On the radius of convergence for multipole expansion Page: 3 of 9
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II. Field due to a Current Filament in Two Sheets of Parallel Plates
The magnetic field due to a single current filament at (xe, yC) in two sheets of infinite
permeability iron is given by'
H(z) = - tanh' (z + coth 7r(z-Zc)(1)
4g 2g 2g (1
where g is the gap distance between iron plates, I is the current on the filament, z = x+ iy
is the location of measuring the field, Z, = x, + iyc, H = Hy + iH, is the magnetic field.
The magnetic field in Eq. (1) can be expanded in multipole as
H = 9 E i(an + P.) ( )zn (2)
where a,,8 coefficients are tabulated in ref. 1.
There are some interesting features in Eq. (2), i.e. when the coil location x, is much
larger than the gap, i.e. 7rx/2g >> 1, the dipole component of the flux density becomes
the familiar equation as,
Bo = 0 , (3)
g
which is independent of xe. The higher multipoles depends on x. exponentially as
Bn ' I g z"e-c/9 (4)
Thus the multipoles would be smaller, when the coils are far away from the center of the
magnet.
III. Radius of Convergence
Since Eq. (1) has a simple pole at z = Zc, we might expect that the radius of
convergence is within the coil radius, 1Z01. The validity of multipole expansion depends
also on the number of multipoles used in the expansion. In the present study, we shall
use 0-18th terms. In reality, we shall see that 0-10h terms are sufficient to represent the
magnetic field inside the radius of convergence. The 12th to the 18th terms in Eq. (2)
hardly improve the result by increasing the radius of convergence.
To simplify the analysis, we shall study the 4 coils configuration as following:
I at (x,,y,) and (x,--y,)
-I at (-Z, y,) and (-x,-y,)
These four coils are located inside two parallel sheets of irons with gap, g = 0.08255 m.2
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Y., Lee S. On the radius of convergence for multipole expansion, report, October 1, 1989; United States. (https://digital.library.unt.edu/ark:/67531/metadc866548/m1/3/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.