SLAC Magnetic Measurements Date: 07-25-1995 Time: 12:48:28 Magnet Name (Serial #): 043 Bar Code Number: 019 Project: PEP II High Energy Ring Sextupoles Test Stand: IR8, S1, W Measurement Coil: # 1 Operator: pdr Run Number: 1 Comment: Standard measurement 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: 07-25-1995 Time: 13:18:44 Magnet Name: 043 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 25.57592 +- 1.193266E-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 0.00006 0.00000 -105.15 0.16 1.1318 0.0024 2 0.00008 0.00000 -64.97 0.11 1.5992 0.0043 3 0.00524 0.00000 30.09 0.00 100.0000 0.0600 4 0.00001 0.00000 -6.27 0.55 0.1054 0.0067 5 0.00001 0.00000 -17.00 0.28 0.1451 0.0031 6 0.00000 0.00000 3.23 6.17 0.0124 0.0079 7 0.00000 0.00000 -11.16 0.97 0.0350 0.0068 8 0.00000 0.00000 6.21 2.19 0.0227 0.0026 9 0.00002 0.00000 -9.84 0.12 0.3268 0.0078 10 0.00000 0.00000 9.12 3.61 0.0107 0.0018 11 0.00000 0.00000 -4.76 3.66 0.0101 0.0047 12 0.00000 0.00000 6.81 2.10 0.0094 0.0040 13 0.00000 0.00000 3.55 8.47 0.0091 0.0018 14 0.00000 0.00000 4.63 4.69 0.0111 0.0042 15 0.00000 0.00000 -5.89 0.22 0.0723 0.0116 16 0.00000 0.00000 1.41 3.18 0.0067 0.0051 The magnetic center is at X = 275.2 +- 1.1 microns Y = 230.4 +- 1.2 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 ^ | N S S --> x N N S Date: 07-25-1995 Time: 13:22:09 Magnet Name: 043 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 50.56212 +- 5.117189E-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 0.00008 0.00000 -101.82 0.32 0.7323 0.0002 2 0.00017 0.00000 -64.85 0.04 1.6208 0.0029 3 0.01025 0.00000 30.09 0.00 100.0000 0.0121 4 0.00001 0.00000 -7.50 0.68 0.1127 0.0038 5 0.00002 0.00000 -17.37 0.57 0.1634 0.0049 6 0.00000 0.00000 -3.20 5.29 0.0117 0.0056 7 0.00000 0.00000 -12.36 0.84 0.0407 0.0019 8 0.00000 0.00000 4.09 0.75 0.0186 0.0077 9 0.00003 0.00000 -9.88 0.08 0.3343 0.0058 10 0.00000 0.00000 0.65 10.96 0.0124 0.0028 11 0.00000 0.00000 -5.85 4.92 0.0124 0.0025 12 0.00000 0.00000 -2.61 5.69 0.0060 0.0009 13 0.00000 0.00000 0.25 8.37 0.0063 0.0028 14 0.00000 0.00000 3.51 5.53 0.0063 0.0030 15 0.00001 0.00000 -5.95 0.47 0.0761 0.0076 16 0.00000 0.00000 -4.28 3.97 0.0071 0.0028 The magnetic center is at X = 279.9 +- 0.6 microns Y = 232.3 +- 0.6 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 ^ | N S S --> x N N S Date: 07-25-1995 Time: 13:25:38 Magnet Name: 043 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 100.559 +- 1.56085E-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 0.00011 0.00000 -97.65 0.37 0.5197 0.0014 2 0.00033 0.00000 -64.57 0.03 1.6135 0.0035 3 0.02036 0.00000 30.09 0.00 100.0000 0.0149 4 0.00002 0.00000 -8.29 0.52 0.1108 0.0020 5 0.00003 0.00000 -17.21 0.27 0.1653 0.0033 6 0.00000 0.00000 -13.01 6.65 0.0045 0.0016 7 0.00001 0.00000 -11.60 0.63 0.0348 0.0017 8 0.00000 0.00000 3.44 1.05 0.0167 0.0019 9 0.00007 0.00000 -9.86 0.09 0.3308 0.0016 10 0.00000 0.00000 7.13 8.63 0.0037 0.0014 11 0.00000 0.00000 -6.43 2.68 0.0051 0.0028 12 0.00000 0.00000 -0.95 1.18 0.0132 0.0019 13 0.00000 0.00000 4.02 4.88 0.0095 0.0021 14 0.00000 0.00000 1.59 2.16 0.0077 0.0014 15 0.00001 0.00000 -5.73 0.21 0.0710 0.0074 16 0.00000 0.00000 0.06 2.11 0.0058 0.0014 The magnetic center is at X = 280.9 +- 0.7 microns Y = 228.5 +- 0.6 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 ^ | N S S --> x N N S Date: 07-25-1995 Time: 13:32:21 Magnet Name: 043 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 150.543 +- 2.498368E-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 0.00014 0.00000 -95.75 0.22 0.4630 0.0016 2 0.00049 0.00000 -64.46 0.16 1.6127 0.0062 3 0.03043 0.00001 30.09 0.00 100.0000 0.0289 4 0.00003 0.00000 -8.79 0.29 0.1073 0.0037 5 0.00005 0.00000 -17.15 0.12 0.1707 0.0019 6 0.00000 0.00000 -2.06 2.48 0.0097 0.0024 7 0.00001 0.00000 -11.86 0.36 0.0379 0.0021 8 0.00001 0.00000 3.91 1.12 0.0189 0.0025 9 0.00010 0.00000 -9.85 0.05 0.3314 0.0031 10 0.00000 0.00000 5.76 10.71 0.0040 0.0015 11 0.00000 0.00000 -5.33 2.16 0.0044 0.0021 12 0.00000 0.00000 -0.98 8.94 0.0031 0.0017 13 0.00000 0.00000 5.84 8.23 0.0040 0.0024 14 0.00000 0.00000 2.30 1.42 0.0080 0.0009 15 0.00002 0.00000 -5.85 0.12 0.0681 0.0022 16 0.00000 0.00000 3.02 2.56 0.0036 0.0016 The magnetic center is at X = 281.6 +- 1.6 microns Y = 227.3 +- 1.8 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 ^ | N S S --> x N N S Date: 07-25-1995 Time: 13:39:04 Magnet Name: 043 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 200.5478 +- 3.18304E-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 0.00017 0.00000 -94.94 0.40 0.4200 0.0016 2 0.00064 0.00000 -64.09 0.07 1.5969 0.0028 3 0.04038 0.00001 30.09 0.00 100.0000 0.0328 4 0.00004 0.00000 -9.93 0.24 0.0973 0.0029 5 0.00007 0.00000 -16.97 0.31 0.1701 0.0030 6 0.00000 0.00000 -1.90 4.46 0.0109 0.0039 7 0.00002 0.00000 -11.58 0.53 0.0387 0.0013 8 0.00001 0.00000 3.34 1.80 0.0200 0.0023 9 0.00013 0.00000 -9.78 0.07 0.3308 0.0046 10 0.00000 0.00000 -3.46 12.14 0.0067 0.0041 11 0.00000 0.00000 -4.27 3.46 0.0071 0.0015 12 0.00000 0.00000 -5.61 4.72 0.0051 0.0009 13 0.00000 0.00000 0.33 3.08 0.0065 0.0013 14 0.00000 0.00000 4.03 1.95 0.0058 0.0008 15 0.00003 0.00000 -5.86 0.20 0.0705 0.0056 16 0.00000 0.00000 -1.86 5.88 0.0039 0.0011 The magnetic center is at X = 281.7 +- 0.7 microns Y = 221.5 +- 0.8 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 ^ | N S S --> x N N S 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)