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Crab Cavities: Speed of Voltage Change (a machine protection issue for LHC [and SPS] )

Crab Cavities: Speed of Voltage Change (a machine protection issue for LHC [and SPS] ). J. Tückmantel, CERN-BE-RF. CCinS WG, 27 Nov 2009. Contents:. • The Problem • Time scales of incidents and equipment • Cavity and RF basics – longitudinal – transversal • Examples

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Crab Cavities: Speed of Voltage Change (a machine protection issue for LHC [and SPS] )

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  1. Crab Cavities:Speed of Voltage Change(a machine protection issue for LHC [and SPS] ) J. Tückmantel, CERN-BE-RF CCinS WG, 27 Nov 2009

  2. Contents: • The Problem • Time scales of incidents and equipment • Cavity and RF basics – longitudinal – transversal • Examples • (Conclusion)

  3. • The Problem • When a crab cavity gets out of control and changes its voltage/phase, the beam may also get out of control: bunch is ‘banged’ by a single CC passage: Δpt,CC/pt,0 ≈ 1(*) • If the speed of change is so fast that the beam dump system – requiring 3 turns (≈ 300 µs) in the worst case – cannot react in time, severe machine damage is possible. • Here we consider only the possible voltage/phase change scenarios the possible aftermath for the beam is not analyzed. (*) The main RF can change rapidly causing much less problems: the large longitudinal beam inertia ‘saves the day’: Δp||/p||,0 ≈ eVcav/Ebeam <<<< 1

  4. – Time scales of ‘incidents’ +  Mains power cut (anywhere): RF power supply has enough stored energy to survive many ms (mains 50 … 300 Hz -> 20 … 7 ms) : no problem + ‘Short’ or … in low power electronics, controllers: Develops >> 1 ms : no problem – RF arcing in high power part (WG, coupler, cavity): Full arc develops within about 1 µs: rely on τF – Operator or control-logics error: ‘instant’ change: rely on τF

  5. – Time scales of equipment changes Any tuner of a (high-powered sc.) cavity is mechanical: it is too slow to change significantly within 300µs (if foreseen) Qext is changed by mechanical means (stepper motor, ….) generally slower than tuner: it is (much) too slow to change significantly in 300µs During the total ‘fast’ incident (300 µs): Δω and Qext are what they were at onset

  6. • Some Cavity and RF basics - longitudinal The proven (longitudinal) model for cavity-klystron-beam Incident (generator) wave Ig Reflected wave Ir Circulator: (1)->(2) Iin=Ig to cavity; (2)->(3) Ir to load; (3)->(1) If RF switch off (Ig=0): (and no beam IB=0) Cavity unloads over R and Z !!! (the coupler sucks)

  7. carry charge q across capacitor C charge q through cavity (R/Q), ω: any resonator: equivalent: (R/Q): Circuit Ω convention: 1 Ωcircuit = 2 linac Ωlinac (or 1 Boussard = 2 Schnell) Dictionary lumped circuit L,C,.. <–––> cavity (R/Q), .. …. spare you the math …..

  8. Even if you do not like it, a reflected wave comes for free … Reactive beam loading compensation: Im(I g,r)=0 Sc. cavity: Qext<<<Q0   1/Qext ± 1/Q0 ≈  1/Qext To get V (steady state = constant quantities) I g,r are (proportional) model quantities, only P are absolute quantities !!! fB: relative bunch form factor: fB=1 for ‘point bunches’ fin ‘proton machine convention’: f=90º for beam on top of RF (max. accel.) IDC: DC beam current;   Δω: cavity detuning wrsp. to machine line = RF drive V: cavity voltage (generally considered real)

  9. There is a Qext optimum enforcing Ir=0 i.e. Pr=0 Assume sc. cavity + reactive beam-loading compensation The choice of Qext (for given IB, V, .. ) is not for free: If too low or too high: reflected power increases in both cases ––> klystron has to deliver this power more (used to heat coffee-water !!!)

  10. Quantities only for steady state, what is it good for ? Driving ‘force’ jumps(*) from one state to another one: RF drive Ig suddenly off, Ig jumps in ampl./phase, …. Linear system: superposition ‘Old’ field decays “exp( )” with natural (field-) time constant τF, ‘new’ field builds up ”1- exp( )” with the same time constant For any resonator τω = Q : τ is the energy decay time !! When fields decay as A=A0*exp(-t/ τF), then energy decay as A2=A02 exp(-2t/ τF)= A02 exp(-t/ τ), τF = 2·τ Cavity (essentially) unloads over coupler: τF = 2·τ=2·Qext/ω (*) transition ‘much’ faster than τF

  11. Double driving ‘force’: klystron Ig and beam IB,DC: Assume ‘sudden’(t=0) ( time-scale << τF) change of drive (ΔI): ‘Old’ drive (t < 0): keeps an equilibrium Voltage VA (complex) ‘New’ drive (t > 0): corresponds to new equilibrium Voltage VB

  12. Sucked from the beam & dumped into load Special case: the klystron is (goes) off, i.e. Ig=0 And Φ is not ‘stabilized’ anymore –> maximum induced voltage Φ –> 90º the so-called RF current IRF (Fourier component)

  13. Loss of Ig (no beam) Step up of Ig to Pg,max Loss of Ig with strong beam Examples without change of phase : drivevoltage Feedback action: to peak & back to new equilibrium

  14. Speed depends on (same )τf but also on Ig,(max), i.e. Pg,(max) Ig = 2 a.u. Ig = 4 a.u. Ig = 6 a.u. Ig = 10 a.u.

  15. Example of 90º phase jump of drive cos(ωt) -> sin(ωt) i.e. Real ––> Imag Complex V Complex V versus time

  16. Speed of change: feedback versus beam Does not depend on Qext, τF !!!! Same rise for same ΔI Steepest rise by feedback P: 0 –> Pmax To keep same enforced (FB) speed of change: speed scales as 1/√Qext Loop gain scales as √Qext (for same’ hardware gain’)

  17. - transverse Generalized Panofsky-Wenzel theorem Deflection requires transverse gradient in longitudinal accelerating voltage (–>Ez) the same Vz gradient = same deflection !! Chose field configuration having x0 that Vz(x0) = 0: Δpx, Bunch centre 90º out of phase (set like this since we want only tilt, no kick for bunch center !!) Δpx, Vz 90º out of phase  no longitudinal beam-cavity interaction ( if beam really at x0)

  18.  Bunch Center (==Ib), Vz in phase !!! Bad news (for RF installation): worst phase angle for parasitic longitudinal interaction ( for x ≠ 0) Good news (for machine protection): the beam drives a transverse voltage with phase for tilting the bunch, NOT kicking the whole bunch !

  19. For highly relativistic beam: longitudinal Analogue definition: transverse voltage Dipole (=crab) mode: Beam passing at offset x sees (only magnitudes, forget 90º phase factor ‘i’ here)

  20. …analogue Cavity geometry constant - indep. of excitation - indep. of cav. material (Cu, iron, superc., ..) Only perfect for a chosen x0 (R/Q): Circuit Ω convention: 1 Ωcircuit = 2 linac Ωlinac (or 1 Boussard = 2 Schnell)

  21. oufffffff Power finite even for x ––> 0 Possibility: Renormalize I and P (J=xI gets dimension [A·m] !) Factor 1/x in Ig,r ; if x ––> 0 ??? Currents are proportional to real waves, power is ‘absolute’

  22. Transverse impedance: Beam drives  Ig=0, ϕ=90º Longitudinal impedance of dipole mode at offset x0

  23. Additional power due to ‘wrong’ frequency: δω – reactive beam loading compensation not perfect – fight a mechanical (*) cavity oscillation = sideband (microphonics, ponderomotive oscillations) + assume δV << V Feedback action with gain gFB (not shown explicitly here, delay=0; realistic delay gFB ≤ 100) for several BW detuning but g still larger (*) Perturbations over the RF input or beam and their combat over the same RF are on same ‘footing’ : neutral wrsp. Qext

  24. (Intermediate) Summary of facts for several BW detuning … at injection x0 is “not so perfect” …

  25. Examples ‘Given’ are ( = LHC 800 MHz ‘test cavity’, others similar) (R/Q)t=60 Ωcircuit (=120 Ωlinac); Vt=2.5 MV; Ib=0.6 A (neglect bunch form factor < 1 at 800 MHz, it helps) x == 0 not possible in real life: allow (limited) deviation |x|max Assume: guaranteed |x| ≤ 0.2 mm (=200 µm!) … at injection x is “not so perfect” … (maybe larger than ‘coast’-xmax) Qext,opt = 107 (τF=4000µs: field decay to 93% in 300µs) Pmax.opt= 5 kW

  26. Assume gFB=100; δf=3 kHz (β-tron f); δVt=2.5kV=10-3 Vt

  27. Assume gFB=100; δf=3 kHz (β-tron f); δVt=2.5kV=10-3 Vt |x|max = 4 mm <–> Pmax=100kW (P=30 kW @ 0.2 mm) Only previously ‘critical’ items:

  28. Thank you for listening!

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