SLAC Magnetic Measurements Date: 07-12-1995 Time: 09:13:03 Magnet Name (Serial #): 089 Bar Code Number: 116 Project: PEP II High Energy Ring Sextupoles Test Stand: IR8, S1 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: 07-12-1995 Time: 09:43:26 Magnet Name: 089 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 25.63964 +- 1.86386E-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 0.00004 0.00000 -105.83 0.65 1.1116 0.0043 2 0.00003 0.00000 -83.64 0.13 0.8133 0.0027 3 0.00369 0.00000 30.27 0.00 100.0000 0.0346 4 0.00000 0.00000 -8.45 0.50 0.1131 0.0039 5 0.00001 0.00000 -17.43 0.81 0.1354 0.0020 6 0.00000 0.00000 11.25 9.73 0.0102 0.0070 7 0.00000 0.00000 -4.17 13.48 0.0591 0.0273 8 0.00000 0.00000 3.55 5.92 0.0183 0.0146 9 0.00001 0.00000 -10.11 0.32 0.3330 0.0568 10 0.00000 0.00000 -1.14 10.16 0.0145 0.0117 11 0.00000 0.00000 -4.50 10.30 0.0048 0.0021 12 0.00000 0.00000 1.53 7.09 0.0079 0.0031 13 0.00000 0.00000 3.13 6.59 0.0100 0.0074 14 0.00000 0.00000 -5.66 3.83 0.0094 0.0055 15 0.00000 0.00000 -5.73 0.81 0.0745 0.0015 16 0.00000 0.00000 -6.19 3.70 0.0235 0.0170 The magnetic center is at X = 40.2 +- 0.8 microns Y = 178.0 +- 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: 07-12-1995 Time: 09:46:58 Magnet Name: 089 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 50.61993 +- 5.805143E-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 0.00000 -101.52 0.28 0.7151 0.0050 2 0.00006 0.00000 -84.91 0.09 0.8184 0.0142 3 0.00722 0.00000 30.27 0.00 100.0000 0.0767 4 0.00001 0.00000 -5.41 0.79 0.1282 0.0096 5 0.00001 0.00000 -17.73 0.34 0.1487 0.0044 6 0.00000 0.00000 3.52 6.36 0.0115 0.0048 7 0.00000 0.00000 -15.19 4.26 0.0393 0.0144 8 0.00000 0.00000 2.27 2.20 0.0115 0.0040 9 0.00003 0.00000 -9.85 0.56 0.3542 0.0117 10 0.00000 0.00000 9.38 2.45 0.0040 0.0029 11 0.00000 0.00000 3.40 4.08 0.0067 0.0020 12 0.00000 0.00000 -5.75 6.62 0.0047 0.0024 13 0.00000 0.00000 -3.27 6.22 0.0063 0.0033 14 0.00000 0.00000 6.53 2.87 0.0073 0.0018 15 0.00001 0.00000 -5.86 0.34 0.0733 0.0051 16 0.00000 0.00000 1.32 7.04 0.0078 0.0028 The magnetic center is at X = 32.5 +- 0.8 microns Y = 180.8 +- 3.1 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 ^ | N S S --> x N N S Date: 07-12-1995 Time: 09:50:34 Magnet Name: 089 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 100.6077 +- 1.188822E-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 -96.84 0.22 0.5262 0.0014 2 0.00012 0.00000 -84.90 0.07 0.8105 0.0048 3 0.01442 0.00002 30.27 0.00 100.0000 0.2183 4 0.00002 0.00000 -6.53 0.29 0.1180 0.0011 5 0.00002 0.00000 -17.57 0.15 0.1573 0.0055 6 0.00000 0.00000 9.06 2.19 0.0097 0.0022 7 0.00001 0.00000 -12.62 0.45 0.0365 0.0047 8 0.00000 0.00000 3.15 2.60 0.0114 0.0023 9 0.00007 0.00000 -9.34 0.08 0.4897 0.0040 10 0.00000 0.00000 12.01 2.02 0.0057 0.0026 11 0.00000 0.00000 -1.29 8.46 0.0054 0.0012 12 0.00000 0.00000 8.01 5.56 0.0017 0.0008 13 0.00000 0.00000 -5.02 7.26 0.0028 0.0012 14 0.00000 0.00000 5.66 0.62 0.0063 0.0024 15 0.00001 0.00000 -5.81 0.07 0.0653 0.0013 16 0.00000 0.00000 8.89 1.53 0.0030 0.0017 The magnetic center is at X = 32.2 +- 0.5 microns Y = 179.0 +- 1.1 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 ^ | N S S --> x N N S Date: 07-12-1995 Time: 09:57:31 Magnet Name: 089 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 150.5911 +- 4.195793E-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 0.00010 0.00000 -94.71 0.11 0.4717 0.0012 2 0.00017 0.00000 -85.42 0.08 0.7984 0.0011 3 0.02171 0.00001 30.27 0.00 100.0000 0.0488 4 0.00002 0.00000 -4.88 0.13 0.1118 0.0013 5 0.00004 0.00000 -17.33 0.14 0.1710 0.0011 6 0.00001 0.00000 -21.93 0.84 0.0272 0.0023 7 0.00001 0.00000 -12.51 0.41 0.0377 0.0018 8 0.00000 0.00000 1.50 0.59 0.0134 0.0010 9 0.00014 0.00000 -9.12 0.03 0.6346 0.0016 10 0.00000 0.00000 -0.36 17.13 0.0050 0.0024 11 0.00000 0.00000 -11.22 3.30 0.0017 0.0006 12 0.00000 0.00000 2.85 0.30 0.0125 0.0014 13 0.00000 0.00000 0.15 4.28 0.0020 0.0013 14 0.00000 0.00000 0.85 7.03 0.0033 0.0014 15 0.00002 0.00000 -5.68 0.04 0.0726 0.0005 16 0.00000 0.00000 -3.56 1.97 0.0033 0.0013 The magnetic center is at X = 28.5 +- 0.5 microns Y = 176.9 +- 0.3 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: 07-12-1995 Time: 10:04:28 Magnet Name: 089 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 200.5874 +- 3.449735E-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 -93.69 0.20 0.4194 0.0025 2 0.00025 0.00000 -85.31 0.19 0.7892 0.0090 3 0.03223 0.00004 30.21 0.00 100.0000 0.1776 4 0.00004 0.00000 -6.18 0.72 0.1125 0.0033 5 0.00006 0.00000 -17.70 0.19 0.1719 0.0036 6 0.00000 0.00000 3.17 15.29 0.0103 0.0052 7 0.00001 0.00000 -13.31 0.71 0.0320 0.0043 8 0.00001 0.00000 0.58 0.66 0.0233 0.0025 9 0.00030 0.00001 -9.98 0.08 0.9408 0.0387 10 0.00000 0.00000 1.70 14.57 0.0058 0.0021 11 0.00000 0.00000 10.86 2.70 0.0034 0.0012 12 0.00000 0.00000 5.26 11.33 0.0052 0.0005 13 0.00000 0.00000 0.10 9.73 0.0038 0.0013 14 0.00000 0.00000 5.21 1.79 0.0062 0.0016 15 0.00002 0.00000 -6.51 0.54 0.0759 0.0069 16 0.00000 0.00000 -2.89 2.90 0.0033 0.0015 The magnetic center is at X = 28.8 +- 1.2 microns Y = 174.7 +- 2.0 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)