Calculation of pulsed kicker magnetic field attenuation inside beam chambers. Page: 1 of 10
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LS-291
CALCULATION OF PULSED KICKER MAGNETIC FIELD ATTENUATION
INSIDE BEAM CHAMBERS
S. H. Kim
January 8, 2001
1. Introduction and Summary
The ceramic beam chambers in the sections of the kicker magnets for the beam injection and
extraction in the Advanced Photon Source (APS) are made of alumina. The inner surface of the
ceramic chamber is coated with a conductive paste. The choice of coating thickness is intended
to reduce the shielding of the pulsed kicker magnetic field while containing the electromagnetic
fields due to the beam bunches inside the chamber, and minimize the Ohmic heating due to the
fields on the chamber [1]. The thin coating generally does not give a uniform surface resistivity
for typical dimensions of the ceramic chambers in use. The chamber cross section is a circular or
an elliptic shape. The chamber or its wall thickness refers to the conductive coating in the
following sections.
This note calculates the penetration of the kicker magnetic field inside the beam chamber.
The kicker field is assumed to be a half-sine pulse and be spatially uniform over the chamber
dimensions. The purpose of the calculation is to be able to deduce the average surface resistivity
of a chamber by fitting the measured magnetic field data with the calculation inside the chamber.
In the following section, assuming that the coating thickness d is much smaller than the classical
skin depth S, the penetrated field inside the chamber is calculated by subtracting the shielding
field due to the eddy currents. In Section 3, for the kicker fields parallel and perpendicular to the
axis of a circular beam chamber, the fields inside the chamber with an arbitrary wall thickness
are calculated. For both directions of the kicker fields, the approximations made for d << S
achieve the same results as given in Section 2. For elliptic chambers, the calculations for the
vector potentials are not completed because of the tedious approximation procedure with
Mathieu functions. Instead, the results in Section 2 and the time constants calculated for the
elliptic geometries in Table 1 could be used for the purpose of this note.
2. Simplified Solution
The two-dimensional cross section of a circular cylinder is depicted in Fig. 1. The
conducting-wall thickness, d = b - a, is greatly exaggerated compared to its inner and outer radii,
a and b, and the main body of the chamber (ceramic) is not shown. The kicker magnetic field
By(t) is applied perpendicular to the chamber axis. Neglecting the eddy current distributions near
both ends of the chamber, the eddy currents in the chamber are assumed to be parallel to its axis.
For a simple calculation of the eddy current shielding, we will follow the textbook by Smythe
[2]. The eddy currents due to the inducing kicker field By(t'), at time t'before the present time t,
reduce the field inside the chamber at time t. The eddy currents have been decaying out
exponentially from the time t'with a time constant t. Then the field inside the chamber at time t,
Bi(t), may be calculated by subtracting the shielding field due to the eddy currents from the kicker
field By(t):
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Kim, S. H. Calculation of pulsed kicker magnetic field attenuation inside beam chambers., report, April 6, 2001; Illinois. (https://digital.library.unt.edu/ark:/67531/metadc736256/m1/1/: accessed April 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.