SLAC Magnetic Measurements Date: 06-26-1995 Time: 13:10:30 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: 2 Comment: Check standardization 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: 13:30:36 Magnet Name: 000 Run Number: 2 FIELD HARMONICS Magnet Current, Imag = 50.5527 +- 5.226518E-04 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 -104.56 0.23 0.7273 0.0036 2 0.00010 0.00000 -52.96 0.07 0.9532 0.0020 3 0.01027 0.00000 30.05 0.00 100.0000 0.0388 4 0.00002 0.00000 -3.65 0.44 0.1613 0.0007 5 0.00002 0.00000 -15.97 0.22 0.1734 0.0020 6 0.00000 0.00000 1.14 7.37 0.0125 0.0056 7 0.00000 0.00000 -21.31 1.82 0.0474 0.0119 8 0.00000 0.00000 -6.73 16.04 0.0355 0.0022 9 0.00003 0.00000 -10.29 0.30 0.2999 0.0107 10 0.00000 0.00000 0.25 10.17 0.0074 0.0030 11 0.00000 0.00000 -5.49 0.85 0.0386 0.0071 12 0.00001 0.00000 -3.02 2.98 0.1166 0.0114 13 0.00001 0.00000 -1.25 1.58 0.0927 0.0145 14 0.00000 0.00000 9.12 0.46 0.0345 0.0021 15 0.00001 0.00000 -6.13 0.43 0.0938 0.0046 16 0.00000 0.00000 -1.47 1.71 0.0112 0.0061 The magnetic center is at X = 205.7 +- 0.5 microns Y = 58.7 +- 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 ^ | N S S --> x N N S Date: 06-26-1995 Time: 13:34:14 Magnet Name: 000 Run Number: 2 FIELD HARMONICS Magnet Current, Imag = 100.5468 +- 1.092809E-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 -101.64 0.26 0.5401 0.0008 2 0.00019 0.00000 -52.40 0.10 0.9382 0.0035 3 0.02038 0.00001 30.06 0.00 100.0000 0.0512 4 0.00003 0.00000 -4.29 0.17 0.1527 0.0011 5 0.00004 0.00000 -16.78 0.32 0.1778 0.0029 6 0.00000 0.00000 24.11 1.19 0.0226 0.0071 7 0.00001 0.00000 20.69 0.58 0.0369 0.0126 8 0.00001 0.00000 16.49 0.86 0.0647 0.0041 9 0.00006 0.00000 -9.70 0.15 0.2928 0.0032 10 0.00000 0.00000 -2.59 3.40 0.0090 0.0027 11 0.00001 0.00000 -7.11 0.75 0.0412 0.0022 12 0.00003 0.00001 -7.01 2.38 0.1285 0.0393 13 0.00002 0.00000 -3.08 0.78 0.0891 0.0154 14 0.00001 0.00000 6.80 0.97 0.0265 0.0081 15 0.00002 0.00000 -7.16 0.09 0.0802 0.0065 16 0.00000 0.00000 0.69 5.02 0.0077 0.0049 The magnetic center is at X = 203.5 +- 0.8 microns Y = 53.8 +- 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 ^ | N S S --> x N N S Date: 06-26-1995 Time: 13:37:51 Magnet Name: 000 Run Number: 2 FIELD HARMONICS Magnet Current, Imag = 150.5292 +- 1.476303E-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 0.30 0.4857 0.0020 2 0.00028 0.00000 -52.17 0.11 0.9278 0.0020 3 0.03045 0.00001 30.06 0.00 100.0000 0.0523 4 0.00005 0.00000 -4.02 0.28 0.1679 0.0053 5 0.00006 0.00000 -15.66 0.18 0.1814 0.0036 6 0.00000 0.00000 -5.98 2.92 0.0045 0.0007 7 0.00003 0.00000 -11.40 20.72 0.0895 0.0088 8 0.00003 0.00000 -17.34 2.55 0.1109 0.0160 9 0.00008 0.00001 -9.79 0.14 0.2672 0.0173 10 0.00001 0.00000 9.89 1.73 0.0206 0.0048 11 0.00001 0.00000 -5.61 0.19 0.0301 0.0034 12 0.00006 0.00000 -0.93 0.65 0.2016 0.0037 13 0.00002 0.00000 1.98 1.49 0.0603 0.0072 14 0.00002 0.00000 6.85 1.27 0.0724 0.0139 15 0.00003 0.00000 -5.89 0.41 0.0978 0.0019 16 0.00000 0.00000 4.53 4.10 0.0030 0.0019 The magnetic center is at X = 201.7 +- 0.5 microns Y = 51.6 +- 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 ^ | N S S --> x N N S Date: 06-26-1995 Time: 13:41:28 Magnet Name: 000 Run Number: 2 FIELD HARMONICS Magnet Current, Imag = 200.5305 +- 2.001433E-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 -101.36 0.11 0.4335 0.0011 2 0.00038 0.00000 -52.24 0.05 0.9320 0.0024 3 0.04041 0.00001 30.05 0.00 100.0000 0.0461 4 0.00006 0.00000 -2.34 0.19 0.1564 0.0062 5 0.00007 0.00000 -15.07 0.12 0.1693 0.0016 6 0.00001 0.00000 -1.74 2.01 0.0200 0.0019 7 0.00004 0.00000 -17.93 0.99 0.1036 0.0041 8 0.00004 0.00001 -9.65 1.53 0.0991 0.0201 9 0.00012 0.00001 -10.33 0.19 0.2930 0.0202 10 0.00001 0.00000 7.17 14.41 0.0194 0.0034 11 0.00001 0.00000 -4.34 0.76 0.0233 0.0064 12 0.00008 0.00000 4.42 1.15 0.1938 0.0102 13 0.00000 0.00000 -0.29 8.53 0.0109 0.0041 14 0.00003 0.00000 10.47 0.83 0.0773 0.0104 15 0.00003 0.00000 -5.54 0.09 0.0788 0.0025 16 0.00000 0.00000 6.57 0.99 0.0110 0.0040 The magnetic center is at X = 202.5 +- 0.5 microns Y = 52.3 +- 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)