STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA

More documents

Recommendations

Info

V (t) = Voe −iΩt and integrating over the delta function yields e −iΩt = where KIoe − ω 2Q L (t−Tr) 2i ℓ n=−∞ 95 (4.35) e iω(t−Tr) e n ω ω −i(Ω+ω) to 2Q −iω(t−Tr) n −i(Ω−ω) to L 2Q − e e L toωk(Rd/Qo)M K ≡ ∗ 2Vb and M ∗ is given by Eq. (4.19). The upper limit of the summation, ℓ, is given by ℓ = t − Tr Equation (4.36) takes the form of a geometrical series with ℓ n=−∞ ω z± = 2QL to e nz± = e (ℓ+1)z± e z± − 1 − i(Ω ± ω) (4.36) (4.37) (4.38) (4.39) (4.40) Explicitly summing the terms, and after a fair amount of algebra, the sum can be written in the compact form where 1 Io = Ke iΩTr ξ sin(ωto) 1 − 2ξ cos(ωto) + ξ2 (4.41) ξ ≡ e ωto 2Q L e −iΩto (4.42)
Equation (4.41) is a dispersion relation between Io and Ω which must, in general, be solved numerically [14, 72]. Consider a perturbative solution by treating the case when K ≪ 1. The frequency can be approximated to first-order in K by [67] 96 Ω = a + bK (4.43) Plugging Eq. (4.43) into Eq. (4.41) and expanding exponentials to first order in K yields a and b. The constants are given by It follows that Ω is given by a = − iω ∓ ω (4.44) 2QL b = ∓ 1 2to Ω = ∓ω − iω ∓ 2QL e ωTr 2Q L e ∓iωTr (4.45) Io e 2to ωTr 2QL e ∓iωTr K (4.46) The instability develops when the imaginary part of Ω goes to zero. That is, where Ith = − Im(Ω) = − ω 2QL 1 − Io 2Vb Ith k(Rd/Qo)QLM ∗ sin(ωTr)e ωTr 2Q L (4.47) (4.48) For the assumptions used in the derivation, namely that the change in the HOM voltage is negligible on the time scale of a single recirculation, the exponential in the denominator of Eq. (4.48) can be neglected and the result is in perfect agreement with Eq. (4.21) derived in Section 4.2.
Page 1 and 2:
STUDIES OF ENERGY RECOVERY LINACS A
Page 3 and 4:
DEDICATION To my wife Danielle and
Page 5 and 6:
2 CEBAF with Energy Recovery . . .
Page 7 and 8:
5.2 HOM Power . . . . . . . . . . .
Page 9 and 10:
ACKNOWLEDGMENTS First and foremost
Page 11 and 12:
6.1 Summary of the measured effects
Page 13 and 14:
2.10 Illustration of quadrupole sca
Page 15 and 16:
5.1 Successive frames in time (prog
Page 17 and 18:
6.8 A plot of 1/Qeff versus average
Page 19 and 20:
ABSTRACT An energy recovering linac
Page 21 and 22:
CHAPTER 1 Introduction An increasin
Page 23 and 24:
FIG. 1.1: Schematic of a generic li
Page 25 and 26:
FIG. 1.2: A CEBAF 5-cell cavity wit
Page 27 and 28:
The solution to Eq. (1.3) is U(t) =
Page 29 and 30:
y reducing the impedance of HOMs, a
Page 31 and 32:
Despite its success, this method of
Page 33 and 34:
design parameters, most notably ach
Page 35 and 36:
1.4.2 Machine Optics The second cat
Page 37 and 38:
analytic model elucidates many impo
Page 39 and 40:
CHAPTER 2 CEBAF with Energy Recover
Page 41 and 42:
FIG. 2.1: Energy versus average cur
Page 43 and 44:
FIG. 2.3: Additional hardware insta
Page 45 and 46:
FIG. 2.4: A picture of the energy r
Page 47 and 48:
dipoles and beam diagnostics such a
Page 49 and 50:
FIG. 2.7: Horizontal (red) and vert
Page 51 and 52:
FIG. 2.8: Illustration of the cryom
Page 53 and 54:
linac and θNL is the RF phase. The
Page 55 and 56:
2.4 Transverse Emittance One of the
Page 57 and 58:
where σ2 is the rms beam size meas
Page 59 and 60:
eams. The effects of varying the qu
Page 61 and 62:
FIG. 2.12: A typical wire scan near
Page 63 and 64: quadratic fit and a multiple regres
Page 65 and 66: ting the data is difficult. Without
Page 67 and 68: primary source of error is measurin
Page 69 and 70: identified, although the phase dela
Page 71 and 72: TABLE 2.3: Comparison of Twiss para
Page 73 and 74: the results of the fits. The vertic
Page 75 and 76: FIG. 2.18: Schematic illustrating t
Page 77 and 78: FIG. 2.19: The GASK signal measured
Page 79 and 80: FIG. 2.20: The measured normalized
Page 81 and 82: CHAPTER 3 The Jefferson Laboratory
Page 83 and 84: FIG. 3.1: Schematic of the 10 kW FE
Page 85 and 86: FIG. 3.2: Layout of the DC photocat
Page 87 and 88: accelerating gradient at the front
Page 89 and 90: eason for making the endloops achro
Page 91 and 92: FIG. 3.7: Illustration of path leng
Page 93 and 94: 3.5 Longitudinal Dynamics This sect
Page 95 and 96: FIG. 3.9: The effect of a thin focu
Page 97 and 98: Under the constraint that each orde
Page 99 and 100: form of beam breakup not only occur
Page 101 and 102: 4.1 The Pillbox Cavity Although the
Page 103 and 104: FIG. 4.2: Electric field (red) and
Page 105 and 106: where the full 4×4 transfer matrix
Page 107 and 108: The threshold is inversely proporti
Page 109 and 110: 4.3 BBU Simulation Codes: Particle
Page 111 and 112: 6. The second pass beam bunch then
Page 113: which excites it. The BBU instabili
Page 117 and 118: FIG. 4.4: Output from MATBBU showin
Page 119 and 120: FIG. 4.5: Setup for measuring cavit
Page 121 and 122: Consequently, depending on the exte
Page 123 and 124: The projection of the beam displace
Page 125 and 126: TABLE 4.1: Experimental measurement
Page 127 and 128: FIG. 4.10: A plot showing the effec
Page 129 and 130: these cryomodules. Modes from these
Page 131 and 132: CHAPTER 5 Experimental Measurements
Page 133 and 134: threshold current - preferably with
Page 135 and 136: occurred at approximately 2 mA of a
Page 137 and 138: FIG. 5.5: FFT of a pure 2106.007 MH
Page 139 and 140: FIG. 5.6: Illustration to show the
Page 141 and 142: 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 and 194:
FIG. 7.7: The threshold current as
Page 195 and 196:
FIG. 7.8: Threshold current versus
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
show all

STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA

Create successful ePaper yourself

Delete template?

Save as template?