SLAC Magnetic Measurements Date: 07-12-1995 Time: 12:23:46 Magnet Name (Serial #): 078 Bar Code Number: 124 Project: PEP II High Energy Ring Sextupoles Test Stand: IR8, S1, N 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: 07-12-1995 Time: 12:54:08 Magnet Name: 078 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 25.61472 +- 1.045125E-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 -104.35 0.72 0.9790 0.0022 2 0.00002 0.00000 83.77 0.20 0.6879 0.0080 3 0.00363 0.00000 30.27 0.00 100.0000 0.1097 4 0.00000 0.00000 -11.00 1.60 0.0878 0.0102 5 0.00001 0.00000 -17.04 0.67 0.1473 0.0041 6 0.00000 0.00000 6.20 3.98 0.0082 0.0023 7 0.00000 0.00000 -11.09 1.58 0.0405 0.0024 8 0.00000 0.00000 -2.15 1.32 0.0122 0.0023 9 0.00001 0.00000 -9.53 0.14 0.3377 0.0032 10 0.00000 0.00000 10.65 5.27 0.0091 0.0032 11 0.00000 0.00000 1.47 6.76 0.0146 0.0038 12 0.00000 0.00000 6.13 5.62 0.0083 0.0025 13 0.00000 0.00000 0.38 6.47 0.0114 0.0032 14 0.00000 0.00000 -3.01 6.21 0.0119 0.0080 15 0.00000 0.00000 -5.61 0.63 0.0753 0.0060 16 0.00000 0.00000 -0.77 6.62 0.0118 0.0012 The magnetic center is at X = -33.3 +- 1.1 microns Y = 150.7 +- 1.8 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: 12:57:40 Magnet Name: 078 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 50.59655 +- 8.59897E-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 0.00004 0.00000 -99.95 0.34 0.6239 0.0014 2 0.00005 0.00000 82.69 0.08 0.6971 0.0023 3 0.00710 0.00000 30.27 0.00 100.0000 0.0188 4 0.00001 0.00000 -9.62 0.41 0.0811 0.0041 5 0.00001 0.00000 -17.11 0.26 0.1700 0.0018 6 0.00000 0.00000 6.28 6.19 0.0046 0.0029 7 0.00000 0.00000 -12.00 0.36 0.0402 0.0034 8 0.00000 0.00000 -2.17 1.23 0.0137 0.0021 9 0.00003 0.00000 -9.49 0.03 0.3680 0.0029 10 0.00000 0.00000 6.16 9.96 0.0041 0.0018 11 0.00000 0.00000 -1.16 3.40 0.0114 0.0031 12 0.00000 0.00000 -2.71 8.54 0.0049 0.0034 13 0.00000 0.00000 -0.41 3.97 0.0066 0.0045 14 0.00000 0.00000 -4.37 4.61 0.0036 0.0020 15 0.00000 0.00000 -5.52 0.29 0.0664 0.0062 16 0.00000 0.00000 1.71 6.44 0.0055 0.0019 The magnetic center is at X = -39.5 +- 0.4 microns Y = 151.4 +- 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: 07-12-1995 Time: 13:01:17 Magnet Name: 078 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 100.5879 +- 8.742401E-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 0.00006 0.00000 -94.95 0.17 0.4461 0.0059 2 0.00010 0.00000 82.23 0.28 0.6935 0.0067 3 0.01423 0.00003 30.27 0.00 100.0000 0.3458 4 0.00001 0.00000 -8.84 1.74 0.0851 0.0100 5 0.00003 0.00000 -17.07 0.07 0.1826 0.0053 6 0.00000 0.00000 9.45 3.28 0.0129 0.0099 7 0.00001 0.00000 -12.76 1.10 0.0407 0.0028 8 0.00000 0.00000 -2.57 2.13 0.0125 0.0020 9 0.00007 0.00000 -9.11 0.06 0.4658 0.0019 10 0.00000 0.00000 -8.45 9.86 0.0048 0.0025 11 0.00000 0.00000 3.66 7.64 0.0043 0.0016 12 0.00000 0.00000 -3.38 8.37 0.0047 0.0034 13 0.00000 0.00000 -4.70 3.38 0.0067 0.0034 14 0.00000 0.00000 2.28 2.68 0.0057 0.0008 15 0.00001 0.00000 -5.94 0.11 0.0679 0.0010 16 0.00000 0.00000 0.44 9.54 0.0048 0.0029 The magnetic center is at X = -41.7 +- 1.5 microns Y = 149.9 +- 1.5 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: 07-12-1995 Time: 13:08:14 Magnet Name: 078 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 150.5728 +- 7.64307E-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 0.00009 0.00000 -92.11 0.82 0.4118 0.0086 2 0.00015 0.00000 81.80 0.17 0.6892 0.0126 3 0.02159 0.00010 30.27 0.00 100.0000 0.6701 4 0.00002 0.00000 -10.19 3.09 0.0792 0.0087 5 0.00004 0.00000 -16.83 0.23 0.1981 0.0085 6 0.00000 0.00000 17.36 3.66 0.0124 0.0034 7 0.00001 0.00000 -14.41 1.55 0.0376 0.0011 8 0.00000 0.00000 -5.81 2.77 0.0180 0.0024 9 0.00013 0.00000 -8.79 0.02 0.5922 0.0082 10 0.00000 0.00000 0.38 9.93 0.0055 0.0032 11 0.00000 0.00000 -11.61 5.81 0.0054 0.0026 12 0.00000 0.00000 -1.16 4.54 0.0035 0.0020 13 0.00000 0.00000 -2.91 1.05 0.0087 0.0024 14 0.00000 0.00000 4.29 5.08 0.0044 0.0026 15 0.00002 0.00000 -5.98 0.27 0.0702 0.0031 16 0.00000 0.00000 -5.35 2.80 0.0045 0.0022 The magnetic center is at X = -43.7 +- 1.2 microns Y = 148.4 +- 2.7 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 ^ | N S S --> x N N S Date: 07-12-1995 Time: 13:15:10 Magnet Name: 078 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 200.5708 +- 1.999746E-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 -91.22 0.07 0.3701 0.0006 2 0.00021 0.00000 81.70 0.08 0.7109 0.0027 3 0.02886 0.00002 30.28 0.00 100.0000 0.0817 4 0.00003 0.00000 -6.57 0.26 0.0921 0.0019 5 0.00006 0.00000 -17.10 0.14 0.1916 0.0016 6 0.00000 0.00000 10.58 9.23 0.0030 0.0012 7 0.00001 0.00000 -13.11 0.69 0.0365 0.0042 8 0.00001 0.00000 -0.82 0.25 0.0205 0.0020 9 0.00022 0.00000 -8.62 0.03 0.7518 0.0056 10 0.00000 0.00000 0.51 16.65 0.0063 0.0029 11 0.00000 0.00000 4.57 6.59 0.0018 0.0010 12 0.00000 0.00000 -7.15 11.77 0.0068 0.0013 13 0.00000 0.00000 2.14 6.99 0.0022 0.0006 14 0.00000 0.00000 3.45 6.44 0.0038 0.0014 15 0.00002 0.00000 -5.89 0.16 0.0674 0.0029 16 0.00000 0.00000 0.10 8.69 0.0039 0.0037 The magnetic center is at X = -45.5 +- 0.5 microns Y = 152.9 +- 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 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)