SLAC Magnetic Measurements Date: 08-29-1995 Time: 09:56:15 Magnet Name (Serial #): 112 Bar Code Number: 157 Project: PEP II High Energy Ring Sextupoles Test Stand: IR8, S1, W Measurement Coil: # 1 Operator: dan Run Number: 3 Comment: cleaned brushes 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: 08-29-1995 Time: 10:26:36 Magnet Name: 112 Run Number: 3 FIELD HARMONICS Magnet Current, Imag = 25.62763 +- 3.828334E-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 -104.61 0.37 0.9879 0.0028 2 0.00003 0.00000 26.89 0.34 0.4752 0.0044 3 0.00527 0.00001 -29.90 0.00 100.0000 0.3416 4 0.00001 0.00000 21.20 37.86 0.1004 0.0060 5 0.00001 0.00000 18.18 0.06 0.1079 0.0023 6 0.00000 0.00000 -22.67 3.04 0.0175 0.0089 7 0.00000 0.00000 13.28 0.85 0.0268 0.0022 8 0.00000 0.00000 -16.13 4.12 0.0101 0.0031 9 0.00002 0.00000 10.02 0.06 0.3351 0.0048 10 0.00000 0.00000 4.59 6.56 0.0046 0.0015 11 0.00000 0.00000 9.50 6.01 0.0048 0.0010 12 0.00000 0.00000 -2.68 10.04 0.0039 0.0009 13 0.00000 0.00000 -8.34 2.44 0.0068 0.0026 14 0.00000 0.00000 -0.00 3.07 0.0043 0.0016 15 0.00000 0.00000 6.26 0.10 0.0720 0.0008 16 0.00000 0.00000 -2.62 4.32 0.0056 0.0024 The magnetic center is at X = 86.0 +- 1.1 microns Y = 63.0 +- 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: 08-29-1995 Time: 10:30:00 Magnet Name: 112 Run Number: 3 FIELD HARMONICS Magnet Current, Imag = 50.60566 +- 2.707757E-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 -111.80 0.02 0.3564 0.0006 2 0.00005 0.00000 28.58 0.17 0.4974 0.0027 3 0.01027 0.00000 -29.90 0.00 100.0000 0.0321 4 0.00001 0.00000 41.58 1.21 0.0853 0.0076 5 0.00001 0.00000 18.45 0.03 0.1227 0.0020 6 0.00000 0.00000 -19.49 7.56 0.0166 0.0043 7 0.00000 0.00000 13.46 0.62 0.0296 0.0010 8 0.00000 0.00000 -12.27 3.33 0.0084 0.0017 9 0.00003 0.00000 10.04 0.06 0.3323 0.0014 10 0.00000 0.00000 6.31 1.58 0.0040 0.0020 11 0.00000 0.00000 10.02 6.26 0.0021 0.0013 12 0.00000 0.00000 -5.98 5.12 0.0023 0.0013 13 0.00000 0.00000 -3.92 9.09 0.0031 0.0009 14 0.00000 0.00000 -6.89 1.67 0.0054 0.0016 15 0.00001 0.00000 6.15 0.07 0.0717 0.0017 16 0.00000 0.00000 -2.30 7.05 0.0029 0.0016 The magnetic center is at X = 93.8 +- 0.6 microns Y = 60.5 +- 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: 08-29-1995 Time: 10:33:29 Magnet Name: 112 Run Number: 3 FIELD HARMONICS Magnet Current, Imag = 100.5982 +- 4.755318E-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 -176.17 0.44 0.0780 0.0034 2 0.00010 0.00000 29.01 0.12 0.5140 0.0023 3 0.02039 0.00000 -29.91 0.00 100.0000 0.0094 4 0.00002 0.00000 40.26 1.21 0.0901 0.0015 5 0.00003 0.00000 18.63 0.16 0.1286 0.0008 6 0.00000 0.00000 -11.39 23.51 0.0218 0.0040 7 0.00001 0.00000 13.69 0.63 0.0313 0.0015 8 0.00000 0.00000 -14.87 0.51 0.0068 0.0018 9 0.00007 0.00000 10.04 0.07 0.3347 0.0031 10 0.00000 0.00000 8.71 1.97 0.0035 0.0005 11 0.00000 0.00000 -4.52 10.53 0.0032 0.0011 12 0.00000 0.00000 -0.18 0.53 0.0099 0.0006 13 0.00000 0.00000 2.82 9.88 0.0024 0.0011 14 0.00000 0.00000 -1.88 1.21 0.0037 0.0004 15 0.00002 0.00000 6.15 0.07 0.0737 0.0006 16 0.00000 0.00000 -1.27 1.59 0.0028 0.0012 The magnetic center is at X = 97.8 +- 0.5 microns Y = 61.1 +- 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: 08-29-1995 Time: 10:40:12 Magnet Name: 112 Run Number: 3 FIELD HARMONICS Magnet Current, Imag = 150.5854 +- 4.574641E-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 118.50 0.77 0.1255 0.0019 2 0.00016 0.00000 28.91 0.13 0.5156 0.0017 3 0.03047 0.00001 -29.91 0.00 100.0000 0.0243 4 0.00003 0.00000 40.35 0.81 0.0857 0.0027 5 0.00004 0.00000 18.47 0.11 0.1337 0.0036 6 0.00000 0.00000 -17.56 1.52 0.0090 0.0046 7 0.00001 0.00000 13.71 0.51 0.0309 0.0012 8 0.00000 0.00000 -12.23 1.67 0.0079 0.0022 9 0.00010 0.00000 10.04 0.03 0.3339 0.0044 10 0.00000 0.00000 7.37 3.36 0.0035 0.0014 11 0.00000 0.00000 -0.18 8.29 0.0029 0.0004 12 0.00000 0.00000 -0.19 13.83 0.0034 0.0008 13 0.00000 0.00000 -9.31 1.95 0.0029 0.0015 14 0.00000 0.00000 -4.71 1.31 0.0031 0.0012 15 0.00002 0.00000 6.07 0.05 0.0698 0.0004 16 0.00000 0.00000 -3.19 3.20 0.0028 0.0015 The magnetic center is at X = 97.9 +- 0.4 microns Y = 61.6 +- 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: 08-29-1995 Time: 10:46:55 Magnet Name: 112 Run Number: 3 FIELD HARMONICS Magnet Current, Imag = 200.5766 +- 4.761366E-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 107.45 0.79 0.1683 0.0008 2 0.00021 0.00000 28.94 0.17 0.5147 0.0013 3 0.04046 0.00001 -29.91 0.00 100.0000 0.0402 4 0.00004 0.00000 39.85 0.50 0.0902 0.0031 5 0.00005 0.00000 18.49 0.14 0.1292 0.0021 6 0.00001 0.00000 -25.48 3.11 0.0124 0.0038 7 0.00001 0.00000 13.03 0.71 0.0294 0.0017 8 0.00000 0.00000 -14.84 1.05 0.0077 0.0015 9 0.00014 0.00000 10.00 0.04 0.3343 0.0008 10 0.00000 0.00000 3.46 12.75 0.0043 0.0016 11 0.00000 0.00000 -4.26 7.52 0.0028 0.0005 12 0.00000 0.00000 -9.18 1.31 0.0028 0.0004 13 0.00000 0.00000 -7.13 2.11 0.0047 0.0014 14 0.00000 0.00000 -2.99 0.99 0.0042 0.0008 15 0.00003 0.00000 6.08 0.05 0.0701 0.0018 16 0.00000 0.00000 2.92 6.62 0.0027 0.0009 The magnetic center is at X = 97.8 +- 0.4 microns Y = 61.4 +- 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 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)