SLAC Magnetic Measurements Date: 08-29-1995 Time: 08:00:03 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: 2 Comment: missalined while taking wide bar out 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: 08:30:32 Magnet Name: 112 Run Number: 2 FIELD HARMONICS Magnet Current, Imag = 25.67281 +- 3.604266E-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 0.00075 0.00001 -91.22 0.00 15.8070 0.2166 2 0.00076 0.00001 -0.92 0.02 16.0096 0.2246 3 0.00472 0.00001 -29.91 0.00 100.0000 0.3325 4 0.00033 0.00000 -44.85 0.00 6.9217 0.0963 5 0.00013 0.00000 -17.87 0.01 2.6525 0.0405 6 0.00001 0.00000 -0.13 0.24 0.1266 0.0039 7 0.00001 0.00000 -12.76 0.21 0.1164 0.0048 8 0.00000 0.00000 0.52 0.45 0.0828 0.0012 9 0.00002 0.00000 10.00 0.16 0.3402 0.0101 10 0.00000 0.00000 0.28 0.41 0.0892 0.0034 11 0.00000 0.00000 8.04 0.37 0.0658 0.0066 12 0.00000 0.00000 4.70 8.29 0.0069 0.0033 13 0.00000 0.00000 6.85 0.74 0.0130 0.0056 14 0.00000 0.00000 5.47 10.57 0.0117 0.0016 15 0.00000 0.00000 6.09 0.30 0.0739 0.0059 16 0.00000 0.00000 0.97 2.73 0.0056 0.0022 The magnetic center is at X = -115.3 +- 2.8 microns Y = 3590.9 +- 50.4 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: 08-29-1995 Time: 08:33:56 Magnet Name: 112 Run Number: 2 FIELD HARMONICS Magnet Current, Imag = 50.652 +- 3.845214E-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 0.00126 0.00000 -90.82 0.01 13.4846 0.0165 2 0.00133 0.00000 -0.94 0.00 14.3001 0.0207 3 0.00932 0.00000 -29.91 0.00 100.0000 0.0498 4 0.00057 0.00000 -44.89 0.00 6.1641 0.0084 5 0.00022 0.00000 -17.88 0.01 2.3368 0.0033 6 0.00001 0.00000 -0.25 0.05 0.1099 0.0028 7 0.00001 0.00000 -12.91 0.25 0.1043 0.0027 8 0.00001 0.00000 0.75 0.38 0.0728 0.0052 9 0.00003 0.00000 10.07 0.09 0.3352 0.0020 10 0.00001 0.00000 0.33 0.28 0.0831 0.0057 11 0.00001 0.00000 8.26 0.29 0.0566 0.0019 12 0.00000 0.00000 -1.49 6.95 0.0033 0.0014 13 0.00000 0.00000 7.64 1.48 0.0106 0.0024 14 0.00000 0.00000 11.81 0.92 0.0131 0.0017 15 0.00001 0.00000 6.18 0.19 0.0715 0.0008 16 0.00000 0.00000 -0.79 2.77 0.0058 0.0017 The magnetic center is at X = -105.3 +- 0.3 microns Y = 3207.4 +- 4.7 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: 08-29-1995 Time: 08:37:26 Magnet Name: 112 Run Number: 2 FIELD HARMONICS Magnet Current, Imag = 100.6414 +- 3.48931E-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 0.00007 -118.57 16.56 0.3592 0.3670 2 0.00019 0.00007 -18.17 5.20 0.9534 0.3619 3 0.02036 0.00005 -29.90 0.00 100.0000 0.3488 4 0.00005 0.00004 42.32 0.79 0.2488 0.1871 5 0.00002 0.00000 8.40 17.46 0.0978 0.0184 6 0.00000 0.00000 -26.88 2.78 0.0119 0.0063 7 0.00001 0.00000 13.37 0.71 0.0271 0.0035 8 0.00000 0.00000 15.01 3.92 0.0074 0.0020 9 0.00007 0.00000 9.96 0.13 0.3370 0.0043 10 0.00000 0.00000 2.25 5.28 0.0062 0.0080 11 0.00000 0.00000 5.14 5.38 0.0033 0.0022 12 0.00000 0.00000 -0.92 1.40 0.0086 0.0014 13 0.00000 0.00000 0.41 7.45 0.0036 0.0027 14 0.00000 0.00000 2.76 7.21 0.0073 0.0074 15 0.00002 0.00000 6.16 0.12 0.0753 0.0034 16 0.00000 0.00000 -3.86 3.18 0.0076 0.0069 The magnetic center is at X = -126.8 +- 57.4 microns Y = 172.3 +- 69.3 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: 08-29-1995 Time: 08:44:09 Magnet Name: 112 Run Number: 2 FIELD HARMONICS Magnet Current, Imag = 150.6248 +- 1.997037E-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 0.00335 0.00001 -90.47 0.01 12.0357 0.0336 2 0.00369 0.00001 -0.98 0.00 13.2638 0.0361 3 0.02781 0.00001 -29.91 0.00 100.0000 0.0734 4 0.00159 0.00000 -44.91 0.00 5.7098 0.0163 5 0.00060 0.00000 -17.89 0.01 2.1464 0.0067 6 0.00003 0.00000 -0.25 0.13 0.0965 0.0030 7 0.00003 0.00000 -12.80 0.17 0.0919 0.0014 8 0.00002 0.00000 0.78 0.17 0.0667 0.0012 9 0.00009 0.00000 10.03 0.03 0.3350 0.0013 10 0.00002 0.00000 0.10 0.11 0.0718 0.0014 11 0.00001 0.00000 8.19 0.18 0.0495 0.0027 12 0.00000 0.00000 -12.20 1.49 0.0032 0.0030 13 0.00000 0.00000 5.90 2.23 0.0098 0.0022 14 0.00000 0.00000 6.06 10.88 0.0114 0.0024 15 0.00002 0.00000 6.05 0.08 0.0656 0.0012 16 0.00000 0.00000 2.05 2.27 0.0038 0.0020 The magnetic center is at X = -101.6 +- 0.6 microns Y = 2974.8 +- 8.1 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 ^ | S N N --> x S S N Date: 08-29-1995 Time: 08:50:51 Magnet Name: 112 Run Number: 2 FIELD HARMONICS Magnet Current, Imag = 200.6221 +- 8.217418E-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 0.00454 0.00005 -90.45 0.01 12.3066 0.1389 2 0.00500 0.00005 -0.96 0.01 13.5503 0.1418 3 0.03687 0.00005 -29.91 0.00 100.0000 0.1824 4 0.00215 0.00002 -44.91 0.01 5.8325 0.0643 5 0.00081 0.00001 -17.89 0.02 2.2081 0.0256 6 0.00004 0.00000 -0.16 0.29 0.1035 0.0009 7 0.00004 0.00000 -12.64 0.46 0.0959 0.0048 8 0.00002 0.00000 0.64 0.19 0.0671 0.0024 9 0.00012 0.00000 10.02 0.01 0.3372 0.0022 10 0.00003 0.00000 0.28 0.13 0.0745 0.0041 11 0.00002 0.00000 8.22 0.24 0.0556 0.0019 12 0.00000 0.00000 2.87 7.24 0.0013 0.0014 13 0.00000 0.00000 6.79 0.93 0.0102 0.0007 14 0.00000 0.00000 11.82 0.79 0.0096 0.0003 15 0.00003 0.00000 6.08 0.06 0.0714 0.0016 16 0.00000 0.00000 0.64 1.21 0.0025 0.0009 The magnetic center is at X = -102.2 +- 1.8 microns Y = 3039.1 +- 31.8 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)