90 Ë 4 Superconducting magnetic levitation
P=Js^2 Rs=J^2 s^1
휎훿=
J^2 s
k 1¡v
휎
(^12)
, (4.4)
whereJsis surface current density andRsis surface resistance.
Clearly,Pproportionally changes withv
(^12)
, and from Eq. (4.1), it can be obtained
thatFDis changing proportionally withv−
(^12)
. The drag forceFDis smaller with a
faster speed휐. At very low frequency, the magnetic flux distribution is uniform in
the guideway, and the induced current depends directly on the speed.Pchanges
proportionally withv^2 , i.e.FD∼v. If the skin effect is ignored at a very high speed,
the ratio of lift forceFLand drag forceFDcan be expressed as
FL
FD= v
v 0, (4.5)
wherev 0 is a parameter depending on frequencies, the guideway dimensions, material
parameters, and structure. Clearly, the drag forceFDis smaller for faster speed.
Experimental research [31, 32] proved the correctness of Eq. (4.5).
Guderjahn et al. [33] found that at high velocities, the levitation force can be
calculated as if the guideway had infinite conductivity, and they also get similar
results.
Coffey et al. [34] confirmed that at finite velocities, the resistivity of the guideway
reduces the actual lift forceFLand produces a drag forceFD, and they have found
the important results: (1) the lift forceFLapproaches the magnetic image forceFi
asymptotically (at speeds of∼80 km/h for magnets of 0.5 m square,FL≈ 0. 8 Fi); (2)
the drag force at high speeds (>80 km/h) decreases asv−^1 /^2 ; (3) the suspension height
is adjustable (150 mm seems reasonable); (4) superconducting magnets of modest
dimensions are practical and necessary for this application.
Reitz [31] reported the calculation results of the lift and drag forces of induced
eddy currents and their associated fields for several magnet geometries. The ratio of
lift to drag is found to be independent of coil geometry, but the velocity dependence
of the lift is greatly affected by the geometry. The ratio of lift to coil weight can be as
high as 2000 for a superconducting coil moving at 483 km/h (300 mph) at 10 cm above
a conducting plate.
On the basis of the above analysis, Miericke and Urankar [35] derived the exact
analytical expressions for liftFLand dragFDforces on moving flat rectangular current-
carrying coils above and below an infinite conducting sheet track of arbitrary thick-
ness. Subsequently, they developed approximate expressions [36] for liftFLand drag
FDforces in these systems. The numerical computations on the basis of approximate
formulae were done with an ordinary computer, and the results agreed so well with
costly computer calculations on the basis of the exact expressions.
From here, we see that the results only agree with experiment in specific con-
ditions. Hieronymus et al. [37] measured the lift and drag forces exerted by various