SLAC Magnetic Measurements Date: 07-14-1995 Time: 08:28:55 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: 2 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: 08:59:21 Magnet Name: 104 Run Number: 2 FIELD HARMONICS Magnet Current, Imag = 25.59674 +- 1.386329E-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 -112.62 1.24 0.8025 0.0193 2 0.00005 0.00000 16.35 0.25 1.0067 0.0104 3 0.00525 0.00000 -29.92 0.00 100.0000 0.0633 4 0.00001 0.00000 41.43 0.73 0.1086 0.0053 5 0.00001 0.00000 18.58 0.44 0.1427 0.0065 6 0.00000 0.00000 -17.16 16.26 0.0172 0.0047 7 0.00000 0.00000 13.73 0.67 0.0336 0.0049 8 0.00000 0.00000 -14.05 6.25 0.0219 0.0086 9 0.00002 0.00000 10.22 0.13 0.3326 0.0093 10 0.00000 0.00000 -6.25 4.50 0.0138 0.0112 11 0.00000 0.00000 -3.81 10.88 0.0153 0.0106 12 0.00000 0.00000 7.12 7.47 0.0121 0.0032 13 0.00000 0.00000 3.72 7.59 0.0161 0.0134 14 0.00000 0.00000 -7.77 1.64 0.0110 0.0043 15 0.00000 0.00000 6.68 1.34 0.0780 0.0091 16 0.00000 0.00000 -4.46 3.93 0.0101 0.0084 The magnetic center is at X = 122.1 +- 2.1 microns Y = 190.1 +- 2.2 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: 07-14-1995 Time: 09:02:54 Magnet Name: 104 Run Number: 2 FIELD HARMONICS Magnet Current, Imag = 50.57954 +- 2.008674E-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 -125.26 0.22 0.2789 0.0089 2 0.00011 0.00000 17.75 0.29 1.0290 0.0024 3 0.01027 0.00000 -29.91 0.00 100.0000 0.0340 4 0.00001 0.00000 40.49 0.34 0.1107 0.0085 5 0.00002 0.00000 18.91 0.62 0.1513 0.0046 6 0.00000 0.00000 -27.84 1.43 0.0144 0.0044 7 0.00000 0.00000 13.37 0.59 0.0398 0.0024 8 0.00000 0.00000 -19.60 1.37 0.0117 0.0032 9 0.00003 0.00000 10.11 0.10 0.3309 0.0043 10 0.00000 0.00000 2.18 9.33 0.0067 0.0044 11 0.00000 0.00000 -6.96 8.76 0.0072 0.0054 12 0.00000 0.00000 2.38 8.91 0.0057 0.0023 13 0.00000 0.00000 2.59 9.77 0.0072 0.0008 14 0.00000 0.00000 -1.62 7.43 0.0114 0.0052 15 0.00001 0.00000 6.02 0.12 0.0706 0.0037 16 0.00000 0.00000 0.71 4.44 0.0103 0.0112 The magnetic center is at X = 134.1 +- 2.0 microns Y = 188.0 +- 1.4 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 ^ | S N N --> x S S N Date: 07-14-1995 Time: 09:06:30 Magnet Name: 104 Run Number: 2 FIELD HARMONICS Magnet Current, Imag = 100.5732 +- 4.620117E-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 157.10 1.80 0.1065 0.0037 2 0.00021 0.00000 18.31 0.11 1.0416 0.0018 3 0.02038 0.00000 -29.92 0.00 100.0000 0.0310 4 0.00002 0.00000 39.37 0.60 0.1020 0.0031 5 0.00003 0.00000 18.39 0.03 0.1615 0.0074 6 0.00000 0.00000 -14.17 24.95 0.0178 0.0043 7 0.00001 0.00000 12.98 1.03 0.0424 0.0092 8 0.00000 0.00000 -11.91 7.04 0.0126 0.0071 9 0.00007 0.00000 10.14 0.04 0.3354 0.0033 10 0.00000 0.00000 -1.15 12.82 0.0079 0.0041 11 0.00000 0.00000 4.11 11.50 0.0073 0.0034 12 0.00000 0.00000 -1.05 5.50 0.0083 0.0034 13 0.00000 0.00000 9.39 0.60 0.0090 0.0053 14 0.00000 0.00000 -4.81 10.34 0.0069 0.0038 15 0.00002 0.00000 6.17 0.14 0.0776 0.0055 16 0.00000 0.00000 1.12 5.22 0.0111 0.0121 The magnetic center is at X = 139.4 +- 0.7 microns Y = 187.6 +- 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: 07-14-1995 Time: 09:13:27 Magnet Name: 104 Run Number: 2 FIELD HARMONICS Magnet Current, Imag = 150.5621 +- 5.86624E-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 115.62 0.39 0.1634 0.0012 2 0.00032 0.00000 18.71 0.06 1.0448 0.0034 3 0.03048 0.00002 -29.92 0.00 100.0000 0.0789 4 0.00003 0.00000 39.87 0.38 0.1030 0.0050 5 0.00005 0.00000 18.62 0.14 0.1618 0.0022 6 0.00000 0.00000 -28.54 0.74 0.0095 0.0012 7 0.00001 0.00000 13.58 0.24 0.0399 0.0007 8 0.00000 0.00000 -17.85 0.35 0.0137 0.0020 9 0.00010 0.00000 10.12 0.03 0.3331 0.0020 10 0.00000 0.00000 -3.86 3.44 0.0029 0.0015 11 0.00000 0.00000 13.09 1.83 0.0031 0.0012 12 0.00000 0.00000 13.34 1.08 0.0067 0.0015 13 0.00000 0.00000 9.78 1.11 0.0051 0.0022 14 0.00000 0.00000 -4.90 9.52 0.0047 0.0005 15 0.00002 0.00000 6.11 0.05 0.0700 0.0011 16 0.00000 0.00000 4.33 1.86 0.0025 0.0012 The magnetic center is at X = 142.5 +- 0.6 microns Y = 186.2 +- 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 ^ | S N N --> x S S N Date: 07-14-1995 Time: 09:20:25 Magnet Name: 104 Run Number: 2 FIELD HARMONICS Magnet Current, Imag = 200.5588 +- 3.587617E-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 0.00008 0.00000 108.08 0.30 0.1962 0.0013 2 0.00042 0.00000 19.36 0.04 1.0452 0.0012 3 0.04046 0.00001 -29.91 0.00 100.0000 0.0323 4 0.00004 0.00000 39.07 0.23 0.1005 0.0015 5 0.00006 0.00000 18.57 0.02 0.1592 0.0024 6 0.00001 0.00000 0.39 29.39 0.0135 0.0017 7 0.00002 0.00000 13.79 0.18 0.0405 0.0023 8 0.00001 0.00000 -17.70 0.73 0.0135 0.0011 9 0.00014 0.00000 10.15 0.02 0.3362 0.0010 10 0.00000 0.00000 -15.41 3.70 0.0034 0.0011 11 0.00000 0.00000 12.46 0.80 0.0037 0.0004 12 0.00000 0.00000 11.43 1.12 0.0022 0.0004 13 0.00000 0.00000 10.56 0.91 0.0059 0.0025 14 0.00000 0.00000 -10.22 1.70 0.0037 0.0012 15 0.00003 0.00000 6.06 0.04 0.0716 0.0004 16 0.00000 0.00000 -5.51 8.61 0.0023 0.0008 The magnetic center is at X = 146.7 +- 0.3 microns Y = 183.0 +- 0.3 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 ^ | 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)