SLAC Magnetic Measurements Date: 07-07-1995 Time: 12:11:37 Magnet Name (Serial #): 042 Bar Code Number: 008 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: 07-07-1995 Time: 12:42:32 Magnet Name: 042 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 25.57033 +- 1.429353E-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 -113.88 0.70 0.8176 0.0117 2 0.00006 0.00000 -13.09 0.47 1.0518 0.0032 3 0.00525 0.00000 -29.90 0.00 100.0000 0.0601 4 0.00001 0.00000 38.68 0.59 0.1735 0.0106 5 0.00001 0.00000 19.09 1.22 0.1363 0.0090 6 0.00000 0.00000 11.74 21.23 0.0215 0.0085 7 0.00000 0.00000 15.50 1.07 0.0254 0.0069 8 0.00000 0.00000 17.95 3.22 0.0315 0.0130 9 0.00002 0.00000 10.05 0.10 0.3226 0.0089 10 0.00000 0.00000 3.91 7.77 0.0162 0.0061 11 0.00000 0.00000 -7.72 7.05 0.0216 0.0045 12 0.00000 0.00000 -3.56 7.52 0.0227 0.0083 13 0.00000 0.00000 -6.34 6.17 0.0199 0.0143 14 0.00000 0.00000 -4.10 5.49 0.0181 0.0095 15 0.00000 0.00000 6.72 0.88 0.0780 0.0152 16 0.00000 0.00000 -9.23 0.92 0.0090 0.0031 The magnetic center is at X = -104.2 +- 3.5 microns Y = 211.8 +- 1.8 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 Date: 07-07-1995 Time: 12:46:04 Magnet Name: 042 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 50.56 +- 1.587601E-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 -130.52 3.28 0.2990 0.0099 2 0.00011 0.00000 -14.30 0.16 1.0365 0.0044 3 0.01026 0.00000 -29.90 0.00 100.0000 0.0275 4 0.00002 0.00000 40.04 0.49 0.1537 0.0042 5 0.00002 0.00000 17.52 1.36 0.1632 0.0115 6 0.00000 0.00000 -13.44 23.88 0.0184 0.0084 7 0.00001 0.00000 13.02 1.74 0.0490 0.0061 8 0.00000 0.00000 8.89 17.20 0.0193 0.0073 9 0.00003 0.00000 10.12 0.09 0.3299 0.0021 10 0.00000 0.00000 0.18 6.14 0.0132 0.0076 11 0.00000 0.00000 0.26 10.39 0.0088 0.0050 12 0.00000 0.00000 -1.83 8.75 0.0088 0.0024 13 0.00000 0.00000 2.33 10.38 0.0095 0.0058 14 0.00000 0.00000 4.56 9.97 0.0086 0.0031 15 0.00001 0.00000 6.35 0.55 0.0764 0.0056 16 0.00000 0.00000 -2.37 7.48 0.0104 0.0035 The magnetic center is at X = -111.3 +- 1.2 microns Y = 204.2 +- 1.1 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: 07-07-1995 Time: 12:49:41 Magnet Name: 042 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 100.5545 +- 1.378674E-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 162.65 2.87 0.1094 0.0041 2 0.00021 0.00000 -14.60 0.07 1.0437 0.0042 3 0.02038 0.00001 -29.91 0.00 100.0000 0.0579 4 0.00003 0.00000 40.91 0.35 0.1474 0.0077 5 0.00004 0.00000 18.43 0.29 0.1752 0.0028 6 0.00000 0.00000 -0.33 28.75 0.0218 0.0019 7 0.00001 0.00000 13.80 0.93 0.0465 0.0025 8 0.00000 0.00000 17.53 1.07 0.0213 0.0014 9 0.00007 0.00000 10.10 0.06 0.3320 0.0030 10 0.00000 0.00000 7.50 2.52 0.0112 0.0071 11 0.00000 0.00000 2.34 10.88 0.0070 0.0043 12 0.00000 0.00000 2.05 1.14 0.0122 0.0060 13 0.00000 0.00000 0.12 11.40 0.0093 0.0070 14 0.00000 0.00000 7.97 2.27 0.0032 0.0013 15 0.00001 0.00000 6.38 0.32 0.0714 0.0062 16 0.00000 0.00000 1.80 6.91 0.0036 0.0017 The magnetic center is at X = -114.3 +- 0.7 microns Y = 204.4 +- 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: 07-07-1995 Time: 12:56:38 Magnet Name: 042 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 150.5378 +- 3.596691E-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 120.31 1.09 0.1609 0.0040 2 0.00032 0.00000 -14.59 0.17 1.0453 0.0018 3 0.03046 0.00001 -29.91 0.00 100.0000 0.0591 4 0.00004 0.00000 40.49 0.47 0.1472 0.0036 5 0.00005 0.00000 18.27 0.28 0.1748 0.0098 6 0.00000 0.00000 12.91 23.02 0.0106 0.0024 7 0.00001 0.00000 13.89 0.62 0.0311 0.0032 8 0.00000 0.00000 17.70 0.82 0.0164 0.0049 9 0.00010 0.00000 10.15 0.07 0.3245 0.0033 10 0.00000 0.00000 2.99 6.72 0.0081 0.0026 11 0.00000 0.00000 9.83 3.08 0.0042 0.0010 12 0.00000 0.00000 0.64 13.09 0.0092 0.0043 13 0.00000 0.00000 5.08 10.23 0.0049 0.0015 14 0.00000 0.00000 4.32 7.94 0.0064 0.0030 15 0.00002 0.00000 6.04 0.13 0.0711 0.0034 16 0.00000 0.00000 -4.11 5.28 0.0052 0.0038 The magnetic center is at X = -114.3 +- 1.2 microns Y = 204.8 +- 0.8 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 Date: 07-07-1995 Time: 13:03:34 Magnet Name: 042 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 200.545 +- 1.555187E-02 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 110.24 0.76 0.1914 0.0023 2 0.00042 0.00000 -14.82 0.07 1.0386 0.0017 3 0.04045 0.00001 -29.90 0.00 100.0000 0.0324 4 0.00006 0.00000 40.70 0.44 0.1414 0.0043 5 0.00007 0.00000 18.14 0.15 0.1704 0.0018 6 0.00000 0.00000 -28.50 1.28 0.0106 0.0014 7 0.00002 0.00000 13.96 0.51 0.0398 0.0021 8 0.00001 0.00000 18.18 2.69 0.0139 0.0057 9 0.00013 0.00000 10.12 0.05 0.3301 0.0023 10 0.00000 0.00000 3.05 6.87 0.0070 0.0030 11 0.00000 0.00000 3.91 11.69 0.0041 0.0010 12 0.00000 0.00000 3.70 10.80 0.0033 0.0010 13 0.00000 0.00000 1.70 9.93 0.0054 0.0015 14 0.00000 0.00000 9.73 1.01 0.0050 0.0028 15 0.00003 0.00000 6.35 0.44 0.0731 0.0035 16 0.00000 0.00000 -0.04 8.84 0.0027 0.0011 The magnetic center is at X = -115.2 +- 0.6 microns Y = 202.6 +- 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)