SLAC Magnetic Measurements Date: 09-01-1995 Time: 09:46:16 Magnet Name (Serial #): 023 Bar Code Number: 172 Project: PEP II High Energy Ring Sextupoles Test Stand: IR8, S1, W Measurement Coil: # 1 Operator: dan Run Number: 1 Comment: none 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: 09-01-1995 Time: 10:17:54 Magnet Name: 023 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 25.65732 +- 9.503153E-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 -107.23 0.15 0.9473 0.0009 2 0.00010 0.00000 24.13 0.01 1.9509 0.0027 3 0.00527 0.00000 -29.89 0.00 100.0000 0.0164 4 0.00001 0.00000 38.01 0.16 0.1543 0.0027 5 0.00001 0.00000 18.91 0.23 0.0975 0.0024 6 0.00000 0.00000 -23.46 2.33 0.0117 0.0015 7 0.00000 0.00000 12.37 0.62 0.0318 0.0015 8 0.00000 0.00000 -16.14 0.27 0.0249 0.0023 9 0.00002 0.00000 10.17 0.05 0.3298 0.0012 10 0.00000 0.00000 11.45 1.62 0.0028 0.0018 11 0.00000 0.00000 -6.09 10.05 0.0059 0.0030 12 0.00000 0.00000 4.46 8.12 0.0037 0.0022 13 0.00000 0.00000 -0.68 4.00 0.0042 0.0025 14 0.00000 0.00000 -9.69 0.98 0.0107 0.0028 15 0.00000 0.00000 6.26 0.10 0.0683 0.0025 16 0.00000 0.00000 -0.59 7.76 0.0049 0.0023 The magnetic center is at X = 326.7 +- 0.5 microns Y = 291.4 +- 0.4 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 ^ | S N N --> x S S N Date: 09-01-1995 Time: 10:21:18 Magnet Name: 023 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 50.63448 +- 1.166392E-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 0.00003 0.00000 -120.26 0.39 0.2897 0.0023 2 0.00020 0.00000 23.97 0.03 1.9420 0.0026 3 0.01029 0.00001 -29.89 0.00 100.0000 0.0885 4 0.00002 0.00000 38.34 0.11 0.1496 0.0011 5 0.00001 0.00000 18.80 0.30 0.1138 0.0019 6 0.00000 0.00000 -22.38 1.40 0.0138 0.0019 7 0.00000 0.00000 12.73 0.17 0.0324 0.0010 8 0.00000 0.00000 -16.31 0.54 0.0252 0.0006 9 0.00003 0.00000 10.14 0.02 0.3318 0.0020 10 0.00000 0.00000 -4.74 10.11 0.0025 0.0014 11 0.00000 0.00000 3.33 12.18 0.0047 0.0034 12 0.00000 0.00000 1.82 9.93 0.0038 0.0022 13 0.00000 0.00000 -1.34 7.56 0.0054 0.0023 14 0.00000 0.00000 -9.82 1.00 0.0095 0.0012 15 0.00001 0.00000 6.09 0.02 0.0718 0.0030 16 0.00000 0.00000 1.14 7.71 0.0024 0.0016 The magnetic center is at X = 323.6 +- 0.5 microns Y = 291.9 +- 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 ^ | S N N --> x S S N Date: 09-01-1995 Time: 10:24:48 Magnet Name: 023 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 100.6267 +- 1.607993E-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 129.94 0.69 0.1192 0.0011 2 0.00040 0.00000 23.92 0.03 1.9391 0.0020 3 0.02041 0.00000 -29.90 0.00 100.0000 0.0308 4 0.00003 0.00000 38.79 0.16 0.1499 0.0014 5 0.00002 0.00000 18.75 0.19 0.1216 0.0014 6 0.00000 0.00000 -25.63 0.59 0.0219 0.0014 7 0.00001 0.00000 12.48 0.82 0.0330 0.0011 8 0.00000 0.00000 -16.30 0.31 0.0245 0.0005 9 0.00007 0.00000 10.17 0.02 0.3335 0.0015 10 0.00000 0.00000 11.27 2.18 0.0056 0.0021 11 0.00000 0.00000 -6.88 12.42 0.0055 0.0024 12 0.00000 0.00000 3.03 0.72 0.0072 0.0023 13 0.00000 0.00000 2.53 4.36 0.0040 0.0030 14 0.00000 0.00000 -9.44 0.88 0.0078 0.0007 15 0.00001 0.00000 6.15 0.06 0.0698 0.0021 16 0.00000 0.00000 -1.16 2.13 0.0033 0.0013 The magnetic center is at X = 322.5 +- 0.5 microns Y = 292.1 +- 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 ^ | S N N --> x S S N Date: 09-01-1995 Time: 10:31:31 Magnet Name: 023 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 150.6083 +- 6.663655E-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 0.00000 103.62 0.18 0.2190 0.0009 2 0.00059 0.00000 23.91 0.02 1.9427 0.0010 3 0.03051 0.00000 -29.90 0.00 100.0000 0.0096 4 0.00005 0.00000 38.66 0.15 0.1514 0.0016 5 0.00004 0.00000 18.89 0.08 0.1264 0.0015 6 0.00000 0.00000 -22.27 0.33 0.0140 0.0010 7 0.00001 0.00000 12.32 0.30 0.0332 0.0018 8 0.00001 0.00000 -16.50 0.26 0.0246 0.0007 9 0.00010 0.00000 10.16 0.02 0.3327 0.0015 10 0.00000 0.00000 4.72 6.70 0.0029 0.0020 11 0.00000 0.00000 -0.92 14.03 0.0021 0.0009 12 0.00000 0.00000 12.97 0.28 0.0059 0.0007 13 0.00000 0.00000 1.74 4.22 0.0030 0.0022 14 0.00000 0.00000 -8.67 0.53 0.0072 0.0011 15 0.00002 0.00000 6.18 0.06 0.0678 0.0005 16 0.00000 0.00000 -0.23 4.09 0.0017 0.0010 The magnetic center is at X = 323.0 +- 0.3 microns Y = 292.8 +- 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 ^ | S N N --> x S S N Date: 09-01-1995 Time: 10:38:14 Magnet Name: 023 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 200.6034 +- 3.118267E-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 99.55 0.20 0.2574 0.0005 2 0.00078 0.00000 23.77 0.02 1.9353 0.0025 3 0.04049 0.00001 -29.90 0.00 100.0000 0.0414 4 0.00006 0.00000 38.80 0.13 0.1520 0.0021 5 0.00005 0.00000 19.09 0.11 0.1228 0.0018 6 0.00001 0.00000 -24.32 1.07 0.0141 0.0010 7 0.00001 0.00000 12.46 1.01 0.0349 0.0021 8 0.00001 0.00000 -16.90 0.37 0.0237 0.0011 9 0.00014 0.00000 10.16 0.03 0.3341 0.0020 10 0.00000 0.00000 5.82 10.09 0.0022 0.0013 11 0.00000 0.00000 -8.09 10.50 0.0033 0.0016 12 0.00000 0.00000 3.07 9.65 0.0032 0.0011 13 0.00000 0.00000 2.36 6.63 0.0041 0.0009 14 0.00000 0.00000 -8.67 0.96 0.0080 0.0014 15 0.00003 0.00000 6.24 0.14 0.0694 0.0005 16 0.00000 0.00000 2.84 7.18 0.0023 0.0006 The magnetic center is at X = 320.4 +- 0.5 microns Y = 293.2 +- 0.4 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 ^ | 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)