SLAC Magnetic Measurements Date: 08-02-1995 Time: 12:08:46 Magnet Name (Serial #): 026 Bar Code Number: 114 Project: PEP II High Energy Ring Sextupoles Test Stand: IR8, S1, N Measurement Coil: # 1 Operator: bda 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: 08-02-1995 Time: 12:39:07 Magnet Name: 026 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 25.5665 +- 2.161002E-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 0.00002 -99.93 7.30 1.0613 0.3298 2 0.00002 0.00000 4.99 1.01 0.3474 0.0132 3 0.00526 0.00000 -29.93 0.00 100.0000 0.0740 4 0.00000 0.00000 39.89 0.23 0.0827 0.0074 5 0.00001 0.00000 17.03 2.24 0.1205 0.0329 6 0.00000 0.00000 -19.21 3.38 0.0136 0.0085 7 0.00000 0.00000 12.89 0.94 0.0316 0.0055 8 0.00000 0.00000 -6.71 16.66 0.0110 0.0048 9 0.00002 0.00000 10.01 0.17 0.3350 0.0076 10 0.00000 0.00000 -0.22 5.11 0.0082 0.0079 11 0.00000 0.00000 7.77 6.89 0.0111 0.0054 12 0.00000 0.00000 7.79 5.22 0.0072 0.0053 13 0.00000 0.00000 -4.03 9.33 0.0070 0.0036 14 0.00000 0.00000 3.86 8.87 0.0115 0.0101 15 0.00000 0.00000 6.36 0.34 0.0646 0.0123 16 0.00000 0.00000 2.91 5.59 0.0150 0.0152 The magnetic center is at X = 13.5 +- 2.8 microns Y = 76.8 +- 3.0 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 ^ | S N N --> x S S N Date: 08-02-1995 Time: 12:42:31 Magnet Name: 026 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 50.55452 +- 2.804758E-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 0.00010 0.00000 -79.16 0.38 0.9835 0.0209 2 0.00004 0.00000 6.98 0.75 0.3455 0.0060 3 0.01028 0.00000 -29.93 0.00 100.0000 0.0202 4 0.00001 0.00000 40.08 3.21 0.0988 0.0082 5 0.00002 0.00000 17.76 1.06 0.1625 0.0217 6 0.00000 0.00000 -14.23 9.21 0.0158 0.0071 7 0.00000 0.00000 10.12 2.37 0.0391 0.0152 8 0.00000 0.00000 -3.40 13.69 0.0107 0.0060 9 0.00003 0.00000 10.06 0.19 0.3248 0.0057 10 0.00000 0.00000 -0.83 9.55 0.0180 0.0066 11 0.00000 0.00000 -6.45 10.36 0.0100 0.0040 12 0.00000 0.00000 -1.71 5.15 0.0131 0.0066 13 0.00000 0.00000 1.39 7.47 0.0243 0.0061 14 0.00000 0.00000 0.68 7.56 0.0154 0.0084 15 0.00001 0.00000 6.04 0.58 0.0823 0.0110 16 0.00000 0.00000 5.94 2.92 0.0090 0.0008 The magnetic center is at X = 18.7 +- 2.0 microns Y = 75.2 +- 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 ^ | S N N --> x S S N Date: 08-02-1995 Time: 12:46:01 Magnet Name: 026 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 100.5504 +- 3.284226E-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 129.68 0.77 0.0917 0.0009 2 0.00007 0.00000 6.36 0.28 0.3629 0.0008 3 0.02042 0.00000 -29.94 0.00 100.0000 0.0226 4 0.00002 0.00000 39.01 0.24 0.0933 0.0022 5 0.00003 0.00000 18.47 0.10 0.1632 0.0013 6 0.00000 0.00000 -25.70 0.75 0.0157 0.0018 7 0.00001 0.00000 12.46 0.29 0.0383 0.0017 8 0.00000 0.00000 0.39 21.62 0.0059 0.0005 9 0.00007 0.00000 10.09 0.03 0.3327 0.0016 10 0.00000 0.00000 12.16 0.97 0.0035 0.0018 11 0.00000 0.00000 -4.15 8.53 0.0041 0.0008 12 0.00000 0.00000 0.59 0.73 0.0079 0.0018 13 0.00000 0.00000 3.09 3.43 0.0032 0.0021 14 0.00000 0.00000 2.32 8.58 0.0027 0.0010 15 0.00002 0.00000 6.13 0.08 0.0749 0.0009 16 0.00000 0.00000 -3.87 3.56 0.0026 0.0007 The magnetic center is at X = 17.9 +- 0.8 microns Y = 79.4 +- 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: 08-02-1995 Time: 12:52:44 Magnet Name: 026 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 150.5359 +- 2.364608E-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 0.00005 0.00001 66.50 63.96 0.1562 0.0489 2 0.00011 0.00000 6.45 0.16 0.3549 0.0056 3 0.03052 0.00001 -29.94 0.00 100.0000 0.0299 4 0.00003 0.00000 39.18 0.23 0.0926 0.0024 5 0.00005 0.00000 18.26 0.13 0.1696 0.0036 6 0.00000 0.00000 -17.47 3.52 0.0080 0.0020 7 0.00001 0.00000 12.61 0.12 0.0400 0.0042 8 0.00000 0.00000 -17.23 1.76 0.0053 0.0015 9 0.00010 0.00000 10.10 0.04 0.3326 0.0018 10 0.00000 0.00000 7.17 2.66 0.0028 0.0006 11 0.00000 0.00000 5.13 9.55 0.0028 0.0019 12 0.00000 0.00000 6.30 10.93 0.0037 0.0021 13 0.00000 0.00000 2.59 8.67 0.0032 0.0014 14 0.00000 0.00000 8.12 5.90 0.0026 0.0006 15 0.00002 0.00000 6.16 0.10 0.0716 0.0005 16 0.00000 0.00000 -2.13 5.58 0.0017 0.0009 The magnetic center is at X = 17.8 +- 0.5 microns Y = 77.6 +- 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 ^ | S N N --> x S S N Date: 08-02-1995 Time: 12:59:27 Magnet Name: 026 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 200.5361 +- 7.299209E-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 40.01 8.84 0.0496 0.0022 2 0.00014 0.00000 6.44 0.41 0.3492 0.0032 3 0.04051 0.00001 -29.94 0.00 100.0000 0.0198 4 0.00004 0.00000 40.05 0.66 0.0920 0.0057 5 0.00007 0.00000 18.45 0.09 0.1639 0.0066 6 0.00000 0.00000 -18.52 5.66 0.0096 0.0031 7 0.00002 0.00000 12.33 0.48 0.0412 0.0074 8 0.00000 0.00000 -1.57 16.39 0.0050 0.0028 9 0.00014 0.00000 10.09 0.04 0.3344 0.0029 10 0.00000 0.00000 7.07 11.69 0.0037 0.0010 11 0.00000 0.00000 0.06 10.10 0.0033 0.0012 12 0.00000 0.00000 7.34 0.80 0.0028 0.0011 13 0.00000 0.00000 2.94 7.03 0.0068 0.0033 14 0.00000 0.00000 6.55 2.79 0.0046 0.0012 15 0.00003 0.00000 6.08 0.15 0.0760 0.0056 16 0.00000 0.00000 1.05 7.18 0.0032 0.0020 The magnetic center is at X = 17.5 +- 1.1 microns Y = 76.4 +- 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 ^ | 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)