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TABLE 7.1: Parameters used for simulations to study the effect of a bunch-by-bunch feedback system on the BBU threshold current. Parameter Value ω 2106.007 MHz QL 6.11 × 106 (R/Q) 29.9 Ω Vb 39 MV Tr 433.200 ns M34 −5.1 m M p 34 1.1 m 13.7 m M k 34 and the behavior of the system in the pseudo-stable regime (M F B > 0) where the feedback system acts to increase, rather than decrease, the threshold current. 7.2.4 Simulation Results A number of simulations were performed to ascertain some of the parametric dependencies in the pseudo-stable regime. All simulations were performed with the parameters in Table 7.1. The remaining free parameters are the feedback time delay (td) and the feedback gain (g). As discussed in the previous section, the effect of feedback time delay is the most important issue for determining if sufficient suppression can be achieved if the bunch corrected/kicked uses an error signal derived from a different bunch. Simulation results showing the dependence of the threshold current on the time delay (for g = 1) are given in Fig. 7.8. The value of threshold current oscillates and as the time delay increases, the maximum achievable threshold decreases according to a power law. The functional dependence of the maximally achieved threshold current (represented by black markers in Fig. 7.8) as a function of the time delay is shown in the log-log plot in Fig. 7.9. The data is fit with a straight line of slope 175 −0.93. Thus, for the parameters used in these studies, the maximum threshold
FIG. 7.8: Threshold current versus the feedback time delay. As the time delay gets longer the maximum achievable threshold decreases according to a power law (see Fig. 7.9). Black circles mark the highest threshold current for each period of the oscillation. possible by implementing a bunch-by-bunch feedback system scales as t −0.93 d . Figure 7.10 shows the threshold current as a function of the feedback gain for several different values of time delay. The values of td were chosen such that they correspond to the maximum achievable threshold current (i.e. the black markers in Fig. 7.8). In the region for which the analytic model is valid (g < 0.35) the simulations show excellent agreement, save for the case of the longest time delay (423 µs) where the perturbative treatment of the problem begins to fail. From a practical point of view, for td < 423 µs and for g = 1, the threshold current can be increased with a feedback system. Not surprisingly, the best suppression occurs when td = 0 and the feedback is truly on a bunch by bunch basis. To achieve an order of magnitude increase in the threshold current, from 2.1 mA to 21 mA, requires a feedback time delay of less than 30 µs. Conversely, for delays greater 176 than 423 µs, the threshold current becomes completely ineffective, independent of
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STUDIES OF ENERGY RECOVERY LINACS A
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DEDICATION To my wife Danielle and
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2 CEBAF with Energy Recovery . . .
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5.2 HOM Power . . . . . . . . . . .
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ACKNOWLEDGMENTS First and foremost
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6.1 Summary of the measured effects
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2.10 Illustration of quadrupole sca
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5.1 Successive frames in time (prog
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6.8 A plot of 1/Qeff versus average
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ABSTRACT An energy recovering linac
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CHAPTER 1 Introduction An increasin
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FIG. 1.1: Schematic of a generic li
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FIG. 1.2: A CEBAF 5-cell cavity wit
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The solution to Eq. (1.3) is U(t) =
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y reducing the impedance of HOMs, a
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Despite its success, this method of
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design parameters, most notably ach
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1.4.2 Machine Optics The second cat
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analytic model elucidates many impo
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CHAPTER 2 CEBAF with Energy Recover
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FIG. 2.1: Energy versus average cur
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FIG. 2.3: Additional hardware insta
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FIG. 2.4: A picture of the energy r
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dipoles and beam diagnostics such a
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FIG. 2.7: Horizontal (red) and vert
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FIG. 2.8: Illustration of the cryom
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linac and θNL is the RF phase. The
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2.4 Transverse Emittance One of the
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where σ2 is the rms beam size meas
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eams. The effects of varying the qu
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FIG. 2.12: A typical wire scan near
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quadratic fit and a multiple regres
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ting the data is difficult. Without
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primary source of error is measurin
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identified, although the phase dela
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TABLE 2.3: Comparison of Twiss para
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the results of the fits. The vertic
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FIG. 2.18: Schematic illustrating t
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FIG. 2.19: The GASK signal measured
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FIG. 2.20: The measured normalized
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CHAPTER 3 The Jefferson Laboratory
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FIG. 3.1: Schematic of the 10 kW FE
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FIG. 3.2: Layout of the DC photocat
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accelerating gradient at the front
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eason for making the endloops achro
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FIG. 3.7: Illustration of path leng
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3.5 Longitudinal Dynamics This sect
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FIG. 3.9: The effect of a thin focu
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Under the constraint that each orde
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form of beam breakup not only occur
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4.1 The Pillbox Cavity Although the
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FIG. 4.2: Electric field (red) and
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where the full 4×4 transfer matrix
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The threshold is inversely proporti
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4.3 BBU Simulation Codes: Particle
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6. The second pass beam bunch then
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which excites it. The BBU instabili
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Equation (4.41) is a dispersion rel
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FIG. 4.4: Output from MATBBU showin
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FIG. 4.5: Setup for measuring cavit
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Consequently, depending on the exte
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The projection of the beam displace
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TABLE 4.1: Experimental measurement
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FIG. 4.10: A plot showing the effec
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these cryomodules. Modes from these
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CHAPTER 5 Experimental Measurements
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threshold current - preferably with
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occurred at approximately 2 mA of a
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FIG. 5.5: FFT of a pure 2106.007 MH
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FIG. 5.6: Illustration to show the
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5.4 Measuring the Threshold Current
Page 143 and 144: for the HOM-beam system and is deri
Page 145 and 146: FIG. 5.10: Schematic of the experim
Page 147 and 148: FIG. 5.12: A plot of 1/Qeff versus
Page 149 and 150: measured HOMs in zone 3, a BTF meas
Page 151 and 152: FIG. 5.16: HOM voltage measured fro
Page 153 and 154: FIG. 5.18: A plot of the three valu
Page 155 and 156: the beam’s response in regions wh
Page 157 and 158: CHAPTER 6 BBU Suppression: Beam Opt
Page 159 and 160: FIG. 6.1: Schematic of a FODO cell
Page 161 and 162: plane [85]. Equations (6.7) and (6.
Page 163 and 164: 6.2.3 Discussion The method of poin
Page 165 and 166: FIG. 6.3: Beam envelopes (horizonta
Page 167 and 168: FIG. 6.6: Beam position monitor rea
Page 169 and 170: FIG. 6.8: A plot of 1/Qeff versus a
Page 171 and 172: ⎛ ⎞ ⎜ ⎝ 0 0 0 0 0 −1/K 0
Page 173 and 174: FIG. 6.11: A plot of 1/Qeff versus
Page 175 and 176: FIG. 6.12: Threshold current for no
Page 177 and 178: FIG. 6.14: Threshold current utiliz
Page 179 and 180: TABLE 6.1: Summary of the measured
Page 181 and 182: CHAPTER 7 BBU Suppression: Feedback
Page 183 and 184: FIG. 7.1: A schematic of the feedba
Page 185 and 186: FIG. 7.4: A coaxial 3-stub tuner us
Page 187 and 188: All of these considerations are con
Page 189 and 190: FIG. 7.6: Generic layout for a feed
Page 191 and 192: in Section 4.2.1, however, in the p
Page 193: FIG. 7.7: The threshold current as
Page 197 and 198: FIG. 7.10: The threshold current as
Page 199 and 200: CHAPTER 8 Conclusions The work pres
Page 201 and 202: le were experimentally measured. Du
Page 203 and 204: APPENDIX A The Pillbox Cavity Start
Page 205 and 206: FIG. A.1: A pillbox cavity exhibiti
Page 207 and 208: Ez(ρ, φ) = ψ(ρ, φ) = E0Jm(γρ
Page 209 and 210: FIG. B.1: Relationship of the S-par
Page 211 and 212: FIG. C.1: Impedance and frequency o
Page 213 and 214: FIG. C.5: Impedance and frequency o
Page 215 and 216: BIBLIOGRAPHY [1] M. Tigner, Nuovo C
Page 217 and 218: [22] L. Merminga, in Proceedings of
Page 219 and 220: [50] C. Hernandez-Garcia et al., in
Page 221 and 222: [79] L. Merminga et al., in Proceed
Page 223 and 224: VITA Christopher D. Tennant Christo
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