Halbach array

 ( Magnetic Levitation ) 

The crucial point is that the flux will cancel below the plane and reinforce itself above the plane. In fact, any magnetization pattern where the components of magnetization are $\backslash pi/2$ out of phase with each other will result in a one-sided flux. The mathematical transform that shifts the phase of all components of some function by $\backslash pi/2$ is called a Hilbert transform; the components of the magnetization vector can therefore be any Hilbert-transform pair (the simplest of which is simply $\backslash sin(x)\; \backslash cos(y)$, as shown in the diagram above).

The field on the non-cancelling side of the ideal, continuously varying, infinite array is of the form

$F(x,\; y)\; =\; F\_0\; e^\{ikx\}\; e^\{-ky\},$

$F(x,\; y)$ is the field in the form $F\_x\; +\; i\; F\_y$,

$F\_0$ is the magnitude of the field at the surface of the array,

$k$ is the wavenumber (i.e., the spatial frequency) $2\backslash pi\; /\; \backslash lambda.$

• The field is twice as large on the side on which the flux is confined (in the idealized case).
• There is no stray field produced (in the ideal case) on the opposite side. This helps with field confinement, usually a problem in the design of magnetic structures.

Thus they have a number of applications, ranging from flat refrigerator magnets through industrial applications such as the brushless DC motor, , magnetic drug targeting to high-tech applications such as wiggler magnets used in particle accelerators and free-electron lasers.

The Inductrack maglev train and Inductrack rocket-launch system utilize the Halbach array to lift the train by repelling loops of wire in the track. Flat flexible (not hard ceramic ferrite) refrigerator magnets are created with a Halbach magnetization pattern for a stronger holding force when attached to a flat ferromagnetic surface (e.g. a fridge door) than the holding force from a uniform magnetization. They're made from powdered ferrite mixed in a flexible binder (e.g. plastic or rubber) that is exposed to a Halbach magnetization field pattern as it is Extrusion, permanently giving the ferrite particles in the magnetic compound this one-sided flux distribution (which can be viewed with magnetic viewing film).

Scaling up this design and adding a top sheet gives a wiggler magnet, used in and free-electron lasers. Wiggler magnets wiggle, or oscillate, an electron beam perpendicular to the magnetic field. As the electrons are undergoing acceleration, they radiate electromagnetic energy in their flight direction, and as they interact with the light already emitted, photons along its line are emitted in phase, resulting in a "laser-like" monochromatic and coherent beam.

The design shown above is usually known as a Halbach wiggler. The magnetization vectors in the magnetized sheets rotate in the opposite senses to each other; above, the top sheet's magnetization vector rotates clockwise, and the bottom sheet's magnetization vector rotates counter-clockwise. This design is chosen so that the x components of the magnetic fields from the sheets cancel, and the y components reinforce, so that the field is given by

$H\_y\; \backslash approx\; \backslash cos(kx),$

where k is the wavenumber of the magnetic sheet given by the spacing between magnetic blocks with the same magnetization vector.

This arrangement allows the field to effectively be turned on and off above or below the plane of the rods, depending on the rotation of the rods. Such a device makes an efficient mechanical magnetic latch requiring no power. A detailed study of this arrangement has shown that each rod becomes a subject to a strong torque from its neighboring rods when rotated. However, a simple and efficient solution, providing both stabilization and the ability to rotate each rod alternately, is simply to provide an equal-gearing arrangement on each rod, as shown in the figure.

The direction of magnetization within the ferromagnetic material, in plane perpendicular to the axis of the cylinder, is given by

$M\; =\; M\_r\backslash left\backslash cos\backslash left((k,$

where M_r is the ferromagnetic remanence (A/m). A positive value of k − 1 gives an internal magnetic field, and a negative one 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 either the bore of the cylinder or the outside of the cylinder. Of course, the ideal case of infinite length is not realizable, and in practice the finite length of the cylinders produces end effects, which introduce non-uniformities in the field. The difficulty of manufacturing a cylinder with a continuously varying magnetization also usually leads to the design being broken into segments.

• Brushless motors or alternators typically use cylindrical designs in which all the flux is confined to the centre of the bore (such as k = 4 above, a 6-pole rotor) with the AC coils also contained within the bore. Such self-shielding motor or alternator designs are more efficient and produce higher torque or output than conventional motor or alternator designs.
• Magnetic-coupling devices transmit torque through magnetically transparent barriers (that is, the barrier is non-magnetic or is magnetic but 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 this configuration is the only system for infinitely long cylinders that produces a torque. 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.
• Cylindrical Halbach Arrays are used in Portable MRI Scanners. They offer the potential for a relatively lightweight, low to mid-field system with no cryogenics, a small fringe field, and no electrical power requirements or heat dissipation needs. The reduced stray fields also enhance safety and minimize interference with surrounding electronic devices.

$H\; =\; M\_r\; \backslash ln\backslash left(\backslash frac\{R\_\backslash text\{o\}\}\{R\_\backslash text\{i\}\}\backslash right)\; \backslash widehat\{y\},$

where the inner and outer cylinder radii are R_i and R_o 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. However, care has to be taken not to produce a field that exceeds the coercivity of the permanent magnets used, as this can result in demagnetization of the cylinder and the production of a much lower field than intended.

This cylindrical design is only one class of designs that produce 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.

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 6-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 the designs shown in (A) and (B) are closely related to the k = 2 Halbach cylinder. Other very simple designs for a uniform field include separated magnets with soft iron return paths, as shown in figure (C).

In recent years, these Halbach dipoles have been used to conduct low-field NMR experiments. Compared to commercially available (Bruker Minispec) standard plate geometries (C) of permanent magnets, they, as explained above, offer a huge bore diameter, while still having a reasonably homogeneous field.

Because of the symmetry of the arrangement along the cylinder's axis, the problem can be treated as two-dimensional. Work in plane-polar coordinates $(r,\; \backslash theta)$ with associated unit vectors $\backslash hat\backslash mathbf\{r\}$ and $\backslash hat\backslash boldsymbol\{\backslash theta\}$, and let the cylinder have radial extent . Then the Magnetization in the cylinder walls, which has magnitude $M\_0$, rotates smoothly as

$\backslash mathbf\{M\}\_\backslash mathrm\{cyl\}\; =\; M\_0\; (\backslash cos\backslash theta\backslash ,\; \backslash hat\backslash mathbf\{r\}\; +\; \backslash sin\backslash theta\backslash ,\; \backslash hat\backslash boldsymbol\{\backslash theta\}),$

By definition, the auxiliary Magnetic field $\backslash mathbf\{H\}$ is related to the magnetisation and magnetic flux density $\backslash mathbf\{B\}$ by $\backslash mathbf\{B\}\; =\; \backslash mu\_0\; (\backslash mathbf\{H\}\; +\; \backslash mathbf\{M\})$. Using Gauss' law $\backslash nabla\backslash cdot\backslash mathbf\{B\}\; =\; 0$, this is equivalently

Since the problem is static there are no free currents and all time derivatives vanish, so Ampère's law additionally requires $\backslash nabla\backslash times\backslash mathbf\{H\}=0\; \backslash implies\; \backslash mathbf\{H\}\; =\; \backslash nabla\backslash varphi$, where $\backslash varphi$ is the magnetic scalar potential (up to a sign under some definitions). Substituting this back into the previous Equation governing $\backslash mathbf\{H\}$ and $\backslash mathbf\{M\}$, we find that we need to solve

which has the form of Poisson's equation.

Consider now the boundary conditions at the cylinder-air interfaces $r\; =\; r\_\backslash mathrm\{i\}$ and $r\; =\; r\_\backslash mathrm\{o\}$. Integrating $\backslash nabla\; \backslash times\; \backslash mathbf\{H\}\; =\; 0$ over a small loop straddling the boundary and applying Stokes' theorem requires that the parallel component of $\backslash mathbf\{H\}$ is continuous. This in turn requires that $\backslash varphi$ is continuous across the boundary. (More properly this implies that $\backslash varphi$ must differ by a constant across the boundary, but since the physical quantities we are interested in depend on gradients of this potential, we can arbitrarily set the constant to zero for convenience.) To obtain a second set of conditions, integrate Equation across a small volume straddling the boundary and apply the divergence theorem to find

$\backslash left\backslash frac\{\backslash partial\; =\; \backslash pm\; \backslash mathbf\{M\}\; \backslash cdot\; \backslash hat\{\backslash mathbf\{r\}\},$

In plane-polar coordinates, the divergence of a vector field $\backslash mathbf\{F\}\; =\; F\_r\backslash hat\backslash mathbf\{r\}\; +\; F\_\backslash theta\; \backslash hat\backslash boldsymbol\{\backslash theta\}$ is given by

Similarly, the gradient of a scalar field $f$ is given by

Combining these two relations, the Laplace operator $\backslash nabla^2\; f\; =\; \backslash nabla\backslash cdot(\backslash nabla\; f)$ becomes

Using Equation , the magnetisation divergence in the cylinder walls is

Look for a particular solution of this equation in the cylinder walls. With the benefit of hindsight, consider $\backslash varphi\_\backslash mathrm\{p\}\; =\; r\; \backslash ln\; r\; \backslash cos\backslash theta$, because then we have

Now consider the homogeneous equation for Equation , namely $\backslash nabla^2\; \backslash varphi\_\backslash mathrm\{h\}\; =\; 0$. This has the form of Laplace's equation. Through the method of separation of variables, it can be shown that the general homogeneous solution whose gradient is periodic in $\backslash theta$ (such that all the physical quantities are single-valued) is given by

$\backslash varphi\_\backslash mathrm\{h\}\; =\; A\_0\; +\; B\_0\; \backslash theta\; +\; C\_0\; \backslash ln\; r\; +\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash left(A\_n\; r^n\; +\; B\_n\; r^\{-n\}\backslash right)\backslash cos\; n\backslash theta\; +\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash left(C\_n\; r^n\; +\; D\_n\; r^\{-n\}\backslash right)\backslash sin\; n\backslash theta,$

The continuity conditions give

at the inner boundary and

at the outer boundary. The potential gradient has non-vanishing radial component $\backslash beta\backslash cos\backslash theta\; -\; M\_0\; \backslash ln\; r\; \backslash cos\backslash theta\; -\; M\_0\; \backslash cos\backslash theta$ in the cylinder walls and $\backslash alpha\backslash cos\backslash theta$ in the bore, and so the conditions on the potential derivative become

$(\backslash beta\backslash cos\backslash theta\; -\; M\_0\; \backslash ln\; r\_\backslash mathrm\{i\}\; \backslash cos\backslash theta\; -\; M\_0\backslash cos\backslash theta)\; -\; \backslash alpha\; \backslash cos\backslash theta\; =\; -M\_0\backslash cos\backslash theta\; \backslash implies\; \backslash beta\; -\; M\_0\; \backslash ln\; r\_\backslash mathrm\{i\}\; -\; \backslash alpha\; =\; 0\; \backslash implies\; \backslash alpha\; =\; \backslash beta\; -\; M\_0\; \backslash ln\; r\_\backslash mathrm\{i\}$

$0\; -\; (\backslash beta\backslash cos\backslash theta\; -\; M\_0\; \backslash ln\; r\_\backslash mathrm\{0\}\; \backslash cos\backslash theta\; -\; M\_0\backslash cos\backslash theta)\; =\; M\_0\backslash cos\backslash theta\; \backslash implies\; \backslash beta\; -\; M\_0\; \backslash ln\; r\_\backslash mathrm\{o\}\; =\; 0$

$B\_\backslash mathrm\{bore\}\; =\; \backslash left|\backslash mathbf\{B\}\_\backslash mathrm\{bore\}\backslash right|\; =\; \backslash sqrt\{\backslash cos^2\backslash theta\; +\; \backslash sin^2\backslash theta\}\backslash frac\{M\_0\}\{\backslash mu\_0\}\; \backslash ln\backslash left(\backslash frac\{r\_\backslash mathrm\{o\}\}\{r\_\backslash mathrm\{i\}\}\backslash right)\; =\; \backslash frac\{M\_0\}\{\backslash mu\_0\}\; \backslash ln\backslash left(\backslash frac\{r\_\backslash mathrm\{o\}\}\{r\_\backslash mathrm\{i\}\}\backslash right),$

The equation for the field in a Halbach sphere is Permanent magnet based sources of magnetic field .

$B\; =\; \backslash frac\{4\}\{3\}\; B\_0\; \backslash ln\backslash left(\backslash frac\{R\_\backslash text\{o\}\}\{R\_\backslash text\{i\}\}\backslash right).$

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. Using this method, as well as soft pole pieces within the design, 4.5 T in a working volume of 20 mm³ was achieved, and this was increased further to 5 T in 2002, although over a smaller working volume of 0.05 mm³. As hard materials are temperature-dependent, refrigeration of the entire magnet array can increase the field within the working area further. This group also reported development of a 5.16 T Halbach dipole cylinder in 2003.

• Neodymium magnet
• Permanent magnet
• Strong focusing
• Inductrack uses Halbach arrays to generate strong fields for maglev trains
• Helmholtz coil can give very even magnetic fields

• Passive levitation of a shaft
• Electric model aircraft motor
• magnet levitated vehicle switching system