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

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

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

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.

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+1 2 + 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

Lorentz force on a single cavity, no feedback Mechanical system System linearization No damping and no external force 𝑥 𝑦 = 0 −1 1 0 𝑥 𝑦 Eigenvalues ±𝑖, circle with radius 1 in the phase space Adding perturbation, Lorentz force 𝑥 𝑦 = 0 −1 1 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 3 𝜔 2 +1 2 𝜔 =𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 𝑡ℎ𝑒 𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐𝑎𝑙 𝑏𝑎𝑛𝑑𝑤𝑖𝑑𝑡ℎ 𝑡𝑜 𝑡ℎ𝑒 𝑚𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝑜𝑠𝑐𝑖𝑙𝑙𝑎𝑡𝑖𝑜𝑛 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 Ramona Leewe, Ken Fong

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 2 𝜔 3 𝜔 2 +1 2 ±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 3 𝜔 2 +1 2 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

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

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 2 𝜔 3 𝜔 2 +1 2 ±𝑖 λ 1,2 ≈−𝛼±𝑖𝛽 for 𝛼=0 𝑎=0 2. 𝑑𝛼 𝑑𝑎 𝑎=0 =𝑑≠0; 𝑑=−2Λ 𝑣 𝑓 2 𝜔 3 𝜔 2 +1 2 3. 𝐿1 ​ 𝑎=0 =− Λ 𝑣 𝑓 2 2 𝜔 3 𝜔 2 +4 <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

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

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

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

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

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

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

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

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

Thank you for your consideration!

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