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



            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)
--------+--------  --------+--------  --------+--------
  25.554    0.005   5.21932  0.00147  204.2453   0.0700
  50.540    0.004  10.18962  0.00188  201.6131   0.0411
 100.538    0.005  20.22851  0.00557  201.2036   0.0563
 150.518    0.003  30.22464  0.01282  200.8048   0.0853
 200.521    0.003  40.11952  0.00613  200.0768   0.0307
 150.526    0.002  30.50621  0.00592  202.6639   0.0394
 100.540    0.005  20.48394  0.00212  203.7393   0.0233
  50.533    0.000  10.39281  0.00337  205.6631   0.0667
  25.546    0.001   5.35273  0.00178  209.5326   0.0700


           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)