SLAC Magnetic Measurements
Date: 04-20-1999
Time: 11:13:33

Project: PEP2INT
Magnet Type: PMSLICES
Magnet Name (Serial #): SK1_2
Measurement System: 5334
Operator: mks
Run Number: 12
Comment: 6th tuning step.

Temperature Measurements
Table Top Thermistor, T =  19.99655  deg C
Slice Thermistor # 1, T =  20.60938  deg C
Slice Thermistor # 2, T =  20.50034  deg C

Temperature Measurements
Table Top Thermistor, T =  20.02087  deg C
Slice Thermistor # 1, T =  20.50809  deg C
Slice Thermistor # 2, T =  20.45773  deg C


           SUMMARY OF THE CALCULATIONS AND CONVENTIONS USED


Field expansion (Bryant, CERN 92-05, p. 55)
Br = Sum Bref (r / rref)^n-1 [-an cos(n th) + bn sin(n th)]
Bt = Sum Bref (r / rref)^n-1 [ an sin(n th) + bn cos(n th)]
For flux calculations, we re-express Bt
Bt = - Sum Brefn (r / rref)^n-1 sin[n (th - THspole)]

Flux from the origin to a wire bundle of N turns at R
Flux = Int_0^R Bt L N dr
     = - Sum Brefn L N rref (1 / n) (R / rref)^n sin[n (th - THspole)]
Fluxn = - Brefn L N rref (1 / n) (R / rref)^n sin[n (th - THspole)]

Integrator output
The integrated coil voltage VT is sampled and Fourier analyzed
The FFT gives VTn and PhiVTn in the formula
VTn(i) = VTn cos(n 2pi i / N + PhiVTn),   VT(i) = Sum(VTn(i))
VTn(th) = VTn cos(n th + PhiVTn)

Equating the flux to the integrator output gives
Brefn L N rref (1 / n) (R / rref)^n = VTn
BLn|_rref = n VTn / (N rref (R / rref)^n)     <<<---

-sin[n (th - THspole)] = cos(n th + PhiVTn)
                      = -sin(n th + PhiVTn - pi / 2)
- n THspole = PhiVTn - pi / 2
THspole = - (PhiVTn - pi / 2) / n             <<<---

For our planar bucking coil, we have:
Nm turns at Rm (main winding)
Nd turns at Rd and -Rd (dipole bucking)
Nq turns at Rq and -Rq (quadrupole bucking)
various return windings at R = 0
The sin[n (th - THspole)] dependence must be included for -R.
At -R, sin[n (th + pi - THspole)] = cos(n pi) sin[n (th - THspole)]
= (-1)^n sin[n (th - THspole)].
VTn = BLn|_rref rref (1 / n) { Nm (Rm / rref)^n
  - Nd (Rd / rref)^n + Nd (-Rd / rref)^n
  - Nq (Rq / rref)^n - Nq (-Rq / rref)^n }
Set rref = Rm, take out Nm
VTn = BLn|_Rm Rm (1 / n) Nm { 1
  + (-1 + (-1)^n) (Nd / Nm) (Rd / Rm)^n
  + (-1 - (-1)^n) (Nq / Nm) (Rq / Rm)^n }
BLn|_Rm = n VTn / (Rm Nm) { 1
  + (-1 + (-1)^n) (Nd / Nm) (Rd / Rm)^n
  + (-1 - (-1)^n) (Nq / Nm) (Rq / Rm)^n }^(-1)
Define BFn = { 1
  + (-1 + (-1)^n) (Nd / Nm) (Rd / Rm)^n
  + (-1 - (-1)^n) (Nq / Nm) (Rq / Rm)^n }
BLn|_Rm = n VTn / (Rm Nm BFn)
THspole = - (PhiVTn - pi / 2) / 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 The Magnetic Center:
In the quadrupole's frame, Bx' = G * y', By' = G * x'.
In the coil's frame (unprimed frame) the magnetic center is at (x0, y0).
In the coil's frame,
     Bx = G * (y - y0), By = G * (x - x0).
This gives the magnetic center in terms of the measured
dipole field.
     x0 = - By / G, y0 = - Bx / G
In terms of the measured integrated strengths,
     Xcenter = - (1/GL) * BL1 * sin(THspole1)
     Ycenter = - (1/GL) * BL1 * cos(THspole1)