SLAC Magnetic Measurements Date: 07-13-1995 Time: 08:04:51 Magnet Name (Serial #): 104 Bar Code Number: 027 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: 07-13-1995 Time: 08:35:18 Magnet Name: 104 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 25.63227 +- 2.240066E-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 -112.05 0.11 0.8132 0.0006 2 0.00005 0.00000 -2.30 0.03 1.2438 0.0023 3 0.00383 0.00000 -29.74 0.00 100.0000 0.0558 4 0.00000 0.00000 43.05 0.16 0.1132 0.0016 5 0.00001 0.00000 19.02 0.12 0.1456 0.0020 6 0.00000 0.00000 28.11 0.30 0.0122 0.0012 7 0.00000 0.00000 12.29 0.46 0.0355 0.0020 8 0.00000 0.00000 -20.71 0.32 0.0188 0.0017 9 0.00001 0.00000 10.44 0.06 0.3378 0.0030 10 0.00000 0.00000 11.87 2.09 0.0057 0.0003 11 0.00000 0.00000 6.99 1.12 0.0060 0.0021 12 0.00000 0.00000 -3.26 9.85 0.0042 0.0016 13 0.00000 0.00000 -5.71 11.26 0.0080 0.0034 14 0.00000 0.00000 -0.04 11.02 0.0110 0.0020 15 0.00000 0.00000 6.19 0.07 0.0758 0.0020 16 0.00000 0.00000 3.30 4.60 0.0070 0.0020 The magnetic center is at X = -22.4 +- 0.3 microns Y = 278.2 +- 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 ^ | S N N --> x S S N Date: 07-13-1995 Time: 08:38:50 Magnet Name: 104 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 50.61197 +- 3.13646E-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 -124.99 0.28 0.2920 0.0043 2 0.00009 0.00000 -1.37 0.13 1.2449 0.0162 3 0.00750 0.00003 -29.74 0.00 100.0000 0.5592 4 0.00001 0.00000 40.87 0.58 0.1154 0.0111 5 0.00001 0.00000 18.82 0.12 0.1545 0.0015 6 0.00000 0.00000 27.78 0.99 0.0148 0.0040 7 0.00000 0.00000 13.10 0.41 0.0361 0.0009 8 0.00000 0.00000 -21.63 0.53 0.0174 0.0014 9 0.00003 0.00000 10.50 0.03 0.3613 0.0019 10 0.00000 0.00000 6.60 12.37 0.0039 0.0011 11 0.00000 0.00000 9.20 1.70 0.0034 0.0012 12 0.00000 0.00000 -8.98 3.09 0.0041 0.0022 13 0.00000 0.00000 1.04 11.98 0.0059 0.0018 14 0.00000 0.00000 5.64 10.45 0.0070 0.0019 15 0.00001 0.00000 6.31 0.04 0.0745 0.0008 16 0.00000 0.00000 3.23 1.42 0.0055 0.0014 The magnetic center is at X = -13.4 +- 1.3 microns Y = 279.0 +- 3.6 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: 07-13-1995 Time: 08:42:27 Magnet Name: 104 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 100.6058 +- 3.044212E-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 157.73 0.92 0.0999 0.0011 2 0.00018 0.00000 -0.94 0.03 1.2298 0.0039 3 0.01493 0.00001 -29.74 0.00 100.0000 0.1110 4 0.00002 0.00000 39.96 0.05 0.1082 0.0021 5 0.00002 0.00000 18.86 0.08 0.1568 0.0019 6 0.00000 0.00000 15.20 24.49 0.0096 0.0010 7 0.00001 0.00000 13.22 0.28 0.0365 0.0021 8 0.00000 0.00000 0.71 21.68 0.0224 0.0032 9 0.00007 0.00000 10.72 0.05 0.4358 0.0027 10 0.00000 0.00000 8.89 4.89 0.0022 0.0010 11 0.00000 0.00000 -3.64 12.38 0.0018 0.0011 12 0.00000 0.00000 -6.71 2.98 0.0017 0.0005 13 0.00000 0.00000 -6.19 10.24 0.0061 0.0022 14 0.00000 0.00000 5.94 10.16 0.0079 0.0021 15 0.00001 0.00000 6.20 0.03 0.0732 0.0014 16 0.00000 0.00000 3.65 2.72 0.0036 0.0008 The magnetic center is at X = -9.1 +- 0.3 microns Y = 275.8 +- 0.9 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: 07-13-1995 Time: 08:49:25 Magnet Name: 104 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 150.5838 +- 2.717104E-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 116.02 0.34 0.1659 0.0024 2 0.00028 0.00000 -1.06 0.03 1.2385 0.0052 3 0.02240 0.00005 -29.73 0.00 100.0000 0.3153 4 0.00003 0.00000 40.79 0.18 0.1205 0.0042 5 0.00004 0.00000 18.89 0.25 0.1673 0.0013 6 0.00000 0.00000 28.49 0.52 0.0211 0.0020 7 0.00001 0.00000 12.84 0.09 0.0388 0.0006 8 0.00001 0.00000 -21.21 0.50 0.0226 0.0009 9 0.00012 0.00000 10.88 0.05 0.5369 0.0017 10 0.00000 0.00000 2.15 2.32 0.0055 0.0010 11 0.00000 0.00000 7.32 2.00 0.0036 0.0008 12 0.00000 0.00000 -0.78 0.54 0.0093 0.0011 13 0.00000 0.00000 5.00 10.95 0.0041 0.0004 14 0.00000 0.00000 -12.10 0.45 0.0078 0.0012 15 0.00002 0.00000 6.20 0.07 0.0730 0.0030 16 0.00000 0.00000 -0.54 0.63 0.0066 0.0020 The magnetic center is at X = -10.3 +- 0.3 microns Y = 277.7 +- 1.2 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: 07-13-1995 Time: 08:56:23 Magnet Name: 104 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 200.5841 +- 5.421795E-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 107.80 0.89 0.1997 0.0036 2 0.00037 0.00000 -0.92 0.02 1.2270 0.0127 3 0.02997 0.00020 -29.73 0.00 100.0000 0.9583 4 0.00003 0.00000 40.13 0.25 0.1154 0.0028 5 0.00005 0.00000 18.74 0.17 0.1691 0.0025 6 0.00000 0.00000 23.72 1.87 0.0124 0.0032 7 0.00001 0.00000 12.33 0.25 0.0375 0.0007 8 0.00001 0.00000 -21.87 0.13 0.0269 0.0006 9 0.00021 0.00000 10.99 0.06 0.6855 0.0148 10 0.00000 0.00000 0.01 0.80 0.0074 0.0013 11 0.00000 0.00000 -7.74 2.74 0.0021 0.0005 12 0.00000 0.00000 -9.21 1.11 0.0061 0.0006 13 0.00000 0.00000 -1.05 7.99 0.0029 0.0014 14 0.00000 0.00000 -10.27 1.99 0.0039 0.0009 15 0.00002 0.00000 6.29 0.20 0.0520 0.0053 16 0.00000 0.00000 -1.17 1.93 0.0023 0.0010 The magnetic center is at X = -8.8 +- 0.3 microns Y = 275.2 +- 2.8 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 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)