Advanced Photon Source

A U.S. Department of Energy, Office of Science,
Office of Basic Energy Sciences national synchrotron x-ray research facility

LS Note 166

A Radial Coil Probe for Quadrupole Magnet Measurements

S. H. Kim

Introduction

This note describes a rotating coil probe of "radial-winding geometry" for the measurements of the magnetic center, quadrupole fields and multipole coefficients of quadrupole magnets. The active length of the coil is longer than the magnet length, so that all the measurements will be integrated values along the longitudinal direction of the magnets. Errors of the measurements due to the fabrication tolerances of the coil are discussed.

Multipole Coefficients

The magnetic field in the aperture of a magnet may be expressed in the two-dimensional Cartesian or cylindrical coordinate system:

where z = x + iy = r exp (i),

and the multipole coefficients for the two different definitions are related by

Here R is a reference radius and b₁ = 1.0 cm^-1 for quadrupole magnets. The reference angle for a normal quadrupole magnet is defined as a₁ = 0 or ₁ = 0. For the storage ring quadrupole Q4 of the APS, B_ob₁ = C₂/R = 18.9 T/m at 7 Gev.

For a quadrupole magnet with effective length L, the total flux linkage for coil A at an angular position in Fig. 1 is

When the two probe coils, coil A and coil B in Fig. 1, are connected in series such that their flux linkages are opposed to each other, the measurement sensitivity of higher multipoles, n>2, could be increased by rejected the dipole and main (quadrupole) field components of the flux linkages.

The flux linkage for the two coils at is

where

represents the sensitivity of the measurements for each multipole coefficient. From Eq. (6), the conditions for the rejection of dipole and quadrupole components are given by

Table 1 lists two examples of the coil locations for N_B/N_A = 2 and their R_fac. Each multipole coefficient obtained from the Fourier transformation of the measureed data should be divided by the measurement sensitivity, R_fac, listed in Table 1 in order to obtain the relative magnitude of the multipole coefficient at radius r_A1.

Magnetic Center and Quadrupole Field Integral

Figure 2 shows two coordinate systems, the xy-coordinate system with its origin at the magnetic center (MC) and the x'y' at the cylinder rotation axis (CR). The location of the CR with respect to the MC is Z_o = r_o exp (i_o). If the x'y'-coordinate system is displaced from the xy-coordinate system, the expression of the magnetic field at point, p, with respect to the two coordinate systems can be expressed as

By using the following relation,

Eq. (8) becomes

Equation (9) shows that the dipole field component in the x'y'-coordinate system consists of the dipole and all other higher multipole components in the xy-coordinate system. If we write up to the sextupole terms in both sides of Eq. (9), we find

For a typical quadrupole magnet, C_n(n2)/C₂<1, the azimuthal field componet in the x'y'-coordinate system due to the main quadrupole field only is

The flux linkage for coil A in Fig. 1 is

The off-center of the cylinder rotation axis with respect to the magnetic center can be found from the first term of Eq. (12). The second term of Eq. (12) gives the quadrupole field integral, which does not depend on the off-axis distance Z_o. When Z_o is not zero, it should be noted, however, that there are other correction terms in the quadrupole field integral measurements such as the 2C₃exp(-i3₃)Z_oZ'/R² term of Eq. (10).

For L = 0.6m, N_A = 20 turns, C₂/R = 19 T/M, r_A2 + r_A1 = 50 mm and = 3 for = t, the induced voltage from coil A is

Equations (12) and (13) indicate that the magnetic center-detecting coil shoul have radii of r_A2 = r_A1 to minimize the sin(2t) term of the equations.

Coil Position Errors

In Fig. 3, it is assumed that the plane of coil A is displaced from the cylinder rotation axis by r. When the angulator position of r_A2 is used as the reference angular position for the measurements, the errors of radial and angular positions of r_A2 and r_A1 are (r_A, 0) and (r_A, ), where = r (r_A2 + r_A1)/R_A2r_A1. Then the flux linkage of coil A is given by

In Eq. (14) the D and Q terms are used for the measurements of the magnetic center axis and quadrupole field integral, respectively. Relative errors, r/r_A2, r₂/r_A2 and r₁/r_A2, in Eq. (14), do not make the procedures for the detection of the magnetic center axis particularly difficult. In the Fourier analysis of the measurement data, however, one should keep in mind the angular phase error due to sin ' term. For a relative error of radial positions of 1x10^-3 (for case #1 coil geometry in Table 1, = 3 x 10^-3), the error of the Q term in Eq. (14) would be less than 5 x 10^-3.

Relative errors of multipole coefficients due to coil position tolerances for coil A and coil B in Table 1 are listed in Table 2. It is assumed that relative radial position errors of the coils with respect to r_A2 and relative position errors of the coil planes from its rotation axis are 1 10^-3.