Investigatory Project

SUMMARY
Magnetic levitation is the process of levitating an object by exploiting magnetic fields. If the magnetic force of attraction is used, it is known as magnetic suspension. If magnetic repulsion is used, it is known as magnetic levitation.
In the past, magnetic levitation was attempted by using permanent magnets.
Earnshaw’s theorem however, proves that this is mathematically impossible.
There exists no arrangement of static magnets of charges that can stably levitate an object. There are however means of circumventing this theorem by altering its basic assumptions. The following conditions are exceptions to Earnshaw’s theorem: • Diamagnetism: occurs in materials which have a relative permeability less than one. The result is that is eddy currents are induced in a diamagnetic material, it will repel magnetic flux.
• The Meissner Effect: occurs in superconductors. Superconductors have zero internal resistance. As such induced currents tend to persist, and as a result the magnetic field they cause will persist as well. • Oscillation: when an A current is passed through an electromagnet, it behaves like a diamagnetic material.
• Rotation: employed by the Levitron, it uses gyroscopic motion to overcome levitation instability.
• Feedback: used in conjunction with electromagnets to dynamically adjust magnetic flux in order to maintain levitation.
Each of the above conditions provides solutions to the problem of magnetic levitation. The focus of this thesis is the feedback technique. Feedback with electromagnets can be divided into magnetic suspension and levitation.
Magnetic suspension works via the force of attraction between an electromagnet and some object. If the object gets too close to the electromagnet, the current in the electromagnet must be reduced. If the object gets too far, the current to the electromagnet must be increased. Thus the information which must be sensed is the position of the levitating object. The position can then be used to determine how much current the electromagnet must receive. To prevent oscillations however, the rate of change of position must used as well. The position information can easily be differentiated to acquire the speed information required.
Electromagnetic levitation works via the magnetic force of repulsion. Using repulsion though makes a much more difficult control problem. The levitating object is now able to move in any direction, meaning that the control problem has shifted from one dimension to three. There is much interest in levitation due to its possible applications in high speed transport technology. These applications can be broadly referred to as MagLev, which stands for magnetic levitation. A system which more closely resembles the work done in this thesis project is the “MagLev cradle”. The MagLev cradle is a system designed by Bill Beaty. It is able to levitate a small rod magnet for a few seconds at a time. This system suffers from serious instability. As such levitation can only be maintained for a few seconds.
The MagLev cradle utilizes an arrangement of up to 12 electromagnets and their control circuits in a “v” configuration to levitate a bar magnet. The MagLev cradle uses rapid switching circuits to control current to the electromagnets. If the bar magnet falls too close to the electromagnet, the circuit switches on, thus applying more repelling force. If the bar magnet rises too high above the electromagnet, it turns off, thus removing the repelling force.
The system developed for this thesis uses the position sensing technique employed by the magnetic cradle. Hall Effect sensors are placed on each of the electromagnets in the system. Each electromagnet and its current control circuitry operates as an independent system to levitate part of a bar magnet.
The Hall effect sensor is a device that senses magnetic flux. It is also capable of detecting the magnetic flux orientation. It is placed on an electromagnet to sense the presence of the bar magnet we wish to levitate. The circuitry is configured such that is magnetic flux is detected; the system will energize the electromagnet in order to make the net magnetic flux with the hall effect sensor zero. Therefore this system electronically simulates the Meissner effect by repelling both north and south poles of a magnet. Experiments were also done to investigate various configurations of electromagnets in order to achieve stable magnetic levitation.
This current control circuit for the electromagnets used an opamp summer circuit and a power amplification stage (sink/source transistor circuit). Initial tests revealed that besides position sensing, speed information was required as well.
This was achieved by adding a phase lead circuit, which negated the phase lag caused by the electromagnet (an inductive load) and the control circuitry.
Different configurations of electromagnets were used to attempt to levitate a bar magnet. The main problem that was soon identified was that of keeping the levitating bar magnet in the area above the electromagnets. Despite moving the electromagnets closer and further apart, the bar magnet could not be effectively trapped above the electromagnets. The bar magnet has a tendency to “slide” off the ends, as the end magnets cannot react quickly enough to movements in the bar magnet. Thus current system lacks the control circuitry required to achieve stable electromagnetic levitation.
At present, pairs of electromagnets can effectively levitate part of a bar magnet which is supported at one end. If the necessary control circuit required to effectively hold the levitating bar magnet in position above the electromagnet can be designed, then a working system can be quickly realised.

1. INTRODUCTION TO MAGNETIC LEVITATION
Magnetic levitation is the process of levitating an object by exploiting magnetic fields. In other words, it is overcoming the gravitational force on an object by applying a counteracting magnetic field. Either the magnetic force of repulsion or attraction can be used. In the case of magnetic attraction, the experiment is known as magnetic suspension. Using magnetic repulsion, it becomes magnetic levitation. In the past, magnetic levitation was attempted by using permanent magnets.
Attempts were made to find the correct arrangement of permanent magnets to levitate another smaller magnet, or to suspend a magnet or some other object made of a ferrous material. It was however, mathematically proven by Earnshaw that a static arrangement of permanent magnets or charges could not stably magnetically levitate an object
Apart from permanent magnets, other ways to produce magnetic fields can also be used to perform levitation. One of these is an electrodynamic system, which exploits Lenz’s law. When a magnet is moving relative to a conductor in close proximity, a current is induced within the conductor. This induced current will cause an opposing magnetic field. This opposing magnetic field can be used to levitate a magnet. This means of overcoming the restrictions identified by
Earnshaw is referred to as oscillation.
Electrodynamic magnetic levitation also results from an effect observed in superconductors. This effect was observed by Meissner and is known as the
Meissner effect. This is a special case of diamagnetism.
This thesis will mainly deal with electromagnetic levitation using feedback techniques to attain stable levitation of a bar magnet. 2. THE EARNSHAW THEOREM
Earnshaw’s theorem basically proves that a static magnet cannot be levitated by any arrangement of permanent magnets or charges. This can be simply proved as follows:
“The static force as a function of position F(x) acting on any body in vacuum due to gravitation, electrostatic and magnetostatic fields will always be divergenceless. divF =
0. At a point of equilibrium the force is zero. If the equilibrium is stable the force must point in towards the point of equilibrium on some small sphere around the point.
However, by Gauss' theorem, s ∫ F(x).dS = v ∫divF. dV
The integral of the radial component of the force over the surface must be equal to the integral of the divergence of the force over the volume inside which is zero.” – (Philip
Gibbs and Andre Geim, March 1997)
This theorem though makes certain assumptions. Thus the result can be circumvented under certain conditions. The exceptions to Earnshaw’s theorem are as follows:
2.1 QUANTUM THEORY
Firstly this theorem only takes into account classical physics and not quantum mechanics. At the atomic level there is a type of levitation occurring through forces of repulsion between particles. This effect is so small however, that it is not generally considered as magnetic levitation. 2.2 ROTATION
This property is used in the patented magnetic levitation display called the
Levitron. The Levitron uses an arrangement of static permanent magnets to levitate a smaller magnet. The system overcomes the instability described in
Earnshaw’s theorem by rotating the levitating magnet at high speed.
2.3 DIAMAGNETISM
Earnshaw’s theorem doesn’t apply to diamagnetic materials, because they have a relative permeability less than one. This means that they don’t behave like regular magnets, as they will tend to repel any magnetic flux.
2.4 MEISSNER EFFECT
A special case of diamagnetism is observed in conductors cooled to below their critical temperature (typically close to 0 K). Below this temperature, they become superconductors, with an internal resistance of zero. They attain a relative permeability of zero, making them the perfect diamagnetic material.
This allows them to maintain their repelling magnetic field as long as a foreign source of magnetic flux is present.
2.5 FEEDBACK SYSTEMS
The position of the levitating magnet can be sensed and used to control the field strength of an electromagnet. Thus the tendency for instability can be removed by constantly correcting the magnetic field strength of the electromagnets to keep a permanent magnet levitated. 2.6 OSCILLATION
Passing an alternating current through an electromagnet causes eddy currents to flow within its core. These currents according to Lenz’s law will flow such that they repel a nearby magnetic field. Thus, it causes the electromagnet to behave like a diamagnetic material. http://www.rrsg.ee.uct.ac.za/theses/ug_projects/williams_ugthesis.pdf