SLAC Magnetic Measurements Date: 08-01-1995 Time: 09:31:29 Magnet Name (Serial #): 137 Bar Code Number: 133 Project: PEP II High Energy Ring Sextupoles Test Stand: IR8, S1, W Measurement Coil: # 1 Operator: bda 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: 08-01-1995 Time: 10:01:48 Magnet Name: 137 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 25.63777 +- 1.488897E-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 0.00001 -81.78 0.75 2.6677 0.1116 2 0.00005 0.00000 76.81 0.48 0.9178 0.0256 3 0.00527 0.00000 30.11 0.00 100.0000 0.0323 4 0.00000 0.00000 -9.02 3.20 0.0886 0.0190 5 0.00000 0.00000 -19.42 1.54 0.0934 0.0054 6 0.00000 0.00000 -12.97 11.62 0.0129 0.0078 7 0.00000 0.00000 -18.69 3.95 0.0469 0.0164 8 0.00000 0.00000 -11.80 4.49 0.0266 0.0154 9 0.00002 0.00000 -10.23 0.41 0.3285 0.0063 10 0.00000 0.00000 -3.38 8.45 0.0288 0.0220 11 0.00000 0.00000 1.47 10.07 0.0189 0.0098 12 0.00000 0.00000 -4.12 4.08 0.0401 0.0175 13 0.00000 0.00000 -5.65 6.07 0.0405 0.0089 14 0.00000 0.00000 -2.64 7.98 0.0310 0.0272 15 0.00000 0.00000 -6.55 1.33 0.0903 0.0197 16 0.00000 0.00000 0.34 7.69 0.0180 0.0089 The magnetic center is at X = -91.5 +- 4.0 microns Y = 184.5 +- 5.4 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: 08-01-1995 Time: 10:05:12 Magnet Name: 137 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 50.62315 +- 2.880188E-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 0.00003 -89.94 4.53 0.9973 0.2441 2 0.00009 0.00000 76.20 0.11 0.8833 0.0020 3 0.01029 0.00000 30.11 0.00 100.0000 0.0352 4 0.00001 0.00000 -8.66 0.83 0.0864 0.0125 5 0.00001 0.00000 -16.37 0.19 0.1278 0.0118 6 0.00000 0.00000 2.12 2.30 0.0112 0.0111 7 0.00000 0.00000 -12.27 0.57 0.0357 0.0120 8 0.00000 0.00000 -3.85 5.71 0.0148 0.0055 9 0.00003 0.00000 -9.96 0.04 0.3299 0.0014 10 0.00000 0.00000 -5.70 7.93 0.0132 0.0141 11 0.00000 0.00000 -3.05 7.21 0.0098 0.0046 12 0.00000 0.00000 4.68 10.50 0.0052 0.0038 13 0.00000 0.00000 8.26 5.54 0.0069 0.0042 14 0.00000 0.00000 -0.46 6.69 0.0073 0.0041 15 0.00001 0.00000 -5.07 1.63 0.0635 0.0204 16 0.00000 0.00000 -1.57 5.94 0.0053 0.0059 The magnetic center is at X = -91.9 +- 0.7 microns Y = 175.6 +- 0.5 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: 08-01-1995 Time: 10:08:42 Magnet Name: 137 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 100.6148 +- 1.693068E-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 0.00021 0.00000 -80.63 0.76 1.0249 0.0164 2 0.00018 0.00000 75.86 0.13 0.8951 0.0056 3 0.02043 0.00001 30.10 0.00 100.0000 0.0503 4 0.00002 0.00000 -9.16 1.33 0.0775 0.0024 5 0.00003 0.00000 -16.22 0.40 0.1367 0.0084 6 0.00000 0.00000 6.02 19.68 0.0070 0.0022 7 0.00001 0.00000 -11.36 0.58 0.0348 0.0048 8 0.00000 0.00000 -4.09 2.86 0.0160 0.0038 9 0.00007 0.00000 -9.90 0.05 0.3353 0.0040 10 0.00000 0.00000 2.37 11.26 0.0091 0.0025 11 0.00000 0.00000 -5.76 6.15 0.0073 0.0012 12 0.00000 0.00000 -1.15 2.63 0.0088 0.0049 13 0.00000 0.00000 -0.69 4.71 0.0061 0.0023 14 0.00000 0.00000 -1.28 2.06 0.0076 0.0030 15 0.00002 0.00000 -5.98 0.32 0.0743 0.0076 16 0.00000 0.00000 -0.55 5.40 0.0046 0.0022 The magnetic center is at X = -95.2 +- 1.0 microns Y = 176.9 +- 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: 08-01-1995 Time: 10:15:24 Magnet Name: 137 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 150.6049 +- 9.444675E-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 0.00022 0.00004 -83.20 2.93 0.7346 0.1184 2 0.00027 0.00000 75.81 0.22 0.8797 0.0040 3 0.03053 0.00001 30.10 0.00 100.0000 0.0244 4 0.00003 0.00000 -8.75 0.99 0.0876 0.0030 5 0.00004 0.00000 -16.91 0.41 0.1322 0.0028 6 0.00000 0.00000 -3.79 5.03 0.0075 0.0031 7 0.00001 0.00000 -12.87 1.25 0.0258 0.0054 8 0.00000 0.00000 -6.02 1.27 0.0136 0.0031 9 0.00010 0.00000 -9.93 0.02 0.3274 0.0010 10 0.00000 0.00000 6.04 9.31 0.0055 0.0020 11 0.00000 0.00000 7.28 1.90 0.0032 0.0007 12 0.00000 0.00000 4.46 10.31 0.0068 0.0028 13 0.00000 0.00000 2.21 9.08 0.0024 0.0007 14 0.00000 0.00000 -3.00 1.75 0.0071 0.0037 15 0.00002 0.00000 -5.99 0.22 0.0719 0.0021 16 0.00000 0.00000 0.20 3.30 0.0038 0.0024 The magnetic center is at X = -93.8 +- 1.4 microns Y = 173.7 +- 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: 08-01-1995 Time: 10:22:07 Magnet Name: 137 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 200.604 +- 3.237328E-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 0.00023 0.00003 -84.67 2.14 0.5725 0.0853 2 0.00036 0.00000 75.91 0.04 0.8778 0.0025 3 0.04054 0.00001 30.10 0.00 100.0000 0.0511 4 0.00003 0.00000 -8.94 0.41 0.0851 0.0040 5 0.00005 0.00000 -16.60 0.25 0.1300 0.0031 6 0.00000 0.00000 3.84 9.79 0.0042 0.0014 7 0.00001 0.00000 -12.00 1.40 0.0276 0.0027 8 0.00001 0.00000 -5.91 1.68 0.0127 0.0004 9 0.00013 0.00000 -9.91 0.03 0.3304 0.0010 10 0.00000 0.00000 -4.84 12.56 0.0049 0.0027 11 0.00000 0.00000 -2.23 4.03 0.0042 0.0022 12 0.00000 0.00000 -5.31 9.19 0.0039 0.0013 13 0.00000 0.00000 0.67 7.15 0.0047 0.0014 14 0.00000 0.00000 -3.78 3.48 0.0046 0.0016 15 0.00003 0.00000 -5.99 0.13 0.0735 0.0040 16 0.00000 0.00000 -0.07 8.73 0.0039 0.0017 The magnetic center is at X = -93.1 +- 0.4 microns Y = 173.6 +- 0.5 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)