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frequency-domain study of acceleration & beam loading based on a circuit model by raquel fandos 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 Motivation Low vg structures Dispersion is not negligible Accurate model Beam Loading & Acceleration Introduction An accelerating structure cell 1 A series of coupled resonant circuits cell 2 cell 3 … … Matching elements: Zin=Zmatch 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) Scheme of the analysis Beam & RF pulse params. Struct. params. f0 vg Q RQ Circuit params. R L C k (coupling ) PSPICE Cell to Cell Transfer Function Signal proc. RF response Beam loading H ij Rt , Lt (match. elements) S-params. Filling time Power & Grad Due to insufficient accuracy in PSPICE the analysis was performed using scripts that work with all the signals in the frequency domain. From structure parameters to circuit elements Cell i 2R L 0Q Q C 0 R , f 0 , Q, v g k (circuit differential equations) (from PhD thesis of C.D. Nantista, SLAC) 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. 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 Vin Rt I s11 20 log 10 in Vin Rt I in s21 20 log 10 Vout Vg Rt : matching impedance Information extracted from the circuit model … … Voltage Amplitude & Phase as functions of frequency at the output of every cell n Vn Vampn exp jVphasen Hij(f) 1.46 1.00 Transfer Functions from input cell i to output cell j H ij Vj 0.50 Vi 0 11.88GHz 11.90GHz V(R27:2)/v(r01:2) 11.92GHz 11.94GHz 11.96GHz 11.98GHz f(GHz) Frequency 12.00GHz 12.02GHz 12.04GHz 12.06G Working in the frequency domain V(f) v(t ) FT 1 (V ( f )) f 0 f min f0 f0 f 0 f max 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 AV(f) 1 v(t ) Re( FT ( AV ( f ))) f min f f max RF pulse response calculation Envelope of the Input RF pulse FT RFIN cell1 RF1 cell2 RF2 cell3 RF3 … t RFIN RF1 H 21 RFi RFIN H n1 V H n1 n V1 cell1 RFIN cell1 cell2 Transfer function from the input to cell n Vn Vampn exp jVphasen H 31 Voltage signal at the output of cell n RF2 Beam response. The beam signal. 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 ( Nbunches ) and the bunch spacing T0 - 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 N qe I0 Vbeam BEAM Beam signal (v) … T FT T0 t T0 f(Hz) 1 T0 Nbunches Beam response BEAM1 BEAM 3 BEAM 2 cell1 cell2 cell3 RESP3 RESP2 BEAM1 BEAM 5 BEAM 4 cell4 … RESP5 RESP4 cell1 cell2 cell3 p BEAM i BEAM i 1 exp j c i RESPi BEAM n H n i BEAM 2 cell2 cell3 n 1 BEAM 3 cell3 RESP4 Example: G241 Parameters: – f0 = 11.994GHz – = 120deg – 26 cells Cell First Middle Last vg/c[%] 1.66 1.19 0.83 Q 6100 6177 6265 R’/Q[Linac kOhm/m] 14.6 16.2 17.9 G241 phase advance Phase advance (degrees) Phase advance (degrees) Phase value @ nominal frequency Cell number f(GHz) Very sensitive to changes in matching elements. Example: 0.01% change in the output Lt Phase advance (degrees) Nominal =120deg. Matched to 119deg. Phase value @ nominal frequency Cell number G241 S-parameters s11 s21 s22 0 -5 -10 S params (dB) -15 -20 -25 -30 -35 -40 -45 -50 11.8 11.85 11.9 11.95 f(GHz) 12 12.05 12.1 Hz G241. Group delay 11.850GHz 11.900GHz 11.950GHz 12.000GHz 12.050GHz 12.100GHz 12.15 Frequency 65.15ns @ f0 (62 ns from difference model) G241. RF pulse response RF pulse at cell 70ns 40ns G241. Unloaded Power and Electric Field along the structure. Circuit Model Difference model based on HFSS results 150 127.7 118 100 63.8 50 29.5 0 0 5 10 15 cell number 20 25 cell number 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. G241.Beam loading 300 bunches separated by 6 cycles G241.Beam loading Loaded and unloaded electric field along the structure Circuit model Electric field (MV/m) Difference model based on HFSS results cell number cell number G241.Beam loading compensation 70ns 7ns Filling time=65.15ns G241.Beam loading RF pulse and beam response along the structure Thanks ¿ ?