SLAC Magnetic Measurements Date: 07-12-1995 Time: 09:13:03 Magnet Name (Serial #): 089 Bar Code Number: 116 Project: PEP II High Energy Ring Sextupoles Test Stand: IR8, S1 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 INTEGRATED SEXTUPOLE STRENGTH VS CURRENT Coil: Radius = .044882 m, # Turns = 20 Average: # Rotations/Measurement = 10 , # Measurements = 4 Imag sigImag SL sigSL SL/I sigSL/I (A) (A) (T/m) (T/m) (T/m/kA) (T/m/kA) --------+-------- --------+-------- --------+-------- 25.640 0.002 3.66768 0.00090 143.0474 0.0365 50.620 0.001 7.16787 0.00389 141.6018 0.0768 100.608 0.001 14.32043 0.02210 142.3393 0.2197 125.587 0.001 17.91935 0.00754 142.6846 0.0600 150.591 0.004 21.55821 0.00744 143.1573 0.0496 175.573 0.001 29.42336 0.54192 167.5851 3.0866 200.587 0.003 31.99528 0.04018 159.5079 0.2003 150.597 0.004 24.36652 0.04032 161.7995 0.2678 100.617 0.003 16.28215 0.00445 161.8225 0.0445 50.624 0.003 8.23027 0.00316 162.5773 0.0633 25.637 0.001 4.22710 0.00231 164.8855 0.0905 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)