SLAC Magnetic Measurements Date: 06-26-1995 Time: 14:07:19 Magnet Name (Serial #): 000 Bar Code Number: 001 Project: PEP II High Energy Ring Sextupoles Test Stand: IR8, S1 Measurement Coil: # 1 Operator: pdr Run Number: 3 Comment: Repeat run, check repeatability Standardization Currents (A): 250.0 10.0 200.0 10.0 200.0 10.0 200.0 10.0 Test Currents (A): 25.0 50.0 100.0 150.0 200.0 150.0 100.0 50.0 25.0 Harmonics Measurements Date: 06-26-1995 Time: 14:27:25 Magnet Name: 000 Run Number: 3 FIELD HARMONICS Magnet Current, Imag = 50.54048 +- 4.314477E-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 -104.68 0.10 0.7285 0.0019 2 0.00010 0.00000 -52.76 0.08 0.9613 0.0023 3 0.01026 0.00000 30.05 0.00 100.0000 0.0261 4 0.00002 0.00000 -3.93 0.66 0.1598 0.0061 5 0.00002 0.00000 -16.11 0.05 0.1716 0.0025 6 0.00000 0.00000 8.67 5.13 0.0146 0.0057 7 0.00000 0.00000 -20.76 0.29 0.0415 0.0193 8 0.00000 0.00000 6.33 16.02 0.0248 0.0053 9 0.00003 0.00000 -10.26 0.46 0.2954 0.0128 10 0.00000 0.00000 3.65 6.75 0.0088 0.0016 11 0.00000 0.00000 -5.08 1.20 0.0463 0.0100 12 0.00001 0.00000 -4.08 5.28 0.0866 0.0359 13 0.00001 0.00000 -1.88 1.85 0.0802 0.0045 14 0.00000 0.00000 10.97 1.16 0.0380 0.0113 15 0.00001 0.00000 -6.13 0.19 0.0853 0.0118 16 0.00000 0.00000 2.95 3.66 0.0126 0.0066 The magnetic center is at X = 207.9 +- 0.5 microns Y = 57.7 +- 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 ^ | N S S --> x N N S Date: 06-26-1995 Time: 14:31:03 Magnet Name: 000 Run Number: 3 FIELD HARMONICS Magnet Current, Imag = 100.5376 +- 5.115215E-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 0.00011 0.00000 -101.74 0.17 0.5430 0.0017 2 0.00019 0.00000 -52.21 0.12 0.9449 0.0060 3 0.02037 0.00001 30.06 0.00 100.0000 0.0389 4 0.00003 0.00000 -4.41 0.21 0.1543 0.0022 5 0.00004 0.00000 -16.64 0.31 0.1757 0.0037 6 0.00001 0.00000 21.17 2.67 0.0254 0.0076 7 0.00001 0.00000 13.54 5.35 0.0294 0.0134 8 0.00001 0.00000 6.39 16.68 0.0429 0.0041 9 0.00006 0.00000 -9.15 0.43 0.2772 0.0075 10 0.00000 0.00000 -10.02 2.76 0.0141 0.0037 11 0.00001 0.00000 -9.31 1.11 0.0566 0.0057 12 0.00004 0.00000 -8.85 2.59 0.1801 0.0165 13 0.00002 0.00000 -4.68 1.73 0.1117 0.0086 14 0.00001 0.00000 7.45 0.78 0.0511 0.0097 15 0.00002 0.00000 -7.00 0.39 0.0759 0.0134 16 0.00000 0.00000 2.71 1.26 0.0157 0.0048 The magnetic center is at X = 205.4 +- 1.3 microns Y = 52.8 +- 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 ^ | N S S --> x N N S Date: 06-26-1995 Time: 14:34:40 Magnet Name: 000 Run Number: 3 FIELD HARMONICS Magnet Current, Imag = 150.5175 +- 3.11341E-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 0.00015 0.00000 -100.97 0.23 0.4898 0.0024 2 0.00028 0.00000 -51.96 0.01 0.9324 0.0022 3 0.03044 0.00001 30.06 0.00 100.0000 0.0600 4 0.00005 0.00000 -3.53 0.46 0.1679 0.0025 5 0.00005 0.00000 -15.62 0.09 0.1789 0.0019 6 0.00000 0.00000 0.62 10.39 0.0053 0.0030 7 0.00002 0.00000 0.36 23.92 0.0725 0.0133 8 0.00004 0.00001 -16.68 2.22 0.1196 0.0287 9 0.00007 0.00000 -9.76 0.21 0.2421 0.0127 10 0.00000 0.00000 8.62 4.35 0.0137 0.0069 11 0.00002 0.00000 -5.80 0.50 0.0544 0.0098 12 0.00007 0.00001 -3.21 1.00 0.2304 0.0238 13 0.00003 0.00000 1.37 1.28 0.1040 0.0061 14 0.00002 0.00001 8.07 0.50 0.0723 0.0172 15 0.00003 0.00000 -6.13 0.40 0.1116 0.0014 16 0.00000 0.00000 1.75 8.79 0.0097 0.0019 The magnetic center is at X = 203.1 +- 0.5 microns Y = 50.4 +- 0.1 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 ^ | N S S --> x N N S Date: 06-26-1995 Time: 14:38:17 Magnet Name: 000 Run Number: 3 FIELD HARMONICS Magnet Current, Imag = 200.5206 +- 3.245095E-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 0.00018 0.00000 -101.60 0.17 0.4339 0.0007 2 0.00038 0.00000 -52.08 0.06 0.9361 0.0031 3 0.04041 0.00001 30.05 0.00 100.0000 0.0216 4 0.00006 0.00000 -2.03 0.20 0.1545 0.0040 5 0.00007 0.00000 -14.94 0.20 0.1647 0.0015 6 0.00001 0.00000 -0.10 0.83 0.0249 0.0022 7 0.00004 0.00000 -17.23 0.86 0.0960 0.0043 8 0.00004 0.00001 -8.07 0.49 0.0934 0.0132 9 0.00012 0.00000 -10.51 0.19 0.2907 0.0113 10 0.00001 0.00000 14.24 1.08 0.0209 0.0036 11 0.00001 0.00000 -1.01 1.54 0.0223 0.0109 12 0.00008 0.00001 5.10 0.97 0.2031 0.0246 13 0.00001 0.00000 1.90 8.99 0.0191 0.0101 14 0.00003 0.00000 11.11 0.25 0.0708 0.0038 15 0.00003 0.00000 -5.32 0.13 0.0748 0.0053 16 0.00001 0.00000 6.76 0.36 0.0124 0.0063 The magnetic center is at X = 203.7 +- 0.7 microns Y = 51.4 +- 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 ^ | N S S --> x N N S 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)