SLAC Magnetic Measurements Date: 09-06-1995 Time: 10:20:12 Magnet Name (Serial #): 121 Bar Code Number: 066 Project: PEP II High Energy Ring Sextupoles Test Stand: IR8, S1, N Measurement Coil: # 1 Operator: dan Run Number: 1 Comment: none 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: 09-06-1995 Time: 10:50:31 Magnet Name: 121 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 25.59727 +- 9.863657E-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 0.00005 0.00000 -99.89 0.73 0.9697 0.0062 2 0.00005 0.00000 11.79 0.35 1.0046 0.0028 3 0.00526 0.00000 -29.92 0.00 100.0000 0.0179 4 0.00001 0.00000 40.93 1.71 0.0952 0.0100 5 0.00001 0.00000 19.58 0.59 0.1167 0.0058 6 0.00000 0.00000 -13.66 17.75 0.0214 0.0063 7 0.00000 0.00000 15.66 3.68 0.0259 0.0079 8 0.00000 0.00000 -9.46 18.06 0.0269 0.0082 9 0.00002 0.00000 9.97 0.49 0.3274 0.0146 10 0.00000 0.00000 -0.40 8.69 0.0142 0.0041 11 0.00000 0.00000 0.75 10.73 0.0233 0.0101 12 0.00000 0.00000 -3.36 6.80 0.0200 0.0095 13 0.00000 0.00000 7.41 6.89 0.0200 0.0089 14 0.00000 0.00000 -6.74 4.73 0.0239 0.0113 15 0.00000 0.00000 6.17 0.22 0.0738 0.0093 16 0.00000 0.00000 -3.66 7.00 0.0142 0.0055 The magnetic center is at X = 90.2 +- 2.6 microns Y = 206.6 +- 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 ^ | S N N --> x S S N Date: 09-06-1995 Time: 10:53:55 Magnet Name: 121 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 50.57775 +- 8.045575E-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 0.00004 0.00000 -106.71 0.37 0.3698 0.0039 2 0.00010 0.00000 12.94 0.08 1.0188 0.0025 3 0.01028 0.00000 -29.92 0.00 100.0000 0.0120 4 0.00001 0.00000 39.59 1.05 0.0934 0.0056 5 0.00001 0.00000 19.76 0.07 0.1282 0.0037 6 0.00000 0.00000 -15.94 16.75 0.0084 0.0020 7 0.00000 0.00000 14.42 0.67 0.0292 0.0019 8 0.00000 0.00000 -19.23 1.17 0.0162 0.0041 9 0.00003 0.00000 10.19 0.07 0.3314 0.0072 10 0.00000 0.00000 3.20 6.18 0.0087 0.0039 11 0.00000 0.00000 6.93 10.37 0.0104 0.0030 12 0.00000 0.00000 -0.76 9.14 0.0064 0.0013 13 0.00000 0.00000 9.85 2.37 0.0080 0.0047 14 0.00000 0.00000 -3.62 9.42 0.0048 0.0011 15 0.00001 0.00000 5.95 0.31 0.0699 0.0042 16 0.00000 0.00000 0.64 2.23 0.0063 0.0046 The magnetic center is at X = 99.8 +- 0.6 microns Y = 205.7 +- 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: 09-06-1995 Time: 10:57:24 Magnet Name: 121 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 100.576 +- 1.395392E-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 -158.37 0.31 0.0811 0.0025 2 0.00021 0.00000 13.69 0.05 1.0267 0.0005 3 0.02042 0.00000 -29.93 0.00 100.0000 0.0237 4 0.00002 0.00000 38.41 0.45 0.0945 0.0014 5 0.00003 0.00000 19.35 0.16 0.1321 0.0023 6 0.00000 0.00000 -28.36 1.06 0.0152 0.0013 7 0.00001 0.00000 13.85 0.51 0.0297 0.0029 8 0.00000 0.00000 -18.75 0.63 0.0144 0.0020 9 0.00007 0.00000 10.18 0.01 0.3309 0.0028 10 0.00000 0.00000 8.27 5.35 0.0032 0.0014 11 0.00000 0.00000 -0.04 12.55 0.0055 0.0025 12 0.00000 0.00000 1.10 1.45 0.0091 0.0016 13 0.00000 0.00000 9.41 0.48 0.0077 0.0026 14 0.00000 0.00000 -11.86 0.55 0.0056 0.0031 15 0.00002 0.00000 6.09 0.05 0.0752 0.0016 16 0.00000 0.00000 -2.38 4.25 0.0024 0.0005 The magnetic center is at X = 106.0 +- 0.4 microns Y = 204.6 +- 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 ^ | S N N --> x S S N Date: 09-06-1995 Time: 11:04:07 Magnet Name: 121 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 150.5638 +- 7.877881E-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 0.00003 0.00000 125.92 1.24 0.1073 0.0005 2 0.00031 0.00000 13.92 0.07 1.0204 0.0006 3 0.03052 0.00000 -29.92 0.00 100.0000 0.0101 4 0.00003 0.00000 38.15 0.16 0.0930 0.0020 5 0.00004 0.00000 19.33 0.07 0.1354 0.0017 6 0.00000 0.00000 -28.03 1.24 0.0072 0.0015 7 0.00001 0.00000 13.71 0.32 0.0295 0.0008 8 0.00001 0.00000 -19.23 0.92 0.0170 0.0009 9 0.00010 0.00000 10.12 0.04 0.3303 0.0012 10 0.00000 0.00000 -0.92 2.99 0.0016 0.0009 11 0.00000 0.00000 2.26 10.76 0.0016 0.0005 12 0.00000 0.00000 13.63 0.18 0.0067 0.0005 13 0.00000 0.00000 8.79 1.05 0.0066 0.0023 14 0.00000 0.00000 -11.00 0.71 0.0043 0.0011 15 0.00002 0.00000 6.07 0.03 0.0702 0.0010 16 0.00000 0.00000 -0.00 5.01 0.0021 0.0005 The magnetic center is at X = 106.9 +- 0.5 microns Y = 202.5 +- 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 Date: 09-06-1995 Time: 11:10:50 Magnet Name: 121 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 200.5632 +- 3.252127E-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 0.00005 0.00000 114.58 0.47 0.1306 0.0009 2 0.00041 0.00000 13.99 0.05 1.0213 0.0015 3 0.04052 0.00001 -29.93 0.00 100.0000 0.0182 4 0.00004 0.00000 38.61 0.31 0.0930 0.0012 5 0.00005 0.00000 19.40 0.17 0.1309 0.0014 6 0.00000 0.00000 -26.80 2.05 0.0065 0.0009 7 0.00001 0.00000 13.70 0.23 0.0290 0.0018 8 0.00001 0.00000 -19.52 0.79 0.0144 0.0008 9 0.00013 0.00000 10.10 0.04 0.3323 0.0023 10 0.00000 0.00000 0.07 17.12 0.0033 0.0006 11 0.00000 0.00000 -2.85 10.16 0.0034 0.0011 12 0.00000 0.00000 4.54 10.43 0.0023 0.0012 13 0.00000 0.00000 7.76 1.25 0.0051 0.0019 14 0.00000 0.00000 2.75 9.62 0.0051 0.0021 15 0.00003 0.00000 6.07 0.08 0.0695 0.0020 16 0.00000 0.00000 1.47 6.57 0.0024 0.0008 The magnetic center is at X = 107.5 +- 0.4 microns Y = 202.4 +- 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)