ken Fong

7. Stability analysis of the linearized system
Eigenvalue analysis
Growth rate
System is stable, if real part of eigenvalues is negative
λ 1,2 ≈−2Λ 𝑣 𝑓 2 𝑎 1+ 𝑎 𝜔 𝜔 ±𝑖
Linearized system has an unstable spiral center at (0,0) for 𝑎<0 and a stable spiral center for 𝑎>0
𝑅≅−2Λ 𝑣 𝑓 2 𝑎 𝜔 𝜔
Lorentz force constant Λ
Squared driving voltage 𝑣 𝑓 2
Amount of detuning 𝑎= 𝜔 0 −𝜔 𝜔
Max. growth rate 𝜔 = 3 ≅1.7
Ramona Leewe, Ken Fong

8. Oscillation growth\decay rate of the nonlinear system
simulation results of the nonlinear system
Pumping is due lag in RF voltage wrt mechanical movement
RF voltage fluctuation depends on its detuning angle
Ramona Leewe, Ken Fong

9. Limit cycle existence of the nonlinear system by proof of Poincare- Andronov-Hopf bifurcation
Conditions for limit cycle existence
Lorentz force system
Has a pair of purely imaginary eigenvalues
During a parametric change, the critical point changes from a stable to an unstable spiral or vice versa
Whether the bifurcation is sub or supercritical is determined by the sign of the first Lyapunov coefficient
λ 1,2 ≈−2Λ 𝑣 𝑓 2 𝑎 1+ 𝑎 𝜔 𝜔 ±𝑖
λ 1,2 ≈−𝛼±𝑖𝛽
for 𝛼=0 𝑎=0
𝑑𝛼 𝑑𝑎 𝑎=0 =𝑑≠0; 𝑑=−2Λ 𝑣 𝑓 2 𝜔 𝜔
3. 𝐿1 ​ 𝑎=0 =− Λ 𝑣 𝑓 𝜔 𝜔 <0
What is a limit cycle? If we have a two dimensional system (as in our specific case, x and y the position and the velocity of the cavity wall movement), a limit cycle is closed orbit in the phase space, this will become a bit more clear in the next couple slides. Now we are looking a the existence of limit cycles
Supercritical, stable limit cycle
Ramona Leewe, Ken Fong

10. Supercritical Hopf Bifurcation with a stable limit cycle
Nonlinear system
Linearized system
Ramona Leewe, Ken Fong

11. Supercritical Hopf Bifurcation with a stable limit cycle
Unstable within the cone 𝑎>0 , small initial conditions
Considering damping
Cone position moves with increasing damping coefficient along the ‘a-Axis’
Increases system stability
In a linear system when c is equal zero (a=0), the system would run in a circle, but the nonlinearity brings the system states down to zero
Ramona Leewe, Ken Fong

12. Lyapunov stability/ nonlinear system
(Lyapunov functions may be considered as energy functions)
Mechanical cavity system with Lorentz force/ different 𝛼
Specific case 𝛼=1.5°
General stable system
𝑑𝑉(𝑥) 𝑑𝑡 >0
𝑑𝑉(𝑥) 𝑑𝑡 ≤0
unstable
𝑑𝑉(𝑥) 𝑑𝑡 <0
stable
Nonlinear systems are
Ramona Leewe, Ken Fong

13. 𝒙 +𝒄 𝒙 + 𝜴 𝟐 𝒙=− 𝜦 𝒗 𝟐 − 𝒗 𝟎 𝟐 Simulations Matlab/simulation equations
Damping included
4 state variables
Voltage signal split into in phase and in quadrature
𝒙 +𝒄 𝒙 + 𝜴 𝟐 𝒙=− 𝜦 𝒗 𝟐 − 𝒗 𝟎 𝟐
𝑥 =−𝑦
𝑦 =𝑐𝑦− Ω 2 𝑥− Λ 𝑣 𝑖 2 + 𝑣 𝑞 2 − 𝑣 0 2
𝑣 𝑖 =− 𝜔 𝑣 𝑖 − 𝑎+𝑥 𝑣 𝑞 − 𝑣 𝑖
𝑣 𝑞 =− 𝜔 𝑎+𝑥 𝑣 𝑖 + 𝑣 𝑞
Ramona Leewe, Ken Fong

14. Simulation results Stable: 𝛂<𝟎, 𝑑𝑒𝑡𝑢𝑛𝑖𝑛𝑔 𝑎𝑛𝑔𝑙𝑒 −1.5°
Unstable: 𝜶>𝟎, 𝑑𝑒𝑡𝑢𝑛𝑖𝑛𝑔 𝑎𝑛𝑔𝑙𝑒 1.5°
Ramona Leewe, Ken Fong

15. Conditions for Oscillations in RF cavity due to Lorentz force
High electric field ~10𝑀𝑉/𝑚
Insufficient rigidity in RF resonator
Low mechanical damping
Bandwidth of RF roughly double of mechanical mode oscillation
Poor voltage regulation
Driven frequency > RF resonance frequency
Slow growth rate
Non zero initial conditions
High Q cavity Nb
Metal
300Hz/160Hz
Vector sum
CW-operation
Microphonics
Ramona Leewe, Ken Fong

16. What can we do to suppress these instabilities?
High electric field ~10𝑀𝑉/𝑚
Insufficient rigidity in RF resonator
Low mechanical damping
Bandwidth of RF roughly double of mechanical mode oscillation
Poor voltage regulation
Driven frequency > RF resonance frequency
Slow growth rate
Non zero initial conditions
Nothing
Strengthen mechanically
Add damping
Avoid vector sum
Tune for 𝜔< 𝜔 0
Pulsed-operation
Active Microphonics feedback
Active Lorentz force oscillation suppression
Ramona Leewe, Ken Fong

17. Conclusion and lookout
Field oscillation can occur at high field gradients
It has a slow rise time
First observed at TRIUMF’s e-LINAC in summer 2018
It is suppressed under certain conditions
Active Lorentz force oscillation suppression feasible?
Lorentz force affects the cavity acceleration (not the position)
What variable could be measured?
Phase lag between the mechanical detune and the electrical response
𝑥 =𝑦+ 𝐹 𝑦 =𝑥+𝐺 , 𝐹=𝑓𝑒𝑒𝑑𝑏𝑎𝑐𝑘,
piezo is affecting the position, not a trivial cancellation feedback problem
A deeper analysis of the presented results with respect to a feedback tuning system will be necessary
Ramona Leewe, Ken Fong

18. Thank you for your consideration!

19. Simulation results (increased detuning angle)
Unstable: 𝜶>𝟎 𝑑𝑒𝑡𝑢𝑛𝑖𝑛𝑔 𝑎𝑛𝑔𝑙𝑒 5°
Ramona Leewe, Ken Fong