SLAC Magnetic Measurements
Date: 08-11-1995
Time: 12:49:01

Magnet Name (Serial #): 070
Bar Code Number: 071
Project: PEP II High Energy Ring Sextupoles
Test Stand: IR8, S1, W
Measurement Coil: # 1
Operator: b
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



            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)
--------+--------  --------+--------  --------+--------
  25.592    0.002   5.20582  0.00092  203.4121   0.0380
  50.579    0.003  10.17764  0.00193  201.2224   0.0406
 100.566    0.008  20.21634  0.00426  201.0251   0.0454
 125.555    0.007  25.22065  0.00600  200.8730   0.0491
 150.560    0.009  30.20462  0.00327  200.6155   0.0246
 175.544    0.011  35.18084  0.00425  200.4101   0.0275
 200.556    0.009  40.11303  0.00275  200.0092   0.0165
 150.561    0.011  30.44081  0.00478  202.1825   0.0349
 100.577    0.002  20.43121  0.00185  203.1409   0.0190
  50.571    0.004  10.36030  0.00118  204.8645   0.0283
  25.585    0.007   5.32299  0.00206  208.0553   0.0977


           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)