SLAC Magnetic Measurements Date: 07-17-1995 Time: 07:15:11 Magnet Name (Serial #): 098 Bar Code Number: 014 Project: PEP II High Energy Ring Sextupoles Test Stand: IR8, S1, W Measurement Coil: # 1 Operator: bmd Run Number: 5 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: 07-17-1995 Time: 07:46:46 Magnet Name: 098 Run Number: 5 FIELD HARMONICS Magnet Current, Imag = 25.64802 +- 9.706113E-04 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 -106.93 1.47 1.0995 0.0160 2 0.00007 0.00000 -59.37 0.17 1.3135 0.0114 3 0.00528 0.00000 30.10 0.00 100.0000 0.0682 4 0.00001 0.00000 -5.91 1.02 0.1256 0.0078 5 0.00001 0.00000 -18.38 4.37 0.1494 0.0218 6 0.00000 0.00000 -0.84 3.98 0.0142 0.0068 7 0.00000 0.00000 -4.42 17.45 0.0555 0.0131 8 0.00000 0.00000 5.65 8.16 0.0152 0.0023 9 0.00002 0.00000 -9.95 0.17 0.3322 0.0030 10 0.00000 0.00000 -0.83 9.92 0.0082 0.0055 11 0.00000 0.00000 0.68 8.74 0.0145 0.0118 12 0.00000 0.00000 -6.26 4.38 0.0116 0.0055 13 0.00000 0.00000 -1.86 5.17 0.0071 0.0028 14 0.00000 0.00000 3.42 2.15 0.0082 0.0032 15 0.00000 0.00000 -5.76 0.52 0.0750 0.0042 16 0.00000 0.00000 -3.47 8.50 0.0090 0.0066 The magnetic center is at X = 258.4 +- 2.4 microns Y = 141.8 +- 2.0 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 ^ | N S S --> x N N S Date: 07-17-1995 Time: 07:50:17 Magnet Name: 098 Run Number: 5 FIELD HARMONICS Magnet Current, Imag = 50.62901 +- 1.474313E-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 -104.88 0.67 0.7103 0.0031 2 0.00014 0.00000 -59.69 0.11 1.3259 0.0078 3 0.01028 0.00000 30.10 0.00 100.0000 0.0653 4 0.00001 0.00000 -6.84 1.22 0.1407 0.0127 5 0.00002 0.00000 -16.38 1.77 0.1546 0.0027 6 0.00000 0.00000 -2.95 16.50 0.0157 0.0084 7 0.00000 0.00000 -11.66 4.58 0.0389 0.0092 8 0.00000 0.00000 6.31 14.24 0.0171 0.0053 9 0.00003 0.00000 -9.95 0.11 0.3342 0.0085 10 0.00000 0.00000 -8.16 0.57 0.0105 0.0050 11 0.00000 0.00000 -4.01 10.44 0.0101 0.0074 12 0.00000 0.00000 0.95 5.85 0.0091 0.0080 13 0.00000 0.00000 -2.39 7.92 0.0111 0.0056 14 0.00000 0.00000 -0.12 6.94 0.0126 0.0070 15 0.00001 0.00000 -5.61 0.26 0.0770 0.0040 16 0.00000 0.00000 2.76 4.76 0.0175 0.0105 The magnetic center is at X = 259.3 +- 1.6 microns Y = 146.0 +- 1.3 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 ^ | N S S --> x N N S Date: 07-17-1995 Time: 07:53:54 Magnet Name: 098 Run Number: 5 FIELD HARMONICS Magnet Current, Imag = 100.6213 +- 4.426818E-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 0.00010 0.00000 -101.00 0.49 0.5098 0.0016 2 0.00027 0.00000 -59.40 0.08 1.3183 0.0020 3 0.02040 0.00001 30.09 0.00 100.0000 0.0576 4 0.00003 0.00000 -6.03 0.41 0.1280 0.0028 5 0.00003 0.00000 -16.30 2.06 0.1648 0.0158 6 0.00000 0.00000 14.63 2.42 0.0067 0.0025 7 0.00001 0.00000 -12.59 4.38 0.0424 0.0184 8 0.00000 0.00000 8.04 0.57 0.0147 0.0018 9 0.00007 0.00000 -9.86 0.06 0.3350 0.0008 10 0.00000 0.00000 -10.58 1.81 0.0033 0.0013 11 0.00000 0.00000 -3.76 4.43 0.0053 0.0032 12 0.00000 0.00000 -1.65 0.80 0.0097 0.0013 13 0.00000 0.00000 -6.33 1.68 0.0059 0.0022 14 0.00000 0.00000 2.92 0.76 0.0072 0.0008 15 0.00002 0.00000 -5.77 0.11 0.0770 0.0031 16 0.00000 0.00000 -0.23 0.62 0.0065 0.0057 The magnetic center is at X = 259.3 +- 0.5 microns Y = 142.5 +- 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 ^ | N S S --> x N N S Date: 07-17-1995 Time: 08:00:52 Magnet Name: 098 Run Number: 5 FIELD HARMONICS Magnet Current, Imag = 150.6026 +- 1.59416E-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 0.00014 0.00000 -99.23 0.41 0.4510 0.0022 2 0.00040 0.00000 -59.12 0.04 1.3265 0.0041 3 0.03049 0.00001 30.09 0.00 100.0000 0.0482 4 0.00004 0.00000 -6.14 0.37 0.1354 0.0025 5 0.00005 0.00001 -16.44 1.46 0.1583 0.0231 6 0.00000 0.00000 5.11 2.73 0.0142 0.0047 7 0.00001 0.00000 -10.78 6.31 0.0340 0.0159 8 0.00000 0.00000 7.04 0.69 0.0153 0.0029 9 0.00010 0.00000 -9.87 0.04 0.3308 0.0029 10 0.00000 0.00000 -8.58 4.36 0.0057 0.0024 11 0.00000 0.00000 -5.10 6.96 0.0051 0.0020 12 0.00000 0.00000 -5.45 10.48 0.0042 0.0035 13 0.00000 0.00000 -0.22 6.95 0.0041 0.0013 14 0.00000 0.00000 4.21 1.88 0.0078 0.0016 15 0.00002 0.00000 -5.79 0.23 0.0684 0.0024 16 0.00000 0.00000 -1.12 4.94 0.0040 0.0010 The magnetic center is at X = 262.3 +- 0.8 microns Y = 140.8 +- 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 ^ | N S S --> x N N S Date: 07-17-1995 Time: 08:07:50 Magnet Name: 098 Run Number: 5 FIELD HARMONICS Magnet Current, Imag = 200.597 +- 3.959288E-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 0.00016 0.00000 -98.58 0.23 0.3982 0.0010 2 0.00054 0.00000 -58.86 0.06 1.3328 0.0015 3 0.04047 0.00001 30.10 0.00 100.0000 0.0484 4 0.00005 0.00000 -6.46 0.26 0.1293 0.0022 5 0.00007 0.00000 -17.10 0.24 0.1606 0.0034 6 0.00000 0.00000 4.67 1.98 0.0101 0.0012 7 0.00001 0.00000 -13.07 0.60 0.0339 0.0028 8 0.00001 0.00000 8.04 0.41 0.0163 0.0027 9 0.00014 0.00000 -9.88 0.05 0.3354 0.0007 10 0.00000 0.00000 -7.29 13.90 0.0021 0.0008 11 0.00000 0.00000 4.83 5.26 0.0020 0.0005 12 0.00000 0.00000 1.15 9.83 0.0023 0.0008 13 0.00000 0.00000 -2.36 5.47 0.0025 0.0013 14 0.00000 0.00000 5.65 1.43 0.0054 0.0011 15 0.00003 0.00000 -5.92 0.07 0.0716 0.0011 16 0.00000 0.00000 -4.68 8.56 0.0032 0.0015 The magnetic center is at X = 264.8 +- 0.4 microns Y = 139.1 +- 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 ^ | N S S --> x N N S 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)