1. Stephen Molloy RF Group ESS Accelerator Division
RF Systems Design
Stephen Molloy
RF Group
ESS Accelerator Division
AD Seminarino
17/02/2012

2. Outline Some basic concepts Steady-state analysis Transient
(Hopefully not *too* basic…)
Steady-state analysis
Optimising a cavity
Optimising the linac
Transient
Filling a cavity
Commissioning the machine
Protecting the machine

3. RF System Concepts

4. Lumped elements: RF cavity
Parallel LCR circuit, where L, C, & R, depend on geometry & material.
Resonant with a certain quality factor, Q0.

5. Lumped elements: RF system
Generator current after transformation by the coupler
Transmission line impedance seen from “the other side” of the transformer.
Note it is in parallel with the cavity resistance, R.
Note that loaded R & Q both scale in the same way when shunted by the coupler.
Therefore R/Q is unchanged.
R/Q is a function of the geometry only, and so the circuit resistance, (R/Q)QL, is set by choosing the coupler loading.

6. Optimising a cavity for RF power
Equivalent circuit allows tuning of parameters
Loaded quality factor, QL
Transformer ratio of the coupler
Location and dimensions of coupler & conductor
Frequency
Inductance & capacitance
Dimensions of the cavity
Coupling to beam, R/Q
Also the inductance & capacitance
Cavity dimensions

7. Optimising the coupling
How best to squeeze RF into the cavity?
Minimise QL to speed power transfer from klystron?
Maximise QL to improve efficiency of the cavity?
Match voltages excited by klystron & beam
Requires a specific value for QL
For a specific forward power…
Thus, steady state signals
are equal

8. Tuning the frequency: Why use the wrong frequency?
Vcav= Vforward + Vreflected
Vbeam
Ibeam φb
Vforward=Vg/2
Vg= Vcav - Vbeam
Vcav
A non-zero synchronous phase angle will always lead to reflected power, unless…

9. Break the phase relationship
Driving a resonator off-resonance leads to a drop in the amplitude and a rotation of the phase of the excited signal.
The higher power required to achieve the same cavity field could be easily compensated by the elimination of the reflected power

10. Tuning the frequency: Why use the wrong frequency?
Ibeam Vg ψ φb ψ
Vforward
Vbeam
Vcav
Forward voltage can be made equal to the cavity voltage  no reflected power!

11. Linac & cavity optimisation
For a single cavity
Reflected power can be eliminated
Correctly choose:
Detuning
QL due to the coupler
For a linac, it is not so simple
Detuning is easy
Forgetting about Lorenz detuning for the moment
Coupler
Prohibitively expensive to design individual couplers for each cavity
So, optimise the QL for the total reflected power

12. An aside: Beam cavity coupling
Coupling composed of 2 signals
Cavity field vector (depends on position)
Cavity phase (depends on time)
Magic
Integration by parts (twice)
Cosine is an even function
Sine is an odd function
π phase advance per cell
Five-cell cavity
Magic!
See ESS Tech Note: ESS/AD/0025

13. Discussion
Velocity bandwidth may be approximated by the closest zeros of the cosine:
That the optimum β is greater than β0 is a well known phenomenon.
This curve agrees very well with simulation/measurement.
R/Q depends on square of V.
β=β0 may seem problematic as the cosine will go to zero, however the denominator also goes to zero. In this limit:

14. Additional spatial harmonics?
2nd term is negligible
Result is the same as for 1 spatial harmonic
No advantage in velocity bandwidth
12.5% improved acceleration
With no increase in peak voltage!

15. Transit-time factor conclusions
Note assumptions:
Fixed cell length
No significant velocity change
π-mode cavity
Observed voltage dependent on lots of things
Cavity β, particle β, peak voltage, frequency, etc.
Velocity bandwidth depends….
Only on the number of cells!
Increase effective voltage:
Increase number of cells
Increase 1st order spatial component
Add additional components to maintain reasonable peak field

16. Optimising the SC linac

17. Goals, technique, assumptions
Minimise the total reflected power
Vary the QL’s, and sum the reflected powers
Nominal beam  50 mA, 2.8 ms
Each section has a single QL
Spoke, medium/high beta
Each cavity detuned optimally
Velocity dependence of impedance included
Theoretical for elliptical cavities
Spoke based on field profile from S. Bousson

18. Result of optimisation
Note the large reflected power from the spoke cavities

19. Why are the spokes problematic?
R/Q

20. Spoke reflected power Fixes: Redesign spokes for a lower beam velocity
Begin spoke section at a higher beam energy
Use multiple coupler designs in the spoke section

21. Re-optimise

22. Dynamic (not steady state) performace

23. Klystron control & linac commissioning
Choose klystron current to achieve correct phase & amplitude
Vg + Vbeam = Vcav
Only in steady-state!
Must ensure that phase & amplitude are correct at beam arrival
Vforward must change phase at beam arrival
Due to synchronous phase angle
In addition:
How much power is reflected when commissioning with low current beam?

25. Beam trip!
In reality, LLRF would detect the incorrect cavity amplitude & phase, and the large reflected power, and act to prevent this.

26. Dynamic effects – work in progress
Nominal beam
Control klystron to achieve required RF conditions
Commissioning
Shorten RF pulses to match beam duration
Lower peak current will cause problems
QL matching done for 50 mA
Preferable to run with same bunch charge
Machine faults
How much power can we reflect back to the loads?
Klystrons tripped by MPS within a pulse?

27. Conclusions Steady-state analysis Transient analysis
Linac optimised using 5 families of couplers
Mismatches between the voltage profile and R/Q profile are simple to fix
Reflected power per cavity reduced to <10 kW
Transient analysis
A work in progress…
Reflected energy/pulse calculated for all cavities
Begun investigating:
Commissioning strategies
Fault scenarios