Halbach cylinder

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A Halbach cylinder is a magnetized cylinder composed of ferromagnetic material producing (in the idealised case) a magnetic field confined entirely within the cylinder with zero field outside. The cylinders can also be magnetized such that the magnetic field is entirely outside the cylinder, with zero field inside. Several magnetization distributions are shown below:

A ferromagnetic cylinder showing various magnetization patterns and magnetic field.

The direction of magnetization within the ferromagnetic material is given by

$M = M_{r} (\sin(k\phi)\hat{\rho}-\cos(k\phi)\hat{\phi})$

where M_r is the ferromagnetic remanence (T/m). A choice of +k gives an internal magnetic field and -k gives an external magnetic field.

Ideally, these structures would be created from an infinite length cylinder of magnetic material with the direction of magnetization continuously varying. The magnetic flux produced by this ideal design would be perfectly uniform and be entirely confined to the bore of the cylinder. Of course, the ideal case of infinite length is not realisable and in practice the finite length of the cylinders produces end effects which introduce non-uniformities in the field within the bore. The difficulty of manufacturing a cylinder with a continuously varying magnetization also usually leads to the design being broken into segments.

These cylindrical structures are used in devices such as brushless AC motors, magnetic couplings and high field cylinders. Both brushless motors and coupling devices use multipole field arrangements:

Brushless motors typically use cylindrical designs in which all the flux is confined to the centre of the bore (such as k = 3 above, a six pole rotor) with the AC coils also contained within the bore. Such self-shielding motors designs are more efficient and produce higher torque than conventional motor designs.

Magnetic coupling devices transmit torque through magnetically transparent barriers (that is the barrier is non-magnetic or is magnetic but is not affected by an applied magnetic field), for instance between sealed containers or pressurised vessels. The optimal torque couplings consists of a pair of coaxially nested cylinders with opposite +k and -k flux magnetization patterns, as -k magnetization patterns produce fields entirely external to the cylinder. In the lowest energy state, the outer flux of the inner cylinder exactly matches the internal flux of the outer cylinder. Rotating one cylinder relative to the other from this state results in a restoring torque.

1 Uniform Field
2 High Uniform Field Designs
- 2.1 Varying the field
3 References

[edit] Uniform Field

Uniform field inside Halbach cylinder

For the special case of k = 2, the field inside the bore is uniform, and is given by:

$H = M_{r} \ln\left(\frac{R_{o}}{R_{i}}\right)\hat{y}$

where the inner and outer cylinder radii are R_o and R_i, respectively. H is in the y direction. This is the simplest form of the Halbach cylinder, and it can be seen that if the ratio of outer to inner radii is greater than e the flux inside the bore actually exceeds the remanence of the magnetic material used to create the cylinder.

This cylindrical design is only one class of design which produces a uniform field inside a cavity within an array of permanent magnets. Other classes of design include wedge designs, proposed by Abele and Jensen in which wedges of magnetized material are arranged to provide uniform field within cavities inside the design as shown below.

Three designs producing uniform magnetic fields within their central air gap

The direction of magnetization of the wedges in (A) can be calculated using a set of rules given by Abele, and allows for great freedom in the shape of the cavity. Another class of design is the magnetic mangle (B), proposed by Coey and Cugat, in which uniformly magnetized rods are arranged such that their magnetization matches that of a Halbach cylinder, as shown for a six rod design. This design greatly increases access to the region of uniform field, at the expense of the volume of uniform field being smaller than in the cylindrical designs (although this area can be made larger by increasing the number of component rods). Rotating the rods relative to each other results in many possibilities including a dynamically variable field and various dipolar configurations. It can be seen that these designs are closely related to the k=1 Halbach cylinder. Other very simple designs for a uniform fields include separated magnets with soft iron return paths, as shown in figure (C).

If a k = 2 cylinder is cut and unrolled flat, the resulting structure is a Halbach array.

[edit] High Uniform Field Designs

If the two dimensional magnetic distribution patterns of the Halbach cylinder are extended to three dimensions, the result is the Halbach sphere. These designs have an extremely uniform field within the interior of the design, as they are not affected by the 'end effects' prevalent in the finite length cylinder design. The magnitude of the uniform field for a sphere also increases to 4/3 the amount for the ideal cylindrical design with the same inner and outer radii. However, being spherical, access to the region of uniform field is usually restricted to a narrow hole at the top and bottom of the design.

Higher fields are possible by optimising the spherical design to take account of the fact that it is composed of point dipoles (and not line dipoles). This results in the stretching of the sphere to an elliptical shape and having a non-uniform distribution of magnetization over the component parts of the sphere. Using this method, as well as soft pole pieces within the design, 4.5 T in a working volume of 20 mm³ was achieved by Bloch et al. in 1998 and this was increased further to 5 T in 2000, although over a smaller working area of 0.05 mm . As hard materials are temperature dependent, refrigeration of the entire magnet array can increase the field within the working area further as shown by Kumada et al. This group also reported development of a 5.16 T Halbach dipole cylinder in 2003.

[edit] Varying the field

Halbach cylinders give a static field. However cylinders can be nested, and by rotating one cylinder relative to the other, cancellation of the field and adjustment of the direction can be achieved. ^[1]

[edit] References

^ Tip Magazine: Magnets, Markets, and Magic Cylinders The Industrial Physicist by Michael Coey and Denis Weaire

K. Halbach, Nuclear Instruments and Methods, 169, 1, (1980)
J. M. D. Coey and T.R. Ní Mhíocháin, "Permanent Magnets", High Magnetic Fields: Science and Technology, Volume 1, ed. F. Herlach and N. Miura, World Scientific Publishing, p25 - 47 (2003)
O. Cugat and F. Bloch, "4-Tesla Permanent Magnetic Flux Source", Proc. 15th International Workshop on Rare Earth Magnets and Their Applications, published by MAT INFO, Dresden 1998, p807 (1998)