SLAC Magnetic Measurements Date: 09-27-1995 Time: 08:08:42 Magnet Name (Serial #): 024 Bar Code Number: 140 Project: PEP II High Energy Ring Sextupoles Test Stand: IR8, S1, W 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: 09-27-1995 Time: 08:39:06 Magnet Name: 024 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 25.66434 +- 1.981967E-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 -105.08 0.21 0.8197 0.0014 2 0.00006 0.00000 35.43 0.07 1.1566 0.0019 3 0.00527 0.00000 -29.92 0.00 100.0000 0.0311 4 0.00001 0.00000 37.33 0.60 0.1254 0.0033 5 0.00001 0.00000 18.61 0.26 0.1037 0.0007 6 0.00000 0.00000 -25.34 1.43 0.0147 0.0018 7 0.00000 0.00000 13.89 0.98 0.0258 0.0014 8 0.00000 0.00000 -14.34 0.62 0.0158 0.0017 9 0.00002 0.00000 10.13 0.12 0.3318 0.0076 10 0.00000 0.00000 1.47 9.62 0.0028 0.0012 11 0.00000 0.00000 -0.46 11.22 0.0053 0.0016 12 0.00000 0.00000 1.08 13.07 0.0037 0.0009 13 0.00000 0.00000 -5.16 6.88 0.0058 0.0024 14 0.00000 0.00000 -7.81 1.02 0.0053 0.0009 15 0.00000 0.00000 6.13 0.07 0.0737 0.0018 16 0.00000 0.00000 4.81 4.31 0.0041 0.0030 The magnetic center is at X = 245.2 +- 0.5 microns Y = 85.1 +- 0.6 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: 09-27-1995 Time: 08:42:31 Magnet Name: 024 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 50.63983 +- 2.901983E-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 -114.36 0.72 0.2228 0.0025 2 0.00012 0.00000 35.65 0.04 1.1411 0.0008 3 0.01029 0.00000 -29.92 0.00 100.0000 0.0170 4 0.00001 0.00000 38.00 0.25 0.1162 0.0033 5 0.00001 0.00000 18.79 0.19 0.1229 0.0024 6 0.00000 0.00000 -25.11 1.01 0.0139 0.0026 7 0.00000 0.00000 13.54 0.43 0.0261 0.0012 8 0.00000 0.00000 -14.53 0.70 0.0147 0.0019 9 0.00003 0.00000 10.06 0.02 0.3306 0.0018 10 0.00000 0.00000 -2.14 9.43 0.0023 0.0016 11 0.00000 0.00000 3.84 12.42 0.0030 0.0001 12 0.00000 0.00000 3.18 3.29 0.0027 0.0005 13 0.00000 0.00000 -5.25 2.05 0.0051 0.0010 14 0.00000 0.00000 -7.93 1.71 0.0050 0.0014 15 0.00001 0.00000 6.06 0.03 0.0703 0.0026 16 0.00000 0.00000 2.16 5.37 0.0027 0.0015 The magnetic center is at X = 242.6 +- 0.2 microns Y = 82.1 +- 0.3 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: 09-27-1995 Time: 08:46:01 Magnet Name: 024 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 100.6301 +- 1.97587E-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 110.41 0.60 0.1022 0.0010 2 0.00023 0.00000 35.71 0.05 1.1431 0.0029 3 0.02041 0.00000 -29.92 0.00 100.0000 0.0182 4 0.00002 0.00000 37.86 0.15 0.1168 0.0027 5 0.00003 0.00000 18.67 0.11 0.1277 0.0012 6 0.00000 0.00000 -27.33 0.44 0.0234 0.0014 7 0.00001 0.00000 13.20 0.37 0.0259 0.0013 8 0.00000 0.00000 -16.48 1.09 0.0157 0.0021 9 0.00007 0.00000 10.08 0.04 0.3306 0.0029 10 0.00000 0.00000 14.26 2.09 0.0067 0.0028 11 0.00000 0.00000 -4.69 11.41 0.0033 0.0011 12 0.00000 0.00000 2.14 0.88 0.0059 0.0011 13 0.00000 0.00000 -7.57 2.03 0.0045 0.0015 14 0.00000 0.00000 -7.73 0.96 0.0056 0.0019 15 0.00002 0.00000 6.22 0.06 0.0751 0.0013 16 0.00000 0.00000 -5.09 2.14 0.0025 0.0009 The magnetic center is at X = 243.2 +- 0.6 microns Y = 81.7 +- 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: 09-27-1995 Time: 08:52:44 Magnet Name: 024 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 150.5986 +- 5.213133E-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 0.00000 93.96 0.36 0.2011 0.0004 2 0.00035 0.00000 35.75 0.03 1.1445 0.0015 3 0.03050 0.00002 -29.92 0.00 100.0000 0.0756 4 0.00004 0.00000 37.70 0.10 0.1178 0.0016 5 0.00004 0.00000 18.74 0.13 0.1306 0.0010 6 0.00000 0.00000 -25.29 0.52 0.0140 0.0005 7 0.00001 0.00000 13.25 0.53 0.0224 0.0010 8 0.00000 0.00000 -14.05 0.54 0.0150 0.0009 9 0.00010 0.00000 10.01 0.03 0.3280 0.0014 10 0.00000 0.00000 6.06 0.86 0.0053 0.0006 11 0.00000 0.00000 -8.24 3.27 0.0016 0.0008 12 0.00000 0.00000 7.26 12.67 0.0050 0.0005 13 0.00000 0.00000 -5.51 1.00 0.0037 0.0011 14 0.00000 0.00000 -7.45 0.75 0.0057 0.0010 15 0.00002 0.00000 6.18 0.06 0.0702 0.0021 16 0.00000 0.00000 1.41 2.14 0.0036 0.0010 The magnetic center is at X = 243.6 +- 0.3 microns Y = 81.5 +- 0.3 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: 09-27-1995 Time: 08:59:28 Magnet Name: 024 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 200.615 +- 3.920804E-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 0.00009 0.00000 90.93 0.64 0.2271 0.0015 2 0.00046 0.00000 35.94 0.02 1.1351 0.0012 3 0.04041 0.00001 -29.92 0.00 100.0000 0.0235 4 0.00005 0.00000 37.68 0.10 0.1162 0.0025 5 0.00005 0.00000 18.64 0.09 0.1279 0.0008 6 0.00001 0.00000 -27.22 1.11 0.0148 0.0007 7 0.00001 0.00000 13.32 0.57 0.0247 0.0011 8 0.00001 0.00000 -13.65 0.55 0.0150 0.0020 9 0.00013 0.00000 10.04 0.03 0.3292 0.0013 10 0.00000 0.00000 -0.68 10.45 0.0020 0.0010 11 0.00000 0.00000 -6.13 11.66 0.0024 0.0005 12 0.00000 0.00000 11.39 4.41 0.0026 0.0007 13 0.00000 0.00000 -3.94 2.39 0.0027 0.0011 14 0.00000 0.00000 -8.89 1.02 0.0046 0.0012 15 0.00003 0.00000 6.06 0.13 0.0716 0.0009 16 0.00000 0.00000 -4.79 8.16 0.0026 0.0019 The magnetic center is at X = 242.1 +- 0.3 microns Y = 79.2 +- 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 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)