SLAC Magnetic Measurements Date: 08-03-1995 Time: 09:51:28 Magnet Name (Serial #): 144 Bar Code Number: 180 Project: PEP II High Energy Ring Sextupoles Test Stand: IR8, S1, N 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: 08-03-1995 Time: 10:21:47 Magnet Name: 144 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 25.59374 +- 1.638151E-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 -99.27 0.12 1.0617 0.0026 2 0.00002 0.00000 46.86 0.04 0.3426 0.0036 3 0.00526 0.00000 30.07 0.00 100.0000 0.0296 4 0.00001 0.00000 -8.06 0.44 0.1315 0.0016 5 0.00001 0.00000 -17.98 0.23 0.1192 0.0024 6 0.00000 0.00000 -1.30 8.50 0.0035 0.0019 7 0.00000 0.00000 -13.20 0.32 0.0392 0.0029 8 0.00000 0.00000 -13.27 1.15 0.0075 0.0018 9 0.00002 0.00000 -9.96 0.05 0.3293 0.0022 10 0.00000 0.00000 -15.00 1.50 0.0058 0.0014 11 0.00000 0.00000 -9.87 3.80 0.0045 0.0017 12 0.00000 0.00000 0.29 13.00 0.0037 0.0008 13 0.00000 0.00000 -4.58 10.08 0.0044 0.0018 14 0.00000 0.00000 -6.56 2.94 0.0055 0.0018 15 0.00000 0.00000 -5.95 0.07 0.0730 0.0011 16 0.00000 0.00000 -3.75 5.10 0.0057 0.0017 The magnetic center is at X = -76.7 +- 0.8 microns Y = 5.0 +- 0.1 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: 08-03-1995 Time: 10:25:11 Magnet Name: 144 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 50.5747 +- 1.497147E-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 -97.10 0.16 0.6589 0.0008 2 0.00004 0.00000 46.70 0.13 0.3750 0.0033 3 0.01027 0.00000 30.07 0.00 100.0000 0.0391 4 0.00001 0.00000 -6.85 0.24 0.1287 0.0014 5 0.00001 0.00000 -17.90 0.11 0.1358 0.0032 6 0.00000 0.00000 -1.69 1.16 0.0051 0.0015 7 0.00000 0.00000 -13.40 0.27 0.0396 0.0028 8 0.00000 0.00000 -10.43 2.12 0.0058 0.0006 9 0.00003 0.00000 -9.91 0.01 0.3277 0.0028 10 0.00000 0.00000 -14.99 1.62 0.0030 0.0014 11 0.00000 0.00000 -6.32 4.41 0.0023 0.0011 12 0.00000 0.00000 -1.16 11.26 0.0038 0.0013 13 0.00000 0.00000 3.14 10.06 0.0032 0.0019 14 0.00000 0.00000 -4.35 0.68 0.0061 0.0022 15 0.00001 0.00000 -5.95 0.09 0.0727 0.0009 16 0.00000 0.00000 5.63 2.67 0.0015 0.0007 The magnetic center is at X = -84.0 +- 0.7 microns Y = 5.0 +- 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 ^ | N S S --> x N N S Date: 08-03-1995 Time: 10:28:41 Magnet Name: 144 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 100.5695 +- 8.887672E-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 0.00009 0.00000 -94.18 0.08 0.4567 0.0014 2 0.00008 0.00000 45.95 0.11 0.3893 0.0008 3 0.02040 0.00000 30.07 0.00 100.0000 0.0130 4 0.00003 0.00000 -6.76 0.19 0.1241 0.0010 5 0.00003 0.00000 -17.83 0.25 0.1423 0.0022 6 0.00000 0.00000 -26.91 1.89 0.0081 0.0019 7 0.00001 0.00000 -13.54 0.27 0.0381 0.0016 8 0.00000 0.00000 -12.12 0.95 0.0044 0.0010 9 0.00007 0.00000 -9.93 0.02 0.3278 0.0015 10 0.00000 0.00000 -15.29 1.59 0.0061 0.0018 11 0.00000 0.00000 -7.98 1.93 0.0026 0.0011 12 0.00000 0.00000 -3.16 1.42 0.0052 0.0018 13 0.00000 0.00000 -7.84 1.37 0.0043 0.0006 14 0.00000 0.00000 -4.69 0.87 0.0048 0.0011 15 0.00002 0.00000 -5.89 0.07 0.0740 0.0019 16 0.00000 0.00000 -4.96 2.74 0.0016 0.0014 The magnetic center is at X = -87.3 +- 0.2 microns Y = 2.9 +- 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: 08-03-1995 Time: 10:35:23 Magnet Name: 144 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 150.5596 +- 3.322547E-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 0.00012 0.00000 -92.42 0.13 0.3892 0.0011 2 0.00012 0.00000 45.00 0.12 0.3963 0.0015 3 0.03050 0.00000 30.07 0.00 100.0000 0.0172 4 0.00004 0.00000 -6.47 0.09 0.1226 0.0012 5 0.00004 0.00000 -17.86 0.13 0.1472 0.0023 6 0.00000 0.00000 1.75 7.58 0.0046 0.0031 7 0.00001 0.00000 -13.20 0.34 0.0380 0.0011 8 0.00000 0.00000 -11.89 2.35 0.0038 0.0011 9 0.00010 0.00000 -9.93 0.04 0.3282 0.0018 10 0.00000 0.00000 -5.92 13.36 0.0032 0.0011 11 0.00000 0.00000 0.90 5.04 0.0012 0.0005 12 0.00000 0.00000 -0.01 14.47 0.0070 0.0019 13 0.00000 0.00000 -8.11 1.18 0.0033 0.0010 14 0.00000 0.00000 -4.07 2.74 0.0043 0.0017 15 0.00002 0.00000 -5.90 0.05 0.0713 0.0003 16 0.00000 0.00000 0.39 7.15 0.0015 0.0007 The magnetic center is at X = -88.9 +- 0.3 microns Y = -0.0 +- 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 ^ | N S S --> x N N S Date: 08-03-1995 Time: 10:42:06 Magnet Name: 144 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 200.5564 +- 8.708525E-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 0.00013 0.00000 -91.42 0.18 0.3263 0.0008 2 0.00016 0.00000 44.46 0.11 0.3993 0.0015 3 0.04050 0.00000 30.07 0.00 100.0000 0.0031 4 0.00005 0.00000 -6.57 0.15 0.1177 0.0008 5 0.00006 0.00000 -17.80 0.14 0.1415 0.0024 6 0.00000 0.00000 0.73 2.52 0.0038 0.0008 7 0.00001 0.00000 -13.23 0.07 0.0358 0.0013 8 0.00000 0.00000 -9.75 1.71 0.0038 0.0016 9 0.00013 0.00000 -9.96 0.04 0.3281 0.0012 10 0.00000 0.00000 -13.41 1.35 0.0044 0.0016 11 0.00000 0.00000 4.59 5.37 0.0012 0.0002 12 0.00000 0.00000 -7.16 7.43 0.0022 0.0006 13 0.00000 0.00000 -6.87 1.60 0.0033 0.0002 14 0.00000 0.00000 -4.21 1.68 0.0051 0.0016 15 0.00003 0.00000 -5.94 0.09 0.0722 0.0012 16 0.00000 0.00000 0.33 10.37 0.0018 0.0012 The magnetic center is at X = -89.6 +- 0.3 microns Y = -1.7 +- 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 ^ | 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)