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RF Systems Design. Stephen Molloy RF Group ESS Accelerator Division. AD Seminarino 17 / 02/2012. Outline. Some basic concepts (Hopefully not *too* basic…) Steady-state analysis Optimising a cavity Optimising the linac Transient Filling a cavity Commissioning the machine

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rf systems design

RF Systems Design

Stephen Molloy

RF Group

ESS Accelerator Division

AD Seminarino


  • Some basic concepts
        • (Hopefully not *too* basic…)
  • Steady-state analysis
    • Optimising a cavity
    • Optimising the linac
  • Transient
    • Filling a cavity
    • Commissioning the machine
    • Protecting the machine
lumped elements rf cavity
Lumped elements: RF cavity

Parallel LCR circuit, where L, C, & R, depend on geometry & material.

Resonant with a certain quality factor, Q0.

lumped elements rf system
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.

optimising a cavity for rf power
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
optimising the coupling
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

tuning the frequency why use the wrong frequency
Tuning the frequency:Why use the wrong frequency?

Vcav= Vforward + Vreflected





Vg= Vcav - Vbeam


A non-zero synchronous phase angle will always lead to reflected power, unless…

break the phase relationship
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

tuning the frequency why use the wrong frequency1
Tuning the frequency:Why use the wrong frequency?









Forward voltage can be made equal to the cavity voltage  no reflected power!

linac cavity optimisation
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
an aside beam cavity coupling
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


See ESS Tech Note: ESS/AD/0025


β=β0 may seem problematic as the cosine will go to zero, however the denominator also goes to zero. In this limit:

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.

additional spatial harmonics
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!
transit time factor conclusions
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
goals technique assumptions
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
result of optimisation
Result of optimisation

Note the large reflected power from the spoke cavities

spoke reflected power
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
klystron control linac commissioning
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
  • Vforwardmust change phase at beam arrival
      • Due to synchronous phase angle
  • In addition:
      • How much power is reflected when commissioning with low current beam?

Beam trip!

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

dynamic effects work in progress
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?
  • Steady-state 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