SLAC Magnetic Measurements Date: 07-19-1995 Time: 06:04:40 Magnet Name (Serial #): 132 Bar Code Number: 185 Project: PEP II High Energy Ring Sextupoles Test Stand: IR8, S1, N Measurement Coil: # 1 Operator: bmd Run Number: 1 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: 07-19-1995 Time: 06:35:09 Magnet Name: 132 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 25.62735 +- .0010776 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 -111.78 1.84 0.8502 0.0139 2 0.00003 0.00000 -5.58 1.10 0.5623 0.0112 3 0.00527 0.00000 -29.91 0.00 100.0000 0.0655 4 0.00001 0.00000 35.49 3.71 0.1115 0.0490 5 0.00001 0.00000 20.86 1.66 0.1429 0.0565 6 0.00000 0.00000 -5.40 20.45 0.0591 0.0344 7 0.00000 0.00000 10.98 6.32 0.0534 0.0525 8 0.00000 0.00000 1.26 18.03 0.0566 0.0466 9 0.00002 0.00000 9.77 0.53 0.3178 0.0156 10 0.00000 0.00000 -0.58 9.38 0.0186 0.0043 11 0.00000 0.00000 2.71 10.39 0.0211 0.0081 12 0.00000 0.00000 0.57 6.99 0.0147 0.0068 13 0.00000 0.00000 0.52 7.70 0.0167 0.0117 14 0.00000 0.00000 0.22 9.10 0.0117 0.0028 15 0.00000 0.00000 6.32 0.66 0.0738 0.0209 16 0.00000 0.00000 5.88 7.26 0.0312 0.0119 The magnetic center is at X = -24.4 +- 4.8 microns Y = 123.8 +- 2.6 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: 07-19-1995 Time: 06:38:34 Magnet Name: 132 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 50.60843 +- 2.682265E-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 -120.93 1.01 0.2933 0.0032 2 0.00006 0.00000 -4.27 0.38 0.5562 0.0039 3 0.01029 0.00000 -29.91 0.00 100.0000 0.0425 4 0.00001 0.00000 36.82 0.93 0.0954 0.0064 5 0.00001 0.00000 18.57 1.18 0.1410 0.0220 6 0.00000 0.00000 -14.76 19.26 0.0318 0.0281 7 0.00001 0.00000 11.61 2.22 0.0488 0.0266 8 0.00001 0.00000 -2.47 19.07 0.0495 0.0379 9 0.00003 0.00000 10.12 0.18 0.3341 0.0055 10 0.00000 0.00000 5.01 8.62 0.0135 0.0050 11 0.00000 0.00000 3.52 10.82 0.0107 0.0059 12 0.00000 0.00000 -0.75 10.19 0.0100 0.0078 13 0.00000 0.00000 -1.29 9.83 0.0104 0.0112 14 0.00000 0.00000 5.09 10.38 0.0155 0.0084 15 0.00001 0.00000 5.53 0.51 0.0612 0.0108 16 0.00000 0.00000 4.90 6.53 0.0120 0.0060 The magnetic center is at X = -18.6 +- 1.6 microns Y = 123.4 +- 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 ^ | S N N --> x S S N Date: 07-19-1995 Time: 06:42:03 Magnet Name: 132 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 100.6004 +- 2.571567E-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 157.75 0.74 0.0848 0.0022 2 0.00012 0.00000 -4.01 0.06 0.5675 0.0019 3 0.02044 0.00000 -29.91 0.00 100.0000 0.0340 4 0.00002 0.00000 37.43 0.54 0.1064 0.0009 5 0.00003 0.00000 19.61 0.44 0.1381 0.0074 6 0.00000 0.00000 -28.05 2.15 0.0202 0.0093 7 0.00001 0.00000 14.17 0.72 0.0389 0.0061 8 0.00000 0.00000 8.65 16.17 0.0164 0.0112 9 0.00007 0.00000 10.09 0.03 0.3302 0.0037 10 0.00000 0.00000 0.56 7.29 0.0040 0.0010 11 0.00000 0.00000 7.61 7.44 0.0045 0.0023 12 0.00000 0.00000 -1.28 2.55 0.0069 0.0011 13 0.00000 0.00000 4.75 3.03 0.0065 0.0051 14 0.00000 0.00000 3.46 8.32 0.0041 0.0033 15 0.00002 0.00000 5.97 0.25 0.0739 0.0014 16 0.00000 0.00000 -2.61 3.18 0.0051 0.0039 The magnetic center is at X = -17.8 +- 0.3 microns Y = 126.1 +- 0.4 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: 07-19-1995 Time: 06:48:47 Magnet Name: 132 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 150.5813 +- 1.562387E-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 109.50 0.51 0.1452 0.0007 2 0.00018 0.00000 -3.86 0.12 0.5877 0.0011 3 0.03054 0.00001 -29.91 0.00 100.0000 0.0268 4 0.00003 0.00000 38.20 0.29 0.1132 0.0013 5 0.00004 0.00000 19.25 0.05 0.1372 0.0008 6 0.00000 0.00000 -26.14 2.22 0.0050 0.0011 7 0.00001 0.00000 13.87 0.29 0.0363 0.0015 8 0.00000 0.00000 17.54 1.12 0.0113 0.0010 9 0.00010 0.00000 10.13 0.02 0.3310 0.0004 10 0.00000 0.00000 9.57 4.24 0.0033 0.0020 11 0.00000 0.00000 0.80 11.09 0.0013 0.0010 12 0.00000 0.00000 -6.64 12.14 0.0052 0.0010 13 0.00000 0.00000 -0.92 7.09 0.0019 0.0013 14 0.00000 0.00000 6.19 8.74 0.0044 0.0015 15 0.00002 0.00000 6.14 0.04 0.0706 0.0007 16 0.00000 0.00000 2.13 5.96 0.0014 0.0006 The magnetic center is at X = -17.7 +- 0.5 microns Y = 130.7 +- 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: 07-19-1995 Time: 06:55:30 Magnet Name: 132 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 200.5806 +- 3.235484E-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 0.00000 100.63 0.43 0.1739 0.0008 2 0.00024 0.00000 -3.68 0.02 0.6029 0.0019 3 0.04054 0.00001 -29.91 0.00 100.0000 0.0329 4 0.00005 0.00000 38.28 0.28 0.1190 0.0023 5 0.00006 0.00000 19.25 0.12 0.1363 0.0014 6 0.00000 0.00000 2.79 26.14 0.0052 0.0013 7 0.00001 0.00000 13.58 0.87 0.0342 0.0018 8 0.00000 0.00000 17.44 1.01 0.0109 0.0028 9 0.00013 0.00000 10.11 0.05 0.3306 0.0010 10 0.00000 0.00000 10.42 2.40 0.0035 0.0009 11 0.00000 0.00000 12.47 2.65 0.0019 0.0010 12 0.00000 0.00000 6.73 11.66 0.0031 0.0008 13 0.00000 0.00000 2.31 0.75 0.0029 0.0010 14 0.00000 0.00000 5.46 9.92 0.0046 0.0023 15 0.00003 0.00000 6.08 0.04 0.0708 0.0017 16 0.00000 0.00000 -1.52 3.92 0.0016 0.0003 The magnetic center is at X = -17.3 +- 0.1 microns Y = 134.2 +- 0.4 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 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)