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**Stephen Molloy RF Group ESS Accelerator Division**

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

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**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

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RF System Concepts

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**Lumped elements: RF cavity**

Parallel LCR circuit, where L, C, & R, depend on geometry & material. Resonant with a certain quality factor, Q0.

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**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.

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**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

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**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

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**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…

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**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

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**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!

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**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

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**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

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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:

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**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!

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**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

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**Optimising the SC linac**

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**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

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**Result of optimisation**

Note the large reflected power from the spoke cavities

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**Why are the spokes problematic?**

R/Q

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**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

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Re-optimise

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**Dynamic (not steady state) performace**

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**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?

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

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**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?

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**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

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