SLAC Magnetic Measurements Date: 09-27-1995 Time: 09:45:36 Magnet Name (Serial #): 123 Bar Code Number: 187 Project: PEP II High Energy Ring Sextupoles Test Stand: IR8, S1, N 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: 09-27-1995 Time: 10:15:55 Magnet Name: 123 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 25.62288 +- 2.379434E-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 0.00005 0.00000 -108.59 0.53 0.8706 0.0048 2 0.00006 0.00000 26.97 0.23 1.0987 0.0105 3 0.00526 0.00000 -29.90 0.00 100.0000 0.0138 4 0.00000 0.00000 40.19 2.56 0.0930 0.0129 5 0.00001 0.00000 18.82 0.65 0.1099 0.0054 6 0.00000 0.00000 -12.44 21.33 0.0245 0.0084 7 0.00000 0.00000 14.27 2.00 0.0269 0.0041 8 0.00000 0.00000 -17.60 2.56 0.0251 0.0036 9 0.00002 0.00000 10.13 0.08 0.3306 0.0046 10 0.00000 0.00000 3.74 12.00 0.0113 0.0036 11 0.00000 0.00000 3.50 9.42 0.0113 0.0063 12 0.00000 0.00000 -2.40 8.16 0.0201 0.0055 13 0.00000 0.00000 -2.19 5.30 0.0149 0.0068 14 0.00000 0.00000 -2.01 7.37 0.0180 0.0091 15 0.00000 0.00000 6.10 0.48 0.0705 0.0056 16 0.00000 0.00000 -0.66 5.94 0.0163 0.0077 The magnetic center is at X = 199.3 +- 2.2 microns Y = 145.2 +- 2.1 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 ^ | S N N --> x S S N Date: 09-27-1995 Time: 10:19:21 Magnet Name: 123 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 50.60597 +- 1.115127E-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 -117.02 1.13 0.2931 0.0048 2 0.00011 0.00000 27.90 0.24 1.1108 0.0036 3 0.01029 0.00000 -29.91 0.00 100.0000 0.0304 4 0.00001 0.00000 38.48 0.95 0.0896 0.0074 5 0.00001 0.00000 18.93 0.22 0.1308 0.0057 6 0.00000 0.00000 11.84 15.28 0.0077 0.0048 7 0.00000 0.00000 12.65 3.62 0.0221 0.0046 8 0.00000 0.00000 5.82 14.21 0.0169 0.0091 9 0.00003 0.00000 10.11 0.18 0.3202 0.0035 10 0.00000 0.00000 10.44 5.52 0.0212 0.0077 11 0.00000 0.00000 3.64 10.70 0.0084 0.0013 12 0.00000 0.00000 1.23 7.92 0.0097 0.0013 13 0.00000 0.00000 4.74 10.27 0.0088 0.0040 14 0.00000 0.00000 -7.38 2.22 0.0085 0.0051 15 0.00001 0.00000 5.92 0.29 0.0718 0.0023 16 0.00000 0.00000 0.28 7.95 0.0077 0.0056 The magnetic center is at X = 206.2 +- 1.4 microns Y = 140.1 +- 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 ^ | S N N --> x S S N Date: 09-27-1995 Time: 10:22:50 Magnet Name: 123 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 100.6028 +- 1.031535E-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 0.00001 0.00000 151.87 2.72 0.0695 0.0024 2 0.00023 0.00000 28.26 0.07 1.1173 0.0024 3 0.02042 0.00000 -29.91 0.00 100.0000 0.0211 4 0.00002 0.00000 36.81 0.55 0.0981 0.0035 5 0.00003 0.00000 18.61 0.61 0.1366 0.0013 6 0.00000 0.00000 -28.04 2.34 0.0235 0.0037 7 0.00001 0.00000 13.43 0.91 0.0302 0.0035 8 0.00000 0.00000 -11.80 2.14 0.0166 0.0079 9 0.00007 0.00000 10.08 0.11 0.3321 0.0023 10 0.00000 0.00000 4.21 7.55 0.0083 0.0021 11 0.00000 0.00000 8.41 10.13 0.0071 0.0019 12 0.00000 0.00000 3.62 3.02 0.0086 0.0046 13 0.00000 0.00000 6.14 9.96 0.0123 0.0044 14 0.00000 0.00000 -8.01 3.69 0.0058 0.0024 15 0.00001 0.00000 6.11 0.32 0.0732 0.0039 16 0.00000 0.00000 3.08 6.74 0.0102 0.0032 The magnetic center is at X = 209.1 +- 0.6 microns Y = 138.3 +- 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: 09-27-1995 Time: 10:29:34 Magnet Name: 123 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 150.582 +- 5.788066E-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 0.00005 0.00000 104.50 0.86 0.1523 0.0060 2 0.00034 0.00000 28.61 0.12 1.1172 0.0026 3 0.03051 0.00000 -29.91 0.00 100.0000 0.0228 4 0.00003 0.00000 36.82 0.35 0.0993 0.0048 5 0.00004 0.00000 18.62 0.38 0.1378 0.0040 6 0.00000 0.00000 -26.54 2.12 0.0122 0.0029 7 0.00001 0.00000 13.12 0.88 0.0303 0.0027 8 0.00000 0.00000 -15.89 1.39 0.0148 0.0017 9 0.00010 0.00000 10.15 0.05 0.3270 0.0029 10 0.00000 0.00000 8.28 8.85 0.0065 0.0025 11 0.00000 0.00000 -6.75 13.21 0.0049 0.0016 12 0.00000 0.00000 6.58 10.97 0.0087 0.0019 13 0.00000 0.00000 4.02 10.34 0.0051 0.0022 14 0.00000 0.00000 -8.21 2.49 0.0066 0.0017 15 0.00002 0.00000 6.58 0.29 0.0697 0.0046 16 0.00000 0.00000 1.82 6.50 0.0028 0.0021 The magnetic center is at X = 210.8 +- 0.8 microns Y = 135.8 +- 1.0 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: 09-27-1995 Time: 10:36:16 Magnet Name: 123 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 200.5845 +- 5.347144E-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 97.93 0.08 0.1767 0.0031 2 0.00046 0.00000 28.39 0.06 1.1288 0.0024 3 0.04051 0.00000 -29.91 0.00 100.0000 0.0103 4 0.00004 0.00000 36.79 0.34 0.0987 0.0039 5 0.00005 0.00000 18.50 0.07 0.1339 0.0017 6 0.00001 0.00000 -26.94 0.77 0.0124 0.0017 7 0.00001 0.00000 12.92 1.22 0.0306 0.0041 8 0.00001 0.00000 -9.97 12.19 0.0182 0.0086 9 0.00013 0.00000 10.12 0.08 0.3314 0.0035 10 0.00001 0.00000 -3.19 13.46 0.0167 0.0079 11 0.00000 0.00000 5.07 8.24 0.0041 0.0025 12 0.00000 0.00000 -1.35 13.43 0.0035 0.0007 13 0.00000 0.00000 3.58 9.70 0.0045 0.0011 14 0.00000 0.00000 -8.72 4.54 0.0057 0.0019 15 0.00003 0.00000 6.17 0.13 0.0708 0.0021 16 0.00000 0.00000 -3.52 7.66 0.0034 0.0007 The magnetic center is at X = 211.9 +- 0.5 microns Y = 138.8 +- 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 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)