SLAC Magnetic Measurements Date: 08-03-1995 Time: 07:57:26 Magnet Name (Serial #): 016 Bar Code Number: 128 Project: PEP II High Energy Ring Sextupoles Test Stand: IR8, S1, N Measurement Coil: # 1 Operator: bda 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: 08-03-1995 Time: 08:27:47 Magnet Name: 016 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 25.62136 +- 3.250325E-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 0.00013 0.00001 -82.17 1.38 2.4081 0.1919 2 0.00008 0.00000 -4.98 0.42 1.4727 0.0173 3 0.00526 0.00000 -29.87 0.00 100.0000 0.0517 4 0.00001 0.00000 39.27 2.91 0.1330 0.0255 5 0.00001 0.00000 18.58 2.98 0.1436 0.0527 6 0.00000 0.00000 -6.68 21.69 0.0342 0.0183 7 0.00000 0.00000 15.00 6.03 0.0723 0.0074 8 0.00000 0.00000 13.69 4.99 0.0218 0.0080 9 0.00002 0.00000 10.32 0.42 0.3408 0.0264 10 0.00000 0.00000 2.17 9.26 0.0272 0.0145 11 0.00000 0.00000 10.62 3.55 0.0217 0.0098 12 0.00000 0.00000 6.03 6.20 0.0303 0.0137 13 0.00000 0.00000 -1.59 7.33 0.0346 0.0062 14 0.00000 0.00000 4.01 6.93 0.0258 0.0128 15 0.00000 0.00000 5.63 0.94 0.0725 0.0230 16 0.00000 0.00000 -2.87 6.67 0.0311 0.0277 The magnetic center is at X = -57.2 +- 4.8 microns Y = 325.5 +- 3.9 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: 08-03-1995 Time: 08:31:11 Magnet Name: 016 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 50.60182 +- 5.217489E-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 0.00003 -97.55 16.10 0.7005 0.3342 2 0.00016 0.00000 -5.21 0.18 1.5488 0.0122 3 0.01026 0.00000 -29.87 0.00 100.0000 0.0386 4 0.00001 0.00000 39.00 0.63 0.1255 0.0096 5 0.00002 0.00000 16.72 1.28 0.1654 0.0075 6 0.00000 0.00000 -26.30 1.29 0.0158 0.0070 7 0.00000 0.00000 11.88 1.31 0.0388 0.0060 8 0.00000 0.00000 1.20 20.63 0.0238 0.0078 9 0.00003 0.00000 10.11 0.06 0.3259 0.0107 10 0.00000 0.00000 0.90 3.24 0.0103 0.0072 11 0.00000 0.00000 -3.30 6.26 0.0107 0.0053 12 0.00000 0.00000 3.83 3.46 0.0121 0.0091 13 0.00000 0.00000 3.73 5.84 0.0127 0.0062 14 0.00000 0.00000 4.85 9.87 0.0080 0.0048 15 0.00001 0.00000 5.85 0.36 0.0673 0.0075 16 0.00000 0.00000 1.06 6.74 0.0106 0.0082 The magnetic center is at X = -62.9 +- 2.2 microns Y = 341.8 +- 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 ^ | S N N --> x S S N Date: 08-03-1995 Time: 08:34:41 Magnet Name: 016 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 100.5973 +- 6.594514E-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 0.00003 -142.53 35.16 0.1759 0.1458 2 0.00032 0.00000 -5.21 0.14 1.5570 0.0025 3 0.02038 0.00001 -29.88 0.00 100.0000 0.0492 4 0.00003 0.00000 39.14 0.72 0.1237 0.0054 5 0.00003 0.00000 18.32 0.21 0.1688 0.0023 6 0.00000 0.00000 -27.32 0.79 0.0238 0.0021 7 0.00001 0.00000 13.94 0.56 0.0430 0.0020 8 0.00000 0.00000 10.93 19.13 0.0188 0.0009 9 0.00007 0.00000 10.18 0.03 0.3330 0.0007 10 0.00000 0.00000 12.04 1.89 0.0082 0.0077 11 0.00000 0.00000 3.16 10.46 0.0028 0.0015 12 0.00000 0.00000 1.49 3.38 0.0126 0.0041 13 0.00000 0.00000 6.25 9.18 0.0069 0.0058 14 0.00000 0.00000 4.18 9.78 0.0057 0.0011 15 0.00001 0.00000 6.45 0.48 0.0676 0.0078 16 0.00000 0.00000 1.71 4.15 0.0038 0.0026 The magnetic center is at X = -63.2 +- 1.7 microns Y = 343.7 +- 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: 08-03-1995 Time: 08:41:24 Magnet Name: 016 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 150.5808 +- 9.279031E-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 126.42 0.45 0.1306 0.0008 2 0.00048 0.00000 -5.32 0.04 1.5606 0.0008 3 0.03046 0.00000 -29.88 0.00 100.0000 0.0196 4 0.00004 0.00000 39.74 0.36 0.1286 0.0018 5 0.00005 0.00000 18.15 0.07 0.1772 0.0013 6 0.00000 0.00000 -26.59 0.75 0.0159 0.0017 7 0.00001 0.00000 13.16 0.24 0.0421 0.0011 8 0.00001 0.00000 21.65 0.22 0.0201 0.0012 9 0.00010 0.00000 10.16 0.01 0.3328 0.0010 10 0.00000 0.00000 4.39 3.25 0.0051 0.0016 11 0.00000 0.00000 -2.50 7.86 0.0028 0.0008 12 0.00000 0.00000 5.97 11.52 0.0052 0.0017 13 0.00000 0.00000 6.22 6.06 0.0029 0.0016 14 0.00000 0.00000 5.06 10.23 0.0069 0.0023 15 0.00002 0.00000 6.10 0.09 0.0706 0.0006 16 0.00000 0.00000 -4.22 3.74 0.0018 0.0005 The magnetic center is at X = -64.7 +- 0.5 microns Y = 344.2 +- 0.2 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: 08-03-1995 Time: 08:48:07 Magnet Name: 016 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 200.5721 +- 5.991067E-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 0.00002 23.74 89.18 0.0887 0.0525 2 0.00063 0.00000 -5.39 0.05 1.5623 0.0045 3 0.04043 0.00001 -29.88 0.00 100.0000 0.0421 4 0.00005 0.00000 40.08 0.26 0.1315 0.0053 5 0.00007 0.00000 18.34 0.12 0.1738 0.0015 6 0.00001 0.00000 -26.14 2.13 0.0173 0.0045 7 0.00002 0.00000 12.46 0.55 0.0443 0.0021 8 0.00001 0.00000 21.40 0.47 0.0225 0.0034 9 0.00013 0.00000 10.10 0.01 0.3317 0.0011 10 0.00000 0.00000 7.38 3.23 0.0055 0.0023 11 0.00000 0.00000 -5.67 3.91 0.0025 0.0002 12 0.00000 0.00000 4.23 3.72 0.0028 0.0011 13 0.00000 0.00000 2.87 2.46 0.0064 0.0037 14 0.00000 0.00000 5.52 10.08 0.0068 0.0019 15 0.00003 0.00000 6.00 0.10 0.0756 0.0027 16 0.00000 0.00000 -3.92 5.54 0.0028 0.0017 The magnetic center is at X = -65.6 +- 0.6 microns Y = 344.4 +- 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 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)