# Magnetic levitation

Levitating pyrolytic carbon

Magnetic levitation, maglev, or magnetic suspension is a method by which an object is suspended with no support other than magnetic fields. Magnetic pressure is used to counteract the effects of the gravitational and any other accelerations.

Earnshaw's theorem proves that using only ferromagnetic or paramagnetic materials it is impossible to stably levitate against gravity, but servomechanisms, the use of diamagnetic materials, superconduction, or systems involving eddy currents permit this to occur.

In some cases the lifting force is provided by magnetic levitation, but there is a mechanical support bearing little load that provides stability. This is termed pseudo-levitation.

Magnetic levitation is used for maglev trains, magnetic bearings and for product display purposes.

## Lift

Magnetic materials and systems are able to attract or press each other apart or together with a force dependent on the magnetic field and the area of the magnets, and a magnetic pressure can then be defined.

The magnetic pressure of a magnetic field on a superconductor can be calculated by:

$P_{mag} = \frac {B^2} {2 \mu_0}$

where $P_{mag}$ is the force per unit area in pascals, $B$ is the magnetic field just above the superconductor in teslas, and $\mu_0$ = 4π×10−7 N·A−2 is the permeability of the vacuum.[1]

## Stability

### Static

Static stability means that any small displacement away from a stable equilibrium causes a net force to push it back to the equilibrium point.

Earnshaw's theorem proved conclusively that it is not possible to levitate stably using only static, macroscopic, paramagnetic fields. The forces acting on any paramagnetic object in any combinations of gravitational, electrostatic, and magnetostatic fields will make the object's position, at best, unstable along at least one axis, and it can be unstable equilibrium along all axes. However, several possibilities exist to make levitation viable, for example, the use of electronic stabilization or diamagnetic materials (since relative magnetic permeability is less than one[2]); it can be shown that diamagnetic materials are stable along at least one axis, and can be stable along all axes. Conductors can have a relative permeability to alternating magnetic fields of below one, so some configurations using simple AC driven electromagnets are self stable.

### Dynamic stability

Dynamic stability occurs when the levitation system is able to damp out any vibration-like motion that may occur.

Magnetic fields are conservative forces and therefore in principle have no built-in damping, and in practice many of the levitation schemes are under-damped and in some cases negatively damped.[3] This can permit vibration modes to exist that can cause the item to leave the stable region.

Damping of motion is done in a number of ways:

• external mechanical damping (in the support), such as dashpots, air drag etc.
• eddy current damping (conductive metal influenced by field)
• tuned mass dampers in the levitated object
• electromagnets controlled by electronics

## Methods

Mechanical constraint (in this case the lateral restrictions created by a box) can permit pseudo-levitation of permanent magnets

For successful levitation and control of all 6 axes (3 spatial and 3 rotational) a combination of permanent magnets and electromagnets or diamagnets or superconductors as well as attractive and repulsive fields can be used. From Earnshaw's theorem at least one stable axis must be present for the system to levitate successfully, but the other axes can be stabilised using ferromagnetism.

The primary ones used in maglev trains are servo-stabilized electromagnetic suspension (EMS), electrodynamic suspension (EDS).

### Mechanical constraint (pseudo-levitation)

With a small amount of mechanical constraint for stability, achieving pseudo-levitation is a relatively straightforward process.

If two magnets are mechanically constrained along a single vertical axis, for example, and arranged to repel each other strongly, this will act to levitate one of the magnets above the other.

Another geometry is where the magnets are attracted, but constrained from touching by a tensile member, such as a string or cable.

Another example is the Zippe-type centrifuge where a cylinder is suspended under an attractive magnet, and stabilized by a needle bearing from below.

### Servomechanisms

The Transrapid system uses servomechanisms to pull the train up from underneath the track and maintains a constant gap while travelling at high speed

The attraction from a fixed strength magnet decreases with increased distance, and increases at closer distances. This is unstable. For a stable system, the opposite is needed, variations from a stable position should push it back to the target position.

Stable magnetic levitation can be achieved by measuring the position and speed of the object being levitated, and using a feedback loop which continuously adjusts one or more electromagnets to correct the object's motion, thus forming a servomechanism.

Many systems use magnetic attraction pulling upwards against gravity for these kinds of systems as this gives some inherent lateral stability, but some use a combination of magnetic attraction and magnetic repulsion to push upwards.

Either system represents examples of ElectroMagnetic Suspension (EMS). For a very simple example, some tabletop levitation demonstrations use this principle, and the object cuts a beam of light to measure the position of the object. The electromagnet is above the object being levitated; the electromagnet is turned off whenever the object gets too close, and turned back on when it falls further away. Such a simple system is not very robust; far more effective control systems exist, but this illustrates the basic idea.

EMS magnetic levitation trains are based on this kind of levitation: The train wraps around the track, and is pulled upwards from below. The servo controls keep it safely at a constant distance from the track.

### Induced currents

These schemes work due to repulsion due to Lenz's law. When a conductor is presented with a time-varying magnetic field electrical currents in the conductor are set up which create a magnetic field that causes a repulsive effect.

#### Relative motion between conductors and magnets

If one moves a base made of a very good electrical conductor such as copper, aluminium or silver close to a magnet, an (eddy) current will be induced in the conductor that will oppose the changes in the field and create an opposite field that will repel the magnet (Lenz's law). At a sufficiently high rate of movement, a suspended magnet will levitate on the metal, or vice versa with suspended metal. Litz wire made of wire thinner than the skin depth for the frequencies seen by the metal works much more efficiently than solid conductors.

An especially technologically interesting case of this comes when one uses a Halbach array instead of a single pole permanent magnet, as this almost doubles the field strength, which in turn almost doubles the strength of the eddy currents. The net effect is to more than triple the lift force. Using two opposed Halbach arrays increases the field even further.[4]

Halbach arrays are also well-suited to magnetic levitation and stabilisation of gyroscopes and electric motor and generator spindles.

#### Oscillating electromagnetic fields

A conductor can be levitated above an electromagnet (or vice versa) with an alternating current flowing through it. This causes any regular conductor to behave like a diamagnet, due to the eddy currents generated in the conductor.[5][6] Since the eddy currents create their own fields which oppose the magnetic field, the conductive object is repelled from the electromagnet, and most of the field lines of the magnetic field will no longer penetrate the conductive object.

This effect requires non-ferromagnetic but highly conductive materials like aluminium or copper, as the ferromagnetic ones are also strongly attracted to the electromagnet (although at high frequencies the field can still be expelled) and tend to have a higher resistivity giving lower eddy currents. Again, litz wire gives the best results.

The effect can be used for stunts such as levitating a telephone book by concealing an aluminium plate within it.

At high frequencies (a few tens of kilohertz or so) and kilowatt powers small quantities of metals can be levitated and melted using levitation melting without the risk of the metal being contaminated by the crucible.[7]

One source of oscillating magnetic field that is used is the linear induction motor. This can be used to levitate as well as provide propulsion.

### Diamagnetism

Diamagnetism is the property of an object which causes it to create a magnetic field in opposition to an externally applied magnetic field, thus causing a repulsive effect. Specifically, an external magnetic field alters the orbital velocity of electrons around their nuclei, thus changing the magnetic dipole moment. According to Lenz's law, this opposes the external field. Diamagnets are materials with a magnetic permeability less than μ0 (a relative permeability less than 1). Consequently, diamagnetism is a form of magnetism that is only exhibited by a substance in the presence of an externally applied magnetic field. It is generally quite a weak effect in most materials, although superconductors exhibit a strong effect. Diamagnetic materials cause lines of magnetic flux to curve away from the material, and superconductors can exclude them completely (except for a very thin layer at the surface).

#### Direct diamagnetic levitation

A live frog levitates inside a 32 mm diameter vertical bore of a Bitter solenoid in a magnetic field of about 16 teslas

A substance that is diamagnetic repels a magnetic field. All materials have diamagnetic properties, but the effect is very weak, and is usually overcome by the object's paramagnetic or ferromagnetic properties, which act in the opposite manner. Any material in which the diamagnetic component is strongest will be repelled by a magnet.

Earnshaw's theorem does not apply to diamagnets. These behave in the opposite manner to normal magnets owing to their relative permeability of μr < 1 (i.e. negative magnetic susceptibility).

Diamagnetic levitation can be used to levitate very light pieces of pyrolytic graphite or bismuth above a moderately strong permanent magnet. As water is predominantly diamagnetic, this technique has been used to levitate water droplets and even live animals, such as a grasshopper, frog and a mouse.[8] However, the magnetic fields required for this are very high, typically in the range of 16 teslas, and therefore create significant problems if ferromagnetic materials are nearby.

The minimum criterion for diamagnetic levitation is $B \frac{dB}{dz} = \mu_0 \, \rho \, \frac{g}{\chi}$, where:

• $\chi$ is the magnetic susceptibility
• $\rho$ is the density of the material
• $g$ is the local gravitational acceleration (−9.8 m/s2 on Earth)
• $\mu_0$ is the permeability of free space
• $B$ is the magnetic field
• $\frac{dB}{dz}$ is the rate of change of the magnetic field along the vertical axis.

Assuming ideal conditions along the z-direction of solenoid magnet:

• Water levitates at $B \frac{dB}{dz} \approx 1400\ \mathrm{T^2/m}$
• Graphite levitates at $B \frac{dB}{dz} \approx 375\ \mathrm{T^2/m}.$

#### Diamagnetically stabilized levitation

A permanent magnet can be stably suspended by various configurations of strong permanent magnets and strong diamagnets. When using superconducting magnets, the levitation of a permanent magnet can even be stabilized by the small diamagnetism of water in human fingers.[9]

A superconductor levitating a permanent magnet

### Superconductors

Superconductors may be considered perfect diamagnets (μr = 0), and have the additional property of completely expelling magnetic fields due to the Meissner effect when the superconductivity initially forms. The levitation of the magnet is further stabilized due to flux pinning within the superconductor; this tends to stop the superconductor leaving the magnetic field, even if the levitated system is inverted.

These principles are exploited by EDS (Electrodynamic Suspension), superconducting bearings, flywheels, etc.

In trains, a very strong magnetic field is required to levitate a massive train. The JR–Maglev trains have superconducting magnetic coils, but the JR–Maglev levitation is not by the Meissner effect.

Levitron is an example of spin stabilized magnetic levitation

### Rotational stabilization

A magnet with a toroidal field can be levitated against gravity when gyroscopically stabilized by spinning it in a second toroidal field created by a base ring of magnet(s). However, it will only remain stable while the rate of precession is between both upper and lower critical thresholds—the region of stability is quite narrow both spatially and in the required rate of precession. The first discovery of this phenomenon was by Roy M. Harrigan, a Vermont inventor who patented a levitation device in 1983 based upon it.[10] Several devices using rotational stabilization (such as the popular Levitron branded levitating top toy) have been developed citing this patent. Non-commercial devices have been created for university research laboratories, generally using magnets too powerful for safe public interaction.

### Strong focusing

Earnshaw's theory strictly only applies to static fields. Alternating magnetic fields, even purely alternating attractive fields,[11] can induce stability and confine a trajectory through a magnetic field to give a levitation effect.

This is used in particle accelerators to confine and lift charged particles, and has been proposed for maglev trains also.[11]

## Uses

### Maglev transportation

Maglev, or magnetic levitation, is a system of transportation that suspends, guides and propels vehicles, predominantly trains, using magnetic levitation from a very large number of magnets for lift and propulsion. This method has the potential to be faster, quieter and smoother than wheeled mass transit systems. The technology has the potential to exceed 6,400 km/h (4,000 mi/h) if deployed in an evacuated tunnel.[12] If not deployed in an evacuated tube the power needed for levitation is usually not a particularly large percentage and most of the power needed is used to overcome air drag, as with any other high speed train.

The highest recorded speed of a maglev train is 581 kilometers per hour (361 mph), achieved in Japan in 2003,[13] 6 km/h faster than the conventional TGV speed record.

### Levitation melting

Electromagnetic levitation (EML), patented by Muck in 1923,[14] is one of the oldest levitation techniques used for containerless experiments.[15] The technique enables the levitation of an object using electromagnets. A typical EML coil has reversed winding of upper and lower sections energized by a radio frequency power supply.

## History

• 1839 Earnshaw's theorem showed electrostatic levitation was impossible, later theorem was extended to magnetostatic levitation by others
• 1912 Emile Bachelet awarded a patent in March 1912 for his “levitating transmitting apparatus” (patent no. 1,020,942) for electromagnetic suspension system
• 1933 Superdiamagnetism Walter Meissner and Robert Ochsenfeld (the Meissner effect)
• 1934 Hermann Kemper “monorail vehicle with no wheels attached.” Reich Patent number 643316
• 1939 Braunbeck’s extension showed that magnetic levitation is possible with diamagnetic materials
• 1939 Bedford, Peer, and Tonks aluminum plate placed on two concentric cylindrical coils shows 6-axis stable levitation.[16]
• 1961 James R. Powell and BNL colleague Gordon Danby electrodynamic levitation using superconducting magnets
• 1970s Spin stabilized magnetic levitation Roy M. Harrigan
• 1974 Magnetic river Eric Laithwaite and others
• 1979 transrapid train carried passengers
• 1984 Low speed maglev shuttle in Birmingham Eric Laithwaite and others
• 1997 Diamagnetically levitated live frog Andre Geim[8]
• 1999 Inductrack permanent magnet electrodynamic levitation (General Atomics)
• 2005 homopolar electrodynamic bearing[17]

## References

1. ^ Lecture 19 MIT 8.02 Electricity and Magnetism, Spring 2002
2. ^ Braunbeck, W. Free suspension of bodies in electric and magnetic fields, Zeitschrift für Physik, 112, 11, pp753-763 (1939)
3. ^ A Review of Dynamic Stability of Repulsive-Force Maglev Suspension Systems- Y. Cai and D.M.Rote
4. ^ [1]
5. ^ Eddy current magnetic levitation, models and experiments by Marc T. Thompson
6. ^ Levitated Ball-Levitating a 1 cm aluminum sphere
7. ^ "Magnetic levitation of liquid metals", Journal of Fluid Mechanics 117, pages 27-43, by A. J. Mestel
8. ^ a b "The Frog That Learned to Fly". Radboud University Nijmegen. Retrieved 19 October 2010. For Geim's account of diamagnetic levitation, see Geim, Andrey. "Everyone's Magnetism PDF (688 KB). Physics Today. September 1998. pp. 36–39. Retrieved 19 October 2010. For the experiment with Berry, see Berry, M. V.; Geim, Andre. (1997). "Of flying frogs and levitrons" PDF (228 KB). European Journal of Physics 18: 307–313. Retrieved 19 October 2010.
9. ^ Diamagnetically stabilized magnet levitation
10. ^ US patent 4382245, Roy M. Harrigan, "Levitation device", issued 1983-05-03
11. ^ a b Attractive levitation for high-speed ground transport with largeguideway clearance and alternating-gradient stabilization Hull, J.R. Magnetics, IEEE Transactions on Volume 25, Issue 5, Sep 1989 Page(s):3272 - 3274 Digital Object Identifier 10.1109/20.42275
12. ^ http://www.popsci.com/scitech/article/2004-04/trans-atlantic-maglev
13. ^ http://www.guardian.co.uk/world/2003/dec/03/japan.justinmccurry
14. ^ O. Muck. German patent no. 42204 (Oct. 30, 1923)
15. ^ Paul C. Nordine, J. K. Richard Weber, and Johan G. Abadie (2000), "Properties of high-temperature melts using levitation", Pure and Applied Chemistry 72: 2127–2136, doi:10.1351/pac200072112127
16. ^ linear Electric Machines- A Personal View ERIC R. LAITHWAITE, FELLOW, IEEE, PROCEEDINGS OF THE IEEE, VOL. 63, NO. 2, FEBRUARY 1975
17. ^ "Design and Analysis of a Novel Low Loss Homopolar Electrodynamic Bearing." Lembke, Torbjörn. PhD Thesis. Stockholm: Universitetsservice US AB, 2005. Print. ISBN 91-7178-032-7