SLAC Magnetic Measurements Date: 08-14-1995 Time: 06:40:16 Magnet Name (Serial #): 098 Bar Code Number: 14 Project: PEP II High Energy Ring Sextupoles Test Stand: IR8, S1, W Measurement Coil: # 1 Operator: b Run Number: 9 Comment: monref 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: 08-14-1995 Time: 07:10:41 Magnet Name: 098 Run Number: 9 FIELD HARMONICS Magnet Current, Imag = 25.63099 +- 1.410441E-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 -99.37 0.16 1.1797 0.0011 2 0.00006 0.00000 -51.60 0.11 1.1531 0.0024 3 0.00527 0.00000 30.07 0.00 100.0000 0.0261 4 0.00001 0.00000 -6.12 0.51 0.1223 0.0028 5 0.00001 0.00000 -16.53 0.18 0.1452 0.0011 6 0.00000 0.00000 1.30 1.58 0.0186 0.0012 7 0.00000 0.00000 -11.64 0.74 0.0382 0.0029 8 0.00000 0.00000 12.75 0.69 0.0179 0.0014 9 0.00002 0.00000 -9.91 0.09 0.3266 0.0041 10 0.00000 0.00000 14.18 2.39 0.0057 0.0019 11 0.00000 0.00000 8.11 1.81 0.0111 0.0023 12 0.00000 0.00000 11.59 0.75 0.0126 0.0015 13 0.00000 0.00000 -9.09 1.39 0.0129 0.0019 14 0.00000 0.00000 2.88 1.24 0.0049 0.0021 15 0.00000 0.00000 -5.92 0.07 0.0733 0.0015 16 0.00000 0.00000 4.90 2.55 0.0038 0.0012 The magnetic center is at X = 251.9 +- 0.6 microns Y = 59.1 +- 1.0 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: 08-14-1995 Time: 07:14:06 Magnet Name: 098 Run Number: 9 FIELD HARMONICS Magnet Current, Imag = 50.61001 +- 4.239116E-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 -99.09 0.61 0.7513 0.0038 2 0.00012 0.00000 -51.74 0.14 1.1449 0.0130 3 0.01030 0.00002 30.07 0.00 100.0000 0.2614 4 0.00001 0.00000 -5.02 0.44 0.1401 0.0147 5 0.00002 0.00000 -16.24 0.42 0.1665 0.0077 6 0.00000 0.00000 2.70 7.48 0.0248 0.0136 7 0.00000 0.00000 -10.41 2.08 0.0428 0.0080 8 0.00000 0.00000 10.72 4.42 0.0262 0.0067 9 0.00003 0.00000 -9.90 0.21 0.3373 0.0068 10 0.00000 0.00000 3.93 12.87 0.0212 0.0132 11 0.00000 0.00000 -0.27 6.11 0.0103 0.0048 12 0.00000 0.00000 11.91 2.43 0.0151 0.0072 13 0.00000 0.00000 -9.22 2.58 0.0242 0.0101 14 0.00000 0.00000 1.88 2.20 0.0096 0.0045 15 0.00001 0.00000 -5.61 0.77 0.0784 0.0221 16 0.00000 0.00000 1.82 3.99 0.0133 0.0054 The magnetic center is at X = 249.8 +- 2.9 microns Y = 59.8 +- 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 ^ | N S S --> x N N S Date: 08-14-1995 Time: 07:17:35 Magnet Name: 098 Run Number: 9 FIELD HARMONICS Magnet Current, Imag = 100.6023 +- 3.415978E-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 -97.14 0.37 0.5430 0.0039 2 0.00023 0.00000 -51.59 0.17 1.1386 0.0049 3 0.02039 0.00000 30.07 0.00 100.0000 0.0139 4 0.00003 0.00000 -5.82 1.17 0.1275 0.0065 5 0.00003 0.00000 -16.54 0.54 0.1637 0.0071 6 0.00000 0.00000 -3.81 13.45 0.0122 0.0047 7 0.00001 0.00000 -12.52 0.82 0.0341 0.0081 8 0.00000 0.00000 8.32 5.57 0.0202 0.0064 9 0.00007 0.00000 -10.01 0.12 0.3272 0.0047 10 0.00000 0.00000 -0.58 11.43 0.0152 0.0100 11 0.00000 0.00000 7.85 2.23 0.0094 0.0036 12 0.00000 0.00000 2.97 10.70 0.0122 0.0043 13 0.00000 0.00000 -10.23 1.36 0.0139 0.0059 14 0.00000 0.00000 -2.80 7.05 0.0070 0.0027 15 0.00001 0.00000 -5.84 0.19 0.0689 0.0038 16 0.00000 0.00000 -3.66 5.41 0.0054 0.0021 The magnetic center is at X = 248.8 +- 1.1 microns Y = 58.2 +- 1.5 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: 08-14-1995 Time: 07:24:18 Magnet Name: 098 Run Number: 9 FIELD HARMONICS Magnet Current, Imag = 150.586 +- 4.784397E-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 -95.69 0.36 0.4753 0.0023 2 0.00035 0.00000 -51.36 0.13 1.1378 0.0018 3 0.03048 0.00001 30.07 0.00 100.0000 0.0320 4 0.00004 0.00000 -5.97 0.53 0.1227 0.0042 5 0.00005 0.00000 -16.89 0.13 0.1723 0.0035 6 0.00001 0.00000 1.99 1.30 0.0177 0.0054 7 0.00001 0.00000 -11.81 0.18 0.0396 0.0008 8 0.00001 0.00000 12.51 1.23 0.0235 0.0059 9 0.00010 0.00000 -9.87 0.05 0.3349 0.0044 10 0.00000 0.00000 11.55 4.87 0.0159 0.0069 11 0.00000 0.00000 6.95 1.78 0.0119 0.0044 12 0.00001 0.00000 11.22 0.80 0.0190 0.0032 13 0.00000 0.00000 -11.15 2.10 0.0109 0.0040 14 0.00000 0.00000 0.88 3.68 0.0075 0.0023 15 0.00002 0.00000 -6.19 0.26 0.0696 0.0012 16 0.00000 0.00000 0.75 2.83 0.0070 0.0021 The magnetic center is at X = 249.1 +- 0.5 microns Y = 56.2 +- 1.1 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: 08-14-1995 Time: 07:31:01 Magnet Name: 098 Run Number: 9 FIELD HARMONICS Magnet Current, Imag = 200.5856 +- 3.229001E-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 0.00017 0.00000 -94.52 0.33 0.4214 0.0016 2 0.00045 0.00000 -51.30 0.10 1.1245 0.0027 3 0.04046 0.00001 30.07 0.00 100.0000 0.0245 4 0.00005 0.00000 -6.10 0.44 0.1220 0.0020 5 0.00007 0.00000 -16.63 0.11 0.1637 0.0054 6 0.00001 0.00000 -0.64 1.40 0.0161 0.0026 7 0.00001 0.00000 -11.88 0.48 0.0339 0.0015 8 0.00001 0.00000 10.95 0.67 0.0200 0.0013 9 0.00013 0.00000 -9.86 0.03 0.3314 0.0020 10 0.00000 0.00000 6.51 13.76 0.0075 0.0038 11 0.00000 0.00000 7.26 0.37 0.0115 0.0018 12 0.00000 0.00000 9.96 0.29 0.0117 0.0024 13 0.00001 0.00000 -10.11 0.34 0.0137 0.0025 14 0.00000 0.00000 3.05 1.03 0.0042 0.0017 15 0.00003 0.00000 -5.79 0.09 0.0752 0.0022 16 0.00000 0.00000 0.55 3.84 0.0023 0.0011 The magnetic center is at X = 246.3 +- 0.6 microns Y = 55.0 +- 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 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)