SLAC Magnetic Measurements Date: 07-13-1995 Time: 06:12:52 Magnet Name (Serial #): 119 Bar Code Number: 016 Project: PEP II High Energy Ring Sextupoles Test Stand: IR8, S1 Measurement Coil: # 1 Operator: bmd Run Number: 1 Comment: Standardization Currents (A): 250.0 10.0 200.0 10.0 200.0 10.0 200.0 10.0 Test Currents (A): 25.0 50.0 100.0 125.0 150.0 175.0 200.0 150.0 100.0 50.0 25.0 Harmonics Measurements Date: 07-13-1995 Time: 06:44:01 Magnet Name: 119 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 25.66416 +- 9.088842E-04 A Coil Radius, Rcoil = .044882 m Coil # Turns, Nturns = 20 Average, # Rotations/Measurement = 10 , # Measurements = 4 Nmain = 3 N BLn sigBLn THspole sigTH BLn/BLN sBLn/BLN (Tm) (Tm) (deg) (deg) (%) (%) --- --------+-------- --------+-------- --------+-------- 1 0.00003 0.00000 -110.84 1.80 0.7664 0.0226 2 0.00007 0.00000 29.01 0.23 1.8891 0.0359 3 0.00396 0.00002 -29.74 0.00 100.0000 0.8201 4 0.00000 0.00000 -1.56 40.94 0.1026 0.0182 5 0.00000 0.00000 24.29 1.75 0.1108 0.0267 6 0.00000 0.00000 8.38 20.63 0.0508 0.0140 7 0.00000 0.00000 10.34 4.54 0.0599 0.0105 8 0.00000 0.00000 -5.23 9.17 0.0397 0.0187 9 0.00001 0.00000 10.39 0.57 0.3128 0.0097 10 0.00000 0.00000 1.03 13.66 0.0323 0.0122 11 0.00000 0.00000 1.92 10.14 0.0342 0.0138 12 0.00000 0.00000 4.17 5.62 0.0310 0.0147 13 0.00000 0.00000 -2.90 4.81 0.0308 0.0124 14 0.00000 0.00000 -3.06 9.70 0.0381 0.0136 15 0.00000 0.00000 6.95 0.85 0.0815 0.0275 16 0.00000 0.00000 6.16 4.20 0.0256 0.0123 The magnetic center is at X = 359.6 +- 7.1 microns Y = 224.6 +- 5.1 microns The magnetic center is measured relative to the coil axis effectively at the longitudinal center of the magnet. Pitch and yaw are not seen by the coil. View from the non-lead end of the magnet: y ^ | S N N --> x S S N Date: 07-13-1995 Time: 06:47:33 Magnet Name: 119 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 50.64346 +- 1.141031E-03 A Coil Radius, Rcoil = .044882 m Coil # Turns, Nturns = 20 Average, # Rotations/Measurement = 10 , # Measurements = 4 Nmain = 3 N BLn sigBLn THspole sigTH BLn/BLN sBLn/BLN (Tm) (Tm) (deg) (deg) (%) (%) --- --------+-------- --------+-------- --------+-------- 1 0.00002 0.00000 -117.58 14.24 0.2442 0.0358 2 0.00014 0.00001 27.77 1.30 1.8156 0.0785 3 0.00784 0.00011 -29.74 0.01 100.0000 2.0381 4 0.00001 0.00001 -13.34 32.61 0.1768 0.0845 5 0.00001 0.00000 28.38 5.12 0.1589 0.0509 6 0.00001 0.00000 6.85 20.63 0.1068 0.0441 7 0.00001 0.00000 9.91 4.73 0.0996 0.0270 8 0.00001 0.00000 -1.68 10.01 0.0656 0.0392 9 0.00003 0.00000 11.04 0.82 0.3395 0.0606 10 0.00000 0.00000 8.82 11.20 0.0539 0.0318 11 0.00000 0.00000 9.22 4.17 0.0585 0.0301 12 0.00000 0.00000 2.58 5.03 0.0474 0.0252 13 0.00000 0.00000 -4.08 3.97 0.0515 0.0303 14 0.00000 0.00000 -3.18 9.05 0.0527 0.0288 15 0.00001 0.00000 6.32 0.77 0.0756 0.0467 16 0.00000 0.00000 4.22 3.82 0.0449 0.0336 The magnetic center is at X = 335.9 +- 17.9 microns Y = 230.6 +- 18.2 microns The magnetic center is measured relative to the coil axis effectively at the longitudinal center of the magnet. Pitch and yaw are not seen by the coil. View from the non-lead end of the magnet: y ^ | S N N --> x S S N Date: 07-13-1995 Time: 06:51:10 Magnet Name: 119 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 100.6304 +- 3.323799E-03 A Coil Radius, Rcoil = .044882 m Coil # Turns, Nturns = 20 Average, # Rotations/Measurement = 10 , # Measurements = 4 Nmain = 3 N BLn sigBLn THspole sigTH BLn/BLN sBLn/BLN (Tm) (Tm) (deg) (deg) (%) (%) --- --------+-------- --------+-------- --------+-------- 1 0.00001 0.00000 90.12 96.42 0.0788 0.0270 2 0.00030 0.00001 28.46 0.84 1.8594 0.0619 3 0.01599 0.00013 -29.73 0.02 100.0000 1.1671 4 0.00002 0.00001 17.64 32.12 0.1459 0.0378 5 0.00002 0.00000 21.08 8.53 0.1040 0.0260 6 0.00002 0.00001 2.93 23.79 0.1062 0.0344 7 0.00001 0.00000 5.63 17.69 0.0741 0.0299 8 0.00001 0.00000 -0.44 13.24 0.0609 0.0225 9 0.00017 0.00004 10.76 0.38 1.0464 0.2393 10 0.00001 0.00001 -4.41 9.05 0.0646 0.0410 11 0.00001 0.00001 -10.32 6.12 0.0462 0.0381 12 0.00000 0.00000 -2.86 8.81 0.0304 0.0305 13 0.00001 0.00000 3.48 4.29 0.0375 0.0192 14 0.00001 0.00000 -4.90 1.41 0.0436 0.0161 15 0.00002 0.00000 -4.75 8.96 0.1236 0.0117 16 0.00001 0.00000 0.11 8.64 0.0432 0.0267 The magnetic center is at X = 349.6 +- 13.4 microns Y = 227.7 +- 12.7 microns The magnetic center is measured relative to the coil axis effectively at the longitudinal center of the magnet. Pitch and yaw are not seen by the coil. View from the non-lead end of the magnet: y ^ | S N N --> x S S N Date: 07-13-1995 Time: 06:58:08 Magnet Name: 119 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 150.6065 +- 1.699113E-03 A Coil Radius, Rcoil = .044882 m Coil # Turns, Nturns = 20 Average, # Rotations/Measurement = 10 , # Measurements = 4 Nmain = 3 N BLn sigBLn THspole sigTH BLn/BLN sBLn/BLN (Tm) (Tm) (deg) (deg) (%) (%) --- --------+-------- --------+-------- --------+-------- 1 0.00007 0.00002 -3.01 62.38 0.3138 0.1007 2 0.00037 0.00003 25.53 2.90 1.6614 0.1558 3 0.02235 0.00043 -29.71 0.03 100.0000 2.7429 4 0.00007 0.00004 -25.09 17.11 0.2940 0.1684 5 0.00007 0.00002 -10.78 22.46 0.2972 0.0897 6 0.00004 0.00002 12.57 18.30 0.1615 0.0673 7 0.00006 0.00003 17.87 5.72 0.2471 0.1397 8 0.00004 0.00003 5.32 10.40 0.1599 0.1254 9 0.00011 0.00009 8.69 2.55 0.4728 0.4129 10 0.00003 0.00002 -5.52 5.86 0.1229 0.0961 11 0.00002 0.00001 -2.58 9.27 0.0998 0.0607 12 0.00002 0.00001 5.51 7.20 0.0923 0.0605 13 0.00003 0.00002 -0.69 4.15 0.1138 0.0872 14 0.00002 0.00001 -0.20 6.51 0.1034 0.0579 15 0.00007 0.00002 -2.29 8.07 0.3336 0.1116 16 0.00004 0.00003 8.53 3.90 0.1701 0.1462 The magnetic center is at X = 290.0 +- 36.1 microns Y = 234.3 +- 36.6 microns The magnetic center is measured relative to the coil axis effectively at the longitudinal center of the magnet. Pitch and yaw are not seen by the coil. View from the non-lead end of the magnet: y ^ | S N N --> x S S N Date: 07-13-1995 Time: 07:05:07 Magnet Name: 119 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 200.6004 +- 8.066619E-04 A Coil Radius, Rcoil = .044882 m Coil # Turns, Nturns = 20 Average, # Rotations/Measurement = 10 , # Measurements = 4 Nmain = 3 N BLn sigBLn THspole sigTH BLn/BLN sBLn/BLN (Tm) (Tm) (deg) (deg) (%) (%) --- --------+-------- --------+-------- --------+-------- 1 0.00005 0.00002 79.68 7.20 0.1591 0.0588 2 0.00055 0.00003 27.53 0.77 1.9090 0.1066 3 0.02874 0.00057 -29.71 0.01 100.0000 2.8076 4 0.00006 0.00002 -20.90 36.17 0.2055 0.0672 5 0.00004 0.00002 28.37 2.66 0.1454 0.0747 6 0.00003 0.00001 10.82 21.11 0.0952 0.0387 7 0.00003 0.00000 12.59 3.56 0.0898 0.0109 8 0.00002 0.00001 -5.56 5.41 0.0800 0.0275 9 0.00007 0.00003 4.49 12.21 0.2591 0.1040 10 0.00003 0.00001 -3.56 7.04 0.0935 0.0370 11 0.00003 0.00002 3.58 10.84 0.0928 0.0682 12 0.00002 0.00002 6.23 6.16 0.0790 0.0544 13 0.00002 0.00001 -0.79 4.80 0.0776 0.0453 14 0.00002 0.00002 2.61 5.28 0.0847 0.0664 15 0.00008 0.00002 -2.16 4.67 0.2899 0.0744 16 0.00002 0.00001 -2.50 9.08 0.0725 0.0364 The magnetic center is at X = 351.2 +- 20.7 microns Y = 245.3 +- 16.6 microns The magnetic center is measured relative to the coil axis effectively at the longitudinal center of the magnet. Pitch and yaw are not seen by the coil. View from the non-lead end of the magnet: y ^ | S N N --> x S S N SUMMARY OF THE CALCULATIONS AND CONVENTIONS USED Field Expansion: The expansion of the radial and azimuthal field in polar coordinates is Br(r,th) = Sum Bn (r/rref)^(n-1) cos(n(th-thspole)) Bth(r,th) = -Sum Bn (r/rref)^(n-1) sin(n(th-thspole)) Our convention is to set rref = Rcoil. Thspole is the angle of the first magnetic south pole with respect to the horizontal, measured ccw by a shaft encoder in the system. Coil Voltage: The coil voltage from each field harmonic is Vn(th) = Nturns * velocity * Brn(Rcoil,th) * L L is the magnet effective length, or BLn is the integrated field strength of the n'th harmonic. At the coil radius, the radial field as a function of angle is, Brn(Rcoil,th) * L = BLn * cos(n*(th - thspole)) The coil voltage is Vn(th) = Nturns * velocity * BLn * cos(n*(th - thspole)) = Nturns * Rcoil * ang_freq * BLn * cos(n*(th - THspole)) An FFT of the coil voltage gives Vn and PhiVn according to the formula Vn(i) = Vn * cos(n*2pi*i/N + PhiVn) Multipole Field Calculations: To find the multipole field magnitudes and phases, the measured voltage harmonics are related to their values calculated from the field harmonics: Nturns * Rcoil * ang_freq * BLn = Vn -n * thspole = PhiVn Or, BLn = Vn / (Nturns * Rcoil * ang_freq) thspole = -PhiVn / n Harmonic Strength Ratios: The main field, denoted by capital N, is the field harmonic with the largest strength at the coil radius. The field strength ratio is defined by Rn = BLn / BLN It gives the ratio of each harmonic field strength to the main field strength at the coil radius. Calculation Of SL: The sextupole strength S is defined by the vertical field on the x-axis (median plane). It is the quadratic term in the Taylor expansion. By(x) = 1/2 S x^2 On the x-axis, By(x) = Bth(r=x,th=0) From the expression for Bth above, By(x) = - B3 (x/rref)^2 sin(-3 thspole) Take rref = Rcoil, thspole = pi/6, then at x=Rcoil, By(Rcoil) = B3 = 1/2 S Rcoil^2 So, S = 2 B3 / Rcoil^2 The integrated sextupole strength is SL = 2 BL3 / Rcoil^2 Calculation Of The Sextupole Center: In the sextupole's frame, Bx' = S * x' * y' By' = 1/2 * S * (x'^2 - y'^2). In the coil's frame (unprimed frame) the magnetic center is at (x0, y0). In the coil's frame, Bx = S * (x - x0) * (y - y0) By = 1/2 * S * [(x - x0)^2 - (y - y0)^2]. The magnetic center can be found in terms of the measured quadrupole field. Compare Br and Btheta on the x-axis to Bxquad = -S * (x0 * y + y0 * x) Byquad = -S * (x0 * x - y0 * y). Evaluate at x = Rcoil, y = 0 and compare to Br(theta = 0) = Bx, Btheta(theta = 0) = By. In terms of the measured integrated strengths, this gives Xcenter = x0 = - (1/(SL * Rcoil)) * BL2 * sin(2 * THspole2) Ycenter = y0 = - (1/(SL * Rcoil)) * BL2 * cos(2 * THspole2)