1. frequency-domain study of acceleration & beam loading based on a circuit model by raquel fandos

2. Outline Motivation Introduction Scheme of the analysis From structure parameters to circuit elements Information extracted from the circuit model RF response calculation Beam response calculation Example: G241 –Phase advance –Power and electric field –S parameters –Group delay –RF response –Beam loading

3. Motivation Low vg structures Dispersion is not negligible Accurate model Beam Loading & Acceleration

4. Introduction An accelerating structure Matching elements: Zin=Zmatch cell 1 cell 2 cell 3 … Input matching cell Output matching cell Tapered structure  - vg, Q & R/Q vary  R, C, L & k vary. - Input & output have different matching parameters (Rt & Lt) … A series of coupled resonant circuits

5. Scheme of the analysis Due to insufficient accuracy in PSPICE the analysis was performed using scripts that work with all the signals in the frequency domain. S-params. Filling time Power & Grad RF response Beam loading Struct. params. Circuit params. PSPICE Cell to Cell Transfer Function Signal proc. Beam & RF pulse params.

6. From structure parameters to circuit elements When the structure is tapered, vg, Q and R/Q vary along the structure, and so do R, C, L and k from cell to cell. (circuit differential equations) Cell i (from PhD thesis of C.D. Nantista, SLAC)

7. Information extracted from the circuit model … Directly in PSPICE we can measure: - Filling time - Voltage (prop. to electric field) and power flow along the structure - S parameters … : matching impedance

8. Information extracted from the circuit model Voltage Amplitude & Phase as functions of frequency at the output of every cell n Transfer Functions from input cell i to output cell j … … Hij(f) f(GHz)

9. Working in the frequency domain f V(f) In order to have a reasonable number of samples in the pass band, we need to store a lot of zeros  Solution: Work in baseband f AV(f)

10. RF pulse response calculation cell1cell2cell3 t Envelope of the Input RF pulse … Transfer function from the input to cell n FT Voltage signal at the output of cell n cell1 cell2

11. The beam in the time domain can be assumed to be a Dirac train, therefore its FT is a sinc signal centered in f0 and with - a width that depends on the number of bunches ( ) and the bunch spacing - an amplitude Vbeam that depends on the charge per bunch. The voltage amplitude that corresponds to a certain bunch charge is estimated in the PSPICE circuit model from the response in voltage to a current Dirac signal of value Beam response. The beam signal. … t Beam signal (v) FT Vbeam f(Hz)

12. Beam response cell1cell2cell3 … cell1cell2cell3 cell2cell3 cell4

13. Example: G241 –f0 = 11.994GHz – = 120deg –26 cells CellFirstMiddleLast vg/c[%]1.661.190.83 Q610061776265 R’/Q[Linac kOhm/m] 14.616.217.9 Parameters:

14. G241 phase advance Nominal =120deg. Matched to 119deg. Very sensitive to changes in matching elements. Example: 0.01% change in the output Lt f(GHz) Phase advance (degrees) Cell number Phase value @ nominal frequency Phase advance (degrees)

15. G241 S-parameters f(GHz) S params (dB)

16. G241. Group delay 65.15ns @ f0 (62 ns from difference model)

17. G241. RF pulse response RF pulse at cell 70ns 40ns

18. G241. Unloaded Power and Electric Field along the structure. 63.8 29.5 127.7 118 Electric field (MV/m) Power (MW) HFSS data for the first, last and middle cell were available. 2nd order polynomial interpolation used for the rest. cell number Circuit Model Difference model based on HFSS results cell number

19. 300 bunches separated by 6 cycles G241.Beam loading

20. Difference model based on HFSS results Circuit model G241.Beam loading cell number Electric field (MV/m) Loaded and unloaded electric field along the structure

21. 70ns 7ns Filling time=65.15ns G241.Beam loading compensation

22. G241.Beam loading RF pulse and beam response along the structure

23. Thanks ¿ ?