# Why it’s so difficult to model maglev systems: (Borcherts and Davis, “Force on a Coil Moving over a Conducting Surface Including Edge and Channel Effects, Journal of Applied Physics (1972))

The complexity of equations (like the one seen above) for the magnetic field produced by a real current-carrying loop with finite dimensions is why the perfect dipole approximation had to be made in the course of my research. Potentially, future work would involve using the equations for the “real” case and attempting to model the motion of trains that use these coils to levitate.

# Future Considerations

In order to expand upon the work I’ve done with this project thus far, I would consider modeling the actual motion of a maglev train, that is, its motion relative to the height above the track. I would begin by attempting to solve the second-order differential equation associated with this motion, based off of Newton’s Second Law, given by the equation:

(1) However, this equation would still only apply under the assumption that the superconducting coils on board the train act as a perfect dipole. Furthermore, I could consider the effects of the magnetic forces that guide the train along the tracks, and how this force affects the magnetic lift and drag forces on the train in that case.

# References & Mathematica Code

Here are the various references that I consulted while researching the methods of magnetic levitation:

Jayawant, B. V. “Electrodynamic Suspension and Levitation Techniques.” Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 416.1851 (1988): 245-320. Web. 25 Apr. 2012. <http://www.jstor.org/stable/pdfplus/2398062.pdf?acceptTC=true>.

Kraftmahker, Yaakov. “Maglev for Students.” European Journal of Physics. 29. (2008): 663-669. Web. 25 Apr. 2012. <http://www.rose-hulman.edu/~moloney/Ph425/ProjectPDFs/ejp_jul_08_maglev_0143-0807_29_4_001.pdf>.

Reitz, John R. “Forces on Moving Magnets due to Eddy Currents.” Journal of Applied Physics. 41. (1970): 2067-2071. Web. 25 Apr. 2012. <http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=JAPIAU000041000005002067000001&idtype=cvips&doi=10.1063/1.1659166&prog=normal>.

Rossing, Thomas D., and John R. Hull. “Magnetic Levitation.” Physics Teacher. (1991): 552-562. Web. 25 Apr. 2012. <http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=PHTEAH000029000009000552000001&idtype=cvips&doi=10.1119/1.2343425&prog=normal>.

Here is a link to the Mathematica code used to produce the various graphs and plots in my project:

# Conclusion

Initially, I sought out to study the interaction of magnetic fields and moving charges that allows Maglev trains to function as one of the world’s leading competitors in future transportation. Conceptually, their supposed high efficiency was not something unclear or doubtable: through magnetic levitation, the methods involved in EMS and EDS virtually eliminate all ground drag forces that moving vehicles typically face. However, though these machines seem to be “free of all forces,” it now seems that forces – and the delicate balance of interacting forces – is extremely important in determining whether or not a maglev system will work properly and efficiency.

I found that there are many variables that influence how a maglev system will function, and that different types of variables affect different types of magnetic levitation. Although the levitation force in the EMS system is entirely attractive and not very dependent upon the speed of the moving train, the EDS system relies upon repulsive forces and speed drives the levitation. I’ve revisited inductance and its quantitative – rather than qualitative – value beyond the simplicities of Lenz’s Law. In the non-ideal physics world, there are real forces that affect our system that we can’t always “assume away.” Although I did approximate the superconducting coil configuration with that of a magnetic dipole, I would have compared my outcome with that of a real rectangular coil with finite dimensions if an equation for the magnetic field or force was available.

To extend upon this project, I would like to do several things: (i) determine an exact analytical equation for the magnetic force that would allow a superconducting coil with physical dimensions to levitate; (ii) compare my result with similar results, and vary some other values besides speed (current, number of loops, area of loops, etc.); (iii) determine which settings are optimal for a real maglev train to attain maximum efficiency; and (iv) introduce the concept of a guidance force (the force that stabilizes the train in the horizontal direction), and analyze how this additional force impacts the lift, magnetic drag, and the ratio between the two.

Lastly, I think it would be most valuable to conduct experiments on my own, testing the relationships proposed by physicists in the past, and searching to develop a model to describe the exact relationship between the speed of the train (or some scale model) and the levitation force, magnetic drag force, and aerodynamic drag force that result. Ideally, then, I could derive an expression for the ideal height (the height that maximizes ) of the maglev train at any given speed.

# Results IV – Graph of Forces and Final Analysis

Using the equation for the lifting force on a magnetic dipole by a conducting plane, as well as the relationship between lifting force and drag force (which happens to apply for all coil configurations) given by:

(1) (stated by Jayawant, using the same variables as previously)

I produced the following graph in Mathematica, a plot of the ratio of the force and the ideal image force with respect to speed, ignoring the constant term out front: where the blue line represents the lifting force , the red line represents the drag force , and the yellow line represents the ratio of the lifting force and the drag force .

From this graph, it appears that our theory for the limiting case (as ) holds true, since the lifting force starts off proportional to , but then quickly becomes asymptotic toward the ideal image force as speed continues to increase. Meanwhile, the drag force is proportional to at low velocities, and is actually greater than the lifting force within this low speed interval. However, at around , the lifting force surpasses the drag force, as the lifting force continues to rise while the drag force reaches a peak. Then, as speed increases, the drag force appears to fall off as . Additionally, for this magnetic dipole approximation, the ratio of lifting force to drag force remains fairly linear and very steep.

If I were able to compare these results with those for a rectangular coil of wire – a configuration more appropriate to maglev trains – I would expect the same patterns to appear, though the lifting force might take longer to approach the limiting value of the ideal image force, since the rectangular coil situation represents a real case rather than an ideal case. Also, the drag peak might vary, thus causing a change in the train speed at which the lifting force surpasses the drag force. Because the ratio of lifting force to drag force is one measure of the efficiency of a maglev system (i.e., because it measured some desired outcome with respect to an undesired outcome, both dependent upon the speed of the train), I would expect the ratio for this perfect dipole case to always be greater than any nonideal case: thus, the slope of the ratio curve should be lower for real EDS maglev systems.

# Results III – Calculating Lift and Drag

Considering the lifting magnetic force on the train’s superconducting coils, we know that at extremely low speeds, the magnetic force of repulsion will likely not be great enough to levitate the train. However, as the speed of the train increases, the lifting force should increase proportionally, as the magnetic repulsion force increases. It would be nice to have a compact equation directly relating the speed of the train to the lifting force; regrettably, I could not discover any such equation. There is, however, a method by which one can estimate this force. Just as there was the “Method of Images” to solve problems with conducting planes in electrostatics, there is a similar method for moving sources of magnetic fields. As the superconducting coils on the train pass over the conducting coils in the track, the repulsive force generated by the opposing magnetic fields between the two loops can be thought of as an equal and opposite force being created by an image of the superconducting coil located an equal distance away on the opposite side of  (underneath) the track. Then, as the train moves forward some distance, a new image is created underneath the track, and the previous image begins to move away from the track, further into the ground. The speed at which the images recede from the plane is given by , where is the permeability of free space, is the conductivity of the conducting plate (coils) in the track, and is the thickness of the conducting plate. Though most of the force created by the images is repulsive, there is still the electromagnetic drag force created by the remaining currents in the passed loops, which can be thought of as coming from the previous images. This successive line of images is known as the “wake of images.” Figure 2. The “wake of images” that appears on the opposite side of a conducting plane, caused by the eddy currents that are established by the moving coil above. (a) When the speed of the superconducting coils is low, the previous images move away from the plane (and the real coils) relatively slowly, allowing the eddy currents to produce a force on the coil in the direction opposite its motion. (b) When the speed of the superconducting coils is high, the images move away relatively quickly, allowing the lifting force to approach the ideal image force. (Reproduced from Rossing.)

As the train increases speed, the distance between successive images appears to increase, until the point at which the lifting force is best approximated by the repulsive force that would be produced if an exact image of the superconducting coils existed an equal distance beneath the tracks. This force is called the ideal image force, and is essentially the limiting value of the lifting force on the moving object.

In order to quantify the lifting force, I attempted to use the following equation, given in Jayawant:

(1) which relates the ratio of the lifting force and the ideal image force to the speed of the moving coils (speed of the train) and the recession speed of the coil’s images inside the conducting plane, and is related to the dimensions of the coil. However, when graphing this function (the ratio of to in Mathematica, the output did not seem to model the behavior that I predicted; the function did not increase steadily with increasing speed, nor did it asymptotically approach a value of 1.0. Thus, I followed an assumption suggested by Kraftmakher, and replaced this equation for the lifting force on a rectangular coil with the simplified equation for the lifting force on a magnetic dipole, given by Reitz as:

(2) where is the magnetic dipole moment of the train’s coils, is the height of the train’s coils above the conducting plate (track coils), is the speed of the train, and is the recession speed of the coil’s images. It is clear that in the limiting case where , the lifting force approaches the ideal image force for the dipole configuration, given as the constant out front in the right side of the equation. With this relationship, I will be able to approximate what the lifting and drag forces would be on a coil of superconducting wire inside the moving train.

# Results II – How & Why EDS Works

The induction process of the EDS system is depicted in the diagram below: Figure 1. The principle of electrodynamic suspension. (a) As the upper current-carrying wire loop approaches the lower stationary conducting loop, the magnetic flux through the lower loop increases with time. The induced EMF in the lower loop, and the lagging induced current, are graphed with respect to time below. (b) As the smaller upper loop passes by the larger lower loop, there is no change in magnetic flux through the lower loop, so no additional EMF is induced. Instead, the built-up current in the loop begins to decline, due to the resistance of the wire. (c) As the upper loop moves away from the lower loop, an opposite EMF is induced, with the same magnitude as before, but negative. As a result, due to the loss in current during (b), there is a net negative current (the shaded region) flowing in the lower loop after the upper loop passes. (Reproduced from Jayawant.)

As seen in the figure, there remains an induced current in the lower loop (the conducting track) after the upper loop (on board the train) passes by, flowing in the opposite direction as the current that was initially induced. Thus, this process produces two forces: a lifting force (caused by the opposing magnetic fields generated by the upper and lower loops) and a drag force (caused by the residual current in the lower loop after the upper loop passes, which presents an attractive force on the upper loop in the direction opposite its motion).

From now on, I will focus on finding the relationships between the speed of the train (or, rather, the superconducting coils inside of it), the distance between the train and the tracks, and the magnetic lifting force and drag force that allow the EDS system to function.

# Results I – EMS & EDS

So, after having done extensive research into the various types of maglev technologies, I have found that the methods of electromagnetic suspension (EMS) and electrodynamic suspension (EDS) are the most developed and practiced types of magnetic levitation.

As explained before, EMS involves the electromagnets on board the train being attracted to the metallic track from underneath. The current running through the electromagnets is constantly adjusted to maintain a steady distance between the train and the track. One of the easiest ways to model the behavior of these types of maglev systems is by using the equation for magnetic pressure, given as , from which we can derive the amount of force that the bottom of the train feels towards the track in a given area. However, the main component of this system that changes over time is the current flowing through the electromagnet, which would be extremely difficult to study analytically, and so I will focus on the other main maglev method.

In the EDS system, the electromagnets on board the train induce in a conducting guide way (the track) eddy currents that generate an opposing magnetic field, creating a repulsive force that levitates the train as it moves along the track. This phenomenon is explained by a combination of Faraday’s Law and Lenz’s Law. As the train’s superconducting magnets (or current-carrying wires) move along the track, the magnetic field produced by these will move relative to the conducting track, thus generating eddy currents within the track itself. However, these eddy currents flow in a direction such that a magnetic field is produced that opposes the change in magnetic field entering the conducting track. Thus, the magnetic field produced by the train’s electromagnets or coils and the field produced by the track oppose one another, and create the repulsive force that causes the train to levitate.

A simplified model of the interaction between the two opposing magnetic fields is given in the following vector plot. Notice how the two fields interact, opposing one another, resulting in a repulsive force between the two coils that are generating the fields. As the top coil is pulled downwards by gravity, the two fields will interact much more, increasing the repulsion force, and thus levitating the train containing the superconducting coils. # Intro To Preliminary Results

As it turns out, the equations for magnetic fields and forces that I was hoping to plot in Mathematica are too complicated for Mathematica to handle. I am unsure whether this drawback just applies to my Student version of Mathematica, or if my personal computer is to blame. However, I took a slight deviation to begin my studies with a simpler equation to model, the equation for the magnetic force between two plates of some finite area, given by:

(1) where is the magnetic flux density (magnetic field strength) at a given distance above the bottom magnet, is the common area between the two plates, and is the permeability of free space. In order to easily visualize this effect, I generalized the magnetic flux density by replacing it with a constant multiple times the inverse of distance cubed (which is the relationship between magnetic field strength and distance), producing the following graph in Mathematica: Clearly, the force falls off extremely quickly with respect to distance, so I am expecting my other graphs to have a range of many fractions of a meter. This outcome is exactly as I would expect (ignoring any fringing effects near the edges), since the magnetic force between the two magnets decreases rapidly with distance; in my future calculations, I will expect the force on the top magnet to increase as the two get closer together, and decrease as the magnets separate, hopefully leading to a steady distance between the two over time. As soon as I can get Mathematica to accept my more complicated equations and graphs, I will post them.

# Project Outline

There are three main methods that allow Maglev technologies to operate.

The first, electromagnetic suspension (EMS), involves electromagnets on the levitating object that are oriented toward the rail from below. However, since magnetic attraction varies inversely with the cube of distance, minor changes in the distance from the rail cause the lifting force to vary greatly. This is why the EMS method is typically considered unstable; it requires a feedback control to vary the current passing through the electromagnet, to continuously adjust the magnetic field so that the object maintains a constant height above the track. Then, to propel the object forward, some additional method must be employed, whether it involves the use of a propeller, an engine, or propulsion coils, the latter of which is explained in the section below.

The second method, called electrodynamic suspension (EDS), requires that both the levitating object and the track exert a magnetic field, so that the object is levitated by the repulsive force between the two fields. For this case, the magnetic field produced by the object can originate from either superconducting magnets or permanent magnets; the field produced by the track is an induced magnetic field created by current-carrying wires inside the track. One benefit of this method is that no feedback control system is necessary for the object to remain at a constant height above the track. Then, to make the object move forward along the track, propulsion coils carrying AC current generate a continuously-changing magnetic field that exerts a force on the magnets in the object. The frequency of the AC current is coordinated such that the field always repels the magnets in the object, sending it forward.

The third method, magnetodynamic suspension (MDS), employs the attractive magnetic force of a permanent magnet near a metal track to hold it in place. Then, propulsion is attained by the same methods as in the EMS system. However, since this method has not been fully developed and employed widely, I will focus my attention on the two previous methods of magnetic levitation.

In order to achieve stability while the object (a maglev train) is moving forward along the track, the EMS method uses the feedback control to adjust the magnetic field strength produced by the train’s electromagnets by altering the current running through them. Meanwhile, in an EDS system, no feedback control is necessary because, as the distance between the train and the track decreases, the magnetic force exerted on the train by the track increases, and vice-versa, until the train remains at a stable height above the track.

Additional effects that allow maglev trains to remain at a steady height above the track include the Meissner effect, and magnetic flux trapping. If the train contains a superconductor while the track consists of permanent magnets, the magnetic field created by the track can be expelled from the object’s electromagnet by cooling the superconductor below its superconducting transition temperature (also called the critical temperature). This expulsion of magnetic fields from a superconductor while it’s below its critical temperature is the Meissner effect. However, while most of the magnetic field is expelled from the superconductor, some of it still passes through, thus causing the superconductor to be both attracted to and repelled away from the track. The superconductor – and consequentially, the train itself – is essentially “trapped in space” above the track, yet free to move along the track with zero ground friction. This phenomenon is called magnetic flux trapping. The Meissner effect and magnetic flux trapping are further explained and demonstrated in the following videos:

How Superconducting Levitation Works

Maglev Trains

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For my project, I will model the magnetic field produced by the superconducting magnet on the train, as well as the field produced by the tracks, and potentially, a combination of how the two fields interact. With this I will model the force of repulsion between the train and the tracks, which should display how the force varies as the distance between the train and the track changes. Then, I hope to create a resultant graph showing how the train’s distance from the track changes (or, as I hope to expect, does not change).

One equation that I am considering using to model the behavior of a magnetically levitating object is that for the magnetic field (or magnetic flux density) above the supporting magnet, given by:

(1) (Jayawant, Electromagnetic Suspension and Levitation)

where M is the intrinsic magnetization, z is the height above the magnet, x is the position in the x-direction, y is the position in the y-direction, and 2a is the depth of the supported magnet.

Additionally, I will consider the equation for the force of repulsion on the object by the supporting magnet, given by the equation:

(2) (Jayawant, Electromagnetic Suspension and Levitation)

where h is the gap length, 2d is the magnet depth (given as 2a in Equation 1), M is the intrinsic magnetization, R is the length of the magnet along the x-axis (the direction of motion), P is the length of the magnet along the y-axis, µ0 is the magnetic permeability of free space, j and k refer to the rows and columns of a superimposed repulsion force matrix (which acts as a “grid” for any specified piece of the magnet), and p and q are infinitesimally small lengths that make up the dimensions of some piece of the magnet in the xy-plane (and are intended to be as large as possible).

I may need to manipulate these equations a bit before employing them in my studies, but I will make sure that whatever changes I make will still model the system as accurately as my assumptions will allow.

The goal of my project is to develop graphs for the magnetic fields produced by the object and the track, and, as I mentioned before, I would like to model the interaction of these fields as well. Graphs of the repulsion force and resulting position of the object with respect to the track will be necessary to analyze the relationship between the two variables. I am expecting my plots to show a strong relationship between force and distance from the track, such that the distance steadies out to a constant height over a length of time. As for the plot of the magnetic fields, I expect that they will interact such that the repulsive force acting on the object will allow the object to remain above the track at a reasonable distance. I may also look into magnetic pressure, and how this variable might explain how and why the object levitates.

References:

Jayawant, B. V. “Electromagnetic Suspension and Levitation.” Reports on Progress in Physics. 44.4 (1981): 413-474. 3 Apr. 2012. <http://www.maglev.ir/eng/documents/papers/journals/IMT_JP_56.pdf>.