SLAC Magnetic Measurements Date: 07-11-1995 Time: 06:59:27 Magnet Name (Serial #): 030 Bar Code Number: 106 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: 07:29:54 Magnet Name: 030 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 25.64734 +- 2.032649E-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 -115.81 1.05 0.8628 0.0137 2 0.00007 0.00000 27.97 0.24 1.2362 0.0196 3 0.00526 0.00000 -29.91 0.00 100.0000 0.0264 4 0.00001 0.00000 32.91 1.29 0.1272 0.0184 5 0.00001 0.00000 21.15 0.89 0.1188 0.0102 6 0.00000 0.00000 7.32 20.10 0.0226 0.0021 7 0.00000 0.00000 4.33 17.47 0.0364 0.0076 8 0.00001 0.00000 1.67 11.43 0.0952 0.0515 9 0.00002 0.00000 10.17 0.82 0.3230 0.0199 10 0.00000 0.00000 1.66 7.84 0.0905 0.0752 11 0.00000 0.00000 -2.14 12.25 0.0194 0.0121 12 0.00000 0.00000 0.09 10.61 0.0214 0.0125 13 0.00000 0.00000 1.22 8.76 0.0171 0.0077 14 0.00000 0.00000 -0.77 7.80 0.0285 0.0112 15 0.00000 0.00000 4.81 1.69 0.0766 0.0361 16 0.00000 0.00000 7.24 2.33 0.0411 0.0131 The magnetic center is at X = 229.8 +- 3.9 microns Y = 155.4 +- 3.1 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: 07-11-1995 Time: 07:33:26 Magnet Name: 030 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 50.62892 +- 3.585642E-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 -128.49 1.00 0.3278 0.0048 2 0.00012 0.00000 27.29 0.16 1.2127 0.0040 3 0.01028 0.00000 -29.91 0.00 100.0000 0.0158 4 0.00001 0.00000 34.71 0.73 0.1058 0.0060 5 0.00001 0.00000 19.81 0.32 0.1334 0.0082 6 0.00000 0.00000 -2.48 19.78 0.0096 0.0025 7 0.00000 0.00000 13.88 1.02 0.0439 0.0115 8 0.00000 0.00000 -0.13 15.19 0.0435 0.0250 9 0.00003 0.00000 10.12 0.22 0.3345 0.0178 10 0.00001 0.00000 -0.62 9.81 0.0501 0.0260 11 0.00000 0.00000 -0.21 11.39 0.0143 0.0038 12 0.00000 0.00000 -0.26 5.99 0.0137 0.0042 13 0.00000 0.00000 -2.12 10.32 0.0148 0.0058 14 0.00000 0.00000 -0.45 4.68 0.0151 0.0128 15 0.00001 0.00000 5.83 1.43 0.0722 0.0226 16 0.00000 0.00000 3.98 6.73 0.0120 0.0032 The magnetic center is at X = 221.8 +- 1.2 microns Y = 157.7 +- 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: 07:37:03 Magnet Name: 030 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 100.6204 +- 2.433873E-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 -88.04 153.21 0.1249 0.0048 2 0.00024 0.00000 27.57 0.10 1.1884 0.0067 3 0.02040 0.00001 -29.92 0.00 100.0000 0.0399 4 0.00002 0.00000 36.52 0.73 0.1119 0.0037 5 0.00003 0.00000 19.42 0.13 0.1306 0.0054 6 0.00000 0.00000 -14.20 20.21 0.0161 0.0041 7 0.00001 0.00000 14.17 1.82 0.0289 0.0044 8 0.00000 0.00000 -18.10 3.48 0.0160 0.0098 9 0.00007 0.00000 10.17 0.09 0.3334 0.0092 10 0.00000 0.00000 4.01 8.65 0.0202 0.0084 11 0.00000 0.00000 10.94 2.49 0.0087 0.0031 12 0.00000 0.00000 3.31 2.60 0.0068 0.0011 13 0.00000 0.00000 -2.97 7.08 0.0066 0.0026 14 0.00000 0.00000 -0.59 6.65 0.0044 0.0024 15 0.00001 0.00000 6.58 0.44 0.0677 0.0129 16 0.00000 0.00000 -2.40 3.57 0.0094 0.0023 The magnetic center is at X = 218.8 +- 1.3 microns Y = 152.4 +- 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: 07-11-1995 Time: 07:44:00 Magnet Name: 030 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 150.5986 +- 3.179911E-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 134.36 0.55 0.1387 0.0029 2 0.00036 0.00000 27.55 0.04 1.1932 0.0023 3 0.03050 0.00001 -29.92 0.00 100.0000 0.0322 4 0.00003 0.00000 35.06 0.19 0.1123 0.0039 5 0.00004 0.00000 19.33 0.22 0.1337 0.0015 6 0.00000 0.00000 18.62 4.70 0.0027 0.0013 7 0.00001 0.00000 13.31 0.47 0.0306 0.0046 8 0.00001 0.00000 -7.49 15.45 0.0242 0.0103 9 0.00010 0.00000 10.12 0.08 0.3358 0.0042 10 0.00001 0.00000 1.46 11.30 0.0228 0.0061 11 0.00000 0.00000 10.25 5.29 0.0063 0.0013 12 0.00000 0.00000 -6.32 12.14 0.0066 0.0033 13 0.00000 0.00000 0.57 9.89 0.0052 0.0019 14 0.00000 0.00000 -6.48 4.03 0.0031 0.0018 15 0.00002 0.00000 6.00 0.24 0.0710 0.0033 16 0.00000 0.00000 3.17 4.23 0.0040 0.0014 The magnetic center is at X = 219.6 +- 0.5 microns Y = 153.2 +- 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 ^ | S N N --> x S S N Date: 07-11-1995 Time: 07:50:57 Magnet Name: 030 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 200.5923 +- 4.932467E-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 118.02 0.96 0.1582 0.0026 2 0.00048 0.00000 27.73 0.07 1.1905 0.0026 3 0.04047 0.00002 -29.92 0.00 100.0000 0.0594 4 0.00004 0.00000 35.41 0.48 0.1074 0.0021 5 0.00005 0.00000 19.52 0.12 0.1312 0.0011 6 0.00000 0.00000 10.67 22.38 0.0061 0.0016 7 0.00001 0.00000 13.58 0.68 0.0318 0.0018 8 0.00000 0.00000 -15.46 1.14 0.0084 0.0019 9 0.00013 0.00000 10.11 0.05 0.3332 0.0028 10 0.00000 0.00000 -5.23 4.63 0.0032 0.0019 11 0.00000 0.00000 5.77 10.42 0.0039 0.0009 12 0.00000 0.00000 0.95 12.86 0.0030 0.0015 13 0.00000 0.00000 -6.80 2.98 0.0053 0.0016 14 0.00000 0.00000 -9.81 2.90 0.0028 0.0022 15 0.00003 0.00000 6.09 0.21 0.0679 0.0065 16 0.00000 0.00000 4.26 3.24 0.0027 0.0025 The magnetic center is at X = 220.1 +- 0.6 microns Y = 151.5 +- 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 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)