Magnetic Levitation Train 1

[Main] [Project Deliverables ] [Progress Log] [Methods/Research ] [References/Links]

Halbach Array Equations

The equations used for the analysis of the Inductrack have been obtained from documents written by Dr. Post, physicist at Lawrence Livermore National Laboratory. The wavelength of the magnetic field, lamda m, shown in figure 1 is 0.05 m. The train’s velocity v [m/sec.], creates a magnetic field frequency of omega radians/sec below the array given by

omega = 2*pi*v/lamda radians/sec (1)

The remanent magnetic field Br equals 1.21 Tesla for grade 38 NdFeB magnets. The thickness of the magnets 0.012 m is represented by d m. M is equal to 4, the number of permanent magnets used per wavelength, lamda. The equation for the highest strength of the magnetic field at the surface of the lower side of the Halbach array is given by

Bo = Br (1 – exp(-4*pi*x/lamda)[(sin(pi/M))/( pi/M)] = 0.8482 Tesla (2)

Using x as the horizontal component and y as the vertical component of the magnetic field, the equations for the magnetic fields of a planar Halbach array are given by

Bx = Bo sin((2*pi/lamda)x) exp[(-4*pi*x/lamda)(y1-y)] Tesla (3)
By = Bo cos((2*pi/lamda)x) exp[(-4*pi*y/lamda)(y1-y)] Tesla (4)

It is noted that y1 equal to 0.01 m, is a reference distance in meters from the bottom of the magnets and the center of the top inductors of the track. The Bx and By equations are graphed in figure 1. The sinusoidal magnetic fields passing over the track are responsible for creating the frequency omega radians/sec.

Figure 1 - Bx & By Sinusoidal Fields