SLAC Magnetic Measurements Date: 07-14-1995 Time: 06:23:31 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: 4 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: 07-14-1995 Time: 06:54:07 Magnet Name: 098 Run Number: FIELD HARMONICS Magnet Current, Imag = 25.63312 +- 1.693374E-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 -107.41 0.17 1.1039 0.0016 2 0.00007 0.00000 -56.95 0.09 1.3603 0.0028 3 0.00528 0.00000 30.12 0.00 100.0000 0.0436 4 0.00001 0.00000 -6.33 0.61 0.1287 0.0049 5 0.00001 0.00000 -16.67 0.35 0.1484 0.0032 6 0.00000 0.00000 3.24 2.02 0.0148 0.0052 7 0.00000 0.00000 -11.68 0.94 0.0378 0.0095 8 0.00000 0.00000 7.49 3.11 0.0185 0.0032 9 0.00002 0.00000 -9.77 0.13 0.3366 0.0039 10 0.00000 0.00000 -1.62 5.34 0.0044 0.0037 11 0.00000 0.00000 1.68 10.47 0.0085 0.0018 12 0.00000 0.00000 0.49 7.43 0.0092 0.0017 13 0.00000 0.00000 2.49 9.37 0.0088 0.0018 14 0.00000 0.00000 1.35 5.66 0.0072 0.0034 15 0.00000 0.00000 -5.76 0.06 0.0708 0.0039 16 0.00000 0.00000 3.88 3.21 0.0061 0.0021 The magnetic center is at X = 279.1 +- 0.7 microns Y = 123.6 +- 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 ^ | N S S --> x N N S Date: 07-14-1995 Time: 06:57:39 Magnet Name: 098 Run Number: FIELD HARMONICS Magnet Current, Imag = 50.61283 +- 2.564155E-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 -104.49 0.17 0.7102 0.0021 2 0.00014 0.00000 -57.51 0.01 1.3567 0.0023 3 0.01028 0.00001 30.12 0.00 100.0000 0.0867 4 0.00001 0.00000 -5.60 0.20 0.1339 0.0025 5 0.00002 0.00000 -16.88 0.19 0.1566 0.0046 6 0.00000 0.00000 5.40 2.97 0.0142 0.0028 7 0.00000 0.00000 -12.82 0.50 0.0396 0.0011 8 0.00000 0.00000 7.87 2.52 0.0182 0.0054 9 0.00003 0.00000 -9.92 0.07 0.3321 0.0043 10 0.00000 0.00000 1.45 10.15 0.0053 0.0035 11 0.00000 0.00000 7.79 6.75 0.0030 0.0013 12 0.00000 0.00000 1.99 4.56 0.0039 0.0010 13 0.00000 0.00000 3.07 8.93 0.0045 0.0012 14 0.00000 0.00000 4.67 1.13 0.0072 0.0023 15 0.00001 0.00000 -5.77 0.14 0.0715 0.0013 16 0.00000 0.00000 -6.31 4.35 0.0043 0.0018 The magnetic center is at X = 275.9 +- 0.5 microns Y = 128.8 +- 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 ^ | N S S --> x N N S Date: 07-14-1995 Time: 07:01:16 Magnet Name: 098 Run Number: FIELD HARMONICS Magnet Current, Imag = 100.6055 +- 1.62724E-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 0.00011 0.00000 -101.47 0.28 0.5172 0.0012 2 0.00027 0.00000 -57.30 0.04 1.3458 0.0040 3 0.02040 0.00001 30.12 0.00 100.0000 0.0443 4 0.00003 0.00000 -5.99 0.12 0.1268 0.0022 5 0.00003 0.00000 -16.98 0.15 0.1649 0.0017 6 0.00000 0.00000 13.91 8.07 0.0033 0.0015 7 0.00001 0.00000 -12.94 1.43 0.0380 0.0022 8 0.00000 0.00000 7.90 1.37 0.0147 0.0013 9 0.00007 0.00000 -9.87 0.01 0.3340 0.0025 10 0.00000 0.00000 -2.25 12.65 0.0044 0.0007 11 0.00000 0.00000 -4.08 3.87 0.0042 0.0010 12 0.00000 0.00000 0.97 4.58 0.0058 0.0021 13 0.00000 0.00000 -1.79 2.71 0.0025 0.0011 14 0.00000 0.00000 2.84 1.87 0.0043 0.0017 15 0.00001 0.00000 -5.68 0.09 0.0731 0.0030 16 0.00000 0.00000 3.11 3.45 0.0033 0.0015 The magnetic center is at X = 274.6 +- 0.8 microns Y = 125.7 +- 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 ^ | N S S --> x N N S Date: 07-14-1995 Time: 07:08:13 Magnet Name: 098 Run Number: FIELD HARMONICS Magnet Current, Imag = 150.5815 +- 1.795325E-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 -100.18 0.28 0.4540 0.0027 2 0.00041 0.00000 -57.14 0.04 1.3426 0.0017 3 0.03047 0.00001 30.12 0.00 100.0000 0.0502 4 0.00004 0.00000 -6.12 0.27 0.1302 0.0035 5 0.00005 0.00000 -16.93 0.17 0.1676 0.0012 6 0.00000 0.00000 3.14 0.79 0.0112 0.0018 7 0.00001 0.00000 -11.92 0.66 0.0374 0.0021 8 0.00001 0.00000 7.94 0.60 0.0164 0.0016 9 0.00010 0.00000 -9.82 0.04 0.3346 0.0009 10 0.00000 0.00000 -1.65 10.50 0.0019 0.0011 11 0.00000 0.00000 -5.67 5.08 0.0030 0.0008 12 0.00000 0.00000 14.18 0.54 0.0031 0.0023 13 0.00000 0.00000 -7.41 5.17 0.0037 0.0009 14 0.00000 0.00000 3.82 1.31 0.0091 0.0024 15 0.00002 0.00000 -5.75 0.23 0.0693 0.0034 16 0.00000 0.00000 0.96 1.57 0.0035 0.0002 The magnetic center is at X = 274.6 +- 0.4 microns Y = 123.9 +- 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 ^ | N S S --> x N N S Date: 07-14-1995 Time: 07:15:11 Magnet Name: 098 Run Number: FIELD HARMONICS Magnet Current, Imag = 200.5833 +- 3.204264E-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 -99.62 0.13 0.4041 0.0014 2 0.00054 0.00000 -57.00 0.05 1.3408 0.0025 3 0.04046 0.00001 30.12 0.00 100.0000 0.0468 4 0.00005 0.00000 -6.49 0.28 0.1254 0.0009 5 0.00007 0.00000 -16.92 0.13 0.1635 0.0023 6 0.00000 0.00000 4.33 2.79 0.0071 0.0016 7 0.00001 0.00000 -11.97 0.10 0.0350 0.0036 8 0.00001 0.00000 7.66 1.37 0.0158 0.0011 9 0.00014 0.00000 -9.84 0.05 0.3342 0.0024 10 0.00000 0.00000 -0.61 13.17 0.0060 0.0003 11 0.00000 0.00000 -2.18 6.78 0.0028 0.0003 12 0.00000 0.00000 1.34 3.31 0.0046 0.0019 13 0.00000 0.00000 -2.07 5.33 0.0042 0.0003 14 0.00000 0.00000 3.27 1.02 0.0062 0.0012 15 0.00003 0.00000 -5.70 0.11 0.0729 0.0009 16 0.00000 0.00000 3.81 8.63 0.0037 0.0010 The magnetic center is at X = 274.9 +- 0.6 microns Y = 122.4 +- 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 ^ | 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)