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2005-39

An Electrodynamic Wheel with a Split-Guideway Capable

of Simultaneously Creating Suspension, Thrust andGuidance Forces

University of Wisconsin-MadisonCollege of Engineering

Wisconsin Power Electronics Research Center

2559D Engineering Hall1415 Engineering DriveMadison WI 53706-1691

2005 Confidential

Research Report

J. Bird, T. A. Lipo

Dept. of Elect. & Comp. Engr.University of Wisconsin-Madison

1415 Engineering DriveMadison, WI 53706

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zero guidance position. Also, as the two conducting sheets or ladders can be very close there does

not need to be a large amount of flux facing the guideway that is not used.

RadialMagnet

ShuntMagnet

y

z

x

v

Figure 1 Single Rotor with SheetGuideway (Unstable)

Figure 2 Halbach Rotor Figure 3 EDW with Ladder Guideway(Unstable)

Figure 4 EDW with Split-Guideway Sheet Topology

y

z

x v

Figure 5 EDW with Split-Guideway Ladder Topology

VN

S

DragForce

Track

Magnet

S

1I

2I

(a) Vehicle magnets at centre position, no guidance force

(I2=I1). But lift and thrust forces are still created

VN

S

Drag

Force

Magnet

S

Guidance

Force1

I

2I

(b) Offset vehicle magnets create guidance, lift and

thrust force. (I2> I1)Figure 6 Induced Current Path in Split-Guideway Ladder Topology

3 The 3D Mathematical Model of the EDW and Guideway

The calculation of the forces in all three dimensions for the geometry shown in Figure 4 and 5 is not

trivial. Almost all finite element analysis (FEA) programs are incapable of having both rotational

and translational motion simultaneously. Also due to the large rotor geometry, but thin guideway

structure, the computational requirements for a 3D transient eddy current computation is substantial.

In order to enable both the rotational and translational motion to be computed accurately a fast 3D

analytic model of the rotor magnets was created using equivalent current sheets, while the guideway

was modelled using dynamic circuit theory [14]. Only the ladder guideway structure is analyzedhere. Due to insufficient space, only an overview of the mathematical model is presented.

3.1 Halbach Rotor Model

The field created at any point in 3D space by the Halbach rotor was computed using the Biot-Savart

law and equivalent current sheets [15]. For instance, the radial magnet was created by first

determining the field created by a radial filament, as shown in Figure 7, and then integrated in the

radial direction to obtain the field for a current sheet, shown in Figure 8. This current sheet model is

equivalent to a radially magnetised magnet piece. Similarly, the shunt magnet was obtained by

integrating an azimuthally directed current filament, shown in Figure 9, along the azimuthal direction

so as to obtain a current sheet model of the Halbach shunt magnet, Figure 10. By summing up all the

radial and shunt magnets for a full Halbach rotor, the field at any point in 3D space can be obtained.

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An example of the radial and z directed field components above a 4 pole-pair Halbach rotor and the

validation of the radial field component, using Ansofts Maxwell 3D, is shown in Figure 11 to 13.

z

e

ar

y

s

b

b x

1

2

3

4

( , , )P r z

1

1dl

2dl

2z

z

e

ar

y

s

b

b x

1

2

3

4

e

dr

0M

3

4

2

( , , )P r z

z

y

b

bx

1

23

4inr

or

a

( , , )P r z

1dl

2dlo

r

z

y

b

bx

1

2

3

4in

r

( , , )P r z

2dl4

S

e

1

23

4

1d

Figure 7 Radial Magnet

Filament

Figure 8 Equivalent Current

Sheet Model for a RadialMagnet Piece

Figure 9 Azimuthal Magnet

Filament

Figure 10 Equivalent

Current Sheet Model for aShunt Magnet Piece

0

50

100

150

200

250

300

-0.2

-0.1

0

0.1

0.2

-1

-0.5

0

0.5

1

Theta [degrees]z-axis [m]

0 50 100 150 2000.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

Matlab

Maxwell

Figure 11 Halbach Rotor Field [T] in

the Radial Direction

Figure 12 Halbach Rotor Field [T] in z-

direction

Figure 13 Radial Magnet Flux Density [T]

at z=20mm and r=105mm

3.1 Guideway Model

The guideway was modelled by using dynamic circuit theory [14] in which the guideway is

represented using lumped parameters:

[ ] [ ]de M i R idt

= + (1) dedt= (2)

where [M] is the inductance matrix, [R] is the resistance matrix and e is a voltage column matrix

computed from the changing Halbach rotor flux,, over each guideway loop. Equation (1) was

solved for the current, i. The self and mutual inductance terms were computed using exact analytic

equations for rectangular conducting bars [16] and using the concept of partial inductances [17].

3.3 Computation of the ForcesThe forces in the air-gap were computed by integrating Maxwells stress tensors over a flat surface in

the air-gap between the rotor and the guideway. The field components in the air-gap are composed

of the field due to the magnets, Brotorand the field due to the induced guideway currents, Bguideway:( , , ) ( , , ) ( , , )rotor guidewayy z x y z x y z= +B B B (3)

The guideway field was computed using Biot-Savarts law (using the same method that was used tofind the magnet filament field). The stress tensor force equations are: [18]

Thrust Force:0

1x y x

SF B B dxdz

= [N] (4)

Lift Force: ( )2 2 20

1

2y y x z

SF B B B dxdz

= [N] (5)

Guidance Force: 0

1z y zSF B B dxdz

=

[N] (6)

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results for when the EDW is rotated at 1000RPM, which corresponds to 10.4ms-1

circumferential

velocity, and also moved translationally at 8m/s is shown in Figures 18 to 20. As expected the forces

decrease significantly because the relative change in flux seen by the track decreases. As a last

example the simulation results for the case when the rotor has 6000RPM and is moving at 40ms -1is

also shown.

4 Conclusion

Preliminary results from a 3D Matlab model of a Halbach rotor rotating and moving translationally

over an aluminium split-guideway ladder topology have been presented. The Matlab model has been

validated using magnetostatic FEA models in Ansoft Maxwell 3D and transient FEA models in

Magsoft 3D. Using a split-guideway structure it was confirmed that suspension, thrust and guidance

force can be simultaneously obtained. Unlike the FEA model the Matlab model enables transient

translational motion to be also modelled. The Matlab model could be further improved byaccounting for the skin effect. Its fast computational speed, relative to FEA, will enable future

realistic design and control studies to be conducted. The only intrinsic damping for the EDW comesfrom the aerodynamic drag; thus, control of multiple EDWs will be essential.

5 Acknowledgements

The authors would like to acknowledge the support provided by the member companies of the

Wisconsin Electric Machines and Power Electronics Consortium (WEMPEC) at the University ofWisconsin-Madison. Also the authors would like to gratefully thank the Magsoft and Ansoft

Corporation for the use of their FEA software.

6 References

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[14] J. He, D.M. Rote, H.T. Coffey,Applications of the dynamic circuit theory to maglev suspension systems.IEEE T.Magn., 1993. 29(6): p. 4153.

[15] D.K. Cheng,Field and Wave Electromagnetics. 1989: Addison-Wesley Publishing.

[16] C. Hoer, C. Love,Exact inductance equations for rectangular conductors with applications to more complicatedgeometries.Journal of Research of the National Bureau of Standards, 1965. 69C(2): p. 127.

[17] E.B. Rosa, The self and mutual inductances of linear conductors.Bull. of the Bureau of Stand., 1908. 4(2): p. 301.

[18] D.J. Griffiths,Introduction to Electrodynamics. 3rd ed. 1999: Prentice Hall.