1. 1 Simulation for power overhead and cavity field estimation Shin Michizono (KEK) Performance (rf power and max. cavity field) @35 MV/m 24 cav. operation @31.5 MV/m 24 cav. Operation @33.5 MV/m 26 cav. operation @31.5 MV/m 26 cav. operation

2. 2 Assumption Number of cavities: 24(8-8-8) or 26(9-8-9) Ql=~3.3e6 (optimum loaded Q depending on the set-cavity gradient and beam current) Beam current=9.6 mA (corresponding to +1%) Cavity gradient=35 MV/m,31.5 MV/m 33.5MV/m, 31.5 MV/m Microphonics 10Hz(rms) FB gain=50 Ql preciseness : 3%rms (Random selections at each simulation) Waveguide coupling preciseness : 3%rms  (0.2 dBrms) (random selections at each simulation) Beam fluctuation (bunch by bunch) : 1%rms (But this does not affect rf power because this random current fluctuations can not be compensated by the FB.) Lorentz force detuning reduction 97% Vector sum control Bottom up rf power calculation with variations of Ql/rf distribution/beam current.

3. 3 Procedure 1. Set parameters: 1-1.Number of cavities: 24 or 26 1-2.Cavity gradient:35 MV/m, 31.5 MV/m, 33.5 MV/m 2. Random selection of: 2-1. loaded Q of 24 (or 26) cavities 2-2. rf distribution ratio of 24 (or 26) cavities 3. Calculation start (24 or 26 cav. Vector sum) 4. Calculation of rf power, cavity gradient distribution The obtained rf power is the power required for the set cavity gradient (such as 35 MV/m). 5. Back to 1. and repeat 500 times 6. Statistical investigation of rf power and maximum cavity fields

4. 4 Cavity field variation (during rf pulse) Pf [MW] (sum of cavity inputs.) Detuning during rf pulse (microphonics+Lorentz force) Cavity phase variation (during rf pulse) Ql variation 3%rms (random values) Beam current fluctuation during rf pulse (but no extra rf power is necessary because FB can not suppress this fast random variation) Example of simulation Cavity # Maximum cavity field (due to vector sum)

5. 5 Simulation 4 kinds of simulations 1. 24 cav. 35 MV/m 2. 24 cav. 31.5 MV/m 3. 26 cav. 33.5 MV/m 4. 26 cav. 31.5 MV/m Each 500 simulation (~same order to linac rf units) 500 times

6. 6 Histograms of cavity power and maximum cavity gradient @24 cav. system 31.5 MV/m 24 cav. 35 MV/m 24 cav. These value is the sum of the cavity input (w/o rf losses and so on.) Maximum field gradient in 500 times simulation is >40 MV/m Histogram of RF power Histogram of max. cavity field 38.8 MV/m: Mean value of max. cavity field with 500 times simulations

7. 7 Histograms of cavity power and maximum cavity gradient @26 cav. system Maximum field gradient in 500 times simulation is >40 MV/m Histogram of RF powerHistogram of max. cavity field 33.5 MV/m 26 cav. 31.5 MV/m 26 cav.

8. 8 Summary *I add 17%(12% (13%-1%parameter vatriation)+5% extra FB margin) in order to include waveguide loss etc.. FB margin of 5% is necessary for suppression of the perturbations. (We do not need this margin if we give up FB and operate only with FF.) ** mean value of the maximum gradient by 500 times simulations *** maximum gradient in 500 times simulations >10 MW will be necessary for the FB with 26 cavities system at maximum field gradient operation (33.5 MV/m) In case of vector sum operation, some cavities with higher Ql or higher power rf input have 10%-20% higher rf fields.

9. 9 Histograms of cavity power and maximum cavity gradient @26 cav. System (33.5 MV/m) Ql,distribution error 3%rms + microphonics+LFD 11% higher (ave.) or 21% higher (max.) field 5.1% more power is necessary. Only Ql variation 5% higher (ave.) or 10% higher (max.) field by 3%(rms) loaded Q distribution 4.4% more power is necessary.

10. 10 Histograms of cavity power and maximum cavity gradient @26 cav. System (33.5 MV/m) Ql and rf distribution error 3%rms 11% higher (ave.) or 23% higher (max.) field. 4.3% more power is necessary. Only LFD+microphonics variation 0.6% higher (ave.) or 1% higher (max.) field ->negligible small But 2.9% more power is necessary.

11. 11 Ql error v.s. max. cavity gradient in case of the 2 cavities Only Ql variation FB gain=50 and 5 deg.off-crest beam -> -0.1 deg=5 deg/50 (This can be compensated with proper FF.) 10% error in loaded Q induces 4% higher cavity field

12. 12 Rf distribution error v.s. max. cavity gradient in case of the 2 cavities Only rf distribution variation 10% error in rf distribution induces 8.5% higher cavity field

13. 13 Summary (2) * I add 17%(12% (13%-1%parameter vatriation)+5% extra FB margin) in order to include waveguide loss etc.. FB margin of 5% is necessary for suppression of the perturbations. (We do not need this margin if we give up FB and operate only with FF.) ** mean value of the maximum gradient by 500 times simulations *** maximum gradient in 500 times simulations Each components of the power loss are calculated. –Parameter variation (Ql+rf distribution): ~4% 3% rms loaded Q and rf distribution control requires 4% additional power. –Detuning (Lorentz force detuning + microphonics): ~3% –Total (detuning, parameter variation, beam current): ~5% Higher gradient during vector sum will become another problem. (9.03/8.59-1)*100 (8.59 MW: ideal cavity input)

14. 14 Summary (3) I include in the simulation 1.10 Hz microphonics 2. 3% Lorentz force detuning 3. +1% beam current 4. Loaded Q variation 3% rms 5. Rf distribution variation 3% rms The results show 1. 1% power loss by 1% beam 2. 3% detuning effects (p.10,p.13) 3. 4% due to parameter change (p.9,p.10,p.13) Agree well with the total rf overhead.