IMPROVING THE DESIGN AND ANALYSIS OF SUPERCONDUCTING MAGNETS FOR PARTICLE ACCELERATORS
Thesis by Ramesh Gupta
CONTENTS (in pdf)
1. REVIEW OF THE FIELD . . . . . (in pdf) . . . . . . . . . . . 1
1.1. Introduction . . . . . . . . . . . . . . . . . . . . . ………… . . . . . . . . 1
1.2. Physics Potentials and Goals of RHIC . . . . ………… . . . . . 2
1.3. Overview of RHIC Machine . . . . . . . . . . . . ………… . . . . 5
1.4. Superconducting Magnets . . . . . . . . . . . . . . . ………… . . . 8
1.4.1. Introduction to the Magnet Geometry . . . . . . . . . . 8
1.4.2. Superconducting Cable . . . . . . . . . . . . . . . . . . . . 10
1.4.3. Cryogenic System . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.4. Mechanical Design . . . . . . . . . . . . . . . .. . . . . . . . 14
1.4.5. Magnetic Design . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.6. Magnet Construction . . . . . . . . . . . . . . . . . . . . . . 16
1.4.7. Magnet Measurements . . . . . . . . . . . . . . . . . . . . 17
1.5. Magnetic Field Analysis in Accelerator Magnets . . . (in pdf) . . . 22
1.5.1. Basic Electromagnetic Field Equations . .. . . . . . . . . . 22
1.5.2. Field Harmonic Definitions . . . . . . . . . . . . . . . . . . . . 26
1.5.3. Analytic Expressions for Accelerator Magnets . . . . . 31
1.5.3.1. Field and Vector Potential due to a Line Current . . . . .. . . 31
1.5.3.2. Line Current in a Cylindrical Iron Cavity . . . . . . ….. . . .. . 36
1.5.3.3. Line Current in a Cylindrical Iron Shell . . . . . . . . . . …… . 40
1.5.3.4. Field and Harmonics due to Current Blocks in Air . . . . . . 44
1.5.3.5. Field Harmonics due to Current Blocks in a Cylindrical Iron Shell . . . 47
1.5.3.6. COS(mq) Current Distribution for Ideal Fields . . . ………………. . . . . 48
1.5.3.7. COS(mq) Current Distribution in a Cylindrical Iron Shell . . ………… . 54
1.5.3.8. Intersecting Circles with a Constant Current Density for Ideal Fields…57
1.5.4. Complex Variable Method in 2d Magnetic Field Calculations …… 60
1.5.4.1. Field due to an array of Line Currents . . . . …… . . . . . . . 62
1.5.4.2. Beth’s Current Sheet Theorem . . . . . . . . . . . ……… . . . . 63
1.5.4.3. Example — Cos(mq) current distribution . . . . . …… . . . . 65
1.6. Methods Investigated for Improving Field Quality . . . . . ………….. . . . . . . 66
1.6.1. Improvements in the Computational and Analysis Methods …… . 66
1.6.2. Field Quality Improvements through Yoke Design . . …. . . . . . . . 67
1.6.3. Field Quality Improvements through Coil Design . . . . …… . . . . . 67
1.6.4. Field Quality Improvements after Construction . . . . . ……. . . . . . 68
1.6.5. Optimized Cross section Designs . . . . . . . …………. . . . . . . . . . . 69
2. IMPROVEMENTS IN THE COMPUTATIONAL AND ANALYSIS METHODS …(in pdf)… 70
2.1. Introduction . . . . . . . . . . . . …………………………… . . . . . . . . . . . . . . . . . 70
2.2. Computer Aided Crosssection Measurement and Analysis . . . . ……. . . . 71
2.3. IMPROVEMENTS IN THE POISSON GROUP CODES . . . … . . . . . 78
2.3.1. Upgraded AUTOMESH — Input Method No. 1 . . . . … . . . . . . 79
2.3.2. Upgraded AUTOMESH — Input Method No. 2 . . . . . … . . . . . 83
2.3.3. Upgraded AUTOMESH — Input Method No. 3 . . . . . . … . . . . 88
2.4. Conclusions on the Improvements in the Computational and Analysis Methods . . . . . . 91
3. FIELD QUALITY IMPROVEMENTS THROUGH YOKE DESIGN . . . (in pdf) . . . 92
3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.2. Reduction in Saturation Induced Allowed Harmonics . . . . . . . . . . . 102
3.2.1. Varying the yoke inner radius . . . . . . . . . . . . . . . . . . . 103
3.2.2. Varying the yoke outer radius . . . . . . . . . . . . . . . . . . . 111
3.2.3. Varying the location of the helium bypass hole in the yoke . . . . . 116
3.2.4. Additional Saturation suppressor holes in the iron yoke . . . . . . . 123
3.2.5. Yokeyoke alignment keys . . . . . . . . . . . . . . . . . . . . . 129
3.2.6. Yoke collaring keys . . . . . . . . . . . . . . . . . . . . . . . . 132
3.2.7. Tooth at the midplane of the yoke aperture . . . . . . . . . . . . . 135
3.2.8. Cutout or Bump in the iron aperture . . . . . . . . . . . . . . . . 139
3.2.9. Elliptical iron aperture . . . . . . . . . . . . . . . . . . . . . . 145
3.2.10. Two radius aperture yoke . . . . . . . . . . . . . . . . . . . . . 149
3.3. Saturation Induced Allowed Harmonics in RHIC Arc Dipoles . . . . . . . 153
3.4. Reduction in the Saturation-induced Non-allowed Harmonics . . . . . . . 169
3.4.1. b1 saturation — Cross talk . . . . . . . . . . . . . . . . . . . . . 170
3.4.2. a1 saturation — Cryostat and other sources . . . . . . . . . . . . 176
3.5. a1 Saturation in SSC Dipole Magnets . . . . . . . . . . . . . . . . . . 182
3.5.1. da1 variation with axial position within a magnet . . . . . . . . . . 182
3.5.2. Magnet to magnet variations in the integral da1 . . . . . . . . . . 185
3.5.3. Compensation of the saturation induced a1 in SSC magnets . . . . . 188
3.6. a1 Saturation in RHIC Dipole Magnets . . . . . . . . . . . . . . . . . 190
3.6.1. Magnet to magnet variation in a1 saturation . . . . . . . . . . . . 190
3.6.2. Reduction in a1 saturation in RHIC dipoles . . . . . . . . . . . . 194
3.7. Conclusions on the Field Quality Improvements through Yoke Design . . . 200
4. FIELD QUALITY IMPROVEMENTS THROUGH COIL DESIGN . . . (in pdf) . . . 206
4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
4.2. Sources of Harmonics Allowed by the Magnet Geometry . . . . . . . . . 209
4.3. Reduction in the Allowed Harmonics through Wedges . . . . . . . . . . 210
4.4. Reduction in the Allowed Harmonics in RHIC Arc Dipoles by Changing the Midplane Gap . .212
4.5. Reduction in b3 in RHIC Quadrupoles with Midplane Gaps . . . . . . . . 214
4.6. Coil Cross-section Iterations without Changing Wedges . . . . . . . . . . 216
4.7. Conclusions on the Field Quality Improvements through Coil Design . . . . 219
5. FIELD QUALITY IMPROVEMENTS AFTER CONSTRUCTION . . . (in pdf) . . . 220
5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
5.2. Tuning Shims in Magnet Body for Extra High Field Quality . . . (in pdf) . . . 222
5.2.1. Tuning Shims in the RHIC Interaction Region Quadrupoles . .. . . . 222
5.2.2. Tuning Shim and the Magnet Design . . . . . . . . . . . . . . . . 224
5.2.3. Procedure for Implementing the Tuning Shim Correction . . . . . . 227
5.2.4. Calculations for Tuning Shim Corrections . . . . . . . . . . . . . 229
5.2.4.1. Approximate Analytic Expressions for Low Field Estimate . . . . . . 230
5.2.4.2. Numerical Calculations for Low Field Correction . . . . . . . 233
5.2.4.3. Numerical Calculations for High Field Correction . . . . . . . 243
5.2.5. Symmetries in the Harmonics Generated by Tuning Shims . . . . . . . . . 249
5.2.6. Independent and Coupled Changes in Harmonics Correction . . . . . . 250
5.2.7. Comparison with the Measurements . . . . . . . . . . . . . . . . 250
5.2.8. Tuning Shim Correction Vs. External Correctors . . . . . . . . . . 255
5.3. Tuning Yoke Length at Magnet Ends for Field Correction . . . . . . . . . 258
5.3.1. Yoke Length for Integral a1 Correction . . . . . . . . . . . . . . . 260
5.3.2. Yoke Length for Integral Transfer Function Correction . . . . . . . 262
5.4. Conclusions on the Field Quality Improvements after Construction . . . 264
6. OPTIMIZED CROSS SECTION DESIGNS . . . . . . . . (in pdf) . . . . . . . 265
6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
6.2. SSC 50 mm Aperture Collider Dipole Magnet Crosssection . . . (in pdf) . . . . 266
6.2.1. Coil Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
6.2.2. Low Field Harmonics . . . . . . . . . . . . . . . . . . . . . . . 268
6.2.3. Iron Yoke Design . . . . . . . . . . . . . . . . . . . . . . . . . 270
6.2.4. Expected Quench Performance . . . . . . . . . . . . . . . . . . . 278
6.2.5. Effect of Manufacturing Errors on the Allowed Harmononics . . . 280
6.2.6. Stored Energy and Inductance Calculations . . . . . . . . . . . . . 282
6.2.7. Lorentz Force Calculations . . . . . . . . . . . . . . . . . . . . 283
6.2.8. Summary of the Design . . . . . . . . . . . . . . . . . . . . . . 284
6.3. RHIC 130 mm Aperture Interaction Region Quadrupole Cross-section . . . (in pdf) . . . 286
6.3.1. Basic Construction . . . . . . . . . . . . . . . . . . . . . . . . 286
6.3.2. Coil Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . 287
6.3.3. Yoke Cross Section . . . . . . . . . . . . . . . . . . . . . . . . 293
6.3.4. Expected Quench Performance . . . . . . . . . . . . . . . . . . . 300
6.4. Conclusions on the Optimized Cross Section Designs . . . . . . . . . . . 301
7. CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK . . . (in pdf) . . . 302
References . . . . . . . (in pdf) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
List of Figures . . . . (in pdf) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
List of Tables . . . . . (in pdf) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
COPIES OF PUBLICATIONS . . . .(link to a few papers). . . . . 332
