SLAC Magnetic Measurements Date: 07-10-1995 Time: 08:57:17 Magnet Name (Serial #): 086 Bar Code Number: 119 Project: PEP II High Energy Ring Sextupoles Test Stand: IR8, S1, W Measurement Coil: # 1 Operator: pdr Run Number: 1 Comment: Standard Measurement 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: 07-10-1995 Time: 09:27:42 Magnet Name: 086 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 25.62897 +- 2.809929E-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 -114.81 0.59 0.7399 0.0085 2 0.00006 0.00000 30.69 0.48 1.0867 0.0219 3 0.00526 0.00000 -29.90 0.00 100.0000 0.0487 4 0.00000 0.00000 41.56 2.07 0.0787 0.0108 5 0.00001 0.00000 19.72 0.40 0.1454 0.0028 6 0.00000 0.00000 22.44 8.95 0.0274 0.0121 7 0.00000 0.00000 11.93 4.90 0.0531 0.0208 8 0.00000 0.00000 0.67 14.80 0.0645 0.0449 9 0.00002 0.00000 9.56 0.37 0.3267 0.0241 10 0.00000 0.00000 7.03 7.06 0.0805 0.0434 11 0.00000 0.00000 -1.82 10.23 0.0195 0.0099 12 0.00000 0.00000 0.37 8.02 0.0288 0.0101 13 0.00000 0.00000 4.53 7.43 0.0161 0.0040 14 0.00000 0.00000 -0.56 6.45 0.0198 0.0093 15 0.00001 0.00000 6.51 0.80 0.0996 0.0451 16 0.00000 0.00000 -2.62 7.13 0.0309 0.0095 The magnetic center is at X = 214.1 +- 4.7 microns Y = 116.8 +- 4.3 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: 07-10-1995 Time: 09:31:13 Magnet Name: 086 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 50.61071 +- 2.572861E-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 -136.82 3.06 0.2053 0.0089 2 0.00012 0.00000 29.80 0.27 1.1628 0.0083 3 0.01026 0.00000 -29.90 0.00 100.0000 0.0246 4 0.00001 0.00000 41.35 1.54 0.1212 0.0101 5 0.00002 0.00000 18.55 0.90 0.1586 0.0047 6 0.00000 0.00000 2.84 24.96 0.0106 0.0055 7 0.00000 0.00000 11.56 0.97 0.0427 0.0020 8 0.00000 0.00000 -3.30 14.53 0.0117 0.0070 9 0.00003 0.00000 10.07 0.09 0.3297 0.0051 10 0.00000 0.00000 6.42 2.71 0.0100 0.0053 11 0.00000 0.00000 1.10 12.35 0.0097 0.0059 12 0.00000 0.00000 8.18 3.20 0.0111 0.0051 13 0.00000 0.00000 0.24 8.36 0.0064 0.0052 14 0.00000 0.00000 -2.75 8.82 0.0068 0.0026 15 0.00001 0.00000 6.34 0.13 0.0671 0.0051 16 0.00000 0.00000 3.53 2.59 0.0071 0.0063 The magnetic center is at X = 225.1 +- 2.0 microns Y = 132.0 +- 2.3 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: 07-10-1995 Time: 09:34:50 Magnet Name: 086 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 100.5996 +- 2.936271E-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 118.13 1.07 0.1711 0.0059 2 0.00024 0.00000 29.99 0.01 1.1709 0.0014 3 0.02037 0.00001 -29.90 0.00 100.0000 0.0686 4 0.00002 0.00000 41.13 0.07 0.1195 0.0014 5 0.00003 0.00000 18.55 0.14 0.1584 0.0047 6 0.00000 0.00000 -28.21 0.90 0.0200 0.0028 7 0.00001 0.00000 12.24 0.26 0.0422 0.0038 8 0.00000 0.00000 -16.26 2.74 0.0140 0.0008 9 0.00007 0.00000 10.11 0.04 0.3359 0.0011 10 0.00000 0.00000 6.08 6.13 0.0044 0.0030 11 0.00000 0.00000 -0.39 13.86 0.0040 0.0026 12 0.00000 0.00000 0.76 0.72 0.0081 0.0034 13 0.00000 0.00000 3.15 7.89 0.0042 0.0020 14 0.00000 0.00000 -5.58 3.33 0.0051 0.0021 15 0.00002 0.00000 6.03 0.27 0.0785 0.0062 16 0.00000 0.00000 0.48 3.37 0.0047 0.0028 The magnetic center is at X = 227.5 +- 0.3 microns Y = 131.5 +- 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 Date: 07-10-1995 Time: 09:41:46 Magnet Name: 086 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 150.5797 +- .0038923 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 102.53 0.30 0.2585 0.0010 2 0.00036 0.00000 29.95 0.05 1.1697 0.0007 3 0.03043 0.00001 -29.90 0.00 100.0000 0.0557 4 0.00004 0.00000 40.68 0.21 0.1191 0.0010 5 0.00005 0.00000 18.59 0.09 0.1578 0.0009 6 0.00000 0.00000 -27.33 2.30 0.0070 0.0004 7 0.00001 0.00000 13.00 0.30 0.0384 0.0006 8 0.00000 0.00000 -14.77 0.46 0.0134 0.0007 9 0.00010 0.00000 10.11 0.02 0.3335 0.0012 10 0.00000 0.00000 3.77 4.66 0.0037 0.0006 11 0.00000 0.00000 -2.22 13.10 0.0025 0.0005 12 0.00000 0.00000 -0.41 14.09 0.0067 0.0007 13 0.00000 0.00000 -7.18 3.00 0.0027 0.0013 14 0.00000 0.00000 -8.72 1.53 0.0051 0.0014 15 0.00002 0.00000 6.14 0.06 0.0697 0.0013 16 0.00000 0.00000 -0.61 2.24 0.0019 0.0006 The magnetic center is at X = 227.1 +- 0.3 microns Y = 131.6 +- 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 Date: 07-10-1995 Time: 09:48:43 Magnet Name: 086 Run Number: 1 FIELD HARMONICS Magnet Current, Imag = 200.5785 +- 7.604195E-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 97.98 0.29 0.2889 0.0007 2 0.00047 0.00000 29.93 0.03 1.1677 0.0010 3 0.04040 0.00000 -29.90 0.00 100.0000 0.0124 4 0.00005 0.00000 40.58 0.22 0.1232 0.0013 5 0.00006 0.00000 18.72 0.03 0.1543 0.0012 6 0.00000 0.00000 1.22 28.40 0.0096 0.0011 7 0.00002 0.00000 13.18 0.29 0.0399 0.0005 8 0.00001 0.00000 -14.76 0.96 0.0147 0.0016 9 0.00013 0.00000 10.13 0.03 0.3336 0.0019 10 0.00000 0.00000 3.96 4.70 0.0025 0.0011 11 0.00000 0.00000 11.88 2.25 0.0030 0.0012 12 0.00000 0.00000 0.55 14.31 0.0029 0.0010 13 0.00000 0.00000 -4.31 10.29 0.0045 0.0005 14 0.00000 0.00000 -9.73 1.47 0.0058 0.0010 15 0.00003 0.00000 6.20 0.03 0.0703 0.0015 16 0.00000 0.00000 -1.21 8.09 0.0024 0.0017 The magnetic center is at X = 226.6 +- 0.2 microns Y = 131.6 +- 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 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)