SLAC Magnetic Measurements Date: 08-11-1995 Time: 11:08:12 Magnet Name (Serial #): 146 Bar Code Number: 147 Project: PEP II High Energy Ring Sextupoles Test Stand: IR8, S1, W 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: 08-11-1995 Time: 11:38:32 Magnet Name: 146 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 25.60955 +- 1.313087E-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 0.00000 -110.13 0.06 0.8568 0.0016 2 0.00008 0.00000 31.38 0.04 1.5622 0.0020 3 0.00526 0.00000 -29.90 0.00 100.0000 0.0357 4 0.00001 0.00000 33.68 0.97 0.1063 0.0051 5 0.00001 0.00000 18.79 0.15 0.1219 0.0013 6 0.00000 0.00000 -11.00 9.12 0.0083 0.0038 7 0.00000 0.00000 13.42 0.65 0.0321 0.0028 8 0.00000 0.00000 -15.18 0.48 0.0189 0.0031 9 0.00002 0.00000 10.10 0.08 0.3253 0.0051 10 0.00000 0.00000 6.55 4.99 0.0031 0.0018 11 0.00000 0.00000 7.26 5.98 0.0048 0.0010 12 0.00000 0.00000 -5.88 7.89 0.0025 0.0015 13 0.00000 0.00000 -2.51 7.17 0.0040 0.0013 14 0.00000 0.00000 -8.52 0.70 0.0080 0.0014 15 0.00000 0.00000 6.10 0.03 0.0730 0.0010 16 0.00000 0.00000 -7.25 1.98 0.0048 0.0018 The magnetic center is at X = 311.7 +- 0.5 microns Y = 160.5 +- 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: 08-11-1995 Time: 11:41:56 Magnet Name: 146 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 50.5935 +- 2.64392E-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 -135.85 0.17 0.2582 0.0013 2 0.00016 0.00000 31.44 0.08 1.5481 0.0037 3 0.01028 0.00000 -29.90 0.00 100.0000 0.0328 4 0.00001 0.00000 34.68 0.36 0.1055 0.0084 5 0.00001 0.00000 18.86 0.08 0.1362 0.0007 6 0.00000 0.00000 3.40 24.54 0.0119 0.0037 7 0.00000 0.00000 13.87 0.25 0.0319 0.0016 8 0.00000 0.00000 -16.60 0.84 0.0198 0.0009 9 0.00003 0.00000 10.11 0.04 0.3285 0.0025 10 0.00000 0.00000 9.12 4.92 0.0036 0.0012 11 0.00000 0.00000 -7.42 12.42 0.0029 0.0014 12 0.00000 0.00000 -1.38 6.70 0.0023 0.0012 13 0.00000 0.00000 -3.24 6.88 0.0040 0.0018 14 0.00000 0.00000 -6.98 0.87 0.0067 0.0016 15 0.00001 0.00000 6.10 0.05 0.0729 0.0016 16 0.00000 0.00000 -4.73 2.17 0.0021 0.0010 The magnetic center is at X = 309.2 +- 0.9 microns Y = 158.3 +- 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: 08-11-1995 Time: 11:45:25 Magnet Name: 146 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 100.5899 +- 3.610443E-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 132.53 0.15 0.1901 0.0008 2 0.00031 0.00000 31.38 0.04 1.5386 0.0019 3 0.02042 0.00000 -29.90 0.00 100.0000 0.0317 4 0.00002 0.00000 35.63 0.38 0.0926 0.0027 5 0.00003 0.00000 18.70 0.08 0.1450 0.0011 6 0.00000 0.00000 -24.30 0.31 0.0173 0.0036 7 0.00001 0.00000 13.19 0.44 0.0353 0.0015 8 0.00000 0.00000 -15.23 0.31 0.0224 0.0017 9 0.00007 0.00000 10.09 0.04 0.3309 0.0025 10 0.00000 0.00000 12.42 3.01 0.0051 0.0006 11 0.00000 0.00000 1.01 8.56 0.0016 0.0007 12 0.00000 0.00000 2.35 0.93 0.0077 0.0016 13 0.00000 0.00000 2.00 1.33 0.0052 0.0018 14 0.00000 0.00000 -7.65 1.12 0.0061 0.0018 15 0.00002 0.00000 6.10 0.05 0.0744 0.0009 16 0.00000 0.00000 -0.49 3.56 0.0017 0.0008 The magnetic center is at X = 307.0 +- 0.4 microns Y = 158.0 +- 0.5 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: 08-11-1995 Time: 11:52:07 Magnet Name: 146 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 150.5739 +- 5.726469E-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 0.00008 0.00000 112.47 0.20 0.2748 0.0011 2 0.00047 0.00000 31.19 0.04 1.5486 0.0024 3 0.03051 0.00001 -29.90 0.00 100.0000 0.0636 4 0.00003 0.00000 35.25 0.66 0.0955 0.0032 5 0.00005 0.00000 18.91 0.04 0.1486 0.0010 6 0.00000 0.00000 -13.86 10.01 0.0058 0.0025 7 0.00001 0.00000 13.99 0.21 0.0358 0.0013 8 0.00001 0.00000 -15.18 0.45 0.0203 0.0016 9 0.00010 0.00000 10.15 0.01 0.3309 0.0003 10 0.00000 0.00000 1.72 9.42 0.0029 0.0008 11 0.00000 0.00000 -7.31 7.95 0.0011 0.0005 12 0.00000 0.00000 13.12 1.25 0.0053 0.0006 13 0.00000 0.00000 1.87 6.52 0.0029 0.0010 14 0.00000 0.00000 -7.19 0.91 0.0073 0.0015 15 0.00002 0.00000 6.15 0.05 0.0708 0.0010 16 0.00000 0.00000 -1.46 7.41 0.0023 0.0007 The magnetic center is at X = 307.9 +- 0.5 microns Y = 161.1 +- 0.5 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: 08-11-1995 Time: 11:58:49 Magnet Name: 146 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 200.5746 +- 9.84272E-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 108.64 0.41 0.2938 0.0010 2 0.00063 0.00000 30.73 0.06 1.5581 0.0023 3 0.04050 0.00001 -29.90 0.00 100.0000 0.0191 4 0.00004 0.00000 36.68 0.63 0.0994 0.0069 5 0.00006 0.00000 18.89 0.06 0.1438 0.0025 6 0.00000 0.00000 -8.22 10.61 0.0078 0.0044 7 0.00001 0.00000 13.78 0.34 0.0352 0.0013 8 0.00001 0.00000 -15.09 1.37 0.0188 0.0037 9 0.00013 0.00000 10.17 0.05 0.3312 0.0021 10 0.00000 0.00000 -3.32 12.63 0.0024 0.0004 11 0.00000 0.00000 2.85 11.54 0.0026 0.0017 12 0.00000 0.00000 6.82 3.07 0.0025 0.0011 13 0.00000 0.00000 0.35 1.18 0.0032 0.0017 14 0.00000 0.00000 -9.34 0.39 0.0066 0.0008 15 0.00003 0.00000 6.19 0.09 0.0705 0.0008 16 0.00000 0.00000 -2.61 9.18 0.0038 0.0018 The magnetic center is at X = 307.2 +- 0.6 microns Y = 167.0 +- 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)