STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA
STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA
VR = ω2 2c Io fRF /h (Rd/Qo) yc,1 91 (4.25) where VR is the real component of the HOM voltage and yc,1 is the first pass dis- placement of the bunch through the cavity. The real component of the voltage corresponds to the electric field and is the means by which the beam bunch couples to the HOM. 3. The bunch is deflected by the HOM excited by the passage of previous bunches according to ∆y ′ c,1 = VI where VI is the imaginary component of the voltage and corresponds to the Vb (4.26) magnetic field. The location of the kick, depending on the simulation code and the manner in which the input file is configured, can occur immediately before or after the location of the accelerating cavity where the energy gain is implemented. After its passage, the bunch coordinates are stored in an array. The array contains all bunches present in the linac on the first pass and in the recirculation pass. 4. Before the arrival of a recirculated bunch, the HOM voltage decays according to ⎛ ⎜ ⎝ VR VI ⎞ ⎟ ⎠ t+dt ωdt − 2Q = e L ⎛ ⎜ ⎝ cos(ω dt) − sin(ω dt) sin(ω dt) cos(ω dt) ⎞ ⎛ ⎟ ⎜ ⎠ ⎝ VR VI ⎞ ⎟ ⎠ t (4.27) where dt is the time interval between an injected bunch into the linac and a re- circulated bunch. 5. The first pass beam is propagated from the cavity through the recirculator and back to the cavity according to a user-input transfer matrix.
6. The second pass beam bunch then induces a voltage in the cavity according to Eq. (4.25) where yc,1 is replaced by the second pass displacement. 7. The bunch coordinates and the HOM voltages (real and imaginary) are written to a data file for off-line analysis. 8. The cavity voltage is allowed to decay according to Eq. (4.27) where now dt re- presents the time interval between the recirculated bunch and the next injected bunch. Steps 2 through 8 are repeated until the simulation time exceeds the specified run time. The simulations also utilize a binary search algorithm to automatically search for the threshold current. 4.3.2 TDBBU The first particle tracking BBU code developed was written in FORTRAN and called TDBBU [70, 71]. The code can handle transverse coupled motion as well as arbitrary mode polarizations. However, until the derivation of the two- dimensional threshold current formula, Eq. (4.21), the effect of mode polarization was not investigated in any detail. The code was used extensively at Jefferson Laboratory to study BBU in the CEBAF accelerator and in the IR FEL Demo. 4.3.3 ERLBBU The most recent code at Jefferson Laboratory was written in C++ in 2005 and is called ERLBBU [65]. In addition to being able to handle coupled transverse optics and arbitrary mode polarizations, it runs an order of magnitude or more faster than either TDBBU or MATBBU (discussed in Section 4.5.1). ERLBBU was 92
- 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: 4.3 BBU Simulation Codes: Particle
- 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
6. The second pass beam bunch then induces a voltage in the cavity according to<br />
Eq. (4.25) where yc,1 is replaced by the second pass displacement.<br />
7. The bunch coordinates and the HOM voltages (real and imaginary) are written<br />
to a data file for off-line analysis.<br />
8. The cavity voltage is allowed to decay according to Eq. (4.27) where now dt re-<br />
presents the time interval between the recirculated bunch and the next injected<br />
bunch.<br />
Steps 2 through 8 are repeated until the simulation time exceeds the specified run<br />
time. The simulations also utilize a binary search algorithm to automatically search<br />
for the threshold current.<br />
4.3.2 TDBBU<br />
The first particle tracking BBU code developed was written in FORTRAN<br />
and called TDBBU [70, 71]. The code can handle transverse coupled motion as<br />
well as arbitrary mode polarizations. However, until the derivation of the two-<br />
dimensional threshold current formula, Eq. (4.21), the effect of mode polarization<br />
was not investigated in any detail. The code was used extensively at Jefferson<br />
Laboratory to study BBU in the CEBAF accelerator and in the IR FEL Demo.<br />
4.3.3 ERLBBU<br />
The most recent code at Jefferson Laboratory was written in C++ in 2005<br />
and is called ERLBBU [65]. In addition to being able to handle coupled transverse<br />
optics and arbitrary mode polarizations, it runs an order of magnitude or more<br />
faster than either TDBBU or M<strong>AT</strong>BBU (discussed in Section 4.5.1). ERLBBU was<br />
92
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