Beam Based Alignment Procedure with Local Bumps

Beam Based Alignment Procedure with local bumps

The applied method is based on the well known fact that if the strength of a single quadrupole q in the ring is changed, the resulting difference in the closed orbit y(s) is proportional to the original offset yq of the beam at q.

The equation for the resulting difference orbit is:

The difference orbit is thus given by the closed orbit formula for a single kick, but calculated with the perturbed optics including k(s).

Figure 1: Illustration of the beam-based alignment technique applied to the quadrupoles with adjacent BPMs

From the measured difference orbit the kick and thus yqcan be easily determined and compared to the nominal orbit ybpmin the BPM adjacent to the quadrupole, yielding the offset between BPM and quadrupole axis. The precision of the method is very much improved by taking difference orbit data for several local beam positions yqvaried with an orbit bump. The principle of the method is illustrated in Figure 1. The error of the nominal position ybpmfor which the beam goes through the centre of the quadrupole is then given by the resolution of the BPM system.

Simulation: Case-1, with fixed misalignment error:

In all the cases the quadrupole strength is increased by a factor of 10%. No reading errors for the Monitors. For BBA the of the difference orbit is considered for better accuracy. The monitors inside the bump area are excluded for the difference orbit. The difference between the orbits that of the bumped orbit and the bumped orbit with 10% increased quadrupole strength is considered.

Defocussing Quadrupole at South West

ONLY fixed misalignment of dx=200um, dy= -150um QD at SW. Sext. ON
QD Bump(mm) / MON.003(mm) / (mm) / ^2(mm^2) / A / B / h(um)
-0.70 / -0.804598 / 0.1632118 / 0.0266381 / 0.0254 / -0.010900 / 214.57
-0.60 / -0.6929825 / 0.1440576 / 0.0207526 / Bx / 6.400000
-0.50 / -0.5814152 / 0.1251815 / 0.0156704 / Dkl / 0.100000
-0.40 / -0.4698956 / 0.1065772 / 0.0113587
-0.30 / -0.3584226 / 0.0882389 / 0.0077861
-0.20 / -0.2469958 / 0.0701605 / 0.0049225
-0.10 / -0.1356144 / 0.0523364 / 0.0027391
0.10 / 0.0869494 / 0.0174385 / 0.0003041
0.20 / 0.1982571 / 0.0003162 / 0.0000001
0.30 / 0.3094529 / 0.0165227 / 0.000273
0.40 / 0.4206354 / 0.0331572 / 0.0010994
0.50 / 0.5317583 / 0.0495651 / 0.0024567
0.60 / 0.6428403 / 0.0657548 / 0.0043237
0.70 / 0.753882 / 0.0817294 / 0.0066797

Defocussing Quadrupole at South West

ONLY fixed misalignment of dx=200um, dy= -150um QD at SW. Sext. ON
QD Bump(mm) / MON.003(mm) / (mm) / ^2(mm^2) / A / B / h(um)
-0.70 / -0.6605692 / 0.076788 / 0.0058964 / 0.0226 / 0.006800 / -150.44
-0.60 / -0.5677196 / 0.0628236 / 0.0039468 / By / 23.750000
-0.50 / -0.4748884 / 0.0488621 / 0.0023875 / Dkl / 0.100000
-0.40 / -0.3820865 / 0.0349056 / 0.0012184
-0.30 / -0.2892429 / 0.0209428 / 0.0004386
-0.20 / -0.1964127 / 0.0069857 / 0.0000488
-0.10 / -0.1035879 / 0.0069785 / 0.0000487
0.10 / 0.0820637 / 0.0348956 / 0.0012177
0.20 / 0.1748925 / 0.0488559 / 0.0023869
0.30 / 0.2677183 / 0.0628164 / 0.0039459
0.40 / 0.3605609 / 0.0767796 / 0.0058951
0.50 / 0.4533643 / 0.0907375 / 0.0082333
0.60 / 0.5461947 / 0.1046996 / 0.010962
0.70 / 0.6390433 / 0.1186651 / 0.0140814

In this case a defocussing quadrupole at south west is misaligned with a fixed misalignment of 200 and -150m in the horizontal and vertical plane respectively, while all other magnets are not misaligned. The local bumps are varied with a step of 100m from -0.7mm to 0.7mm at the centre of the Qd using 4 correctors. The Qd strength is increased by 10% to take the difference orbit. On fitting a parabola the estimated misalignment for BPM to quadrupole is found to be 215 and -150m, which is a very good agreement. The difference orbit is almost zero at the centre of the quadrupole.

Simulation: Case-2, with fixed misalignment error:

Focusing Quadrupole at South West

ONLY fixed misalignment of dx=-150um, dy= 200um Q4A at SW. Sext. ON
Q4A Bump(mm) / MON.003(mm) / (mm) / ^2(mm^2) / A / B / h(um)
-0.70 / -0.7577623 / 0.1012522 / 0.010252 / 0.0275 / 0.008200 / -149.09
-0.60 / -0.6574532 / 0.0843594 / 0.0071165 / Bx / 6.800000
-0.50 / -0.5571708 / 0.0675107 / 0.0045577 / Dkl / 0.100000
-0.40 / -0.4569149 / 0.050707 / 0.0025712
-0.30 / -0.3566851 / 0.033947 / 0.0011524
-0.20 / -0.2564812 / 0.0172308 / 0.0002969
-0.10 / -0.1563028 / 0.0005477 / 0.0000003
0.10 / 0.0439782 / 0.0326665 / 0.0010671
0.20 / 0.1440813 / 0.0492148 / 0.0024221
0.30 / 0.24416 / 0.0657221 / 0.0043194
0.40 / 0.3441984 / 0.0821864 / 0.0067546
0.50 / 0.4442447 / 0.0986154 / 0.009725
0.60 / 0.5442513 / 0.1150022 / 0.0132255
0.70 / 0.6442343 / 0.1313499 / 0.0172528

Focusing Quadrupole at South West

ONLY fixed misalignment of dx=-150um, dy= 200um Q4A at SW. Sext. ON
Q4A Bump(mm) / MON.003(mm) / (mm) / ^2(mm^2) / A / B / h(um)
-0.70 / -0.7410897 / 0.0748679 / 0.0056052 / 0.0064 / -0.002500 / 195.31
-0.60 / -0.6354637 / 0.066447 / 0.0044152 / Bx / 6.800000
-0.50 / -0.5298412 / 0.0580267 / 0.0033671 / Dkl / 0.100000
-0.40 / -0.4242179 / 0.0496054 / 0.0024607
-0.30 / -0.3185944 / 0.0411849 / 0.0016962
-0.20 / -0.2129705 / 0.0327643 / 0.0010735
-0.10 / -0.107347 / 0.0243434 / 0.0005926
0.10 / 0.1039002 / 0.0075033 / 0.0000563
0.20 / 0.2095237 / 0.0009487 / 0.0000009
0.30 / 0.3151474 / 0.0093434 / 0.0000873
0.40 / 0.4207706 / 0.0177623 / 0.0003155
0.50 / 0.5263944 / 0.026184 / 0.0006856
0.60 / 0.6320176 / 0.0346049 / 0.0011975
0.70 / 0.7376402 / 0.0430256 / 0.0018512

In this case a focussing quadrupole at south west is misaligned with a fixed misalignment of -150 and 200m in the horizontal and vertical plane respectively, while all other magnets are not misaligned. The local bumps are varied with a step of 100m from -0.7mm to 0.7mm at the centre of the Qf using 4 correctors. The Qf strength is increased by 10% to take the difference orbit. On fitting a parabola the estimated misalignment for BPM to quadrupole is found to be -149 and 195m, which is a very good agreement. The difference orbit is almost zero at the centre of the quadrupole.

Simulation: Case-3, including all misalignment error in random:

Defocussing Quadrupole at South West

Random Number 81775600

QD Bump(mm) / MON.003(mm) / (mm) / ^2(mm^2) / A / B / h(um)
-0.70 / -0.8291337 / 0.1580047 / 0.0249655 / 0.023 / -0.009500 / 206.52
-0.60 / -0.7180654 / 0.1399543 / 0.0195872 / Bx / 6.400000
-0.50 / -0.6070369 / 0.122154 / 0.0149216 / Dkl / 0.100000
-0.40 / -0.4960481 / 0.1045988 / 0.0109409
-0.35 / -0.4405685 / 0.0959109 / 0.0091989
-0.30 / -0.3850985 / 0.0872829 / 0.0076183
-0.25 / -0.3296382 / 0.0787134 / 0.0061958
-0.20 / -0.2741875 / 0.0702019 / 0.0049283
-0.15 / -0.2187462 / 0.0617471 / 0.0038127
-0.10 / -0.1633144 / 0.0533498 / 0.0028462
-0.05 / -0.107892 / 0.0450089 / 0.0020258
0.05 / 0.0029251 / 0.0284921 / 0.0008118
0.10 / 0.0583199 / 0.0203175 / 0.0004128
0.15 / 0.1137056 / 0.0121943 / 0.0001487
0.20 / 0.1690822 / 0.0041231 / 0.000017
0.25 / 0.2244499 / 0.0038859 / 0.0000151
0.30 / 0.2798087 / 0.0118533 / 0.0001405
0.35 / 0.3351587 / 0.0197661 / 0.0003907
0.40 / 0.3904999 / 0.0276261 / 0.0007632
0.50 / 0.5011563 / 0.0431972 / 0.001866
0.60 / 0.6117783 / 0.0585679 / 0.0034302
0.70 / 0.7223664 / 0.0737442 / 0.0054382

All ERRORs (dx=175um, dy= 80um QD at SW). Sext. ON
QD Bump(mm) / MON.003(mm) / (mm) / ^2(mm^2) / A / B / h(um)
-0.70 / -0.6614974 / 0.1126046 / 0.0126798 / 0.0228 / -0.003800 / 83.33
-0.60 / -0.5686166 / 0.0985814 / 0.0097183 / By / 23.750000
-0.50 / -0.4757808 / 0.084566 / 0.0071514 / Dkl / 0.100000
-0.40 / -0.382877 / 0.0705415 / 0.0049761
-0.30 / -0.2900275 / 0.0565261 / 0.0031952
-0.20 / -0.1971466 / 0.04251 / 0.0018071
-0.10 / -0.1042798 / 0.0285061 / 0.0008126
0.10 / 0.0814242 / 0.0018708 / 0.0000035
0.20 / 0.1742676 / 0.0137441 / 0.0001889
0.30 / 0.267112 / 0.0277056 / 0.0007676
0.40 / 0.359954 / 0.0417085 / 0.0017396
0.50 / 0.4528047 / 0.0557252 / 0.0031053
0.60 / 0.5456078 / 0.0697395 / 0.0048636
0.70 / 0.638431 / 0.0837598 / 0.0070157

In this case a realistic situation is considered. All the magnets are misaligned and with random errors as generally it is. The QD is randomly misaligned with x=175m, and y=80m. On BBA study it is found out to 206m and 83m in horizontal and vertical plane respectively with bump amplitudes of ±0.7m at the centre of QD.