Halbach Arrays and Magnetic Levitation

A Halbach array is a cleverly engineered arrangement of permanent magnets in which the magnetization direction rotates progressively from one magnet to the next. The result is striking: the magnetic field concentrates strongly on one side of the array while nearly canceling itself on the other. This asymmetry, first identified by John Mallinson in 1973 and fully developed by Klaus Halbach at Lawrence Berkeley National Laboratory in 1980, underlies some of the most exciting modern applications in transportation, medical imaging, and precision engineering.

Array Structure and Magnetization Rotation

In its simplest form, a Halbach array consists of a row of square or rectangular permanent magnets whose magnetization directions follow the repeating sequence:

$$uparrow rightarrow downarrow leftarrow uparrow rightarrow downarrow leftarrow$$

More precisely, for a magnet at position $x$ along the array and a spatial period $lambda_H$, the magnetization angle is:

$$theta(x) = frac{2pi x}{lambda_H}$$

This progressive rotation causes the magnetic field components to add constructively on the strong side and destructively on the weak side. The underlying mechanism is a kind of spatial Fourier filtering: the rotating dipoles generate only odd-order spatial harmonics on one face, while the fundamental and all harmonics cancel on the opposite face.

Field Distribution: Exponential Decay

Above the strong side of the array, at height $y$, the magnetic flux density decays exponentially:

$$B(y) = B_0 cdot e^{-2pi y / lambda_H}$$

Here $B_0$ is the field amplitude at the array surface ($y = 0$) and $lambda_H$ is the spatial wavelength of the array. A critical design insight follows: longer spatial periods sustain a meaningful field at greater distances, which matters when sizing the air gap in a levitation system. Conversely, a shorter period gives a sharper field gradient close to the surface.

The Inductrack Principle: Levitation via Eddy Currents

When a Halbach array moves at velocity $v$ over a conducting track (aluminum or copper sheet, or a ladder of transverse conductors), the time-varying field induces eddy currents in the conductor. By Lenz’s law, these currents oppose the change in flux, creating a repulsive lift force that pushes the array away from the track. Richard Post and Dimitri Ryutov at Lawrence Livermore National Laboratory named this scheme Inductrack and demonstrated its feasibility for passive, room-temperature magnetic levitation without superconductors.

At high speed the lift force takes the form:

$$F_{text{lift}} = frac{B_0^2 cdot lambda_H cdot w}{2mu_0} cdot frac{v^2}{v^2 + v_c^2}$$

As $v gg v_c$, the lift force saturates toward a maximum set by the array geometry and surface field. Equally important is the lift-to-drag ratio, which grows linearly with velocity:

$$frac{F_{text{lift}}}{F_{text{drag}}} = frac{v}{v_c}$$

At low speeds the drag dominates and the system is energetically costly; at high speeds the ratio improves substantially. This is why Inductrack is best suited to high-speed transport — the same physics that limits performance at slow speeds is negligible at Hyperloop velocities (~300 m/s).

The Halbach Cylinder: Uniform Interior Field

Wrapping the magnetization rotation around a full cylinder — so that the angle $theta$ completes $2pi$ around the circumference — produces a Halbach cylinder. Its interior contains a remarkably uniform transverse field, while the exterior field vanishes identically (in the ideal continuous limit). This geometry is the physical basis for compact, cryogen-free MRI magnets: the field uniformity across the imaging volume determines image quality, and the absence of external fringe fields eliminates interference with nearby equipment and reduces the exclusion zone around the scanner.

Comparison with Simple Magnet Arrays

A conventional row of parallel-magnetized magnets distributes flux roughly equally above and below. The same number of magnets arranged as a Halbach array increases the strong-side field amplitude by roughly 40% and suppresses the weak-side field by more than 95%. In practical terms, a Halbach-based levitation system achieves the same lift force with fewer (and lighter) magnets — a decisive advantage for any weight-sensitive vehicle. The improvement in field concentration also reduces magnetic stray fields, lowering the electromagnetic interference footprint of the system.

References

1. Halbach, K. (1980). “Design of permanent multipole magnets with oriented rare earth cobalt material.” Nuclear Instruments and Methods, 169(1), 1–10.
2. Post, R. F., & Ryutov, D. D. (2000). “The Inductrack: A simpler approach to magnetic levitation.” IEEE Transactions on Applied Superconductivity, 10(1), 901–904.
3. Mallinson, J. C. (1973). “One-sided fluxes — A magnetic curiosity?” IEEE Transactions on Magnetics, 9(4), 678–682.
4. Trumper, D. L., Williams, M. E., & Nguyen, T. H. (1993). “Magnet arrays for synchronous machines.” IEEE IAS Annual Meeting, 9–18.