SLAC Magnetic Measurements Date: 07-11-1995 Time: 13:02:30 Magnet Name (Serial #): 135 Bar Code Number: 134 Project: PEP II High Energy Ring Sextupoles Test Stand: IR8, S1, W Measurement Coil: # 1 Operator: bmd 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: 07-11-1995 Time: 13:32:53 Magnet Name: 135 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 25.59723 +- 1.136446E-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 -115.03 0.12 0.7667 0.0028 2 0.00008 0.00000 32.47 0.11 1.6056 0.0054 3 0.00525 0.00000 -29.89 0.00 100.0000 0.0718 4 0.00001 0.00000 36.93 0.39 0.1404 0.0047 5 0.00001 0.00000 19.74 0.32 0.1184 0.0044 6 0.00000 0.00000 -13.18 20.92 0.0127 0.0019 7 0.00000 0.00000 13.77 1.37 0.0233 0.0118 8 0.00000 0.00000 -16.83 1.45 0.0236 0.0058 9 0.00002 0.00000 10.06 0.06 0.3263 0.0125 10 0.00000 0.00000 -9.01 5.10 0.0042 0.0022 11 0.00000 0.00000 0.20 8.53 0.0091 0.0033 12 0.00000 0.00000 -0.80 9.72 0.0084 0.0025 13 0.00000 0.00000 0.52 8.10 0.0103 0.0036 14 0.00000 0.00000 -2.32 7.15 0.0086 0.0048 15 0.00000 0.00000 6.13 0.13 0.0740 0.0031 16 0.00000 0.00000 -4.48 4.71 0.0056 0.0021 The magnetic center is at X = 326.4 +- 1.2 microns Y = 152.6 +- 1.4 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: 07-11-1995 Time: 13:36:25 Magnet Name: 135 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 50.58067 +- 1.575686E-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 -141.16 0.41 0.2437 0.0005 2 0.00016 0.00000 32.41 0.05 1.5931 0.0014 3 0.01027 0.00000 -29.89 0.00 100.0000 0.0224 4 0.00001 0.00000 37.64 0.32 0.1259 0.0011 5 0.00001 0.00000 19.16 0.25 0.1304 0.0022 6 0.00000 0.00000 -4.67 20.16 0.0069 0.0040 7 0.00000 0.00000 13.83 0.75 0.0295 0.0020 8 0.00000 0.00000 -15.46 0.92 0.0193 0.0038 9 0.00003 0.00000 10.06 0.06 0.3260 0.0017 10 0.00000 0.00000 -1.20 11.79 0.0042 0.0014 11 0.00000 0.00000 9.27 4.87 0.0051 0.0025 12 0.00000 0.00000 -4.20 7.22 0.0050 0.0025 13 0.00000 0.00000 -3.39 9.88 0.0042 0.0007 14 0.00000 0.00000 -5.99 1.76 0.0063 0.0029 15 0.00001 0.00000 6.13 0.23 0.0706 0.0012 16 0.00000 0.00000 -0.13 6.00 0.0025 0.0018 The magnetic center is at X = 323.5 +- 0.4 microns Y = 152.1 +- 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 ^ | S N N --> x S S N Date: 07-11-1995 Time: 13:40:01 Magnet Name: 135 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 100.5729 +- 5.560918E-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 131.39 0.29 0.1748 0.0012 2 0.00033 0.00000 32.31 0.05 1.5948 0.0037 3 0.02040 0.00001 -29.90 0.00 100.0000 0.0419 4 0.00003 0.00000 38.06 0.31 0.1284 0.0013 5 0.00003 0.00000 19.08 0.09 0.1345 0.0016 6 0.00000 0.00000 -26.82 1.13 0.0121 0.0013 7 0.00001 0.00000 12.84 0.46 0.0305 0.0016 8 0.00000 0.00000 -14.69 0.92 0.0186 0.0012 9 0.00007 0.00000 10.08 0.03 0.3316 0.0013 10 0.00000 0.00000 6.60 13.93 0.0047 0.0013 11 0.00000 0.00000 2.96 10.93 0.0031 0.0013 12 0.00000 0.00000 0.72 2.41 0.0043 0.0007 13 0.00000 0.00000 11.88 0.45 0.0033 0.0016 14 0.00000 0.00000 -8.28 1.44 0.0076 0.0013 15 0.00002 0.00000 6.23 0.05 0.0737 0.0012 16 0.00000 0.00000 -2.32 3.16 0.0023 0.0009 The magnetic center is at X = 323.3 +- 0.8 microns Y = 153.4 +- 0.7 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: 07-11-1995 Time: 13:46:57 Magnet Name: 135 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 150.5602 +- 4.59249E-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 110.78 0.41 0.2522 0.0012 2 0.00049 0.00000 32.11 0.02 1.6009 0.0025 3 0.03048 0.00001 -29.90 0.00 100.0000 0.0530 4 0.00004 0.00000 38.25 0.23 0.1321 0.0008 5 0.00004 0.00000 19.02 0.21 0.1416 0.0010 6 0.00000 0.00000 5.43 21.03 0.0034 0.0022 7 0.00001 0.00000 13.04 0.19 0.0315 0.0009 8 0.00001 0.00000 -14.65 0.40 0.0191 0.0016 9 0.00010 0.00000 10.11 0.02 0.3315 0.0024 10 0.00000 0.00000 -0.98 13.10 0.0037 0.0011 11 0.00000 0.00000 6.28 12.42 0.0023 0.0010 12 0.00000 0.00000 -7.47 12.90 0.0084 0.0018 13 0.00000 0.00000 0.53 11.43 0.0028 0.0014 14 0.00000 0.00000 -8.59 0.69 0.0058 0.0008 15 0.00002 0.00000 6.24 0.07 0.0698 0.0003 16 0.00000 0.00000 1.74 1.66 0.0013 0.0008 The magnetic center is at X = 323.5 +- 0.5 microns Y = 156.3 +- 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: 07-11-1995 Time: 13:53:53 Magnet Name: 135 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 200.5591 +- 4.903499E-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 106.11 0.24 0.2772 0.0007 2 0.00065 0.00000 32.07 0.02 1.6078 0.0016 3 0.04046 0.00001 -29.90 0.00 100.0000 0.0441 4 0.00005 0.00000 38.41 0.21 0.1348 0.0023 5 0.00005 0.00000 18.95 0.18 0.1341 0.0018 6 0.00000 0.00000 -23.33 3.25 0.0044 0.0022 7 0.00001 0.00000 13.37 1.02 0.0316 0.0020 8 0.00001 0.00000 -14.86 0.36 0.0166 0.0016 9 0.00013 0.00000 10.11 0.03 0.3321 0.0011 10 0.00000 0.00000 -5.95 12.25 0.0046 0.0019 11 0.00000 0.00000 10.67 3.56 0.0029 0.0011 12 0.00000 0.00000 -6.94 12.48 0.0040 0.0007 13 0.00000 0.00000 0.35 13.29 0.0049 0.0020 14 0.00000 0.00000 -9.23 0.30 0.0066 0.0007 15 0.00003 0.00000 6.27 0.07 0.0699 0.0013 16 0.00000 0.00000 1.50 8.95 0.0016 0.0006 The magnetic center is at X = 324.7 +- 0.3 microns Y = 157.4 +- 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 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)