STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA
STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA
FIG. 6.2: A plot of the threshold current versus the change in quadrupole strength showing the effect of point-to-point focusing. At approximately 250 G the change in phase advance makes the M34 element of the recirculation matrix from the cavity back to itself equal to zero before changing sign and leading to a negative threshold current. back to itself. For a quadrupole strength of approximately 250 G the phase advance from cavity 7 back to itself is equal to nπ, where n is an integer. Therefore M34 can be expressed as 143 M34 ∝ sin(∆ψ) = sin(nπ + δ) = (−1) n δ (6.9) where δ is the variation of the phase advance from nπ. It follows that for small variations, δ is proportional to the change of the quadrupole strength. For a change in the quadrupole strength of +300, the BTF measurements yielded a negative threshold. This indicates that M34 changed its sign and the product M ∗ sin(ωTr) is positive.
6.2.3 Discussion The method of point-to-point focusing proved to be a straightforward and effec- tive method to suppress BBU much of the time. This is due to the fact that there is only a single dangerous mode that prohibits operation with 10 mA of beam current (see Table 4.2). One of the attractive features of this method is that the beam optics remain decoupled transversely and that the beam envelopes are minimally effected. There are, however, some limitations to this method. For an extended linac containing many dangerous modes, it may not be advantageous to modify the phase advance. While one mode may be stabilized, in all likelihood the resulting change in phase advance will have harmful effects on other modes which were previously not a threat for BBU. Although this was never investigated in earnest, there is evidence to suggest that this situation occurred in the FEL when the threshold current became as low as 400 µA. Adequate suppression was generated for the HOMs in cavity 7, but the resulting change in phase advance caused an order of magnitude increase in the M34 element of the recirculation matrix from cavity 1 back to itself, from which BBU was facilitated at an even lower threshold current. Therefore care should be taken when applying this method to large-scale ERLs with extended linacs. 6.3 Local Reflection The idea behind implementing a local reflector is to map a BBU-induced vertical kick into the horizontal plane, and likewise to map a BBU-induced horizontal kick into the vertical plane. The transport matrix describing a reflection about a plane at 45 ◦ to the horizontal or vertical axis takes the following form, where each element represents a 2 × 2 matrix 144
- Page 111 and 112: 6. The second pass beam bunch then
- Page 113 and 114: which excites it. The BBU instabili
- Page 115 and 116: Equation (4.41) is a dispersion rel
- 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: plane [85]. Equations (6.7) and (6.
- 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
FIG. 6.2: A plot of the threshold current versus the change in quadrupole strength<br />
showing the effect of point-to-point focusing. At approximately 250 G the change in<br />
phase advance makes the M34 element of the recirculation matrix from the cavity back<br />
to itself equal to zero before changing sign and leading to a negative threshold current.<br />
back to itself. For a quadrupole strength of approximately 250 G the phase advance<br />
from cavity 7 back to itself is equal to nπ, where n is an integer. Therefore M34 can<br />
be expressed as<br />
143<br />
M34 ∝ sin(∆ψ) = sin(nπ + δ) = (−1) n δ (6.9)<br />
where δ is the variation of the phase advance from nπ. It follows that for small<br />
variations, δ is proportional to the change of the quadrupole strength. For a change<br />
in the quadrupole strength of +300, the BTF measurements yielded a negative<br />
threshold. This indicates that M34 changed its sign and the product M ∗ sin(ωTr) is<br />
positive.
Change language