SLAC Magnetic Measurements Date: 07-10-1995 Time: 14:41:43 Magnet Name (Serial #): 020 Bar Code Number: 017 Project: PEP II High Energy Ring Sextupoles Test Stand: IR8, S1, W Measurement Coil: # 1 Operator: pdr Run Number: 1 Comment: Standard Measurement 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: 07-10-1995 Time: 15:12:05 Magnet Name: 020 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 25.57384 +- 1.276832E-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 -104.21 0.27 1.1806 0.0021 2 0.00008 0.00000 -59.64 0.09 1.5993 0.0055 3 0.00525 0.00000 30.10 0.00 100.0000 0.0402 4 0.00001 0.00000 -5.44 1.97 0.1220 0.0050 5 0.00001 0.00000 -17.01 0.26 0.1201 0.0049 6 0.00000 0.00000 1.74 10.42 0.0099 0.0082 7 0.00000 0.00000 -12.28 1.58 0.0301 0.0046 8 0.00000 0.00000 6.76 5.04 0.0214 0.0064 9 0.00002 0.00000 -9.85 0.17 0.3317 0.0098 10 0.00000 0.00000 -1.86 11.81 0.0127 0.0030 11 0.00000 0.00000 -4.82 6.38 0.0124 0.0026 12 0.00000 0.00000 -4.69 6.63 0.0070 0.0043 13 0.00000 0.00000 2.11 7.28 0.0080 0.0035 14 0.00000 0.00000 1.24 3.24 0.0095 0.0035 15 0.00000 0.00000 -5.83 0.23 0.0749 0.0023 16 0.00000 0.00000 0.96 6.27 0.0052 0.0015 The magnetic center is at X = 313.0 +- 1.2 microns Y = 175.5 +- 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 ^ | N S S --> x N N S Date: 07-10-1995 Time: 15:15:37 Magnet Name: 020 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 50.55783 +- 9.19708E-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 0.00008 0.00000 -101.54 0.16 0.7952 0.0032 2 0.00017 0.00000 -59.44 0.07 1.6130 0.0022 3 0.01027 0.00000 30.10 0.00 100.0000 0.0125 4 0.00001 0.00000 -6.26 0.43 0.1298 0.0053 5 0.00001 0.00000 -17.59 0.38 0.1305 0.0023 6 0.00000 0.00000 3.58 11.47 0.0072 0.0005 7 0.00000 0.00000 -13.34 1.13 0.0299 0.0022 8 0.00000 0.00000 8.63 1.11 0.0168 0.0040 9 0.00003 0.00000 -9.85 0.04 0.3330 0.0017 10 0.00000 0.00000 -8.71 8.57 0.0065 0.0020 11 0.00000 0.00000 4.33 7.23 0.0033 0.0010 12 0.00000 0.00000 -2.29 7.45 0.0036 0.0034 13 0.00000 0.00000 7.14 4.76 0.0052 0.0024 14 0.00000 0.00000 3.79 0.63 0.0107 0.0053 15 0.00001 0.00000 -5.84 0.15 0.0734 0.0038 16 0.00000 0.00000 2.81 6.93 0.0037 0.0028 The magnetic center is at X = 316.9 +- 0.6 microns Y = 174.9 +- 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: 07-10-1995 Time: 15:19:13 Magnet Name: 020 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 100.5541 +- 1.330723E-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 0.00012 0.00000 -99.04 0.44 0.5963 0.0029 2 0.00033 0.00000 -59.46 0.06 1.6205 0.0016 3 0.02039 0.00001 30.10 0.00 100.0000 0.0622 4 0.00003 0.00000 -5.28 0.74 0.1363 0.0063 5 0.00003 0.00000 -17.39 0.36 0.1321 0.0024 6 0.00000 0.00000 16.32 4.21 0.0126 0.0068 7 0.00001 0.00000 -13.67 1.11 0.0276 0.0020 8 0.00000 0.00000 7.66 1.28 0.0164 0.0032 9 0.00007 0.00000 -9.93 0.07 0.3305 0.0017 10 0.00000 0.00000 -5.07 12.83 0.0053 0.0035 11 0.00000 0.00000 -5.51 7.89 0.0026 0.0014 12 0.00000 0.00000 0.95 3.02 0.0066 0.0022 13 0.00000 0.00000 1.10 5.34 0.0027 0.0006 14 0.00000 0.00000 2.51 1.65 0.0070 0.0023 15 0.00001 0.00000 -5.86 0.19 0.0691 0.0023 16 0.00000 0.00000 -2.77 2.36 0.0029 0.0011 The magnetic center is at X = 318.3 +- 0.5 microns Y = 175.9 +- 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 ^ | N S S --> x N N S Date: 07-10-1995 Time: 15:26:10 Magnet Name: 020 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 150.5378 +- 2.454982E-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 -97.34 0.33 0.5399 0.0018 2 0.00050 0.00000 -59.52 0.06 1.6275 0.0019 3 0.03046 0.00000 30.10 0.00 100.0000 0.0228 4 0.00004 0.00000 -5.78 0.71 0.1398 0.0044 5 0.00004 0.00000 -17.45 0.23 0.1358 0.0029 6 0.00000 0.00000 6.76 6.34 0.0080 0.0050 7 0.00001 0.00000 -13.46 0.31 0.0274 0.0035 8 0.00001 0.00000 7.95 0.62 0.0175 0.0016 9 0.00010 0.00000 -9.90 0.02 0.3310 0.0031 10 0.00000 0.00000 -7.21 12.11 0.0060 0.0017 11 0.00000 0.00000 -0.92 9.22 0.0034 0.0012 12 0.00000 0.00000 0.08 13.17 0.0064 0.0012 13 0.00000 0.00000 -3.61 3.43 0.0040 0.0011 14 0.00000 0.00000 3.77 1.74 0.0050 0.0012 15 0.00002 0.00000 -5.74 0.12 0.0708 0.0048 16 0.00000 0.00000 -0.46 7.70 0.0031 0.0014 The magnetic center is at X = 319.3 +- 0.5 microns Y = 177.3 +- 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 ^ | N S S --> x N N S Date: 07-10-1995 Time: 15:33:06 Magnet Name: 020 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 200.5429 +- 1.200072E-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 0.00020 0.00000 -96.69 0.20 0.4965 0.0006 2 0.00066 0.00000 -59.86 0.06 1.6416 0.0018 3 0.04043 0.00001 30.10 0.00 100.0000 0.0302 4 0.00006 0.00000 -6.15 0.20 0.1448 0.0017 5 0.00006 0.00000 -17.30 0.16 0.1376 0.0022 6 0.00000 0.00000 8.09 4.83 0.0061 0.0029 7 0.00001 0.00000 -13.65 0.93 0.0280 0.0018 8 0.00001 0.00000 6.57 0.88 0.0202 0.0006 9 0.00013 0.00000 -9.93 0.08 0.3337 0.0012 10 0.00000 0.00000 -2.10 12.82 0.0069 0.0022 11 0.00000 0.00000 -2.63 10.85 0.0021 0.0010 12 0.00000 0.00000 -3.28 7.10 0.0035 0.0009 13 0.00000 0.00000 -0.82 7.17 0.0030 0.0014 14 0.00000 0.00000 4.23 1.24 0.0049 0.0026 15 0.00003 0.00000 -5.79 0.15 0.0716 0.0010 16 0.00000 0.00000 -2.82 8.63 0.0052 0.0017 The magnetic center is at X = 319.9 +- 0.5 microns Y = 182.6 +- 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 ^ | 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)