STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA
STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA
with Eq. (7.4). It follows that KF B ≡ etoωk(R/Q)M eff 2(c/e)pb where M eff takes the place of M ∗ in Eq. (4.37) and is given by 171 (7.5) M eff = M34 + gM p 34M k 34e iωtd (7.6) After explicitly performing the sum, a modified dispersion relation is found 1 Io = KF Be iΩTr ξ sin(ωto) 1 − 2ξ cos(ωto) + ξ2 (7.7) where ξ is given by Eq. (4.42). After assuming a perturbative solution and expanding in Ω, the threshold current is found to be where 2Vb Ith = − (7.8) k(R/Q)QLM F B M F B = M34e ωTr 2Q L sin(ωTr) + gM p 34M k 34e ω(Tr+t d ) 2Q L sin(ω(Tr + td)) (7.9) As expected, with no feedback g = 0, the threshold current derived in Section 4.2 (and using an alternative method in Section 4.4) is recovered. For a physically viable solution, the threshold current must be a positive quan- tity. This condition requires that M F B < 0 for Eq. (7.8) to be valid. Equation (7.8) is based on a perturbative treatment of the problem. Thus, for the perturbative solu- tions to be valid, the gain, g, must be less than |M34/M p 34M 34 k | for sin(ω(Tr +td)) > 0 and greater than |M34/M p 34M 34 k | for sin(ω(Tr + td)) < 0. When these conditions are not met, the system is said to be in a pseudo-stable regime, where the negative threshold current implies beam stability. As discussed
in Section 4.2.1, however, in the pseudo-stable regime the threshold can be on the order of Amperes. For the 10 mA FEL Upgrade, and even for the proposed 100 mA ERL-based drivers, this represents, for all practical purposes, a stable system. Because it is assumed that the pickup can generate a position signal for each beam bunch and likewise, that the kicker can impulsively kick each bunch indepen- dently, this represents an idealized model. In reality, the signal produced by a single bunch through a pickup-amplifier-kicker system will affect more than a single bunch [95]. This model does not take these effects into account, nevertheless, important insights can be gained about the performance of a feedback system. 7.2.3 BBU Code with Feedback The ultimate goal of the feedback system is to put the system in the pseudo- stable regime, effectively pushing the threshold current to several Amperes. While the analytic models provide insights into the behavior of the system in the regime where M F B < 0, the region of greatest interest is the pseudo-stable regime, for which the analytic model can offer no information. Therefore it is necessary to investigate this region with numerical methods using computer simulation codes. Initial studies were performed in 2003 by modifying the code TDBBU to in- clude a simple feedback system for the case of td = 0. Results from those simulations indicated that an unstable system could be stabilized by implementing such a feed- back [94]. A code to simulate beam dynamics in a two-pass machine for a cavity containing a single HOM which is assumed to be oriented either purely horizontally or vertically was developed. The code was written using Igor Pro so that generating input files, executing the code and post-run analysis could be performed with the same program. 172 The tracking algorithm is the same as described in Section 4.3.1 except that
- 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: FIG. 7.6: Generic layout for a feed
- 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
in Section 4.2.1, however, in the pseudo-stable regime the threshold can be on the<br />
order of Amperes. For the 10 mA FEL Upgrade, and even for the proposed 100 mA<br />
ERL-based drivers, this represents, for all practical purposes, a stable system.<br />
Because it is assumed that the pickup can generate a position signal for each<br />
beam bunch and likewise, that the kicker can impulsively kick each bunch indepen-<br />
dently, this represents an idealized model. In reality, the signal produced by a single<br />
bunch through a pickup-amplifier-kicker system will affect more than a single bunch<br />
[95]. This model does not take these effects into account, nevertheless, important<br />
insights can be gained about the performance of a feedback system.<br />
7.2.3 BBU Code with Feedback<br />
The ultimate goal of the feedback system is to put the system in the pseudo-<br />
stable regime, effectively pushing the threshold current to several Amperes. While<br />
the analytic models provide insights into the behavior of the system in the regime<br />
where M F B < 0, the region of greatest interest is the pseudo-stable regime, for which<br />
the analytic model can offer no information. Therefore it is necessary to investigate<br />
this region with numerical methods using computer simulation codes.<br />
Initial studies were performed in 2003 by modifying the code TDBBU to in-<br />
clude a simple feedback system for the case of td = 0. Results from those simulations<br />
indicated that an unstable system could be stabilized by implementing such a feed-<br />
back [94].<br />
A code to simulate beam dynamics in a two-pass machine for a cavity containing<br />
a single HOM which is assumed to be oriented either purely horizontally or vertically<br />
was developed. The code was written using Igor Pro so that generating input files,<br />
executing the code and post-run analysis could be performed with the same program.<br />
172<br />
The tracking algorithm is the same as described in Section 4.3.1 except that
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