SLAC Magnetic Measurements Date: 08-17-1995 Time: 12:15:35 Magnet Name (Serial #): 015 Bar Code Number: 168 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: 08-17-1995 Time: 12:45:56 Magnet Name: 015 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 25.5802 +- 2.63154E-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 -104.98 0.86 0.9278 0.0087 2 0.00006 0.00000 28.03 0.46 1.1481 0.0166 3 0.00525 0.00000 -29.91 0.00 100.0000 0.0378 4 0.00001 0.00000 36.04 2.55 0.1028 0.0118 5 0.00001 0.00000 17.60 1.51 0.1026 0.0103 6 0.00000 0.00000 -2.83 15.36 0.0380 0.0127 7 0.00000 0.00000 10.83 3.34 0.0379 0.0166 8 0.00000 0.00000 -2.34 12.63 0.0415 0.0099 9 0.00002 0.00000 9.91 0.22 0.3297 0.0101 10 0.00000 0.00000 7.15 12.92 0.0250 0.0107 11 0.00000 0.00000 4.45 8.93 0.0273 0.0082 12 0.00000 0.00000 4.50 11.07 0.0286 0.0062 13 0.00000 0.00000 -2.64 4.40 0.0747 0.0503 14 0.00000 0.00000 0.97 5.63 0.0168 0.0087 15 0.00001 0.00000 3.19 2.92 0.0972 0.0215 16 0.00000 0.00000 -1.50 6.00 0.0166 0.0100 The magnetic center is at X = 213.7 +- 3.8 microns Y = 143.8 +- 4.0 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: 08-17-1995 Time: 12:49:20 Magnet Name: 015 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 50.56406 +- 6.59462E-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 -113.21 0.82 0.2715 0.0099 2 0.00012 0.00000 27.62 0.21 1.1322 0.0024 3 0.01027 0.00000 -29.92 0.00 100.0000 0.0264 4 0.00001 0.00000 39.12 0.98 0.1029 0.0031 5 0.00001 0.00000 18.28 0.13 0.1212 0.0021 6 0.00000 0.00000 -16.93 2.12 0.0186 0.0041 7 0.00000 0.00000 12.90 0.95 0.0315 0.0025 8 0.00000 0.00000 -16.10 0.13 0.0191 0.0050 9 0.00003 0.00000 10.08 0.07 0.3309 0.0094 10 0.00000 0.00000 -7.07 8.05 0.0051 0.0052 11 0.00000 0.00000 -5.53 5.65 0.0057 0.0057 12 0.00000 0.00000 1.66 9.34 0.0129 0.0175 13 0.00000 0.00000 -7.69 5.12 0.0082 0.0103 14 0.00000 0.00000 -5.82 8.30 0.0096 0.0087 15 0.00001 0.00000 6.12 0.11 0.0670 0.0068 16 0.00000 0.00000 -1.34 4.73 0.0055 0.0009 The magnetic center is at X = 208.7 +- 1.1 microns Y = 144.9 +- 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 ^ | S N N --> x S S N Date: 08-17-1995 Time: 12:52:50 Magnet Name: 015 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 100.5562 +- 6.16898E-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 126.67 0.96 0.0792 0.0038 2 0.00023 0.00000 27.70 0.11 1.1297 0.0052 3 0.02040 0.00000 -29.92 0.00 100.0000 0.0123 4 0.00002 0.00000 39.00 1.05 0.1042 0.0015 5 0.00003 0.00000 18.18 0.22 0.1284 0.0023 6 0.00000 0.00000 -18.93 2.64 0.0182 0.0065 7 0.00001 0.00000 12.13 1.28 0.0318 0.0079 8 0.00000 0.00000 -14.70 1.30 0.0184 0.0054 9 0.00007 0.00000 10.14 0.05 0.3300 0.0041 10 0.00000 0.00000 7.64 7.68 0.0078 0.0037 11 0.00000 0.00000 2.94 11.55 0.0031 0.0012 12 0.00000 0.00000 1.07 2.10 0.0118 0.0023 13 0.00000 0.00000 -5.11 4.91 0.0138 0.0105 14 0.00000 0.00000 -5.51 1.70 0.0091 0.0043 15 0.00001 0.00000 6.53 0.57 0.0684 0.0060 16 0.00000 0.00000 -0.99 2.30 0.0083 0.0045 The magnetic center is at X = 208.7 +- 1.1 microns Y = 144.0 +- 1.1 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: 08-17-1995 Time: 12:59:33 Magnet Name: 015 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 150.5475 +- 6.663904E-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 97.48 1.27 0.1839 0.0030 2 0.00034 0.00000 27.67 0.04 1.1302 0.0014 3 0.03049 0.00000 -29.92 0.00 100.0000 0.0209 4 0.00003 0.00000 39.28 0.12 0.1012 0.0033 5 0.00004 0.00000 18.23 0.30 0.1327 0.0016 6 0.00000 0.00000 -13.18 0.55 0.0154 0.0022 7 0.00001 0.00000 12.34 1.00 0.0293 0.0030 8 0.00001 0.00000 -15.22 0.90 0.0166 0.0019 9 0.00010 0.00000 10.12 0.08 0.3280 0.0028 10 0.00000 0.00000 4.90 8.19 0.0031 0.0027 11 0.00000 0.00000 -2.72 3.85 0.0047 0.0026 12 0.00000 0.00000 12.72 1.42 0.0074 0.0028 13 0.00000 0.00000 0.30 1.88 0.0063 0.0024 14 0.00000 0.00000 -9.82 1.60 0.0065 0.0027 15 0.00002 0.00000 6.10 0.18 0.0690 0.0015 16 0.00000 0.00000 1.73 2.18 0.0055 0.0023 The magnetic center is at X = 208.6 +- 0.3 microns Y = 144.3 +- 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: 08-17-1995 Time: 13:06:16 Magnet Name: 015 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 200.5572 +- 4.241731E-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 93.43 0.17 0.2152 0.0011 2 0.00045 0.00000 27.51 0.08 1.1199 0.0023 3 0.04047 0.00001 -29.92 0.00 100.0000 0.0249 4 0.00004 0.00000 39.21 0.51 0.1018 0.0036 5 0.00005 0.00000 18.34 0.11 0.1294 0.0027 6 0.00001 0.00000 -14.82 2.55 0.0130 0.0033 7 0.00001 0.00000 11.89 0.15 0.0288 0.0019 8 0.00001 0.00000 -15.41 1.07 0.0143 0.0017 9 0.00013 0.00000 10.07 0.08 0.3292 0.0006 10 0.00000 0.00000 -8.18 12.59 0.0049 0.0015 11 0.00000 0.00000 -2.30 10.75 0.0026 0.0005 12 0.00000 0.00000 -3.94 11.26 0.0016 0.0008 13 0.00000 0.00000 -2.14 1.74 0.0051 0.0015 14 0.00000 0.00000 -7.30 1.06 0.0117 0.0028 15 0.00003 0.00000 6.01 0.10 0.0707 0.0010 16 0.00000 0.00000 -5.48 3.49 0.0026 0.0016 The magnetic center is at X = 205.9 +- 0.6 microns Y = 144.1 +- 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 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)